# Properties

 Label 507.4.b.i.337.1 Level $507$ Weight $4$ Character 507.337 Analytic conductor $29.914$ Analytic rank $0$ Dimension $10$ 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: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648$$ x^10 + 70*x^8 + 1645*x^6 + 14700*x^4 + 44100*x^2 + 27648 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{5}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$-5.36472i$$ of defining polynomial Character $$\chi$$ $$=$$ 507.337 Dual form 507.4.b.i.337.10

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.36472i q^{2} +3.00000 q^{3} -20.7803 q^{4} +2.69631i q^{5} -16.0942i q^{6} -15.2025i q^{7} +68.5626i q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-5.36472i q^{2} +3.00000 q^{3} -20.7803 q^{4} +2.69631i q^{5} -16.0942i q^{6} -15.2025i q^{7} +68.5626i q^{8} +9.00000 q^{9} +14.4650 q^{10} +66.8848i q^{11} -62.3408 q^{12} -81.5570 q^{14} +8.08894i q^{15} +201.577 q^{16} -4.16354 q^{17} -48.2825i q^{18} +26.0850i q^{19} -56.0301i q^{20} -45.6074i q^{21} +358.819 q^{22} -47.3242 q^{23} +205.688i q^{24} +117.730 q^{25} +27.0000 q^{27} +315.911i q^{28} +257.007 q^{29} +43.3949 q^{30} +206.242i q^{31} -532.906i q^{32} +200.655i q^{33} +22.3362i q^{34} +40.9906 q^{35} -187.022 q^{36} -175.686i q^{37} +139.939 q^{38} -184.866 q^{40} +156.463i q^{41} -244.671 q^{42} -51.9845 q^{43} -1389.88i q^{44} +24.2668i q^{45} +253.881i q^{46} +354.222i q^{47} +604.732 q^{48} +111.885 q^{49} -631.588i q^{50} -12.4906 q^{51} -10.4723 q^{53} -144.848i q^{54} -180.342 q^{55} +1042.32 q^{56} +78.2550i q^{57} -1378.77i q^{58} -445.114i q^{59} -168.090i q^{60} +119.696 q^{61} +1106.43 q^{62} -136.822i q^{63} -1246.28 q^{64} +1076.46 q^{66} -22.4078i q^{67} +86.5195 q^{68} -141.973 q^{69} -219.903i q^{70} +285.207i q^{71} +617.064i q^{72} -740.989i q^{73} -942.507 q^{74} +353.190 q^{75} -542.053i q^{76} +1016.81 q^{77} -547.679 q^{79} +543.516i q^{80} +81.0000 q^{81} +839.378 q^{82} +603.056i q^{83} +947.734i q^{84} -11.2262i q^{85} +278.882i q^{86} +771.021 q^{87} -4585.80 q^{88} -215.668i q^{89} +130.185 q^{90} +983.409 q^{92} +618.726i q^{93} +1900.31 q^{94} -70.3333 q^{95} -1598.72i q^{96} +1447.50i q^{97} -600.233i q^{98} +601.964i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 30 q^{3} - 60 q^{4} + 90 q^{9}+O(q^{10})$$ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 $$10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36} - 1380 q^{38} + 2000 q^{40} - 180 q^{42} + 740 q^{43} + 1500 q^{48} - 1550 q^{49} - 630 q^{51} + 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} - 3140 q^{64} + 1740 q^{66} + 1200 q^{68} + 360 q^{69} - 4380 q^{74} - 2880 q^{75} - 4320 q^{77} + 1100 q^{79} + 810 q^{81} + 4780 q^{82} + 2970 q^{87} - 6340 q^{88} - 720 q^{90} - 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100})$$ 10 * q + 30 * q^3 - 60 * q^4 + 90 * q^9 - 80 * q^10 - 180 * q^12 - 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 - 960 * q^25 + 270 * q^27 + 990 * q^29 - 240 * q^30 - 120 * q^35 - 540 * q^36 - 1380 * q^38 + 2000 * q^40 - 180 * q^42 + 740 * q^43 + 1500 * q^48 - 1550 * q^49 - 630 * q^51 + 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 - 3140 * q^64 + 1740 * q^66 + 1200 * q^68 + 360 * q^69 - 4380 * q^74 - 2880 * q^75 - 4320 * q^77 + 1100 * q^79 + 810 * q^81 + 4780 * q^82 + 2970 * q^87 - 6340 * q^88 - 720 * q^90 - 1740 * q^92 + 6460 * q^94 + 2760 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 5.36472i − 1.89672i −0.317201 0.948358i $$-0.602743\pi$$
0.317201 0.948358i $$-0.397257\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −20.7803 −2.59753
$$5$$ 2.69631i 0.241165i 0.992703 + 0.120583i $$0.0384763\pi$$
−0.992703 + 0.120583i $$0.961524\pi$$
$$6$$ − 16.0942i − 1.09507i
$$7$$ − 15.2025i − 0.820856i −0.911893 0.410428i $$-0.865379\pi$$
0.911893 0.410428i $$-0.134621\pi$$
$$8$$ 68.5626i 3.03007i
$$9$$ 9.00000 0.333333
$$10$$ 14.4650 0.457423
$$11$$ 66.8848i 1.83332i 0.399666 + 0.916661i $$0.369126\pi$$
−0.399666 + 0.916661i $$0.630874\pi$$
$$12$$ −62.3408 −1.49969
$$13$$ 0 0
$$14$$ −81.5570 −1.55693
$$15$$ 8.08894i 0.139237i
$$16$$ 201.577 3.14965
$$17$$ −4.16354 −0.0594004 −0.0297002 0.999559i $$-0.509455\pi$$
−0.0297002 + 0.999559i $$0.509455\pi$$
$$18$$ − 48.2825i − 0.632239i
$$19$$ 26.0850i 0.314963i 0.987522 + 0.157482i $$0.0503375\pi$$
−0.987522 + 0.157482i $$0.949662\pi$$
$$20$$ − 56.0301i − 0.626436i
$$21$$ − 45.6074i − 0.473921i
$$22$$ 358.819 3.47729
$$23$$ −47.3242 −0.429034 −0.214517 0.976720i $$-0.568818\pi$$
−0.214517 + 0.976720i $$0.568818\pi$$
$$24$$ 205.688i 1.74941i
$$25$$ 117.730 0.941839
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 315.911i 2.13220i
$$29$$ 257.007 1.64569 0.822845 0.568266i $$-0.192386\pi$$
0.822845 + 0.568266i $$0.192386\pi$$
$$30$$ 43.3949 0.264093
$$31$$ 206.242i 1.19491i 0.801903 + 0.597455i $$0.203821\pi$$
−0.801903 + 0.597455i $$0.796179\pi$$
$$32$$ − 532.906i − 2.94392i
$$33$$ 200.655i 1.05847i
$$34$$ 22.3362i 0.112666i
$$35$$ 40.9906 0.197962
$$36$$ −187.022 −0.865845
$$37$$ − 175.686i − 0.780611i −0.920685 0.390305i $$-0.872369\pi$$
0.920685 0.390305i $$-0.127631\pi$$
$$38$$ 139.939 0.597396
$$39$$ 0 0
$$40$$ −184.866 −0.730748
$$41$$ 156.463i 0.595984i 0.954568 + 0.297992i $$0.0963169\pi$$
−0.954568 + 0.297992i $$0.903683\pi$$
$$42$$ −244.671 −0.898895
$$43$$ −51.9845 −0.184362 −0.0921809 0.995742i $$-0.529384\pi$$
