# Properties

 Label 8568.2.a.bj.1.3 Level $8568$ Weight $2$ Character 8568.1 Self dual yes Analytic conductor $68.416$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8568,2,Mod(1,8568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8568, 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("8568.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8568 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8568.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: $$68.4158244518$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.5225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 8x^{2} + x + 11$$ x^4 - x^3 - 8*x^2 + x + 11 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 952) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.88301$$ of defining polynomial Character $$\chi$$ $$=$$ 8568.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.78180 q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q+1.78180 q^{5} -1.00000 q^{7} -3.23607 q^{11} +2.47214 q^{13} -1.00000 q^{17} +7.76602 q^{19} +5.32962 q^{23} -1.82519 q^{25} -10.5657 q^{29} -9.64137 q^{31} -1.78180 q^{35} -1.62142 q^{37} -2.82519 q^{41} +0.587035 q^{43} -12.1871 q^{47} +1.00000 q^{49} -3.40321 q^{53} -5.76602 q^{55} +7.32962 q^{59} -3.82310 q^{61} +4.40485 q^{65} +0.394361 q^{67} +2.15137 q^{71} -14.3572 q^{73} +3.23607 q^{77} +4.18710 q^{79} -8.42108 q^{83} -1.78180 q^{85} +17.0446 q^{89} -2.47214 q^{91} +13.8375 q^{95} -7.72398 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{5} - 4 q^{7}+O(q^{10})$$ 4 * q + q^5 - 4 * q^7 $$4 q + q^{5} - 4 q^{7} - 4 q^{11} - 8 q^{13} - 4 q^{17} + 14 q^{19} - 8 q^{23} + 11 q^{25} - 4 q^{29} + 5 q^{31} - q^{35} - 4 q^{37} + 7 q^{41} + 19 q^{43} - 8 q^{47} + 4 q^{49} - 5 q^{53} - 6 q^{55} - 23 q^{61} + 8 q^{65} + 15 q^{67} - 2 q^{71} - 5 q^{73} + 4 q^{77} - 24 q^{79} - 10 q^{83} - q^{85} + 16 q^{89} + 8 q^{91} - 22 q^{95} - 15 q^{97}+O(q^{100})$$ 4 * q + q^5 - 4 * q^7 - 4 * q^11 - 8 * q^13 - 4 * q^17 + 14 * q^19 - 8 * q^23 + 11 * q^25 - 4 * q^29 + 5 * q^31 - q^35 - 4 * q^37 + 7 * q^41 + 19 * q^43 - 8 * q^47 + 4 * q^49 - 5 * q^53 - 6 * q^55 - 23 * q^61 + 8 * q^65 + 15 * q^67 - 2 * q^71 - 5 * q^73 + 4 * q^77 - 24 * q^79 - 10 * q^83 - q^85 + 16 * q^89 + 8 * q^91 - 22 * q^95 - 15 * 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
$$5$$ 1.78180 0.796845 0.398422 0.917202i $$-0.369558\pi$$
0.398422 + 0.917202i $$0.369558\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.23607 −0.975711 −0.487856 0.872924i $$-0.662221\pi$$
−0.487856 + 0.872924i $$0.662221\pi$$
$$12$$ 0 0
$$13$$ 2.47214 0.685647 0.342824 0.939400i $$-0.388617\pi$$
0.342824 + 0.939400i $$0.388617\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ 7.76602 1.78165 0.890824 0.454349i $$-0.150128\pi$$
0.890824 + 0.454349i $$0.150128\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 5.32962 1.11130 0.555651 0.831415i $$-0.312469\pi$$
0.555651 + 0.831415i $$0.312469\pi$$
$$24$$ 0 0
$$25$$ −1.82519 −0.365039
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −10.5657 −1.96200 −0.980999 0.194010i $$-0.937850\pi$$
−0.980999 + 0.194010i $$0.937850\pi$$
$$30$$ 0 0
$$31$$ −9.64137 −1.73164 −0.865821 0.500354i $$-0.833203\pi$$
−0.865821 + 0.500354i $$0.833203\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.78180 −0.301179
$$36$$ 0 0
$$37$$ −1.62142 −0.266559 −0.133280 0.991078i $$-0.542551\pi$$
−0.133280 + 0.991078i $$0.542551\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.82519 −0.441221 −0.220610 0.975362i $$-0.570805\pi$$
−0.220610 + 0.975362i $$0.570805\pi$$
$$42$$ 0 0
$$43$$ 0.587035 0.0895219 0.0447610 0.998998i $$-0.485747\pi$$
0.0447610 + 0.998998i $$0.485747\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −12.1871 −1.77767 −0.888836 0.458226i $$-0.848485\pi$$
−0.888836 + 0.458226i $$0.848485\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −3.40321 −0.467468 −0.233734 0.972301i $$-0.575094\pi$$
−0.233734 + 0.972301i $$0.575094\pi$$
$$54$$ 0 0
$$55$$ −5.76602 −0.777490
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 7.32962 0.954235 0.477118 0.878839i $$-0.341682\pi$$
