# Properties

 Label 7600.2.a.bj.1.2 Level $7600$ Weight $2$ Character 7600.1 Self dual yes Analytic conductor $60.686$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7600 = 2^{4} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7600.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: $$60.6863055362$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 3$$ x^3 - x^2 - 4*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1900) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.713538$$ of defining polynomial Character $$\chi$$ $$=$$ 7600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.286462 q^{3} -0.286462 q^{7} -2.91794 q^{9} +O(q^{10})$$ $$q-0.286462 q^{3} -0.286462 q^{7} -2.91794 q^{9} -4.26819 q^{11} +3.20440 q^{13} +0.286462 q^{17} +1.00000 q^{19} +0.0820605 q^{21} -0.936212 q^{23} +1.69527 q^{27} +2.26819 q^{29} +4.18613 q^{31} +1.22267 q^{33} +8.67699 q^{37} -0.917939 q^{39} +1.08206 q^{41} -4.42708 q^{43} +0.759053 q^{47} -6.91794 q^{49} -0.0820605 q^{51} +4.42708 q^{53} -0.286462 q^{57} +4.70050 q^{59} +10.1861 q^{61} +0.835879 q^{63} -9.82284 q^{67} +0.268189 q^{69} -4.83588 q^{71} +10.4726 q^{73} +1.22267 q^{77} -5.10407 q^{79} +8.26819 q^{81} -15.7355 q^{83} -0.649750 q^{87} -9.75382 q^{89} -0.917939 q^{91} -1.19917 q^{93} -9.61320 q^{97} +12.4543 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} - 2 q^{7} + q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 - 2 * q^7 + q^9 $$3 q - 2 q^{3} - 2 q^{7} + q^{9} + q^{11} + q^{13} + 2 q^{17} + 3 q^{19} + 10 q^{21} - 8 q^{23} - 11 q^{27} - 7 q^{29} - 11 q^{31} + 10 q^{33} - 5 q^{37} + 7 q^{39} + 13 q^{41} - 11 q^{43} - 19 q^{47} - 11 q^{49} - 10 q^{51} + 11 q^{53} - 2 q^{57} + 6 q^{59} + 7 q^{61} - 17 q^{63} - 3 q^{67} - 13 q^{69} + 5 q^{71} + 9 q^{73} + 10 q^{77} + 18 q^{79} + 11 q^{81} - 3 q^{83} - 6 q^{87} + 7 q^{91} + 13 q^{93} - 3 q^{97}+O(q^{100})$$ 3 * q - 2 * q^3 - 2 * q^7 + q^9 + q^11 + q^13 + 2 * q^17 + 3 * q^19 + 10 * q^21 - 8 * q^23 - 11 * q^27 - 7 * q^29 - 11 * q^31 + 10 * q^33 - 5 * q^37 + 7 * q^39 + 13 * q^41 - 11 * q^43 - 19 * q^47 - 11 * q^49 - 10 * q^51 + 11 * q^53 - 2 * q^57 + 6 * q^59 + 7 * q^61 - 17 * q^63 - 3 * q^67 - 13 * q^69 + 5 * q^71 + 9 * q^73 + 10 * q^77 + 18 * q^79 + 11 * q^81 - 3 * q^83 - 6 * q^87 + 7 * q^91 + 13 * q^93 - 3 * q^97

## 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
$$3$$ −0.286462 −0.165389 −0.0826945 0.996575i $$-0.526353\pi$$
−0.0826945 + 0.996575i $$0.526353\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.286462 −0.108272 −0.0541362 0.998534i $$-0.517241\pi$$
−0.0541362 + 0.998534i $$0.517241\pi$$
$$8$$ 0 0
$$9$$ −2.91794 −0.972646
$$10$$ 0 0
$$11$$ −4.26819 −1.28691 −0.643454 0.765485i $$-0.722499\pi$$
−0.643454 + 0.765485i $$0.722499\pi$$
$$12$$ 0 0
$$13$$ 3.20440 0.888741 0.444371 0.895843i $$-0.353427\pi$$
0.444371 + 0.895843i $$0.353427\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0.286462 0.0694773 0.0347386 0.999396i $$-0.488940\pi$$
0.0347386 + 0.999396i $$0.488940\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0.0820605 0.0179071
$$22$$ 0 0
$$23$$ −0.936212 −0.195214 −0.0976069 0.995225i $$-0.531119\pi$$
−0.0976069 + 0.995225i $$0.531119\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.69527 0.326254
$$28$$ 0 0
$$29$$ 2.26819 0.421192 0.210596 0.977573i $$-0.432460\pi$$
0.210596 + 0.977573i $$0.432460\pi$$
$$30$$ 0 0
$$31$$ 4.18613 0.751851 0.375925 0.926650i $$-0.377325\pi$$
0.375925 + 0.926650i $$0.377325\pi$$
$$32$$ 0 0
$$33$$ 1.22267 0.212840
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.67699 1.42649 0.713244 0.700915i $$-0.247225\pi$$
0.713244 + 0.700915i $$0.247225\pi$$
$$38$$ 0 0
$$39$$ −0.917939 −0.146988
$$40$$ 0 0
$$41$$ 1.08206 0.168989 0.0844947 0.996424i $$-0.473072\pi$$
0.0844947 + 0.996424i $$0.473072\pi$$
$$42$$ 0 0
$$43$$ −4.42708 −0.675123 −0.337561 0.941304i $$-0.609602\pi$$
−0.337561 + 0.941304i $$0.609602\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0.759053 0.110719 0.0553596 0.998466i $$-0.482369\pi$$
0.0553596 + 0.998466i $$0.482369\pi$$
$$48$$ 0 0
$$49$$ −6.91794 −0.988277
$$50$$ 0 0
$$51$$ −0.0820605 −0.0114908
$$52$$ 0 0
$$53$$ 4.42708 0.608106 0.304053 0.952655i $$-0.401660\pi$$
