# Properties

 Label 4368.2.a.bq.1.3 Level $4368$ Weight $2$ Character 4368.1 Self dual yes Analytic conductor $34.879$ 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] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4368.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: $$34.8786556029$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 273) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.470683$$ of defining polynomial Character $$\chi$$ $$=$$ 4368.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.77846 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.77846 q^{5} +1.00000 q^{7} +1.00000 q^{9} +6.49828 q^{11} -1.00000 q^{13} +1.77846 q^{15} -2.94137 q^{17} +4.83709 q^{19} +1.00000 q^{21} +5.77846 q^{23} -1.83709 q^{25} +1.00000 q^{27} -2.83709 q^{29} -6.27674 q^{31} +6.49828 q^{33} +1.77846 q^{35} +9.55691 q^{37} -1.00000 q^{39} -3.05863 q^{41} -2.71982 q^{43} +1.77846 q^{45} +8.71982 q^{47} +1.00000 q^{49} -2.94137 q^{51} +6.39400 q^{53} +11.5569 q^{55} +4.83709 q^{57} -1.55691 q^{59} +3.88273 q^{61} +1.00000 q^{63} -1.77846 q^{65} -5.67418 q^{67} +5.77846 q^{69} -10.0552 q^{71} -15.8337 q^{73} -1.83709 q^{75} +6.49828 q^{77} +1.28018 q^{79} +1.00000 q^{81} +2.83709 q^{83} -5.23109 q^{85} -2.83709 q^{87} -7.66119 q^{89} -1.00000 q^{91} -6.27674 q^{93} +8.60256 q^{95} -17.7164 q^{97} +6.49828 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 $$3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 3 q^{15} - 8 q^{17} + 7 q^{19} + 3 q^{21} + 9 q^{23} + 2 q^{25} + 3 q^{27} - q^{29} + 7 q^{31} + 2 q^{33} - 3 q^{35} + 12 q^{37} - 3 q^{39} - 10 q^{41} + q^{43} - 3 q^{45} + 17 q^{47} + 3 q^{49} - 8 q^{51} - 5 q^{53} + 18 q^{55} + 7 q^{57} + 12 q^{59} + 10 q^{61} + 3 q^{63} + 3 q^{65} - 2 q^{67} + 9 q^{69} + 4 q^{71} - 5 q^{73} + 2 q^{75} + 2 q^{77} + 13 q^{79} + 3 q^{81} + q^{83} + 16 q^{85} - q^{87} - 13 q^{89} - 3 q^{91} + 7 q^{93} + 15 q^{95} - 9 q^{97} + 2 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 3 * q^15 - 8 * q^17 + 7 * q^19 + 3 * q^21 + 9 * q^23 + 2 * q^25 + 3 * q^27 - q^29 + 7 * q^31 + 2 * q^33 - 3 * q^35 + 12 * q^37 - 3 * q^39 - 10 * q^41 + q^43 - 3 * q^45 + 17 * q^47 + 3 * q^49 - 8 * q^51 - 5 * q^53 + 18 * q^55 + 7 * q^57 + 12 * q^59 + 10 * q^61 + 3 * q^63 + 3 * q^65 - 2 * q^67 + 9 * q^69 + 4 * q^71 - 5 * q^73 + 2 * q^75 + 2 * q^77 + 13 * q^79 + 3 * q^81 + q^83 + 16 * q^85 - q^87 - 13 * q^89 - 3 * q^91 + 7 * q^93 + 15 * q^95 - 9 * q^97 + 2 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 1.77846 0.795350 0.397675 0.917526i $$-0.369817\pi$$
0.397675 + 0.917526i $$0.369817\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 6.49828 1.95931 0.979653 0.200700i $$-0.0643217\pi$$
0.979653 + 0.200700i $$0.0643217\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 1.77846 0.459196
$$16$$ 0 0
$$17$$ −2.94137 −0.713386 −0.356693 0.934222i $$-0.616096\pi$$
−0.356693 + 0.934222i $$0.616096\pi$$
$$18$$ 0 0
$$19$$ 4.83709 1.10970 0.554852 0.831949i $$-0.312775\pi$$
0.554852 + 0.831949i $$0.312775\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 5.77846 1.20489 0.602446 0.798160i $$-0.294193\pi$$
0.602446 + 0.798160i $$0.294193\pi$$
$$24$$ 0 0
$$25$$ −1.83709 −0.367418
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −2.83709 −0.526834 −0.263417 0.964682i $$-0.584850\pi$$
−0.263417 + 0.964682i $$0.584850\pi$$
$$30$$ 0 0
$$31$$ −6.27674 −1.12734 −0.563668 0.826002i $$-0.690610\pi$$
−0.563668 + 0.826002i $$0.690610\pi$$
$$32$$ 0 0
$$33$$ 6.49828 1.13121
$$34$$ 0 0
$$35$$ 1.77846 0.300614
$$36$$ 0 0
$$37$$ 9.55691 1.57115 0.785574 0.618768i $$-0.212368\pi$$
0.785574 + 0.618768i $$0.212368\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −3.05863 −0.477678 −0.238839 0.971059i $$-0.576767\pi$$
−0.238839 + 0.971059i $$0.576767\pi$$
$$42$$ 0 0
$$43$$ −2.71982 −0.414769 −0.207385 0.978259i $$-0.566495\pi$$
−0.207385 + 0.978259i $$0.566495\pi$$
$$44$$ 0 0
$$45$$ 1.77846 0.265117
$$46$$ 0 0
$$47$$ 8.71982 1.27192 0.635959 0.771723i $$-0.280605\pi$$
0.635959 + 0.771723i $$0.280605\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −2.94137 −0.411874
$$52$$ 0 0
$$53$$ 6.39400 0.878284 0.439142 0.898418i $$-0.355282\pi$$
0.439142 + 0.898418i $$0.355282\pi$$
$$54$$ 0 0
$$55$$ 11.5569 1.55833
