Properties

 Label 9576.2.a.bl.1.2 Level $9576$ Weight $2$ Character 9576.1 Self dual yes Analytic conductor $76.465$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9576.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: $$76.4647449756$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.82843 q^{5} +1.00000 q^{7} +O(q^{10})$$ $$q+2.82843 q^{5} +1.00000 q^{7} -2.00000 q^{11} -0.828427 q^{13} -4.00000 q^{17} +1.00000 q^{19} -4.82843 q^{23} +3.00000 q^{25} -3.65685 q^{29} +6.82843 q^{31} +2.82843 q^{35} -3.17157 q^{37} -2.00000 q^{41} -9.65685 q^{43} +6.48528 q^{47} +1.00000 q^{49} -6.00000 q^{53} -5.65685 q^{55} -4.00000 q^{59} -6.00000 q^{61} -2.34315 q^{65} -11.3137 q^{67} -2.34315 q^{71} +10.0000 q^{73} -2.00000 q^{77} +9.17157 q^{79} +4.34315 q^{83} -11.3137 q^{85} +0.343146 q^{89} -0.828427 q^{91} +2.82843 q^{95} -0.343146 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^7 $$2 q + 2 q^{7} - 4 q^{11} + 4 q^{13} - 8 q^{17} + 2 q^{19} - 4 q^{23} + 6 q^{25} + 4 q^{29} + 8 q^{31} - 12 q^{37} - 4 q^{41} - 8 q^{43} - 4 q^{47} + 2 q^{49} - 12 q^{53} - 8 q^{59} - 12 q^{61} - 16 q^{65} - 16 q^{71} + 20 q^{73} - 4 q^{77} + 24 q^{79} + 20 q^{83} + 12 q^{89} + 4 q^{91} - 12 q^{97}+O(q^{100})$$ 2 * q + 2 * q^7 - 4 * q^11 + 4 * q^13 - 8 * q^17 + 2 * q^19 - 4 * q^23 + 6 * q^25 + 4 * q^29 + 8 * q^31 - 12 * q^37 - 4 * q^41 - 8 * q^43 - 4 * q^47 + 2 * q^49 - 12 * q^53 - 8 * q^59 - 12 * q^61 - 16 * q^65 - 16 * q^71 + 20 * q^73 - 4 * q^77 + 24 * q^79 + 20 * q^83 + 12 * q^89 + 4 * q^91 - 12 * 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 0
$$4$$ 0 0
$$5$$ 2.82843 1.26491 0.632456 0.774597i $$-0.282047\pi$$
0.632456 + 0.774597i $$0.282047\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −0.828427 −0.229764 −0.114882 0.993379i $$-0.536649\pi$$
−0.114882 + 0.993379i $$0.536649\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.82843 −1.00680 −0.503398 0.864054i $$-0.667917\pi$$
−0.503398 + 0.864054i $$0.667917\pi$$
$$24$$ 0 0
$$25$$ 3.00000 0.600000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −3.65685 −0.679061 −0.339530 0.940595i $$-0.610268\pi$$
−0.339530 + 0.940595i $$0.610268\pi$$
$$30$$ 0 0
$$31$$ 6.82843 1.22642 0.613211 0.789919i $$-0.289878\pi$$
0.613211 + 0.789919i $$0.289878\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.82843 0.478091
$$36$$ 0 0
$$37$$ −3.17157 −0.521403 −0.260702 0.965419i $$-0.583954\pi$$
−0.260702 + 0.965419i $$0.583954\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −9.65685 −1.47266 −0.736328 0.676625i $$-0.763442\pi$$
−0.736328 + 0.676625i $$0.763442\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.48528 0.945976 0.472988 0.881069i $$-0.343175\pi$$
0.472988 + 0.881069i $$0.343175\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ −5.65685 −0.762770
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2.34315 −0.290631
$$66$$ 0 0
$$67$$ −11.3137 −1.38219 −0.691095 0.722764i $$-0.742871\pi$$
−0.691095 + 0.722764i $$0.742871\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −2.34315 −0.278080 −0.139040 0.990287i $$-0.544402\pi$$
−0.139040 + 0.990287i $$0.544402\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −2.00000 −0.227921
$$78$$ 0 0
$$79$$ 9.17157 1.03188 0.515941 0.856624i $$-0.327442\pi$$
0.515941 + 0.856624i $$0.327442\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 4.34315 0.476722 0.238361 0.971177i $$-0.423390\pi$$
0.238361 + 0.971177i $$0.423390\pi$$
$$84$$ 0 0
$$85$$ −11.3137 −1.22714
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0.343146 0.0363734 0.0181867 0.999835i $$-0.494211\pi$$
0.0181867 + 0.999835i $$0.494211\pi$$
$$90$$ 0 0
$$91$$ −0.828427 −0.0868428
