# Properties

 Label 768.4.a.p.1.2 Level $768$ Weight $4$ Character 768.1 Self dual yes Analytic conductor $45.313$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.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: $$45.3134668844$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 768.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +18.4222 q^{5} -22.4222 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +18.4222 q^{5} -22.4222 q^{7} +9.00000 q^{9} -53.6888 q^{11} -7.15559 q^{13} +55.2666 q^{15} +39.6888 q^{17} -125.689 q^{19} -67.2666 q^{21} -99.1556 q^{23} +214.378 q^{25} +27.0000 q^{27} -205.800 q^{29} -147.489 q^{31} -161.066 q^{33} -413.066 q^{35} -125.689 q^{37} -21.4668 q^{39} -506.444 q^{41} +413.689 q^{43} +165.800 q^{45} +313.911 q^{47} +159.755 q^{49} +119.066 q^{51} +44.3331 q^{53} -989.066 q^{55} -377.066 q^{57} +324.000 q^{59} -324.000 q^{61} -201.800 q^{63} -131.822 q^{65} +464.266 q^{67} -297.467 q^{69} -1052.84 q^{71} +1022.27 q^{73} +643.133 q^{75} +1203.82 q^{77} +602.910 q^{79} +81.0000 q^{81} +15.8217 q^{83} +731.156 q^{85} -617.400 q^{87} +381.378 q^{89} +160.444 q^{91} -442.466 q^{93} -2315.47 q^{95} +659.154 q^{97} -483.199 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 8 q^{5} - 16 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 8 * q^5 - 16 * q^7 + 18 * q^9 $$2 q + 6 q^{3} + 8 q^{5} - 16 q^{7} + 18 q^{9} + 8 q^{11} - 72 q^{13} + 24 q^{15} - 36 q^{17} - 136 q^{19} - 48 q^{21} - 256 q^{23} + 198 q^{25} + 54 q^{27} - 152 q^{29} + 80 q^{31} + 24 q^{33} - 480 q^{35} - 136 q^{37} - 216 q^{39} - 436 q^{41} + 712 q^{43} + 72 q^{45} + 224 q^{47} - 142 q^{49} - 108 q^{51} - 344 q^{53} - 1632 q^{55} - 408 q^{57} + 648 q^{59} - 648 q^{61} - 144 q^{63} + 544 q^{65} - 456 q^{67} - 768 q^{69} - 2048 q^{71} + 660 q^{73} + 594 q^{75} + 1600 q^{77} - 496 q^{79} + 162 q^{81} - 776 q^{83} + 1520 q^{85} - 456 q^{87} + 532 q^{89} - 256 q^{91} + 240 q^{93} - 2208 q^{95} - 1220 q^{97} + 72 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 8 * q^5 - 16 * q^7 + 18 * q^9 + 8 * q^11 - 72 * q^13 + 24 * q^15 - 36 * q^17 - 136 * q^19 - 48 * q^21 - 256 * q^23 + 198 * q^25 + 54 * q^27 - 152 * q^29 + 80 * q^31 + 24 * q^33 - 480 * q^35 - 136 * q^37 - 216 * q^39 - 436 * q^41 + 712 * q^43 + 72 * q^45 + 224 * q^47 - 142 * q^49 - 108 * q^51 - 344 * q^53 - 1632 * q^55 - 408 * q^57 + 648 * q^59 - 648 * q^61 - 144 * q^63 + 544 * q^65 - 456 * q^67 - 768 * q^69 - 2048 * q^71 + 660 * q^73 + 594 * q^75 + 1600 * q^77 - 496 * q^79 + 162 * q^81 - 776 * q^83 + 1520 * q^85 - 456 * q^87 + 532 * q^89 - 256 * q^91 + 240 * q^93 - 2208 * q^95 - 1220 * q^97 + 72 * 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
