# Properties

 Label 2800.2.a.bi.1.2 Level $2800$ Weight $2$ Character 2800.1 Self dual yes Analytic conductor $22.358$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(1,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$22.3581125660$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 2800.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.56155 q^{3} -1.00000 q^{7} -0.561553 q^{9} +O(q^{10})$$ $$q+1.56155 q^{3} -1.00000 q^{7} -0.561553 q^{9} +1.56155 q^{11} -0.438447 q^{13} +0.438447 q^{17} +7.12311 q^{19} -1.56155 q^{21} +3.12311 q^{23} -5.56155 q^{27} +6.68466 q^{29} +2.43845 q^{33} -6.00000 q^{37} -0.684658 q^{39} +5.12311 q^{41} +0.876894 q^{43} -8.68466 q^{47} +1.00000 q^{49} +0.684658 q^{51} +5.12311 q^{53} +11.1231 q^{57} +4.00000 q^{59} +15.3693 q^{61} +0.561553 q^{63} +10.2462 q^{67} +4.87689 q^{69} -8.00000 q^{71} +12.2462 q^{73} -1.56155 q^{77} +2.43845 q^{79} -7.00000 q^{81} +4.00000 q^{83} +10.4384 q^{87} -1.12311 q^{89} +0.438447 q^{91} -5.80776 q^{97} -0.876894 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^7 + 3 * q^9 $$2 q - q^{3} - 2 q^{7} + 3 q^{9} - q^{11} - 5 q^{13} + 5 q^{17} + 6 q^{19} + q^{21} - 2 q^{23} - 7 q^{27} + q^{29} + 9 q^{33} - 12 q^{37} + 11 q^{39} + 2 q^{41} + 10 q^{43} - 5 q^{47} + 2 q^{49} - 11 q^{51} + 2 q^{53} + 14 q^{57} + 8 q^{59} + 6 q^{61} - 3 q^{63} + 4 q^{67} + 18 q^{69} - 16 q^{71} + 8 q^{73} + q^{77} + 9 q^{79} - 14 q^{81} + 8 q^{83} + 25 q^{87} + 6 q^{89} + 5 q^{91} + 9 q^{97} - 10 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^7 + 3 * q^9 - q^11 - 5 * q^13 + 5 * q^17 + 6 * q^19 + q^21 - 2 * q^23 - 7 * q^27 + q^29 + 9 * q^33 - 12 * q^37 + 11 * q^39 + 2 * q^41 + 10 * q^43 - 5 * q^47 + 2 * q^49 - 11 * q^51 + 2 * q^53 + 14 * q^57 + 8 * q^59 + 6 * q^61 - 3 * q^63 + 4 * q^67 + 18 * q^69 - 16 * q^71 + 8 * q^73 + q^77 + 9 * q^79 - 14 * q^81 + 8 * q^83 + 25 * q^87 + 6 * q^89 + 5 * q^91 + 9 * q^97 - 10 * q^99

## 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
$$3$$ 1.56155 0.901563 0.450781 0.892634i $$-0.351145\pi$$
0.450781 + 0.892634i $$0.351145\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ −0.561553 −0.187184
$$10$$ 0 0
$$11$$ 1.56155 0.470826 0.235413 0.971895i $$-0.424356\pi$$
0.235413 + 0.971895i $$0.424356\pi$$
$$12$$ 0 0
$$13$$ −0.438447 −0.121603 −0.0608017 0.998150i $$-0.519366\pi$$
−0.0608017 + 0.998150i $$0.519366\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0.438447 0.106339 0.0531695 0.998586i $$-0.483068\pi$$
0.0531695 + 0.998586i $$0.483068\pi$$
$$18$$ 0 0
$$19$$ 7.12311 1.63415 0.817076 0.576530i $$-0.195593\pi$$
0.817076 + 0.576530i $$0.195593\pi$$
$$20$$ 0 0
$$21$$ −1.56155 −0.340759
$$22$$ 0 0
$$23$$ 3.12311 0.651213 0.325606 0.945505i $$-0.394432\pi$$
0.325606 + 0.945505i $$0.394432\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.56155 −1.07032
$$28$$ 0 0
$$29$$ 6.68466 1.24131 0.620655 0.784084i $$-0.286867\pi$$
0.620655 + 0.784084i $$0.286867\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 2.43845 0.424479
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ −0.684658 −0.109633
$$40$$ 0 0
$$41$$ 5.12311 0.800095 0.400047 0.916494i $$-0.368994\pi$$
0.400047 + 0.916494i $$0.368994\pi$$
$$42$$ 0 0
$$43$$ 0.876894 0.133725 0.0668626 0.997762i $$-0.478701\pi$$
0.0668626 + 0.997762i $$0.478701\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −8.68466 −1.26679 −0.633394 0.773830i $$-0.718339\pi$$
−0.633394 + 0.773830i $$0.718339\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0.684658 0.0958714
$$52$$ 0 0
$$53$$ 5.12311 0.703713 0.351856 0.936054i $$-0.385551\pi$$
0.351856 + 0.936054i $$0.385551\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 11.1231 1.47329
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 15.3693 1.96784 0.983920 0.178611i $$-0.0571605\pi$$
0.983920 + 0.178611i $$0.0571605\pi$$
$$62$$ 0 0
$$63$$ 0.561553 0.0707490
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 10.2462 1.25177 0.625887 0.779914i $$-0.284737\pi$$
0.625887 + 0.779914i $$0.284737\pi$$
$$68$$ 0 0
$$69$$ 4.87689 0.587109
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 12.2462 1.43331 0.716655 0.697428i $$-0.245672\pi$$
