# Properties

 Label 507.4.a.j.1.1 Level $507$ Weight $4$ Character 507.1 Self dual yes Analytic conductor $29.914$ Analytic rank $1$ 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] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, 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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.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: $$29.9139683729$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 507.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.21898 q^{2} -3.00000 q^{3} +19.2377 q^{4} +5.83936 q^{5} +15.6569 q^{6} +31.3139 q^{7} -58.6495 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-5.21898 q^{2} -3.00000 q^{3} +19.2377 q^{4} +5.83936 q^{5} +15.6569 q^{6} +31.3139 q^{7} -58.6495 q^{8} +9.00000 q^{9} -30.4755 q^{10} +16.2773 q^{11} -57.7132 q^{12} -163.426 q^{14} -17.5181 q^{15} +152.189 q^{16} -54.0000 q^{17} -46.9708 q^{18} -66.3500 q^{19} +112.336 q^{20} -93.9416 q^{21} -84.9510 q^{22} -182.853 q^{23} +175.949 q^{24} -90.9019 q^{25} -27.0000 q^{27} +602.408 q^{28} -164.853 q^{29} +91.4264 q^{30} +58.9055 q^{31} -325.073 q^{32} -48.8319 q^{33} +281.825 q^{34} +182.853 q^{35} +173.140 q^{36} +110.366 q^{37} +346.279 q^{38} -342.475 q^{40} -55.0357 q^{41} +490.279 q^{42} -113.147 q^{43} +313.139 q^{44} +52.5542 q^{45} +954.305 q^{46} -514.089 q^{47} -456.566 q^{48} +637.559 q^{49} +474.415 q^{50} +162.000 q^{51} +242.559 q^{53} +140.912 q^{54} +95.0490 q^{55} -1836.54 q^{56} +199.050 q^{57} +860.364 q^{58} +265.036 q^{59} -337.008 q^{60} -468.098 q^{61} -307.426 q^{62} +281.825 q^{63} +479.042 q^{64} +254.853 q^{66} -852.919 q^{67} -1038.84 q^{68} +548.559 q^{69} -954.305 q^{70} -165.619 q^{71} -527.846 q^{72} -315.325 q^{73} -576.000 q^{74} +272.706 q^{75} -1276.42 q^{76} +509.706 q^{77} +479.608 q^{79} +888.684 q^{80} +81.0000 q^{81} +287.230 q^{82} -574.235 q^{83} -1807.22 q^{84} -315.325 q^{85} +590.512 q^{86} +494.559 q^{87} -954.657 q^{88} -66.7144 q^{89} -274.279 q^{90} -3517.68 q^{92} -176.716 q^{93} +2683.02 q^{94} -387.441 q^{95} +975.220 q^{96} +1438.25 q^{97} -3327.40 q^{98} +146.496 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} + 26 q^{4} + 36 q^{9}+O(q^{10})$$ 4 * q - 12 * q^3 + 26 * q^4 + 36 * q^9 $$4 q - 12 q^{3} + 26 q^{4} + 36 q^{9} - 20 q^{10} - 78 q^{12} - 348 q^{14} + 354 q^{16} - 216 q^{17} - 136 q^{22} - 120 q^{23} + 44 q^{25} - 108 q^{27} - 48 q^{29} + 60 q^{30} + 120 q^{35} + 234 q^{36} + 468 q^{38} - 1268 q^{40} + 1044 q^{42} - 1064 q^{43} - 1062 q^{48} + 716 q^{49} + 648 q^{51} - 864 q^{53} + 584 q^{55} - 3372 q^{56} - 2280 q^{61} - 924 q^{62} + 1050 q^{64} + 408 q^{66} - 1404 q^{68} + 360 q^{69} - 2304 q^{74} - 132 q^{75} + 816 q^{77} + 288 q^{79} + 324 q^{81} + 28 q^{82} + 144 q^{87} - 2392 q^{88} - 180 q^{90} - 8568 q^{92} + 6656 q^{94} - 3384 q^{95}+O(q^{100})$$ 4 * q - 12 * q^3 + 26 * q^4 + 36 * q^9 - 20 * q^10 - 78 * q^12 - 348 * q^14 + 354 * q^16 - 216 * q^17 - 136 * q^22 - 120 * q^23 + 44 * q^25 - 108 * q^27 - 48 * q^29 + 60 * q^30 + 120 * q^35 + 234 * q^36 + 468 * q^38 - 1268 * q^40 + 1044 * q^42 - 1064 * q^43 - 1062 * q^48 + 716 * q^49 + 648 * q^51 - 864 * q^53 + 584 * q^55 - 3372 * q^56 - 2280 * q^61 - 924 * q^62 + 1050 * q^64 + 408 * q^66 - 1404 * q^68 + 360 * q^69 - 2304 * q^74 - 132 * q^75 + 816 * q^77 + 288 * q^79 + 324 * q^81 + 28 * q^82 + 144 * q^87 - 2392 * q^88 - 180 * q^90 - 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 $$0.873935\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 19.2377 2.40472
$$5$$ 5.83936 0.522288 0.261144 0.965300i $$-0.415900\pi$$
0.261144 + 0.965300i $$0.415900\pi$$
$$6$$ 15.6569 1.06532
$$7$$ 31.3139 1.69079 0.845395 0.534141i $$-0.179365\pi$$
0.845395 + 0.534141i $$0.179365\pi$$
$$8$$ −58.6495 −2.59197
$$9$$ 9.00000 0.333333
$$10$$ −30.4755 −0.963719
$$11$$ 16.2773 0.446163 0.223082 0.974800i $$-0.428388\pi$$
0.223082 + 0.974800i $$0.428388\pi$$
$$12$$ −57.7132 −1.38836
$$13$$ 0 0
$$14$$ −163.426 −3.11983
$$15$$ −17.5181 −0.301543
$$16$$ 152.189 2.37795
$$17$$ −54.0000 −0.770407 −0.385204 0.922832i $$-0.625869\pi$$
−0.385204 + 0.922832i $$0.625869\pi$$
$$18$$ −46.9708 −0.615063
$$19$$ −66.3500 −0.801144 −0.400572 0.916265i $$-0.631189\pi$$
−0.400572 + 0.916265i $$0.631189\pi$$
$$20$$ 112.336 1.25595
$$21$$ −93.9416 −0.976178
$$22$$ −84.9510 −0.823255
$$23$$ −182.853 −1.65772 −0.828858 0.559459i $$-0.811009\pi$$
−0.828858 + 0.559459i $$0.811009\pi$$
$$24$$ 175.949 1.49647
$$25$$ −90.9019 −0.727215
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 602.408 4.06587
$$29$$ −164.853 −1.05560 −0.527800 0.849369i $$-0.676983\pi$$
−0.527800 + 0.849369i $$0.676983\pi$$
$$30$$ 91.4264 0.556404
$$31$$ 58.9055 0.341282 0.170641 0.985333i $$-0.445416\pi$$
0.170641 + 0.985333i $$0.445416\pi$$
$$32$$ −325.073 −1.79579
$$33$$ −48.8319 −0.257592
$$34$$ 281.825 1.42155
$$35$$ 182.853 0.883079
$$36$$ 173.140 0.801572
$$37$$ 110.366 0.490382 0.245191 0.969475i $$-0.421149\pi$$