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2.82843 0.290191
$$96$$ 0 0
$$97$$ −0.343146 −0.0348412 −0.0174206 0.999848i $$-0.505545\pi$$
−0.0174206 + 0.999848i $$0.505545\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.48528 −0.446302 −0.223151 0.974784i $$-0.571634\pi$$
−0.223151 + 0.974784i $$0.571634\pi$$
$$102$$ 0 0
$$103$$ −2.82843 −0.278693 −0.139347 0.990244i $$-0.544500\pi$$
−0.139347 + 0.990244i $$0.544500\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 13.6569 1.32026 0.660129 0.751152i $$-0.270502\pi$$
0.660129 + 0.751152i $$0.270502\pi$$
$$108$$ 0 0
$$109$$ −16.1421 −1.54614 −0.773068 0.634323i $$-0.781279\pi$$
−0.773068 + 0.634323i $$0.781279\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ −13.6569 −1.27351
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −5.65685 −0.505964
$$126$$ 0 0
$$127$$ −8.48528 −0.752947 −0.376473 0.926427i $$-0.622863\pi$$
−0.376473 + 0.926427i $$0.622863\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −4.34315 −0.379462 −0.189731 0.981836i $$-0.560762\pi$$
−0.189731 + 0.981836i $$0.560762\pi$$
$$132$$ 0 0
$$133$$ 1.00000 0.0867110
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −20.9706 −1.79164 −0.895818 0.444421i $$-0.853409\pi$$
−0.895818 + 0.444421i $$0.853409\pi$$
$$138$$ 0 0
$$139$$ 1.65685 0.140533 0.0702663 0.997528i $$-0.477615\pi$$
0.0702663 + 0.997528i $$0.477615\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 1.65685 0.138553
$$144$$ 0 0
$$145$$ −10.3431 −0.858952
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −6.82843 −0.559407 −0.279703 0.960086i $$-0.590236\pi$$
−0.279703 + 0.960086i $$0.590236\pi$$
$$150$$ 0 0
$$151$$ 4.48528 0.365007 0.182504 0.983205i $$-0.441580\pi$$
0.182504 + 0.983205i $$0.441580\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 19.3137 1.55131
$$156$$ 0 0
$$157$$ 18.9706 1.51402 0.757008 0.653406i $$-0.226660\pi$$
0.757008 + 0.653406i $$0.226660\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.82843 −0.380533
$$162$$ 0 0
$$163$$ −0.686292 −0.0537545 −0.0268772 0.999639i $$-0.508556\pi$$
−0.0268772 + 0.999639i $$0.508556\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −11.3137 −0.875481 −0.437741 0.899101i $$-0.644221\pi$$
−0.437741 + 0.899101i $$0.644221\pi$$
$$168$$ 0 0
$$169$$ −12.3137 −0.947208
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 0 0
$$175$$ 3.00000 0.226779
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −21.6569 −1.61871 −0.809355 0.587320i $$-0.800183\pi$$
−0.809355 + 0.587320i $$0.800183\pi$$
$$180$$ 0 0
$$181$$ −11.1716 −0.830376 −0.415188 0.909736i $$-0.636284\pi$$
−0.415188 + 0.909736i $$0.636284\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −8.97056 −0.659529
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.8284 −1.21766 −0.608831 0.793300i $$-0.708361\pi$$
−0.608831 + 0.793300i $$0.708361\pi$$
$$192$$ 0 0
$$193$$ −14.9706 −1.07760 −0.538802 0.842432i $$-0.681123\pi$$
−0.538802 + 0.842432i $$0.681123\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −7.51472 −0.535402 −0.267701 0.963502i $$-0.586264\pi$$
−0.267701 + 0.963502i $$0.586264\pi$$
$$198$$ 0 0
$$199$$ 5.65685 0.401004 0.200502 0.979693i $$-0.435743\pi$$
0.200502 + 0.979693i $$0.435743\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −3.65685 −0.256661
$$204$$ 0 0
$$205$$ −5.65685 −0.395092
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ 24.9706 1.71904 0.859522 0.511098i $$-0.170761\pi$$
0.859522 + 0.511098i $$0.170761\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −27.3137 −1.86278
$$216$$ 0 0
$$217$$ 6.82843 0.463544
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 3.31371 0.222904
$$222$$ 0 0
$$223$$ 10.8284 0.725125 0.362563 0.931959i $$-0.381902\pi$$