$$56$$ 0 0
$$57$$ 4.83709 0.640688
$$58$$ 0 0
$$59$$ −1.55691 −0.202693 −0.101346 0.994851i $$-0.532315\pi$$
−0.101346 + 0.994851i $$0.532315\pi$$
$$60$$ 0 0
$$61$$ 3.88273 0.497133 0.248567 0.968615i $$-0.420041\pi$$
0.248567 + 0.968615i $$0.420041\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −1.77846 −0.220590
$$66$$ 0 0
$$67$$ −5.67418 −0.693211 −0.346606 0.938011i $$-0.612666\pi$$
−0.346606 + 0.938011i $$0.612666\pi$$
$$68$$ 0 0
$$69$$ 5.77846 0.695644
$$70$$ 0 0
$$71$$ −10.0552 −1.19333 −0.596666 0.802490i $$-0.703508\pi$$
−0.596666 + 0.802490i $$0.703508\pi$$
$$72$$ 0 0
$$73$$ −15.8337 −1.85319 −0.926594 0.376062i $$-0.877278\pi$$
−0.926594 + 0.376062i $$0.877278\pi$$
$$74$$ 0 0
$$75$$ −1.83709 −0.212129
$$76$$ 0 0
$$77$$ 6.49828 0.740548
$$78$$ 0 0
$$79$$ 1.28018 0.144031 0.0720155 0.997404i $$-0.477057\pi$$
0.0720155 + 0.997404i $$0.477057\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 2.83709 0.311411 0.155706 0.987804i $$-0.450235\pi$$
0.155706 + 0.987804i $$0.450235\pi$$
$$84$$ 0 0
$$85$$ −5.23109 −0.567392
$$86$$ 0 0
$$87$$ −2.83709 −0.304168
$$88$$ 0 0
$$89$$ −7.66119 −0.812085 −0.406042 0.913854i $$-0.633091\pi$$
−0.406042 + 0.913854i $$0.633091\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ −6.27674 −0.650867
$$94$$ 0 0
$$95$$ 8.60256 0.882604
$$96$$ 0 0
$$97$$ −17.7164 −1.79883 −0.899413 0.437099i $$-0.856006\pi$$
−0.899413 + 0.437099i $$0.856006\pi$$
$$98$$ 0 0
$$99$$ 6.49828 0.653102
$$100$$ 0 0
$$101$$ 6.73281 0.669940 0.334970 0.942229i $$-0.391274\pi$$
0.334970 + 0.942229i $$0.391274\pi$$
$$102$$ 0 0
$$103$$ −10.8793 −1.07197 −0.535984 0.844228i $$-0.680059\pi$$
−0.535984 + 0.844228i $$0.680059\pi$$
$$104$$ 0 0
$$105$$ 1.77846 0.173560
$$106$$ 0 0
$$107$$ 8.49828 0.821560 0.410780 0.911735i $$-0.365256\pi$$
0.410780 + 0.911735i $$0.365256\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 9.55691 0.907102
$$112$$ 0 0
$$113$$ 14.6026 1.37369 0.686847 0.726802i $$-0.258994\pi$$
0.686847 + 0.726802i $$0.258994\pi$$
$$114$$ 0 0
$$115$$ 10.2767 0.958311
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ −2.94137 −0.269635
$$120$$ 0 0
$$121$$ 31.2277 2.83888
$$122$$ 0 0
$$123$$ −3.05863 −0.275788
$$124$$ 0 0
$$125$$ −12.1595 −1.08758
$$126$$ 0 0
$$127$$ 9.88273 0.876951 0.438475 0.898743i $$-0.355519\pi$$
0.438475 + 0.898743i $$0.355519\pi$$
$$128$$ 0 0
$$129$$ −2.71982 −0.239467
$$130$$ 0 0
$$131$$ −3.76547 −0.328990 −0.164495 0.986378i $$-0.552600\pi$$
−0.164495 + 0.986378i $$0.552600\pi$$
$$132$$ 0 0
$$133$$ 4.83709 0.419429
$$134$$ 0 0
$$135$$ 1.77846 0.153065
$$136$$ 0 0
$$137$$ −16.9966 −1.45211 −0.726057 0.687634i $$-0.758649\pi$$
−0.726057 + 0.687634i $$0.758649\pi$$
$$138$$ 0 0
$$139$$ 8.55348 0.725496 0.362748 0.931887i $$-0.381839\pi$$
0.362748 + 0.931887i $$0.381839\pi$$
$$140$$ 0 0
$$141$$ 8.71982 0.734342
$$142$$ 0 0
$$143$$ −6.49828 −0.543414
$$144$$ 0 0
$$145$$ −5.04564 −0.419018
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ −15.5569 −1.27447 −0.637236 0.770669i $$-0.719922\pi$$
−0.637236 + 0.770669i $$0.719922\pi$$
$$150$$ 0 0
$$151$$ −4.99656 −0.406614 −0.203307 0.979115i $$-0.565169\pi$$
−0.203307 + 0.979115i $$0.565169\pi$$
$$152$$ 0 0
$$153$$ −2.94137 −0.237795
$$154$$ 0 0
$$155$$ −11.1629 −0.896626
$$156$$ 0 0
$$157$$ −18.7880 −1.49945 −0.749723 0.661752i $$-0.769813\pi$$
−0.749723 + 0.661752i $$0.769813\pi$$
$$158$$ 0 0
$$159$$ 6.39400 0.507078
$$160$$ 0 0
$$161$$ 5.77846 0.455406
$$162$$ 0 0
$$163$$ 9.88273 0.774075 0.387038 0.922064i $$-0.373498\pi$$
0.387038 + 0.922064i $$0.373498\pi$$
$$164$$ 0 0
$$165$$ 11.5569 0.899705
$$166$$ 0 0
$$167$$ 7.04564 0.545208 0.272604 0.962126i $$-0.412115\pi$$
0.272604 + 0.962126i $$0.412115\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 4.83709 0.369902
$$172$$ 0 0
$$173$$ −14.2897 −1.08643 −0.543214 0.839594i $$-0.682793\pi$$
−0.543214 + 0.839594i $$0.682793\pi$$
$$174$$ 0 0
$$175$$ −1.83709 −0.138871
$$176$$ 0 0
$$177$$ −1.55691 −0.117025
$$178$$ 0 0
$$179$$ 8.65775 0.647111 0.323555 0.946209i $$-0.395122\pi$$
0.323555 + 0.946209i $$0.395122\pi$$
$$180$$ 0 0
$$181$$ 26.3449 1.95820 0.979101 0.203373i $$-0.0651904\pi$$
0.979101 + 0.203373i $$0.0651904\pi$$
$$182$$ 0 0
$$183$$ 3.88273 0.287020
$$184$$ 0 0
