# Properties

 Label 256.4.b.e.129.1 Level $256$ Weight $4$ Character 256.129 Analytic conductor $15.104$ 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] = [256,4,Mod(129,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("256.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 256.b (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: $$15.1044889615$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 129.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 256.129 Dual form 256.4.b.e.129.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-8.00000i q^{3} +10.0000i q^{5} +16.0000 q^{7} -37.0000 q^{9} +O(q^{10})$$ $$q-8.00000i q^{3} +10.0000i q^{5} +16.0000 q^{7} -37.0000 q^{9} -40.0000i q^{11} -50.0000i q^{13} +80.0000 q^{15} -30.0000 q^{17} -40.0000i q^{19} -128.000i q^{21} +48.0000 q^{23} +25.0000 q^{25} +80.0000i q^{27} -34.0000i q^{29} -320.000 q^{31} -320.000 q^{33} +160.000i q^{35} -310.000i q^{37} -400.000 q^{39} -410.000 q^{41} +152.000i q^{43} -370.000i q^{45} +416.000 q^{47} -87.0000 q^{49} +240.000i q^{51} +410.000i q^{53} +400.000 q^{55} -320.000 q^{57} -200.000i q^{59} +30.0000i q^{61} -592.000 q^{63} +500.000 q^{65} -776.000i q^{67} -384.000i q^{69} +400.000 q^{71} +630.000 q^{73} -200.000i q^{75} -640.000i q^{77} +1120.00 q^{79} -359.000 q^{81} -552.000i q^{83} -300.000i q^{85} -272.000 q^{87} +326.000 q^{89} -800.000i q^{91} +2560.00i q^{93} +400.000 q^{95} -110.000 q^{97} +1480.00i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 32 q^{7} - 74 q^{9}+O(q^{10})$$ 2 * q + 32 * q^7 - 74 * q^9 $$2 q + 32 q^{7} - 74 q^{9} + 160 q^{15} - 60 q^{17} + 96 q^{23} + 50 q^{25} - 640 q^{31} - 640 q^{33} - 800 q^{39} - 820 q^{41} + 832 q^{47} - 174 q^{49} + 800 q^{55} - 640 q^{57} - 1184 q^{63} + 1000 q^{65} + 800 q^{71} + 1260 q^{73} + 2240 q^{79} - 718 q^{81} - 544 q^{87} + 652 q^{89} + 800 q^{95} - 220 q^{97}+O(q^{100})$$ 2 * q + 32 * q^7 - 74 * q^9 + 160 * q^15 - 60 * q^17 + 96 * q^23 + 50 * q^25 - 640 * q^31 - 640 * q^33 - 800 * q^39 - 820 * q^41 + 832 * q^47 - 174 * q^49 + 800 * q^55 - 640 * q^57 - 1184 * q^63 + 1000 * q^65 + 800 * q^71 + 1260 * q^73 + 2240 * q^79 - 718 * q^81 - 544 * q^87 + 652 * q^89 + 800 * q^95 - 220 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$255$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$-0.720356\pi$$
0.638285 0.769800i $$-0.279644\pi$$
$$4$$ 0 0
$$5$$ 10.0000i 0.894427i 0.894427 + 0.447214i $$0.147584\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ 16.0000 0.863919 0.431959 0.901893i $$-0.357822\pi$$
0.431959 + 0.901893i $$0.357822\pi$$
$$8$$ 0 0
$$9$$ −37.0000 −1.37037
$$10$$ 0 0
$$11$$ − 40.0000i − 1.09640i −0.836346 0.548202i $$-0.815312\pi$$
0.836346 0.548202i $$-0.184688\pi$$
$$12$$ 0 0
$$13$$ − 50.0000i − 1.06673i −0.845885 0.533366i $$-0.820927\pi$$
0.845885 0.533366i $$-0.179073\pi$$
$$14$$ 0 0
$$15$$ 80.0000 1.37706
$$16$$ 0 0
$$17$$ −30.0000 −0.428004 −0.214002 0.976833i $$-0.568650\pi$$
−0.214002 + 0.976833i $$0.568650\pi$$
$$18$$ 0 0
$$19$$ − 40.0000i − 0.482980i −0.970403 0.241490i $$-0.922364\pi$$
0.970403 0.241490i $$-0.0776362\pi$$
$$20$$ 0 0
$$21$$ − 128.000i − 1.33009i
$$22$$ 0 0
$$23$$ 48.0000 0.435161 0.217580 0.976042i $$-0.430184\pi$$
0.217580 + 0.976042i $$0.430184\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 80.0000i 0.570222i
$$28$$ 0 0
$$29$$ − 34.0000i − 0.217712i −0.994058 0.108856i $$-0.965281\pi$$
0.994058 0.108856i $$-0.0347187\pi$$
$$30$$ 0 0
$$31$$ −320.000 −1.85399 −0.926995 0.375073i $$-0.877617\pi$$
−0.926995 + 0.375073i $$0.877617\pi$$
$$32$$ 0 0
$$33$$ −320.000 −1.68803
$$34$$ 0 0
$$35$$ 160.000i 0.772712i
$$36$$ 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 $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 152.000i 0.539065i 0.962991 + 0.269532i $$0.0868691\pi$$
−0.962991 + 0.269532i $$0.913131\pi$$
$$44$$ 0 0
$$45$$ − 370.000i − 1.22570i
$$46$$ 0 0
$$47$$ 416.000 1.29106 0.645530 0.763735i $$-0.276636\pi$$
0.645530 + 0.763735i $$0.276636\pi$$
$$48$$ 0 0
$$49$$ −87.0000 −0.253644
$$50$$ 0 0
$$51$$ 240.000i 0.658955i
$$52$$ 0 0