0.477118 + 0.878839i $$0.341682\pi$$
$$60$$ 0 0
$$61$$ −3.82310 −0.489498 −0.244749 0.969586i $$-0.578706\pi$$
−0.244749 + 0.969586i $$0.578706\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.40485 0.546354
$$66$$ 0 0
$$67$$ 0.394361 0.0481788 0.0240894 0.999710i $$-0.492331\pi$$
0.0240894 + 0.999710i $$0.492331\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 2.15137 0.255321 0.127660 0.991818i $$-0.459253\pi$$
0.127660 + 0.991818i $$0.459253\pi$$
$$72$$ 0 0
$$73$$ −14.3572 −1.68039 −0.840194 0.542286i $$-0.817559\pi$$
−0.840194 + 0.542286i $$0.817559\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.23607 0.368784
$$78$$ 0 0
$$79$$ 4.18710 0.471086 0.235543 0.971864i $$-0.424313\pi$$
0.235543 + 0.971864i $$0.424313\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −8.42108 −0.924334 −0.462167 0.886793i $$-0.652928\pi$$
−0.462167 + 0.886793i $$0.652928\pi$$
$$84$$ 0 0
$$85$$ −1.78180 −0.193263
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 17.0446 1.80672 0.903361 0.428880i $$-0.141092\pi$$
0.903361 + 0.428880i $$0.141092\pi$$
$$90$$ 0 0
$$91$$ −2.47214 −0.259150
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 13.8375 1.41970
$$96$$ 0 0
$$97$$ −7.72398 −0.784251 −0.392126 0.919912i $$-0.628260\pi$$
−0.392126 + 0.919912i $$0.628260\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 13.6834 1.36155 0.680775 0.732492i $$-0.261643\pi$$
0.680775 + 0.732492i $$0.261643\pi$$
$$102$$ 0 0
$$103$$ 6.90854 0.680718 0.340359 0.940295i $$-0.389451\pi$$
0.340359 + 0.940295i $$0.389451\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 14.9868 1.44883 0.724413 0.689366i $$-0.242111\pi$$
0.724413 + 0.689366i $$0.242111\pi$$
$$108$$ 0 0
$$109$$ −19.3874 −1.85698 −0.928490 0.371358i $$-0.878892\pi$$
−0.928490 + 0.371358i $$0.878892\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.26971 −0.589805 −0.294902 0.955527i $$-0.595287\pi$$
−0.294902 + 0.955527i $$0.595287\pi$$
$$114$$ 0 0
$$115$$ 9.49631 0.885536
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 1.00000 0.0916698
$$120$$ 0 0
$$121$$ −0.527864 −0.0479876
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −12.1611 −1.08772
$$126$$ 0 0
$$127$$ 0.362807 0.0321939 0.0160970 0.999870i $$-0.494876\pi$$
0.0160970 + 0.999870i $$0.494876\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1.94218 0.169689 0.0848446 0.996394i $$-0.472961\pi$$
0.0848446 + 0.996394i $$0.472961\pi$$
$$132$$ 0 0
$$133$$ −7.76602 −0.673400
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −5.17481 −0.442114 −0.221057 0.975261i $$-0.570951\pi$$
−0.221057 + 0.975261i $$0.570951\pi$$
$$138$$ 0 0
$$139$$ 4.13559 0.350776 0.175388 0.984499i $$-0.443882\pi$$
0.175388 + 0.984499i $$0.443882\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ −18.8259 −1.56341
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2.17013 0.177784 0.0888921 0.996041i $$-0.471667\pi$$
0.0888921 + 0.996041i $$0.471667\pi$$
$$150$$ 0 0
$$151$$ 3.05917 0.248952 0.124476 0.992223i $$-0.460275\pi$$
0.124476 + 0.992223i $$0.460275\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −17.1790 −1.37985
$$156$$ 0 0
$$157$$ 9.95313 0.794346 0.397173 0.917744i $$-0.369991\pi$$
0.397173 + 0.917744i $$0.369991\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −5.32962 −0.420033
$$162$$ 0 0
$$163$$ 15.6613 1.22669 0.613345 0.789815i $$-0.289824\pi$$
0.613345 + 0.789815i $$0.289824\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 8.11908 0.628273 0.314137 0.949378i $$-0.398285\pi$$
0.314137 + 0.949378i $$0.398285\pi$$
$$168$$ 0 0
$$169$$ −6.88854 −0.529888
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −12.3640 −0.940018 −0.470009 0.882662i $$-0.655749\pi$$
−0.470009 + 0.882662i $$0.655749\pi$$
$$174$$ 0 0
$$175$$ 1.82519 0.137972
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 18.1233 1.35460 0.677298 0.735709i $$-0.263151\pi$$
0.677298 + 0.735709i $$0.263151\pi$$
$$180$$ 0 0
$$181$$ −18.4143 −1.36873 −0.684363 0.729142i $$-0.739919\pi$$
−0.684363 + 0.729142i $$0.739919\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −2.88904 −0.212406
