# Properties

 Label 7600.2.a.by.1.1 Level $7600$ Weight $2$ Character 7600.1 Self dual yes Analytic conductor $60.686$ Analytic rank $0$ 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: $$0$$ 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 1.1 Root $$2.19869$$ 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)

 $$f(q)$$ $$=$$ $$q-1.19869 q^{3} -1.19869 q^{7} -1.56314 q^{9} +O(q^{10})$$ $$q-1.19869 q^{3} -1.19869 q^{7} -1.56314 q^{9} +5.86718 q^{11} -0.364448 q^{13} +1.19869 q^{17} +1.00000 q^{19} +1.43686 q^{21} +8.23163 q^{23} +5.46980 q^{27} -7.86718 q^{29} -7.30404 q^{31} -7.03293 q^{33} +7.13828 q^{37} +0.436861 q^{39} +2.43686 q^{41} +7.39738 q^{43} +13.7014 q^{47} -5.56314 q^{49} -1.43686 q^{51} -7.39738 q^{53} -1.19869 q^{57} -12.8606 q^{59} -1.30404 q^{61} +1.87372 q^{63} -11.9330 q^{67} -9.86718 q^{69} -2.12628 q^{71} +2.50273 q^{73} -7.03293 q^{77} +7.74090 q^{79} -1.86718 q^{81} +3.02093 q^{83} +9.43032 q^{87} -5.68942 q^{89} +0.436861 q^{91} +8.75529 q^{93} +1.09334 q^{97} -9.17122 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} + 2 q^{7} + q^{9}+O(q^{10})$$ 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
$$3$$ −1.19869 −0.692065 −0.346032 0.938223i $$-0.612471\pi$$
−0.346032 + 0.938223i $$0.612471\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.19869 −0.453063 −0.226531 0.974004i $$-0.572739\pi$$
−0.226531 + 0.974004i $$0.572739\pi$$
$$8$$ 0 0
$$9$$ −1.56314 −0.521046
$$10$$ 0 0
$$11$$ 5.86718 1.76902 0.884510 0.466521i $$-0.154493\pi$$
0.884510 + 0.466521i $$0.154493\pi$$
$$12$$ 0 0
$$13$$ −0.364448 −0.101080 −0.0505399 0.998722i $$-0.516094\pi$$
−0.0505399 + 0.998722i $$0.516094\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.19869 0.290725 0.145363 0.989378i $$-0.453565\pi$$
0.145363 + 0.989378i $$0.453565\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 1.43686 0.313549
$$22$$ 0 0
$$23$$ 8.23163 1.71641 0.858206 0.513305i $$-0.171579\pi$$
0.858206 + 0.513305i $$0.171579\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.46980 1.05266
$$28$$ 0 0
$$29$$ −7.86718 −1.46090 −0.730449 0.682967i $$-0.760689\pi$$
−0.730449 + 0.682967i $$0.760689\pi$$
$$30$$ 0 0
$$31$$ −7.30404 −1.31184 −0.655922 0.754829i $$-0.727720\pi$$
−0.655922 + 0.754829i $$0.727720\pi$$
$$32$$ 0 0
$$33$$ −7.03293 −1.22428
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.13828 1.17353 0.586763 0.809759i $$-0.300402\pi$$
0.586763 + 0.809759i $$0.300402\pi$$
$$38$$ 0 0
$$39$$ 0.436861 0.0699537
$$40$$ 0 0
$$41$$ 2.43686 0.380574 0.190287 0.981729i $$-0.439058\pi$$
0.190287 + 0.981729i $$0.439058\pi$$
$$42$$ 0 0
$$43$$ 7.39738 1.12809 0.564045 0.825744i $$-0.309244\pi$$
0.564045 + 0.825744i $$0.309244\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 13.7014 1.99856 0.999279 0.0379717i $$-0.0120897\pi$$
0.999279 + 0.0379717i $$0.0120897\pi$$
$$48$$ 0 0
$$49$$ −5.56314 −0.794734
$$50$$ 0 0
$$51$$ −1.43686 −0.201201
$$52$$ 0 0
$$53$$ −7.39738 −1.01611 −0.508054 0.861325i $$-0.669635\pi$$
−0.508054 + 0.861325i $$0.669635\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −1.19869 −0.158771
$$58$$ 0 0
$$59$$ −12.8606 −1.67431 −0.837156 0.546964i $$-0.815783\pi$$
−0.837156 + 0.546964i $$0.815783\pi$$
$$60$$ 0 0
$$61$$ −1.30404 −0.166965 −0.0834825 0.996509i $$-0.526604\pi$$
−0.0834825 + 0.996509i $$0.526604\pi$$
$$62$$ 0 0
$$63$$ 1.87372 0.236067
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −11.9330 −1.45785 −0.728927 0.684592i $$-0.759981\pi$$
−0.728927 + 0.684592i $$0.759981\pi$$
$$68$$ 0 0
$$69$$ −9.86718 −1.18787
$$70$$ 0 0
$$71$$ −2.12628 −0.252343 −0.126171 0.992008i $$-0.540269\pi$$
−0.126171 + 0.992008i $$0.540269\pi$$
$$72$$ 0 0
$$73$$ 2.50273 0.292922 0.146461 0.989216i $$-0.453212\pi$$
0.146461 + 0.989216i $$0.453212\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −7.03293 −0.801477
$$78$$ 0 0
$$79$$ 7.74090 0.870919 0.435460 0.900208i $$-0.356586\pi$$
0.435460 + 0.900208i $$0.356586\pi$$
$$80$$ 0 0
$$81$$ −1.86718 −0.207464
$$82$$ 0 0
$$83$$ 3.02093 0.331590 0.165795 0.986160i $$-0.446981\pi$$
