# Properties

 Label 507.4.a.k.1.4 Level $507$ Weight $4$ Character 507.1 Self dual yes Analytic conductor $29.914$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.9139683729$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - 2x^{3} - 13x^{2} + 14x - 2$$ x^4 - 2*x^3 - 13*x^2 + 14*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 1.4 Root $$4.29360$$ of defining polynomial Character $$\chi$$ $$=$$ 507.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.12311 q^{2} -3.00000 q^{3} +9.00000 q^{4} -3.05006 q^{5} -12.3693 q^{6} -6.68324 q^{7} +4.12311 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+4.12311 q^{2} -3.00000 q^{3} +9.00000 q^{4} -3.05006 q^{5} -12.3693 q^{6} -6.68324 q^{7} +4.12311 q^{8} +9.00000 q^{9} -12.5757 q^{10} +32.2500 q^{11} -27.0000 q^{12} -27.5557 q^{14} +9.15018 q^{15} -55.0000 q^{16} -28.8586 q^{17} +37.1080 q^{18} -101.194 q^{19} -27.4505 q^{20} +20.0497 q^{21} +132.970 q^{22} -118.990 q^{23} -12.3693 q^{24} -115.697 q^{25} -27.0000 q^{27} -60.1492 q^{28} +160.111 q^{29} +37.7271 q^{30} -38.0705 q^{31} -259.756 q^{32} -96.7499 q^{33} -118.987 q^{34} +20.3843 q^{35} +81.0000 q^{36} -327.568 q^{37} -417.233 q^{38} -12.5757 q^{40} +56.0846 q^{41} +82.6671 q^{42} +127.879 q^{43} +290.250 q^{44} -27.4505 q^{45} -490.608 q^{46} -517.983 q^{47} +165.000 q^{48} -298.334 q^{49} -477.032 q^{50} +86.5757 q^{51} -695.546 q^{53} -111.324 q^{54} -98.3643 q^{55} -27.5557 q^{56} +303.581 q^{57} +660.156 q^{58} +656.523 q^{59} +82.3516 q^{60} +701.304 q^{61} -156.969 q^{62} -60.1492 q^{63} -631.000 q^{64} -398.910 q^{66} -57.1750 q^{67} -259.727 q^{68} +356.970 q^{69} +84.0465 q^{70} +309.226 q^{71} +37.1080 q^{72} -389.711 q^{73} -1350.60 q^{74} +347.091 q^{75} -910.744 q^{76} -215.534 q^{77} +901.820 q^{79} +167.753 q^{80} +81.0000 q^{81} +231.243 q^{82} +687.095 q^{83} +180.448 q^{84} +88.0203 q^{85} +527.257 q^{86} -480.334 q^{87} +132.970 q^{88} +1070.54 q^{89} -113.181 q^{90} -1070.91 q^{92} +114.211 q^{93} -2135.70 q^{94} +308.647 q^{95} +779.267 q^{96} +1754.48 q^{97} -1230.06 q^{98} +290.250 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} + 36 q^{4} + 36 q^{9}+O(q^{10})$$ 4 * q - 12 * q^3 + 36 * q^4 + 36 * q^9 $$4 q - 12 q^{3} + 36 q^{4} + 36 q^{9} - 136 q^{10} - 108 q^{12} + 204 q^{14} - 220 q^{16} - 144 q^{17} - 68 q^{22} - 276 q^{23} - 120 q^{25} - 108 q^{27} + 12 q^{29} + 408 q^{30} - 804 q^{35} + 324 q^{36} - 612 q^{38} - 136 q^{40} - 612 q^{42} + 940 q^{43} + 660 q^{48} + 692 q^{49} + 432 q^{51} - 2268 q^{53} + 892 q^{55} + 204 q^{56} + 320 q^{61} - 2856 q^{62} - 2524 q^{64} + 204 q^{66} - 1296 q^{68} + 828 q^{69} - 3060 q^{74} + 360 q^{75} - 2976 q^{77} + 8 q^{79} + 324 q^{81} + 68 q^{82} - 36 q^{87} - 68 q^{88} - 1224 q^{90} - 2484 q^{92} - 5372 q^{94} - 108 q^{95}+O(q^{100})$$ 4 * q - 12 * q^3 + 36 * q^4 + 36 * q^9 - 136 * q^10 - 108 * q^12 + 204 * q^14 - 220 * q^16 - 144 * q^17 - 68 * q^22 - 276 * q^23 - 120 * q^25 - 108 * q^27 + 12 * q^29 + 408 * q^30 - 804 * q^35 + 324 * q^36 - 612 * q^38 - 136 * q^40 - 612 * q^42 + 940 * q^43 + 660 * q^48 + 692 * q^49 + 432 * q^51 - 2268 * q^53 + 892 * q^55 + 204 * q^56 + 320 * q^61 - 2856 * q^62 - 2524 * q^64 + 204 * q^66 - 1296 * q^68 + 828 * q^69 - 3060 * q^74 + 360 * q^75 - 2976 * q^77 + 8 * q^79 + 324 * q^81 + 68 * q^82 - 36 * q^87 - 68 * q^88 - 1224 * q^90 - 2484 * q^92 - 5372 * q^94 - 108 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.12311 1.45774 0.728869 0.684653i $$-0.240046\pi$$
0.728869 + 0.684653i $$0.240046\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 9.00000 1.12500
$$5$$ −3.05006 −0.272806 −0.136403 0.990653i $$-0.543554\pi$$
−0.136403 + 0.990653i $$0.543554\pi$$
$$6$$ −12.3693 −0.841625
$$7$$ −6.68324 −0.360861 −0.180431 0.983588i $$-0.557749\pi$$
−0.180431 + 0.983588i $$0.557749\pi$$
$$8$$ 4.12311 0.182217
$$9$$ 9.00000 0.333333
$$10$$ −12.5757 −0.397679
$$11$$ 32.2500 0.883975 0.441988 0.897021i $$-0.354273\pi$$
0.441988 + 0.897021i $$0.354273\pi$$
$$12$$ −27.0000 −0.649519
$$13$$ 0 0
$$14$$ −27.5557 −0.526041
$$15$$ 9.15018 0.157504
$$16$$ −55.0000 −0.859375
$$17$$ −28.8586 −0.411720 −0.205860 0.978582i $$-0.565999\pi$$
−0.205860 + 0.978582i $$0.565999\pi$$
$$18$$ 37.1080 0.485913
$$19$$ −101.194 −1.22187 −0.610933 0.791682i $$-0.709206\pi$$
−0.610933 + 0.791682i $$0.709206\pi$$
$$20$$ −27.4505 −0.306906
$$21$$ 20.0497 0.208343
$$22$$ 132.970 1.28860
$$23$$ −118.990 −1.07874 −0.539372 0.842067i $$-0.681338\pi$$
−0.539372 + 0.842067i $$0.681338\pi$$
$$24$$ −12.3693 −0.105203
$$25$$ −115.697 −0.925577
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ −60.1492 −0.405969
$$29$$ 160.111 1.02524 0.512620 0.858616i $$-0.328675\pi$$
0.512620 + 0.858616i $$0.328675\pi$$
$$30$$ 37.7271 0.229600
$$31$$ −38.0705 −0.220570 −0.110285 0.993900i $$-0.535176\pi$$
−0.110285 + 0.993900i $$0.535176\pi$$
$$32$$ −259.756 −1.43496
$$33$$ −96.7499 −0.510363
$$34$$ −118.987 −0.600179
$$35$$ 20.3843 0.0984449
$$36$$ 81.0000 0.375000
$$37$$ −327.568 −1.45545 −0.727727 0.685866i $$-0.759423\pi$$
−0.727727 + 0.685866i $$0.759423\pi$$
$$38$$ −417.233 −1.78116
$$39$$ 0 0
$$40$$ −12.5757 −0.0497099
$$41$$ 56.0846 0.213633 0.106816 0.994279i $$-0.465934\pi$$
