# Properties

 Label 80.4.a.b.1.1 Level $80$ Weight $4$ Character 80.1 Self dual yes Analytic conductor $4.720$ 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] = [80,4,Mod(1,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("80.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 80.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: $$4.72015280046$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 80.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.00000 q^{3} +5.00000 q^{5} -16.0000 q^{7} -11.0000 q^{9} +O(q^{10})$$ $$q-4.00000 q^{3} +5.00000 q^{5} -16.0000 q^{7} -11.0000 q^{9} -36.0000 q^{11} -42.0000 q^{13} -20.0000 q^{15} -110.000 q^{17} +116.000 q^{19} +64.0000 q^{21} -16.0000 q^{23} +25.0000 q^{25} +152.000 q^{27} +198.000 q^{29} -240.000 q^{31} +144.000 q^{33} -80.0000 q^{35} -258.000 q^{37} +168.000 q^{39} +442.000 q^{41} +292.000 q^{43} -55.0000 q^{45} -392.000 q^{47} -87.0000 q^{49} +440.000 q^{51} +142.000 q^{53} -180.000 q^{55} -464.000 q^{57} +348.000 q^{59} -570.000 q^{61} +176.000 q^{63} -210.000 q^{65} -692.000 q^{67} +64.0000 q^{69} -168.000 q^{71} -134.000 q^{73} -100.000 q^{75} +576.000 q^{77} -784.000 q^{79} -311.000 q^{81} -564.000 q^{83} -550.000 q^{85} -792.000 q^{87} +1034.00 q^{89} +672.000 q^{91} +960.000 q^{93} +580.000 q^{95} -382.000 q^{97} +396.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −4.00000 −0.769800 −0.384900 0.922958i $$-0.625764\pi$$
−0.384900 + 0.922958i $$0.625764\pi$$
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −16.0000 −0.863919 −0.431959 0.901893i $$-0.642178\pi$$
−0.431959 + 0.901893i $$0.642178\pi$$
$$8$$ 0 0
$$9$$ −11.0000 −0.407407
$$10$$ 0 0
$$11$$ −36.0000 −0.986764 −0.493382 0.869813i $$-0.664240\pi$$
−0.493382 + 0.869813i $$0.664240\pi$$
$$12$$ 0 0
$$13$$ −42.0000 −0.896054 −0.448027 0.894020i $$-0.647873\pi$$
−0.448027 + 0.894020i $$0.647873\pi$$
$$14$$ 0 0
$$15$$ −20.0000 −0.344265
$$16$$ 0 0
$$17$$ −110.000 −1.56935 −0.784674 0.619909i $$-0.787170\pi$$
−0.784674 + 0.619909i $$0.787170\pi$$
$$18$$ 0 0
$$19$$ 116.000 1.40064 0.700322 0.713827i $$-0.253040\pi$$
0.700322 + 0.713827i $$0.253040\pi$$
$$20$$ 0 0
$$21$$ 64.0000 0.665045
$$22$$ 0 0
$$23$$ −16.0000 −0.145054 −0.0725268 0.997366i $$-0.523106\pi$$
−0.0725268 + 0.997366i $$0.523106\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 152.000 1.08342
$$28$$ 0 0
$$29$$ 198.000 1.26785 0.633925 0.773394i $$-0.281443\pi$$
0.633925 + 0.773394i $$0.281443\pi$$
$$30$$ 0 0
$$31$$ −240.000 −1.39049 −0.695246 0.718772i $$-0.744705\pi$$
−0.695246 + 0.718772i $$0.744705\pi$$
$$32$$ 0 0
$$33$$ 144.000 0.759612
$$34$$ 0 0
$$35$$ −80.0000 −0.386356
$$36$$ 0 0
$$37$$ −258.000 −1.14635 −0.573175 0.819433i $$-0.694288\pi$$
−0.573175 + 0.819433i $$0.694288\pi$$
$$38$$ 0 0
$$39$$ 168.000 0.689783
$$40$$ 0 0
$$41$$ 442.000 1.68363 0.841815 0.539767i $$-0.181488\pi$$
0.841815 + 0.539767i $$0.181488\pi$$
$$42$$ 0 0
$$43$$ 292.000 1.03557 0.517786 0.855510i $$-0.326756\pi$$
0.517786 + 0.855510i $$0.326756\pi$$
$$44$$ 0 0
$$45$$ −55.0000 −0.182198
$$46$$ 0 0
$$47$$ −392.000 −1.21658 −0.608288 0.793716i $$-0.708143\pi$$
−0.608288 + 0.793716i $$0.708143\pi$$
$$48$$ 0 0
$$49$$ −87.0000 −0.253644
$$50$$ 0 0
$$51$$ 440.000 1.20808
$$52$$ 0 0
$$53$$ 142.000 0.368023 0.184011 0.982924i $$-0.441092\pi$$
0.184011 + 0.982924i $$0.441092\pi$$
$$54$$ 0 0
$$55$$ −180.000 −0.441294
$$56$$ 0 0
$$57$$ −464.000 −1.07822
$$58$$ 0 0
$$59$$ 348.000 0.767894 0.383947 0.923355i $$-0.374565\pi$$
0.383947 + 0.923355i $$0.374565\pi$$
$$60$$ 0 0
$$61$$ −570.000 −1.19641 −0.598205 0.801343i $$-0.704119\pi$$
−0.598205 + 0.801343i $$0.704119\pi$$
$$62$$ 0 0
$$63$$ 176.000 0.351967
$$64$$ 0 0
$$65$$ −210.000 −0.400728
$$66$$ 0 0
$$67$$ −692.000 −1.26181 −0.630905 0.775860i $$-0.717316\pi$$
−0.630905 + 0.775860i $$0.717316\pi$$
$$68$$ 0 0
$$69$$ 64.0000 0.111662
$$70$$ 0 0
$$71$$ −168.000 −0.280816 −0.140408 0.990094i $$-0.544841\pi$$
−0.140408 + 0.990094i $$0.544841\pi$$
$$72$$ 0 0
$$73$$ −134.000 −0.214843 −0.107421 0.994214i $$-0.534259\pi$$
−0.107421 + 0.994214i $$0.534259\pi$$
$$74$$ 0 0
$$75$$ −100.000 −0.153960
$$76$$ 0 0
$$77$$ 576.000 0.852484
$$78$$ 0 0
$$79$$ −784.000 −1.11654 −0.558271 0.829658i $$-0.688535\pi$$
−0.558271 + 0.829658i $$0.688535\pi$$
