# Properties

 Label 800.4.c.a.449.2 Level $800$ Weight $4$ Character 800.449 Analytic conductor $47.202$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(449,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 800.c (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$47.2015280046$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 449.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 800.449 Dual form 800.4.c.a.449.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+8.00000i q^{3} -16.0000i q^{7} -37.0000 q^{9} +O(q^{10})$$ $$q+8.00000i q^{3} -16.0000i q^{7} -37.0000 q^{9} -40.0000 q^{11} -50.0000i q^{13} +30.0000i q^{17} -40.0000 q^{19} +128.000 q^{21} +48.0000i q^{23} -80.0000i q^{27} +34.0000 q^{29} +320.000 q^{31} -320.000i q^{33} -310.000i q^{37} +400.000 q^{39} +410.000 q^{41} +152.000i q^{43} +416.000i q^{47} +87.0000 q^{49} -240.000 q^{51} -410.000i q^{53} -320.000i q^{57} +200.000 q^{59} +30.0000 q^{61} +592.000i q^{63} -776.000i q^{67} -384.000 q^{69} +400.000 q^{71} -630.000i q^{73} +640.000i q^{77} +1120.00 q^{79} -359.000 q^{81} +552.000i q^{83} +272.000i q^{87} +326.000 q^{89} -800.000 q^{91} +2560.00i q^{93} +110.000i q^{97} +1480.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 74 q^{9}+O(q^{10})$$ 2 * q - 74 * q^9 $$2 q - 74 q^{9} - 80 q^{11} - 80 q^{19} + 256 q^{21} + 68 q^{29} + 640 q^{31} + 800 q^{39} + 820 q^{41} + 174 q^{49} - 480 q^{51} + 400 q^{59} + 60 q^{61} - 768 q^{69} + 800 q^{71} + 2240 q^{79} - 718 q^{81} + 652 q^{89} - 1600 q^{91} + 2960 q^{99}+O(q^{100})$$ 2 * q - 74 * q^9 - 80 * q^11 - 80 * q^19 + 256 * q^21 + 68 * q^29 + 640 * q^31 + 800 * q^39 + 820 * q^41 + 174 * q^49 - 480 * q^51 + 400 * q^59 + 60 * q^61 - 768 * q^69 + 800 * q^71 + 2240 * q^79 - 718 * q^81 + 652 * q^89 - 1600 * q^91 + 2960 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 8.00000i 1.53960i 0.638285 + 0.769800i $$0.279644\pi$$
−0.638285 + 0.769800i $$0.720356\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 16.0000i − 0.863919i −0.901893 0.431959i $$-0.857822\pi$$
0.901893 0.431959i $$-0.142178\pi$$
$$8$$ 0 0
$$9$$ −37.0000 −1.37037
$$10$$ 0 0
$$11$$ −40.0000 −1.09640 −0.548202 0.836346i $$-0.684688\pi$$
−0.548202 + 0.836346i $$0.684688\pi$$
$$12$$ 0 0
$$13$$ − 50.0000i − 1.06673i −0.845885 0.533366i $$-0.820927\pi$$
0.845885 0.533366i $$-0.179073\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 30.0000i 0.428004i 0.976833 + 0.214002i $$0.0686499\pi$$
−0.976833 + 0.214002i $$0.931350\pi$$
$$18$$ 0 0
$$19$$ −40.0000 −0.482980 −0.241490 0.970403i $$-0.577636\pi$$
−0.241490 + 0.970403i $$0.577636\pi$$
$$20$$ 0 0
$$21$$ 128.000 1.33009
$$22$$ 0 0
$$23$$ 48.0000i 0.435161i 0.976042 + 0.217580i $$0.0698164\pi$$
−0.976042 + 0.217580i $$0.930184\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 80.0000i − 0.570222i
$$28$$ 0 0
$$29$$ 34.0000 0.217712 0.108856 0.994058i $$-0.465281\pi$$
0.108856 + 0.994058i $$0.465281\pi$$
$$30$$ 0 0
$$31$$ 320.000 1.85399 0.926995 0.375073i $$-0.122383\pi$$
0.926995 + 0.375073i $$0.122383\pi$$
$$32$$ 0 0
$$33$$ − 320.000i − 1.68803i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 310.000i − 1.37740i −0.725048 0.688698i $$-0.758182\pi$$
0.725048 0.688698i $$-0.241818\pi$$
$$38$$ 0 0
$$39$$ 400.000 1.64234
$$40$$ 0 0
$$41$$ 410.000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 152.000i 0.539065i 0.962991 + 0.269532i $$0.0868691\pi$$
−0.962991 + 0.269532i $$0.913131\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 416.000i 1.29106i 0.763735 + 0.645530i $$0.223364\pi$$
−0.763735 + 0.645530i $$0.776636\pi$$
$$48$$ 0 0
$$49$$ 87.0000 0.253644
$$50$$ 0 0
$$51$$ −240.000 −0.658955
$$52$$ 0 0
$$53$$ − 410.000i − 1.06260i −0.847184 0.531300i $$-0.821704\pi$$
0.847184 0.531300i $$-0.178296\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 320.000i − 0.743597i
$$58$$ 0 0
$$59$$ 200.000 0.441318 0.220659 0.975351i $$-0.429179\pi$$
0.220659 + 0.975351i $$0.429179\pi$$
$$60$$ 0 0
$$61$$ 30.0000 0.0629690 0.0314845 0.999504i $$-0.489977\pi$$
0.0314845 + 0.999504i $$0.489977\pi$$
$$62$$ 0 0
$$63$$ 592.000i 1.18389i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 776.000i − 1.41498i −0.706725 0.707489i $$-0.749828\pi$$
0.706725 0.707489i $$-0.250172\pi$$
$$68$$ 0 0
$$69$$ −384.000 −0.669973
$$70$$ 0 0