0.106816 + 0.994279i $$0.465934\pi$$
$$42$$ 82.6671 0.303710
$$43$$ 127.879 0.453519 0.226759 0.973951i $$-0.427187\pi$$
0.226759 + 0.973951i $$0.427187\pi$$
$$44$$ 290.250 0.994472
$$45$$ −27.4505 −0.0909352
$$46$$ −490.608 −1.57253
$$47$$ −517.983 −1.60757 −0.803783 0.594923i $$-0.797183\pi$$
−0.803783 + 0.594923i $$0.797183\pi$$
$$48$$ 165.000 0.496160
$$49$$ −298.334 −0.869779
$$50$$ −477.032 −1.34925
$$51$$ 86.5757 0.237706
$$52$$ 0 0
$$53$$ −695.546 −1.80265 −0.901326 0.433141i $$-0.857405\pi$$
−0.901326 + 0.433141i $$0.857405\pi$$
$$54$$ −111.324 −0.280542
$$55$$ −98.3643 −0.241153
$$56$$ −27.5557 −0.0657551
$$57$$ 303.581 0.705445
$$58$$ 660.156 1.49453
$$59$$ 656.523 1.44868 0.724339 0.689444i $$-0.242145\pi$$
0.724339 + 0.689444i $$0.242145\pi$$
$$60$$ 82.3516 0.177192
$$61$$ 701.304 1.47201 0.736007 0.676974i $$-0.236709\pi$$
0.736007 + 0.676974i $$0.236709\pi$$
$$62$$ −156.969 −0.321533
$$63$$ −60.1492 −0.120287
$$64$$ −631.000 −1.23242
$$65$$ 0 0
$$66$$ −398.910 −0.743976
$$67$$ −57.1750 −0.104254 −0.0521271 0.998640i $$-0.516600\pi$$
−0.0521271 + 0.998640i $$0.516600\pi$$
$$68$$ −259.727 −0.463184
$$69$$ 356.970 0.622814
$$70$$ 84.0465 0.143507
$$71$$ 309.226 0.516879 0.258440 0.966027i $$-0.416792\pi$$
0.258440 + 0.966027i $$0.416792\pi$$
$$72$$ 37.1080 0.0607391
$$73$$ −389.711 −0.624826 −0.312413 0.949946i $$-0.601137\pi$$
−0.312413 + 0.949946i $$0.601137\pi$$
$$74$$ −1350.60 −2.12167
$$75$$ 347.091 0.534382
$$76$$ −910.744 −1.37460
$$77$$ −215.534 −0.318992
$$78$$ 0 0
$$79$$ 901.820 1.28434 0.642169 0.766563i $$-0.278035\pi$$
0.642169 + 0.766563i $$0.278035\pi$$
$$80$$ 167.753 0.234442
$$81$$ 81.0000 0.111111
$$82$$ 231.243 0.311421
$$83$$ 687.095 0.908657 0.454328 0.890834i $$-0.349879\pi$$
0.454328 + 0.890834i $$0.349879\pi$$
$$84$$ 180.448 0.234386
$$85$$ 88.0203 0.112319
$$86$$ 527.257 0.661111
$$87$$ −480.334 −0.591922
$$88$$ 132.970 0.161076
$$89$$ 1070.54 1.27502 0.637510 0.770442i $$-0.279964\pi$$
0.637510 + 0.770442i $$0.279964\pi$$
$$90$$ −113.181 −0.132560
$$91$$ 0 0
$$92$$ −1070.91 −1.21359
$$93$$ 114.211 0.127346
$$94$$ −2135.70 −2.34341
$$95$$ 308.647 0.333332
$$96$$ 779.267 0.828475
$$97$$ 1754.48 1.83650 0.918251 0.395998i $$-0.129601\pi$$
0.918251 + 0.395998i $$0.129601\pi$$
$$98$$ −1230.06 −1.26791
$$99$$ 290.250 0.294658
$$100$$ −1041.27 −1.04127
$$101$$ −640.840 −0.631346 −0.315673 0.948868i $$-0.602230\pi$$
−0.315673 + 0.948868i $$0.602230\pi$$
$$102$$ 356.961 0.346514
$$103$$ −693.153 −0.663091 −0.331546 0.943439i $$-0.607570\pi$$
−0.331546 + 0.943439i $$0.607570\pi$$
$$104$$ 0 0
$$105$$ −61.1528 −0.0568372
$$106$$ −2867.81 −2.62779
$$107$$ −405.676 −0.366525 −0.183262 0.983064i $$-0.558666\pi$$
−0.183262 + 0.983064i $$0.558666\pi$$
$$108$$ −243.000 −0.216506
$$109$$ −479.516 −0.421370 −0.210685 0.977554i $$-0.567569\pi$$
−0.210685 + 0.977554i $$0.567569\pi$$
$$110$$ −405.566 −0.351538
$$111$$ 982.704 0.840307
$$112$$ 367.578 0.310115
$$113$$ −1547.20 −1.28804 −0.644021 0.765008i $$-0.722735\pi$$
−0.644021 + 0.765008i $$0.722735\pi$$
$$114$$ 1251.70 1.02835
$$115$$ 362.926 0.294288
$$116$$ 1441.00 1.15339
$$117$$ 0 0
$$118$$ 2706.91 2.11179
$$119$$ 192.869 0.148574
$$120$$ 37.7271 0.0287000
$$121$$ −290.940 −0.218588
$$122$$ 2891.55 2.14581
$$123$$ −168.254 −0.123341
$$124$$ −342.634 −0.248141
$$125$$ 734.140 0.525308
$$126$$ −248.001 −0.175347
$$127$$ −2495.15 −1.74338 −0.871690 0.490059i $$-0.836975\pi$$
−0.871690 + 0.490059i $$0.836975\pi$$
$$128$$ −523.634 −0.361587
$$129$$ −383.636 −0.261839
$$130$$ 0 0
$$131$$ 43.7571 0.0291838 0.0145919 0.999894i $$-0.495355\pi$$
0.0145919 + 0.999894i $$0.495355\pi$$
$$132$$ −870.749 −0.574159
$$133$$ 676.303 0.440924
$$134$$ −235.739 −0.151975
$$135$$ 82.3516 0.0525015
$$136$$ −118.987 −0.0750224
$$137$$ −206.194 −0.128586 −0.0642932 0.997931i $$-0.520479\pi$$
−0.0642932 + 0.997931i $$0.520479\pi$$
$$138$$ 1471.83 0.907899
$$139$$ −100.000 −0.0610208 −0.0305104 0.999534i $$-0.509713\pi$$
−0.0305104 + 0.999534i $$0.509713\pi$$
$$140$$ 183.459 0.110751
$$141$$ 1553.95 0.928128
$$142$$ 1274.97 0.753474
$$143$$ 0 0
$$144$$ −495.000 −0.286458
$$145$$ −488.349 −0.279691
$$146$$ −1606.82 −0.910832
$$147$$ 895.003 0.502167
$$148$$ −2948.11 −1.63739
$$149$$ 380.451 0.209179 0.104590 0.994515i $$-0.466647\pi$$
0.104590 + 0.994515i $$0.466647\pi$$
$$150$$ 1431.09 0.778989
$$151$$ −1517.45 −0.817805 −0.408902 0.912578i $$-0.634088\pi$$
−0.408902 + 0.912578i $$0.634088\pi$$
$$152$$ −417.233 −0.222645
$$153$$ −259.727 −0.137240
$$154$$ −888.671 −0.465007
$$155$$ 116.117 0.0601726
$$156$$ 0 0
$$157$$ 1450.16 0.737166 0.368583 0.929595i $$-0.379843\pi$$
0.368583 + 0.929595i $$0.379843\pi$$
$$158$$ 3718.30 1.87223
$$159$$ 2086.64 1.04076
$$160$$ 792.270 0.391465
$$161$$ 795.239 0.389277
$$162$$ 333.972 0.161971
$$163$$ 2342.36 1.12557 0.562785 0.826603i $$-0.309730\pi$$
0.562785 + 0.826603i $$0.309730\pi$$
$$164$$ 504.762 0.240337
$$165$$ 295.093 0.139230
$$166$$ 2832.97 1.32458
$$167$$ −40.0731 −0.0185686 −0.00928428 0.999957i $$-0.502955\pi$$
−0.00928428 + 0.999957i $$0.502955\pi$$
$$168$$ 82.6671 0.0379637
$$169$$ 0 0
$$170$$ 362.917 0.163732
$$171$$ −910.744 −0.407289
$$172$$ 1150.91 0.510208