$$53$$ 410.000i 1.06260i 0.847184 + 0.531300i $$0.178296\pi$$
−0.847184 + 0.531300i $$0.821704\pi$$
$$54$$ 0 0
$$55$$ 400.000 0.980654
$$56$$ 0 0
$$57$$ −320.000 −0.743597
$$58$$ 0 0
$$59$$ − 200.000i − 0.441318i −0.975351 0.220659i $$-0.929179\pi$$
0.975351 0.220659i $$-0.0708208\pi$$
$$60$$ 0 0
$$61$$ 30.0000i 0.0629690i 0.999504 + 0.0314845i $$0.0100235\pi$$
−0.999504 + 0.0314845i $$0.989977\pi$$
$$62$$ 0 0
$$63$$ −592.000 −1.18389
$$64$$ 0 0
$$65$$ 500.000 0.954113
$$66$$ 0 0
$$67$$ − 776.000i − 1.41498i −0.706725 0.707489i $$-0.749828\pi$$
0.706725 0.707489i $$-0.250172\pi$$
$$68$$ 0 0
$$69$$ − 384.000i − 0.669973i
$$70$$ 0 0
$$71$$ 400.000 0.668609 0.334305 0.942465i $$-0.391499\pi$$
0.334305 + 0.942465i $$0.391499\pi$$
$$72$$ 0 0
$$73$$ 630.000 1.01008 0.505041 0.863096i $$-0.331478\pi$$
0.505041 + 0.863096i $$0.331478\pi$$
$$74$$ 0 0
$$75$$ − 200.000i − 0.307920i
$$76$$ 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 $$-0.881069\pi$$
0.931008 0.364999i $$-0.118931\pi$$
$$84$$ 0 0
$$85$$ − 300.000i − 0.382818i
$$86$$ 0 0
$$87$$ −272.000 −0.335189
$$88$$ 0 0
$$89$$ 326.000 0.388269 0.194134 0.980975i $$-0.437810\pi$$
0.194134 + 0.980975i $$0.437810\pi$$
$$90$$ 0 0
$$91$$ − 800.000i − 0.921569i
$$92$$ 0 0
$$93$$ 2560.00i 2.85440i
$$94$$ 0 0
$$95$$ 400.000 0.431991
$$96$$ 0 0
$$97$$ −110.000 −0.115142 −0.0575712 0.998341i $$-0.518336\pi$$
−0.0575712 + 0.998341i $$0.518336\pi$$
$$98$$ 0 0
$$99$$ 1480.00i 1.50248i
$$100$$ 0 0
$$101$$ 1098.00i 1.08173i 0.841108 + 0.540867i $$0.181904\pi$$
−0.841108 + 0.540867i $$0.818096\pi$$
$$102$$ 0 0
$$103$$ −48.0000 −0.0459183 −0.0229591 0.999736i $$-0.507309\pi$$
−0.0229591 + 0.999736i $$0.507309\pi$$
$$104$$ 0 0
$$105$$ 1280.00 1.18967
$$106$$ 0 0
$$107$$ 664.000i 0.599919i 0.953952 + 0.299959i $$0.0969731\pi$$
−0.953952 + 0.299959i $$0.903027\pi$$
$$108$$ 0 0
$$109$$ − 370.000i − 0.325134i −0.986698 0.162567i $$-0.948023\pi$$
0.986698 0.162567i $$-0.0519773\pi$$
$$110$$ 0 0
$$111$$ −2480.00 −2.12064
$$112$$ 0 0
$$113$$ 1490.00 1.24042 0.620210 0.784436i $$-0.287047\pi$$
0.620210 + 0.784436i $$0.287047\pi$$
$$114$$ 0 0
$$115$$ 480.000i 0.389219i
$$116$$ 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
$$125$$ 1500.00i 1.07331i
$$126$$ 0 0
$$127$$ 1024.00 0.715475 0.357737 0.933822i $$-0.383548\pi$$
0.357737 + 0.933822i $$0.383548\pi$$
$$128$$ 0 0
$$129$$ 1216.00 0.829944
$$130$$ 0 0
$$131$$ − 1160.00i − 0.773662i −0.922151 0.386831i $$-0.873570\pi$$
0.922151 0.386831i $$-0.126430\pi$$
$$132$$ 0 0
$$133$$ − 640.000i − 0.417256i
$$134$$ 0 0
$$135$$ −800.000 −0.510022
$$136$$ 0 0
$$137$$ −570.000 −0.355463 −0.177731 0.984079i $$-0.556876\pi$$
−0.177731 + 0.984079i $$0.556876\pi$$
$$138$$ 0 0
$$139$$ − 1960.00i − 1.19601i −0.801493 0.598004i $$-0.795961\pi$$
0.801493 0.598004i $$-0.204039\pi$$
$$140$$ 0 0
$$141$$ − 3328.00i − 1.98772i
$$142$$ 0 0
$$143$$ −2000.00 −1.16957
$$144$$ 0 0
$$145$$ 340.000 0.194727
$$146$$ 0 0
$$147$$ 696.000i 0.390511i
$$148$$ 0 0
$$149$$ 2010.00i 1.10514i 0.833467 + 0.552569i $$0.186352\pi$$
−0.833467 + 0.552569i $$0.813648\pi$$
$$150$$ 0 0
$$151$$ −720.000 −0.388032 −0.194016 0.980998i $$-0.562151\pi$$
−0.194016 + 0.980998i $$0.562151\pi$$
$$152$$ 0 0
$$153$$ 1110.00 0.586524
$$154$$ 0 0
$$155$$ − 3200.00i − 1.65826i
$$156$$ 0 0
$$157$$ 1790.00i 0.909921i 0.890512 + 0.454960i $$0.150347\pi$$
−0.890512 + 0.454960i $$0.849653\pi$$
$$158$$ 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 $$0.0937348\pi$$
−0.956954 + 0.290239i $$0.906265\pi$$
$$164$$ 0 0
$$165$$ − 3200.00i − 1.50982i
$$166$$ 0 0
$$167$$ 2896.00 1.34191 0.670956 0.741497i $$-0.265884\pi$$
0.670956 + 0.741497i $$0.265884\pi$$
$$168$$ 0 0
$$169$$ −303.000 −0.137915
$$170$$ 0 0
$$171$$ 1480.00i 0.661862i
$$172$$ 0 0
$$173$$ 750.000i 0.329604i 0.986327 + 0.164802i $$0.0526985\pi$$
−0.986327 + 0.164802i $$0.947302\pi$$
$$174$$ 0 0
$$175$$ 400.000 0.172784
$$176$$ 0 0
$$177$$ −1600.00 −0.679454
$$178$$ 0 0
$$179$$ − 2280.00i − 0.952040i −0.879434 0.476020i $$-0.842079\pi$$
0.879434 0.476020i $$-0.157921\pi$$
$$180$$ 0 0
$$181$$ 442.000i 0.181512i 0.995873 + 0.0907558i $$0.0289283\pi$$