0.245191 + 0.969475i $$0.421149\pi$$
$$38$$ 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 $$0.533426\pi$$
$$42$$ 490.279 1.80123
$$43$$ −113.147 −0.401274 −0.200637 0.979666i $$-0.564301\pi$$
−0.200637 + 0.979666i $$0.564301\pi$$
$$44$$ 313.139 1.07290
$$45$$ 52.5542 0.174096
$$46$$ 954.305 3.05880
$$47$$ −514.089 −1.59548 −0.797740 0.603001i $$-0.793971\pi$$
−0.797740 + 0.603001i $$0.793971\pi$$
$$48$$ −456.566 −1.37291
$$49$$ 637.559 1.85877
$$50$$ 474.415 1.34185
$$51$$ 162.000 0.444795
$$52$$ 0 0
$$53$$ 242.559 0.628641 0.314321 0.949317i $$-0.398223\pi$$
0.314321 + 0.949317i $$0.398223\pi$$
$$54$$ 140.912 0.355107
$$55$$ 95.0490 0.233026
$$56$$ −1836.54 −4.38247
$$57$$ 199.050 0.462541
$$58$$ 860.364 1.94778
$$59$$ 265.036 0.584825 0.292413 0.956292i $$-0.405542\pi$$
0.292413 + 0.956292i $$0.405542\pi$$
$$60$$ −337.008 −0.725126
$$61$$ −468.098 −0.982522 −0.491261 0.871013i $$-0.663464\pi$$
−0.491261 + 0.871013i $$0.663464\pi$$
$$62$$ −307.426 −0.629729
$$63$$ 281.825 0.563597
$$64$$ 479.042 0.935628
$$65$$ 0 0
$$66$$ 254.853 0.475306
$$67$$ −852.919 −1.55523 −0.777617 0.628739i $$-0.783572\pi$$
−0.777617 + 0.628739i $$0.783572\pi$$
$$68$$ −1038.84 −1.85261
$$69$$ 548.559 0.957083
$$70$$ −954.305 −1.62945
$$71$$ −165.619 −0.276836 −0.138418 0.990374i $$-0.544202\pi$$
−0.138418 + 0.990374i $$0.544202\pi$$
$$72$$ −527.846 −0.863989
$$73$$ −315.325 −0.505562 −0.252781 0.967524i $$-0.581345\pi$$
−0.252781 + 0.967524i $$0.581345\pi$$
$$74$$ −576.000 −0.904846
$$75$$ 272.706 0.419858
$$76$$ −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
$$81$$ 81.0000 0.111111
$$82$$ 287.230 0.386820
$$83$$ −574.235 −0.759404 −0.379702 0.925109i $$-0.623973\pi$$
−0.379702 + 0.925109i $$0.623973\pi$$
$$84$$ −1807.22 −2.34743
$$85$$ −315.325 −0.402374
$$86$$ 590.512 0.740426
$$87$$ 494.559 0.609451
$$88$$ −954.657 −1.15644
$$89$$ −66.7144 −0.0794575 −0.0397287 0.999211i $$-0.512649\pi$$
−0.0397287 + 0.999211i $$0.512649\pi$$
$$90$$ −274.279 −0.321240
$$91$$ 0 0
$$92$$ −3517.68 −3.98634
$$93$$ −176.716 −0.197039
$$94$$ 2683.02 2.94396
$$95$$ −387.441 −0.418428
$$96$$ 975.220 1.03680
$$97$$ 1438.25 1.50549 0.752744 0.658313i $$-0.228730\pi$$
0.752744 + 0.658313i $$0.228730\pi$$
$$98$$ −3327.40 −3.42978
$$99$$ 146.496 0.148721
$$100$$ −1748.75 −1.74875
$$101$$ −896.264 −0.882986 −0.441493 0.897265i $$-0.645551\pi$$
−0.441493 + 0.897265i $$0.645551\pi$$
$$102$$ −845.475 −0.820730
$$103$$ −22.2644 −0.0212988 −0.0106494 0.999943i $$-0.503390\pi$$
−0.0106494 + 0.999943i $$0.503390\pi$$
$$104$$ 0 0
$$105$$ −548.559 −0.509846
$$106$$ −1265.91 −1.15996
$$107$$ −351.441 −0.317524 −0.158762 0.987317i $$-0.550750\pi$$
−0.158762 + 0.987317i $$0.550750\pi$$
$$108$$ −519.419 −0.462788
$$109$$ −967.008 −0.849748 −0.424874 0.905252i $$-0.639682\pi$$
−0.424874 + 0.905252i $$0.639682\pi$$
$$110$$ −496.059 −0.429976
$$111$$ −331.099 −0.283122
$$112$$ 4765.62 4.02061
$$113$$ 48.2943 0.0402048 0.0201024 0.999798i $$-0.493601\pi$$
0.0201024 + 0.999798i $$0.493601\pi$$
$$114$$ −1038.84 −0.853474
$$115$$ −1067.74 −0.865805
$$116$$ −3171.40 −2.53842
$$117$$ 0 0
$$118$$ −1383.22 −1.07911
$$119$$ −1690.95 −1.30260
$$120$$ 1027.43 0.781590
$$121$$ −1066.05 −0.800938
$$122$$ 2442.99 1.81294
$$123$$ 165.107 0.121034
$$124$$ 1133.21 0.820686
$$125$$ −1260.73 −0.902104
$$126$$ −1470.84 −1.03994
$$127$$ −1763.02 −1.23183 −0.615916 0.787812i $$-0.711214\pi$$
−0.615916 + 0.787812i $$0.711214\pi$$
$$128$$ 100.479 0.0693844
$$129$$ 339.441 0.231676
$$130$$ 0 0
$$131$$ 955.970 0.637584 0.318792 0.947825i $$-0.396723\pi$$
0.318792 + 0.947825i $$0.396723\pi$$
$$132$$ −939.416 −0.619437
$$133$$ −2077.68 −1.35457
$$134$$ 4451.37 2.86970
$$135$$ −157.663 −0.100514
$$136$$ 3167.07 1.99687
$$137$$ −15.9215 −0.00992894 −0.00496447 0.999988i $$-0.501580\pi$$
−0.00496447 + 0.999988i $$0.501580\pi$$
$$138$$ −2862.92 −1.76600
$$139$$ 2074.26 1.26573 0.632866 0.774261i $$-0.281878\pi$$
0.632866 + 0.774261i $$0.281878\pi$$
$$140$$ 3517.68 2.12356
$$141$$ 1542.27 0.921151
$$142$$ 864.362 0.510815
$$143$$ 0 0
$$144$$ 1369.70 0.792649
$$145$$ −962.635 −0.551327
$$146$$ 1645.68 0.932857
$$147$$ −1912.68 −1.07316
$$148$$ 2123.20 1.17923
$$149$$ 2764.08 1.51975 0.759873 0.650071i $$-0.225261\pi$$
0.759873 + 0.650071i $$0.225261\pi$$
$$150$$ −1423.25 −0.774717
$$151$$ 1618.46 0.872239 0.436120 0.899889i $$-0.356352\pi$$
0.436120 + 0.899889i $$0.356352\pi$$
$$152$$ 3891.40 2.07654
$$153$$ −486.000 −0.256802
$$154$$ −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
$$159$$ −727.676 −0.362946
$$160$$ −1898.22 −0.937921
$$161$$ −5725.83 −2.80285
$$162$$ −422.737 −0.205021
$$163$$ −233.201 −0.112060 −0.0560299 0.998429i $$-0.517844\pi$$
−0.0560299 + 0.998429i $$0.517844\pi$$
$$164$$ −1058.76 −0.504119
$$165$$ −285.147 −0.134537
$$166$$ 2996.92 1.40124
$$167$$ −215.405 −0.0998118 −0.0499059 0.998754i $$-0.515892\pi$$
−0.0499059 + 0.998754i $$0.515892\pi$$
$$168$$ 5509.63 2.53022