−0.0921809 + 0.995742i $$0.529384\pi$$
$$44$$ − 1389.88i − 4.76211i
$$45$$ 24.2668i 0.0803885i
$$46$$ 253.881i 0.813755i
$$47$$ 354.222i 1.09933i 0.835384 + 0.549666i $$0.185245\pi$$
−0.835384 + 0.549666i $$0.814755\pi$$
$$48$$ 604.732 1.81845
$$49$$ 111.885 0.326196
$$50$$ − 631.588i − 1.78640i
$$51$$ −12.4906 −0.0342948
$$52$$ 0 0
$$53$$ −10.4723 −0.0271412 −0.0135706 0.999908i $$-0.504320\pi$$
−0.0135706 + 0.999908i $$0.504320\pi$$
$$54$$ − 144.848i − 0.365023i
$$55$$ −180.342 −0.442134
$$56$$ 1042.32 2.48725
$$57$$ 78.2550i 0.181844i
$$58$$ − 1378.77i − 3.12141i
$$59$$ − 445.114i − 0.982185i −0.871107 0.491092i $$-0.836598\pi$$
0.871107 0.491092i $$-0.163402\pi$$
$$60$$ − 168.090i − 0.361673i
$$61$$ 119.696 0.251238 0.125619 0.992079i $$-0.459908\pi$$
0.125619 + 0.992079i $$0.459908\pi$$
$$62$$ 1106.43 2.26640
$$63$$ − 136.822i − 0.273619i
$$64$$ −1246.28 −2.43413
$$65$$ 0 0
$$66$$ 1076.46 2.00761
$$67$$ − 22.4078i − 0.0408589i −0.999791 0.0204294i $$-0.993497\pi$$
0.999791 0.0204294i $$-0.00650334\pi$$
$$68$$ 86.5195 0.154295
$$69$$ −141.973 −0.247703
$$70$$ − 219.903i − 0.375478i
$$71$$ 285.207i 0.476731i 0.971176 + 0.238365i $$0.0766116\pi$$
−0.971176 + 0.238365i $$0.923388\pi$$
$$72$$ 617.064i 1.01002i
$$73$$ − 740.989i − 1.18803i −0.804454 0.594015i $$-0.797542\pi$$
0.804454 0.594015i $$-0.202458\pi$$
$$74$$ −942.507 −1.48060
$$75$$ 353.190 0.543771
$$76$$ − 542.053i − 0.818128i
$$77$$ 1016.81 1.50489
$$78$$ 0 0
$$79$$ −547.679 −0.779983 −0.389992 0.920818i $$-0.627522\pi$$
−0.389992 + 0.920818i $$0.627522\pi$$
$$80$$ 543.516i 0.759586i
$$81$$ 81.0000 0.111111
$$82$$ 839.378 1.13041
$$83$$ 603.056i 0.797518i 0.917056 + 0.398759i $$0.130559\pi$$
−0.917056 + 0.398759i $$0.869441\pi$$
$$84$$ 947.734i 1.23103i
$$85$$ − 11.2262i − 0.0143253i
$$86$$ 278.882i 0.349682i
$$87$$ 771.021 0.950139
$$88$$ −4585.80 −5.55509
$$89$$ − 215.668i − 0.256863i −0.991718 0.128431i $$-0.959006\pi$$
0.991718 0.128431i $$-0.0409942\pi$$
$$90$$ 130.185 0.152474
$$91$$ 0 0
$$92$$ 983.409 1.11443
$$93$$ 618.726i 0.689881i
$$94$$ 1900.31 2.08512
$$95$$ −70.3333 −0.0759583
$$96$$ − 1598.72i − 1.69967i
$$97$$ 1447.50i 1.51517i 0.652735 + 0.757586i $$0.273622\pi$$
−0.652735 + 0.757586i $$0.726378\pi$$
$$98$$ − 600.233i − 0.618700i
$$99$$ 601.964i 0.611107i
$$100$$ −2446.46 −2.44646
$$101$$ 883.450 0.870362 0.435181 0.900343i $$-0.356684\pi$$
0.435181 + 0.900343i $$0.356684\pi$$
$$102$$ 67.0087i 0.0650476i
$$103$$ −1251.74 −1.19745 −0.598726 0.800954i $$-0.704326\pi$$
−0.598726 + 0.800954i $$0.704326\pi$$
$$104$$ 0 0
$$105$$ 122.972 0.114293
$$106$$ 56.1812i 0.0514792i
$$107$$ −341.614 −0.308645 −0.154323 0.988021i $$-0.549319\pi$$
−0.154323 + 0.988021i $$0.549319\pi$$
$$108$$ −561.067 −0.499896
$$109$$ 775.177i 0.681179i 0.940212 + 0.340589i $$0.110627\pi$$
−0.940212 + 0.340589i $$0.889373\pi$$
$$110$$ 967.487i 0.838603i
$$111$$ − 527.058i − 0.450686i
$$112$$ − 3064.47i − 2.58541i
$$113$$ 1279.05 1.06480 0.532402 0.846492i $$-0.321290\pi$$
0.532402 + 0.846492i $$0.321290\pi$$
$$114$$ 419.816 0.344907
$$115$$ − 127.601i − 0.103468i
$$116$$ −5340.67 −4.27473
$$117$$ 0 0
$$118$$ −2387.91 −1.86293
$$119$$ 63.2961i 0.0487592i
$$120$$ −554.599 −0.421898
$$121$$ −3142.58 −2.36107
$$122$$ − 642.137i − 0.476528i
$$123$$ 469.388i 0.344091i
$$124$$ − 4285.77i − 3.10382i
$$125$$ 654.476i 0.468305i
$$126$$ −734.013 −0.518977
$$127$$ 1113.82 0.778233 0.389117 0.921188i $$-0.372780\pi$$
0.389117 + 0.921188i $$0.372780\pi$$
$$128$$ 2422.68i 1.67294i
$$129$$ −155.953 −0.106441
$$130$$ 0 0
$$131$$ 2100.12 1.40068 0.700339 0.713811i $$-0.253032\pi$$
0.700339 + 0.713811i $$0.253032\pi$$
$$132$$ − 4169.65i − 2.74941i
$$133$$ 396.556 0.258540
$$134$$ −120.211 −0.0774977
$$135$$ 72.8004i 0.0464123i
$$136$$ − 285.463i − 0.179987i
$$137$$ 1205.02i 0.751471i 0.926727 + 0.375736i $$0.122610\pi$$
−0.926727 + 0.375736i $$0.877390\pi$$
$$138$$ 761.643i 0.469822i
$$139$$ 322.890 0.197030 0.0985149 0.995136i $$-0.468591\pi$$
0.0985149 + 0.995136i $$0.468591\pi$$
$$140$$ −851.796 −0.514213
$$141$$ 1062.67i 0.634700i
$$142$$ 1530.06 0.904223
$$143$$ 0 0
$$144$$ 1814.20 1.04988
$$145$$ 692.971i 0.396884i
$$146$$ −3975.20 −2.25336
$$147$$ 335.655 0.188329
$$148$$ 3650.80i 2.02766i
$$149$$ 1128.86i 0.620669i 0.950627 + 0.310335i $$0.100441\pi$$
−0.950627 + 0.310335i $$0.899559\pi$$
$$150$$ − 1894.77i − 1.03138i
$$151$$ 2940.44i 1.58470i 0.610066 + 0.792350i $$0.291143\pi$$
−0.610066 + 0.792350i $$0.708857\pi$$
$$152$$ −1788.46 −0.954361
$$153$$ −37.4719 −0.0198001
$$154$$ − 5454.93i − 2.85436i
$$155$$ −556.093 −0.288171
$$156$$ 0 0
$$157$$ 629.388 0.319940 0.159970 0.987122i $$-0.448860\pi$$
0.159970 + 0.987122i $$0.448860\pi$$
$$158$$ 2938.15i 1.47941i
$$159$$ −31.4170 −0.0156700
$$160$$ 1436.88 0.709972
$$161$$ 719.444i 0.352175i
$$162$$ − 434.543i − 0.210746i
$$163$$ 394.912i 0.189766i 0.995488 + 0.0948832i $$0.0302478\pi$$
−0.995488 + 0.0948832i $$0.969752\pi$$
$$164$$ − 3251.33i − 1.54809i
$$165$$ −541.027 −0.255266
$$166$$ 3235.23 1.51267
$$167$$ 151.860i 0.0703669i 0.999381 + 0.0351834i $$0.0112015\pi$$
−0.999381 + 0.0351834i $$0.988798\pi$$
$$168$$ 3126.96 1.43601
$$169$$ 0 0
$$170$$ −60.2255 −0.0271711
$$171$$ 234.765i 0.104988i
$$172$$ 1080.25 0.478886
$$173$$ −538.813 −0.236793 −0.118397 0.992966i $$-0.537775\pi$$
−0.118397 + 0.992966i $$0.537775\pi$$
