# Properties

 Label 405.4.a.a.1.1 Level $405$ Weight $4$ Character 405.1 Self dual yes Analytic conductor $23.896$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(1,405)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("405.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 405.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$23.8957735523$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 405.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.00000 q^{2} +17.0000 q^{4} +5.00000 q^{5} +9.00000 q^{7} -45.0000 q^{8} +O(q^{10})$$ $$q-5.00000 q^{2} +17.0000 q^{4} +5.00000 q^{5} +9.00000 q^{7} -45.0000 q^{8} -25.0000 q^{10} -8.00000 q^{11} +43.0000 q^{13} -45.0000 q^{14} +89.0000 q^{16} -122.000 q^{17} -59.0000 q^{19} +85.0000 q^{20} +40.0000 q^{22} -213.000 q^{23} +25.0000 q^{25} -215.000 q^{26} +153.000 q^{28} +224.000 q^{29} -36.0000 q^{31} -85.0000 q^{32} +610.000 q^{34} +45.0000 q^{35} +206.000 q^{37} +295.000 q^{38} -225.000 q^{40} +413.000 q^{41} -392.000 q^{43} -136.000 q^{44} +1065.00 q^{46} -311.000 q^{47} -262.000 q^{49} -125.000 q^{50} +731.000 q^{52} -377.000 q^{53} -40.0000 q^{55} -405.000 q^{56} -1120.00 q^{58} +337.000 q^{59} +40.0000 q^{61} +180.000 q^{62} -287.000 q^{64} +215.000 q^{65} +348.000 q^{67} -2074.00 q^{68} -225.000 q^{70} +62.0000 q^{71} -1214.00 q^{73} -1030.00 q^{74} -1003.00 q^{76} -72.0000 q^{77} -294.000 q^{79} +445.000 q^{80} -2065.00 q^{82} +534.000 q^{83} -610.000 q^{85} +1960.00 q^{86} +360.000 q^{88} -810.000 q^{89} +387.000 q^{91} -3621.00 q^{92} +1555.00 q^{94} -295.000 q^{95} -928.000 q^{97} +1310.00 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.00000 −1.76777 −0.883883 0.467707i $$-0.845080\pi$$
−0.883883 + 0.467707i $$0.845080\pi$$
$$3$$ 0 0
$$4$$ 17.0000 2.12500
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ 9.00000 0.485954 0.242977 0.970032i $$-0.421876\pi$$
0.242977 + 0.970032i $$0.421876\pi$$
$$8$$ −45.0000 −1.98874
$$9$$ 0 0
$$10$$ −25.0000 −0.790569
$$11$$ −8.00000 −0.219281 −0.109640 0.993971i $$-0.534970\pi$$
−0.109640 + 0.993971i $$0.534970\pi$$
$$12$$ 0 0
$$13$$ 43.0000 0.917389 0.458694 0.888594i $$-0.348317\pi$$
0.458694 + 0.888594i $$0.348317\pi$$
$$14$$ −45.0000 −0.859054
$$15$$ 0 0
$$16$$ 89.0000 1.39062
$$17$$ −122.000 −1.74055 −0.870275 0.492566i $$-0.836059\pi$$
−0.870275 + 0.492566i $$0.836059\pi$$
$$18$$ 0 0
$$19$$ −59.0000 −0.712396 −0.356198 0.934410i $$-0.615927\pi$$
−0.356198 + 0.934410i $$0.615927\pi$$
$$20$$ 85.0000 0.950329
$$21$$ 0 0
$$22$$ 40.0000 0.387638
$$23$$ −213.000 −1.93102 −0.965512 0.260357i $$-0.916160\pi$$
−0.965512 + 0.260357i $$0.916160\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ −215.000 −1.62173
$$27$$ 0 0
$$28$$ 153.000 1.03265
$$29$$ 224.000 1.43434 0.717168 0.696900i $$-0.245438\pi$$
0.717168 + 0.696900i $$0.245438\pi$$
$$30$$ 0 0
$$31$$ −36.0000 −0.208574 −0.104287 0.994547i $$-0.533256\pi$$
−0.104287 + 0.994547i $$0.533256\pi$$
$$32$$ −85.0000 −0.469563
$$33$$ 0 0
$$34$$ 610.000 3.07689
$$35$$ 45.0000 0.217325
$$36$$ 0 0
$$37$$ 206.000 0.915302 0.457651 0.889132i $$-0.348691\pi$$
0.457651 + 0.889132i $$0.348691\pi$$
$$38$$ 295.000 1.25935
$$39$$ 0 0
$$40$$ −225.000 −0.889391
$$41$$ 413.000 1.57316 0.786582 0.617485i $$-0.211848\pi$$
0.786582 + 0.617485i $$0.211848\pi$$
$$42$$ 0 0
$$43$$ −392.000 −1.39022 −0.695110 0.718904i $$-0.744644\pi$$
−0.695110 + 0.718904i $$0.744644\pi$$
$$44$$ −136.000 −0.465972
$$45$$ 0 0
$$46$$ 1065.00 3.41360
$$47$$ −311.000 −0.965192 −0.482596 0.875843i $$-0.660306\pi$$
−0.482596 + 0.875843i $$0.660306\pi$$
$$48$$ 0 0
$$49$$ −262.000 −0.763848
$$50$$ −125.000 −0.353553
$$51$$ 0 0
$$52$$ 731.000 1.94945
$$53$$ −377.000 −0.977074 −0.488537 0.872543i $$-0.662469\pi$$
−0.488537 + 0.872543i $$0.662469\pi$$
$$54$$ 0 0
$$55$$ −40.0000 −0.0980654
$$56$$ −405.000 −0.966436
$$57$$ 0 0
$$58$$ −1120.00 −2.53557
$$59$$ 337.000 0.743621 0.371811 0.928309i $$-0.378737\pi$$
0.371811 + 0.928309i $$0.378737\pi$$
$$60$$ 0 0
$$61$$ 40.0000 0.0839586 0.0419793 0.999118i $$-0.486634\pi$$
0.0419793 + 0.999118i $$0.486634\pi$$
$$62$$ 180.000 0.368710
$$63$$ 0 0
$$64$$ −287.000 −0.560547
$$65$$ 215.000 0.410269
$$66$$ 0 0
$$67$$ 348.000 0.634552 0.317276 0.948333i $$-0.397232\pi$$
0.317276 + 0.948333i $$0.397232\pi$$
$$68$$ −2074.00 −3.69867
$$69$$ 0 0
$$70$$ −225.000 −0.384181
$$71$$ 62.0000 0.103634 0.0518172 0.998657i $$-0.483499\pi$$
0.0518172 + 0.998657i $$0.483499\pi$$
$$72$$ 0 0
$$73$$ −1214.00 −1.94641 −0.973205 0.229939i $$-0.926147\pi$$