0.716655 + 0.697428i $$0.245672\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1.56155 −0.177955
$$78$$ 0 0
$$79$$ 2.43845 0.274347 0.137173 0.990547i $$-0.456198\pi$$
0.137173 + 0.990547i $$0.456198\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 10.4384 1.11912
$$88$$ 0 0
$$89$$ −1.12311 −0.119049 −0.0595245 0.998227i $$-0.518958\pi$$
−0.0595245 + 0.998227i $$0.518958\pi$$
$$90$$ 0 0
$$91$$ 0.438447 0.0459618
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −5.80776 −0.589689 −0.294845 0.955545i $$-0.595268\pi$$
−0.294845 + 0.955545i $$0.595268\pi$$
$$98$$ 0 0
$$99$$ −0.876894 −0.0881312
$$100$$ 0 0
$$101$$ −16.2462 −1.61656 −0.808279 0.588799i $$-0.799601\pi$$
−0.808279 + 0.588799i $$0.799601\pi$$
$$102$$ 0 0
$$103$$ 5.56155 0.547996 0.273998 0.961730i $$-0.411654\pi$$
0.273998 + 0.961730i $$0.411654\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 13.3693 1.29246 0.646230 0.763142i $$-0.276345\pi$$
0.646230 + 0.763142i $$0.276345\pi$$
$$108$$ 0 0
$$109$$ 5.31534 0.509117 0.254559 0.967057i $$-0.418070\pi$$
0.254559 + 0.967057i $$0.418070\pi$$
$$110$$ 0 0
$$111$$ −9.36932 −0.889296
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0.246211 0.0227622
$$118$$ 0 0
$$119$$ −0.438447 −0.0401924
$$120$$ 0 0
$$121$$ −8.56155 −0.778323
$$122$$ 0 0
$$123$$ 8.00000 0.721336
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −6.24621 −0.554262 −0.277131 0.960832i $$-0.589384\pi$$
−0.277131 + 0.960832i $$0.589384\pi$$
$$128$$ 0 0
$$129$$ 1.36932 0.120562
$$130$$ 0 0
$$131$$ 0.876894 0.0766146 0.0383073 0.999266i $$-0.487803\pi$$
0.0383073 + 0.999266i $$0.487803\pi$$
$$132$$ 0 0
$$133$$ −7.12311 −0.617652
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 17.1231 1.46293 0.731463 0.681881i $$-0.238838\pi$$
0.731463 + 0.681881i $$0.238838\pi$$
$$138$$ 0 0
$$139$$ 15.1231 1.28273 0.641363 0.767238i $$-0.278369\pi$$
0.641363 + 0.767238i $$0.278369\pi$$
$$140$$ 0 0
$$141$$ −13.5616 −1.14209
$$142$$ 0 0
$$143$$ −0.684658 −0.0572540
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.56155 0.128795
$$148$$ 0 0
$$149$$ 12.2462 1.00325 0.501624 0.865086i $$-0.332736\pi$$
0.501624 + 0.865086i $$0.332736\pi$$
$$150$$ 0 0
$$151$$ 6.93087 0.564026 0.282013 0.959411i $$-0.408998\pi$$
0.282013 + 0.959411i $$0.408998\pi$$
$$152$$ 0 0
$$153$$ −0.246211 −0.0199050
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −20.2462 −1.61582 −0.807912 0.589303i $$-0.799402\pi$$
−0.807912 + 0.589303i $$0.799402\pi$$
$$158$$ 0 0
$$159$$ 8.00000 0.634441
$$160$$ 0 0
$$161$$ −3.12311 −0.246135
$$162$$ 0 0
$$163$$ −7.12311 −0.557925 −0.278962 0.960302i $$-0.589990\pi$$
−0.278962 + 0.960302i $$0.589990\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −6.93087 −0.536327 −0.268163 0.963373i $$-0.586417\pi$$
−0.268163 + 0.963373i $$0.586417\pi$$
$$168$$ 0 0
$$169$$ −12.8078 −0.985213
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 0 0
$$173$$ 4.43845 0.337449 0.168724 0.985663i $$-0.446035\pi$$
0.168724 + 0.985663i $$0.446035\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.24621 0.469494
$$178$$ 0 0
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ −17.6155 −1.30935 −0.654676 0.755910i $$-0.727195\pi$$
−0.654676 + 0.755910i $$0.727195\pi$$
$$182$$ 0 0
$$183$$ 24.0000 1.77413
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.684658 0.0500672
$$188$$ 0 0
$$189$$ 5.56155 0.404543
$$190$$ 0 0
$$191$$ 13.5616 0.981280 0.490640 0.871363i $$-0.336763\pi$$
0.490640 + 0.871363i $$0.336763\pi$$
$$192$$ 0 0
$$193$$ −19.3693 −1.39423 −0.697117 0.716957i $$-0.745534\pi$$
−0.697117 + 0.716957i $$0.745534\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1.12311 −0.0800180 −0.0400090 0.999199i $$-0.512739\pi$$
−0.0400090 + 0.999199i $$0.512739\pi$$
$$198$$ 0 0
$$199$$ 1.75379 0.124323 0.0621614 0.998066i $$-0.480201\pi$$
0.0621614 + 0.998066i $$0.480201\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ 0 0
$$203$$ −6.68466 −0.469171
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −1.75379 −0.121897
$$208$$ 0 0
$$209$$ 11.1231 0.769401
$$210$$ 0 0
