# Properties

 Label 800.2.c.a.449.2 Level $800$ Weight $2$ Character 800.449 Analytic conductor $6.388$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,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, 2, names="a")

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$6.38803216170$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 160) 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 800.2.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)

 $$f(q)$$ $$=$$ $$q+2.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+2.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +6.00000i q^{13} +2.00000i q^{17} -8.00000 q^{19} +4.00000 q^{21} +6.00000i q^{23} +4.00000i q^{27} +2.00000 q^{29} +4.00000 q^{31} -8.00000i q^{33} +2.00000i q^{37} -12.0000 q^{39} -10.0000 q^{41} +2.00000i q^{43} -2.00000i q^{47} +3.00000 q^{49} -4.00000 q^{51} -2.00000i q^{53} -16.0000i q^{57} +2.00000 q^{61} +2.00000i q^{63} -6.00000i q^{67} -12.0000 q^{69} -12.0000 q^{71} -10.0000i q^{73} +8.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} +10.0000i q^{83} +4.00000i q^{87} +6.00000 q^{89} +12.0000 q^{91} +8.00000i q^{93} +10.0000i q^{97} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 8 q^{11} - 16 q^{19} + 8 q^{21} + 4 q^{29} + 8 q^{31} - 24 q^{39} - 20 q^{41} + 6 q^{49} - 8 q^{51} + 4 q^{61} - 24 q^{69} - 24 q^{71} + 16 q^{79} - 22 q^{81} + 12 q^{89} + 24 q^{91} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 - 8 * q^11 - 16 * q^19 + 8 * q^21 + 4 * q^29 + 8 * q^31 - 24 * q^39 - 20 * q^41 + 6 * q^49 - 8 * q^51 + 4 * q^61 - 24 * q^69 - 24 * q^71 + 16 * q^79 - 22 * q^81 + 12 * q^89 + 24 * q^91 + 8 * 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
$$3$$ 2.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 0 0
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 0 0
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4.00000i 0.769800i
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ − 8.00000i − 1.39262i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ −12.0000 −1.92154
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 2.00000i 0.304997i 0.988304 + 0.152499i $$0.0487319\pi$$
−0.988304 + 0.152499i $$0.951268\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 2.00000i − 0.291730i −0.989305 0.145865i $$-0.953403\pi$$
0.989305 0.145865i $$-0.0465965\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ − 2.00000i − 0.274721i −0.990521 0.137361i $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 16.0000i − 2.11925i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 2.00000i 0.251976i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 6.00000i − 0.733017i −0.930415 0.366508i $$-0.880553\pi$$
0.930415 0.366508i $$-0.119447\pi$$
$$68$$ 0 0
$$69$$ −12.0000 −1.44463
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 8.00000i 0.911685i
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 10.0000i 1.09764i 0.835940 + 0.548821i $$0.184923\pi$$
−0.835940 + 0.548821i $$0.815077\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 4.00000i 0.428845i
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ 0 0
$$93$$ 8.00000i 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 0 0
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ − 2.00000i − 0.197066i −0.995134 0.0985329i $$-0.968585\pi$$
0.995134 0.0985329i $$-0.0314150\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 6.00000i − 0.580042i −0.957020 0.290021i $$-0.906338\pi$$
0.957020 0.290021i $$-0.0936623\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 6.00000i − 0.554700i
$$118$$ 0 0
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ − 20.0000i − 1.80334i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 6.00000i 0.532414i 0.963916 + 0.266207i $$0.0857705\pi$$
−0.963916 + 0.266207i $$0.914230\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 0 0