−0.973205 + 0.229939i $$0.926147\pi$$
$$74$$ −1030.00 −1.61804
$$75$$ 0 0
$$76$$ −1003.00 −1.51384
$$77$$ −72.0000 −0.106561
$$78$$ 0 0
$$79$$ −294.000 −0.418704 −0.209352 0.977840i $$-0.567135\pi$$
−0.209352 + 0.977840i $$0.567135\pi$$
$$80$$ 445.000 0.621906
$$81$$ 0 0
$$82$$ −2065.00 −2.78099
$$83$$ 534.000 0.706194 0.353097 0.935587i $$-0.385129\pi$$
0.353097 + 0.935587i $$0.385129\pi$$
$$84$$ 0 0
$$85$$ −610.000 −0.778398
$$86$$ 1960.00 2.45758
$$87$$ 0 0
$$88$$ 360.000 0.436092
$$89$$ −810.000 −0.964717 −0.482359 0.875974i $$-0.660220\pi$$
−0.482359 + 0.875974i $$0.660220\pi$$
$$90$$ 0 0
$$91$$ 387.000 0.445809
$$92$$ −3621.00 −4.10343
$$93$$ 0 0
$$94$$ 1555.00 1.70623
$$95$$ −295.000 −0.318593
$$96$$ 0 0
$$97$$ −928.000 −0.971383 −0.485691 0.874130i $$-0.661432\pi$$
−0.485691 + 0.874130i $$0.661432\pi$$
$$98$$ 1310.00 1.35031
$$99$$ 0 0
$$100$$ 425.000 0.425000
$$101$$ −996.000 −0.981245 −0.490622 0.871372i $$-0.663231\pi$$
−0.490622 + 0.871372i $$0.663231\pi$$
$$102$$ 0 0
$$103$$ −433.000 −0.414221 −0.207110 0.978318i $$-0.566406\pi$$
−0.207110 + 0.978318i $$0.566406\pi$$
$$104$$ −1935.00 −1.82445
$$105$$ 0 0
$$106$$ 1885.00 1.72724
$$107$$ −1686.00 −1.52329 −0.761644 0.647996i $$-0.775607\pi$$
−0.761644 + 0.647996i $$0.775607\pi$$
$$108$$ 0 0
$$109$$ 656.000 0.576453 0.288227 0.957562i $$-0.406934\pi$$
0.288227 + 0.957562i $$0.406934\pi$$
$$110$$ 200.000 0.173357
$$111$$ 0 0
$$112$$ 801.000 0.675780
$$113$$ 1018.00 0.847481 0.423741 0.905784i $$-0.360717\pi$$
0.423741 + 0.905784i $$0.360717\pi$$
$$114$$ 0 0
$$115$$ −1065.00 −0.863581
$$116$$ 3808.00 3.04796
$$117$$ 0 0
$$118$$ −1685.00 −1.31455
$$119$$ −1098.00 −0.845828
$$120$$ 0 0
$$121$$ −1267.00 −0.951916
$$122$$ −200.000 −0.148419
$$123$$ 0 0
$$124$$ −612.000 −0.443220
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −1361.00 −0.950939 −0.475469 0.879732i $$-0.657722\pi$$
−0.475469 + 0.879732i $$0.657722\pi$$
$$128$$ 2115.00 1.46048
$$129$$ 0 0
$$130$$ −1075.00 −0.725260
$$131$$ 1911.00 1.27454 0.637270 0.770640i $$-0.280063\pi$$
0.637270 + 0.770640i $$0.280063\pi$$
$$132$$ 0 0
$$133$$ −531.000 −0.346192
$$134$$ −1740.00 −1.12174
$$135$$ 0 0
$$136$$ 5490.00 3.46150
$$137$$ 654.000 0.407847 0.203923 0.978987i $$-0.434631\pi$$
0.203923 + 0.978987i $$0.434631\pi$$
$$138$$ 0 0
$$139$$ −733.000 −0.447282 −0.223641 0.974672i $$-0.571794\pi$$
−0.223641 + 0.974672i $$0.571794\pi$$
$$140$$ 765.000 0.461816
$$141$$ 0 0
$$142$$ −310.000 −0.183202
$$143$$ −344.000 −0.201166
$$144$$ 0 0
$$145$$ 1120.00 0.641455
$$146$$ 6070.00 3.44080
$$147$$ 0 0
$$148$$ 3502.00 1.94502
$$149$$ −1126.00 −0.619097 −0.309549 0.950884i $$-0.600178\pi$$
−0.309549 + 0.950884i $$0.600178\pi$$
$$150$$ 0 0
$$151$$ −2546.00 −1.37212 −0.686061 0.727544i $$-0.740662\pi$$
−0.686061 + 0.727544i $$0.740662\pi$$
$$152$$ 2655.00 1.41677
$$153$$ 0 0
$$154$$ 360.000 0.188374
$$155$$ −180.000 −0.0932771
$$156$$ 0 0
$$157$$ 1223.00 0.621694 0.310847 0.950460i $$-0.399387\pi$$
0.310847 + 0.950460i $$0.399387\pi$$
$$158$$ 1470.00 0.740170
$$159$$ 0 0
$$160$$ −425.000 −0.209995
$$161$$ −1917.00 −0.938390
$$162$$ 0 0
$$163$$ 3176.00 1.52616 0.763078 0.646306i $$-0.223687\pi$$
0.763078 + 0.646306i $$0.223687\pi$$
$$164$$ 7021.00 3.34298
$$165$$ 0 0
$$166$$ −2670.00 −1.24839
$$167$$ 132.000 0.0611645 0.0305822 0.999532i $$-0.490264\pi$$
0.0305822 + 0.999532i $$0.490264\pi$$
$$168$$ 0 0
$$169$$ −348.000 −0.158398
$$170$$ 3050.00 1.37603
$$171$$ 0 0
$$172$$ −6664.00 −2.95422
$$173$$ −993.000 −0.436395 −0.218198 0.975905i $$-0.570018\pi$$
−0.218198 + 0.975905i $$0.570018\pi$$
$$174$$ 0 0
$$175$$ 225.000 0.0971909
$$176$$ −712.000 −0.304938
$$177$$ 0 0
$$178$$ 4050.00 1.70540
$$179$$ −3101.00 −1.29486 −0.647429 0.762126i $$-0.724156\pi$$
−0.647429 + 0.762126i $$0.724156\pi$$
$$180$$ 0 0
$$181$$ 2846.00 1.16874 0.584369 0.811488i $$-0.301342\pi$$
0.584369 + 0.811488i $$0.301342\pi$$
$$182$$ −1935.00 −0.788086
$$183$$ 0 0
$$184$$ 9585.00 3.84030
$$185$$ 1030.00 0.409336
$$186$$ 0 0
$$187$$ 976.000 0.381669
$$188$$ −5287.00 −2.05103
$$189$$ 0 0
$$190$$ 1475.00 0.563199
$$191$$ 3080.00 1.16681 0.583406 0.812181i $$-0.301720\pi$$
0.583406 + 0.812181i $$0.301720\pi$$
$$192$$ 0 0
$$193$$ 2588.00 0.965224 0.482612 0.875834i $$-0.339688\pi$$
0.482612 + 0.875834i $$0.339688\pi$$
$$194$$ 4640.00 1.71718
$$195$$ 0 0
$$196$$ −4454.00 −1.62318
$$197$$ −1335.00 −0.482816 −0.241408 0.970424i $$-0.577609\pi$$
−0.241408 + 0.970424i $$0.577609\pi$$