$$211$$ −14.0540 −0.967516 −0.483758 0.875202i $$-0.660728\pi$$
−0.483758 + 0.875202i $$0.660728\pi$$
$$212$$ 0 0
$$213$$ −12.4924 −0.855967
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 19.1231 1.29222
$$220$$ 0 0
$$221$$ −0.192236 −0.0129312
$$222$$ 0 0
$$223$$ −2.43845 −0.163291 −0.0816453 0.996661i $$-0.526017\pi$$
−0.0816453 + 0.996661i $$0.526017\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 11.3153 0.751026 0.375513 0.926817i $$-0.377467\pi$$
0.375513 + 0.926817i $$0.377467\pi$$
$$228$$ 0 0
$$229$$ 10.8769 0.718765 0.359383 0.933190i $$-0.382987\pi$$
0.359383 + 0.933190i $$0.382987\pi$$
$$230$$ 0 0
$$231$$ −2.43845 −0.160438
$$232$$ 0 0
$$233$$ −5.12311 −0.335626 −0.167813 0.985819i $$-0.553670\pi$$
−0.167813 + 0.985819i $$0.553670\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 3.80776 0.247341
$$238$$ 0 0
$$239$$ −19.8078 −1.28126 −0.640629 0.767851i $$-0.721326\pi$$
−0.640629 + 0.767851i $$0.721326\pi$$
$$240$$ 0 0
$$241$$ −4.24621 −0.273523 −0.136761 0.990604i $$-0.543669\pi$$
−0.136761 + 0.990604i $$0.543669\pi$$
$$242$$ 0 0
$$243$$ 5.75379 0.369106
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3.12311 −0.198718
$$248$$ 0 0
$$249$$ 6.24621 0.395838
$$250$$ 0 0
$$251$$ 8.87689 0.560305 0.280152 0.959956i $$-0.409615\pi$$
0.280152 + 0.959956i $$0.409615\pi$$
$$252$$ 0 0
$$253$$ 4.87689 0.306608
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 10.4924 0.654499 0.327250 0.944938i $$-0.393878\pi$$
0.327250 + 0.944938i $$0.393878\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ −3.75379 −0.232354
$$262$$ 0 0
$$263$$ −12.8769 −0.794023 −0.397012 0.917814i $$-0.629953\pi$$
−0.397012 + 0.917814i $$0.629953\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −1.75379 −0.107330
$$268$$ 0 0
$$269$$ −20.7386 −1.26446 −0.632228 0.774782i $$-0.717860\pi$$
−0.632228 + 0.774782i $$0.717860\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ 0.684658 0.0414374
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0.246211 0.0147934 0.00739670 0.999973i $$-0.497646\pi$$
0.00739670 + 0.999973i $$0.497646\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.4384 0.742016 0.371008 0.928630i $$-0.379012\pi$$
0.371008 + 0.928630i $$0.379012\pi$$
$$282$$ 0 0
$$283$$ −11.3153 −0.672627 −0.336314 0.941750i $$-0.609180\pi$$
−0.336314 + 0.941750i $$0.609180\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −5.12311 −0.302407
$$288$$ 0 0
$$289$$ −16.8078 −0.988692
$$290$$ 0 0
$$291$$ −9.06913 −0.531642
$$292$$ 0 0
$$293$$ 2.68466 0.156839 0.0784197 0.996920i $$-0.475013\pi$$
0.0784197 + 0.996920i $$0.475013\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −8.68466 −0.503935
$$298$$ 0 0
$$299$$ −1.36932 −0.0791896
$$300$$ 0 0
$$301$$ −0.876894 −0.0505434
$$302$$ 0 0
$$303$$ −25.3693 −1.45743
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −19.3153 −1.10238 −0.551192 0.834378i $$-0.685827\pi$$
−0.551192 + 0.834378i $$0.685827\pi$$
$$308$$ 0 0
$$309$$ 8.68466 0.494053
$$310$$ 0 0
$$311$$ −31.6155 −1.79275 −0.896376 0.443294i $$-0.853810\pi$$
−0.896376 + 0.443294i $$0.853810\pi$$
$$312$$ 0 0
$$313$$ 22.3002 1.26048 0.630241 0.776400i $$-0.282956\pi$$
0.630241 + 0.776400i $$0.282956\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −10.4924 −0.589313 −0.294657 0.955603i $$-0.595205\pi$$
−0.294657 + 0.955603i $$0.595205\pi$$
$$318$$ 0 0
$$319$$ 10.4384 0.584441
$$320$$ 0 0
$$321$$ 20.8769 1.16523
$$322$$ 0 0
$$323$$ 3.12311 0.173774
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 8.30019 0.459001
$$328$$ 0 0
$$329$$ 8.68466 0.478801
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 0 0
$$333$$ 3.36932 0.184637
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.50758 0.0821230 0.0410615 0.999157i $$-0.486926\pi$$
0.0410615 + 0.999157i $$0.486926\pi$$
$$338$$ 0 0
$$339$$ 21.8617 1.18737
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 7.12311 0.382388 0.191194 0.981552i $$-0.438764\pi$$
0.191194 + 0.981552i $$0.438764\pi$$
$$348$$ 0 0
$$349$$ 10.4924 0.561646 0.280823 0.959760i $$-0.409393\pi$$
0.280823 + 0.959760i $$0.409393\pi$$
$$350$$ 0 0
$$351$$ 2.43845 0.130155
