# Properties

 Label 9800.2.a.bt.1.2 Level $9800$ Weight $2$ Character 9800.1 Self dual yes Analytic conductor $78.253$ 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] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, 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("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9800 = 2^{3} \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9800.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: $$78.2533939809$$ 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 1400) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 9800.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.56155 q^{3} -0.561553 q^{9} +O(q^{10})$$ $$q+1.56155 q^{3} -0.561553 q^{9} -6.12311 q^{11} -2.00000 q^{13} -1.56155 q^{17} -3.56155 q^{19} +1.43845 q^{23} -5.56155 q^{27} +3.43845 q^{29} +9.12311 q^{31} -9.56155 q^{33} +8.80776 q^{37} -3.12311 q^{39} +2.43845 q^{41} +6.56155 q^{43} +8.24621 q^{47} -2.43845 q^{51} -1.12311 q^{53} -5.56155 q^{57} -11.3693 q^{59} -11.1231 q^{61} +7.87689 q^{67} +2.24621 q^{69} +1.68466 q^{71} -6.43845 q^{73} -5.68466 q^{79} -7.00000 q^{81} +1.31534 q^{83} +5.36932 q^{87} -9.80776 q^{89} +14.2462 q^{93} -6.00000 q^{97} +3.43845 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 + 3 * q^9 $$2 q - q^{3} + 3 q^{9} - 4 q^{11} - 4 q^{13} + q^{17} - 3 q^{19} + 7 q^{23} - 7 q^{27} + 11 q^{29} + 10 q^{31} - 15 q^{33} - 3 q^{37} + 2 q^{39} + 9 q^{41} + 9 q^{43} - 9 q^{51} + 6 q^{53} - 7 q^{57} + 2 q^{59} - 14 q^{61} + 24 q^{67} - 12 q^{69} - 9 q^{71} - 17 q^{73} + q^{79} - 14 q^{81} + 15 q^{83} - 14 q^{87} + q^{89} + 12 q^{93} - 12 q^{97} + 11 q^{99}+O(q^{100})$$ 2 * q - q^3 + 3 * q^9 - 4 * q^11 - 4 * q^13 + q^17 - 3 * q^19 + 7 * q^23 - 7 * q^27 + 11 * q^29 + 10 * q^31 - 15 * q^33 - 3 * q^37 + 2 * q^39 + 9 * q^41 + 9 * q^43 - 9 * q^51 + 6 * q^53 - 7 * q^57 + 2 * q^59 - 14 * q^61 + 24 * q^67 - 12 * q^69 - 9 * q^71 - 17 * q^73 + q^79 - 14 * q^81 + 15 * q^83 - 14 * q^87 + q^89 + 12 * q^93 - 12 * q^97 + 11 * 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
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −0.561553 −0.187184
$$10$$ 0 0
$$11$$ −6.12311 −1.84619 −0.923093 0.384577i $$-0.874347\pi$$
−0.923093 + 0.384577i $$0.874347\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.56155 −0.378732 −0.189366 0.981907i $$-0.560643\pi$$
−0.189366 + 0.981907i $$0.560643\pi$$
$$18$$ 0 0
$$19$$ −3.56155 −0.817076 −0.408538 0.912741i $$-0.633961\pi$$
−0.408538 + 0.912741i $$0.633961\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 1.43845 0.299937 0.149968 0.988691i $$-0.452083\pi$$
0.149968 + 0.988691i $$0.452083\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.56155 −1.07032
$$28$$ 0 0
$$29$$ 3.43845 0.638504 0.319252 0.947670i $$-0.396568\pi$$
0.319252 + 0.947670i $$0.396568\pi$$
$$30$$ 0 0
$$31$$ 9.12311 1.63856 0.819279 0.573395i $$-0.194374\pi$$
0.819279 + 0.573395i $$0.194374\pi$$
$$32$$ 0 0
$$33$$ −9.56155 −1.66445
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.80776 1.44799 0.723994 0.689807i $$-0.242304\pi$$
0.723994 + 0.689807i $$0.242304\pi$$
$$38$$ 0 0
$$39$$ −3.12311 −0.500097
$$40$$ 0 0
$$41$$ 2.43845 0.380821 0.190411 0.981705i $$-0.439018\pi$$
0.190411 + 0.981705i $$0.439018\pi$$
$$42$$ 0 0
$$43$$ 6.56155 1.00063 0.500314 0.865844i $$-0.333218\pi$$
0.500314 + 0.865844i $$0.333218\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.24621 1.20283 0.601417 0.798935i $$-0.294603\pi$$
0.601417 + 0.798935i $$0.294603\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −2.43845 −0.341451
$$52$$ 0 0
$$53$$ −1.12311 −0.154270 −0.0771352 0.997021i $$-0.524577\pi$$
−0.0771352 + 0.997021i $$0.524577\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −5.56155 −0.736646
$$58$$ 0 0
$$59$$ −11.3693 −1.48016 −0.740079 0.672519i $$-0.765212\pi$$
−0.740079 + 0.672519i $$0.765212\pi$$
$$60$$ 0 0