−0.995873 + 0.0907558i $$0.971072\pi$$
$$182$$ 0 0
$$183$$ 240.000 0.0969471
$$184$$ 0 0
$$185$$ 3100.00 1.23198
$$186$$ 0 0
$$187$$ 1200.00i 0.469266i
$$188$$ 0 0
$$189$$ 1280.00i 0.492626i
$$190$$ 0 0
$$191$$ 1920.00 0.727363 0.363681 0.931523i $$-0.381520\pi$$
0.363681 + 0.931523i $$0.381520\pi$$
$$192$$ 0 0
$$193$$ −5070.00 −1.89091 −0.945457 0.325746i $$-0.894385\pi$$
−0.945457 + 0.325746i $$0.894385\pi$$
$$194$$ 0 0
$$195$$ − 4000.00i − 1.46895i
$$196$$ 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 $$-0.323239\pi$$
0.527208 + 0.849736i $$0.323239\pi$$
$$200$$ 0 0
$$201$$ −6208.00 −2.17850
$$202$$ 0 0
$$203$$ − 544.000i − 0.188085i
$$204$$ 0 0
$$205$$ − 4100.00i − 1.39686i
$$206$$ 0 0
$$207$$ −1776.00 −0.596331
$$208$$ 0 0
$$209$$ −1600.00 −0.529542
$$210$$ 0 0
$$211$$ − 40.0000i − 0.0130508i −0.999979 0.00652539i $$-0.997923\pi$$
0.999979 0.00652539i $$-0.00207711\pi$$
$$212$$ 0 0
$$213$$ − 3200.00i − 1.02939i
$$214$$ 0 0
$$215$$ −1520.00 −0.482154
$$216$$ 0 0
$$217$$ −5120.00 −1.60170
$$218$$ 0 0
$$219$$ − 5040.00i − 1.55512i
$$220$$ 0 0
$$221$$ 1500.00i 0.456565i
$$222$$ 0 0
$$223$$ −4288.00 −1.28765 −0.643824 0.765173i $$-0.722653\pi$$
−0.643824 + 0.765173i $$0.722653\pi$$
$$224$$ 0 0
$$225$$ −925.000 −0.274074
$$226$$ 0 0
$$227$$ 6456.00i 1.88766i 0.330425 + 0.943832i $$0.392808\pi$$
−0.330425 + 0.943832i $$0.607192\pi$$
$$228$$ 0 0
$$229$$ 1066.00i 0.307613i 0.988101 + 0.153806i $$0.0491532\pi$$
−0.988101 + 0.153806i $$0.950847\pi$$
$$230$$ 0 0
$$231$$ −5120.00 −1.45832
$$232$$ 0 0
$$233$$ 5910.00 1.66170 0.830852 0.556494i $$-0.187854\pi$$
0.830852 + 0.556494i $$0.187854\pi$$
$$234$$ 0 0
$$235$$ 4160.00i 1.15476i
$$236$$ 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
$$245$$ − 870.000i − 0.226866i
$$246$$ 0 0
$$247$$ −2000.00 −0.515210
$$248$$ 0 0
$$249$$ −4416.00 −1.12391
$$250$$ 0 0
$$251$$ 6840.00i 1.72007i 0.510237 + 0.860034i $$0.329558\pi$$
−0.510237 + 0.860034i $$0.670442\pi$$
$$252$$ 0 0
$$253$$ − 1920.00i − 0.477112i
$$254$$ 0 0
$$255$$ −2400.00 −0.589388
$$256$$ 0 0
$$257$$ 4610.00 1.11893 0.559463 0.828855i $$-0.311007\pi$$
0.559463 + 0.828855i $$0.311007\pi$$
$$258$$ 0 0
$$259$$ − 4960.00i − 1.18996i
$$260$$ 0 0
$$261$$ 1258.00i 0.298346i
$$262$$ 0 0
$$263$$ −4848.00 −1.13666 −0.568328 0.822802i $$-0.692409\pi$$
−0.568328 + 0.822802i $$0.692409\pi$$
$$264$$ 0 0
$$265$$ −4100.00 −0.950419
$$266$$ 0 0
$$267$$ − 2608.00i − 0.597779i
$$268$$ 0 0
$$269$$ 5550.00i 1.25795i 0.777424 + 0.628977i $$0.216526\pi$$
−0.777424 + 0.628977i $$0.783474\pi$$
$$270$$ 0 0
$$271$$ 480.000 0.107594 0.0537969 0.998552i $$-0.482868\pi$$
0.0537969 + 0.998552i $$0.482868\pi$$
$$272$$ 0 0
$$273$$ −6400.00 −1.41885
$$274$$ 0 0
$$275$$ − 1000.00i − 0.219281i
$$276$$ 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 $$-0.387165\pi$$
0.347103 + 0.937827i $$0.387165\pi$$
$$282$$ 0 0
$$283$$ 2168.00i 0.455386i 0.973733 + 0.227693i $$0.0731183\pi$$
−0.973733 + 0.227693i $$0.926882\pi$$
$$284$$ 0 0
$$285$$ − 3200.00i − 0.665093i
$$286$$ 0 0
$$287$$ −6560.00 −1.34921
$$288$$ 0 0
$$289$$ −4013.00 −0.816813
$$290$$ 0 0
$$291$$ 880.000i 0.177273i
$$292$$ 0 0
$$293$$ − 2070.00i − 0.412733i −0.978475 0.206366i $$-0.933836\pi$$
0.978475 0.206366i $$-0.0661639\pi$$
$$294$$ 0 0
$$295$$ 2000.00 0.394727
$$296$$ 0 0
$$297$$ 3200.00 0.625195
$$298$$ 0 0
$$299$$ − 2400.00i − 0.464199i
$$300$$ 0 0
$$301$$ 2432.00i 0.465708i
$$302$$ 0 0
$$303$$ 8784.00 1.66544
$$304$$ 0 0
$$305$$ −300.000 −0.0563211
$$306$$ 0 0
$$307$$ − 1896.00i − 0.352477i −0.984347 0.176238i $$-0.943607\pi$$
0.984347 0.176238i $$-0.0563930\pi$$
$$308$$ 0 0
$$309$$ 384.000i 0.0706958i
$$310$$ 0 0
$$311$$ −1680.00 −0.306315 −0.153158 0.988202i $$-0.548944\pi$$
−0.153158 + 0.988202i $$0.548944\pi$$
$$312$$ 0 0
$$313$$ −970.000 −0.175168 −0.0875841 0.996157i $$-0.527915\pi$$
−0.0875841 + 0.996157i $$0.527915\pi$$
$$314$$ 0 0
$$315$$ − 5920.00i − 1.05890i
$$316$$ 0 0
$$317$$ 7230.00i 1.28100i 0.767958 + 0.640500i $$0.221273\pi$$
−0.767958 + 0.640500i $$0.778727\pi$$
$$318$$ 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
