# Properties

 Label 1008.6.a.bd.1.2 Level $1008$ Weight $6$ Character 1008.1 Self dual yes Analytic conductor $161.667$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,6,Mod(1,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1008.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1008.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: $$161.666890371$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{345})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 86$$ x^2 - x - 86 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 56) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-8.78709$$ of defining polynomial Character $$\chi$$ $$=$$ 1008.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+14.7225 q^{5} +49.0000 q^{7} +O(q^{10})$$ $$q+14.7225 q^{5} +49.0000 q^{7} +58.5549 q^{11} +1179.39 q^{13} -1496.55 q^{17} -498.838 q^{19} -1889.34 q^{23} -2908.25 q^{25} -1914.54 q^{29} -794.577 q^{31} +721.404 q^{35} +2987.93 q^{37} -11941.3 q^{41} -9820.19 q^{43} -19636.0 q^{47} +2401.00 q^{49} +19875.0 q^{53} +862.077 q^{55} +35838.6 q^{59} +49975.9 q^{61} +17363.6 q^{65} -48176.2 q^{67} +77179.1 q^{71} -59667.3 q^{73} +2869.19 q^{77} -60743.1 q^{79} -46134.2 q^{83} -22033.1 q^{85} -78668.7 q^{89} +57790.2 q^{91} -7344.15 q^{95} -43573.3 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 82 q^{5} + 98 q^{7}+O(q^{10})$$ 2 * q - 82 * q^5 + 98 * q^7 $$2 q - 82 q^{5} + 98 q^{7} + 340 q^{11} + 910 q^{13} - 3216 q^{17} + 674 q^{19} - 1104 q^{23} + 3322 q^{25} - 8064 q^{29} + 6212 q^{31} - 4018 q^{35} - 8512 q^{37} + 1304 q^{41} + 10004 q^{43} - 12748 q^{47} + 4802 q^{49} + 11220 q^{53} - 26360 q^{55} - 12018 q^{59} + 102738 q^{61} + 43420 q^{65} - 24136 q^{67} + 89720 q^{71} - 55588 q^{73} + 16660 q^{77} - 48824 q^{79} + 35782 q^{83} + 144276 q^{85} + 18300 q^{89} + 44590 q^{91} - 120784 q^{95} - 69984 q^{97}+O(q^{100})$$ 2 * q - 82 * q^5 + 98 * q^7 + 340 * q^11 + 910 * q^13 - 3216 * q^17 + 674 * q^19 - 1104 * q^23 + 3322 * q^25 - 8064 * q^29 + 6212 * q^31 - 4018 * q^35 - 8512 * q^37 + 1304 * q^41 + 10004 * q^43 - 12748 * q^47 + 4802 * q^49 + 11220 * q^53 - 26360 * q^55 - 12018 * q^59 + 102738 * q^61 + 43420 * q^65 - 24136 * q^67 + 89720 * q^71 - 55588 * q^73 + 16660 * q^77 - 48824 * q^79 + 35782 * q^83 + 144276 * q^85 + 18300 * q^89 + 44590 * q^91 - 120784 * q^95 - 69984 * q^97

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 14.7225 0.263365 0.131682 0.991292i $$-0.457962\pi$$
0.131682 + 0.991292i $$0.457962\pi$$
$$6$$ 0 0
$$7$$ 49.0000 0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 58.5549 0.145909 0.0729545 0.997335i $$-0.476757\pi$$
0.0729545 + 0.997335i $$0.476757\pi$$
$$12$$ 0 0
$$13$$ 1179.39 1.93553 0.967765 0.251853i $$-0.0810400\pi$$
0.967765 + 0.251853i $$0.0810400\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1496.55 −1.25594 −0.627972 0.778236i $$-0.716115\pi$$
−0.627972 + 0.778236i $$0.716115\pi$$
$$18$$ 0 0
$$19$$ −498.838 −0.317012 −0.158506 0.987358i $$-0.550668\pi$$
−0.158506 + 0.987358i $$0.550668\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1889.34 −0.744716 −0.372358 0.928089i $$-0.621451\pi$$
−0.372358 + 0.928089i $$0.621451\pi$$
$$24$$ 0 0
$$25$$ −2908.25 −0.930639
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −1914.54 −0.422737 −0.211369 0.977406i $$-0.567792\pi$$
−0.211369 + 0.977406i $$0.567792\pi$$
$$30$$ 0 0
$$31$$ −794.577 −0.148502 −0.0742509 0.997240i $$-0.523657\pi$$
−0.0742509 + 0.997240i $$0.523657\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 721.404 0.0995424
$$36$$ 0 0
$$37$$ 2987.93 0.358811 0.179406 0.983775i $$-0.442583\pi$$
0.179406 + 0.983775i $$0.442583\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −11941.3 −1.10941 −0.554704 0.832047i $$-0.687169\pi$$
−0.554704 + 0.832047i $$0.687169\pi$$
$$42$$ 0 0
$$43$$ −9820.19 −0.809933 −0.404966 0.914332i $$-0.632717\pi$$
−0.404966 + 0.914332i $$0.632717\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −19636.0 −1.29660 −0.648302 0.761383i $$-0.724521\pi$$
−0.648302 + 0.761383i $$0.724521\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 19875.0 0.971889 0.485945 0.873990i $$-0.338476\pi$$
0.485945 + 0.873990i $$0.338476\pi$$
$$54$$ 0 0