0.362563 + 0.931959i $$0.381902\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 11.3137 0.750917 0.375459 0.926839i $$-0.377485\pi$$
0.375459 + 0.926839i $$0.377485\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 11.3137 0.741186 0.370593 0.928795i $$-0.379155\pi$$
0.370593 + 0.928795i $$0.379155\pi$$
$$234$$ 0 0
$$235$$ 18.3431 1.19657
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1.51472 0.0979790 0.0489895 0.998799i $$-0.484400\pi$$
0.0489895 + 0.998799i $$0.484400\pi$$
$$240$$ 0 0
$$241$$ 26.9706 1.73733 0.868663 0.495403i $$-0.164980\pi$$
0.868663 + 0.495403i $$0.164980\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 2.82843 0.180702
$$246$$ 0 0
$$247$$ −0.828427 −0.0527116
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 10.9706 0.692456 0.346228 0.938150i $$-0.387462\pi$$
0.346228 + 0.938150i $$0.387462\pi$$
$$252$$ 0 0
$$253$$ 9.65685 0.607121
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −0.343146 −0.0214048 −0.0107024 0.999943i $$-0.503407\pi$$
−0.0107024 + 0.999943i $$0.503407\pi$$
$$258$$ 0 0
$$259$$ −3.17157 −0.197072
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 32.1421 1.98197 0.990984 0.133977i $$-0.0427747\pi$$
0.990984 + 0.133977i $$0.0427747\pi$$
$$264$$ 0 0
$$265$$ −16.9706 −1.04249
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −10.3431 −0.628301 −0.314151 0.949373i $$-0.601720\pi$$
−0.314151 + 0.949373i $$0.601720\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −6.00000 −0.361814
$$276$$ 0 0
$$277$$ 7.65685 0.460056 0.230028 0.973184i $$-0.426118\pi$$
0.230028 + 0.973184i $$0.426118\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −21.3137 −1.27147 −0.635735 0.771908i $$-0.719303\pi$$
−0.635735 + 0.771908i $$0.719303\pi$$
$$282$$ 0 0
$$283$$ 1.65685 0.0984898 0.0492449 0.998787i $$-0.484319\pi$$
0.0492449 + 0.998787i $$0.484319\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −2.00000 −0.118056
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 0 0
$$295$$ −11.3137 −0.658710
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ −9.65685 −0.556612
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −16.9706 −0.971732
$$306$$ 0 0
$$307$$ −16.9706 −0.968561 −0.484281 0.874913i $$-0.660919\pi$$
−0.484281 + 0.874913i $$0.660919\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −16.1421 −0.915337 −0.457668 0.889123i $$-0.651315\pi$$
−0.457668 + 0.889123i $$0.651315\pi$$
$$312$$ 0 0
$$313$$ 28.6274 1.61812 0.809059 0.587728i $$-0.199977\pi$$
0.809059 + 0.587728i $$0.199977\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 11.6569 0.654714 0.327357 0.944901i $$-0.393842\pi$$
0.327357 + 0.944901i $$0.393842\pi$$
$$318$$ 0 0
$$319$$ 7.31371 0.409489
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −4.00000 −0.222566
$$324$$ 0 0
$$325$$ −2.48528 −0.137859
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 6.48528 0.357545
$$330$$ 0 0
$$331$$ 31.3137 1.72116 0.860579 0.509318i $$-0.170102\pi$$
0.860579 + 0.509318i $$0.170102\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −32.0000 −1.74835
$$336$$ 0 0
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −13.6569 −0.739560
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 33.3137 1.78837 0.894187 0.447694i $$-0.147755\pi$$
0.894187 + 0.447694i $$0.147755\pi$$
$$348$$ 0 0
$$349$$ 12.6274 0.675930 0.337965 0.941159i $$-0.390261\pi$$
0.337965 + 0.941159i $$0.390261\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 4.97056 0.264556 0.132278 0.991213i $$-0.457771\pi$$
0.132278 + 0.991213i $$0.457771\pi$$
$$354$$ 0 0
$$355$$ −6.62742 −0.351747
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −16.8284 −0.888170 −0.444085 0.895985i $$-0.646471\pi$$
