# Properties

 Label 675.4.b.l.649.6 Level $675$ Weight $4$ Character 675.649 Analytic conductor $39.826$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 675.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.8262892539$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.2033649216.1 Defining polynomial: $$x^{6} + 47x^{4} + 541x^{2} + 36$$ x^6 + 47*x^4 + 541*x^2 + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 135) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.6 Root $$5.20067i$$ of defining polynomial Character $$\chi$$ $$=$$ 675.649 Dual form 675.4.b.l.649.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.20067i q^{2} -19.0470 q^{4} -24.4013i q^{7} -57.4517i q^{8} +O(q^{10})$$ $$q+5.20067i q^{2} -19.0470 q^{4} -24.4013i q^{7} -57.4517i q^{8} +28.9839 q^{11} -65.3919i q^{13} +126.903 q^{14} +146.411 q^{16} +68.1718i q^{17} -104.424 q^{19} +150.736i q^{22} +154.807i q^{23} +340.082 q^{26} +464.772i q^{28} -205.658 q^{29} -18.2497 q^{31} +301.824i q^{32} -354.539 q^{34} +337.613i q^{37} -543.076i q^{38} +195.969 q^{41} +334.882i q^{43} -552.055 q^{44} -805.098 q^{46} -5.00398i q^{47} -252.425 q^{49} +1245.52i q^{52} +319.965i q^{53} -1401.90 q^{56} -1069.56i q^{58} +430.611 q^{59} +594.581 q^{61} -94.9106i q^{62} -398.396 q^{64} -195.876i q^{67} -1298.47i q^{68} -425.955 q^{71} +929.193i q^{73} -1755.82 q^{74} +1988.96 q^{76} -707.245i q^{77} -24.4296 q^{79} +1019.17i q^{82} +545.859i q^{83} -1741.61 q^{86} -1665.17i q^{88} -84.1332 q^{89} -1595.65 q^{91} -2948.60i q^{92} +26.0241 q^{94} -827.613i q^{97} -1312.78i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 46 q^{4}+O(q^{10})$$ 6 * q - 46 * q^4 $$6 q - 46 q^{4} + 76 q^{11} + 216 q^{14} + 382 q^{16} - 374 q^{19} + 832 q^{26} - 320 q^{29} + 454 q^{31} - 34 q^{34} - 676 q^{41} - 3272 q^{44} - 2850 q^{46} + 394 q^{49} - 2508 q^{56} - 280 q^{59} + 1190 q^{61} + 1836 q^{64} - 1204 q^{71} - 5756 q^{74} + 1050 q^{76} - 1258 q^{79} - 7460 q^{86} - 4308 q^{89} - 880 q^{91} + 2216 q^{94}+O(q^{100})$$ 6 * q - 46 * q^4 + 76 * q^11 + 216 * q^14 + 382 * q^16 - 374 * q^19 + 832 * q^26 - 320 * q^29 + 454 * q^31 - 34 * q^34 - 676 * q^41 - 3272 * q^44 - 2850 * q^46 + 394 * q^49 - 2508 * q^56 - 280 * q^59 + 1190 * q^61 + 1836 * q^64 - 1204 * q^71 - 5756 * q^74 + 1050 * q^76 - 1258 * q^79 - 7460 * q^86 - 4308 * q^89 - 880 * q^91 + 2216 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.20067i 1.83871i 0.393424 + 0.919357i $$0.371291\pi$$
−0.393424 + 0.919357i $$0.628709\pi$$
$$3$$ 0 0
$$4$$ −19.0470 −2.38087
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 24.4013i − 1.31755i −0.752341 0.658774i $$-0.771075\pi$$
0.752341 0.658774i $$-0.228925\pi$$
$$8$$ − 57.4517i − 2.53903i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 28.9839 0.794451 0.397226 0.917721i $$-0.369973\pi$$
0.397226 + 0.917721i $$0.369973\pi$$
$$12$$ 0 0
$$13$$ − 65.3919i − 1.39511i −0.716530 0.697556i $$-0.754271\pi$$
0.716530 0.697556i $$-0.245729\pi$$
$$14$$ 126.903 2.42260
$$15$$ 0 0
$$16$$ 146.411 2.28768
$$17$$ 68.1718i 0.972593i 0.873794 + 0.486296i $$0.161653\pi$$
−0.873794 + 0.486296i $$0.838347\pi$$
$$18$$ 0 0
$$19$$ −104.424 −1.26087 −0.630435 0.776242i $$-0.717124\pi$$
−0.630435 + 0.776242i $$0.717124\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 150.736i 1.46077i
$$23$$ 154.807i 1.40345i 0.712446 + 0.701727i $$0.247587\pi$$
−0.712446 + 0.701727i $$0.752413\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 340.082 2.56521
$$27$$ 0 0
$$28$$ 464.772i 3.13691i
$$29$$ −205.658 −1.31689 −0.658443 0.752631i $$-0.728785\pi$$
−0.658443 + 0.752631i $$0.728785\pi$$
$$30$$ 0 0
$$31$$ −18.2497 −0.105734 −0.0528668 0.998602i $$-0.516836\pi$$
−0.0528668 + 0.998602i $$0.516836\pi$$
$$32$$ 301.824i 1.66736i
$$33$$ 0 0
$$34$$ −354.539 −1.78832
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 337.613i 1.50009i 0.661387 + 0.750044i $$0.269968\pi$$
−0.661387 + 0.750044i $$0.730032\pi$$
$$38$$ − 543.076i − 2.31838i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 195.969 0.746469 0.373234 0.927737i $$-0.378249\pi$$
0.373234 + 0.927737i $$0.378249\pi$$
$$42$$ 0 0
$$43$$ 334.882i 1.18765i 0.804594 + 0.593826i $$0.202383\pi$$
−0.804594 + 0.593826i $$0.797617\pi$$
$$44$$ −552.055 −1.89149
$$45$$ 0 0
$$46$$ −805.098 −2.58055
$$47$$ − 5.00398i − 0.0155299i −0.999970 0.00776496i $$-0.997528\pi$$
0.999970 0.00776496i $$-0.00247169\pi$$
$$48$$ 0 0
$$49$$ −252.425 −0.735934
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1245.52i 3.32158i
$$53$$ 319.965i 0.829256i 0.909991 + 0.414628i $$0.136088\pi$$
−0.909991 + 0.414628i $$0.863912\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1401.90 −3.34529
$$57$$ 0 0
$$58$$ − 1069.56i − 2.42138i
$$59$$ 430.611 0.950182 0.475091 0.879937i $$-0.342415\pi$$
0.475091 + 0.879937i $$0.342415\pi$$