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 18.4222 1.64773 0.823866 0.566785i $$-0.191813\pi$$
0.823866 + 0.566785i $$0.191813\pi$$
$$6$$ 0 0
$$7$$ −22.4222 −1.21069 −0.605343 0.795965i $$-0.706964\pi$$
−0.605343 + 0.795965i $$0.706964\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −53.6888 −1.47162 −0.735809 0.677190i $$-0.763198\pi$$
−0.735809 + 0.677190i $$0.763198\pi$$
$$12$$ 0 0
$$13$$ −7.15559 −0.152662 −0.0763309 0.997083i $$-0.524321\pi$$
−0.0763309 + 0.997083i $$0.524321\pi$$
$$14$$ 0 0
$$15$$ 55.2666 0.951319
$$16$$ 0 0
$$17$$ 39.6888 0.566233 0.283116 0.959086i $$-0.408632\pi$$
0.283116 + 0.959086i $$0.408632\pi$$
$$18$$ 0 0
$$19$$ −125.689 −1.51763 −0.758816 0.651306i $$-0.774222\pi$$
−0.758816 + 0.651306i $$0.774222\pi$$
$$20$$ 0 0
$$21$$ −67.2666 −0.698989
$$22$$ 0 0
$$23$$ −99.1556 −0.898929 −0.449465 0.893298i $$-0.648385\pi$$
−0.449465 + 0.893298i $$0.648385\pi$$
$$24$$ 0 0
$$25$$ 214.378 1.71502
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −205.800 −1.31780 −0.658898 0.752232i $$-0.728977\pi$$
−0.658898 + 0.752232i $$0.728977\pi$$
$$30$$ 0 0
$$31$$ −147.489 −0.854508 −0.427254 0.904132i $$-0.640519\pi$$
−0.427254 + 0.904132i $$0.640519\pi$$
$$32$$ 0 0
$$33$$ −161.066 −0.849639
$$34$$ 0 0
$$35$$ −413.066 −1.99489
$$36$$ 0 0
$$37$$ −125.689 −0.558463 −0.279231 0.960224i $$-0.590080\pi$$
−0.279231 + 0.960224i $$0.590080\pi$$
$$38$$ 0 0
$$39$$ −21.4668 −0.0881393
$$40$$ 0 0
$$41$$ −506.444 −1.92910 −0.964552 0.263892i $$-0.914994\pi$$
−0.964552 + 0.263892i $$0.914994\pi$$
$$42$$ 0 0
$$43$$ 413.689 1.46714 0.733569 0.679615i $$-0.237853\pi$$
0.733569 + 0.679615i $$0.237853\pi$$
$$44$$ 0 0
$$45$$ 165.800 0.549244
$$46$$ 0 0
$$47$$ 313.911 0.974226 0.487113 0.873339i $$-0.338050\pi$$
0.487113 + 0.873339i $$0.338050\pi$$
$$48$$ 0 0
$$49$$ 159.755 0.465759
$$50$$ 0 0
$$51$$ 119.066 0.326914
$$52$$ 0 0
$$53$$ 44.3331 0.114898 0.0574492 0.998348i $$-0.481703\pi$$
0.0574492 + 0.998348i $$0.481703\pi$$
$$54$$ 0 0
$$55$$ −989.066 −2.42483
$$56$$ 0 0
$$57$$ −377.066 −0.876205
$$58$$ 0 0