$$325$$ − 1250.00i − 0.213346i
$$326$$ 0 0
$$327$$ −2960.00 −0.500576
$$328$$ 0 0
$$329$$ 6656.00 1.11537
$$330$$ 0 0
$$331$$ − 5800.00i − 0.963132i −0.876410 0.481566i $$-0.840068\pi$$
0.876410 0.481566i $$-0.159932\pi$$
$$332$$ 0 0
$$333$$ 11470.0i 1.88754i
$$334$$ 0 0
$$335$$ 7760.00 1.26559
$$336$$ 0 0
$$337$$ −1870.00 −0.302271 −0.151136 0.988513i $$-0.548293\pi$$
−0.151136 + 0.988513i $$0.548293\pi$$
$$338$$ 0 0
$$339$$ − 11920.0i − 1.90975i
$$340$$ 0 0
$$341$$ 12800.0i 2.03272i
$$342$$ 0 0
$$343$$ −6880.00 −1.08305
$$344$$ 0 0
$$345$$ 3840.00 0.599242
$$346$$ 0 0
$$347$$ 376.000i 0.0581693i 0.999577 + 0.0290846i $$0.00925923\pi$$
−0.999577 + 0.0290846i $$0.990741\pi$$
$$348$$ 0 0
$$349$$ − 7586.00i − 1.16352i −0.813360 0.581761i $$-0.802364\pi$$
0.813360 0.581761i $$-0.197636\pi$$
$$350$$ 0 0
$$351$$ 4000.00 0.608274
$$352$$ 0 0
$$353$$ 2530.00 0.381468 0.190734 0.981642i $$-0.438913\pi$$
0.190734 + 0.981642i $$0.438913\pi$$
$$354$$ 0 0
$$355$$ 4000.00i 0.598022i
$$356$$ 0 0
$$357$$ 3840.00i 0.569284i
$$358$$ 0 0
$$359$$ 9680.00 1.42309 0.711547 0.702638i $$-0.247995\pi$$
0.711547 + 0.702638i $$0.247995\pi$$
$$360$$ 0 0
$$361$$ 5259.00 0.766730
$$362$$ 0 0
$$363$$ 2152.00i 0.311159i
$$364$$ 0 0
$$365$$ 6300.00i 0.903444i
$$366$$ 0 0
$$367$$ −2784.00 −0.395977 −0.197989 0.980204i $$-0.563441\pi$$
−0.197989 + 0.980204i $$0.563441\pi$$
$$368$$ 0 0
$$369$$ 15170.0 2.14016
$$370$$ 0 0
$$371$$ 6560.00i 0.918001i
$$372$$ 0 0
$$373$$ − 7910.00i − 1.09803i −0.835813 0.549014i $$-0.815003\pi$$
0.835813 0.549014i $$-0.184997\pi$$
$$374$$ 0 0
$$375$$ 12000.0 1.65247
$$376$$ 0 0
$$377$$ −1700.00 −0.232240
$$378$$ 0 0
$$379$$ 1720.00i 0.233115i 0.993184 + 0.116557i $$0.0371859\pi$$
−0.993184 + 0.116557i $$0.962814\pi$$
$$380$$ 0 0
$$381$$ − 8192.00i − 1.10155i
$$382$$ 0 0
$$383$$ −11008.0 −1.46862 −0.734311 0.678813i $$-0.762495\pi$$
−0.734311 + 0.678813i $$0.762495\pi$$
$$384$$ 0 0
$$385$$ 6400.00 0.847206
$$386$$ 0 0
$$387$$ − 5624.00i − 0.738718i
$$388$$ 0 0
$$389$$ 12330.0i 1.60708i 0.595248 + 0.803542i $$0.297054\pi$$
−0.595248 + 0.803542i $$0.702946\pi$$
$$390$$ 0 0
$$391$$ −1440.00 −0.186250
$$392$$ 0 0
$$393$$ −9280.00 −1.19113
$$394$$ 0 0
$$395$$ 11200.0i 1.42667i
$$396$$ 0 0
$$397$$ − 4370.00i − 0.552453i −0.961093 0.276227i $$-0.910916\pi$$
0.961093 0.276227i $$-0.0890841\pi$$
$$398$$ 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
$$405$$ − 3590.00i − 0.440466i
$$406$$ 0 0
$$407$$ −12400.0 −1.51018
$$408$$ 0 0
$$409$$ 9110.00 1.10137 0.550685 0.834713i $$-0.314366\pi$$
0.550685 + 0.834713i $$0.314366\pi$$
$$410$$ 0 0
$$411$$ 4560.00i 0.547271i
$$412$$ 0 0
$$413$$ − 3200.00i − 0.381263i
$$414$$ 0 0
$$415$$ 5520.00 0.652930
$$416$$ 0 0
$$417$$ −15680.0 −1.84137
$$418$$ 0 0
$$419$$ − 7880.00i − 0.918767i −0.888238 0.459383i $$-0.848070\pi$$
0.888238 0.459383i $$-0.151930\pi$$
$$420$$ 0 0
$$421$$ 5290.00i 0.612396i 0.951968 + 0.306198i $$0.0990570\pi$$
−0.951968 + 0.306198i $$0.900943\pi$$
$$422$$ 0 0
$$423$$ −15392.0 −1.76923
$$424$$ 0 0
$$425$$ −750.000 −0.0856008
$$426$$ 0 0
$$427$$ 480.000i 0.0544001i
$$428$$ 0 0
$$429$$ 16000.0i 1.80067i
$$430$$ 0 0
$$431$$ −13920.0 −1.55569 −0.777845 0.628456i $$-0.783687\pi$$
−0.777845 + 0.628456i $$0.783687\pi$$
$$432$$ 0 0
$$433$$ 4930.00 0.547161 0.273580 0.961849i $$-0.411792\pi$$
0.273580 + 0.961849i $$0.411792\pi$$
$$434$$ 0 0
$$435$$ − 2720.00i − 0.299802i
$$436$$ 0 0
$$437$$ − 1920.00i − 0.210174i
$$438$$ 0 0
$$439$$ −10640.0 −1.15676 −0.578382 0.815766i $$-0.696316\pi$$
−0.578382 + 0.815766i $$0.696316\pi$$
$$440$$ 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
$$445$$ 3260.00i 0.347278i
$$446$$ 0 0
$$447$$ 16080.0 1.70147
$$448$$ 0 0
$$449$$ 12850.0 1.35062 0.675311 0.737533i $$-0.264010\pi$$
0.675311 + 0.737533i $$0.264010\pi$$
$$450$$ 0 0
$$451$$ 16400.0i 1.71230i
$$452$$ 0 0
$$453$$ 5760.00i 0.597414i
$$454$$ 0 0
$$455$$ 8000.00 0.824276
$$456$$ 0 0
$$457$$ −10490.0 −1.07375 −0.536873 0.843663i $$-0.680394\pi$$
−0.536873 + 0.843663i $$0.680394\pi$$
$$458$$ 0 0
$$459$$ − 2400.00i − 0.244058i
$$460$$ 0 0
$$461$$ 11118.0i 1.12325i 0.827393 + 0.561624i $$0.189823\pi$$