$$185$$ 16.9966 1.24961
$$186$$ 0 0
$$187$$ −19.1138 −1.39774
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −6.61555 −0.478684 −0.239342 0.970935i $$-0.576932\pi$$
−0.239342 + 0.970935i $$0.576932\pi$$
$$192$$ 0 0
$$193$$ 11.8827 0.855338 0.427669 0.903935i $$-0.359335\pi$$
0.427669 + 0.903935i $$0.359335\pi$$
$$194$$ 0 0
$$195$$ −1.77846 −0.127358
$$196$$ 0 0
$$197$$ 11.5569 0.823396 0.411698 0.911320i $$-0.364936\pi$$
0.411698 + 0.911320i $$0.364936\pi$$
$$198$$ 0 0
$$199$$ −0.996562 −0.0706444 −0.0353222 0.999376i $$-0.511246\pi$$
−0.0353222 + 0.999376i $$0.511246\pi$$
$$200$$ 0 0
$$201$$ −5.67418 −0.400226
$$202$$ 0 0
$$203$$ −2.83709 −0.199125
$$204$$ 0 0
$$205$$ −5.43965 −0.379921
$$206$$ 0 0
$$207$$ 5.77846 0.401631
$$208$$ 0 0
$$209$$ 31.4328 2.17425
$$210$$ 0 0
$$211$$ −26.8302 −1.84707 −0.923534 0.383516i $$-0.874713\pi$$
−0.923534 + 0.383516i $$0.874713\pi$$
$$212$$ 0 0
$$213$$ −10.0552 −0.688971
$$214$$ 0 0
$$215$$ −4.83709 −0.329887
$$216$$ 0 0
$$217$$ −6.27674 −0.426093
$$218$$ 0 0
$$219$$ −15.8337 −1.06994
$$220$$ 0 0
$$221$$ 2.94137 0.197858
$$222$$ 0 0
$$223$$ 2.92838 0.196099 0.0980493 0.995182i $$-0.468740\pi$$
0.0980493 + 0.995182i $$0.468740\pi$$
$$224$$ 0 0
$$225$$ −1.83709 −0.122473
$$226$$ 0 0
$$227$$ 9.79145 0.649881 0.324941 0.945734i $$-0.394656\pi$$
0.324941 + 0.945734i $$0.394656\pi$$
$$228$$ 0 0
$$229$$ 3.88273 0.256578 0.128289 0.991737i $$-0.459051\pi$$
0.128289 + 0.991737i $$0.459051\pi$$
$$230$$ 0 0
$$231$$ 6.49828 0.427556
$$232$$ 0 0
$$233$$ −15.8337 −1.03730 −0.518649 0.854988i $$-0.673565\pi$$
−0.518649 + 0.854988i $$0.673565\pi$$
$$234$$ 0 0
$$235$$ 15.5078 1.01162
$$236$$ 0 0
$$237$$ 1.28018 0.0831564
$$238$$ 0 0
$$239$$ 6.94137 0.449000 0.224500 0.974474i $$-0.427925\pi$$
0.224500 + 0.974474i $$0.427925\pi$$
$$240$$ 0 0
$$241$$ 7.28018 0.468957 0.234479 0.972121i $$-0.424662\pi$$
0.234479 + 0.972121i $$0.424662\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 1.77846 0.113621
$$246$$ 0 0
$$247$$ −4.83709 −0.307777
$$248$$ 0 0
$$249$$ 2.83709 0.179793
$$250$$ 0 0
$$251$$ −23.3224 −1.47210 −0.736048 0.676930i $$-0.763310\pi$$
−0.736048 + 0.676930i $$0.763310\pi$$
$$252$$ 0 0
$$253$$ 37.5500 2.36075
$$254$$ 0 0
$$255$$ −5.23109 −0.327584
$$256$$ 0 0
$$257$$ 15.4948 0.966542 0.483271 0.875471i $$-0.339449\pi$$
0.483271 + 0.875471i $$0.339449\pi$$
$$258$$ 0 0
$$259$$ 9.55691 0.593838
$$260$$ 0 0
$$261$$ −2.83709 −0.175611
$$262$$ 0 0
$$263$$ −3.42666 −0.211297 −0.105648 0.994404i $$-0.533692\pi$$
−0.105648 + 0.994404i $$0.533692\pi$$
$$264$$ 0 0
$$265$$ 11.3715 0.698543
$$266$$ 0 0
$$267$$ −7.66119 −0.468857
$$268$$ 0 0
$$269$$ −0.172462 −0.0105152 −0.00525759 0.999986i $$-0.501674\pi$$
−0.00525759 + 0.999986i $$0.501674\pi$$
$$270$$ 0 0
$$271$$ 25.9931 1.57897 0.789485 0.613770i $$-0.210348\pi$$
0.789485 + 0.613770i $$0.210348\pi$$
$$272$$ 0 0
$$273$$ −1.00000 −0.0605228
$$274$$ 0 0
$$275$$ −11.9379 −0.719884
$$276$$ 0 0
$$277$$ 3.72326 0.223709 0.111855 0.993725i $$-0.464321\pi$$
0.111855 + 0.993725i $$0.464321\pi$$
$$278$$ 0 0
$$279$$ −6.27674 −0.375778
$$280$$ 0 0
$$281$$ −21.5500 −1.28557 −0.642784 0.766048i $$-0.722221\pi$$
−0.642784 + 0.766048i $$0.722221\pi$$
$$282$$ 0 0
$$283$$ 31.8759 1.89482 0.947412 0.320018i $$-0.103689\pi$$
0.947412 + 0.320018i $$0.103689\pi$$
$$284$$ 0 0
$$285$$ 8.60256 0.509572
$$286$$ 0 0
$$287$$ −3.05863 −0.180545
$$288$$ 0 0
$$289$$ −8.34836 −0.491080
$$290$$ 0 0
$$291$$ −17.7164 −1.03855
$$292$$ 0 0
$$293$$ 3.42666 0.200188 0.100094 0.994978i $$-0.468086\pi$$
0.100094 + 0.994978i $$0.468086\pi$$
$$294$$ 0 0
$$295$$ −2.76891 −0.161212
$$296$$ 0 0
$$297$$ 6.49828 0.377069
$$298$$ 0 0
$$299$$ −5.77846 −0.334177
$$300$$ 0 0
$$301$$ −2.71982 −0.156768
$$302$$ 0 0
$$303$$ 6.73281 0.386790
$$304$$ 0 0
$$305$$ 6.90528 0.395395
$$306$$ 0 0
$$307$$ 3.39744 0.193902 0.0969511 0.995289i $$-0.469091\pi$$
0.0969511 + 0.995289i $$0.469091\pi$$
$$308$$ 0 0
$$309$$ −10.8793 −0.618902
$$310$$ 0 0
$$311$$ 0.443086 0.0251251 0.0125625 0.999921i $$-0.496001\pi$$
0.0125625 + 0.999921i $$0.496001\pi$$
$$312$$ 0 0
$$313$$ −19.8827 −1.12384 −0.561919 0.827192i $$-0.689937\pi$$
−0.561919 + 0.827192i $$0.689937\pi$$
$$314$$ 0 0
$$315$$ 1.77846 0.100205
$$316$$ 0 0