0.165795 + 0.986160i $$0.446981\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 9.43032 1.01104
$$88$$ 0 0
$$89$$ −5.68942 −0.603077 −0.301539 0.953454i $$-0.597500\pi$$
−0.301539 + 0.953454i $$0.597500\pi$$
$$90$$ 0 0
$$91$$ 0.436861 0.0457954
$$92$$ 0 0
$$93$$ 8.75529 0.907881
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1.09334 0.111012 0.0555061 0.998458i $$-0.482323\pi$$
0.0555061 + 0.998458i $$0.482323\pi$$
$$98$$ 0 0
$$99$$ −9.17122 −0.921742
$$100$$ 0 0
$$101$$ 10.8672 1.08132 0.540662 0.841240i $$-0.318174\pi$$
0.540662 + 0.841240i $$0.318174\pi$$
$$102$$ 0 0
$$103$$ 9.74090 0.959799 0.479900 0.877323i $$-0.340673\pi$$
0.479900 + 0.877323i $$0.340673\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −16.3634 −1.58191 −0.790953 0.611877i $$-0.790415\pi$$
−0.790953 + 0.611877i $$0.790415\pi$$
$$108$$ 0 0
$$109$$ 9.29749 0.890538 0.445269 0.895397i $$-0.353108\pi$$
0.445269 + 0.895397i $$0.353108\pi$$
$$110$$ 0 0
$$111$$ −8.55660 −0.812156
$$112$$ 0 0
$$113$$ −2.96707 −0.279118 −0.139559 0.990214i $$-0.544568\pi$$
−0.139559 + 0.990214i $$0.544568\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0.569683 0.0526672
$$118$$ 0 0
$$119$$ −1.43686 −0.131717
$$120$$ 0 0
$$121$$ 23.4238 2.12943
$$122$$ 0 0
$$123$$ −2.92104 −0.263382
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 11.3579 1.00785 0.503926 0.863747i $$-0.331889\pi$$
0.503926 + 0.863747i $$0.331889\pi$$
$$128$$ 0 0
$$129$$ −8.86718 −0.780711
$$130$$ 0 0
$$131$$ 1.56314 0.136572 0.0682861 0.997666i $$-0.478247\pi$$
0.0682861 + 0.997666i $$0.478247\pi$$
$$132$$ 0 0
$$133$$ −1.19869 −0.103940
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −15.8397 −1.35328 −0.676639 0.736315i $$-0.736564\pi$$
−0.676639 + 0.736315i $$0.736564\pi$$
$$138$$ 0 0
$$139$$ 19.8606 1.68456 0.842278 0.539043i $$-0.181214\pi$$
0.842278 + 0.539043i $$0.181214\pi$$
$$140$$ 0 0
$$141$$ −16.4238 −1.38313
$$142$$ 0 0
$$143$$ −2.13828 −0.178812
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 6.66849 0.550007
$$148$$ 0 0
$$149$$ 2.68942 0.220326 0.110163 0.993914i $$-0.464863\pi$$
0.110163 + 0.993914i $$0.464863\pi$$
$$150$$ 0 0
$$151$$ −17.1647 −1.39684 −0.698421 0.715688i $$-0.746113\pi$$
−0.698421 + 0.715688i $$0.746113\pi$$
$$152$$ 0 0
$$153$$ −1.87372 −0.151481
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −16.3634 −1.30594 −0.652969 0.757384i $$-0.726477\pi$$
−0.652969 + 0.757384i $$0.726477\pi$$
$$158$$ 0 0
$$159$$ 8.86718 0.703213
$$160$$ 0 0
$$161$$ −9.86718 −0.777643
$$162$$ 0 0
$$163$$ 8.28549 0.648970 0.324485 0.945891i $$-0.394809\pi$$
0.324485 + 0.945891i $$0.394809\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 7.49727 0.580156 0.290078 0.957003i $$-0.406319\pi$$
0.290078 + 0.957003i $$0.406319\pi$$
$$168$$ 0 0
$$169$$ −12.8672 −0.989783
$$170$$ 0 0
$$171$$ −1.56314 −0.119536
$$172$$ 0 0
$$173$$ −14.7464 −1.12114 −0.560572 0.828105i $$-0.689419\pi$$
−0.560572 + 0.828105i $$0.689419\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 15.4159 1.15873
$$178$$ 0 0
$$179$$ −13.2526 −0.990543 −0.495271 0.868738i $$-0.664931\pi$$
−0.495271 + 0.868738i $$0.664931\pi$$
$$180$$ 0 0
$$181$$ 16.7344 1.24385 0.621927 0.783075i $$-0.286350\pi$$
0.621927 + 0.783075i $$0.286350\pi$$
$$182$$ 0 0
$$183$$ 1.56314 0.115551
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 7.03293 0.514299
$$188$$ 0 0
$$189$$ −6.55660 −0.476922
$$190$$ 0 0
$$191$$ 4.13282 0.299041 0.149520 0.988759i $$-0.452227\pi$$
0.149520 + 0.988759i $$0.452227\pi$$
$$192$$ 0 0
$$193$$ 7.08680 0.510119 0.255060 0.966925i $$-0.417905\pi$$
0.255060 + 0.966925i $$0.417905\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 20.4753 1.45880 0.729401 0.684087i $$-0.239799\pi$$
0.729401 + 0.684087i $$0.239799\pi$$
$$198$$ 0 0
$$199$$ −8.74090 −0.619626 −0.309813 0.950798i $$-0.600266\pi$$
−0.309813 + 0.950798i $$0.600266\pi$$
$$200$$ 0 0
$$201$$ 14.3040 1.00893
$$202$$ 0 0
$$203$$ 9.43032 0.661878
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −12.8672 −0.894331
