# Properties

 Label 1875.4.a.g.1.2 Level $1875$ Weight $4$ Character 1875.1 Self dual yes Analytic conductor $110.629$ Analytic rank $1$ Dimension $14$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1875.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$110.628581261$$ Analytic rank: $$1$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \cdots + 42064$$ x^14 - 81*x^12 - 7*x^11 + 2512*x^10 + 517*x^9 - 36970*x^8 - 12987*x^7 + 257291*x^6 + 125779*x^5 - 718713*x^4 - 371750*x^3 + 579848*x^2 + 394896*x + 42064 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 5^{3}$$ Twist minimal: no (minimal twist has level 75) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-4.79301$$ of defining polynomial Character $$\chi$$ $$=$$ 1875.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.79301 q^{2} +3.00000 q^{3} +14.9730 q^{4} -14.3790 q^{6} +0.140520 q^{7} -33.4216 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-4.79301 q^{2} +3.00000 q^{3} +14.9730 q^{4} -14.3790 q^{6} +0.140520 q^{7} -33.4216 q^{8} +9.00000 q^{9} +47.3504 q^{11} +44.9190 q^{12} +34.3424 q^{13} -0.673515 q^{14} +40.4065 q^{16} +16.5164 q^{17} -43.1371 q^{18} +91.0020 q^{19} +0.421560 q^{21} -226.951 q^{22} -161.582 q^{23} -100.265 q^{24} -164.604 q^{26} +27.0000 q^{27} +2.10400 q^{28} -12.3327 q^{29} -333.038 q^{31} +73.7042 q^{32} +142.051 q^{33} -79.1633 q^{34} +134.757 q^{36} -281.402 q^{37} -436.174 q^{38} +103.027 q^{39} +125.610 q^{41} -2.02054 q^{42} -529.821 q^{43} +708.977 q^{44} +774.463 q^{46} +75.0713 q^{47} +121.220 q^{48} -342.980 q^{49} +49.5492 q^{51} +514.208 q^{52} -249.426 q^{53} -129.411 q^{54} -4.69641 q^{56} +273.006 q^{57} +59.1108 q^{58} -268.564 q^{59} +369.848 q^{61} +1596.26 q^{62} +1.26468 q^{63} -676.517 q^{64} -680.854 q^{66} -176.506 q^{67} +247.300 q^{68} -484.745 q^{69} +722.104 q^{71} -300.795 q^{72} -2.44271 q^{73} +1348.77 q^{74} +1362.57 q^{76} +6.65368 q^{77} -493.811 q^{78} +607.662 q^{79} +81.0000 q^{81} -602.049 q^{82} -1268.55 q^{83} +6.31201 q^{84} +2539.44 q^{86} -36.9981 q^{87} -1582.53 q^{88} -1337.53 q^{89} +4.82579 q^{91} -2419.36 q^{92} -999.114 q^{93} -359.818 q^{94} +221.113 q^{96} -1431.21 q^{97} +1643.91 q^{98} +426.154 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9}+O(q^{10})$$ 14 * q + 42 * q^3 + 50 * q^4 - 27 * q^7 + 21 * q^8 + 126 * q^9 $$14 q + 42 q^{3} + 50 q^{4} - 27 q^{7} + 21 q^{8} + 126 q^{9} - 33 q^{11} + 150 q^{12} - 188 q^{13} - 287 q^{14} + 82 q^{16} - 146 q^{17} - 184 q^{19} - 81 q^{21} - 277 q^{22} - 164 q^{23} + 63 q^{24} + 103 q^{26} + 378 q^{27} + 224 q^{28} + 252 q^{29} - 889 q^{31} - 422 q^{32} - 99 q^{33} - 230 q^{34} + 450 q^{36} - 642 q^{37} - 1833 q^{38} - 564 q^{39} - 164 q^{41} - 861 q^{42} - 696 q^{43} - 6 q^{44} - 419 q^{46} + 92 q^{47} + 246 q^{48} + 81 q^{49} - 438 q^{51} - 1956 q^{52} - 949 q^{53} - 2380 q^{56} - 552 q^{57} - 1041 q^{58} - 81 q^{59} - 496 q^{61} - 1454 q^{62} - 243 q^{63} - 3903 q^{64} - 831 q^{66} - 1926 q^{67} - 685 q^{68} - 492 q^{69} + 2498 q^{71} + 189 q^{72} + 1026 q^{73} - 707 q^{74} - 5704 q^{76} - 5434 q^{77} + 309 q^{78} - 695 q^{79} + 1134 q^{81} + 886 q^{82} - 5315 q^{83} + 672 q^{84} + 3997 q^{86} + 756 q^{87} - 2969 q^{88} - 1424 q^{89} - 1194 q^{91} - 1607 q^{92} - 2667 q^{93} - 629 q^{94} - 1266 q^{96} - 291 q^{97} + 1018 q^{98} - 297 q^{99}+O(q^{100})$$ 14 * q + 42 * q^3 + 50 * q^4 - 27 * q^7 + 21 * q^8 + 126 * q^9 - 33 * q^11 + 150 * q^12 - 188 * q^13 - 287 * q^14 + 82 * q^16 - 146 * q^17 - 184 * q^19 - 81 * q^21 - 277 * q^22 - 164 * q^23 + 63 * q^24 + 103 * q^26 + 378 * q^27 + 224 * q^28 + 252 * q^29 - 889 * q^31 - 422 * q^32 - 99 * q^33 - 230 * q^34 + 450 * q^36 - 642 * q^37 - 1833 * q^38 - 564 * q^39 - 164 * q^41 - 861 * q^42 - 696 * q^43 - 6 * q^44 - 419 * q^46 + 92 * q^47 + 246 * q^48 + 81 * q^49 - 438 * q^51 - 1956 * q^52 - 949 * q^53 - 2380 * q^56 - 552 * q^57 - 1041 * q^58 - 81 * q^59 - 496 * q^61 - 1454 * q^62 - 243 * q^63 - 3903 * q^64 - 831 * q^66 - 1926 * q^67 - 685 * q^68 - 492 * q^69 + 2498 * q^71 + 189 * q^72 + 1026 * q^73 - 707 * q^74 - 5704 * q^76 - 5434 * q^77 + 309 * q^78 - 695 * q^79 + 1134 * q^81 + 886 * q^82 - 5315 * q^83 + 672 * q^84 + 3997 * q^86 + 756 * q^87 - 2969 * q^88 - 1424 * q^89 - 1194 * q^91 - 1607 * q^92 - 2667 * q^93 - 629 * q^94 - 1266 * q^96 - 291 * q^97 + 1018 * q^98 - 297 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.79301 −1.69459 −0.847293 0.531125i $$-0.821769\pi$$
−0.847293 + 0.531125i $$0.821769\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 14.9730 1.87162
$$5$$ 0 0
$$6$$ −14.3790 −0.978370
$$7$$ 0.140520 0.00758737 0.00379368 0.999993i $$-0.498792\pi$$
0.00379368 + 0.999993i $$0.498792\pi$$
$$8$$ −33.4216 −1.47704
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 47.3504 1.29788 0.648940 0.760839i $$-0.275212\pi$$
0.648940 + 0.760839i $$0.275212\pi$$
$$12$$ 44.9190 1.08058
$$13$$ 34.3424 0.732682 0.366341 0.930481i $$-0.380610\pi$$
0.366341 + 0.930481i $$0.380610\pi$$
$$14$$ −0.673515 −0.0128575
$$15$$ 0 0
$$16$$ 40.4065 0.631352
$$17$$ 16.5164 0.235636 0.117818 0.993035i $$-0.462410\pi$$