$$71$$ 400.000 0.668609 0.334305 0.942465i $$-0.391499\pi$$
0.334305 + 0.942465i $$0.391499\pi$$
$$72$$ 0 0
$$73$$ − 630.000i − 1.01008i −0.863096 0.505041i $$-0.831478\pi$$
0.863096 0.505041i $$-0.168522\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 640.000i 0.947205i
$$78$$ 0 0
$$79$$ 1120.00 1.59506 0.797531 0.603278i $$-0.206139\pi$$
0.797531 + 0.603278i $$0.206139\pi$$
$$80$$ 0 0
$$81$$ −359.000 −0.492455
$$82$$ 0 0
$$83$$ 552.000i 0.729998i 0.931008 + 0.364999i $$0.118931\pi$$
−0.931008 + 0.364999i $$0.881069\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 272.000i 0.335189i
$$88$$ 0 0
$$89$$ 326.000 0.388269 0.194134 0.980975i $$-0.437810\pi$$
0.194134 + 0.980975i $$0.437810\pi$$
$$90$$ 0 0
$$91$$ −800.000 −0.921569
$$92$$ 0 0
$$93$$ 2560.00i 2.85440i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 110.000i 0.115142i 0.998341 + 0.0575712i $$0.0183356\pi$$
−0.998341 + 0.0575712i $$0.981664\pi$$
$$98$$ 0 0
$$99$$ 1480.00 1.50248
$$100$$ 0 0
$$101$$ −1098.00 −1.08173 −0.540867 0.841108i $$-0.681904\pi$$
−0.540867 + 0.841108i $$0.681904\pi$$
$$102$$ 0 0
$$103$$ − 48.0000i − 0.0459183i −0.999736 0.0229591i $$-0.992691\pi$$
0.999736 0.0229591i $$-0.00730876\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 664.000i − 0.599919i −0.953952 0.299959i $$-0.903027\pi$$
0.953952 0.299959i $$-0.0969731\pi$$
$$108$$ 0 0
$$109$$ 370.000 0.325134 0.162567 0.986698i $$-0.448023\pi$$
0.162567 + 0.986698i $$0.448023\pi$$
$$110$$ 0 0
$$111$$ 2480.00 2.12064
$$112$$ 0 0
$$113$$ 1490.00i 1.24042i 0.784436 + 0.620210i $$0.212953\pi$$
−0.784436 + 0.620210i $$0.787047\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1850.00i 1.46182i
$$118$$ 0 0
$$119$$ 480.000 0.369761
$$120$$ 0 0
$$121$$ 269.000 0.202104
$$122$$ 0 0
$$123$$ 3280.00i 2.40445i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1024.00i 0.715475i 0.933822 + 0.357737i $$0.116452\pi$$
−0.933822 + 0.357737i $$0.883548\pi$$
$$128$$ 0 0
$$129$$ −1216.00 −0.829944
$$130$$ 0 0
$$131$$ 1160.00 0.773662 0.386831 0.922151i $$-0.373570\pi$$
0.386831 + 0.922151i $$0.373570\pi$$
$$132$$ 0 0
$$133$$ 640.000i 0.417256i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 570.000i − 0.355463i −0.984079 0.177731i $$-0.943124\pi$$
0.984079 0.177731i $$-0.0568758\pi$$
$$138$$ 0 0
$$139$$ 1960.00 1.19601 0.598004 0.801493i $$-0.295961\pi$$
0.598004 + 0.801493i $$0.295961\pi$$
$$140$$ 0 0
$$141$$ −3328.00 −1.98772
$$142$$ 0 0
$$143$$ 2000.00i 1.16957i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 696.000i 0.390511i
$$148$$ 0 0
$$149$$ 2010.00 1.10514 0.552569 0.833467i $$-0.313648\pi$$
0.552569 + 0.833467i $$0.313648\pi$$
$$150$$ 0 0
$$151$$ −720.000 −0.388032 −0.194016 0.980998i $$-0.562151\pi$$
−0.194016 + 0.980998i $$0.562151\pi$$
$$152$$ 0 0
$$153$$ − 1110.00i − 0.586524i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 1790.00i − 0.909921i −0.890512 0.454960i $$-0.849653\pi$$
0.890512 0.454960i $$-0.150347\pi$$
$$158$$ 0 0
$$159$$ 3280.00 1.63598
$$160$$ 0 0
$$161$$ 768.000 0.375943
$$162$$ 0 0
$$163$$ − 1208.00i − 0.580478i −0.956954 0.290239i $$-0.906265\pi$$
0.956954 0.290239i $$-0.0937348\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 2896.00i − 1.34191i −0.741497 0.670956i $$-0.765884\pi$$
0.741497 0.670956i $$-0.234116\pi$$
$$168$$ 0 0
$$169$$ −303.000 −0.137915
$$170$$ 0 0
$$171$$ 1480.00 0.661862
$$172$$ 0 0
$$173$$ 750.000i 0.329604i 0.986327 + 0.164802i $$0.0526985\pi$$
−0.986327 + 0.164802i $$0.947302\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1600.00i 0.679454i
$$178$$ 0 0
$$179$$ −2280.00 −0.952040 −0.476020 0.879434i $$-0.657921\pi$$
−0.476020 + 0.879434i $$0.657921\pi$$
$$180$$ 0 0
$$181$$ −442.000 −0.181512 −0.0907558 0.995873i $$-0.528928\pi$$
−0.0907558 + 0.995873i $$0.528928\pi$$
$$182$$ 0 0
$$183$$ 240.000i 0.0969471i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 1200.00i − 0.469266i
$$188$$ 0 0
$$189$$ −1280.00 −0.492626
$$190$$ 0 0
$$191$$ −1920.00 −0.727363 −0.363681 0.931523i $$-0.618480\pi$$
−0.363681 + 0.931523i $$0.618480\pi$$
$$192$$ 0 0
$$193$$ − 5070.00i − 1.89091i −0.325746 0.945457i $$-0.605615\pi$$
0.325746 0.945457i $$-0.394385\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 1910.00i − 0.690771i −0.938461 0.345385i $$-0.887748\pi$$
0.938461 0.345385i $$-0.112252\pi$$