$$208$$ 0 0
$$209$$ 5.86718 0.405841
$$210$$ 0 0
$$211$$ 18.7344 1.28973 0.644863 0.764298i $$-0.276914\pi$$
0.644863 + 0.764298i $$0.276914\pi$$
$$212$$ 0 0
$$213$$ 2.54875 0.174638
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 8.75529 0.594348
$$218$$ 0 0
$$219$$ −3.00000 −0.202721
$$220$$ 0 0
$$221$$ −0.436861 −0.0293864
$$222$$ 0 0
$$223$$ −4.27110 −0.286014 −0.143007 0.989722i $$-0.545677\pi$$
−0.143007 + 0.989722i $$0.545677\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 21.5631 1.43120 0.715598 0.698512i $$-0.246154\pi$$
0.715598 + 0.698512i $$0.246154\pi$$
$$228$$ 0 0
$$229$$ 27.4172 1.81178 0.905891 0.423511i $$-0.139203\pi$$
0.905891 + 0.423511i $$0.139203\pi$$
$$230$$ 0 0
$$231$$ 8.43032 0.554674
$$232$$ 0 0
$$233$$ 7.86718 0.515396 0.257698 0.966226i $$-0.417036\pi$$
0.257698 + 0.966226i $$0.417036\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −9.27895 −0.602732
$$238$$ 0 0
$$239$$ −9.30404 −0.601828 −0.300914 0.953651i $$-0.597292\pi$$
−0.300914 + 0.953651i $$0.597292\pi$$
$$240$$ 0 0
$$241$$ −21.9056 −1.41106 −0.705531 0.708679i $$-0.749291\pi$$
−0.705531 + 0.708679i $$0.749291\pi$$
$$242$$ 0 0
$$243$$ −14.1712 −0.909084
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −0.364448 −0.0231893
$$248$$ 0 0
$$249$$ −3.62116 −0.229482
$$250$$ 0 0
$$251$$ 7.43032 0.468997 0.234499 0.972116i $$-0.424655\pi$$
0.234499 + 0.972116i $$0.424655\pi$$
$$252$$ 0 0
$$253$$ 48.2964 3.03637
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 5.05387 0.315251 0.157626 0.987499i $$-0.449616\pi$$
0.157626 + 0.987499i $$0.449616\pi$$
$$258$$ 0 0
$$259$$ −8.55660 −0.531681
$$260$$ 0 0
$$261$$ 12.2975 0.761196
$$262$$ 0 0
$$263$$ 17.7673 1.09558 0.547789 0.836617i $$-0.315470\pi$$
0.547789 + 0.836617i $$0.315470\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.81986 0.417368
$$268$$ 0 0
$$269$$ 9.17776 0.559578 0.279789 0.960062i $$-0.409736\pi$$
0.279789 + 0.960062i $$0.409736\pi$$
$$270$$ 0 0
$$271$$ 20.7278 1.25912 0.629562 0.776950i $$-0.283234\pi$$
0.629562 + 0.776950i $$0.283234\pi$$
$$272$$ 0 0
$$273$$ −0.523661 −0.0316934
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 24.5895 1.47744 0.738721 0.674012i $$-0.235430\pi$$
0.738721 + 0.674012i $$0.235430\pi$$
$$278$$ 0 0
$$279$$ 11.4172 0.683532
$$280$$ 0 0
$$281$$ 3.73436 0.222773 0.111386 0.993777i $$-0.464471\pi$$
0.111386 + 0.993777i $$0.464471\pi$$
$$282$$ 0 0
$$283$$ 13.0318 0.774663 0.387332 0.921941i $$-0.373397\pi$$
0.387332 + 0.921941i $$0.373397\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −2.92104 −0.172424
$$288$$ 0 0
$$289$$ −15.5631 −0.915479
$$290$$ 0 0
$$291$$ −1.31058 −0.0768277
$$292$$ 0 0
$$293$$ 19.9001 1.16258 0.581288 0.813698i $$-0.302549\pi$$
0.581288 + 0.813698i $$0.302549\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 32.0923 1.86218
$$298$$ 0 0
$$299$$ −3.00000 −0.173494
$$300$$ 0 0
$$301$$ −8.86718 −0.511096
$$302$$ 0 0
$$303$$ −13.0264 −0.748347
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −8.02093 −0.457779 −0.228889 0.973452i $$-0.573509\pi$$
−0.228889 + 0.973452i $$0.573509\pi$$
$$308$$ 0 0
$$309$$ −11.6763 −0.664243
$$310$$ 0 0
$$311$$ 16.1778 0.917357 0.458678 0.888602i $$-0.348323\pi$$
0.458678 + 0.888602i $$0.348323\pi$$
$$312$$ 0 0
$$313$$ −7.13828 −0.403480 −0.201740 0.979439i $$-0.564660\pi$$
−0.201740 + 0.979439i $$0.564660\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3.96052 0.222445 0.111223 0.993796i $$-0.464523\pi$$
0.111223 + 0.993796i $$0.464523\pi$$
$$318$$ 0 0
$$319$$ −46.1581 −2.58436
$$320$$ 0 0
$$321$$ 19.6146 1.09478
$$322$$ 0 0
$$323$$ 1.19869 0.0666970
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −11.1448 −0.616310
$$328$$ 0 0
$$329$$ −16.4238 −0.905472
$$330$$ 0 0
$$331$$ 5.94852 0.326960 0.163480 0.986547i $$-0.447728\pi$$
0.163480 + 0.986547i $$0.447728\pi$$
$$332$$ 0 0
$$333$$ −11.1581 −0.611462
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8.90666 0.485176 0.242588 0.970129i $$-0.422004\pi$$
0.242588 + 0.970129i $$0.422004\pi$$