0.117818 + 0.993035i $$0.462410\pi$$
$$18$$ −43.1371 −0.564862
$$19$$ 91.0020 1.09881 0.549403 0.835558i $$-0.314855\pi$$
0.549403 + 0.835558i $$0.314855\pi$$
$$20$$ 0 0
$$21$$ 0.421560 0.00438057
$$22$$ −226.951 −2.19937
$$23$$ −161.582 −1.46487 −0.732436 0.680835i $$-0.761617\pi$$
−0.732436 + 0.680835i $$0.761617\pi$$
$$24$$ −100.265 −0.852771
$$25$$ 0 0
$$26$$ −164.604 −1.24159
$$27$$ 27.0000 0.192450
$$28$$ 2.10400 0.0142007
$$29$$ −12.3327 −0.0789698 −0.0394849 0.999220i $$-0.512572\pi$$
−0.0394849 + 0.999220i $$0.512572\pi$$
$$30$$ 0 0
$$31$$ −333.038 −1.92953 −0.964765 0.263115i $$-0.915250\pi$$
−0.964765 + 0.263115i $$0.915250\pi$$
$$32$$ 73.7042 0.407162
$$33$$ 142.051 0.749332
$$34$$ −79.1633 −0.399306
$$35$$ 0 0
$$36$$ 134.757 0.623875
$$37$$ −281.402 −1.25033 −0.625166 0.780492i $$-0.714969\pi$$
−0.625166 + 0.780492i $$0.714969\pi$$
$$38$$ −436.174 −1.86202
$$39$$ 103.027 0.423014
$$40$$ 0 0
$$41$$ 125.610 0.478462 0.239231 0.970963i $$-0.423105\pi$$
0.239231 + 0.970963i $$0.423105\pi$$
$$42$$ −2.02054 −0.00742325
$$43$$ −529.821 −1.87900 −0.939500 0.342549i $$-0.888710\pi$$
−0.939500 + 0.342549i $$0.888710\pi$$
$$44$$ 708.977 2.42914
$$45$$ 0 0
$$46$$ 774.463 2.48235
$$47$$ 75.0713 0.232985 0.116492 0.993192i $$-0.462835\pi$$
0.116492 + 0.993192i $$0.462835\pi$$
$$48$$ 121.220 0.364511
$$49$$ −342.980 −0.999942
$$50$$ 0 0
$$51$$ 49.5492 0.136045
$$52$$ 514.208 1.37130
$$53$$ −249.426 −0.646438 −0.323219 0.946324i $$-0.604765\pi$$
−0.323219 + 0.946324i $$0.604765\pi$$
$$54$$ −129.411 −0.326123
$$55$$ 0 0
$$56$$ −4.69641 −0.0112069
$$57$$ 273.006 0.634395
$$58$$ 59.1108 0.133821
$$59$$ −268.564 −0.592611 −0.296305 0.955093i $$-0.595755\pi$$
−0.296305 + 0.955093i $$0.595755\pi$$
$$60$$ 0 0
$$61$$ 369.848 0.776297 0.388149 0.921597i $$-0.373115\pi$$
0.388149 + 0.921597i $$0.373115\pi$$
$$62$$ 1596.26 3.26975
$$63$$ 1.26468 0.00252912
$$64$$ −676.517 −1.32132
$$65$$ 0 0
$$66$$ −680.854 −1.26981
$$67$$ −176.506 −0.321846 −0.160923 0.986967i $$-0.551447\pi$$
−0.160923 + 0.986967i $$0.551447\pi$$
$$68$$ 247.300 0.441022
$$69$$ −484.745 −0.845745
$$70$$ 0 0
$$71$$ 722.104 1.20701 0.603507 0.797358i $$-0.293770\pi$$
0.603507 + 0.797358i $$0.293770\pi$$
$$72$$ −300.795 −0.492347
$$73$$ −2.44271 −0.00391641 −0.00195820 0.999998i $$-0.500623\pi$$
−0.00195820 + 0.999998i $$0.500623\pi$$
$$74$$ 1348.77 2.11879
$$75$$ 0 0
$$76$$ 1362.57 2.05655
$$77$$ 6.65368 0.00984750
$$78$$ −493.811 −0.716834
$$79$$ 607.662 0.865409 0.432705 0.901536i $$-0.357559\pi$$
0.432705 + 0.901536i $$0.357559\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ −602.049 −0.810795
$$83$$ −1268.55 −1.67761 −0.838807 0.544429i $$-0.816747\pi$$
−0.838807 + 0.544429i $$0.816747\pi$$
$$84$$ 6.31201 0.00819878
$$85$$ 0 0
$$86$$ 2539.44 3.18413
$$87$$ −36.9981 −0.0455933
$$88$$ −1582.53 −1.91702
$$89$$ −1337.53 −1.59301 −0.796507 0.604629i $$-0.793321\pi$$
−0.796507 + 0.604629i $$0.793321\pi$$
$$90$$ 0 0
$$91$$ 4.82579 0.00555913
$$92$$ −2419.36 −2.74169
$$93$$ −999.114 −1.11401
$$94$$ −359.818 −0.394812
$$95$$ 0 0
$$96$$ 221.113 0.235075
$$97$$ −1431.21 −1.49812 −0.749059 0.662503i $$-0.769494\pi$$
−0.749059 + 0.662503i $$0.769494\pi$$
$$98$$ 1643.91 1.69449
$$99$$ 426.154 0.432627
$$100$$ 0 0
$$101$$ −374.102 −0.368559 −0.184280 0.982874i $$-0.558995\pi$$
−0.184280 + 0.982874i $$0.558995\pi$$
$$102$$ −237.490 −0.230539
$$103$$ −2017.87 −1.93035 −0.965177 0.261597i $$-0.915751\pi$$
−0.965177 + 0.261597i $$0.915751\pi$$
$$104$$ −1147.78 −1.08220
$$105$$ 0 0
$$106$$ 1195.50 1.09545
$$107$$ 1679.71 1.51761 0.758804 0.651319i $$-0.225784\pi$$
0.758804 + 0.651319i $$0.225784\pi$$
$$108$$ 404.271 0.360194
$$109$$ 20.2283 0.0177754 0.00888772 0.999961i $$-0.497171\pi$$
0.00888772 + 0.999961i $$0.497171\pi$$
$$110$$ 0 0
$$111$$ −844.207 −0.721879
$$112$$ 5.67792 0.00479030
$$113$$ −642.520 −0.534896 −0.267448 0.963572i $$-0.586180\pi$$
−0.267448 + 0.963572i $$0.586180\pi$$
$$114$$ −1308.52 −1.07504
$$115$$ 0 0
$$116$$ −184.657 −0.147802
$$117$$ 309.081 0.244227
$$118$$ 1287.23 1.00423
$$119$$ 2.32088 0.00178786
$$120$$ 0 0
$$121$$ 911.063 0.684495
$$122$$ −1772.69 −1.31550
$$123$$ 376.829 0.276240
$$124$$ −4986.58 −3.61135
$$125$$ 0 0
$$126$$ −6.06163 −0.00428582
$$127$$ −1185.91 −0.828603 −0.414302 0.910140i $$-0.635974\pi$$
−0.414302 + 0.910140i $$0.635974\pi$$
$$128$$ 2652.92 1.83193
$$129$$ −1589.46 −1.08484
$$130$$ 0 0
$$131$$ −223.094 −0.148793 −0.0743963 0.997229i $$-0.523703\pi$$
−0.0743963 + 0.997229i $$0.523703\pi$$
$$132$$ 2126.93 1.40247
$$133$$ 12.7876 0.00833704
$$134$$ 845.997 0.545396
$$135$$ 0 0
$$136$$ −552.005 −0.348045
$$137$$ −1579.55 −0.985035 −0.492517 0.870303i $$-0.663923\pi$$
−0.492517 + 0.870303i $$0.663923\pi$$
$$138$$ 2323.39 1.43319
$$139$$ 1565.25 0.955128 0.477564 0.878597i $$-0.341520\pi$$
0.477564 + 0.878597i $$0.341520\pi$$
$$140$$ 0 0
$$141$$ 225.214 0.134514
$$142$$ −3461.05 −2.04539
$$143$$ 1626.13 0.950934
$$144$$ 363.659 0.210451
$$145$$ 0 0
$$146$$ 11.7080 0.00663669
$$147$$ −1028.94 −0.577317
$$148$$ −4213.43 −2.34015