$$198$$ 0 0
$$199$$ −2960.00 −1.05442 −0.527208 0.849736i $$-0.676761\pi$$
−0.527208 + 0.849736i $$0.676761\pi$$
$$200$$ 0 0
$$201$$ 6208.00 2.17850
$$202$$ 0 0
$$203$$ − 544.000i − 0.188085i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 1776.00i − 0.596331i
$$208$$ 0 0
$$209$$ 1600.00 0.529542
$$210$$ 0 0
$$211$$ 40.0000 0.0130508 0.00652539 0.999979i $$-0.497923\pi$$
0.00652539 + 0.999979i $$0.497923\pi$$
$$212$$ 0 0
$$213$$ 3200.00i 1.02939i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 5120.00i − 1.60170i
$$218$$ 0 0
$$219$$ 5040.00 1.55512
$$220$$ 0 0
$$221$$ 1500.00 0.456565
$$222$$ 0 0
$$223$$ 4288.00i 1.28765i 0.765173 + 0.643824i $$0.222653\pi$$
−0.765173 + 0.643824i $$0.777347\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 6456.00i 1.88766i 0.330425 + 0.943832i $$0.392808\pi$$
−0.330425 + 0.943832i $$0.607192\pi$$
$$228$$ 0 0
$$229$$ 1066.00 0.307613 0.153806 0.988101i $$-0.450847\pi$$
0.153806 + 0.988101i $$0.450847\pi$$
$$230$$ 0 0
$$231$$ −5120.00 −1.45832
$$232$$ 0 0
$$233$$ − 5910.00i − 1.66170i −0.556494 0.830852i $$-0.687854\pi$$
0.556494 0.830852i $$-0.312146\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8960.00i 2.45576i
$$238$$ 0 0
$$239$$ 3360.00 0.909374 0.454687 0.890651i $$-0.349751\pi$$
0.454687 + 0.890651i $$0.349751\pi$$
$$240$$ 0 0
$$241$$ 3970.00 1.06112 0.530561 0.847647i $$-0.321981\pi$$
0.530561 + 0.847647i $$0.321981\pi$$
$$242$$ 0 0
$$243$$ − 5032.00i − 1.32841i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2000.00i 0.515210i
$$248$$ 0 0
$$249$$ −4416.00 −1.12391
$$250$$ 0 0
$$251$$ 6840.00 1.72007 0.860034 0.510237i $$-0.170442\pi$$
0.860034 + 0.510237i $$0.170442\pi$$
$$252$$ 0 0
$$253$$ − 1920.00i − 0.477112i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 4610.00i − 1.11893i −0.828855 0.559463i $$-0.811007\pi$$
0.828855 0.559463i $$-0.188993\pi$$
$$258$$ 0 0
$$259$$ −4960.00 −1.18996
$$260$$ 0 0
$$261$$ −1258.00 −0.298346
$$262$$ 0 0
$$263$$ − 4848.00i − 1.13666i −0.822802 0.568328i $$-0.807591\pi$$
0.822802 0.568328i $$-0.192409\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 2608.00i 0.597779i
$$268$$ 0 0
$$269$$ −5550.00 −1.25795 −0.628977 0.777424i $$-0.716526\pi$$
−0.628977 + 0.777424i $$0.716526\pi$$
$$270$$ 0 0
$$271$$ −480.000 −0.107594 −0.0537969 0.998552i $$-0.517132\pi$$
−0.0537969 + 0.998552i $$0.517132\pi$$
$$272$$ 0 0
$$273$$ − 6400.00i − 1.41885i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 1030.00i − 0.223418i −0.993741 0.111709i $$-0.964368\pi$$
0.993741 0.111709i $$-0.0356324\pi$$
$$278$$ 0 0
$$279$$ −11840.0 −2.54065
$$280$$ 0 0
$$281$$ −3270.00 −0.694206 −0.347103 0.937827i $$-0.612835\pi$$
−0.347103 + 0.937827i $$0.612835\pi$$
$$282$$ 0 0
$$283$$ 2168.00i 0.455386i 0.973733 + 0.227693i $$0.0731183\pi$$
−0.973733 + 0.227693i $$0.926882\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 6560.00i − 1.34921i
$$288$$ 0 0
$$289$$ 4013.00 0.816813
$$290$$ 0 0
$$291$$ −880.000 −0.177273
$$292$$ 0 0
$$293$$ 2070.00i 0.412733i 0.978475 + 0.206366i $$0.0661639\pi$$
−0.978475 + 0.206366i $$0.933836\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3200.00i 0.625195i
$$298$$ 0 0
$$299$$ 2400.00 0.464199
$$300$$ 0 0
$$301$$ 2432.00 0.465708
$$302$$ 0 0
$$303$$ − 8784.00i − 1.66544i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 1896.00i − 0.352477i −0.984347 0.176238i $$-0.943607\pi$$
0.984347 0.176238i $$-0.0563930\pi$$
$$308$$ 0 0
$$309$$ 384.000 0.0706958
$$310$$ 0 0
$$311$$ −1680.00 −0.306315 −0.153158 0.988202i $$-0.548944\pi$$
−0.153158 + 0.988202i $$0.548944\pi$$
$$312$$ 0 0
$$313$$ 970.000i 0.175168i 0.996157 + 0.0875841i $$0.0279146\pi$$
−0.996157 + 0.0875841i $$0.972085\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 7230.00i − 1.28100i −0.767958 0.640500i $$-0.778727\pi$$
0.767958 0.640500i $$-0.221273\pi$$
$$318$$ 0 0
$$319$$ −1360.00 −0.238700
$$320$$ 0 0
$$321$$ 5312.00 0.923635
$$322$$ 0 0
$$323$$ − 1200.00i − 0.206718i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 2960.00i 0.500576i
$$328$$ 0 0
$$329$$ 6656.00 1.11537
$$330$$ 0 0
$$331$$ −5800.00 −0.963132 −0.481566 0.876410i $$-0.659932\pi$$
−0.481566 + 0.876410i $$0.659932\pi$$
$$332$$ 0 0
$$333$$ 11470.0i 1.88754i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1870.00i 0.302271i 0.988513 + 0.151136i $$0.0482930\pi$$
