# Properties

 Label 507.4.b.e.337.4 Level $507$ Weight $4$ Character 507.337 Analytic conductor $29.914$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-17})$$ Defining polynomial: $$x^{4} - 17x^{2} + 289$$ x^4 - 17*x^2 + 289 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.4 Root $$3.57071 + 2.06155i$$ of defining polynomial Character $$\chi$$ $$=$$ 507.337 Dual form 507.4.b.e.337.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} -3.05006i q^{5} -12.3693i q^{6} +6.68324i q^{7} -4.12311i q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+4.12311i q^{2} -3.00000 q^{3} -9.00000 q^{4} -3.05006i q^{5} -12.3693i q^{6} +6.68324i q^{7} -4.12311i q^{8} +9.00000 q^{9} +12.5757 q^{10} -32.2500i q^{11} +27.0000 q^{12} -27.5557 q^{14} +9.15018i q^{15} -55.0000 q^{16} +28.8586 q^{17} +37.1080i q^{18} -101.194i q^{19} +27.4505i q^{20} -20.0497i q^{21} +132.970 q^{22} +118.990 q^{23} +12.3693i q^{24} +115.697 q^{25} -27.0000 q^{27} -60.1492i q^{28} +160.111 q^{29} -37.7271 q^{30} -38.0705i q^{31} -259.756i q^{32} +96.7499i q^{33} +118.987i q^{34} +20.3843 q^{35} -81.0000 q^{36} +327.568i q^{37} +417.233 q^{38} -12.5757 q^{40} +56.0846i q^{41} +82.6671 q^{42} -127.879 q^{43} +290.250i q^{44} -27.4505i q^{45} +490.608i q^{46} +517.983i q^{47} +165.000 q^{48} +298.334 q^{49} +477.032i q^{50} -86.5757 q^{51} -695.546 q^{53} -111.324i q^{54} -98.3643 q^{55} +27.5557 q^{56} +303.581i q^{57} +660.156i q^{58} -656.523i q^{59} -82.3516i q^{60} +701.304 q^{61} +156.969 q^{62} +60.1492i q^{63} +631.000 q^{64} -398.910 q^{66} -57.1750i q^{67} -259.727 q^{68} -356.970 q^{69} +84.0465i q^{70} +309.226i q^{71} -37.1080i q^{72} +389.711i q^{73} -1350.60 q^{74} -347.091 q^{75} +910.744i q^{76} +215.534 q^{77} +901.820 q^{79} +167.753i q^{80} +81.0000 q^{81} -231.243 q^{82} +687.095i q^{83} +180.448i q^{84} -88.0203i q^{85} -527.257i q^{86} -480.334 q^{87} -132.970 q^{88} -1070.54i q^{89} +113.181 q^{90} -1070.91 q^{92} +114.211i q^{93} -2135.70 q^{94} -308.647 q^{95} +779.267i q^{96} +1754.48i q^{97} +1230.06i q^{98} -290.250i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} - 36 q^{4} + 36 q^{9}+O(q^{10})$$ 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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.12311i 1.45774i 0.684653 + 0.728869i $$0.259954\pi$$
−0.684653 + 0.728869i $$0.740046\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ −9.00000 −1.12500
$$5$$ − 3.05006i − 0.272806i −0.990653 0.136403i $$-0.956446\pi$$
0.990653 0.136403i $$-0.0435541\pi$$
$$6$$ − 12.3693i − 0.841625i
$$7$$ 6.68324i 0.360861i 0.983588 + 0.180431i $$0.0577491\pi$$
−0.983588 + 0.180431i $$0.942251\pi$$
$$8$$ − 4.12311i − 0.182217i
$$9$$ 9.00000 0.333333
$$10$$ 12.5757 0.397679
$$11$$ − 32.2500i − 0.883975i −0.897021 0.441988i $$-0.854273\pi$$
0.897021 0.441988i $$-0.145727\pi$$
$$12$$ 27.0000 0.649519
$$13$$ 0 0
$$14$$ −27.5557 −0.526041
$$15$$ 9.15018i 0.157504i
$$16$$ −55.0000 −0.859375
$$17$$ 28.8586 0.411720 0.205860 0.978582i $$-0.434001\pi$$
0.205860 + 0.978582i $$0.434001\pi$$
$$18$$ 37.1080i 0.485913i
$$19$$ − 101.194i − 1.22187i −0.791682 0.610933i $$-0.790794\pi$$
0.791682 0.610933i $$-0.209206\pi$$
$$20$$ 27.4505i 0.306906i
$$21$$ − 20.0497i − 0.208343i
$$22$$ 132.970 1.28860
$$23$$ 118.990 1.07874 0.539372 0.842067i $$-0.318662\pi$$
0.539372 + 0.842067i $$0.318662\pi$$
$$24$$ 12.3693i 0.105203i
$$25$$ 115.697 0.925577
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ − 60.1492i − 0.405969i
$$29$$ 160.111 1.02524 0.512620 0.858616i $$-0.328675\pi$$
0.512620 + 0.858616i $$0.328675\pi$$
$$30$$ −37.7271 −0.229600
$$31$$ − 38.0705i − 0.220570i −0.993900 0.110285i $$-0.964824\pi$$
0.993900 0.110285i $$-0.0351763\pi$$
$$32$$ − 259.756i − 1.43496i
$$33$$ 96.7499i 0.510363i
$$34$$ 118.987i 0.600179i
$$35$$ 20.3843 0.0984449
$$36$$ −81.0000 −0.375000
$$37$$ 327.568i 1.45545i 0.685866 + 0.727727i $$0.259423\pi$$
−0.685866 + 0.727727i $$0.740577\pi$$
$$38$$ 417.233 1.78116
$$39$$ 0 0
$$40$$ −12.5757 −0.0497099
$$41$$ 56.0846i 0.213633i 0.994279 + 0.106816i $$0.0340657\pi$$
−0.994279 + 0.106816i $$0.965934\pi$$
$$42$$ 82.6671 0.303710
$$43$$ −127.879 −0.453519 −0.226759 0.973951i $$-0.572813\pi$$
−0.226759 + 0.973951i $$0.572813\pi$$
$$44$$ 290.250i 0.994472i
$$45$$ − 27.4505i − 0.0909352i
$$46$$ 490.608i 1.57253i
$$47$$ 517.983i 1.60757i 0.594923 + 0.803783i $$0.297183\pi$$
−0.594923 + 0.803783i $$0.702817\pi$$
$$48$$ 165.000 0.496160
$$49$$ 298.334 0.869779
$$50$$ 477.032i 1.34925i
$$51$$ −86.5757 −0.237706
$$52$$ 0 0
$$53$$ −695.546 −1.80265 −0.901326 0.433141i $$-0.857405\pi$$
−0.901326 + 0.433141i $$0.857405\pi$$
$$54$$ − 111.324i − 0.280542i
$$55$$ −98.3643 −0.241153
$$56$$ 27.5557 0.0657551
$$57$$ 303.581i 0.705445i
$$58$$ 660.156i 1.49453i
$$59$$ − 656.523i − 1.44868i −0.689444 0.724339i $$-0.742145\pi$$
0.689444 0.724339i $$-0.257855\pi$$
$$60$$ − 82.3516i − 0.177192i
$$61$$ 701.304 1.47201 0.736007 0.676974i $$-0.236709\pi$$
0.736007 + 0.676974i $$0.236709\pi$$
$$62$$ 156.969 0.321533
$$63$$ 60.1492i 0.120287i
$$64$$ 631.000 1.23242
$$65$$ 0 0
$$66$$ −398.910 −0.743976
$$67$$ − 57.1750i − 0.104254i −0.998640 0.0521271i $$-0.983400\pi$$
0.998640 0.0521271i $$-0.0166001\pi$$
$$68$$ −259.727 −0.463184
