# Properties

 Label 400.6.c.j.49.1 Level $400$ Weight $6$ Character 400.49 Analytic conductor $64.154$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.1535279252$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 400.49 Dual form 400.6.c.j.49.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.00000i q^{3} -192.000i q^{7} +227.000 q^{9} +O(q^{10})$$ $$q-4.00000i q^{3} -192.000i q^{7} +227.000 q^{9} +148.000 q^{11} -286.000i q^{13} -1678.00i q^{17} +1060.00 q^{19} -768.000 q^{21} +2976.00i q^{23} -1880.00i q^{27} +3410.00 q^{29} +2448.00 q^{31} -592.000i q^{33} +182.000i q^{37} -1144.00 q^{39} -9398.00 q^{41} -1244.00i q^{43} +12088.0i q^{47} -20057.0 q^{49} -6712.00 q^{51} -23846.0i q^{53} -4240.00i q^{57} -20020.0 q^{59} +32302.0 q^{61} -43584.0i q^{63} -60972.0i q^{67} +11904.0 q^{69} +32648.0 q^{71} +38774.0i q^{73} -28416.0i q^{77} -33360.0 q^{79} +47641.0 q^{81} +16716.0i q^{83} -13640.0i q^{87} -101370. q^{89} -54912.0 q^{91} -9792.00i q^{93} -119038. i q^{97} +33596.0 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 454 q^{9} + O(q^{10})$$ $$2 q + 454 q^{9} + 296 q^{11} + 2120 q^{19} - 1536 q^{21} + 6820 q^{29} + 4896 q^{31} - 2288 q^{39} - 18796 q^{41} - 40114 q^{49} - 13424 q^{51} - 40040 q^{59} + 64604 q^{61} + 23808 q^{69} + 65296 q^{71} - 66720 q^{79} + 95282 q^{81} - 202740 q^{89} - 109824 q^{91} + 67192 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 4.00000i − 0.256600i −0.991735 0.128300i $$-0.959048\pi$$
0.991735 0.128300i $$-0.0409521\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 192.000i − 1.48100i −0.672054 0.740502i $$-0.734588\pi$$
0.672054 0.740502i $$-0.265412\pi$$
$$8$$ 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
$$13$$ − 286.000i − 0.469362i −0.972072 0.234681i $$-0.924595\pi$$
0.972072 0.234681i $$-0.0754045\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 1678.00i − 1.40822i −0.710092 0.704109i $$-0.751347\pi$$
0.710092 0.704109i $$-0.248653\pi$$
$$18$$ 0 0
$$19$$ 1060.00 0.673631 0.336815 0.941571i $$-0.390650\pi$$
0.336815 + 0.941571i $$0.390650\pi$$
$$20$$ 0 0
$$21$$ −768.000 −0.380026
$$22$$ 0 0
$$23$$ 2976.00i 1.17304i 0.809934 + 0.586521i $$0.199503\pi$$
−0.809934 + 0.586521i $$0.800497\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1880.00i − 0.496305i
$$28$$ 0 0
$$29$$ 3410.00 0.752938 0.376469 0.926429i $$-0.377138\pi$$
0.376469 + 0.926429i $$0.377138\pi$$
$$30$$ 0 0
$$31$$ 2448.00 0.457517 0.228758 0.973483i $$-0.426533\pi$$
0.228758 + 0.973483i $$0.426533\pi$$
$$32$$ 0 0
$$33$$ − 592.000i − 0.0946317i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 182.000i 0.0218558i 0.999940 + 0.0109279i $$0.00347853\pi$$
−0.999940 + 0.0109279i $$0.996521\pi$$
$$38$$ 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
$$43$$ − 1244.00i − 0.102600i −0.998683 0.0513002i $$-0.983663\pi$$
0.998683 0.0513002i $$-0.0163365\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12088.0i 0.798196i 0.916908 + 0.399098i $$0.130677\pi$$
−0.916908 + 0.399098i $$0.869323\pi$$
$$48$$ 0 0
$$49$$ −20057.0 −1.19337
$$50$$ 0 0
$$51$$ −6712.00 −0.361349
$$52$$ 0 0
$$53$$ − 23846.0i − 1.16607i −0.812446 0.583037i $$-0.801864\pi$$
0.812446 0.583037i $$-0.198136\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 4240.00i − 0.172854i
$$58$$ 0 0
$$59$$ −20020.0 −0.748745 −0.374373 0.927278i $$-0.622142\pi$$
−0.374373 + 0.927278i $$0.622142\pi$$
$$60$$ 0 0
$$61$$ 32302.0 1.11149 0.555744 0.831353i $$-0.312433\pi$$
0.555744 + 0.831353i $$0.312433\pi$$
$$62$$ 0 0
$$63$$ − 43584.0i − 1.38349i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 60972.0i − 1.65937i −0.558231 0.829685i $$-0.688520\pi$$
0.558231 0.829685i $$-0.311480\pi$$
$$68$$ 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
$$73$$ 38774.0i 0.851596i 0.904818 + 0.425798i $$0.140007\pi$$
