# Properties

 Label 80.6.a.e.1.1 Level $80$ Weight $6$ Character 80.1 Self dual yes Analytic conductor $12.831$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 80.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.8307055850$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.00000 q^{3} +25.0000 q^{5} -192.000 q^{7} -227.000 q^{9} +O(q^{10})$$ $$q+4.00000 q^{3} +25.0000 q^{5} -192.000 q^{7} -227.000 q^{9} +148.000 q^{11} +286.000 q^{13} +100.000 q^{15} -1678.00 q^{17} -1060.00 q^{19} -768.000 q^{21} -2976.00 q^{23} +625.000 q^{25} -1880.00 q^{27} -3410.00 q^{29} +2448.00 q^{31} +592.000 q^{33} -4800.00 q^{35} +182.000 q^{37} +1144.00 q^{39} -9398.00 q^{41} +1244.00 q^{43} -5675.00 q^{45} +12088.0 q^{47} +20057.0 q^{49} -6712.00 q^{51} +23846.0 q^{53} +3700.00 q^{55} -4240.00 q^{57} +20020.0 q^{59} +32302.0 q^{61} +43584.0 q^{63} +7150.00 q^{65} -60972.0 q^{67} -11904.0 q^{69} +32648.0 q^{71} -38774.0 q^{73} +2500.00 q^{75} -28416.0 q^{77} +33360.0 q^{79} +47641.0 q^{81} -16716.0 q^{83} -41950.0 q^{85} -13640.0 q^{87} +101370. q^{89} -54912.0 q^{91} +9792.00 q^{93} -26500.0 q^{95} -119038. q^{97} -33596.0 q^{99} +O(q^{100})$$

## 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 0.256600 0.128300 0.991735i $$-0.459048\pi$$
0.128300 + 0.991735i $$0.459048\pi$$
$$4$$ 0 0
$$5$$ 25.0000 0.447214
$$6$$ 0 0
$$7$$ −192.000 −1.48100 −0.740502 0.672054i $$-0.765412\pi$$
−0.740502 + 0.672054i $$0.765412\pi$$
$$8$$ 0 0
$$9$$ −227.000 −0.934156
$$10$$ 0 0
$$11$$ 148.000 0.368791 0.184395 0.982852i $$-0.440967\pi$$
0.184395 + 0.982852i $$0.440967\pi$$
$$12$$ 0 0
$$13$$ 286.000 0.469362 0.234681 0.972072i $$-0.424595\pi$$
0.234681 + 0.972072i $$0.424595\pi$$
$$14$$ 0 0
$$15$$ 100.000 0.114755
$$16$$ 0 0
$$17$$ −1678.00 −1.40822 −0.704109 0.710092i $$-0.748653\pi$$
−0.704109 + 0.710092i $$0.748653\pi$$
$$18$$ 0 0
$$19$$ −1060.00 −0.673631 −0.336815 0.941571i $$-0.609350\pi$$
−0.336815 + 0.941571i $$0.609350\pi$$
$$20$$ 0 0
$$21$$ −768.000 −0.380026
$$22$$ 0 0
$$23$$ −2976.00 −1.17304 −0.586521 0.809934i $$-0.699503\pi$$
−0.586521 + 0.809934i $$0.699503\pi$$
$$24$$ 0 0
$$25$$ 625.000 0.200000
$$26$$ 0 0
$$27$$ −1880.00 −0.496305
$$28$$ 0 0
$$29$$ −3410.00 −0.752938 −0.376469 0.926429i $$-0.622862\pi$$
−0.376469 + 0.926429i $$0.622862\pi$$
$$30$$ 0 0
$$31$$ 2448.00 0.457517 0.228758 0.973483i $$-0.426533\pi$$
0.228758 + 0.973483i $$0.426533\pi$$
$$32$$ 0 0
$$33$$ 592.000 0.0946317
$$34$$ 0 0
$$35$$ −4800.00 −0.662325
$$36$$ 0 0
$$37$$ 182.000 0.0218558 0.0109279 0.999940i $$-0.496521\pi$$
0.0109279 + 0.999940i $$0.496521\pi$$
$$38$$ 0 0
$$39$$ 1144.00 0.120438
$$40$$ 0 0
$$41$$ −9398.00 −0.873124 −0.436562 0.899674i $$-0.643804\pi$$
−0.436562 + 0.899674i $$0.643804\pi$$
$$42$$ 0 0
$$43$$ 1244.00 0.102600 0.0513002 0.998683i $$-0.483663\pi$$
0.0513002 + 0.998683i $$0.483663\pi$$
$$44$$ 0 0
$$45$$ −5675.00 −0.417767
$$46$$ 0 0
$$47$$ 12088.0 0.798196 0.399098 0.916908i $$-0.369323\pi$$
0.399098 + 0.916908i $$0.369323\pi$$
$$48$$ 0 0
$$49$$ 20057.0 1.19337
$$50$$ 0 0
$$51$$ −6712.00 −0.361349
$$52$$ 0 0
$$53$$ 23846.0 1.16607 0.583037 0.812446i $$-0.301864\pi$$
0.583037 + 0.812446i $$0.301864\pi$$
$$54$$ 0 0
$$55$$ 3700.00 0.164928
$$56$$ 0 0
$$57$$ −4240.00 −0.172854
$$58$$ 0 0
$$59$$ 20020.0 0.748745 0.374373 0.927278i $$-0.377858\pi$$
0.374373 + 0.927278i $$0.377858\pi$$
$$60$$ 0 0
$$61$$ 32302.0 1.11149 0.555744 0.831353i $$-0.312433\pi$$
0.555744 + 0.831353i $$0.312433\pi$$
$$62$$ 0 0
$$63$$ 43584.0 1.38349
$$64$$ 0 0
$$65$$ 7150.00 0.209905
$$66$$ 0 0
$$67$$ −60972.0 −1.65937 −0.829685 0.558231i $$-0.811480\pi$$
−0.829685 + 0.558231i $$0.811480\pi$$
$$68$$ 0 0
$$69$$ −11904.0 −0.301003
$$70$$ 0 0
$$71$$ 32648.0 0.768618 0.384309 0.923204i $$-0.374440\pi$$
0.384309 + 0.923204i $$0.374440\pi$$
$$72$$ 0 0
$$73$$ −38774.0 −0.851596 −0.425798 0.904818i $$-0.640007\pi$$
−0.425798 + 0.904818i $$0.640007\pi$$
$$74$$ 0 0
$$75$$ 2500.00 0.0513200
$$76$$ 0 0
$$77$$ −28416.0 −0.546180
$$78$$ 0 0
$$79$$ 33360.0 0.601393 0.300696 0.953720i $$-0.402781\pi$$
0.300696 + 0.953720i $$0.402781\pi$$
$$80$$ 0 0
$$81$$ 47641.0 0.806805
$$82$$ 0 0
$$83$$ −16716.0 −0.266340 −0.133170 0.991093i $$-0.542516\pi$$
−0.133170 + 0.991093i $$0.542516\pi$$
$$84$$ 0 0
$$85$$ −41950.0 −0.629774
$$86$$ 0 0