$$169$$ 0 0
$$170$$ 1645.68 0.742456
$$171$$ −597.150 −0.267048
$$172$$ −2176.69 −0.964950
$$173$$ −1383.15 −0.607854 −0.303927 0.952695i $$-0.598298\pi$$
−0.303927 + 0.952695i $$0.598298\pi$$
$$174$$ −2581.09 −1.12455
$$175$$ −2846.49 −1.22957
$$176$$ 2477.22 1.06095
$$177$$ −795.107 −0.337649
$$178$$ 348.181 0.146614
$$179$$ 3642.79 1.52109 0.760545 0.649285i $$-0.224932\pi$$
0.760545 + 0.649285i $$0.224932\pi$$
$$180$$ 1011.02 0.418652
$$181$$ 2621.97 1.07674 0.538369 0.842709i $$-0.319041\pi$$
0.538369 + 0.842709i $$0.319041\pi$$
$$182$$ 0 0
$$183$$ 1404.29 0.567259
$$184$$ 10724.2 4.29674
$$185$$ 644.469 0.256121
$$186$$ 922.279 0.363574
$$187$$ −878.975 −0.343727
$$188$$ −9889.91 −3.83668
$$189$$ −845.475 −0.325393
$$190$$ 2022.05 0.772078
$$191$$ −3419.32 −1.29536 −0.647679 0.761913i $$-0.724260\pi$$
−0.647679 + 0.761913i $$0.724260\pi$$
$$192$$ −1437.12 −0.540185
$$193$$ 1698.39 0.633435 0.316718 0.948520i $$-0.397419\pi$$
0.316718 + 0.948520i $$0.397419\pi$$
$$194$$ −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 $$0.363861\pi$$
$$198$$ −764.559 −0.274418
$$199$$ 900.981 0.320949 0.160474 0.987040i $$-0.448698\pi$$
0.160474 + 0.987040i $$0.448698\pi$$
$$200$$ 5331.35 1.88492
$$201$$ 2558.76 0.897915
$$202$$ 4677.58 1.62928
$$203$$ −5162.18 −1.78480
$$204$$ 3116.51 1.06961
$$205$$ −321.373 −0.109491
$$206$$ 116.197 0.0393002
$$207$$ −1645.68 −0.552572
$$208$$ 0 0
$$209$$ −1080.00 −0.357441
$$210$$ 2862.92 0.940762
$$211$$ 431.019 0.140628 0.0703142 0.997525i $$-0.477600\pi$$
0.0703142 + 0.997525i $$0.477600\pi$$
$$212$$ 4666.28 1.51170
$$213$$ 496.857 0.159831
$$214$$ 1834.17 0.585892
$$215$$ −660.706 −0.209580
$$216$$ 1583.54 0.498824
$$217$$ 1844.56 0.577036
$$218$$ 5046.79 1.56794
$$219$$ 945.976 0.291886
$$220$$ 1828.53 0.560361
$$221$$ 0 0
$$222$$ 1728.00 0.522413
$$223$$ −4104.30 −1.23249 −0.616243 0.787556i $$-0.711346\pi$$
−0.616243 + 0.787556i $$0.711346\pi$$
$$224$$ −10179.3 −3.03631
$$225$$ −818.117 −0.242405
$$226$$ −252.047 −0.0741854
$$227$$ 1809.11 0.528963 0.264482 0.964391i $$-0.414799\pi$$
0.264482 + 0.964391i $$0.414799\pi$$
$$228$$ 3829.27 1.11228
$$229$$ −5249.33 −1.51478 −0.757392 0.652961i $$-0.773527\pi$$
−0.757392 + 0.652961i $$0.773527\pi$$
$$230$$ 5572.53 1.59757
$$231$$ −1529.12 −0.435535
$$232$$ 9668.54 2.73608
$$233$$ −2808.88 −0.789768 −0.394884 0.918731i $$-0.629215\pi$$
−0.394884 + 0.918731i $$0.629215\pi$$
$$234$$ 0 0
$$235$$ −3001.95 −0.833300
$$236$$ 5098.69 1.40634
$$237$$ −1438.82 −0.394353
$$238$$ 8825.03 2.40354
$$239$$ 6712.01 1.81659 0.908293 0.418334i $$-0.137386\pi$$
0.908293 + 0.418334i $$0.137386\pi$$
$$240$$ −2666.05 −0.717054
$$241$$ 2519.11 0.673321 0.336661 0.941626i $$-0.390703\pi$$
0.336661 + 0.941626i $$0.390703\pi$$
$$242$$ 5563.69 1.47788
$$243$$ −243.000 −0.0641500
$$244$$ −9005.15 −2.36269
$$245$$ 3722.93 0.970814
$$246$$ −861.691 −0.223331
$$247$$ 0 0
$$248$$ −3454.78 −0.884591
$$249$$ 1722.71 0.438442
$$250$$ 6579.71 1.66455
$$251$$ 828.000 0.208219 0.104109 0.994566i $$-0.466801\pi$$
0.104109 + 0.994566i $$0.466801\pi$$
$$252$$ 5421.67 1.35529
$$253$$ −2976.35 −0.739612
$$254$$ 9201.16 2.27296
$$255$$ 945.976 0.232311
$$256$$ −4356.73 −1.06366
$$257$$ −5840.76 −1.41765 −0.708826 0.705383i $$-0.750775\pi$$
−0.708826 + 0.705383i $$0.750775\pi$$
$$258$$ −1771.54 −0.427485
$$259$$ 3456.00 0.829133
$$260$$ 0 0
$$261$$ −1483.68 −0.351867
$$262$$ −4989.19 −1.17646
$$263$$ −4064.06 −0.952854 −0.476427 0.879214i $$-0.658068\pi$$
−0.476427 + 0.879214i $$0.658068\pi$$
$$264$$ 2863.97 0.667671
$$265$$ 1416.39 0.328332
$$266$$ 10843.3 2.49943
$$267$$ 200.143 0.0458748
$$268$$ −16408.2 −3.73990
$$269$$ 1845.44 0.418285 0.209142 0.977885i $$-0.432933\pi$$
0.209142 + 0.977885i $$0.432933\pi$$
$$270$$ 822.838 0.185468
$$271$$ −2106.78 −0.472242 −0.236121 0.971724i $$-0.575876\pi$$
−0.236121 + 0.971724i $$0.575876\pi$$
$$272$$ −8218.19 −1.83199
$$273$$ 0 0
$$274$$ 83.0939 0.0183208
$$275$$ −1479.64 −0.324457
$$276$$ 10553.0 2.30151
$$277$$ −4781.94 −1.03725 −0.518626 0.855001i $$-0.673556\pi$$
−0.518626 + 0.855001i $$0.673556\pi$$
$$278$$ −10825.5 −2.33551
$$279$$ 530.149 0.113761
$$280$$ −10724.2 −2.28891
$$281$$ 5865.81 1.24528 0.622642 0.782507i $$-0.286059\pi$$
0.622642 + 0.782507i $$0.286059\pi$$
$$282$$ −8049.06 −1.69970
$$283$$ −6407.02 −1.34579 −0.672894 0.739739i $$-0.734949\pi$$
−0.672894 + 0.739739i $$0.734949\pi$$
$$284$$ −3186.14 −0.665713
$$285$$ 1162.32 0.241579
$$286$$ 0 0
$$287$$ −1723.38 −0.354453
$$288$$ −2925.66 −0.598598
$$289$$ −1997.00 −0.406473
$$290$$ 5023.97 1.01730
$$291$$ −4314.75 −0.869194
$$292$$ −6066.15 −1.21573
$$293$$ −3010.24 −0.600204 −0.300102 0.953907i $$-0.597021\pi$$
−0.300102 + 0.953907i $$0.597021\pi$$
$$294$$ 9982.21 1.98019
$$295$$ 1547.64 0.305447
$$296$$ −6472.94 −1.27105
$$297$$ −439.487 −0.0858641
$$298$$ −14425.7 −2.80422
$$299$$ 0 0
$$300$$ 5246.24 1.00964
$$301$$ −3543.07 −0.678470
$$302$$ −8446.69 −1.60944