$$186$$ 0 0
$$187$$ 3.23607 0.236645
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −10.7013 −0.774318 −0.387159 0.922013i $$-0.626543\pi$$
−0.387159 + 0.922013i $$0.626543\pi$$
$$192$$ 0 0
$$193$$ −1.51254 −0.108875 −0.0544376 0.998517i $$-0.517337\pi$$
−0.0544376 + 0.998517i $$0.517337\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 7.27180 0.518094 0.259047 0.965865i $$-0.416591\pi$$
0.259047 + 0.965865i $$0.416591\pi$$
$$198$$ 0 0
$$199$$ −23.1832 −1.64341 −0.821706 0.569912i $$-0.806977\pi$$
−0.821706 + 0.569912i $$0.806977\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 10.5657 0.741566
$$204$$ 0 0
$$205$$ −5.03393 −0.351585
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −25.1314 −1.73837
$$210$$ 0 0
$$211$$ 11.1535 0.767836 0.383918 0.923367i $$-0.374575\pi$$
0.383918 + 0.923367i $$0.374575\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 1.04598 0.0713351
$$216$$ 0 0
$$217$$ 9.64137 0.654499
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −2.47214 −0.166294
$$222$$ 0 0
$$223$$ −15.5636 −1.04222 −0.521108 0.853491i $$-0.674481\pi$$
−0.521108 + 0.853491i $$0.674481\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −15.3649 −1.01980 −0.509902 0.860232i $$-0.670318\pi$$
−0.509902 + 0.860232i $$0.670318\pi$$
$$228$$ 0 0
$$229$$ −14.9085 −0.985184 −0.492592 0.870260i $$-0.663950\pi$$
−0.492592 + 0.870260i $$0.663950\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −12.9085 −0.845666 −0.422833 0.906208i $$-0.638964\pi$$
−0.422833 + 0.906208i $$0.638964\pi$$
$$234$$ 0 0
$$235$$ −21.7150 −1.41653
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −7.05917 −0.456620 −0.228310 0.973589i $$-0.573320\pi$$
−0.228310 + 0.973589i $$0.573320\pi$$
$$240$$ 0 0
$$241$$ −22.0309 −1.41914 −0.709568 0.704637i $$-0.751110\pi$$
−0.709568 + 0.704637i $$0.751110\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 1.78180 0.113835
$$246$$ 0 0
$$247$$ 19.1987 1.22158
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −28.1267 −1.77534 −0.887671 0.460479i $$-0.847678\pi$$
−0.887671 + 0.460479i $$0.847678\pi$$
$$252$$ 0 0
$$253$$ −17.2470 −1.08431
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −9.04459 −0.564186 −0.282093 0.959387i $$-0.591029\pi$$
−0.282093 + 0.959387i $$0.591029\pi$$
$$258$$ 0 0
$$259$$ 1.62142 0.100750
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −14.6592 −0.903927 −0.451964 0.892036i $$-0.649276\pi$$
−0.451964 + 0.892036i $$0.649276\pi$$
$$264$$ 0 0
$$265$$ −6.06384 −0.372499
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 9.75926 0.595032 0.297516 0.954717i $$-0.403842\pi$$
0.297516 + 0.954717i $$0.403842\pi$$
$$270$$ 0 0
$$271$$ 15.1602 0.920918 0.460459 0.887681i $$-0.347685\pi$$
0.460459 + 0.887681i $$0.347685\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 5.90645 0.356172
$$276$$ 0 0
$$277$$ 10.7324 0.644846 0.322423 0.946596i $$-0.395503\pi$$
0.322423 + 0.946596i $$0.395503\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2.06245 0.123036 0.0615179 0.998106i $$-0.480406\pi$$
0.0615179 + 0.998106i $$0.480406\pi$$
$$282$$ 0 0
$$283$$ −15.4364 −0.917597 −0.458798 0.888540i $$-0.651720\pi$$
−0.458798 + 0.888540i $$0.651720\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.82519 0.166766
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −2.28503 −0.133493 −0.0667465 0.997770i $$-0.521262\pi$$
−0.0667465 + 0.997770i $$0.521262\pi$$
$$294$$ 0 0
$$295$$ 13.0599 0.760377
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 13.1755 0.761961
$$300$$ 0 0
$$301$$ −0.587035 −0.0338361
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −6.81200 −0.390054
$$306$$ 0 0
$$307$$ −20.6277 −1.17728 −0.588642 0.808394i $$-0.700337\pi$$
−0.588642 + 0.808394i $$0.700337\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −18.2250 −1.03344 −0.516721 0.856154i $$-0.672848\pi$$
−0.516721 + 0.856154i $$0.672848\pi$$
$$312$$ 0 0
$$313$$ −15.6609 −0.885205 −0.442602 0.896718i $$-0.645945\pi$$
−0.442602 + 0.896718i $$0.645945\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −1.75508 −0.0985750 −0.0492875 0.998785i $$-0.515695\pi$$