$$133$$ 16.0000i 1.38738i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.00000i 0.170872i 0.996344 + 0.0854358i $$0.0272282\pi$$
−0.996344 + 0.0854358i $$0.972772\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ 0 0
$$143$$ − 24.0000i − 2.00698i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 6.00000i 0.494872i
$$148$$ 0 0
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 0 0
$$153$$ − 2.00000i − 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000i 0.798087i 0.916932 + 0.399043i $$0.130658\pi$$
−0.916932 + 0.399043i $$0.869342\pi$$
$$158$$ 0 0
$$159$$ 4.00000 0.317221
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ 0 0
$$163$$ − 6.00000i − 0.469956i −0.972001 0.234978i $$-0.924498\pi$$
0.972001 0.234978i $$-0.0755019\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 14.0000i 1.08335i 0.840587 + 0.541676i $$0.182210\pi$$
−0.840587 + 0.541676i $$0.817790\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 8.00000 0.611775
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 0 0
$$183$$ 4.00000i 0.295689i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 8.00000i − 0.585018i
$$188$$ 0 0
$$189$$ 8.00000 0.581914
$$190$$ 0 0
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 0 0
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 22.0000i − 1.56744i −0.621117 0.783718i $$-0.713321\pi$$
0.621117 0.783718i $$-0.286679\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ 0 0
$$203$$ − 4.00000i − 0.280745i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 6.00000i − 0.417029i
$$208$$ 0 0
$$209$$ 32.0000 2.21349
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ − 24.0000i − 1.64445i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 8.00000i − 0.543075i
$$218$$ 0 0
$$219$$ 20.0000 1.35147
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ − 2.00000i − 0.133930i −0.997755 0.0669650i $$-0.978668\pi$$
0.997755 0.0669650i $$-0.0213316\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 18.0000i 1.19470i 0.801980 + 0.597351i $$0.203780\pi$$
−0.801980 + 0.597351i $$0.796220\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ −16.0000 −1.05272
$$232$$ 0 0
$$233$$ − 2.00000i − 0.131024i −0.997852 0.0655122i $$-0.979132\pi$$
0.997852 0.0655122i $$-0.0208681\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 16.0000i 1.03931i
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ 0 0
$$243$$ − 10.0000i − 0.641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 48.0000i − 3.05417i
$$248$$ 0 0
$$249$$ −20.0000 −1.26745
$$250$$ 0 0
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 0 0
$$253$$ − 24.0000i − 1.50887i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 26.0000i 1.62184i 0.585160 + 0.810918i $$0.301032\pi$$
−0.585160 + 0.810918i $$0.698968\pi$$
$$258$$ 0 0
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ − 2.00000i − 0.123325i −0.998097 0.0616626i $$-0.980360\pi$$
0.998097 0.0616626i $$-0.0196403\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 12.0000i 0.734388i
$$268$$ 0 0
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 28.0000 1.70088 0.850439 0.526073i $$-0.176336\pi$$
0.850439 + 0.526073i $$0.176336\pi$$
$$272$$ 0 0
$$273$$ 24.0000i 1.45255i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 22.0000i − 1.32185i −0.750451 0.660926i $$-0.770164\pi$$
0.750451 0.660926i $$-0.229836\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ 26.0000i 1.54554i 0.634686 + 0.772770i $$0.281129\pi$$
−0.634686 + 0.772770i $$0.718871\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 20.0000i 1.18056i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −20.0000 −1.17242
$$292$$ 0 0
$$293$$ − 18.0000i − 1.05157i −0.850617 0.525786i $$-0.823771\pi$$
0.850617 0.525786i $$-0.176229\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 16.0000i − 0.928414i
$$298$$ 0 0
$$299$$ −36.0000 −2.08193
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 0 0