$$69$$ −356.970 −0.622814
$$70$$ 84.0465i 0.143507i
$$71$$ 309.226i 0.516879i 0.966027 + 0.258440i $$0.0832083\pi$$
−0.966027 + 0.258440i $$0.916792\pi$$
$$72$$ − 37.1080i − 0.0607391i
$$73$$ 389.711i 0.624826i 0.949946 + 0.312413i $$0.101137\pi$$
−0.949946 + 0.312413i $$0.898863\pi$$
$$74$$ −1350.60 −2.12167
$$75$$ −347.091 −0.534382
$$76$$ 910.744i 1.37460i
$$77$$ 215.534 0.318992
$$78$$ 0 0
$$79$$ 901.820 1.28434 0.642169 0.766563i $$-0.278035\pi$$
0.642169 + 0.766563i $$0.278035\pi$$
$$80$$ 167.753i 0.234442i
$$81$$ 81.0000 0.111111
$$82$$ −231.243 −0.311421
$$83$$ 687.095i 0.908657i 0.890834 + 0.454328i $$0.150121\pi$$
−0.890834 + 0.454328i $$0.849879\pi$$
$$84$$ 180.448i 0.234386i
$$85$$ − 88.0203i − 0.112319i
$$86$$ − 527.257i − 0.661111i
$$87$$ −480.334 −0.591922
$$88$$ −132.970 −0.161076
$$89$$ − 1070.54i − 1.27502i −0.770442 0.637510i $$-0.779964\pi$$
0.770442 0.637510i $$-0.220036\pi$$
$$90$$ 113.181 0.132560
$$91$$ 0 0
$$92$$ −1070.91 −1.21359
$$93$$ 114.211i 0.127346i
$$94$$ −2135.70 −2.34341
$$95$$ −308.647 −0.333332
$$96$$ 779.267i 0.828475i
$$97$$ 1754.48i 1.83650i 0.395998 + 0.918251i $$0.370399\pi$$
−0.395998 + 0.918251i $$0.629601\pi$$
$$98$$ 1230.06i 1.26791i
$$99$$ − 290.250i − 0.294658i
$$100$$ −1041.27 −1.04127
$$101$$ 640.840 0.631346 0.315673 0.948868i $$-0.397770\pi$$
0.315673 + 0.948868i $$0.397770\pi$$
$$102$$ − 356.961i − 0.346514i
$$103$$ 693.153 0.663091 0.331546 0.943439i $$-0.392430\pi$$
0.331546 + 0.943439i $$0.392430\pi$$
$$104$$ 0 0
$$105$$ −61.1528 −0.0568372
$$106$$ − 2867.81i − 2.62779i
$$107$$ −405.676 −0.366525 −0.183262 0.983064i $$-0.558666\pi$$
−0.183262 + 0.983064i $$0.558666\pi$$
$$108$$ 243.000 0.216506
$$109$$ − 479.516i − 0.421370i −0.977554 0.210685i $$-0.932431\pi$$
0.977554 0.210685i $$-0.0675694\pi$$
$$110$$ − 405.566i − 0.351538i
$$111$$ − 982.704i − 0.840307i
$$112$$ − 367.578i − 0.310115i
$$113$$ −1547.20 −1.28804 −0.644021 0.765008i $$-0.722735\pi$$
−0.644021 + 0.765008i $$0.722735\pi$$
$$114$$ −1251.70 −1.02835
$$115$$ − 362.926i − 0.294288i
$$116$$ −1441.00 −1.15339
$$117$$ 0 0
$$118$$ 2706.91 2.11179
$$119$$ 192.869i 0.148574i
$$120$$ 37.7271 0.0287000
$$121$$ 290.940 0.218588
$$122$$ 2891.55i 2.14581i
$$123$$ − 168.254i − 0.123341i
$$124$$ 342.634i 0.248141i
$$125$$ − 734.140i − 0.525308i
$$126$$ −248.001 −0.175347
$$127$$ 2495.15 1.74338 0.871690 0.490059i $$-0.163025\pi$$
0.871690 + 0.490059i $$0.163025\pi$$
$$128$$ 523.634i 0.361587i
$$129$$ 383.636 0.261839
$$130$$ 0 0
$$131$$ 43.7571 0.0291838 0.0145919 0.999894i $$-0.495355\pi$$
0.0145919 + 0.999894i $$0.495355\pi$$
$$132$$ − 870.749i − 0.574159i
$$133$$ 676.303 0.440924
$$134$$ 235.739 0.151975
$$135$$ 82.3516i 0.0525015i
$$136$$ − 118.987i − 0.0750224i
$$137$$ 206.194i 0.128586i 0.997931 + 0.0642932i $$0.0204793\pi$$
−0.997931 + 0.0642932i $$0.979521\pi$$
$$138$$ − 1471.83i − 0.907899i
$$139$$ −100.000 −0.0610208 −0.0305104 0.999534i $$-0.509713\pi$$
−0.0305104 + 0.999534i $$0.509713\pi$$
$$140$$ −183.459 −0.110751
$$141$$ − 1553.95i − 0.928128i
$$142$$ −1274.97 −0.753474
$$143$$ 0 0
$$144$$ −495.000 −0.286458
$$145$$ − 488.349i − 0.279691i
$$146$$ −1606.82 −0.910832
$$147$$ −895.003 −0.502167
$$148$$ − 2948.11i − 1.63739i
$$149$$ 380.451i 0.209179i 0.994515 + 0.104590i $$0.0333529\pi$$
−0.994515 + 0.104590i $$0.966647\pi$$
$$150$$ − 1431.09i − 0.778989i
$$151$$ 1517.45i 0.817805i 0.912578 + 0.408902i $$0.134088\pi$$
−0.912578 + 0.408902i $$0.865912\pi$$
$$152$$ −417.233 −0.222645
$$153$$ 259.727 0.137240
$$154$$ 888.671i 0.465007i
$$155$$ −116.117 −0.0601726
$$156$$ 0 0
$$157$$ 1450.16 0.737166 0.368583 0.929595i $$-0.379843\pi$$
0.368583 + 0.929595i $$0.379843\pi$$
$$158$$ 3718.30i 1.87223i
$$159$$ 2086.64 1.04076
$$160$$ −792.270 −0.391465
$$161$$ 795.239i 0.389277i
$$162$$ 333.972i 0.161971i
$$163$$ − 2342.36i − 1.12557i −0.826603 0.562785i $$-0.809730\pi$$
0.826603 0.562785i $$-0.190270\pi$$
$$164$$ − 504.762i − 0.240337i
$$165$$ 295.093 0.139230
$$166$$ −2832.97 −1.32458
$$167$$ 40.0731i 0.0185686i 0.999957 + 0.00928428i $$0.00295532\pi$$
−0.999957 + 0.00928428i $$0.997045\pi$$
$$168$$ −82.6671 −0.0379637
$$169$$ 0 0
$$170$$ 362.917 0.163732
$$171$$ − 910.744i − 0.407289i
$$172$$ 1150.91 0.510208
$$173$$ −1909.54 −0.839189 −0.419594 0.907712i $$-0.637828\pi$$
−0.419594 + 0.907712i $$0.637828\pi$$
$$174$$ − 1980.47i − 0.862868i
$$175$$ 773.232i 0.334005i
$$176$$ 1773.75i 0.759666i
$$177$$ 1969.57i 0.836395i
$$178$$ 4413.94 1.85864
$$179$$ 509.959 0.212939 0.106470 0.994316i $$-0.466045\pi$$
0.106470 + 0.994316i $$0.466045\pi$$
$$180$$ 247.055i 0.102302i
$$181$$ 2136.88 0.877531 0.438766 0.898602i $$-0.355416\pi$$
0.438766 + 0.898602i $$0.355416\pi$$
$$182$$ 0 0
$$183$$ −2103.91 −0.849867
$$184$$ − 490.608i − 0.196566i
$$185$$ 999.101 0.397056
$$186$$ −470.906 −0.185637
$$187$$ − 930.688i − 0.363950i
$$188$$ − 4661.85i − 1.80851i
$$189$$ − 180.448i − 0.0694478i
$$190$$ − 1272.58i − 0.485911i
$$191$$ 4057.74 1.53721 0.768607 0.639721i $$-0.220950\pi$$
0.768607 + 0.639721i $$0.220950\pi$$
$$192$$ −1893.00 −0.711539
$$193$$ − 873.394i − 0.325742i −0.986647 0.162871i $$-0.947925\pi$$
0.986647 0.162871i $$-0.0520755\pi$$
$$194$$ −7233.92 −2.67714
$$195$$ 0 0