−0.0492875 + 0.998785i $$0.515695\pi$$
$$318$$ 0 0
$$319$$ 34.1913 1.91434
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −7.76602 −0.432113
$$324$$ 0 0
$$325$$ −4.51212 −0.250288
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 12.1871 0.671897
$$330$$ 0 0
$$331$$ −14.2829 −0.785059 −0.392530 0.919739i $$-0.628400\pi$$
−0.392530 + 0.919739i $$0.628400\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0.702671 0.0383910
$$336$$ 0 0
$$337$$ 24.9290 1.35797 0.678983 0.734154i $$-0.262421\pi$$
0.678983 + 0.734154i $$0.262421\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 31.2001 1.68958
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −35.1934 −1.88928 −0.944640 0.328110i $$-0.893588\pi$$
−0.944640 + 0.328110i $$0.893588\pi$$
$$348$$ 0 0
$$349$$ 2.71029 0.145079 0.0725394 0.997366i $$-0.476890\pi$$
0.0725394 + 0.997366i $$0.476890\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 12.3580 0.657749 0.328874 0.944374i $$-0.393331\pi$$
0.328874 + 0.944374i $$0.393331\pi$$
$$354$$ 0 0
$$355$$ 3.83331 0.203451
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 5.04942 0.266498 0.133249 0.991083i $$-0.457459\pi$$
0.133249 + 0.991083i $$0.457459\pi$$
$$360$$ 0 0
$$361$$ 41.3111 2.17427
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −25.5817 −1.33901
$$366$$ 0 0
$$367$$ 25.9566 1.35492 0.677461 0.735559i $$-0.263080\pi$$
0.677461 + 0.735559i $$0.263080\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 3.40321 0.176686
$$372$$ 0 0
$$373$$ −8.23890 −0.426594 −0.213297 0.976987i $$-0.568420\pi$$
−0.213297 + 0.976987i $$0.568420\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −26.1198 −1.34524
$$378$$ 0 0
$$379$$ 11.7755 0.604866 0.302433 0.953171i $$-0.402201\pi$$
0.302433 + 0.953171i $$0.402201\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 23.2827 1.18969 0.594846 0.803839i $$-0.297213\pi$$
0.594846 + 0.803839i $$0.297213\pi$$
$$384$$ 0 0
$$385$$ 5.76602 0.293864
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 11.3717 0.576566 0.288283 0.957545i $$-0.406916\pi$$
0.288283 + 0.957545i $$0.406916\pi$$
$$390$$ 0 0
$$391$$ −5.32962 −0.269530
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 7.46058 0.375382
$$396$$ 0 0
$$397$$ 25.0645 1.25795 0.628977 0.777424i $$-0.283474\pi$$
0.628977 + 0.777424i $$0.283474\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −7.49631 −0.374348 −0.187174 0.982327i $$-0.559933\pi$$
−0.187174 + 0.982327i $$0.559933\pi$$
$$402$$ 0 0
$$403$$ −23.8348 −1.18730
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 5.24701 0.260085
$$408$$ 0 0
$$409$$ −11.4657 −0.566941 −0.283470 0.958981i $$-0.591486\pi$$
−0.283470 + 0.958981i $$0.591486\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −7.32962 −0.360667
$$414$$ 0 0
$$415$$ −15.0047 −0.736550
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −3.40998 −0.166588 −0.0832942 0.996525i $$-0.526544\pi$$
−0.0832942 + 0.996525i $$0.526544\pi$$
$$420$$ 0 0
$$421$$ 13.9524 0.679998 0.339999 0.940426i $$-0.389573\pi$$
0.339999 + 0.940426i $$0.389573\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1.82519 0.0885349
$$426$$ 0 0
$$427$$ 3.82310 0.185013
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −27.7702 −1.33764 −0.668822 0.743423i $$-0.733201\pi$$
−0.668822 + 0.743423i $$0.733201\pi$$
$$432$$ 0 0
$$433$$ −16.1913 −0.778103 −0.389052 0.921216i $$-0.627197\pi$$
−0.389052 + 0.921216i $$0.627197\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 41.3899 1.97995
$$438$$ 0 0
$$439$$ −14.5266 −0.693318 −0.346659 0.937991i $$-0.612684\pi$$
−0.346659 + 0.937991i $$0.612684\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −33.5993 −1.59635 −0.798176 0.602424i $$-0.794202\pi$$
−0.798176 + 0.602424i $$0.794202\pi$$
$$444$$ 0 0
$$445$$ 30.3700 1.43968
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 13.7191 0.647447 0.323723 0.946152i $$-0.395065\pi$$
0.323723 + 0.946152i $$0.395065\pi$$
$$450$$ 0 0
$$451$$ 9.14252 0.430504
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −4.40485 −0.206503