$$352$$ 0 0
$$353$$ −5.80776 −0.309116 −0.154558 0.987984i $$-0.549395\pi$$
−0.154558 + 0.987984i $$0.549395\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −0.684658 −0.0362360
$$358$$ 0 0
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ 31.7386 1.67045
$$362$$ 0 0
$$363$$ −13.3693 −0.701707
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.68466 −0.453335 −0.226668 0.973972i $$-0.572783\pi$$
−0.226668 + 0.973972i $$0.572783\pi$$
$$368$$ 0 0
$$369$$ −2.87689 −0.149765
$$370$$ 0 0
$$371$$ −5.12311 −0.265978
$$372$$ 0 0
$$373$$ −4.63068 −0.239768 −0.119884 0.992788i $$-0.538252\pi$$
−0.119884 + 0.992788i $$0.538252\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −2.93087 −0.150947
$$378$$ 0 0
$$379$$ 16.4924 0.847159 0.423579 0.905859i $$-0.360773\pi$$
0.423579 + 0.905859i $$0.360773\pi$$
$$380$$ 0 0
$$381$$ −9.75379 −0.499702
$$382$$ 0 0
$$383$$ 6.24621 0.319166 0.159583 0.987184i $$-0.448985\pi$$
0.159583 + 0.987184i $$0.448985\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −0.492423 −0.0250312
$$388$$ 0 0
$$389$$ −24.9309 −1.26405 −0.632023 0.774950i $$-0.717775\pi$$
−0.632023 + 0.774950i $$0.717775\pi$$
$$390$$ 0 0
$$391$$ 1.36932 0.0692493
$$392$$ 0 0
$$393$$ 1.36932 0.0690729
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −27.5616 −1.38327 −0.691637 0.722245i $$-0.743110\pi$$
−0.691637 + 0.722245i $$0.743110\pi$$
$$398$$ 0 0
$$399$$ −11.1231 −0.556852
$$400$$ 0 0
$$401$$ 31.5616 1.57611 0.788054 0.615606i $$-0.211089\pi$$
0.788054 + 0.615606i $$0.211089\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −9.36932 −0.464420
$$408$$ 0 0
$$409$$ 6.49242 0.321030 0.160515 0.987033i $$-0.448685\pi$$
0.160515 + 0.987033i $$0.448685\pi$$
$$410$$ 0 0
$$411$$ 26.7386 1.31892
$$412$$ 0 0
$$413$$ −4.00000 −0.196827
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 23.6155 1.15646
$$418$$ 0 0
$$419$$ −26.2462 −1.28221 −0.641106 0.767453i $$-0.721524\pi$$
−0.641106 + 0.767453i $$0.721524\pi$$
$$420$$ 0 0
$$421$$ −2.68466 −0.130842 −0.0654211 0.997858i $$-0.520839\pi$$
−0.0654211 + 0.997858i $$0.520839\pi$$
$$422$$ 0 0
$$423$$ 4.87689 0.237123
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −15.3693 −0.743773
$$428$$ 0 0
$$429$$ −1.06913 −0.0516181
$$430$$ 0 0
$$431$$ 19.8078 0.954106 0.477053 0.878874i $$-0.341705\pi$$
0.477053 + 0.878874i $$0.341705\pi$$
$$432$$ 0 0
$$433$$ −8.24621 −0.396288 −0.198144 0.980173i $$-0.563491\pi$$
−0.198144 + 0.980173i $$0.563491\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 22.2462 1.06418
$$438$$ 0 0
$$439$$ −9.36932 −0.447173 −0.223587 0.974684i $$-0.571777\pi$$
−0.223587 + 0.974684i $$0.571777\pi$$
$$440$$ 0 0
$$441$$ −0.561553 −0.0267406
$$442$$ 0 0
$$443$$ −2.63068 −0.124988 −0.0624938 0.998045i $$-0.519905\pi$$
−0.0624938 + 0.998045i $$0.519905\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 19.1231 0.904492
$$448$$ 0 0
$$449$$ −1.80776 −0.0853137 −0.0426568 0.999090i $$-0.513582\pi$$
−0.0426568 + 0.999090i $$0.513582\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 0 0
$$453$$ 10.8229 0.508505
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17.1231 0.800985 0.400493 0.916300i $$-0.368839\pi$$
0.400493 + 0.916300i $$0.368839\pi$$
$$458$$ 0 0
$$459$$ −2.43845 −0.113817
$$460$$ 0 0
$$461$$ −13.1231 −0.611204 −0.305602 0.952159i $$-0.598858\pi$$
−0.305602 + 0.952159i $$0.598858\pi$$
$$462$$ 0 0
$$463$$ 12.4924 0.580572 0.290286 0.956940i $$-0.406250\pi$$
0.290286 + 0.956940i $$0.406250\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 22.4384 1.03833 0.519164 0.854675i $$-0.326243\pi$$
0.519164 + 0.854675i $$0.326243\pi$$
$$468$$ 0 0
$$469$$ −10.2462 −0.473126
$$470$$ 0 0
$$471$$ −31.6155 −1.45677
$$472$$ 0 0
$$473$$ 1.36932 0.0629613
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −2.87689 −0.131724
$$478$$ 0 0
$$479$$ −4.87689 −0.222831 −0.111415 0.993774i $$-0.535538\pi$$
−0.111415 + 0.993774i $$0.535538\pi$$
$$480$$ 0 0
$$481$$ 2.63068 0.119949
$$482$$ 0 0
$$483$$ −4.87689 −0.221906
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −3.12311 −0.141521 −0.0707607 0.997493i $$-0.522543\pi$$