$$87$$ −13640.0 −0.193204
$$88$$ 0 0
$$89$$ 101370. 1.35655 0.678273 0.734810i $$-0.262729\pi$$
0.678273 + 0.734810i $$0.262729\pi$$
$$90$$ 0 0
$$91$$ −54912.0 −0.695126
$$92$$ 0 0
$$93$$ 9792.00 0.117399
$$94$$ 0 0
$$95$$ −26500.0 −0.301257
$$96$$ 0 0
$$97$$ −119038. −1.28457 −0.642283 0.766468i $$-0.722013\pi$$
−0.642283 + 0.766468i $$0.722013\pi$$
$$98$$ 0 0
$$99$$ −33596.0 −0.344508
$$100$$ 0 0
$$101$$ −89898.0 −0.876893 −0.438446 0.898757i $$-0.644471\pi$$
−0.438446 + 0.898757i $$0.644471\pi$$
$$102$$ 0 0
$$103$$ 19504.0 0.181147 0.0905734 0.995890i $$-0.471130\pi$$
0.0905734 + 0.995890i $$0.471130\pi$$
$$104$$ 0 0
$$105$$ −19200.0 −0.169953
$$106$$ 0 0
$$107$$ −158292. −1.33659 −0.668297 0.743895i $$-0.732976\pi$$
−0.668297 + 0.743895i $$0.732976\pi$$
$$108$$ 0 0
$$109$$ 36830.0 0.296917 0.148459 0.988919i $$-0.452569\pi$$
0.148459 + 0.988919i $$0.452569\pi$$
$$110$$ 0 0
$$111$$ 728.000 0.00560821
$$112$$ 0 0
$$113$$ 11186.0 0.0824098 0.0412049 0.999151i $$-0.486880\pi$$
0.0412049 + 0.999151i $$0.486880\pi$$
$$114$$ 0 0
$$115$$ −74400.0 −0.524600
$$116$$ 0 0
$$117$$ −64922.0 −0.438457
$$118$$ 0 0
$$119$$ 322176. 2.08557
$$120$$ 0 0
$$121$$ −139147. −0.863993
$$122$$ 0 0
$$123$$ −37592.0 −0.224044
$$124$$ 0 0
$$125$$ 15625.0 0.0894427
$$126$$ 0 0
$$127$$ −70552.0 −0.388150 −0.194075 0.980987i $$-0.562171\pi$$
−0.194075 + 0.980987i $$0.562171\pi$$
$$128$$ 0 0
$$129$$ 4976.00 0.0263273
$$130$$ 0 0
$$131$$ −76452.0 −0.389234 −0.194617 0.980879i $$-0.562346\pi$$
−0.194617 + 0.980879i $$0.562346\pi$$
$$132$$ 0 0
$$133$$ 203520. 0.997650
$$134$$ 0 0
$$135$$ −47000.0 −0.221954
$$136$$ 0 0
$$137$$ −144918. −0.659661 −0.329831 0.944040i $$-0.606992\pi$$
−0.329831 + 0.944040i $$0.606992\pi$$
$$138$$ 0 0
$$139$$ −112220. −0.492644 −0.246322 0.969188i $$-0.579222\pi$$
−0.246322 + 0.969188i $$0.579222\pi$$
$$140$$ 0 0
$$141$$ 48352.0 0.204817
$$142$$ 0 0
$$143$$ 42328.0 0.173096
$$144$$ 0 0
$$145$$ −85250.0 −0.336724
$$146$$ 0 0
$$147$$ 80228.0 0.306219
$$148$$ 0 0
$$149$$ 403750. 1.48986 0.744932 0.667140i $$-0.232482\pi$$
0.744932 + 0.667140i $$0.232482\pi$$
$$150$$ 0 0
$$151$$ 446648. 1.59413 0.797064 0.603895i $$-0.206385\pi$$
0.797064 + 0.603895i $$0.206385\pi$$
$$152$$ 0 0
$$153$$ 380906. 1.31550
$$154$$ 0 0
$$155$$ 61200.0 0.204608
$$156$$ 0 0
$$157$$ −262258. −0.849141 −0.424570 0.905395i $$-0.639575\pi$$
−0.424570 + 0.905395i $$0.639575\pi$$
$$158$$ 0 0
$$159$$ 95384.0 0.299215
$$160$$ 0 0
$$161$$ 571392. 1.73728
$$162$$ 0 0
$$163$$ 154564. 0.455658 0.227829 0.973701i $$-0.426837\pi$$
0.227829 + 0.973701i $$0.426837\pi$$
$$164$$ 0 0
$$165$$ 14800.0 0.0423206
$$166$$ 0 0
$$167$$ −396672. −1.10063 −0.550314 0.834958i $$-0.685492\pi$$
−0.550314 + 0.834958i $$0.685492\pi$$
$$168$$ 0 0
$$169$$ −289497. −0.779700
$$170$$ 0 0
$$171$$ 240620. 0.629276
$$172$$ 0 0
$$173$$ −573474. −1.45680 −0.728398 0.685155i $$-0.759735\pi$$
−0.728398 + 0.685155i $$0.759735\pi$$
$$174$$ 0 0
$$175$$ −120000. −0.296201
$$176$$ 0 0
$$177$$ 80080.0 0.192128
$$178$$ 0 0
$$179$$ 594460. 1.38672 0.693362 0.720589i $$-0.256129\pi$$
0.693362 + 0.720589i $$0.256129\pi$$
$$180$$ 0 0
$$181$$ −107098. −0.242988 −0.121494 0.992592i $$-0.538769\pi$$
−0.121494 + 0.992592i $$0.538769\pi$$
$$182$$ 0 0
$$183$$ 129208. 0.285208
$$184$$ 0 0
$$185$$ 4550.00 0.00977422
$$186$$ 0 0
$$187$$ −248344. −0.519337
$$188$$ 0 0
$$189$$ 360960. 0.735029
$$190$$ 0 0
$$191$$ −469552. −0.931323 −0.465661 0.884963i $$-0.654184\pi$$
−0.465661 + 0.884963i $$0.654184\pi$$
$$192$$ 0 0
$$193$$ 52706.0 0.101851 0.0509257 0.998702i $$-0.483783\pi$$
0.0509257 + 0.998702i $$0.483783\pi$$
$$194$$ 0 0
$$195$$ 28600.0 0.0538616
$$196$$ 0 0
$$197$$ 455862. 0.836889 0.418444 0.908242i $$-0.362575\pi$$
0.418444 + 0.908242i $$0.362575\pi$$
$$198$$ 0 0
$$199$$ −865000. −1.54840 −0.774200 0.632940i $$-0.781848\pi$$
−0.774200 + 0.632940i $$0.781848\pi$$
$$200$$ 0 0
$$201$$ −243888. −0.425795
$$202$$ 0 0
$$203$$ 654720. 1.11510
$$204$$ 0 0
$$205$$ −234950. −0.390473
$$206$$ 0 0
$$207$$ 675552. 1.09580
$$208$$ 0 0
$$209$$ −156880. −0.248429
$$210$$ 0 0
$$211$$ −1.10565e6 −1.70967 −0.854835 0.518900i $$-0.826342\pi$$
−0.854835 + 0.518900i $$0.826342\pi$$
$$212$$ 0 0
$$213$$ 130592. 0.197228
$$214$$ 0 0
$$215$$ 31100.0 0.0458843
$$216$$ 0 0
$$217$$ −470016. −0.677584