−0.988513 + 0.151136i $$0.951707\pi$$
$$338$$ 0 0
$$339$$ −11920.0 −1.90975
$$340$$ 0 0
$$341$$ −12800.0 −2.03272
$$342$$ 0 0
$$343$$ − 6880.00i − 1.08305i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 376.000i − 0.0581693i −0.999577 0.0290846i $$-0.990741\pi$$
0.999577 0.0290846i $$-0.00925923\pi$$
$$348$$ 0 0
$$349$$ 7586.00 1.16352 0.581761 0.813360i $$-0.302364\pi$$
0.581761 + 0.813360i $$0.302364\pi$$
$$350$$ 0 0
$$351$$ −4000.00 −0.608274
$$352$$ 0 0
$$353$$ 2530.00i 0.381468i 0.981642 + 0.190734i $$0.0610868\pi$$
−0.981642 + 0.190734i $$0.938913\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 3840.00i 0.569284i
$$358$$ 0 0
$$359$$ −9680.00 −1.42309 −0.711547 0.702638i $$-0.752005\pi$$
−0.711547 + 0.702638i $$0.752005\pi$$
$$360$$ 0 0
$$361$$ −5259.00 −0.766730
$$362$$ 0 0
$$363$$ 2152.00i 0.311159i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 2784.00i − 0.395977i −0.980204 0.197989i $$-0.936559\pi$$
0.980204 0.197989i $$-0.0634409\pi$$
$$368$$ 0 0
$$369$$ −15170.0 −2.14016
$$370$$ 0 0
$$371$$ −6560.00 −0.918001
$$372$$ 0 0
$$373$$ 7910.00i 1.09803i 0.835813 + 0.549014i $$0.184997\pi$$
−0.835813 + 0.549014i $$0.815003\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 1700.00i − 0.232240i
$$378$$ 0 0
$$379$$ −1720.00 −0.233115 −0.116557 0.993184i $$-0.537186\pi$$
−0.116557 + 0.993184i $$0.537186\pi$$
$$380$$ 0 0
$$381$$ −8192.00 −1.10155
$$382$$ 0 0
$$383$$ 11008.0i 1.46862i 0.678813 + 0.734311i $$0.262495\pi$$
−0.678813 + 0.734311i $$0.737505\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 5624.00i − 0.738718i
$$388$$ 0 0
$$389$$ 12330.0 1.60708 0.803542 0.595248i $$-0.202946\pi$$
0.803542 + 0.595248i $$0.202946\pi$$
$$390$$ 0 0
$$391$$ −1440.00 −0.186250
$$392$$ 0 0
$$393$$ 9280.00i 1.19113i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 4370.00i 0.552453i 0.961093 + 0.276227i $$0.0890841\pi$$
−0.961093 + 0.276227i $$0.910916\pi$$
$$398$$ 0 0
$$399$$ −5120.00 −0.642408
$$400$$ 0 0
$$401$$ 3298.00 0.410709 0.205354 0.978688i $$-0.434165\pi$$
0.205354 + 0.978688i $$0.434165\pi$$
$$402$$ 0 0
$$403$$ − 16000.0i − 1.97771i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 12400.0i 1.51018i
$$408$$ 0 0
$$409$$ 9110.00 1.10137 0.550685 0.834713i $$-0.314366\pi$$
0.550685 + 0.834713i $$0.314366\pi$$
$$410$$ 0 0
$$411$$ 4560.00 0.547271
$$412$$ 0 0
$$413$$ − 3200.00i − 0.381263i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 15680.0i 1.84137i
$$418$$ 0 0
$$419$$ −7880.00 −0.918767 −0.459383 0.888238i $$-0.651930\pi$$
−0.459383 + 0.888238i $$0.651930\pi$$
$$420$$ 0 0
$$421$$ −5290.00 −0.612396 −0.306198 0.951968i $$-0.599057\pi$$
−0.306198 + 0.951968i $$0.599057\pi$$
$$422$$ 0 0
$$423$$ − 15392.0i − 1.76923i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 480.000i − 0.0544001i
$$428$$ 0 0
$$429$$ −16000.0 −1.80067
$$430$$ 0 0
$$431$$ 13920.0 1.55569 0.777845 0.628456i $$-0.216313\pi$$
0.777845 + 0.628456i $$0.216313\pi$$
$$432$$ 0 0
$$433$$ 4930.00i 0.547161i 0.961849 + 0.273580i $$0.0882080\pi$$
−0.961849 + 0.273580i $$0.911792\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 1920.00i − 0.210174i
$$438$$ 0 0
$$439$$ 10640.0 1.15676 0.578382 0.815766i $$-0.303684\pi$$
0.578382 + 0.815766i $$0.303684\pi$$
$$440$$ 0 0
$$441$$ −3219.00 −0.347587
$$442$$ 0 0
$$443$$ − 9288.00i − 0.996131i −0.867139 0.498066i $$-0.834044\pi$$
0.867139 0.498066i $$-0.165956\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 16080.0i 1.70147i
$$448$$ 0 0
$$449$$ −12850.0 −1.35062 −0.675311 0.737533i $$-0.735990\pi$$
−0.675311 + 0.737533i $$0.735990\pi$$
$$450$$ 0 0
$$451$$ −16400.0 −1.71230
$$452$$ 0 0
$$453$$ − 5760.00i − 0.597414i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 10490.0i − 1.07375i −0.843663 0.536873i $$-0.819606\pi$$
0.843663 0.536873i $$-0.180394\pi$$
$$458$$ 0 0
$$459$$ 2400.00 0.244058
$$460$$ 0 0
$$461$$ 11118.0 1.12325 0.561624 0.827393i $$-0.310177\pi$$
0.561624 + 0.827393i $$0.310177\pi$$
$$462$$ 0 0
$$463$$ 5792.00i 0.581376i 0.956818 + 0.290688i $$0.0938842\pi$$
−0.956818 + 0.290688i $$0.906116\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 2216.00i − 0.219581i −0.993955 0.109790i $$-0.964982\pi$$
0.993955 0.109790i $$-0.0350180\pi$$
$$468$$ 0 0
$$469$$ −12416.0 −1.22243