−0.904818 + 0.425798i $$0.859993\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 28416.0i − 0.546180i
$$78$$ 0 0
$$79$$ −33360.0 −0.601393 −0.300696 0.953720i $$-0.597219\pi$$
−0.300696 + 0.953720i $$0.597219\pi$$
$$80$$ 0 0
$$81$$ 47641.0 0.806805
$$82$$ 0 0
$$83$$ 16716.0i 0.266340i 0.991093 + 0.133170i $$0.0425157\pi$$
−0.991093 + 0.133170i $$0.957484\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 13640.0i − 0.193204i
$$88$$ 0 0
$$89$$ −101370. −1.35655 −0.678273 0.734810i $$-0.737271\pi$$
−0.678273 + 0.734810i $$0.737271\pi$$
$$90$$ 0 0
$$91$$ −54912.0 −0.695126
$$92$$ 0 0
$$93$$ − 9792.00i − 0.117399i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 119038.i − 1.28457i −0.766468 0.642283i $$-0.777987\pi$$
0.766468 0.642283i $$-0.222013\pi$$
$$98$$ 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
$$103$$ − 19504.0i − 0.181147i −0.995890 0.0905734i $$-0.971130\pi$$
0.995890 0.0905734i $$-0.0288700\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 158292.i − 1.33659i −0.743895 0.668297i $$-0.767024\pi$$
0.743895 0.668297i $$-0.232976\pi$$
$$108$$ 0 0
$$109$$ −36830.0 −0.296917 −0.148459 0.988919i $$-0.547431\pi$$
−0.148459 + 0.988919i $$0.547431\pi$$
$$110$$ 0 0
$$111$$ 728.000 0.00560821
$$112$$ 0 0
$$113$$ − 11186.0i − 0.0824098i −0.999151 0.0412049i $$-0.986880\pi$$
0.999151 0.0412049i $$-0.0131196\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 64922.0i − 0.438457i
$$118$$ 0 0
$$119$$ −322176. −2.08557
$$120$$ 0 0
$$121$$ −139147. −0.863993
$$122$$ 0 0
$$123$$ 37592.0i 0.224044i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 70552.0i − 0.388150i −0.980987 0.194075i $$-0.937829\pi$$
0.980987 0.194075i $$-0.0621706\pi$$
$$128$$ 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.i − 0.997650i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 144918.i − 0.659661i −0.944040 0.329831i $$-0.893008\pi$$
0.944040 0.329831i $$-0.106992\pi$$
$$138$$ 0 0
$$139$$ 112220. 0.492644 0.246322 0.969188i $$-0.420778\pi$$
0.246322 + 0.969188i $$0.420778\pi$$
$$140$$ 0 0
$$141$$ 48352.0 0.204817
$$142$$ 0 0
$$143$$ − 42328.0i − 0.173096i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 80228.0i 0.306219i
$$148$$ 0 0
$$149$$ −403750. −1.48986 −0.744932 0.667140i $$-0.767518\pi$$
−0.744932 + 0.667140i $$0.767518\pi$$
$$150$$ 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.i − 1.31550i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 262258.i − 0.849141i −0.905395 0.424570i $$-0.860425\pi$$
0.905395 0.424570i $$-0.139575\pi$$
$$158$$ 0 0
$$159$$ −95384.0 −0.299215
$$160$$ 0 0
$$161$$ 571392. 1.73728
$$162$$ 0 0
$$163$$ − 154564.i − 0.455658i −0.973701 0.227829i $$-0.926837\pi$$
0.973701 0.227829i $$-0.0731628\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 396672.i − 1.10063i −0.834958 0.550314i $$-0.814508\pi$$
0.834958 0.550314i $$-0.185492\pi$$
$$168$$ 0 0
$$169$$ 289497. 0.779700
$$170$$ 0 0
$$171$$ 240620. 0.629276
$$172$$ 0 0
$$173$$ 573474.i 1.45680i 0.685155 + 0.728398i $$0.259735\pi$$
−0.685155 + 0.728398i $$0.740265\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 80080.0i 0.192128i
$$178$$ 0 0
$$179$$ −594460. −1.38672 −0.693362 0.720589i $$-0.743871\pi$$
−0.693362 + 0.720589i $$0.743871\pi$$
$$180$$ 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.i − 0.285208i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 248344.i − 0.519337i
$$188$$ 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
$$193$$ − 52706.0i − 0.101851i −0.998702 0.0509257i $$-0.983783\pi$$
0.998702 0.0509257i $$-0.0162172\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 455862.i 0.836889i 0.908242 + 0.418444i $$0.137425\pi$$
−0.908242 + 0.418444i $$0.862575\pi$$
$$198$$ 0 0
$$199$$ 865000. 1.54840 0.774200 0.632940i $$-0.218152\pi$$
0.774200 + 0.632940i $$0.218152\pi$$
$$200$$ 0 0
$$201$$ −243888. −0.425795
$$202$$ 0 0