−0.444085 + 0.895985i $$0.646471\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 28.2843 1.48047
$$366$$ 0 0
$$367$$ 28.2843 1.47643 0.738213 0.674567i $$-0.235670\pi$$
0.738213 + 0.674567i $$0.235670\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ 12.8284 0.664231 0.332115 0.943239i $$-0.392238\pi$$
0.332115 + 0.943239i $$0.392238\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 3.02944 0.156024
$$378$$ 0 0
$$379$$ 0.686292 0.0352524 0.0176262 0.999845i $$-0.494389\pi$$
0.0176262 + 0.999845i $$0.494389\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −17.6569 −0.902223 −0.451112 0.892468i $$-0.648972\pi$$
−0.451112 + 0.892468i $$0.648972\pi$$
$$384$$ 0 0
$$385$$ −5.65685 −0.288300
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −27.7990 −1.40946 −0.704732 0.709473i $$-0.748933\pi$$
−0.704732 + 0.709473i $$0.748933\pi$$
$$390$$ 0 0
$$391$$ 19.3137 0.976736
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 25.9411 1.30524
$$396$$ 0 0
$$397$$ −5.31371 −0.266687 −0.133344 0.991070i $$-0.542571\pi$$
−0.133344 + 0.991070i $$0.542571\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.68629 −0.134147 −0.0670735 0.997748i $$-0.521366\pi$$
−0.0670735 + 0.997748i $$0.521366\pi$$
$$402$$ 0 0
$$403$$ −5.65685 −0.281788
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 6.34315 0.314418
$$408$$ 0 0
$$409$$ −14.9706 −0.740247 −0.370123 0.928983i $$-0.620685\pi$$
−0.370123 + 0.928983i $$0.620685\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −4.00000 −0.196827
$$414$$ 0 0
$$415$$ 12.2843 0.603011
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −6.68629 −0.326647 −0.163323 0.986573i $$-0.552221\pi$$
−0.163323 + 0.986573i $$0.552221\pi$$
$$420$$ 0 0
$$421$$ −11.8579 −0.577917 −0.288958 0.957342i $$-0.593309\pi$$
−0.288958 + 0.957342i $$0.593309\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −12.0000 −0.582086
$$426$$ 0 0
$$427$$ −6.00000 −0.290360
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 21.6569 1.04317 0.521587 0.853198i $$-0.325340\pi$$
0.521587 + 0.853198i $$0.325340\pi$$
$$432$$ 0 0
$$433$$ 5.02944 0.241699 0.120850 0.992671i $$-0.461438\pi$$
0.120850 + 0.992671i $$0.461438\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −4.82843 −0.230975
$$438$$ 0 0
$$439$$ 18.8284 0.898632 0.449316 0.893373i $$-0.351668\pi$$
0.449316 + 0.893373i $$0.351668\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −32.6274 −1.55018 −0.775088 0.631854i $$-0.782294\pi$$
−0.775088 + 0.631854i $$0.782294\pi$$
$$444$$ 0 0
$$445$$ 0.970563 0.0460091
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −7.65685 −0.361349 −0.180675 0.983543i $$-0.557828\pi$$
−0.180675 + 0.983543i $$0.557828\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −2.34315 −0.109848
$$456$$ 0 0
$$457$$ −2.68629 −0.125659 −0.0628297 0.998024i $$-0.520012\pi$$
−0.0628297 + 0.998024i $$0.520012\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −37.4558 −1.74449 −0.872246 0.489067i $$-0.837337\pi$$
−0.872246 + 0.489067i $$0.837337\pi$$
$$462$$ 0 0
$$463$$ 13.6569 0.634688 0.317344 0.948311i $$-0.397209\pi$$
0.317344 + 0.948311i $$0.397209\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 4.34315 0.200977 0.100488 0.994938i $$-0.467959\pi$$
0.100488 + 0.994938i $$0.467959\pi$$
$$468$$ 0 0
$$469$$ −11.3137 −0.522419
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 19.3137 0.888045
$$474$$ 0 0
$$475$$ 3.00000 0.137649
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −33.7990 −1.54432 −0.772158 0.635431i $$-0.780822\pi$$
−0.772158 + 0.635431i $$0.780822\pi$$
$$480$$ 0 0
$$481$$ 2.62742 0.119800