$$149$$ 192.635 0.105915 0.0529574 0.998597i $$-0.483135\pi$$
0.0529574 + 0.998597i $$0.483135\pi$$
$$150$$ 0 0
$$151$$ 1052.83 0.567406 0.283703 0.958912i $$-0.408437\pi$$
0.283703 + 0.958912i $$0.408437\pi$$
$$152$$ −3041.44 −1.62298
$$153$$ 148.648 0.0785454
$$154$$ −31.8912 −0.0166874
$$155$$ 0 0
$$156$$ 1542.62 0.791723
$$157$$ 2045.26 1.03968 0.519838 0.854265i $$-0.325992\pi$$
0.519838 + 0.854265i $$0.325992\pi$$
$$158$$ −2912.53 −1.46651
$$159$$ −748.277 −0.373221
$$160$$ 0 0
$$161$$ −22.7054 −0.0111145
$$162$$ −388.234 −0.188287
$$163$$ 848.013 0.407494 0.203747 0.979024i $$-0.434688\pi$$
0.203747 + 0.979024i $$0.434688\pi$$
$$164$$ 1880.75 0.895500
$$165$$ 0 0
$$166$$ 6080.20 2.84286
$$167$$ 1972.45 0.913968 0.456984 0.889475i $$-0.348930\pi$$
0.456984 + 0.889475i $$0.348930\pi$$
$$168$$ −14.0892 −0.00647028
$$169$$ −1017.60 −0.463177
$$170$$ 0 0
$$171$$ 819.018 0.366268
$$172$$ −7933.01 −3.51678
$$173$$ 2707.65 1.18993 0.594967 0.803750i $$-0.297165\pi$$
0.594967 + 0.803750i $$0.297165\pi$$
$$174$$ 177.332 0.0772617
$$175$$ 0 0
$$176$$ 1913.26 0.819419
$$177$$ −805.692 −0.342144
$$178$$ 6410.82 2.69950
$$179$$ 3132.25 1.30791 0.653953 0.756535i $$-0.273109\pi$$
0.653953 + 0.756535i $$0.273109\pi$$
$$180$$ 0 0
$$181$$ 317.051 0.130200 0.0651000 0.997879i $$-0.479263\pi$$
0.0651000 + 0.997879i $$0.479263\pi$$
$$182$$ −23.1301 −0.00942042
$$183$$ 1109.54 0.448195
$$184$$ 5400.32 2.16368
$$185$$ 0 0
$$186$$ 4788.77 1.88779
$$187$$ 782.059 0.305828
$$188$$ 1124.04 0.436059
$$189$$ 3.79404 0.00146019
$$190$$ 0 0
$$191$$ 1679.46 0.636237 0.318119 0.948051i $$-0.396949\pi$$
0.318119 + 0.948051i $$0.396949\pi$$
$$192$$ −2029.55 −0.762866
$$193$$ −1449.62 −0.540653 −0.270326 0.962769i $$-0.587132\pi$$
−0.270326 + 0.962769i $$0.587132\pi$$
$$194$$ 6859.82 2.53869
$$195$$ 0 0
$$196$$ −5135.44 −1.87152
$$197$$ −1060.62 −0.383586 −0.191793 0.981435i $$-0.561430\pi$$
−0.191793 + 0.981435i $$0.561430\pi$$
$$198$$ −2042.56 −0.733124
$$199$$ −183.077 −0.0652159 −0.0326080 0.999468i $$-0.510381\pi$$
−0.0326080 + 0.999468i $$0.510381\pi$$
$$200$$ 0 0
$$201$$ −529.519 −0.185818
$$202$$ 1793.07 0.624556
$$203$$ −1.73299 −0.000599173 0
$$204$$ 741.900 0.254624
$$205$$ 0 0
$$206$$ 9671.67 3.27115
$$207$$ −1454.23 −0.488291
$$208$$ 1387.66 0.462580
$$209$$ 4308.98 1.42612
$$210$$ 0 0
$$211$$ −4904.73 −1.60026 −0.800131 0.599825i $$-0.795237\pi$$
−0.800131 + 0.599825i $$0.795237\pi$$
$$212$$ −3734.65 −1.20989
$$213$$ 2166.31 0.696870
$$214$$ −8050.90 −2.57172
$$215$$ 0 0
$$216$$ −902.384 −0.284257
$$217$$ −46.7985 −0.0146400
$$218$$ −96.9547 −0.0301220
$$219$$ −7.32813 −0.00226114
$$220$$ 0 0
$$221$$ 567.212 0.172646
$$222$$ 4046.30 1.22329
$$223$$ −4837.42 −1.45263 −0.726317 0.687360i $$-0.758770\pi$$
−0.726317 + 0.687360i $$0.758770\pi$$
$$224$$ 10.3569 0.00308929
$$225$$ 0 0
$$226$$ 3079.61 0.906428
$$227$$ −44.8176 −0.0131042 −0.00655209 0.999979i $$-0.502086\pi$$
−0.00655209 + 0.999979i $$0.502086\pi$$
$$228$$ 4087.72 1.18735
$$229$$ −4159.78 −1.20038 −0.600188 0.799859i $$-0.704908\pi$$
−0.600188 + 0.799859i $$0.704908\pi$$
$$230$$ 0 0
$$231$$ 19.9610 0.00568546
$$232$$ 412.179 0.116642
$$233$$ 2945.40 0.828152 0.414076 0.910242i $$-0.364105\pi$$
0.414076 + 0.910242i $$0.364105\pi$$
$$234$$ −1481.43 −0.413864
$$235$$ 0 0
$$236$$ −4021.21 −1.10914
$$237$$ 1822.99 0.499644
$$238$$ −11.1240 −0.00302968
$$239$$ 2174.54 0.588533 0.294266 0.955723i $$-0.404925\pi$$
0.294266 + 0.955723i $$0.404925\pi$$
$$240$$ 0 0
$$241$$ 2300.69 0.614939 0.307470 0.951558i $$-0.400518\pi$$
0.307470 + 0.951558i $$0.400518\pi$$
$$242$$ −4366.74 −1.15994
$$243$$ 243.000 0.0641500
$$244$$ 5537.72 1.45294
$$245$$ 0 0
$$246$$ −1806.15 −0.468112
$$247$$ 3125.23 0.805074
$$248$$ 11130.7 2.85000
$$249$$ −3805.66 −0.968571
$$250$$ 0 0
$$251$$ 3368.55 0.847096 0.423548 0.905874i $$-0.360785\pi$$
0.423548 + 0.905874i $$0.360785\pi$$
$$252$$ 18.9360 0.00473357
$$253$$ −7650.95 −1.90123
$$254$$ 5684.09 1.40414
$$255$$ 0 0
$$256$$ −7303.36 −1.78305
$$257$$ 404.905 0.0982772 0.0491386 0.998792i $$-0.484352\pi$$
0.0491386 + 0.998792i $$0.484352\pi$$
$$258$$ 7618.33 1.83836
$$259$$ −39.5427 −0.00948672
$$260$$ 0 0
$$261$$ −110.994 −0.0263233
$$262$$ 1069.29 0.252142
$$263$$ −1514.70 −0.355135 −0.177567 0.984109i $$-0.556823\pi$$
−0.177567 + 0.984109i $$0.556823\pi$$
$$264$$ −4747.59 −1.10679
$$265$$ 0 0
$$266$$ −61.2912 −0.0141278
$$267$$ −4012.60 −0.919727
$$268$$ −2642.83 −0.602374
$$269$$ −1891.88 −0.428811 −0.214405 0.976745i $$-0.568781\pi$$
−0.214405 + 0.976745i $$0.568781\pi$$
$$270$$ 0 0
$$271$$ −2094.31 −0.469448 −0.234724 0.972062i $$-0.575419\pi$$
−0.234724 + 0.972062i $$0.575419\pi$$
$$272$$ 667.370 0.148769
$$273$$ 14.4774 0.00320956
$$274$$ 7570.79 1.66923
$$275$$ 0 0
$$276$$ −7258.08 −1.58292
$$277$$ −1492.73 −0.323789 −0.161894 0.986808i $$-0.551760\pi$$
−0.161894 + 0.986808i $$0.551760\pi$$
$$278$$ −7502.27 −1.61855
$$279$$ −2997.34 −0.643176
$$280$$ 0 0
$$281$$ 2069.49 0.439344 0.219672 0.975574i $$-0.429501\pi$$
0.219672 + 0.975574i $$0.429501\pi$$
$$282$$ −1079.45 −0.227945