$$174$$ − 4136.31i − 1.80214i
$$175$$ − 1789.78i − 0.773114i
$$176$$ 13482.5i 5.77432i
$$177$$ − 1335.34i − 0.567065i
$$178$$ −1157.00 −0.487196
$$179$$ 2220.80 0.927319 0.463659 0.886014i $$-0.346536\pi$$
0.463659 + 0.886014i $$0.346536\pi$$
$$180$$ − 504.271i − 0.208812i
$$181$$ 3822.78 1.56986 0.784932 0.619582i $$-0.212698\pi$$
0.784932 + 0.619582i $$0.212698\pi$$
$$182$$ 0 0
$$183$$ 359.089 0.145052
$$184$$ − 3244.67i − 1.30000i
$$185$$ 473.704 0.188256
$$186$$ 3319.30 1.30851
$$187$$ − 278.478i − 0.108900i
$$188$$ − 7360.84i − 2.85555i
$$189$$ − 410.467i − 0.157974i
$$190$$ 377.319i 0.144071i
$$191$$ 3464.19 1.31236 0.656178 0.754606i $$-0.272172\pi$$
0.656178 + 0.754606i $$0.272172\pi$$
$$192$$ −3738.83 −1.40535
$$193$$ − 4697.40i − 1.75195i −0.482357 0.875975i $$-0.660219\pi$$
0.482357 0.875975i $$-0.339781\pi$$
$$194$$ 7765.46 2.87385
$$195$$ 0 0
$$196$$ −2325.00 −0.847304
$$197$$ − 2887.89i − 1.04443i −0.852813 0.522217i $$-0.825105\pi$$
0.852813 0.522217i $$-0.174895\pi$$
$$198$$ 3229.37 1.15910
$$199$$ −63.0092 −0.0224453 −0.0112226 0.999937i $$-0.503572\pi$$
−0.0112226 + 0.999937i $$0.503572\pi$$
$$200$$ 8071.87i 2.85384i
$$201$$ − 67.2233i − 0.0235899i
$$202$$ − 4739.47i − 1.65083i
$$203$$ − 3907.14i − 1.35087i
$$204$$ 259.558 0.0890820
$$205$$ −421.872 −0.143731
$$206$$ 6715.24i 2.27123i
$$207$$ −425.918 −0.143011
$$208$$ 0 0
$$209$$ −1744.69 −0.577429
$$210$$ − 659.710i − 0.216782i
$$211$$ −1049.70 −0.342484 −0.171242 0.985229i $$-0.554778\pi$$
−0.171242 + 0.985229i $$0.554778\pi$$
$$212$$ 217.618 0.0705002
$$213$$ 855.622i 0.275241i
$$214$$ 1832.66i 0.585412i
$$215$$ − 140.166i − 0.0444617i
$$216$$ 1851.19i 0.583137i
$$217$$ 3135.39 0.980848
$$218$$ 4158.61 1.29200
$$219$$ − 2222.97i − 0.685909i
$$220$$ 3747.56 1.14846
$$221$$ 0 0
$$222$$ −2827.52 −0.854823
$$223$$ 2313.49i 0.694722i 0.937732 + 0.347361i $$0.112922\pi$$
−0.937732 + 0.347361i $$0.887078\pi$$
$$224$$ −8101.49 −2.41653
$$225$$ 1059.57 0.313946
$$226$$ − 6861.74i − 2.01963i
$$227$$ 3799.46i 1.11092i 0.831543 + 0.555460i $$0.187458\pi$$
−0.831543 + 0.555460i $$0.812542\pi$$
$$228$$ − 1626.16i − 0.472347i
$$229$$ 4321.07i 1.24692i 0.781856 + 0.623459i $$0.214273\pi$$
−0.781856 + 0.623459i $$0.785727\pi$$
$$230$$ −684.543 −0.196250
$$231$$ 3050.44 0.868850
$$232$$ 17621.1i 4.98655i
$$233$$ −5279.77 −1.48450 −0.742251 0.670122i $$-0.766242\pi$$
−0.742251 + 0.670122i $$0.766242\pi$$
$$234$$ 0 0
$$235$$ −955.094 −0.265121
$$236$$ 9249.59i 2.55126i
$$237$$ −1643.04 −0.450323
$$238$$ 339.566 0.0924823
$$239$$ 1547.92i 0.418939i 0.977815 + 0.209469i $$0.0671737\pi$$
−0.977815 + 0.209469i $$0.932826\pi$$
$$240$$ 1630.55i 0.438547i
$$241$$ 4918.01i 1.31451i 0.753669 + 0.657255i $$0.228282\pi$$
−0.753669 + 0.657255i $$0.771718\pi$$
$$242$$ 16859.1i 4.47828i
$$243$$ 243.000 0.0641500
$$244$$ −2487.32 −0.652600
$$245$$ 301.677i 0.0786671i
$$246$$ 2518.14 0.652644
$$247$$ 0 0
$$248$$ −14140.5 −3.62066
$$249$$ 1809.17i 0.460447i
$$250$$ 3511.08 0.888241
$$251$$ −1155.78 −0.290646 −0.145323 0.989384i $$-0.546422\pi$$
−0.145323 + 0.989384i $$0.546422\pi$$
$$252$$ 2843.20i 0.710734i
$$253$$ − 3165.27i − 0.786556i
$$254$$ − 5975.34i − 1.47609i
$$255$$ − 33.6786i − 0.00827073i
$$256$$ 3026.80 0.738965
$$257$$ −2351.95 −0.570859 −0.285429 0.958400i $$-0.592136\pi$$
−0.285429 + 0.958400i $$0.592136\pi$$
$$258$$ 836.647i 0.201889i
$$259$$ −2670.86 −0.640769
$$260$$ 0 0
$$261$$ 2313.06 0.548563
$$262$$ − 11266.6i − 2.65669i
$$263$$ 5521.88 1.29465 0.647326 0.762213i $$-0.275887\pi$$
0.647326 + 0.762213i $$0.275887\pi$$
$$264$$ −13757.4 −3.20723
$$265$$ − 28.2367i − 0.00654553i
$$266$$ − 2127.41i − 0.490376i
$$267$$ − 647.004i − 0.148300i
$$268$$ 465.639i 0.106132i
$$269$$ 3916.95 0.887810 0.443905 0.896074i $$-0.353593\pi$$
0.443905 + 0.896074i $$0.353593\pi$$
$$270$$ 390.554 0.0880310
$$271$$ 2777.53i 0.622593i 0.950313 + 0.311297i $$0.100763\pi$$
−0.950313 + 0.311297i $$0.899237\pi$$
$$272$$ −839.276 −0.187090
$$273$$ 0 0
$$274$$ 6464.58 1.42533
$$275$$ 7874.34i 1.72669i
$$276$$ 2950.23 0.643416
$$277$$ −6583.08 −1.42794 −0.713969 0.700177i $$-0.753104\pi$$
−0.713969 + 0.700177i $$0.753104\pi$$
$$278$$ − 1732.21i − 0.373710i
$$279$$ 1856.18i 0.398303i
$$280$$ 2810.42i 0.599839i
$$281$$ − 2871.66i − 0.609640i −0.952410 0.304820i $$-0.901404\pi$$
0.952410 0.304820i $$-0.0985963\pi$$
$$282$$ 5700.92 1.20385
$$283$$ −7518.04 −1.57916 −0.789578 0.613651i $$-0.789700\pi$$
−0.789578 + 0.613651i $$0.789700\pi$$
$$284$$ − 5926.69i − 1.23832i
$$285$$ −211.000 −0.0438546
$$286$$ 0 0
$$287$$ 2378.62 0.489217
$$288$$ − 4796.16i − 0.981307i
$$289$$ −4895.66 −0.996472
$$290$$ 3717.60 0.752776
$$291$$ 4342.51i 0.874785i
$$292$$ 15397.9i 3.08595i
$$293$$ − 4506.57i − 0.898555i −0.893392 0.449278i $$-0.851681\pi$$
0.893392 0.449278i $$-0.148319\pi$$
$$294$$ − 1800.70i − 0.357207i
$$295$$ 1200.17 0.236869
$$296$$ 12045.5 2.36530
$$297$$ 1805.89i 0.352823i
$$298$$ 6056.02 1.17723
$$299$$ 0 0
$$300$$ −7339.38 −1.41246
$$301$$ 790.292i 0.151335i
$$302$$ 15774.7 3.00573
$$303$$ 2650.35 0.502504
$$304$$ 5258.15i 0.992024i
$$305$$ 322.738i 0.0605900i
$$306$$ 201.026i 0.0375552i
$$307$$ − 9538.89i − 1.77333i −0.462409 0.886667i $$-0.653015\pi$$
0.462409 0.886667i $$-0.346985\pi$$
$$308$$ −21129.7 −3.90901
$$309$$ −3755.22 −0.691350
$$310$$ 2983.29i 0.546578i
$$311$$ 7466.28 1.36133 0.680666 0.732594i $$-0.261691\pi$$