−0.0707607 + 0.997493i $$0.522543\pi$$
$$488$$ 0 0
$$489$$ −11.1231 −0.503004
$$490$$ 0 0
$$491$$ 41.1771 1.85830 0.929148 0.369708i $$-0.120542\pi$$
0.929148 + 0.369708i $$0.120542\pi$$
$$492$$ 0 0
$$493$$ 2.93087 0.132000
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.00000 0.358849
$$498$$ 0 0
$$499$$ −41.1771 −1.84334 −0.921670 0.387976i $$-0.873174\pi$$
−0.921670 + 0.387976i $$0.873174\pi$$
$$500$$ 0 0
$$501$$ −10.8229 −0.483532
$$502$$ 0 0
$$503$$ 38.9309 1.73584 0.867921 0.496703i $$-0.165456\pi$$
0.867921 + 0.496703i $$0.165456\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −20.0000 −0.888231
$$508$$ 0 0
$$509$$ −11.7538 −0.520978 −0.260489 0.965477i $$-0.583884\pi$$
−0.260489 + 0.965477i $$0.583884\pi$$
$$510$$ 0 0
$$511$$ −12.2462 −0.541740
$$512$$ 0 0
$$513$$ −39.6155 −1.74907
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −13.5616 −0.596436
$$518$$ 0 0
$$519$$ 6.93087 0.304231
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 0 0
$$523$$ 40.4924 1.77061 0.885305 0.465011i $$-0.153950\pi$$
0.885305 + 0.465011i $$0.153950\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −13.2462 −0.575922
$$530$$ 0 0
$$531$$ −2.24621 −0.0974773
$$532$$ 0 0
$$533$$ −2.24621 −0.0972942
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −31.2311 −1.34772
$$538$$ 0 0
$$539$$ 1.56155 0.0672608
$$540$$ 0 0
$$541$$ −37.8078 −1.62548 −0.812741 0.582625i $$-0.802026\pi$$
−0.812741 + 0.582625i $$0.802026\pi$$
$$542$$ 0 0
$$543$$ −27.5076 −1.18046
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −2.24621 −0.0960411 −0.0480205 0.998846i $$-0.515291\pi$$
−0.0480205 + 0.998846i $$0.515291\pi$$
$$548$$ 0 0
$$549$$ −8.63068 −0.368349
$$550$$ 0 0
$$551$$ 47.6155 2.02849
$$552$$ 0 0
$$553$$ −2.43845 −0.103693
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 13.1231 0.556044 0.278022 0.960575i $$-0.410321\pi$$
0.278022 + 0.960575i $$0.410321\pi$$
$$558$$ 0 0
$$559$$ −0.384472 −0.0162614
$$560$$ 0 0
$$561$$ 1.06913 0.0451387
$$562$$ 0 0
$$563$$ −28.0000 −1.18006 −0.590030 0.807382i $$-0.700884\pi$$
−0.590030 + 0.807382i $$0.700884\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 7.00000 0.293972
$$568$$ 0 0
$$569$$ −30.9848 −1.29895 −0.649476 0.760382i $$-0.725012\pi$$
−0.649476 + 0.760382i $$0.725012\pi$$
$$570$$ 0 0
$$571$$ −40.4924 −1.69456 −0.847278 0.531150i $$-0.821760\pi$$
−0.847278 + 0.531150i $$0.821760\pi$$
$$572$$ 0 0
$$573$$ 21.1771 0.884685
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 24.0540 1.00138 0.500690 0.865627i $$-0.333080\pi$$
0.500690 + 0.865627i $$0.333080\pi$$
$$578$$ 0 0
$$579$$ −30.2462 −1.25699
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 0 0
$$583$$ 8.00000 0.331326
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −26.2462 −1.08330 −0.541649 0.840605i $$-0.682200\pi$$
−0.541649 + 0.840605i $$0.682200\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −1.75379 −0.0721412
$$592$$ 0 0
$$593$$ 27.5616 1.13182 0.565909 0.824468i $$-0.308525\pi$$
0.565909 + 0.824468i $$0.308525\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 2.73863 0.112085
$$598$$ 0 0
$$599$$ 11.8078 0.482452 0.241226 0.970469i $$-0.422450\pi$$
0.241226 + 0.970469i $$0.422450\pi$$
$$600$$ 0 0
$$601$$ 6.49242 0.264831 0.132416 0.991194i $$-0.457727\pi$$
0.132416 + 0.991194i $$0.457727\pi$$
$$602$$ 0 0
$$603$$ −5.75379 −0.234312
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −42.0540 −1.70692 −0.853459 0.521160i $$-0.825500\pi$$
−0.853459 + 0.521160i $$0.825500\pi$$
$$608$$ 0 0
$$609$$ −10.4384 −0.422987
$$610$$ 0 0
$$611$$ 3.80776 0.154046
$$612$$ 0 0
$$613$$ −40.7386 −1.64542 −0.822709 0.568463i $$-0.807538\pi$$
−0.822709 + 0.568463i $$0.807538\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −32.2462 −1.29818 −0.649092 0.760710i $$-0.724851\pi$$
−0.649092 + 0.760710i $$0.724851\pi$$
$$618$$ 0 0
$$619$$ −32.1080 −1.29053 −0.645264 0.763960i $$-0.723253\pi$$
−0.645264 + 0.763960i $$0.723253\pi$$
$$620$$ 0 0
$$621$$ −17.3693 −0.697007
$$622$$ 0 0
$$623$$ 1.12311 0.0449963
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 17.3693 0.693664
$$628$$ 0 0