0.304053 + 0.952655i $$0.401660\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −0.286462 −0.0379428
$$58$$ 0 0
$$59$$ 4.70050 0.611953 0.305976 0.952039i $$-0.401017\pi$$
0.305976 + 0.952039i $$0.401017\pi$$
$$60$$ 0 0
$$61$$ 10.1861 1.30420 0.652100 0.758133i $$-0.273888\pi$$
0.652100 + 0.758133i $$0.273888\pi$$
$$62$$ 0 0
$$63$$ 0.835879 0.105311
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.82284 −1.20005 −0.600025 0.799981i $$-0.704843\pi$$
−0.600025 + 0.799981i $$0.704843\pi$$
$$68$$ 0 0
$$69$$ 0.268189 0.0322862
$$70$$ 0 0
$$71$$ −4.83588 −0.573913 −0.286957 0.957944i $$-0.592644\pi$$
−0.286957 + 0.957944i $$0.592644\pi$$
$$72$$ 0 0
$$73$$ 10.4726 1.22572 0.612862 0.790190i $$-0.290018\pi$$
0.612862 + 0.790190i $$0.290018\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.22267 0.139337
$$78$$ 0 0
$$79$$ −5.10407 −0.574253 −0.287126 0.957893i $$-0.592700\pi$$
−0.287126 + 0.957893i $$0.592700\pi$$
$$80$$ 0 0
$$81$$ 8.26819 0.918688
$$82$$ 0 0
$$83$$ −15.7355 −1.72720 −0.863600 0.504177i $$-0.831796\pi$$
−0.863600 + 0.504177i $$0.831796\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −0.649750 −0.0696605
$$88$$ 0 0
$$89$$ −9.75382 −1.03390 −0.516951 0.856015i $$-0.672933\pi$$
−0.516951 + 0.856015i $$0.672933\pi$$
$$90$$ 0 0
$$91$$ −0.917939 −0.0962262
$$92$$ 0 0
$$93$$ −1.19917 −0.124348
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −9.61320 −0.976073 −0.488037 0.872823i $$-0.662287\pi$$
−0.488037 + 0.872823i $$0.662287\pi$$
$$98$$ 0 0
$$99$$ 12.4543 1.25171
$$100$$ 0 0
$$101$$ 0.731811 0.0728179 0.0364089 0.999337i $$-0.488408\pi$$
0.0364089 + 0.999337i $$0.488408\pi$$
$$102$$ 0 0
$$103$$ 3.10407 0.305853 0.152926 0.988238i $$-0.451130\pi$$
0.152926 + 0.988238i $$0.451130\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −14.1731 −1.37016 −0.685082 0.728466i $$-0.740234\pi$$
−0.685082 + 0.728466i $$0.740234\pi$$
$$108$$ 0 0
$$109$$ −9.61844 −0.921279 −0.460640 0.887587i $$-0.652380\pi$$
−0.460640 + 0.887587i $$0.652380\pi$$
$$110$$ 0 0
$$111$$ −2.48563 −0.235925
$$112$$ 0 0
$$113$$ 8.77733 0.825701 0.412851 0.910799i $$-0.364533\pi$$
0.412851 + 0.910799i $$0.364533\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −9.35025 −0.864431
$$118$$ 0 0
$$119$$ −0.0820605 −0.00752248
$$120$$ 0 0
$$121$$ 7.21744 0.656131
$$122$$ 0 0
$$123$$ −0.309969 −0.0279490
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −6.77209 −0.600926 −0.300463 0.953793i $$-0.597141\pi$$
−0.300463 + 0.953793i $$0.597141\pi$$
$$128$$ 0 0
$$129$$ 1.26819 0.111658
$$130$$ 0 0
$$131$$ 2.91794 0.254942 0.127471 0.991842i $$-0.459314\pi$$
0.127471 + 0.991842i $$0.459314\pi$$
$$132$$ 0 0
$$133$$ −0.286462 −0.0248394
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −14.4360 −1.23335 −0.616677 0.787216i $$-0.711522\pi$$
−0.616677 + 0.787216i $$0.711522\pi$$
$$138$$ 0 0
$$139$$ 2.29950 0.195041 0.0975205 0.995234i $$-0.468909\pi$$
0.0975205 + 0.995234i $$0.468909\pi$$
$$140$$ 0 0
$$141$$ −0.217440 −0.0183117
$$142$$ 0 0
$$143$$ −13.6770 −1.14373
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.98173 0.163450
$$148$$ 0 0
$$149$$ 6.75382 0.553294 0.276647 0.960972i $$-0.410777\pi$$
0.276647 + 0.960972i $$0.410777\pi$$
$$150$$ 0 0
$$151$$ 11.8866 0.967320 0.483660 0.875256i $$-0.339307\pi$$
0.483660 + 0.875256i $$0.339307\pi$$
$$152$$ 0 0
$$153$$ −0.835879 −0.0675768
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −14.1731 −1.13114 −0.565568 0.824702i $$-0.691343\pi$$
−0.565568 + 0.824702i $$0.691343\pi$$
$$158$$ 0 0
$$159$$ −1.26819 −0.100574
$$160$$ 0 0
$$161$$ 0.268189 0.0211363
$$162$$ 0 0
$$163$$ −7.89443 −0.618340 −0.309170 0.951007i $$-0.600051\pi$$
−0.309170 + 0.951007i $$0.600051\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −20.4726 −1.58422 −0.792108 0.610381i $$-0.791017\pi$$
−0.792108 + 0.610381i $$0.791017\pi$$
$$168$$ 0 0
$$169$$ −2.73181 −0.210139
$$170$$ 0 0
$$171$$ −2.91794 −0.223140
$$172$$ 0 0
$$173$$ −24.0492 −1.82843 −0.914215 0.405229i $$-0.867192\pi$$
−0.914215 + 0.405229i $$0.867192\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −1.34651 −0.101210
$$178$$ 0 0
$$179$$ −18.6718 −1.39559 −0.697796 0.716296i $$-0.745836\pi$$
−0.697796 + 0.716296i $$0.745836\pi$$