$$338$$ 0 0
$$339$$ 3.55660 0.193168
$$340$$ 0 0
$$341$$ −42.8541 −2.32068
$$342$$ 0 0
$$343$$ 15.0593 0.813127
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 18.5961 0.998290 0.499145 0.866519i $$-0.333648\pi$$
0.499145 + 0.866519i $$0.333648\pi$$
$$348$$ 0 0
$$349$$ −14.3040 −0.765678 −0.382839 0.923815i $$-0.625054\pi$$
−0.382839 + 0.923815i $$0.625054\pi$$
$$350$$ 0 0
$$351$$ −1.99346 −0.106403
$$352$$ 0 0
$$353$$ −12.7673 −0.679534 −0.339767 0.940510i $$-0.610348\pi$$
−0.339767 + 0.940510i $$0.610348\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 1.72235 0.0911566
$$358$$ 0 0
$$359$$ −2.69596 −0.142287 −0.0711437 0.997466i $$-0.522665\pi$$
−0.0711437 + 0.997466i $$0.522665\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −28.0779 −1.47371
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −3.43140 −0.179118 −0.0895589 0.995982i $$-0.528546\pi$$
−0.0895589 + 0.995982i $$0.528546\pi$$
$$368$$ 0 0
$$369$$ −3.80915 −0.198297
$$370$$ 0 0
$$371$$ 8.86718 0.460361
$$372$$ 0 0
$$373$$ 15.5697 0.806168 0.403084 0.915163i $$-0.367938\pi$$
0.403084 + 0.915163i $$0.367938\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 2.86718 0.147667
$$378$$ 0 0
$$379$$ 0.164672 0.00845864 0.00422932 0.999991i $$-0.498654\pi$$
0.00422932 + 0.999991i $$0.498654\pi$$
$$380$$ 0 0
$$381$$ −13.6146 −0.697498
$$382$$ 0 0
$$383$$ −22.3514 −1.14210 −0.571051 0.820915i $$-0.693464\pi$$
−0.571051 + 0.820915i $$0.693464\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −11.5631 −0.587787
$$388$$ 0 0
$$389$$ −9.34243 −0.473680 −0.236840 0.971549i $$-0.576112\pi$$
−0.236840 + 0.971549i $$0.576112\pi$$
$$390$$ 0 0
$$391$$ 9.86718 0.499005
$$392$$ 0 0
$$393$$ −1.87372 −0.0945167
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 4.83533 0.242678 0.121339 0.992611i $$-0.461281\pi$$
0.121339 + 0.992611i $$0.461281\pi$$
$$398$$ 0 0
$$399$$ 1.43686 0.0719330
$$400$$ 0 0
$$401$$ 37.4622 1.87077 0.935386 0.353629i $$-0.115053\pi$$
0.935386 + 0.353629i $$0.115053\pi$$
$$402$$ 0 0
$$403$$ 2.66194 0.132601
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 41.8816 2.07599
$$408$$ 0 0
$$409$$ 9.42377 0.465976 0.232988 0.972480i $$-0.425150\pi$$
0.232988 + 0.972480i $$0.425150\pi$$
$$410$$ 0 0
$$411$$ 18.9869 0.936555
$$412$$ 0 0
$$413$$ 15.4159 0.758568
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −23.8068 −1.16582
$$418$$ 0 0
$$419$$ −27.4303 −1.34006 −0.670029 0.742335i $$-0.733718\pi$$
−0.670029 + 0.742335i $$0.733718\pi$$
$$420$$ 0 0
$$421$$ 18.4303 0.898239 0.449119 0.893472i $$-0.351738\pi$$
0.449119 + 0.893472i $$0.351738\pi$$
$$422$$ 0 0
$$423$$ −21.4172 −1.04134
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1.56314 0.0756456
$$428$$ 0 0
$$429$$ 2.56314 0.123750
$$430$$ 0 0
$$431$$ 15.5117 0.747170 0.373585 0.927596i $$-0.378128\pi$$
0.373585 + 0.927596i $$0.378128\pi$$
$$432$$ 0 0
$$433$$ −31.8870 −1.53239 −0.766196 0.642607i $$-0.777853\pi$$
−0.766196 + 0.642607i $$0.777853\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 8.23163 0.393772
$$438$$ 0 0
$$439$$ −14.7278 −0.702920 −0.351460 0.936203i $$-0.614315\pi$$
−0.351460 + 0.936203i $$0.614315\pi$$
$$440$$ 0 0
$$441$$ 8.69596 0.414093
$$442$$ 0 0
$$443$$ 6.46087 0.306965 0.153483 0.988151i $$-0.450951\pi$$
0.153483 + 0.988151i $$0.450951\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −3.22378 −0.152480
$$448$$ 0 0
$$449$$ 21.2910 1.00478 0.502391 0.864641i $$-0.332454\pi$$
0.502391 + 0.864641i $$0.332454\pi$$
$$450$$ 0 0
$$451$$ 14.2975 0.673243
$$452$$ 0 0
$$453$$ 20.5751 0.966705
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 19.6465 0.919023 0.459512 0.888172i $$-0.348024\pi$$
0.459512 + 0.888172i $$0.348024\pi$$
$$458$$ 0 0
$$459$$ 6.55660 0.306036
$$460$$ 0 0
$$461$$ 1.16467 0.0542442 0.0271221 0.999632i $$-0.491366\pi$$
0.0271221 + 0.999632i $$0.491366\pi$$
$$462$$ 0 0
$$463$$ −5.15375 −0.239515 −0.119758 0.992803i $$-0.538212\pi$$
−0.119758 + 0.992803i $$0.538212\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −4.17122 −0.193021 −0.0965104 0.995332i $$-0.530768\pi$$