$$283$$ 3398.90 0.713935 0.356967 0.934117i $$-0.383811\pi$$
0.356967 + 0.934117i $$0.383811\pi$$
$$284$$ 10812.1 2.25908
$$285$$ 0 0
$$286$$ −7794.05 −1.61144
$$287$$ 17.6507 0.00363026
$$288$$ 663.338 0.135721
$$289$$ −4640.21 −0.944476
$$290$$ 0 0
$$291$$ −4293.64 −0.864939
$$292$$ −36.5747 −0.00733004
$$293$$ 7677.60 1.53082 0.765410 0.643543i $$-0.222536\pi$$
0.765410 + 0.643543i $$0.222536\pi$$
$$294$$ 4931.73 0.978314
$$295$$ 0 0
$$296$$ 9404.93 1.84679
$$297$$ 1278.46 0.249777
$$298$$ −923.304 −0.179482
$$299$$ −5549.09 −1.07329
$$300$$ 0 0
$$301$$ −74.4505 −0.0142567
$$302$$ −5046.24 −0.961518
$$303$$ −1122.31 −0.212788
$$304$$ 3677.07 0.693732
$$305$$ 0 0
$$306$$ −712.470 −0.133102
$$307$$ −4507.90 −0.838043 −0.419022 0.907976i $$-0.637627\pi$$
−0.419022 + 0.907976i $$0.637627\pi$$
$$308$$ 99.6255 0.0184308
$$309$$ −6053.61 −1.11449
$$310$$ 0 0
$$311$$ 7065.19 1.28820 0.644100 0.764941i $$-0.277232\pi$$
0.644100 + 0.764941i $$0.277232\pi$$
$$312$$ −3443.34 −0.624809
$$313$$ 2063.81 0.372694 0.186347 0.982484i $$-0.440335\pi$$
0.186347 + 0.982484i $$0.440335\pi$$
$$314$$ −9802.94 −1.76182
$$315$$ 0 0
$$316$$ 9098.52 1.61972
$$317$$ 8596.49 1.52311 0.761557 0.648098i $$-0.224435\pi$$
0.761557 + 0.648098i $$0.224435\pi$$
$$318$$ 3586.50 0.632456
$$319$$ −583.959 −0.102493
$$320$$ 0 0
$$321$$ 5039.14 0.876192
$$322$$ 108.827 0.0188345
$$323$$ 1503.03 0.258918
$$324$$ 1212.81 0.207958
$$325$$ 0 0
$$326$$ −4064.54 −0.690533
$$327$$ 60.6850 0.0102627
$$328$$ −4198.08 −0.706708
$$329$$ 10.5490 0.00176774
$$330$$ 0 0
$$331$$ 259.628 0.0431131 0.0215566 0.999768i $$-0.493138\pi$$
0.0215566 + 0.999768i $$0.493138\pi$$
$$332$$ −18994.1 −3.13986
$$333$$ −2532.62 −0.416777
$$334$$ −9453.97 −1.54880
$$335$$ 0 0
$$336$$ 17.0338 0.00276568
$$337$$ 6721.19 1.08643 0.543215 0.839594i $$-0.317207\pi$$
0.543215 + 0.839594i $$0.317207\pi$$
$$338$$ 4877.38 0.784894
$$339$$ −1927.56 −0.308822
$$340$$ 0 0
$$341$$ −15769.5 −2.50430
$$342$$ −3925.57 −0.620673
$$343$$ −96.3940 −0.0151743
$$344$$ 17707.5 2.77536
$$345$$ 0 0
$$346$$ −12977.8 −2.01645
$$347$$ −7139.56 −1.10453 −0.552264 0.833669i $$-0.686236\pi$$
−0.552264 + 0.833669i $$0.686236\pi$$
$$348$$ −553.972 −0.0853334
$$349$$ −11473.4 −1.75976 −0.879879 0.475198i $$-0.842376\pi$$
−0.879879 + 0.475198i $$0.842376\pi$$
$$350$$ 0 0
$$351$$ 927.244 0.141005
$$352$$ 3489.92 0.528448
$$353$$ 59.6020 0.00898667 0.00449333 0.999990i $$-0.498570\pi$$
0.00449333 + 0.999990i $$0.498570\pi$$
$$354$$ 3861.69 0.579793
$$355$$ 0 0
$$356$$ −20026.9 −2.98152
$$357$$ 6.96265 0.00103222
$$358$$ −15012.9 −2.21636
$$359$$ −7801.88 −1.14698 −0.573492 0.819211i $$-0.694412\pi$$
−0.573492 + 0.819211i $$0.694412\pi$$
$$360$$ 0 0
$$361$$ 1422.37 0.207373
$$362$$ −1519.63 −0.220635
$$363$$ 2733.19 0.395193
$$364$$ 72.2565 0.0104046
$$365$$ 0 0
$$366$$ −5318.06 −0.759506
$$367$$ −1583.29 −0.225197 −0.112599 0.993641i $$-0.535917\pi$$
−0.112599 + 0.993641i $$0.535917\pi$$
$$368$$ −6528.94 −0.924850
$$369$$ 1130.49 0.159487
$$370$$ 0 0
$$371$$ −35.0493 −0.00490477
$$372$$ −14959.7 −2.08502
$$373$$ −11181.2 −1.55212 −0.776059 0.630661i $$-0.782784\pi$$
−0.776059 + 0.630661i $$0.782784\pi$$
$$374$$ −3748.42 −0.518252
$$375$$ 0 0
$$376$$ −2509.01 −0.344128
$$377$$ −423.534 −0.0578598
$$378$$ −18.1849 −0.00247442
$$379$$ −11723.0 −1.58885 −0.794423 0.607365i $$-0.792227\pi$$
−0.794423 + 0.607365i $$0.792227\pi$$
$$380$$ 0 0
$$381$$ −3557.73 −0.478394
$$382$$ −8049.67 −1.07816
$$383$$ −9272.02 −1.23702 −0.618509 0.785778i $$-0.712263\pi$$
−0.618509 + 0.785778i $$0.712263\pi$$
$$384$$ 7958.77 1.05767
$$385$$ 0 0
$$386$$ 6948.05 0.916183
$$387$$ −4768.39 −0.626333
$$388$$ −21429.5 −2.80391
$$389$$ −1341.15 −0.174805 −0.0874023 0.996173i $$-0.527857\pi$$
−0.0874023 + 0.996173i $$0.527857\pi$$
$$390$$ 0 0
$$391$$ −2668.74 −0.345177
$$392$$ 11463.0 1.47696
$$393$$ −669.283 −0.0859054
$$394$$ 5083.59 0.650019
$$395$$ 0 0
$$396$$ 6380.80 0.809715
$$397$$ 11366.3 1.43692 0.718459 0.695569i $$-0.244848\pi$$
0.718459 + 0.695569i $$0.244848\pi$$
$$398$$ 877.490 0.110514
$$399$$ 38.3628 0.00481339
$$400$$ 0 0
$$401$$ −12421.7 −1.54690 −0.773451 0.633856i $$-0.781471\pi$$
−0.773451 + 0.633856i $$0.781471\pi$$
$$402$$ 2537.99 0.314884
$$403$$ −11437.3 −1.41373
$$404$$ −5601.42 −0.689805
$$405$$ 0 0
$$406$$ 8.30625 0.00101535
$$407$$ −13324.5 −1.62278
$$408$$ −1656.02 −0.200944
$$409$$ 12917.8 1.56172 0.780860 0.624706i $$-0.214781\pi$$
0.780860 + 0.624706i $$0.214781\pi$$
$$410$$ 0 0
$$411$$ −4738.64 −0.568710
$$412$$ −30213.5 −3.61290
$$413$$ −37.7386 −0.00449636
$$414$$ 6970.16 0.827451
$$415$$ 0 0
$$416$$ 2531.18 0.298320
$$417$$ 4695.75 0.551443
$$418$$ −20653.0 −2.41668
$$419$$ −4162.80 −0.485361 −0.242680 0.970106i $$-0.578027\pi$$
−0.242680 + 0.970106i $$0.578027\pi$$
$$420$$ 0 0
$$421$$ −14947.3 −1.73037 −0.865186 0.501451i $$-0.832800\pi$$
−0.865186 + 0.501451i $$0.832800\pi$$
$$422$$ 23508.4 2.71178
$$423$$ 675.642 0.0776615
$$424$$ 8336.21 0.954817
$$425$$ 0 0
$$426$$ −10383.2 −1.18091
$$427$$ 51.9710 0.00589005
$$428$$ 25150.4 2.84039