$$59$$ 324.000 0.714936 0.357468 0.933925i $$-0.383640\pi$$
0.357468 + 0.933925i $$0.383640\pi$$
$$60$$ 0 0
$$61$$ −324.000 −0.680065 −0.340032 0.940414i $$-0.610438\pi$$
−0.340032 + 0.940414i $$0.610438\pi$$
$$62$$ 0 0
$$63$$ −201.800 −0.403562
$$64$$ 0 0
$$65$$ −131.822 −0.251546
$$66$$ 0 0
$$67$$ 464.266 0.846554 0.423277 0.906000i $$-0.360880\pi$$
0.423277 + 0.906000i $$0.360880\pi$$
$$68$$ 0 0
$$69$$ −297.467 −0.518997
$$70$$ 0 0
$$71$$ −1052.84 −1.75985 −0.879927 0.475109i $$-0.842409\pi$$
−0.879927 + 0.475109i $$0.842409\pi$$
$$72$$ 0 0
$$73$$ 1022.27 1.63900 0.819501 0.573078i $$-0.194251\pi$$
0.819501 + 0.573078i $$0.194251\pi$$
$$74$$ 0 0
$$75$$ 643.133 0.990168
$$76$$ 0 0
$$77$$ 1203.82 1.78167
$$78$$ 0 0
$$79$$ 602.910 0.858642 0.429321 0.903152i $$-0.358753\pi$$
0.429321 + 0.903152i $$0.358753\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 15.8217 0.0209236 0.0104618 0.999945i $$-0.496670\pi$$
0.0104618 + 0.999945i $$0.496670\pi$$
$$84$$ 0 0
$$85$$ 731.156 0.933000
$$86$$ 0 0
$$87$$ −617.400 −0.760830
$$88$$ 0 0
$$89$$ 381.378 0.454224 0.227112 0.973869i $$-0.427072\pi$$
0.227112 + 0.973869i $$0.427072\pi$$
$$90$$ 0 0
$$91$$ 160.444 0.184825
$$92$$ 0 0
$$93$$ −442.466 −0.493350
$$94$$ 0 0
$$95$$ −2315.47 −2.50065
$$96$$ 0 0
$$97$$ 659.154 0.689969 0.344984 0.938608i $$-0.387884\pi$$
0.344984 + 0.938608i $$0.387884\pi$$
$$98$$ 0 0
$$99$$ −483.199 −0.490539
$$100$$ 0 0
$$101$$ −498.510 −0.491125 −0.245562 0.969381i $$-0.578973\pi$$
−0.245562 + 0.969381i $$0.578973\pi$$
$$102$$ 0 0
$$103$$ 196.821 0.188285 0.0941425 0.995559i $$-0.469989\pi$$
0.0941425 + 0.995559i $$0.469989\pi$$
$$104$$ 0 0
$$105$$ −1239.20 −1.15175
$$106$$ 0 0
$$107$$ 359.378 0.324695 0.162347 0.986734i $$-0.448093\pi$$
0.162347 + 0.986734i $$0.448093\pi$$
$$108$$ 0 0
$$109$$ −1969.73 −1.73088 −0.865441 0.501011i $$-0.832962\pi$$
−0.865441 + 0.501011i $$0.832962\pi$$
$$110$$ 0 0
$$111$$ −377.066 −0.322429
$$112$$ 0 0
$$113$$ −693.643 −0.577456 −0.288728 0.957411i $$-0.593232\pi$$
−0.288728 + 0.957411i $$0.593232\pi$$
$$114$$ 0 0
$$115$$ −1826.66 −1.48119
$$116$$ 0 0
$$117$$ −64.4003 −0.0508873
$$118$$ 0 0
$$119$$ −889.911 −0.685529
$$120$$ 0 0
$$121$$ 1551.49 1.16566
$$122$$ 0 0
$$123$$ −1519.33 −1.11377
$$124$$ 0 0
$$125$$ 1646.53 1.17816
$$126$$ 0 0
$$127$$ −2656.78 −1.85631 −0.928153 0.372199i $$-0.878604\pi$$