$$80$$ 0 0
$$81$$ −311.000 −0.426612
$$82$$ 0 0
$$83$$ −564.000 −0.745868 −0.372934 0.927858i $$-0.621648\pi$$
−0.372934 + 0.927858i $$0.621648\pi$$
$$84$$ 0 0
$$85$$ −550.000 −0.701834
$$86$$ 0 0
$$87$$ −792.000 −0.975992
$$88$$ 0 0
$$89$$ 1034.00 1.23150 0.615752 0.787940i $$-0.288852\pi$$
0.615752 + 0.787940i $$0.288852\pi$$
$$90$$ 0 0
$$91$$ 672.000 0.774118
$$92$$ 0 0
$$93$$ 960.000 1.07040
$$94$$ 0 0
$$95$$ 580.000 0.626387
$$96$$ 0 0
$$97$$ −382.000 −0.399858 −0.199929 0.979810i $$-0.564071\pi$$
−0.199929 + 0.979810i $$0.564071\pi$$
$$98$$ 0 0
$$99$$ 396.000 0.402015
$$100$$ 0 0
$$101$$ −674.000 −0.664015 −0.332007 0.943277i $$-0.607726\pi$$
−0.332007 + 0.943277i $$0.607726\pi$$
$$102$$ 0 0
$$103$$ 992.000 0.948977 0.474489 0.880262i $$-0.342633\pi$$
0.474489 + 0.880262i $$0.342633\pi$$
$$104$$ 0 0
$$105$$ 320.000 0.297417
$$106$$ 0 0
$$107$$ 500.000 0.451746 0.225873 0.974157i $$-0.427477\pi$$
0.225873 + 0.974157i $$0.427477\pi$$
$$108$$ 0 0
$$109$$ 1046.00 0.919162 0.459581 0.888136i $$-0.348000\pi$$
0.459581 + 0.888136i $$0.348000\pi$$
$$110$$ 0 0
$$111$$ 1032.00 0.882460
$$112$$ 0 0
$$113$$ −558.000 −0.464533 −0.232266 0.972652i $$-0.574614\pi$$
−0.232266 + 0.972652i $$0.574614\pi$$
$$114$$ 0 0
$$115$$ −80.0000 −0.0648699
$$116$$ 0 0
$$117$$ 462.000 0.365059
$$118$$ 0 0
$$119$$ 1760.00 1.35579
$$120$$ 0 0
$$121$$ −35.0000 −0.0262960
$$122$$ 0 0
$$123$$ −1768.00 −1.29606
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 328.000 0.229176 0.114588 0.993413i $$-0.463445\pi$$
0.114588 + 0.993413i $$0.463445\pi$$
$$128$$ 0 0
$$129$$ −1168.00 −0.797183
$$130$$ 0 0
$$131$$ 212.000 0.141393 0.0706967 0.997498i $$-0.477478\pi$$
0.0706967 + 0.997498i $$0.477478\pi$$
$$132$$ 0 0
$$133$$ −1856.00 −1.21004
$$134$$ 0 0
$$135$$ 760.000 0.484521
$$136$$ 0 0
$$137$$ 1434.00 0.894269 0.447135 0.894467i $$-0.352444\pi$$
0.447135 + 0.894467i $$0.352444\pi$$
$$138$$ 0 0
$$139$$ −2196.00 −1.34002 −0.670008 0.742354i $$-0.733709\pi$$
−0.670008 + 0.742354i $$0.733709\pi$$
$$140$$ 0 0
$$141$$ 1568.00 0.936521
$$142$$ 0 0
$$143$$ 1512.00 0.884194
$$144$$ 0 0
$$145$$ 990.000 0.567000
$$146$$ 0 0
$$147$$ 348.000 0.195255
$$148$$ 0 0
$$149$$ −2418.00 −1.32946 −0.664732 0.747081i $$-0.731454\pi$$
−0.664732 + 0.747081i $$0.731454\pi$$
$$150$$ 0 0
$$151$$ −3672.00 −1.97896 −0.989481 0.144666i $$-0.953789\pi$$
−0.989481 + 0.144666i $$0.953789\pi$$
$$152$$ 0 0
$$153$$ 1210.00 0.639364
$$154$$ 0 0
$$155$$ −1200.00 −0.621847
$$156$$ 0 0
$$157$$ 358.000 0.181984 0.0909921 0.995852i $$-0.470996\pi$$
0.0909921 + 0.995852i $$0.470996\pi$$
$$158$$ 0 0
$$159$$ −568.000 −0.283304
$$160$$ 0 0
$$161$$ 256.000 0.125314
$$162$$ 0 0
$$163$$ −2564.00 −1.23207 −0.616037 0.787717i $$-0.711263\pi$$
−0.616037 + 0.787717i $$0.711263\pi$$
$$164$$ 0 0
$$165$$ 720.000 0.339709
$$166$$ 0 0
$$167$$ 3056.00 1.41605 0.708025 0.706187i $$-0.249586\pi$$
0.708025 + 0.706187i $$0.249586\pi$$
$$168$$ 0 0
$$169$$ −433.000 −0.197087
$$170$$ 0 0
$$171$$ −1276.00 −0.570633
$$172$$ 0 0
$$173$$ −234.000 −0.102836 −0.0514182 0.998677i $$-0.516374\pi$$
−0.0514182 + 0.998677i $$0.516374\pi$$
$$174$$ 0 0
$$175$$ −400.000 −0.172784
$$176$$ 0 0
$$177$$ −1392.00 −0.591125
$$178$$ 0 0
$$179$$ −524.000 −0.218802 −0.109401 0.993998i $$-0.534893\pi$$
−0.109401 + 0.993998i $$0.534893\pi$$
$$180$$ 0 0
$$181$$ −1138.00 −0.467331 −0.233665 0.972317i $$-0.575072\pi$$
−0.233665 + 0.972317i $$0.575072\pi$$
$$182$$ 0 0
$$183$$ 2280.00 0.920997
$$184$$ 0 0
$$185$$ −1290.00 −0.512663
$$186$$ 0 0
$$187$$ 3960.00 1.54858
$$188$$ 0 0
$$189$$ −2432.00 −0.935989
$$190$$ 0 0
$$191$$ −1520.00 −0.575829 −0.287915 0.957656i $$-0.592962\pi$$
−0.287915 + 0.957656i $$0.592962\pi$$
$$192$$ 0 0
$$193$$ −2142.00 −0.798884 −0.399442 0.916759i $$-0.630796\pi$$
−0.399442 + 0.916759i $$0.630796\pi$$
$$194$$ 0 0
$$195$$ 840.000 0.308480
$$196$$ 0 0
$$197$$ −2306.00 −0.833988 −0.416994 0.908909i $$-0.636916\pi$$
−0.416994 + 0.908909i $$0.636916\pi$$
$$198$$ 0 0
$$199$$ −3288.00 −1.17126 −0.585628 0.810580i $$-0.699152\pi$$
−0.585628 + 0.810580i $$0.699152\pi$$
$$200$$ 0 0
$$201$$ 2768.00 0.971342
$$202$$ 0 0
$$203$$ −3168.00 −1.09532
$$204$$ 0 0
$$205$$ 2210.00 0.752942
$$206$$ 0 0
$$207$$ 176.000 0.0590959
$$208$$ 0 0