$$317$$ 9.46563 0.531643 0.265821 0.964022i $$-0.414357\pi$$
0.265821 + 0.964022i $$0.414357\pi$$
$$318$$ 0 0
$$319$$ −18.4362 −1.03223
$$320$$ 0 0
$$321$$ 8.49828 0.474328
$$322$$ 0 0
$$323$$ −14.2277 −0.791648
$$324$$ 0 0
$$325$$ 1.83709 0.101903
$$326$$ 0 0
$$327$$ 10.0000 0.553001
$$328$$ 0 0
$$329$$ 8.71982 0.480739
$$330$$ 0 0
$$331$$ 27.4328 1.50784 0.753921 0.656965i $$-0.228160\pi$$
0.753921 + 0.656965i $$0.228160\pi$$
$$332$$ 0 0
$$333$$ 9.55691 0.523716
$$334$$ 0 0
$$335$$ −10.0913 −0.551346
$$336$$ 0 0
$$337$$ −20.2767 −1.10454 −0.552272 0.833664i $$-0.686239\pi$$
−0.552272 + 0.833664i $$0.686239\pi$$
$$338$$ 0 0
$$339$$ 14.6026 0.793102
$$340$$ 0 0
$$341$$ −40.7880 −2.20879
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 10.2767 0.553281
$$346$$ 0 0
$$347$$ −16.7328 −0.898265 −0.449132 0.893465i $$-0.648267\pi$$
−0.449132 + 0.893465i $$0.648267\pi$$
$$348$$ 0 0
$$349$$ 22.3940 1.19872 0.599362 0.800478i $$-0.295421\pi$$
0.599362 + 0.800478i $$0.295421\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ −27.1690 −1.44606 −0.723031 0.690816i $$-0.757251\pi$$
−0.723031 + 0.690816i $$0.757251\pi$$
$$354$$ 0 0
$$355$$ −17.8827 −0.949117
$$356$$ 0 0
$$357$$ −2.94137 −0.155674
$$358$$ 0 0
$$359$$ 29.0586 1.53366 0.766828 0.641853i $$-0.221834\pi$$
0.766828 + 0.641853i $$0.221834\pi$$
$$360$$ 0 0
$$361$$ 4.39744 0.231444
$$362$$ 0 0
$$363$$ 31.2277 1.63903
$$364$$ 0 0
$$365$$ −28.1595 −1.47393
$$366$$ 0 0
$$367$$ 4.44309 0.231927 0.115964 0.993253i $$-0.463004\pi$$
0.115964 + 0.993253i $$0.463004\pi$$
$$368$$ 0 0
$$369$$ −3.05863 −0.159226
$$370$$ 0 0
$$371$$ 6.39400 0.331960
$$372$$ 0 0
$$373$$ 31.3415 1.62280 0.811400 0.584491i $$-0.198706\pi$$
0.811400 + 0.584491i $$0.198706\pi$$
$$374$$ 0 0
$$375$$ −12.1595 −0.627912
$$376$$ 0 0
$$377$$ 2.83709 0.146118
$$378$$ 0 0
$$379$$ 11.5569 0.593639 0.296819 0.954934i $$-0.404074\pi$$
0.296819 + 0.954934i $$0.404074\pi$$
$$380$$ 0 0
$$381$$ 9.88273 0.506308
$$382$$ 0 0
$$383$$ 25.6673 1.31154 0.655769 0.754962i $$-0.272345\pi$$
0.655769 + 0.754962i $$0.272345\pi$$
$$384$$ 0 0
$$385$$ 11.5569 0.588995
$$386$$ 0 0
$$387$$ −2.71982 −0.138256
$$388$$ 0 0
$$389$$ −24.2277 −1.22839 −0.614195 0.789154i $$-0.710519\pi$$
−0.614195 + 0.789154i $$0.710519\pi$$
$$390$$ 0 0
$$391$$ −16.9966 −0.859553
$$392$$ 0 0
$$393$$ −3.76547 −0.189943
$$394$$ 0 0
$$395$$ 2.27674 0.114555
$$396$$ 0 0
$$397$$ −14.3680 −0.721111 −0.360555 0.932738i $$-0.617413\pi$$
−0.360555 + 0.932738i $$0.617413\pi$$
$$398$$ 0 0
$$399$$ 4.83709 0.242157
$$400$$ 0 0
$$401$$ −21.8827 −1.09277 −0.546386 0.837534i $$-0.683997\pi$$
−0.546386 + 0.837534i $$0.683997\pi$$
$$402$$ 0 0
$$403$$ 6.27674 0.312667
$$404$$ 0 0
$$405$$ 1.77846 0.0883722
$$406$$ 0 0
$$407$$ 62.1035 3.07836
$$408$$ 0 0
$$409$$ 7.39057 0.365440 0.182720 0.983165i $$-0.441510\pi$$
0.182720 + 0.983165i $$0.441510\pi$$
$$410$$ 0 0
$$411$$ −16.9966 −0.838379
$$412$$ 0 0
$$413$$ −1.55691 −0.0766107
$$414$$ 0 0
$$415$$ 5.04564 0.247681
$$416$$ 0 0
$$417$$ 8.55348 0.418866
$$418$$ 0 0
$$419$$ −33.7846 −1.65048 −0.825242 0.564779i $$-0.808961\pi$$
−0.825242 + 0.564779i $$0.808961\pi$$
$$420$$ 0 0
$$421$$ −35.5500 −1.73260 −0.866301 0.499522i $$-0.833509\pi$$
−0.866301 + 0.499522i $$0.833509\pi$$
$$422$$ 0 0
$$423$$ 8.71982 0.423972
$$424$$ 0 0
$$425$$ 5.40356 0.262111
$$426$$ 0 0
$$427$$ 3.88273 0.187899
$$428$$ 0 0
$$429$$ −6.49828 −0.313740
$$430$$ 0 0
$$431$$ −23.7294 −1.14300 −0.571502 0.820601i $$-0.693639\pi$$
−0.571502 + 0.820601i $$0.693639\pi$$
$$432$$ 0 0
$$433$$ −5.55691 −0.267048 −0.133524 0.991046i $$-0.542629\pi$$
−0.133524 + 0.991046i $$0.542629\pi$$
$$434$$ 0 0
$$435$$ −5.04564 −0.241920
$$436$$ 0 0
$$437$$ 27.9509 1.33707
$$438$$ 0 0
$$439$$ −5.32926 −0.254352 −0.127176 0.991880i $$-0.540591\pi$$
−0.127176 + 0.991880i $$0.540591\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −5.33537 −0.253491 −0.126746 0.991935i $$-0.540453\pi$$
−0.126746 + 0.991935i $$0.540453\pi$$
$$444$$ 0 0
$$445$$ −13.6251 −0.645892
$$446$$ 0 0
$$447$$ −15.5569 −0.735817
$$448$$ 0 0
$$449$$ 26.2277 1.23776 0.618880 0.785486i $$-0.287587\pi$$
0.618880 + 0.785486i $$0.287587\pi$$
$$450$$ 0 0
$$451$$ −19.8759 −0.935918
$$452$$ 0 0
$$453$$ −4.99656 −0.234759
$$454$$ 0 0