−0.928153 + 0.372199i $$0.878604\pi$$
$$128$$ 0 0
$$129$$ 1241.07 0.847053
$$130$$ 0 0
$$131$$ 615.734 0.410664 0.205332 0.978692i $$-0.434173\pi$$
0.205332 + 0.978692i $$0.434173\pi$$
$$132$$ 0 0
$$133$$ 2818.22 1.83737
$$134$$ 0 0
$$135$$ 497.400 0.317106
$$136$$ 0 0
$$137$$ −613.290 −0.382459 −0.191230 0.981545i $$-0.561247\pi$$
−0.191230 + 0.981545i $$0.561247\pi$$
$$138$$ 0 0
$$139$$ −1899.29 −1.15896 −0.579480 0.814987i $$-0.696744\pi$$
−0.579480 + 0.814987i $$0.696744\pi$$
$$140$$ 0 0
$$141$$ 941.733 0.562469
$$142$$ 0 0
$$143$$ 384.175 0.224660
$$144$$ 0 0
$$145$$ −3791.29 −2.17137
$$146$$ 0 0
$$147$$ 479.266 0.268906
$$148$$ 0 0
$$149$$ 976.377 0.536832 0.268416 0.963303i $$-0.413500\pi$$
0.268416 + 0.963303i $$0.413500\pi$$
$$150$$ 0 0
$$151$$ −683.132 −0.368162 −0.184081 0.982911i $$-0.558931\pi$$
−0.184081 + 0.982911i $$0.558931\pi$$
$$152$$ 0 0
$$153$$ 357.199 0.188744
$$154$$ 0 0
$$155$$ −2717.07 −1.40800
$$156$$ 0 0
$$157$$ 511.109 0.259815 0.129907 0.991526i $$-0.458532\pi$$
0.129907 + 0.991526i $$0.458532\pi$$
$$158$$ 0 0
$$159$$ 132.999 0.0663366
$$160$$ 0 0
$$161$$ 2223.29 1.08832
$$162$$ 0 0
$$163$$ 2425.95 1.16574 0.582869 0.812566i $$-0.301930\pi$$
0.582869 + 0.812566i $$0.301930\pi$$
$$164$$ 0 0
$$165$$ −2967.20 −1.39998
$$166$$ 0 0
$$167$$ −337.332 −0.156309 −0.0781544 0.996941i $$-0.524903\pi$$
−0.0781544 + 0.996941i $$0.524903\pi$$
$$168$$ 0 0
$$169$$ −2145.80 −0.976694
$$170$$ 0 0
$$171$$ −1131.20 −0.505877
$$172$$ 0 0
$$173$$ 2648.29 1.16385 0.581924 0.813243i $$-0.302300\pi$$
0.581924 + 0.813243i $$0.302300\pi$$
$$174$$ 0 0
$$175$$ −4806.82 −2.07635
$$176$$ 0 0
$$177$$ 972.000 0.412768
$$178$$ 0 0
$$179$$ 2907.29 1.21397 0.606986 0.794713i $$-0.292379\pi$$
0.606986 + 0.794713i $$0.292379\pi$$
$$180$$ 0 0
$$181$$ −3682.80 −1.51238 −0.756188 0.654354i $$-0.772940\pi$$
−0.756188 + 0.654354i $$0.772940\pi$$
$$182$$ 0 0
$$183$$ −972.000 −0.392636
$$184$$ 0 0
$$185$$ −2315.47 −0.920197
$$186$$ 0 0
$$187$$ −2130.85 −0.833277
$$188$$ 0 0
$$189$$ −605.400 −0.232996
$$190$$ 0 0
$$191$$ −1279.56 −0.484740 −0.242370 0.970184i $$-0.577925\pi$$
−0.242370 + 0.970184i $$0.577925\pi$$
$$192$$ 0 0
$$193$$ −4836.84 −1.80396 −0.901978 0.431783i $$-0.857885\pi$$
−0.901978 + 0.431783i $$0.857885\pi$$
$$194$$ 0 0
$$195$$ −395.465 −0.145230
$$196$$ 0 0
$$197$$ 2869.31 1.03771 0.518857 0.854861i $$-0.326358\pi$$