$$209$$ −4176.00 −1.38211
$$210$$ 0 0
$$211$$ 3876.00 1.26462 0.632310 0.774715i $$-0.282107\pi$$
0.632310 + 0.774715i $$0.282107\pi$$
$$212$$ 0 0
$$213$$ 672.000 0.216172
$$214$$ 0 0
$$215$$ 1460.00 0.463122
$$216$$ 0 0
$$217$$ 3840.00 1.20127
$$218$$ 0 0
$$219$$ 536.000 0.165386
$$220$$ 0 0
$$221$$ 4620.00 1.40622
$$222$$ 0 0
$$223$$ 5688.00 1.70806 0.854028 0.520226i $$-0.174152\pi$$
0.854028 + 0.520226i $$0.174152\pi$$
$$224$$ 0 0
$$225$$ −275.000 −0.0814815
$$226$$ 0 0
$$227$$ 2796.00 0.817520 0.408760 0.912642i $$-0.365961\pi$$
0.408760 + 0.912642i $$0.365961\pi$$
$$228$$ 0 0
$$229$$ 4446.00 1.28297 0.641485 0.767136i $$-0.278319\pi$$
0.641485 + 0.767136i $$0.278319\pi$$
$$230$$ 0 0
$$231$$ −2304.00 −0.656243
$$232$$ 0 0
$$233$$ 2522.00 0.709106 0.354553 0.935036i $$-0.384633\pi$$
0.354553 + 0.935036i $$0.384633\pi$$
$$234$$ 0 0
$$235$$ −1960.00 −0.544069
$$236$$ 0 0
$$237$$ 3136.00 0.859515
$$238$$ 0 0
$$239$$ −816.000 −0.220848 −0.110424 0.993885i $$-0.535221\pi$$
−0.110424 + 0.993885i $$0.535221\pi$$
$$240$$ 0 0
$$241$$ −5422.00 −1.44922 −0.724609 0.689160i $$-0.757980\pi$$
−0.724609 + 0.689160i $$0.757980\pi$$
$$242$$ 0 0
$$243$$ −2860.00 −0.755017
$$244$$ 0 0
$$245$$ −435.000 −0.113433
$$246$$ 0 0
$$247$$ −4872.00 −1.25505
$$248$$ 0 0
$$249$$ 2256.00 0.574169
$$250$$ 0 0
$$251$$ 5900.00 1.48368 0.741842 0.670575i $$-0.233952\pi$$
0.741842 + 0.670575i $$0.233952\pi$$
$$252$$ 0 0
$$253$$ 576.000 0.143134
$$254$$ 0 0
$$255$$ 2200.00 0.540272
$$256$$ 0 0
$$257$$ 5250.00 1.27426 0.637132 0.770754i $$-0.280120\pi$$
0.637132 + 0.770754i $$0.280120\pi$$
$$258$$ 0 0
$$259$$ 4128.00 0.990353
$$260$$ 0 0
$$261$$ −2178.00 −0.516532
$$262$$ 0 0
$$263$$ −6240.00 −1.46302 −0.731511 0.681829i $$-0.761185\pi$$
−0.731511 + 0.681829i $$0.761185\pi$$
$$264$$ 0 0
$$265$$ 710.000 0.164585
$$266$$ 0 0
$$267$$ −4136.00 −0.948012
$$268$$ 0 0
$$269$$ −714.000 −0.161834 −0.0809170 0.996721i $$-0.525785\pi$$
−0.0809170 + 0.996721i $$0.525785\pi$$
$$270$$ 0 0
$$271$$ −2144.00 −0.480586 −0.240293 0.970700i $$-0.577243\pi$$
−0.240293 + 0.970700i $$0.577243\pi$$
$$272$$ 0 0
$$273$$ −2688.00 −0.595916
$$274$$ 0 0
$$275$$ −900.000 −0.197353
$$276$$ 0 0
$$277$$ −4466.00 −0.968722 −0.484361 0.874868i $$-0.660948\pi$$
−0.484361 + 0.874868i $$0.660948\pi$$
$$278$$ 0 0
$$279$$ 2640.00 0.566497
$$280$$ 0 0
$$281$$ −5302.00 −1.12559 −0.562795 0.826596i $$-0.690274\pi$$
−0.562795 + 0.826596i $$0.690274\pi$$
$$282$$ 0 0
$$283$$ 6932.00 1.45606 0.728029 0.685546i $$-0.240436\pi$$
0.728029 + 0.685546i $$0.240436\pi$$
$$284$$ 0 0
$$285$$ −2320.00 −0.482193
$$286$$ 0 0
$$287$$ −7072.00 −1.45452
$$288$$ 0 0
$$289$$ 7187.00 1.46285
$$290$$ 0 0
$$291$$ 1528.00 0.307811
$$292$$ 0 0
$$293$$ −4034.00 −0.804330 −0.402165 0.915567i $$-0.631742\pi$$
−0.402165 + 0.915567i $$0.631742\pi$$
$$294$$ 0 0
$$295$$ 1740.00 0.343413
$$296$$ 0 0
$$297$$ −5472.00 −1.06908
$$298$$ 0 0
$$299$$ 672.000 0.129976
$$300$$ 0 0
$$301$$ −4672.00 −0.894650
$$302$$ 0 0
$$303$$ 2696.00 0.511159
$$304$$ 0 0
$$305$$ −2850.00 −0.535051
$$306$$ 0 0
$$307$$ 3836.00 0.713134 0.356567 0.934270i $$-0.383947\pi$$
0.356567 + 0.934270i $$0.383947\pi$$
$$308$$ 0 0
$$309$$ −3968.00 −0.730523
$$310$$ 0 0
$$311$$ −664.000 −0.121067 −0.0605337 0.998166i $$-0.519280\pi$$
−0.0605337 + 0.998166i $$0.519280\pi$$
$$312$$ 0 0
$$313$$ 2986.00 0.539229 0.269615 0.962968i $$-0.413104\pi$$
0.269615 + 0.962968i $$0.413104\pi$$
$$314$$ 0 0
$$315$$ 880.000 0.157404
$$316$$ 0 0
$$317$$ 2726.00 0.482989 0.241494 0.970402i $$-0.422362\pi$$
0.241494 + 0.970402i $$0.422362\pi$$
$$318$$ 0 0
$$319$$ −7128.00 −1.25107
$$320$$ 0 0
$$321$$ −2000.00 −0.347754
$$322$$ 0 0
$$323$$ −12760.0 −2.19810
$$324$$ 0 0
$$325$$ −1050.00 −0.179211
$$326$$ 0 0
$$327$$ −4184.00 −0.707571
$$328$$ 0 0
$$329$$ 6272.00 1.05102
$$330$$ 0 0
$$331$$ 9212.00 1.52972 0.764860 0.644197i $$-0.222808\pi$$
0.764860 + 0.644197i $$0.222808\pi$$
$$332$$ 0 0
$$333$$ 2838.00 0.467031
$$334$$ 0 0
$$335$$ −3460.00 −0.564298
$$336$$ 0 0
$$337$$ −3278.00 −0.529864 −0.264932 0.964267i $$-0.585349\pi$$
−0.264932 + 0.964267i $$0.585349\pi$$
$$338$$ 0 0
$$339$$ 2232.00 0.357598
$$340$$ 0 0
$$341$$ 8640.00 1.37209
$$342$$ 0 0
$$343$$ 6880.00 1.08305
$$344$$ 0 0
$$345$$ 320.000 0.0499369
$$346$$ 0 0