$$218$$ 0 0
$$219$$ −155096. −0.218520
$$220$$ 0 0
$$221$$ −479908. −0.660963
$$222$$ 0 0
$$223$$ −1.12158e6 −1.51031 −0.755156 0.655545i $$-0.772439\pi$$
−0.755156 + 0.655545i $$0.772439\pi$$
$$224$$ 0 0
$$225$$ −141875. −0.186831
$$226$$ 0 0
$$227$$ 23348.0 0.0300736 0.0150368 0.999887i $$-0.495213\pi$$
0.0150368 + 0.999887i $$0.495213\pi$$
$$228$$ 0 0
$$229$$ −596010. −0.751043 −0.375522 0.926814i $$-0.622536\pi$$
−0.375522 + 0.926814i $$0.622536\pi$$
$$230$$ 0 0
$$231$$ −113664. −0.140150
$$232$$ 0 0
$$233$$ −485334. −0.585667 −0.292834 0.956163i $$-0.594598\pi$$
−0.292834 + 0.956163i $$0.594598\pi$$
$$234$$ 0 0
$$235$$ 302200. 0.356964
$$236$$ 0 0
$$237$$ 133440. 0.154317
$$238$$ 0 0
$$239$$ 48880.0 0.0553524 0.0276762 0.999617i $$-0.491189\pi$$
0.0276762 + 0.999617i $$0.491189\pi$$
$$240$$ 0 0
$$241$$ −110798. −0.122882 −0.0614411 0.998111i $$-0.519570\pi$$
−0.0614411 + 0.998111i $$0.519570\pi$$
$$242$$ 0 0
$$243$$ 647404. 0.703331
$$244$$ 0 0
$$245$$ 501425. 0.533692
$$246$$ 0 0
$$247$$ −303160. −0.316176
$$248$$ 0 0
$$249$$ −66864.0 −0.0683430
$$250$$ 0 0
$$251$$ 1.64375e6 1.64684 0.823419 0.567434i $$-0.192064\pi$$
0.823419 + 0.567434i $$0.192064\pi$$
$$252$$ 0 0
$$253$$ −440448. −0.432607
$$254$$ 0 0
$$255$$ −167800. −0.161600
$$256$$ 0 0
$$257$$ 1.30624e6 1.23365 0.616823 0.787102i $$-0.288419\pi$$
0.616823 + 0.787102i $$0.288419\pi$$
$$258$$ 0 0
$$259$$ −34944.0 −0.0323685
$$260$$ 0 0
$$261$$ 774070. 0.703362
$$262$$ 0 0
$$263$$ −2.12834e6 −1.89736 −0.948682 0.316231i $$-0.897583\pi$$
−0.948682 + 0.316231i $$0.897583\pi$$
$$264$$ 0 0
$$265$$ 596150. 0.521484
$$266$$ 0 0
$$267$$ 405480. 0.348090
$$268$$ 0 0
$$269$$ −1.44109e6 −1.21426 −0.607128 0.794604i $$-0.707679\pi$$
−0.607128 + 0.794604i $$0.707679\pi$$
$$270$$ 0 0
$$271$$ 93248.0 0.0771288 0.0385644 0.999256i $$-0.487722\pi$$
0.0385644 + 0.999256i $$0.487722\pi$$
$$272$$ 0 0
$$273$$ −219648. −0.178370
$$274$$ 0 0
$$275$$ 92500.0 0.0737581
$$276$$ 0 0
$$277$$ −110298. −0.0863711 −0.0431855 0.999067i $$-0.513751\pi$$
−0.0431855 + 0.999067i $$0.513751\pi$$
$$278$$ 0 0
$$279$$ −555696. −0.427392
$$280$$ 0 0
$$281$$ −192198. −0.145205 −0.0726027 0.997361i $$-0.523131\pi$$
−0.0726027 + 0.997361i $$0.523131\pi$$
$$282$$ 0 0
$$283$$ 331884. 0.246332 0.123166 0.992386i $$-0.460695\pi$$
0.123166 + 0.992386i $$0.460695\pi$$
$$284$$ 0 0
$$285$$ −106000. −0.0773025
$$286$$ 0 0
$$287$$ 1.80442e6 1.29310
$$288$$ 0 0
$$289$$ 1.39583e6 0.983076
$$290$$ 0 0
$$291$$ −476152. −0.329620
$$292$$ 0 0
$$293$$ 2.19481e6 1.49358 0.746788 0.665063i $$-0.231595\pi$$
0.746788 + 0.665063i $$0.231595\pi$$
$$294$$ 0 0
$$295$$ 500500. 0.334849
$$296$$ 0 0
$$297$$ −278240. −0.183033
$$298$$ 0 0
$$299$$ −851136. −0.550581
$$300$$ 0 0
$$301$$ −238848. −0.151952
$$302$$ 0 0
$$303$$ −359592. −0.225011
$$304$$ 0 0
$$305$$ 807550. 0.497073
$$306$$ 0 0
$$307$$ 2.37751e6 1.43971 0.719857 0.694123i $$-0.244207\pi$$
0.719857 + 0.694123i $$0.244207\pi$$
$$308$$ 0 0
$$309$$ 78016.0 0.0464823
$$310$$ 0 0
$$311$$ 2.37305e6 1.39125 0.695626 0.718405i $$-0.255127\pi$$
0.695626 + 0.718405i $$0.255127\pi$$
$$312$$ 0 0
$$313$$ −1.42941e6 −0.824702 −0.412351 0.911025i $$-0.635292\pi$$
−0.412351 + 0.911025i $$0.635292\pi$$
$$314$$ 0 0
$$315$$ 1.08960e6 0.618715
$$316$$ 0 0
$$317$$ 2.12462e6 1.18750 0.593750 0.804650i $$-0.297647\pi$$
0.593750 + 0.804650i $$0.297647\pi$$
$$318$$ 0 0
$$319$$ −504680. −0.277677
$$320$$ 0 0
$$321$$ −633168. −0.342970
$$322$$ 0 0
$$323$$ 1.77868e6 0.948618
$$324$$ 0 0
$$325$$ 178750. 0.0938723
$$326$$ 0 0
$$327$$ 147320. 0.0761890
$$328$$ 0 0
$$329$$ −2.32090e6 −1.18213
$$330$$ 0 0
$$331$$ −3.09985e6 −1.55515 −0.777573 0.628793i $$-0.783549\pi$$
−0.777573 + 0.628793i $$0.783549\pi$$
$$332$$ 0 0
$$333$$ −41314.0 −0.0204168
$$334$$ 0 0
$$335$$ −1.52430e6 −0.742093
$$336$$ 0 0
$$337$$ 2.40008e6 1.15120 0.575601 0.817731i $$-0.304768\pi$$
0.575601 + 0.817731i $$0.304768\pi$$
$$338$$ 0 0
$$339$$ 44744.0 0.0211464
$$340$$ 0 0
$$341$$ 362304. 0.168728
$$342$$ 0 0
$$343$$ −624000. −0.286384
$$344$$ 0 0
$$345$$ −297600. −0.134612
$$346$$ 0 0
$$347$$ −1.77741e6 −0.792436 −0.396218 0.918156i $$-0.629678\pi$$
−0.396218 + 0.918156i $$0.629678\pi$$
$$348$$ 0 0
$$349$$ −2.14805e6 −0.944019 −0.472010 0.881593i $$-0.656471\pi$$