$$196$$ −2685.01 −0.978502
$$197$$ 4147.25i 1.49989i 0.661498 + 0.749947i $$0.269921\pi$$
−0.661498 + 0.749947i $$0.730079\pi$$
$$198$$ 1196.73 0.429535
$$199$$ 2404.06 0.856379 0.428189 0.903689i $$-0.359152\pi$$
0.428189 + 0.903689i $$0.359152\pi$$
$$200$$ − 477.032i − 0.168656i
$$201$$ 171.525i 0.0601912i
$$202$$ 2642.25i 0.920337i
$$203$$ 1070.06i 0.369969i
$$204$$ 779.181 0.267420
$$205$$ 171.061 0.0582802
$$206$$ 2857.94i 0.966613i
$$207$$ 1070.91 0.359582
$$208$$ 0 0
$$209$$ −3263.50 −1.08010
$$210$$ − 252.140i − 0.0828538i
$$211$$ 3868.85 1.26229 0.631144 0.775665i $$-0.282586\pi$$
0.631144 + 0.775665i $$0.282586\pi$$
$$212$$ 6259.91 2.02798
$$213$$ − 927.679i − 0.298420i
$$214$$ − 1672.64i − 0.534297i
$$215$$ 390.037i 0.123722i
$$216$$ 111.324i 0.0350677i
$$217$$ 254.434 0.0795950
$$218$$ 1977.09 0.614246
$$219$$ − 1169.13i − 0.360743i
$$220$$ 885.279 0.271298
$$221$$ 0 0
$$222$$ 4051.79 1.22495
$$223$$ 2813.55i 0.844885i 0.906390 + 0.422443i $$0.138827\pi$$
−0.906390 + 0.422443i $$0.861173\pi$$
$$224$$ 1736.01 0.517822
$$225$$ 1041.27 0.308526
$$226$$ − 6379.29i − 1.87763i
$$227$$ − 4518.37i − 1.32112i −0.750772 0.660561i $$-0.770318\pi$$
0.750772 0.660561i $$-0.229682\pi$$
$$228$$ − 2732.23i − 0.793625i
$$229$$ 1305.27i 0.376658i 0.982106 + 0.188329i $$0.0603070\pi$$
−0.982106 + 0.188329i $$0.939693\pi$$
$$230$$ 1496.38 0.428994
$$231$$ −646.603 −0.184170
$$232$$ − 660.156i − 0.186816i
$$233$$ 3360.55 0.944879 0.472440 0.881363i $$-0.343373\pi$$
0.472440 + 0.881363i $$0.343373\pi$$
$$234$$ 0 0
$$235$$ 1579.88 0.438553
$$236$$ 5908.71i 1.62976i
$$237$$ −2705.46 −0.741513
$$238$$ −795.219 −0.216581
$$239$$ − 4737.17i − 1.28210i −0.767499 0.641050i $$-0.778499\pi$$
0.767499 0.641050i $$-0.221501\pi$$
$$240$$ − 503.260i − 0.135355i
$$241$$ − 4785.28i − 1.27903i −0.768778 0.639516i $$-0.779135\pi$$
0.768778 0.639516i $$-0.220865\pi$$
$$242$$ 1199.58i 0.318643i
$$243$$ −243.000 −0.0641500
$$244$$ −6311.74 −1.65601
$$245$$ − 909.937i − 0.237281i
$$246$$ 693.729 0.179799
$$247$$ 0 0
$$248$$ −156.969 −0.0401916
$$249$$ − 2061.29i − 0.524613i
$$250$$ 3026.94 0.765762
$$251$$ 3273.86 0.823284 0.411642 0.911346i $$-0.364955\pi$$
0.411642 + 0.911346i $$0.364955\pi$$
$$252$$ − 541.343i − 0.135323i
$$253$$ − 3837.42i − 0.953584i
$$254$$ 10287.8i 2.54139i
$$255$$ 264.061i 0.0648476i
$$256$$ 2889.00 0.705322
$$257$$ 6545.81 1.58878 0.794390 0.607408i $$-0.207791\pi$$
0.794390 + 0.607408i $$0.207791\pi$$
$$258$$ 1581.77i 0.381693i
$$259$$ −2189.22 −0.525217
$$260$$ 0 0
$$261$$ 1441.00 0.341746
$$262$$ 180.415i 0.0425424i
$$263$$ 88.2014 0.0206796 0.0103398 0.999947i $$-0.496709\pi$$
0.0103398 + 0.999947i $$0.496709\pi$$
$$264$$ 398.910 0.0929970
$$265$$ 2121.46i 0.491773i
$$266$$ 2788.47i 0.642752i
$$267$$ 3211.61i 0.736133i
$$268$$ 514.575i 0.117286i
$$269$$ −4527.60 −1.02622 −0.513109 0.858324i $$-0.671506\pi$$
−0.513109 + 0.858324i $$0.671506\pi$$
$$270$$ −339.544 −0.0765334
$$271$$ − 8321.82i − 1.86537i −0.360695 0.932684i $$-0.617460\pi$$
0.360695 0.932684i $$-0.382540\pi$$
$$272$$ −1587.22 −0.353821
$$273$$ 0 0
$$274$$ −850.160 −0.187445
$$275$$ − 3731.23i − 0.818187i
$$276$$ 3212.73 0.700665
$$277$$ 2881.31 0.624986 0.312493 0.949920i $$-0.398836\pi$$
0.312493 + 0.949920i $$0.398836\pi$$
$$278$$ − 412.311i − 0.0889523i
$$279$$ − 342.634i − 0.0735232i
$$280$$ − 84.0465i − 0.0179384i
$$281$$ 2817.99i 0.598247i 0.954214 + 0.299123i $$0.0966942\pi$$
−0.954214 + 0.299123i $$0.903306\pi$$
$$282$$ 6407.10 1.35297
$$283$$ −264.601 −0.0555792 −0.0277896 0.999614i $$-0.508847\pi$$
−0.0277896 + 0.999614i $$0.508847\pi$$
$$284$$ − 2783.04i − 0.581489i
$$285$$ 925.941 0.192449
$$286$$ 0 0
$$287$$ −374.827 −0.0770918
$$288$$ − 2337.80i − 0.478320i
$$289$$ −4080.18 −0.830487
$$290$$ 2013.52 0.407716
$$291$$ − 5263.45i − 1.06031i
$$292$$ − 3507.40i − 0.702929i
$$293$$ − 4292.52i − 0.855877i −0.903808 0.427938i $$-0.859240\pi$$
0.903808 0.427938i $$-0.140760\pi$$
$$294$$ − 3690.19i − 0.732028i
$$295$$ −2002.43 −0.395208
$$296$$ 1350.60 0.265209
$$297$$ 870.749i 0.170121i
$$298$$ −1568.64 −0.304929
$$299$$ 0 0
$$300$$ 3123.82 0.601180
$$301$$ − 854.643i − 0.163657i
$$302$$ −6256.62 −1.19214
$$303$$ −1922.52 −0.364508
$$304$$ 5565.66i 1.05004i
$$305$$ − 2139.02i − 0.401573i
$$306$$ 1070.88i 0.200060i
$$307$$ − 7026.26i − 1.30622i −0.757263 0.653110i $$-0.773464\pi$$
0.757263 0.653110i $$-0.226536\pi$$
$$308$$ −1939.81 −0.358866
$$309$$ −2079.46 −0.382836
$$310$$ − 478.763i − 0.0877159i
$$311$$ 1133.21 0.206618 0.103309 0.994649i $$-0.467057\pi$$
0.103309 + 0.994649i $$0.467057\pi$$
$$312$$ 0 0
$$313$$ −5285.95 −0.954566 −0.477283 0.878750i $$-0.658378\pi$$
−0.477283 + 0.878750i $$0.658378\pi$$
$$314$$ 5979.15i 1.07459i
$$315$$ 183.459 0.0328150
$$316$$ −8116.38 −1.44488
$$317$$ − 4782.16i − 0.847296i −0.905827 0.423648i $$-0.860749\pi$$
0.905827 0.423648i $$-0.139251\pi$$
$$318$$ 8603.43i 1.51716i
$$319$$ − 5163.59i − 0.906286i
$$320$$ − 1924.59i − 0.336212i
$$321$$ 1217.03 0.211613
$$322$$ −3278.85 −0.567464
$$323$$ − 2920.31i − 0.503066i
$$324$$ −729.000 −0.125000
$$325$$ 0 0
$$326$$ 9657.80 1.64079
$$327$$ 1438.55i 0.243278i
$$328$$ 231.243 0.0389276
$$329$$ −3461.81 −0.580108
$$330$$ 1216.70i 0.202961i