$$347$$ −4956.00 −0.766721 −0.383360 0.923599i $$-0.625233\pi$$
−0.383360 + 0.923599i $$0.625233\pi$$
$$348$$ 0 0
$$349$$ 4678.00 0.717500 0.358750 0.933434i $$-0.383203\pi$$
0.358750 + 0.933434i $$0.383203\pi$$
$$350$$ 0 0
$$351$$ −6384.00 −0.970805
$$352$$ 0 0
$$353$$ 1890.00 0.284970 0.142485 0.989797i $$-0.454491\pi$$
0.142485 + 0.989797i $$0.454491\pi$$
$$354$$ 0 0
$$355$$ −840.000 −0.125585
$$356$$ 0 0
$$357$$ −7040.00 −1.04369
$$358$$ 0 0
$$359$$ 6472.00 0.951474 0.475737 0.879588i $$-0.342181\pi$$
0.475737 + 0.879588i $$0.342181\pi$$
$$360$$ 0 0
$$361$$ 6597.00 0.961802
$$362$$ 0 0
$$363$$ 140.000 0.0202427
$$364$$ 0 0
$$365$$ −670.000 −0.0960806
$$366$$ 0 0
$$367$$ −1960.00 −0.278777 −0.139389 0.990238i $$-0.544514\pi$$
−0.139389 + 0.990238i $$0.544514\pi$$
$$368$$ 0 0
$$369$$ −4862.00 −0.685923
$$370$$ 0 0
$$371$$ −2272.00 −0.317942
$$372$$ 0 0
$$373$$ 8750.00 1.21463 0.607316 0.794460i $$-0.292246\pi$$
0.607316 + 0.794460i $$0.292246\pi$$
$$374$$ 0 0
$$375$$ −500.000 −0.0688530
$$376$$ 0 0
$$377$$ −8316.00 −1.13606
$$378$$ 0 0
$$379$$ 380.000 0.0515021 0.0257510 0.999668i $$-0.491802\pi$$
0.0257510 + 0.999668i $$0.491802\pi$$
$$380$$ 0 0
$$381$$ −1312.00 −0.176419
$$382$$ 0 0
$$383$$ 9688.00 1.29252 0.646258 0.763119i $$-0.276333\pi$$
0.646258 + 0.763119i $$0.276333\pi$$
$$384$$ 0 0
$$385$$ 2880.00 0.381243
$$386$$ 0 0
$$387$$ −3212.00 −0.421900
$$388$$ 0 0
$$389$$ 3870.00 0.504413 0.252207 0.967673i $$-0.418844\pi$$
0.252207 + 0.967673i $$0.418844\pi$$
$$390$$ 0 0
$$391$$ 1760.00 0.227639
$$392$$ 0 0
$$393$$ −848.000 −0.108845
$$394$$ 0 0
$$395$$ −3920.00 −0.499333
$$396$$ 0 0
$$397$$ 1622.00 0.205053 0.102526 0.994730i $$-0.467307\pi$$
0.102526 + 0.994730i $$0.467307\pi$$
$$398$$ 0 0
$$399$$ 7424.00 0.931491
$$400$$ 0 0
$$401$$ 9906.00 1.23362 0.616811 0.787112i $$-0.288424\pi$$
0.616811 + 0.787112i $$0.288424\pi$$
$$402$$ 0 0
$$403$$ 10080.0 1.24596
$$404$$ 0 0
$$405$$ −1555.00 −0.190787
$$406$$ 0 0
$$407$$ 9288.00 1.13118
$$408$$ 0 0
$$409$$ −4214.00 −0.509459 −0.254730 0.967012i $$-0.581986\pi$$
−0.254730 + 0.967012i $$0.581986\pi$$
$$410$$ 0 0
$$411$$ −5736.00 −0.688409
$$412$$ 0 0
$$413$$ −5568.00 −0.663398
$$414$$ 0 0
$$415$$ −2820.00 −0.333562
$$416$$ 0 0
$$417$$ 8784.00 1.03155
$$418$$ 0 0
$$419$$ 7012.00 0.817562 0.408781 0.912632i $$-0.365954\pi$$
0.408781 + 0.912632i $$0.365954\pi$$
$$420$$ 0 0
$$421$$ −1602.00 −0.185455 −0.0927277 0.995692i $$-0.529559\pi$$
−0.0927277 + 0.995692i $$0.529559\pi$$
$$422$$ 0 0
$$423$$ 4312.00 0.495642
$$424$$ 0 0
$$425$$ −2750.00 −0.313870
$$426$$ 0 0
$$427$$ 9120.00 1.03360
$$428$$ 0 0
$$429$$ −6048.00 −0.680653
$$430$$ 0 0
$$431$$ 3584.00 0.400546 0.200273 0.979740i $$-0.435817\pi$$
0.200273 + 0.979740i $$0.435817\pi$$
$$432$$ 0 0
$$433$$ −3470.00 −0.385121 −0.192561 0.981285i $$-0.561679\pi$$
−0.192561 + 0.981285i $$0.561679\pi$$
$$434$$ 0 0
$$435$$ −3960.00 −0.436477
$$436$$ 0 0
$$437$$ −1856.00 −0.203168
$$438$$ 0 0
$$439$$ 3416.00 0.371382 0.185691 0.982608i $$-0.440548\pi$$
0.185691 + 0.982608i $$0.440548\pi$$
$$440$$ 0 0
$$441$$ 957.000 0.103337
$$442$$ 0 0
$$443$$ −9708.00 −1.04118 −0.520588 0.853808i $$-0.674287\pi$$
−0.520588 + 0.853808i $$0.674287\pi$$
$$444$$ 0 0
$$445$$ 5170.00 0.550745
$$446$$ 0 0
$$447$$ 9672.00 1.02342
$$448$$ 0 0
$$449$$ −10366.0 −1.08954 −0.544768 0.838587i $$-0.683382\pi$$
−0.544768 + 0.838587i $$0.683382\pi$$
$$450$$ 0 0
$$451$$ −15912.0 −1.66135
$$452$$ 0 0
$$453$$ 14688.0 1.52340
$$454$$ 0 0
$$455$$ 3360.00 0.346196
$$456$$ 0 0
$$457$$ −16742.0 −1.71369 −0.856847 0.515572i $$-0.827580\pi$$
−0.856847 + 0.515572i $$0.827580\pi$$
$$458$$ 0 0
$$459$$ −16720.0 −1.70027
$$460$$ 0 0
$$461$$ −1258.00 −0.127095 −0.0635476 0.997979i $$-0.520241\pi$$
−0.0635476 + 0.997979i $$0.520241\pi$$
$$462$$ 0 0
$$463$$ −13528.0 −1.35788 −0.678941 0.734193i $$-0.737561\pi$$
−0.678941 + 0.734193i $$0.737561\pi$$
$$464$$ 0 0
$$465$$ 4800.00 0.478698
$$466$$ 0 0
$$467$$ −6916.00 −0.685298 −0.342649 0.939463i $$-0.611324\pi$$
−0.342649 + 0.939463i $$0.611324\pi$$
$$468$$ 0 0
$$469$$ 11072.0 1.09010
$$470$$ 0 0
$$471$$ −1432.00 −0.140091
$$472$$ 0 0
$$473$$ −10512.0 −1.02187
$$474$$ 0 0
$$475$$ 2900.00 0.280129
$$476$$ 0 0
$$477$$ −1562.00 −0.149935
$$478$$ 0 0