$$60$$ 0 0
$$61$$ 594.581 1.24800 0.624002 0.781422i $$-0.285505\pi$$
0.624002 + 0.781422i $$0.285505\pi$$
$$62$$ − 94.9106i − 0.194414i
$$63$$ 0 0
$$64$$ −398.396 −0.778118
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 195.876i − 0.357166i −0.983925 0.178583i $$-0.942849\pi$$
0.983925 0.178583i $$-0.0571513\pi$$
$$68$$ − 1298.47i − 2.31562i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −425.955 −0.711994 −0.355997 0.934487i $$-0.615859\pi$$
−0.355997 + 0.934487i $$0.615859\pi$$
$$72$$ 0 0
$$73$$ 929.193i 1.48978i 0.667188 + 0.744889i $$0.267498\pi$$
−0.667188 + 0.744889i $$0.732502\pi$$
$$74$$ −1755.82 −2.75824
$$75$$ 0 0
$$76$$ 1988.96 3.00197
$$77$$ − 707.245i − 1.04673i
$$78$$ 0 0
$$79$$ −24.4296 −0.0347917 −0.0173959 0.999849i $$-0.505538\pi$$
−0.0173959 + 0.999849i $$0.505538\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 1019.17i 1.37254i
$$83$$ 545.859i 0.721877i 0.932590 + 0.360938i $$0.117544\pi$$
−0.932590 + 0.360938i $$0.882456\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1741.61 −2.18375
$$87$$ 0 0
$$88$$ − 1665.17i − 2.01714i
$$89$$ −84.1332 −0.100203 −0.0501017 0.998744i $$-0.515955\pi$$
−0.0501017 + 0.998744i $$0.515955\pi$$
$$90$$ 0 0
$$91$$ −1595.65 −1.83813
$$92$$ − 2948.60i − 3.34144i
$$93$$ 0 0
$$94$$ 26.0241 0.0285551
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 827.613i − 0.866303i −0.901321 0.433152i $$-0.857401\pi$$
0.901321 0.433152i $$-0.142599\pi$$
$$98$$ − 1312.78i − 1.35317i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −823.576 −0.811375 −0.405688 0.914012i $$-0.632968\pi$$
−0.405688 + 0.914012i $$0.632968\pi$$
$$102$$ 0 0
$$103$$ 1171.19i 1.12040i 0.828359 + 0.560198i $$0.189275\pi$$
−0.828359 + 0.560198i $$0.810725\pi$$
$$104$$ −3756.87 −3.54223
$$105$$ 0 0
$$106$$ −1664.03 −1.52477
$$107$$ 1023.21i 0.924460i 0.886760 + 0.462230i $$0.152951\pi$$
−0.886760 + 0.462230i $$0.847049\pi$$
$$108$$ 0 0
$$109$$ 403.647 0.354700 0.177350 0.984148i $$-0.443247\pi$$
0.177350 + 0.984148i $$0.443247\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 3572.63i − 3.01413i
$$113$$ − 1082.20i − 0.900931i −0.892794 0.450465i $$-0.851258\pi$$
0.892794 0.450465i $$-0.148742\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 3917.16 3.13534
$$117$$ 0 0
$$118$$ 2239.46i 1.74711i
$$119$$ 1663.48 1.28144
$$120$$ 0 0
$$121$$ −490.935 −0.368847
$$122$$ 3092.22i 2.29473i
$$123$$ 0 0
$$124$$ 347.601 0.251738
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 774.132i 0.540890i 0.962735 + 0.270445i $$0.0871709\pi$$
−0.962735 + 0.270445i $$0.912829\pi$$
$$128$$ 342.664i 0.236621i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1214.04 0.809702 0.404851 0.914383i $$-0.367323\pi$$
0.404851 + 0.914383i $$0.367323\pi$$
$$132$$ 0 0
$$133$$ 2548.09i 1.66126i
$$134$$ 1018.69 0.656726
$$135$$ 0 0
$$136$$ 3916.58 2.46944
$$137$$ 2300.15i 1.43441i 0.696860 + 0.717207i $$0.254580\pi$$
−0.696860 + 0.717207i $$0.745420\pi$$
$$138$$ 0 0
$$139$$ 1355.93 0.827396 0.413698 0.910414i $$-0.364237\pi$$
0.413698 + 0.910414i $$0.364237\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 2215.25i − 1.30915i
$$143$$ − 1895.31i − 1.10835i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −4832.43 −2.73928
$$147$$ 0 0
$$148$$ − 6430.51i − 3.57152i
$$149$$ 259.845 0.142868 0.0714340 0.997445i $$-0.477242\pi$$
0.0714340 + 0.997445i $$0.477242\pi$$
$$150$$ 0 0
$$151$$ −508.304 −0.273941 −0.136971 0.990575i $$-0.543737\pi$$
−0.136971 + 0.990575i $$0.543737\pi$$
$$152$$ 5999.34i 3.20139i
$$153$$ 0 0
$$154$$ 3678.15 1.92463
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 23.3052i 0.0118468i 0.999982 + 0.00592342i $$0.00188549\pi$$
−0.999982 + 0.00592342i $$0.998115\pi$$
$$158$$ − 127.050i − 0.0639721i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3777.49 1.84912
$$162$$ 0 0
$$163$$ − 4032.10i − 1.93754i −0.247964 0.968769i $$-0.579762\pi$$
0.247964 0.968769i $$-0.420238\pi$$
$$164$$ −3732.62 −1.77725
$$165$$ 0 0
$$166$$ −2838.83 −1.32733
$$167$$ − 671.911i − 0.311341i −0.987809 0.155671i $$-0.950246\pi$$
0.987809 0.155671i $$-0.0497539\pi$$
$$168$$ 0 0
$$169$$ −2079.10 −0.946337
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 6378.49i − 2.82765i
$$173$$ − 1633.53i − 0.717889i −0.933359 0.358944i $$-0.883137\pi$$
0.933359 0.358944i $$-0.116863\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4243.57 1.81745
$$177$$ 0 0
$$178$$ − 437.549i − 0.184245i
$$179$$ 341.260 0.142497 0.0712485 0.997459i $$-0.477302\pi$$
0.0712485 + 0.997459i $$0.477302\pi$$
$$180$$ 0 0
$$181$$ −1695.92 −0.696447 −0.348223 0.937412i $$-0.613215\pi$$
−0.348223 + 0.937412i $$0.613215\pi$$
$$182$$ − 8298.45i − 3.37979i
$$183$$ 0 0
$$184$$ 8893.90 3.56341
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1975.88i 0.772678i