$$55$$ 862.077 0.0384272
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 35838.6 1.34036 0.670180 0.742199i $$-0.266217\pi$$
0.670180 + 0.742199i $$0.266217\pi$$
$$60$$ 0 0
$$61$$ 49975.9 1.71964 0.859818 0.510601i $$-0.170577\pi$$
0.859818 + 0.510601i $$0.170577\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 17363.6 0.509750
$$66$$ 0 0
$$67$$ −48176.2 −1.31113 −0.655565 0.755139i $$-0.727569\pi$$
−0.655565 + 0.755139i $$0.727569\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 77179.1 1.81699 0.908497 0.417891i $$-0.137230\pi$$
0.908497 + 0.417891i $$0.137230\pi$$
$$72$$ 0 0
$$73$$ −59667.3 −1.31048 −0.655238 0.755422i $$-0.727432\pi$$
−0.655238 + 0.755422i $$0.727432\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2869.19 0.0551484
$$78$$ 0 0
$$79$$ −60743.1 −1.09504 −0.547519 0.836793i $$-0.684428\pi$$
−0.547519 + 0.836793i $$0.684428\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −46134.2 −0.735068 −0.367534 0.930010i $$-0.619798\pi$$
−0.367534 + 0.930010i $$0.619798\pi$$
$$84$$ 0 0
$$85$$ −22033.1 −0.330771
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −78668.7 −1.05275 −0.526377 0.850251i $$-0.676450\pi$$
−0.526377 + 0.850251i $$0.676450\pi$$
$$90$$ 0 0
$$91$$ 57790.2 0.731562
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −7344.15 −0.0834897
$$96$$ 0 0
$$97$$ −43573.3 −0.470209 −0.235104 0.971970i $$-0.575543\pi$$
−0.235104 + 0.971970i $$0.575543\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 164188. 1.60154 0.800772 0.598970i $$-0.204423\pi$$
0.800772 + 0.598970i $$0.204423\pi$$
$$102$$ 0 0
$$103$$ −164547. −1.52825 −0.764127 0.645065i $$-0.776830\pi$$
−0.764127 + 0.645065i $$0.776830\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −185426. −1.56571 −0.782854 0.622206i $$-0.786237\pi$$
−0.782854 + 0.622206i $$0.786237\pi$$
$$108$$ 0 0
$$109$$ 190063. 1.53225 0.766127 0.642689i $$-0.222181\pi$$
0.766127 + 0.642689i $$0.222181\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 116023. 0.854769 0.427384 0.904070i $$-0.359435\pi$$
0.427384 + 0.904070i $$0.359435\pi$$
$$114$$ 0 0
$$115$$ −27815.9 −0.196132
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −73331.2 −0.474702
$$120$$ 0 0
$$121$$ −157622. −0.978711
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −88824.6 −0.508462
$$126$$ 0 0
$$127$$ −71785.5 −0.394936 −0.197468 0.980309i $$-0.563272\pi$$
−0.197468 + 0.980309i $$0.563272\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2016.22 −0.0102650 −0.00513250 0.999987i $$-0.501634\pi$$
−0.00513250 + 0.999987i $$0.501634\pi$$
$$132$$ 0 0
$$133$$ −24443.1 −0.119819
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 243904. 1.11024 0.555120 0.831770i $$-0.312672\pi$$
0.555120 + 0.831770i $$0.312672\pi$$
$$138$$ 0 0
$$139$$ −209413. −0.919319 −0.459660 0.888095i $$-0.652029\pi$$
−0.459660 + 0.888095i $$0.652029\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 69059.3 0.282411
$$144$$ 0 0
$$145$$ −28186.9 −0.111334
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −22954.3 −0.0847028 −0.0423514 0.999103i $$-0.513485\pi$$
−0.0423514 + 0.999103i $$0.513485\pi$$
$$150$$ 0 0
$$151$$ 276737. 0.987700 0.493850 0.869547i $$-0.335589\pi$$
0.493850 + 0.869547i $$0.335589\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −11698.2 −0.0391101
$$156$$ 0 0
$$157$$ 276692. 0.895874 0.447937 0.894065i $$-0.352159\pi$$
0.447937 + 0.894065i $$0.352159\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −92577.7 −0.281476
$$162$$ 0 0
$$163$$ −274258. −0.808518 −0.404259 0.914645i $$-0.632471\pi$$
−0.404259 + 0.914645i $$0.632471\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −215716. −0.598537 −0.299268 0.954169i $$-0.596743\pi$$
−0.299268 + 0.954169i $$0.596743\pi$$
$$168$$ 0 0
$$169$$ 1.01967e6 2.74628
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −86252.7 −0.219108 −0.109554 0.993981i $$-0.534942\pi$$
−0.109554 + 0.993981i $$0.534942\pi$$
$$174$$ 0 0
$$175$$ −142504. −0.351749
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −435822. −1.01666 −0.508331 0.861162i $$-0.669738\pi$$
−0.508331 + 0.861162i $$0.669738\pi$$
$$180$$ 0 0
$$181$$ 56972.2 0.129261 0.0646303 0.997909i $$-0.479413\pi$$