$$479$$ −1728.00 −0.164832 −0.0824158 0.996598i $$-0.526264\pi$$
−0.0824158 + 0.996598i $$0.526264\pi$$
$$480$$ 0 0
$$481$$ 10836.0 1.02719
$$482$$ 0 0
$$483$$ −1024.00 −0.0964671
$$484$$ 0 0
$$485$$ −1910.00 −0.178822
$$486$$ 0 0
$$487$$ −16656.0 −1.54981 −0.774903 0.632080i $$-0.782201\pi$$
−0.774903 + 0.632080i $$0.782201\pi$$
$$488$$ 0 0
$$489$$ 10256.0 0.948451
$$490$$ 0 0
$$491$$ 1084.00 0.0996339 0.0498169 0.998758i $$-0.484136\pi$$
0.0498169 + 0.998758i $$0.484136\pi$$
$$492$$ 0 0
$$493$$ −21780.0 −1.98970
$$494$$ 0 0
$$495$$ 1980.00 0.179787
$$496$$ 0 0
$$497$$ 2688.00 0.242602
$$498$$ 0 0
$$499$$ −5804.00 −0.520687 −0.260343 0.965516i $$-0.583836\pi$$
−0.260343 + 0.965516i $$0.583836\pi$$
$$500$$ 0 0
$$501$$ −12224.0 −1.09008
$$502$$ 0 0
$$503$$ −10512.0 −0.931823 −0.465911 0.884831i $$-0.654273\pi$$
−0.465911 + 0.884831i $$0.654273\pi$$
$$504$$ 0 0
$$505$$ −3370.00 −0.296956
$$506$$ 0 0
$$507$$ 1732.00 0.151718
$$508$$ 0 0
$$509$$ −4314.00 −0.375667 −0.187834 0.982201i $$-0.560147\pi$$
−0.187834 + 0.982201i $$0.560147\pi$$
$$510$$ 0 0
$$511$$ 2144.00 0.185607
$$512$$ 0 0
$$513$$ 17632.0 1.51749
$$514$$ 0 0
$$515$$ 4960.00 0.424396
$$516$$ 0 0
$$517$$ 14112.0 1.20047
$$518$$ 0 0
$$519$$ 936.000 0.0791635
$$520$$ 0 0
$$521$$ −1190.00 −0.100067 −0.0500334 0.998748i $$-0.515933\pi$$
−0.0500334 + 0.998748i $$0.515933\pi$$
$$522$$ 0 0
$$523$$ 3780.00 0.316038 0.158019 0.987436i $$-0.449489\pi$$
0.158019 + 0.987436i $$0.449489\pi$$
$$524$$ 0 0
$$525$$ 1600.00 0.133009
$$526$$ 0 0
$$527$$ 26400.0 2.18217
$$528$$ 0 0
$$529$$ −11911.0 −0.978959
$$530$$ 0 0
$$531$$ −3828.00 −0.312846
$$532$$ 0 0
$$533$$ −18564.0 −1.50862
$$534$$ 0 0
$$535$$ 2500.00 0.202027
$$536$$ 0 0
$$537$$ 2096.00 0.168434
$$538$$ 0 0
$$539$$ 3132.00 0.250287
$$540$$ 0 0
$$541$$ −11002.0 −0.874331 −0.437165 0.899381i $$-0.644018\pi$$
−0.437165 + 0.899381i $$0.644018\pi$$
$$542$$ 0 0
$$543$$ 4552.00 0.359751
$$544$$ 0 0
$$545$$ 5230.00 0.411062
$$546$$ 0 0
$$547$$ −5908.00 −0.461806 −0.230903 0.972977i $$-0.574168\pi$$
−0.230903 + 0.972977i $$0.574168\pi$$
$$548$$ 0 0
$$549$$ 6270.00 0.487426
$$550$$ 0 0
$$551$$ 22968.0 1.77581
$$552$$ 0 0
$$553$$ 12544.0 0.964602
$$554$$ 0 0
$$555$$ 5160.00 0.394648
$$556$$ 0 0
$$557$$ 14806.0 1.12630 0.563151 0.826354i $$-0.309589\pi$$
0.563151 + 0.826354i $$0.309589\pi$$
$$558$$ 0 0
$$559$$ −12264.0 −0.927928
$$560$$ 0 0
$$561$$ −15840.0 −1.19210
$$562$$ 0 0
$$563$$ 684.000 0.0512028 0.0256014 0.999672i $$-0.491850\pi$$
0.0256014 + 0.999672i $$0.491850\pi$$
$$564$$ 0 0
$$565$$ −2790.00 −0.207745
$$566$$ 0 0
$$567$$ 4976.00 0.368558
$$568$$ 0 0
$$569$$ −2582.00 −0.190234 −0.0951169 0.995466i $$-0.530323\pi$$
−0.0951169 + 0.995466i $$0.530323\pi$$
$$570$$ 0 0
$$571$$ 2540.00 0.186157 0.0930785 0.995659i $$-0.470329\pi$$
0.0930785 + 0.995659i $$0.470329\pi$$
$$572$$ 0 0
$$573$$ 6080.00 0.443273
$$574$$ 0 0
$$575$$ −400.000 −0.0290107
$$576$$ 0 0
$$577$$ 22786.0 1.64401 0.822005 0.569480i $$-0.192856\pi$$
0.822005 + 0.569480i $$0.192856\pi$$
$$578$$ 0 0
$$579$$ 8568.00 0.614981
$$580$$ 0 0
$$581$$ 9024.00 0.644369
$$582$$ 0 0
$$583$$ −5112.00 −0.363152
$$584$$ 0 0
$$585$$ 2310.00 0.163259
$$586$$ 0 0
$$587$$ −7884.00 −0.554357 −0.277178 0.960818i $$-0.589399\pi$$
−0.277178 + 0.960818i $$0.589399\pi$$
$$588$$ 0 0
$$589$$ −27840.0 −1.94758
$$590$$ 0 0
$$591$$ 9224.00 0.642005
$$592$$ 0 0
$$593$$ −21902.0 −1.51671 −0.758354 0.651843i $$-0.773996\pi$$
−0.758354 + 0.651843i $$0.773996\pi$$
$$594$$ 0 0
$$595$$ 8800.00 0.606327
$$596$$ 0 0
$$597$$ 13152.0 0.901634
$$598$$ 0 0
$$599$$ −15080.0 −1.02863 −0.514317 0.857600i $$-0.671955\pi$$
−0.514317 + 0.857600i $$0.671955\pi$$
$$600$$ 0 0
$$601$$ −19702.0 −1.33721 −0.668603 0.743619i $$-0.733108\pi$$
−0.668603 + 0.743619i $$0.733108\pi$$
$$602$$ 0 0
$$603$$ 7612.00 0.514071
$$604$$ 0 0
$$605$$ −175.000 −0.0117599
$$606$$ 0 0
$$607$$ −7320.00 −0.489472 −0.244736 0.969590i $$-0.578701\pi$$
−0.244736 + 0.969590i $$0.578701\pi$$
$$608$$ 0 0
$$609$$ 12672.0 0.843178
$$610$$ 0 0
$$611$$ 16464.0 1.09012
$$612$$ 0 0
$$613$$ 24350.0 1.60438 0.802192 0.597066i $$-0.203667\pi$$
0.802192 + 0.597066i $$0.203667\pi$$
$$614$$ 0 0
$$615$$ −8840.00 −0.579615
$$616$$ 0 0
$$617$$ 19546.0 1.27535 0.637676 0.770305i $$-0.279896\pi$$