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −0.970563 −0.0440710
$$486$$ 0 0
$$487$$ 21.1716 0.959376 0.479688 0.877439i $$-0.340750\pi$$
0.479688 + 0.877439i $$0.340750\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 3.65685 0.165032 0.0825158 0.996590i $$-0.473705\pi$$
0.0825158 + 0.996590i $$0.473705\pi$$
$$492$$ 0 0
$$493$$ 14.6274 0.658786
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −2.34315 −0.105104
$$498$$ 0 0
$$499$$ −9.65685 −0.432300 −0.216150 0.976360i $$-0.569350\pi$$
−0.216150 + 0.976360i $$0.569350\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 2.48528 0.110813 0.0554066 0.998464i $$-0.482354\pi$$
0.0554066 + 0.998464i $$0.482354\pi$$
$$504$$ 0 0
$$505$$ −12.6863 −0.564533
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −8.34315 −0.369803 −0.184902 0.982757i $$-0.559197\pi$$
−0.184902 + 0.982757i $$0.559197\pi$$
$$510$$ 0 0
$$511$$ 10.0000 0.442374
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ −12.9706 −0.570445
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 35.6569 1.56216 0.781078 0.624434i $$-0.214670\pi$$
0.781078 + 0.624434i $$0.214670\pi$$
$$522$$ 0 0
$$523$$ 1.65685 0.0724492 0.0362246 0.999344i $$-0.488467\pi$$
0.0362246 + 0.999344i $$0.488467\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −27.3137 −1.18980
$$528$$ 0 0
$$529$$ 0.313708 0.0136395
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 1.65685 0.0717663
$$534$$ 0 0
$$535$$ 38.6274 1.67001
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ −12.3431 −0.530673 −0.265337 0.964156i $$-0.585483\pi$$
−0.265337 + 0.964156i $$0.585483\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −45.6569 −1.95572
$$546$$ 0 0
$$547$$ 26.6274 1.13851 0.569253 0.822162i $$-0.307232\pi$$
0.569253 + 0.822162i $$0.307232\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −3.65685 −0.155787
$$552$$ 0 0
$$553$$ 9.17157 0.390015
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −0.201010 −0.00851707 −0.00425854 0.999991i $$-0.501356\pi$$
−0.00425854 + 0.999991i $$0.501356\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −20.6863 −0.871823 −0.435912 0.899989i $$-0.643574\pi$$
−0.435912 + 0.899989i $$0.643574\pi$$
$$564$$ 0 0
$$565$$ 5.65685 0.237986
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −20.3431 −0.852829 −0.426415 0.904528i $$-0.640224\pi$$
−0.426415 + 0.904528i $$0.640224\pi$$
$$570$$ 0 0
$$571$$ −11.0294 −0.461568 −0.230784 0.973005i $$-0.574129\pi$$
−0.230784 + 0.973005i $$0.574129\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −14.4853 −0.604078
$$576$$ 0 0
$$577$$ 12.6274 0.525686 0.262843 0.964839i $$-0.415340\pi$$
0.262843 + 0.964839i $$0.415340\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 4.34315 0.180184
$$582$$ 0 0
$$583$$ 12.0000 0.496989
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0.627417 0.0258963 0.0129481 0.999916i $$-0.495878\pi$$
0.0129481 + 0.999916i $$0.495878\pi$$
$$588$$ 0 0
$$589$$ 6.82843 0.281360
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 44.2843 1.81854 0.909269 0.416210i $$-0.136642\pi$$
0.909269 + 0.416210i $$0.136642\pi$$
$$594$$ 0 0
$$595$$ −11.3137 −0.463817
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −29.9411 −1.22336 −0.611681 0.791105i $$-0.709506\pi$$
−0.611681 + 0.791105i $$0.709506\pi$$
$$600$$ 0 0
$$601$$ −32.6274 −1.33090 −0.665450 0.746442i $$-0.731760\pi$$
−0.665450 + 0.746442i $$0.731760\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −19.7990 −0.804943
$$606$$ 0 0
$$607$$ 13.8579 0.562473 0.281237 0.959638i $$-0.409255\pi$$
0.281237 + 0.959638i $$0.409255\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −5.37258 −0.217351
$$612$$ 0 0
$$613$$ −28.6274 −1.15625 −0.578125 0.815948i $$-0.696215\pi$$