0.0646303 + 0.997909i $$0.479413\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 43989.9 0.0944981
$$186$$ 0 0
$$187$$ −87630.7 −0.183253
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −350077. −0.694353 −0.347177 0.937800i $$-0.612860\pi$$
−0.347177 + 0.937800i $$0.612860\pi$$
$$192$$ 0 0
$$193$$ −623227. −1.20435 −0.602175 0.798364i $$-0.705699\pi$$
−0.602175 + 0.798364i $$0.705699\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −223265. −0.409879 −0.204939 0.978775i $$-0.565700\pi$$
−0.204939 + 0.978775i $$0.565700\pi$$
$$198$$ 0 0
$$199$$ −756375. −1.35396 −0.676978 0.736004i $$-0.736711\pi$$
−0.676978 + 0.736004i $$0.736711\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −93812.7 −0.159780
$$204$$ 0 0
$$205$$ −175806. −0.292179
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −29209.4 −0.0462549
$$210$$ 0 0
$$211$$ 30644.3 0.0473853 0.0236927 0.999719i $$-0.492458\pi$$
0.0236927 + 0.999719i $$0.492458\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −144578. −0.213308
$$216$$ 0 0
$$217$$ −38934.3 −0.0561284
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1.76503e6 −2.43092
$$222$$ 0 0
$$223$$ −461375. −0.621286 −0.310643 0.950527i $$-0.600544\pi$$
−0.310643 + 0.950527i $$0.600544\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 346439. 0.446234 0.223117 0.974792i $$-0.428377\pi$$
0.223117 + 0.974792i $$0.428377\pi$$
$$228$$ 0 0
$$229$$ −521650. −0.657341 −0.328671 0.944445i $$-0.606601\pi$$
−0.328671 + 0.944445i $$0.606601\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −154120. −0.185981 −0.0929907 0.995667i $$-0.529643\pi$$
−0.0929907 + 0.995667i $$0.529643\pi$$
$$234$$ 0 0
$$235$$ −289091. −0.341480
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −478940. −0.542358 −0.271179 0.962529i $$-0.587414\pi$$
−0.271179 + 0.962529i $$0.587414\pi$$
$$240$$ 0 0
$$241$$ 86551.8 0.0959917 0.0479958 0.998848i $$-0.484717\pi$$
0.0479958 + 0.998848i $$0.484717\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 35348.8 0.0376235
$$246$$ 0 0
$$247$$ −588326. −0.613586
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1.90322e6 −1.90679 −0.953397 0.301719i $$-0.902439\pi$$
−0.953397 + 0.301719i $$0.902439\pi$$
$$252$$ 0 0
$$253$$ −110630. −0.108661
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 569922. 0.538248 0.269124 0.963105i $$-0.413266\pi$$
0.269124 + 0.963105i $$0.413266\pi$$
$$258$$ 0 0
$$259$$ 146408. 0.135618
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 1.29853e6 1.15761 0.578806 0.815465i $$-0.303519\pi$$
0.578806 + 0.815465i $$0.303519\pi$$
$$264$$ 0 0
$$265$$ 292610. 0.255961
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 53721.5 0.0452655 0.0226328 0.999744i $$-0.492795\pi$$
0.0226328 + 0.999744i $$0.492795\pi$$
$$270$$ 0 0
$$271$$ 758053. 0.627013 0.313506 0.949586i $$-0.398496\pi$$
0.313506 + 0.949586i $$0.398496\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −170292. −0.135789
$$276$$ 0 0
$$277$$ 2.07598e6 1.62564 0.812819 0.582516i $$-0.197932\pi$$
0.812819 + 0.582516i $$0.197932\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2.61195e6 −1.97333 −0.986664 0.162770i $$-0.947957\pi$$
−0.986664 + 0.162770i $$0.947957\pi$$
$$282$$ 0 0
$$283$$ −992734. −0.736829 −0.368414 0.929662i $$-0.620099\pi$$
−0.368414 + 0.929662i $$0.620099\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −585123. −0.419317
$$288$$ 0 0
$$289$$ 819820. 0.577396
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −371728. −0.252962 −0.126481 0.991969i $$-0.540368\pi$$
−0.126481 + 0.991969i $$0.540368\pi$$
$$294$$ 0 0
$$295$$ 527635. 0.353003
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −2.22827e6 −1.44142
$$300$$ 0 0
$$301$$ −481189. −0.306126
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 735772. 0.452891
$$306$$ 0 0
$$307$$ 2.83906e6 1.71921 0.859606 0.510958i $$-0.170709\pi$$
0.859606 + 0.510958i $$0.170709\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 759362. 0.445193 0.222596 0.974911i $$-0.428547\pi$$
0.222596 + 0.974911i $$0.428547\pi$$
$$312$$ 0 0
$$313$$ −1.80191e6 −1.03961 −0.519806 0.854284i $$-0.673996\pi$$
−0.519806 + 0.854284i $$0.673996\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 441899. 0.246987 0.123494 0.992345i $$-0.460590\pi$$