−0.0965104 + 0.995332i $$0.530768\pi$$
$$468$$ 0 0
$$469$$ 14.3040 0.660499
$$470$$ 0 0
$$471$$ 19.6146 0.903794
$$472$$ 0 0
$$473$$ 43.4018 1.99561
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 11.5631 0.529440
$$478$$ 0 0
$$479$$ 2.35552 0.107626 0.0538132 0.998551i $$-0.482862\pi$$
0.0538132 + 0.998551i $$0.482862\pi$$
$$480$$ 0 0
$$481$$ −2.60153 −0.118620
$$482$$ 0 0
$$483$$ 11.8277 0.538179
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 19.9056 0.902008 0.451004 0.892522i $$-0.351066\pi$$
0.451004 + 0.892522i $$0.351066\pi$$
$$488$$ 0 0
$$489$$ −9.93175 −0.449129
$$490$$ 0 0
$$491$$ −7.90557 −0.356773 −0.178387 0.983960i $$-0.557088\pi$$
−0.178387 + 0.983960i $$0.557088\pi$$
$$492$$ 0 0
$$493$$ −9.43032 −0.424720
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.54875 0.114327
$$498$$ 0 0
$$499$$ 35.7147 1.59881 0.799405 0.600792i $$-0.205148\pi$$
0.799405 + 0.600792i $$0.205148\pi$$
$$500$$ 0 0
$$501$$ −8.98691 −0.401506
$$502$$ 0 0
$$503$$ −30.7991 −1.37327 −0.686633 0.727004i $$-0.740912\pi$$
−0.686633 + 0.727004i $$0.740912\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 15.4238 0.684994
$$508$$ 0 0
$$509$$ 17.5631 0.778472 0.389236 0.921138i $$-0.372739\pi$$
0.389236 + 0.921138i $$0.372739\pi$$
$$510$$ 0 0
$$511$$ −3.00000 −0.132712
$$512$$ 0 0
$$513$$ 5.46980 0.241497
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 80.3887 3.53549
$$518$$ 0 0
$$519$$ 17.6763 0.775905
$$520$$ 0 0
$$521$$ −24.6081 −1.07810 −0.539050 0.842274i $$-0.681217\pi$$
−0.539050 + 0.842274i $$0.681217\pi$$
$$522$$ 0 0
$$523$$ −43.7147 −1.91151 −0.955756 0.294162i $$-0.904960\pi$$
−0.955756 + 0.294162i $$0.904960\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8.75529 −0.381386
$$528$$ 0 0
$$529$$ 44.7597 1.94607
$$530$$ 0 0
$$531$$ 20.1030 0.872394
$$532$$ 0 0
$$533$$ −0.888109 −0.0384683
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 15.8857 0.685520
$$538$$ 0 0
$$539$$ −32.6399 −1.40590
$$540$$ 0 0
$$541$$ −8.38538 −0.360516 −0.180258 0.983619i $$-0.557693\pi$$
−0.180258 + 0.983619i $$0.557693\pi$$
$$542$$ 0 0
$$543$$ −20.0593 −0.860828
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 26.3040 1.12468 0.562340 0.826906i $$-0.309901\pi$$
0.562340 + 0.826906i $$0.309901\pi$$
$$548$$ 0 0
$$549$$ 2.03839 0.0869965
$$550$$ 0 0
$$551$$ −7.86718 −0.335153
$$552$$ 0 0
$$553$$ −9.27895 −0.394581
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −2.66740 −0.113021 −0.0565107 0.998402i $$-0.517998\pi$$
−0.0565107 + 0.998402i $$0.517998\pi$$
$$558$$ 0 0
$$559$$ −2.69596 −0.114027
$$560$$ 0 0
$$561$$ −8.43032 −0.355928
$$562$$ 0 0
$$563$$ 37.7751 1.59203 0.796016 0.605276i $$-0.206937\pi$$
0.796016 + 0.605276i $$0.206937\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 2.23817 0.0939943
$$568$$ 0 0
$$569$$ 38.8606 1.62912 0.814561 0.580078i $$-0.196978\pi$$
0.814561 + 0.580078i $$0.196978\pi$$
$$570$$ 0 0
$$571$$ −4.17122 −0.174560 −0.0872800 0.996184i $$-0.527817\pi$$
−0.0872800 + 0.996184i $$0.527817\pi$$
$$572$$ 0 0
$$573$$ −4.95398 −0.206955
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −39.3413 −1.63780 −0.818901 0.573935i $$-0.805416\pi$$
−0.818901 + 0.573935i $$0.805416\pi$$
$$578$$ 0 0
$$579$$ −8.49489 −0.353035
$$580$$ 0 0
$$581$$ −3.62116 −0.150231
$$582$$ 0 0
$$583$$ −43.4018 −1.79752
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −17.8672 −0.737457 −0.368729 0.929537i $$-0.620207\pi$$
−0.368729 + 0.929537i $$0.620207\pi$$
$$588$$ 0 0
$$589$$ −7.30404 −0.300958
$$590$$ 0 0
$$591$$ −24.5435 −1.00959
$$592$$ 0 0
$$593$$ 13.0803 0.537142 0.268571 0.963260i $$-0.413449\pi$$
0.268571 + 0.963260i $$0.413449\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 10.4776 0.428821
$$598$$ 0 0
$$599$$ −10.7278 −0.438326 −0.219163 0.975688i $$-0.570333\pi$$
−0.219163 + 0.975688i $$0.570333\pi$$
$$600$$ 0 0
$$601$$ 39.8026 1.62358 0.811791 0.583948i $$-0.198493\pi$$
0.811791 + 0.583948i $$0.198493\pi$$
$$602$$ 0 0
$$603$$ 18.6530 0.759609