−0.578125 + 0.815948i $$0.696215\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.0000 0.483102 0.241551 0.970388i $$-0.422344\pi$$
0.241551 + 0.970388i $$0.422344\pi$$
$$618$$ 0 0
$$619$$ 37.9411 1.52498 0.762491 0.646998i $$-0.223976\pi$$
0.762491 + 0.646998i $$0.223976\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0.343146 0.0137478
$$624$$ 0 0
$$625$$ −31.0000 −1.24000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 12.6863 0.505836
$$630$$ 0 0
$$631$$ −47.5980 −1.89485 −0.947423 0.319984i $$-0.896322\pi$$
−0.947423 + 0.319984i $$0.896322\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −24.0000 −0.952411
$$636$$ 0 0
$$637$$ −0.828427 −0.0328235
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 4.34315 0.171544 0.0857720 0.996315i $$-0.472664\pi$$
0.0857720 + 0.996315i $$0.472664\pi$$
$$642$$ 0 0
$$643$$ −24.2843 −0.957678 −0.478839 0.877903i $$-0.658942\pi$$
−0.478839 + 0.877903i $$0.658942\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −21.7990 −0.857007 −0.428503 0.903540i $$-0.640959\pi$$
−0.428503 + 0.903540i $$0.640959\pi$$
$$648$$ 0 0
$$649$$ 8.00000 0.314027
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 19.1127 0.747938 0.373969 0.927441i $$-0.377997\pi$$
0.373969 + 0.927441i $$0.377997\pi$$
$$654$$ 0 0
$$655$$ −12.2843 −0.479986
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −16.2843 −0.634345 −0.317173 0.948368i $$-0.602733\pi$$
−0.317173 + 0.948368i $$0.602733\pi$$
$$660$$ 0 0
$$661$$ 9.79899 0.381137 0.190568 0.981674i $$-0.438967\pi$$
0.190568 + 0.981674i $$0.438967\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2.82843 0.109682
$$666$$ 0 0
$$667$$ 17.6569 0.683676
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 12.0000 0.463255
$$672$$ 0 0
$$673$$ 9.02944 0.348059 0.174030 0.984740i $$-0.444321\pi$$
0.174030 + 0.984740i $$0.444321\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −31.6569 −1.21667 −0.608336 0.793680i $$-0.708163\pi$$
−0.608336 + 0.793680i $$0.708163\pi$$
$$678$$ 0 0
$$679$$ −0.343146 −0.0131687
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 5.65685 0.216454 0.108227 0.994126i $$-0.465483\pi$$
0.108227 + 0.994126i $$0.465483\pi$$
$$684$$ 0 0
$$685$$ −59.3137 −2.26626
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 4.97056 0.189363
$$690$$ 0 0
$$691$$ 15.3137 0.582561 0.291280 0.956638i $$-0.405919\pi$$
0.291280 + 0.956638i $$0.405919\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4.68629 0.177761
$$696$$ 0 0
$$697$$ 8.00000 0.303022
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 45.4558 1.71684 0.858422 0.512945i $$-0.171445\pi$$
0.858422 + 0.512945i $$0.171445\pi$$
$$702$$ 0 0
$$703$$ −3.17157 −0.119618
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −4.48528 −0.168686
$$708$$ 0 0
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −32.9706 −1.23476
$$714$$ 0 0
$$715$$ 4.68629 0.175257
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −24.1421 −0.900350 −0.450175 0.892940i $$-0.648638\pi$$
−0.450175 + 0.892940i $$0.648638\pi$$
$$720$$ 0 0
$$721$$ −2.82843 −0.105336
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −10.9706 −0.407436
$$726$$ 0 0
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 38.6274 1.42869
$$732$$ 0 0
$$733$$ 12.6274 0.466404 0.233202 0.972428i $$-0.425080\pi$$
0.233202 + 0.972428i $$0.425080\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 22.6274 0.833492
$$738$$ 0 0
$$739$$ −18.6274 −0.685221 −0.342610 0.939478i $$-0.611311\pi$$
−0.342610 + 0.939478i $$0.611311\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −28.2843 −1.03765 −0.518825 0.854881i $$-0.673630\pi$$
−0.518825 + 0.854881i $$0.673630\pi$$