$$331$$ 8669.98i 1.43971i 0.694122 + 0.719857i $$0.255793\pi$$
−0.694122 + 0.719857i $$0.744207\pi$$
$$332$$ − 6183.86i − 1.02224i
$$333$$ 2948.11i 0.485152i
$$334$$ −165.226 −0.0270681
$$335$$ −174.387 −0.0284412
$$336$$ 1102.73i 0.179045i
$$337$$ −8526.59 −1.37826 −0.689129 0.724639i $$-0.742007\pi$$
−0.689129 + 0.724639i $$0.742007\pi$$
$$338$$ 0 0
$$339$$ 4641.61 0.743651
$$340$$ 792.183i 0.126359i
$$341$$ −1227.77 −0.194978
$$342$$ 3755.10 0.593720
$$343$$ 4286.19i 0.674731i
$$344$$ 527.257i 0.0826389i
$$345$$ 1088.78i 0.169907i
$$346$$ − 7873.23i − 1.22332i
$$347$$ −12581.5 −1.94643 −0.973213 0.229907i $$-0.926158\pi$$
−0.973213 + 0.229907i $$0.926158\pi$$
$$348$$ 4323.01 0.665913
$$349$$ 8961.18i 1.37444i 0.726447 + 0.687222i $$0.241170\pi$$
−0.726447 + 0.687222i $$0.758830\pi$$
$$350$$ −3188.12 −0.486892
$$351$$ 0 0
$$352$$ −8377.11 −1.26847
$$353$$ − 5357.99i − 0.807868i −0.914788 0.403934i $$-0.867643\pi$$
0.914788 0.403934i $$-0.132357\pi$$
$$354$$ −8120.74 −1.21924
$$355$$ 943.159 0.141007
$$356$$ 9634.84i 1.43440i
$$357$$ − 578.606i − 0.0857790i
$$358$$ 2102.61i 0.310409i
$$359$$ 2705.40i 0.397731i 0.980027 + 0.198866i $$0.0637257\pi$$
−0.980027 + 0.198866i $$0.936274\pi$$
$$360$$ −113.181 −0.0165700
$$361$$ −3381.19 −0.492957
$$362$$ 8810.59i 1.27921i
$$363$$ −872.820 −0.126202
$$364$$ 0 0
$$365$$ 1188.64 0.170456
$$366$$ − 8674.65i − 1.23888i
$$367$$ 10473.8 1.48972 0.744858 0.667223i $$-0.232517\pi$$
0.744858 + 0.667223i $$0.232517\pi$$
$$368$$ −6544.45 −0.927046
$$369$$ 504.762i 0.0712110i
$$370$$ 4119.40i 0.578804i
$$371$$ − 4648.50i − 0.650507i
$$372$$ − 1027.90i − 0.143264i
$$373$$ −12763.0 −1.77170 −0.885850 0.463973i $$-0.846424\pi$$
−0.885850 + 0.463973i $$0.846424\pi$$
$$374$$ 3837.32 0.530544
$$375$$ 2202.42i 0.303287i
$$376$$ 2135.70 0.292926
$$377$$ 0 0
$$378$$ 744.004 0.101237
$$379$$ − 2318.02i − 0.314166i −0.987585 0.157083i $$-0.949791\pi$$
0.987585 0.157083i $$-0.0502090\pi$$
$$380$$ 2777.82 0.374998
$$381$$ −7485.46 −1.00654
$$382$$ 16730.5i 2.24086i
$$383$$ 1983.34i 0.264606i 0.991209 + 0.132303i $$0.0422372\pi$$
−0.991209 + 0.132303i $$0.957763\pi$$
$$384$$ − 1570.90i − 0.208763i
$$385$$ − 657.392i − 0.0870229i
$$386$$ 3601.10 0.474847
$$387$$ −1150.91 −0.151173
$$388$$ − 15790.3i − 2.06607i
$$389$$ 3244.51 0.422887 0.211444 0.977390i $$-0.432184\pi$$
0.211444 + 0.977390i $$0.432184\pi$$
$$390$$ 0 0
$$391$$ 3433.88 0.444140
$$392$$ − 1230.06i − 0.158489i
$$393$$ −131.271 −0.0168493
$$394$$ −17099.5 −2.18645
$$395$$ − 2750.60i − 0.350374i
$$396$$ 2612.25i 0.331491i
$$397$$ − 3759.72i − 0.475302i −0.971351 0.237651i $$-0.923622\pi$$
0.971351 0.237651i $$-0.0763775\pi$$
$$398$$ 9912.20i 1.24838i
$$399$$ −2028.91 −0.254568
$$400$$ −6363.34 −0.795418
$$401$$ 1997.55i 0.248760i 0.992235 + 0.124380i $$0.0396941\pi$$
−0.992235 + 0.124380i $$0.960306\pi$$
$$402$$ −707.216 −0.0877431
$$403$$ 0 0
$$404$$ −5767.56 −0.710264
$$405$$ − 247.055i − 0.0303117i
$$406$$ −4411.98 −0.539318
$$407$$ 10564.1 1.28659
$$408$$ 356.961i 0.0433142i
$$409$$ − 5195.23i − 0.628087i −0.949409 0.314044i $$-0.898316\pi$$
0.949409 0.314044i $$-0.101684\pi$$
$$410$$ 705.304i 0.0849573i
$$411$$ − 618.582i − 0.0742394i
$$412$$ −6238.38 −0.745977
$$413$$ 4387.70 0.522772
$$414$$ 4415.48i 0.524176i
$$415$$ 2095.68 0.247887
$$416$$ 0 0
$$417$$ 300.000 0.0352304
$$418$$ − 13455.7i − 1.57450i
$$419$$ −6822.11 −0.795422 −0.397711 0.917511i $$-0.630195\pi$$
−0.397711 + 0.917511i $$0.630195\pi$$
$$420$$ 550.376 0.0639419
$$421$$ 7537.70i 0.872601i 0.899801 + 0.436300i $$0.143712\pi$$
−0.899801 + 0.436300i $$0.856288\pi$$
$$422$$ 15951.7i 1.84009i
$$423$$ 4661.85i 0.535855i
$$424$$ 2867.81i 0.328474i
$$425$$ 3338.85 0.381078
$$426$$ 3824.92 0.435019
$$427$$ 4686.99i 0.531192i
$$428$$ 3651.08 0.412340
$$429$$ 0 0
$$430$$ −1608.16 −0.180355
$$431$$ − 13404.2i − 1.49805i −0.662544 0.749023i $$-0.730523\pi$$
0.662544 0.749023i $$-0.269477\pi$$
$$432$$ 1485.00 0.165387
$$433$$ 17715.9 1.96622 0.983110 0.183014i $$-0.0585852\pi$$
0.983110 + 0.183014i $$0.0585852\pi$$
$$434$$ 1049.06i 0.116029i
$$435$$ 1465.05i 0.161480i
$$436$$ 4315.64i 0.474041i
$$437$$ − 12041.1i − 1.31808i
$$438$$ 4820.46 0.525869
$$439$$ −7163.47 −0.778801 −0.389401 0.921068i $$-0.627318\pi$$
−0.389401 + 0.921068i $$0.627318\pi$$
$$440$$ 405.566i 0.0439423i
$$441$$ 2685.01 0.289926
$$442$$ 0 0
$$443$$ −10169.2 −1.09064 −0.545321 0.838227i $$-0.683592\pi$$
−0.545321 + 0.838227i $$0.683592\pi$$
$$444$$ 8844.33i 0.945346i
$$445$$ −3265.20 −0.347833
$$446$$ −11600.6 −1.23162
$$447$$ − 1141.35i − 0.120770i
$$448$$ 4217.13i 0.444733i
$$449$$ 17142.5i 1.80179i 0.434037 + 0.900895i $$0.357089\pi$$
−0.434037 + 0.900895i $$0.642911\pi$$
$$450$$ 4293.28i 0.449750i
$$451$$ 1808.73 0.188846
$$452$$ 13924.8 1.44905
$$453$$ − 4552.36i − 0.472160i
$$454$$ 18629.7 1.92585
$$455$$ 0 0
$$456$$ 1251.70 0.128544
$$457$$ 14091.1i 1.44235i 0.692750 + 0.721177i $$0.256399\pi$$
−0.692750 + 0.721177i $$0.743601\pi$$
$$458$$ −5381.76 −0.549068
$$459$$ −779.181 −0.0792355
$$460$$ 3266.34i 0.331074i
$$461$$ − 2922.22i − 0.295231i −0.989045 0.147616i $$-0.952840\pi$$
0.989045 0.147616i $$-0.0471598\pi$$
$$462$$ − 2666.01i − 0.268472i
$$463$$ 2072.61i 0.208040i 0.994575 + 0.104020i $$0.0331706\pi$$