$$61$$ −11.1231 −1.42417 −0.712084 0.702094i $$-0.752248\pi$$
−0.712084 + 0.702094i $$0.752248\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.87689 0.962316 0.481158 0.876634i $$-0.340216\pi$$
0.481158 + 0.876634i $$0.340216\pi$$
$$68$$ 0 0
$$69$$ 2.24621 0.270412
$$70$$ 0 0
$$71$$ 1.68466 0.199932 0.0999661 0.994991i $$-0.468127\pi$$
0.0999661 + 0.994991i $$0.468127\pi$$
$$72$$ 0 0
$$73$$ −6.43845 −0.753563 −0.376782 0.926302i $$-0.622969\pi$$
−0.376782 + 0.926302i $$0.622969\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −5.68466 −0.639574 −0.319787 0.947489i $$-0.603611\pi$$
−0.319787 + 0.947489i $$0.603611\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 1.31534 0.144377 0.0721887 0.997391i $$-0.477002\pi$$
0.0721887 + 0.997391i $$0.477002\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 5.36932 0.575651
$$88$$ 0 0
$$89$$ −9.80776 −1.03962 −0.519810 0.854282i $$-0.673997\pi$$
−0.519810 + 0.854282i $$0.673997\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 14.2462 1.47726
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ 0 0
$$99$$ 3.43845 0.345577
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 1.12311 0.110663 0.0553314 0.998468i $$-0.482378\pi$$
0.0553314 + 0.998468i $$0.482378\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 7.80776 0.754805 0.377403 0.926049i $$-0.376817\pi$$
0.377403 + 0.926049i $$0.376817\pi$$
$$108$$ 0 0
$$109$$ 19.9309 1.90903 0.954516 0.298161i $$-0.0963733\pi$$
0.954516 + 0.298161i $$0.0963733\pi$$
$$110$$ 0 0
$$111$$ 13.7538 1.30545
$$112$$ 0 0
$$113$$ 17.2462 1.62239 0.811194 0.584778i $$-0.198818\pi$$
0.811194 + 0.584778i $$0.198818\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.12311 0.103831
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 26.4924 2.40840
$$122$$ 0 0
$$123$$ 3.80776 0.343335
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 10.8078 0.959034 0.479517 0.877533i $$-0.340812\pi$$
0.479517 + 0.877533i $$0.340812\pi$$
$$128$$ 0 0
$$129$$ 10.2462 0.902129
$$130$$ 0 0
$$131$$ 18.2462 1.59418 0.797089 0.603861i $$-0.206372\pi$$
0.797089 + 0.603861i $$0.206372\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −14.6847 −1.25460 −0.627298 0.778780i $$-0.715839\pi$$
−0.627298 + 0.778780i $$0.715839\pi$$
$$138$$ 0 0
$$139$$ 7.31534 0.620479 0.310240 0.950658i $$-0.399591\pi$$
0.310240 + 0.950658i $$0.399591\pi$$
$$140$$ 0 0
$$141$$ 12.8769 1.08443
$$142$$ 0 0
$$143$$ 12.2462 1.02408
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 13.4384 1.10092 0.550460 0.834861i $$-0.314452\pi$$
0.550460 + 0.834861i $$0.314452\pi$$
$$150$$ 0 0
$$151$$ −1.43845 −0.117059 −0.0585296 0.998286i $$-0.518641\pi$$
−0.0585296 + 0.998286i $$0.518641\pi$$
$$152$$ 0 0
$$153$$ 0.876894 0.0708927
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 8.49242 0.677769 0.338885 0.940828i $$-0.389950\pi$$
0.338885 + 0.940828i $$0.389950\pi$$
$$158$$ 0 0
$$159$$ −1.75379 −0.139084
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 14.9309 1.16948 0.584738 0.811222i $$-0.301197\pi$$
0.584738 + 0.811222i $$0.301197\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −21.6155 −1.67266 −0.836330 0.548227i $$-0.815303\pi$$
−0.836330 + 0.548227i $$0.815303\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 0 0
$$173$$ 18.4924 1.40595 0.702976 0.711213i $$-0.251854\pi$$
0.702976 + 0.711213i $$0.251854\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −17.7538 −1.33446
$$178$$ 0 0
$$179$$ −20.6847 −1.54604 −0.773022 0.634379i $$-0.781256\pi$$
−0.773022 + 0.634379i $$0.781256\pi$$
$$180$$ 0 0
$$181$$ 11.1231 0.826774 0.413387 0.910555i $$-0.364346\pi$$
0.413387 + 0.910555i $$0.364346\pi$$
$$182$$ 0 0
$$183$$ −17.3693 −1.28398
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.56155 0.699210
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 25.3693 1.83566 0.917830 0.396974i $$-0.129940\pi$$
0.917830 + 0.396974i $$0.129940\pi$$
$$192$$ 0 0