−0.472010 + 0.881593i $$0.656471\pi$$
$$350$$ 0 0
$$351$$ −537680. −0.232946
$$352$$ 0 0
$$353$$ −661854. −0.282700 −0.141350 0.989960i $$-0.545144\pi$$
−0.141350 + 0.989960i $$0.545144\pi$$
$$354$$ 0 0
$$355$$ 816200. 0.343737
$$356$$ 0 0
$$357$$ 1.28870e6 0.535159
$$358$$ 0 0
$$359$$ 259320. 0.106194 0.0530970 0.998589i $$-0.483091\pi$$
0.0530970 + 0.998589i $$0.483091\pi$$
$$360$$ 0 0
$$361$$ −1.35250e6 −0.546222
$$362$$ 0 0
$$363$$ −556588. −0.221701
$$364$$ 0 0
$$365$$ −969350. −0.380845
$$366$$ 0 0
$$367$$ 1.49993e6 0.581307 0.290653 0.956828i $$-0.406127\pi$$
0.290653 + 0.956828i $$0.406127\pi$$
$$368$$ 0 0
$$369$$ 2.13335e6 0.815634
$$370$$ 0 0
$$371$$ −4.57843e6 −1.72696
$$372$$ 0 0
$$373$$ −2.23807e6 −0.832918 −0.416459 0.909154i $$-0.636729\pi$$
−0.416459 + 0.909154i $$0.636729\pi$$
$$374$$ 0 0
$$375$$ 62500.0 0.0229510
$$376$$ 0 0
$$377$$ −975260. −0.353400
$$378$$ 0 0
$$379$$ −3.15934e6 −1.12979 −0.564896 0.825162i $$-0.691084\pi$$
−0.564896 + 0.825162i $$0.691084\pi$$
$$380$$ 0 0
$$381$$ −282208. −0.0995994
$$382$$ 0 0
$$383$$ −342216. −0.119207 −0.0596037 0.998222i $$-0.518984\pi$$
−0.0596037 + 0.998222i $$0.518984\pi$$
$$384$$ 0 0
$$385$$ −710400. −0.244259
$$386$$ 0 0
$$387$$ −282388. −0.0958449
$$388$$ 0 0
$$389$$ 88470.0 0.0296430 0.0148215 0.999890i $$-0.495282\pi$$
0.0148215 + 0.999890i $$0.495282\pi$$
$$390$$ 0 0
$$391$$ 4.99373e6 1.65190
$$392$$ 0 0
$$393$$ −305808. −0.0998775
$$394$$ 0 0
$$395$$ 834000. 0.268951
$$396$$ 0 0
$$397$$ −5.45674e6 −1.73763 −0.868814 0.495138i $$-0.835117\pi$$
−0.868814 + 0.495138i $$0.835117\pi$$
$$398$$ 0 0
$$399$$ 814080. 0.255997
$$400$$ 0 0
$$401$$ 4.04680e6 1.25676 0.628378 0.777908i $$-0.283719\pi$$
0.628378 + 0.777908i $$0.283719\pi$$
$$402$$ 0 0
$$403$$ 700128. 0.214741
$$404$$ 0 0
$$405$$ 1.19102e6 0.360814
$$406$$ 0 0
$$407$$ 26936.0 0.00806022
$$408$$ 0 0
$$409$$ −2.71207e6 −0.801664 −0.400832 0.916151i $$-0.631279\pi$$
−0.400832 + 0.916151i $$0.631279\pi$$
$$410$$ 0 0
$$411$$ −579672. −0.169269
$$412$$ 0 0
$$413$$ −3.84384e6 −1.10889
$$414$$ 0 0
$$415$$ −417900. −0.119111
$$416$$ 0 0
$$417$$ −448880. −0.126413
$$418$$ 0 0
$$419$$ −3.71746e6 −1.03445 −0.517227 0.855848i $$-0.673036\pi$$
−0.517227 + 0.855848i $$0.673036\pi$$
$$420$$ 0 0
$$421$$ 3.55250e6 0.976853 0.488426 0.872605i $$-0.337571\pi$$
0.488426 + 0.872605i $$0.337571\pi$$
$$422$$ 0 0
$$423$$ −2.74398e6 −0.745640
$$424$$ 0 0
$$425$$ −1.04875e6 −0.281643
$$426$$ 0 0
$$427$$ −6.20198e6 −1.64612
$$428$$ 0 0
$$429$$ 169312. 0.0444165
$$430$$ 0 0
$$431$$ 4.06205e6 1.05330 0.526650 0.850082i $$-0.323448\pi$$
0.526650 + 0.850082i $$0.323448\pi$$
$$432$$ 0 0
$$433$$ 7.26287e6 1.86161 0.930804 0.365518i $$-0.119108\pi$$
0.930804 + 0.365518i $$0.119108\pi$$
$$434$$ 0 0
$$435$$ −341000. −0.0864035
$$436$$ 0 0
$$437$$ 3.15456e6 0.790197
$$438$$ 0 0
$$439$$ 5.41028e6 1.33986 0.669928 0.742426i $$-0.266325\pi$$
0.669928 + 0.742426i $$0.266325\pi$$
$$440$$ 0 0
$$441$$ −4.55294e6 −1.11480
$$442$$ 0 0
$$443$$ 6.51524e6 1.57733 0.788663 0.614826i $$-0.210774\pi$$
0.788663 + 0.614826i $$0.210774\pi$$
$$444$$ 0 0
$$445$$ 2.53425e6 0.606666
$$446$$ 0 0
$$447$$ 1.61500e6 0.382299
$$448$$ 0 0
$$449$$ −509950. −0.119375 −0.0596873 0.998217i $$-0.519010\pi$$
−0.0596873 + 0.998217i $$0.519010\pi$$
$$450$$ 0 0
$$451$$ −1.39090e6 −0.322000
$$452$$ 0 0
$$453$$ 1.78659e6 0.409053
$$454$$ 0 0
$$455$$ −1.37280e6 −0.310870
$$456$$ 0 0
$$457$$ 1.22084e6 0.273444 0.136722 0.990609i $$-0.456343\pi$$
0.136722 + 0.990609i $$0.456343\pi$$
$$458$$ 0 0
$$459$$ 3.15464e6 0.698905
$$460$$ 0 0
$$461$$ −4.07210e6 −0.892413 −0.446207 0.894930i $$-0.647225\pi$$
−0.446207 + 0.894930i $$0.647225\pi$$
$$462$$ 0 0
$$463$$ −2.02294e6 −0.438561 −0.219280 0.975662i $$-0.570371\pi$$
−0.219280 + 0.975662i $$0.570371\pi$$
$$464$$ 0 0
$$465$$ 244800. 0.0525024
$$466$$ 0 0
$$467$$ −3.25097e6 −0.689797 −0.344898 0.938640i $$-0.612087\pi$$
−0.344898 + 0.938640i $$0.612087\pi$$
$$468$$ 0 0
$$469$$ 1.17066e7 2.45753
$$470$$ 0 0
$$471$$ −1.04903e6 −0.217890
$$472$$ 0 0
$$473$$ 184112. 0.0378381
$$474$$ 0 0
$$475$$ −662500. −0.134726
$$476$$ 0 0
$$477$$ −5.41304e6 −1.08929
$$478$$ 0 0
$$479$$ 3.27936e6 0.653056 0.326528 0.945188i $$-0.394121\pi$$
0.326528 + 0.945188i $$0.394121\pi$$
$$480$$ 0 0
$$481$$ 52052.0 0.0102583