$$456$$ 0 0
$$457$$ 9.18874 0.429831 0.214916 0.976633i $$-0.431052\pi$$
0.214916 + 0.976633i $$0.431052\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 27.3338 1.27306 0.636531 0.771251i $$-0.280369\pi$$
0.636531 + 0.771251i $$0.280369\pi$$
$$462$$ 0 0
$$463$$ 7.36191 0.342137 0.171069 0.985259i $$-0.445278\pi$$
0.171069 + 0.985259i $$0.445278\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 28.4805 1.31792 0.658960 0.752178i $$-0.270997\pi$$
0.658960 + 0.752178i $$0.270997\pi$$
$$468$$ 0 0
$$469$$ −0.394361 −0.0182099
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −1.89968 −0.0873476
$$474$$ 0 0
$$475$$ −14.1745 −0.650370
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −7.72259 −0.352854 −0.176427 0.984314i $$-0.556454\pi$$
−0.176427 + 0.984314i $$0.556454\pi$$
$$480$$ 0 0
$$481$$ −4.00836 −0.182766
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −13.7626 −0.624927
$$486$$ 0 0
$$487$$ 29.0172 1.31490 0.657448 0.753500i $$-0.271636\pi$$
0.657448 + 0.753500i $$0.271636\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −14.2250 −0.641964 −0.320982 0.947085i $$-0.604013\pi$$
−0.320982 + 0.947085i $$0.604013\pi$$
$$492$$ 0 0
$$493$$ 10.5657 0.475855
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −2.15137 −0.0965021
$$498$$ 0 0
$$499$$ −21.6971 −0.971294 −0.485647 0.874155i $$-0.661416\pi$$
−0.485647 + 0.874155i $$0.661416\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 26.4700 1.18024 0.590120 0.807316i $$-0.299081\pi$$
0.590120 + 0.807316i $$0.299081\pi$$
$$504$$ 0 0
$$505$$ 24.3811 1.08494
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −37.3380 −1.65498 −0.827488 0.561483i $$-0.810231\pi$$
−0.827488 + 0.561483i $$0.810231\pi$$
$$510$$ 0 0
$$511$$ 14.3572 0.635127
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 12.3096 0.542427
$$516$$ 0 0
$$517$$ 39.4383 1.73449
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −40.7287 −1.78436 −0.892179 0.451681i $$-0.850824\pi$$
−0.892179 + 0.451681i $$0.850824\pi$$
$$522$$ 0 0
$$523$$ 29.7056 1.29894 0.649468 0.760389i $$-0.274992\pi$$
0.649468 + 0.760389i $$0.274992\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 9.64137 0.419985
$$528$$ 0 0
$$529$$ 5.40485 0.234993
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −6.98426 −0.302522
$$534$$ 0 0
$$535$$ 26.7034 1.15449
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −3.23607 −0.139387
$$540$$ 0 0
$$541$$ −45.4631 −1.95461 −0.977305 0.211835i $$-0.932056\pi$$
−0.977305 + 0.211835i $$0.932056\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −34.5445 −1.47972
$$546$$ 0 0
$$547$$ −10.3113 −0.440879 −0.220440 0.975401i $$-0.570749\pi$$
−0.220440 + 0.975401i $$0.570749\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −82.0534 −3.49559
$$552$$ 0 0
$$553$$ −4.18710 −0.178054
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −16.8770 −0.715101 −0.357550 0.933894i $$-0.616388\pi$$
−0.357550 + 0.933894i $$0.616388\pi$$
$$558$$ 0 0
$$559$$ 1.45123 0.0613805
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 35.6197 1.50119 0.750597 0.660761i $$-0.229766\pi$$
0.750597 + 0.660761i $$0.229766\pi$$
$$564$$ 0 0
$$565$$ −11.1714 −0.469983
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −32.2578 −1.35232 −0.676159 0.736755i $$-0.736357\pi$$
−0.676159 + 0.736755i $$0.736357\pi$$
$$570$$ 0 0
$$571$$ 35.4946 1.48540 0.742702 0.669622i $$-0.233544\pi$$
0.742702 + 0.669622i $$0.233544\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −9.72758 −0.405668
$$576$$ 0 0
$$577$$ 33.1356 1.37945 0.689726 0.724071i $$-0.257731\pi$$
0.689726 + 0.724071i $$0.257731\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 8.42108 0.349365
$$582$$ 0 0
$$583$$ 11.0130 0.456113
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 35.1397 1.45037 0.725186 0.688553i $$-0.241754\pi$$
0.725186 + 0.688553i $$0.241754\pi$$
$$588$$ 0 0
$$589$$ −74.8751 −3.08518
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 30.0729 1.23495 0.617474 0.786591i $$-0.288156\pi$$
0.617474 + 0.786591i $$0.288156\pi$$
$$594$$ 0 0