$$629$$ −2.63068 −0.104892
$$630$$ 0 0
$$631$$ 11.8078 0.470060 0.235030 0.971988i $$-0.424481\pi$$
0.235030 + 0.971988i $$0.424481\pi$$
$$632$$ 0 0
$$633$$ −21.9460 −0.872276
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −0.438447 −0.0173719
$$638$$ 0 0
$$639$$ 4.49242 0.177717
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ 1.56155 0.0615816 0.0307908 0.999526i $$-0.490197\pi$$
0.0307908 + 0.999526i $$0.490197\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 36.4924 1.43467 0.717333 0.696731i $$-0.245363\pi$$
0.717333 + 0.696731i $$0.245363\pi$$
$$648$$ 0 0
$$649$$ 6.24621 0.245185
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 33.2311 1.30043 0.650216 0.759750i $$-0.274678\pi$$
0.650216 + 0.759750i $$0.274678\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −6.87689 −0.268293
$$658$$ 0 0
$$659$$ −9.17708 −0.357488 −0.178744 0.983896i $$-0.557203\pi$$
−0.178744 + 0.983896i $$0.557203\pi$$
$$660$$ 0 0
$$661$$ −5.12311 −0.199266 −0.0996329 0.995024i $$-0.531767\pi$$
−0.0996329 + 0.995024i $$0.531767\pi$$
$$662$$ 0 0
$$663$$ −0.300187 −0.0116583
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 20.8769 0.808357
$$668$$ 0 0
$$669$$ −3.80776 −0.147217
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ −31.8617 −1.22818 −0.614090 0.789236i $$-0.710477\pi$$
−0.614090 + 0.789236i $$0.710477\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −4.93087 −0.189509 −0.0947544 0.995501i $$-0.530207\pi$$
−0.0947544 + 0.995501i $$0.530207\pi$$
$$678$$ 0 0
$$679$$ 5.80776 0.222882
$$680$$ 0 0
$$681$$ 17.6695 0.677097
$$682$$ 0 0
$$683$$ −6.73863 −0.257847 −0.128923 0.991655i $$-0.541152\pi$$
−0.128923 + 0.991655i $$0.541152\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 16.9848 0.648012
$$688$$ 0 0
$$689$$ −2.24621 −0.0855738
$$690$$ 0 0
$$691$$ 24.4924 0.931736 0.465868 0.884854i $$-0.345742\pi$$
0.465868 + 0.884854i $$0.345742\pi$$
$$692$$ 0 0
$$693$$ 0.876894 0.0333105
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.24621 0.0850813
$$698$$ 0 0
$$699$$ −8.00000 −0.302588
$$700$$ 0 0
$$701$$ 28.9309 1.09270 0.546352 0.837556i $$-0.316016\pi$$
0.546352 + 0.837556i $$0.316016\pi$$
$$702$$ 0 0
$$703$$ −42.7386 −1.61192
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 16.2462 0.611002
$$708$$ 0 0
$$709$$ 27.1771 1.02066 0.510328 0.859980i $$-0.329524\pi$$
0.510328 + 0.859980i $$0.329524\pi$$
$$710$$ 0 0
$$711$$ −1.36932 −0.0513534
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −30.9309 −1.15513
$$718$$ 0 0
$$719$$ −8.38447 −0.312688 −0.156344 0.987703i $$-0.549971\pi$$
−0.156344 + 0.987703i $$0.549971\pi$$
$$720$$ 0 0
$$721$$ −5.56155 −0.207123
$$722$$ 0 0
$$723$$ −6.63068 −0.246598
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 52.4924 1.94684 0.973418 0.229035i $$-0.0735572\pi$$
0.973418 + 0.229035i $$0.0735572\pi$$
$$728$$ 0 0
$$729$$ 29.9848 1.11055
$$730$$ 0 0
$$731$$ 0.384472 0.0142202
$$732$$ 0 0
$$733$$ −6.68466 −0.246903 −0.123452 0.992351i $$-0.539396\pi$$
−0.123452 + 0.992351i $$0.539396\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ −34.9309 −1.28495 −0.642476 0.766305i $$-0.722093\pi$$
−0.642476 + 0.766305i $$0.722093\pi$$
$$740$$ 0 0
$$741$$ −4.87689 −0.179157
$$742$$ 0 0
$$743$$ −32.9848 −1.21010 −0.605048 0.796189i $$-0.706846\pi$$
−0.605048 + 0.796189i $$0.706846\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −2.24621 −0.0821846
$$748$$ 0 0
$$749$$ −13.3693 −0.488504
$$750$$ 0 0
$$751$$ −17.0691 −0.622861 −0.311431 0.950269i $$-0.600808\pi$$
−0.311431 + 0.950269i $$0.600808\pi$$
$$752$$ 0 0
$$753$$ 13.8617 0.505150
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −39.3693 −1.43090 −0.715451 0.698663i $$-0.753779\pi$$
−0.715451 + 0.698663i $$0.753779\pi$$
$$758$$ 0 0
$$759$$ 7.61553 0.276426
$$760$$ 0 0
$$761$$ 48.2462 1.74892 0.874462 0.485094i $$-0.161215\pi$$
0.874462 + 0.485094i $$0.161215\pi$$
$$762$$ 0 0
$$763$$ −5.31534 −0.192428
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −1.75379 −0.0633256
$$768$$ 0 0
$$769$$ −42.4924 −1.53232 −0.766158 0.642652i $$-0.777834\pi$$
−0.766158 + 0.642652i $$0.777834\pi$$