$$482$$ 0 0
$$483$$ 2.28557e6 0.445786
$$484$$ 0 0
$$485$$ −2.97595e6 −0.574475
$$486$$ 0 0
$$487$$ 8.53197e6 1.63015 0.815074 0.579357i $$-0.196696\pi$$
0.815074 + 0.579357i $$0.196696\pi$$
$$488$$ 0 0
$$489$$ 618256. 0.116922
$$490$$ 0 0
$$491$$ −1.51265e6 −0.283162 −0.141581 0.989927i $$-0.545219\pi$$
−0.141581 + 0.989927i $$0.545219\pi$$
$$492$$ 0 0
$$493$$ 5.72198e6 1.06030
$$494$$ 0 0
$$495$$ −839900. −0.154069
$$496$$ 0 0
$$497$$ −6.26842e6 −1.13833
$$498$$ 0 0
$$499$$ 6.49190e6 1.16713 0.583567 0.812065i $$-0.301657\pi$$
0.583567 + 0.812065i $$0.301657\pi$$
$$500$$ 0 0
$$501$$ −1.58669e6 −0.282421
$$502$$ 0 0
$$503$$ −8.61770e6 −1.51870 −0.759349 0.650684i $$-0.774482\pi$$
−0.759349 + 0.650684i $$0.774482\pi$$
$$504$$ 0 0
$$505$$ −2.24745e6 −0.392158
$$506$$ 0 0
$$507$$ −1.15799e6 −0.200071
$$508$$ 0 0
$$509$$ 2.67323e6 0.457343 0.228671 0.973504i $$-0.426562\pi$$
0.228671 + 0.973504i $$0.426562\pi$$
$$510$$ 0 0
$$511$$ 7.44461e6 1.26122
$$512$$ 0 0
$$513$$ 1.99280e6 0.334326
$$514$$ 0 0
$$515$$ 487600. 0.0810113
$$516$$ 0 0
$$517$$ 1.78902e6 0.294367
$$518$$ 0 0
$$519$$ −2.29390e6 −0.373814
$$520$$ 0 0
$$521$$ 6.18500e6 0.998264 0.499132 0.866526i $$-0.333652\pi$$
0.499132 + 0.866526i $$0.333652\pi$$
$$522$$ 0 0
$$523$$ 6.89452e6 1.10217 0.551087 0.834448i $$-0.314213\pi$$
0.551087 + 0.834448i $$0.314213\pi$$
$$524$$ 0 0
$$525$$ −480000. −0.0760051
$$526$$ 0 0
$$527$$ −4.10774e6 −0.644283
$$528$$ 0 0
$$529$$ 2.42023e6 0.376026
$$530$$ 0 0
$$531$$ −4.54454e6 −0.699445
$$532$$ 0 0
$$533$$ −2.68783e6 −0.409811
$$534$$ 0 0
$$535$$ −3.95730e6 −0.597743
$$536$$ 0 0
$$537$$ 2.37784e6 0.355834
$$538$$ 0 0
$$539$$ 2.96844e6 0.440104
$$540$$ 0 0
$$541$$ 155502. 0.0228425 0.0114212 0.999935i $$-0.496364\pi$$
0.0114212 + 0.999935i $$0.496364\pi$$
$$542$$ 0 0
$$543$$ −428392. −0.0623508
$$544$$ 0 0
$$545$$ 920750. 0.132785
$$546$$ 0 0
$$547$$ −1.26544e7 −1.80831 −0.904157 0.427201i $$-0.859500\pi$$
−0.904157 + 0.427201i $$0.859500\pi$$
$$548$$ 0 0
$$549$$ −7.33255e6 −1.03830
$$550$$ 0 0
$$551$$ 3.61460e6 0.507202
$$552$$ 0 0
$$553$$ −6.40512e6 −0.890665
$$554$$ 0 0
$$555$$ 18200.0 0.00250807
$$556$$ 0 0
$$557$$ −7.07786e6 −0.966638 −0.483319 0.875444i $$-0.660569\pi$$
−0.483319 + 0.875444i $$0.660569\pi$$
$$558$$ 0 0
$$559$$ 355784. 0.0481567
$$560$$ 0 0
$$561$$ −993376. −0.133262
$$562$$ 0 0
$$563$$ −846636. −0.112571 −0.0562854 0.998415i $$-0.517926\pi$$
−0.0562854 + 0.998415i $$0.517926\pi$$
$$564$$ 0 0
$$565$$ 279650. 0.0368548
$$566$$ 0 0
$$567$$ −9.14707e6 −1.19488
$$568$$ 0 0
$$569$$ 4.96041e6 0.642299 0.321149 0.947029i $$-0.395931\pi$$
0.321149 + 0.947029i $$0.395931\pi$$
$$570$$ 0 0
$$571$$ −8.96505e6 −1.15070 −0.575351 0.817907i $$-0.695134\pi$$
−0.575351 + 0.817907i $$0.695134\pi$$
$$572$$ 0 0
$$573$$ −1.87821e6 −0.238978
$$574$$ 0 0
$$575$$ −1.86000e6 −0.234608
$$576$$ 0 0
$$577$$ −2.86080e6 −0.357724 −0.178862 0.983874i $$-0.557242\pi$$
−0.178862 + 0.983874i $$0.557242\pi$$
$$578$$ 0 0
$$579$$ 210824. 0.0261351
$$580$$ 0 0
$$581$$ 3.20947e6 0.394451
$$582$$ 0 0
$$583$$ 3.52921e6 0.430037
$$584$$ 0 0
$$585$$ −1.62305e6 −0.196084
$$586$$ 0 0
$$587$$ 6.74027e6 0.807387 0.403694 0.914894i $$-0.367726\pi$$
0.403694 + 0.914894i $$0.367726\pi$$
$$588$$ 0 0
$$589$$ −2.59488e6 −0.308197
$$590$$ 0 0
$$591$$ 1.82345e6 0.214746
$$592$$ 0 0
$$593$$ −1.78609e6 −0.208578 −0.104289 0.994547i $$-0.533257\pi$$
−0.104289 + 0.994547i $$0.533257\pi$$
$$594$$ 0 0
$$595$$ 8.05440e6 0.932697
$$596$$ 0 0
$$597$$ −3.46000e6 −0.397320
$$598$$ 0 0
$$599$$ −4.94620e6 −0.563254 −0.281627 0.959524i $$-0.590874\pi$$
−0.281627 + 0.959524i $$0.590874\pi$$
$$600$$ 0 0
$$601$$ −4.58100e6 −0.517337 −0.258669 0.965966i $$-0.583284\pi$$
−0.258669 + 0.965966i $$0.583284\pi$$
$$602$$ 0 0
$$603$$ 1.38406e7 1.55011
$$604$$ 0 0
$$605$$ −3.47868e6 −0.386390
$$606$$ 0 0
$$607$$ −7.07999e6 −0.779940 −0.389970 0.920828i $$-0.627515\pi$$
−0.389970 + 0.920828i $$0.627515\pi$$
$$608$$ 0 0
$$609$$ 2.61888e6 0.286136
$$610$$ 0 0
$$611$$ 3.45717e6 0.374643
$$612$$ 0 0
$$613$$ 5.09609e6 0.547754 0.273877 0.961765i $$-0.411694\pi$$
0.273877 + 0.961765i $$0.411694\pi$$
$$614$$ 0 0
$$615$$ −939800. −0.100195
$$616$$ 0 0
$$617$$ −1.30003e7 −1.37480 −0.687400 0.726279i $$-0.741248\pi$$
−0.687400 + 0.726279i $$0.741248\pi$$