$$198$$ 0 0
$$199$$ −5204.00 −1.85378 −0.926889 0.375336i $$-0.877527\pi$$
−0.926889 + 0.375336i $$0.877527\pi$$
$$200$$ −1125.00 −0.397748
$$201$$ 0 0
$$202$$ 4980.00 1.73461
$$203$$ 2016.00 0.697022
$$204$$ 0 0
$$205$$ 2065.00 0.703541
$$206$$ 2165.00 0.732246
$$207$$ 0 0
$$208$$ 3827.00 1.27574
$$209$$ 472.000 0.156215
$$210$$ 0 0
$$211$$ 1637.00 0.534103 0.267051 0.963682i $$-0.413951\pi$$
0.267051 + 0.963682i $$0.413951\pi$$
$$212$$ −6409.00 −2.07628
$$213$$ 0 0
$$214$$ 8430.00 2.69282
$$215$$ −1960.00 −0.621725
$$216$$ 0 0
$$217$$ −324.000 −0.101357
$$218$$ −3280.00 −1.01904
$$219$$ 0 0
$$220$$ −680.000 −0.208389
$$221$$ −5246.00 −1.59676
$$222$$ 0 0
$$223$$ −4480.00 −1.34530 −0.672652 0.739959i $$-0.734845\pi$$
−0.672652 + 0.739959i $$0.734845\pi$$
$$224$$ −765.000 −0.228186
$$225$$ 0 0
$$226$$ −5090.00 −1.49815
$$227$$ −3736.00 −1.09237 −0.546183 0.837666i $$-0.683920\pi$$
−0.546183 + 0.837666i $$0.683920\pi$$
$$228$$ 0 0
$$229$$ −1380.00 −0.398223 −0.199111 0.979977i $$-0.563806\pi$$
−0.199111 + 0.979977i $$0.563806\pi$$
$$230$$ 5325.00 1.52661
$$231$$ 0 0
$$232$$ −10080.0 −2.85252
$$233$$ −2904.00 −0.816512 −0.408256 0.912867i $$-0.633863\pi$$
−0.408256 + 0.912867i $$0.633863\pi$$
$$234$$ 0 0
$$235$$ −1555.00 −0.431647
$$236$$ 5729.00 1.58020
$$237$$ 0 0
$$238$$ 5490.00 1.49523
$$239$$ −5966.00 −1.61468 −0.807340 0.590087i $$-0.799094\pi$$
−0.807340 + 0.590087i $$0.799094\pi$$
$$240$$ 0 0
$$241$$ −3218.00 −0.860123 −0.430061 0.902800i $$-0.641508\pi$$
−0.430061 + 0.902800i $$0.641508\pi$$
$$242$$ 6335.00 1.68277
$$243$$ 0 0
$$244$$ 680.000 0.178412
$$245$$ −1310.00 −0.341603
$$246$$ 0 0
$$247$$ −2537.00 −0.653544
$$248$$ 1620.00 0.414799
$$249$$ 0 0
$$250$$ −625.000 −0.158114
$$251$$ 6123.00 1.53976 0.769881 0.638187i $$-0.220315\pi$$
0.769881 + 0.638187i $$0.220315\pi$$
$$252$$ 0 0
$$253$$ 1704.00 0.423437
$$254$$ 6805.00 1.68104
$$255$$ 0 0
$$256$$ −8279.00 −2.02124
$$257$$ −1398.00 −0.339318 −0.169659 0.985503i $$-0.554267\pi$$
−0.169659 + 0.985503i $$0.554267\pi$$
$$258$$ 0 0
$$259$$ 1854.00 0.444795
$$260$$ 3655.00 0.871821
$$261$$ 0 0
$$262$$ −9555.00 −2.25309
$$263$$ −3211.00 −0.752847 −0.376423 0.926448i $$-0.622846\pi$$
−0.376423 + 0.926448i $$0.622846\pi$$
$$264$$ 0 0
$$265$$ −1885.00 −0.436961
$$266$$ 2655.00 0.611987
$$267$$ 0 0
$$268$$ 5916.00 1.34842
$$269$$ 4018.00 0.910713 0.455356 0.890309i $$-0.349512\pi$$
0.455356 + 0.890309i $$0.349512\pi$$
$$270$$ 0 0
$$271$$ 2314.00 0.518692 0.259346 0.965784i $$-0.416493\pi$$
0.259346 + 0.965784i $$0.416493\pi$$
$$272$$ −10858.0 −2.42045
$$273$$ 0 0
$$274$$ −3270.00 −0.720978
$$275$$ −200.000 −0.0438562
$$276$$ 0 0
$$277$$ −4347.00 −0.942909 −0.471455 0.881890i $$-0.656271\pi$$
−0.471455 + 0.881890i $$0.656271\pi$$
$$278$$ 3665.00 0.790691
$$279$$ 0 0
$$280$$ −2025.00 −0.432203
$$281$$ 1551.00 0.329270 0.164635 0.986355i $$-0.447355\pi$$
0.164635 + 0.986355i $$0.447355\pi$$
$$282$$ 0 0
$$283$$ 4380.00 0.920014 0.460007 0.887915i $$-0.347847\pi$$
0.460007 + 0.887915i $$0.347847\pi$$
$$284$$ 1054.00 0.220223
$$285$$ 0 0
$$286$$ 1720.00 0.355614
$$287$$ 3717.00 0.764486
$$288$$ 0 0
$$289$$ 9971.00 2.02951
$$290$$ −5600.00 −1.13394
$$291$$ 0 0
$$292$$ −20638.0 −4.13612
$$293$$ 5049.00 1.00671 0.503354 0.864080i $$-0.332099\pi$$
0.503354 + 0.864080i $$0.332099\pi$$
$$294$$ 0 0
$$295$$ 1685.00 0.332558
$$296$$ −9270.00 −1.82030
$$297$$ 0 0
$$298$$ 5630.00 1.09442
$$299$$ −9159.00 −1.77150
$$300$$ 0 0
$$301$$ −3528.00 −0.675583
$$302$$ 12730.0 2.42559
$$303$$ 0 0
$$304$$ −5251.00 −0.990676
$$305$$ 200.000 0.0375474
$$306$$ 0 0
$$307$$ 5428.00 1.00910 0.504548 0.863384i $$-0.331659\pi$$
0.504548 + 0.863384i $$0.331659\pi$$
$$308$$ −1224.00 −0.226441
$$309$$ 0 0
$$310$$ 900.000 0.164892
$$311$$ −18.0000 −0.00328195 −0.00164097 0.999999i $$-0.500522\pi$$
−0.00164097 + 0.999999i $$0.500522\pi$$
$$312$$ 0 0
$$313$$ −2116.00 −0.382119 −0.191060 0.981578i $$-0.561192\pi$$
−0.191060 + 0.981578i $$0.561192\pi$$
$$314$$ −6115.00 −1.09901
$$315$$ 0 0
$$316$$ −4998.00 −0.889745
$$317$$ 4415.00 0.782243 0.391122 0.920339i $$-0.372087\pi$$
0.391122 + 0.920339i $$0.372087\pi$$
$$318$$ 0 0
$$319$$ −1792.00 −0.314523
$$320$$ −1435.00 −0.250684
$$321$$ 0 0
$$322$$ 9585.00 1.65885
$$323$$ 7198.00 1.23996
$$324$$ 0 0
$$325$$ 1075.00 0.183478
$$326$$ −15880.0 −2.69789
$$327$$ 0 0
$$328$$ −18585.0 −3.12861
$$329$$ −2799.00 −0.469039
$$330$$ 0 0
$$331$$ 3480.00 0.577879 0.288940 0.957347i $$-0.406697\pi$$
