# Properties

 Label 256.4.b.c.129.2 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 129.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 256.129 Dual form 256.4.b.c.129.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+8.00000i q^{3} +10.0000i q^{5} -16.0000 q^{7} -37.0000 q^{9} +O(q^{10})$$ $$q+8.00000i q^{3} +10.0000i q^{5} -16.0000 q^{7} -37.0000 q^{9} +40.0000i q^{11} -50.0000i q^{13} -80.0000 q^{15} -30.0000 q^{17} +40.0000i q^{19} -128.000i q^{21} -48.0000 q^{23} +25.0000 q^{25} -80.0000i q^{27} -34.0000i q^{29} +320.000 q^{31} -320.000 q^{33} -160.000i q^{35} -310.000i q^{37} +400.000 q^{39} -410.000 q^{41} -152.000i q^{43} -370.000i q^{45} -416.000 q^{47} -87.0000 q^{49} -240.000i q^{51} +410.000i q^{53} -400.000 q^{55} -320.000 q^{57} +200.000i q^{59} +30.0000i q^{61} +592.000 q^{63} +500.000 q^{65} +776.000i q^{67} -384.000i q^{69} -400.000 q^{71} +630.000 q^{73} +200.000i q^{75} -640.000i q^{77} -1120.00 q^{79} -359.000 q^{81} +552.000i q^{83} -300.000i q^{85} +272.000 q^{87} +326.000 q^{89} +800.000i q^{91} +2560.00i q^{93} -400.000 q^{95} -110.000 q^{97} -1480.00i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{7} - 74 q^{9}+O(q^{10})$$ 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 $$0.279644\pi$$
−0.638285 + 0.769800i $$0.720356\pi$$
$$4$$ 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 $$-0.642178\pi$$
−0.431959 + 0.901893i $$0.642178\pi$$
$$8$$ 0 0
$$9$$ −37.0000 −1.37037
$$10$$ 0 0
$$11$$ 40.0000i 1.09640i 0.836346 + 0.548202i $$0.184688\pi$$
−0.836346 + 0.548202i $$0.815312\pi$$
$$12$$ 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 $$0.0776362\pi$$
−0.970403 + 0.241490i $$0.922364\pi$$
$$20$$ 0 0
$$21$$ − 128.000i − 1.33009i
$$22$$ 0 0
$$23$$ −48.0000 −0.435161 −0.217580 0.976042i $$-0.569816\pi$$
−0.217580 + 0.976042i $$0.569816\pi$$
$$24$$ 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 $$-0.122383\pi$$
0.926995 + 0.375073i $$0.122383\pi$$
$$32$$ 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 $$-0.913131\pi$$
0.962991 0.269532i $$-0.0868691\pi$$
$$44$$ 0 0
$$45$$ − 370.000i − 1.22570i
$$46$$ 0 0
$$47$$ −416.000 −1.29106 −0.645530 0.763735i $$-0.723364\pi$$
−0.645530 + 0.763735i $$0.723364\pi$$
$$48$$ 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 $$0.0708208\pi$$
−0.975351 + 0.220659i $$0.929179\pi$$
$$60$$ 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 $$0.250172\pi$$
−0.706725 + 0.707489i $$0.749828\pi$$
$$68$$ 0 0
$$69$$ − 384.000i − 0.669973i
$$70$$ 0 0
$$71$$ −400.000 −0.668609 −0.334305 0.942465i $$-0.608501\pi$$
−0.334305 + 0.942465i $$0.608501\pi$$
$$72$$ 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 $$-0.793861\pi$$
−0.797531 + 0.603278i $$0.793861\pi$$
$$80$$ 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
$$85$$ − 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 $$-0.492691\pi$$
0.0229591 + 0.999736i $$0.492691\pi$$
$$104$$ 0 0
$$105$$ 1280.00 1.18967
$$106$$ 0 0
$$107$$ − 664.000i − 0.599919i −0.953952 0.299959i $$-0.903027\pi$$
0.953952 0.299959i $$-0.0969731\pi$$
$$108$$ 0 0
$$109$$ − 370.000i − 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 $$-0.616452\pi$$
−0.357737 + 0.933822i $$0.616452\pi$$
$$128$$ 0 0
$$129$$ 1216.00 0.829944
$$130$$ 0 0
$$131$$ 1160.00i 0.773662i 0.922151 + 0.386831i $$0.126430\pi$$
−0.922151 + 0.386831i $$0.873570\pi$$