$$744$$ 0 0
$$745$$ −19.3137 −0.707600
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 13.6569 0.499011
$$750$$ 0 0
$$751$$ 5.85786 0.213757 0.106878 0.994272i $$-0.465914\pi$$
0.106878 + 0.994272i $$0.465914\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 12.6863 0.461701
$$756$$ 0 0
$$757$$ −12.6274 −0.458951 −0.229476 0.973314i $$-0.573701\pi$$
−0.229476 + 0.973314i $$0.573701\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 15.3137 0.555121 0.277561 0.960708i $$-0.410474\pi$$
0.277561 + 0.960708i $$0.410474\pi$$
$$762$$ 0 0
$$763$$ −16.1421 −0.584385
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 3.31371 0.119651
$$768$$ 0 0
$$769$$ 20.6274 0.743844 0.371922 0.928264i $$-0.378699\pi$$
0.371922 + 0.928264i $$0.378699\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −26.2843 −0.945380 −0.472690 0.881229i $$-0.656717\pi$$
−0.472690 + 0.881229i $$0.656717\pi$$
$$774$$ 0 0
$$775$$ 20.4853 0.735853
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2.00000 −0.0716574
$$780$$ 0 0
$$781$$ 4.68629 0.167689
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 53.6569 1.91510
$$786$$ 0 0
$$787$$ 41.2548 1.47058 0.735288 0.677755i $$-0.237047\pi$$
0.735288 + 0.677755i $$0.237047\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 2.00000 0.0711118
$$792$$ 0 0
$$793$$ 4.97056 0.176510
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 4.34315 0.153842 0.0769211 0.997037i $$-0.475491\pi$$
0.0769211 + 0.997037i $$0.475491\pi$$
$$798$$ 0 0
$$799$$ −25.9411 −0.917731
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −20.0000 −0.705785
$$804$$ 0 0
$$805$$ −13.6569 −0.481341
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −26.3431 −0.926176 −0.463088 0.886312i $$-0.653259\pi$$
−0.463088 + 0.886312i $$0.653259\pi$$
$$810$$ 0 0
$$811$$ −0.970563 −0.0340811 −0.0170405 0.999855i $$-0.505424\pi$$
−0.0170405 + 0.999855i $$0.505424\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −1.94113 −0.0679947
$$816$$ 0 0
$$817$$ −9.65685 −0.337851
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −44.4853 −1.55255 −0.776274 0.630396i $$-0.782892\pi$$
−0.776274 + 0.630396i $$0.782892\pi$$
$$822$$ 0 0
$$823$$ 0.970563 0.0338317 0.0169158 0.999857i $$-0.494615\pi$$
0.0169158 + 0.999857i $$0.494615\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 10.3431 0.359666 0.179833 0.983697i $$-0.442444\pi$$
0.179833 + 0.983697i $$0.442444\pi$$
$$828$$ 0 0
$$829$$ −46.4853 −1.61450 −0.807250 0.590209i $$-0.799045\pi$$
−0.807250 + 0.590209i $$0.799045\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −4.00000 −0.138592
$$834$$ 0 0
$$835$$ −32.0000 −1.10741
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 16.2843 0.562195 0.281098 0.959679i $$-0.409301\pi$$
0.281098 + 0.959679i $$0.409301\pi$$
$$840$$ 0 0
$$841$$ −15.6274 −0.538876
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −34.8284 −1.19813
$$846$$ 0 0
$$847$$ −7.00000 −0.240523
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 15.3137 0.524947
$$852$$ 0 0
$$853$$ 10.2843 0.352127 0.176063 0.984379i $$-0.443664\pi$$
0.176063 + 0.984379i $$0.443664\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 46.9706 1.60448 0.802242 0.596999i $$-0.203640\pi$$
0.802242 + 0.596999i $$0.203640\pi$$
$$858$$ 0 0
$$859$$ 2.62742 0.0896463 0.0448232 0.998995i $$-0.485728\pi$$
0.0448232 + 0.998995i $$0.485728\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −20.2843 −0.690485 −0.345242 0.938514i $$-0.612203\pi$$
−0.345242 + 0.938514i $$0.612203\pi$$
$$864$$ 0 0
$$865$$ −39.5980 −1.34637
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −18.3431 −0.622249
$$870$$ 0 0
$$871$$ 9.37258 0.317578