0.288940 + 0.957347i $$0.406697\pi$$
$$332$$ 9078.00 1.50066
$$333$$ 0 0
$$334$$ −660.000 −0.108125
$$335$$ 1740.00 0.283780
$$336$$ 0 0
$$337$$ 6322.00 1.02190 0.510951 0.859610i $$-0.329293\pi$$
0.510951 + 0.859610i $$0.329293\pi$$
$$338$$ 1740.00 0.280010
$$339$$ 0 0
$$340$$ −10370.0 −1.65409
$$341$$ 288.000 0.0457363
$$342$$ 0 0
$$343$$ −5445.00 −0.857150
$$344$$ 17640.0 2.76478
$$345$$ 0 0
$$346$$ 4965.00 0.771445
$$347$$ −10034.0 −1.55232 −0.776158 0.630539i $$-0.782834\pi$$
−0.776158 + 0.630539i $$0.782834\pi$$
$$348$$ 0 0
$$349$$ −2510.00 −0.384978 −0.192489 0.981299i $$-0.561656\pi$$
−0.192489 + 0.981299i $$0.561656\pi$$
$$350$$ −1125.00 −0.171811
$$351$$ 0 0
$$352$$ 680.000 0.102966
$$353$$ 3726.00 0.561799 0.280899 0.959737i $$-0.409367\pi$$
0.280899 + 0.959737i $$0.409367\pi$$
$$354$$ 0 0
$$355$$ 310.000 0.0463467
$$356$$ −13770.0 −2.05002
$$357$$ 0 0
$$358$$ 15505.0 2.28901
$$359$$ −10710.0 −1.57452 −0.787259 0.616622i $$-0.788501\pi$$
−0.787259 + 0.616622i $$0.788501\pi$$
$$360$$ 0 0
$$361$$ −3378.00 −0.492492
$$362$$ −14230.0 −2.06606
$$363$$ 0 0
$$364$$ 6579.00 0.947344
$$365$$ −6070.00 −0.870461
$$366$$ 0 0
$$367$$ −2160.00 −0.307224 −0.153612 0.988131i $$-0.549091\pi$$
−0.153612 + 0.988131i $$0.549091\pi$$
$$368$$ −18957.0 −2.68533
$$369$$ 0 0
$$370$$ −5150.00 −0.723610
$$371$$ −3393.00 −0.474813
$$372$$ 0 0
$$373$$ 3394.00 0.471138 0.235569 0.971858i $$-0.424305\pi$$
0.235569 + 0.971858i $$0.424305\pi$$
$$374$$ −4880.00 −0.674703
$$375$$ 0 0
$$376$$ 13995.0 1.91951
$$377$$ 9632.00 1.31584
$$378$$ 0 0
$$379$$ 9031.00 1.22399 0.611994 0.790863i $$-0.290368\pi$$
0.611994 + 0.790863i $$0.290368\pi$$
$$380$$ −5015.00 −0.677011
$$381$$ 0 0
$$382$$ −15400.0 −2.06265
$$383$$ −10305.0 −1.37483 −0.687416 0.726264i $$-0.741255\pi$$
−0.687416 + 0.726264i $$0.741255\pi$$
$$384$$ 0 0
$$385$$ −360.000 −0.0476553
$$386$$ −12940.0 −1.70629
$$387$$ 0 0
$$388$$ −15776.0 −2.06419
$$389$$ −3480.00 −0.453581 −0.226790 0.973944i $$-0.572823\pi$$
−0.226790 + 0.973944i $$0.572823\pi$$
$$390$$ 0 0
$$391$$ 25986.0 3.36104
$$392$$ 11790.0 1.51909
$$393$$ 0 0
$$394$$ 6675.00 0.853507
$$395$$ −1470.00 −0.187250
$$396$$ 0 0
$$397$$ 3706.00 0.468511 0.234255 0.972175i $$-0.424735\pi$$
0.234255 + 0.972175i $$0.424735\pi$$
$$398$$ 26020.0 3.27705
$$399$$ 0 0
$$400$$ 2225.00 0.278125
$$401$$ 2679.00 0.333623 0.166812 0.985989i $$-0.446653\pi$$
0.166812 + 0.985989i $$0.446653\pi$$
$$402$$ 0 0
$$403$$ −1548.00 −0.191343
$$404$$ −16932.0 −2.08514
$$405$$ 0 0
$$406$$ −10080.0 −1.23217
$$407$$ −1648.00 −0.200708
$$408$$ 0 0
$$409$$ −12499.0 −1.51109 −0.755545 0.655097i $$-0.772628\pi$$
−0.755545 + 0.655097i $$0.772628\pi$$
$$410$$ −10325.0 −1.24370
$$411$$ 0 0
$$412$$ −7361.00 −0.880220
$$413$$ 3033.00 0.361366
$$414$$ 0 0
$$415$$ 2670.00 0.315820
$$416$$ −3655.00 −0.430772
$$417$$ 0 0
$$418$$ −2360.00 −0.276152
$$419$$ −928.000 −0.108200 −0.0541000 0.998536i $$-0.517229\pi$$
−0.0541000 + 0.998536i $$0.517229\pi$$
$$420$$ 0 0
$$421$$ 7570.00 0.876340 0.438170 0.898892i $$-0.355627\pi$$
0.438170 + 0.898892i $$0.355627\pi$$
$$422$$ −8185.00 −0.944170
$$423$$ 0 0
$$424$$ 16965.0 1.94314
$$425$$ −3050.00 −0.348110
$$426$$ 0 0
$$427$$ 360.000 0.0408000
$$428$$ −28662.0 −3.23699
$$429$$ 0 0
$$430$$ 9800.00 1.09907
$$431$$ −2460.00 −0.274928 −0.137464 0.990507i $$-0.543895\pi$$
−0.137464 + 0.990507i $$0.543895\pi$$
$$432$$ 0 0
$$433$$ 1648.00 0.182905 0.0914525 0.995809i $$-0.470849\pi$$
0.0914525 + 0.995809i $$0.470849\pi$$
$$434$$ 1620.00 0.179176
$$435$$ 0 0
$$436$$ 11152.0 1.22496
$$437$$ 12567.0 1.37565
$$438$$ 0 0
$$439$$ 15826.0 1.72058 0.860289 0.509807i $$-0.170283\pi$$
0.860289 + 0.509807i $$0.170283\pi$$
$$440$$ 1800.00 0.195026
$$441$$ 0 0
$$442$$ 26230.0 2.82270
$$443$$ 12774.0 1.37000 0.685001 0.728542i $$-0.259802\pi$$
0.685001 + 0.728542i $$0.259802\pi$$
$$444$$ 0 0
$$445$$ −4050.00 −0.431435
$$446$$ 22400.0 2.37819
$$447$$ 0 0
$$448$$ −2583.00 −0.272400
$$449$$ 8875.00 0.932822 0.466411 0.884568i $$-0.345547\pi$$
0.466411 + 0.884568i $$0.345547\pi$$
$$450$$ 0 0
$$451$$ −3304.00 −0.344965
$$452$$ 17306.0 1.80090
$$453$$ 0 0
$$454$$ 18680.0 1.93105
$$455$$ 1935.00 0.199372
$$456$$ 0 0
$$457$$ 11524.0 1.17958 0.589792 0.807555i $$-0.299210\pi$$
0.589792 + 0.807555i $$0.299210\pi$$
$$458$$ 6900.00 0.703965
$$459$$ 0 0
$$460$$ −18105.0 −1.83511
$$461$$ −8544.00 −0.863197 −0.431598 0.902066i $$-0.642050\pi$$
−0.431598 + 0.902066i $$0.642050\pi$$
$$462$$ 0 0