$$429$$ 4878.38 0.549022
$$430$$ 0 0
$$431$$ −12762.9 −1.42638 −0.713188 0.700973i $$-0.752749\pi$$
−0.713188 + 0.700973i $$0.752749\pi$$
$$432$$ 1090.98 0.121504
$$433$$ 777.497 0.0862913 0.0431456 0.999069i $$-0.486262\pi$$
0.0431456 + 0.999069i $$0.486262\pi$$
$$434$$ 224.306 0.0248088
$$435$$ 0 0
$$436$$ 302.879 0.0332689
$$437$$ −14704.2 −1.60961
$$438$$ 35.1239 0.00383170
$$439$$ 9936.45 1.08028 0.540138 0.841577i $$-0.318372\pi$$
0.540138 + 0.841577i $$0.318372\pi$$
$$440$$ 0 0
$$441$$ −3086.82 −0.333314
$$442$$ −2718.66 −0.292564
$$443$$ 7139.86 0.765745 0.382873 0.923801i $$-0.374935\pi$$
0.382873 + 0.923801i $$0.374935\pi$$
$$444$$ −12640.3 −1.35109
$$445$$ 0 0
$$446$$ 23185.8 2.46162
$$447$$ 577.906 0.0611499
$$448$$ −95.0642 −0.0100254
$$449$$ 10438.2 1.09712 0.548560 0.836111i $$-0.315176\pi$$
0.548560 + 0.836111i $$0.315176\pi$$
$$450$$ 0 0
$$451$$ 5947.67 0.620986
$$452$$ −9620.45 −1.00112
$$453$$ 3158.50 0.327592
$$454$$ 214.811 0.0222062
$$455$$ 0 0
$$456$$ −9124.31 −0.937029
$$457$$ 5039.76 0.515864 0.257932 0.966163i $$-0.416959\pi$$
0.257932 + 0.966163i $$0.416959\pi$$
$$458$$ 19937.9 2.03414
$$459$$ 445.943 0.0453482
$$460$$ 0 0
$$461$$ 1832.68 0.185155 0.0925775 0.995705i $$-0.470489\pi$$
0.0925775 + 0.995705i $$0.470489\pi$$
$$462$$ −95.6736 −0.00963450
$$463$$ 1887.49 0.189458 0.0947290 0.995503i $$-0.469802\pi$$
0.0947290 + 0.995503i $$0.469802\pi$$
$$464$$ −498.321 −0.0498577
$$465$$ 0 0
$$466$$ −14117.3 −1.40337
$$467$$ −3944.31 −0.390837 −0.195419 0.980720i $$-0.562607\pi$$
−0.195419 + 0.980720i $$0.562607\pi$$
$$468$$ 4627.87 0.457101
$$469$$ −24.8027 −0.00244196
$$470$$ 0 0
$$471$$ 6135.77 0.600257
$$472$$ 8975.85 0.875311
$$473$$ −25087.3 −2.43872
$$474$$ −8737.60 −0.846690
$$475$$ 0 0
$$476$$ 34.7506 0.00334620
$$477$$ −2244.83 −0.215479
$$478$$ −10422.6 −0.997320
$$479$$ −10759.1 −1.02630 −0.513150 0.858299i $$-0.671522\pi$$
−0.513150 + 0.858299i $$0.671522\pi$$
$$480$$ 0 0
$$481$$ −9664.02 −0.916095
$$482$$ −11027.2 −1.04207
$$483$$ −68.1163 −0.00641698
$$484$$ 13641.3 1.28112
$$485$$ 0 0
$$486$$ −1164.70 −0.108708
$$487$$ 3191.34 0.296948 0.148474 0.988916i $$-0.452564\pi$$
0.148474 + 0.988916i $$0.452564\pi$$
$$488$$ −12360.9 −1.14662
$$489$$ 2544.04 0.235267
$$490$$ 0 0
$$491$$ 933.900 0.0858377 0.0429189 0.999079i $$-0.486334\pi$$
0.0429189 + 0.999079i $$0.486334\pi$$
$$492$$ 5642.25 0.517017
$$493$$ −203.692 −0.0186082
$$494$$ −14979.3 −1.36427
$$495$$ 0 0
$$496$$ −13456.9 −1.21821
$$497$$ 101.470 0.00915806
$$498$$ 18240.6 1.64133
$$499$$ 5499.10 0.493333 0.246667 0.969100i $$-0.420665\pi$$
0.246667 + 0.969100i $$0.420665\pi$$
$$500$$ 0 0
$$501$$ 5917.34 0.527679
$$502$$ −16145.5 −1.43548
$$503$$ −11172.5 −0.990374 −0.495187 0.868787i $$-0.664900\pi$$
−0.495187 + 0.868787i $$0.664900\pi$$
$$504$$ −42.2677 −0.00373562
$$505$$ 0 0
$$506$$ 36671.1 3.22180
$$507$$ −3052.80 −0.267416
$$508$$ −17756.6 −1.55083
$$509$$ −8576.81 −0.746877 −0.373439 0.927655i $$-0.621821\pi$$
−0.373439 + 0.927655i $$0.621821\pi$$
$$510$$ 0 0
$$511$$ −0.343250 −2.97152e−5 0
$$512$$ 13781.7 1.18960
$$513$$ 2457.05 0.211465
$$514$$ −1940.71 −0.166539
$$515$$ 0 0
$$516$$ −23799.0 −2.03041
$$517$$ 3554.66 0.302386
$$518$$ 189.529 0.0160761
$$519$$ 8122.94 0.687008
$$520$$ 0 0
$$521$$ −19732.4 −1.65929 −0.829647 0.558289i $$-0.811458\pi$$
−0.829647 + 0.558289i $$0.811458\pi$$
$$522$$ 531.997 0.0446071
$$523$$ 16353.7 1.36730 0.683651 0.729809i $$-0.260391\pi$$
0.683651 + 0.729809i $$0.260391\pi$$
$$524$$ −3340.39 −0.278484
$$525$$ 0 0
$$526$$ 7259.98 0.601807
$$527$$ −5500.59 −0.454667
$$528$$ 5739.79 0.473092
$$529$$ 13941.6 1.14585
$$530$$ 0 0
$$531$$ −2417.08 −0.197537
$$532$$ 191.469 0.0156038
$$533$$ 4313.73 0.350560
$$534$$ 19232.5 1.55856
$$535$$ 0 0
$$536$$ 5899.13 0.475380
$$537$$ 9396.75 0.755120
$$538$$ 9067.82 0.726657
$$539$$ −16240.3 −1.29781
$$540$$ 0 0
$$541$$ 14238.7 1.13155 0.565776 0.824559i $$-0.308577\pi$$
0.565776 + 0.824559i $$0.308577\pi$$
$$542$$ 10038.1 0.795521
$$543$$ 951.153 0.0751710
$$544$$ 1217.33 0.0959421
$$545$$ 0 0
$$546$$ −69.3903 −0.00543888
$$547$$ 7018.76 0.548630 0.274315 0.961640i $$-0.411549\pi$$
0.274315 + 0.961640i $$0.411549\pi$$
$$548$$ −23650.5 −1.84361
$$549$$ 3328.63 0.258766
$$550$$ 0 0
$$551$$ −1122.30 −0.0867725
$$552$$ 16201.0 1.24920
$$553$$ 85.3887 0.00656618
$$554$$ 7154.68 0.548688
$$555$$ 0 0
$$556$$ 23436.5 1.78764
$$557$$ 6144.57 0.467421 0.233711 0.972306i $$-0.424913\pi$$
0.233711 + 0.972306i $$0.424913\pi$$
$$558$$ 14366.3 1.08992
$$559$$ −18195.3 −1.37671
$$560$$ 0 0
$$561$$ 2346.18 0.176570
$$562$$ −9919.12 −0.744507
$$563$$ 17169.5 1.28527 0.642635 0.766172i $$-0.277841\pi$$
0.642635 + 0.766172i $$0.277841\pi$$
$$564$$ 3372.12 0.251759
$$565$$ 0 0
$$566$$ −16291.0 −1.20982
$$567$$ 11.3821 0.000843041 0
$$568$$ −24133.9 −1.78281
$$569$$ −15575.2 −1.14754 −0.573768 0.819018i $$-0.694519\pi$$
−0.573768 + 0.819018i $$0.694519\pi$$
$$570$$ 0 0
$$571$$ −21207.6 −1.55431 −0.777156 0.629308i $$-0.783339\pi$$
−0.777156 + 0.629308i $$0.783339\pi$$
$$572$$ 24348.0 1.77979