−0.827393 + 0.561624i $$0.810177\pi$$
$$462$$ 0 0
$$463$$ −5792.00 −0.581376 −0.290688 0.956818i $$-0.593884\pi$$
−0.290688 + 0.956818i $$0.593884\pi$$
$$464$$ 0 0
$$465$$ −25600.0 −2.55306
$$466$$ 0 0
$$467$$ − 2216.00i − 0.219581i −0.993955 0.109790i $$-0.964982\pi$$
0.993955 0.109790i $$-0.0350180\pi$$
$$468$$ 0 0
$$469$$ − 12416.0i − 1.22243i
$$470$$ 0 0
$$471$$ 14320.0 1.40091
$$472$$ 0 0
$$473$$ 6080.00 0.591033
$$474$$ 0 0
$$475$$ − 1000.00i − 0.0965961i
$$476$$ 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
$$485$$ − 1100.00i − 0.102986i
$$486$$ 0 0
$$487$$ 13264.0 1.23419 0.617094 0.786890i $$-0.288310\pi$$
0.617094 + 0.786890i $$0.288310\pi$$
$$488$$ 0 0
$$489$$ 9664.00 0.893704
$$490$$ 0 0
$$491$$ − 4840.00i − 0.444860i −0.974949 0.222430i $$-0.928601\pi$$
0.974949 0.222430i $$-0.0713988\pi$$
$$492$$ 0 0
$$493$$ 1020.00i 0.0931815i
$$494$$ 0 0
$$495$$ −14800.0 −1.34386
$$496$$ 0 0
$$497$$ 6400.00 0.577624
$$498$$ 0 0
$$499$$ − 19560.0i − 1.75476i −0.479795 0.877381i $$-0.659289\pi$$
0.479795 0.877381i $$-0.340711\pi$$
$$500$$ 0 0
$$501$$ − 23168.0i − 2.06601i
$$502$$ 0 0
$$503$$ −528.000 −0.0468039 −0.0234019 0.999726i $$-0.507450\pi$$
−0.0234019 + 0.999726i $$0.507450\pi$$
$$504$$ 0 0
$$505$$ −10980.0 −0.967532
$$506$$ 0 0
$$507$$ 2424.00i 0.212335i
$$508$$ 0 0
$$509$$ − 19554.0i − 1.70278i −0.524532 0.851391i $$-0.675760\pi$$
0.524532 0.851391i $$-0.324240\pi$$
$$510$$ 0 0
$$511$$ 10080.0 0.872628
$$512$$ 0 0
$$513$$ 3200.00 0.275406
$$514$$ 0 0
$$515$$ − 480.000i − 0.0410705i
$$516$$ 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 $$-0.720025\pi$$
−0.637485 + 0.770463i $$0.720025\pi$$
$$522$$ 0 0
$$523$$ 10968.0i 0.917012i 0.888691 + 0.458506i $$0.151615\pi$$
−0.888691 + 0.458506i $$0.848385\pi$$
$$524$$ 0 0
$$525$$ − 3200.00i − 0.266018i
$$526$$ 0 0
$$527$$ 9600.00 0.793515
$$528$$ 0 0
$$529$$ −9863.00 −0.810635
$$530$$ 0 0
$$531$$ 7400.00i 0.604770i
$$532$$ 0 0
$$533$$ 20500.0i 1.66595i
$$534$$ 0 0
$$535$$ −6640.00 −0.536584
$$536$$ 0 0
$$537$$ −18240.0 −1.46576
$$538$$ 0 0
$$539$$ 3480.00i 0.278097i
$$540$$ 0 0
$$541$$ − 6722.00i − 0.534198i −0.963669 0.267099i $$-0.913935\pi$$
0.963669 0.267099i $$-0.0860651\pi$$
$$542$$ 0 0
$$543$$ 3536.00 0.279455
$$544$$ 0 0
$$545$$ 3700.00 0.290808
$$546$$ 0 0
$$547$$ − 20424.0i − 1.59647i −0.602348 0.798233i $$-0.705768\pi$$
0.602348 0.798233i $$-0.294232\pi$$
$$548$$ 0 0
$$549$$ − 1110.00i − 0.0862908i
$$550$$ 0 0
$$551$$ −1360.00 −0.105151
$$552$$ 0 0
$$553$$ 17920.0 1.37800
$$554$$ 0 0
$$555$$ − 24800.0i − 1.89676i
$$556$$ 0 0
$$557$$ − 6610.00i − 0.502827i −0.967880 0.251414i $$-0.919105\pi$$
0.967880 0.251414i $$-0.0808954\pi$$
$$558$$ 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 $$0.0323665\pi$$
−0.994835 + 0.101507i $$0.967633\pi$$
$$564$$ 0 0
$$565$$ 14900.0i 1.10946i
$$566$$ 0 0
$$567$$ −5744.00 −0.425441
$$568$$ 0 0
$$569$$ −3530.00 −0.260080 −0.130040 0.991509i $$-0.541511\pi$$
−0.130040 + 0.991509i $$0.541511\pi$$
$$570$$ 0 0
$$571$$ − 13640.0i − 0.999678i −0.866118 0.499839i $$-0.833392\pi$$
0.866118 0.499839i $$-0.166608\pi$$
$$572$$ 0 0
$$573$$ − 15360.0i − 1.11985i
$$574$$ 0 0
$$575$$ 1200.00 0.0870321
$$576$$ 0 0
$$577$$ −6270.00 −0.452380 −0.226190 0.974083i $$-0.572627\pi$$
−0.226190 + 0.974083i $$0.572627\pi$$
$$578$$ 0 0
$$579$$ 40560.0i 2.91125i
$$580$$ 0 0
$$581$$ − 8832.00i − 0.630659i
$$582$$ 0 0
$$583$$ 16400.0 1.16504
$$584$$ 0 0
$$585$$ −18500.0 −1.30749
$$586$$ 0 0
$$587$$ − 8616.00i − 0.605827i −0.953018 0.302913i $$-0.902041\pi$$
0.953018 0.302913i $$-0.0979593\pi$$
$$588$$ 0 0
$$589$$ 12800.0i 0.895441i
$$590$$ 0 0
$$591$$ −15280.0 −1.06351
$$592$$ 0 0
$$593$$ 5490.00 0.380181 0.190090 0.981767i $$-0.439122\pi$$
0.190090 + 0.981767i $$0.439122\pi$$
$$594$$ 0 0
$$595$$ − 4800.00i − 0.330724i
$$596$$ 0 0
$$597$$ − 23680.0i − 1.62338i
$$598$$ 0 0
$$599$$ −15440.0 −1.05319 −0.526595 0.850116i $$-0.676532\pi$$
−0.526595 + 0.850116i $$0.676532\pi$$
$$600$$ 0 0
$$601$$ −8890.00 −0.603379 −0.301689 0.953406i $$-0.597551\pi$$
−0.301689 + 0.953406i $$0.597551\pi$$