$$463$$ 2523.00 0.253248 0.126624 0.991951i $$-0.459586\pi$$
0.126624 + 0.991951i $$0.459586\pi$$
$$464$$ 19936.0 1.99462
$$465$$ 0 0
$$466$$ 14520.0 1.44340
$$467$$ −2902.00 −0.287556 −0.143778 0.989610i $$-0.545925\pi$$
−0.143778 + 0.989610i $$0.545925\pi$$
$$468$$ 0 0
$$469$$ 3132.00 0.308363
$$470$$ 7775.00 0.763051
$$471$$ 0 0
$$472$$ −15165.0 −1.47887
$$473$$ 3136.00 0.304849
$$474$$ 0 0
$$475$$ −1475.00 −0.142479
$$476$$ −18666.0 −1.79738
$$477$$ 0 0
$$478$$ 29830.0 2.85438
$$479$$ 4362.00 0.416085 0.208043 0.978120i $$-0.433291\pi$$
0.208043 + 0.978120i $$0.433291\pi$$
$$480$$ 0 0
$$481$$ 8858.00 0.839688
$$482$$ 16090.0 1.52050
$$483$$ 0 0
$$484$$ −21539.0 −2.02282
$$485$$ −4640.00 −0.434416
$$486$$ 0 0
$$487$$ 5723.00 0.532513 0.266257 0.963902i $$-0.414213\pi$$
0.266257 + 0.963902i $$0.414213\pi$$
$$488$$ −1800.00 −0.166972
$$489$$ 0 0
$$490$$ 6550.00 0.603875
$$491$$ −13339.0 −1.22603 −0.613015 0.790071i $$-0.710043\pi$$
−0.613015 + 0.790071i $$0.710043\pi$$
$$492$$ 0 0
$$493$$ −27328.0 −2.49653
$$494$$ 12685.0 1.15531
$$495$$ 0 0
$$496$$ −3204.00 −0.290048
$$497$$ 558.000 0.0503616
$$498$$ 0 0
$$499$$ −19637.0 −1.76167 −0.880835 0.473424i $$-0.843018\pi$$
−0.880835 + 0.473424i $$0.843018\pi$$
$$500$$ 2125.00 0.190066
$$501$$ 0 0
$$502$$ −30615.0 −2.72194
$$503$$ −5416.00 −0.480094 −0.240047 0.970761i $$-0.577163\pi$$
−0.240047 + 0.970761i $$0.577163\pi$$
$$504$$ 0 0
$$505$$ −4980.00 −0.438826
$$506$$ −8520.00 −0.748538
$$507$$ 0 0
$$508$$ −23137.0 −2.02074
$$509$$ −6110.00 −0.532065 −0.266032 0.963964i $$-0.585713\pi$$
−0.266032 + 0.963964i $$0.585713\pi$$
$$510$$ 0 0
$$511$$ −10926.0 −0.945867
$$512$$ 24475.0 2.11260
$$513$$ 0 0
$$514$$ 6990.00 0.599836
$$515$$ −2165.00 −0.185245
$$516$$ 0 0
$$517$$ 2488.00 0.211648
$$518$$ −9270.00 −0.786294
$$519$$ 0 0
$$520$$ −9675.00 −0.815917
$$521$$ 20375.0 1.71333 0.856665 0.515873i $$-0.172532\pi$$
0.856665 + 0.515873i $$0.172532\pi$$
$$522$$ 0 0
$$523$$ −19010.0 −1.58939 −0.794693 0.607011i $$-0.792368\pi$$
−0.794693 + 0.607011i $$0.792368\pi$$
$$524$$ 32487.0 2.70840
$$525$$ 0 0
$$526$$ 16055.0 1.33086
$$527$$ 4392.00 0.363033
$$528$$ 0 0
$$529$$ 33202.0 2.72886
$$530$$ 9425.00 0.772445
$$531$$ 0 0
$$532$$ −9027.00 −0.735658
$$533$$ 17759.0 1.44320
$$534$$ 0 0
$$535$$ −8430.00 −0.681235
$$536$$ −15660.0 −1.26196
$$537$$ 0 0
$$538$$ −20090.0 −1.60993
$$539$$ 2096.00 0.167497
$$540$$ 0 0
$$541$$ 3288.00 0.261298 0.130649 0.991429i $$-0.458294\pi$$
0.130649 + 0.991429i $$0.458294\pi$$
$$542$$ −11570.0 −0.916926
$$543$$ 0 0
$$544$$ 10370.0 0.817298
$$545$$ 3280.00 0.257798
$$546$$ 0 0
$$547$$ 3256.00 0.254509 0.127255 0.991870i $$-0.459383\pi$$
0.127255 + 0.991870i $$0.459383\pi$$
$$548$$ 11118.0 0.866674
$$549$$ 0 0
$$550$$ 1000.00 0.0775275
$$551$$ −13216.0 −1.02182
$$552$$ 0 0
$$553$$ −2646.00 −0.203471
$$554$$ 21735.0 1.66684
$$555$$ 0 0
$$556$$ −12461.0 −0.950475
$$557$$ 213.000 0.0162031 0.00810153 0.999967i $$-0.497421\pi$$
0.00810153 + 0.999967i $$0.497421\pi$$
$$558$$ 0 0
$$559$$ −16856.0 −1.27537
$$560$$ 4005.00 0.302218
$$561$$ 0 0
$$562$$ −7755.00 −0.582073
$$563$$ 17388.0 1.30163 0.650814 0.759237i $$-0.274428\pi$$
0.650814 + 0.759237i $$0.274428\pi$$
$$564$$ 0 0
$$565$$ 5090.00 0.379005
$$566$$ −21900.0 −1.62637
$$567$$ 0 0
$$568$$ −2790.00 −0.206102
$$569$$ −4353.00 −0.320716 −0.160358 0.987059i $$-0.551265\pi$$
−0.160358 + 0.987059i $$0.551265\pi$$
$$570$$ 0 0
$$571$$ −1924.00 −0.141010 −0.0705052 0.997511i $$-0.522461\pi$$
−0.0705052 + 0.997511i $$0.522461\pi$$
$$572$$ −5848.00 −0.427478
$$573$$ 0 0
$$574$$ −18585.0 −1.35143
$$575$$ −5325.00 −0.386205
$$576$$ 0 0
$$577$$ 16832.0 1.21443 0.607214 0.794538i $$-0.292287\pi$$
0.607214 + 0.794538i $$0.292287\pi$$
$$578$$ −49855.0 −3.58771
$$579$$ 0 0
$$580$$ 19040.0 1.36309
$$581$$ 4806.00 0.343178
$$582$$ 0 0
$$583$$ 3016.00 0.214254
$$584$$ 54630.0 3.87090
$$585$$ 0 0
$$586$$ −25245.0 −1.77963
$$587$$ −2106.00 −0.148082 −0.0740408 0.997255i $$-0.523590\pi$$
−0.0740408 + 0.997255i $$0.523590\pi$$
$$588$$ 0 0
$$589$$ 2124.00 0.148587
$$590$$ −8425.00 −0.587884
$$591$$ 0 0
$$592$$ 18334.0 1.27284
$$593$$ 4694.00 0.325058 0.162529 0.986704i $$-0.448035\pi$$
0.162529 + 0.986704i $$0.448035\pi$$
$$594$$ 0 0
$$595$$ −5490.00 −0.378266
$$596$$ −19142.0 −1.31558
$$597$$ 0 0
$$598$$ 45795.0 3.13160
$$599$$ 13226.0 0.902170 0.451085 0.892481i $$-0.351037\pi$$
0.451085 + 0.892481i $$0.351037\pi$$
$$600$$ 0 0
$$601$$ −13291.0 −0.902082 −0.451041 0.892503i $$-0.648947\pi$$