$$193$$ −20.6155 −1.48394 −0.741969 0.670434i $$-0.766108\pi$$
−0.741969 + 0.670434i $$0.766108\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0.561553 0.0400090 0.0200045 0.999800i $$-0.493632\pi$$
0.0200045 + 0.999800i $$0.493632\pi$$
$$198$$ 0 0
$$199$$ 4.87689 0.345714 0.172857 0.984947i $$-0.444700\pi$$
0.172857 + 0.984947i $$0.444700\pi$$
$$200$$ 0 0
$$201$$ 12.3002 0.867588
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −0.807764 −0.0561435
$$208$$ 0 0
$$209$$ 21.8078 1.50847
$$210$$ 0 0
$$211$$ 6.43845 0.443241 0.221620 0.975133i $$-0.428865\pi$$
0.221620 + 0.975133i $$0.428865\pi$$
$$212$$ 0 0
$$213$$ 2.63068 0.180251
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −10.0540 −0.679385
$$220$$ 0 0
$$221$$ 3.12311 0.210083
$$222$$ 0 0
$$223$$ 25.3693 1.69886 0.849428 0.527705i $$-0.176947\pi$$
0.849428 + 0.527705i $$0.176947\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −11.3693 −0.754608 −0.377304 0.926089i $$-0.623149\pi$$
−0.377304 + 0.926089i $$0.623149\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −0.561553 −0.0367885 −0.0183943 0.999831i $$-0.505855\pi$$
−0.0183943 + 0.999831i $$0.505855\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −8.87689 −0.576616
$$238$$ 0 0
$$239$$ −4.00000 −0.258738 −0.129369 0.991596i $$-0.541295\pi$$
−0.129369 + 0.991596i $$0.541295\pi$$
$$240$$ 0 0
$$241$$ 12.0540 0.776465 0.388232 0.921562i $$-0.373086\pi$$
0.388232 + 0.921562i $$0.373086\pi$$
$$242$$ 0 0
$$243$$ 5.75379 0.369106
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 7.12311 0.453232
$$248$$ 0 0
$$249$$ 2.05398 0.130165
$$250$$ 0 0
$$251$$ 24.6847 1.55808 0.779041 0.626973i $$-0.215706\pi$$
0.779041 + 0.626973i $$0.215706\pi$$
$$252$$ 0 0
$$253$$ −8.80776 −0.553739
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −12.2462 −0.763898 −0.381949 0.924183i $$-0.624747\pi$$
−0.381949 + 0.924183i $$0.624747\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −1.93087 −0.119518
$$262$$ 0 0
$$263$$ 1.68466 0.103880 0.0519402 0.998650i $$-0.483459\pi$$
0.0519402 + 0.998650i $$0.483459\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −15.3153 −0.937284
$$268$$ 0 0
$$269$$ −28.2462 −1.72220 −0.861101 0.508434i $$-0.830225\pi$$
−0.861101 + 0.508434i $$0.830225\pi$$
$$270$$ 0 0
$$271$$ 0.246211 0.0149563 0.00747813 0.999972i $$-0.497620\pi$$
0.00747813 + 0.999972i $$0.497620\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.4924 1.35144 0.675719 0.737159i $$-0.263833\pi$$
0.675719 + 0.737159i $$0.263833\pi$$
$$278$$ 0 0
$$279$$ −5.12311 −0.306712
$$280$$ 0 0
$$281$$ 7.43845 0.443741 0.221870 0.975076i $$-0.428784\pi$$
0.221870 + 0.975076i $$0.428784\pi$$
$$282$$ 0 0
$$283$$ 1.31534 0.0781889 0.0390945 0.999236i $$-0.487553\pi$$
0.0390945 + 0.999236i $$0.487553\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −14.5616 −0.856562
$$290$$ 0 0
$$291$$ −9.36932 −0.549239
$$292$$ 0 0
$$293$$ 30.0000 1.75262 0.876309 0.481749i $$-0.159998\pi$$
0.876309 + 0.481749i $$0.159998\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 34.0540 1.97601
$$298$$ 0 0
$$299$$ −2.87689 −0.166375
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −16.6847 −0.952244 −0.476122 0.879379i $$-0.657958\pi$$
−0.476122 + 0.879379i $$0.657958\pi$$
$$308$$ 0 0
$$309$$ 1.75379 0.0997696
$$310$$ 0 0
$$311$$ −11.1231 −0.630733 −0.315367 0.948970i $$-0.602128\pi$$
−0.315367 + 0.948970i $$0.602128\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −8.31534 −0.467036 −0.233518 0.972352i $$-0.575024\pi$$
−0.233518 + 0.972352i $$0.575024\pi$$
$$318$$ 0 0
$$319$$ −21.0540 −1.17880
$$320$$ 0 0
$$321$$ 12.1922 0.680504
$$322$$ 0 0
$$323$$ 5.56155 0.309453
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 31.1231 1.72111
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 27.0000 1.48405 0.742027 0.670370i $$-0.233865\pi$$
0.742027 + 0.670370i $$0.233865\pi$$
$$332$$ 0 0
$$333$$ −4.94602 −0.271040