$$173$$ 1909.54 0.839189 0.419594 0.907712i $$-0.362172\pi$$
0.419594 + 0.907712i $$0.362172\pi$$
$$174$$ −1980.47 −0.862868
$$175$$ 773.232 0.334005
$$176$$ −1773.75 −0.759666
$$177$$ −1969.57 −0.836395
$$178$$ 4413.94 1.85864
$$179$$ −509.959 −0.212939 −0.106470 0.994316i $$-0.533955\pi$$
−0.106470 + 0.994316i $$0.533955\pi$$
$$180$$ −247.055 −0.102302
$$181$$ −2136.88 −0.877531 −0.438766 0.898602i $$-0.644584\pi$$
−0.438766 + 0.898602i $$0.644584\pi$$
$$182$$ 0 0
$$183$$ −2103.91 −0.849867
$$184$$ −490.608 −0.196566
$$185$$ 999.101 0.397056
$$186$$ 470.906 0.185637
$$187$$ −930.688 −0.363950
$$188$$ −4661.85 −1.80851
$$189$$ 180.448 0.0694478
$$190$$ 1272.58 0.485911
$$191$$ 4057.74 1.53721 0.768607 0.639721i $$-0.220950\pi$$
0.768607 + 0.639721i $$0.220950\pi$$
$$192$$ 1893.00 0.711539
$$193$$ 873.394 0.325742 0.162871 0.986647i $$-0.447925\pi$$
0.162871 + 0.986647i $$0.447925\pi$$
$$194$$ 7233.92 2.67714
$$195$$ 0 0
$$196$$ −2685.01 −0.978502
$$197$$ 4147.25 1.49989 0.749947 0.661498i $$-0.230079\pi$$
0.749947 + 0.661498i $$0.230079\pi$$
$$198$$ 1196.73 0.429535
$$199$$ −2404.06 −0.856379 −0.428189 0.903689i $$-0.640848\pi$$
−0.428189 + 0.903689i $$0.640848\pi$$
$$200$$ −477.032 −0.168656
$$201$$ 171.525 0.0601912
$$202$$ −2642.25 −0.920337
$$203$$ −1070.06 −0.369969
$$204$$ 779.181 0.267420
$$205$$ −171.061 −0.0582802
$$206$$ −2857.94 −0.966613
$$207$$ −1070.91 −0.359582
$$208$$ 0 0
$$209$$ −3263.50 −1.08010
$$210$$ −252.140 −0.0828538
$$211$$ 3868.85 1.26229 0.631144 0.775665i $$-0.282586\pi$$
0.631144 + 0.775665i $$0.282586\pi$$
$$212$$ −6259.91 −2.02798
$$213$$ −927.679 −0.298420
$$214$$ −1672.64 −0.534297
$$215$$ −390.037 −0.123722
$$216$$ −111.324 −0.0350677
$$217$$ 254.434 0.0795950
$$218$$ −1977.09 −0.614246
$$219$$ 1169.13 0.360743
$$220$$ −885.279 −0.271298
$$221$$ 0 0
$$222$$ 4051.79 1.22495
$$223$$ 2813.55 0.844885 0.422443 0.906390i $$-0.361173\pi$$
0.422443 + 0.906390i $$0.361173\pi$$
$$224$$ 1736.01 0.517822
$$225$$ −1041.27 −0.308526
$$226$$ −6379.29 −1.87763
$$227$$ −4518.37 −1.32112 −0.660561 0.750772i $$-0.729682\pi$$
−0.660561 + 0.750772i $$0.729682\pi$$
$$228$$ 2732.23 0.793625
$$229$$ −1305.27 −0.376658 −0.188329 0.982106i $$-0.560307\pi$$
−0.188329 + 0.982106i $$0.560307\pi$$
$$230$$ 1496.38 0.428994
$$231$$ 646.603 0.184170
$$232$$ 660.156 0.186816
$$233$$ −3360.55 −0.944879 −0.472440 0.881363i $$-0.656627\pi$$
−0.472440 + 0.881363i $$0.656627\pi$$
$$234$$ 0 0
$$235$$ 1579.88 0.438553
$$236$$ 5908.71 1.62976
$$237$$ −2705.46 −0.741513
$$238$$ 795.219 0.216581
$$239$$ −4737.17 −1.28210 −0.641050 0.767499i $$-0.721501\pi$$
−0.641050 + 0.767499i $$0.721501\pi$$
$$240$$ −503.260 −0.135355
$$241$$ 4785.28 1.27903 0.639516 0.768778i $$-0.279135\pi$$
0.639516 + 0.768778i $$0.279135\pi$$
$$242$$ −1199.58 −0.318643
$$243$$ −243.000 −0.0641500
$$244$$ 6311.74 1.65601
$$245$$ 909.937 0.237281
$$246$$ −693.729 −0.179799
$$247$$ 0 0
$$248$$ −156.969 −0.0401916
$$249$$ −2061.29 −0.524613
$$250$$ 3026.94 0.765762
$$251$$ −3273.86 −0.823284 −0.411642 0.911346i $$-0.635045\pi$$
−0.411642 + 0.911346i $$0.635045\pi$$
$$252$$ −541.343 −0.135323
$$253$$ −3837.42 −0.953584
$$254$$ −10287.8 −2.54139
$$255$$ −264.061 −0.0648476
$$256$$ 2889.00 0.705322
$$257$$ −6545.81 −1.58878 −0.794390 0.607408i $$-0.792209\pi$$
−0.794390 + 0.607408i $$0.792209\pi$$
$$258$$ −1581.77 −0.381693
$$259$$ 2189.22 0.525217
$$260$$ 0 0
$$261$$ 1441.00 0.341746
$$262$$ 180.415 0.0425424
$$263$$ 88.2014 0.0206796 0.0103398 0.999947i $$-0.496709\pi$$
0.0103398 + 0.999947i $$0.496709\pi$$
$$264$$ −398.910 −0.0929970
$$265$$ 2121.46 0.491773
$$266$$ 2788.47 0.642752
$$267$$ −3211.61 −0.736133
$$268$$ −514.575 −0.117286
$$269$$ −4527.60 −1.02622 −0.513109 0.858324i $$-0.671506\pi$$
−0.513109 + 0.858324i $$0.671506\pi$$
$$270$$ 339.544 0.0765334
$$271$$ 8321.82 1.86537 0.932684 0.360695i $$-0.117460\pi$$
0.932684 + 0.360695i $$0.117460\pi$$
$$272$$ 1587.22 0.353821
$$273$$ 0 0
$$274$$ −850.160 −0.187445
$$275$$ −3731.23 −0.818187
$$276$$ 3212.73 0.700665
$$277$$ −2881.31 −0.624986 −0.312493 0.949920i $$-0.601164\pi$$
−0.312493 + 0.949920i $$0.601164\pi$$
$$278$$ −412.311 −0.0889523
$$279$$ −342.634 −0.0735232
$$280$$ 84.0465 0.0179384
$$281$$ −2817.99 −0.598247 −0.299123 0.954214i $$-0.596694\pi$$
−0.299123 + 0.954214i $$0.596694\pi$$
$$282$$ 6407.10 1.35297
$$283$$ 264.601 0.0555792 0.0277896 0.999614i $$-0.491153\pi$$
0.0277896 + 0.999614i $$0.491153\pi$$
$$284$$ 2783.04 0.581489
$$285$$ −925.941 −0.192449
$$286$$ 0 0
$$287$$ −374.827 −0.0770918
$$288$$ −2337.80 −0.478320
$$289$$ −4080.18 −0.830487
$$290$$ −2013.52 −0.407716
$$291$$ −5263.45 −1.06031
$$292$$ −3507.40 −0.702929
$$293$$ 4292.52 0.855877 0.427938 0.903808i $$-0.359240\pi$$
0.427938 + 0.903808i $$0.359240\pi$$
$$294$$ 3690.19 0.732028
$$295$$ −2002.43 −0.395208
$$296$$ −1350.60 −0.265209
$$297$$ −870.749 −0.170121
$$298$$ 1568.64 0.304929
$$299$$ 0 0
$$300$$ 3123.82 0.601180
$$301$$ −854.643 −0.163657
$$302$$ −6256.62 −1.19214
$$303$$ 1922.52 0.364508
$$304$$ 5565.66 1.05004
$$305$$ −2139.02 −0.401573
$$306$$ −1070.88 −0.200060
$$307$$ 7026.26 1.30622 0.653110 0.757263i $$-0.273464\pi$$
0.653110 + 0.757263i $$0.273464\pi$$
$$308$$ −1939.81 −0.358866