$$203$$ − 654720.i − 1.11510i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 675552.i 1.09580i
$$208$$ 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.i − 0.197228i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 470016.i − 0.677584i
$$218$$ 0 0
$$219$$ 155096. 0.218520
$$220$$ 0 0
$$221$$ −479908. −0.660963
$$222$$ 0 0
$$223$$ 1.12158e6i 1.51031i 0.655545 + 0.755156i $$0.272439\pi$$
−0.655545 + 0.755156i $$0.727561\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 23348.0i 0.0300736i 0.999887 + 0.0150368i $$0.00478654\pi$$
−0.999887 + 0.0150368i $$0.995213\pi$$
$$228$$ 0 0
$$229$$ 596010. 0.751043 0.375522 0.926814i $$-0.377464\pi$$
0.375522 + 0.926814i $$0.377464\pi$$
$$230$$ 0 0
$$231$$ −113664. −0.140150
$$232$$ 0 0
$$233$$ 485334.i 0.585667i 0.956163 + 0.292834i $$0.0945982\pi$$
−0.956163 + 0.292834i $$0.905402\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 133440.i 0.154317i
$$238$$ 0 0
$$239$$ −48880.0 −0.0553524 −0.0276762 0.999617i $$-0.508811\pi$$
−0.0276762 + 0.999617i $$0.508811\pi$$
$$240$$ 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.i − 0.703331i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 303160.i − 0.316176i
$$248$$ 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.i 0.432607i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 1.30624e6i 1.23365i 0.787102 + 0.616823i $$0.211581\pi$$
−0.787102 + 0.616823i $$0.788419\pi$$
$$258$$ 0 0
$$259$$ 34944.0 0.0323685
$$260$$ 0 0
$$261$$ 774070. 0.703362
$$262$$ 0 0
$$263$$ 2.12834e6i 1.89736i 0.316231 + 0.948682i $$0.397583\pi$$
−0.316231 + 0.948682i $$0.602417\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 405480.i 0.348090i
$$268$$ 0 0
$$269$$ 1.44109e6 1.21426 0.607128 0.794604i $$-0.292321\pi$$
0.607128 + 0.794604i $$0.292321\pi$$
$$270$$ 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.i 0.178370i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 110298.i − 0.0863711i −0.999067 0.0431855i $$-0.986249\pi$$
0.999067 0.0431855i $$-0.0137507\pi$$
$$278$$ 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.i − 0.246332i −0.992386 0.123166i $$-0.960695\pi$$
0.992386 0.123166i $$-0.0393047\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1.80442e6i 1.29310i
$$288$$ 0 0
$$289$$ −1.39583e6 −0.983076
$$290$$ 0 0
$$291$$ −476152. −0.329620
$$292$$ 0 0
$$293$$ − 2.19481e6i − 1.49358i −0.665063 0.746788i $$-0.731595\pi$$
0.665063 0.746788i $$-0.268405\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 278240.i − 0.183033i
$$298$$ 0 0
$$299$$ 851136. 0.550581
$$300$$ 0 0
$$301$$ −238848. −0.151952
$$302$$ 0 0
$$303$$ 359592.i 0.225011i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.37751e6i 1.43971i 0.694123 + 0.719857i $$0.255793\pi$$
−0.694123 + 0.719857i $$0.744207\pi$$
$$308$$ 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
$$313$$ 1.42941e6i 0.824702i 0.911025 + 0.412351i $$0.135292\pi$$
−0.911025 + 0.412351i $$0.864708\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.12462e6i 1.18750i 0.804650 + 0.593750i $$0.202353\pi$$
−0.804650 + 0.593750i $$0.797647\pi$$
$$318$$ 0 0
$$319$$ 504680. 0.277677
$$320$$ 0 0
$$321$$ −633168. −0.342970
$$322$$ 0 0
$$323$$ − 1.77868e6i − 0.948618i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 147320.i 0.0761890i
$$328$$ 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
$$333$$ 41314.0i 0.0204168i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.40008e6i 1.15120i 0.817731 + 0.575601i $$0.195232\pi$$
−0.817731 + 0.575601i $$0.804768\pi$$
$$338$$ 0 0
$$339$$ −44744.0 −0.0211464
$$340$$ 0 0
$$341$$ 362304. 0.168728
$$342$$ 0 0
$$343$$ 624000.i 0.286384i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 1.77741e6i − 0.792436i −0.918156 0.396218i $$-0.870322\pi$$
0.918156 0.396218i $$-0.129678\pi$$
$$348$$ 0 0
$$349$$ 2.14805e6 0.944019 0.472010 0.881593i $$-0.343529\pi$$