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 6.68466 0.364137 0.182068 0.983286i $$-0.441721\pi$$
0.182068 + 0.983286i $$0.441721\pi$$
$$338$$ 0 0
$$339$$ 26.9309 1.46268
$$340$$ 0 0
$$341$$ −55.8617 −3.02508
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −18.6155 −0.999334 −0.499667 0.866218i $$-0.666544\pi$$
−0.499667 + 0.866218i $$0.666544\pi$$
$$348$$ 0 0
$$349$$ 21.1231 1.13069 0.565347 0.824853i $$-0.308742\pi$$
0.565347 + 0.824853i $$0.308742\pi$$
$$350$$ 0 0
$$351$$ 11.1231 0.593707
$$352$$ 0 0
$$353$$ 28.2462 1.50339 0.751697 0.659509i $$-0.229236\pi$$
0.751697 + 0.659509i $$0.229236\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −2.31534 −0.122199 −0.0610995 0.998132i $$-0.519461\pi$$
−0.0610995 + 0.998132i $$0.519461\pi$$
$$360$$ 0 0
$$361$$ −6.31534 −0.332386
$$362$$ 0 0
$$363$$ 41.3693 2.17133
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −34.2462 −1.78764 −0.893819 0.448428i $$-0.851984\pi$$
−0.893819 + 0.448428i $$0.851984\pi$$
$$368$$ 0 0
$$369$$ −1.36932 −0.0712838
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −23.0540 −1.19369 −0.596845 0.802357i $$-0.703579\pi$$
−0.596845 + 0.802357i $$0.703579\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.87689 −0.354178
$$378$$ 0 0
$$379$$ 32.8617 1.68799 0.843997 0.536348i $$-0.180196\pi$$
0.843997 + 0.536348i $$0.180196\pi$$
$$380$$ 0 0
$$381$$ 16.8769 0.864629
$$382$$ 0 0
$$383$$ 3.75379 0.191810 0.0959048 0.995391i $$-0.469426\pi$$
0.0959048 + 0.995391i $$0.469426\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −3.68466 −0.187302
$$388$$ 0 0
$$389$$ 6.56155 0.332684 0.166342 0.986068i $$-0.446804\pi$$
0.166342 + 0.986068i $$0.446804\pi$$
$$390$$ 0 0
$$391$$ −2.24621 −0.113596
$$392$$ 0 0
$$393$$ 28.4924 1.43725
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −5.50758 −0.276417 −0.138209 0.990403i $$-0.544134\pi$$
−0.138209 + 0.990403i $$0.544134\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1.49242 0.0745280 0.0372640 0.999305i $$-0.488136\pi$$
0.0372640 + 0.999305i $$0.488136\pi$$
$$402$$ 0 0
$$403$$ −18.2462 −0.908909
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −53.9309 −2.67325
$$408$$ 0 0
$$409$$ 25.8078 1.27611 0.638056 0.769990i $$-0.279739\pi$$
0.638056 + 0.769990i $$0.279739\pi$$
$$410$$ 0 0
$$411$$ −22.9309 −1.13110
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 11.4233 0.559401
$$418$$ 0 0
$$419$$ −1.80776 −0.0883151 −0.0441575 0.999025i $$-0.514060\pi$$
−0.0441575 + 0.999025i $$0.514060\pi$$
$$420$$ 0 0
$$421$$ 33.9309 1.65369 0.826845 0.562430i $$-0.190134\pi$$
0.826845 + 0.562430i $$0.190134\pi$$
$$422$$ 0 0
$$423$$ −4.63068 −0.225152
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 19.1231 0.923272
$$430$$ 0 0
$$431$$ −30.2462 −1.45691 −0.728454 0.685094i $$-0.759761\pi$$
−0.728454 + 0.685094i $$0.759761\pi$$
$$432$$ 0 0
$$433$$ −34.5464 −1.66019 −0.830097 0.557619i $$-0.811715\pi$$
−0.830097 + 0.557619i $$0.811715\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −5.12311 −0.245071
$$438$$ 0 0
$$439$$ −1.12311 −0.0536029 −0.0268015 0.999641i $$-0.508532\pi$$
−0.0268015 + 0.999641i $$0.508532\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 18.4384 0.876037 0.438019 0.898966i $$-0.355680\pi$$
0.438019 + 0.898966i $$0.355680\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 20.9848 0.992549
$$448$$ 0 0
$$449$$ −14.3693 −0.678130 −0.339065 0.940763i $$-0.610111\pi$$
−0.339065 + 0.940763i $$0.610111\pi$$
$$450$$ 0 0
$$451$$ −14.9309 −0.703067
$$452$$ 0 0
$$453$$ −2.24621 −0.105536
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −37.1080 −1.73584 −0.867918 0.496707i $$-0.834542\pi$$
−0.867918 + 0.496707i $$0.834542\pi$$
$$458$$ 0 0
$$459$$ 8.68466 0.405365
$$460$$ 0 0
$$461$$ 26.4924 1.23388 0.616938 0.787012i $$-0.288373\pi$$
0.616938 + 0.787012i $$0.288373\pi$$
$$462$$ 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