$$303$$ 2688.79 0.509792
$$304$$ −10097.7 −1.90508
$$305$$ −2733.39 −0.513159
$$306$$ 2536.42 0.473849
$$307$$ −3341.84 −0.621266 −0.310633 0.950530i $$-0.600541\pi$$
−0.310633 + 0.950530i $$0.600541\pi$$
$$308$$ 9805.59 1.81404
$$309$$ 66.7931 0.0122968
$$310$$ −1795.17 −0.328900
$$311$$ 8755.20 1.59634 0.798170 0.602432i $$-0.205801\pi$$
0.798170 + 0.602432i $$0.205801\pi$$
$$312$$ 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
$$315$$ 1645.68 0.294360
$$316$$ 9226.57 1.64252
$$317$$ −1940.43 −0.343802 −0.171901 0.985114i $$-0.554991\pi$$
−0.171901 + 0.985114i $$0.554991\pi$$
$$318$$ 3797.72 0.669704
$$319$$ −2683.36 −0.470970
$$320$$ 2797.29 0.488667
$$321$$ 1054.32 0.183323
$$322$$ 29883.0 5.17178
$$323$$ 3582.90 0.617207
$$324$$ 1558.26 0.267191
$$325$$ 0 0
$$326$$ 1217.07 0.206771
$$327$$ 2901.02 0.490602
$$328$$ 3227.82 0.543373
$$329$$ −16098.1 −2.69762
$$330$$ 1488.18 0.248247
$$331$$ −7402.13 −1.22918 −0.614589 0.788848i $$-0.710678\pi$$
−0.614589 + 0.788848i $$0.710678\pi$$
$$332$$ −11047.0 −1.82615
$$333$$ 993.298 0.163461
$$334$$ 1124.20 0.184171
$$335$$ −4980.50 −0.812280
$$336$$ −14296.9 −2.32130
$$337$$ −5494.05 −0.888071 −0.444035 0.896009i $$-0.646454\pi$$
−0.444035 + 0.896009i $$0.646454\pi$$
$$338$$ 0 0
$$339$$ −144.883 −0.0232122
$$340$$ −6066.15 −0.967597
$$341$$ 958.823 0.152267
$$342$$ 3116.51 0.492754
$$343$$ 9223.77 1.45200
$$344$$ 6636.03 1.04009
$$345$$ 3203.23 0.499873
$$346$$ 7218.62 1.12160
$$347$$ 3410.76 0.527664 0.263832 0.964569i $$-0.415014\pi$$
0.263832 + 0.964569i $$0.415014\pi$$
$$348$$ 9514.19 1.46556
$$349$$ −12629.8 −1.93712 −0.968562 0.248773i $$-0.919973\pi$$
−0.968562 + 0.248773i $$0.919973\pi$$
$$350$$ 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 $$0.427829\pi$$
$$354$$ 4149.65 0.623026
$$355$$ −967.109 −0.144588
$$356$$ −1283.43 −0.191073
$$357$$ 5072.85 0.752055
$$358$$ −19011.7 −2.80670
$$359$$ 8942.30 1.31464 0.657321 0.753611i $$-0.271690\pi$$
0.657321 + 0.753611i $$0.271690\pi$$
$$360$$ −3082.28 −0.451251
$$361$$ −2456.68 −0.358168
$$362$$ −13684.0 −1.98678
$$363$$ 3198.15 0.462422
$$364$$ 0 0
$$365$$ −1841.30 −0.264049
$$366$$ −7328.98 −1.04670
$$367$$ −4735.26 −0.673511 −0.336756 0.941592i $$-0.609330\pi$$
−0.336756 + 0.941592i $$0.609330\pi$$
$$368$$ −27828.1 −3.94196
$$369$$ −495.321 −0.0698792
$$370$$ −3363.47 −0.472590
$$371$$ 7595.45 1.06290
$$372$$ −3399.62 −0.473823
$$373$$ −8304.01 −1.15272 −0.576361 0.817195i $$-0.695528\pi$$
−0.576361 + 0.817195i $$0.695528\pi$$
$$374$$ 4587.35 0.634241
$$375$$ 3782.18 0.520830
$$376$$ 30151.1 4.13543
$$377$$ 0 0
$$378$$ 4412.51 0.600411
$$379$$ 4088.11 0.554069 0.277035 0.960860i $$-0.410648\pi$$
0.277035 + 0.960860i $$0.410648\pi$$
$$380$$ −7453.50 −1.00620
$$381$$ 5289.06 0.711198
$$382$$ 17845.4 2.39018
$$383$$ 13951.5 1.86132 0.930662 0.365879i $$-0.119232\pi$$
0.930662 + 0.365879i $$0.119232\pi$$
$$384$$ −301.438 −0.0400591
$$385$$ 2976.35 0.393997
$$386$$ −8863.88 −1.16881
$$387$$ −1018.32 −0.133758
$$388$$ 27668.7 3.62027
$$389$$ −2804.26 −0.365506 −0.182753 0.983159i $$-0.558501\pi$$
−0.182753 + 0.983159i $$0.558501\pi$$
$$390$$ 0 0
$$391$$ 9874.06 1.27712
$$392$$ −37392.5 −4.81787
$$393$$ −2867.91 −0.368109
$$394$$ −11970.9 −1.53067
$$395$$ 2800.60 0.356743
$$396$$ 2818.25 0.357632
$$397$$ 6556.18 0.828830 0.414415 0.910088i $$-0.363986\pi$$
0.414415 + 0.910088i $$0.363986\pi$$
$$398$$ −4702.20 −0.592211
$$399$$ 6233.03 0.782059
$$400$$ −13834.2 −1.72928
$$401$$ −4730.95 −0.589157 −0.294579 0.955627i $$-0.595179\pi$$
−0.294579 + 0.955627i $$0.595179\pi$$
$$402$$ −13354.1 −1.65682
$$403$$ 0 0
$$404$$ −17242.1 −2.12333
$$405$$ 472.988 0.0580320
$$406$$ 26941.3 3.29329
$$407$$ 1796.47 0.218790
$$408$$ −9501.22 −1.15289
$$409$$ −12314.4 −1.48878 −0.744389 0.667746i $$-0.767259\pi$$
−0.744389 + 0.667746i $$0.767259\pi$$
$$410$$ 1677.24 0.202032
$$411$$ 47.7645 0.00573247
$$412$$ −428.316 −0.0512175
$$413$$ 8299.29 0.988817
$$414$$ 8588.75 1.01960
$$415$$ −3353.16 −0.396627
$$416$$ 0 0
$$417$$ −6222.79 −0.730771
$$418$$ 5636.50 0.659546
$$419$$ −5499.85 −0.641254 −0.320627 0.947206i $$-0.603894\pi$$
−0.320627 + 0.947206i $$0.603894\pi$$
$$420$$ −10553.0 −1.22604
$$421$$ 12629.7 1.46208 0.731039 0.682336i $$-0.239036\pi$$
0.731039 + 0.682336i $$0.239036\pi$$
$$422$$ −2249.48 −0.259486
$$423$$ −4626.80 −0.531827
$$424$$ −14225.9 −1.62942
$$425$$ 4908.70 0.560252
$$426$$ −2593.09 −0.294919
$$427$$ −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 $$0.368371\pi$$
$$432$$ −4109.09 −0.457636
$$433$$ 6062.68 0.672873 0.336436 0.941706i $$-0.390778\pi$$
0.336436 + 0.941706i $$0.390778\pi$$
$$434$$ −9626.71 −1.06474
$$435$$ 2887.90 0.318309
$$436$$ −18603.0 −2.04340
$$437$$ 12132.3 1.32807
$$438$$ −4937.03 −0.538585
$$439$$ 11864.3 1.28986 0.644932 0.764240i $$-0.276886\pi$$
0.644932 + 0.764240i $$0.276886\pi$$
$$440$$ −5574.58 −0.603995
$$441$$ 5738.03 0.619590
$$442$$ 0 0
$$443$$ 10560.5 1.13261 0.566303 0.824197i $$-0.308373\pi$$