$$188$$ 95.3107i 0.0369747i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 726.451 0.275205 0.137603 0.990488i $$-0.456060\pi$$
0.137603 + 0.990488i $$0.456060\pi$$
$$192$$ 0 0
$$193$$ 4247.26i 1.58406i 0.610479 + 0.792032i $$0.290977\pi$$
−0.610479 + 0.792032i $$0.709023\pi$$
$$194$$ 4304.14 1.59288
$$195$$ 0 0
$$196$$ 4807.94 1.75216
$$197$$ 2678.52i 0.968713i 0.874871 + 0.484357i $$0.160946\pi$$
−0.874871 + 0.484357i $$0.839054\pi$$
$$198$$ 0 0
$$199$$ −1486.48 −0.529517 −0.264759 0.964315i $$-0.585292\pi$$
−0.264759 + 0.964315i $$0.585292\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 4283.15i − 1.49189i
$$203$$ 5018.32i 1.73506i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −6090.97 −2.06009
$$207$$ 0 0
$$208$$ − 9574.12i − 3.19157i
$$209$$ −3026.62 −1.00170
$$210$$ 0 0
$$211$$ −4827.41 −1.57504 −0.787519 0.616291i $$-0.788635\pi$$
−0.787519 + 0.616291i $$0.788635\pi$$
$$212$$ − 6094.37i − 1.97435i
$$213$$ 0 0
$$214$$ −5321.37 −1.69982
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 445.317i 0.139309i
$$218$$ 2099.23i 0.652193i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4457.88 1.35688
$$222$$ 0 0
$$223$$ 2774.48i 0.833153i 0.909101 + 0.416576i $$0.136770\pi$$
−0.909101 + 0.416576i $$0.863230\pi$$
$$224$$ 7364.91 2.19683
$$225$$ 0 0
$$226$$ 5628.19 1.65655
$$227$$ − 5101.34i − 1.49158i −0.666184 0.745788i $$-0.732073\pi$$
0.666184 0.745788i $$-0.267927\pi$$
$$228$$ 0 0
$$229$$ 4097.83 1.18250 0.591249 0.806489i $$-0.298635\pi$$
0.591249 + 0.806489i $$0.298635\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 11815.4i 3.34361i
$$233$$ − 357.613i − 0.100549i −0.998735 0.0502747i $$-0.983990\pi$$
0.998735 0.0502747i $$-0.0160097\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −8201.83 −2.26226
$$237$$ 0 0
$$238$$ 8651.22i 2.35620i
$$239$$ −351.682 −0.0951818 −0.0475909 0.998867i $$-0.515154\pi$$
−0.0475909 + 0.998867i $$0.515154\pi$$
$$240$$ 0 0
$$241$$ −6165.53 −1.64795 −0.823976 0.566624i $$-0.808249\pi$$
−0.823976 + 0.566624i $$0.808249\pi$$
$$242$$ − 2553.19i − 0.678204i
$$243$$ 0 0
$$244$$ −11325.0 −2.97134
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6828.50i 1.75906i
$$248$$ 1048.48i 0.268461i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −3245.53 −0.816160 −0.408080 0.912946i $$-0.633802\pi$$
−0.408080 + 0.912946i $$0.633802\pi$$
$$252$$ 0 0
$$253$$ 4486.90i 1.11498i
$$254$$ −4026.00 −0.994543
$$255$$ 0 0
$$256$$ −4969.25 −1.21320
$$257$$ 3552.19i 0.862178i 0.902309 + 0.431089i $$0.141871\pi$$
−0.902309 + 0.431089i $$0.858129\pi$$
$$258$$ 0 0
$$259$$ 8238.22 1.97644
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 6313.81i 1.48881i
$$263$$ − 4416.59i − 1.03551i −0.855530 0.517754i $$-0.826768\pi$$
0.855530 0.517754i $$-0.173232\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −13251.8 −3.05458
$$267$$ 0 0
$$268$$ 3730.85i 0.850366i
$$269$$ −3419.93 −0.775155 −0.387578 0.921837i $$-0.626688\pi$$
−0.387578 + 0.921837i $$0.626688\pi$$
$$270$$ 0 0
$$271$$ 716.407 0.160585 0.0802927 0.996771i $$-0.474415\pi$$
0.0802927 + 0.996771i $$0.474415\pi$$
$$272$$ 9981.12i 2.22498i
$$273$$ 0 0
$$274$$ −11962.3 −2.63748
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 657.529i − 0.142625i −0.997454 0.0713124i $$-0.977281\pi$$
0.997454 0.0713124i $$-0.0227187\pi$$
$$278$$ 7051.72i 1.52135i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1513.91 −0.321397 −0.160698 0.987004i $$-0.551375\pi$$
−0.160698 + 0.987004i $$0.551375\pi$$
$$282$$ 0 0
$$283$$ 3906.38i 0.820532i 0.911966 + 0.410266i $$0.134564\pi$$
−0.911966 + 0.410266i $$0.865436\pi$$
$$284$$ 8113.16 1.69517
$$285$$ 0 0
$$286$$ 9856.89 2.03794
$$287$$ − 4781.91i − 0.983509i
$$288$$ 0 0
$$289$$ 265.611 0.0540629
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 17698.3i − 3.54697i
$$293$$ 8048.76i 1.60483i 0.596770 + 0.802413i $$0.296451\pi$$
−0.596770 + 0.802413i $$0.703549\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 19396.4 3.80877
$$297$$ 0 0
$$298$$ 1351.37i 0.262693i
$$299$$ 10123.1 1.95797
$$300$$ 0 0
$$301$$ 8171.57 1.56479
$$302$$ − 2643.52i − 0.503700i
$$303$$ 0 0
$$304$$ −15288.9 −2.88447
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 101.564i − 0.0188814i −0.999955 0.00944068i $$-0.996995\pi$$
0.999955 0.00944068i $$-0.00300511\pi$$
$$308$$ 13470.9i 2.49213i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 7684.59 1.40113 0.700567 0.713586i $$-0.252930\pi$$
0.700567 + 0.713586i $$0.252930\pi$$
$$312$$ 0 0
$$313$$ − 1345.15i − 0.242915i −0.992597 0.121457i $$-0.961243\pi$$
0.992597 0.121457i $$-0.0387568\pi$$
$$314$$ −121.202 −0.0217830
$$315$$ 0 0
$$316$$ 465.310 0.0828346
$$317$$ 7622.33i 1.35051i 0.737583 + 0.675257i $$0.235967\pi$$
−0.737583 + 0.675257i $$0.764033\pi$$
$$318$$ 0 0
$$319$$ −5960.76 −1.04620
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 19645.5i 3.40000i