$$467$$ 33.1231 1.53275 0.766377 0.642391i $$-0.222057\pi$$
0.766377 + 0.642391i $$0.222057\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 13.2614 0.611052
$$472$$ 0 0
$$473$$ −40.1771 −1.84734
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0.630683 0.0288770
$$478$$ 0 0
$$479$$ −8.73863 −0.399278 −0.199639 0.979869i $$-0.563977\pi$$
−0.199639 + 0.979869i $$0.563977\pi$$
$$480$$ 0 0
$$481$$ −17.6155 −0.803199
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 31.4384 1.42461 0.712306 0.701869i $$-0.247651\pi$$
0.712306 + 0.701869i $$0.247651\pi$$
$$488$$ 0 0
$$489$$ 23.3153 1.05436
$$490$$ 0 0
$$491$$ 8.31534 0.375266 0.187633 0.982239i $$-0.439918\pi$$
0.187633 + 0.982239i $$0.439918\pi$$
$$492$$ 0 0
$$493$$ −5.36932 −0.241822
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 16.4924 0.738302 0.369151 0.929369i $$-0.379648\pi$$
0.369151 + 0.929369i $$0.379648\pi$$
$$500$$ 0 0
$$501$$ −33.7538 −1.50801
$$502$$ 0 0
$$503$$ −36.2462 −1.61614 −0.808069 0.589087i $$-0.799487\pi$$
−0.808069 + 0.589087i $$0.799487\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −14.0540 −0.624159
$$508$$ 0 0
$$509$$ −26.0000 −1.15243 −0.576215 0.817298i $$-0.695471\pi$$
−0.576215 + 0.817298i $$0.695471\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 19.8078 0.874534
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −50.4924 −2.22065
$$518$$ 0 0
$$519$$ 28.8769 1.26755
$$520$$ 0 0
$$521$$ 31.5616 1.38274 0.691368 0.722502i $$-0.257008\pi$$
0.691368 + 0.722502i $$0.257008\pi$$
$$522$$ 0 0
$$523$$ −40.6847 −1.77902 −0.889508 0.456920i $$-0.848953\pi$$
−0.889508 + 0.456920i $$0.848953\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −14.2462 −0.620575
$$528$$ 0 0
$$529$$ −20.9309 −0.910038
$$530$$ 0 0
$$531$$ 6.38447 0.277062
$$532$$ 0 0
$$533$$ −4.87689 −0.211242
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −32.3002 −1.39386
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −20.4233 −0.878066 −0.439033 0.898471i $$-0.644679\pi$$
−0.439033 + 0.898471i $$0.644679\pi$$
$$542$$ 0 0
$$543$$ 17.3693 0.745389
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −15.7386 −0.672935 −0.336468 0.941695i $$-0.609232\pi$$
−0.336468 + 0.941695i $$0.609232\pi$$
$$548$$ 0 0
$$549$$ 6.24621 0.266582
$$550$$ 0 0
$$551$$ −12.2462 −0.521706
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 5.43845 0.230434 0.115217 0.993340i $$-0.463244\pi$$
0.115217 + 0.993340i $$0.463244\pi$$
$$558$$ 0 0
$$559$$ −13.1231 −0.555048
$$560$$ 0 0
$$561$$ 14.9309 0.630382
$$562$$ 0 0
$$563$$ −1.12311 −0.0473333 −0.0236666 0.999720i $$-0.507534\pi$$
−0.0236666 + 0.999720i $$0.507534\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 45.9848 1.92778 0.963892 0.266292i $$-0.0857984\pi$$
0.963892 + 0.266292i $$0.0857984\pi$$
$$570$$ 0 0
$$571$$ 39.6847 1.66075 0.830376 0.557204i $$-0.188126\pi$$
0.830376 + 0.557204i $$0.188126\pi$$
$$572$$ 0 0
$$573$$ 39.6155 1.65496
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −24.5464 −1.02188 −0.510940 0.859616i $$-0.670703\pi$$
−0.510940 + 0.859616i $$0.670703\pi$$
$$578$$ 0 0
$$579$$ −32.1922 −1.33786
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 6.87689 0.284812
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 37.1771 1.53446 0.767231 0.641371i $$-0.221634\pi$$
0.767231 + 0.641371i $$0.221634\pi$$
$$588$$ 0 0
$$589$$ −32.4924 −1.33883
$$590$$ 0 0
$$591$$ 0.876894 0.0360706
$$592$$ 0 0
$$593$$ −24.6847 −1.01368 −0.506839 0.862041i $$-0.669186\pi$$
−0.506839 + 0.862041i $$0.669186\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 7.61553 0.311683
$$598$$ 0 0
$$599$$ −29.3002 −1.19717 −0.598587 0.801058i $$-0.704271\pi$$
−0.598587 + 0.801058i $$0.704271\pi$$
$$600$$ 0 0
$$601$$ 42.0540 1.71542 0.857709 0.514136i $$-0.171887\pi$$
0.857709 + 0.514136i $$0.171887\pi$$
$$602$$ 0 0
$$603$$ −4.42329 −0.180130
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 12.6307 0.512664 0.256332 0.966589i $$-0.417486\pi$$