$$770$$ 0 0
$$771$$ 16.3845 0.590072
$$772$$ 0 0
$$773$$ −36.9309 −1.32831 −0.664156 0.747594i $$-0.731209\pi$$
−0.664156 + 0.747594i $$0.731209\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 9.36932 0.336122
$$778$$ 0 0
$$779$$ 36.4924 1.30748
$$780$$ 0 0
$$781$$ −12.4924 −0.447014
$$782$$ 0 0
$$783$$ −37.1771 −1.32860
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −49.1771 −1.75297 −0.876487 0.481426i $$-0.840119\pi$$
−0.876487 + 0.481426i $$0.840119\pi$$
$$788$$ 0 0
$$789$$ −20.1080 −0.715862
$$790$$ 0 0
$$791$$ −14.0000 −0.497783
$$792$$ 0 0
$$793$$ −6.73863 −0.239296
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −24.0540 −0.852036 −0.426018 0.904715i $$-0.640084\pi$$
−0.426018 + 0.904715i $$0.640084\pi$$
$$798$$ 0 0
$$799$$ −3.80776 −0.134709
$$800$$ 0 0
$$801$$ 0.630683 0.0222841
$$802$$ 0 0
$$803$$ 19.1231 0.674840
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −32.3845 −1.13999
$$808$$ 0 0
$$809$$ 16.5464 0.581740 0.290870 0.956763i $$-0.406055\pi$$
0.290870 + 0.956763i $$0.406055\pi$$
$$810$$ 0 0
$$811$$ −19.6155 −0.688794 −0.344397 0.938824i $$-0.611917\pi$$
−0.344397 + 0.938824i $$0.611917\pi$$
$$812$$ 0 0
$$813$$ 24.9848 0.876257
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 6.24621 0.218527
$$818$$ 0 0
$$819$$ −0.246211 −0.00860332
$$820$$ 0 0
$$821$$ −21.4233 −0.747678 −0.373839 0.927494i $$-0.621959\pi$$
−0.373839 + 0.927494i $$0.621959\pi$$
$$822$$ 0 0
$$823$$ −36.4924 −1.27205 −0.636023 0.771670i $$-0.719422\pi$$
−0.636023 + 0.771670i $$0.719422\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −5.36932 −0.186709 −0.0933547 0.995633i $$-0.529759\pi$$
−0.0933547 + 0.995633i $$0.529759\pi$$
$$828$$ 0 0
$$829$$ 34.8769 1.21132 0.605662 0.795722i $$-0.292908\pi$$
0.605662 + 0.795722i $$0.292908\pi$$
$$830$$ 0 0
$$831$$ 0.384472 0.0133372
$$832$$ 0 0
$$833$$ 0.438447 0.0151913
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 28.8769 0.996941 0.498471 0.866907i $$-0.333895\pi$$
0.498471 + 0.866907i $$0.333895\pi$$
$$840$$ 0 0
$$841$$ 15.6847 0.540850
$$842$$ 0 0
$$843$$ 19.4233 0.668974
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 8.56155 0.294178
$$848$$ 0 0
$$849$$ −17.6695 −0.606416
$$850$$ 0 0
$$851$$ −18.7386 −0.642352
$$852$$ 0 0
$$853$$ 7.26137 0.248624 0.124312 0.992243i $$-0.460328\pi$$
0.124312 + 0.992243i $$0.460328\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 15.7538 0.538139 0.269070 0.963121i $$-0.413284\pi$$
0.269070 + 0.963121i $$0.413284\pi$$
$$858$$ 0 0
$$859$$ 16.4924 0.562714 0.281357 0.959603i $$-0.409215\pi$$
0.281357 + 0.959603i $$0.409215\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ −25.7538 −0.876669 −0.438335 0.898812i $$-0.644431\pi$$
−0.438335 + 0.898812i $$0.644431\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −26.2462 −0.891368
$$868$$ 0 0
$$869$$ 3.80776 0.129170
$$870$$ 0 0
$$871$$ −4.49242 −0.152220
$$872$$ 0 0
$$873$$ 3.26137 0.110381
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 40.2462 1.35902 0.679509 0.733667i $$-0.262193\pi$$
0.679509 + 0.733667i $$0.262193\pi$$
$$878$$ 0 0
$$879$$ 4.19224 0.141401
$$880$$ 0 0
$$881$$ −11.8617 −0.399632 −0.199816 0.979833i $$-0.564034\pi$$
−0.199816 + 0.979833i $$0.564034\pi$$
$$882$$ 0 0
$$883$$ −8.49242 −0.285793 −0.142896 0.989738i $$-0.545642\pi$$
−0.142896 + 0.989738i $$0.545642\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 20.4924 0.688068 0.344034 0.938957i $$-0.388206\pi$$
0.344034 + 0.938957i $$0.388206\pi$$
$$888$$ 0 0
$$889$$ 6.24621 0.209491
$$890$$ 0 0
$$891$$ −10.9309 −0.366198
$$892$$ 0 0
$$893$$ −61.8617 −2.07012
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −2.13826 −0.0713944
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 2.24621 0.0748321
$$902$$ 0 0
$$903$$ −1.36932 −0.0455680
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −24.1080 −0.800491 −0.400246 0.916408i $$-0.631075\pi$$
−0.400246 + 0.916408i $$0.631075\pi$$
$$908$$ 0 0
$$909$$ 9.12311 0.302594
$$910$$ 0 0
$$911$$ 28.4924 0.943996 0.471998 0.881600i $$-0.343533\pi$$
0.471998 + 0.881600i $$0.343533\pi$$
$$912$$ 0 0
$$913$$ 6.24621 0.206719