$$303$$ 28.0000i 1.60856i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 22.0000i − 1.25561i −0.778372 0.627803i $$-0.783954\pi$$
0.778372 0.627803i $$-0.216046\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 28.0000 1.58773 0.793867 0.608091i $$-0.208065\pi$$
0.793867 + 0.608091i $$0.208065\pi$$
$$312$$ 0 0
$$313$$ 6.00000i 0.339140i 0.985518 + 0.169570i $$0.0542379\pi$$
−0.985518 + 0.169570i $$0.945762\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.00000i 0.112331i 0.998421 + 0.0561656i $$0.0178875\pi$$
−0.998421 + 0.0561656i $$0.982113\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ − 16.0000i − 0.890264i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 28.0000i 1.54840i
$$328$$ 0 0
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 0 0
$$333$$ − 2.00000i − 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 14.0000i − 0.762629i −0.924445 0.381314i $$-0.875472\pi$$
0.924445 0.381314i $$-0.124528\pi$$
$$338$$ 0 0
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ − 20.0000i − 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 18.0000i 0.966291i 0.875540 + 0.483145i $$0.160506\pi$$
−0.875540 + 0.483145i $$0.839494\pi$$
$$348$$ 0 0
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ −24.0000 −1.28103
$$352$$ 0 0
$$353$$ 30.0000i 1.59674i 0.602168 + 0.798369i $$0.294304\pi$$
−0.602168 + 0.798369i $$0.705696\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 8.00000i 0.423405i
$$358$$ 0 0
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 0 0
$$363$$ 10.0000i 0.524864i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 18.0000i − 0.939592i −0.882775 0.469796i $$-0.844327\pi$$
0.882775 0.469796i $$-0.155673\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ − 10.0000i − 0.517780i −0.965907 0.258890i $$-0.916643\pi$$
0.965907 0.258890i $$-0.0833568\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000i 0.618031i
$$378$$ 0 0
$$379$$ −24.0000 −1.23280 −0.616399 0.787434i $$-0.711409\pi$$
−0.616399 + 0.787434i $$0.711409\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 0 0
$$383$$ 14.0000i 0.715367i 0.933843 + 0.357683i $$0.116433\pi$$
−0.933843 + 0.357683i $$0.883567\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 2.00000i − 0.101666i
$$388$$ 0 0
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ 0 0
$$393$$ 8.00000i 0.403547i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 26.0000i 1.30490i 0.757831 + 0.652451i $$0.226259\pi$$
−0.757831 + 0.652451i $$0.773741\pi$$
$$398$$ 0 0
$$399$$ −32.0000 −1.60200
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 24.0000i 1.19553i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 8.00000i − 0.396545i
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ −4.00000 −0.197305
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 32.0000i 1.56705i
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 0 0
$$423$$ 2.00000i 0.0972433i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 4.00000i − 0.193574i
$$428$$ 0 0
$$429$$ 48.0000 2.31746
$$430$$ 0 0
$$431$$ −20.0000 −0.963366 −0.481683 0.876346i $$-0.659974\pi$$
−0.481683 + 0.876346i $$0.659974\pi$$
$$432$$ 0 0
$$433$$ 22.0000i 1.05725i 0.848855 + 0.528626i $$0.177293\pi$$
−0.848855 + 0.528626i $$0.822707\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 48.0000i − 2.29615i
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 18.0000i 0.855206i 0.903967 + 0.427603i $$0.140642\pi$$
−0.903967 + 0.427603i $$0.859358\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 20.0000i − 0.945968i
$$448$$ 0 0
$$449$$ −22.0000 −1.03824 −0.519122 0.854700i $$-0.673741\pi$$
−0.519122 + 0.854700i $$0.673741\pi$$
$$450$$ 0 0
$$451$$ 40.0000 1.88353
$$452$$ 0 0
$$453$$ 40.0000i 1.87936i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 6.00000i − 0.280668i −0.990104 0.140334i $$-0.955182\pi$$
0.990104 0.140334i $$-0.0448177\pi$$
$$458$$ 0 0