$$132$$ 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 $$0.204039\pi$$
−0.801493 + 0.598004i $$0.795961\pi$$
$$140$$ 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 $$-0.437849\pi$$
0.194016 + 0.980998i $$0.437849\pi$$
$$152$$ 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 $$-0.906265\pi$$
0.956954 0.290239i $$-0.0937348\pi$$
$$164$$ 0 0
$$165$$ − 3200.00i − 1.50982i
$$166$$ 0 0
$$167$$ −2896.00 −1.34191 −0.670956 0.741497i $$-0.734116\pi$$
−0.670956 + 0.741497i $$0.734116\pi$$
$$168$$ 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 $$0.157921\pi$$
−0.879434 + 0.476020i $$0.842079\pi$$
$$180$$ 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 $$-0.618480\pi$$
−0.363681 + 0.931523i $$0.618480\pi$$
$$192$$ 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 $$-0.676761\pi$$
−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
$$205$$ − 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 $$0.00207711\pi$$
−0.999979 + 0.00652539i $$0.997923\pi$$
$$212$$ 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 $$-0.277347\pi$$
0.643824 + 0.765173i $$0.277347\pi$$
$$224$$ 0 0
$$225$$ −925.000 −0.274074
$$226$$ 0 0
$$227$$ − 6456.00i − 1.88766i −0.330425 0.943832i $$-0.607192\pi$$
0.330425 0.943832i $$-0.392808\pi$$
$$228$$ 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 $$-0.650249\pi$$
−0.454687 + 0.890651i $$0.650249\pi$$
$$240$$ 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 $$-0.670442\pi$$
0.510237 0.860034i $$-0.329558\pi$$
$$252$$ 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 $$-0.307591\pi$$
0.568328 + 0.822802i $$0.307591\pi$$
$$264$$ 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 $$-0.517132\pi$$
−0.0537969 + 0.998552i $$0.517132\pi$$
$$272$$ 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 $$-0.926882\pi$$
0.973733 0.227693i $$-0.0731183\pi$$
$$284$$ 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 $$0.0563930\pi$$
−0.984347 + 0.176238i $$0.943607\pi$$
$$308$$ 0 0
$$309$$ 384.000i 0.0706958i
$$310$$ 0 0
$$311$$ 1680.00 0.306315 0.153158 0.988202i $$-0.451056\pi$$
0.153158 + 0.988202i $$0.451056\pi$$
$$312$$ 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 $$0.159932\pi$$
−0.876410 + 0.481566i $$0.840068\pi$$
$$332$$ 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 $$-0.990741\pi$$
0.999577 0.0290846i $$-0.00925923\pi$$
$$348$$ 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 $$-0.752005\pi$$
−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
$$365$$ 6300.00i 0.903444i
$$366$$ 0 0
$$367$$ 2784.00 0.395977 0.197989 0.980204i $$-0.436559\pi$$
0.197989 + 0.980204i $$0.436559\pi$$
$$368$$ 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 $$-0.962814\pi$$
0.993184 0.116557i $$-0.0371859\pi$$
$$380$$ 0 0
$$381$$ − 8192.00i − 1.10155i
$$382$$ 0 0
$$383$$ 11008.0 1.46862 0.734311 0.678813i $$-0.237505\pi$$
0.734311 + 0.678813i $$0.237505\pi$$
$$384$$ 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 $$0.151930\pi$$
−0.888238 + 0.459383i $$0.848070\pi$$
$$420$$ 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 $$-0.216313\pi$$
0.777845 + 0.628456i $$0.216313\pi$$
$$432$$ 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 $$-0.303684\pi$$
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 $$0.165956\pi$$
−0.867139 + 0.498066i $$0.834044\pi$$
$$444$$ 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 $$-0.406116\pi$$