0.123494 + 0.992345i $$0.460590\pi$$
$$318$$ 0 0
$$319$$ −112106. −0.0616811
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 746538. 0.398149
$$324$$ 0 0
$$325$$ −3.42997e6 −1.80128
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −962162. −0.490070
$$330$$ 0 0
$$331$$ 3.15779e6 1.58421 0.792107 0.610383i $$-0.208984\pi$$
0.792107 + 0.610383i $$0.208984\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −709275. −0.345305
$$336$$ 0 0
$$337$$ −1.66612e6 −0.799158 −0.399579 0.916699i $$-0.630844\pi$$
−0.399579 + 0.916699i $$0.630844\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −46526.4 −0.0216677
$$342$$ 0 0
$$343$$ 117649. 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −3.41953e6 −1.52455 −0.762277 0.647251i $$-0.775919\pi$$
−0.762277 + 0.647251i $$0.775919\pi$$
$$348$$ 0 0
$$349$$ −3.18396e6 −1.39928 −0.699638 0.714497i $$-0.746655\pi$$
−0.699638 + 0.714497i $$0.746655\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −3.02506e6 −1.29210 −0.646051 0.763294i $$-0.723581\pi$$
−0.646051 + 0.763294i $$0.723581\pi$$
$$354$$ 0 0
$$355$$ 1.13627e6 0.478532
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1.12353e6 −0.460098 −0.230049 0.973179i $$-0.573889\pi$$
−0.230049 + 0.973179i $$0.573889\pi$$
$$360$$ 0 0
$$361$$ −2.22726e6 −0.899504
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −878453. −0.345133
$$366$$ 0 0
$$367$$ −3.39115e6 −1.31426 −0.657131 0.753777i $$-0.728230\pi$$
−0.657131 + 0.753777i $$0.728230\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 973873. 0.367340
$$372$$ 0 0
$$373$$ 1.33823e6 0.498033 0.249017 0.968499i $$-0.419893\pi$$
0.249017 + 0.968499i $$0.419893\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −2.25800e6 −0.818221
$$378$$ 0 0
$$379$$ −3.11403e6 −1.11359 −0.556795 0.830650i $$-0.687969\pi$$
−0.556795 + 0.830650i $$0.687969\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −334533. −0.116531 −0.0582656 0.998301i $$-0.518557\pi$$
−0.0582656 + 0.998301i $$0.518557\pi$$
$$384$$ 0 0
$$385$$ 42241.8 0.0145241
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −1.59871e6 −0.535667 −0.267834 0.963465i $$-0.586308\pi$$
−0.267834 + 0.963465i $$0.586308\pi$$
$$390$$ 0 0
$$391$$ 2.82750e6 0.935322
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −894292. −0.288394
$$396$$ 0 0
$$397$$ −2.20789e6 −0.703074 −0.351537 0.936174i $$-0.614341\pi$$
−0.351537 + 0.936174i $$0.614341\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −1.89272e6 −0.587793 −0.293897 0.955837i $$-0.594952\pi$$
−0.293897 + 0.955837i $$0.594952\pi$$
$$402$$ 0 0
$$403$$ −937118. −0.287430
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 174958. 0.0523537
$$408$$ 0 0
$$409$$ 1.14418e6 0.338210 0.169105 0.985598i $$-0.445912\pi$$
0.169105 + 0.985598i $$0.445912\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 1.75609e6 0.506608
$$414$$ 0 0
$$415$$ −679212. −0.193591
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 5.17298e6 1.43948 0.719740 0.694244i $$-0.244261\pi$$
0.719740 + 0.694244i $$0.244261\pi$$
$$420$$ 0 0
$$421$$ −1.40960e6 −0.387606 −0.193803 0.981040i $$-0.562082\pi$$
−0.193803 + 0.981040i $$0.562082\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 4.35235e6 1.16883
$$426$$ 0 0
$$427$$ 2.44882e6 0.649961
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6.17076e6 1.60009 0.800047 0.599938i $$-0.204808\pi$$
0.800047 + 0.599938i $$0.204808\pi$$
$$432$$ 0 0
$$433$$ 387046. 0.0992071 0.0496036 0.998769i $$-0.484204\pi$$
0.0496036 + 0.998769i $$0.484204\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 942475. 0.236084
$$438$$ 0 0
$$439$$ −3.32974e6 −0.824611 −0.412305 0.911046i $$-0.635276\pi$$
−0.412305 + 0.911046i $$0.635276\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −1.44003e6 −0.348627 −0.174314 0.984690i $$-0.555771\pi$$
−0.174314 + 0.984690i $$0.555771\pi$$
$$444$$ 0 0
$$445$$ −1.15820e6 −0.277258
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −4.28977e6 −1.00420 −0.502098 0.864811i $$-0.667438\pi$$
−0.502098 + 0.864811i $$0.667438\pi$$
$$450$$ 0 0
$$451$$ −699222. −0.161873
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 850819. 0.192667
$$456$$ 0 0
$$457$$ −5.27889e6 −1.18237 −0.591183 0.806537i $$-0.701339\pi$$