$$602$$ 0 0
$$603$$ 28712.0i 1.93904i
$$604$$ 0 0
$$605$$ − 2690.00i − 0.180767i
$$606$$ 0 0
$$607$$ −23744.0 −1.58771 −0.793854 0.608108i $$-0.791929\pi$$
−0.793854 + 0.608108i $$0.791929\pi$$
$$608$$ 0 0
$$609$$ −4352.00 −0.289576
$$610$$ 0 0
$$611$$ − 20800.0i − 1.37721i
$$612$$ 0 0
$$613$$ 15210.0i 1.00216i 0.865400 + 0.501082i $$0.167064\pi$$
−0.865400 + 0.501082i $$0.832936\pi$$
$$614$$ 0 0
$$615$$ −32800.0 −2.15061
$$616$$ 0 0
$$617$$ 12630.0 0.824092 0.412046 0.911163i $$-0.364814\pi$$
0.412046 + 0.911163i $$0.364814\pi$$
$$618$$ 0 0
$$619$$ 11160.0i 0.724650i 0.932052 + 0.362325i $$0.118017\pi$$
−0.932052 + 0.362325i $$0.881983\pi$$
$$620$$ 0 0
$$621$$ 3840.00i 0.248138i
$$622$$ 0 0
$$623$$ 5216.00 0.335433
$$624$$ 0 0
$$625$$ −11875.0 −0.760000
$$626$$ 0 0
$$627$$ 12800.0i 0.815284i
$$628$$ 0 0
$$629$$ 9300.00i 0.589531i
$$630$$ 0 0
$$631$$ 13040.0 0.822685 0.411342 0.911481i $$-0.365060\pi$$
0.411342 + 0.911481i $$0.365060\pi$$
$$632$$ 0 0
$$633$$ −320.000 −0.0200930
$$634$$ 0 0
$$635$$ 10240.0i 0.639940i
$$636$$ 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 $$-0.956052\pi$$
0.990484 0.137628i $$-0.0439478\pi$$
$$644$$ 0 0
$$645$$ 12160.0i 0.742325i
$$646$$ 0 0
$$647$$ 2064.00 0.125416 0.0627080 0.998032i $$-0.480026\pi$$
0.0627080 + 0.998032i $$0.480026\pi$$
$$648$$ 0 0
$$649$$ −8000.00 −0.483864
$$650$$ 0 0
$$651$$ 40960.0i 2.46597i
$$652$$ 0 0
$$653$$ 4270.00i 0.255893i 0.991781 + 0.127946i $$0.0408386\pi$$
−0.991781 + 0.127946i $$0.959161\pi$$
$$654$$ 0 0
$$655$$ 11600.0 0.691984
$$656$$ 0 0
$$657$$ −23310.0 −1.38419
$$658$$ 0 0
$$659$$ 19800.0i 1.17041i 0.810886 + 0.585204i $$0.198985\pi$$
−0.810886 + 0.585204i $$0.801015\pi$$
$$660$$ 0 0
$$661$$ − 27110.0i − 1.59524i −0.603157 0.797622i $$-0.706091\pi$$
0.603157 0.797622i $$-0.293909\pi$$
$$662$$ 0 0
$$663$$ 12000.0 0.702928
$$664$$ 0 0
$$665$$ 6400.00 0.373205
$$666$$ 0 0
$$667$$ − 1632.00i − 0.0947396i
$$668$$ 0 0
$$669$$ 34304.0i 1.98247i
$$670$$ 0 0
$$671$$ 1200.00 0.0690395
$$672$$ 0 0
$$673$$ 32210.0 1.84488 0.922440 0.386140i $$-0.126192\pi$$
0.922440 + 0.386140i $$0.126192\pi$$
$$674$$ 0 0
$$675$$ 2000.00i 0.114044i
$$676$$ 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
$$685$$ − 5700.00i − 0.317935i
$$686$$ 0 0
$$687$$ 8528.00 0.473600
$$688$$ 0 0
$$689$$ 20500.0 1.13351
$$690$$ 0 0
$$691$$ − 12520.0i − 0.689267i −0.938737 0.344633i $$-0.888003\pi$$
0.938737 0.344633i $$-0.111997\pi$$
$$692$$ 0 0
$$693$$ 23680.0i 1.29802i
$$694$$ 0 0
$$695$$ 19600.0 1.06974
$$696$$ 0 0
$$697$$ 12300.0 0.668430
$$698$$ 0 0
$$699$$ − 47280.0i − 2.55836i
$$700$$ 0 0
$$701$$ 11550.0i 0.622307i 0.950360 + 0.311154i $$0.100715\pi$$
−0.950360 + 0.311154i $$0.899285\pi$$
$$702$$ 0 0
$$703$$ −12400.0 −0.665256
$$704$$ 0 0
$$705$$ 33280.0 1.77787
$$706$$ 0 0
$$707$$ 17568.0i 0.934530i
$$708$$ 0 0
$$709$$ 34154.0i 1.80914i 0.426325 + 0.904570i $$0.359808\pi$$
−0.426325 + 0.904570i $$0.640192\pi$$
$$710$$ 0 0
$$711$$ −41440.0 −2.18582
$$712$$ 0 0
$$713$$ −15360.0 −0.806783
$$714$$ 0 0
$$715$$ − 20000.0i − 1.04609i
$$716$$ 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
$$725$$ − 850.000i − 0.0435424i
$$726$$ 0 0
$$727$$ 10416.0 0.531373 0.265686 0.964060i $$-0.414401\pi$$
0.265686 + 0.964060i $$0.414401\pi$$
$$728$$ 0 0
$$729$$ 30563.0 1.55276
$$730$$ 0 0
$$731$$ − 4560.00i − 0.230722i
$$732$$ 0 0
$$733$$ 14750.0i 0.743252i 0.928383 + 0.371626i $$0.121200\pi$$
−0.928383 + 0.371626i $$0.878800\pi$$
$$734$$ 0 0
$$735$$ −6960.00 −0.349284
$$736$$ 0 0
$$737$$ −31040.0 −1.55139
$$738$$ 0 0
$$739$$ 2360.00i 0.117475i 0.998273 + 0.0587375i $$0.0187075\pi$$
−0.998273 + 0.0587375i $$0.981293\pi$$
$$740$$ 0 0
$$741$$ 16000.0i 0.793218i
$$742$$ 0 0
$$743$$ 32208.0 1.59031 0.795153 0.606409i $$-0.207391\pi$$
0.795153 + 0.606409i $$0.207391\pi$$
$$744$$ 0 0
$$745$$ −20100.0 −0.988466
$$746$$ 0 0
$$747$$ 20424.0i 1.00037i
$$748$$ 0 0
$$749$$ 10624.0i 0.518281i
$$750$$ 0 0
$$751$$ 36640.0 1.78031 0.890155 0.455658i $$-0.150596\pi$$
0.890155 + 0.455658i $$0.150596\pi$$
$$752$$ 0 0
$$753$$ 54720.0 2.64822
$$754$$ 0 0
$$755$$ − 7200.00i − 0.347066i