−0.994575 + 0.104020i $$0.966829\pi$$
$$464$$ −8806.13 −0.881065
$$465$$ 348.352 0.0347407
$$466$$ 13855.9i 1.37739i
$$467$$ 2664.19 0.263992 0.131996 0.991250i $$-0.457861\pi$$
0.131996 + 0.991250i $$0.457861\pi$$
$$468$$ 0 0
$$469$$ 382.114 0.0376213
$$470$$ 6514.01i 0.639295i
$$471$$ −4350.47 −0.425603
$$472$$ −2706.91 −0.263974
$$473$$ 4124.08i 0.400899i
$$474$$ − 11154.9i − 1.08093i
$$475$$ − 11707.8i − 1.13093i
$$476$$ − 1735.82i − 0.167145i
$$477$$ −6259.91 −0.600884
$$478$$ 19531.8 1.86897
$$479$$ 5220.70i 0.497995i 0.968504 + 0.248998i $$0.0801011\pi$$
−0.968504 + 0.248998i $$0.919899\pi$$
$$480$$ 2376.81 0.226013
$$481$$ 0 0
$$482$$ 19730.2 1.86449
$$483$$ − 2385.72i − 0.224749i
$$484$$ −2618.46 −0.245911
$$485$$ 5351.28 0.501008
$$486$$ − 1001.91i − 0.0935139i
$$487$$ − 12224.6i − 1.13747i −0.822520 0.568737i $$-0.807432\pi$$
0.822520 0.568737i $$-0.192568\pi$$
$$488$$ − 2891.55i − 0.268226i
$$489$$ 7027.08i 0.649848i
$$490$$ 3751.77 0.345893
$$491$$ −19653.2 −1.80639 −0.903195 0.429231i $$-0.858784\pi$$
−0.903195 + 0.429231i $$0.858784\pi$$
$$492$$ 1514.29i 0.138759i
$$493$$ 4620.59 0.422111
$$494$$ 0 0
$$495$$ −885.279 −0.0803845
$$496$$ 2093.88i 0.189552i
$$497$$ −2066.63 −0.186522
$$498$$ 8498.90 0.764749
$$499$$ − 11713.6i − 1.05084i −0.850842 0.525422i $$-0.823907\pi$$
0.850842 0.525422i $$-0.176093\pi$$
$$500$$ 6607.26i 0.590972i
$$501$$ − 120.219i − 0.0107206i
$$502$$ 13498.5i 1.20013i
$$503$$ 13003.3 1.15266 0.576332 0.817216i $$-0.304483\pi$$
0.576332 + 0.817216i $$0.304483\pi$$
$$504$$ 248.001 0.0219184
$$505$$ − 1954.60i − 0.172235i
$$506$$ 15822.1 1.39008
$$507$$ 0 0
$$508$$ −22456.4 −1.96130
$$509$$ − 5328.93i − 0.464049i −0.972710 0.232024i $$-0.925465\pi$$
0.972710 0.232024i $$-0.0745349\pi$$
$$510$$ −1088.75 −0.0945308
$$511$$ −2604.54 −0.225475
$$512$$ 16100.7i 1.38976i
$$513$$ 2732.23i 0.235148i
$$514$$ 26989.1i 2.31603i
$$515$$ − 2114.16i − 0.180895i
$$516$$ −3452.72 −0.294569
$$517$$ 16704.9 1.42105
$$518$$ − 9026.37i − 0.765629i
$$519$$ 5728.62 0.484506
$$520$$ 0 0
$$521$$ −11700.3 −0.983876 −0.491938 0.870630i $$-0.663711\pi$$
−0.491938 + 0.870630i $$0.663711\pi$$
$$522$$ 5941.41i 0.498177i
$$523$$ −4535.04 −0.379165 −0.189583 0.981865i $$-0.560714\pi$$
−0.189583 + 0.981865i $$0.560714\pi$$
$$524$$ −393.814 −0.0328318
$$525$$ − 2319.70i − 0.192838i
$$526$$ 363.664i 0.0301454i
$$527$$ − 1098.66i − 0.0908128i
$$528$$ − 5321.24i − 0.438594i
$$529$$ 1991.62 0.163690
$$530$$ −8746.98 −0.716877
$$531$$ − 5908.71i − 0.482893i
$$532$$ −6086.73 −0.496040
$$533$$ 0 0
$$534$$ −13241.8 −1.07309
$$535$$ 1237.33i 0.0999900i
$$536$$ −235.739 −0.0189969
$$537$$ −1529.88 −0.122940
$$538$$ − 18667.8i − 1.49596i
$$539$$ − 9621.27i − 0.768863i
$$540$$ − 741.164i − 0.0590641i
$$541$$ 5184.89i 0.412044i 0.978547 + 0.206022i $$0.0660519\pi$$
−0.978547 + 0.206022i $$0.933948\pi$$
$$542$$ 34311.7 2.71922
$$543$$ −6410.64 −0.506643
$$544$$ − 7496.18i − 0.590801i
$$545$$ −1462.55 −0.114952
$$546$$ 0 0
$$547$$ 5609.12 0.438443 0.219222 0.975675i $$-0.429648\pi$$
0.219222 + 0.975675i $$0.429648\pi$$
$$548$$ − 1855.75i − 0.144660i
$$549$$ 6311.74 0.490671
$$550$$ 15384.2 1.19270
$$551$$ − 16202.3i − 1.25271i
$$552$$ 1471.83i 0.113487i
$$553$$ 6027.08i 0.463468i
$$554$$ 11879.9i 0.911066i
$$555$$ −2997.30 −0.229241
$$556$$ 900.000 0.0686484
$$557$$ 20150.5i 1.53286i 0.642326 + 0.766432i $$0.277970\pi$$
−0.642326 + 0.766432i $$0.722030\pi$$
$$558$$ 1412.72 0.107178
$$559$$ 0 0
$$560$$ −1121.14 −0.0846011
$$561$$ 2792.06i 0.210127i
$$562$$ −11618.9 −0.872087
$$563$$ −16292.2 −1.21960 −0.609800 0.792556i $$-0.708750\pi$$
−0.609800 + 0.792556i $$0.708750\pi$$
$$564$$ 13985.5i 1.04414i
$$565$$ 4719.06i 0.351385i
$$566$$ − 1090.98i − 0.0810200i
$$567$$ 541.343i 0.0400957i
$$568$$ 1274.97 0.0941843
$$569$$ −10460.5 −0.770700 −0.385350 0.922770i $$-0.625919\pi$$
−0.385350 + 0.922770i $$0.625919\pi$$
$$570$$ 3817.75i 0.280541i
$$571$$ −2225.96 −0.163141 −0.0815705 0.996668i $$-0.525994\pi$$
−0.0815705 + 0.996668i $$0.525994\pi$$
$$572$$ 0 0
$$573$$ −12173.2 −0.887511
$$574$$ − 1545.45i − 0.112380i
$$575$$ 13766.8 0.998461
$$576$$ 5679.00 0.410807
$$577$$ 4686.23i 0.338112i 0.985606 + 0.169056i $$0.0540718\pi$$
−0.985606 + 0.169056i $$0.945928\pi$$
$$578$$ − 16823.0i − 1.21063i
$$579$$ 2620.18i 0.188067i
$$580$$ 4395.14i 0.314652i
$$581$$ −4592.03 −0.327899
$$582$$ 21701.8 1.54565
$$583$$ 22431.3i 1.59350i
$$584$$ 1606.82 0.113854
$$585$$ 0 0
$$586$$ 17698.5 1.24764
$$587$$ 12090.6i 0.850138i 0.905161 + 0.425069i $$0.139750\pi$$
−0.905161 + 0.425069i $$0.860250\pi$$
$$588$$ 8055.03 0.564938
$$589$$ −3852.50 −0.269507
$$590$$ − 8256.25i − 0.576109i
$$591$$ − 12441.7i − 0.865964i
$$592$$ − 18016.2i − 1.25078i
$$593$$ − 6135.97i − 0.424914i −0.977170 0.212457i $$-0.931853\pi$$
0.977170 0.212457i $$-0.0681466\pi$$
$$594$$ −3590.19 −0.247992
$$595$$ 588.261 0.0405317
$$596$$ − 3424.06i − 0.235327i
$$597$$ −7212.18 −0.494431
$$598$$ 0 0
$$599$$ −6198.80 −0.422831 −0.211416 0.977396i $$-0.567807\pi$$
−0.211416 + 0.977396i $$0.567807\pi$$
$$600$$ 1431.09i 0.0973737i
$$601$$ 18345.4 1.24513 0.622565 0.782568i $$-0.286091\pi$$
0.622565 + 0.782568i $$0.286091\pi$$
$$602$$ 3523.79 0.238569