0.518857 + 0.854861i $$0.326358\pi$$
$$198$$ 0 0
$$199$$ −652.242 −0.232343 −0.116171 0.993229i $$-0.537062\pi$$
−0.116171 + 0.993229i $$0.537062\pi$$
$$200$$ 0 0
$$201$$ 1392.80 0.488758
$$202$$ 0 0
$$203$$ 4614.49 1.59544
$$204$$ 0 0
$$205$$ −9329.82 −3.17865
$$206$$ 0 0
$$207$$ −892.400 −0.299643
$$208$$ 0 0
$$209$$ 6748.08 2.23337
$$210$$ 0 0
$$211$$ 537.511 0.175373 0.0876866 0.996148i $$-0.472053\pi$$
0.0876866 + 0.996148i $$0.472053\pi$$
$$212$$ 0 0
$$213$$ −3158.53 −1.01605
$$214$$ 0 0
$$215$$ 7621.06 2.41745
$$216$$ 0 0
$$217$$ 3307.02 1.03454
$$218$$ 0 0
$$219$$ 3066.80 0.946278
$$220$$ 0 0
$$221$$ −283.997 −0.0864421
$$222$$ 0 0
$$223$$ 4041.98 1.21377 0.606885 0.794790i $$-0.292419\pi$$
0.606885 + 0.794790i $$0.292419\pi$$
$$224$$ 0 0
$$225$$ 1929.40 0.571674
$$226$$ 0 0
$$227$$ 3070.22 0.897699 0.448850 0.893607i $$-0.351834\pi$$
0.448850 + 0.893607i $$0.351834\pi$$
$$228$$ 0 0
$$229$$ 205.110 0.0591881 0.0295940 0.999562i $$-0.490579\pi$$
0.0295940 + 0.999562i $$0.490579\pi$$
$$230$$ 0 0
$$231$$ 3611.47 1.02864
$$232$$ 0 0
$$233$$ 13.3776 0.00376137 0.00188068 0.999998i $$-0.499401\pi$$
0.00188068 + 0.999998i $$0.499401\pi$$
$$234$$ 0 0
$$235$$ 5782.93 1.60526
$$236$$ 0 0
$$237$$ 1808.73 0.495737
$$238$$ 0 0
$$239$$ −4327.99 −1.17136 −0.585679 0.810543i $$-0.699172\pi$$
−0.585679 + 0.810543i $$0.699172\pi$$
$$240$$ 0 0
$$241$$ −1508.31 −0.403150 −0.201575 0.979473i $$-0.564606\pi$$
−0.201575 + 0.979473i $$0.564606\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 2943.04 0.767446
$$246$$ 0 0
$$247$$ 899.378 0.231684
$$248$$ 0 0
$$249$$ 47.4652 0.0120803
$$250$$ 0 0
$$251$$ −4871.47 −1.22504 −0.612518 0.790456i $$-0.709843\pi$$
−0.612518 + 0.790456i $$0.709843\pi$$
$$252$$ 0 0
$$253$$ 5323.55 1.32288
$$254$$ 0 0
$$255$$ 2193.47 0.538668
$$256$$ 0 0
$$257$$ −1665.55 −0.404258 −0.202129 0.979359i $$-0.564786\pi$$
−0.202129 + 0.979359i $$0.564786\pi$$
$$258$$ 0 0
$$259$$ 2818.22 0.676122
$$260$$ 0 0
$$261$$ −1852.20 −0.439265
$$262$$ 0 0
$$263$$ −7167.64 −1.68052 −0.840258 0.542188i $$-0.817596\pi$$
−0.840258 + 0.542188i $$0.817596\pi$$
$$264$$ 0 0
$$265$$ 816.713 0.189322
$$266$$ 0 0
$$267$$ 1144.13 0.262246
$$268$$ 0 0
$$269$$ 5453.84 1.23616 0.618079 0.786116i $$-0.287911\pi$$
0.618079 + 0.786116i $$0.287911\pi$$
$$270$$ 0 0
$$271$$ 5416.20 1.21406 0.607031 0.794678i $$-0.292360\pi$$
0.607031 + 0.794678i $$0.292360\pi$$