$$595$$ 1.78180 0.0730466
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −20.6464 −0.843591 −0.421796 0.906691i $$-0.638600\pi$$
−0.421796 + 0.906691i $$0.638600\pi$$
$$600$$ 0 0
$$601$$ −20.5961 −0.840134 −0.420067 0.907493i $$-0.637993\pi$$
−0.420067 + 0.907493i $$0.637993\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −0.940548 −0.0382387
$$606$$ 0 0
$$607$$ −14.9199 −0.605582 −0.302791 0.953057i $$-0.597918\pi$$
−0.302791 + 0.953057i $$0.597918\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −30.1282 −1.21886
$$612$$ 0 0
$$613$$ −24.9575 −1.00802 −0.504011 0.863697i $$-0.668143\pi$$
−0.504011 + 0.863697i $$0.668143\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 7.19596 0.289698 0.144849 0.989454i $$-0.453730\pi$$
0.144849 + 0.989454i $$0.453730\pi$$
$$618$$ 0 0
$$619$$ −17.7770 −0.714517 −0.357258 0.934006i $$-0.616288\pi$$
−0.357258 + 0.934006i $$0.616288\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −17.0446 −0.682877
$$624$$ 0 0
$$625$$ −12.5427 −0.501708
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 1.62142 0.0646501
$$630$$ 0 0
$$631$$ 35.3428 1.40698 0.703488 0.710708i $$-0.251625\pi$$
0.703488 + 0.710708i $$0.251625\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0.646450 0.0256536
$$636$$ 0 0
$$637$$ 2.47214 0.0979496
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 3.66153 0.144622 0.0723108 0.997382i $$-0.476963\pi$$
0.0723108 + 0.997382i $$0.476963\pi$$
$$642$$ 0 0
$$643$$ 29.6783 1.17040 0.585199 0.810890i $$-0.301016\pi$$
0.585199 + 0.810890i $$0.301016\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −28.9758 −1.13916 −0.569579 0.821937i $$-0.692894\pi$$
−0.569579 + 0.821937i $$0.692894\pi$$
$$648$$ 0 0
$$649$$ −23.7191 −0.931058
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −19.2003 −0.751367 −0.375684 0.926748i $$-0.622592\pi$$
−0.375684 + 0.926748i $$0.622592\pi$$
$$654$$ 0 0
$$655$$ 3.46058 0.135216
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 5.13350 0.199973 0.0999865 0.994989i $$-0.468120\pi$$
0.0999865 + 0.994989i $$0.468120\pi$$
$$660$$ 0 0
$$661$$ −34.8839 −1.35683 −0.678413 0.734681i $$-0.737332\pi$$
−0.678413 + 0.734681i $$0.737332\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −13.8375 −0.536595
$$666$$ 0 0
$$667$$ −56.3111 −2.18037
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 12.3718 0.477609
$$672$$ 0 0
$$673$$ −46.2976 −1.78464 −0.892320 0.451403i $$-0.850924\pi$$
−0.892320 + 0.451403i $$0.850924\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 8.29450 0.318784 0.159392 0.987215i $$-0.449047\pi$$
0.159392 + 0.987215i $$0.449047\pi$$
$$678$$ 0 0
$$679$$ 7.72398 0.296419
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 43.0198 1.64611 0.823053 0.567964i $$-0.192269\pi$$
0.823053 + 0.567964i $$0.192269\pi$$
$$684$$ 0 0
$$685$$ −9.22047 −0.352296
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −8.41321 −0.320518
$$690$$ 0 0
$$691$$ −36.3817 −1.38403 −0.692013 0.721885i $$-0.743276\pi$$
−0.692013 + 0.721885i $$0.743276\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 7.36880 0.279514
$$696$$ 0 0
$$697$$ 2.82519 0.107012
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −18.5920 −0.702208 −0.351104 0.936336i $$-0.614194\pi$$
−0.351104 + 0.936336i $$0.614194\pi$$
$$702$$ 0 0
$$703$$ −12.5920 −0.474915
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −13.6834 −0.514618
$$708$$ 0 0
$$709$$ 11.0871 0.416384 0.208192 0.978088i $$-0.433242\pi$$
0.208192 + 0.978088i $$0.433242\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −51.3849 −1.92438
$$714$$ 0 0
$$715$$ −14.2544 −0.533084
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −0.357237 −0.0133227 −0.00666135 0.999978i $$-0.502120\pi$$
−0.00666135 + 0.999978i $$0.502120\pi$$
$$720$$ 0 0
$$721$$ −6.90854 −0.257287
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 19.2844 0.716205
$$726$$ 0 0
$$727$$ −3.07614 −0.114088 −0.0570439 0.998372i $$-0.518167\pi$$
−0.0570439 + 0.998372i $$0.518167\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −0.587035 −0.0217123
$$732$$ 0 0