0.680666 + 0.732594i $$0.261691\pi$$
$$312$$ 0 0
$$313$$ −1821.65 −0.328964 −0.164482 0.986380i $$-0.552595\pi$$
−0.164482 + 0.986380i $$0.552595\pi$$
$$314$$ − 3376.49i − 0.606836i
$$315$$ 368.915 0.0659874
$$316$$ 11380.9 2.02603
$$317$$ 3125.14i 0.553708i 0.960912 + 0.276854i $$0.0892918\pi$$
−0.960912 + 0.276854i $$0.910708\pi$$
$$318$$ 168.543i 0.0297215i
$$319$$ 17189.9i 3.01708i
$$320$$ − 3360.35i − 0.587029i
$$321$$ −1024.84 −0.178196
$$322$$ 3859.62 0.667976
$$323$$ − 108.606i − 0.0187090i
$$324$$ −1683.20 −0.288615
$$325$$ 0 0
$$326$$ 2118.60 0.359933
$$327$$ 2325.53i 0.393279i
$$328$$ −10727.5 −1.80587
$$329$$ 5385.05 0.902394
$$330$$ 2902.46i 0.484167i
$$331$$ − 1553.67i − 0.257999i −0.991645 0.128999i $$-0.958823\pi$$
0.991645 0.128999i $$-0.0411765\pi$$
$$332$$ − 12531.7i − 2.07158i
$$333$$ − 1581.17i − 0.260204i
$$334$$ 814.686 0.133466
$$335$$ 60.4183 0.00985375
$$336$$ − 9193.42i − 1.49269i
$$337$$ 3190.43 0.515709 0.257855 0.966184i $$-0.416984\pi$$
0.257855 + 0.966184i $$0.416984\pi$$
$$338$$ 0 0
$$339$$ 3837.15 0.614764
$$340$$ 233.283i 0.0372105i
$$341$$ −13794.5 −2.19065
$$342$$ 1259.45 0.199132
$$343$$ − 6915.37i − 1.08862i
$$344$$ − 3564.19i − 0.558629i
$$345$$ − 382.802i − 0.0597373i
$$346$$ 2890.59i 0.449130i
$$347$$ −5718.68 −0.884712 −0.442356 0.896840i $$-0.645857\pi$$
−0.442356 + 0.896840i $$0.645857\pi$$
$$348$$ −16022.0 −2.46802
$$349$$ 3328.46i 0.510511i 0.966874 + 0.255256i $$0.0821596\pi$$
−0.966874 + 0.255256i $$0.917840\pi$$
$$350$$ −9601.70 −1.46638
$$351$$ 0 0
$$352$$ 35643.4 5.39715
$$353$$ − 12306.5i − 1.85555i −0.373142 0.927774i $$-0.621719\pi$$
0.373142 0.927774i $$-0.378281\pi$$
$$354$$ −7163.74 −1.07556
$$355$$ −769.008 −0.114971
$$356$$ 4481.64i 0.667210i
$$357$$ 189.888i 0.0281511i
$$358$$ − 11914.0i − 1.75886i
$$359$$ − 8539.97i − 1.25549i −0.778418 0.627747i $$-0.783977\pi$$
0.778418 0.627747i $$-0.216023\pi$$
$$360$$ −1663.80 −0.243583
$$361$$ 6178.57 0.900798
$$362$$ − 20508.2i − 2.97759i
$$363$$ −9427.74 −1.36316
$$364$$ 0 0
$$365$$ 1997.94 0.286512
$$366$$ − 1926.41i − 0.275123i
$$367$$ 2496.65 0.355107 0.177553 0.984111i $$-0.443182\pi$$
0.177553 + 0.984111i $$0.443182\pi$$
$$368$$ −9539.49 −1.35130
$$369$$ 1408.16i 0.198661i
$$370$$ − 2541.29i − 0.357069i
$$371$$ 159.205i 0.0222790i
$$372$$ − 12857.3i − 1.79199i
$$373$$ −1142.91 −0.158653 −0.0793264 0.996849i $$-0.525277\pi$$
−0.0793264 + 0.996849i $$0.525277\pi$$
$$374$$ −1493.96 −0.206552
$$375$$ 1963.43i 0.270376i
$$376$$ −24286.4 −3.33105
$$377$$ 0 0
$$378$$ −2202.04 −0.299632
$$379$$ − 12181.8i − 1.65102i −0.564384 0.825512i $$-0.690886\pi$$
0.564384 0.825512i $$-0.309114\pi$$
$$380$$ 1461.54 0.197304
$$381$$ 3341.46 0.449313
$$382$$ − 18584.4i − 2.48917i
$$383$$ − 10180.6i − 1.35824i −0.734029 0.679118i $$-0.762362\pi$$
0.734029 0.679118i $$-0.237638\pi$$
$$384$$ 7268.04i 0.965874i
$$385$$ 2741.65i 0.362928i
$$386$$ −25200.3 −3.32295
$$387$$ −467.860 −0.0614539
$$388$$ − 30079.5i − 3.93571i
$$389$$ −5845.83 −0.761941 −0.380971 0.924587i $$-0.624410\pi$$
−0.380971 + 0.924587i $$0.624410\pi$$
$$390$$ 0 0
$$391$$ 197.036 0.0254848
$$392$$ 7671.13i 0.988395i
$$393$$ 6300.37 0.808681
$$394$$ −15492.7 −1.98100
$$395$$ − 1476.71i − 0.188105i
$$396$$ − 12509.0i − 1.58737i
$$397$$ 2500.92i 0.316166i 0.987426 + 0.158083i $$0.0505313\pi$$
−0.987426 + 0.158083i $$0.949469\pi$$
$$398$$ 338.027i 0.0425723i
$$399$$ 1189.67 0.149268
$$400$$ 23731.7 2.96646
$$401$$ 9189.25i 1.14436i 0.820127 + 0.572181i $$0.193902\pi$$
−0.820127 + 0.572181i $$0.806098\pi$$
$$402$$ −360.634 −0.0447433
$$403$$ 0 0
$$404$$ −18358.3 −2.26080
$$405$$ 218.401i 0.0267962i
$$406$$ −20960.7 −2.56223
$$407$$ 11750.7 1.43111
$$408$$ − 856.390i − 0.103916i
$$409$$ 9214.38i 1.11399i 0.830516 + 0.556995i $$0.188046\pi$$
−0.830516 + 0.556995i $$0.811954\pi$$
$$410$$ 2263.23i 0.272617i
$$411$$ 3615.05i 0.433862i
$$412$$ 26011.5 3.11042
$$413$$ −6766.83 −0.806232
$$414$$ 2284.93i 0.271252i
$$415$$ −1626.03 −0.192334
$$416$$ 0 0
$$417$$ 968.669 0.113755
$$418$$ 9359.78i 1.09522i
$$419$$ −6494.41 −0.757214 −0.378607 0.925558i $$-0.623597\pi$$
−0.378607 + 0.925558i $$0.623597\pi$$
$$420$$ −2555.39 −0.296881
$$421$$ 3059.56i 0.354190i 0.984194 + 0.177095i $$0.0566699\pi$$
−0.984194 + 0.177095i $$0.943330\pi$$
$$422$$ 5631.33i 0.649595i
$$423$$ 3188.00i 0.366444i
$$424$$ − 718.010i − 0.0822398i
$$425$$ −490.173 −0.0559456
$$426$$ 4590.18 0.522054
$$427$$ − 1819.68i − 0.206230i
$$428$$ 7098.82 0.801716
$$429$$ 0 0
$$430$$ −751.954 −0.0843313
$$431$$ 7937.05i 0.887040i 0.896264 + 0.443520i $$0.146271\pi$$
−0.896264 + 0.443520i $$0.853729\pi$$
$$432$$ 5442.59 0.606150
$$433$$ −7294.37 −0.809573 −0.404786 0.914411i $$-0.632654\pi$$
−0.404786 + 0.914411i $$0.632654\pi$$
$$434$$ − 16820.5i − 1.86039i
$$435$$ 2078.91i 0.229141i
$$436$$ − 16108.4i − 1.76938i
$$437$$ − 1234.45i − 0.135130i
$$438$$ −11925.6 −1.30098
$$439$$ −15214.7 −1.65412 −0.827059 0.562115i $$-0.809988\pi$$
−0.827059 + 0.562115i $$0.809988\pi$$
$$440$$ − 12364.7i − 1.33970i
$$441$$ 1006.97 0.108732
$$442$$ 0 0
$$443$$ 1517.05 0.162703 0.0813515 0.996685i $$-0.474076\pi$$
0.0813515 + 0.996685i $$0.474076\pi$$
$$444$$ 10952.4i 1.17067i
$$445$$ 581.509 0.0619464
$$446$$ 12411.3 1.31769
$$447$$ 3386.58i 0.358343i
$$448$$ 18946.5i 1.99807i
$$449$$ 705.247i 0.0741262i 0.999313 + 0.0370631i $$0.0118002\pi$$