$$603$$ − 514.575i − 0.0347514i
$$604$$ − 13657.1i − 0.920030i
$$605$$ − 887.384i − 0.0596319i
$$606$$ − 7926.75i − 0.531357i
$$607$$ 10388.1 0.694631 0.347315 0.937748i $$-0.387093\pi$$
0.347315 + 0.937748i $$0.387093\pi$$
$$608$$ −26285.7 −1.75333
$$609$$ − 3210.19i − 0.213602i
$$610$$ 8819.40 0.585389
$$611$$ 0 0
$$612$$ −2337.54 −0.154395
$$613$$ − 804.480i − 0.0530060i −0.999649 0.0265030i $$-0.991563\pi$$
0.999649 0.0265030i $$-0.00843715\pi$$
$$614$$ 28970.0 1.90413
$$615$$ −513.184 −0.0336481
$$616$$ − 888.671i − 0.0581259i
$$617$$ 15218.3i 0.992973i 0.868044 + 0.496486i $$0.165377\pi$$
−0.868044 + 0.496486i $$0.834623\pi$$
$$618$$ − 8573.83i − 0.558074i
$$619$$ 11462.5i 0.744291i 0.928174 + 0.372145i $$0.121378\pi$$
−0.928174 + 0.372145i $$0.878622\pi$$
$$620$$ 1045.05 0.0676942
$$621$$ −3212.73 −0.207605
$$622$$ 4672.33i 0.301195i
$$623$$ 7154.66 0.460105
$$624$$ 0 0
$$625$$ 12223.0 0.782270
$$626$$ − 21794.5i − 1.39151i
$$627$$ 9790.49 0.623596
$$628$$ −13051.4 −0.829311
$$629$$ 9453.14i 0.599239i
$$630$$ 756.419i 0.0478356i
$$631$$ 4468.68i 0.281926i 0.990015 + 0.140963i $$0.0450199\pi$$
−0.990015 + 0.140963i $$0.954980\pi$$
$$632$$ − 3718.30i − 0.234028i
$$633$$ −11606.6 −0.728783
$$634$$ 19717.3 1.23514
$$635$$ − 7610.37i − 0.475603i
$$636$$ −18779.7 −1.17086
$$637$$ 0 0
$$638$$ 21290.0 1.32113
$$639$$ 2783.04i 0.172293i
$$640$$ 1597.12 0.0986430
$$641$$ −6142.36 −0.378484 −0.189242 0.981930i $$-0.560603\pi$$
−0.189242 + 0.981930i $$0.560603\pi$$
$$642$$ 5017.93i 0.308477i
$$643$$ 20738.2i 1.27190i 0.771729 + 0.635951i $$0.219392\pi$$
−0.771729 + 0.635951i $$0.780608\pi$$
$$644$$ − 7157.15i − 0.437937i
$$645$$ − 1170.11i − 0.0714312i
$$646$$ 12040.7 0.733339
$$647$$ −852.757 −0.0518166 −0.0259083 0.999664i $$-0.508248\pi$$
−0.0259083 + 0.999664i $$0.508248\pi$$
$$648$$ − 333.972i − 0.0202464i
$$649$$ −21172.8 −1.28060
$$650$$ 0 0
$$651$$ −763.303 −0.0459542
$$652$$ 21081.3i 1.26627i
$$653$$ −7345.75 −0.440217 −0.220108 0.975475i $$-0.570641\pi$$
−0.220108 + 0.975475i $$0.570641\pi$$
$$654$$ −5931.28 −0.354635
$$655$$ − 133.462i − 0.00796151i
$$656$$ − 3084.65i − 0.183591i
$$657$$ 3507.40i 0.208275i
$$658$$ − 14273.4i − 0.845645i
$$659$$ 12540.7 0.741297 0.370648 0.928773i $$-0.379136\pi$$
0.370648 + 0.928773i $$0.379136\pi$$
$$660$$ −2655.84 −0.156634
$$661$$ 2242.95i 0.131983i 0.997820 + 0.0659915i $$0.0210210\pi$$
−0.997820 + 0.0659915i $$0.978979\pi$$
$$662$$ −35747.3 −2.09873
$$663$$ 0 0
$$664$$ 2832.97 0.165573
$$665$$ − 2062.76i − 0.120287i
$$666$$ −12155.4 −0.707224
$$667$$ 19051.7 1.10597
$$668$$ − 360.658i − 0.0208896i
$$669$$ − 8440.66i − 0.487795i
$$670$$ − 719.016i − 0.0414597i
$$671$$ − 22617.0i − 1.30122i
$$672$$ −5208.03 −0.298964
$$673$$ 4776.46 0.273579 0.136790 0.990600i $$-0.456322\pi$$
0.136790 + 0.990600i $$0.456322\pi$$
$$674$$ − 35156.0i − 2.00914i
$$675$$ −3123.82 −0.178127
$$676$$ 0 0
$$677$$ −7933.57 −0.450387 −0.225193 0.974314i $$-0.572301\pi$$
−0.225193 + 0.974314i $$0.572301\pi$$
$$678$$ 19137.9i 1.08405i
$$679$$ −11725.6 −0.662723
$$680$$ −362.917 −0.0204665
$$681$$ 13555.1i 0.762750i
$$682$$ − 5062.23i − 0.284227i
$$683$$ − 23573.8i − 1.32068i −0.750967 0.660340i $$-0.770412\pi$$
0.750967 0.660340i $$-0.229588\pi$$
$$684$$ 8196.70i 0.458200i
$$685$$ 628.904 0.0350791
$$686$$ −17672.4 −0.983581
$$687$$ − 3915.81i − 0.217463i
$$688$$ 7033.32 0.389743
$$689$$ 0 0
$$690$$ −4489.15 −0.247680
$$691$$ − 12543.8i − 0.690575i −0.938497 0.345288i $$-0.887781\pi$$
0.938497 0.345288i $$-0.112219\pi$$
$$692$$ 17185.9 0.944087
$$693$$ 1939.81 0.106331
$$694$$ − 51874.8i − 2.83738i
$$695$$ 305.006i 0.0166468i
$$696$$ 1980.47i 0.107858i
$$697$$ 1618.52i 0.0879568i
$$698$$ −36947.9 −2.00358
$$699$$ −10081.6 −0.545526
$$700$$ − 6959.09i − 0.375755i
$$701$$ 581.786 0.0313463 0.0156731 0.999877i $$-0.495011\pi$$
0.0156731 + 0.999877i $$0.495011\pi$$
$$702$$ 0 0
$$703$$ 33147.9 1.77837
$$704$$ − 20349.7i − 1.08943i
$$705$$ −4739.64 −0.253199
$$706$$ 22091.6 1.17766
$$707$$ 4282.89i 0.227828i
$$708$$ − 17726.1i − 0.940944i
$$709$$ 20742.0i 1.09871i 0.835590 + 0.549353i $$0.185126\pi$$
−0.835590 + 0.549353i $$0.814874\pi$$
$$710$$ 3888.74i 0.205552i
$$711$$ 8116.38 0.428113
$$712$$ −4413.94 −0.232331
$$713$$ − 4530.01i − 0.237938i
$$714$$ 2385.66 0.125043
$$715$$ 0 0
$$716$$ −4589.63 −0.239556
$$717$$ 14211.5i 0.740221i
$$718$$ −11154.6 −0.579788
$$719$$ −25350.2 −1.31489 −0.657443 0.753504i $$-0.728362\pi$$
−0.657443 + 0.753504i $$0.728362\pi$$
$$720$$ 1509.78i 0.0781474i
$$721$$ 4632.51i 0.239284i
$$722$$ − 13941.0i − 0.718602i
$$723$$ 14355.8i 0.738449i
$$724$$ −19231.9 −0.987223
$$725$$ 18524.4 0.948938
$$726$$ − 3598.73i − 0.183969i
$$727$$ −33428.2 −1.70534 −0.852672 0.522447i $$-0.825019\pi$$
−0.852672 + 0.522447i $$0.825019\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 4900.90i 0.248480i
$$731$$ −3690.39 −0.186722
$$732$$ 18935.2 0.956101
$$733$$ − 3842.67i − 0.193632i −0.995302 0.0968160i $$-0.969134\pi$$
0.995302 0.0968160i $$-0.0308658\pi$$
$$734$$ 43184.4i 2.17162i
$$735$$ 2729.81i 0.136994i
$$736$$ − 30908.3i − 1.54796i
$$737$$ −1843.89 −0.0921582
$$738$$ −2081.19 −0.103807
$$739$$ − 29029.6i − 1.44502i −0.691359 0.722511i $$-0.742988\pi$$