$$180$$ 0 0
$$181$$ −3.53638 −0.262857 −0.131428 0.991326i $$-0.541956\pi$$
−0.131428 + 0.991326i $$0.541956\pi$$
$$182$$ 0 0
$$183$$ −2.91794 −0.215700
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1.22267 −0.0894108
$$188$$ 0 0
$$189$$ −0.485629 −0.0353243
$$190$$ 0 0
$$191$$ 14.2682 1.03241 0.516205 0.856465i $$-0.327344\pi$$
0.516205 + 0.856465i $$0.327344\pi$$
$$192$$ 0 0
$$193$$ −8.18089 −0.588874 −0.294437 0.955671i $$-0.595132\pi$$
−0.294437 + 0.955671i $$0.595132\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.6404 0.900595 0.450297 0.892879i $$-0.351318\pi$$
0.450297 + 0.892879i $$0.351318\pi$$
$$198$$ 0 0
$$199$$ 4.10407 0.290930 0.145465 0.989363i $$-0.453532\pi$$
0.145465 + 0.989363i $$0.453532\pi$$
$$200$$ 0 0
$$201$$ 2.81387 0.198475
$$202$$ 0 0
$$203$$ −0.649750 −0.0456035
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 2.73181 0.189874
$$208$$ 0 0
$$209$$ −4.26819 −0.295237
$$210$$ 0 0
$$211$$ −1.53638 −0.105769 −0.0528843 0.998601i $$-0.516841\pi$$
−0.0528843 + 0.998601i $$0.516841\pi$$
$$212$$ 0 0
$$213$$ 1.38530 0.0949189
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −1.19917 −0.0814048
$$218$$ 0 0
$$219$$ −3.00000 −0.202721
$$220$$ 0 0
$$221$$ 0.917939 0.0617473
$$222$$ 0 0
$$223$$ −1.40880 −0.0943404 −0.0471702 0.998887i $$-0.515020\pi$$
−0.0471702 + 0.998887i $$0.515020\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −22.9179 −1.52112 −0.760559 0.649269i $$-0.775075\pi$$
−0.760559 + 0.649269i $$0.775075\pi$$
$$228$$ 0 0
$$229$$ 3.78513 0.250128 0.125064 0.992149i $$-0.460086\pi$$
0.125064 + 0.992149i $$0.460086\pi$$
$$230$$ 0 0
$$231$$ −0.350250 −0.0230447
$$232$$ 0 0
$$233$$ 2.26819 0.148594 0.0742970 0.997236i $$-0.476329\pi$$
0.0742970 + 0.997236i $$0.476329\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1.46212 0.0949750
$$238$$ 0 0
$$239$$ 2.18613 0.141409 0.0707045 0.997497i $$-0.477475\pi$$
0.0707045 + 0.997497i $$0.477475\pi$$
$$240$$ 0 0
$$241$$ 19.9907 1.28771 0.643857 0.765146i $$-0.277333\pi$$
0.643857 + 0.765146i $$0.277333\pi$$
$$242$$ 0 0
$$243$$ −7.45432 −0.478195
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.20440 0.203891
$$248$$ 0 0
$$249$$ 4.50764 0.285660
$$250$$ 0 0
$$251$$ −1.35025 −0.0852270 −0.0426135 0.999092i $$-0.513568\pi$$
−0.0426135 + 0.999092i $$0.513568\pi$$
$$252$$ 0 0
$$253$$ 3.99593 0.251222
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −11.9582 −0.745933 −0.372967 0.927845i $$-0.621659\pi$$
−0.372967 + 0.927845i $$0.621659\pi$$
$$258$$ 0 0
$$259$$ −2.48563 −0.154449
$$260$$ 0 0
$$261$$ −6.61844 −0.409671
$$262$$ 0 0
$$263$$ 8.31370 0.512645 0.256322 0.966591i $$-0.417489\pi$$
0.256322 + 0.966591i $$0.417489\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 2.79410 0.170996
$$268$$ 0 0
$$269$$ −5.02201 −0.306197 −0.153099 0.988211i $$-0.548925\pi$$
−0.153099 + 0.988211i $$0.548925\pi$$
$$270$$ 0 0
$$271$$ −6.96869 −0.423318 −0.211659 0.977344i $$-0.567887\pi$$
−0.211659 + 0.977344i $$0.567887\pi$$
$$272$$ 0 0
$$273$$ 0.262955 0.0159148
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −12.7083 −0.763568 −0.381784 0.924252i $$-0.624690\pi$$
−0.381784 + 0.924252i $$0.624690\pi$$
$$278$$ 0 0
$$279$$ −12.2149 −0.731285
$$280$$ 0 0
$$281$$ −16.5364 −0.986478 −0.493239 0.869894i $$-0.664187\pi$$
−0.493239 + 0.869894i $$0.664187\pi$$
$$282$$ 0 0
$$283$$ 26.1548 1.55474 0.777371 0.629042i $$-0.216553\pi$$
0.777371 + 0.629042i $$0.216553\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −0.309969 −0.0182969
$$288$$ 0 0
$$289$$ −16.9179 −0.995173
$$290$$ 0 0
$$291$$ 2.75382 0.161432
$$292$$ 0 0
$$293$$ −3.95449 −0.231023 −0.115512 0.993306i $$-0.536851\pi$$
−0.115512 + 0.993306i $$0.536851\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −7.23571 −0.419859
$$298$$ 0 0
$$299$$ −3.00000 −0.173494
$$300$$ 0 0
$$301$$ 1.26819 0.0730972
$$302$$ 0 0
$$303$$ −0.209636 −0.0120433
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 20.7355 1.18344 0.591720 0.806144i $$-0.298449\pi$$
0.591720 + 0.806144i $$0.298449\pi$$
$$308$$ 0 0
$$309$$ −0.889198 −0.0505847
$$310$$ 0 0
$$311$$ 1.97799 0.112162 0.0560808 0.998426i $$-0.482140\pi$$
0.0560808 + 0.998426i $$0.482140\pi$$
$$312$$ 0 0