−0.451041 + 0.892503i $$0.648947\pi$$
$$602$$ 17640.0 1.19427
$$603$$ 0 0
$$604$$ −43282.0 −2.91576
$$605$$ −6335.00 −0.425710
$$606$$ 0 0
$$607$$ 6040.00 0.403881 0.201941 0.979398i $$-0.435275\pi$$
0.201941 + 0.979398i $$0.435275\pi$$
$$608$$ 5015.00 0.334515
$$609$$ 0 0
$$610$$ −1000.00 −0.0663751
$$611$$ −13373.0 −0.885456
$$612$$ 0 0
$$613$$ −23.0000 −0.00151543 −0.000757717 1.00000i $$-0.500241\pi$$
−0.000757717 1.00000i $$0.500241\pi$$
$$614$$ −27140.0 −1.78385
$$615$$ 0 0
$$616$$ 3240.00 0.211921
$$617$$ −3018.00 −0.196921 −0.0984604 0.995141i $$-0.531392\pi$$
−0.0984604 + 0.995141i $$0.531392\pi$$
$$618$$ 0 0
$$619$$ −9439.00 −0.612901 −0.306450 0.951887i $$-0.599141\pi$$
−0.306450 + 0.951887i $$0.599141\pi$$
$$620$$ −3060.00 −0.198214
$$621$$ 0 0
$$622$$ 90.0000 0.00580172
$$623$$ −7290.00 −0.468808
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 10580.0 0.675498
$$627$$ 0 0
$$628$$ 20791.0 1.32110
$$629$$ −25132.0 −1.59313
$$630$$ 0 0
$$631$$ −9800.00 −0.618275 −0.309138 0.951017i $$-0.600040\pi$$
−0.309138 + 0.951017i $$0.600040\pi$$
$$632$$ 13230.0 0.832692
$$633$$ 0 0
$$634$$ −22075.0 −1.38282
$$635$$ −6805.00 −0.425273
$$636$$ 0 0
$$637$$ −11266.0 −0.700746
$$638$$ 8960.00 0.556003
$$639$$ 0 0
$$640$$ 10575.0 0.653146
$$641$$ 23442.0 1.44447 0.722233 0.691649i $$-0.243116\pi$$
0.722233 + 0.691649i $$0.243116\pi$$
$$642$$ 0 0
$$643$$ 31308.0 1.92017 0.960083 0.279715i $$-0.0902398\pi$$
0.960083 + 0.279715i $$0.0902398\pi$$
$$644$$ −32589.0 −1.99408
$$645$$ 0 0
$$646$$ −35990.0 −2.19196
$$647$$ 712.000 0.0432637 0.0216318 0.999766i $$-0.493114\pi$$
0.0216318 + 0.999766i $$0.493114\pi$$
$$648$$ 0 0
$$649$$ −2696.00 −0.163062
$$650$$ −5375.00 −0.324346
$$651$$ 0 0
$$652$$ 53992.0 3.24308
$$653$$ 31478.0 1.88642 0.943208 0.332203i $$-0.107792\pi$$
0.943208 + 0.332203i $$0.107792\pi$$
$$654$$ 0 0
$$655$$ 9555.00 0.569992
$$656$$ 36757.0 2.18768
$$657$$ 0 0
$$658$$ 13995.0 0.829152
$$659$$ −16121.0 −0.952936 −0.476468 0.879192i $$-0.658083\pi$$
−0.476468 + 0.879192i $$0.658083\pi$$
$$660$$ 0 0
$$661$$ −19160.0 −1.12744 −0.563720 0.825966i $$-0.690630\pi$$
−0.563720 + 0.825966i $$0.690630\pi$$
$$662$$ −17400.0 −1.02156
$$663$$ 0 0
$$664$$ −24030.0 −1.40444
$$665$$ −2655.00 −0.154822
$$666$$ 0 0
$$667$$ −47712.0 −2.76974
$$668$$ 2244.00 0.129975
$$669$$ 0 0
$$670$$ −8700.00 −0.501657
$$671$$ −320.000 −0.0184105
$$672$$ 0 0
$$673$$ −13422.0 −0.768767 −0.384383 0.923174i $$-0.625586\pi$$
−0.384383 + 0.923174i $$0.625586\pi$$
$$674$$ −31610.0 −1.80649
$$675$$ 0 0
$$676$$ −5916.00 −0.336595
$$677$$ 25905.0 1.47062 0.735310 0.677731i $$-0.237036\pi$$
0.735310 + 0.677731i $$0.237036\pi$$
$$678$$ 0 0
$$679$$ −8352.00 −0.472048
$$680$$ 27450.0 1.54803
$$681$$ 0 0
$$682$$ −1440.00 −0.0808511
$$683$$ −9246.00 −0.517992 −0.258996 0.965878i $$-0.583392\pi$$
−0.258996 + 0.965878i $$0.583392\pi$$
$$684$$ 0 0
$$685$$ 3270.00 0.182395
$$686$$ 27225.0 1.51524
$$687$$ 0 0
$$688$$ −34888.0 −1.93327
$$689$$ −16211.0 −0.896357
$$690$$ 0 0
$$691$$ 25039.0 1.37848 0.689239 0.724534i $$-0.257945\pi$$
0.689239 + 0.724534i $$0.257945\pi$$
$$692$$ −16881.0 −0.927340
$$693$$ 0 0
$$694$$ 50170.0 2.74413
$$695$$ −3665.00 −0.200031
$$696$$ 0 0
$$697$$ −50386.0 −2.73817
$$698$$ 12550.0 0.680551
$$699$$ 0 0
$$700$$ 3825.00 0.206531
$$701$$ 32930.0 1.77425 0.887125 0.461530i $$-0.152699\pi$$
0.887125 + 0.461530i $$0.152699\pi$$
$$702$$ 0 0
$$703$$ −12154.0 −0.652058
$$704$$ 2296.00 0.122917
$$705$$ 0 0
$$706$$ −18630.0 −0.993129
$$707$$ −8964.00 −0.476840
$$708$$ 0 0
$$709$$ 1882.00 0.0996897 0.0498448 0.998757i $$-0.484127\pi$$
0.0498448 + 0.998757i $$0.484127\pi$$
$$710$$ −1550.00 −0.0819302
$$711$$ 0 0
$$712$$ 36450.0 1.91857
$$713$$ 7668.00 0.402761
$$714$$ 0 0
$$715$$ −1720.00 −0.0899641
$$716$$ −52717.0 −2.75157
$$717$$ 0 0
$$718$$ 53550.0 2.78338
$$719$$ −1962.00 −0.101767 −0.0508833 0.998705i $$-0.516204\pi$$
−0.0508833 + 0.998705i $$0.516204\pi$$
$$720$$ 0 0
$$721$$ −3897.00 −0.201292
$$722$$ 16890.0 0.870610
$$723$$ 0 0
$$724$$ 48382.0 2.48357
$$725$$ 5600.00 0.286867
$$726$$ 0 0
$$727$$ −13741.0 −0.700998 −0.350499 0.936563i $$-0.613988\pi$$
−0.350499 + 0.936563i $$0.613988\pi$$
$$728$$ −17415.0 −0.886597
$$729$$ 0 0
$$730$$ 30350.0 1.53877
$$731$$ 47824.0 2.41975
$$732$$ 0 0
$$733$$ −32458.0 −1.63556 −0.817779 0.575533i $$-0.804795\pi$$
−0.817779 + 0.575533i $$0.804795\pi$$
$$734$$ 10800.0 0.543100
$$735$$ 0 0
$$736$$ 18105.0 0.906738
$$737$$ −2784.00 −0.139145
$$738$$ 0 0