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 43.0318 1.74661 0.873304 0.487175i $$-0.161973\pi$$
0.873304 + 0.487175i $$0.161973\pi$$
$$608$$ 0 0
$$609$$ −11.3040 −0.458063
$$610$$ 0 0
$$611$$ −4.99346 −0.202014
$$612$$ 0 0
$$613$$ 15.4722 0.624915 0.312458 0.949932i $$-0.398848\pi$$
0.312458 + 0.949932i $$0.398848\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −35.5237 −1.43013 −0.715064 0.699059i $$-0.753603\pi$$
−0.715064 + 0.699059i $$0.753603\pi$$
$$618$$ 0 0
$$619$$ 15.0449 0.604707 0.302354 0.953196i $$-0.402228\pi$$
0.302354 + 0.953196i $$0.402228\pi$$
$$620$$ 0 0
$$621$$ 45.0253 1.80680
$$622$$ 0 0
$$623$$ 6.81986 0.273232
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −7.03293 −0.280868
$$628$$ 0 0
$$629$$ 8.55660 0.341174
$$630$$ 0 0
$$631$$ −35.6399 −1.41880 −0.709402 0.704805i $$-0.751035\pi$$
−0.709402 + 0.704805i $$0.751035\pi$$
$$632$$ 0 0
$$633$$ −22.4567 −0.892574
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.02748 0.0803315
$$638$$ 0 0
$$639$$ 3.32367 0.131482
$$640$$ 0 0
$$641$$ 6.98691 0.275966 0.137983 0.990435i $$-0.455938\pi$$
0.137983 + 0.990435i $$0.455938\pi$$
$$642$$ 0 0
$$643$$ 3.74744 0.147785 0.0738924 0.997266i $$-0.476458\pi$$
0.0738924 + 0.997266i $$0.476458\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 34.1658 1.34319 0.671597 0.740916i $$-0.265609\pi$$
0.671597 + 0.740916i $$0.265609\pi$$
$$648$$ 0 0
$$649$$ −75.4556 −2.96189
$$650$$ 0 0
$$651$$ −10.4949 −0.411327
$$652$$ 0 0
$$653$$ 13.5446 0.530041 0.265020 0.964243i $$-0.414621\pi$$
0.265020 + 0.964243i $$0.414621\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −3.91211 −0.152626
$$658$$ 0 0
$$659$$ 22.6334 0.881671 0.440836 0.897588i $$-0.354682\pi$$
0.440836 + 0.897588i $$0.354682\pi$$
$$660$$ 0 0
$$661$$ −3.96161 −0.154089 −0.0770443 0.997028i $$-0.524548\pi$$
−0.0770443 + 0.997028i $$0.524548\pi$$
$$662$$ 0 0
$$663$$ 0.523661 0.0203373
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −64.7597 −2.50750
$$668$$ 0 0
$$669$$ 5.11973 0.197940
$$670$$ 0 0
$$671$$ −7.65102 −0.295365
$$672$$ 0 0
$$673$$ −28.9605 −1.11635 −0.558173 0.829725i $$-0.688497\pi$$
−0.558173 + 0.829725i $$0.688497\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −24.7518 −0.951290 −0.475645 0.879637i $$-0.657785\pi$$
−0.475645 + 0.879637i $$0.657785\pi$$
$$678$$ 0 0
$$679$$ −1.31058 −0.0502955
$$680$$ 0 0
$$681$$ −25.8475 −0.990480
$$682$$ 0 0
$$683$$ 40.0593 1.53283 0.766414 0.642347i $$-0.222039\pi$$
0.766414 + 0.642347i $$0.222039\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −32.8648 −1.25387
$$688$$ 0 0
$$689$$ 2.69596 0.102708
$$690$$ 0 0
$$691$$ −35.2162 −1.33969 −0.669843 0.742503i $$-0.733639\pi$$
−0.669843 + 0.742503i $$0.733639\pi$$
$$692$$ 0 0
$$693$$ 10.9935 0.417607
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.92104 0.110642
$$698$$ 0 0
$$699$$ −9.43032 −0.356687
$$700$$ 0 0
$$701$$ −38.1283 −1.44008 −0.720042 0.693930i $$-0.755878\pi$$
−0.720042 + 0.693930i $$0.755878\pi$$
$$702$$ 0 0
$$703$$ 7.13828 0.269225
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −13.0264 −0.489908
$$708$$ 0 0
$$709$$ −6.08789 −0.228635 −0.114318 0.993444i $$-0.536468\pi$$
−0.114318 + 0.993444i $$0.536468\pi$$
$$710$$ 0 0
$$711$$ −12.1001 −0.453789
$$712$$ 0 0
$$713$$ −60.1241 −2.25167
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 11.1527 0.416504
$$718$$ 0 0
$$719$$ 12.1647 0.453666 0.226833 0.973934i $$-0.427163\pi$$
0.226833 + 0.973934i $$0.427163\pi$$
$$720$$ 0 0
$$721$$ −11.6763 −0.434849
$$722$$ 0 0
$$723$$ 26.2580 0.976546
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −25.6859 −0.952639 −0.476320 0.879272i $$-0.658029\pi$$
−0.476320 + 0.879272i $$0.658029\pi$$
$$728$$ 0 0
$$729$$ 22.5884 0.836609
$$730$$ 0 0
$$731$$ 8.86718 0.327964
$$732$$ 0 0
$$733$$ 43.4502 1.60487 0.802434 0.596741i $$-0.203538\pi$$
0.802434 + 0.596741i $$0.203538\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −70.0133 −2.57897
$$738$$ 0 0
$$739$$ 43.4172 1.59713 0.798564 0.601910i $$-0.205593\pi$$