$$455$$ −1.77846 −0.0833754
$$456$$ 0 0
$$457$$ 11.4396 0.535124 0.267562 0.963541i $$-0.413782\pi$$
0.267562 + 0.963541i $$0.413782\pi$$
$$458$$ 0 0
$$459$$ −2.94137 −0.137291
$$460$$ 0 0
$$461$$ −28.0552 −1.30666 −0.653330 0.757073i $$-0.726629\pi$$
−0.653330 + 0.757073i $$0.726629\pi$$
$$462$$ 0 0
$$463$$ 10.5604 0.490781 0.245391 0.969424i $$-0.421084\pi$$
0.245391 + 0.969424i $$0.421084\pi$$
$$464$$ 0 0
$$465$$ −11.1629 −0.517668
$$466$$ 0 0
$$467$$ −16.6776 −0.771748 −0.385874 0.922551i $$-0.626100\pi$$
−0.385874 + 0.922551i $$0.626100\pi$$
$$468$$ 0 0
$$469$$ −5.67418 −0.262009
$$470$$ 0 0
$$471$$ −18.7880 −0.865706
$$472$$ 0 0
$$473$$ −17.6742 −0.812660
$$474$$ 0 0
$$475$$ −8.88617 −0.407726
$$476$$ 0 0
$$477$$ 6.39400 0.292761
$$478$$ 0 0
$$479$$ −4.06819 −0.185880 −0.0929401 0.995672i $$-0.529626\pi$$
−0.0929401 + 0.995672i $$0.529626\pi$$
$$480$$ 0 0
$$481$$ −9.55691 −0.435758
$$482$$ 0 0
$$483$$ 5.77846 0.262929
$$484$$ 0 0
$$485$$ −31.5078 −1.43070
$$486$$ 0 0
$$487$$ 0.443086 0.0200781 0.0100391 0.999950i $$-0.496804\pi$$
0.0100391 + 0.999950i $$0.496804\pi$$
$$488$$ 0 0
$$489$$ 9.88273 0.446913
$$490$$ 0 0
$$491$$ −20.7328 −0.935659 −0.467829 0.883819i $$-0.654964\pi$$
−0.467829 + 0.883819i $$0.654964\pi$$
$$492$$ 0 0
$$493$$ 8.34492 0.375836
$$494$$ 0 0
$$495$$ 11.5569 0.519445
$$496$$ 0 0
$$497$$ −10.0552 −0.451037
$$498$$ 0 0
$$499$$ 13.9931 0.626418 0.313209 0.949684i $$-0.398596\pi$$
0.313209 + 0.949684i $$0.398596\pi$$
$$500$$ 0 0
$$501$$ 7.04564 0.314776
$$502$$ 0 0
$$503$$ −5.67418 −0.252999 −0.126500 0.991967i $$-0.540374\pi$$
−0.126500 + 0.991967i $$0.540374\pi$$
$$504$$ 0 0
$$505$$ 11.9740 0.532837
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ −6.22154 −0.275765 −0.137883 0.990449i $$-0.544030\pi$$
−0.137883 + 0.990449i $$0.544030\pi$$
$$510$$ 0 0
$$511$$ −15.8337 −0.700440
$$512$$ 0 0
$$513$$ 4.83709 0.213563
$$514$$ 0 0
$$515$$ −19.3484 −0.852591
$$516$$ 0 0
$$517$$ 56.6639 2.49207
$$518$$ 0 0
$$519$$ −14.2897 −0.627249
$$520$$ 0 0
$$521$$ −10.2897 −0.450801 −0.225401 0.974266i $$-0.572369\pi$$
−0.225401 + 0.974266i $$0.572369\pi$$
$$522$$ 0 0
$$523$$ 2.32582 0.101701 0.0508505 0.998706i $$-0.483807\pi$$
0.0508505 + 0.998706i $$0.483807\pi$$
$$524$$ 0 0
$$525$$ −1.83709 −0.0801772
$$526$$ 0 0
$$527$$ 18.4622 0.804226
$$528$$ 0 0
$$529$$ 10.3906 0.451764
$$530$$ 0 0
$$531$$ −1.55691 −0.0675643
$$532$$ 0 0
$$533$$ 3.05863 0.132484
$$534$$ 0 0
$$535$$ 15.1138 0.653428
$$536$$ 0 0
$$537$$ 8.65775 0.373610
$$538$$ 0 0
$$539$$ 6.49828 0.279901
$$540$$ 0 0
$$541$$ 5.55691 0.238910 0.119455 0.992840i $$-0.461885\pi$$
0.119455 + 0.992840i $$0.461885\pi$$
$$542$$ 0 0
$$543$$ 26.3449 1.13057
$$544$$ 0 0
$$545$$ 17.7846 0.761807
$$546$$ 0 0
$$547$$ 5.48873 0.234681 0.117341 0.993092i $$-0.462563\pi$$
0.117341 + 0.993092i $$0.462563\pi$$
$$548$$ 0 0
$$549$$ 3.88273 0.165711
$$550$$ 0 0
$$551$$ −13.7233 −0.584631
$$552$$ 0 0
$$553$$ 1.28018 0.0544386
$$554$$ 0 0
$$555$$ 16.9966 0.721464
$$556$$ 0 0
$$557$$ −22.9897 −0.974104 −0.487052 0.873373i $$-0.661928\pi$$
−0.487052 + 0.873373i $$0.661928\pi$$
$$558$$ 0 0
$$559$$ 2.71982 0.115036
$$560$$ 0 0
$$561$$ −19.1138 −0.806986
$$562$$ 0 0
$$563$$ 32.7620 1.38075 0.690377 0.723449i $$-0.257444\pi$$
0.690377 + 0.723449i $$0.257444\pi$$
$$564$$ 0 0
$$565$$ 25.9700 1.09257
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 36.1526 1.51560 0.757798 0.652489i $$-0.226275\pi$$
0.757798 + 0.652489i $$0.226275\pi$$
$$570$$ 0 0
$$571$$ −6.48529 −0.271401 −0.135700 0.990750i $$-0.543328\pi$$
−0.135700 + 0.990750i $$0.543328\pi$$
$$572$$ 0 0
$$573$$ −6.61555 −0.276368
$$574$$ 0 0
$$575$$ −10.6155 −0.442699
$$576$$ 0 0
$$577$$ 3.99312 0.166236 0.0831180 0.996540i $$-0.473512\pi$$
0.0831180 + 0.996540i $$0.473512\pi$$
$$578$$ 0 0
$$579$$ 11.8827 0.493830
$$580$$ 0 0
$$581$$ 2.83709 0.117702
$$582$$ 0 0
$$583$$ 41.5500 1.72083
$$584$$ 0 0
$$585$$ −1.77846 −0.0735302
$$586$$ 0 0
$$587$$ −4.16635 −0.171964 −0.0859818 0.996297i $$-0.527403\pi$$
−0.0859818 + 0.996297i $$0.527403\pi$$
$$588$$ 0 0
$$589$$ −30.3611 −1.25101
$$590$$ 0 0
$$591$$ 11.5569 0.475388
$$592$$ 0 0
$$593$$ 16.1303 0.662390 0.331195 0.943562i $$-0.392548\pi$$
0.331195 + 0.943562i $$0.392548\pi$$
$$594$$ 0 0