$$272$$ 0 0
$$273$$ 481.332 0.106709
$$274$$ 0 0
$$275$$ −11509.7 −2.52385
$$276$$ 0 0
$$277$$ 2648.75 0.574542 0.287271 0.957849i $$-0.407252\pi$$
0.287271 + 0.957849i $$0.407252\pi$$
$$278$$ 0 0
$$279$$ −1327.40 −0.284836
$$280$$ 0 0
$$281$$ 6664.57 1.41486 0.707429 0.706784i $$-0.249855\pi$$
0.707429 + 0.706784i $$0.249855\pi$$
$$282$$ 0 0
$$283$$ −5630.84 −1.18275 −0.591376 0.806396i $$-0.701415\pi$$
−0.591376 + 0.806396i $$0.701415\pi$$
$$284$$ 0 0
$$285$$ −6946.40 −1.44375
$$286$$ 0 0
$$287$$ 11355.6 2.33554
$$288$$ 0 0
$$289$$ −3337.80 −0.679381
$$290$$ 0 0
$$291$$ 1977.46 0.398354
$$292$$ 0 0
$$293$$ −908.374 −0.181119 −0.0905593 0.995891i $$-0.528865\pi$$
−0.0905593 + 0.995891i $$0.528865\pi$$
$$294$$ 0 0
$$295$$ 5968.79 1.17802
$$296$$ 0 0
$$297$$ −1449.60 −0.283213
$$298$$ 0 0
$$299$$ 709.517 0.137232
$$300$$ 0 0
$$301$$ −9275.82 −1.77624
$$302$$ 0 0
$$303$$ −1495.53 −0.283551
$$304$$ 0 0
$$305$$ −5968.79 −1.12056
$$306$$ 0 0
$$307$$ −414.671 −0.0770896 −0.0385448 0.999257i $$-0.512272\pi$$
−0.0385448 + 0.999257i $$0.512272\pi$$
$$308$$ 0 0
$$309$$ 590.463 0.108706
$$310$$ 0 0
$$311$$ −1615.91 −0.294629 −0.147315 0.989090i $$-0.547063\pi$$
−0.147315 + 0.989090i $$0.547063\pi$$
$$312$$ 0 0
$$313$$ −8479.33 −1.53125 −0.765623 0.643289i $$-0.777569\pi$$
−0.765623 + 0.643289i $$0.777569\pi$$
$$314$$ 0 0
$$315$$ −3717.60 −0.664962
$$316$$ 0 0
$$317$$ 6774.73 1.20034 0.600169 0.799873i $$-0.295100\pi$$
0.600169 + 0.799873i $$0.295100\pi$$
$$318$$ 0 0
$$319$$ 11049.2 1.93929
$$320$$ 0 0
$$321$$ 1078.13 0.187463
$$322$$ 0 0
$$323$$ −4988.44 −0.859332
$$324$$ 0 0
$$325$$ −1534.00 −0.261818
$$326$$ 0 0
$$327$$ −5909.20 −0.999325
$$328$$ 0 0
$$329$$ −7038.57 −1.17948
$$330$$ 0 0
$$331$$ 9292.36 1.54306 0.771532 0.636191i $$-0.219491\pi$$
0.771532 + 0.636191i $$0.219491\pi$$
$$332$$ 0 0
$$333$$ −1131.20 −0.186154
$$334$$ 0 0
$$335$$ 8552.80 1.39489
$$336$$ 0 0
$$337$$ 6563.78 1.06098 0.530492 0.847690i $$-0.322007\pi$$
0.530492 + 0.847690i $$0.322007\pi$$
$$338$$ 0 0
$$339$$ −2080.93 −0.333394
$$340$$ 0 0
$$341$$ 7918.49 1.25751
$$342$$ 0 0
$$343$$ 4108.75 0.646798
$$344$$ 0 0
$$345$$ −5479.99 −0.855168
$$346$$ 0 0
$$347$$ −3870.93 −0.598855 −0.299427 0.954119i $$-0.596796\pi$$
−0.299427 + 0.954119i $$0.596796\pi$$
$$348$$ 0 0
$$349$$ −3474.57 −0.532922 −0.266461 0.963846i $$-0.585854\pi$$
−0.266461 + 0.963846i $$0.585854\pi$$