$$573$$ 5038.37 0.367332
$$574$$ −84.5999 −0.00615180
$$575$$ 0 0
$$576$$ −6088.65 −0.440441
$$577$$ −2071.25 −0.149441 −0.0747205 0.997205i $$-0.523806\pi$$
−0.0747205 + 0.997205i $$0.523806\pi$$
$$578$$ 22240.6 1.60050
$$579$$ −4348.86 −0.312146
$$580$$ 0 0
$$581$$ −178.257 −0.0127287
$$582$$ 20579.5 1.46571
$$583$$ −11810.4 −0.839000
$$584$$ 81.6394 0.00578470
$$585$$ 0 0
$$586$$ −36798.9 −2.59411
$$587$$ 20730.6 1.45766 0.728828 0.684697i $$-0.240065\pi$$
0.728828 + 0.684697i $$0.240065\pi$$
$$588$$ −15406.3 −1.08052
$$589$$ −30307.1 −2.12018
$$590$$ 0 0
$$591$$ −3181.87 −0.221463
$$592$$ −11370.5 −0.789399
$$593$$ −1924.00 −0.133236 −0.0666182 0.997779i $$-0.521221\pi$$
−0.0666182 + 0.997779i $$0.521221\pi$$
$$594$$ −6127.68 −0.423269
$$595$$ 0 0
$$596$$ 2884.33 0.198233
$$597$$ −549.230 −0.0376524
$$598$$ 26596.9 1.81878
$$599$$ −8088.34 −0.551720 −0.275860 0.961198i $$-0.588963\pi$$
−0.275860 + 0.961198i $$0.588963\pi$$
$$600$$ 0 0
$$601$$ 19953.8 1.35430 0.677149 0.735846i $$-0.263215\pi$$
0.677149 + 0.735846i $$0.263215\pi$$
$$602$$ 356.842 0.0241592
$$603$$ −1588.56 −0.107282
$$604$$ 15764.0 1.06197
$$605$$ 0 0
$$606$$ 5379.22 0.360588
$$607$$ −3633.68 −0.242976 −0.121488 0.992593i $$-0.538767\pi$$
−0.121488 + 0.992593i $$0.538767\pi$$
$$608$$ 6707.23 0.447392
$$609$$ −5.19897 −0.000345933 0
$$610$$ 0 0
$$611$$ 2578.13 0.170704
$$612$$ 2225.70 0.147007
$$613$$ −1581.91 −0.104230 −0.0521148 0.998641i $$-0.516596\pi$$
−0.0521148 + 0.998641i $$0.516596\pi$$
$$614$$ 21606.4 1.42014
$$615$$ 0 0
$$616$$ −222.377 −0.0145452
$$617$$ 4454.03 0.290620 0.145310 0.989386i $$-0.453582\pi$$
0.145310 + 0.989386i $$0.453582\pi$$
$$618$$ 29015.0 1.88860
$$619$$ −2630.26 −0.170790 −0.0853951 0.996347i $$-0.527215\pi$$
−0.0853951 + 0.996347i $$0.527215\pi$$
$$620$$ 0 0
$$621$$ −4362.70 −0.281915
$$622$$ −33863.6 −2.18297
$$623$$ −187.950 −0.0120868
$$624$$ 4162.97 0.267071
$$625$$ 0 0
$$626$$ −9891.85 −0.631562
$$627$$ 12927.0 0.823370
$$628$$ 30623.6 1.94588
$$629$$ −4647.75 −0.294623
$$630$$ 0 0
$$631$$ −16896.4 −1.06599 −0.532993 0.846120i $$-0.678933\pi$$
−0.532993 + 0.846120i $$0.678933\pi$$
$$632$$ −20309.1 −1.27825
$$633$$ −14714.2 −0.923912
$$634$$ −41203.1 −2.58105
$$635$$ 0 0
$$636$$ −11203.9 −0.698530
$$637$$ −11778.8 −0.732640
$$638$$ 2798.92 0.173684
$$639$$ 6498.93 0.402338
$$640$$ 0 0
$$641$$ 28804.5 1.77490 0.887448 0.460908i $$-0.152476\pi$$
0.887448 + 0.460908i $$0.152476\pi$$
$$642$$ −24152.7 −1.48478
$$643$$ 10219.9 0.626801 0.313400 0.949621i $$-0.398532\pi$$
0.313400 + 0.949621i $$0.398532\pi$$
$$644$$ −339.968 −0.0208022
$$645$$ 0 0
$$646$$ −7204.03 −0.438759
$$647$$ −1792.38 −0.108912 −0.0544558 0.998516i $$-0.517342\pi$$
−0.0544558 + 0.998516i $$0.517342\pi$$
$$648$$ −2707.15 −0.164116
$$649$$ −12716.6 −0.769138
$$650$$ 0 0
$$651$$ −140.396 −0.00845243
$$652$$ 12697.3 0.762675
$$653$$ −13405.4 −0.803359 −0.401679 0.915780i $$-0.631573\pi$$
−0.401679 + 0.915780i $$0.631573\pi$$
$$654$$ −290.864 −0.0173910
$$655$$ 0 0
$$656$$ 5075.44 0.302077
$$657$$ −21.9844 −0.00130547
$$658$$ −50.5616 −0.00299559
$$659$$ −15441.6 −0.912778 −0.456389 0.889780i $$-0.650858\pi$$
−0.456389 + 0.889780i $$0.650858\pi$$
$$660$$ 0 0
$$661$$ 4356.59 0.256356 0.128178 0.991751i $$-0.459087\pi$$
0.128178 + 0.991751i $$0.459087\pi$$
$$662$$ −1244.40 −0.0730589
$$663$$ 1701.64 0.0996774
$$664$$ 42397.2 2.47791
$$665$$ 0 0
$$666$$ 12138.9 0.706265
$$667$$ 1992.74 0.115681
$$668$$ 29533.4 1.71060
$$669$$ −14512.3 −0.838679
$$670$$ 0 0
$$671$$ 17512.4 1.00754
$$672$$ 31.0707 0.00178360
$$673$$ −17617.2 −1.00905 −0.504527 0.863396i $$-0.668333\pi$$
−0.504527 + 0.863396i $$0.668333\pi$$
$$674$$ −32214.8 −1.84105
$$675$$ 0 0
$$676$$ −15236.5 −0.866894
$$677$$ 3294.63 0.187036 0.0935178 0.995618i $$-0.470189\pi$$
0.0935178 + 0.995618i $$0.470189\pi$$
$$678$$ 9238.83 0.523326
$$679$$ −201.114 −0.0113668
$$680$$ 0 0
$$681$$ −134.453 −0.00756570
$$682$$ 75583.4 4.24375
$$683$$ 4829.50 0.270565 0.135282 0.990807i $$-0.456806\pi$$
0.135282 + 0.990807i $$0.456806\pi$$
$$684$$ 12263.2 0.685517
$$685$$ 0 0
$$686$$ 462.018 0.0257142
$$687$$ −12479.3 −0.693037
$$688$$ −21408.2 −1.18631
$$689$$ −8565.87 −0.473634
$$690$$ 0 0
$$691$$ 23330.5 1.28442 0.642211 0.766528i $$-0.278017\pi$$
0.642211 + 0.766528i $$0.278017\pi$$
$$692$$ 40541.5 2.22711
$$693$$ 59.8831 0.00328250
$$694$$ 34220.0 1.87172
$$695$$ 0 0
$$696$$ 1236.54 0.0673432
$$697$$ 2074.62 0.112743
$$698$$ 54992.0 2.98206
$$699$$ 8836.19 0.478134
$$700$$ 0 0
$$701$$ 27619.7 1.48813 0.744066 0.668106i $$-0.232895\pi$$
0.744066 + 0.668106i $$0.232895\pi$$
$$702$$ −4444.30 −0.238945
$$703$$ −25608.2 −1.37387
$$704$$ −32033.4 −1.71492
$$705$$ 0 0
$$706$$ −285.673 −0.0152287
$$707$$ −52.5688 −0.00279640
$$708$$ −12063.6 −0.640365
$$709$$ 25819.3 1.36765 0.683826 0.729645i $$-0.260315\pi$$
0.683826 + 0.729645i $$0.260315\pi$$
$$710$$ 0 0
$$711$$ 5468.96 0.288470
$$712$$ 44702.6 2.35295
$$713$$ 53812.8 2.82651
$$714$$ −33.3721 −0.00174919
$$715$$ 0 0
$$716$$ 46899.1 2.44791
$$717$$ 6523.62 0.339790
$$718$$ 37394.5 1.94366