$$459$$ −8.00000 −0.373408
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ 38.0000i 1.76601i 0.469364 + 0.883005i $$0.344483\pi$$
−0.469364 + 0.883005i $$0.655517\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 6.00000i − 0.277647i −0.990317 0.138823i $$-0.955668\pi$$
0.990317 0.138823i $$-0.0443321\pi$$
$$468$$ 0 0
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ −20.0000 −0.921551
$$472$$ 0 0
$$473$$ − 8.00000i − 0.367840i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 2.00000i 0.0915737i
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 0 0
$$483$$ 24.0000i 1.09204i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 38.0000i 1.72194i 0.508652 + 0.860972i $$0.330144\pi$$
−0.508652 + 0.860972i $$0.669856\pi$$
$$488$$ 0 0
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 0 0
$$493$$ 4.00000i 0.180151i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 24.0000i 1.07655i
$$498$$ 0 0
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ 0 0
$$501$$ −28.0000 −1.25095
$$502$$ 0 0
$$503$$ 14.0000i 0.624229i 0.950044 + 0.312115i $$0.101037\pi$$
−0.950044 + 0.312115i $$0.898963\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 46.0000i − 2.04293i
$$508$$ 0 0
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ 0 0
$$511$$ −20.0000 −0.884748
$$512$$ 0 0
$$513$$ − 32.0000i − 1.41283i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 8.00000i 0.351840i
$$518$$ 0 0
$$519$$ −12.0000 −0.526742
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ − 22.0000i − 0.961993i −0.876723 0.480996i $$-0.840275\pi$$
0.876723 0.480996i $$-0.159725\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8.00000i 0.348485i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 60.0000i − 2.59889i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −12.0000 −0.516877
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 0 0
$$543$$ − 36.0000i − 1.54491i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 14.0000i − 0.598597i −0.954160 0.299298i $$-0.903247\pi$$
0.954160 0.299298i $$-0.0967526\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −16.0000 −0.681623
$$552$$ 0 0
$$553$$ − 16.0000i − 0.680389i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 46.0000i − 1.94908i −0.224208 0.974541i $$-0.571980\pi$$
0.224208 0.974541i $$-0.428020\pi$$
$$558$$ 0 0
$$559$$ −12.0000 −0.507546
$$560$$ 0 0
$$561$$ 16.0000 0.675521
$$562$$ 0 0
$$563$$ − 30.0000i − 1.26435i −0.774826 0.632175i $$-0.782163\pi$$
0.774826 0.632175i $$-0.217837\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 22.0000i 0.923913i
$$568$$ 0 0
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ − 8.00000i − 0.334205i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 38.0000i − 1.58196i −0.611842 0.790980i $$-0.709571\pi$$
0.611842 0.790980i $$-0.290429\pi$$
$$578$$ 0 0
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ 20.0000 0.829740
$$582$$ 0 0
$$583$$ 8.00000i 0.331326i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 10.0000i 0.412744i 0.978474 + 0.206372i $$0.0661657\pi$$
−0.978474 + 0.206372i $$0.933834\pi$$
$$588$$ 0 0
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 44.0000 1.80992
$$592$$ 0 0
$$593$$ − 18.0000i − 0.739171i −0.929197 0.369586i $$-0.879500\pi$$
0.929197 0.369586i $$-0.120500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 32.0000i 1.30967i
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 6.00000i 0.244339i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 2.00000i − 0.0811775i −0.999176 0.0405887i $$-0.987077\pi$$
0.999176 0.0405887i $$-0.0129233\pi$$
$$608$$ 0 0
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ 22.0000i 0.888572i 0.895885 + 0.444286i $$0.146543\pi$$
−0.895885 + 0.444286i $$0.853457\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 30.0000i − 1.20775i −0.797077 0.603877i $$-0.793622\pi$$
0.797077 0.603877i $$-0.206378\pi$$