0.637676 + 0.770305i $$0.279896\pi$$
$$618$$ 0 0
$$619$$ −3476.00 −0.225706 −0.112853 0.993612i $$-0.535999\pi$$
−0.112853 + 0.993612i $$0.535999\pi$$
$$620$$ 0 0
$$621$$ −2432.00 −0.157154
$$622$$ 0 0
$$623$$ −16544.0 −1.06392
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 16704.0 1.06394
$$628$$ 0 0
$$629$$ 28380.0 1.79902
$$630$$ 0 0
$$631$$ −21880.0 −1.38039 −0.690197 0.723621i $$-0.742476\pi$$
−0.690197 + 0.723621i $$0.742476\pi$$
$$632$$ 0 0
$$633$$ −15504.0 −0.973505
$$634$$ 0 0
$$635$$ 1640.00 0.102490
$$636$$ 0 0
$$637$$ 3654.00 0.227279
$$638$$ 0 0
$$639$$ 1848.00 0.114406
$$640$$ 0 0
$$641$$ 20994.0 1.29362 0.646812 0.762649i $$-0.276102\pi$$
0.646812 + 0.762649i $$0.276102\pi$$
$$642$$ 0 0
$$643$$ 18204.0 1.11648 0.558239 0.829680i $$-0.311477\pi$$
0.558239 + 0.829680i $$0.311477\pi$$
$$644$$ 0 0
$$645$$ −5840.00 −0.356511
$$646$$ 0 0
$$647$$ 2064.00 0.125416 0.0627080 0.998032i $$-0.480026\pi$$
0.0627080 + 0.998032i $$0.480026\pi$$
$$648$$ 0 0
$$649$$ −12528.0 −0.757730
$$650$$ 0 0
$$651$$ −15360.0 −0.924740
$$652$$ 0 0
$$653$$ 9942.00 0.595805 0.297902 0.954596i $$-0.403713\pi$$
0.297902 + 0.954596i $$0.403713\pi$$
$$654$$ 0 0
$$655$$ 1060.00 0.0632330
$$656$$ 0 0
$$657$$ 1474.00 0.0875285
$$658$$ 0 0
$$659$$ −24236.0 −1.43263 −0.716313 0.697779i $$-0.754172\pi$$
−0.716313 + 0.697779i $$0.754172\pi$$
$$660$$ 0 0
$$661$$ 17614.0 1.03647 0.518234 0.855239i $$-0.326590\pi$$
0.518234 + 0.855239i $$0.326590\pi$$
$$662$$ 0 0
$$663$$ −18480.0 −1.08251
$$664$$ 0 0
$$665$$ −9280.00 −0.541147
$$666$$ 0 0
$$667$$ −3168.00 −0.183906
$$668$$ 0 0
$$669$$ −22752.0 −1.31486
$$670$$ 0 0
$$671$$ 20520.0 1.18057
$$672$$ 0 0
$$673$$ 13058.0 0.747918 0.373959 0.927445i $$-0.378000\pi$$
0.373959 + 0.927445i $$0.378000\pi$$
$$674$$ 0 0
$$675$$ 3800.00 0.216685
$$676$$ 0 0
$$677$$ −33186.0 −1.88396 −0.941980 0.335668i $$-0.891038\pi$$
−0.941980 + 0.335668i $$0.891038\pi$$
$$678$$ 0 0
$$679$$ 6112.00 0.345445
$$680$$ 0 0
$$681$$ −11184.0 −0.629327
$$682$$ 0 0
$$683$$ 31716.0 1.77684 0.888418 0.459035i $$-0.151805\pi$$
0.888418 + 0.459035i $$0.151805\pi$$
$$684$$ 0 0
$$685$$ 7170.00 0.399929
$$686$$ 0 0
$$687$$ −17784.0 −0.987630
$$688$$ 0 0
$$689$$ −5964.00 −0.329768
$$690$$ 0 0
$$691$$ 2084.00 0.114731 0.0573655 0.998353i $$-0.481730\pi$$
0.0573655 + 0.998353i $$0.481730\pi$$
$$692$$ 0 0
$$693$$ −6336.00 −0.347308
$$694$$ 0 0
$$695$$ −10980.0 −0.599274
$$696$$ 0 0
$$697$$ −48620.0 −2.64220
$$698$$ 0 0
$$699$$ −10088.0 −0.545870
$$700$$ 0 0
$$701$$ −7418.00 −0.399678 −0.199839 0.979829i $$-0.564042\pi$$
−0.199839 + 0.979829i $$0.564042\pi$$
$$702$$ 0 0
$$703$$ −29928.0 −1.60563
$$704$$ 0 0
$$705$$ 7840.00 0.418825
$$706$$ 0 0
$$707$$ 10784.0 0.573655
$$708$$ 0 0
$$709$$ −18242.0 −0.966280 −0.483140 0.875543i $$-0.660504\pi$$
−0.483140 + 0.875543i $$0.660504\pi$$
$$710$$ 0 0
$$711$$ 8624.00 0.454888
$$712$$ 0 0
$$713$$ 3840.00 0.201696
$$714$$ 0 0
$$715$$ 7560.00 0.395424
$$716$$ 0 0
$$717$$ 3264.00 0.170009
$$718$$ 0 0
$$719$$ −3024.00 −0.156851 −0.0784257 0.996920i $$-0.524989\pi$$
−0.0784257 + 0.996920i $$0.524989\pi$$
$$720$$ 0 0
$$721$$ −15872.0 −0.819839
$$722$$ 0 0
$$723$$ 21688.0 1.11561
$$724$$ 0 0
$$725$$ 4950.00 0.253570
$$726$$ 0 0
$$727$$ 26176.0 1.33537 0.667685 0.744444i $$-0.267285\pi$$
0.667685 + 0.744444i $$0.267285\pi$$
$$728$$ 0 0
$$729$$ 19837.0 1.00782
$$730$$ 0 0
$$731$$ −32120.0 −1.62517
$$732$$ 0 0
$$733$$ −17818.0 −0.897848 −0.448924 0.893570i $$-0.648193\pi$$
−0.448924 + 0.893570i $$0.648193\pi$$
$$734$$ 0 0
$$735$$ 1740.00 0.0873209
$$736$$ 0 0
$$737$$ 24912.0 1.24511
$$738$$ 0 0
$$739$$ 22052.0 1.09769 0.548847 0.835923i $$-0.315067\pi$$
0.548847 + 0.835923i $$0.315067\pi$$
$$740$$ 0 0
$$741$$ 19488.0 0.966140
$$742$$ 0 0
$$743$$ −15840.0 −0.782117 −0.391059 0.920366i $$-0.627891\pi$$
−0.391059 + 0.920366i $$0.627891\pi$$
$$744$$ 0 0
$$745$$ −12090.0 −0.594555
$$746$$ 0 0
$$747$$ 6204.00 0.303872
$$748$$ 0 0
$$749$$ −8000.00 −0.390272
$$750$$ 0 0
$$751$$ −21024.0 −1.02154 −0.510770 0.859717i $$-0.670640\pi$$
−0.510770 + 0.859717i $$0.670640\pi$$
$$752$$ 0 0
$$753$$ −23600.0 −1.14214
$$754$$ 0 0
$$755$$ −18360.0 −0.885018
$$756$$ 0 0
$$757$$ −38034.0 −1.82612 −0.913058 0.407831i $$-0.866285\pi$$
−0.913058 + 0.407831i $$0.866285\pi$$