$$595$$ −5.23109 −0.214454
$$596$$ 0 0
$$597$$ −0.996562 −0.0407866
$$598$$ 0 0
$$599$$ 33.2372 1.35804 0.679018 0.734122i $$-0.262406\pi$$
0.679018 + 0.734122i $$0.262406\pi$$
$$600$$ 0 0
$$601$$ −28.9897 −1.18251 −0.591257 0.806483i $$-0.701368\pi$$
−0.591257 + 0.806483i $$0.701368\pi$$
$$602$$ 0 0
$$603$$ −5.67418 −0.231070
$$604$$ 0 0
$$605$$ 55.5370 2.25790
$$606$$ 0 0
$$607$$ 25.7846 1.04656 0.523282 0.852160i $$-0.324708\pi$$
0.523282 + 0.852160i $$0.324708\pi$$
$$608$$ 0 0
$$609$$ −2.83709 −0.114965
$$610$$ 0 0
$$611$$ −8.71982 −0.352766
$$612$$ 0 0
$$613$$ −7.43965 −0.300485 −0.150242 0.988649i $$-0.548005\pi$$
−0.150242 + 0.988649i $$0.548005\pi$$
$$614$$ 0 0
$$615$$ −5.43965 −0.219348
$$616$$ 0 0
$$617$$ −2.67074 −0.107520 −0.0537600 0.998554i $$-0.517121\pi$$
−0.0537600 + 0.998554i $$0.517121\pi$$
$$618$$ 0 0
$$619$$ 4.46907 0.179627 0.0898134 0.995959i $$-0.471373\pi$$
0.0898134 + 0.995959i $$0.471373\pi$$
$$620$$ 0 0
$$621$$ 5.77846 0.231881
$$622$$ 0 0
$$623$$ −7.66119 −0.306939
$$624$$ 0 0
$$625$$ −12.4396 −0.497586
$$626$$ 0 0
$$627$$ 31.4328 1.25530
$$628$$ 0 0
$$629$$ −28.1104 −1.12083
$$630$$ 0 0
$$631$$ −23.0878 −0.919113 −0.459556 0.888149i $$-0.651992\pi$$
−0.459556 + 0.888149i $$0.651992\pi$$
$$632$$ 0 0
$$633$$ −26.8302 −1.06641
$$634$$ 0 0
$$635$$ 17.5760 0.697483
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 0 0
$$639$$ −10.0552 −0.397777
$$640$$ 0 0
$$641$$ 41.8268 1.65206 0.826029 0.563627i $$-0.190595\pi$$
0.826029 + 0.563627i $$0.190595\pi$$
$$642$$ 0 0
$$643$$ −3.11383 −0.122797 −0.0613987 0.998113i $$-0.519556\pi$$
−0.0613987 + 0.998113i $$0.519556\pi$$
$$644$$ 0 0
$$645$$ −4.83709 −0.190460
$$646$$ 0 0
$$647$$ −30.4362 −1.19657 −0.598285 0.801283i $$-0.704151\pi$$
−0.598285 + 0.801283i $$0.704151\pi$$
$$648$$ 0 0
$$649$$ −10.1173 −0.397137
$$650$$ 0 0
$$651$$ −6.27674 −0.246005
$$652$$ 0 0
$$653$$ 3.99312 0.156263 0.0781315 0.996943i $$-0.475105\pi$$
0.0781315 + 0.996943i $$0.475105\pi$$
$$654$$ 0 0
$$655$$ −6.69672 −0.261663
$$656$$ 0 0
$$657$$ −15.8337 −0.617730
$$658$$ 0 0
$$659$$ −32.6578 −1.27217 −0.636083 0.771621i $$-0.719446\pi$$
−0.636083 + 0.771621i $$0.719446\pi$$
$$660$$ 0 0
$$661$$ 43.7355 1.70111 0.850557 0.525883i $$-0.176265\pi$$
0.850557 + 0.525883i $$0.176265\pi$$
$$662$$ 0 0
$$663$$ 2.94137 0.114233
$$664$$ 0 0
$$665$$ 8.60256 0.333593
$$666$$ 0 0
$$667$$ −16.3940 −0.634778
$$668$$ 0 0
$$669$$ 2.92838 0.113218
$$670$$ 0 0
$$671$$ 25.2311 0.974036
$$672$$ 0 0
$$673$$ −9.63198 −0.371285 −0.185643 0.982617i $$-0.559437\pi$$
−0.185643 + 0.982617i $$0.559437\pi$$
$$674$$ 0 0
$$675$$ −1.83709 −0.0707096
$$676$$ 0 0
$$677$$ 16.4914 0.633816 0.316908 0.948456i $$-0.397355\pi$$
0.316908 + 0.948456i $$0.397355\pi$$
$$678$$ 0 0
$$679$$ −17.7164 −0.679892
$$680$$ 0 0
$$681$$ 9.79145 0.375209
$$682$$ 0 0
$$683$$ −41.9311 −1.60445 −0.802224 0.597024i $$-0.796350\pi$$
−0.802224 + 0.597024i $$0.796350\pi$$
$$684$$ 0 0
$$685$$ −30.2277 −1.15494
$$686$$ 0 0
$$687$$ 3.88273 0.148136
$$688$$ 0 0
$$689$$ −6.39400 −0.243592
$$690$$ 0 0
$$691$$ −38.1786 −1.45238 −0.726191 0.687493i $$-0.758711\pi$$
−0.726191 + 0.687493i $$0.758711\pi$$
$$692$$ 0 0
$$693$$ 6.49828 0.246849
$$694$$ 0 0
$$695$$ 15.2120 0.577024
$$696$$ 0 0
$$697$$ 8.99656 0.340769
$$698$$ 0 0
$$699$$ −15.8337 −0.598884
$$700$$ 0 0
$$701$$ 16.5957 0.626810 0.313405 0.949620i $$-0.398530\pi$$
0.313405 + 0.949620i $$0.398530\pi$$
$$702$$ 0 0
$$703$$ 46.2277 1.74351
$$704$$ 0 0
$$705$$ 15.5078 0.584059
$$706$$ 0 0
$$707$$ 6.73281 0.253214
$$708$$ 0 0
$$709$$ 21.7655 0.817419 0.408710 0.912664i $$-0.365979\pi$$
0.408710 + 0.912664i $$0.365979\pi$$
$$710$$ 0 0
$$711$$ 1.28018 0.0480104
$$712$$ 0 0
$$713$$ −36.2699 −1.35832
$$714$$ 0 0
$$715$$ −11.5569 −0.432204
$$716$$ 0 0
$$717$$ 6.94137 0.259230
$$718$$ 0 0
$$719$$ −36.7880 −1.37196 −0.685981 0.727620i $$-0.740627\pi$$
−0.685981 + 0.727620i $$0.740627\pi$$
$$720$$ 0 0
$$721$$ −10.8793 −0.405166
$$722$$ 0 0
$$723$$ 7.28018 0.270753
$$724$$ 0 0
$$725$$ 5.21199 0.193568
$$726$$ 0 0
$$727$$ 32.3189 1.19864 0.599322 0.800508i $$-0.295437\pi$$
0.599322 + 0.800508i $$0.295437\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ −12.7198 −0.469817 −0.234909 0.972017i $$-0.575479\pi$$