−0.999313 + 0.0370631i $$0.988200\pi$$
$$450$$ − 5684.30i − 0.595467i
$$451$$ −10465.0 −1.09263
$$452$$ −26579.0 −2.76586
$$453$$ 8821.33i 0.914927i
$$454$$ 20383.1 2.10710
$$455$$ 0 0
$$456$$ −5365.37 −0.551001
$$457$$ − 7277.73i − 0.744940i −0.928044 0.372470i $$-0.878511\pi$$
0.928044 0.372470i $$-0.121489\pi$$
$$458$$ 23181.4 2.36505
$$459$$ −112.416 −0.0114316
$$460$$ 2651.58i 0.268762i
$$461$$ 1961.88i 0.198208i 0.995077 + 0.0991041i $$0.0315977\pi$$
−0.995077 + 0.0991041i $$0.968402\pi$$
$$462$$ − 16364.8i − 1.64796i
$$463$$ 10374.1i 1.04131i 0.853768 + 0.520653i $$0.174311\pi$$
−0.853768 + 0.520653i $$0.825689\pi$$
$$464$$ 51806.8 5.18334
$$465$$ −1668.28 −0.166376
$$466$$ 28324.5i 2.81568i
$$467$$ −8788.92 −0.870883 −0.435442 0.900217i $$-0.643408\pi$$
−0.435442 + 0.900217i $$0.643408\pi$$
$$468$$ 0 0
$$469$$ −340.653 −0.0335392
$$470$$ 5123.82i 0.502860i
$$471$$ 1888.16 0.184718
$$472$$ 30518.2 2.97609
$$473$$ − 3476.97i − 0.337995i
$$474$$ 8814.44i 0.854136i
$$475$$ 3070.98i 0.296645i
$$476$$ − 1315.31i − 0.126654i
$$477$$ −94.2510 −0.00904707
$$478$$ 8304.14 0.794608
$$479$$ 11141.7i 1.06279i 0.847125 + 0.531394i $$0.178332\pi$$
−0.847125 + 0.531394i $$0.821668\pi$$
$$480$$ 4310.65 0.409903
$$481$$ 0 0
$$482$$ 26383.8 2.49325
$$483$$ 2158.33i 0.203328i
$$484$$ 65303.7 6.13295
$$485$$ −3902.92 −0.365407
$$486$$ − 1303.63i − 0.121674i
$$487$$ − 19640.5i − 1.82750i −0.406273 0.913752i $$-0.633172\pi$$
0.406273 0.913752i $$-0.366828\pi$$
$$488$$ 8206.69i 0.761269i
$$489$$ 1184.74i 0.109562i
$$490$$ 1618.41 0.149209
$$491$$ 3410.31 0.313453 0.156726 0.987642i $$-0.449906\pi$$
0.156726 + 0.987642i $$0.449906\pi$$
$$492$$ − 9754.00i − 0.893789i
$$493$$ −1070.06 −0.0977546
$$494$$ 0 0
$$495$$ −1623.08 −0.147378
$$496$$ 41573.8i 3.76354i
$$497$$ 4335.86 0.391327
$$498$$ 9705.68 0.873338
$$499$$ − 5032.44i − 0.451469i −0.974189 0.225735i $$-0.927522\pi$$
0.974189 0.225735i $$-0.0724782\pi$$
$$500$$ − 13600.2i − 1.21644i
$$501$$ 455.580i 0.0406263i
$$502$$ 6200.44i 0.551274i
$$503$$ 17189.4 1.52373 0.761866 0.647735i $$-0.224284\pi$$
0.761866 + 0.647735i $$0.224284\pi$$
$$504$$ 9380.89 0.829083
$$505$$ 2382.06i 0.209901i
$$506$$ −16980.8 −1.49187
$$507$$ 0 0
$$508$$ −23145.5 −2.02149
$$509$$ − 930.560i − 0.0810341i −0.999179 0.0405170i $$-0.987099\pi$$
0.999179 0.0405170i $$-0.0129005\pi$$
$$510$$ −180.676 −0.0156872
$$511$$ −11264.9 −0.975201
$$512$$ 3143.50i 0.271337i
$$513$$ 704.295i 0.0606148i
$$514$$ 12617.6i 1.08276i
$$515$$ − 3375.08i − 0.288784i
$$516$$ 3240.75 0.276485
$$517$$ −23692.1 −2.01543
$$518$$ 14328.4i 1.21536i
$$519$$ −1616.44 −0.136713
$$520$$ 0 0
$$521$$ −9869.60 −0.829933 −0.414966 0.909837i $$-0.636207\pi$$
−0.414966 + 0.909837i $$0.636207\pi$$
$$522$$ − 12408.9i − 1.04047i
$$523$$ −21420.6 −1.79093 −0.895466 0.445129i $$-0.853158\pi$$
−0.895466 + 0.445129i $$0.853158\pi$$
$$524$$ −43641.1 −3.63831
$$525$$ − 5369.35i − 0.446358i
$$526$$ − 29623.4i − 2.45559i
$$527$$ − 858.697i − 0.0709781i
$$528$$ 40447.4i 3.33380i
$$529$$ −9927.42 −0.815930
$$530$$ −151.482 −0.0124150
$$531$$ − 4006.03i − 0.327395i
$$532$$ −8240.54 −0.671565
$$533$$ 0 0
$$534$$ −3471.00 −0.281283
$$535$$ − 921.097i − 0.0744345i
$$536$$ 1536.34 0.123805
$$537$$ 6662.39 0.535388
$$538$$ − 21013.4i − 1.68392i
$$539$$ 7483.41i 0.598021i
$$540$$ − 1512.81i − 0.120558i
$$541$$ − 7771.50i − 0.617602i −0.951127 0.308801i $$-0.900072\pi$$
0.951127 0.308801i $$-0.0999277\pi$$
$$542$$ 14900.7 1.18088
$$543$$ 11468.3 0.906361
$$544$$ 2218.78i 0.174870i
$$545$$ −2090.12 −0.164277
$$546$$ 0 0
$$547$$ −15577.5 −1.21763 −0.608817 0.793310i $$-0.708356\pi$$
−0.608817 + 0.793310i $$0.708356\pi$$
$$548$$ − 25040.6i − 1.95197i
$$549$$ 1077.27 0.0837461
$$550$$ 42243.7 3.27505
$$551$$ 6704.02i 0.518332i
$$552$$ − 9734.01i − 0.750556i
$$553$$ 8326.07i 0.640254i
$$554$$ 35316.4i 2.70840i
$$555$$ 1421.11 0.108690
$$556$$ −6709.74 −0.511792
$$557$$ 22804.5i 1.73475i 0.497652 + 0.867377i $$0.334196\pi$$
−0.497652 + 0.867377i $$0.665804\pi$$
$$558$$ 9957.89 0.755468
$$559$$ 0 0
$$560$$ 8262.78 0.623511
$$561$$ − 835.433i − 0.0628734i
$$562$$ −15405.7 −1.15631
$$563$$ 3517.41 0.263306 0.131653 0.991296i $$-0.457972\pi$$
0.131653 + 0.991296i $$0.457972\pi$$
$$564$$ − 22082.5i − 1.64865i
$$565$$ 3448.71i 0.256794i
$$566$$ 40332.2i 2.99521i
$$567$$ − 1231.40i − 0.0912062i
$$568$$ −19554.6 −1.44453
$$569$$ 6093.44 0.448946 0.224473 0.974480i $$-0.427934\pi$$
0.224473 + 0.974480i $$0.427934\pi$$
$$570$$ 1131.96i 0.0831797i
$$571$$ −10460.2 −0.766630 −0.383315 0.923618i $$-0.625218\pi$$
−0.383315 + 0.923618i $$0.625218\pi$$
$$572$$ 0 0
$$573$$ 10392.6 0.757689
$$574$$ − 12760.6i − 0.927906i
$$575$$ −5571.47 −0.404081
$$576$$ −11216.5 −0.811378
$$577$$ − 9648.19i − 0.696117i −0.937473 0.348058i $$-0.886841\pi$$
0.937473 0.348058i $$-0.113159\pi$$
$$578$$ 26263.9i 1.89002i
$$579$$ − 14092.2i − 1.01149i
$$580$$ − 14400.1i − 1.03092i
$$581$$ 9167.93 0.654647
$$582$$ 23296.4 1.65922
$$583$$ − 700.440i − 0.0497586i
$$584$$ 50804.1 3.59981
$$585$$ 0 0
$$586$$ −24176.5 −1.70430
$$587$$ 2170.73i 0.152633i 0.997084 + 0.0763166i $$0.0243160\pi$$
−0.997084 + 0.0763166i $$0.975684\pi$$
$$588$$ −6975.01 −0.489191
$$589$$ −5379.82 −0.376353
$$590$$ − 6438.56i − 0.449274i
$$591$$ − 8663.66i − 0.603004i
$$592$$ − 35414.3i − 2.45865i
$$593$$ − 22885.9i − 1.58484i −0.609975 0.792421i $$-0.708820\pi$$