0.566303 + 0.824197i $$0.308373\pi$$
$$444$$ −6369.60 −0.680829
$$445$$ −389.569 −0.0414997
$$446$$ 21420.3 2.27417
$$447$$ −8292.24 −0.877426
$$448$$ 15000.6 1.58195
$$449$$ −12659.7 −1.33062 −0.665308 0.746569i $$-0.731700\pi$$
−0.665308 + 0.746569i $$0.731700\pi$$
$$450$$ 4269.74 0.447283
$$451$$ −895.834 −0.0935325
$$452$$ 929.072 0.0966812
$$453$$ −4855.37 −0.503587
$$454$$ −9441.69 −0.976037
$$455$$ 0 0
$$456$$ −11674.2 −1.19889
$$457$$ −1544.24 −0.158067 −0.0790336 0.996872i $$-0.525183\pi$$
−0.0790336 + 0.996872i $$0.525183\pi$$
$$458$$ 27396.1 2.79506
$$459$$ 1458.00 0.148265
$$460$$ −20541.0 −2.08202
$$461$$ −13196.8 −1.33327 −0.666635 0.745384i $$-0.732266\pi$$
−0.666635 + 0.745384i $$0.732266\pi$$
$$462$$ 7980.43 0.803643
$$463$$ 16309.2 1.63705 0.818524 0.574472i $$-0.194792\pi$$
0.818524 + 0.574472i $$0.194792\pi$$
$$464$$ −25088.7 −2.51016
$$465$$ −1031.91 −0.102911
$$466$$ 14659.5 1.45727
$$467$$ −14260.8 −1.41308 −0.706541 0.707672i $$-0.749745\pi$$
−0.706541 + 0.707672i $$0.749745\pi$$
$$468$$ 0 0
$$469$$ −26708.2 −2.62957
$$470$$ 15667.1 1.53760
$$471$$ 3329.91 0.325763
$$472$$ −15544.2 −1.51585
$$473$$ −1841.73 −0.179034
$$474$$ 7509.19 0.727655
$$475$$ 6031.34 0.582604
$$476$$ −32530.0 −3.13238
$$477$$ 2183.03 0.209547
$$478$$ −35029.9 −3.35194
$$479$$ 18011.5 1.71809 0.859046 0.511899i $$-0.171058\pi$$
0.859046 + 0.511899i $$0.171058\pi$$
$$480$$ 5694.66 0.541509
$$481$$ 0 0
$$482$$ −13147.2 −1.24240
$$483$$ 17177.5 1.61823
$$484$$ −20508.4 −1.92603
$$485$$ 8398.46 0.786298
$$486$$ 1268.21 0.118369
$$487$$ −14043.3 −1.30670 −0.653351 0.757055i $$-0.726637\pi$$
−0.653351 + 0.757055i $$0.726637\pi$$
$$488$$ 27453.7 2.54666
$$489$$ 699.604 0.0646977
$$490$$ −19429.9 −1.79133
$$491$$ −12966.1 −1.19176 −0.595878 0.803075i $$-0.703196\pi$$
−0.595878 + 0.803075i $$0.703196\pi$$
$$492$$ 3176.29 0.291053
$$493$$ 8902.06 0.813242
$$494$$ 0 0
$$495$$ 855.441 0.0776752
$$496$$ 8964.75 0.811551
$$497$$ −5186.17 −0.468072
$$498$$ −8990.76 −0.809007
$$499$$ −10215.5 −0.916453 −0.458227 0.888835i $$-0.651515\pi$$
−0.458227 + 0.888835i $$0.651515\pi$$
$$500$$ −24253.6 −2.16930
$$501$$ 646.216 0.0576264
$$502$$ −4321.31 −0.384203
$$503$$ −16632.0 −1.47432 −0.737161 0.675717i $$-0.763834\pi$$
−0.737161 + 0.675717i $$0.763834\pi$$
$$504$$ −16528.9 −1.46082
$$505$$ −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 $$0.269132\pi$$
$$510$$ −4937.03 −0.428657
$$511$$ −9874.06 −0.854799
$$512$$ 21933.9 1.89326
$$513$$ 1791.45 0.154180
$$514$$ 30482.8 2.61584
$$515$$ −130.009 −0.0111241
$$516$$ 6530.08 0.557114
$$517$$ −8367.99 −0.711845
$$518$$ −18036.8 −1.52991
$$519$$ 4149.44 0.350945
$$520$$ 0 0
$$521$$ 2680.23 0.225380 0.112690 0.993630i $$-0.464053\pi$$
0.112690 + 0.993630i $$0.464053\pi$$
$$522$$ 7743.27 0.649260
$$523$$ −2410.38 −0.201527 −0.100764 0.994910i $$-0.532129\pi$$
−0.100764 + 0.994910i $$0.532129\pi$$
$$524$$ 18390.7 1.53321
$$525$$ 8539.47 0.709892
$$526$$ 21210.2 1.75819
$$527$$ −3180.90 −0.262926
$$528$$ −7431.67 −0.612541
$$529$$ 21268.2 1.74802
$$530$$ −7392.09 −0.605834
$$531$$ 2385.32 0.194942
$$532$$ −39969.8 −3.25735
$$533$$ 0 0
$$534$$ −1044.54 −0.0846476
$$535$$ −2052.19 −0.165839
$$536$$ 50023.3 4.03111
$$537$$ −10928.4 −0.878202
$$538$$ −9631.32 −0.771814
$$539$$ 10377.7 0.829315
$$540$$ −3033.07 −0.241709
$$541$$ −9969.58 −0.792284 −0.396142 0.918189i $$-0.629651\pi$$
−0.396142 + 0.918189i $$0.629651\pi$$
$$542$$ 10995.2 0.871375
$$543$$ −7865.91 −0.621655
$$544$$ 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
$$549$$ −4212.88 −0.327507
$$550$$ 7722.20 0.598683
$$551$$ 10938.0 0.845688
$$552$$ −32172.7 −2.48073
$$553$$ 15018.4 1.15488
$$554$$ 24956.8 1.91393
$$555$$ −1933.41 −0.147871
$$556$$ 39904.2 3.04373
$$557$$ 3800.83 0.289132 0.144566 0.989495i $$-0.453821\pi$$
0.144566 + 0.989495i $$0.453821\pi$$
$$558$$ −2766.84 −0.209910
$$559$$ 0 0
$$560$$ 27828.1 2.09992
$$561$$ 2636.92 0.198451
$$562$$ −30613.5 −2.29778
$$563$$ −15750.2 −1.17903 −0.589513 0.807759i $$-0.700680\pi$$
−0.589513 + 0.807759i $$0.700680\pi$$
$$564$$ 29669.7 2.21511
$$565$$ 282.007 0.0209985
$$566$$ 33438.1 2.48323
$$567$$ 2536.42 0.187866
$$568$$ 9713.48 0.717550
$$569$$ 17753.2 1.30800 0.654002 0.756493i $$-0.273089\pi$$
0.654002 + 0.756493i $$0.273089\pi$$
$$570$$ −6066.15 −0.445759
$$571$$ 25293.1 1.85374 0.926868 0.375388i $$-0.122491\pi$$
0.926868 + 0.375388i $$0.122491\pi$$
$$572$$ 0 0
$$573$$ 10258.0 0.747875
$$574$$ 8994.29 0.654032
$$575$$ 16621.7 1.20552
$$576$$ 4311.37 0.311876
$$577$$ 18488.8 1.33396 0.666982 0.745073i $$-0.267586\pi$$
0.666982 + 0.745073i $$0.267586\pi$$
$$578$$ 10422.3 0.750018
$$579$$ −5095.18 −0.365714
$$580$$ −18518.9 −1.32579
$$581$$ −17981.5 −1.28399
$$582$$ 22518.6 1.60383
$$583$$ 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 $$0.290802\pi$$
$$588$$ −36795.6 −2.58065
$$589$$ −3908.38 −0.273416
$$590$$ −8077.09 −0.563608
$$591$$ −6881.16 −0.478940
$$592$$ 16796.5 1.16610