$$309$$ 2079.46 0.382836
$$310$$ 478.763 0.0877159
$$311$$ −1133.21 −0.206618 −0.103309 0.994649i $$-0.532943\pi$$
−0.103309 + 0.994649i $$0.532943\pi$$
$$312$$ 0 0
$$313$$ −5285.95 −0.954566 −0.477283 0.878750i $$-0.658378\pi$$
−0.477283 + 0.878750i $$0.658378\pi$$
$$314$$ 5979.15 1.07459
$$315$$ 183.459 0.0328150
$$316$$ 8116.38 1.44488
$$317$$ −4782.16 −0.847296 −0.423648 0.905827i $$-0.639251\pi$$
−0.423648 + 0.905827i $$0.639251\pi$$
$$318$$ 8603.43 1.51716
$$319$$ 5163.59 0.906286
$$320$$ 1924.59 0.336212
$$321$$ 1217.03 0.211613
$$322$$ 3278.85 0.567464
$$323$$ 2920.31 0.503066
$$324$$ 729.000 0.125000
$$325$$ 0 0
$$326$$ 9657.80 1.64079
$$327$$ 1438.55 0.243278
$$328$$ 231.243 0.0389276
$$329$$ 3461.81 0.580108
$$330$$ 1216.70 0.202961
$$331$$ 8669.98 1.43971 0.719857 0.694122i $$-0.244207\pi$$
0.719857 + 0.694122i $$0.244207\pi$$
$$332$$ 6183.86 1.02224
$$333$$ −2948.11 −0.485152
$$334$$ −165.226 −0.0270681
$$335$$ 174.387 0.0284412
$$336$$ −1102.73 −0.179045
$$337$$ 8526.59 1.37826 0.689129 0.724639i $$-0.257993\pi$$
0.689129 + 0.724639i $$0.257993\pi$$
$$338$$ 0 0
$$339$$ 4641.61 0.743651
$$340$$ 792.183 0.126359
$$341$$ −1227.77 −0.194978
$$342$$ −3755.10 −0.593720
$$343$$ 4286.19 0.674731
$$344$$ 527.257 0.0826389
$$345$$ −1088.78 −0.169907
$$346$$ 7873.23 1.22332
$$347$$ −12581.5 −1.94643 −0.973213 0.229907i $$-0.926158\pi$$
−0.973213 + 0.229907i $$0.926158\pi$$
$$348$$ −4323.01 −0.665913
$$349$$ −8961.18 −1.37444 −0.687222 0.726447i $$-0.741170\pi$$
−0.687222 + 0.726447i $$0.741170\pi$$
$$350$$ 3188.12 0.486892
$$351$$ 0 0
$$352$$ −8377.11 −1.26847
$$353$$ −5357.99 −0.807868 −0.403934 0.914788i $$-0.632357\pi$$
−0.403934 + 0.914788i $$0.632357\pi$$
$$354$$ −8120.74 −1.21924
$$355$$ −943.159 −0.141007
$$356$$ 9634.84 1.43440
$$357$$ −578.606 −0.0857790
$$358$$ −2102.61 −0.310409
$$359$$ −2705.40 −0.397731 −0.198866 0.980027i $$-0.563726\pi$$
−0.198866 + 0.980027i $$0.563726\pi$$
$$360$$ −113.181 −0.0165700
$$361$$ 3381.19 0.492957
$$362$$ −8810.59 −1.27921
$$363$$ 872.820 0.126202
$$364$$ 0 0
$$365$$ 1188.64 0.170456
$$366$$ −8674.65 −1.23888
$$367$$ 10473.8 1.48972 0.744858 0.667223i $$-0.232517\pi$$
0.744858 + 0.667223i $$0.232517\pi$$
$$368$$ 6544.45 0.927046
$$369$$ 504.762 0.0712110
$$370$$ 4119.40 0.578804
$$371$$ 4648.50 0.650507
$$372$$ 1027.90 0.143264
$$373$$ −12763.0 −1.77170 −0.885850 0.463973i $$-0.846424\pi$$
−0.885850 + 0.463973i $$0.846424\pi$$
$$374$$ −3837.32 −0.530544
$$375$$ −2202.42 −0.303287
$$376$$ −2135.70 −0.292926
$$377$$ 0 0
$$378$$ 744.004 0.101237
$$379$$ −2318.02 −0.314166 −0.157083 0.987585i $$-0.550209\pi$$
−0.157083 + 0.987585i $$0.550209\pi$$
$$380$$ 2777.82 0.374998
$$381$$ 7485.46 1.00654
$$382$$ 16730.5 2.24086
$$383$$ 1983.34 0.264606 0.132303 0.991209i $$-0.457763\pi$$
0.132303 + 0.991209i $$0.457763\pi$$
$$384$$ 1570.90 0.208763
$$385$$ 657.392 0.0870229
$$386$$ 3601.10 0.474847
$$387$$ 1150.91 0.151173
$$388$$ 15790.3 2.06607
$$389$$ −3244.51 −0.422887 −0.211444 0.977390i $$-0.567816\pi$$
−0.211444 + 0.977390i $$0.567816\pi$$
$$390$$ 0 0
$$391$$ 3433.88 0.444140
$$392$$ −1230.06 −0.158489
$$393$$ −131.271 −0.0168493
$$394$$ 17099.5 2.18645
$$395$$ −2750.60 −0.350374
$$396$$ 2612.25 0.331491
$$397$$ 3759.72 0.475302 0.237651 0.971351i $$-0.423622\pi$$
0.237651 + 0.971351i $$0.423622\pi$$
$$398$$ −9912.20 −1.24838
$$399$$ −2028.91 −0.254568
$$400$$ 6363.34 0.795418
$$401$$ −1997.55 −0.248760 −0.124380 0.992235i $$-0.539694\pi$$
−0.124380 + 0.992235i $$0.539694\pi$$
$$402$$ 707.216 0.0877431
$$403$$ 0 0
$$404$$ −5767.56 −0.710264
$$405$$ −247.055 −0.0303117
$$406$$ −4411.98 −0.539318
$$407$$ −10564.1 −1.28659
$$408$$ 356.961 0.0433142
$$409$$ −5195.23 −0.628087 −0.314044 0.949409i $$-0.601684\pi$$
−0.314044 + 0.949409i $$0.601684\pi$$
$$410$$ −705.304 −0.0849573
$$411$$ 618.582 0.0742394
$$412$$ −6238.38 −0.745977
$$413$$ −4387.70 −0.522772
$$414$$ −4415.48 −0.524176
$$415$$ −2095.68 −0.247887
$$416$$ 0 0
$$417$$ 300.000 0.0352304
$$418$$ −13455.7 −1.57450
$$419$$ −6822.11 −0.795422 −0.397711 0.917511i $$-0.630195\pi$$
−0.397711 + 0.917511i $$0.630195\pi$$
$$420$$ −550.376 −0.0639419
$$421$$ 7537.70 0.872601 0.436300 0.899801i $$-0.356288\pi$$
0.436300 + 0.899801i $$0.356288\pi$$
$$422$$ 15951.7 1.84009
$$423$$ −4661.85 −0.535855
$$424$$ −2867.81 −0.328474
$$425$$ 3338.85 0.381078
$$426$$ −3824.92 −0.435019
$$427$$ −4686.99 −0.531192
$$428$$ −3651.08 −0.412340
$$429$$ 0 0
$$430$$ −1608.16 −0.180355
$$431$$ −13404.2 −1.49805 −0.749023 0.662544i $$-0.769477\pi$$
−0.749023 + 0.662544i $$0.769477\pi$$
$$432$$ 1485.00 0.165387
$$433$$ −17715.9 −1.96622 −0.983110 0.183014i $$-0.941415\pi$$
−0.983110 + 0.183014i $$0.941415\pi$$
$$434$$ 1049.06 0.116029
$$435$$ 1465.05 0.161480
$$436$$ −4315.64 −0.474041
$$437$$ 12041.1 1.31808
$$438$$ 4820.46 0.525869
$$439$$ 7163.47 0.778801 0.389401 0.921068i $$-0.372682\pi$$
0.389401 + 0.921068i $$0.372682\pi$$
$$440$$ −405.566 −0.0439423
$$441$$ −2685.01 −0.289926
$$442$$ 0 0
$$443$$ −10169.2 −1.09064 −0.545321 0.838227i $$-0.683592\pi$$
−0.545321 + 0.838227i $$0.683592\pi$$
$$444$$ 8844.33 0.945346
$$445$$ −3265.20 −0.347833
$$446$$ 11600.6 1.23162
$$447$$ −1141.35 −0.120770