$$350$$ 0 0
$$351$$ −193.201 −0.0293798
$$352$$ 0 0
$$353$$ −4308.58 −0.649639 −0.324820 0.945776i $$-0.605304\pi$$
−0.324820 + 0.945776i $$0.605304\pi$$
$$354$$ 0 0
$$355$$ −19395.7 −2.89977
$$356$$ 0 0
$$357$$ −2669.73 −0.395791
$$358$$ 0 0
$$359$$ 8161.19 1.19981 0.599904 0.800072i $$-0.295205\pi$$
0.599904 + 0.800072i $$0.295205\pi$$
$$360$$ 0 0
$$361$$ 8938.68 1.30320
$$362$$ 0 0
$$363$$ 4654.47 0.672992
$$364$$ 0 0
$$365$$ 18832.4 2.70064
$$366$$ 0 0
$$367$$ −4427.66 −0.629760 −0.314880 0.949132i $$-0.601964\pi$$
−0.314880 + 0.949132i $$0.601964\pi$$
$$368$$ 0 0
$$369$$ −4558.00 −0.643035
$$370$$ 0 0
$$371$$ −994.045 −0.139106
$$372$$ 0 0
$$373$$ −11278.5 −1.56562 −0.782812 0.622258i $$-0.786215\pi$$
−0.782812 + 0.622258i $$0.786215\pi$$
$$374$$ 0 0
$$375$$ 4939.60 0.680213
$$376$$ 0 0
$$377$$ 1472.62 0.201177
$$378$$ 0 0
$$379$$ 709.683 0.0961846 0.0480923 0.998843i $$-0.484686\pi$$
0.0480923 + 0.998843i $$0.484686\pi$$
$$380$$ 0 0
$$381$$ −7970.33 −1.07174
$$382$$ 0 0
$$383$$ 1233.69 0.164592 0.0822962 0.996608i $$-0.473775\pi$$
0.0822962 + 0.996608i $$0.473775\pi$$
$$384$$ 0 0
$$385$$ 22177.1 2.93571
$$386$$ 0 0
$$387$$ 3723.20 0.489046
$$388$$ 0 0
$$389$$ 6830.06 0.890226 0.445113 0.895474i $$-0.353164\pi$$
0.445113 + 0.895474i $$0.353164\pi$$
$$390$$ 0 0
$$391$$ −3935.37 −0.509003
$$392$$ 0 0
$$393$$ 1847.20 0.237097
$$394$$ 0 0
$$395$$ 11106.9 1.41481
$$396$$ 0 0
$$397$$ 11289.7 1.42724 0.713618 0.700535i $$-0.247055\pi$$
0.713618 + 0.700535i $$0.247055\pi$$
$$398$$ 0 0
$$399$$ 8454.66 1.06081
$$400$$ 0 0
$$401$$ −3055.59 −0.380521 −0.190261 0.981734i $$-0.560933\pi$$
−0.190261 + 0.981734i $$0.560933\pi$$
$$402$$ 0 0
$$403$$ 1055.37 0.130451
$$404$$ 0 0
$$405$$ 1492.20 0.183081
$$406$$ 0 0
$$407$$ 6748.08 0.821843
$$408$$ 0 0
$$409$$ 4089.01 0.494349 0.247175 0.968971i $$-0.420498\pi$$
0.247175 + 0.968971i $$0.420498\pi$$
$$410$$ 0 0
$$411$$ −1839.87 −0.220813
$$412$$ 0 0
$$413$$ −7264.79 −0.865562
$$414$$ 0 0
$$415$$ 291.471 0.0344765
$$416$$ 0 0
$$417$$ −5697.86 −0.669126
$$418$$ 0 0
$$419$$ 15397.0 1.79520 0.897602 0.440806i $$-0.145307\pi$$
0.897602 + 0.440806i $$0.145307\pi$$
$$420$$ 0 0
$$421$$ −1034.45 −0.119753 −0.0598766 0.998206i $$-0.519071\pi$$
−0.0598766 + 0.998206i $$0.519071\pi$$
$$422$$ 0 0
$$423$$ 2825.20 0.324742
$$424$$ 0 0
$$425$$ 8508.40 0.971101
$$426$$ 0 0