$$739$$ 19612.0 0.976237 0.488118 0.872777i $$-0.337683\pi$$
0.488118 + 0.872777i $$0.337683\pi$$
$$740$$ 17510.0 0.869838
$$741$$ 0 0
$$742$$ 16965.0 0.839359
$$743$$ 36736.0 1.81388 0.906940 0.421259i $$-0.138412\pi$$
0.906940 + 0.421259i $$0.138412\pi$$
$$744$$ 0 0
$$745$$ −5630.00 −0.276869
$$746$$ −16970.0 −0.832863
$$747$$ 0 0
$$748$$ 16592.0 0.811048
$$749$$ −15174.0 −0.740248
$$750$$ 0 0
$$751$$ −3746.00 −0.182015 −0.0910076 0.995850i $$-0.529009\pi$$
−0.0910076 + 0.995850i $$0.529009\pi$$
$$752$$ −27679.0 −1.34222
$$753$$ 0 0
$$754$$ −48160.0 −2.32611
$$755$$ −12730.0 −0.613632
$$756$$ 0 0
$$757$$ 5725.00 0.274873 0.137436 0.990511i $$-0.456114\pi$$
0.137436 + 0.990511i $$0.456114\pi$$
$$758$$ −45155.0 −2.16372
$$759$$ 0 0
$$760$$ 13275.0 0.633599
$$761$$ 37323.0 1.77787 0.888934 0.458035i $$-0.151447\pi$$
0.888934 + 0.458035i $$0.151447\pi$$
$$762$$ 0 0
$$763$$ 5904.00 0.280130
$$764$$ 52360.0 2.47947
$$765$$ 0 0
$$766$$ 51525.0 2.43038
$$767$$ 14491.0 0.682190
$$768$$ 0 0
$$769$$ −24586.0 −1.15292 −0.576459 0.817126i $$-0.695566\pi$$
−0.576459 + 0.817126i $$0.695566\pi$$
$$770$$ 1800.00 0.0842435
$$771$$ 0 0
$$772$$ 43996.0 2.05110
$$773$$ −3078.00 −0.143219 −0.0716093 0.997433i $$-0.522813\pi$$
−0.0716093 + 0.997433i $$0.522813\pi$$
$$774$$ 0 0
$$775$$ −900.000 −0.0417148
$$776$$ 41760.0 1.93183
$$777$$ 0 0
$$778$$ 17400.0 0.801825
$$779$$ −24367.0 −1.12072
$$780$$ 0 0
$$781$$ −496.000 −0.0227251
$$782$$ −129930. −5.94154
$$783$$ 0 0
$$784$$ −23318.0 −1.06223
$$785$$ 6115.00 0.278030
$$786$$ 0 0
$$787$$ 41038.0 1.85876 0.929382 0.369120i $$-0.120341\pi$$
0.929382 + 0.369120i $$0.120341\pi$$
$$788$$ −22695.0 −1.02598
$$789$$ 0 0
$$790$$ 7350.00 0.331014
$$791$$ 9162.00 0.411837
$$792$$ 0 0
$$793$$ 1720.00 0.0770227
$$794$$ −18530.0 −0.828218
$$795$$ 0 0
$$796$$ −88468.0 −3.93928
$$797$$ 9362.00 0.416084 0.208042 0.978120i $$-0.433291\pi$$
0.208042 + 0.978120i $$0.433291\pi$$
$$798$$ 0 0
$$799$$ 37942.0 1.67996
$$800$$ −2125.00 −0.0939126
$$801$$ 0 0
$$802$$ −13395.0 −0.589768
$$803$$ 9712.00 0.426811
$$804$$ 0 0
$$805$$ −9585.00 −0.419661
$$806$$ 7740.00 0.338250
$$807$$ 0 0
$$808$$ 44820.0 1.95144
$$809$$ −45115.0 −1.96064 −0.980321 0.197411i $$-0.936747\pi$$
−0.980321 + 0.197411i $$0.936747\pi$$
$$810$$ 0 0
$$811$$ −13512.0 −0.585044 −0.292522 0.956259i $$-0.594494\pi$$
−0.292522 + 0.956259i $$0.594494\pi$$
$$812$$ 34272.0 1.48117
$$813$$ 0 0
$$814$$ 8240.00 0.354806
$$815$$ 15880.0 0.682518
$$816$$ 0 0
$$817$$ 23128.0 0.990387
$$818$$ 62495.0 2.67125
$$819$$ 0 0
$$820$$ 35105.0 1.49502
$$821$$ −4530.00 −0.192568 −0.0962839 0.995354i $$-0.530696\pi$$
−0.0962839 + 0.995354i $$0.530696\pi$$
$$822$$ 0 0
$$823$$ 30884.0 1.30808 0.654039 0.756461i $$-0.273073\pi$$
0.654039 + 0.756461i $$0.273073\pi$$
$$824$$ 19485.0 0.823777
$$825$$ 0 0
$$826$$ −15165.0 −0.638811
$$827$$ 12088.0 0.508272 0.254136 0.967168i $$-0.418209\pi$$
0.254136 + 0.967168i $$0.418209\pi$$
$$828$$ 0 0
$$829$$ −14112.0 −0.591230 −0.295615 0.955307i $$-0.595525\pi$$
−0.295615 + 0.955307i $$0.595525\pi$$
$$830$$ −13350.0 −0.558295
$$831$$ 0 0
$$832$$ −12341.0 −0.514239
$$833$$ 31964.0 1.32952
$$834$$ 0 0
$$835$$ 660.000 0.0273536
$$836$$ 8024.00 0.331957
$$837$$ 0 0
$$838$$ 4640.00 0.191272
$$839$$ 3860.00 0.158834 0.0794172 0.996841i $$-0.474694\pi$$
0.0794172 + 0.996841i $$0.474694\pi$$
$$840$$ 0 0
$$841$$ 25787.0 1.05732
$$842$$ −37850.0 −1.54917
$$843$$ 0 0
$$844$$ 27829.0 1.13497
$$845$$ −1740.00 −0.0708377
$$846$$ 0 0
$$847$$ −11403.0 −0.462588
$$848$$ −33553.0 −1.35874
$$849$$ 0 0
$$850$$ 15250.0 0.615377
$$851$$ −43878.0 −1.76747
$$852$$ 0 0
$$853$$ 2862.00 0.114880 0.0574402 0.998349i $$-0.481706\pi$$
0.0574402 + 0.998349i $$0.481706\pi$$
$$854$$ −1800.00 −0.0721250
$$855$$ 0 0
$$856$$ 75870.0 3.02942
$$857$$ −32534.0 −1.29678 −0.648390 0.761308i $$-0.724557\pi$$
−0.648390 + 0.761308i $$0.724557\pi$$
$$858$$ 0 0
$$859$$ −500.000 −0.0198600 −0.00993002 0.999951i $$-0.503161\pi$$
−0.00993002 + 0.999951i $$0.503161\pi$$
$$860$$ −33320.0 −1.32117
$$861$$ 0 0
$$862$$ 12300.0 0.486009
$$863$$ −36595.0 −1.44346 −0.721731 0.692173i $$-0.756653\pi$$
−0.721731 + 0.692173i $$0.756653\pi$$
$$864$$ 0 0
$$865$$ −4965.00 −0.195162
$$866$$ −8240.00 −0.323333
$$867$$ 0 0
$$868$$ −5508.00 −0.215384
$$869$$ 2352.00 0.0918137
$$870$$ 0 0
$$871$$ 14964.0 0.582131
$$872$$ −29520.0 −1.14641
$$873$$ 0 0
$$874$$ −62835.0 −2.43184
$$875$$ 1125.00 0.0434651
$$876$$ 0 0
$$877$$ 40143.0 1.54565 0.772824 0.634621i $$-0.218844\pi$$