$$323$$ − 7118.78i − 1.22631i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 20969.6 3.56258
$$327$$ 0 0
$$328$$ − 11258.7i − 1.89531i
$$329$$ −122.104 −0.0204614
$$330$$ 0 0
$$331$$ −6585.09 −1.09350 −0.546751 0.837295i $$-0.684136\pi$$
−0.546751 + 0.837295i $$0.684136\pi$$
$$332$$ − 10397.0i − 1.71870i
$$333$$ 0 0
$$334$$ 3494.39 0.572468
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2946.94i 0.476351i 0.971222 + 0.238175i $$0.0765493\pi$$
−0.971222 + 0.238175i $$0.923451\pi$$
$$338$$ − 10812.7i − 1.74004i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −528.947 −0.0840002
$$342$$ 0 0
$$343$$ − 2210.14i − 0.347919i
$$344$$ 19239.5 3.01548
$$345$$ 0 0
$$346$$ 8495.44 1.31999
$$347$$ 8493.48i 1.31399i 0.753896 + 0.656994i $$0.228172\pi$$
−0.753896 + 0.656994i $$0.771828\pi$$
$$348$$ 0 0
$$349$$ −5646.54 −0.866053 −0.433027 0.901381i $$-0.642554\pi$$
−0.433027 + 0.901381i $$0.642554\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 8748.03i 1.32464i
$$353$$ 1221.93i 0.184240i 0.995748 + 0.0921202i $$0.0293644\pi$$
−0.995748 + 0.0921202i $$0.970636\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 1602.48 0.238571
$$357$$ 0 0
$$358$$ 1774.78i 0.262011i
$$359$$ −4151.44 −0.610319 −0.305160 0.952301i $$-0.598710\pi$$
−0.305160 + 0.952301i $$0.598710\pi$$
$$360$$ 0 0
$$361$$ 4045.41 0.589796
$$362$$ − 8819.93i − 1.28057i
$$363$$ 0 0
$$364$$ 30392.3 4.37635
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 7038.71i 1.00114i 0.865696 + 0.500569i $$0.166876\pi$$
−0.865696 + 0.500569i $$0.833124\pi$$
$$368$$ 22665.5i 3.21065i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 7807.58 1.09259
$$372$$ 0 0
$$373$$ − 7119.57i − 0.988303i −0.869376 0.494152i $$-0.835479\pi$$
0.869376 0.494152i $$-0.164521\pi$$
$$374$$ −10275.9 −1.42073
$$375$$ 0 0
$$376$$ −287.487 −0.0394309
$$377$$ 13448.4i 1.83720i
$$378$$ 0 0
$$379$$ −3372.29 −0.457053 −0.228526 0.973538i $$-0.573391\pi$$
−0.228526 + 0.973538i $$0.573391\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 3778.03i 0.506024i
$$383$$ − 3958.63i − 0.528138i −0.964504 0.264069i $$-0.914935\pi$$
0.964504 0.264069i $$-0.0850646\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −22088.6 −2.91264
$$387$$ 0 0
$$388$$ 15763.5i 2.06256i
$$389$$ 9654.01 1.25830 0.629148 0.777285i $$-0.283404\pi$$
0.629148 + 0.777285i $$0.283404\pi$$
$$390$$ 0 0
$$391$$ −10553.4 −1.36499
$$392$$ 14502.3i 1.86856i
$$393$$ 0 0
$$394$$ −13930.1 −1.78119
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 10928.3i − 1.38155i −0.723068 0.690776i $$-0.757269\pi$$
0.723068 0.690776i $$-0.242731\pi$$
$$398$$ − 7730.70i − 0.973631i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −4085.57 −0.508787 −0.254393 0.967101i $$-0.581876\pi$$
−0.254393 + 0.967101i $$0.581876\pi$$
$$402$$ 0 0
$$403$$ 1193.38i 0.147510i
$$404$$ 15686.6 1.93178
$$405$$ 0 0
$$406$$ −26098.7 −3.19028
$$407$$ 9785.34i 1.19175i
$$408$$ 0 0
$$409$$ −10156.3 −1.22786 −0.613930 0.789361i $$-0.710412\pi$$
−0.613930 + 0.789361i $$0.710412\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 22307.6i − 2.66752i
$$413$$ − 10507.5i − 1.25191i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 19736.9 2.32615
$$417$$ 0 0
$$418$$ − 15740.4i − 1.84184i
$$419$$ 15878.8 1.85139 0.925693 0.378275i $$-0.123483\pi$$
0.925693 + 0.378275i $$0.123483\pi$$
$$420$$ 0 0
$$421$$ −2279.85 −0.263926 −0.131963 0.991255i $$-0.542128\pi$$
−0.131963 + 0.991255i $$0.542128\pi$$
$$422$$ − 25105.8i − 2.89604i
$$423$$ 0 0
$$424$$ 18382.5 2.10551
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 14508.6i − 1.64431i
$$428$$ − 19489.0i − 2.20102i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 1947.38 0.217638 0.108819 0.994062i $$-0.465293\pi$$
0.108819 + 0.994062i $$0.465293\pi$$
$$432$$ 0 0
$$433$$ 12636.2i 1.40244i 0.712946 + 0.701219i $$0.247361\pi$$
−0.712946 + 0.701219i $$0.752639\pi$$
$$434$$ −2315.95 −0.256150
$$435$$ 0 0
$$436$$ −7688.25 −0.844496
$$437$$ − 16165.6i − 1.76957i
$$438$$ 0 0
$$439$$ 15849.8 1.72317 0.861585 0.507614i $$-0.169472\pi$$
0.861585 + 0.507614i $$0.169472\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 23184.0i 2.49491i
$$443$$ 17455.6i 1.87210i 0.351872 + 0.936048i $$0.385545\pi$$
−0.351872 + 0.936048i $$0.614455\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −14429.2 −1.53193
$$447$$ 0 0
$$448$$ 9721.41i 1.02521i
$$449$$ −16068.1 −1.68887 −0.844435 0.535658i $$-0.820064\pi$$
−0.844435 + 0.535658i $$0.820064\pi$$
$$450$$ 0 0
$$451$$ 5679.94 0.593033
$$452$$ 20612.7i 2.14500i
$$453$$ 0 0
$$454$$ 26530.4 2.74258
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 11891.7i − 1.21722i −0.793468 0.608612i $$-0.791727\pi$$
0.793468 0.608612i $$-0.208273\pi$$