$$758$$ 0 0
$$759$$ −2304.00 −0.110184
$$760$$ 0 0
$$761$$ 37802.0 1.80069 0.900343 0.435182i $$-0.143316\pi$$
0.900343 + 0.435182i $$0.143316\pi$$
$$762$$ 0 0
$$763$$ −16736.0 −0.794081
$$764$$ 0 0
$$765$$ 6050.00 0.285932
$$766$$ 0 0
$$767$$ −14616.0 −0.688075
$$768$$ 0 0
$$769$$ 15042.0 0.705369 0.352684 0.935742i $$-0.385269\pi$$
0.352684 + 0.935742i $$0.385269\pi$$
$$770$$ 0 0
$$771$$ −21000.0 −0.980929
$$772$$ 0 0
$$773$$ 5950.00 0.276852 0.138426 0.990373i $$-0.455796\pi$$
0.138426 + 0.990373i $$0.455796\pi$$
$$774$$ 0 0
$$775$$ −6000.00 −0.278099
$$776$$ 0 0
$$777$$ −16512.0 −0.762374
$$778$$ 0 0
$$779$$ 51272.0 2.35816
$$780$$ 0 0
$$781$$ 6048.00 0.277099
$$782$$ 0 0
$$783$$ 30096.0 1.37362
$$784$$ 0 0
$$785$$ 1790.00 0.0813858
$$786$$ 0 0
$$787$$ −23364.0 −1.05824 −0.529121 0.848546i $$-0.677478\pi$$
−0.529121 + 0.848546i $$0.677478\pi$$
$$788$$ 0 0
$$789$$ 24960.0 1.12624
$$790$$ 0 0
$$791$$ 8928.00 0.401319
$$792$$ 0 0
$$793$$ 23940.0 1.07205
$$794$$ 0 0
$$795$$ −2840.00 −0.126697
$$796$$ 0 0
$$797$$ 19846.0 0.882034 0.441017 0.897499i $$-0.354618\pi$$
0.441017 + 0.897499i $$0.354618\pi$$
$$798$$ 0 0
$$799$$ 43120.0 1.90923
$$800$$ 0 0
$$801$$ −11374.0 −0.501724
$$802$$ 0 0
$$803$$ 4824.00 0.211999
$$804$$ 0 0
$$805$$ 1280.00 0.0560423
$$806$$ 0 0
$$807$$ 2856.00 0.124580
$$808$$ 0 0
$$809$$ 24762.0 1.07613 0.538063 0.842905i $$-0.319156\pi$$
0.538063 + 0.842905i $$0.319156\pi$$
$$810$$ 0 0
$$811$$ −16644.0 −0.720653 −0.360327 0.932826i $$-0.617335\pi$$
−0.360327 + 0.932826i $$0.617335\pi$$
$$812$$ 0 0
$$813$$ 8576.00 0.369955
$$814$$ 0 0
$$815$$ −12820.0 −0.551000
$$816$$ 0 0
$$817$$ 33872.0 1.45047
$$818$$ 0 0
$$819$$ −7392.00 −0.315381
$$820$$ 0 0
$$821$$ 3182.00 0.135265 0.0676325 0.997710i $$-0.478455\pi$$
0.0676325 + 0.997710i $$0.478455\pi$$
$$822$$ 0 0
$$823$$ 7504.00 0.317829 0.158914 0.987292i $$-0.449201\pi$$
0.158914 + 0.987292i $$0.449201\pi$$
$$824$$ 0 0
$$825$$ 3600.00 0.151922
$$826$$ 0 0
$$827$$ −12604.0 −0.529969 −0.264984 0.964253i $$-0.585367\pi$$
−0.264984 + 0.964253i $$0.585367\pi$$
$$828$$ 0 0
$$829$$ 12230.0 0.512383 0.256191 0.966626i $$-0.417532\pi$$
0.256191 + 0.966626i $$0.417532\pi$$
$$830$$ 0 0
$$831$$ 17864.0 0.745722
$$832$$ 0 0
$$833$$ 9570.00 0.398056
$$834$$ 0 0
$$835$$ 15280.0 0.633277
$$836$$ 0 0
$$837$$ −36480.0 −1.50649
$$838$$ 0 0
$$839$$ −9656.00 −0.397333 −0.198666 0.980067i $$-0.563661\pi$$
−0.198666 + 0.980067i $$0.563661\pi$$
$$840$$ 0 0
$$841$$ 14815.0 0.607446
$$842$$ 0 0
$$843$$ 21208.0 0.866480
$$844$$ 0 0
$$845$$ −2165.00 −0.0881400
$$846$$ 0 0
$$847$$ 560.000 0.0227176
$$848$$ 0 0
$$849$$ −27728.0 −1.12087
$$850$$ 0 0
$$851$$ 4128.00 0.166282
$$852$$ 0 0
$$853$$ 5806.00 0.233052 0.116526 0.993188i $$-0.462824\pi$$
0.116526 + 0.993188i $$0.462824\pi$$
$$854$$ 0 0
$$855$$ −6380.00 −0.255195
$$856$$ 0 0
$$857$$ −39094.0 −1.55826 −0.779128 0.626865i $$-0.784338\pi$$
−0.779128 + 0.626865i $$0.784338\pi$$
$$858$$ 0 0
$$859$$ 18876.0 0.749756 0.374878 0.927074i $$-0.377685\pi$$
0.374878 + 0.927074i $$0.377685\pi$$
$$860$$ 0 0
$$861$$ 28288.0 1.11969
$$862$$ 0 0
$$863$$ −32296.0 −1.27389 −0.636946 0.770909i $$-0.719803\pi$$
−0.636946 + 0.770909i $$0.719803\pi$$
$$864$$ 0 0
$$865$$ −1170.00 −0.0459898
$$866$$ 0 0
$$867$$ −28748.0 −1.12611
$$868$$ 0 0
$$869$$ 28224.0 1.10176
$$870$$ 0 0
$$871$$ 29064.0 1.13065
$$872$$ 0 0
$$873$$ 4202.00 0.162905
$$874$$ 0 0
$$875$$ −2000.00 −0.0772712
$$876$$ 0 0
$$877$$ −9578.00 −0.368787 −0.184393 0.982853i $$-0.559032\pi$$
−0.184393 + 0.982853i $$0.559032\pi$$
$$878$$ 0 0
$$879$$ 16136.0 0.619174
$$880$$ 0 0
$$881$$ −41710.0 −1.59506 −0.797529 0.603281i $$-0.793860\pi$$
−0.797529 + 0.603281i $$0.793860\pi$$
$$882$$ 0 0
$$883$$ −2260.00 −0.0861326 −0.0430663 0.999072i $$-0.513713\pi$$
−0.0430663 + 0.999072i $$0.513713\pi$$
$$884$$ 0 0
$$885$$ −6960.00 −0.264359
$$886$$ 0 0
$$887$$ 33696.0 1.27554 0.637768 0.770228i $$-0.279858\pi$$
0.637768 + 0.770228i $$0.279858\pi$$
$$888$$ 0 0
$$889$$ −5248.00 −0.197989
$$890$$ 0 0
$$891$$ 11196.0 0.420965
$$892$$ 0 0
$$893$$ −45472.0 −1.70399
$$894$$ 0 0
$$895$$ −2620.00 −0.0978513
$$896$$ 0 0
$$897$$ −2688.00 −0.100055
$$898$$ 0 0
$$899$$ −47520.0 −1.76294
$$900$$ 0 0
$$901$$ −15620.0 −0.577556
$$902$$ 0 0
$$903$$ 18688.0 0.688702