$$733$$ 17.9930 0.664588 0.332294 0.943176i $$-0.392177\pi$$
0.332294 + 0.943176i $$0.392177\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −1.27618 −0.0470086
$$738$$ 0 0
$$739$$ −8.47140 −0.311625 −0.155813 0.987787i $$-0.549800\pi$$
−0.155813 + 0.987787i $$0.549800\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −23.1587 −0.849612 −0.424806 0.905284i $$-0.639658\pi$$
−0.424806 + 0.905284i $$0.639658\pi$$
$$744$$ 0 0
$$745$$ 3.86674 0.141666
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −14.9868 −0.547605
$$750$$ 0 0
$$751$$ −19.3849 −0.707363 −0.353682 0.935366i $$-0.615070\pi$$
−0.353682 + 0.935366i $$0.615070\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 5.45083 0.198376
$$756$$ 0 0
$$757$$ 48.9669 1.77973 0.889866 0.456222i $$-0.150798\pi$$
0.889866 + 0.456222i $$0.150798\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 25.9354 0.940158 0.470079 0.882624i $$-0.344225\pi$$
0.470079 + 0.882624i $$0.344225\pi$$
$$762$$ 0 0
$$763$$ 19.3874 0.701872
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 18.1198 0.654269
$$768$$ 0 0
$$769$$ −5.37649 −0.193881 −0.0969407 0.995290i $$-0.530906\pi$$
−0.0969407 + 0.995290i $$0.530906\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 3.82642 0.137627 0.0688135 0.997630i $$-0.478079\pi$$
0.0688135 + 0.997630i $$0.478079\pi$$
$$774$$ 0 0
$$775$$ 17.5974 0.632116
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −21.9405 −0.786100
$$780$$ 0 0
$$781$$ −6.96198 −0.249119
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 17.7345 0.632970
$$786$$ 0 0
$$787$$ −1.50487 −0.0536427 −0.0268214 0.999640i $$-0.508539\pi$$
−0.0268214 + 0.999640i $$0.508539\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.26971 0.222925
$$792$$ 0 0
$$793$$ −9.45123 −0.335623
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −28.4448 −1.00757 −0.503783 0.863830i $$-0.668059\pi$$
−0.503783 + 0.863830i $$0.668059\pi$$
$$798$$ 0 0
$$799$$ 12.1871 0.431149
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 46.4610 1.63957
$$804$$ 0 0
$$805$$ −9.49631 −0.334701
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 29.5952 1.04051 0.520255 0.854011i $$-0.325837\pi$$
0.520255 + 0.854011i $$0.325837\pi$$
$$810$$ 0 0
$$811$$ 49.2818 1.73052 0.865259 0.501325i $$-0.167154\pi$$
0.865259 + 0.501325i $$0.167154\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 27.9053 0.977481
$$816$$ 0 0
$$817$$ 4.55892 0.159497
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 39.0114 1.36151 0.680753 0.732513i $$-0.261653\pi$$
0.680753 + 0.732513i $$0.261653\pi$$
$$822$$ 0 0
$$823$$ −36.4772 −1.27152 −0.635758 0.771888i $$-0.719312\pi$$
−0.635758 + 0.771888i $$0.719312\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 7.06846 0.245795 0.122897 0.992419i $$-0.460781\pi$$
0.122897 + 0.992419i $$0.460781\pi$$
$$828$$ 0 0
$$829$$ −19.5952 −0.680568 −0.340284 0.940323i $$-0.610523\pi$$
−0.340284 + 0.940323i $$0.610523\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −1.00000 −0.0346479
$$834$$ 0 0
$$835$$ 14.4666 0.500636
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −5.42576 −0.187318 −0.0936589 0.995604i $$-0.529856\pi$$
−0.0936589 + 0.995604i $$0.529856\pi$$
$$840$$ 0 0
$$841$$ 82.6338 2.84944
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −12.2740 −0.422238
$$846$$ 0 0
$$847$$ 0.527864 0.0181376
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −8.64153 −0.296228
$$852$$ 0 0
$$853$$ 8.52577 0.291917 0.145958 0.989291i $$-0.453373\pi$$
0.145958 + 0.989291i $$0.453373\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −28.9710 −0.989630 −0.494815 0.868998i $$-0.664764\pi$$
−0.494815 + 0.868998i $$0.664764\pi$$
$$858$$ 0 0
$$859$$ 9.67776 0.330201 0.165100 0.986277i $$-0.447205\pi$$
0.165100 + 0.986277i $$0.447205\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −9.67293 −0.329270 −0.164635 0.986355i $$-0.552645\pi$$
−0.164635 + 0.986355i $$0.552645\pi$$
$$864$$ 0 0
$$865$$ −22.0302 −0.749048
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −13.5498 −0.459644
$$870$$ 0 0
$$871$$ 0.974913 0.0330337