$$458$$ 21311.4i 2.17428i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 2802.23 0.283108 0.141554 0.989931i $$-0.454790\pi$$
0.141554 + 0.989931i $$0.454790\pi$$
$$462$$ 0 0
$$463$$ − 12933.3i − 1.29819i −0.760707 0.649096i $$-0.775147\pi$$
0.760707 0.649096i $$-0.224853\pi$$
$$464$$ −30110.6 −3.01261
$$465$$ 0 0
$$466$$ 1859.83 0.184882
$$467$$ 5748.11i 0.569573i 0.958591 + 0.284787i $$0.0919228\pi$$
−0.958591 + 0.284787i $$0.908077\pi$$
$$468$$ 0 0
$$469$$ −4779.65 −0.470583
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 24739.3i − 2.41254i
$$473$$ 9706.17i 0.943531i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −31684.3 −3.05094
$$477$$ 0 0
$$478$$ − 1828.98i − 0.175012i
$$479$$ −11217.3 −1.07000 −0.535002 0.844851i $$-0.679689\pi$$
−0.535002 + 0.844851i $$0.679689\pi$$
$$480$$ 0 0
$$481$$ 22077.2 2.09279
$$482$$ − 32064.9i − 3.03012i
$$483$$ 0 0
$$484$$ 9350.83 0.878177
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 8905.12i 0.828603i 0.910140 + 0.414301i $$0.135974\pi$$
−0.910140 + 0.414301i $$0.864026\pi$$
$$488$$ − 34159.7i − 3.16872i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 6553.10 0.602316 0.301158 0.953574i $$-0.402627\pi$$
0.301158 + 0.953574i $$0.402627\pi$$
$$492$$ 0 0
$$493$$ − 14020.1i − 1.28079i
$$494$$ −35512.8 −3.23440
$$495$$ 0 0
$$496$$ −2671.96 −0.241884
$$497$$ 10393.9i 0.938087i
$$498$$ 0 0
$$499$$ 4610.09 0.413579 0.206789 0.978385i $$-0.433698\pi$$
0.206789 + 0.978385i $$0.433698\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 16878.9i − 1.50069i
$$503$$ − 13069.1i − 1.15850i −0.815151 0.579249i $$-0.803346\pi$$
0.815151 0.579249i $$-0.196654\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −23334.9 −2.05012
$$507$$ 0 0
$$508$$ − 14744.9i − 1.28779i
$$509$$ 15930.8 1.38727 0.693635 0.720327i $$-0.256008\pi$$
0.693635 + 0.720327i $$0.256008\pi$$
$$510$$ 0 0
$$511$$ 22673.6 1.96286
$$512$$ − 23102.1i − 1.99410i
$$513$$ 0 0
$$514$$ −18473.8 −1.58530
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 145.035i − 0.0123378i
$$518$$ 42844.3i 3.63411i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −3654.38 −0.307296 −0.153648 0.988126i $$-0.549102\pi$$
−0.153648 + 0.988126i $$0.549102\pi$$
$$522$$ 0 0
$$523$$ − 5138.66i − 0.429633i −0.976654 0.214816i $$-0.931085\pi$$
0.976654 0.214816i $$-0.0689153\pi$$
$$524$$ −23123.7 −1.92780
$$525$$ 0 0
$$526$$ 22969.2 1.90400
$$527$$ − 1244.11i − 0.102836i
$$528$$ 0 0
$$529$$ −11798.1 −0.969681
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 48533.4i − 3.95524i
$$533$$ − 12814.8i − 1.04141i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −11253.4 −0.906854
$$537$$ 0 0
$$538$$ − 17785.9i − 1.42529i
$$539$$ −7316.27 −0.584664
$$540$$ 0 0
$$541$$ 6932.06 0.550892 0.275446 0.961317i $$-0.411175\pi$$
0.275446 + 0.961317i $$0.411175\pi$$
$$542$$ 3725.80i 0.295271i
$$543$$ 0 0
$$544$$ −20575.9 −1.62166
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 3423.11i 0.267572i 0.991010 + 0.133786i $$0.0427135\pi$$
−0.991010 + 0.133786i $$0.957287\pi$$
$$548$$ − 43810.8i − 3.41516i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 21475.6 1.66042
$$552$$ 0 0
$$553$$ 596.115i 0.0458398i
$$554$$ 3419.59 0.262246
$$555$$ 0 0
$$556$$ −25826.3 −1.96992
$$557$$ − 24489.2i − 1.86291i −0.363856 0.931455i $$-0.618540\pi$$
0.363856 0.931455i $$-0.381460\pi$$
$$558$$ 0 0
$$559$$ 21898.6 1.65691
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 7873.37i − 0.590957i
$$563$$ 10053.1i 0.752552i 0.926508 + 0.376276i $$0.122796\pi$$
−0.926508 + 0.376276i $$0.877204\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −20315.8 −1.50872
$$567$$ 0 0
$$568$$ 24471.8i 1.80777i
$$569$$ −6670.45 −0.491459 −0.245729 0.969338i $$-0.579027\pi$$
−0.245729 + 0.969338i $$0.579027\pi$$
$$570$$ 0 0
$$571$$ 4633.55 0.339594 0.169797 0.985479i $$-0.445689\pi$$
0.169797 + 0.985479i $$0.445689\pi$$
$$572$$ 36099.9i 2.63884i
$$573$$ 0 0
$$574$$ 24869.1 1.80839
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 7045.15i − 0.508307i −0.967164 0.254154i $$-0.918203\pi$$
0.967164 0.254154i $$-0.0817969\pi$$
$$578$$ 1381.35i 0.0994062i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 13319.7 0.951108
$$582$$ 0 0
$$583$$ 9273.82i 0.658804i
$$584$$ 53383.7 3.78259
$$585$$ 0 0
$$586$$ −41859.0 −2.95082
$$587$$ − 8001.06i − 0.562588i −0.959622 0.281294i $$-0.909236\pi$$
0.959622 0.281294i $$-0.0907636\pi$$
$$588$$ 0 0
$$589$$ 1905.71 0.133316
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 49430.4i 3.43172i
$$593$$ − 6747.53i − 0.467265i −0.972325 0.233632i $$-0.924939\pi$$
0.972325 0.233632i $$-0.0750612\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −4949.26 −0.340150
$$597$$ 0 0
$$598$$ 52646.9i 3.60016i
$$599$$ 21547.3 1.46978 0.734890 0.678186i $$-0.237234\pi$$
0.734890 + 0.678186i $$0.237234\pi$$
$$600$$ 0 0
$$601$$ −12155.1 −0.824983 −0.412492 0.910961i $$-0.635341\pi$$