$$593$$ −7991.09 −0.553381 −0.276690 0.960959i $$-0.589238\pi$$
−0.276690 + 0.960959i $$0.589238\pi$$
$$594$$ 2293.68 0.158435
$$595$$ −9874.06 −0.680331
$$596$$ 53174.7 3.65456
$$597$$ −2702.94 −0.185300
$$598$$ 0 0
$$599$$ 10386.5 0.708480 0.354240 0.935154i $$-0.384740\pi$$
0.354240 + 0.935154i $$0.384740\pi$$
$$600$$ −15994.1 −1.08826
$$601$$ −9241.77 −0.627254 −0.313627 0.949546i $$-0.601544\pi$$
−0.313627 + 0.949546i $$0.601544\pi$$
$$602$$ 18491.2 1.25190
$$603$$ −7676.27 −0.518411
$$604$$ 31135.4 2.09749
$$605$$ −6225.04 −0.418320
$$606$$ −14032.8 −0.940663
$$607$$ 18921.5 1.26524 0.632619 0.774463i $$-0.281980\pi$$
0.632619 + 0.774463i $$0.281980\pi$$
$$608$$ 21568.6 1.43869
$$609$$ 15486.5 1.03045
$$610$$ 14265.5 0.946875
$$611$$ 0 0
$$612$$ −9349.54 −0.617537
$$613$$ −17138.1 −1.12921 −0.564603 0.825363i $$-0.690971\pi$$
−0.564603 + 0.825363i $$0.690971\pi$$
$$614$$ 17441.0 1.14635
$$615$$ 964.120 0.0632147
$$616$$ −29894.0 −1.95530
$$617$$ −18825.8 −1.22836 −0.614180 0.789166i $$-0.710513\pi$$
−0.614180 + 0.789166i $$0.710513\pi$$
$$618$$ −348.592 −0.0226900
$$619$$ −1392.83 −0.0904404 −0.0452202 0.998977i $$-0.514399\pi$$
−0.0452202 + 0.998977i $$0.514399\pi$$
$$620$$ 6617.21 0.428635
$$621$$ 4937.03 0.319028
$$622$$ −45693.2 −2.94555
$$623$$ −2089.09 −0.134346
$$624$$ 0 0
$$625$$ 4000.90 0.256057
$$626$$ −10171.4 −0.649412
$$627$$ 3240.00 0.206369
$$628$$ −21353.3 −1.35683
$$629$$ −5959.79 −0.377794
$$630$$ −8588.75 −0.543149
$$631$$ −25488.4 −1.60804 −0.804022 0.594599i $$-0.797311\pi$$
−0.804022 + 0.594599i $$0.797311\pi$$
$$632$$ −28128.8 −1.77041
$$633$$ −1293.06 −0.0811918
$$634$$ 10127.1 0.634379
$$635$$ −10294.9 −0.643371
$$636$$ −13998.8 −0.872783
$$637$$ 0 0
$$638$$ 14004.4 0.869028
$$639$$ −1490.57 −0.0922787
$$640$$ 586.735 0.0362387
$$641$$ −6066.41 −0.373805 −0.186902 0.982379i $$-0.559845\pi$$
−0.186902 + 0.982379i $$0.559845\pi$$
$$642$$ −5502.50 −0.338265
$$643$$ 1598.78 0.0980554 0.0490277 0.998797i $$-0.484388\pi$$
0.0490277 + 0.998797i $$0.484388\pi$$
$$644$$ −110152. −6.74006
$$645$$ 1982.12 0.121001
$$646$$ −18699.1 −1.13886
$$647$$ 23067.2 1.40164 0.700822 0.713336i $$-0.252817\pi$$
0.700822 + 0.713336i $$0.252817\pi$$
$$648$$ −4750.61 −0.287996
$$649$$ 4314.07 0.260928
$$650$$ 0 0
$$651$$ −5533.68 −0.333152
$$652$$ −4486.26 −0.269472
$$653$$ −23743.9 −1.42293 −0.711463 0.702723i $$-0.751967\pi$$
−0.711463 + 0.702723i $$0.751967\pi$$
$$654$$ −15140.4 −0.905253
$$655$$ 5582.25 0.333002
$$656$$ −8375.81 −0.498507
$$657$$ −2837.93 −0.168521
$$658$$ 84015.7 4.97762
$$659$$ −7497.65 −0.443197 −0.221598 0.975138i $$-0.571127\pi$$
−0.221598 + 0.975138i $$0.571127\pi$$
$$660$$ −5485.59 −0.323524
$$661$$ −1255.26 −0.0738638 −0.0369319 0.999318i $$-0.511758\pi$$
−0.0369319 + 0.999318i $$0.511758\pi$$
$$662$$ 38631.5 2.26806
$$663$$ 0 0
$$664$$ 33678.6 1.96835
$$665$$ −12132.3 −0.707474
$$666$$ −5184.00 −0.301615
$$667$$ 30143.8 1.74989
$$668$$ −4143.91 −0.240019
$$669$$ 12312.9 0.711576
$$670$$ 25993.1 1.49881
$$671$$ −7619.38 −0.438365
$$672$$ 30537.9 1.75301
$$673$$ 1505.97 0.0862569 0.0431284 0.999070i $$-0.486268\pi$$
0.0431284 + 0.999070i $$0.486268\pi$$
$$674$$ 28673.3 1.63866
$$675$$ 2454.35 0.139953
$$676$$ 0 0
$$677$$ −16201.4 −0.919751 −0.459876 0.887983i $$-0.652106\pi$$
−0.459876 + 0.887983i $$0.652106\pi$$
$$678$$ 756.140 0.0428310
$$679$$ 45037.2 2.54546
$$680$$ 18493.7 1.04294
$$681$$ −5427.32 −0.305397
$$682$$ −5004.08 −0.280962
$$683$$ 29090.4 1.62974 0.814870 0.579644i $$-0.196808\pi$$
0.814870 + 0.579644i $$0.196808\pi$$
$$684$$ −11487.8 −0.642175
$$685$$ −92.9712 −0.00518576
$$686$$ −48138.7 −2.67922
$$687$$ 15748.0 0.874561
$$688$$ −17219.7 −0.954208
$$689$$ 0 0
$$690$$ −16717.6 −0.922359
$$691$$ 940.952 0.0518025 0.0259012 0.999665i $$-0.491754\pi$$
0.0259012 + 0.999665i $$0.491754\pi$$
$$692$$ −26608.6 −1.46172
$$693$$ 4587.35 0.251456
$$694$$ −17800.7 −0.973639
$$695$$ 12112.4 0.661077
$$696$$ −29005.6 −1.57968
$$697$$ 2971.93 0.161506
$$698$$ 65914.5 3.57436
$$699$$ 8426.65 0.455973
$$700$$ −54760.1 −2.95676
$$701$$ −30713.9 −1.65485 −0.827424 0.561578i $$-0.810195\pi$$
−0.827424 + 0.561578i $$0.810195\pi$$
$$702$$ 0 0
$$703$$ −7322.81 −0.392866
$$704$$ 7797.51 0.417443
$$705$$ 9005.85 0.481106
$$706$$ −15561.8 −0.829571
$$707$$ −28065.5 −1.49294
$$708$$ −15296.1 −0.811951
$$709$$ 25640.5 1.35818 0.679089 0.734056i $$-0.262375\pi$$
0.679089 + 0.734056i $$0.262375\pi$$
$$710$$ 5047.32 0.266792
$$711$$ 4316.47 0.227680
$$712$$ 3912.77 0.205951
$$713$$ −10771.0 −0.565748
$$714$$ −26475.1 −1.38768
$$715$$ 0 0
$$716$$ 70079.1 3.65779
$$717$$ −20136.0 −1.04881
$$718$$ −46669.7 −2.42576
$$719$$ 20842.5 1.08108 0.540538 0.841319i $$-0.318221\pi$$
0.540538 + 0.841319i $$0.318221\pi$$
$$720$$ 7998.16 0.413991
$$721$$ −697.183 −0.0360117
$$722$$ 12821.3 0.660888
$$723$$ −7557.34 −0.388742
$$724$$ 50440.8 2.58925
$$725$$ 14985.4 0.767649
$$726$$ −16691.1 −0.853255