$$448$$ 4217.13 0.444733
$$449$$ −17142.5 −1.80179 −0.900895 0.434037i $$-0.857089\pi$$
−0.900895 + 0.434037i $$0.857089\pi$$
$$450$$ −4293.28 −0.449750
$$451$$ 1808.73 0.188846
$$452$$ −13924.8 −1.44905
$$453$$ 4552.36 0.472160
$$454$$ −18629.7 −1.92585
$$455$$ 0 0
$$456$$ 1251.70 0.128544
$$457$$ 14091.1 1.44235 0.721177 0.692750i $$-0.243601\pi$$
0.721177 + 0.692750i $$0.243601\pi$$
$$458$$ −5381.76 −0.549068
$$459$$ 779.181 0.0792355
$$460$$ 3266.34 0.331074
$$461$$ −2922.22 −0.295231 −0.147616 0.989045i $$-0.547160\pi$$
−0.147616 + 0.989045i $$0.547160\pi$$
$$462$$ 2666.01 0.268472
$$463$$ −2072.61 −0.208040 −0.104020 0.994575i $$-0.533171\pi$$
−0.104020 + 0.994575i $$0.533171\pi$$
$$464$$ −8806.13 −0.881065
$$465$$ −348.352 −0.0347407
$$466$$ −13855.9 −1.37739
$$467$$ −2664.19 −0.263992 −0.131996 0.991250i $$-0.542139\pi$$
−0.131996 + 0.991250i $$0.542139\pi$$
$$468$$ 0 0
$$469$$ 382.114 0.0376213
$$470$$ 6514.01 0.639295
$$471$$ −4350.47 −0.425603
$$472$$ 2706.91 0.263974
$$473$$ 4124.08 0.400899
$$474$$ −11154.9 −1.08093
$$475$$ 11707.8 1.13093
$$476$$ 1735.82 0.167145
$$477$$ −6259.91 −0.600884
$$478$$ −19531.8 −1.86897
$$479$$ −5220.70 −0.497995 −0.248998 0.968504i $$-0.580101\pi$$
−0.248998 + 0.968504i $$0.580101\pi$$
$$480$$ −2376.81 −0.226013
$$481$$ 0 0
$$482$$ 19730.2 1.86449
$$483$$ −2385.72 −0.224749
$$484$$ −2618.46 −0.245911
$$485$$ −5351.28 −0.501008
$$486$$ −1001.91 −0.0935139
$$487$$ −12224.6 −1.13747 −0.568737 0.822520i $$-0.692568\pi$$
−0.568737 + 0.822520i $$0.692568\pi$$
$$488$$ 2891.55 0.268226
$$489$$ −7027.08 −0.649848
$$490$$ 3751.77 0.345893
$$491$$ 19653.2 1.80639 0.903195 0.429231i $$-0.141216\pi$$
0.903195 + 0.429231i $$0.141216\pi$$
$$492$$ −1514.29 −0.138759
$$493$$ −4620.59 −0.422111
$$494$$ 0 0
$$495$$ −885.279 −0.0803845
$$496$$ 2093.88 0.189552
$$497$$ −2066.63 −0.186522
$$498$$ −8498.90 −0.764749
$$499$$ −11713.6 −1.05084 −0.525422 0.850842i $$-0.676093\pi$$
−0.525422 + 0.850842i $$0.676093\pi$$
$$500$$ 6607.26 0.590972
$$501$$ 120.219 0.0107206
$$502$$ −13498.5 −1.20013
$$503$$ 13003.3 1.15266 0.576332 0.817216i $$-0.304483\pi$$
0.576332 + 0.817216i $$0.304483\pi$$
$$504$$ −248.001 −0.0219184
$$505$$ 1954.60 0.172235
$$506$$ −15822.1 −1.39008
$$507$$ 0 0
$$508$$ −22456.4 −1.96130
$$509$$ −5328.93 −0.464049 −0.232024 0.972710i $$-0.574535\pi$$
−0.232024 + 0.972710i $$0.574535\pi$$
$$510$$ −1088.75 −0.0945308
$$511$$ 2604.54 0.225475
$$512$$ 16100.7 1.38976
$$513$$ 2732.23 0.235148
$$514$$ −26989.1 −2.31603
$$515$$ 2114.16 0.180895
$$516$$ −3452.72 −0.294569
$$517$$ −16704.9 −1.42105
$$518$$ 9026.37 0.765629
$$519$$ −5728.62 −0.484506
$$520$$ 0 0
$$521$$ −11700.3 −0.983876 −0.491938 0.870630i $$-0.663711\pi$$
−0.491938 + 0.870630i $$0.663711\pi$$
$$522$$ 5941.41 0.498177
$$523$$ −4535.04 −0.379165 −0.189583 0.981865i $$-0.560714\pi$$
−0.189583 + 0.981865i $$0.560714\pi$$
$$524$$ 393.814 0.0328318
$$525$$ −2319.70 −0.192838
$$526$$ 363.664 0.0301454
$$527$$ 1098.66 0.0908128
$$528$$ 5321.24 0.438594
$$529$$ 1991.62 0.163690
$$530$$ 8746.98 0.716877
$$531$$ 5908.71 0.482893
$$532$$ 6086.73 0.496040
$$533$$ 0 0
$$534$$ −13241.8 −1.07309
$$535$$ 1237.33 0.0999900
$$536$$ −235.739 −0.0189969
$$537$$ 1529.88 0.122940
$$538$$ −18667.8 −1.49596
$$539$$ −9621.27 −0.768863
$$540$$ 741.164 0.0590641
$$541$$ −5184.89 −0.412044 −0.206022 0.978547i $$-0.566052\pi$$
−0.206022 + 0.978547i $$0.566052\pi$$
$$542$$ 34311.7 2.71922
$$543$$ 6410.64 0.506643
$$544$$ 7496.18 0.590801
$$545$$ 1462.55 0.114952
$$546$$ 0 0
$$547$$ 5609.12 0.438443 0.219222 0.975675i $$-0.429648\pi$$
0.219222 + 0.975675i $$0.429648\pi$$
$$548$$ −1855.75 −0.144660
$$549$$ 6311.74 0.490671
$$550$$ −15384.2 −1.19270
$$551$$ −16202.3 −1.25271
$$552$$ 1471.83 0.113487
$$553$$ −6027.08 −0.463468
$$554$$ −11879.9 −0.911066
$$555$$ −2997.30 −0.229241
$$556$$ −900.000 −0.0686484
$$557$$ −20150.5 −1.53286 −0.766432 0.642326i $$-0.777970\pi$$
−0.766432 + 0.642326i $$0.777970\pi$$
$$558$$ −1412.72 −0.107178
$$559$$ 0 0
$$560$$ −1121.14 −0.0846011
$$561$$ 2792.06 0.210127
$$562$$ −11618.9 −0.872087
$$563$$ 16292.2 1.21960 0.609800 0.792556i $$-0.291250\pi$$
0.609800 + 0.792556i $$0.291250\pi$$
$$564$$ 13985.5 1.04414
$$565$$ 4719.06 0.351385
$$566$$ 1090.98 0.0810200
$$567$$ −541.343 −0.0400957
$$568$$ 1274.97 0.0941843
$$569$$ 10460.5 0.770700 0.385350 0.922770i $$-0.374081\pi$$
0.385350 + 0.922770i $$0.374081\pi$$
$$570$$ −3817.75 −0.280541
$$571$$ 2225.96 0.163141 0.0815705 0.996668i $$-0.474006\pi$$
0.0815705 + 0.996668i $$0.474006\pi$$
$$572$$ 0 0
$$573$$ −12173.2 −0.887511
$$574$$ −1545.45 −0.112380
$$575$$ 13766.8 0.998461
$$576$$ −5679.00 −0.410807
$$577$$ 4686.23 0.338112 0.169056 0.985606i $$-0.445928\pi$$
0.169056 + 0.985606i $$0.445928\pi$$
$$578$$ −16823.0 −1.21063
$$579$$ −2620.18 −0.188067
$$580$$ −4395.14 −0.314652
$$581$$ −4592.03 −0.327899
$$582$$ −21701.8 −1.54565
$$583$$ −22431.3 −1.59350
$$584$$ −1606.82 −0.113854
$$585$$ 0 0
$$586$$ 17698.5 1.24764
$$587$$ 12090.6 0.850138 0.425069 0.905161i $$-0.360250\pi$$
0.425069 + 0.905161i $$0.360250\pi$$
$$588$$ 8055.03 0.564938
$$589$$ 3852.50 0.269507
$$590$$ −8256.25 −0.576109
$$591$$ −12441.7 −0.865964