$$618$$ 0 0
$$619$$ −4.84406e6 −0.508139 −0.254070 0.967186i $$-0.581769\pi$$
−0.254070 + 0.967186i $$0.581769\pi$$
$$620$$ 0 0
$$621$$ 5.59488e6 0.582186
$$622$$ 0 0
$$623$$ −1.94630e7 −2.00905
$$624$$ 0 0
$$625$$ 390625. 0.0400000
$$626$$ 0 0
$$627$$ −627520. −0.0637468
$$628$$ 0 0
$$629$$ −305396. −0.0307777
$$630$$ 0 0
$$631$$ −6.22775e6 −0.622670 −0.311335 0.950300i $$-0.600776\pi$$
−0.311335 + 0.950300i $$0.600776\pi$$
$$632$$ 0 0
$$633$$ −4.42261e6 −0.438702
$$634$$ 0 0
$$635$$ −1.76380e6 −0.173586
$$636$$ 0 0
$$637$$ 5.73630e6 0.560123
$$638$$ 0 0
$$639$$ −7.41110e6 −0.718010
$$640$$ 0 0
$$641$$ 1.53280e6 0.147347 0.0736734 0.997282i $$-0.476528\pi$$
0.0736734 + 0.997282i $$0.476528\pi$$
$$642$$ 0 0
$$643$$ 1.74382e7 1.66332 0.831659 0.555287i $$-0.187391\pi$$
0.831659 + 0.555287i $$0.187391\pi$$
$$644$$ 0 0
$$645$$ 124400. 0.0117739
$$646$$ 0 0
$$647$$ 4.25469e6 0.399583 0.199792 0.979838i $$-0.435974\pi$$
0.199792 + 0.979838i $$0.435974\pi$$
$$648$$ 0 0
$$649$$ 2.96296e6 0.276130
$$650$$ 0 0
$$651$$ −1.88006e6 −0.173868
$$652$$ 0 0
$$653$$ 3.01085e6 0.276316 0.138158 0.990410i $$-0.455882\pi$$
0.138158 + 0.990410i $$0.455882\pi$$
$$654$$ 0 0
$$655$$ −1.91130e6 −0.174071
$$656$$ 0 0
$$657$$ 8.80170e6 0.795524
$$658$$ 0 0
$$659$$ 8.11462e6 0.727871 0.363936 0.931424i $$-0.381433\pi$$
0.363936 + 0.931424i $$0.381433\pi$$
$$660$$ 0 0
$$661$$ 2.47370e6 0.220213 0.110107 0.993920i $$-0.464881\pi$$
0.110107 + 0.993920i $$0.464881\pi$$
$$662$$ 0 0
$$663$$ −1.91963e6 −0.169603
$$664$$ 0 0
$$665$$ 5.08800e6 0.446162
$$666$$ 0 0
$$667$$ 1.01482e7 0.883228
$$668$$ 0 0
$$669$$ −4.48630e6 −0.387546
$$670$$ 0 0
$$671$$ 4.78070e6 0.409907
$$672$$ 0 0
$$673$$ 5.77063e6 0.491117 0.245559 0.969382i $$-0.421029\pi$$
0.245559 + 0.969382i $$0.421029\pi$$
$$674$$ 0 0
$$675$$ −1.17500e6 −0.0992610
$$676$$ 0 0
$$677$$ 1.67197e7 1.40203 0.701014 0.713147i $$-0.252731\pi$$
0.701014 + 0.713147i $$0.252731\pi$$
$$678$$ 0 0
$$679$$ 2.28553e7 1.90245
$$680$$ 0 0
$$681$$ 93392.0 0.00771688
$$682$$ 0 0
$$683$$ −7.14532e6 −0.586097 −0.293049 0.956098i $$-0.594670\pi$$
−0.293049 + 0.956098i $$0.594670\pi$$
$$684$$ 0 0
$$685$$ −3.62295e6 −0.295009
$$686$$ 0 0
$$687$$ −2.38404e6 −0.192718
$$688$$ 0 0
$$689$$ 6.81996e6 0.547310
$$690$$ 0 0
$$691$$ 8.78395e6 0.699833 0.349917 0.936781i $$-0.386210\pi$$
0.349917 + 0.936781i $$0.386210\pi$$
$$692$$ 0 0
$$693$$ 6.45043e6 0.510218
$$694$$ 0 0
$$695$$ −2.80550e6 −0.220317
$$696$$ 0 0
$$697$$ 1.57698e7 1.22955
$$698$$ 0 0
$$699$$ −1.94134e6 −0.150282
$$700$$ 0 0
$$701$$ −1.60141e7 −1.23086 −0.615428 0.788193i $$-0.711017\pi$$
−0.615428 + 0.788193i $$0.711017\pi$$
$$702$$ 0 0
$$703$$ −192920. −0.0147228
$$704$$ 0 0
$$705$$ 1.20880e6 0.0915971
$$706$$ 0 0
$$707$$ 1.72604e7 1.29868
$$708$$ 0 0
$$709$$ −1.91354e7 −1.42962 −0.714811 0.699318i $$-0.753487\pi$$
−0.714811 + 0.699318i $$0.753487\pi$$
$$710$$ 0 0
$$711$$ −7.57272e6 −0.561795
$$712$$ 0 0
$$713$$ −7.28525e6 −0.536686
$$714$$ 0 0
$$715$$ 1.05820e6 0.0774110
$$716$$ 0 0
$$717$$ 195520. 0.0142034
$$718$$ 0 0
$$719$$ −1.02934e7 −0.742566 −0.371283 0.928520i $$-0.621082\pi$$
−0.371283 + 0.928520i $$0.621082\pi$$
$$720$$ 0 0
$$721$$ −3.74477e6 −0.268279
$$722$$ 0 0
$$723$$ −443192. −0.0315316
$$724$$ 0 0
$$725$$ −2.13125e6 −0.150588
$$726$$ 0 0
$$727$$ 1.93264e7 1.35618 0.678088 0.734981i $$-0.262809\pi$$
0.678088 + 0.734981i $$0.262809\pi$$
$$728$$ 0 0
$$729$$ −8.98715e6 −0.626330
$$730$$ 0 0
$$731$$ −2.08743e6 −0.144484
$$732$$ 0 0
$$733$$ 5.26197e6 0.361733 0.180866 0.983508i $$-0.442110\pi$$
0.180866 + 0.983508i $$0.442110\pi$$
$$734$$ 0 0
$$735$$ 2.00570e6 0.136945
$$736$$ 0 0
$$737$$ −9.02386e6 −0.611961
$$738$$ 0 0
$$739$$ −2.82944e7 −1.90585 −0.952927 0.303199i $$-0.901945\pi$$
−0.952927 + 0.303199i $$0.901945\pi$$
$$740$$ 0 0
$$741$$ −1.21264e6 −0.0811309
$$742$$ 0 0
$$743$$ −2.09863e7 −1.39464 −0.697321 0.716759i $$-0.745625\pi$$
−0.697321 + 0.716759i $$0.745625\pi$$
$$744$$ 0 0
$$745$$ 1.00938e7 0.666288
$$746$$ 0 0
$$747$$ 3.79453e6 0.248804
$$748$$ 0 0
$$749$$ 3.03921e7 1.97950
$$750$$ 0 0
$$751$$ 1.89668e7 1.22714 0.613572 0.789639i $$-0.289732\pi$$
0.613572 + 0.789639i $$0.289732\pi$$
$$752$$ 0 0
$$753$$ 6.57499e6 0.422579
$$754$$ 0 0
$$755$$ 1.11662e7 0.712915
$$756$$ 0 0