0.290688 + 0.956818i $$0.406116\pi$$
$$464$$ 0 0
$$465$$ −25600.0 −2.55306
$$466$$ 0 0
$$467$$ 2216.00i 0.219581i 0.993955 + 0.109790i $$0.0350180\pi$$
−0.993955 + 0.109790i $$0.964982\pi$$
$$468$$ 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 $$-0.668011\pi$$
−0.503652 + 0.863907i $$0.668011\pi$$
$$480$$ 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 $$-0.711690\pi$$
−0.617094 + 0.786890i $$0.711690\pi$$
$$488$$ 0 0
$$489$$ 9664.00 0.893704
$$490$$ 0 0
$$491$$ 4840.00i 0.444860i 0.974949 + 0.222430i $$0.0713988\pi$$
−0.974949 + 0.222430i $$0.928601\pi$$
$$492$$ 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 $$0.340711\pi$$
−0.479795 + 0.877381i $$0.659289\pi$$
$$500$$ 0 0
$$501$$ − 23168.0i − 2.06601i
$$502$$ 0 0
$$503$$ 528.000 0.0468039 0.0234019 0.999726i $$-0.492550\pi$$
0.0234019 + 0.999726i $$0.492550\pi$$
$$504$$ 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 $$-0.848385\pi$$
0.888691 0.458506i $$-0.151615\pi$$
$$524$$ 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 $$0.294232\pi$$
−0.602348 + 0.798233i $$0.705768\pi$$
$$548$$ 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 $$-0.967633\pi$$
0.994835 0.101507i $$-0.0323665\pi$$
$$564$$ 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 $$0.166608\pi$$
−0.866118 + 0.499839i $$0.833392\pi$$
$$572$$ 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 $$0.0979593\pi$$
−0.953018 + 0.302913i $$0.902041\pi$$
$$588$$ 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 $$-0.323468\pi$$
0.526595 + 0.850116i $$0.323468\pi$$
$$600$$ 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 $$-0.208071\pi$$
0.793854 + 0.608108i $$0.208071\pi$$
$$608$$ 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 $$-0.881983\pi$$
0.932052 0.362325i $$-0.118017\pi$$
$$620$$ 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 $$-0.634940\pi$$
−0.411342 + 0.911481i $$0.634940\pi$$
$$632$$ 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 $$0.0439478\pi$$
−0.990484 + 0.137628i $$0.956052\pi$$
$$644$$ 0 0
$$645$$ 12160.0i 0.742325i
$$646$$ 0 0
$$647$$ −2064.00 −0.125416 −0.0627080 0.998032i $$-0.519974\pi$$
−0.0627080 + 0.998032i $$0.519974\pi$$
$$648$$ 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 $$-0.801015\pi$$
0.810886 0.585204i $$-0.198985\pi$$
$$660$$ 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 $$0.192833\pi$$
−0.822046 + 0.569421i $$0.807167\pi$$
$$684$$ 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 $$0.111997\pi$$
−0.938737 + 0.344633i $$0.888003\pi$$
$$692$$ 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 $$-0.702207\pi$$
−0.593380 + 0.804923i $$0.702207\pi$$
$$720$$ 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 $$-0.585599\pi$$
−0.265686 + 0.964060i $$0.585599\pi$$
$$728$$ 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 $$-0.981293\pi$$
0.998273 0.0587375i $$-0.0187075\pi$$
$$740$$ 0 0
$$741$$ 16000.0i 0.793218i
$$742$$ 0 0
$$743$$ −32208.0 −1.59031 −0.795153 0.606409i $$-0.792609\pi$$
−0.795153 + 0.606409i $$0.792609\pi$$
$$744$$ 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 $$-0.849404\pi$$
−0.890155 + 0.455658i $$0.849404\pi$$
$$752$$ 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 $$-0.839477\pi$$
0.875514 0.483193i $$-0.160523\pi$$
$$788$$ 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 $$0.229019\pi$$
−0.752146 + 0.658997i $$0.770981\pi$$