$$727$$ −263.608 −0.0134480 −0.00672398 0.999977i $$-0.502140\pi$$
−0.00672398 + 0.999977i $$0.502140\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 9609.69 0.487220
$$731$$ 6109.94 0.309144
$$732$$ 27015.4 1.36410
$$733$$ 21835.5 1.10029 0.550146 0.835069i $$-0.314572\pi$$
0.550146 + 0.835069i $$0.314572\pi$$
$$734$$ 24713.2 1.24275
$$735$$ −11168.8 −0.560500
$$736$$ 59440.6 2.97692
$$737$$ −13883.2 −0.693888
$$738$$ 2585.07 0.128940
$$739$$ −19536.4 −0.972476 −0.486238 0.873826i $$-0.661631\pi$$
−0.486238 + 0.873826i $$0.661631\pi$$
$$740$$ 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 $$0.769639\pi$$
$$744$$ 10364.3 0.510719
$$745$$ 16140.5 0.793746
$$746$$ 43338.4 2.12699
$$747$$ −5168.12 −0.253135
$$748$$ −16909.5 −0.826567
$$749$$ −11005.0 −0.536867
$$750$$ −19739.1 −0.961029
$$751$$ −3904.20 −0.189702 −0.0948510 0.995491i $$-0.530237\pi$$
−0.0948510 + 0.995491i $$0.530237\pi$$
$$752$$ −78238.5 −3.79397
$$753$$ −2484.00 −0.120215
$$754$$ 0 0
$$755$$ 9450.74 0.455560
$$756$$ −16265.0 −0.782478
$$757$$ −2900.52 −0.139262 −0.0696308 0.997573i $$-0.522182\pi$$
−0.0696308 + 0.997573i $$0.522182\pi$$
$$758$$ −21335.8 −1.02236
$$759$$ 8929.06 0.427015
$$760$$ 22723.3 1.08455
$$761$$ −33518.7 −1.59665 −0.798325 0.602227i $$-0.794280\pi$$
−0.798325 + 0.602227i $$0.794280\pi$$
$$762$$ −27603.5 −1.31229
$$763$$ −30280.8 −1.43675
$$764$$ −65780.0 −3.11497
$$765$$ −2837.93 −0.134125
$$766$$ −72812.5 −3.43449
$$767$$ 0 0
$$768$$ 13070.2 0.614102
$$769$$ −552.769 −0.0259211 −0.0129606 0.999916i $$-0.504126\pi$$
−0.0129606 + 0.999916i $$0.504126\pi$$
$$770$$ −15533.5 −0.726999
$$771$$ 17522.3 0.818482
$$772$$ 32673.3 1.52323
$$773$$ 36498.8 1.69828 0.849141 0.528166i $$-0.177120\pi$$
0.849141 + 0.528166i $$0.177120\pi$$
$$774$$ 5314.61 0.246809
$$775$$ −5354.62 −0.248185
$$776$$ −84352.8 −3.90218
$$777$$ −10368.0 −0.478700
$$778$$ 14635.4 0.674427
$$779$$ 3651.62 0.167950
$$780$$ 0 0
$$781$$ −2695.83 −0.123514
$$782$$ −51532.5 −2.35652
$$783$$ 4451.03 0.203150
$$784$$ 97029.2 4.42006
$$785$$ −6481.51 −0.294694
$$786$$ 14967.6 0.679231
$$787$$ 4284.79 0.194074 0.0970371 0.995281i $$-0.469063\pi$$
0.0970371 + 0.995281i $$0.469063\pi$$
$$788$$ 44126.0 1.99483
$$789$$ 12192.2 0.550131
$$790$$ −14616.3 −0.658258
$$791$$ 1512.28 0.0679779
$$792$$ −8591.91 −0.385480
$$793$$ 0 0
$$794$$ −34216.6 −1.52935
$$795$$ −4249.16 −0.189562
$$796$$ 17332.8 0.771792
$$797$$ −29538.5 −1.31281 −0.656405 0.754409i $$-0.727924\pi$$
−0.656405 + 0.754409i $$0.727924\pi$$
$$798$$ −32530.0 −1.44305
$$799$$ 27760.8 1.22917
$$800$$ 29549.8 1.30593
$$801$$ −600.430 −0.0264858
$$802$$ 24690.7 1.08711
$$803$$ −5132.65 −0.225563
$$804$$ 49224.7 2.15923
$$805$$ −33435.2 −1.46389
$$806$$ 0 0
$$807$$ −5536.32 −0.241497
$$808$$ 52565.5 2.28867
$$809$$ −895.586 −0.0389211 −0.0194605 0.999811i $$-0.506195\pi$$
−0.0194605 + 0.999811i $$0.506195\pi$$
$$810$$ −2468.51 −0.107080
$$811$$ −20139.7 −0.872011 −0.436006 0.899944i $$-0.643607\pi$$
−0.436006 + 0.899944i $$0.643607\pi$$
$$812$$ −99308.7 −4.29194
$$813$$ 6320.33 0.272649
$$814$$ −9375.73 −0.403709
$$815$$ −1361.75 −0.0585274
$$816$$ 24654.6 1.05770
$$817$$ 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 $$0.380411\pi$$
$$822$$ −249.282 −0.0105775
$$823$$ 12114.5 0.513104 0.256552 0.966530i $$-0.417413\pi$$
0.256552 + 0.966530i $$0.417413\pi$$
$$824$$ 1305.79 0.0552057
$$825$$ 4438.92 0.187325
$$826$$ −43313.8 −1.82455
$$827$$ −31450.9 −1.32243 −0.661217 0.750194i $$-0.729960\pi$$
−0.661217 + 0.750194i $$0.729960\pi$$
$$828$$ −31659.1 −1.32878
$$829$$ 13760.4 0.576499 0.288250 0.957555i $$-0.406927\pi$$
0.288250 + 0.957555i $$0.406927\pi$$
$$830$$ 17500.1 0.731852
$$831$$ 14345.8 0.598858
$$832$$ 0 0
$$833$$ −34428.2 −1.43201
$$834$$ 32476.6 1.34841
$$835$$ −1257.83 −0.0521305
$$836$$ −20776.8 −0.859545
$$837$$ −1590.45 −0.0656797
$$838$$ 28703.6 1.18323
$$839$$ 9846.21 0.405160 0.202580 0.979266i $$-0.435067\pi$$
0.202580 + 0.979266i $$0.435067\pi$$
$$840$$ 32172.7 1.32150
$$841$$ 2787.47 0.114292
$$842$$ −65914.2 −2.69781
$$843$$ −17597.4 −0.718965
$$844$$ 8291.83 0.338171
$$845$$ 0 0
$$846$$ 24147.2 0.981320
$$847$$ −33382.1 −1.35422
$$848$$ 36914.7 1.49488
$$849$$ 19221.1 0.776991
$$850$$ −25618.4 −1.03377
$$851$$ −20180.8 −0.812914
$$852$$ 9558.41 0.384349
$$853$$ 27574.5 1.10684 0.553420 0.832903i $$-0.313323\pi$$
0.553420 + 0.832903i $$0.313323\pi$$
$$854$$ 76499.6 3.06530
$$855$$ −3486.97 −0.139476
$$856$$ 20611.9 0.823013
$$857$$ −8046.95 −0.320745 −0.160373 0.987057i $$-0.551270\pi$$
−0.160373 + 0.987057i $$0.551270\pi$$
$$858$$ 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
$$861$$ 5170.14 0.204644
$$862$$ −37530.5 −1.48294
$$863$$ 4961.16 0.195689 0.0978447 0.995202i $$-0.468805\pi$$
0.0978447 + 0.995202i $$0.468805\pi$$
$$864$$ 8776.98 0.345601
$$865$$ −8076.69 −0.317475
$$866$$ −31641.0 −1.24158
$$867$$ 5991.00 0.234677
$$868$$ 35485.1 1.38761
$$869$$ 7806.72 0.304747
$$870$$ −15071.9 −0.587340