0.798564 + 0.601910i $$0.205593\pi$$
$$740$$ 0 0
$$741$$ 0.436861 0.0160485
$$742$$ 0 0
$$743$$ 26.6709 0.978459 0.489230 0.872155i $$-0.337278\pi$$
0.489230 + 0.872155i $$0.337278\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −4.72214 −0.172774
$$748$$ 0 0
$$749$$ 19.6146 0.716703
$$750$$ 0 0
$$751$$ 1.52674 0.0557114 0.0278557 0.999612i $$-0.491132\pi$$
0.0278557 + 0.999612i $$0.491132\pi$$
$$752$$ 0 0
$$753$$ −8.90666 −0.324577
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 9.38191 0.340991 0.170496 0.985358i $$-0.445463\pi$$
0.170496 + 0.985358i $$0.445463\pi$$
$$758$$ 0 0
$$759$$ −57.8925 −2.10136
$$760$$ 0 0
$$761$$ −6.70251 −0.242966 −0.121483 0.992594i $$-0.538765\pi$$
−0.121483 + 0.992594i $$0.538765\pi$$
$$762$$ 0 0
$$763$$ −11.1448 −0.403470
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 4.68703 0.169239
$$768$$ 0 0
$$769$$ −36.0637 −1.30049 −0.650245 0.759724i $$-0.725334\pi$$
−0.650245 + 0.759724i $$0.725334\pi$$
$$770$$ 0 0
$$771$$ −6.05802 −0.218174
$$772$$ 0 0
$$773$$ 11.6763 0.419968 0.209984 0.977705i $$-0.432659\pi$$
0.209984 + 0.977705i $$0.432659\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 10.2567 0.367958
$$778$$ 0 0
$$779$$ 2.43686 0.0873096
$$780$$ 0 0
$$781$$ −12.4753 −0.446400
$$782$$ 0 0
$$783$$ −43.0318 −1.53783
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 43.1880 1.53949 0.769743 0.638354i $$-0.220384\pi$$
0.769743 + 0.638354i $$0.220384\pi$$
$$788$$ 0 0
$$789$$ −21.2975 −0.758211
$$790$$ 0 0
$$791$$ 3.55660 0.126458
$$792$$ 0 0
$$793$$ 0.475254 0.0168768
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −29.1658 −1.03310 −0.516552 0.856256i $$-0.672785\pi$$
−0.516552 + 0.856256i $$0.672785\pi$$
$$798$$ 0 0
$$799$$ 16.4238 0.581031
$$800$$ 0 0
$$801$$ 8.89335 0.314231
$$802$$ 0 0
$$803$$ 14.6840 0.518186
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −11.0013 −0.387264
$$808$$ 0 0
$$809$$ 2.43032 0.0854454 0.0427227 0.999087i $$-0.486397\pi$$
0.0427227 + 0.999087i $$0.486397\pi$$
$$810$$ 0 0
$$811$$ 42.2779 1.48458 0.742288 0.670081i $$-0.233741\pi$$
0.742288 + 0.670081i $$0.233741\pi$$
$$812$$ 0 0
$$813$$ −24.8462 −0.871396
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7.39738 0.258802
$$818$$ 0 0
$$819$$ −0.682874 −0.0238616
$$820$$ 0 0
$$821$$ 52.5136 1.83274 0.916369 0.400334i $$-0.131106\pi$$
0.916369 + 0.400334i $$0.131106\pi$$
$$822$$ 0 0
$$823$$ −36.3150 −1.26586 −0.632930 0.774209i $$-0.718148\pi$$
−0.632930 + 0.774209i $$0.718148\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 46.3150 1.61053 0.805264 0.592916i $$-0.202023\pi$$
0.805264 + 0.592916i $$0.202023\pi$$
$$828$$ 0 0
$$829$$ 20.0899 0.697750 0.348875 0.937169i $$-0.386564\pi$$
0.348875 + 0.937169i $$0.386564\pi$$
$$830$$ 0 0
$$831$$ −29.4753 −1.02249
$$832$$ 0 0
$$833$$ −6.66849 −0.231049
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −39.9516 −1.38093
$$838$$ 0 0
$$839$$ 10.5930 0.365711 0.182855 0.983140i $$-0.441466\pi$$
0.182855 + 0.983140i $$0.441466\pi$$
$$840$$ 0 0
$$841$$ 32.8925 1.13422
$$842$$ 0 0
$$843$$ −4.47634 −0.154173
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −28.0779 −0.964767
$$848$$ 0 0
$$849$$ −15.6212 −0.536117
$$850$$ 0 0
$$851$$ 58.7597 2.01426
$$852$$ 0 0
$$853$$ −40.0648 −1.37179 −0.685896 0.727700i $$-0.740590\pi$$
−0.685896 + 0.727700i $$0.740590\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 38.3019 1.30837 0.654183 0.756336i $$-0.273012\pi$$
0.654183 + 0.756336i $$0.273012\pi$$
$$858$$ 0 0
$$859$$ 39.9056 1.36156 0.680780 0.732488i $$-0.261641\pi$$
0.680780 + 0.732488i $$0.261641\pi$$
$$860$$ 0 0
$$861$$ 3.50143 0.119328
$$862$$ 0 0
$$863$$ 23.3315 0.794214 0.397107 0.917772i $$-0.370014\pi$$
0.397107 + 0.917772i $$0.370014\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 18.6554 0.633571
$$868$$ 0 0
$$869$$ 45.4172 1.54067
$$870$$ 0 0
$$871$$ 4.34898 0.147359
$$872$$ 0 0
$$873$$ −1.70905 −0.0578426
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 10.4105 0.351537 0.175768 0.984432i $$-0.443759\pi$$