−0.591183 + 0.806537i $$0.701339\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 3.72322e6 0.815955 0.407977 0.912992i $$-0.366234\pi$$
0.407977 + 0.912992i $$0.366234\pi$$
$$462$$ 0 0
$$463$$ −1.04809e6 −0.227220 −0.113610 0.993525i $$-0.536241\pi$$
−0.113610 + 0.993525i $$0.536241\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6.04244e6 1.28210 0.641048 0.767501i $$-0.278500\pi$$
0.641048 + 0.767501i $$0.278500\pi$$
$$468$$ 0 0
$$469$$ −2.36063e6 −0.495560
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −575021. −0.118176
$$474$$ 0 0
$$475$$ 1.45074e6 0.295024
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −2.06868e6 −0.411959 −0.205980 0.978556i $$-0.566038\pi$$
−0.205980 + 0.978556i $$0.566038\pi$$
$$480$$ 0 0
$$481$$ 3.52394e6 0.694490
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −641509. −0.123836
$$486$$ 0 0
$$487$$ 2.82966e6 0.540645 0.270323 0.962770i $$-0.412870\pi$$
0.270323 + 0.962770i $$0.412870\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −4.44554e6 −0.832186 −0.416093 0.909322i $$-0.636601\pi$$
−0.416093 + 0.909322i $$0.636601\pi$$
$$492$$ 0 0
$$493$$ 2.86522e6 0.530934
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 3.78177e6 0.686759
$$498$$ 0 0
$$499$$ 1.34623e6 0.242029 0.121015 0.992651i $$-0.461385\pi$$
0.121015 + 0.992651i $$0.461385\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −4.88909e6 −0.861604 −0.430802 0.902446i $$-0.641769\pi$$
−0.430802 + 0.902446i $$0.641769\pi$$
$$504$$ 0 0
$$505$$ 2.41727e6 0.421790
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −6.55518e6 −1.12148 −0.560738 0.827993i $$-0.689483\pi$$
−0.560738 + 0.827993i $$0.689483\pi$$
$$510$$ 0 0
$$511$$ −2.92370e6 −0.495313
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −2.42254e6 −0.402488
$$516$$ 0 0
$$517$$ −1.14978e6 −0.189186
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 4.77290e6 0.770349 0.385175 0.922844i $$-0.374141\pi$$
0.385175 + 0.922844i $$0.374141\pi$$
$$522$$ 0 0
$$523$$ 828135. 0.132387 0.0661937 0.997807i $$-0.478914\pi$$
0.0661937 + 0.997807i $$0.478914\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1.18913e6 0.186510
$$528$$ 0 0
$$529$$ −2.86673e6 −0.445398
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −1.40835e7 −2.14730
$$534$$ 0 0
$$535$$ −2.72994e6 −0.412352
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 140590. 0.0208441
$$540$$ 0 0
$$541$$ 6.93356e6 1.01851 0.509253 0.860617i $$-0.329922\pi$$
0.509253 + 0.860617i $$0.329922\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 2.79820e6 0.403541
$$546$$ 0 0
$$547$$ −559077. −0.0798920 −0.0399460 0.999202i $$-0.512719\pi$$
−0.0399460 + 0.999202i $$0.512719\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 955047. 0.134013
$$552$$ 0 0
$$553$$ −2.97641e6 −0.413885
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 3.90444e6 0.533237 0.266619 0.963802i $$-0.414094\pi$$
0.266619 + 0.963802i $$0.414094\pi$$
$$558$$ 0 0
$$559$$ −1.15819e7 −1.56765
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −4.02307e6 −0.534918 −0.267459 0.963569i $$-0.586184\pi$$
−0.267459 + 0.963569i $$0.586184\pi$$
$$564$$ 0 0
$$565$$ 1.70815e6 0.225116
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 2.35801e6 0.305327 0.152663 0.988278i $$-0.451215\pi$$
0.152663 + 0.988278i $$0.451215\pi$$
$$570$$ 0 0
$$571$$ 7.76733e6 0.996969 0.498484 0.866899i $$-0.333890\pi$$
0.498484 + 0.866899i $$0.333890\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 5.49467e6 0.693062
$$576$$ 0 0
$$577$$ 1.38363e7 1.73013 0.865066 0.501657i $$-0.167276\pi$$
0.865066 + 0.501657i $$0.167276\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −2.26057e6 −0.277830
$$582$$ 0 0
$$583$$ 1.16378e6 0.141807
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 2.13747e6 0.256038 0.128019 0.991772i $$-0.459138\pi$$
0.128019 + 0.991772i $$0.459138\pi$$
$$588$$ 0 0
$$589$$ 396365. 0.0470768
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 1.11381e7 1.30069 0.650345 0.759639i $$-0.274624\pi$$
0.650345 + 0.759639i $$0.274624\pi$$
$$594$$ 0 0
$$595$$ −1.07962e6 −0.125020
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 1.10044e7 1.25314 0.626572 0.779364i $$-0.284457\pi$$