$$470$$ 0 0
$$471$$ 14320.0 1.40091
$$472$$ 0 0
$$473$$ − 6080.00i − 0.591033i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 15170.0i 1.45616i
$$478$$ 0 0
$$479$$ 10560.0 1.00730 0.503652 0.863907i $$-0.331989\pi$$
0.503652 + 0.863907i $$0.331989\pi$$
$$480$$ 0 0
$$481$$ −15500.0 −1.46931
$$482$$ 0 0
$$483$$ 6144.00i 0.578803i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 13264.0i − 1.23419i −0.786890 0.617094i $$-0.788310\pi$$
0.786890 0.617094i $$-0.211690\pi$$
$$488$$ 0 0
$$489$$ 9664.00 0.893704
$$490$$ 0 0
$$491$$ −4840.00 −0.444860 −0.222430 0.974949i $$-0.571399\pi$$
−0.222430 + 0.974949i $$0.571399\pi$$
$$492$$ 0 0
$$493$$ 1020.00i 0.0931815i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 6400.00i − 0.577624i
$$498$$ 0 0
$$499$$ −19560.0 −1.75476 −0.877381 0.479795i $$-0.840711\pi$$
−0.877381 + 0.479795i $$0.840711\pi$$
$$500$$ 0 0
$$501$$ 23168.0 2.06601
$$502$$ 0 0
$$503$$ − 528.000i − 0.0468039i −0.999726 0.0234019i $$-0.992550\pi$$
0.999726 0.0234019i $$-0.00744975\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 2424.00i − 0.212335i
$$508$$ 0 0
$$509$$ 19554.0 1.70278 0.851391 0.524532i $$-0.175760\pi$$
0.851391 + 0.524532i $$0.175760\pi$$
$$510$$ 0 0
$$511$$ −10080.0 −0.872628
$$512$$ 0 0
$$513$$ 3200.00i 0.275406i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 16640.0i − 1.41552i
$$518$$ 0 0
$$519$$ −6000.00 −0.507458
$$520$$ 0 0
$$521$$ 15162.0 1.27497 0.637485 0.770463i $$-0.279975\pi$$
0.637485 + 0.770463i $$0.279975\pi$$
$$522$$ 0 0
$$523$$ 10968.0i 0.917012i 0.888691 + 0.458506i $$0.151615\pi$$
−0.888691 + 0.458506i $$0.848385\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 9600.00i 0.793515i
$$528$$ 0 0
$$529$$ 9863.00 0.810635
$$530$$ 0 0
$$531$$ −7400.00 −0.604770
$$532$$ 0 0
$$533$$ − 20500.0i − 1.66595i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 18240.0i − 1.46576i
$$538$$ 0 0
$$539$$ −3480.00 −0.278097
$$540$$ 0 0
$$541$$ −6722.00 −0.534198 −0.267099 0.963669i $$-0.586065\pi$$
−0.267099 + 0.963669i $$0.586065\pi$$
$$542$$ 0 0
$$543$$ − 3536.00i − 0.279455i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 20424.0i − 1.59647i −0.602348 0.798233i $$-0.705768\pi$$
0.602348 0.798233i $$-0.294232\pi$$
$$548$$ 0 0
$$549$$ −1110.00 −0.0862908
$$550$$ 0 0
$$551$$ −1360.00 −0.105151
$$552$$ 0 0
$$553$$ − 17920.0i − 1.37800i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 6610.00i 0.502827i 0.967880 + 0.251414i $$0.0808954\pi$$
−0.967880 + 0.251414i $$0.919105\pi$$
$$558$$ 0 0
$$559$$ 7600.00 0.575037
$$560$$ 0 0
$$561$$ 9600.00 0.722482
$$562$$ 0 0
$$563$$ − 2712.00i − 0.203015i −0.994835 0.101507i $$-0.967633\pi$$
0.994835 0.101507i $$-0.0323665\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 5744.00i 0.425441i
$$568$$ 0 0
$$569$$ −3530.00 −0.260080 −0.130040 0.991509i $$-0.541511\pi$$
−0.130040 + 0.991509i $$0.541511\pi$$
$$570$$ 0 0
$$571$$ −13640.0 −0.999678 −0.499839 0.866118i $$-0.666608\pi$$
−0.499839 + 0.866118i $$0.666608\pi$$
$$572$$ 0 0
$$573$$ − 15360.0i − 1.11985i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 6270.00i 0.452380i 0.974083 + 0.226190i $$0.0726271\pi$$
−0.974083 + 0.226190i $$0.927373\pi$$
$$578$$ 0 0
$$579$$ 40560.0 2.91125
$$580$$ 0 0
$$581$$ 8832.00 0.630659
$$582$$ 0 0
$$583$$ 16400.0i 1.16504i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 8616.00i 0.605827i 0.953018 + 0.302913i $$0.0979593\pi$$
−0.953018 + 0.302913i $$0.902041\pi$$
$$588$$ 0 0
$$589$$ −12800.0 −0.895441
$$590$$ 0 0
$$591$$ 15280.0 1.06351
$$592$$ 0 0
$$593$$ 5490.00i 0.380181i 0.981767 + 0.190090i $$0.0608781\pi$$
−0.981767 + 0.190090i $$0.939122\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 23680.0i − 1.62338i
$$598$$ 0 0
$$599$$ 15440.0 1.05319 0.526595 0.850116i $$-0.323468\pi$$
0.526595 + 0.850116i $$0.323468\pi$$
$$600$$ 0 0
$$601$$ 8890.00 0.603379 0.301689 0.953406i $$-0.402449\pi$$
0.301689 + 0.953406i $$0.402449\pi$$
$$602$$ 0 0
$$603$$ 28712.0i 1.93904i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 23744.0i − 1.58771i −0.608108 0.793854i $$-0.708071\pi$$
0.608108 0.793854i $$-0.291929\pi$$
$$608$$ 0 0
$$609$$ 4352.00 0.289576
$$610$$ 0 0
$$611$$ 20800.0 1.37721
$$612$$ 0 0
$$613$$ − 15210.0i − 1.00216i −0.865400 0.501082i $$-0.832936\pi$$