0.472010 + 0.881593i $$0.343529\pi$$
$$350$$ 0 0
$$351$$ −537680. −0.232946
$$352$$ 0 0
$$353$$ 661854.i 0.282700i 0.989960 + 0.141350i $$0.0451443\pi$$
−0.989960 + 0.141350i $$0.954856\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 1.28870e6i 0.535159i
$$358$$ 0 0
$$359$$ −259320. −0.106194 −0.0530970 0.998589i $$-0.516909\pi$$
−0.0530970 + 0.998589i $$0.516909\pi$$
$$360$$ 0 0
$$361$$ −1.35250e6 −0.546222
$$362$$ 0 0
$$363$$ 556588.i 0.221701i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1.49993e6i 0.581307i 0.956828 + 0.290653i $$0.0938726\pi$$
−0.956828 + 0.290653i $$0.906127\pi$$
$$368$$ 0 0
$$369$$ −2.13335e6 −0.815634
$$370$$ 0 0
$$371$$ −4.57843e6 −1.72696
$$372$$ 0 0
$$373$$ 2.23807e6i 0.832918i 0.909154 + 0.416459i $$0.136729\pi$$
−0.909154 + 0.416459i $$0.863271\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 975260.i − 0.353400i
$$378$$ 0 0
$$379$$ 3.15934e6 1.12979 0.564896 0.825162i $$-0.308916\pi$$
0.564896 + 0.825162i $$0.308916\pi$$
$$380$$ 0 0
$$381$$ −282208. −0.0995994
$$382$$ 0 0
$$383$$ 342216.i 0.119207i 0.998222 + 0.0596037i $$0.0189837\pi$$
−0.998222 + 0.0596037i $$0.981016\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 282388.i − 0.0958449i
$$388$$ 0 0
$$389$$ −88470.0 −0.0296430 −0.0148215 0.999890i $$-0.504718\pi$$
−0.0148215 + 0.999890i $$0.504718\pi$$
$$390$$ 0 0
$$391$$ 4.99373e6 1.65190
$$392$$ 0 0
$$393$$ 305808.i 0.0998775i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 5.45674e6i − 1.73763i −0.495138 0.868814i $$-0.664883\pi$$
0.495138 0.868814i $$-0.335117\pi$$
$$398$$ 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.i − 0.214741i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 26936.0i 0.00806022i
$$408$$ 0 0
$$409$$ 2.71207e6 0.801664 0.400832 0.916151i $$-0.368721\pi$$
0.400832 + 0.916151i $$0.368721\pi$$
$$410$$ 0 0
$$411$$ −579672. −0.169269
$$412$$ 0 0
$$413$$ 3.84384e6i 1.10889i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 448880.i − 0.126413i
$$418$$ 0 0
$$419$$ 3.71746e6 1.03445 0.517227 0.855848i $$-0.326964\pi$$
0.517227 + 0.855848i $$0.326964\pi$$
$$420$$ 0 0
$$421$$ 3.55250e6 0.976853 0.488426 0.872605i $$-0.337571\pi$$
0.488426 + 0.872605i $$0.337571\pi$$
$$422$$ 0 0
$$423$$ 2.74398e6i 0.745640i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 6.20198e6i − 1.64612i
$$428$$ 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
$$433$$ − 7.26287e6i − 1.86161i −0.365518 0.930804i $$-0.619108\pi$$
0.365518 0.930804i $$-0.380892\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.15456e6i 0.790197i
$$438$$ 0 0
$$439$$ −5.41028e6 −1.33986 −0.669928 0.742426i $$-0.733675\pi$$
−0.669928 + 0.742426i $$0.733675\pi$$
$$440$$ 0 0
$$441$$ −4.55294e6 −1.11480
$$442$$ 0 0
$$443$$ − 6.51524e6i − 1.57733i −0.614826 0.788663i $$-0.710774\pi$$
0.614826 0.788663i $$-0.289226\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 1.61500e6i 0.382299i
$$448$$ 0 0
$$449$$ 509950. 0.119375 0.0596873 0.998217i $$-0.480990\pi$$
0.0596873 + 0.998217i $$0.480990\pi$$
$$450$$ 0 0
$$451$$ −1.39090e6 −0.322000
$$452$$ 0 0
$$453$$ − 1.78659e6i − 0.409053i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.22084e6i 0.273444i 0.990609 + 0.136722i $$0.0436568\pi$$
−0.990609 + 0.136722i $$0.956343\pi$$
$$458$$ 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
$$463$$ 2.02294e6i 0.438561i 0.975662 + 0.219280i $$0.0703709\pi$$
−0.975662 + 0.219280i $$0.929629\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 3.25097e6i − 0.689797i −0.938640 0.344898i $$-0.887913\pi$$
0.938640 0.344898i $$-0.112087\pi$$
$$468$$ 0 0
$$469$$ −1.17066e7 −2.45753
$$470$$ 0 0
$$471$$ −1.04903e6 −0.217890
$$472$$ 0 0
$$473$$ − 184112.i − 0.0378381i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 5.41304e6i − 1.08929i
$$478$$ 0 0
$$479$$ −3.27936e6 −0.653056 −0.326528 0.945188i $$-0.605879\pi$$