$$812$$ 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 $$-0.432959\pi$$
0.209062 + 0.977902i $$0.432959\pi$$
$$824$$ 0 0
$$825$$ −8000.00 −0.337605
$$826$$ 0 0
$$827$$ 5704.00i 0.239840i 0.992784 + 0.119920i $$0.0382638\pi$$
−0.992784 + 0.119920i $$0.961736\pi$$
$$828$$ 0 0
$$829$$ 27230.0i 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 $$-0.373581\pi$$
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
$$845$$ − 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 $$-0.838681\pi$$
0.874304 0.485380i $$-0.161319\pi$$
$$860$$ 0 0
$$861$$ 52480.0i 2.07725i
$$862$$ 0 0
$$863$$ −22592.0 −0.891125 −0.445562 0.895251i $$-0.646996\pi$$
−0.445562 + 0.895251i $$0.646996\pi$$
$$864$$ 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 $$-0.979654\pi$$
0.997958 0.0638753i $$-0.0203460\pi$$
$$884$$ 0 0
$$885$$ − 16000.0i − 0.607722i
$$886$$ 0 0
$$887$$ 48144.0 1.82245 0.911227 0.411904i $$-0.135136\pi$$
0.911227 + 0.411904i $$0.135136\pi$$
$$888$$ 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 $$-0.904072\pi$$
0.954931 0.296827i $$-0.0959283\pi$$
$$908$$ 0 0
$$909$$ − 40626.0i − 1.48238i
$$910$$ 0 0
$$911$$ 49440.0 1.79805 0.899023 0.437901i $$-0.144278\pi$$
0.899023 + 0.437901i $$0.144278\pi$$
$$912$$ 0 0
$$913$$ −22080.0 −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 $$-0.406813\pi$$
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
$$925$$ − 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 $$0.215097\pi$$
−0.780239 + 0.625481i $$0.784903\pi$$
$$948$$ 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 $$-0.695794\pi$$
−0.577045 + 0.816712i $$0.695794\pi$$
$$968$$ 0 0
$$969$$ 9600.00 0.318263
$$970$$ 0 0
$$971$$ 30760.0i 1.01662i 0.861175 + 0.508309i $$0.169729\pi$$
−0.861175 + 0.508309i $$0.830271\pi$$
$$972$$ 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 $$-0.603178\pi$$
−0.318496 + 0.947924i $$0.603178\pi$$
$$984$$ 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 $$-0.776856\pi$$
−0.764180 + 0.645003i $$0.776856\pi$$
$$992$$ 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.c.129.2 2
4.3 odd 2 256.4.b.e.129.1 2
8.3 odd 2 256.4.b.e.129.2 2
8.5 even 2 inner 256.4.b.c.129.1 2
16.3 odd 4 64.4.a.e.1.1 1
16.5 even 4 32.4.a.c.1.1 yes 1
16.11 odd 4 32.4.a.a.1.1 1
16.13 even 4 64.4.a.a.1.1 1
48.5 odd 4 288.4.a.i.1.1 1
48.11 even 4 288.4.a.h.1.1 1
48.29 odd 4 576.4.a.h.1.1 1
48.35 even 4 576.4.a.g.1.1 1
80.19 odd 4 1600.4.a.e.1.1 1
80.27 even 4 800.4.c.b.449.2 2
80.29 even 4 1600.4.a.bw.1.1 1
80.37 odd 4 800.4.c.a.449.1 2
80.43 even 4 800.4.c.b.449.1 2
80.53 odd 4 800.4.c.a.449.2 2
80.59 odd 4 800.4.a.k.1.1 1
80.69 even 4 800.4.a.a.1.1 1
112.27 even 4 1568.4.a.o.1.1 1
112.69 odd 4 1568.4.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.a.a.1.1 1 16.11 odd 4
32.4.a.c.1.1 yes 1 16.5 even 4
64.4.a.a.1.1 1 16.13 even 4
64.4.a.e.1.1 1 16.3 odd 4
256.4.b.c.129.1 2 8.5 even 2 inner
256.4.b.c.129.2 2 1.1 even 1 trivial
256.4.b.e.129.1 2 4.3 odd 2
256.4.b.e.129.2 2 8.3 odd 2
288.4.a.h.1.1 1 48.11 even 4
288.4.a.i.1.1 1 48.5 odd 4
576.4.a.g.1.1 1 48.35 even 4
576.4.a.h.1.1 1 48.29 odd 4
800.4.a.a.1.1 1 80.69 even 4
800.4.a.k.1.1 1 80.59 odd 4
800.4.c.a.449.1 2 80.37 odd 4
800.4.c.a.449.2 2 80.53 odd 4
800.4.c.b.449.1 2 80.43 even 4
800.4.c.b.449.2 2 80.27 even 4
1568.4.a.c.1.1 1 112.69 odd 4
1568.4.a.o.1.1 1 112.27 even 4
1600.4.a.e.1.1 1 80.19 odd 4
1600.4.a.bw.1.1 1 80.29 even 4