0.626572 + 0.779364i $$0.284457\pi$$
$$600$$ 0 0
$$601$$ 1.28046e6 0.144604 0.0723019 0.997383i $$-0.476965\pi$$
0.0723019 + 0.997383i $$0.476965\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −2.32060e6 −0.257758
$$606$$ 0 0
$$607$$ 2.08327e6 0.229496 0.114748 0.993395i $$-0.463394\pi$$
0.114748 + 0.993395i $$0.463394\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2.31585e7 −2.50962
$$612$$ 0 0
$$613$$ −1.37116e7 −1.47380 −0.736899 0.676003i $$-0.763711\pi$$
−0.736899 + 0.676003i $$0.763711\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1.65172e7 1.74672 0.873361 0.487073i $$-0.161936\pi$$
0.873361 + 0.487073i $$0.161936\pi$$
$$618$$ 0 0
$$619$$ −3.51321e6 −0.368534 −0.184267 0.982876i $$-0.558991\pi$$
−0.184267 + 0.982876i $$0.558991\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −3.85477e6 −0.397904
$$624$$ 0 0
$$625$$ 7.78055e6 0.796728
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −4.47160e6 −0.450647
$$630$$ 0 0
$$631$$ 8.54135e6 0.853991 0.426995 0.904254i $$-0.359572\pi$$
0.426995 + 0.904254i $$0.359572\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −1.05686e6 −0.104012
$$636$$ 0 0
$$637$$ 2.83172e6 0.276504
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 9.95519e6 0.956983 0.478492 0.878092i $$-0.341184\pi$$
0.478492 + 0.878092i $$0.341184\pi$$
$$642$$ 0 0
$$643$$ 3.98313e6 0.379924 0.189962 0.981791i $$-0.439163\pi$$
0.189962 + 0.981791i $$0.439163\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −3.84121e6 −0.360750 −0.180375 0.983598i $$-0.557731\pi$$
−0.180375 + 0.983598i $$0.557731\pi$$
$$648$$ 0 0
$$649$$ 2.09853e6 0.195570
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1.84286e7 −1.69126 −0.845628 0.533772i $$-0.820774\pi$$
−0.845628 + 0.533772i $$0.820774\pi$$
$$654$$ 0 0
$$655$$ −29683.8 −0.00270344
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 2.15854e7 1.93619 0.968093 0.250590i $$-0.0806245\pi$$
0.968093 + 0.250590i $$0.0806245\pi$$
$$660$$ 0 0
$$661$$ 1.44072e7 1.28256 0.641278 0.767308i $$-0.278404\pi$$
0.641278 + 0.767308i $$0.278404\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −359864. −0.0315561
$$666$$ 0 0
$$667$$ 3.61723e6 0.314819
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 2.92634e6 0.250910
$$672$$ 0 0
$$673$$ −5.55230e6 −0.472536 −0.236268 0.971688i $$-0.575924\pi$$
−0.236268 + 0.971688i $$0.575924\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 9.92009e6 0.831848 0.415924 0.909399i $$-0.363458\pi$$
0.415924 + 0.909399i $$0.363458\pi$$
$$678$$ 0 0
$$679$$ −2.13509e6 −0.177722
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −1.01291e7 −0.830846 −0.415423 0.909628i $$-0.636366\pi$$
−0.415423 + 0.909628i $$0.636366\pi$$
$$684$$ 0 0
$$685$$ 3.59088e6 0.292398
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 2.34404e7 1.88112
$$690$$ 0 0
$$691$$ −1.98747e7 −1.58345 −0.791727 0.610875i $$-0.790818\pi$$
−0.791727 + 0.610875i $$0.790818\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −3.08309e6 −0.242116
$$696$$ 0 0
$$697$$ 1.78708e7 1.39336
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.13147e7 1.63826 0.819132 0.573605i $$-0.194455\pi$$
0.819132 + 0.573605i $$0.194455\pi$$
$$702$$ 0 0
$$703$$ −1.49049e6 −0.113747
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.04523e6 0.605327
$$708$$ 0 0
$$709$$ 9.83200e6 0.734559 0.367279 0.930111i $$-0.380289\pi$$
0.367279 + 0.930111i $$0.380289\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 1.50123e6 0.110592
$$714$$ 0 0
$$715$$ 1.01673e6 0.0743771
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −1.80674e7 −1.30338 −0.651692 0.758484i $$-0.725940\pi$$
−0.651692 + 0.758484i $$0.725940\pi$$
$$720$$ 0 0
$$721$$ −8.06278e6 −0.577626
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 5.56797e6 0.393416
$$726$$ 0 0
$$727$$ 1.82003e7 1.27715 0.638576 0.769559i $$-0.279524\pi$$
0.638576 + 0.769559i $$0.279524\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1.46965e7 1.01723
$$732$$ 0 0
$$733$$ 6.70629e6 0.461023 0.230512 0.973070i $$-0.425960\pi$$
0.230512 + 0.973070i $$0.425960\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −2.82095e6 −0.191305
$$738$$ 0 0
$$739$$ −2.16502e7 −1.45831 −0.729156 0.684347i $$-0.760087\pi$$