$$592$$ 18016.2 1.25078
$$593$$ 6135.97 0.424914 0.212457 0.977170i $$-0.431853\pi$$
0.212457 + 0.977170i $$0.431853\pi$$
$$594$$ −3590.19 −0.247992
$$595$$ −588.261 −0.0405317
$$596$$ 3424.06 0.235327
$$597$$ 7212.18 0.494431
$$598$$ 0 0
$$599$$ −6198.80 −0.422831 −0.211416 0.977396i $$-0.567807\pi$$
−0.211416 + 0.977396i $$0.567807\pi$$
$$600$$ 1431.09 0.0973737
$$601$$ 18345.4 1.24513 0.622565 0.782568i $$-0.286091\pi$$
0.622565 + 0.782568i $$0.286091\pi$$
$$602$$ −3523.79 −0.238569
$$603$$ −514.575 −0.0347514
$$604$$ −13657.1 −0.920030
$$605$$ 887.384 0.0596319
$$606$$ 7926.75 0.531357
$$607$$ 10388.1 0.694631 0.347315 0.937748i $$-0.387093\pi$$
0.347315 + 0.937748i $$0.387093\pi$$
$$608$$ 26285.7 1.75333
$$609$$ 3210.19 0.213602
$$610$$ −8819.40 −0.585389
$$611$$ 0 0
$$612$$ −2337.54 −0.154395
$$613$$ −804.480 −0.0530060 −0.0265030 0.999649i $$-0.508437\pi$$
−0.0265030 + 0.999649i $$0.508437\pi$$
$$614$$ 28970.0 1.90413
$$615$$ 513.184 0.0336481
$$616$$ −888.671 −0.0581259
$$617$$ 15218.3 0.992973 0.496486 0.868044i $$-0.334623\pi$$
0.496486 + 0.868044i $$0.334623\pi$$
$$618$$ 8573.83 0.558074
$$619$$ −11462.5 −0.744291 −0.372145 0.928174i $$-0.621378\pi$$
−0.372145 + 0.928174i $$0.621378\pi$$
$$620$$ 1045.05 0.0676942
$$621$$ 3212.73 0.207605
$$622$$ −4672.33 −0.301195
$$623$$ −7154.66 −0.460105
$$624$$ 0 0
$$625$$ 12223.0 0.782270
$$626$$ −21794.5 −1.39151
$$627$$ 9790.49 0.623596
$$628$$ 13051.4 0.829311
$$629$$ 9453.14 0.599239
$$630$$ 756.419 0.0478356
$$631$$ −4468.68 −0.281926 −0.140963 0.990015i $$-0.545020\pi$$
−0.140963 + 0.990015i $$0.545020\pi$$
$$632$$ 3718.30 0.234028
$$633$$ −11606.6 −0.728783
$$634$$ −19717.3 −1.23514
$$635$$ 7610.37 0.475603
$$636$$ 18779.7 1.17086
$$637$$ 0 0
$$638$$ 21290.0 1.32113
$$639$$ 2783.04 0.172293
$$640$$ 1597.12 0.0986430
$$641$$ 6142.36 0.378484 0.189242 0.981930i $$-0.439397\pi$$
0.189242 + 0.981930i $$0.439397\pi$$
$$642$$ 5017.93 0.308477
$$643$$ 20738.2 1.27190 0.635951 0.771729i $$-0.280608\pi$$
0.635951 + 0.771729i $$0.280608\pi$$
$$644$$ 7157.15 0.437937
$$645$$ 1170.11 0.0714312
$$646$$ 12040.7 0.733339
$$647$$ 852.757 0.0518166 0.0259083 0.999664i $$-0.491752\pi$$
0.0259083 + 0.999664i $$0.491752\pi$$
$$648$$ 333.972 0.0202464
$$649$$ 21172.8 1.28060
$$650$$ 0 0
$$651$$ −763.303 −0.0459542
$$652$$ 21081.3 1.26627
$$653$$ −7345.75 −0.440217 −0.220108 0.975475i $$-0.570641\pi$$
−0.220108 + 0.975475i $$0.570641\pi$$
$$654$$ 5931.28 0.354635
$$655$$ −133.462 −0.00796151
$$656$$ −3084.65 −0.183591
$$657$$ −3507.40 −0.208275
$$658$$ 14273.4 0.845645
$$659$$ 12540.7 0.741297 0.370648 0.928773i $$-0.379136\pi$$
0.370648 + 0.928773i $$0.379136\pi$$
$$660$$ 2655.84 0.156634
$$661$$ −2242.95 −0.131983 −0.0659915 0.997820i $$-0.521021\pi$$
−0.0659915 + 0.997820i $$0.521021\pi$$
$$662$$ 35747.3 2.09873
$$663$$ 0 0
$$664$$ 2832.97 0.165573
$$665$$ −2062.76 −0.120287
$$666$$ −12155.4 −0.707224
$$667$$ −19051.7 −1.10597
$$668$$ −360.658 −0.0208896
$$669$$ −8440.66 −0.487795
$$670$$ 719.016 0.0414597
$$671$$ 22617.0 1.30122
$$672$$ −5208.03 −0.298964
$$673$$ −4776.46 −0.273579 −0.136790 0.990600i $$-0.543678\pi$$
−0.136790 + 0.990600i $$0.543678\pi$$
$$674$$ 35156.0 2.00914
$$675$$ 3123.82 0.178127
$$676$$ 0 0
$$677$$ −7933.57 −0.450387 −0.225193 0.974314i $$-0.572301\pi$$
−0.225193 + 0.974314i $$0.572301\pi$$
$$678$$ 19137.9 1.08405
$$679$$ −11725.6 −0.662723
$$680$$ 362.917 0.0204665
$$681$$ 13555.1 0.762750
$$682$$ −5062.23 −0.284227
$$683$$ 23573.8 1.32068 0.660340 0.750967i $$-0.270412\pi$$
0.660340 + 0.750967i $$0.270412\pi$$
$$684$$ −8196.70 −0.458200
$$685$$ 628.904 0.0350791
$$686$$ 17672.4 0.983581
$$687$$ 3915.81 0.217463
$$688$$ −7033.32 −0.389743
$$689$$ 0 0
$$690$$ −4489.15 −0.247680
$$691$$ −12543.8 −0.690575 −0.345288 0.938497i $$-0.612219\pi$$
−0.345288 + 0.938497i $$0.612219\pi$$
$$692$$ 17185.9 0.944087
$$693$$ −1939.81 −0.106331
$$694$$ −51874.8 −2.83738
$$695$$ 305.006 0.0166468
$$696$$ −1980.47 −0.107858
$$697$$ −1618.52 −0.0879568
$$698$$ −36947.9 −2.00358
$$699$$ 10081.6 0.545526
$$700$$ 6959.09 0.375755
$$701$$ −581.786 −0.0313463 −0.0156731 0.999877i $$-0.504989\pi$$
−0.0156731 + 0.999877i $$0.504989\pi$$
$$702$$ 0 0
$$703$$ 33147.9 1.77837
$$704$$ −20349.7 −1.08943
$$705$$ −4739.64 −0.253199
$$706$$ −22091.6 −1.17766
$$707$$ 4282.89 0.227828
$$708$$ −17726.1 −0.940944
$$709$$ −20742.0 −1.09871 −0.549353 0.835590i $$-0.685126\pi$$
−0.549353 + 0.835590i $$0.685126\pi$$
$$710$$ −3888.74 −0.205552
$$711$$ 8116.38 0.428113
$$712$$ 4413.94 0.232331
$$713$$ 4530.01 0.237938
$$714$$ −2385.66 −0.125043
$$715$$ 0 0
$$716$$ −4589.63 −0.239556
$$717$$ 14211.5 0.740221
$$718$$ −11154.6 −0.579788
$$719$$ 25350.2 1.31489 0.657443 0.753504i $$-0.271638\pi$$
0.657443 + 0.753504i $$0.271638\pi$$
$$720$$ 1509.78 0.0781474
$$721$$ 4632.51 0.239284
$$722$$ 13941.0 0.718602
$$723$$ −14355.8 −0.738449
$$724$$ −19231.9 −0.987223
$$725$$ −18524.4 −0.948938
$$726$$ 3598.73 0.183969
$$727$$ 33428.2 1.70534 0.852672 0.522447i $$-0.174981\pi$$
0.852672 + 0.522447i $$0.174981\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 4900.90 0.248480
$$731$$ −3690.39 −0.186722
$$732$$ −18935.2 −0.956101