0.865400 0.501082i $$-0.167064\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12630.0i 0.824092i 0.911163 + 0.412046i $$0.135186\pi$$
−0.911163 + 0.412046i $$0.864814\pi$$
$$618$$ 0 0
$$619$$ −11160.0 −0.724650 −0.362325 0.932052i $$-0.618017\pi$$
−0.362325 + 0.932052i $$0.618017\pi$$
$$620$$ 0 0
$$621$$ 3840.00 0.248138
$$622$$ 0 0
$$623$$ − 5216.00i − 0.335433i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 12800.0i 0.815284i
$$628$$ 0 0
$$629$$ 9300.00 0.589531
$$630$$ 0 0
$$631$$ 13040.0 0.822685 0.411342 0.911481i $$-0.365060\pi$$
0.411342 + 0.911481i $$0.365060\pi$$
$$632$$ 0 0
$$633$$ 320.000i 0.0200930i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 4350.00i − 0.270570i
$$638$$ 0 0
$$639$$ −14800.0 −0.916242
$$640$$ 0 0
$$641$$ −16910.0 −1.04197 −0.520987 0.853565i $$-0.674436\pi$$
−0.520987 + 0.853565i $$0.674436\pi$$
$$642$$ 0 0
$$643$$ 4488.00i 0.275256i 0.990484 + 0.137628i $$0.0439478\pi$$
−0.990484 + 0.137628i $$0.956052\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 2064.00i − 0.125416i −0.998032 0.0627080i $$-0.980026\pi$$
0.998032 0.0627080i $$-0.0199737\pi$$
$$648$$ 0 0
$$649$$ −8000.00 −0.483864
$$650$$ 0 0
$$651$$ 40960.0 2.46597
$$652$$ 0 0
$$653$$ 4270.00i 0.255893i 0.991781 + 0.127946i $$0.0408386\pi$$
−0.991781 + 0.127946i $$0.959161\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 23310.0i 1.38419i
$$658$$ 0 0
$$659$$ 19800.0 1.17041 0.585204 0.810886i $$-0.301015\pi$$
0.585204 + 0.810886i $$0.301015\pi$$
$$660$$ 0 0
$$661$$ 27110.0 1.59524 0.797622 0.603157i $$-0.206091\pi$$
0.797622 + 0.603157i $$0.206091\pi$$
$$662$$ 0 0
$$663$$ 12000.0i 0.702928i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1632.00i 0.0947396i
$$668$$ 0 0
$$669$$ −34304.0 −1.98247
$$670$$ 0 0
$$671$$ −1200.00 −0.0690395
$$672$$ 0 0
$$673$$ 32210.0i 1.84488i 0.386140 + 0.922440i $$0.373808\pi$$
−0.386140 + 0.922440i $$0.626192\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 27190.0i − 1.54357i −0.635884 0.771785i $$-0.719364\pi$$
0.635884 0.771785i $$-0.280636\pi$$
$$678$$ 0 0
$$679$$ 1760.00 0.0994736
$$680$$ 0 0
$$681$$ −51648.0 −2.90625
$$682$$ 0 0
$$683$$ − 20328.0i − 1.13884i −0.822046 0.569421i $$-0.807167\pi$$
0.822046 0.569421i $$-0.192833\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 8528.00i 0.473600i
$$688$$ 0 0
$$689$$ −20500.0 −1.13351
$$690$$ 0 0
$$691$$ 12520.0 0.689267 0.344633 0.938737i $$-0.388003\pi$$
0.344633 + 0.938737i $$0.388003\pi$$
$$692$$ 0 0
$$693$$ − 23680.0i − 1.29802i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 12300.0i 0.668430i
$$698$$ 0 0
$$699$$ 47280.0 2.55836
$$700$$ 0 0
$$701$$ 11550.0 0.622307 0.311154 0.950360i $$-0.399285\pi$$
0.311154 + 0.950360i $$0.399285\pi$$
$$702$$ 0 0
$$703$$ 12400.0i 0.665256i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 17568.0i 0.934530i
$$708$$ 0 0
$$709$$ 34154.0 1.80914 0.904570 0.426325i $$-0.140192\pi$$
0.904570 + 0.426325i $$0.140192\pi$$
$$710$$ 0 0
$$711$$ −41440.0 −2.18582
$$712$$ 0 0
$$713$$ 15360.0i 0.806783i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 26880.0i 1.40007i
$$718$$ 0 0
$$719$$ 22880.0 1.18676 0.593380 0.804923i $$-0.297793\pi$$
0.593380 + 0.804923i $$0.297793\pi$$
$$720$$ 0 0
$$721$$ −768.000 −0.0396696
$$722$$ 0 0
$$723$$ 31760.0i 1.63370i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 10416.0i − 0.531373i −0.964060 0.265686i $$-0.914401\pi$$
0.964060 0.265686i $$-0.0855986\pi$$
$$728$$ 0 0
$$729$$ 30563.0 1.55276
$$730$$ 0 0
$$731$$ −4560.00 −0.230722
$$732$$ 0 0
$$733$$ 14750.0i 0.743252i 0.928383 + 0.371626i $$0.121200\pi$$
−0.928383 + 0.371626i $$0.878800\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 31040.0i 1.55139i
$$738$$ 0 0
$$739$$ 2360.00 0.117475 0.0587375 0.998273i $$-0.481293\pi$$
0.0587375 + 0.998273i $$0.481293\pi$$
$$740$$ 0 0
$$741$$ −16000.0 −0.793218
$$742$$ 0 0
$$743$$ 32208.0i 1.59031i 0.606409 + 0.795153i $$0.292609\pi$$
−0.606409 + 0.795153i $$0.707391\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 20424.0i − 1.00037i
$$748$$ 0 0
$$749$$ −10624.0 −0.518281
$$750$$ 0 0
$$751$$ −36640.0 −1.78031 −0.890155 0.455658i $$-0.849404\pi$$
−0.890155 + 0.455658i $$0.849404\pi$$
$$752$$ 0 0
$$753$$ 54720.0i 2.64822i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 12090.0i 0.580474i 0.956955 + 0.290237i $$0.0937341\pi$$