−0.412492 + 0.910961i $$0.635341\pi$$
$$602$$ 42497.6i 2.87720i
$$603$$ 0 0
$$604$$ 9681.64 0.652219
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 16348.9i − 1.09322i −0.837388 0.546608i $$-0.815919\pi$$
0.837388 0.546608i $$-0.184081\pi$$
$$608$$ − 31517.7i − 2.10232i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −327.220 −0.0216660
$$612$$ 0 0
$$613$$ − 29955.5i − 1.97372i −0.161581 0.986859i $$-0.551659\pi$$
0.161581 0.986859i $$-0.448341\pi$$
$$614$$ 528.202 0.0347174
$$615$$ 0 0
$$616$$ −40632.4 −2.65767
$$617$$ 2159.74i 0.140921i 0.997515 + 0.0704603i $$0.0224468\pi$$
−0.997515 + 0.0704603i $$0.977553\pi$$
$$618$$ 0 0
$$619$$ −22100.8 −1.43507 −0.717535 0.696523i $$-0.754730\pi$$
−0.717535 + 0.696523i $$0.754730\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 39965.0i 2.57629i
$$623$$ 2052.96i 0.132023i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 6995.68 0.446651
$$627$$ 0 0
$$628$$ − 443.893i − 0.0282058i
$$629$$ −23015.7 −1.45898
$$630$$ 0 0
$$631$$ 18360.1 1.15833 0.579164 0.815211i $$-0.303379\pi$$
0.579164 + 0.815211i $$0.303379\pi$$
$$632$$ 1403.52i 0.0883372i
$$633$$ 0 0
$$634$$ −39641.2 −2.48321
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 16506.6i 1.02671i
$$638$$ − 30999.9i − 1.92367i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 21064.5 1.29797 0.648985 0.760802i $$-0.275194\pi$$
0.648985 + 0.760802i $$0.275194\pi$$
$$642$$ 0 0
$$643$$ − 10539.1i − 0.646381i −0.946334 0.323190i $$-0.895245\pi$$
0.946334 0.323190i $$-0.104755\pi$$
$$644$$ −71949.8 −4.40251
$$645$$ 0 0
$$646$$ 37022.4 2.25484
$$647$$ 22553.4i 1.37043i 0.728343 + 0.685213i $$0.240291\pi$$
−0.728343 + 0.685213i $$0.759709\pi$$
$$648$$ 0 0
$$649$$ 12480.8 0.754873
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 76799.4i 4.61303i
$$653$$ − 22624.0i − 1.35582i −0.735147 0.677908i $$-0.762887\pi$$
0.735147 0.677908i $$-0.237113\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 28692.1 1.70768
$$657$$ 0 0
$$658$$ − 635.022i − 0.0376227i
$$659$$ 6376.60 0.376930 0.188465 0.982080i $$-0.439649\pi$$
0.188465 + 0.982080i $$0.439649\pi$$
$$660$$ 0 0
$$661$$ −22097.5 −1.30029 −0.650146 0.759809i $$-0.725292\pi$$
−0.650146 + 0.759809i $$0.725292\pi$$
$$662$$ − 34246.9i − 2.01064i
$$663$$ 0 0
$$664$$ 31360.5 1.83287
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 31837.2i − 1.84819i
$$668$$ 12797.9i 0.741264i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 17233.3 0.991479
$$672$$ 0 0
$$673$$ 24033.0i 1.37653i 0.725461 + 0.688264i $$0.241627\pi$$
−0.725461 + 0.688264i $$0.758373\pi$$
$$674$$ −15326.1 −0.875873
$$675$$ 0 0
$$676$$ 39600.6 2.25311
$$677$$ 179.638i 0.0101980i 0.999987 + 0.00509901i $$0.00162307\pi$$
−0.999987 + 0.00509901i $$0.998377\pi$$
$$678$$ 0 0
$$679$$ −20194.9 −1.14140
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 2750.88i − 0.154452i
$$683$$ 30434.2i 1.70502i 0.522709 + 0.852511i $$0.324921\pi$$
−0.522709 + 0.852511i $$0.675079\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 11494.2 0.639725
$$687$$ 0 0
$$688$$ 49030.5i 2.71696i
$$689$$ 20923.1 1.15691
$$690$$ 0 0
$$691$$ 9792.73 0.539122 0.269561 0.962983i $$-0.413121\pi$$
0.269561 + 0.962983i $$0.413121\pi$$
$$692$$ 31113.7i 1.70920i
$$693$$ 0 0
$$694$$ −44171.8 −2.41605
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 13359.6i 0.726010i
$$698$$ − 29365.8i − 1.59242i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −8130.47 −0.438065 −0.219032 0.975718i $$-0.570290\pi$$
−0.219032 + 0.975718i $$0.570290\pi$$
$$702$$ 0 0
$$703$$ − 35255.0i − 1.89142i
$$704$$ −11547.1 −0.618177
$$705$$ 0 0
$$706$$ −6354.86 −0.338765
$$707$$ 20096.4i 1.06903i
$$708$$ 0 0
$$709$$ 4859.95 0.257432 0.128716 0.991681i $$-0.458914\pi$$
0.128716 + 0.991681i $$0.458914\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 4833.59i 0.254419i
$$713$$ − 2825.17i − 0.148392i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −6499.96 −0.339267
$$717$$ 0 0
$$718$$ − 21590.3i − 1.12220i
$$719$$ 19463.0 1.00952 0.504762 0.863259i $$-0.331580\pi$$
0.504762 + 0.863259i $$0.331580\pi$$
$$720$$ 0 0
$$721$$ 28578.6 1.47618
$$722$$ 21038.8i 1.08447i
$$723$$ 0 0
$$724$$ 32302.2 1.65815
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 2432.66i − 0.124102i −0.998073 0.0620512i $$-0.980236\pi$$
0.998073 0.0620512i $$-0.0197642\pi$$
$$728$$ 91672.8i 4.66706i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −22829.5 −1.15510
$$732$$ 0 0
$$733$$ − 17967.6i − 0.905386i −0.891666 0.452693i $$-0.850463\pi$$
0.891666 0.452693i $$-0.149537\pi$$
$$734$$ −36606.0 −1.84081
$$735$$ 0 0
$$736$$ −46724.4 −2.34006
$$737$$ − 5677.26i − 0.283751i
$$738$$ 0 0
$$739$$ −23473.0 −1.16843 −0.584214 0.811599i $$-0.698597\pi$$
−0.584214 + 0.811599i $$0.698597\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 40604.6i 2.00895i