$$756$$ 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 $$-0.474819\pi$$
0.0790259 + 0.996873i $$0.474819\pi$$
$$762$$ 0 0
$$763$$ − 5920.00i − 0.280889i
$$764$$ 0 0
$$765$$ 11100.0i 0.524603i
$$766$$ 0 0
$$767$$ −10000.0 −0.470768
$$768$$ 0 0
$$769$$ 11506.0 0.539554 0.269777 0.962923i $$-0.413050\pi$$
0.269777 + 0.962923i $$0.413050\pi$$
$$770$$ 0 0
$$771$$ − 36880.0i − 1.72270i
$$772$$ 0 0
$$773$$ − 22230.0i − 1.03436i −0.855878 0.517178i $$-0.826982\pi$$
0.855878 0.517178i $$-0.173018\pi$$
$$774$$ 0 0
$$775$$ −8000.00 −0.370798
$$776$$ 0 0
$$777$$ −39680.0 −1.83206
$$778$$ 0 0
$$779$$ 16400.0i 0.754289i
$$780$$ 0 0
$$781$$ − 16000.0i − 0.733067i
$$782$$ 0 0
$$783$$ 2720.00 0.124144
$$784$$ 0 0
$$785$$ −17900.0 −0.813858
$$786$$ 0 0
$$787$$ 21336.0i 0.966387i 0.875514 + 0.483193i $$0.160523\pi$$
−0.875514 + 0.483193i $$0.839477\pi$$
$$788$$ 0 0
$$789$$ 38784.0i 1.75000i
$$790$$ 0 0
$$791$$ 23840.0 1.07162
$$792$$ 0 0
$$793$$ 1500.00 0.0671709
$$794$$ 0 0
$$795$$ 32800.0i 1.46327i
$$796$$ 0 0
$$797$$ − 7170.00i − 0.318663i −0.987225 0.159332i $$-0.949066\pi$$
0.987225 0.159332i $$-0.0509339\pi$$
$$798$$ 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
$$805$$ 7680.00i 0.336254i
$$806$$ 0 0
$$807$$ 44400.0 1.93675
$$808$$ 0 0
$$809$$ 23654.0 1.02797 0.513987 0.857798i $$-0.328168\pi$$
0.513987 + 0.857798i $$0.328168\pi$$
$$810$$ 0 0
$$811$$ − 30440.0i − 1.31799i −0.752146 0.658997i $$-0.770981\pi$$
0.752146 0.658997i $$-0.229019\pi$$
$$812$$ 0 0
$$813$$ − 3840.00i − 0.165652i
$$814$$ 0 0
$$815$$ −12080.0 −0.519195
$$816$$ 0 0
$$817$$ 6080.00 0.260358
$$818$$ 0 0
$$819$$ 29600.0i 1.26289i
$$820$$ 0 0
$$821$$ 19930.0i 0.847213i 0.905846 + 0.423606i $$0.139236\pi$$
−0.905846 + 0.423606i $$0.860764\pi$$
$$822$$ 0 0
$$823$$ −9872.00 −0.418124 −0.209062 0.977902i $$-0.567041\pi$$
−0.209062 + 0.977902i $$0.567041\pi$$
$$824$$ 0 0
$$825$$ −8000.00 −0.337605
$$826$$ 0 0
$$827$$ − 5704.00i − 0.239840i −0.992784 0.119920i $$-0.961736\pi$$
0.992784 0.119920i $$-0.0382638\pi$$
$$828$$ 0 0
$$829$$ 27230.0i 1.14082i 0.821361 + 0.570408i $$0.193215\pi$$
−0.821361 + 0.570408i $$0.806785\pi$$
$$830$$ 0 0
$$831$$ −8240.00 −0.343974
$$832$$ 0 0
$$833$$ 2610.00 0.108561
$$834$$ 0 0
$$835$$ 28960.0i 1.20024i
$$836$$ 0 0
$$837$$ − 25600.0i − 1.05719i
$$838$$ 0 0
$$839$$ −18800.0 −0.773597 −0.386799 0.922164i $$-0.626419\pi$$
−0.386799 + 0.922164i $$0.626419\pi$$
$$840$$ 0 0
$$841$$ 23233.0 0.952602
$$842$$ 0 0
$$843$$ − 26160.0i − 1.06880i
$$844$$ 0 0
$$845$$ − 3030.00i − 0.123355i
$$846$$ 0 0
$$847$$ −4304.00 −0.174601
$$848$$ 0 0
$$849$$ 17344.0 0.701113
$$850$$ 0 0
$$851$$ − 14880.0i − 0.599389i
$$852$$ 0 0
$$853$$ 12090.0i 0.485292i 0.970115 + 0.242646i $$0.0780153\pi$$
−0.970115 + 0.242646i $$0.921985\pi$$
$$854$$ 0 0
$$855$$ −14800.0 −0.591988
$$856$$ 0 0
$$857$$ 470.000 0.0187338 0.00936692 0.999956i $$-0.497018\pi$$
0.00936692 + 0.999956i $$0.497018\pi$$
$$858$$ 0 0
$$859$$ 24440.0i 0.970759i 0.874304 + 0.485380i $$0.161319\pi$$
−0.874304 + 0.485380i $$0.838681\pi$$
$$860$$ 0 0
$$861$$ 52480.0i 2.07725i
$$862$$ 0 0
$$863$$ 22592.0 0.891125 0.445562 0.895251i $$-0.353004\pi$$
0.445562 + 0.895251i $$0.353004\pi$$
$$864$$ 0 0
$$865$$ −7500.00 −0.294807
$$866$$ 0 0
$$867$$ 32104.0i 1.25757i
$$868$$ 0 0
$$869$$ − 44800.0i − 1.74883i
$$870$$ 0 0
$$871$$ −38800.0 −1.50940
$$872$$ 0 0
$$873$$ 4070.00 0.157788
$$874$$ 0 0
$$875$$ 24000.0i 0.927255i
$$876$$ 0 0
$$877$$ − 17330.0i − 0.667266i −0.942703 0.333633i $$-0.891725\pi$$
0.942703 0.333633i $$-0.108275\pi$$
$$878$$ 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 $$0.0203460\pi$$
−0.997958 + 0.0638753i $$0.979654\pi$$
$$884$$ 0 0
$$885$$ − 16000.0i − 0.607722i
$$886$$ 0 0
$$887$$ −48144.0 −1.82245 −0.911227 0.411904i $$-0.864864\pi$$
−0.911227 + 0.411904i $$0.864864\pi$$
$$888$$ 0 0
$$889$$ 16384.0 0.618112
$$890$$ 0 0
$$891$$ 14360.0i 0.539931i
$$892$$ 0 0
$$893$$ − 16640.0i − 0.623557i
$$894$$ 0 0
$$895$$ 22800.0 0.851531
$$896$$ 0 0
$$897$$ −19200.0 −0.714682
$$898$$ 0 0
$$899$$ 10880.0i 0.403636i
$$900$$ 0 0
$$901$$ − 12300.0i − 0.454797i