$$904$$ 0 0
$$905$$ −5690.00 −0.208997
$$906$$ 0 0
$$907$$ −7756.00 −0.283940 −0.141970 0.989871i $$-0.545344\pi$$
−0.141970 + 0.989871i $$0.545344\pi$$
$$908$$ 0 0
$$909$$ 7414.00 0.270525
$$910$$ 0 0
$$911$$ 5312.00 0.193188 0.0965941 0.995324i $$-0.469205\pi$$
0.0965941 + 0.995324i $$0.469205\pi$$
$$912$$ 0 0
$$913$$ 20304.0 0.735996
$$914$$ 0 0
$$915$$ 11400.0 0.411882
$$916$$ 0 0
$$917$$ −3392.00 −0.122152
$$918$$ 0 0
$$919$$ 23576.0 0.846246 0.423123 0.906072i $$-0.360934\pi$$
0.423123 + 0.906072i $$0.360934\pi$$
$$920$$ 0 0
$$921$$ −15344.0 −0.548971
$$922$$ 0 0
$$923$$ 7056.00 0.251626
$$924$$ 0 0
$$925$$ −6450.00 −0.229270
$$926$$ 0 0
$$927$$ −10912.0 −0.386620
$$928$$ 0 0
$$929$$ −19038.0 −0.672354 −0.336177 0.941799i $$-0.609134\pi$$
−0.336177 + 0.941799i $$0.609134\pi$$
$$930$$ 0 0
$$931$$ −10092.0 −0.355265
$$932$$ 0 0
$$933$$ 2656.00 0.0931978
$$934$$ 0 0
$$935$$ 19800.0 0.692545
$$936$$ 0 0
$$937$$ 20570.0 0.717175 0.358587 0.933496i $$-0.383259\pi$$
0.358587 + 0.933496i $$0.383259\pi$$
$$938$$ 0 0
$$939$$ −11944.0 −0.415099
$$940$$ 0 0
$$941$$ −21386.0 −0.740875 −0.370438 0.928857i $$-0.620792\pi$$
−0.370438 + 0.928857i $$0.620792\pi$$
$$942$$ 0 0
$$943$$ −7072.00 −0.244216
$$944$$ 0 0
$$945$$ −12160.0 −0.418587
$$946$$ 0 0
$$947$$ −38020.0 −1.30463 −0.652315 0.757948i $$-0.726202\pi$$
−0.652315 + 0.757948i $$0.726202\pi$$
$$948$$ 0 0
$$949$$ 5628.00 0.192511
$$950$$ 0 0
$$951$$ −10904.0 −0.371805
$$952$$ 0 0
$$953$$ 20202.0 0.686681 0.343340 0.939211i $$-0.388442\pi$$
0.343340 + 0.939211i $$0.388442\pi$$
$$954$$ 0 0
$$955$$ −7600.00 −0.257519
$$956$$ 0 0
$$957$$ 28512.0 0.963074
$$958$$ 0 0
$$959$$ −22944.0 −0.772576
$$960$$ 0 0
$$961$$ 27809.0 0.933470
$$962$$ 0 0
$$963$$ −5500.00 −0.184045
$$964$$ 0 0
$$965$$ −10710.0 −0.357272
$$966$$ 0 0
$$967$$ −29840.0 −0.992337 −0.496168 0.868226i $$-0.665260\pi$$
−0.496168 + 0.868226i $$0.665260\pi$$
$$968$$ 0 0
$$969$$ 51040.0 1.69210
$$970$$ 0 0
$$971$$ 12476.0 0.412332 0.206166 0.978517i $$-0.433901\pi$$
0.206166 + 0.978517i $$0.433901\pi$$
$$972$$ 0 0
$$973$$ 35136.0 1.15767
$$974$$ 0 0
$$975$$ 4200.00 0.137957
$$976$$ 0 0
$$977$$ −36974.0 −1.21075 −0.605375 0.795940i $$-0.706977\pi$$
−0.605375 + 0.795940i $$0.706977\pi$$
$$978$$ 0 0
$$979$$ −37224.0 −1.21520
$$980$$ 0 0
$$981$$ −11506.0 −0.374473
$$982$$ 0 0
$$983$$ 16368.0 0.531087 0.265543 0.964099i $$-0.414449\pi$$
0.265543 + 0.964099i $$0.414449\pi$$
$$984$$ 0 0
$$985$$ −11530.0 −0.372971
$$986$$ 0 0
$$987$$ −25088.0 −0.809078
$$988$$ 0 0
$$989$$ −4672.00 −0.150213
$$990$$ 0 0
$$991$$ 49552.0 1.58837 0.794183 0.607678i $$-0.207899\pi$$
0.794183 + 0.607678i $$0.207899\pi$$
$$992$$ 0 0
$$993$$ −36848.0 −1.17758
$$994$$ 0 0
$$995$$ −16440.0 −0.523802
$$996$$ 0 0
$$997$$ 24414.0 0.775526 0.387763 0.921759i $$-0.373248\pi$$
0.387763 + 0.921759i $$0.373248\pi$$
$$998$$ 0 0
$$999$$ −39216.0 −1.24198
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 80.4.a.b.1.1 1
3.2 odd 2 720.4.a.d.1.1 1
4.3 odd 2 40.4.a.b.1.1 1
5.2 odd 4 400.4.c.h.49.2 2
5.3 odd 4 400.4.c.h.49.1 2
5.4 even 2 400.4.a.p.1.1 1
8.3 odd 2 320.4.a.e.1.1 1
8.5 even 2 320.4.a.j.1.1 1
12.11 even 2 360.4.a.f.1.1 1
16.3 odd 4 1280.4.d.d.641.2 2
16.5 even 4 1280.4.d.m.641.2 2
16.11 odd 4 1280.4.d.d.641.1 2
16.13 even 4 1280.4.d.m.641.1 2
20.3 even 4 200.4.c.f.49.2 2
20.7 even 4 200.4.c.f.49.1 2
20.19 odd 2 200.4.a.d.1.1 1
28.27 even 2 1960.4.a.e.1.1 1
40.19 odd 2 1600.4.a.bk.1.1 1
40.29 even 2 1600.4.a.q.1.1 1
60.23 odd 4 1800.4.f.d.649.1 2
60.47 odd 4 1800.4.f.d.649.2 2
60.59 even 2 1800.4.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.a.b.1.1 1 4.3 odd 2
80.4.a.b.1.1 1 1.1 even 1 trivial
200.4.a.d.1.1 1 20.19 odd 2
200.4.c.f.49.1 2 20.7 even 4
200.4.c.f.49.2 2 20.3 even 4
320.4.a.e.1.1 1 8.3 odd 2
320.4.a.j.1.1 1 8.5 even 2
360.4.a.f.1.1 1 12.11 even 2
400.4.a.p.1.1 1 5.4 even 2
400.4.c.h.49.1 2 5.3 odd 4
400.4.c.h.49.2 2 5.2 odd 4
720.4.a.d.1.1 1 3.2 odd 2
1280.4.d.d.641.1 2 16.11 odd 4
1280.4.d.d.641.2 2 16.3 odd 4
1280.4.d.m.641.1 2 16.13 even 4
1280.4.d.m.641.2 2 16.5 even 4
1600.4.a.q.1.1 1 40.29 even 2
1600.4.a.bk.1.1 1 40.19 odd 2
1800.4.a.h.1.1 1 60.59 even 2
1800.4.f.d.649.1 2 60.23 odd 4
1800.4.f.d.649.2 2 60.47 odd 4
1960.4.a.e.1.1 1 28.27 even 2