$$719$$ −16339.0 −0.847484 −0.423742 0.905783i $$-0.639284\pi$$
−0.423742 + 0.905783i $$0.639284\pi$$
$$720$$ 0 0
$$721$$ −283.551 −0.0146463
$$722$$ −6817.44 −0.351411
$$723$$ 6902.06 0.355035
$$724$$ 4747.20 0.243685
$$725$$ 0 0
$$726$$ −13100.2 −0.669689
$$727$$ −7287.52 −0.371773 −0.185887 0.982571i $$-0.559516\pi$$
−0.185887 + 0.982571i $$0.559516\pi$$
$$728$$ −161.286 −0.00821106
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −8750.74 −0.442760
$$732$$ 16613.2 0.838853
$$733$$ −12030.1 −0.606197 −0.303098 0.952959i $$-0.598021\pi$$
−0.303098 + 0.952959i $$0.598021\pi$$
$$734$$ 7588.76 0.381616
$$735$$ 0 0
$$736$$ −11909.2 −0.596440
$$737$$ −8357.64 −0.417718
$$738$$ −5418.44 −0.270265
$$739$$ −15163.0 −0.754778 −0.377389 0.926055i $$-0.623178\pi$$
−0.377389 + 0.926055i $$0.623178\pi$$
$$740$$ 0 0
$$741$$ 9375.68 0.464810
$$742$$ 167.992 0.00831155
$$743$$ −22772.8 −1.12443 −0.562216 0.826991i $$-0.690051\pi$$
−0.562216 + 0.826991i $$0.690051\pi$$
$$744$$ 33392.0 1.64545
$$745$$ 0 0
$$746$$ 53591.6 2.63020
$$747$$ −11417.0 −0.559205
$$748$$ 11709.8 0.572394
$$749$$ 236.034 0.0115147
$$750$$ 0 0
$$751$$ −27855.5 −1.35348 −0.676739 0.736223i $$-0.736608\pi$$
−0.676739 + 0.736223i $$0.736608\pi$$
$$752$$ 3033.37 0.147095
$$753$$ 10105.6 0.489071
$$754$$ 2030.01 0.0980484
$$755$$ 0 0
$$756$$ 56.8081 0.00273293
$$757$$ 302.235 0.0145111 0.00725557 0.999974i $$-0.497690\pi$$
0.00725557 + 0.999974i $$0.497690\pi$$
$$758$$ 56188.7 2.69244
$$759$$ −22952.9 −1.09768
$$760$$ 0 0
$$761$$ −4497.19 −0.214222 −0.107111 0.994247i $$-0.534160\pi$$
−0.107111 + 0.994247i $$0.534160\pi$$
$$762$$ 17052.3 0.810680
$$763$$ 2.84249 0.000134869 0
$$764$$ 25146.5 1.19080
$$765$$ 0 0
$$766$$ 44440.9 2.09623
$$767$$ −9223.13 −0.434195
$$768$$ −21910.1 −1.02944
$$769$$ 2438.76 0.114362 0.0571808 0.998364i $$-0.481789\pi$$
0.0571808 + 0.998364i $$0.481789\pi$$
$$770$$ 0 0
$$771$$ 1214.71 0.0567404
$$772$$ −21705.1 −1.01190
$$773$$ −37464.5 −1.74321 −0.871607 0.490205i $$-0.836922\pi$$
−0.871607 + 0.490205i $$0.836922\pi$$
$$774$$ 22855.0 1.06138
$$775$$ 0 0
$$776$$ 47833.4 2.21278
$$777$$ −118.628 −0.00547716
$$778$$ 6428.15 0.296222
$$779$$ 11430.7 0.525736
$$780$$ 0 0
$$781$$ 34191.9 1.56656
$$782$$ 12791.3 0.584932
$$783$$ −332.983 −0.0151978
$$784$$ −13858.6 −0.631315
$$785$$ 0 0
$$786$$ 3207.88 0.145574
$$787$$ 30317.8 1.37321 0.686603 0.727033i $$-0.259101\pi$$
0.686603 + 0.727033i $$0.259101\pi$$
$$788$$ −15880.7 −0.717928
$$789$$ −4544.10 −0.205037
$$790$$ 0 0
$$791$$ −90.2870 −0.00405845
$$792$$ −14242.8 −0.639008
$$793$$ 12701.4 0.568779
$$794$$ −54478.7 −2.43498
$$795$$ 0 0
$$796$$ −2741.21 −0.122060
$$797$$ −23627.1 −1.05008 −0.525041 0.851077i $$-0.675950\pi$$
−0.525041 + 0.851077i $$0.675950\pi$$
$$798$$ −183.874 −0.00815671
$$799$$ 1239.91 0.0548996
$$800$$ 0 0
$$801$$ −12037.8 −0.531005
$$802$$ 59537.2 2.62136
$$803$$ −115.663 −0.00508303
$$804$$ −7928.48 −0.347781
$$805$$ 0 0
$$806$$ 54819.2 2.39569
$$807$$ −5675.65 −0.247574
$$808$$ 12503.1 0.544378
$$809$$ −30566.8 −1.32839 −0.664197 0.747558i $$-0.731226\pi$$
−0.664197 + 0.747558i $$0.731226\pi$$
$$810$$ 0 0
$$811$$ 36976.4 1.60101 0.800504 0.599328i $$-0.204565\pi$$
0.800504 + 0.599328i $$0.204565\pi$$
$$812$$ −25.9481 −0.00112143
$$813$$ −6282.94 −0.271036
$$814$$ 63864.6 2.74994
$$815$$ 0 0
$$816$$ 2002.11 0.0858920
$$817$$ −48214.8 −2.06466
$$818$$ −61915.1 −2.64647
$$819$$ 43.4321 0.00185304
$$820$$ 0 0
$$821$$ −9499.03 −0.403798 −0.201899 0.979406i $$-0.564711\pi$$
−0.201899 + 0.979406i $$0.564711\pi$$
$$822$$ 22712.4 0.963728
$$823$$ −4753.25 −0.201322 −0.100661 0.994921i $$-0.532096\pi$$
−0.100661 + 0.994921i $$0.532096\pi$$
$$824$$ 67440.5 2.85121
$$825$$ 0 0
$$826$$ 180.882 0.00761947
$$827$$ 13224.0 0.556039 0.278020 0.960575i $$-0.410322\pi$$
0.278020 + 0.960575i $$0.410322\pi$$
$$828$$ −21774.2 −0.913897
$$829$$ −10210.7 −0.427782 −0.213891 0.976858i $$-0.568614\pi$$
−0.213891 + 0.976858i $$0.568614\pi$$
$$830$$ 0 0
$$831$$ −4478.19 −0.186939
$$832$$ −23233.2 −0.968109
$$833$$ −5664.80 −0.235623
$$834$$ −22506.8 −0.934469
$$835$$ 0 0
$$836$$ 64518.4 2.66916
$$837$$ −8992.03 −0.371338
$$838$$ 19952.4 0.822485
$$839$$ −6722.41 −0.276619 −0.138310 0.990389i $$-0.544167\pi$$
−0.138310 + 0.990389i $$0.544167\pi$$
$$840$$ 0 0
$$841$$ −24236.9 −0.993764
$$842$$ 71642.6 2.93227
$$843$$ 6208.48 0.253655
$$844$$ −73438.4 −2.99509
$$845$$ 0 0
$$846$$ −3238.36 −0.131604
$$847$$ 128.023 0.00519351
$$848$$ −10078.4 −0.408130
$$849$$ 10196.7 0.412191
$$850$$ 0 0
$$851$$ 45469.4 1.83158
$$852$$ 32436.2 1.30428
$$853$$ −3785.52 −0.151950 −0.0759752 0.997110i $$-0.524207\pi$$
−0.0759752 + 0.997110i $$0.524207\pi$$
$$854$$ −249.098 −0.00998121
$$855$$ 0 0
$$856$$ −56138.8 −2.24157
$$857$$ 13421.7 0.534979 0.267489 0.963561i $$-0.413806\pi$$
0.267489 + 0.963561i $$0.413806\pi$$
$$858$$ −23382.1 −0.930365
$$859$$ 491.965 0.0195409 0.00977045 0.999952i $$-0.496890\pi$$
0.00977045 + 0.999952i $$0.496890\pi$$
$$860$$ 0 0
$$861$$ 52.9520 0.00209593
$$862$$ 61172.8 2.41712
$$863$$ −8088.10 −0.319029 −0.159515 0.987196i $$-0.550993\pi$$