−0.326528 + 0.945188i $$0.605879\pi$$
$$480$$ 0 0
$$481$$ 52052.0 0.0102583
$$482$$ 0 0
$$483$$ − 2.28557e6i − 0.445786i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 8.53197e6i 1.63015i 0.579357 + 0.815074i $$0.303304\pi$$
−0.579357 + 0.815074i $$0.696696\pi$$
$$488$$ 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
$$493$$ − 5.72198e6i − 1.06030i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 6.26842e6i − 1.13833i
$$498$$ 0 0
$$499$$ −6.49190e6 −1.16713 −0.583567 0.812065i $$-0.698343\pi$$
−0.583567 + 0.812065i $$0.698343\pi$$
$$500$$ 0 0
$$501$$ −1.58669e6 −0.282421
$$502$$ 0 0
$$503$$ 8.61770e6i 1.51870i 0.650684 + 0.759349i $$0.274482\pi$$
−0.650684 + 0.759349i $$0.725518\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 1.15799e6i − 0.200071i
$$508$$ 0 0
$$509$$ −2.67323e6 −0.457343 −0.228671 0.973504i $$-0.573438\pi$$
−0.228671 + 0.973504i $$0.573438\pi$$
$$510$$ 0 0
$$511$$ 7.44461e6 1.26122
$$512$$ 0 0
$$513$$ − 1.99280e6i − 0.334326i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1.78902e6i 0.294367i
$$518$$ 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
$$523$$ − 6.89452e6i − 1.10217i −0.834448 0.551087i $$-0.814213\pi$$
0.834448 0.551087i $$-0.185787\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 4.10774e6i − 0.644283i
$$528$$ 0 0
$$529$$ −2.42023e6 −0.376026
$$530$$ 0 0
$$531$$ −4.54454e6 −0.699445
$$532$$ 0 0
$$533$$ 2.68783e6i 0.409811i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 2.37784e6i 0.355834i
$$538$$ 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.i 0.0623508i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 1.26544e7i − 1.80831i −0.427201 0.904157i $$-0.640500\pi$$
0.427201 0.904157i $$-0.359500\pi$$
$$548$$ 0 0
$$549$$ 7.33255e6 1.03830
$$550$$ 0 0
$$551$$ 3.61460e6 0.507202
$$552$$ 0 0
$$553$$ 6.40512e6i 0.890665i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 7.07786e6i − 0.966638i −0.875444 0.483319i $$-0.839431\pi$$
0.875444 0.483319i $$-0.160569\pi$$
$$558$$ 0 0
$$559$$ −355784. −0.0481567
$$560$$ 0 0
$$561$$ −993376. −0.133262
$$562$$ 0 0
$$563$$ 846636.i 0.112571i 0.998415 + 0.0562854i $$0.0179257\pi$$
−0.998415 + 0.0562854i $$0.982074\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 9.14707e6i − 1.19488i
$$568$$ 0 0
$$569$$ −4.96041e6 −0.642299 −0.321149 0.947029i $$-0.604069\pi$$
−0.321149 + 0.947029i $$0.604069\pi$$
$$570$$ 0 0
$$571$$ −8.96505e6 −1.15070 −0.575351 0.817907i $$-0.695134\pi$$
−0.575351 + 0.817907i $$0.695134\pi$$
$$572$$ 0 0
$$573$$ 1.87821e6i 0.238978i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 2.86080e6i − 0.357724i −0.983874 0.178862i $$-0.942758\pi$$
0.983874 0.178862i $$-0.0572415\pi$$
$$578$$ 0 0
$$579$$ −210824. −0.0261351
$$580$$ 0 0
$$581$$ 3.20947e6 0.394451
$$582$$ 0 0
$$583$$ − 3.52921e6i − 0.430037i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 6.74027e6i 0.807387i 0.914894 + 0.403694i $$0.132274\pi$$
−0.914894 + 0.403694i $$0.867726\pi$$
$$588$$ 0 0
$$589$$ 2.59488e6 0.308197
$$590$$ 0 0
$$591$$ 1.82345e6 0.214746
$$592$$ 0 0
$$593$$ 1.78609e6i 0.208578i 0.994547 + 0.104289i $$0.0332566\pi$$
−0.994547 + 0.104289i $$0.966743\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 3.46000e6i − 0.397320i
$$598$$ 0 0
$$599$$ 4.94620e6 0.563254 0.281627 0.959524i $$-0.409126\pi$$
0.281627 + 0.959524i $$0.409126\pi$$
$$600$$ 0 0
$$601$$ −4.58100e6 −0.517337 −0.258669 0.965966i $$-0.583284\pi$$
−0.258669 + 0.965966i $$0.583284\pi$$
$$602$$ 0 0
$$603$$ − 1.38406e7i − 1.55011i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 7.07999e6i − 0.779940i −0.920828 0.389970i $$-0.872485\pi$$
0.920828 0.389970i $$-0.127515\pi$$
$$608$$ 0 0
$$609$$ −2.61888e6 −0.286136
$$610$$ 0 0
$$611$$ 3.45717e6 0.374643
$$612$$ 0 0
$$613$$ − 5.09609e6i − 0.547754i −0.961765 0.273877i $$-0.911694\pi$$