−0.234909 + 0.972017i $$0.575479\pi$$
$$734$$ 0 0
$$735$$ 1.77846 0.0655994
$$736$$ 0 0
$$737$$ −36.8724 −1.35821
$$738$$ 0 0
$$739$$ −23.1982 −0.853361 −0.426681 0.904402i $$-0.640317\pi$$
−0.426681 + 0.904402i $$0.640317\pi$$
$$740$$ 0 0
$$741$$ −4.83709 −0.177695
$$742$$ 0 0
$$743$$ −31.8138 −1.16713 −0.583567 0.812065i $$-0.698344\pi$$
−0.583567 + 0.812065i $$0.698344\pi$$
$$744$$ 0 0
$$745$$ −27.6673 −1.01365
$$746$$ 0 0
$$747$$ 2.83709 0.103804
$$748$$ 0 0
$$749$$ 8.49828 0.310520
$$750$$ 0 0
$$751$$ 0.863070 0.0314939 0.0157469 0.999876i $$-0.494987\pi$$
0.0157469 + 0.999876i $$0.494987\pi$$
$$752$$ 0 0
$$753$$ −23.3224 −0.849915
$$754$$ 0 0
$$755$$ −8.88617 −0.323401
$$756$$ 0 0
$$757$$ −13.3974 −0.486938 −0.243469 0.969909i $$-0.578285\pi$$
−0.243469 + 0.969909i $$0.578285\pi$$
$$758$$ 0 0
$$759$$ 37.5500 1.36298
$$760$$ 0 0
$$761$$ −35.5630 −1.28916 −0.644579 0.764537i $$-0.722967\pi$$
−0.644579 + 0.764537i $$0.722967\pi$$
$$762$$ 0 0
$$763$$ 10.0000 0.362024
$$764$$ 0 0
$$765$$ −5.23109 −0.189131
$$766$$ 0 0
$$767$$ 1.55691 0.0562169
$$768$$ 0 0
$$769$$ −48.3871 −1.74488 −0.872442 0.488717i $$-0.837465\pi$$
−0.872442 + 0.488717i $$0.837465\pi$$
$$770$$ 0 0
$$771$$ 15.4948 0.558033
$$772$$ 0 0
$$773$$ 17.9379 0.645182 0.322591 0.946538i $$-0.395446\pi$$
0.322591 + 0.946538i $$0.395446\pi$$
$$774$$ 0 0
$$775$$ 11.5309 0.414203
$$776$$ 0 0
$$777$$ 9.55691 0.342852
$$778$$ 0 0
$$779$$ −14.7949 −0.530082
$$780$$ 0 0
$$781$$ −65.3415 −2.33810
$$782$$ 0 0
$$783$$ −2.83709 −0.101389
$$784$$ 0 0
$$785$$ −33.4137 −1.19258
$$786$$ 0 0
$$787$$ 3.71639 0.132475 0.0662374 0.997804i $$-0.478901\pi$$
0.0662374 + 0.997804i $$0.478901\pi$$
$$788$$ 0 0
$$789$$ −3.42666 −0.121992
$$790$$ 0 0
$$791$$ 14.6026 0.519207
$$792$$ 0 0
$$793$$ −3.88273 −0.137880
$$794$$ 0 0
$$795$$ 11.3715 0.403304
$$796$$ 0 0
$$797$$ 14.9673 0.530171 0.265085 0.964225i $$-0.414600\pi$$
0.265085 + 0.964225i $$0.414600\pi$$
$$798$$ 0 0
$$799$$ −25.6482 −0.907368
$$800$$ 0 0
$$801$$ −7.66119 −0.270695
$$802$$ 0 0
$$803$$ −102.892 −3.63096
$$804$$ 0 0
$$805$$ 10.2767 0.362207
$$806$$ 0 0
$$807$$ −0.172462 −0.00607094
$$808$$ 0 0
$$809$$ 11.7233 0.412168 0.206084 0.978534i $$-0.433928\pi$$
0.206084 + 0.978534i $$0.433928\pi$$
$$810$$ 0 0
$$811$$ 0.886172 0.0311177 0.0155588 0.999879i $$-0.495047\pi$$
0.0155588 + 0.999879i $$0.495047\pi$$
$$812$$ 0 0
$$813$$ 25.9931 0.911619
$$814$$ 0 0
$$815$$ 17.5760 0.615661
$$816$$ 0 0
$$817$$ −13.1560 −0.460271
$$818$$ 0 0
$$819$$ −1.00000 −0.0349428
$$820$$ 0 0
$$821$$ −53.7846 −1.87709 −0.938547 0.345151i $$-0.887828\pi$$
−0.938547 + 0.345151i $$0.887828\pi$$
$$822$$ 0 0
$$823$$ 35.7655 1.24671 0.623353 0.781941i $$-0.285770\pi$$
0.623353 + 0.781941i $$0.285770\pi$$
$$824$$ 0 0
$$825$$ −11.9379 −0.415625
$$826$$ 0 0
$$827$$ −7.41043 −0.257686 −0.128843 0.991665i $$-0.541126\pi$$
−0.128843 + 0.991665i $$0.541126\pi$$
$$828$$ 0 0
$$829$$ −31.9931 −1.11117 −0.555584 0.831461i $$-0.687505\pi$$
−0.555584 + 0.831461i $$0.687505\pi$$
$$830$$ 0 0
$$831$$ 3.72326 0.129159
$$832$$ 0 0
$$833$$ −2.94137 −0.101912
$$834$$ 0 0
$$835$$ 12.5304 0.433631
$$836$$ 0 0
$$837$$ −6.27674 −0.216956
$$838$$ 0 0
$$839$$ 1.55691 0.0537506 0.0268753 0.999639i $$-0.491444\pi$$
0.0268753 + 0.999639i $$0.491444\pi$$
$$840$$ 0 0
$$841$$ −20.9509 −0.722445
$$842$$ 0 0
$$843$$ −21.5500 −0.742223
$$844$$ 0 0
$$845$$ 1.77846 0.0611808
$$846$$ 0 0
$$847$$ 31.2277 1.07299
$$848$$ 0 0
$$849$$ 31.8759 1.09398
$$850$$ 0 0
$$851$$ 55.2242 1.89306
$$852$$ 0 0
$$853$$ 35.4750 1.21464 0.607320 0.794457i $$-0.292245\pi$$
0.607320 + 0.794457i $$0.292245\pi$$
$$854$$ 0 0
$$855$$ 8.60256 0.294201
$$856$$ 0 0
$$857$$ −2.49828 −0.0853397 −0.0426698 0.999089i $$-0.513586\pi$$
−0.0426698 + 0.999089i $$0.513586\pi$$
$$858$$ 0 0
$$859$$ −52.0122 −1.77463 −0.887317 0.461160i $$-0.847434\pi$$
−0.887317 + 0.461160i $$0.847434\pi$$
$$860$$ 0 0
$$861$$ −3.05863 −0.104238
$$862$$ 0 0
$$863$$ −34.8172 −1.18519 −0.592596 0.805500i $$-0.701897\pi$$
−0.592596 + 0.805500i $$0.701897\pi$$
$$864$$ 0 0
$$865$$ −25.4137 −0.864091
$$866$$ 0 0
$$867$$ −8.34836 −0.283525
$$868$$ 0 0
$$869$$ 8.31894 0.282201
$$870$$ 0 0
$$871$$ 5.67418 0.192262
$$872$$ 0 0
$$873$$ −17.7164 −0.599609
$$874$$ 0 0