$$757$$ −1.08257e7 −0.686617 −0.343309 0.939223i $$-0.611548\pi$$
−0.343309 + 0.939223i $$0.611548\pi$$
$$758$$ 0 0
$$759$$ −1.76179e6 −0.111007
$$760$$ 0 0
$$761$$ 1.90534e7 1.19264 0.596322 0.802745i $$-0.296628\pi$$
0.596322 + 0.802745i $$0.296628\pi$$
$$762$$ 0 0
$$763$$ −7.07136e6 −0.439736
$$764$$ 0 0
$$765$$ 9.52265e6 0.588307
$$766$$ 0 0
$$767$$ 5.72572e6 0.351432
$$768$$ 0 0
$$769$$ −1.57826e7 −0.962415 −0.481208 0.876607i $$-0.659802\pi$$
−0.481208 + 0.876607i $$0.659802\pi$$
$$770$$ 0 0
$$771$$ 5.22497e6 0.316554
$$772$$ 0 0
$$773$$ −2.44049e7 −1.46902 −0.734510 0.678598i $$-0.762588\pi$$
−0.734510 + 0.678598i $$0.762588\pi$$
$$774$$ 0 0
$$775$$ 1.53000e6 0.0915034
$$776$$ 0 0
$$777$$ −139776. −0.00830577
$$778$$ 0 0
$$779$$ 9.96188e6 0.588163
$$780$$ 0 0
$$781$$ 4.83190e6 0.283459
$$782$$ 0 0
$$783$$ 6.41080e6 0.373687
$$784$$ 0 0
$$785$$ −6.55645e6 −0.379747
$$786$$ 0 0
$$787$$ −3.37607e7 −1.94301 −0.971505 0.237019i $$-0.923830\pi$$
−0.971505 + 0.237019i $$0.923830\pi$$
$$788$$ 0 0
$$789$$ −8.51334e6 −0.486864
$$790$$ 0 0
$$791$$ −2.14771e6 −0.122049
$$792$$ 0 0
$$793$$ 9.23837e6 0.521690
$$794$$ 0 0
$$795$$ 2.38460e6 0.133813
$$796$$ 0 0
$$797$$ 2.19885e7 1.22617 0.613083 0.790019i $$-0.289929\pi$$
0.613083 + 0.790019i $$0.289929\pi$$
$$798$$ 0 0
$$799$$ −2.02837e7 −1.12403
$$800$$ 0 0
$$801$$ −2.30110e7 −1.26723
$$802$$ 0 0
$$803$$ −5.73855e6 −0.314061
$$804$$ 0 0
$$805$$ 1.42848e7 0.776935
$$806$$ 0 0
$$807$$ −5.76436e6 −0.311578
$$808$$ 0 0
$$809$$ −2.93597e7 −1.57717 −0.788587 0.614923i $$-0.789187\pi$$
−0.788587 + 0.614923i $$0.789187\pi$$
$$810$$ 0 0
$$811$$ −3.17703e7 −1.69617 −0.848083 0.529863i $$-0.822243\pi$$
−0.848083 + 0.529863i $$0.822243\pi$$
$$812$$ 0 0
$$813$$ 372992. 0.0197912
$$814$$ 0 0
$$815$$ 3.86410e6 0.203777
$$816$$ 0 0
$$817$$ −1.31864e6 −0.0691148
$$818$$ 0 0
$$819$$ 1.24650e7 0.649357
$$820$$ 0 0
$$821$$ −2.71430e6 −0.140540 −0.0702699 0.997528i $$-0.522386\pi$$
−0.0702699 + 0.997528i $$0.522386\pi$$
$$822$$ 0 0
$$823$$ 1.25866e7 0.647753 0.323877 0.946099i $$-0.395014\pi$$
0.323877 + 0.946099i $$0.395014\pi$$
$$824$$ 0 0
$$825$$ 370000. 0.0189263
$$826$$ 0 0
$$827$$ 8.72355e6 0.443537 0.221768 0.975099i $$-0.428817\pi$$
0.221768 + 0.975099i $$0.428817\pi$$
$$828$$ 0 0
$$829$$ −1.06178e7 −0.536597 −0.268299 0.963336i $$-0.586461\pi$$
−0.268299 + 0.963336i $$0.586461\pi$$
$$830$$ 0 0
$$831$$ −441192. −0.0221628
$$832$$ 0 0
$$833$$ −3.36556e7 −1.68053
$$834$$ 0 0
$$835$$ −9.91680e6 −0.492216
$$836$$ 0 0
$$837$$ −4.60224e6 −0.227068
$$838$$ 0 0
$$839$$ −1.67765e7 −0.822805 −0.411403 0.911454i $$-0.634961\pi$$
−0.411403 + 0.911454i $$0.634961\pi$$
$$840$$ 0 0
$$841$$ −8.88305e6 −0.433084
$$842$$ 0 0
$$843$$ −768792. −0.0372597
$$844$$ 0 0
$$845$$ −7.23742e6 −0.348692
$$846$$ 0 0
$$847$$ 2.67162e7 1.27958
$$848$$ 0 0
$$849$$ 1.32754e6 0.0632087
$$850$$ 0 0
$$851$$ −541632. −0.0256378
$$852$$ 0 0
$$853$$ −2.20186e7 −1.03613 −0.518067 0.855340i $$-0.673348\pi$$
−0.518067 + 0.855340i $$0.673348\pi$$
$$854$$ 0 0
$$855$$ 6.01550e6 0.281421
$$856$$ 0 0
$$857$$ 3.16676e7 1.47287 0.736434 0.676510i $$-0.236508\pi$$
0.736434 + 0.676510i $$0.236508\pi$$
$$858$$ 0 0
$$859$$ −1.58064e7 −0.730886 −0.365443 0.930834i $$-0.619082\pi$$
−0.365443 + 0.930834i $$0.619082\pi$$
$$860$$ 0 0
$$861$$ 7.21766e6 0.331809
$$862$$ 0 0
$$863$$ 1.44287e7 0.659476 0.329738 0.944072i $$-0.393040\pi$$
0.329738 + 0.944072i $$0.393040\pi$$
$$864$$ 0 0
$$865$$ −1.43368e7 −0.651499
$$866$$ 0 0
$$867$$ 5.58331e6 0.252257
$$868$$ 0 0
$$869$$ 4.93728e6 0.221788
$$870$$ 0 0
$$871$$ −1.74380e7 −0.778845
$$872$$ 0 0
$$873$$ 2.70216e7 1.19999
$$874$$ 0 0
$$875$$ −3.00000e6 −0.132465
$$876$$ 0 0
$$877$$ 247902. 0.0108838 0.00544191 0.999985i $$-0.498268\pi$$
0.00544191 + 0.999985i $$0.498268\pi$$
$$878$$ 0 0
$$879$$ 8.77922e6 0.383252
$$880$$ 0 0
$$881$$ 4.10268e7 1.78085 0.890426 0.455128i $$-0.150406\pi$$
0.890426 + 0.455128i $$0.150406\pi$$
$$882$$ 0 0
$$883$$ −4.18015e7 −1.80422 −0.902112 0.431503i $$-0.857984\pi$$
−0.902112 + 0.431503i $$0.857984\pi$$
$$884$$ 0 0
$$885$$ 2.00200e6 0.0859223
$$886$$ 0 0
$$887$$ 2.10476e7 0.898241 0.449120 0.893471i $$-0.351737\pi$$
0.449120 + 0.893471i $$0.351737\pi$$
$$888$$ 0 0
$$889$$ 1.35460e7 0.574852
$$890$$ 0 0
$$891$$ 7.05087e6 0.297542
$$892$$ 0 0