0.256332 + 0.966589i $$0.417486\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −16.4924 −0.667212
$$612$$ 0 0
$$613$$ 5.68466 0.229601 0.114801 0.993389i $$-0.463377\pi$$
0.114801 + 0.993389i $$0.463377\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2.80776 −0.113036 −0.0565182 0.998402i $$-0.518000\pi$$
−0.0565182 + 0.998402i $$0.518000\pi$$
$$618$$ 0 0
$$619$$ −4.63068 −0.186123 −0.0930614 0.995660i $$-0.529665\pi$$
−0.0930614 + 0.995660i $$0.529665\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 34.0540 1.35998
$$628$$ 0 0
$$629$$ −13.7538 −0.548399
$$630$$ 0 0
$$631$$ −3.05398 −0.121577 −0.0607884 0.998151i $$-0.519361\pi$$
−0.0607884 + 0.998151i $$0.519361\pi$$
$$632$$ 0 0
$$633$$ 10.0540 0.399610
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −0.946025 −0.0374242
$$640$$ 0 0
$$641$$ −20.5616 −0.812133 −0.406066 0.913844i $$-0.633100\pi$$
−0.406066 + 0.913844i $$0.633100\pi$$
$$642$$ 0 0
$$643$$ 46.7386 1.84319 0.921596 0.388151i $$-0.126886\pi$$
0.921596 + 0.388151i $$0.126886\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0.630683 0.0247947 0.0123974 0.999923i $$-0.496054\pi$$
0.0123974 + 0.999923i $$0.496054\pi$$
$$648$$ 0 0
$$649$$ 69.6155 2.73265
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −23.8617 −0.933782 −0.466891 0.884315i $$-0.654626\pi$$
−0.466891 + 0.884315i $$0.654626\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 3.61553 0.141055
$$658$$ 0 0
$$659$$ 14.4384 0.562442 0.281221 0.959643i $$-0.409261\pi$$
0.281221 + 0.959643i $$0.409261\pi$$
$$660$$ 0 0
$$661$$ −10.4924 −0.408108 −0.204054 0.978960i $$-0.565412\pi$$
−0.204054 + 0.978960i $$0.565412\pi$$
$$662$$ 0 0
$$663$$ 4.87689 0.189403
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 4.94602 0.191511
$$668$$ 0 0
$$669$$ 39.6155 1.53162
$$670$$ 0 0
$$671$$ 68.1080 2.62928
$$672$$ 0 0
$$673$$ 13.5076 0.520679 0.260339 0.965517i $$-0.416166\pi$$
0.260339 + 0.965517i $$0.416166\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −39.3693 −1.51309 −0.756543 0.653944i $$-0.773113\pi$$
−0.756543 + 0.653944i $$0.773113\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −17.7538 −0.680327
$$682$$ 0 0
$$683$$ 25.8769 0.990152 0.495076 0.868850i $$-0.335140\pi$$
0.495076 + 0.868850i $$0.335140\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 21.8617 0.834077
$$688$$ 0 0
$$689$$ 2.24621 0.0855738
$$690$$ 0 0
$$691$$ 21.5616 0.820240 0.410120 0.912032i $$-0.365487\pi$$
0.410120 + 0.912032i $$0.365487\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −3.80776 −0.144229
$$698$$ 0 0
$$699$$ −0.876894 −0.0331672
$$700$$ 0 0
$$701$$ −30.1080 −1.13716 −0.568581 0.822627i $$-0.692507\pi$$
−0.568581 + 0.822627i $$0.692507\pi$$
$$702$$ 0 0
$$703$$ −31.3693 −1.18312
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 12.6307 0.474355 0.237178 0.971466i $$-0.423778\pi$$
0.237178 + 0.971466i $$0.423778\pi$$
$$710$$ 0 0
$$711$$ 3.19224 0.119718
$$712$$ 0 0
$$713$$ 13.1231 0.491464
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −6.24621 −0.233269
$$718$$ 0 0
$$719$$ −21.7538 −0.811279 −0.405640 0.914033i $$-0.632951\pi$$
−0.405640 + 0.914033i $$0.632951\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 18.8229 0.700032
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −39.6155 −1.46926 −0.734629 0.678469i $$-0.762644\pi$$
−0.734629 + 0.678469i $$0.762644\pi$$
$$728$$ 0 0
$$729$$ 29.9848 1.11055
$$730$$ 0 0
$$731$$ −10.2462 −0.378970
$$732$$ 0 0
$$733$$ −5.75379 −0.212521 −0.106261 0.994338i $$-0.533888\pi$$
−0.106261 + 0.994338i $$0.533888\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −48.2311 −1.77661
$$738$$ 0 0
$$739$$ −28.1771 −1.03651 −0.518255 0.855226i $$-0.673418\pi$$
−0.518255 + 0.855226i $$0.673418\pi$$
$$740$$ 0 0
$$741$$ 11.1231 0.408617
$$742$$ 0 0
$$743$$ 30.7386 1.12769 0.563846 0.825880i $$-0.309321\pi$$