$$313$$ −8.67699 −0.490453 −0.245226 0.969466i $$-0.578862\pi$$
−0.245226 + 0.969466i $$0.578862\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.34502 −0.131709 −0.0658546 0.997829i $$-0.520977\pi$$
−0.0658546 + 0.997829i $$0.520977\pi$$
$$318$$ 0 0
$$319$$ −9.68106 −0.542035
$$320$$ 0 0
$$321$$ 4.06005 0.226610
$$322$$ 0 0
$$323$$ 0.286462 0.0159392
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 2.75532 0.152369
$$328$$ 0 0
$$329$$ −0.217440 −0.0119878
$$330$$ 0 0
$$331$$ 22.8579 1.25638 0.628192 0.778059i $$-0.283795\pi$$
0.628192 + 0.778059i $$0.283795\pi$$
$$332$$ 0 0
$$333$$ −25.3189 −1.38747
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −0.386795 −0.0210701 −0.0105350 0.999945i $$-0.503353\pi$$
−0.0105350 + 0.999945i $$0.503353\pi$$
$$338$$ 0 0
$$339$$ −2.51437 −0.136562
$$340$$ 0 0
$$341$$ −17.8672 −0.967563
$$342$$ 0 0
$$343$$ 3.98696 0.215276
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −14.1406 −0.759108 −0.379554 0.925170i $$-0.623923\pi$$
−0.379554 + 0.925170i $$0.623923\pi$$
$$348$$ 0 0
$$349$$ −2.81387 −0.150623 −0.0753115 0.997160i $$-0.523995\pi$$
−0.0753115 + 0.997160i $$0.523995\pi$$
$$350$$ 0 0
$$351$$ 5.43231 0.289955
$$352$$ 0 0
$$353$$ −13.3137 −0.708617 −0.354308 0.935129i $$-0.615284\pi$$
−0.354308 + 0.935129i $$0.615284\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0.0235072 0.00124413
$$358$$ 0 0
$$359$$ −14.1861 −0.748715 −0.374358 0.927284i $$-0.622137\pi$$
−0.374358 + 0.927284i $$0.622137\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −2.06752 −0.108517
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 28.0272 1.46301 0.731505 0.681836i $$-0.238818\pi$$
0.731505 + 0.681836i $$0.238818\pi$$
$$368$$ 0 0
$$369$$ −3.15739 −0.164367
$$370$$ 0 0
$$371$$ −1.26819 −0.0658411
$$372$$ 0 0
$$373$$ −24.3502 −1.26081 −0.630404 0.776267i $$-0.717111\pi$$
−0.630404 + 0.776267i $$0.717111\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7.26819 0.374331
$$378$$ 0 0
$$379$$ −28.8866 −1.48381 −0.741903 0.670507i $$-0.766077\pi$$
−0.741903 + 0.670507i $$0.766077\pi$$
$$380$$ 0 0
$$381$$ 1.93995 0.0993865
$$382$$ 0 0
$$383$$ 10.3398 0.528338 0.264169 0.964476i $$-0.414902\pi$$
0.264169 + 0.964476i $$0.414902\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 12.9179 0.656656
$$388$$ 0 0
$$389$$ 33.9086 1.71924 0.859618 0.510937i $$-0.170702\pi$$
0.859618 + 0.510937i $$0.170702\pi$$
$$390$$ 0 0
$$391$$ −0.268189 −0.0135629
$$392$$ 0 0
$$393$$ −0.835879 −0.0421645
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −33.8866 −1.70072 −0.850361 0.526200i $$-0.823616\pi$$
−0.850361 + 0.526200i $$0.823616\pi$$
$$398$$ 0 0
$$399$$ 0.0820605 0.00410816
$$400$$ 0 0
$$401$$ −10.5051 −0.524598 −0.262299 0.964987i $$-0.584481\pi$$
−0.262299 + 0.964987i $$0.584481\pi$$
$$402$$ 0 0
$$403$$ 13.4140 0.668201
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −37.0350 −1.83576
$$408$$ 0 0
$$409$$ −6.78256 −0.335376 −0.167688 0.985840i $$-0.553630\pi$$
−0.167688 + 0.985840i $$0.553630\pi$$
$$410$$ 0 0
$$411$$ 4.13538 0.203983
$$412$$ 0 0
$$413$$ −1.34651 −0.0662577
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −0.658720 −0.0322576
$$418$$ 0 0
$$419$$ −18.6498 −0.911100 −0.455550 0.890210i $$-0.650557\pi$$
−0.455550 + 0.890210i $$0.650557\pi$$
$$420$$ 0 0
$$421$$ 9.64975 0.470300 0.235150 0.971959i $$-0.424442\pi$$
0.235150 + 0.971959i $$0.424442\pi$$
$$422$$ 0 0
$$423$$ −2.21487 −0.107691
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −2.91794 −0.141209
$$428$$ 0 0
$$429$$ 3.91794 0.189160
$$430$$ 0 0
$$431$$ 33.7758 1.62692 0.813462 0.581618i $$-0.197580\pi$$
0.813462 + 0.581618i $$0.197580\pi$$
$$432$$ 0 0
$$433$$ 1.08986 0.0523755 0.0261878 0.999657i $$-0.491663\pi$$
0.0261878 + 0.999657i $$0.491663\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −0.936212 −0.0447851
$$438$$ 0 0
$$439$$ 12.9687 0.618962 0.309481 0.950906i $$-0.399845\pi$$
0.309481 + 0.950906i $$0.399845\pi$$
$$440$$ 0 0
$$441$$ 20.1861 0.961244
$$442$$ 0 0
$$443$$ 31.9437 1.51769 0.758845 0.651271i $$-0.225764\pi$$
0.758845 + 0.651271i $$0.225764\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −1.93471 −0.0915088
$$448$$ 0 0
$$449$$ −5.05075 −0.238360 −0.119180 0.992873i $$-0.538026\pi$$