0.609975 0.792421i $$-0.291180\pi$$
$$594$$ 9688.11 0.669205
$$595$$ −170.666 −0.0117590
$$596$$ − 23458.0i − 1.61221i
$$597$$ −189.028 −0.0129588
$$598$$ 0 0
$$599$$ −23978.7 −1.63563 −0.817815 0.575482i $$-0.804815\pi$$
−0.817815 + 0.575482i $$0.804815\pi$$
$$600$$ 24215.6i 1.64766i
$$601$$ 12873.2 0.873728 0.436864 0.899528i $$-0.356089\pi$$
0.436864 + 0.899528i $$0.356089\pi$$
$$602$$ 4239.70 0.287039
$$603$$ − 201.670i − 0.0136196i
$$604$$ − 61103.2i − 4.11631i
$$605$$ − 8473.38i − 0.569408i
$$606$$ − 14218.4i − 0.953107i
$$607$$ −7117.15 −0.475908 −0.237954 0.971276i $$-0.576477\pi$$
−0.237954 + 0.971276i $$0.576477\pi$$
$$608$$ 13900.9 0.927227
$$609$$ − 11721.4i − 0.779927i
$$610$$ 1731.40 0.114922
$$611$$ 0 0
$$612$$ 778.675 0.0514315
$$613$$ 173.297i 0.0114183i 0.999984 + 0.00570913i $$0.00181728\pi$$
−0.999984 + 0.00570913i $$0.998183\pi$$
$$614$$ −51173.5 −3.36351
$$615$$ −1265.62 −0.0829830
$$616$$ 69715.5i 4.55993i
$$617$$ − 6102.75i − 0.398197i −0.979979 0.199099i $$-0.936199\pi$$
0.979979 0.199099i $$-0.0638014\pi$$
$$618$$ 20145.7i 1.31129i
$$619$$ 14867.8i 0.965409i 0.875783 + 0.482705i $$0.160346\pi$$
−0.875783 + 0.482705i $$0.839654\pi$$
$$620$$ 11555.8 0.748534
$$621$$ −1277.75 −0.0825676
$$622$$ − 40054.6i − 2.58206i
$$623$$ −3278.69 −0.210847
$$624$$ 0 0
$$625$$ 12951.6 0.828900
$$626$$ 9772.64i 0.623951i
$$627$$ −5234.07 −0.333379
$$628$$ −13078.9 −0.831056
$$629$$ 731.475i 0.0463686i
$$630$$ − 1979.13i − 0.125159i
$$631$$ 17210.6i 1.08580i 0.839796 + 0.542902i $$0.182675\pi$$
−0.839796 + 0.542902i $$0.817325\pi$$
$$632$$ − 37550.3i − 2.36340i
$$633$$ −3149.09 −0.197733
$$634$$ 16765.5 1.05023
$$635$$ 3003.21i 0.187683i
$$636$$ 652.853 0.0407033
$$637$$ 0 0
$$638$$ 92218.9 5.72254
$$639$$ 2566.87i 0.158910i
$$640$$ −6532.30 −0.403456
$$641$$ 12636.0 0.778616 0.389308 0.921108i $$-0.372714\pi$$
0.389308 + 0.921108i $$0.372714\pi$$
$$642$$ 5497.99i 0.337988i
$$643$$ − 9586.37i − 0.587947i −0.955814 0.293973i $$-0.905022\pi$$
0.955814 0.293973i $$-0.0949777\pi$$
$$644$$ − 14950.2i − 0.914786i
$$645$$ − 420.499i − 0.0256700i
$$646$$ −582.641 −0.0354856
$$647$$ −5244.32 −0.318664 −0.159332 0.987225i $$-0.550934\pi$$
−0.159332 + 0.987225i $$0.550934\pi$$
$$648$$ 5553.57i 0.336674i
$$649$$ 29771.4 1.80066
$$650$$ 0 0
$$651$$ 9406.17 0.566293
$$652$$ − 8206.39i − 0.492925i
$$653$$ 18869.0 1.13078 0.565392 0.824822i $$-0.308725\pi$$
0.565392 + 0.824822i $$0.308725\pi$$
$$654$$ 12475.8 0.745938
$$655$$ 5662.59i 0.337795i
$$656$$ 31539.3i 1.87714i
$$657$$ − 6668.90i − 0.396010i
$$658$$ − 28889.3i − 1.71159i
$$659$$ −24299.8 −1.43639 −0.718197 0.695839i $$-0.755033\pi$$
−0.718197 + 0.695839i $$0.755033\pi$$
$$660$$ 11242.7 0.663062
$$661$$ − 29915.0i − 1.76030i −0.474694 0.880151i $$-0.657441\pi$$
0.474694 0.880151i $$-0.342559\pi$$
$$662$$ −8335.03 −0.489350
$$663$$ 0 0
$$664$$ −41347.1 −2.41653
$$665$$ 1069.24i 0.0623508i
$$666$$ −8482.56 −0.493532
$$667$$ −12162.6 −0.706056
$$668$$ − 3155.69i − 0.182780i
$$669$$ 6940.48i 0.401098i
$$670$$ − 324.128i − 0.0186898i
$$671$$ 8005.86i 0.460600i
$$672$$ −24304.5 −1.39519
$$673$$ −15493.7 −0.887426 −0.443713 0.896169i $$-0.646339\pi$$
−0.443713 + 0.896169i $$0.646339\pi$$
$$674$$ − 17115.8i − 0.978154i
$$675$$ 3178.71 0.181257
$$676$$ 0 0
$$677$$ 11729.7 0.665891 0.332945 0.942946i $$-0.391958\pi$$
0.332945 + 0.942946i $$0.391958\pi$$
$$678$$ − 20585.2i − 1.16603i
$$679$$ 22005.6 1.24374
$$680$$ 769.698 0.0434067
$$681$$ 11398.4i 0.641390i
$$682$$ 74003.5i 4.15505i
$$683$$ 1168.42i 0.0654587i 0.999464 + 0.0327294i $$0.0104199\pi$$
−0.999464 + 0.0327294i $$0.989580\pi$$
$$684$$ − 4878.48i − 0.272709i
$$685$$ −3249.10 −0.181229
$$686$$ −37099.1 −2.06480
$$687$$ 12963.2i 0.719909i
$$688$$ −10478.9 −0.580675
$$689$$ 0 0
$$690$$ −2053.63 −0.113305
$$691$$ 32992.9i 1.81637i 0.418574 + 0.908183i $$0.362530\pi$$
−0.418574 + 0.908183i $$0.637470\pi$$
$$692$$ 11196.7 0.615078
$$693$$ 9151.33 0.501631
$$694$$ 30679.2i 1.67805i
$$695$$ 870.612i 0.0475168i
$$696$$ 52863.2i 2.87899i
$$697$$ − 651.438i − 0.0354017i
$$698$$ 17856.3 0.968295
$$699$$ −15839.3 −0.857078
$$700$$ 37192.2i 2.00819i
$$701$$ 14785.8 0.796651 0.398326 0.917244i $$-0.369591\pi$$
0.398326 + 0.917244i $$0.369591\pi$$
$$702$$ 0 0
$$703$$ 4582.77 0.245864
$$704$$ − 83357.0i − 4.46255i
$$705$$ −2865.28 −0.153068
$$706$$ −66021.0 −3.51945
$$707$$ − 13430.6i − 0.714442i
$$708$$ 27748.8i 1.47297i
$$709$$ − 12634.0i − 0.669225i −0.942356 0.334612i $$-0.891395\pi$$
0.942356 0.334612i $$-0.108605\pi$$
$$710$$ 4125.52i 0.218067i
$$711$$ −4929.11 −0.259994
$$712$$ 14786.8 0.778312
$$713$$ − 9760.24i − 0.512656i
$$714$$ 1018.70 0.0533947
$$715$$ 0 0
$$716$$ −46148.7 −2.40874
$$717$$ 4643.75i 0.241874i
$$718$$ −45814.6 −2.38132
$$719$$ 27296.5 1.41584 0.707920 0.706292i $$-0.249634\pi$$
0.707920 + 0.706292i $$0.249634\pi$$
$$720$$ 4891.64i 0.253195i
$$721$$ 19029.5i 0.982936i
$$722$$ − 33146.3i − 1.70856i
$$723$$ 14754.0i 0.758932i
$$724$$ −79438.5 −4.07777
$$725$$ 30257.4 1.54997
$$726$$ 50577.2i 2.58553i
$$727$$ −4658.21 −0.237639 −0.118819 0.992916i $$-0.537911\pi$$
−0.118819 + 0.992916i $$0.537911\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ − 10718.4i − 0.543432i
$$731$$ 216.439 0.0109512
$$732$$ −7461.96 −0.376779
$$733$$ − 166.474i − 0.00838864i −0.999991 0.00419432i $$-0.998665\pi$$