$$618$$ 0 0
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 0 0
$$621$$ −24.0000 −0.963087
$$622$$ 0 0
$$623$$ − 12.0000i − 0.480770i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 64.0000i 2.55591i
$$628$$ 0 0
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 36.0000 1.43314 0.716569 0.697517i $$-0.245712\pi$$
0.716569 + 0.697517i $$0.245712\pi$$
$$632$$ 0 0
$$633$$ 8.00000i 0.317971i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 18.0000i 0.713186i
$$638$$ 0 0
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ − 30.0000i − 1.18308i −0.806274 0.591542i $$-0.798519\pi$$
0.806274 0.591542i $$-0.201481\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 42.0000i − 1.65119i −0.564263 0.825595i $$-0.690840\pi$$
0.564263 0.825595i $$-0.309160\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 0 0
$$653$$ 30.0000i 1.17399i 0.809590 + 0.586995i $$0.199689\pi$$
−0.809590 + 0.586995i $$0.800311\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 10.0000i 0.390137i
$$658$$ 0 0
$$659$$ 16.0000 0.623272 0.311636 0.950202i $$-0.399123\pi$$
0.311636 + 0.950202i $$0.399123\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ 0 0
$$663$$ − 24.0000i − 0.932083i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 12.0000i 0.464642i
$$668$$ 0 0
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ − 10.0000i − 0.385472i −0.981251 0.192736i $$-0.938264\pi$$
0.981251 0.192736i $$-0.0617360\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 26.0000i 0.999261i 0.866239 + 0.499631i $$0.166531\pi$$
−0.866239 + 0.499631i $$0.833469\pi$$
$$678$$ 0 0
$$679$$ 20.0000 0.767530
$$680$$ 0 0
$$681$$ −36.0000 −1.37952
$$682$$ 0 0
$$683$$ 34.0000i 1.30097i 0.759517 + 0.650487i $$0.225435\pi$$
−0.759517 + 0.650487i $$0.774565\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 20.0000i 0.763048i
$$688$$ 0 0
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 12.0000 0.456502 0.228251 0.973602i $$-0.426699\pi$$
0.228251 + 0.973602i $$0.426699\pi$$
$$692$$ 0 0
$$693$$ − 8.00000i − 0.303895i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 20.0000i − 0.757554i
$$698$$ 0 0
$$699$$ 4.00000 0.151294
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ − 16.0000i − 0.603451i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 28.0000i − 1.05305i
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 24.0000i 0.898807i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 16.0000i 0.597531i
$$718$$ 0 0
$$719$$ 40.0000 1.49175 0.745874 0.666087i $$-0.232032\pi$$
0.745874 + 0.666087i $$0.232032\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ 0 0
$$723$$ − 36.0000i − 1.33885i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 2.00000i − 0.0741759i −0.999312 0.0370879i $$-0.988192\pi$$
0.999312 0.0370879i $$-0.0118082\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −4.00000 −0.147945
$$732$$ 0 0
$$733$$ 6.00000i 0.221615i 0.993842 + 0.110808i $$0.0353437\pi$$
−0.993842 + 0.110808i $$0.964656\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 24.0000i 0.884051i
$$738$$ 0 0
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 0 0
$$741$$ 96.0000 3.52665
$$742$$ 0 0
$$743$$ − 42.0000i − 1.54083i −0.637542 0.770415i $$-0.720049\pi$$
0.637542 0.770415i $$-0.279951\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 10.0000i − 0.365881i
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 0 0
$$753$$ − 40.0000i − 1.45768i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.0000i 0.363456i 0.983349 + 0.181728i $$0.0581691\pi$$
−0.983349 + 0.181728i $$0.941831\pi$$
$$758$$ 0 0
$$759$$ 48.0000 1.74229
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ − 28.0000i − 1.01367i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ −52.0000 −1.87273
$$772$$ 0 0
$$773$$ 38.0000i 1.36677i 0.730061 + 0.683383i $$0.239492\pi$$
−0.730061 + 0.683383i $$0.760508\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 8.00000i 0.286998i