−0.956955 + 0.290237i $$0.906266\pi$$
$$758$$ 0 0
$$759$$ 15360.0 0.734562
$$760$$ 0 0
$$761$$ −3318.00 −0.158052 −0.0790259 0.996873i $$-0.525181\pi$$
−0.0790259 + 0.996873i $$0.525181\pi$$
$$762$$ 0 0
$$763$$ − 5920.00i − 0.280889i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 10000.0i − 0.470768i
$$768$$ 0 0
$$769$$ −11506.0 −0.539554 −0.269777 0.962923i $$-0.586950\pi$$
−0.269777 + 0.962923i $$0.586950\pi$$
$$770$$ 0 0
$$771$$ 36880.0 1.72270
$$772$$ 0 0
$$773$$ 22230.0i 1.03436i 0.855878 + 0.517178i $$0.173018\pi$$
−0.855878 + 0.517178i $$0.826982\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 39680.0i − 1.83206i
$$778$$ 0 0
$$779$$ −16400.0 −0.754289
$$780$$ 0 0
$$781$$ −16000.0 −0.733067
$$782$$ 0 0
$$783$$ − 2720.00i − 0.124144i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 21336.0i 0.966387i 0.875514 + 0.483193i $$0.160523\pi$$
−0.875514 + 0.483193i $$0.839477\pi$$
$$788$$ 0 0
$$789$$ 38784.0 1.75000
$$790$$ 0 0
$$791$$ 23840.0 1.07162
$$792$$ 0 0
$$793$$ − 1500.00i − 0.0671709i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 7170.00i 0.318663i 0.987225 + 0.159332i $$0.0509339\pi$$
−0.987225 + 0.159332i $$0.949066\pi$$
$$798$$ 0 0
$$799$$ −12480.0 −0.552579
$$800$$ 0 0
$$801$$ −12062.0 −0.532072
$$802$$ 0 0
$$803$$ 25200.0i 1.10746i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 44400.0i − 1.93675i
$$808$$ 0 0
$$809$$ 23654.0 1.02797 0.513987 0.857798i $$-0.328168\pi$$
0.513987 + 0.857798i $$0.328168\pi$$
$$810$$ 0 0
$$811$$ −30440.0 −1.31799 −0.658997 0.752146i $$-0.729019\pi$$
−0.658997 + 0.752146i $$0.729019\pi$$
$$812$$ 0 0
$$813$$ − 3840.00i − 0.165652i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 6080.00i − 0.260358i
$$818$$ 0 0
$$819$$ 29600.0 1.26289
$$820$$ 0 0
$$821$$ −19930.0 −0.847213 −0.423606 0.905846i $$-0.639236\pi$$
−0.423606 + 0.905846i $$0.639236\pi$$
$$822$$ 0 0
$$823$$ − 9872.00i − 0.418124i −0.977902 0.209062i $$-0.932959\pi$$
0.977902 0.209062i $$-0.0670411\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 5704.00i 0.239840i 0.992784 + 0.119920i $$0.0382638\pi$$
−0.992784 + 0.119920i $$0.961736\pi$$
$$828$$ 0 0
$$829$$ −27230.0 −1.14082 −0.570408 0.821361i $$-0.693215\pi$$
−0.570408 + 0.821361i $$0.693215\pi$$
$$830$$ 0 0
$$831$$ 8240.00 0.343974
$$832$$ 0 0
$$833$$ 2610.00i 0.108561i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 25600.0i − 1.05719i
$$838$$ 0 0
$$839$$ 18800.0 0.773597 0.386799 0.922164i $$-0.373581\pi$$
0.386799 + 0.922164i $$0.373581\pi$$
$$840$$ 0 0
$$841$$ −23233.0 −0.952602
$$842$$ 0 0
$$843$$ − 26160.0i − 1.06880i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 4304.00i − 0.174601i
$$848$$ 0 0
$$849$$ −17344.0 −0.701113
$$850$$ 0 0
$$851$$ 14880.0 0.599389
$$852$$ 0 0
$$853$$ − 12090.0i − 0.485292i −0.970115 0.242646i $$-0.921985\pi$$
0.970115 0.242646i $$-0.0780153\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 470.000i 0.0187338i 0.999956 + 0.00936692i $$0.00298163\pi$$
−0.999956 + 0.00936692i $$0.997018\pi$$
$$858$$ 0 0
$$859$$ −24440.0 −0.970759 −0.485380 0.874304i $$-0.661319\pi$$
−0.485380 + 0.874304i $$0.661319\pi$$
$$860$$ 0 0
$$861$$ 52480.0 2.07725
$$862$$ 0 0
$$863$$ − 22592.0i − 0.891125i −0.895251 0.445562i $$-0.853004\pi$$
0.895251 0.445562i $$-0.146996\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 32104.0i 1.25757i
$$868$$ 0 0
$$869$$ −44800.0 −1.74883
$$870$$ 0 0
$$871$$ −38800.0 −1.50940
$$872$$ 0 0
$$873$$ − 4070.00i − 0.157788i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 17330.0i 0.667266i 0.942703 + 0.333633i $$0.108275\pi$$
−0.942703 + 0.333633i $$0.891725\pi$$
$$878$$ 0 0
$$879$$ −16560.0 −0.635444
$$880$$ 0 0
$$881$$ −31470.0 −1.20346 −0.601732 0.798698i $$-0.705522\pi$$
−0.601732 + 0.798698i $$0.705522\pi$$
$$882$$ 0 0
$$883$$ − 3352.00i − 0.127751i −0.997958 0.0638753i $$-0.979654\pi$$
0.997958 0.0638753i $$-0.0203460\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 48144.0i 1.82245i 0.411904 + 0.911227i $$0.364864\pi$$
−0.411904 + 0.911227i $$0.635136\pi$$
$$888$$ 0 0
$$889$$ 16384.0 0.618112
$$890$$ 0 0
$$891$$ 14360.0 0.539931
$$892$$ 0 0
$$893$$ − 16640.0i − 0.623557i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 19200.0i 0.714682i
$$898$$ 0 0
$$899$$ 10880.0 0.403636
$$900$$ 0 0
$$901$$ 12300.0 0.454797