0.999991 0.00419432i $$-0.00133510\pi$$
$$734$$ − 13393.9i − 0.673537i
$$735$$ 905.031i 0.0454185i
$$736$$ 25219.4i 1.26304i
$$737$$ 1498.74 0.0749074
$$738$$ 7554.41 0.376804
$$739$$ − 12738.1i − 0.634069i −0.948414 0.317035i $$-0.897313\pi$$
0.948414 0.317035i $$-0.102687\pi$$
$$740$$ −9843.70 −0.489002
$$741$$ 0 0
$$742$$ 854.092 0.0422570
$$743$$ − 30724.5i − 1.51705i −0.651641 0.758527i $$-0.725919\pi$$
0.651641 0.758527i $$-0.274081\pi$$
$$744$$ −42421.5 −2.09039
$$745$$ −3043.76 −0.149684
$$746$$ 6131.38i 0.300919i
$$747$$ 5427.50i 0.265839i
$$748$$ 5786.84i 0.282871i
$$749$$ 5193.37i 0.253353i
$$750$$ 10533.2 0.512826
$$751$$ 39538.6 1.92115 0.960575 0.278021i $$-0.0896785\pi$$
0.960575 + 0.278021i $$0.0896785\pi$$
$$752$$ 71403.2i 3.46251i
$$753$$ −3467.34 −0.167805
$$754$$ 0 0
$$755$$ −7928.35 −0.382175
$$756$$ 8529.61i 0.410342i
$$757$$ 23035.1 1.10598 0.552990 0.833188i $$-0.313487\pi$$
0.552990 + 0.833188i $$0.313487\pi$$
$$758$$ −65352.1 −3.13153
$$759$$ − 9495.81i − 0.454119i
$$760$$ − 4822.23i − 0.230159i
$$761$$ − 32454.0i − 1.54594i −0.634445 0.772968i $$-0.718771\pi$$
0.634445 0.772968i $$-0.281229\pi$$
$$762$$ − 17926.0i − 0.852220i
$$763$$ 11784.6 0.559150
$$764$$ −71986.8 −3.40889
$$765$$ − 101.036i − 0.00477511i
$$766$$ −54616.1 −2.57619
$$767$$ 0 0
$$768$$ 9080.40 0.426641
$$769$$ − 32216.2i − 1.51072i −0.655307 0.755362i $$-0.727461\pi$$
0.655307 0.755362i $$-0.272539\pi$$
$$770$$ 14708.2 0.688372
$$771$$ −7055.86 −0.329586
$$772$$ 97613.2i 4.55075i
$$773$$ 2924.60i 0.136081i 0.997683 + 0.0680404i $$0.0216747\pi$$
−0.997683 + 0.0680404i $$0.978325\pi$$
$$774$$ 2509.94i 0.116561i
$$775$$ 24280.9i 1.12541i
$$776$$ −99244.7 −4.59108
$$777$$ −8012.58 −0.369948
$$778$$ 31361.2i 1.44519i
$$779$$ −4081.32 −0.187713
$$780$$ 0 0
$$781$$ −19076.1 −0.874001
$$782$$ − 1057.04i − 0.0483374i
$$783$$ 6939.19 0.316713
$$784$$ 22553.5 1.02740
$$785$$ 1697.03i 0.0771586i
$$786$$ − 33799.8i − 1.53384i
$$787$$ − 25507.1i − 1.15531i −0.816280 0.577656i $$-0.803967\pi$$
0.816280 0.577656i $$-0.196033\pi$$
$$788$$ 60011.1i 2.71295i
$$789$$ 16565.6 0.747468
$$790$$ −7922.16 −0.356782
$$791$$ − 19444.7i − 0.874050i
$$792$$ −41272.2 −1.85170
$$793$$ 0 0
$$794$$ 13416.8 0.599677
$$795$$ − 84.7100i − 0.00377906i
$$796$$ 1309.35 0.0583023
$$797$$ −5448.98 −0.242174 −0.121087 0.992642i $$-0.538638\pi$$
−0.121087 + 0.992642i $$0.538638\pi$$
$$798$$ − 6382.24i − 0.283119i
$$799$$ − 1474.82i − 0.0653008i
$$800$$ − 62739.0i − 2.77270i
$$801$$ − 1941.01i − 0.0856209i
$$802$$ 49297.8 2.17053
$$803$$ 49560.9 2.17804
$$804$$ 1396.92i 0.0612755i
$$805$$ −1939.85 −0.0849324
$$806$$ 0 0
$$807$$ 11750.9 0.512577
$$808$$ 60571.7i 2.63726i
$$809$$ −2453.12 −0.106610 −0.0533048 0.998578i $$-0.516975\pi$$
−0.0533048 + 0.998578i $$0.516975\pi$$
$$810$$ 1171.66 0.0508247
$$811$$ − 5133.85i − 0.222286i −0.993804 0.111143i $$-0.964549\pi$$
0.993804 0.111143i $$-0.0354512\pi$$
$$812$$ 81191.4i 3.50894i
$$813$$ 8332.58i 0.359454i
$$814$$ − 63039.4i − 2.71441i
$$815$$ −1064.81 −0.0457651
$$816$$ −2517.83 −0.108017
$$817$$ − 1356.01i − 0.0580673i
$$818$$ 49432.6 2.11292
$$819$$ 0 0
$$820$$ 8766.61 0.373345
$$821$$ − 23854.0i − 1.01402i −0.861941 0.507009i $$-0.830751\pi$$
0.861941 0.507009i $$-0.169249\pi$$
$$822$$ 19393.8 0.822913
$$823$$ −757.156 −0.0320690 −0.0160345 0.999871i $$-0.505104\pi$$
−0.0160345 + 0.999871i $$0.505104\pi$$
$$824$$ − 85822.6i − 3.62836i
$$825$$ 23623.0i 0.996907i
$$826$$ 36302.2i 1.52919i
$$827$$ − 28621.2i − 1.20345i −0.798702 0.601726i $$-0.794480\pi$$
0.798702 0.601726i $$-0.205520\pi$$
$$828$$ 8850.68 0.371476
$$829$$ −27429.2 −1.14916 −0.574582 0.818447i $$-0.694835\pi$$
−0.574582 + 0.818447i $$0.694835\pi$$
$$830$$ 8723.19i 0.364803i
$$831$$ −19749.2 −0.824421
$$832$$ 0 0
$$833$$ −465.838 −0.0193761
$$834$$ − 5196.64i − 0.215761i
$$835$$ −409.462 −0.0169701
$$836$$ 36255.1 1.49989
$$837$$ 5568.54i 0.229960i
$$838$$ 34840.7i 1.43622i
$$839$$ 4633.62i 0.190668i 0.995445 + 0.0953339i $$0.0303919\pi$$
−0.995445 + 0.0953339i $$0.969608\pi$$
$$840$$ 8431.27i 0.346317i
$$841$$ 41663.6 1.70829
$$842$$ 16413.7 0.671798
$$843$$ − 8614.98i − 0.351976i
$$844$$ 21813.0 0.889614
$$845$$ 0 0
$$846$$ 17102.7 0.695041
$$847$$ 47775.0i 1.93810i
$$848$$ −2110.99 −0.0854853
$$849$$ −22554.1 −0.911726
$$850$$ 2629.64i 0.106113i
$$851$$ 8314.19i 0.334908i
$$852$$ − 17780.1i − 0.714947i
$$853$$ 14854.6i 0.596261i 0.954525 + 0.298131i $$0.0963631\pi$$
−0.954525 + 0.298131i $$0.903637\pi$$
$$854$$ −9762.07 −0.391161
$$855$$ −632.999 −0.0253194
$$856$$ − 23421.9i − 0.935216i
$$857$$ 42799.5 1.70595 0.852977 0.521948i $$-0.174794\pi$$
0.852977 + 0.521948i $$0.174794\pi$$
$$858$$ 0 0
$$859$$ −8246.47 −0.327551 −0.163775 0.986498i $$-0.552367\pi$$
−0.163775 + 0.986498i $$0.552367\pi$$
$$860$$ 2912.70i 0.115491i
$$861$$ 7135.85 0.282450
$$862$$ 42580.1 1.68246
$$863$$ 17695.4i 0.697983i 0.937126 + 0.348991i $$0.113476\pi$$
−0.937126 + 0.348991i $$0.886524\pi$$
$$864$$ − 14388.5i − 0.566558i
$$865$$ − 1452.81i − 0.0571064i
$$866$$ 39132.3i 1.53553i
$$867$$ −14687.0 −0.575313
$$868$$ −65154.2 −2.54779
$$869$$ − 36631.4i − 1.42996i
$$870$$ 11152.8 0.434615
$$871$$ 0 0
$$872$$ −53148.2 −2.06402
$$873$$ 13027.5i 0.505058i
$$874$$ −6622.49 −0.256303
$$875$$ 9949.64 0.384411
$$876$$ 46193.8i 1.78167i
$$877$$ − 14346.8i − 0.552401i −0.961100 0.276200i $$-0.910925\pi$$