0.175768 + 0.984432i $$0.443759\pi$$
$$878$$ 0 0
$$879$$ −23.8541 −0.804578
$$880$$ 0 0
$$881$$ −23.7662 −0.800704 −0.400352 0.916361i $$-0.631112\pi$$
−0.400352 + 0.916361i $$0.631112\pi$$
$$882$$ 0 0
$$883$$ 28.3908 0.955428 0.477714 0.878515i $$-0.341466\pi$$
0.477714 + 0.878515i $$0.341466\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 32.5644 1.09341 0.546703 0.837326i $$-0.315883\pi$$
0.546703 + 0.837326i $$0.315883\pi$$
$$888$$ 0 0
$$889$$ −13.6146 −0.456620
$$890$$ 0 0
$$891$$ −10.9551 −0.367008
$$892$$ 0 0
$$893$$ 13.7014 0.458501
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 3.59607 0.120069
$$898$$ 0 0
$$899$$ 57.4622 1.91647
$$900$$ 0 0
$$901$$ −8.86718 −0.295409
$$902$$ 0 0
$$903$$ 10.6290 0.353711
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −57.2109 −1.89966 −0.949829 0.312771i $$-0.898743\pi$$
−0.949829 + 0.312771i $$0.898743\pi$$
$$908$$ 0 0
$$909$$ −16.9869 −0.563420
$$910$$ 0 0
$$911$$ 42.0617 1.39357 0.696783 0.717282i $$-0.254614\pi$$
0.696783 + 0.717282i $$0.254614\pi$$
$$912$$ 0 0
$$913$$ 17.7243 0.586590
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −1.87372 −0.0618757
$$918$$ 0 0
$$919$$ −5.68287 −0.187461 −0.0937304 0.995598i $$-0.529879\pi$$
−0.0937304 + 0.995598i $$0.529879\pi$$
$$920$$ 0 0
$$921$$ 9.61462 0.316813
$$922$$ 0 0
$$923$$ 0.774918 0.0255067
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −15.2264 −0.500100
$$928$$ 0 0
$$929$$ −30.9289 −1.01474 −0.507372 0.861727i $$-0.669383\pi$$
−0.507372 + 0.861727i $$0.669383\pi$$
$$930$$ 0 0
$$931$$ −5.56314 −0.182325
$$932$$ 0 0
$$933$$ −19.3921 −0.634870
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 44.5357 1.45492 0.727458 0.686152i $$-0.240701\pi$$
0.727458 + 0.686152i $$0.240701\pi$$
$$938$$ 0 0
$$939$$ 8.55660 0.279234
$$940$$ 0 0
$$941$$ −2.08134 −0.0678498 −0.0339249 0.999424i $$-0.510801\pi$$
−0.0339249 + 0.999424i $$0.510801\pi$$
$$942$$ 0 0
$$943$$ 20.0593 0.653221
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 52.7189 1.71313 0.856567 0.516036i $$-0.172593\pi$$
0.856567 + 0.516036i $$0.172593\pi$$
$$948$$ 0 0
$$949$$ −0.912115 −0.0296085
$$950$$ 0 0
$$951$$ −4.74744 −0.153946
$$952$$ 0 0
$$953$$ −3.22071 −0.104329 −0.0521645 0.998639i $$-0.516612\pi$$
−0.0521645 + 0.998639i $$0.516612\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 55.3293 1.78854
$$958$$ 0 0
$$959$$ 18.9869 0.613119
$$960$$ 0 0
$$961$$ 22.3490 0.720935
$$962$$ 0 0
$$963$$ 25.5782 0.824247
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 2.65956 0.0855256 0.0427628 0.999085i $$-0.486384\pi$$
0.0427628 + 0.999085i $$0.486384\pi$$
$$968$$ 0 0
$$969$$ −1.43686 −0.0461586
$$970$$ 0 0
$$971$$ −35.1263 −1.12726 −0.563628 0.826029i $$-0.690595\pi$$
−0.563628 + 0.826029i $$0.690595\pi$$
$$972$$ 0 0
$$973$$ −23.8068 −0.763210
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −18.1668 −0.581209 −0.290604 0.956843i $$-0.593856\pi$$
−0.290604 + 0.956843i $$0.593856\pi$$
$$978$$ 0 0
$$979$$ −33.3808 −1.06686
$$980$$ 0 0
$$981$$ −14.5333 −0.464012
$$982$$ 0 0
$$983$$ 48.4556 1.54549 0.772747 0.634714i $$-0.218882\pi$$
0.772747 + 0.634714i $$0.218882\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 19.6870 0.626645
$$988$$ 0 0
$$989$$ 60.8925 1.93627
$$990$$ 0 0
$$991$$ −29.8157 −0.947127 −0.473563 0.880760i $$-0.657033\pi$$
−0.473563 + 0.880760i $$0.657033\pi$$
$$992$$ 0 0
$$993$$ −7.13044 −0.226278
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −29.6894 −0.940273 −0.470137 0.882594i $$-0.655795\pi$$
−0.470137 + 0.882594i $$0.655795\pi$$
$$998$$ 0 0
$$999$$ 39.0449 1.23533
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.by.1.1 3
4.3 odd 2 1900.2.a.f.1.3 3
5.4 even 2 7600.2.a.bj.1.3 3
20.3 even 4 1900.2.c.g.1749.5 6
20.7 even 4 1900.2.c.g.1749.2 6
20.19 odd 2 1900.2.a.h.1.1 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1900.2.a.f.1.3 3 4.3 odd 2
1900.2.a.h.1.1 yes 3 20.19 odd 2
1900.2.c.g.1749.2 6 20.7 even 4
1900.2.c.g.1749.5 6 20.3 even 4
7600.2.a.bj.1.3 3 5.4 even 2
7600.2.a.by.1.1 3 1.1 even 1 trivial