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −5.65685 −0.191237
$$876$$ 0 0
$$877$$ −30.4853 −1.02941 −0.514707 0.857366i $$-0.672099\pi$$
−0.514707 + 0.857366i $$0.672099\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 20.9706 0.706516 0.353258 0.935526i $$-0.385074\pi$$
0.353258 + 0.935526i $$0.385074\pi$$
$$882$$ 0 0
$$883$$ 11.0294 0.371170 0.185585 0.982628i $$-0.440582\pi$$
0.185585 + 0.982628i $$0.440582\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −38.6274 −1.29698 −0.648491 0.761222i $$-0.724600\pi$$
−0.648491 + 0.761222i $$0.724600\pi$$
$$888$$ 0 0
$$889$$ −8.48528 −0.284587
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 6.48528 0.217022
$$894$$ 0 0
$$895$$ −61.2548 −2.04752
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −24.9706 −0.832815
$$900$$ 0 0
$$901$$ 24.0000 0.799556
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −31.5980 −1.05035
$$906$$ 0 0
$$907$$ −1.37258 −0.0455759 −0.0227879 0.999740i $$-0.507254\pi$$
−0.0227879 + 0.999740i $$0.507254\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −8.68629 −0.287790 −0.143895 0.989593i $$-0.545963\pi$$
−0.143895 + 0.989593i $$0.545963\pi$$
$$912$$ 0 0
$$913$$ −8.68629 −0.287474
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −4.34315 −0.143423
$$918$$ 0 0
$$919$$ 45.6569 1.50608 0.753040 0.657974i $$-0.228586\pi$$
0.753040 + 0.657974i $$0.228586\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 1.94113 0.0638929
$$924$$ 0 0
$$925$$ −9.51472 −0.312842
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 11.3137 0.371191 0.185595 0.982626i $$-0.440579\pi$$
0.185595 + 0.982626i $$0.440579\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 22.6274 0.739996
$$936$$ 0 0
$$937$$ 5.31371 0.173591 0.0867956 0.996226i $$-0.472337\pi$$
0.0867956 + 0.996226i $$0.472337\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 25.5980 0.834470 0.417235 0.908799i $$-0.362999\pi$$
0.417235 + 0.908799i $$0.362999\pi$$
$$942$$ 0 0
$$943$$ 9.65685 0.314470
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −2.97056 −0.0965303 −0.0482652 0.998835i $$-0.515369\pi$$
−0.0482652 + 0.998835i $$0.515369\pi$$
$$948$$ 0 0
$$949$$ −8.28427 −0.268919
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −8.62742 −0.279469 −0.139735 0.990189i $$-0.544625\pi$$
−0.139735 + 0.990189i $$0.544625\pi$$
$$954$$ 0 0
$$955$$ −47.5980 −1.54023
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −20.9706 −0.677175
$$960$$ 0 0
$$961$$ 15.6274 0.504110
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −42.3431 −1.36307
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −45.2548 −1.45230 −0.726148 0.687538i $$-0.758691\pi$$
−0.726148 + 0.687538i $$0.758691\pi$$
$$972$$ 0 0
$$973$$ 1.65685 0.0531163
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −38.0000 −1.21573 −0.607864 0.794041i $$-0.707973\pi$$
−0.607864 + 0.794041i $$0.707973\pi$$
$$978$$ 0 0
$$979$$ −0.686292 −0.0219340
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 19.3137 0.616012 0.308006 0.951384i $$-0.400338\pi$$
0.308006 + 0.951384i $$0.400338\pi$$
$$984$$ 0 0
$$985$$ −21.2548 −0.677235
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 46.6274 1.48267
$$990$$ 0 0
$$991$$ −3.79899 −0.120679 −0.0603394 0.998178i $$-0.519218\pi$$
−0.0603394 + 0.998178i $$0.519218\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 16.0000 0.507234
$$996$$ 0 0
$$997$$ −0.343146 −0.0108675 −0.00543377 0.999985i $$-0.501730\pi$$
−0.00543377 + 0.999985i $$0.501730\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 9576.2.a.bl.1.2 2
3.2 odd 2 9576.2.a.bo.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
9576.2.a.bl.1.2 2 1.1 even 1 trivial
9576.2.a.bo.1.1 yes 2 3.2 odd 2