0.961765 0.273877i $$-0.0883061\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 1.30003e7i − 1.37480i −0.726279 0.687400i $$-0.758752\pi$$
0.726279 0.687400i $$-0.241248\pi$$
$$618$$ 0 0
$$619$$ 4.84406e6 0.508139 0.254070 0.967186i $$-0.418231\pi$$
0.254070 + 0.967186i $$0.418231\pi$$
$$620$$ 0 0
$$621$$ 5.59488e6 0.582186
$$622$$ 0 0
$$623$$ 1.94630e7i 2.00905i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 627520.i − 0.0637468i
$$628$$ 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
$$633$$ 4.42261e6i 0.438702i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 5.73630e6i 0.560123i
$$638$$ 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
$$643$$ − 1.74382e7i − 1.66332i −0.555287 0.831659i $$-0.687391\pi$$
0.555287 0.831659i $$-0.312609\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 4.25469e6i 0.399583i 0.979838 + 0.199792i $$0.0640265\pi$$
−0.979838 + 0.199792i $$0.935974\pi$$
$$648$$ 0 0
$$649$$ −2.96296e6 −0.276130
$$650$$ 0 0
$$651$$ −1.88006e6 −0.173868
$$652$$ 0 0
$$653$$ − 3.01085e6i − 0.276316i −0.990410 0.138158i $$-0.955882\pi$$
0.990410 0.138158i $$-0.0441181\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 8.80170e6i 0.795524i
$$658$$ 0 0
$$659$$ −8.11462e6 −0.727871 −0.363936 0.931424i $$-0.618567\pi$$
−0.363936 + 0.931424i $$0.618567\pi$$
$$660$$ 0 0
$$661$$ 2.47370e6 0.220213 0.110107 0.993920i $$-0.464881\pi$$
0.110107 + 0.993920i $$0.464881\pi$$
$$662$$ 0 0
$$663$$ 1.91963e6i 0.169603i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1.01482e7i 0.883228i
$$668$$ 0 0
$$669$$ 4.48630e6 0.387546
$$670$$ 0 0
$$671$$ 4.78070e6 0.409907
$$672$$ 0 0
$$673$$ − 5.77063e6i − 0.491117i −0.969382 0.245559i $$-0.921029\pi$$
0.969382 0.245559i $$-0.0789714\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 1.67197e7i 1.40203i 0.713147 + 0.701014i $$0.247269\pi$$
−0.713147 + 0.701014i $$0.752731\pi$$
$$678$$ 0 0
$$679$$ −2.28553e7 −1.90245
$$680$$ 0 0
$$681$$ 93392.0 0.00771688
$$682$$ 0 0
$$683$$ 7.14532e6i 0.586097i 0.956098 + 0.293049i $$0.0946698\pi$$
−0.956098 + 0.293049i $$0.905330\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 2.38404e6i − 0.192718i
$$688$$ 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
$$693$$ − 6.45043e6i − 0.510218i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1.57698e7i 1.22955i
$$698$$ 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.i 0.0147228i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1.72604e7i 1.29868i
$$708$$ 0 0
$$709$$ 1.91354e7 1.42962 0.714811 0.699318i $$-0.246513\pi$$
0.714811 + 0.699318i $$0.246513\pi$$
$$710$$ 0 0
$$711$$ −7.57272e6 −0.561795
$$712$$ 0 0
$$713$$ 7.28525e6i 0.536686i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 195520.i 0.0142034i
$$718$$ 0 0
$$719$$ 1.02934e7 0.742566 0.371283 0.928520i $$-0.378918\pi$$
0.371283 + 0.928520i $$0.378918\pi$$
$$720$$ 0 0
$$721$$ −3.74477e6 −0.268279
$$722$$ 0 0
$$723$$ 443192.i 0.0315316i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 1.93264e7i 1.35618i 0.734981 + 0.678088i $$0.237191\pi$$
−0.734981 + 0.678088i $$0.762809\pi$$
$$728$$ 0 0
$$729$$ 8.98715e6 0.626330
$$730$$ 0 0
$$731$$ −2.08743e6 −0.144484
$$732$$ 0 0
$$733$$ − 5.26197e6i − 0.361733i −0.983508 0.180866i $$-0.942110\pi$$
0.983508 0.180866i $$-0.0578902\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 9.02386e6i − 0.611961i
$$738$$ 0 0
$$739$$ 2.82944e7 1.90585 0.952927 0.303199i $$-0.0980548\pi$$
0.952927 + 0.303199i $$0.0980548\pi$$
$$740$$ 0 0
$$741$$ −1.21264e6 −0.0811309
$$742$$ 0 0
$$743$$ 2.09863e7i 1.39464i 0.716759 + 0.697321i $$0.245625\pi$$
−0.716759 + 0.697321i $$0.754375\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 3.79453e6i 0.248804i
$$748$$ 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
$$753$$ − 6.57499e6i − 0.422579i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 1.08257e7i − 0.686617i −0.939223 0.343309i $$-0.888452\pi$$