−0.729156 + 0.684347i $$0.760087\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −1.40437e7 −0.933276 −0.466638 0.884448i $$-0.654535\pi$$
−0.466638 + 0.884448i $$0.654535\pi$$
$$744$$ 0 0
$$745$$ −337945. −0.0223077
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −9.08586e6 −0.591782
$$750$$ 0 0
$$751$$ −8.91381e6 −0.576718 −0.288359 0.957522i $$-0.593110\pi$$
−0.288359 + 0.957522i $$0.593110\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 4.07427e6 0.260125
$$756$$ 0 0
$$757$$ −1.04419e7 −0.662276 −0.331138 0.943582i $$-0.607433\pi$$
−0.331138 + 0.943582i $$0.607433\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −2.96214e7 −1.85414 −0.927072 0.374883i $$-0.877683\pi$$
−0.927072 + 0.374883i $$0.877683\pi$$
$$762$$ 0 0
$$763$$ 9.31307e6 0.579138
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 4.22678e7 2.59431
$$768$$ 0 0
$$769$$ −4.24042e6 −0.258579 −0.129289 0.991607i $$-0.541270\pi$$
−0.129289 + 0.991607i $$0.541270\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 1.37072e7 0.825086 0.412543 0.910938i $$-0.364641\pi$$
0.412543 + 0.910938i $$0.364641\pi$$
$$774$$ 0 0
$$775$$ 2.31083e6 0.138202
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 5.95677e6 0.351696
$$780$$ 0 0
$$781$$ 4.51922e6 0.265116
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 4.07360e6 0.235942
$$786$$ 0 0
$$787$$ −1.02427e7 −0.589491 −0.294745 0.955576i $$-0.595235\pi$$
−0.294745 + 0.955576i $$0.595235\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 5.68513e6 0.323072
$$792$$ 0 0
$$793$$ 5.89413e7 3.32841
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −3.00087e7 −1.67340 −0.836702 0.547658i $$-0.815520\pi$$
−0.836702 + 0.547658i $$0.815520\pi$$
$$798$$ 0 0
$$799$$ 2.93863e7 1.62846
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −3.49381e6 −0.191210
$$804$$ 0 0
$$805$$ −1.36298e6 −0.0741309
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −922081. −0.0495333 −0.0247667 0.999693i $$-0.507884\pi$$
−0.0247667 + 0.999693i $$0.507884\pi$$
$$810$$ 0 0
$$811$$ 2.49323e7 1.33110 0.665549 0.746355i $$-0.268198\pi$$
0.665549 + 0.746355i $$0.268198\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −4.03776e6 −0.212935
$$816$$ 0 0
$$817$$ 4.89868e6 0.256758
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 8.06228e6 0.417446 0.208723 0.977975i $$-0.433069\pi$$
0.208723 + 0.977975i $$0.433069\pi$$
$$822$$ 0 0
$$823$$ 5.68168e6 0.292400 0.146200 0.989255i $$-0.453296\pi$$
0.146200 + 0.989255i $$0.453296\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −976611. −0.0496544 −0.0248272 0.999692i $$-0.507904\pi$$
−0.0248272 + 0.999692i $$0.507904\pi$$
$$828$$ 0 0
$$829$$ −4.94857e6 −0.250088 −0.125044 0.992151i $$-0.539907\pi$$
−0.125044 + 0.992151i $$0.539907\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −3.59323e6 −0.179421
$$834$$ 0 0
$$835$$ −3.17588e6 −0.157633
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −2.03716e7 −0.999125 −0.499562 0.866278i $$-0.666506\pi$$
−0.499562 + 0.866278i $$0.666506\pi$$
$$840$$ 0 0
$$841$$ −1.68457e7 −0.821293
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1.50122e7 0.723273
$$846$$ 0 0
$$847$$ −7.72349e6 −0.369918
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −5.64521e6 −0.267212
$$852$$ 0 0
$$853$$ 1.41579e7 0.666234 0.333117 0.942885i $$-0.391900\pi$$
0.333117 + 0.942885i $$0.391900\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 1.24397e7 0.578575 0.289287 0.957242i $$-0.406582\pi$$
0.289287 + 0.957242i $$0.406582\pi$$
$$858$$ 0 0
$$859$$ −1.72105e7 −0.795810 −0.397905 0.917427i $$-0.630263\pi$$
−0.397905 + 0.917427i $$0.630263\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 2.04374e7 0.934113 0.467057 0.884227i $$-0.345314\pi$$
0.467057 + 0.884227i $$0.345314\pi$$
$$864$$ 0 0
$$865$$ −1.26986e6 −0.0577052
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −3.55681e6 −0.159776
$$870$$ 0 0
$$871$$ −5.68187e7 −2.53773
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −4.35241e6 −0.192181
$$876$$ 0 0
$$877$$ −1.72398e7 −0.756890 −0.378445 0.925624i $$-0.623541\pi$$
−0.378445 + 0.925624i $$0.623541\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 3.86589e7 1.67807 0.839034 0.544080i $$-0.183121\pi$$