−0.119180 + 0.992873i $$0.538026\pi$$
$$450$$ 0 0
$$451$$ −4.61844 −0.217474
$$452$$ 0 0
$$453$$ −3.40507 −0.159984
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 35.0948 1.64166 0.820832 0.571170i $$-0.193510\pi$$
0.820832 + 0.571170i $$0.193510\pi$$
$$458$$ 0 0
$$459$$ 0.485629 0.0226672
$$460$$ 0 0
$$461$$ −27.8866 −1.29881 −0.649405 0.760443i $$-0.724982\pi$$
−0.649405 + 0.760443i $$0.724982\pi$$
$$462$$ 0 0
$$463$$ 28.0037 1.30144 0.650722 0.759316i $$-0.274466\pi$$
0.650722 + 0.759316i $$0.274466\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −17.4543 −0.807690 −0.403845 0.914828i $$-0.632326\pi$$
−0.403845 + 0.914828i $$0.632326\pi$$
$$468$$ 0 0
$$469$$ 2.81387 0.129933
$$470$$ 0 0
$$471$$ 4.06005 0.187077
$$472$$ 0 0
$$473$$ 18.8956 0.868821
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −12.9179 −0.591472
$$478$$ 0 0
$$479$$ −26.0440 −1.18998 −0.594991 0.803733i $$-0.702844\pi$$
−0.594991 + 0.803733i $$0.702844\pi$$
$$480$$ 0 0
$$481$$ 27.8046 1.26778
$$482$$ 0 0
$$483$$ −0.0768261 −0.00349571
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 21.9907 0.996494 0.498247 0.867035i $$-0.333977\pi$$
0.498247 + 0.867035i $$0.333977\pi$$
$$488$$ 0 0
$$489$$ 2.26146 0.102267
$$490$$ 0 0
$$491$$ 33.9907 1.53398 0.766989 0.641660i $$-0.221754\pi$$
0.766989 + 0.641660i $$0.221754\pi$$
$$492$$ 0 0
$$493$$ 0.649750 0.0292633
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1.38530 0.0621390
$$498$$ 0 0
$$499$$ −6.83331 −0.305901 −0.152950 0.988234i $$-0.548877\pi$$
−0.152950 + 0.988234i $$0.548877\pi$$
$$500$$ 0 0
$$501$$ 5.86462 0.262012
$$502$$ 0 0
$$503$$ −34.4685 −1.53688 −0.768438 0.639925i $$-0.778966\pi$$
−0.768438 + 0.639925i $$0.778966\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0.782560 0.0347547
$$508$$ 0 0
$$509$$ 18.9179 0.838523 0.419261 0.907866i $$-0.362289\pi$$
0.419261 + 0.907866i $$0.362289\pi$$
$$510$$ 0 0
$$511$$ −3.00000 −0.132712
$$512$$ 0 0
$$513$$ 1.69527 0.0748478
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −3.23978 −0.142485
$$518$$ 0 0
$$519$$ 6.88920 0.302402
$$520$$ 0 0
$$521$$ −1.62774 −0.0713127 −0.0356563 0.999364i $$-0.511352\pi$$
−0.0356563 + 0.999364i $$0.511352\pi$$
$$522$$ 0 0
$$523$$ 1.16669 0.0510158 0.0255079 0.999675i $$-0.491880\pi$$
0.0255079 + 0.999675i $$0.491880\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1.19917 0.0522365
$$528$$ 0 0
$$529$$ −22.1235 −0.961892
$$530$$ 0 0
$$531$$ −13.7158 −0.595214
$$532$$ 0 0
$$533$$ 3.46736 0.150188
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 5.34875 0.230816
$$538$$ 0 0
$$539$$ 29.5271 1.27182
$$540$$ 0 0
$$541$$ −23.9399 −1.02926 −0.514629 0.857413i $$-0.672070\pi$$
−0.514629 + 0.857413i $$0.672070\pi$$
$$542$$ 0 0
$$543$$ 1.01304 0.0434736
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −14.8139 −0.633395 −0.316698 0.948527i $$-0.602574\pi$$
−0.316698 + 0.948527i $$0.602574\pi$$
$$548$$ 0 0
$$549$$ −29.7225 −1.26853
$$550$$ 0 0
$$551$$ 2.26819 0.0966281
$$552$$ 0 0
$$553$$ 1.46212 0.0621757
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −39.3592 −1.66770 −0.833852 0.551988i $$-0.813869\pi$$
−0.833852 + 0.551988i $$0.813869\pi$$
$$558$$ 0 0
$$559$$ −14.1861 −0.600009
$$560$$ 0 0
$$561$$ 0.350250 0.0147876
$$562$$ 0 0
$$563$$ −9.55722 −0.402789 −0.201394 0.979510i $$-0.564547\pi$$
−0.201394 + 0.979510i $$0.564547\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −2.36852 −0.0994686
$$568$$ 0 0
$$569$$ 21.2995 0.892922 0.446461 0.894803i $$-0.352684\pi$$
0.446461 + 0.894803i $$0.352684\pi$$
$$570$$ 0 0
$$571$$ 17.4543 0.730440 0.365220 0.930921i $$-0.380994\pi$$
0.365220 + 0.930921i $$0.380994\pi$$
$$572$$ 0 0
$$573$$ −4.08729 −0.170749
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −37.2861 −1.55224 −0.776121 0.630584i $$-0.782815\pi$$
−0.776121 + 0.630584i $$0.782815\pi$$
$$578$$ 0 0
$$579$$ 2.34352 0.0973932
$$580$$ 0 0
$$581$$ 4.50764 0.187008
$$582$$ 0 0
$$583$$ −18.8956 −0.782576
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 7.73181 0.319126 0.159563 0.987188i $$-0.448991\pi$$
0.159563 + 0.987188i $$0.448991\pi$$
$$588$$ 0 0
$$589$$ 4.18613 0.172486
$$590$$ 0 0
$$591$$ −3.62101 −0.148948
$$592$$ 0 0