0.961100 0.276200i $$-0.0890754\pi$$
$$878$$ 81622.7i 3.13739i
$$879$$ − 13519.7i − 0.518781i
$$880$$ −36353.0 −1.39257
$$881$$ 2063.51 0.0789121 0.0394560 0.999221i $$-0.487437\pi$$
0.0394560 + 0.999221i $$0.487437\pi$$
$$882$$ − 5402.09i − 0.206233i
$$883$$ 34137.1 1.30103 0.650513 0.759495i $$-0.274554\pi$$
0.650513 + 0.759495i $$0.274554\pi$$
$$884$$ 0 0
$$885$$ 3600.50 0.136756
$$886$$ − 8138.58i − 0.308602i
$$887$$ −22238.5 −0.841821 −0.420910 0.907102i $$-0.638289\pi$$
−0.420910 + 0.907102i $$0.638289\pi$$
$$888$$ 36136.5 1.36561
$$889$$ − 16932.8i − 0.638817i
$$890$$ − 3119.63i − 0.117495i
$$891$$ 5417.67i 0.203702i
$$892$$ − 48075.0i − 1.80456i
$$893$$ −9239.89 −0.346250
$$894$$ 18168.0 0.679676
$$895$$ 5987.96i 0.223637i
$$896$$ 36830.7 1.37325
$$897$$ 0 0
$$898$$ 3783.45 0.140596
$$899$$ 53005.7i 1.96645i
$$900$$ −22018.1 −0.815486
$$901$$ 43.6019 0.00161220
$$902$$ 56141.7i 2.07241i
$$903$$ 2370.88i 0.0873730i
$$904$$ 87694.9i 3.22643i
$$905$$ 10307.4i 0.378597i
$$906$$ 47324.0 1.73536
$$907$$ 44160.3 1.61667 0.808335 0.588723i $$-0.200369\pi$$
0.808335 + 0.588723i $$0.200369\pi$$
$$908$$ − 78953.8i − 2.88565i
$$909$$ 7951.05 0.290121
$$910$$ 0 0
$$911$$ 11916.3 0.433376 0.216688 0.976241i $$-0.430475\pi$$
0.216688 + 0.976241i $$0.430475\pi$$
$$912$$ 15774.4i 0.572745i
$$913$$ −40335.3 −1.46211
$$914$$ −39043.0 −1.41294
$$915$$ 968.215i 0.0349816i
$$916$$ − 89793.0i − 3.23891i
$$917$$ − 31927.1i − 1.14975i
$$918$$ 603.078i 0.0216825i
$$919$$ −20128.9 −0.722516 −0.361258 0.932466i $$-0.617653\pi$$
−0.361258 + 0.932466i $$0.617653\pi$$
$$920$$ 8748.64 0.313515
$$921$$ − 28616.7i − 1.02383i
$$922$$ 10525.0 0.375945
$$923$$ 0 0
$$924$$ −63389.0 −2.25687
$$925$$ − 20683.5i − 0.735210i
$$926$$ 55654.1 1.97506
$$927$$ −11265.7 −0.399151
$$928$$ − 136961.i − 4.84478i
$$929$$ − 31428.4i − 1.10994i −0.831871 0.554969i $$-0.812730\pi$$
0.831871 0.554969i $$-0.187270\pi$$
$$930$$ 8949.86i 0.315567i
$$931$$ 2918.52i 0.102740i
$$932$$ 109715. 3.85605
$$933$$ 22398.9 0.785965
$$934$$ 47150.1i 1.65182i
$$935$$ 750.863 0.0262629
$$936$$ 0 0
$$937$$ 42473.2 1.48083 0.740416 0.672149i $$-0.234629\pi$$
0.740416 + 0.672149i $$0.234629\pi$$
$$938$$ 1827.51i 0.0636144i
$$939$$ −5464.94 −0.189927
$$940$$ 19847.1 0.688661
$$941$$ − 42644.0i − 1.47732i −0.674079 0.738659i $$-0.735459\pi$$
0.674079 0.738659i $$-0.264541\pi$$
$$942$$ − 10129.5i − 0.350357i
$$943$$ − 7404.46i − 0.255697i
$$944$$ − 89725.0i − 3.09354i
$$945$$ 1106.75 0.0380978
$$946$$ −18653.0 −0.641080
$$947$$ − 4282.95i − 0.146966i −0.997296 0.0734831i $$-0.976588\pi$$
0.997296 0.0734831i $$-0.0234115\pi$$
$$948$$ 34142.7 1.16973
$$949$$ 0 0
$$950$$ 16475.0 0.562651
$$951$$ 9375.42i 0.319683i
$$952$$ −4339.74 −0.147744
$$953$$ −40593.9 −1.37982 −0.689908 0.723897i $$-0.742349\pi$$
−0.689908 + 0.723897i $$0.742349\pi$$
$$954$$ 505.630i 0.0171597i
$$955$$ 9340.54i 0.316495i
$$956$$ − 32166.1i − 1.08821i
$$957$$ 51569.6i 1.74191i
$$958$$ 59771.9 2.01581
$$959$$ 18319.2 0.616850
$$960$$ − 10081.1i − 0.338922i
$$961$$ −12744.8 −0.427808
$$962$$ 0 0
$$963$$ −3074.52 −0.102882
$$964$$ − 102198.i − 3.41448i
$$965$$ 12665.7 0.422510
$$966$$ 11578.9 0.385656
$$967$$ 17709.4i 0.588931i 0.955662 + 0.294466i $$0.0951416\pi$$
−0.955662 + 0.294466i $$0.904858\pi$$
$$968$$ − 215464.i − 7.15420i
$$969$$ − 325.818i − 0.0108016i
$$970$$ 20938.1i 0.693074i
$$971$$ 4038.97 0.133488 0.0667440 0.997770i $$-0.478739\pi$$
0.0667440 + 0.997770i $$0.478739\pi$$
$$972$$ −5049.61 −0.166632
$$973$$ − 4908.72i − 0.161733i
$$974$$ −105366. −3.46626
$$975$$ 0 0
$$976$$ 24128.1 0.791312
$$977$$ − 2764.30i − 0.0905197i −0.998975 0.0452598i $$-0.985588\pi$$
0.998975 0.0452598i $$-0.0144116\pi$$
$$978$$ 6355.79 0.207807
$$979$$ 14424.9 0.470912
$$980$$ − 6268.93i − 0.204340i
$$981$$ 6976.59i 0.227060i
$$982$$ − 18295.4i − 0.594531i
$$983$$ 17804.5i 0.577695i 0.957375 + 0.288848i $$0.0932721\pi$$
−0.957375 + 0.288848i $$0.906728\pi$$
$$984$$ −32182.4 −1.04262
$$985$$ 7786.65 0.251881
$$986$$ 5740.57i 0.185413i
$$987$$ 16155.2 0.520997
$$988$$ 0 0
$$989$$ 2460.12 0.0790974
$$990$$ 8707.39i 0.279534i
$$991$$ −8129.69 −0.260593 −0.130297 0.991475i $$-0.541593\pi$$
−0.130297 + 0.991475i $$0.541593\pi$$
$$992$$ 109908. 3.51772
$$993$$ − 4661.02i − 0.148956i
$$994$$ − 23260.7i − 0.742237i
$$995$$ − 169.893i − 0.00541302i
$$996$$ − 37595.0i − 1.19603i
$$997$$ −29452.9 −0.935589 −0.467795 0.883837i $$-0.654951\pi$$
−0.467795 + 0.883837i $$0.654951\pi$$
$$998$$ −26997.7 −0.856309
$$999$$ − 4743.52i − 0.150229i
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.i.337.1 10
13.5 odd 4 507.4.a.r.1.1 10
13.8 odd 4 507.4.a.r.1.10 10
13.9 even 3 39.4.j.c.10.1 yes 10
13.10 even 6 39.4.j.c.4.1 10
13.12 even 2 inner 507.4.b.i.337.10 10
39.5 even 4 1521.4.a.bk.1.10 10
39.8 even 4 1521.4.a.bk.1.1 10
39.23 odd 6 117.4.q.e.82.5 10
39.35 odd 6 117.4.q.e.10.5 10
52.23 odd 6 624.4.bv.h.433.3 10
52.35 odd 6 624.4.bv.h.49.3 10

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.1 10 13.10 even 6
39.4.j.c.10.1 yes 10 13.9 even 3
117.4.q.e.10.5 10 39.35 odd 6
117.4.q.e.82.5 10 39.23 odd 6
507.4.a.r.1.1 10 13.5 odd 4
507.4.a.r.1.10 10 13.8 odd 4
507.4.b.i.337.1 10 1.1 even 1 trivial
507.4.b.i.337.10 10 13.12 even 2 inner
624.4.bv.h.49.3 10 52.35 odd 6
624.4.bv.h.433.3 10 52.23 odd 6
1521.4.a.bk.1.1 10 39.8 even 4
1521.4.a.bk.1.10 10 39.5 even 4