0.563846 + 0.825880i $$0.309321\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −0.738634 −0.0270252
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 2.63068 0.0959950 0.0479975 0.998847i $$-0.484716\pi$$
0.0479975 + 0.998847i $$0.484716\pi$$
$$752$$ 0 0
$$753$$ 38.5464 1.40471
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 25.6847 0.933525 0.466762 0.884383i $$-0.345420\pi$$
0.466762 + 0.884383i $$0.345420\pi$$
$$758$$ 0 0
$$759$$ −13.7538 −0.499231
$$760$$ 0 0
$$761$$ 27.4233 0.994094 0.497047 0.867724i $$-0.334418\pi$$
0.497047 + 0.867724i $$0.334418\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 22.7386 0.821044
$$768$$ 0 0
$$769$$ −20.3002 −0.732043 −0.366022 0.930606i $$-0.619280\pi$$
−0.366022 + 0.930606i $$0.619280\pi$$
$$770$$ 0 0
$$771$$ −19.1231 −0.688702
$$772$$ 0 0
$$773$$ −29.6155 −1.06520 −0.532598 0.846368i $$-0.678784\pi$$
−0.532598 + 0.846368i $$0.678784\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −8.68466 −0.311160
$$780$$ 0 0
$$781$$ −10.3153 −0.369112
$$782$$ 0 0
$$783$$ −19.1231 −0.683404
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −9.75379 −0.347685 −0.173843 0.984773i $$-0.555618\pi$$
−0.173843 + 0.984773i $$0.555618\pi$$
$$788$$ 0 0
$$789$$ 2.63068 0.0936548
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 22.2462 0.789986
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −0.384472 −0.0136187 −0.00680935 0.999977i $$-0.502167\pi$$
−0.00680935 + 0.999977i $$0.502167\pi$$
$$798$$ 0 0
$$799$$ −12.8769 −0.455552
$$800$$ 0 0
$$801$$ 5.50758 0.194601
$$802$$ 0 0
$$803$$ 39.4233 1.39122
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −44.1080 −1.55267
$$808$$ 0 0
$$809$$ −12.4233 −0.436780 −0.218390 0.975862i $$-0.570080\pi$$
−0.218390 + 0.975862i $$0.570080\pi$$
$$810$$ 0 0
$$811$$ −2.38447 −0.0837301 −0.0418651 0.999123i $$-0.513330\pi$$
−0.0418651 + 0.999123i $$0.513330\pi$$
$$812$$ 0 0
$$813$$ 0.384472 0.0134840
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −23.3693 −0.817589
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −11.8617 −0.413978 −0.206989 0.978343i $$-0.566366\pi$$
−0.206989 + 0.978343i $$0.566366\pi$$
$$822$$ 0 0
$$823$$ 16.5616 0.577299 0.288650 0.957435i $$-0.406794\pi$$
0.288650 + 0.957435i $$0.406794\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 7.24621 0.251975 0.125988 0.992032i $$-0.459790\pi$$
0.125988 + 0.992032i $$0.459790\pi$$
$$828$$ 0 0
$$829$$ −14.2462 −0.494791 −0.247396 0.968915i $$-0.579575\pi$$
−0.247396 + 0.968915i $$0.579575\pi$$
$$830$$ 0 0
$$831$$ 35.1231 1.21841
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −50.7386 −1.75378
$$838$$ 0 0
$$839$$ −7.12311 −0.245917 −0.122958 0.992412i $$-0.539238\pi$$
−0.122958 + 0.992412i $$0.539238\pi$$
$$840$$ 0 0
$$841$$ −17.1771 −0.592313
$$842$$ 0 0
$$843$$ 11.6155 0.400060
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 2.05398 0.0704923
$$850$$ 0 0
$$851$$ 12.6695 0.434305
$$852$$ 0 0
$$853$$ −19.1231 −0.654763 −0.327381 0.944892i $$-0.606166\pi$$
−0.327381 + 0.944892i $$0.606166\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −42.0540 −1.43654 −0.718268 0.695766i $$-0.755065\pi$$
−0.718268 + 0.695766i $$0.755065\pi$$
$$858$$ 0 0
$$859$$ 11.1771 0.381357 0.190679 0.981653i $$-0.438931\pi$$
0.190679 + 0.981653i $$0.438931\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −3.19224 −0.108665 −0.0543325 0.998523i $$-0.517303\pi$$
−0.0543325 + 0.998523i $$0.517303\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −22.7386 −0.772244
$$868$$ 0 0
$$869$$ 34.8078 1.18077
$$870$$ 0 0
$$871$$ −15.7538 −0.533797
$$872$$ 0 0
$$873$$ 3.36932 0.114034
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −17.6155 −0.594834 −0.297417 0.954748i $$-0.596125\pi$$
−0.297417 + 0.954748i $$0.596125\pi$$
$$878$$ 0 0
$$879$$ 46.8466 1.58010
$$880$$ 0 0
$$881$$ −8.73863 −0.294412 −0.147206 0.989106i $$-0.547028\pi$$
−0.147206 + 0.989106i $$0.547028\pi$$