$$593$$ −6.74858 −0.277131 −0.138566 0.990353i $$-0.544249\pi$$
−0.138566 + 0.990353i $$0.544249\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −1.17566 −0.0481166
$$598$$ 0 0
$$599$$ 16.9687 0.693322 0.346661 0.937991i $$-0.387315\pi$$
0.346661 + 0.937991i $$0.387315\pi$$
$$600$$ 0 0
$$601$$ 31.7251 1.29409 0.647046 0.762451i $$-0.276004\pi$$
0.647046 + 0.762451i $$0.276004\pi$$
$$602$$ 0 0
$$603$$ 28.6625 1.16723
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −3.84518 −0.156071 −0.0780356 0.996951i $$-0.524865\pi$$
−0.0780356 + 0.996951i $$0.524865\pi$$
$$608$$ 0 0
$$609$$ 0.186129 0.00754232
$$610$$ 0 0
$$611$$ 2.43231 0.0984007
$$612$$ 0 0
$$613$$ −32.1208 −1.29735 −0.648674 0.761066i $$-0.724676\pi$$
−0.648674 + 0.761066i $$0.724676\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 35.2630 1.41963 0.709817 0.704387i $$-0.248778\pi$$
0.709817 + 0.704387i $$0.248778\pi$$
$$618$$ 0 0
$$619$$ −9.29020 −0.373405 −0.186702 0.982417i $$-0.559780\pi$$
−0.186702 + 0.982417i $$0.559780\pi$$
$$620$$ 0 0
$$621$$ −1.58713 −0.0636893
$$622$$ 0 0
$$623$$ 2.79410 0.111943
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 1.22267 0.0488289
$$628$$ 0 0
$$629$$ 2.48563 0.0991085
$$630$$ 0 0
$$631$$ 26.5271 1.05603 0.528013 0.849236i $$-0.322937\pi$$
0.528013 + 0.849236i $$0.322937\pi$$
$$632$$ 0 0
$$633$$ 0.440114 0.0174930
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −22.1679 −0.878322
$$638$$ 0 0
$$639$$ 14.1108 0.558215
$$640$$ 0 0
$$641$$ −7.86462 −0.310634 −0.155317 0.987865i $$-0.549640\pi$$
−0.155317 + 0.987865i $$0.549640\pi$$
$$642$$ 0 0
$$643$$ 1.67176 0.0659277 0.0329638 0.999457i $$-0.489505\pi$$
0.0329638 + 0.999457i $$0.489505\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −38.4909 −1.51323 −0.756616 0.653859i $$-0.773149\pi$$
−0.756616 + 0.653859i $$0.773149\pi$$
$$648$$ 0 0
$$649$$ −20.0626 −0.787527
$$650$$ 0 0
$$651$$ 0.343516 0.0134634
$$652$$ 0 0
$$653$$ −25.9985 −1.01740 −0.508700 0.860944i $$-0.669874\pi$$
−0.508700 + 0.860944i $$0.669874\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −30.5584 −1.19220
$$658$$ 0 0
$$659$$ −46.9594 −1.82928 −0.914639 0.404272i $$-0.867525\pi$$
−0.914639 + 0.404272i $$0.867525\pi$$
$$660$$ 0 0
$$661$$ −35.7225 −1.38944 −0.694722 0.719278i $$-0.744473\pi$$
−0.694722 + 0.719278i $$0.744473\pi$$
$$662$$ 0 0
$$663$$ −0.262955 −0.0102123
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −2.12351 −0.0822225
$$668$$ 0 0
$$669$$ 0.403569 0.0156029
$$670$$ 0 0
$$671$$ −43.4763 −1.67838
$$672$$ 0 0
$$673$$ 27.3450 1.05407 0.527036 0.849843i $$-0.323303\pi$$
0.527036 + 0.849843i $$0.323303\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −39.9944 −1.53711 −0.768555 0.639783i $$-0.779024\pi$$
−0.768555 + 0.639783i $$0.779024\pi$$
$$678$$ 0 0
$$679$$ 2.75382 0.105682
$$680$$ 0 0
$$681$$ 6.56512 0.251576
$$682$$ 0 0
$$683$$ −21.0130 −0.804042 −0.402021 0.915631i $$-0.631692\pi$$
−0.402021 + 0.915631i $$0.631692\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −1.08430 −0.0413685
$$688$$ 0 0
$$689$$ 14.1861 0.540448
$$690$$ 0 0
$$691$$ 10.7445 0.408741 0.204370 0.978894i $$-0.434485\pi$$
0.204370 + 0.978894i $$0.434485\pi$$
$$692$$ 0 0
$$693$$ −3.56769 −0.135525
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0.309969 0.0117409
$$698$$ 0 0
$$699$$ −0.649750 −0.0245758
$$700$$ 0 0
$$701$$ 42.3029 1.59776 0.798879 0.601491i $$-0.205427\pi$$
0.798879 + 0.601491i $$0.205427\pi$$
$$702$$ 0 0
$$703$$ 8.67699 0.327259
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −0.209636 −0.00788417
$$708$$ 0 0
$$709$$ −40.5584 −1.52320 −0.761601 0.648046i $$-0.775586\pi$$
−0.761601 + 0.648046i $$0.775586\pi$$
$$710$$ 0 0
$$711$$ 14.8934 0.558545
$$712$$ 0 0
$$713$$ −3.91911 −0.146772
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −0.626243 −0.0233875
$$718$$ 0 0
$$719$$ −16.8866 −0.629765 −0.314882 0.949131i $$-0.601965\pi$$
−0.314882 + 0.949131i $$0.601965\pi$$
$$720$$ 0 0
$$721$$ −0.889198 −0.0331155
$$722$$ 0 0
$$723$$ −5.72658 −0.212974
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −27.4398 −1.01769 −0.508843 0.860860i $$-0.669926\pi$$
−0.508843 + 0.860860i $$0.669926\pi$$
$$728$$ 0 0
$$729$$ −22.6692 −0.839600