$$902$$ 0 0
$$903$$ 19456.0 0.717005
$$904$$ 0 0
$$905$$ −4420.00 −0.162349
$$906$$ 0 0
$$907$$ 16216.0i 0.593653i 0.954931 + 0.296827i $$0.0959283\pi$$
−0.954931 + 0.296827i $$0.904072\pi$$
$$908$$ 0 0
$$909$$ − 40626.0i − 1.48238i
$$910$$ 0 0
$$911$$ −49440.0 −1.79805 −0.899023 0.437901i $$-0.855722\pi$$
−0.899023 + 0.437901i $$0.855722\pi$$
$$912$$ 0 0
$$913$$ −22080.0 −0.800374
$$914$$ 0 0
$$915$$ 2400.00i 0.0867121i
$$916$$ 0 0
$$917$$ − 18560.0i − 0.668381i
$$918$$ 0 0
$$919$$ −16080.0 −0.577182 −0.288591 0.957452i $$-0.593187\pi$$
−0.288591 + 0.957452i $$0.593187\pi$$
$$920$$ 0 0
$$921$$ −15168.0 −0.542674
$$922$$ 0 0
$$923$$ − 20000.0i − 0.713226i
$$924$$ 0 0
$$925$$ − 7750.00i − 0.275479i
$$926$$ 0 0
$$927$$ 1776.00 0.0629250
$$928$$ 0 0
$$929$$ −11310.0 −0.399428 −0.199714 0.979854i $$-0.564001\pi$$
−0.199714 + 0.979854i $$0.564001\pi$$
$$930$$ 0 0
$$931$$ 3480.00i 0.122505i
$$932$$ 0 0
$$933$$ 13440.0i 0.471603i
$$934$$ 0 0
$$935$$ −12000.0 −0.419724
$$936$$ 0 0
$$937$$ −25130.0 −0.876159 −0.438080 0.898936i $$-0.644341\pi$$
−0.438080 + 0.898936i $$0.644341\pi$$
$$938$$ 0 0
$$939$$ 7760.00i 0.269689i
$$940$$ 0 0
$$941$$ − 22322.0i − 0.773301i −0.922226 0.386651i $$-0.873632\pi$$
0.922226 0.386651i $$-0.126368\pi$$
$$942$$ 0 0
$$943$$ −19680.0 −0.679607
$$944$$ 0 0
$$945$$ −12800.0 −0.440618
$$946$$ 0 0
$$947$$ − 36456.0i − 1.25096i −0.780239 0.625481i $$-0.784903\pi$$
0.780239 0.625481i $$-0.215097\pi$$
$$948$$ 0 0
$$949$$ − 31500.0i − 1.07749i
$$950$$ 0 0
$$951$$ 57840.0 1.97223
$$952$$ 0 0
$$953$$ −40650.0 −1.38172 −0.690862 0.722987i $$-0.742769\pi$$
−0.690862 + 0.722987i $$0.742769\pi$$
$$954$$ 0 0
$$955$$ 19200.0i 0.650573i
$$956$$ 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
$$965$$ − 50700.0i − 1.69129i
$$966$$ 0 0
$$967$$ 34704.0 1.15409 0.577045 0.816712i $$-0.304206\pi$$
0.577045 + 0.816712i $$0.304206\pi$$
$$968$$ 0 0
$$969$$ 9600.00 0.318263
$$970$$ 0 0
$$971$$ − 30760.0i − 1.01662i −0.861175 0.508309i $$-0.830271\pi$$
0.861175 0.508309i $$-0.169729\pi$$
$$972$$ 0 0
$$973$$ − 31360.0i − 1.03325i
$$974$$ 0 0
$$975$$ −10000.0 −0.328468
$$976$$ 0 0
$$977$$ −38110.0 −1.24795 −0.623975 0.781444i $$-0.714483\pi$$
−0.623975 + 0.781444i $$0.714483\pi$$
$$978$$ 0 0
$$979$$ − 13040.0i − 0.425700i
$$980$$ 0 0
$$981$$ 13690.0i 0.445554i
$$982$$ 0 0
$$983$$ 19632.0 0.636992 0.318496 0.947924i $$-0.396822\pi$$
0.318496 + 0.947924i $$0.396822\pi$$
$$984$$ 0 0
$$985$$ 19100.0 0.617844
$$986$$ 0 0
$$987$$ − 53248.0i − 1.71723i
$$988$$ 0 0
$$989$$ 7296.00i 0.234580i
$$990$$ 0 0
$$991$$ 47680.0 1.52836 0.764180 0.645003i $$-0.223144\pi$$
0.764180 + 0.645003i $$0.223144\pi$$
$$992$$ 0 0
$$993$$ −46400.0 −1.48284
$$994$$ 0 0
$$995$$ 29600.0i 0.943099i
$$996$$ 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 256.4.b.e.129.1 2
4.3 odd 2 256.4.b.c.129.2 2
8.3 odd 2 256.4.b.c.129.1 2
8.5 even 2 inner 256.4.b.e.129.2 2
16.3 odd 4 64.4.a.a.1.1 1
16.5 even 4 32.4.a.a.1.1 1
16.11 odd 4 32.4.a.c.1.1 yes 1
16.13 even 4 64.4.a.e.1.1 1
48.5 odd 4 288.4.a.h.1.1 1
48.11 even 4 288.4.a.i.1.1 1
48.29 odd 4 576.4.a.g.1.1 1
48.35 even 4 576.4.a.h.1.1 1
80.19 odd 4 1600.4.a.bw.1.1 1
80.27 even 4 800.4.c.a.449.1 2
80.29 even 4 1600.4.a.e.1.1 1
80.37 odd 4 800.4.c.b.449.2 2
80.43 even 4 800.4.c.a.449.2 2
80.53 odd 4 800.4.c.b.449.1 2
80.59 odd 4 800.4.a.a.1.1 1
80.69 even 4 800.4.a.k.1.1 1
112.27 even 4 1568.4.a.c.1.1 1
112.69 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 16.5 even 4
32.4.a.c.1.1 yes 1 16.11 odd 4
64.4.a.a.1.1 1 16.3 odd 4
64.4.a.e.1.1 1 16.13 even 4
256.4.b.c.129.1 2 8.3 odd 2
256.4.b.c.129.2 2 4.3 odd 2
256.4.b.e.129.1 2 1.1 even 1 trivial
256.4.b.e.129.2 2 8.5 even 2 inner
288.4.a.h.1.1 1 48.5 odd 4
288.4.a.i.1.1 1 48.11 even 4
576.4.a.g.1.1 1 48.29 odd 4
576.4.a.h.1.1 1 48.35 even 4
800.4.a.a.1.1 1 80.59 odd 4
800.4.a.k.1.1 1 80.69 even 4
800.4.c.a.449.1 2 80.27 even 4
800.4.c.a.449.2 2 80.43 even 4
800.4.c.b.449.1 2 80.53 odd 4
800.4.c.b.449.2 2 80.37 odd 4
1568.4.a.c.1.1 1 112.27 even 4
1568.4.a.o.1.1 1 112.69 odd 4
1600.4.a.e.1.1 1 80.29 even 4
1600.4.a.bw.1.1 1 80.19 odd 4