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −0.876894 −0.0289576
$$918$$ 0 0
$$919$$ −40.3002 −1.32938 −0.664690 0.747119i $$-0.731436\pi$$
−0.664690 + 0.747119i $$0.731436\pi$$
$$920$$ 0 0
$$921$$ −30.1619 −0.993869
$$922$$ 0 0
$$923$$ 3.50758 0.115453
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −3.12311 −0.102576
$$928$$ 0 0
$$929$$ 22.1080 0.725338 0.362669 0.931918i $$-0.381866\pi$$
0.362669 + 0.931918i $$0.381866\pi$$
$$930$$ 0 0
$$931$$ 7.12311 0.233450
$$932$$ 0 0
$$933$$ −49.3693 −1.61628
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 55.6695 1.81864 0.909322 0.416094i $$-0.136601\pi$$
0.909322 + 0.416094i $$0.136601\pi$$
$$938$$ 0 0
$$939$$ 34.8229 1.13640
$$940$$ 0 0
$$941$$ 43.8617 1.42985 0.714926 0.699200i $$-0.246460\pi$$
0.714926 + 0.699200i $$0.246460\pi$$
$$942$$ 0 0
$$943$$ 16.0000 0.521032
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ 0 0
$$949$$ −5.36932 −0.174295
$$950$$ 0 0
$$951$$ −16.3845 −0.531303
$$952$$ 0 0
$$953$$ 33.1231 1.07296 0.536481 0.843912i $$-0.319753\pi$$
0.536481 + 0.843912i $$0.319753\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 16.3002 0.526910
$$958$$ 0 0
$$959$$ −17.1231 −0.552934
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −7.50758 −0.241928
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −35.1231 −1.12948 −0.564741 0.825268i $$-0.691024\pi$$
−0.564741 + 0.825268i $$0.691024\pi$$
$$968$$ 0 0
$$969$$ 4.87689 0.156668
$$970$$ 0 0
$$971$$ −49.4773 −1.58780 −0.793901 0.608048i $$-0.791953\pi$$
−0.793901 + 0.608048i $$0.791953\pi$$
$$972$$ 0 0
$$973$$ −15.1231 −0.484825
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −33.2311 −1.06316 −0.531578 0.847009i $$-0.678401\pi$$
−0.531578 + 0.847009i $$0.678401\pi$$
$$978$$ 0 0
$$979$$ −1.75379 −0.0560513
$$980$$ 0 0
$$981$$ −2.98485 −0.0952988
$$982$$ 0 0
$$983$$ 51.4233 1.64015 0.820074 0.572257i $$-0.193932\pi$$
0.820074 + 0.572257i $$0.193932\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 13.5616 0.431669
$$988$$ 0 0
$$989$$ 2.73863 0.0870835
$$990$$ 0 0
$$991$$ −12.4924 −0.396835 −0.198417 0.980118i $$-0.563580\pi$$
−0.198417 + 0.980118i $$0.563580\pi$$
$$992$$ 0 0
$$993$$ −18.7386 −0.594653
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 2.68466 0.0850240 0.0425120 0.999096i $$-0.486464\pi$$
0.0425120 + 0.999096i $$0.486464\pi$$
$$998$$ 0 0
$$999$$ 33.3693 1.05576
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 2800.2.a.bi.1.2 2
4.3 odd 2 175.2.a.f.1.2 2
5.2 odd 4 2800.2.g.t.449.2 4
5.3 odd 4 2800.2.g.t.449.3 4
5.4 even 2 560.2.a.i.1.1 2
12.11 even 2 1575.2.a.p.1.1 2
15.14 odd 2 5040.2.a.bt.1.1 2
20.3 even 4 175.2.b.b.99.1 4
20.7 even 4 175.2.b.b.99.4 4
20.19 odd 2 35.2.a.b.1.1 2
28.27 even 2 1225.2.a.s.1.2 2
35.34 odd 2 3920.2.a.bs.1.2 2
40.19 odd 2 2240.2.a.bh.1.1 2
40.29 even 2 2240.2.a.bd.1.2 2
60.23 odd 4 1575.2.d.e.1324.4 4
60.47 odd 4 1575.2.d.e.1324.1 4
60.59 even 2 315.2.a.e.1.2 2
140.19 even 6 245.2.e.h.116.2 4
140.27 odd 4 1225.2.b.f.99.4 4
140.39 odd 6 245.2.e.i.226.2 4
140.59 even 6 245.2.e.h.226.2 4
140.79 odd 6 245.2.e.i.116.2 4
140.83 odd 4 1225.2.b.f.99.1 4
140.139 even 2 245.2.a.d.1.1 2
220.219 even 2 4235.2.a.m.1.2 2
260.259 odd 2 5915.2.a.l.1.2 2
420.419 odd 2 2205.2.a.x.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.1 2 20.19 odd 2
175.2.a.f.1.2 2 4.3 odd 2
175.2.b.b.99.1 4 20.3 even 4
175.2.b.b.99.4 4 20.7 even 4
245.2.a.d.1.1 2 140.139 even 2
245.2.e.h.116.2 4 140.19 even 6
245.2.e.h.226.2 4 140.59 even 6
245.2.e.i.116.2 4 140.79 odd 6
245.2.e.i.226.2 4 140.39 odd 6
315.2.a.e.1.2 2 60.59 even 2
560.2.a.i.1.1 2 5.4 even 2
1225.2.a.s.1.2 2 28.27 even 2
1225.2.b.f.99.1 4 140.83 odd 4
1225.2.b.f.99.4 4 140.27 odd 4
1575.2.a.p.1.1 2 12.11 even 2
1575.2.d.e.1324.1 4 60.47 odd 4
1575.2.d.e.1324.4 4 60.23 odd 4
2205.2.a.x.1.2 2 420.419 odd 2
2240.2.a.bd.1.2 2 40.29 even 2
2240.2.a.bh.1.1 2 40.19 odd 2
2800.2.a.bi.1.2 2 1.1 even 1 trivial
2800.2.g.t.449.2 4 5.2 odd 4
2800.2.g.t.449.3 4 5.3 odd 4
3920.2.a.bs.1.2 2 35.34 odd 2
4235.2.a.m.1.2 2 220.219 even 2
5040.2.a.bt.1.1 2 15.14 odd 2
5915.2.a.l.1.2 2 260.259 odd 2