$$730$$ 0 0
$$731$$ −1.26819 −0.0469057
$$732$$ 0 0
$$733$$ −14.0078 −0.517390 −0.258695 0.965959i $$-0.583292\pi$$
−0.258695 + 0.965959i $$0.583292\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 41.9257 1.54435
$$738$$ 0 0
$$739$$ 19.7851 0.727808 0.363904 0.931437i $$-0.381444\pi$$
0.363904 + 0.931437i $$0.381444\pi$$
$$740$$ 0 0
$$741$$ −0.917939 −0.0337213
$$742$$ 0 0
$$743$$ −41.8344 −1.53475 −0.767377 0.641196i $$-0.778439\pi$$
−0.767377 + 0.641196i $$0.778439\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 45.9154 1.67996
$$748$$ 0 0
$$749$$ 4.06005 0.148351
$$750$$ 0 0
$$751$$ −48.4983 −1.76973 −0.884865 0.465848i $$-0.845749\pi$$
−0.884865 + 0.465848i $$0.845749\pi$$
$$752$$ 0 0
$$753$$ 0.386795 0.0140956
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 32.2537 1.17228 0.586139 0.810210i $$-0.300647\pi$$
0.586139 + 0.810210i $$0.300647\pi$$
$$758$$ 0 0
$$759$$ −1.14468 −0.0415493
$$760$$ 0 0
$$761$$ −25.6184 −0.928668 −0.464334 0.885660i $$-0.653706\pi$$
−0.464334 + 0.885660i $$0.653706\pi$$
$$762$$ 0 0
$$763$$ 2.75532 0.0997492
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 15.0623 0.543868
$$768$$ 0 0
$$769$$ 42.3096 1.52572 0.762862 0.646561i $$-0.223793\pi$$
0.762862 + 0.646561i $$0.223793\pi$$
$$770$$ 0 0
$$771$$ 3.42558 0.123369
$$772$$ 0 0
$$773$$ −0.889198 −0.0319822 −0.0159911 0.999872i $$-0.505090\pi$$
−0.0159911 + 0.999872i $$0.505090\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0.712038 0.0255442
$$778$$ 0 0
$$779$$ 1.08206 0.0387688
$$780$$ 0 0
$$781$$ 20.6404 0.738573
$$782$$ 0 0
$$783$$ 3.84518 0.137416
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −50.6650 −1.80601 −0.903007 0.429627i $$-0.858645\pi$$
−0.903007 + 0.429627i $$0.858645\pi$$
$$788$$ 0 0
$$789$$ −2.38156 −0.0847858
$$790$$ 0 0
$$791$$ −2.51437 −0.0894007
$$792$$ 0 0
$$793$$ 32.6404 1.15910
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 33.4909 1.18631 0.593154 0.805089i $$-0.297883\pi$$
0.593154 + 0.805089i $$0.297883\pi$$
$$798$$ 0 0
$$799$$ 0.217440 0.00769247
$$800$$ 0 0
$$801$$ 28.4611 1.00562
$$802$$ 0 0
$$803$$ −44.6990 −1.57739
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 1.43861 0.0506416
$$808$$ 0 0
$$809$$ −6.35025 −0.223263 −0.111631 0.993750i $$-0.535608\pi$$
−0.111631 + 0.993750i $$0.535608\pi$$
$$810$$ 0 0
$$811$$ 1.08463 0.0380865 0.0190433 0.999819i $$-0.493938\pi$$
0.0190433 + 0.999819i $$0.493938\pi$$
$$812$$ 0 0
$$813$$ 1.99627 0.0700121
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −4.42708 −0.154884
$$818$$ 0 0
$$819$$ 2.67849 0.0935941
$$820$$ 0 0
$$821$$ −12.3630 −0.431470 −0.215735 0.976452i $$-0.569215\pi$$
−0.215735 + 0.976452i $$0.569215\pi$$
$$822$$ 0 0
$$823$$ −27.0765 −0.943827 −0.471914 0.881645i $$-0.656437\pi$$
−0.471914 + 0.881645i $$0.656437\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 17.0765 0.593808 0.296904 0.954907i $$-0.404046\pi$$
0.296904 + 0.954907i $$0.404046\pi$$
$$828$$ 0 0
$$829$$ −28.5804 −0.992638 −0.496319 0.868140i $$-0.665315\pi$$
−0.496319 + 0.868140i $$0.665315\pi$$
$$830$$ 0 0
$$831$$ 3.64045 0.126286
$$832$$ 0 0
$$833$$ −1.98173 −0.0686628
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 7.09660 0.245294
$$838$$ 0 0
$$839$$ 55.9019 1.92995 0.964974 0.262346i $$-0.0844960\pi$$
0.964974 + 0.262346i $$0.0844960\pi$$
$$840$$ 0 0
$$841$$ −23.8553 −0.822597
$$842$$ 0 0
$$843$$ 4.73705 0.163153
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −2.06752 −0.0710409
$$848$$ 0 0
$$849$$ −7.49236 −0.257137
$$850$$ 0 0
$$851$$ −8.12351 −0.278470
$$852$$ 0 0
$$853$$ −4.93214 −0.168873 −0.0844367 0.996429i $$-0.526909\pi$$
−0.0844367 + 0.996429i $$0.526909\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 39.9411 1.36436 0.682181 0.731183i $$-0.261032\pi$$
0.682181 + 0.731183i $$0.261032\pi$$
$$858$$ 0 0
$$859$$ −1.99070 −0.0679217 −0.0339608 0.999423i $$-0.510812\pi$$
−0.0339608 + 0.999423i $$0.510812\pi$$
$$860$$ 0 0
$$861$$ 0.0887944 0.00302611
$$862$$ 0 0
$$863$$ −31.9817 −1.08867 −0.544335 0.838868i $$-0.683218\pi$$
−0.544335 + 0.838868i $$0.683218\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 4.84635 0.164591
$$868$$ 0 0
$$869$$ 21.7851 0.739010
$$870$$ 0 0
$$871$$ −31.4763 −1.06653