$$882$$ 0 0
$$883$$ 31.4924 1.05980 0.529902 0.848059i $$-0.322229\pi$$
0.529902 + 0.848059i $$0.322229\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 8.38447 0.281523 0.140762 0.990044i $$-0.455045\pi$$
0.140762 + 0.990044i $$0.455045\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 42.8617 1.43592
$$892$$ 0 0
$$893$$ −29.3693 −0.982807
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −4.49242 −0.149998
$$898$$ 0 0
$$899$$ 31.3693 1.04623
$$900$$ 0 0
$$901$$ 1.75379 0.0584272
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 4.00000 0.132818 0.0664089 0.997792i $$-0.478846\pi$$
0.0664089 + 0.997792i $$0.478846\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 22.4233 0.742917 0.371458 0.928450i $$-0.378858\pi$$
0.371458 + 0.928450i $$0.378858\pi$$
$$912$$ 0 0
$$913$$ −8.05398 −0.266548
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 38.8078 1.28015 0.640075 0.768312i $$-0.278903\pi$$
0.640075 + 0.768312i $$0.278903\pi$$
$$920$$ 0 0
$$921$$ −26.0540 −0.858508
$$922$$ 0 0
$$923$$ −3.36932 −0.110902
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −0.630683 −0.0207144
$$928$$ 0 0
$$929$$ 34.3542 1.12712 0.563562 0.826074i $$-0.309431\pi$$
0.563562 + 0.826074i $$0.309431\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −17.3693 −0.568646
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 32.3002 1.05520 0.527601 0.849493i $$-0.323092\pi$$
0.527601 + 0.849493i $$0.323092\pi$$
$$938$$ 0 0
$$939$$ −34.3542 −1.12111
$$940$$ 0 0
$$941$$ −23.3693 −0.761818 −0.380909 0.924613i $$-0.624389\pi$$
−0.380909 + 0.924613i $$0.624389\pi$$
$$942$$ 0 0
$$943$$ 3.50758 0.114222
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −16.4924 −0.535932 −0.267966 0.963428i $$-0.586351\pi$$
−0.267966 + 0.963428i $$0.586351\pi$$
$$948$$ 0 0
$$949$$ 12.8769 0.418002
$$950$$ 0 0
$$951$$ −12.9848 −0.421062
$$952$$ 0 0
$$953$$ −36.1231 −1.17014 −0.585071 0.810982i $$-0.698933\pi$$
−0.585071 + 0.810982i $$0.698933\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −32.8769 −1.06276
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 52.2311 1.68487
$$962$$ 0 0
$$963$$ −4.38447 −0.141288
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −41.3693 −1.33035 −0.665174 0.746689i $$-0.731643\pi$$
−0.665174 + 0.746689i $$0.731643\pi$$
$$968$$ 0 0
$$969$$ 8.68466 0.278991
$$970$$ 0 0
$$971$$ 14.6847 0.471253 0.235627 0.971844i $$-0.424286\pi$$
0.235627 + 0.971844i $$0.424286\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −16.7538 −0.536001 −0.268001 0.963419i $$-0.586363\pi$$
−0.268001 + 0.963419i $$0.586363\pi$$
$$978$$ 0 0
$$979$$ 60.0540 1.91933
$$980$$ 0 0
$$981$$ −11.1922 −0.357341
$$982$$ 0 0
$$983$$ −28.3542 −0.904357 −0.452179 0.891927i $$-0.649353\pi$$
−0.452179 + 0.891927i $$0.649353\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 9.43845 0.300125
$$990$$ 0 0
$$991$$ 48.6695 1.54604 0.773019 0.634383i $$-0.218746\pi$$
0.773019 + 0.634383i $$0.218746\pi$$
$$992$$ 0 0
$$993$$ 42.1619 1.33797
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 4.24621 0.134479 0.0672394 0.997737i $$-0.478581\pi$$
0.0672394 + 0.997737i $$0.478581\pi$$
$$998$$ 0 0
$$999$$ −48.9848 −1.54981
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 9800.2.a.bt.1.2 2
5.4 even 2 9800.2.a.bx.1.1 2
7.6 odd 2 1400.2.a.q.1.1 yes 2
28.27 even 2 2800.2.a.bj.1.2 2
35.13 even 4 1400.2.g.j.449.2 4
35.27 even 4 1400.2.g.j.449.3 4
35.34 odd 2 1400.2.a.o.1.2 2
140.27 odd 4 2800.2.g.v.449.2 4
140.83 odd 4 2800.2.g.v.449.3 4
140.139 even 2 2800.2.a.bo.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.a.o.1.2 2 35.34 odd 2
1400.2.a.q.1.1 yes 2 7.6 odd 2
1400.2.g.j.449.2 4 35.13 even 4
1400.2.g.j.449.3 4 35.27 even 4
2800.2.a.bj.1.2 2 28.27 even 2
2800.2.a.bo.1.1 2 140.139 even 2
2800.2.g.v.449.2 4 140.27 odd 4
2800.2.g.v.449.3 4 140.83 odd 4
9800.2.a.bt.1.2 2 1.1 even 1 trivial
9800.2.a.bx.1.1 2 5.4 even 2