0.772824 + 0.634621i $$0.218844\pi$$
$$878$$ −79130.0 −3.04158
$$879$$ 0 0
$$880$$ −3560.00 −0.136372
$$881$$ 16414.0 0.627698 0.313849 0.949473i $$-0.398381\pi$$
0.313849 + 0.949473i $$0.398381\pi$$
$$882$$ 0 0
$$883$$ −33478.0 −1.27591 −0.637953 0.770076i $$-0.720218\pi$$
−0.637953 + 0.770076i $$0.720218\pi$$
$$884$$ −89182.0 −3.39312
$$885$$ 0 0
$$886$$ −63870.0 −2.42184
$$887$$ −3633.00 −0.137524 −0.0687622 0.997633i $$-0.521905\pi$$
−0.0687622 + 0.997633i $$0.521905\pi$$
$$888$$ 0 0
$$889$$ −12249.0 −0.462113
$$890$$ 20250.0 0.762676
$$891$$ 0 0
$$892$$ −76160.0 −2.85877
$$893$$ 18349.0 0.687599
$$894$$ 0 0
$$895$$ −15505.0 −0.579078
$$896$$ 19035.0 0.709726
$$897$$ 0 0
$$898$$ −44375.0 −1.64901
$$899$$ −8064.00 −0.299165
$$900$$ 0 0
$$901$$ 45994.0 1.70065
$$902$$ 16520.0 0.609818
$$903$$ 0 0
$$904$$ −45810.0 −1.68542
$$905$$ 14230.0 0.522675
$$906$$ 0 0
$$907$$ 20466.0 0.749242 0.374621 0.927178i $$-0.377773\pi$$
0.374621 + 0.927178i $$0.377773\pi$$
$$908$$ −63512.0 −2.32128
$$909$$ 0 0
$$910$$ −9675.00 −0.352443
$$911$$ −15074.0 −0.548215 −0.274108 0.961699i $$-0.588382\pi$$
−0.274108 + 0.961699i $$0.588382\pi$$
$$912$$ 0 0
$$913$$ −4272.00 −0.154855
$$914$$ −57620.0 −2.08523
$$915$$ 0 0
$$916$$ −23460.0 −0.846223
$$917$$ 17199.0 0.619369
$$918$$ 0 0
$$919$$ −36848.0 −1.32264 −0.661318 0.750105i $$-0.730003\pi$$
−0.661318 + 0.750105i $$0.730003\pi$$
$$920$$ 47925.0 1.71744
$$921$$ 0 0
$$922$$ 42720.0 1.52593
$$923$$ 2666.00 0.0950731
$$924$$ 0 0
$$925$$ 5150.00 0.183060
$$926$$ −12615.0 −0.447683
$$927$$ 0 0
$$928$$ −19040.0 −0.673511
$$929$$ 35174.0 1.24222 0.621110 0.783724i $$-0.286682\pi$$
0.621110 + 0.783724i $$0.286682\pi$$
$$930$$ 0 0
$$931$$ 15458.0 0.544163
$$932$$ −49368.0 −1.73509
$$933$$ 0 0
$$934$$ 14510.0 0.508332
$$935$$ 4880.00 0.170688
$$936$$ 0 0
$$937$$ −10092.0 −0.351858 −0.175929 0.984403i $$-0.556293\pi$$
−0.175929 + 0.984403i $$0.556293\pi$$
$$938$$ −15660.0 −0.545114
$$939$$ 0 0
$$940$$ −26435.0 −0.917250
$$941$$ −12910.0 −0.447241 −0.223621 0.974676i $$-0.571788\pi$$
−0.223621 + 0.974676i $$0.571788\pi$$
$$942$$ 0 0
$$943$$ −87969.0 −3.03782
$$944$$ 29993.0 1.03410
$$945$$ 0 0
$$946$$ −15680.0 −0.538901
$$947$$ −48060.0 −1.64914 −0.824572 0.565756i $$-0.808584\pi$$
−0.824572 + 0.565756i $$0.808584\pi$$
$$948$$ 0 0
$$949$$ −52202.0 −1.78561
$$950$$ 7375.00 0.251870
$$951$$ 0 0
$$952$$ 49410.0 1.68213
$$953$$ 6316.00 0.214686 0.107343 0.994222i $$-0.465766\pi$$
0.107343 + 0.994222i $$0.465766\pi$$
$$954$$ 0 0
$$955$$ 15400.0 0.521814
$$956$$ −101422. −3.43119
$$957$$ 0 0
$$958$$ −21810.0 −0.735542
$$959$$ 5886.00 0.198195
$$960$$ 0 0
$$961$$ −28495.0 −0.956497
$$962$$ −44290.0 −1.48437
$$963$$ 0 0
$$964$$ −54706.0 −1.82776
$$965$$ 12940.0 0.431661
$$966$$ 0 0
$$967$$ 7760.00 0.258061 0.129030 0.991641i $$-0.458814\pi$$
0.129030 + 0.991641i $$0.458814\pi$$
$$968$$ 57015.0 1.89311
$$969$$ 0 0
$$970$$ 23200.0 0.767945
$$971$$ 22437.0 0.741542 0.370771 0.928724i $$-0.379093\pi$$
0.370771 + 0.928724i $$0.379093\pi$$
$$972$$ 0 0
$$973$$ −6597.00 −0.217359
$$974$$ −28615.0 −0.941359
$$975$$ 0 0
$$976$$ 3560.00 0.116755
$$977$$ −3506.00 −0.114807 −0.0574037 0.998351i $$-0.518282\pi$$
−0.0574037 + 0.998351i $$0.518282\pi$$
$$978$$ 0 0
$$979$$ 6480.00 0.211544
$$980$$ −22270.0 −0.725907
$$981$$ 0 0
$$982$$ 66695.0 2.16734
$$983$$ −30648.0 −0.994425 −0.497212 0.867629i $$-0.665643\pi$$
−0.497212 + 0.867629i $$0.665643\pi$$
$$984$$ 0 0
$$985$$ −6675.00 −0.215922
$$986$$ 136640. 4.41329
$$987$$ 0 0
$$988$$ −43129.0 −1.38878
$$989$$ 83496.0 2.68455
$$990$$ 0 0
$$991$$ 19232.0 0.616473 0.308236 0.951310i $$-0.400261\pi$$
0.308236 + 0.951310i $$0.400261\pi$$
$$992$$ 3060.00 0.0979386
$$993$$ 0 0
$$994$$ −2790.00 −0.0890276
$$995$$ −26020.0 −0.829035
$$996$$ 0 0
$$997$$ 38415.0 1.22028 0.610138 0.792295i $$-0.291114\pi$$
0.610138 + 0.792295i $$0.291114\pi$$
$$998$$ 98185.0 3.11422
$$999$$ 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 405.4.a.a.1.1 1
3.2 odd 2 405.4.a.b.1.1 yes 1
5.4 even 2 2025.4.a.f.1.1 1
9.2 odd 6 405.4.e.a.271.1 2
9.4 even 3 405.4.e.m.136.1 2
9.5 odd 6 405.4.e.a.136.1 2
9.7 even 3 405.4.e.m.271.1 2
15.14 odd 2 2025.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
405.4.a.a.1.1 1 1.1 even 1 trivial
405.4.a.b.1.1 yes 1 3.2 odd 2
405.4.e.a.136.1 2 9.5 odd 6
405.4.e.a.271.1 2 9.2 odd 6
405.4.e.m.136.1 2 9.4 even 3
405.4.e.m.271.1 2 9.7 even 3
2025.4.a.a.1.1 1 15.14 odd 2
2025.4.a.f.1.1 1 5.4 even 2