$$872$$ 0 0
$$873$$ 28.0507 0.949374
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −22.2917 −0.752737 −0.376369 0.926470i $$-0.622827\pi$$
−0.376369 + 0.926470i $$0.622827\pi$$
$$878$$ 0 0
$$879$$ 1.13281 0.0382087
$$880$$ 0 0
$$881$$ 35.6912 1.20247 0.601233 0.799073i $$-0.294676\pi$$
0.601233 + 0.799073i $$0.294676\pi$$
$$882$$ 0 0
$$883$$ −17.9948 −0.605572 −0.302786 0.953059i $$-0.597917\pi$$
−0.302786 + 0.953059i $$0.597917\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −24.3566 −0.817813 −0.408907 0.912576i $$-0.634090\pi$$
−0.408907 + 0.912576i $$0.634090\pi$$
$$888$$ 0 0
$$889$$ 1.93995 0.0650637
$$890$$ 0 0
$$891$$ −35.2902 −1.18227
$$892$$ 0 0
$$893$$ 0.759053 0.0254007
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0.859386 0.0286941
$$898$$ 0 0
$$899$$ 9.49493 0.316674
$$900$$ 0 0
$$901$$ 1.26819 0.0422495
$$902$$ 0 0
$$903$$ −0.363288 −0.0120895
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −5.73821 −0.190534 −0.0952671 0.995452i $$-0.530371\pi$$
−0.0952671 + 0.995452i $$0.530371\pi$$
$$908$$ 0 0
$$909$$ −2.13538 −0.0708261
$$910$$ 0 0
$$911$$ 46.8291 1.55152 0.775759 0.631029i $$-0.217367\pi$$
0.775759 + 0.631029i $$0.217367\pi$$
$$912$$ 0 0
$$913$$ 67.1623 2.22275
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −0.835879 −0.0276032
$$918$$ 0 0
$$919$$ −2.32151 −0.0765795 −0.0382897 0.999267i $$-0.512191\pi$$
−0.0382897 + 0.999267i $$0.512191\pi$$
$$920$$ 0 0
$$921$$ −5.93995 −0.195728
$$922$$ 0 0
$$923$$ −15.4961 −0.510060
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −9.05748 −0.297487
$$928$$ 0 0
$$929$$ −25.5610 −0.838628 −0.419314 0.907841i $$-0.637729\pi$$
−0.419314 + 0.907841i $$0.637729\pi$$
$$930$$ 0 0
$$931$$ −6.91794 −0.226726
$$932$$ 0 0
$$933$$ −0.566620 −0.0185503
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −25.7501 −0.841219 −0.420609 0.907242i $$-0.638184\pi$$
−0.420609 + 0.907242i $$0.638184\pi$$
$$938$$ 0 0
$$939$$ 2.48563 0.0811154
$$940$$ 0 0
$$941$$ −29.1261 −0.949483 −0.474741 0.880125i $$-0.657458\pi$$
−0.474741 + 0.880125i $$0.657458\pi$$
$$942$$ 0 0
$$943$$ −1.01304 −0.0329891
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 6.21711 0.202029 0.101014 0.994885i $$-0.467791\pi$$
0.101014 + 0.994885i $$0.467791\pi$$
$$948$$ 0 0
$$949$$ 33.5584 1.08935
$$950$$ 0 0
$$951$$ 0.671758 0.0217832
$$952$$ 0 0
$$953$$ 47.8266 1.54925 0.774627 0.632418i $$-0.217937\pi$$
0.774627 + 0.632418i $$0.217937\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 2.77326 0.0896467
$$958$$ 0 0
$$959$$ 4.13538 0.133538
$$960$$ 0 0
$$961$$ −13.4763 −0.434720
$$962$$ 0 0
$$963$$ 41.3562 1.33269
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 37.2301 1.19724 0.598620 0.801033i $$-0.295716\pi$$
0.598620 + 0.801033i $$0.295716\pi$$
$$968$$ 0 0
$$969$$ −0.0820605 −0.00263616
$$970$$ 0 0
$$971$$ −37.8359 −1.21421 −0.607106 0.794621i $$-0.707669\pi$$
−0.607106 + 0.794621i $$0.707669\pi$$
$$972$$ 0 0
$$973$$ −0.658720 −0.0211176
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 55.8684 1.78739 0.893694 0.448678i $$-0.148105\pi$$
0.893694 + 0.448678i $$0.148105\pi$$
$$978$$ 0 0
$$979$$ 41.6311 1.33054
$$980$$ 0 0
$$981$$ 28.0660 0.896079
$$982$$ 0 0
$$983$$ 6.93738 0.221268 0.110634 0.993861i $$-0.464712\pi$$
0.110634 + 0.993861i $$0.464712\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0.0622883 0.00198266
$$988$$ 0 0
$$989$$ 4.14468 0.131793
$$990$$ 0 0
$$991$$ −36.5897 −1.16231 −0.581155 0.813793i $$-0.697399\pi$$
−0.581155 + 0.813793i $$0.697399\pi$$
$$992$$ 0 0
$$993$$ −6.54792 −0.207792
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 33.7538 1.06899 0.534497 0.845170i $$-0.320501\pi$$
0.534497 + 0.845170i $$0.320501\pi$$
$$998$$ 0 0
$$999$$ 14.7098 0.465398
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 7600.2.a.bj.1.2 3
4.3 odd 2 1900.2.a.h.1.2 yes 3
5.4 even 2 7600.2.a.by.1.2 3
20.3 even 4 1900.2.c.g.1749.4 6
20.7 even 4 1900.2.c.g.1749.3 6
20.19 odd 2 1900.2.a.f.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1900.2.a.f.1.2 3 20.19 odd 2
1900.2.a.h.1.2 yes 3 4.3 odd 2
1900.2.c.g.1749.3 6 20.7 even 4
1900.2.c.g.1749.4 6 20.3 even 4
7600.2.a.bj.1.2 3 1.1 even 1 trivial
7600.2.a.by.1.2 3 5.4 even 2