$$893$$ −1.28133e7 −0.537690
$$894$$ 0 0
$$895$$ 1.48615e7 0.620162
$$896$$ 0 0
$$897$$ −3.40454e6 −0.141279
$$898$$ 0 0
$$899$$ −8.34768e6 −0.344482
$$900$$ 0 0
$$901$$ −4.00136e7 −1.64208
$$902$$ 0 0
$$903$$ −955392. −0.0389908
$$904$$ 0 0
$$905$$ −2.67745e6 −0.108668
$$906$$ 0 0
$$907$$ −7.48309e6 −0.302039 −0.151019 0.988531i $$-0.548256\pi$$
−0.151019 + 0.988531i $$0.548256\pi$$
$$908$$ 0 0
$$909$$ 2.04068e7 0.819155
$$910$$ 0 0
$$911$$ 6.63165e6 0.264744 0.132372 0.991200i $$-0.457741\pi$$
0.132372 + 0.991200i $$0.457741\pi$$
$$912$$ 0 0
$$913$$ −2.47397e6 −0.0982239
$$914$$ 0 0
$$915$$ 3.23020e6 0.127549
$$916$$ 0 0
$$917$$ 1.46788e7 0.576457
$$918$$ 0 0
$$919$$ 1.68976e7 0.659990 0.329995 0.943983i $$-0.392953\pi$$
0.329995 + 0.943983i $$0.392953\pi$$
$$920$$ 0 0
$$921$$ 9.51003e6 0.369431
$$922$$ 0 0
$$923$$ 9.33733e6 0.360760
$$924$$ 0 0
$$925$$ 113750. 0.00437116
$$926$$ 0 0
$$927$$ −4.42741e6 −0.169219
$$928$$ 0 0
$$929$$ −1.28653e7 −0.489081 −0.244541 0.969639i $$-0.578637\pi$$
−0.244541 + 0.969639i $$0.578637\pi$$
$$930$$ 0 0
$$931$$ −2.12604e7 −0.803892
$$932$$ 0 0
$$933$$ 9.49219e6 0.356995
$$934$$ 0 0
$$935$$ −6.20860e6 −0.232255
$$936$$ 0 0
$$937$$ 1.06887e7 0.397718 0.198859 0.980028i $$-0.436276\pi$$
0.198859 + 0.980028i $$0.436276\pi$$
$$938$$ 0 0
$$939$$ −5.71766e6 −0.211619
$$940$$ 0 0
$$941$$ 2.82455e7 1.03986 0.519930 0.854209i $$-0.325958\pi$$
0.519930 + 0.854209i $$0.325958\pi$$
$$942$$ 0 0
$$943$$ 2.79684e7 1.02421
$$944$$ 0 0
$$945$$ 9.02400e6 0.328715
$$946$$ 0 0
$$947$$ 1.70892e7 0.619222 0.309611 0.950863i $$-0.399801\pi$$
0.309611 + 0.950863i $$0.399801\pi$$
$$948$$ 0 0
$$949$$ −1.10894e7 −0.399706
$$950$$ 0 0
$$951$$ 8.49849e6 0.304713
$$952$$ 0 0
$$953$$ 2.22259e7 0.792735 0.396367 0.918092i $$-0.370271\pi$$
0.396367 + 0.918092i $$0.370271\pi$$
$$954$$ 0 0
$$955$$ −1.17388e7 −0.416500
$$956$$ 0 0
$$957$$ −2.01872e6 −0.0712519
$$958$$ 0 0
$$959$$ 2.78243e7 0.976961
$$960$$ 0 0
$$961$$ −2.26364e7 −0.790678
$$962$$ 0 0
$$963$$ 3.59323e7 1.24859
$$964$$ 0 0
$$965$$ 1.31765e6 0.0455493
$$966$$ 0 0
$$967$$ −2.41551e7 −0.830696 −0.415348 0.909663i $$-0.636340\pi$$
−0.415348 + 0.909663i $$0.636340\pi$$
$$968$$ 0 0
$$969$$ 7.11472e6 0.243416
$$970$$ 0 0
$$971$$ 5.48313e7 1.86630 0.933149 0.359491i $$-0.117050\pi$$
0.933149 + 0.359491i $$0.117050\pi$$
$$972$$ 0 0
$$973$$ 2.15462e7 0.729608
$$974$$ 0 0
$$975$$ 715000. 0.0240877
$$976$$ 0 0
$$977$$ −1.56612e7 −0.524915 −0.262457 0.964944i $$-0.584533\pi$$
−0.262457 + 0.964944i $$0.584533\pi$$
$$978$$ 0 0
$$979$$ 1.50028e7 0.500281
$$980$$ 0 0
$$981$$ −8.36041e6 −0.277367
$$982$$ 0 0
$$983$$ 1.63420e7 0.539412 0.269706 0.962943i $$-0.413073\pi$$
0.269706 + 0.962943i $$0.413073\pi$$
$$984$$ 0 0
$$985$$ 1.13966e7 0.374268
$$986$$ 0 0
$$987$$ −9.28358e6 −0.303335
$$988$$ 0 0
$$989$$ −3.70214e6 −0.120355
$$990$$ 0 0
$$991$$ −1.37576e7 −0.444997 −0.222498 0.974933i $$-0.571421\pi$$
−0.222498 + 0.974933i $$0.571421\pi$$
$$992$$ 0 0
$$993$$ −1.23994e7 −0.399050
$$994$$ 0 0
$$995$$ −2.16250e7 −0.692466
$$996$$ 0 0
$$997$$ −1.29097e7 −0.411320 −0.205660 0.978624i $$-0.565934\pi$$
−0.205660 + 0.978624i $$0.565934\pi$$
$$998$$ 0 0
$$999$$ −342160. −0.0108471
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 80.6.a.e.1.1 1
3.2 odd 2 720.6.a.a.1.1 1
4.3 odd 2 5.6.a.a.1.1 1
5.2 odd 4 400.6.c.j.49.1 2
5.3 odd 4 400.6.c.j.49.2 2
5.4 even 2 400.6.a.g.1.1 1
8.3 odd 2 320.6.a.j.1.1 1
8.5 even 2 320.6.a.g.1.1 1
12.11 even 2 45.6.a.b.1.1 1
20.3 even 4 25.6.b.a.24.1 2
20.7 even 4 25.6.b.a.24.2 2
20.19 odd 2 25.6.a.a.1.1 1
28.27 even 2 245.6.a.b.1.1 1
44.43 even 2 605.6.a.a.1.1 1
52.51 odd 2 845.6.a.b.1.1 1
60.23 odd 4 225.6.b.e.199.2 2
60.47 odd 4 225.6.b.e.199.1 2
60.59 even 2 225.6.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 1 4.3 odd 2
25.6.a.a.1.1 1 20.19 odd 2
25.6.b.a.24.1 2 20.3 even 4
25.6.b.a.24.2 2 20.7 even 4
45.6.a.b.1.1 1 12.11 even 2
80.6.a.e.1.1 1 1.1 even 1 trivial
225.6.a.f.1.1 1 60.59 even 2
225.6.b.e.199.1 2 60.47 odd 4
225.6.b.e.199.2 2 60.23 odd 4
245.6.a.b.1.1 1 28.27 even 2
320.6.a.g.1.1 1 8.5 even 2
320.6.a.j.1.1 1 8.3 odd 2
400.6.a.g.1.1 1 5.4 even 2
400.6.c.j.49.1 2 5.2 odd 4
400.6.c.j.49.2 2 5.3 odd 4
605.6.a.a.1.1 1 44.43 even 2
720.6.a.a.1.1 1 3.2 odd 2
845.6.a.b.1.1 1 52.51 odd 2