$$733$$ −3842.67 −0.193632 −0.0968160 0.995302i $$-0.530866\pi$$
−0.0968160 + 0.995302i $$0.530866\pi$$
$$734$$ 43184.4 2.17162
$$735$$ −2729.81 −0.136994
$$736$$ 30908.3 1.54796
$$737$$ −1843.89 −0.0921582
$$738$$ 2081.19 0.103807
$$739$$ 29029.6 1.44502 0.722511 0.691359i $$-0.242988\pi$$
0.722511 + 0.691359i $$0.242988\pi$$
$$740$$ 8991.91 0.446688
$$741$$ 0 0
$$742$$ 19166.3 0.948269
$$743$$ −34996.7 −1.72800 −0.864000 0.503492i $$-0.832048\pi$$
−0.864000 + 0.503492i $$0.832048\pi$$
$$744$$ 470.906 0.0232046
$$745$$ −1160.40 −0.0570653
$$746$$ −52623.2 −2.58267
$$747$$ 6183.86 0.302886
$$748$$ −8376.19 −0.409444
$$749$$ 2711.23 0.132265
$$750$$ −9080.82 −0.442113
$$751$$ 10454.1 0.507957 0.253979 0.967210i $$-0.418261\pi$$
0.253979 + 0.967210i $$0.418261\pi$$
$$752$$ 28489.1 1.38150
$$753$$ 9821.58 0.475323
$$754$$ 0 0
$$755$$ 4628.32 0.223102
$$756$$ 1624.03 0.0781287
$$757$$ −28130.4 −1.35062 −0.675308 0.737536i $$-0.735989\pi$$
−0.675308 + 0.737536i $$0.735989\pi$$
$$758$$ −9557.45 −0.457971
$$759$$ 11512.3 0.550552
$$760$$ 1272.58 0.0607388
$$761$$ 21087.0 1.00447 0.502236 0.864731i $$-0.332511\pi$$
0.502236 + 0.864731i $$0.332511\pi$$
$$762$$ 30863.4 1.46727
$$763$$ 3204.72 0.152056
$$764$$ 36519.7 1.72937
$$765$$ 792.183 0.0374398
$$766$$ 8177.54 0.385727
$$767$$ 0 0
$$768$$ −8667.00 −0.407218
$$769$$ 19527.9 0.915728 0.457864 0.889022i $$-0.348615\pi$$
0.457864 + 0.889022i $$0.348615\pi$$
$$770$$ 2710.50 0.126857
$$771$$ 19637.4 0.917283
$$772$$ 7860.55 0.366460
$$773$$ −29352.0 −1.36574 −0.682870 0.730540i $$-0.739269\pi$$
−0.682870 + 0.730540i $$0.739269\pi$$
$$774$$ 4745.31 0.220370
$$775$$ 4404.64 0.204154
$$776$$ 7233.92 0.334643
$$777$$ −6567.65 −0.303234
$$778$$ −13377.5 −0.616459
$$779$$ −5675.42 −0.261031
$$780$$ 0 0
$$781$$ 9972.54 0.456908
$$782$$ 14158.3 0.647440
$$783$$ −4323.01 −0.197307
$$784$$ 16408.4 0.747467
$$785$$ −4423.06 −0.201103
$$786$$ −541.246 −0.0245618
$$787$$ 4463.12 0.202151 0.101076 0.994879i $$-0.467772\pi$$
0.101076 + 0.994879i $$0.467772\pi$$
$$788$$ 37325.2 1.68738
$$789$$ −264.604 −0.0119394
$$790$$ −11341.0 −0.510754
$$791$$ 10340.3 0.464804
$$792$$ 1196.73 0.0536919
$$793$$ 0 0
$$794$$ 15501.7 0.692866
$$795$$ −6364.37 −0.283926
$$796$$ −21636.6 −0.963426
$$797$$ −34785.5 −1.54600 −0.773002 0.634404i $$-0.781246\pi$$
−0.773002 + 0.634404i $$0.781246\pi$$
$$798$$ −8365.40 −0.371093
$$799$$ 14948.2 0.661866
$$800$$ 30053.0 1.32817
$$801$$ 9634.84 0.425007
$$802$$ −8236.09 −0.362627
$$803$$ −12568.2 −0.552330
$$804$$ 1543.72 0.0677152
$$805$$ −2425.53 −0.106197
$$806$$ 0 0
$$807$$ 13582.8 0.592487
$$808$$ −2642.25 −0.115042
$$809$$ −10620.0 −0.461530 −0.230765 0.973010i $$-0.574123\pi$$
−0.230765 + 0.973010i $$0.574123\pi$$
$$810$$ −1018.63 −0.0441866
$$811$$ 5497.87 0.238047 0.119024 0.992891i $$-0.462024\pi$$
0.119024 + 0.992891i $$0.462024\pi$$
$$812$$ −9630.57 −0.416215
$$813$$ −24965.5 −1.07697
$$814$$ −43556.7 −1.87551
$$815$$ −7144.34 −0.307062
$$816$$ −4761.66 −0.204279
$$817$$ −12940.5 −0.554139
$$818$$ −21420.5 −0.915587
$$819$$ 0 0
$$820$$ −1539.55 −0.0655653
$$821$$ 21305.3 0.905678 0.452839 0.891592i $$-0.350411\pi$$
0.452839 + 0.891592i $$0.350411\pi$$
$$822$$ 2550.48 0.108222
$$823$$ 17342.6 0.734537 0.367268 0.930115i $$-0.380293\pi$$
0.367268 + 0.930115i $$0.380293\pi$$
$$824$$ −2857.94 −0.120827
$$825$$ 11193.7 0.472381
$$826$$ −18091.0 −0.762064
$$827$$ 5129.96 0.215703 0.107851 0.994167i $$-0.465603\pi$$
0.107851 + 0.994167i $$0.465603\pi$$
$$828$$ −9638.19 −0.404529
$$829$$ −8471.81 −0.354931 −0.177466 0.984127i $$-0.556790\pi$$
−0.177466 + 0.984127i $$0.556790\pi$$
$$830$$ −8640.72 −0.361354
$$831$$ 8643.93 0.360836
$$832$$ 0 0
$$833$$ 8609.50 0.358105
$$834$$ 1236.93 0.0513566
$$835$$ 122.225 0.00506561
$$836$$ −29371.5 −1.21511
$$837$$ 1027.90 0.0424486
$$838$$ −28128.3 −1.15952
$$839$$ −19155.0 −0.788207 −0.394103 0.919066i $$-0.628945\pi$$
−0.394103 + 0.919066i $$0.628945\pi$$
$$840$$ −252.140 −0.0103567
$$841$$ 1246.67 0.0511160
$$842$$ 31078.7 1.27202
$$843$$ 8453.98 0.345398
$$844$$ 34819.7 1.42007
$$845$$ 0 0
$$846$$ −19221.3 −0.781136
$$847$$ 1944.42 0.0788797
$$848$$ 38255.0 1.54915
$$849$$ −793.804 −0.0320887
$$850$$ 13766.4 0.555512
$$851$$ 38977.3 1.57006
$$852$$ −8349.11 −0.335723
$$853$$ −18075.1 −0.725532 −0.362766 0.931880i $$-0.618168\pi$$
−0.362766 + 0.931880i $$0.618168\pi$$
$$854$$ −19324.9 −0.774339
$$855$$ 2777.82 0.111111
$$856$$ −1672.64 −0.0667871
$$857$$ −21054.6 −0.839219 −0.419609 0.907705i $$-0.637833\pi$$
−0.419609 + 0.907705i $$0.637833\pi$$
$$858$$ 0 0
$$859$$ 920.322 0.0365553 0.0182776 0.999833i $$-0.494182\pi$$
0.0182776 + 0.999833i $$0.494182\pi$$
$$860$$ −3510.33 −0.139188
$$861$$ 1124.48 0.0445090
$$862$$ −55267.0 −2.18376
$$863$$ −19427.5 −0.766304 −0.383152 0.923685i $$-0.625161\pi$$
−0.383152 + 0.923685i $$0.625161\pi$$
$$864$$ 7013.40 0.276158
$$865$$ −5824.21 −0.228935
$$866$$ −73044.7 −2.86623
$$867$$ 12240.5 0.479482
$$868$$ 2289.91 0.0895444
$$869$$ 29083.7 1.13532
$$870$$ 6040.55 0.235395
$$871$$ 0 0
$$872$$ −1977.09 −0.0767808
$$873$$ 15790.3 0.612168
$$874$$ 49646.5 1.92142
$$875$$ −4906.44 −0.189563
$$876$$ 10522.2 0.405836
$$877$$ 14872.2 0.572632 0.286316 0.958135i $$-0.407569\pi$$