$$743$$ 33559.2i 1.65702i 0.559971 + 0.828512i $$0.310812\pi$$
−0.559971 + 0.828512i $$0.689188\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 37026.5 1.81721
$$747$$ 0 0
$$748$$ − 37634.6i − 1.83965i
$$749$$ 24967.6 1.21802
$$750$$ 0 0
$$751$$ −7782.75 −0.378158 −0.189079 0.981962i $$-0.560550\pi$$
−0.189079 + 0.981962i $$0.560550\pi$$
$$752$$ − 732.640i − 0.0355274i
$$753$$ 0 0
$$754$$ −69940.5 −3.37809
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 38154.3i 1.83189i 0.401300 + 0.915946i $$0.368558\pi$$
−0.401300 + 0.915946i $$0.631442\pi$$
$$758$$ − 17538.2i − 0.840390i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −19867.1 −0.946363 −0.473182 0.880965i $$-0.656895\pi$$
−0.473182 + 0.880965i $$0.656895\pi$$
$$762$$ 0 0
$$763$$ − 9849.52i − 0.467335i
$$764$$ −13836.7 −0.655228
$$765$$ 0 0
$$766$$ 20587.5 0.971094
$$767$$ − 28158.5i − 1.32561i
$$768$$ 0 0
$$769$$ 15710.8 0.736730 0.368365 0.929681i $$-0.379918\pi$$
0.368365 + 0.929681i $$0.379918\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 80897.5i − 3.77146i
$$773$$ − 25811.9i − 1.20102i −0.799617 0.600510i $$-0.794964\pi$$
0.799617 0.600510i $$-0.205036\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −47547.8 −2.19957
$$777$$ 0 0
$$778$$ 50207.3i 2.31365i
$$779$$ −20463.9 −0.941201
$$780$$ 0 0
$$781$$ −12345.8 −0.565645
$$782$$ − 54885.0i − 2.50982i
$$783$$ 0 0
$$784$$ −36958.0 −1.68358
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 29242.6i 1.32450i 0.749281 + 0.662252i $$0.230399\pi$$
−0.749281 + 0.662252i $$0.769601\pi$$
$$788$$ − 51017.7i − 2.30638i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −26407.2 −1.18702
$$792$$ 0 0
$$793$$ − 38880.8i − 1.74111i
$$794$$ 56834.6 2.54028
$$795$$ 0 0
$$796$$ 28313.0 1.26071
$$797$$ 32573.7i 1.44771i 0.689955 + 0.723853i $$0.257630\pi$$
−0.689955 + 0.723853i $$0.742370\pi$$
$$798$$ 0 0
$$799$$ 341.130 0.0151043
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 21247.7i − 0.935514i
$$803$$ 26931.6i 1.18356i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −6206.39 −0.271229
$$807$$ 0 0
$$808$$ 47315.8i 2.06010i
$$809$$ −34644.9 −1.50562 −0.752812 0.658236i $$-0.771303\pi$$
−0.752812 + 0.658236i $$0.771303\pi$$
$$810$$ 0 0
$$811$$ −29057.9 −1.25815 −0.629077 0.777343i $$-0.716567\pi$$
−0.629077 + 0.777343i $$0.716567\pi$$
$$812$$ − 95583.9i − 4.13096i
$$813$$ 0 0
$$814$$ −50890.3 −2.19128
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 34969.8i − 1.49748i
$$818$$ − 52819.3i − 2.25768i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −46709.5 −1.98560 −0.992798 0.119802i $$-0.961774\pi$$
−0.992798 + 0.119802i $$0.961774\pi$$
$$822$$ 0 0
$$823$$ − 3468.10i − 0.146890i −0.997299 0.0734450i $$-0.976601\pi$$
0.997299 0.0734450i $$-0.0233993\pi$$
$$824$$ 67286.8 2.84472
$$825$$ 0 0
$$826$$ 54645.9 2.30191
$$827$$ 42454.9i 1.78513i 0.450920 + 0.892564i $$0.351096\pi$$
−0.450920 + 0.892564i $$0.648904\pi$$
$$828$$ 0 0
$$829$$ 3933.47 0.164795 0.0823975 0.996600i $$-0.473742\pi$$
0.0823975 + 0.996600i $$0.473742\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 26051.9i 1.08556i
$$833$$ − 17208.3i − 0.715764i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 57647.9 2.38492
$$837$$ 0 0
$$838$$ 82580.5i 3.40417i
$$839$$ −32959.6 −1.35625 −0.678123 0.734948i $$-0.737206\pi$$
−0.678123 + 0.734948i $$0.737206\pi$$
$$840$$ 0 0
$$841$$ 17906.1 0.734188
$$842$$ − 11856.7i − 0.485285i
$$843$$ 0 0
$$844$$ 91947.6 3.74996
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 11979.5i 0.485974i
$$848$$ 46846.5i 1.89707i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −52264.8 −2.10530
$$852$$ 0 0
$$853$$ 38845.8i 1.55927i 0.626235 + 0.779634i $$0.284595\pi$$
−0.626235 + 0.779634i $$0.715405\pi$$
$$854$$ 75454.3 3.02341
$$855$$ 0 0
$$856$$ 58785.0 2.34723
$$857$$ − 29305.3i − 1.16809i −0.811723 0.584043i $$-0.801470\pi$$
0.811723 0.584043i $$-0.198530\pi$$
$$858$$ 0 0
$$859$$ 909.659 0.0361318 0.0180659 0.999837i $$-0.494249\pi$$
0.0180659 + 0.999837i $$0.494249\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 10127.7i 0.400173i
$$863$$ − 47998.0i − 1.89324i −0.322346 0.946622i $$-0.604471\pi$$
0.322346 0.946622i $$-0.395529\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −65716.6 −2.57868
$$867$$ 0 0
$$868$$ − 8481.94i − 0.331677i
$$869$$ −708.065 −0.0276403
$$870$$ 0 0
$$871$$ −12808.7 −0.498286
$$872$$ − 23190.2i − 0.900594i
$$873$$ 0 0
$$874$$ 84071.7 3.25374
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 3258.01i − 0.125445i −0.998031 0.0627225i $$-0.980022\pi$$
0.998031 0.0627225i $$-0.0199783\pi$$
$$878$$ 82429.8i 3.16842i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 33380.2 1.27651 0.638256 0.769824i $$-0.279656\pi$$
0.638256 + 0.769824i $$0.279656\pi$$
$$882$$ 0 0