0.691359 0.722511i $$-0.257012\pi$$
$$740$$ −8991.91 −0.446688
$$741$$ 0 0
$$742$$ 19166.3 0.948269
$$743$$ − 34996.7i − 1.72800i −0.503492 0.864000i $$-0.667952\pi$$
0.503492 0.864000i $$-0.332048\pi$$
$$744$$ 470.906 0.0232046
$$745$$ 1160.40 0.0570653
$$746$$ − 52623.2i − 2.58267i
$$747$$ 6183.86i 0.302886i
$$748$$ 8376.19i 0.409444i
$$749$$ − 2711.23i − 0.132265i
$$750$$ −9080.82 −0.442113
$$751$$ −10454.1 −0.507957 −0.253979 0.967210i $$-0.581739\pi$$
−0.253979 + 0.967210i $$0.581739\pi$$
$$752$$ − 28489.1i − 1.38150i
$$753$$ −9821.58 −0.475323
$$754$$ 0 0
$$755$$ 4628.32 0.223102
$$756$$ 1624.03i 0.0781287i
$$757$$ −28130.4 −1.35062 −0.675308 0.737536i $$-0.735989\pi$$
−0.675308 + 0.737536i $$0.735989\pi$$
$$758$$ 9557.45 0.457971
$$759$$ 11512.3i 0.550552i
$$760$$ 1272.58i 0.0607388i
$$761$$ − 21087.0i − 1.00447i −0.864731 0.502236i $$-0.832511\pi$$
0.864731 0.502236i $$-0.167489\pi$$
$$762$$ − 30863.4i − 1.46727i
$$763$$ 3204.72 0.152056
$$764$$ −36519.7 −1.72937
$$765$$ − 792.183i − 0.0374398i
$$766$$ −8177.54 −0.385727
$$767$$ 0 0
$$768$$ −8667.00 −0.407218
$$769$$ 19527.9i 0.915728i 0.889022 + 0.457864i $$0.151385\pi$$
−0.889022 + 0.457864i $$0.848615\pi$$
$$770$$ 2710.50 0.126857
$$771$$ −19637.4 −0.917283
$$772$$ 7860.55i 0.366460i
$$773$$ − 29352.0i − 1.36574i −0.730540 0.682870i $$-0.760731\pi$$
0.730540 0.682870i $$-0.239269\pi$$
$$774$$ − 4745.31i − 0.220370i
$$775$$ − 4404.64i − 0.204154i
$$776$$ 7233.92 0.334643
$$777$$ 6567.65 0.303234
$$778$$ 13377.5i 0.616459i
$$779$$ 5675.42 0.261031
$$780$$ 0 0
$$781$$ 9972.54 0.456908
$$782$$ 14158.3i 0.647440i
$$783$$ −4323.01 −0.197307
$$784$$ −16408.4 −0.747467
$$785$$ − 4423.06i − 0.201103i
$$786$$ − 541.246i − 0.0245618i
$$787$$ − 4463.12i − 0.202151i −0.994879 0.101076i $$-0.967772\pi$$
0.994879 0.101076i $$-0.0322284\pi$$
$$788$$ − 37325.2i − 1.68738i
$$789$$ −264.604 −0.0119394
$$790$$ 11341.0 0.510754
$$791$$ − 10340.3i − 0.464804i
$$792$$ −1196.73 −0.0536919
$$793$$ 0 0
$$794$$ 15501.7 0.692866
$$795$$ − 6364.37i − 0.283926i
$$796$$ −21636.6 −0.963426
$$797$$ 34785.5 1.54600 0.773002 0.634404i $$-0.218754\pi$$
0.773002 + 0.634404i $$0.218754\pi$$
$$798$$ − 8365.40i − 0.371093i
$$799$$ 14948.2i 0.661866i
$$800$$ − 30053.0i − 1.32817i
$$801$$ − 9634.84i − 0.425007i
$$802$$ −8236.09 −0.362627
$$803$$ 12568.2 0.552330
$$804$$ − 1543.72i − 0.0677152i
$$805$$ 2425.53 0.106197
$$806$$ 0 0
$$807$$ 13582.8 0.592487
$$808$$ − 2642.25i − 0.115042i
$$809$$ −10620.0 −0.461530 −0.230765 0.973010i $$-0.574123\pi$$
−0.230765 + 0.973010i $$0.574123\pi$$
$$810$$ 1018.63 0.0441866
$$811$$ 5497.87i 0.238047i 0.992891 + 0.119024i $$0.0379764\pi$$
−0.992891 + 0.119024i $$0.962024\pi$$
$$812$$ − 9630.57i − 0.416215i
$$813$$ 24965.5i 1.07697i
$$814$$ 43556.7i 1.87551i
$$815$$ −7144.34 −0.307062
$$816$$ 4761.66 0.204279
$$817$$ 12940.5i 0.554139i
$$818$$ 21420.5 0.915587
$$819$$ 0 0
$$820$$ −1539.55 −0.0655653
$$821$$ 21305.3i 0.905678i 0.891592 + 0.452839i $$0.149589\pi$$
−0.891592 + 0.452839i $$0.850411\pi$$
$$822$$ 2550.48 0.108222
$$823$$ −17342.6 −0.734537 −0.367268 0.930115i $$-0.619707\pi$$
−0.367268 + 0.930115i $$0.619707\pi$$
$$824$$ − 2857.94i − 0.120827i
$$825$$ 11193.7i 0.472381i
$$826$$ 18091.0i 0.762064i
$$827$$ − 5129.96i − 0.215703i −0.994167 0.107851i $$-0.965603\pi$$
0.994167 0.107851i $$-0.0343971\pi$$
$$828$$ −9638.19 −0.404529
$$829$$ 8471.81 0.354931 0.177466 0.984127i $$-0.443210\pi$$
0.177466 + 0.984127i $$0.443210\pi$$
$$830$$ 8640.72i 0.361354i
$$831$$ −8643.93 −0.360836
$$832$$ 0 0
$$833$$ 8609.50 0.358105
$$834$$ 1236.93i 0.0513566i
$$835$$ 122.225 0.00506561
$$836$$ 29371.5 1.21511
$$837$$ 1027.90i 0.0424486i
$$838$$ − 28128.3i − 1.15952i
$$839$$ 19155.0i 0.788207i 0.919066 + 0.394103i $$0.128945\pi$$
−0.919066 + 0.394103i $$0.871055\pi$$
$$840$$ 252.140i 0.0103567i
$$841$$ 1246.67 0.0511160
$$842$$ −31078.7 −1.27202
$$843$$ − 8453.98i − 0.345398i
$$844$$ −34819.7 −1.42007
$$845$$ 0 0
$$846$$ −19221.3 −0.781136
$$847$$ 1944.42i 0.0788797i
$$848$$ 38255.0 1.54915
$$849$$ 793.804 0.0320887
$$850$$ 13766.4i 0.555512i
$$851$$ 38977.3i 1.57006i
$$852$$ 8349.11i 0.335723i
$$853$$ 18075.1i 0.725532i 0.931880 + 0.362766i $$0.118168\pi$$
−0.931880 + 0.362766i $$0.881832\pi$$
$$854$$ −19324.9 −0.774339
$$855$$ −2777.82 −0.111111
$$856$$ 1672.64i 0.0667871i
$$857$$ 21054.6 0.839219 0.419609 0.907705i $$-0.362167\pi$$
0.419609 + 0.907705i $$0.362167\pi$$
$$858$$ 0 0
$$859$$ 920.322 0.0365553 0.0182776 0.999833i $$-0.494182\pi$$
0.0182776 + 0.999833i $$0.494182\pi$$
$$860$$ − 3510.33i − 0.139188i
$$861$$ 1124.48 0.0445090
$$862$$ 55267.0 2.18376
$$863$$ − 19427.5i − 0.766304i −0.923685 0.383152i $$-0.874839\pi$$
0.923685 0.383152i $$-0.125161\pi$$
$$864$$ 7013.40i 0.276158i
$$865$$ 5824.21i 0.228935i
$$866$$ 73044.7i 2.86623i
$$867$$ 12240.5 0.479482
$$868$$ −2289.91 −0.0895444
$$869$$ − 29083.7i − 1.13532i
$$870$$ −6040.55 −0.235395
$$871$$ 0 0
$$872$$ −1977.09 −0.0767808
$$873$$ 15790.3i 0.612168i
$$874$$ 49646.5 1.92142
$$875$$ 4906.44 0.189563
$$876$$ 10522.2i 0.405836i
$$877$$ 14872.2i 0.572632i 0.958135 + 0.286316i $$0.0924306\pi$$
−0.958135 + 0.286316i $$0.907569\pi$$
$$878$$ − 29535.7i − 1.13529i
$$879$$ 12877.6i 0.494141i
$$880$$ 5410.04 0.207241