0.286316 + 0.958135i $$0.407569\pi$$
$$878$$ 29535.7 1.13529
$$879$$ −12877.6 −0.494141
$$880$$ 5410.04 0.207241
$$881$$ −12940.6 −0.494870 −0.247435 0.968905i $$-0.579588\pi$$
−0.247435 + 0.968905i $$0.579588\pi$$
$$882$$ −11070.6 −0.422637
$$883$$ −25585.5 −0.975108 −0.487554 0.873093i $$-0.662111\pi$$
−0.487554 + 0.873093i $$0.662111\pi$$
$$884$$ 0 0
$$885$$ 6007.30 0.228173
$$886$$ −41928.8 −1.58987
$$887$$ 3716.46 0.140684 0.0703418 0.997523i $$-0.477591\pi$$
0.0703418 + 0.997523i $$0.477591\pi$$
$$888$$ 4051.79 0.153118
$$889$$ 16675.7 0.629118
$$890$$ −13462.8 −0.507049
$$891$$ 2612.25 0.0982195
$$892$$ 25322.0 0.950496
$$893$$ 52416.7 1.96423
$$894$$ −4705.92 −0.176051
$$895$$ 1555.40 0.0580910
$$896$$ 3499.58 0.130483
$$897$$ 0 0
$$898$$ −70680.3 −2.62654
$$899$$ −6095.52 −0.226137
$$900$$ −9371.47 −0.347091
$$901$$ 20072.5 0.742187
$$902$$ 7457.57 0.275288
$$903$$ 2563.93 0.0944876
$$904$$ −6379.29 −0.234703
$$905$$ 6517.61 0.239395
$$906$$ 18769.8 0.688285
$$907$$ −12960.4 −0.474469 −0.237235 0.971452i $$-0.576241\pi$$
−0.237235 + 0.971452i $$0.576241\pi$$
$$908$$ −40665.3 −1.48626
$$909$$ −5767.56 −0.210449
$$910$$ 0 0
$$911$$ −36607.1 −1.33134 −0.665668 0.746248i $$-0.731853\pi$$
−0.665668 + 0.746248i $$0.731853\pi$$
$$912$$ −16697.0 −0.606242
$$913$$ 22158.8 0.803230
$$914$$ 58099.3 2.10258
$$915$$ 6417.06 0.231849
$$916$$ −11747.4 −0.423740
$$917$$ −292.440 −0.0105313
$$918$$ 3212.65 0.115505
$$919$$ 20356.3 0.730676 0.365338 0.930875i $$-0.380953\pi$$
0.365338 + 0.930875i $$0.380953\pi$$
$$920$$ 1496.38 0.0536243
$$921$$ −21078.8 −0.754147
$$922$$ −12048.6 −0.430370
$$923$$ 0 0
$$924$$ 5819.43 0.207192
$$925$$ 37898.7 1.34714
$$926$$ −8545.61 −0.303268
$$927$$ −6238.38 −0.221030
$$928$$ −41589.8 −1.47118
$$929$$ −45069.7 −1.59170 −0.795849 0.605495i $$-0.792975\pi$$
−0.795849 + 0.605495i $$0.792975\pi$$
$$930$$ −1436.29 −0.0506428
$$931$$ 30189.6 1.06275
$$932$$ −30244.9 −1.06299
$$933$$ 3399.62 0.119291
$$934$$ −10984.7 −0.384831
$$935$$ 2838.65 0.0992876
$$936$$ 0 0
$$937$$ −6771.10 −0.236075 −0.118037 0.993009i $$-0.537660\pi$$
−0.118037 + 0.993009i $$0.537660\pi$$
$$938$$ 1575.50 0.0548420
$$939$$ 15857.8 0.551119
$$940$$ 14218.9 0.493372
$$941$$ 36690.7 1.27108 0.635538 0.772070i $$-0.280778\pi$$
0.635538 + 0.772070i $$0.280778\pi$$
$$942$$ −17937.4 −0.620417
$$943$$ −6673.51 −0.230455
$$944$$ −36108.8 −1.24496
$$945$$ −550.376 −0.0189457
$$946$$ 17004.0 0.584406
$$947$$ −50861.2 −1.74527 −0.872634 0.488375i $$-0.837590\pi$$
−0.872634 + 0.488375i $$0.837590\pi$$
$$948$$ −24349.1 −0.834202
$$949$$ 0 0
$$950$$ 48272.6 1.64860
$$951$$ 14346.5 0.489187
$$952$$ 795.219 0.0270727
$$953$$ 11855.6 0.402980 0.201490 0.979491i $$-0.435422\pi$$
0.201490 + 0.979491i $$0.435422\pi$$
$$954$$ −25810.3 −0.875931
$$955$$ −12376.4 −0.419361
$$956$$ −42634.5 −1.44236
$$957$$ −15490.8 −0.523245
$$958$$ −21525.5 −0.725947
$$959$$ 1378.04 0.0464019
$$960$$ −5773.76 −0.194112
$$961$$ −28341.6 −0.951349
$$962$$ 0 0
$$963$$ −3651.08 −0.122175
$$964$$ 43067.5 1.43891
$$965$$ −2663.90 −0.0888643
$$966$$ −9836.56 −0.327625
$$967$$ 40661.7 1.35221 0.676107 0.736803i $$-0.263666\pi$$
0.676107 + 0.736803i $$0.263666\pi$$
$$968$$ −1199.58 −0.0398304
$$969$$ −8760.93 −0.290445
$$970$$ −22063.9 −0.730339
$$971$$ 57318.3 1.89437 0.947184 0.320690i $$-0.103915\pi$$
0.947184 + 0.320690i $$0.103915\pi$$
$$972$$ −2187.00 −0.0721688
$$973$$ 668.324 0.0220200
$$974$$ −50403.3 −1.65814
$$975$$ 0 0
$$976$$ −38571.7 −1.26501
$$977$$ −3026.73 −0.0991134 −0.0495567 0.998771i $$-0.515781\pi$$
−0.0495567 + 0.998771i $$0.515781\pi$$
$$978$$ −28973.4 −0.947308
$$979$$ 34524.8 1.12709
$$980$$ 8189.43 0.266941
$$981$$ −4315.64 −0.140457
$$982$$ 81032.3 2.63324
$$983$$ 33942.5 1.10132 0.550659 0.834730i $$-0.314376\pi$$
0.550659 + 0.834730i $$0.314376\pi$$
$$984$$ −693.729 −0.0224749
$$985$$ −12649.3 −0.409179
$$986$$ −19051.2 −0.615327
$$987$$ −10385.4 −0.334925
$$988$$ 0 0
$$989$$ −15216.3 −0.489231
$$990$$ −3650.10 −0.117179
$$991$$ 21637.5 0.693580 0.346790 0.937943i $$-0.387272\pi$$
0.346790 + 0.937943i $$0.387272\pi$$
$$992$$ 9889.02 0.316509
$$993$$ −26009.9 −0.831219
$$994$$ −8520.95 −0.271900
$$995$$ 7332.53 0.233625
$$996$$ −18551.6 −0.590190
$$997$$ 19624.6 0.623388 0.311694 0.950182i $$-0.399104\pi$$
0.311694 + 0.950182i $$0.399104\pi$$
$$998$$ −48296.3 −1.53186
$$999$$ 8844.33 0.280102
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.k.1.4 4
3.2 odd 2 1521.4.a.z.1.1 4
13.2 odd 12 39.4.j.b.4.2 4
13.5 odd 4 507.4.b.e.337.1 4
13.7 odd 12 39.4.j.b.10.2 yes 4
13.8 odd 4 507.4.b.e.337.4 4
13.12 even 2 inner 507.4.a.k.1.1 4
39.2 even 12 117.4.q.d.82.1 4
39.20 even 12 117.4.q.d.10.1 4
39.38 odd 2 1521.4.a.z.1.4 4
52.7 even 12 624.4.bv.c.49.1 4
52.15 even 12 624.4.bv.c.433.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.b.4.2 4 13.2 odd 12
39.4.j.b.10.2 yes 4 13.7 odd 12
117.4.q.d.10.1 4 39.20 even 12
117.4.q.d.82.1 4 39.2 even 12
507.4.a.k.1.1 4 13.12 even 2 inner
507.4.a.k.1.4 4 1.1 even 1 trivial
507.4.b.e.337.1 4 13.5 odd 4
507.4.b.e.337.4 4 13.8 odd 4
624.4.bv.c.49.1 4 52.7 even 12
624.4.bv.c.433.2 4 52.15 even 12
1521.4.a.z.1.1 4 3.2 odd 2
1521.4.a.z.1.4 4 39.38 odd 2