$$875$$ −12.1595 −0.411065
$$876$$ 0 0
$$877$$ −11.2571 −0.380124 −0.190062 0.981772i $$-0.560869\pi$$
−0.190062 + 0.981772i $$0.560869\pi$$
$$878$$ 0 0
$$879$$ 3.42666 0.115578
$$880$$ 0 0
$$881$$ −9.50172 −0.320121 −0.160061 0.987107i $$-0.551169\pi$$
−0.160061 + 0.987107i $$0.551169\pi$$
$$882$$ 0 0
$$883$$ −23.7655 −0.799772 −0.399886 0.916565i $$-0.630950\pi$$
−0.399886 + 0.916565i $$0.630950\pi$$
$$884$$ 0 0
$$885$$ −2.76891 −0.0930757
$$886$$ 0 0
$$887$$ −18.3518 −0.616193 −0.308097 0.951355i $$-0.599692\pi$$
−0.308097 + 0.951355i $$0.599692\pi$$
$$888$$ 0 0
$$889$$ 9.88273 0.331456
$$890$$ 0 0
$$891$$ 6.49828 0.217701
$$892$$ 0 0
$$893$$ 42.1786 1.41145
$$894$$ 0 0
$$895$$ 15.3974 0.514680
$$896$$ 0 0
$$897$$ −5.77846 −0.192937
$$898$$ 0 0
$$899$$ 17.8077 0.593919
$$900$$ 0 0
$$901$$ −18.8071 −0.626556
$$902$$ 0 0
$$903$$ −2.71982 −0.0905101
$$904$$ 0 0
$$905$$ 46.8533 1.55746
$$906$$ 0 0
$$907$$ −35.7164 −1.18594 −0.592972 0.805223i $$-0.702045\pi$$
−0.592972 + 0.805223i $$0.702045\pi$$
$$908$$ 0 0
$$909$$ 6.73281 0.223313
$$910$$ 0 0
$$911$$ 16.2147 0.537216 0.268608 0.963250i $$-0.413436\pi$$
0.268608 + 0.963250i $$0.413436\pi$$
$$912$$ 0 0
$$913$$ 18.4362 0.610149
$$914$$ 0 0
$$915$$ 6.90528 0.228281
$$916$$ 0 0
$$917$$ −3.76547 −0.124347
$$918$$ 0 0
$$919$$ 34.4622 1.13680 0.568401 0.822751i $$-0.307562\pi$$
0.568401 + 0.822751i $$0.307562\pi$$
$$920$$ 0 0
$$921$$ 3.39744 0.111950
$$922$$ 0 0
$$923$$ 10.0552 0.330971
$$924$$ 0 0
$$925$$ −17.5569 −0.577268
$$926$$ 0 0
$$927$$ −10.8793 −0.357323
$$928$$ 0 0
$$929$$ −28.4492 −0.933388 −0.466694 0.884419i $$-0.654555\pi$$
−0.466694 + 0.884419i $$0.654555\pi$$
$$930$$ 0 0
$$931$$ 4.83709 0.158529
$$932$$ 0 0
$$933$$ 0.443086 0.0145060
$$934$$ 0 0
$$935$$ −33.9931 −1.11169
$$936$$ 0 0
$$937$$ 2.91215 0.0951358 0.0475679 0.998868i $$-0.484853\pi$$
0.0475679 + 0.998868i $$0.484853\pi$$
$$938$$ 0 0
$$939$$ −19.8827 −0.648848
$$940$$ 0 0
$$941$$ −59.0096 −1.92366 −0.961828 0.273654i $$-0.911768\pi$$
−0.961828 + 0.273654i $$0.911768\pi$$
$$942$$ 0 0
$$943$$ −17.6742 −0.575551
$$944$$ 0 0
$$945$$ 1.77846 0.0578532
$$946$$ 0 0
$$947$$ −34.8432 −1.13225 −0.566126 0.824319i $$-0.691558\pi$$
−0.566126 + 0.824319i $$0.691558\pi$$
$$948$$ 0 0
$$949$$ 15.8337 0.513982
$$950$$ 0 0
$$951$$ 9.46563 0.306944
$$952$$ 0 0
$$953$$ 23.9578 0.776069 0.388035 0.921645i $$-0.373154\pi$$
0.388035 + 0.921645i $$0.373154\pi$$
$$954$$ 0 0
$$955$$ −11.7655 −0.380722
$$956$$ 0 0
$$957$$ −18.4362 −0.595958
$$958$$ 0 0
$$959$$ −16.9966 −0.548848
$$960$$ 0 0
$$961$$ 8.39744 0.270885
$$962$$ 0 0
$$963$$ 8.49828 0.273853
$$964$$ 0 0
$$965$$ 21.1329 0.680293
$$966$$ 0 0
$$967$$ 37.3155 1.19999 0.599993 0.800005i $$-0.295170\pi$$
0.599993 + 0.800005i $$0.295170\pi$$
$$968$$ 0 0
$$969$$ −14.2277 −0.457058
$$970$$ 0 0
$$971$$ −20.7880 −0.667119 −0.333559 0.942729i $$-0.608250\pi$$
−0.333559 + 0.942729i $$0.608250\pi$$
$$972$$ 0 0
$$973$$ 8.55348 0.274212
$$974$$ 0 0
$$975$$ 1.83709 0.0588340
$$976$$ 0 0
$$977$$ 49.0810 1.57024 0.785120 0.619344i $$-0.212601\pi$$
0.785120 + 0.619344i $$0.212601\pi$$
$$978$$ 0 0
$$979$$ −49.7846 −1.59112
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ 0 0
$$983$$ −10.1855 −0.324865 −0.162433 0.986720i $$-0.551934\pi$$
−0.162433 + 0.986720i $$0.551934\pi$$
$$984$$ 0 0
$$985$$ 20.5535 0.654888
$$986$$ 0 0
$$987$$ 8.71982 0.277555
$$988$$ 0 0
$$989$$ −15.7164 −0.499752
$$990$$ 0 0
$$991$$ −4.34492 −0.138021 −0.0690105 0.997616i $$-0.521984\pi$$
−0.0690105 + 0.997616i $$0.521984\pi$$
$$992$$ 0 0
$$993$$ 27.4328 0.870553
$$994$$ 0 0
$$995$$ −1.77234 −0.0561871
$$996$$ 0 0
$$997$$ 44.9637 1.42401 0.712007 0.702172i $$-0.247786\pi$$
0.712007 + 0.702172i $$0.247786\pi$$
$$998$$ 0 0
$$999$$ 9.55691 0.302367
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 4368.2.a.bq.1.3 3
4.3 odd 2 273.2.a.d.1.2 3
12.11 even 2 819.2.a.j.1.2 3
20.19 odd 2 6825.2.a.bd.1.2 3
28.27 even 2 1911.2.a.n.1.2 3
52.51 odd 2 3549.2.a.t.1.2 3
84.83 odd 2 5733.2.a.bc.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.d.1.2 3 4.3 odd 2
819.2.a.j.1.2 3 12.11 even 2
1911.2.a.n.1.2 3 28.27 even 2
3549.2.a.t.1.2 3 52.51 odd 2
4368.2.a.bq.1.3 3 1.1 even 1 trivial
5733.2.a.bc.1.2 3 84.83 odd 2
6825.2.a.bd.1.2 3 20.19 odd 2