$$881$$ 12940.6 0.494870 0.247435 0.968905i $$-0.420412\pi$$
0.247435 + 0.968905i $$0.420412\pi$$
$$882$$ 11070.6i 0.422637i
$$883$$ 25585.5 0.975108 0.487554 0.873093i $$-0.337889\pi$$
0.487554 + 0.873093i $$0.337889\pi$$
$$884$$ 0 0
$$885$$ 6007.30 0.228173
$$886$$ − 41928.8i − 1.58987i
$$887$$ 3716.46 0.140684 0.0703418 0.997523i $$-0.477591\pi$$
0.0703418 + 0.997523i $$0.477591\pi$$
$$888$$ −4051.79 −0.153118
$$889$$ 16675.7i 0.629118i
$$890$$ − 13462.8i − 0.507049i
$$891$$ − 2612.25i − 0.0982195i
$$892$$ − 25322.0i − 0.950496i
$$893$$ 52416.7 1.96423
$$894$$ 4705.92 0.176051
$$895$$ − 1555.40i − 0.0580910i
$$896$$ −3499.58 −0.130483
$$897$$ 0 0
$$898$$ −70680.3 −2.62654
$$899$$ − 6095.52i − 0.226137i
$$900$$ −9371.47 −0.347091
$$901$$ −20072.5 −0.742187
$$902$$ 7457.57i 0.275288i
$$903$$ 2563.93i 0.0944876i
$$904$$ 6379.29i 0.234703i
$$905$$ − 6517.61i − 0.239395i
$$906$$ 18769.8 0.688285
$$907$$ 12960.4 0.474469 0.237235 0.971452i $$-0.423759\pi$$
0.237235 + 0.971452i $$0.423759\pi$$
$$908$$ 40665.3i 1.48626i
$$909$$ 5767.56 0.210449
$$910$$ 0 0
$$911$$ −36607.1 −1.33134 −0.665668 0.746248i $$-0.731853\pi$$
−0.665668 + 0.746248i $$0.731853\pi$$
$$912$$ − 16697.0i − 0.606242i
$$913$$ 22158.8 0.803230
$$914$$ −58099.3 −2.10258
$$915$$ 6417.06i 0.231849i
$$916$$ − 11747.4i − 0.423740i
$$917$$ 292.440i 0.0105313i
$$918$$ − 3212.65i − 0.115505i
$$919$$ 20356.3 0.730676 0.365338 0.930875i $$-0.380953\pi$$
0.365338 + 0.930875i $$0.380953\pi$$
$$920$$ −1496.38 −0.0536243
$$921$$ 21078.8i 0.754147i
$$922$$ 12048.6 0.430370
$$923$$ 0 0
$$924$$ 5819.43 0.207192
$$925$$ 37898.7i 1.34714i
$$926$$ −8545.61 −0.303268
$$927$$ 6238.38 0.221030
$$928$$ − 41589.8i − 1.47118i
$$929$$ − 45069.7i − 1.59170i −0.605495 0.795849i $$-0.707025\pi$$
0.605495 0.795849i $$-0.292975\pi$$
$$930$$ 1436.29i 0.0506428i
$$931$$ − 30189.6i − 1.06275i
$$932$$ −30244.9 −1.06299
$$933$$ −3399.62 −0.119291
$$934$$ 10984.7i 0.384831i
$$935$$ −2838.65 −0.0992876
$$936$$ 0 0
$$937$$ −6771.10 −0.236075 −0.118037 0.993009i $$-0.537660\pi$$
−0.118037 + 0.993009i $$0.537660\pi$$
$$938$$ 1575.50i 0.0548420i
$$939$$ 15857.8 0.551119
$$940$$ −14218.9 −0.493372
$$941$$ 36690.7i 1.27108i 0.772070 + 0.635538i $$0.219222\pi$$
−0.772070 + 0.635538i $$0.780778\pi$$
$$942$$ − 17937.4i − 0.620417i
$$943$$ 6673.51i 0.230455i
$$944$$ 36108.8i 1.24496i
$$945$$ −550.376 −0.0189457
$$946$$ −17004.0 −0.584406
$$947$$ 50861.2i 1.74527i 0.488375 + 0.872634i $$0.337590\pi$$
−0.488375 + 0.872634i $$0.662410\pi$$
$$948$$ 24349.1 0.834202
$$949$$ 0 0
$$950$$ 48272.6 1.64860
$$951$$ 14346.5i 0.489187i
$$952$$ 795.219 0.0270727
$$953$$ −11855.6 −0.402980 −0.201490 0.979491i $$-0.564578\pi$$
−0.201490 + 0.979491i $$0.564578\pi$$
$$954$$ − 25810.3i − 0.875931i
$$955$$ − 12376.4i − 0.419361i
$$956$$ 42634.5i 1.44236i
$$957$$ 15490.8i 0.523245i
$$958$$ −21525.5 −0.725947
$$959$$ −1378.04 −0.0464019
$$960$$ 5773.76i 0.194112i
$$961$$ 28341.6 0.951349
$$962$$ 0 0
$$963$$ −3651.08 −0.122175
$$964$$ 43067.5i 1.43891i
$$965$$ −2663.90 −0.0888643
$$966$$ 9836.56 0.327625
$$967$$ 40661.7i 1.35221i 0.736803 + 0.676107i $$0.236334\pi$$
−0.736803 + 0.676107i $$0.763666\pi$$
$$968$$ − 1199.58i − 0.0398304i
$$969$$ 8760.93i 0.290445i
$$970$$ 22063.9i 0.730339i
$$971$$ 57318.3 1.89437 0.947184 0.320690i $$-0.103915\pi$$
0.947184 + 0.320690i $$0.103915\pi$$
$$972$$ 2187.00 0.0721688
$$973$$ − 668.324i − 0.0220200i
$$974$$ 50403.3 1.65814
$$975$$ 0 0
$$976$$ −38571.7 −1.26501
$$977$$ − 3026.73i − 0.0991134i −0.998771 0.0495567i $$-0.984219\pi$$
0.998771 0.0495567i $$-0.0157808\pi$$
$$978$$ −28973.4 −0.947308
$$979$$ −34524.8 −1.12709
$$980$$ 8189.43i 0.266941i
$$981$$ − 4315.64i − 0.140457i
$$982$$ − 81032.3i − 2.63324i
$$983$$ − 33942.5i − 1.10132i −0.834730 0.550659i $$-0.814376\pi$$
0.834730 0.550659i $$-0.185624\pi$$
$$984$$ −693.729 −0.0224749
$$985$$ 12649.3 0.409179
$$986$$ 19051.2i 0.615327i
$$987$$ 10385.4 0.334925
$$988$$ 0 0
$$989$$ −15216.3 −0.489231
$$990$$ − 3650.10i − 0.117179i
$$991$$ 21637.5 0.693580 0.346790 0.937943i $$-0.387272\pi$$
0.346790 + 0.937943i $$0.387272\pi$$
$$992$$ −9889.02 −0.316509
$$993$$ − 26009.9i − 0.831219i
$$994$$ − 8520.95i − 0.271900i
$$995$$ − 7332.53i − 0.233625i
$$996$$ 18551.6i 0.590190i
$$997$$ 19624.6 0.623388 0.311694 0.950182i $$-0.399104\pi$$
0.311694 + 0.950182i $$0.399104\pi$$
$$998$$ 48296.3 1.53186
$$999$$ − 8844.33i − 0.280102i
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.b.e.337.4 4
13.5 odd 4 507.4.a.k.1.4 4
13.8 odd 4 507.4.a.k.1.1 4
13.9 even 3 39.4.j.b.10.2 yes 4
13.10 even 6 39.4.j.b.4.2 4
13.12 even 2 inner 507.4.b.e.337.1 4
39.5 even 4 1521.4.a.z.1.1 4
39.8 even 4 1521.4.a.z.1.4 4
39.23 odd 6 117.4.q.d.82.1 4
39.35 odd 6 117.4.q.d.10.1 4
52.23 odd 6 624.4.bv.c.433.2 4
52.35 odd 6 624.4.bv.c.49.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.b.4.2 4 13.10 even 6
39.4.j.b.10.2 yes 4 13.9 even 3
117.4.q.d.10.1 4 39.35 odd 6
117.4.q.d.82.1 4 39.23 odd 6
507.4.a.k.1.1 4 13.8 odd 4
507.4.a.k.1.4 4 13.5 odd 4
507.4.b.e.337.1 4 13.12 even 2 inner
507.4.b.e.337.4 4 1.1 even 1 trivial
624.4.bv.c.49.1 4 52.35 odd 6
624.4.bv.c.433.2 4 52.23 odd 6
1521.4.a.z.1.1 4 39.5 even 4
1521.4.a.z.1.4 4 39.8 even 4