0.939223 0.343309i $$-0.111548\pi$$
$$758$$ 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
$$763$$ 7.07136e6i 0.439736i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 5.72572e6i 0.351432i
$$768$$ 0 0
$$769$$ 1.57826e7 0.962415 0.481208 0.876607i $$-0.340198\pi$$
0.481208 + 0.876607i $$0.340198\pi$$
$$770$$ 0 0
$$771$$ 5.22497e6 0.316554
$$772$$ 0 0
$$773$$ 2.44049e7i 1.46902i 0.678598 + 0.734510i $$0.262588\pi$$
−0.678598 + 0.734510i $$0.737412\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 139776.i − 0.00830577i
$$778$$ 0 0
$$779$$ −9.96188e6 −0.588163
$$780$$ 0 0
$$781$$ 4.83190e6 0.283459
$$782$$ 0 0
$$783$$ − 6.41080e6i − 0.373687i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 3.37607e7i − 1.94301i −0.237019 0.971505i $$-0.576170\pi$$
0.237019 0.971505i $$-0.423830\pi$$
$$788$$ 0 0
$$789$$ 8.51334e6 0.486864
$$790$$ 0 0
$$791$$ −2.14771e6 −0.122049
$$792$$ 0 0
$$793$$ − 9.23837e6i − 0.521690i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 2.19885e7i 1.22617i 0.790019 + 0.613083i $$0.210071\pi$$
−0.790019 + 0.613083i $$0.789929\pi$$
$$798$$ 0 0
$$799$$ 2.02837e7 1.12403
$$800$$ 0 0
$$801$$ −2.30110e7 −1.26723
$$802$$ 0 0
$$803$$ 5.73855e6i 0.314061i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 5.76436e6i − 0.311578i
$$808$$ 0 0
$$809$$ 2.93597e7 1.57717 0.788587 0.614923i $$-0.210813\pi$$
0.788587 + 0.614923i $$0.210813\pi$$
$$810$$ 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.i − 0.0197912i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 1.31864e6i − 0.0691148i
$$818$$ 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
$$823$$ − 1.25866e7i − 0.647753i −0.946099 0.323877i $$-0.895014\pi$$
0.946099 0.323877i $$-0.104986\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 8.72355e6i 0.443537i 0.975099 + 0.221768i $$0.0711828\pi$$
−0.975099 + 0.221768i $$0.928817\pi$$
$$828$$ 0 0
$$829$$ 1.06178e7 0.536597 0.268299 0.963336i $$-0.413539\pi$$
0.268299 + 0.963336i $$0.413539\pi$$
$$830$$ 0 0
$$831$$ −441192. −0.0221628
$$832$$ 0 0
$$833$$ 3.36556e7i 1.68053i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 4.60224e6i − 0.227068i
$$838$$ 0 0
$$839$$ 1.67765e7 0.822805 0.411403 0.911454i $$-0.365039\pi$$
0.411403 + 0.911454i $$0.365039\pi$$
$$840$$ 0 0
$$841$$ −8.88305e6 −0.433084
$$842$$ 0 0
$$843$$ 768792.i 0.0372597i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.67162e7i 1.27958i
$$848$$ 0 0
$$849$$ −1.32754e6 −0.0632087
$$850$$ 0 0
$$851$$ −541632. −0.0256378
$$852$$ 0 0
$$853$$ 2.20186e7i 1.03613i 0.855340 + 0.518067i $$0.173348\pi$$
−0.855340 + 0.518067i $$0.826652\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 3.16676e7i 1.47287i 0.676510 + 0.736434i $$0.263492\pi$$
−0.676510 + 0.736434i $$0.736508\pi$$
$$858$$ 0 0
$$859$$ 1.58064e7 0.730886 0.365443 0.930834i $$-0.380918\pi$$
0.365443 + 0.930834i $$0.380918\pi$$
$$860$$ 0 0
$$861$$ 7.21766e6 0.331809
$$862$$ 0 0
$$863$$ − 1.44287e7i − 0.659476i −0.944072 0.329738i $$-0.893040\pi$$
0.944072 0.329738i $$-0.106960\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 5.58331e6i 0.252257i
$$868$$ 0 0
$$869$$ −4.93728e6 −0.221788
$$870$$ 0 0
$$871$$ −1.74380e7 −0.778845
$$872$$ 0 0
$$873$$ − 2.70216e7i − 1.19999i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 247902.i 0.0108838i 0.999985 + 0.00544191i $$0.00173222\pi$$
−0.999985 + 0.00544191i $$0.998268\pi$$
$$878$$ 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
$$883$$ 4.18015e7i 1.80422i 0.431503 + 0.902112i $$0.357984\pi$$
−0.431503 + 0.902112i $$0.642016\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 2.10476e7i 0.898241i 0.893471 + 0.449120i $$0.148263\pi$$
−0.893471 + 0.449120i $$0.851737\pi$$
$$888$$ 0 0
$$889$$ −1.35460e7 −0.574852
$$890$$ 0 0
$$891$$ 7.05087e6 0.297542
$$892$$ 0 0