$$778$$ 0 0
$$779$$ 80.0000 2.86630
$$780$$ 0 0
$$781$$ 48.0000 1.71758
$$782$$ 0 0
$$783$$ 8.00000i 0.285897i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 14.0000i − 0.499046i −0.968369 0.249523i $$-0.919726\pi$$
0.968369 0.249523i $$-0.0802738\pi$$
$$788$$ 0 0
$$789$$ 4.00000 0.142404
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 0 0
$$793$$ 12.0000i 0.426132i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 30.0000i − 1.06265i −0.847167 0.531327i $$-0.821693\pi$$
0.847167 0.531327i $$-0.178307\pi$$
$$798$$ 0 0
$$799$$ 4.00000 0.141510
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 40.0000i 1.41157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 36.0000i − 1.26726i
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 0 0
$$813$$ 56.0000i 1.96401i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 16.0000i − 0.559769i
$$818$$ 0 0
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ 0 0
$$823$$ 6.00000i 0.209147i 0.994517 + 0.104573i $$0.0333477\pi$$
−0.994517 + 0.104573i $$0.966652\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 50.0000i 1.73867i 0.494223 + 0.869335i $$0.335453\pi$$
−0.494223 + 0.869335i $$0.664547\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ 44.0000 1.52634
$$832$$ 0 0
$$833$$ 6.00000i 0.207888i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 16.0000i 0.553041i
$$838$$ 0 0
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ − 20.0000i − 0.688837i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 10.0000i − 0.343604i
$$848$$ 0 0
$$849$$ −52.0000 −1.78464
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ − 10.0000i − 0.342393i −0.985237 0.171197i $$-0.945237\pi$$
0.985237 0.171197i $$-0.0547634\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 18.0000i 0.614868i 0.951569 + 0.307434i $$0.0994704\pi$$
−0.951569 + 0.307434i $$0.900530\pi$$
$$858$$ 0 0
$$859$$ 16.0000 0.545913 0.272956 0.962026i $$-0.411998\pi$$
0.272956 + 0.962026i $$0.411998\pi$$
$$860$$ 0 0
$$861$$ −40.0000 −1.36320
$$862$$ 0 0
$$863$$ 46.0000i 1.56586i 0.622111 + 0.782929i $$0.286275\pi$$
−0.622111 + 0.782929i $$0.713725\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 26.0000i 0.883006i
$$868$$ 0 0
$$869$$ −32.0000 −1.08553
$$870$$ 0 0
$$871$$ 36.0000 1.21981
$$872$$ 0 0
$$873$$ − 10.0000i − 0.338449i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 10.0000i 0.337676i 0.985644 + 0.168838i $$0.0540015\pi$$
−0.985644 + 0.168838i $$0.945999\pi$$
$$878$$ 0 0
$$879$$ 36.0000 1.21425
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ 2.00000i 0.0673054i 0.999434 + 0.0336527i $$0.0107140\pi$$
−0.999434 + 0.0336527i $$0.989286\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 2.00000i − 0.0671534i −0.999436 0.0335767i $$-0.989310\pi$$
0.999436 0.0335767i $$-0.0106898\pi$$
$$888$$ 0 0
$$889$$ 12.0000 0.402467
$$890$$ 0 0
$$891$$ 44.0000 1.47406
$$892$$ 0 0
$$893$$ 16.0000i 0.535420i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 72.0000i − 2.40401i
$$898$$ 0 0
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ 0 0
$$903$$ 8.00000i 0.266223i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 2.00000i 0.0664089i 0.999449 + 0.0332045i $$0.0105712\pi$$
−0.999449 + 0.0332045i $$0.989429\pi$$
$$908$$ 0 0
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ −28.0000 −0.927681 −0.463841 0.885919i $$-0.653529\pi$$
−0.463841 + 0.885919i $$0.653529\pi$$
$$912$$ 0 0
$$913$$ − 40.0000i − 1.32381i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 8.00000i − 0.264183i
$$918$$ 0 0
$$919$$ −56.0000 −1.84727 −0.923635 0.383274i $$-0.874797\pi$$
−0.923635 + 0.383274i $$0.874797\pi$$
$$920$$ 0 0
$$921$$ 44.0000 1.44985
$$922$$ 0 0
$$923$$ − 72.0000i − 2.36991i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 2.00000i 0.0656886i
$$928$$ 0 0
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ −24.0000 −0.786568