$$902$$ 0 0
$$903$$ 19456.0i 0.717005i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 16216.0i − 0.593653i −0.954931 0.296827i $$-0.904072\pi$$
0.954931 0.296827i $$-0.0959283\pi$$
$$908$$ 0 0
$$909$$ 40626.0 1.48238
$$910$$ 0 0
$$911$$ 49440.0 1.79805 0.899023 0.437901i $$-0.144278\pi$$
0.899023 + 0.437901i $$0.144278\pi$$
$$912$$ 0 0
$$913$$ − 22080.0i − 0.800374i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 18560.0i − 0.668381i
$$918$$ 0 0
$$919$$ 16080.0 0.577182 0.288591 0.957452i $$-0.406813\pi$$
0.288591 + 0.957452i $$0.406813\pi$$
$$920$$ 0 0
$$921$$ 15168.0 0.542674
$$922$$ 0 0
$$923$$ − 20000.0i − 0.713226i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 1776.00i 0.0629250i
$$928$$ 0 0
$$929$$ 11310.0 0.399428 0.199714 0.979854i $$-0.435999\pi$$
0.199714 + 0.979854i $$0.435999\pi$$
$$930$$ 0 0
$$931$$ −3480.00 −0.122505
$$932$$ 0 0
$$933$$ − 13440.0i − 0.471603i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 25130.0i − 0.876159i −0.898936 0.438080i $$-0.855659\pi$$
0.898936 0.438080i $$-0.144341\pi$$
$$938$$ 0 0
$$939$$ −7760.00 −0.269689
$$940$$ 0 0
$$941$$ −22322.0 −0.773301 −0.386651 0.922226i $$-0.626368\pi$$
−0.386651 + 0.922226i $$0.626368\pi$$
$$942$$ 0 0
$$943$$ 19680.0i 0.679607i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 36456.0i − 1.25096i −0.780239 0.625481i $$-0.784903\pi$$
0.780239 0.625481i $$-0.215097\pi$$
$$948$$ 0 0
$$949$$ −31500.0 −1.07749
$$950$$ 0 0
$$951$$ 57840.0 1.97223
$$952$$ 0 0
$$953$$ 40650.0i 1.38172i 0.722987 + 0.690862i $$0.242769\pi$$
−0.722987 + 0.690862i $$0.757231\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 10880.0i − 0.367503i
$$958$$ 0 0
$$959$$ −9120.00 −0.307091
$$960$$ 0 0
$$961$$ 72609.0 2.43728
$$962$$ 0 0
$$963$$ 24568.0i 0.822111i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 34704.0i − 1.15409i −0.816712 0.577045i $$-0.804206\pi$$
0.816712 0.577045i $$-0.195794\pi$$
$$968$$ 0 0
$$969$$ 9600.00 0.318263
$$970$$ 0 0
$$971$$ −30760.0 −1.01662 −0.508309 0.861175i $$-0.669729\pi$$
−0.508309 + 0.861175i $$0.669729\pi$$
$$972$$ 0 0
$$973$$ − 31360.0i − 1.03325i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 38110.0i 1.24795i 0.781444 + 0.623975i $$0.214483\pi$$
−0.781444 + 0.623975i $$0.785517\pi$$
$$978$$ 0 0
$$979$$ −13040.0 −0.425700
$$980$$ 0 0
$$981$$ −13690.0 −0.445554
$$982$$ 0 0
$$983$$ 19632.0i 0.636992i 0.947924 + 0.318496i $$0.103178\pi$$
−0.947924 + 0.318496i $$0.896822\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 53248.0i 1.71723i
$$988$$ 0 0
$$989$$ −7296.00 −0.234580
$$990$$ 0 0
$$991$$ −47680.0 −1.52836 −0.764180 0.645003i $$-0.776856\pi$$
−0.764180 + 0.645003i $$0.776856\pi$$
$$992$$ 0 0
$$993$$ − 46400.0i − 1.48284i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 39690.0i 1.26078i 0.776280 + 0.630389i $$0.217104\pi$$
−0.776280 + 0.630389i $$0.782896\pi$$
$$998$$ 0 0
$$999$$ −24800.0 −0.785423
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 800.4.c.a.449.2 2
4.3 odd 2 800.4.c.b.449.1 2
5.2 odd 4 32.4.a.c.1.1 yes 1
5.3 odd 4 800.4.a.a.1.1 1
5.4 even 2 inner 800.4.c.a.449.1 2
15.2 even 4 288.4.a.i.1.1 1
20.3 even 4 800.4.a.k.1.1 1
20.7 even 4 32.4.a.a.1.1 1
20.19 odd 2 800.4.c.b.449.2 2
35.27 even 4 1568.4.a.c.1.1 1
40.3 even 4 1600.4.a.e.1.1 1
40.13 odd 4 1600.4.a.bw.1.1 1
40.27 even 4 64.4.a.e.1.1 1
40.37 odd 4 64.4.a.a.1.1 1
60.47 odd 4 288.4.a.h.1.1 1
80.27 even 4 256.4.b.e.129.2 2
80.37 odd 4 256.4.b.c.129.1 2
80.67 even 4 256.4.b.e.129.1 2
80.77 odd 4 256.4.b.c.129.2 2
120.77 even 4 576.4.a.h.1.1 1
120.107 odd 4 576.4.a.g.1.1 1
140.27 odd 4 1568.4.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.a.a.1.1 1 20.7 even 4
32.4.a.c.1.1 yes 1 5.2 odd 4
64.4.a.a.1.1 1 40.37 odd 4
64.4.a.e.1.1 1 40.27 even 4
256.4.b.c.129.1 2 80.37 odd 4
256.4.b.c.129.2 2 80.77 odd 4
256.4.b.e.129.1 2 80.67 even 4
256.4.b.e.129.2 2 80.27 even 4
288.4.a.h.1.1 1 60.47 odd 4
288.4.a.i.1.1 1 15.2 even 4
576.4.a.g.1.1 1 120.107 odd 4
576.4.a.h.1.1 1 120.77 even 4
800.4.a.a.1.1 1 5.3 odd 4
800.4.a.k.1.1 1 20.3 even 4
800.4.c.a.449.1 2 5.4 even 2 inner
800.4.c.a.449.2 2 1.1 even 1 trivial
800.4.c.b.449.1 2 4.3 odd 2
800.4.c.b.449.2 2 20.19 odd 2
1568.4.a.c.1.1 1 35.27 even 4
1568.4.a.o.1.1 1 140.27 odd 4
1600.4.a.e.1.1 1 40.3 even 4
1600.4.a.bw.1.1 1 40.13 odd 4