−0.159515 + 0.987196i $$0.550993\pi$$
$$864$$ 1990.01 0.0783583
$$865$$ 0 0
$$866$$ −3726.55 −0.146228
$$867$$ −13920.6 −0.545293
$$868$$ −700.714 −0.0274007
$$869$$ 28773.1 1.12320
$$870$$ 0 0
$$871$$ −6061.64 −0.235811
$$872$$ −676.064 −0.0262551
$$873$$ −12880.9 −0.499373
$$874$$ 70477.7 2.72762
$$875$$ 0 0
$$876$$ −109.724 −0.00423200
$$877$$ 12000.4 0.462058 0.231029 0.972947i $$-0.425791\pi$$
0.231029 + 0.972947i $$0.425791\pi$$
$$878$$ −47625.6 −1.83062
$$879$$ 23032.8 0.883820
$$880$$ 0 0
$$881$$ −2285.38 −0.0873964 −0.0436982 0.999045i $$-0.513914\pi$$
−0.0436982 + 0.999045i $$0.513914\pi$$
$$882$$ 14795.2 0.564830
$$883$$ 19226.6 0.732759 0.366379 0.930466i $$-0.380597\pi$$
0.366379 + 0.930466i $$0.380597\pi$$
$$884$$ 8492.87 0.323129
$$885$$ 0 0
$$886$$ −34221.5 −1.29762
$$887$$ 14952.8 0.566028 0.283014 0.959116i $$-0.408666\pi$$
0.283014 + 0.959116i $$0.408666\pi$$
$$888$$ 28214.8 1.06625
$$889$$ −166.644 −0.00628692
$$890$$ 0 0
$$891$$ 3835.38 0.144209
$$892$$ −72430.6 −2.71879
$$893$$ 6831.64 0.256005
$$894$$ −2769.91 −0.103624
$$895$$ 0 0
$$896$$ 372.789 0.0138996
$$897$$ −16647.3 −0.619662
$$898$$ −50030.2 −1.85916
$$899$$ 4107.26 0.152375
$$900$$ 0 0
$$901$$ −4119.61 −0.152324
$$902$$ −28507.3 −1.05231
$$903$$ −223.352 −0.00823109
$$904$$ 21474.1 0.790064
$$905$$ 0 0
$$906$$ −15138.7 −0.555133
$$907$$ 48744.8 1.78450 0.892251 0.451540i $$-0.149125\pi$$
0.892251 + 0.451540i $$0.149125\pi$$
$$908$$ −671.053 −0.0245261
$$909$$ −3366.92 −0.122853
$$910$$ 0 0
$$911$$ −35064.7 −1.27524 −0.637621 0.770350i $$-0.720081\pi$$
−0.637621 + 0.770350i $$0.720081\pi$$
$$912$$ 11031.2 0.400527
$$913$$ −60066.6 −2.17734
$$914$$ −24155.6 −0.874177
$$915$$ 0 0
$$916$$ −62284.4 −2.24665
$$917$$ −31.3492 −0.00112894
$$918$$ −2137.41 −0.0768465
$$919$$ −20159.6 −0.723617 −0.361808 0.932253i $$-0.617841\pi$$
−0.361808 + 0.932253i $$0.617841\pi$$
$$920$$ 0 0
$$921$$ −13523.7 −0.483845
$$922$$ −8784.07 −0.313761
$$923$$ 24798.8 0.884357
$$924$$ 298.877 0.0106410
$$925$$ 0 0
$$926$$ −9046.76 −0.321053
$$927$$ −18160.8 −0.643451
$$928$$ −908.972 −0.0321535
$$929$$ 22221.8 0.784793 0.392397 0.919796i $$-0.371646\pi$$
0.392397 + 0.919796i $$0.371646\pi$$
$$930$$ 0 0
$$931$$ −31211.9 −1.09874
$$932$$ 44101.4 1.54999
$$933$$ 21195.6 0.743743
$$934$$ 18905.2 0.662308
$$935$$ 0 0
$$936$$ −10330.0 −0.360734
$$937$$ 8123.54 0.283228 0.141614 0.989922i $$-0.454771\pi$$
0.141614 + 0.989922i $$0.454771\pi$$
$$938$$ 118.880 0.00413812
$$939$$ 6191.42 0.215175
$$940$$ 0 0
$$941$$ −41826.3 −1.44899 −0.724494 0.689281i $$-0.757927\pi$$
−0.724494 + 0.689281i $$0.757927\pi$$
$$942$$ −29408.8 −1.01719
$$943$$ −20296.2 −0.700885
$$944$$ −10851.7 −0.374146
$$945$$ 0 0
$$946$$ 120244. 4.13262
$$947$$ 32843.5 1.12700 0.563501 0.826115i $$-0.309454\pi$$
0.563501 + 0.826115i $$0.309454\pi$$
$$948$$ 27295.6 0.935146
$$949$$ −83.8885 −0.00286948
$$950$$ 0 0
$$951$$ 25789.5 0.879370
$$952$$ −77.5678 −0.00264074
$$953$$ −12579.4 −0.427583 −0.213791 0.976879i $$-0.568581\pi$$
−0.213791 + 0.976879i $$0.568581\pi$$
$$954$$ 10759.5 0.365149
$$955$$ 0 0
$$956$$ 32559.4 1.10151
$$957$$ −1751.88 −0.0591746
$$958$$ 51568.8 1.73916
$$959$$ −221.958 −0.00747382
$$960$$ 0 0
$$961$$ 81123.3 2.72308
$$962$$ 46319.8 1.55240
$$963$$ 15117.4 0.505870
$$964$$ 34448.2 1.15093
$$965$$ 0 0
$$966$$ 326.482 0.0108741
$$967$$ 41883.1 1.39283 0.696417 0.717637i $$-0.254777\pi$$
0.696417 + 0.717637i $$0.254777\pi$$
$$968$$ −30449.2 −1.01103
$$969$$ 4509.08 0.149487
$$970$$ 0 0
$$971$$ 55390.3 1.83065 0.915323 0.402720i $$-0.131935\pi$$
0.915323 + 0.402720i $$0.131935\pi$$
$$972$$ 3638.44 0.120065
$$973$$ 219.949 0.00724691
$$974$$ −15296.2 −0.503204
$$975$$ 0 0
$$976$$ 14944.2 0.490117
$$977$$ −35463.4 −1.16128 −0.580642 0.814159i $$-0.697198\pi$$
−0.580642 + 0.814159i $$0.697198\pi$$
$$978$$ −12193.6 −0.398680
$$979$$ −63332.8 −2.06754
$$980$$ 0 0
$$981$$ 182.055 0.00592515
$$982$$ −4476.20 −0.145459
$$983$$ −34670.9 −1.12495 −0.562477 0.826813i $$-0.690151\pi$$
−0.562477 + 0.826813i $$0.690151\pi$$
$$984$$ −12594.2 −0.408018
$$985$$ 0 0
$$986$$ 976.298 0.0315331
$$987$$ 31.6471 0.00102060
$$988$$ 46794.0 1.50680
$$989$$ 85609.3 2.75250
$$990$$ 0 0
$$991$$ −37280.3 −1.19500 −0.597501 0.801868i $$-0.703840\pi$$
−0.597501 + 0.801868i $$0.703840\pi$$
$$992$$ −24546.3 −0.785631
$$993$$ 778.884 0.0248914
$$994$$ −486.347 −0.0155191
$$995$$ 0 0
$$996$$ −56982.2 −1.81280
$$997$$ −34383.2 −1.09220 −0.546101 0.837719i $$-0.683889\pi$$
−0.546101 + 0.837719i $$0.683889\pi$$
$$998$$ −26357.3 −0.835996
$$999$$ −7597.86 −0.240626
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 1875.4.a.g.1.2 14
5.4 even 2 1875.4.a.f.1.13 14
25.6 even 5 75.4.g.b.61.7 yes 28
25.21 even 5 75.4.g.b.16.7 28
75.56 odd 10 225.4.h.a.136.1 28
75.71 odd 10 225.4.h.a.91.1 28

By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.g.b.16.7 28 25.21 even 5
75.4.g.b.61.7 yes 28 25.6 even 5
225.4.h.a.91.1 28 75.71 odd 10
225.4.h.a.136.1 28 75.56 odd 10
1875.4.a.f.1.13 14 5.4 even 2
1875.4.a.g.1.2 14 1.1 even 1 trivial