$$932$$ 0 0
$$933$$ 56.0000i 1.83336i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 34.0000i 1.11073i 0.831606 + 0.555366i $$0.187422\pi$$
−0.831606 + 0.555366i $$0.812578\pi$$
$$938$$ 0 0
$$939$$ −12.0000 −0.391605
$$940$$ 0 0
$$941$$ 38.0000 1.23876 0.619382 0.785090i $$-0.287383\pi$$
0.619382 + 0.785090i $$0.287383\pi$$
$$942$$ 0 0
$$943$$ − 60.0000i − 1.95387i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 38.0000i − 1.23483i −0.786636 0.617417i $$-0.788179\pi$$
0.786636 0.617417i $$-0.211821\pi$$
$$948$$ 0 0
$$949$$ 60.0000 1.94768
$$950$$ 0 0
$$951$$ −4.00000 −0.129709
$$952$$ 0 0
$$953$$ − 58.0000i − 1.87880i −0.342817 0.939402i $$-0.611381\pi$$
0.342817 0.939402i $$-0.388619\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 16.0000i − 0.517207i
$$958$$ 0 0
$$959$$ 4.00000 0.129167
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 6.00000i 0.193347i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 22.0000i 0.707472i 0.935345 + 0.353736i $$0.115089\pi$$
−0.935345 + 0.353736i $$0.884911\pi$$
$$968$$ 0 0
$$969$$ 32.0000 1.02799
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 0 0
$$973$$ − 32.0000i − 1.02587i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ 0 0
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ 0 0
$$983$$ 46.0000i 1.46717i 0.679597 + 0.733586i $$0.262155\pi$$
−0.679597 + 0.733586i $$0.737845\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 8.00000i − 0.254643i
$$988$$ 0 0
$$989$$ −12.0000 −0.381578
$$990$$ 0 0
$$991$$ −44.0000 −1.39771 −0.698853 0.715265i $$-0.746306\pi$$
−0.698853 + 0.715265i $$0.746306\pi$$
$$992$$ 0 0
$$993$$ 8.00000i 0.253872i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 18.0000i 0.570066i 0.958518 + 0.285033i $$0.0920045\pi$$
−0.958518 + 0.285033i $$0.907995\pi$$
$$998$$ 0 0
$$999$$ −8.00000 −0.253109
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 800.2.c.a.449.2 2
3.2 odd 2 7200.2.f.w.6049.1 2
4.3 odd 2 800.2.c.b.449.1 2
5.2 odd 4 800.2.a.i.1.1 1
5.3 odd 4 160.2.a.a.1.1 1
5.4 even 2 inner 800.2.c.a.449.1 2
8.3 odd 2 1600.2.c.c.449.2 2
8.5 even 2 1600.2.c.f.449.1 2
12.11 even 2 7200.2.f.g.6049.2 2
15.2 even 4 7200.2.a.bp.1.1 1
15.8 even 4 1440.2.a.i.1.1 1
15.14 odd 2 7200.2.f.w.6049.2 2
20.3 even 4 160.2.a.b.1.1 yes 1
20.7 even 4 800.2.a.a.1.1 1
20.19 odd 2 800.2.c.b.449.2 2
35.13 even 4 7840.2.a.w.1.1 1
40.3 even 4 320.2.a.b.1.1 1
40.13 odd 4 320.2.a.e.1.1 1
40.19 odd 2 1600.2.c.c.449.1 2
40.27 even 4 1600.2.a.t.1.1 1
40.29 even 2 1600.2.c.f.449.2 2
40.37 odd 4 1600.2.a.e.1.1 1
60.23 odd 4 1440.2.a.l.1.1 1
60.47 odd 4 7200.2.a.l.1.1 1
60.59 even 2 7200.2.f.g.6049.1 2
80.3 even 4 1280.2.d.b.641.2 2
80.13 odd 4 1280.2.d.h.641.1 2
80.43 even 4 1280.2.d.b.641.1 2
80.53 odd 4 1280.2.d.h.641.2 2
120.53 even 4 2880.2.a.d.1.1 1
120.83 odd 4 2880.2.a.o.1.1 1
140.83 odd 4 7840.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.a.a.1.1 1 5.3 odd 4
160.2.a.b.1.1 yes 1 20.3 even 4
320.2.a.b.1.1 1 40.3 even 4
320.2.a.e.1.1 1 40.13 odd 4
800.2.a.a.1.1 1 20.7 even 4
800.2.a.i.1.1 1 5.2 odd 4
800.2.c.a.449.1 2 5.4 even 2 inner
800.2.c.a.449.2 2 1.1 even 1 trivial
800.2.c.b.449.1 2 4.3 odd 2
800.2.c.b.449.2 2 20.19 odd 2
1280.2.d.b.641.1 2 80.43 even 4
1280.2.d.b.641.2 2 80.3 even 4
1280.2.d.h.641.1 2 80.13 odd 4
1280.2.d.h.641.2 2 80.53 odd 4
1440.2.a.i.1.1 1 15.8 even 4
1440.2.a.l.1.1 1 60.23 odd 4
1600.2.a.e.1.1 1 40.37 odd 4
1600.2.a.t.1.1 1 40.27 even 4
1600.2.c.c.449.1 2 40.19 odd 2
1600.2.c.c.449.2 2 8.3 odd 2
1600.2.c.f.449.1 2 8.5 even 2
1600.2.c.f.449.2 2 40.29 even 2
2880.2.a.d.1.1 1 120.53 even 4
2880.2.a.o.1.1 1 120.83 odd 4
7200.2.a.l.1.1 1 60.47 odd 4
7200.2.a.bp.1.1 1 15.2 even 4
7200.2.f.g.6049.1 2 60.59 even 2
7200.2.f.g.6049.2 2 12.11 even 2
7200.2.f.w.6049.1 2 3.2 odd 2
7200.2.f.w.6049.2 2 15.14 odd 2
7840.2.a.e.1.1 1 140.83 odd 4
7840.2.a.w.1.1 1 35.13 even 4