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 12.1611 0.411121
$$876$$ 0 0
$$877$$ −40.8080 −1.37799 −0.688995 0.724766i $$-0.741948\pi$$
−0.688995 + 0.724766i $$0.741948\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −0.544255 −0.0183364 −0.00916821 0.999958i $$-0.502918\pi$$
−0.00916821 + 0.999958i $$0.502918\pi$$
$$882$$ 0 0
$$883$$ 7.31126 0.246043 0.123022 0.992404i $$-0.460742\pi$$
0.123022 + 0.992404i $$0.460742\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −13.8600 −0.465374 −0.232687 0.972552i $$-0.574752\pi$$
−0.232687 + 0.972552i $$0.574752\pi$$
$$888$$ 0 0
$$889$$ −0.362807 −0.0121682
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −94.6453 −3.16718
$$894$$ 0 0
$$895$$ 32.2920 1.07940
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 101.868 3.39748
$$900$$ 0 0
$$901$$ 3.40321 0.113378
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −32.8106 −1.09066
$$906$$ 0 0
$$907$$ 50.8411 1.68815 0.844075 0.536225i $$-0.180150\pi$$
0.844075 + 0.536225i $$0.180150\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −25.8459 −0.856311 −0.428156 0.903705i $$-0.640836\pi$$
−0.428156 + 0.903705i $$0.640836\pi$$
$$912$$ 0 0
$$913$$ 27.2512 0.901883
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −1.94218 −0.0641365
$$918$$ 0 0
$$919$$ 44.8471 1.47937 0.739684 0.672954i $$-0.234975\pi$$
0.739684 + 0.672954i $$0.234975\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 5.31848 0.175060
$$924$$ 0 0
$$925$$ 2.95940 0.0973044
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 44.2402 1.45147 0.725737 0.687972i $$-0.241499\pi$$
0.725737 + 0.687972i $$0.241499\pi$$
$$930$$ 0 0
$$931$$ 7.76602 0.254521
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 5.76602 0.188569
$$936$$ 0 0
$$937$$ −41.2813 −1.34860 −0.674300 0.738457i $$-0.735555\pi$$
−0.674300 + 0.738457i $$0.735555\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 11.1816 0.364509 0.182254 0.983251i $$-0.441661\pi$$
0.182254 + 0.983251i $$0.441661\pi$$
$$942$$ 0 0
$$943$$ −15.0572 −0.490330
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −18.5810 −0.603802 −0.301901 0.953339i $$-0.597621\pi$$
−0.301901 + 0.953339i $$0.597621\pi$$
$$948$$ 0 0
$$949$$ −35.4930 −1.15215
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −13.3359 −0.431993 −0.215997 0.976394i $$-0.569300\pi$$
−0.215997 + 0.976394i $$0.569300\pi$$
$$954$$ 0 0
$$955$$ −19.0675 −0.617011
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 5.17481 0.167103
$$960$$ 0 0
$$961$$ 61.9561 1.99858
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −2.69505 −0.0867567
$$966$$ 0 0
$$967$$ 10.9877 0.353341 0.176670 0.984270i $$-0.443467\pi$$
0.176670 + 0.984270i $$0.443467\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −47.3831 −1.52059 −0.760297 0.649575i $$-0.774947\pi$$
−0.760297 + 0.649575i $$0.774947\pi$$
$$972$$ 0 0
$$973$$ −4.13559 −0.132581
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −1.49148 −0.0477166 −0.0238583 0.999715i $$-0.507595\pi$$
−0.0238583 + 0.999715i $$0.507595\pi$$
$$978$$ 0 0
$$979$$ −55.1574 −1.76284
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 53.9441 1.72055 0.860275 0.509830i $$-0.170292\pi$$
0.860275 + 0.509830i $$0.170292\pi$$
$$984$$ 0 0
$$985$$ 12.9569 0.412841
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 3.12867 0.0994860
$$990$$ 0 0
$$991$$ −51.2401 −1.62769 −0.813847 0.581079i $$-0.802631\pi$$
−0.813847 + 0.581079i $$0.802631\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −41.3077 −1.30954
$$996$$ 0 0
$$997$$ −12.0544 −0.381766 −0.190883 0.981613i $$-0.561135\pi$$
−0.190883 + 0.981613i $$0.561135\pi$$
$$998$$ 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 8568.2.a.bj.1.3 4
3.2 odd 2 952.2.a.g.1.4 4
12.11 even 2 1904.2.a.q.1.1 4
21.20 even 2 6664.2.a.o.1.1 4
24.5 odd 2 7616.2.a.bj.1.1 4
24.11 even 2 7616.2.a.bp.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.g.1.4 4 3.2 odd 2
1904.2.a.q.1.1 4 12.11 even 2
6664.2.a.o.1.1 4 21.20 even 2
7616.2.a.bj.1.1 4 24.5 odd 2
7616.2.a.bp.1.4 4 24.11 even 2
8568.2.a.bj.1.3 4 1.1 even 1 trivial