0.839034 + 0.544080i $$0.183121\pi$$
$$882$$ 0 0
$$883$$ 3.29454e7 1.42198 0.710990 0.703203i $$-0.248247\pi$$
0.710990 + 0.703203i $$0.248247\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 3.61545e7 1.54295 0.771477 0.636258i $$-0.219518\pi$$
0.771477 + 0.636258i $$0.219518\pi$$
$$888$$ 0 0
$$889$$ −3.51749e6 −0.149272
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 9.79516e6 0.411039
$$894$$ 0 0
$$895$$ −6.41641e6 −0.267753
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1.52125e6 0.0627772
$$900$$ 0 0
$$901$$ −2.97440e7 −1.22064
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 838774. 0.0340427
$$906$$ 0 0
$$907$$ 4.15620e7 1.67756 0.838781 0.544469i $$-0.183269\pi$$
0.838781 + 0.544469i $$0.183269\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −9.55917e6 −0.381614 −0.190807 0.981628i $$-0.561110\pi$$
−0.190807 + 0.981628i $$0.561110\pi$$
$$912$$ 0 0
$$913$$ −2.70138e6 −0.107253
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −98794.6 −0.00387980
$$918$$ 0 0
$$919$$ 1.20453e7 0.470469 0.235234 0.971939i $$-0.424414\pi$$
0.235234 + 0.971939i $$0.424414\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 9.10244e7 3.51685
$$924$$ 0 0
$$925$$ −8.68963e6 −0.333924
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 2.42707e7 0.922661 0.461331 0.887228i $$-0.347372\pi$$
0.461331 + 0.887228i $$0.347372\pi$$
$$930$$ 0 0
$$931$$ −1.19771e6 −0.0452874
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −1.29015e6 −0.0482625
$$936$$ 0 0
$$937$$ 2.02014e7 0.751679 0.375840 0.926685i $$-0.377354\pi$$
0.375840 + 0.926685i $$0.377354\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 6.68084e6 0.245956 0.122978 0.992409i $$-0.460756\pi$$
0.122978 + 0.992409i $$0.460756\pi$$
$$942$$ 0 0
$$943$$ 2.25612e7 0.826195
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −3.19373e7 −1.15724 −0.578619 0.815598i $$-0.696408\pi$$
−0.578619 + 0.815598i $$0.696408\pi$$
$$948$$ 0 0
$$949$$ −7.03712e7 −2.53647
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −2.26242e7 −0.806938 −0.403469 0.914993i $$-0.632196\pi$$
−0.403469 + 0.914993i $$0.632196\pi$$
$$954$$ 0 0
$$955$$ −5.15402e6 −0.182868
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 1.19513e7 0.419631
$$960$$ 0 0
$$961$$ −2.79978e7 −0.977947
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −9.17547e6 −0.317183
$$966$$ 0 0
$$967$$ 2.38201e7 0.819178 0.409589 0.912270i $$-0.365672\pi$$
0.409589 + 0.912270i $$0.365672\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1.87074e7 0.636746 0.318373 0.947965i $$-0.396864\pi$$
0.318373 + 0.947965i $$0.396864\pi$$
$$972$$ 0 0
$$973$$ −1.02612e7 −0.347470
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −1.50783e7 −0.505376 −0.252688 0.967548i $$-0.581315\pi$$
−0.252688 + 0.967548i $$0.581315\pi$$
$$978$$ 0 0
$$979$$ −4.60644e6 −0.153606
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 2.38312e6 0.0786614 0.0393307 0.999226i $$-0.487477\pi$$
0.0393307 + 0.999226i $$0.487477\pi$$
$$984$$ 0 0
$$985$$ −3.28703e6 −0.107948
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.85537e7 0.603170
$$990$$ 0 0
$$991$$ 4.86331e7 1.57307 0.786535 0.617546i $$-0.211873\pi$$
0.786535 + 0.617546i $$0.211873\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −1.11357e7 −0.356584
$$996$$ 0 0
$$997$$ 1.73445e7 0.552617 0.276309 0.961069i $$-0.410889\pi$$
0.276309 + 0.961069i $$0.410889\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1008.6.a.bd.1.2 2
3.2 odd 2 112.6.a.i.1.2 2
4.3 odd 2 504.6.a.i.1.2 2
12.11 even 2 56.6.a.e.1.1 2
21.20 even 2 784.6.a.u.1.1 2
24.5 odd 2 448.6.a.t.1.1 2
24.11 even 2 448.6.a.v.1.2 2
84.11 even 6 392.6.i.j.177.2 4
84.23 even 6 392.6.i.j.361.2 4
84.47 odd 6 392.6.i.i.361.1 4
84.59 odd 6 392.6.i.i.177.1 4
84.83 odd 2 392.6.a.d.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.a.e.1.1 2 12.11 even 2
112.6.a.i.1.2 2 3.2 odd 2
392.6.a.d.1.2 2 84.83 odd 2
392.6.i.i.177.1 4 84.59 odd 6
392.6.i.i.361.1 4 84.47 odd 6
392.6.i.j.177.2 4 84.11 even 6
392.6.i.j.361.2 4 84.23 even 6
448.6.a.t.1.1 2 24.5 odd 2
448.6.a.v.1.2 2 24.11 even 2
504.6.a.i.1.2 2 4.3 odd 2
784.6.a.u.1.1 2 21.20 even 2
1008.6.a.bd.1.2 2 1.1 even 1 trivial