$$883$$ 33714.6i 1.28492i 0.766319 + 0.642460i $$0.222086\pi$$
−0.766319 + 0.642460i $$0.777914\pi$$
$$884$$ −84909.2 −3.23055
$$885$$ 0 0
$$886$$ −90780.6 −3.44225
$$887$$ − 6218.80i − 0.235408i −0.993049 0.117704i $$-0.962447\pi$$
0.993049 0.117704i $$-0.0375534\pi$$
$$888$$ 0 0
$$889$$ 18889.8 0.712649
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 52845.5i − 1.98363i
$$893$$ 522.537i 0.0195812i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 8361.46 0.311760
$$897$$ 0 0
$$898$$ − 83565.1i − 3.10535i
$$899$$ 3753.19 0.139239
$$900$$ 0 0
$$901$$ −21812.6 −0.806529
$$902$$ 29539.5i 1.09042i
$$903$$ 0 0
$$904$$ −62174.4 −2.28749
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 22878.6i 0.837566i 0.908086 + 0.418783i $$0.137543\pi$$
−0.908086 + 0.418783i $$0.862457\pi$$
$$908$$ 97165.0i 3.55125i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 30144.8 1.09631 0.548157 0.836376i $$-0.315330\pi$$
0.548157 + 0.836376i $$0.315330\pi$$
$$912$$ 0 0
$$913$$ 15821.1i 0.573496i
$$914$$ 61844.9 2.23813
$$915$$ 0 0
$$916$$ −78051.2 −2.81538
$$917$$ − 29624.1i − 1.06682i
$$918$$ 0 0
$$919$$ −3803.52 −0.136525 −0.0682625 0.997667i $$-0.521746\pi$$
−0.0682625 + 0.997667i $$0.521746\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 14573.5i 0.520555i
$$923$$ 27854.0i 0.993312i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 67262.0 2.38700
$$927$$ 0 0
$$928$$ − 62072.5i − 2.19572i
$$929$$ 125.985 0.00444934 0.00222467 0.999998i $$-0.499292\pi$$
0.00222467 + 0.999998i $$0.499292\pi$$
$$930$$ 0 0
$$931$$ 26359.3 0.927918
$$932$$ 6811.45i 0.239395i
$$933$$ 0 0
$$934$$ −29894.0 −1.04728
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 28107.9i − 0.979984i −0.871727 0.489992i $$-0.837000\pi$$
0.871727 0.489992i $$-0.163000\pi$$
$$938$$ − 24857.4i − 0.865268i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 49194.9 1.70426 0.852130 0.523330i $$-0.175311\pi$$
0.852130 + 0.523330i $$0.175311\pi$$
$$942$$ 0 0
$$943$$ 30337.3i 1.04763i
$$944$$ 63046.3 2.17371
$$945$$ 0 0
$$946$$ −50478.6 −1.73488
$$947$$ − 14498.0i − 0.497490i −0.968569 0.248745i $$-0.919982\pi$$
0.968569 0.248745i $$-0.0800181\pi$$
$$948$$ 0 0
$$949$$ 60761.7 2.07841
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 95569.8i − 3.25361i
$$953$$ − 3201.79i − 0.108831i −0.998518 0.0544155i $$-0.982670\pi$$
0.998518 0.0544155i $$-0.0173296\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 6698.48 0.226616
$$957$$ 0 0
$$958$$ − 58337.6i − 1.96743i
$$959$$ 56126.7 1.88991
$$960$$ 0 0
$$961$$ −29457.9 −0.988820
$$962$$ 114816.i 3.84805i
$$963$$ 0 0
$$964$$ 117435. 3.92356
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 57781.9i − 1.92155i −0.277326 0.960776i $$-0.589448\pi$$
0.277326 0.960776i $$-0.410552\pi$$
$$968$$ 28205.1i 0.936513i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −2611.60 −0.0863133 −0.0431566 0.999068i $$-0.513741\pi$$
−0.0431566 + 0.999068i $$0.513741\pi$$
$$972$$ 0 0
$$973$$ − 33086.4i − 1.09013i
$$974$$ −46312.6 −1.52356
$$975$$ 0 0
$$976$$ 87053.4 2.85503
$$977$$ − 33598.3i − 1.10021i −0.835096 0.550104i $$-0.814588\pi$$
0.835096 0.550104i $$-0.185412\pi$$
$$978$$ 0 0
$$979$$ −2438.51 −0.0796067
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 34080.5i 1.10749i
$$983$$ 39484.8i 1.28115i 0.767895 + 0.640575i $$0.221304\pi$$
−0.767895 + 0.640575i $$0.778696\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 72913.7 2.35501
$$987$$ 0 0
$$988$$ − 130062.i − 4.18809i
$$989$$ −51841.9 −1.66681
$$990$$ 0 0
$$991$$ 39918.6 1.27957 0.639786 0.768553i $$-0.279023\pi$$
0.639786 + 0.768553i $$0.279023\pi$$
$$992$$ − 5508.20i − 0.176296i
$$993$$ 0 0
$$994$$ −54055.2 −1.72487
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 25670.3i 0.815432i 0.913109 + 0.407716i $$0.133675\pi$$
−0.913109 + 0.407716i $$0.866325\pi$$
$$998$$ 23975.6i 0.760454i
$$999$$ 0 0
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 675.4.b.l.649.6 6
3.2 odd 2 675.4.b.k.649.1 6
5.2 odd 4 135.4.a.f.1.1 3
5.3 odd 4 675.4.a.r.1.3 3
5.4 even 2 inner 675.4.b.l.649.1 6
15.2 even 4 135.4.a.g.1.3 yes 3
15.8 even 4 675.4.a.q.1.1 3
15.14 odd 2 675.4.b.k.649.6 6
20.7 even 4 2160.4.a.bm.1.1 3
45.2 even 12 405.4.e.r.271.1 6
45.7 odd 12 405.4.e.t.271.3 6
45.22 odd 12 405.4.e.t.136.3 6
45.32 even 12 405.4.e.r.136.1 6
60.47 odd 4 2160.4.a.be.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.1 3 5.2 odd 4
135.4.a.g.1.3 yes 3 15.2 even 4
405.4.e.r.136.1 6 45.32 even 12
405.4.e.r.271.1 6 45.2 even 12
405.4.e.t.136.3 6 45.22 odd 12
405.4.e.t.271.3 6 45.7 odd 12
675.4.a.q.1.1 3 15.8 even 4
675.4.a.r.1.3 3 5.3 odd 4
675.4.b.k.649.1 6 3.2 odd 2
675.4.b.k.649.6 6 15.14 odd 2
675.4.b.l.649.1 6 5.4 even 2 inner
675.4.b.l.649.6 6 1.1 even 1 trivial
2160.4.a.be.1.1 3 60.47 odd 4
2160.4.a.bm.1.1 3 20.7 even 4