$$871$$ 0 0
$$872$$ 56714.5 2.20252
$$873$$ 12944.3 0.501829
$$874$$ −63318.2 −2.45054
$$875$$ −39478.3 −1.52527
$$876$$ 18198.4 0.701904
$$877$$ 1386.66 0.0533913 0.0266957 0.999644i $$-0.491502\pi$$
0.0266957 + 0.999644i $$0.491502\pi$$
$$878$$ −61919.3 −2.38004
$$879$$ 9030.71 0.346528
$$880$$ 14465.4 0.554123
$$881$$ 9030.36 0.345335 0.172668 0.984980i $$-0.444761\pi$$
0.172668 + 0.984980i $$0.444761\pi$$
$$882$$ −29946.6 −1.14326
$$883$$ 15512.7 0.591216 0.295608 0.955309i $$-0.404478\pi$$
0.295608 + 0.955309i $$0.404478\pi$$
$$884$$ 0 0
$$885$$ −4642.91 −0.176350
$$886$$ −55115.0 −2.08987
$$887$$ 7431.21 0.281303 0.140651 0.990059i $$-0.455080\pi$$
0.140651 + 0.990059i $$0.455080\pi$$
$$888$$ 19418.8 0.733843
$$889$$ −55207.0 −2.08277
$$890$$ 2033.15 0.0765747
$$891$$ 1318.46 0.0495737
$$892$$ −78957.5 −2.96378
$$893$$ 34109.8 1.27821
$$894$$ 43277.0 1.61902
$$895$$ 21271.6 0.794447
$$896$$ 3146.40 0.117315
$$897$$ 0 0
$$898$$ 66070.5 2.45524
$$899$$ −9710.74 −0.360257
$$900$$ −15738.7 −0.582916
$$901$$ −13098.2 −0.484310
$$902$$ 4675.34 0.172585
$$903$$ 10629.2 0.391715
$$904$$ −2832.44 −0.104210
$$905$$ 15310.6 0.562367
$$906$$ 25340.1 0.929213
$$907$$ 10550.2 0.386234 0.193117 0.981176i $$-0.438140\pi$$
0.193117 + 0.981176i $$0.438140\pi$$
$$908$$ 34803.1 1.27201
$$909$$ −8066.38 −0.294329
$$910$$ 0 0
$$911$$ 35703.3 1.29847 0.649234 0.760589i $$-0.275090\pi$$
0.649234 + 0.760589i $$0.275090\pi$$
$$912$$ 30293.2 1.09990
$$913$$ −9347.01 −0.338818
$$914$$ 8059.38 0.291664
$$915$$ 8200.18 0.296273
$$916$$ −100985. −3.64263
$$917$$ 29935.1 1.07802
$$918$$ −7609.27 −0.273577
$$919$$ −42896.6 −1.53975 −0.769873 0.638197i $$-0.779681\pi$$
−0.769873 + 0.638197i $$0.779681\pi$$
$$920$$ 62622.6 2.24414
$$921$$ 10025.5 0.358688
$$922$$ 68874.0 2.46013
$$923$$ 0 0
$$924$$ −29416.8 −1.04734
$$925$$ −10032.5 −0.356613
$$926$$ −85117.5 −3.02066
$$927$$ −200.379 −0.00709959
$$928$$ 53589.3 1.89564
$$929$$ 10366.7 0.366114 0.183057 0.983102i $$-0.441401\pi$$
0.183057 + 0.983102i $$0.441401\pi$$
$$930$$ 5385.52 0.189890
$$931$$ −42302.0 −1.48914
$$932$$ −54036.6 −1.89917
$$933$$ −26265.6 −0.921648
$$934$$ 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
$$939$$ −5846.79 −0.203198
$$940$$ −57750.7 −2.00385
$$941$$ −37089.1 −1.28488 −0.642438 0.766337i $$-0.722077\pi$$
−0.642438 + 0.766337i $$0.722077\pi$$
$$942$$ −17378.7 −0.601093
$$943$$ 10063.4 0.347519
$$944$$ 40335.4 1.39068
$$945$$ −4937.03 −0.169949
$$946$$ 9611.96 0.330351
$$947$$ −23458.2 −0.804952 −0.402476 0.915430i $$-0.631850\pi$$
−0.402476 + 0.915430i $$0.631850\pi$$
$$948$$ −27679.7 −0.948307
$$949$$ 0 0
$$950$$ −31477.5 −1.07501
$$951$$ 5821.28 0.198494
$$952$$ 99173.4 3.37629
$$953$$ 34695.5 1.17933 0.589663 0.807649i $$-0.299260\pi$$
0.589663 + 0.807649i $$0.299260\pi$$
$$954$$ −11393.2 −0.386654
$$955$$ −19966.6 −0.676550
$$956$$ 129124. 4.36838
$$957$$ 8050.09 0.271915
$$958$$ −94001.6 −3.17020
$$959$$ −498.563 −0.0167878
$$960$$ −8391.88 −0.282132
$$961$$ −26321.1 −0.883527
$$962$$ 0 0
$$963$$ −3162.97 −0.105841
$$964$$ 48462.1 1.61915
$$965$$ 9917.53 0.330836
$$966$$ −89649.0 −2.98593
$$967$$ −6289.66 −0.209164 −0.104582 0.994516i $$-0.533351\pi$$
−0.104582 + 0.994516i $$0.533351\pi$$
$$968$$ 62523.3 2.07601
$$969$$ −10748.7 −0.356345
$$970$$ −43831.4 −1.45087
$$971$$ −20185.9 −0.667145 −0.333573 0.942724i $$-0.608254\pi$$
−0.333573 + 0.942724i $$0.608254\pi$$
$$972$$ −4674.77 −0.154263
$$973$$ 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 $$0.757886\pi$$
$$978$$ −3651.22 −0.119379
$$979$$ −1085.93 −0.0354510
$$980$$ 71620.8 2.33453
$$981$$ −8703.07 −0.283249
$$982$$ 67669.9 2.19901
$$983$$ −8835.11 −0.286670 −0.143335 0.989674i $$-0.545783\pi$$
−0.143335 + 0.989674i $$0.545783\pi$$
$$984$$ −9683.46 −0.313717
$$985$$ 13393.9 0.433263
$$986$$ −46459.6 −1.50058
$$987$$ 48294.3 1.55747
$$988$$ 0 0
$$989$$ 20689.3 0.665198
$$990$$ −4464.53 −0.143325
$$991$$ 34915.1 1.11919 0.559594 0.828767i $$-0.310957\pi$$
0.559594 + 0.828767i $$0.310957\pi$$
$$992$$ −19148.6 −0.612872
$$993$$ 22206.4 0.709666
$$994$$ 27066.5 0.863680
$$995$$ 5261.15 0.167628
$$996$$ 33141.0 1.05433
$$997$$ 37962.8 1.20591 0.602956 0.797774i $$-0.293989\pi$$
0.602956 + 0.797774i $$0.293989\pi$$
$$998$$ 53314.6 1.69103
$$999$$ −2979.89 −0.0943740
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 507.4.a.j.1.1 4
3.2 odd 2 1521.4.a.x.1.4 4
13.5 odd 4 39.4.b.a.25.4 yes 4
13.8 odd 4 39.4.b.a.25.1 4
13.12 even 2 inner 507.4.a.j.1.4 4
39.5 even 4 117.4.b.d.64.1 4
39.8 even 4 117.4.b.d.64.4 4
39.38 odd 2 1521.4.a.x.1.1 4
52.31 even 4 624.4.c.e.337.2 4
52.47 even 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 13.8 odd 4
39.4.b.a.25.4 yes 4 13.5 odd 4
117.4.b.d.64.1 4 39.5 even 4
117.4.b.d.64.4 4 39.8 even 4
507.4.a.j.1.1 4 1.1 even 1 trivial
507.4.a.j.1.4 4 13.12 even 2 inner
624.4.c.e.337.2 4 52.31 even 4
624.4.c.e.337.3 4 52.47 even 4
1521.4.a.x.1.1 4 39.38 odd 2
1521.4.a.x.1.4 4 3.2 odd 2