$$893$$ 1.28133e7i 0.537690i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 3.40454e6i − 0.141279i
$$898$$ 0 0
$$899$$ 8.34768e6 0.344482
$$900$$ 0 0
$$901$$ −4.00136e7 −1.64208
$$902$$ 0 0
$$903$$ 955392.i 0.0389908i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 7.48309e6i − 0.302039i −0.988531 0.151019i $$-0.951744\pi$$
0.988531 0.151019i $$-0.0482556\pi$$
$$908$$ 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
$$913$$ 2.47397e6i 0.0982239i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1.46788e7i 0.576457i
$$918$$ 0 0
$$919$$ −1.68976e7 −0.659990 −0.329995 0.943983i $$-0.607047\pi$$
−0.329995 + 0.943983i $$0.607047\pi$$
$$920$$ 0 0
$$921$$ 9.51003e6 0.369431
$$922$$ 0 0
$$923$$ − 9.33733e6i − 0.360760i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 4.42741e6i − 0.169219i
$$928$$ 0 0
$$929$$ 1.28653e7 0.489081 0.244541 0.969639i $$-0.421363\pi$$
0.244541 + 0.969639i $$0.421363\pi$$
$$930$$ 0 0
$$931$$ −2.12604e7 −0.803892
$$932$$ 0 0
$$933$$ − 9.49219e6i − 0.356995i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 1.06887e7i 0.397718i 0.980028 + 0.198859i $$0.0637236\pi$$
−0.980028 + 0.198859i $$0.936276\pi$$
$$938$$ 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
$$943$$ − 2.79684e7i − 1.02421i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1.70892e7i 0.619222i 0.950863 + 0.309611i $$0.100199\pi$$
−0.950863 + 0.309611i $$0.899801\pi$$
$$948$$ 0 0
$$949$$ 1.10894e7 0.399706
$$950$$ 0 0
$$951$$ 8.49849e6 0.304713
$$952$$ 0 0
$$953$$ − 2.22259e7i − 0.792735i −0.918092 0.396367i $$-0.870271\pi$$
0.918092 0.396367i $$-0.129729\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 2.01872e6i − 0.0712519i
$$958$$ 0 0
$$959$$ −2.78243e7 −0.976961
$$960$$ 0 0
$$961$$ −2.26364e7 −0.790678
$$962$$ 0 0
$$963$$ − 3.59323e7i − 1.24859i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 2.41551e7i − 0.830696i −0.909663 0.415348i $$-0.863660\pi$$
0.909663 0.415348i $$-0.136340\pi$$
$$968$$ 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
$$973$$ − 2.15462e7i − 0.729608i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 1.56612e7i − 0.524915i −0.964944 0.262457i $$-0.915467\pi$$
0.964944 0.262457i $$-0.0845329\pi$$
$$978$$ 0 0
$$979$$ −1.50028e7 −0.500281
$$980$$ 0 0
$$981$$ −8.36041e6 −0.277367
$$982$$ 0 0
$$983$$ − 1.63420e7i − 0.539412i −0.962943 0.269706i $$-0.913073\pi$$
0.962943 0.269706i $$-0.0869266\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 9.28358e6i − 0.303335i
$$988$$ 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
$$993$$ 1.23994e7i 0.399050i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 1.29097e7i − 0.411320i −0.978624 0.205660i $$-0.934066\pi$$
0.978624 0.205660i $$-0.0659341\pi$$
$$998$$ 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 400.6.c.j.49.1 2
4.3 odd 2 25.6.b.a.24.2 2
5.2 odd 4 400.6.a.g.1.1 1
5.3 odd 4 80.6.a.e.1.1 1
5.4 even 2 inner 400.6.c.j.49.2 2
12.11 even 2 225.6.b.e.199.1 2
15.8 even 4 720.6.a.a.1.1 1
20.3 even 4 5.6.a.a.1.1 1
20.7 even 4 25.6.a.a.1.1 1
20.19 odd 2 25.6.b.a.24.1 2
40.3 even 4 320.6.a.j.1.1 1
40.13 odd 4 320.6.a.g.1.1 1
60.23 odd 4 45.6.a.b.1.1 1
60.47 odd 4 225.6.a.f.1.1 1
60.59 even 2 225.6.b.e.199.2 2
140.83 odd 4 245.6.a.b.1.1 1
220.43 odd 4 605.6.a.a.1.1 1
260.103 even 4 845.6.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 1 20.3 even 4
25.6.a.a.1.1 1 20.7 even 4
25.6.b.a.24.1 2 20.19 odd 2
25.6.b.a.24.2 2 4.3 odd 2
45.6.a.b.1.1 1 60.23 odd 4
80.6.a.e.1.1 1 5.3 odd 4
225.6.a.f.1.1 1 60.47 odd 4
225.6.b.e.199.1 2 12.11 even 2
225.6.b.e.199.2 2 60.59 even 2
245.6.a.b.1.1 1 140.83 odd 4
320.6.a.g.1.1 1 40.13 odd 4
320.6.a.j.1.1 1 40.3 even 4
400.6.a.g.1.1 1 5.2 odd 4
400.6.c.j.49.1 2 1.1 even 1 trivial
400.6.c.j.49.2 2 5.4 even 2 inner
605.6.a.a.1.1 1 220.43 odd 4
720.6.a.a.1.1 1 15.8 even 4
845.6.a.b.1.1 1 260.103 even 4