$$427$$ 7264.79 0.823344
$$428$$ 0 0
$$429$$ 1152.53 0.129707
$$430$$ 0 0
$$431$$ 4943.86 0.552523 0.276261 0.961083i $$-0.410904\pi$$
0.276261 + 0.961083i $$0.410904\pi$$
$$432$$ 0 0
$$433$$ 337.202 0.0374247 0.0187124 0.999825i $$-0.494043\pi$$
0.0187124 + 0.999825i $$0.494043\pi$$
$$434$$ 0 0
$$435$$ −11373.9 −1.25364
$$436$$ 0 0
$$437$$ 12462.7 1.36424
$$438$$ 0 0
$$439$$ −4493.93 −0.488573 −0.244286 0.969703i $$-0.578554\pi$$
−0.244286 + 0.969703i $$0.578554\pi$$
$$440$$ 0 0
$$441$$ 1437.80 0.155253
$$442$$ 0 0
$$443$$ 4292.26 0.460341 0.230171 0.973150i $$-0.426072\pi$$
0.230171 + 0.973150i $$0.426072\pi$$
$$444$$ 0 0
$$445$$ 7025.82 0.748440
$$446$$ 0 0
$$447$$ 2929.13 0.309940
$$448$$ 0 0
$$449$$ −4167.96 −0.438081 −0.219040 0.975716i $$-0.570293\pi$$
−0.219040 + 0.975716i $$0.570293\pi$$
$$450$$ 0 0
$$451$$ 27190.4 2.83890
$$452$$ 0 0
$$453$$ −2049.40 −0.212559
$$454$$ 0 0
$$455$$ 2955.73 0.304543
$$456$$ 0 0
$$457$$ 301.643 0.0308759 0.0154380 0.999881i $$-0.495086\pi$$
0.0154380 + 0.999881i $$0.495086\pi$$
$$458$$ 0 0
$$459$$ 1071.60 0.108971
$$460$$ 0 0
$$461$$ −9611.88 −0.971085 −0.485542 0.874213i $$-0.661378\pi$$
−0.485542 + 0.874213i $$0.661378\pi$$
$$462$$ 0 0
$$463$$ 13251.0 1.33008 0.665041 0.746807i $$-0.268414\pi$$
0.665041 + 0.746807i $$0.268414\pi$$
$$464$$ 0 0
$$465$$ −8151.20 −0.812909
$$466$$ 0 0
$$467$$ 4432.00 0.439161 0.219581 0.975594i $$-0.429531\pi$$
0.219581 + 0.975594i $$0.429531\pi$$
$$468$$ 0 0
$$469$$ −10409.9 −1.02491
$$470$$ 0 0
$$471$$ 1533.33 0.150004
$$472$$ 0 0
$$473$$ −22210.5 −2.15907
$$474$$ 0 0
$$475$$ −26944.9 −2.60277
$$476$$ 0 0
$$477$$ 398.998 0.0382995
$$478$$ 0 0
$$479$$ −9076.49 −0.865794 −0.432897 0.901443i $$-0.642509\pi$$
−0.432897 + 0.901443i $$0.642509\pi$$
$$480$$ 0 0
$$481$$ 899.378 0.0852559
$$482$$ 0 0
$$483$$ 6669.86 0.628342
$$484$$ 0 0
$$485$$ 12143.1 1.13688
$$486$$ 0 0
$$487$$ 3343.89 0.311142 0.155571 0.987825i $$-0.450278\pi$$
0.155571 + 0.987825i $$0.450278\pi$$
$$488$$ 0 0
$$489$$ 7277.86 0.673040
$$490$$ 0 0
$$491$$ 2423.73 0.222773 0.111386 0.993777i $$-0.464471\pi$$
0.111386 + 0.993777i $$0.464471\pi$$
$$492$$ 0 0
$$493$$ −8167.95 −0.746179
$$494$$ 0 0
$$495$$ −8901.60 −0.808277
$$496$$ 0 0
$$497$$ 23607.1 2.13063
$$498$$ 0 0
$$499$$ −811.819 −0.0728296 −0.0364148 0.999337i $$-0.511594\pi$$
−0.0364148 + 0.999337i $$0.511594\pi$$
$$500$$ 0 0
$$501$$ −1012.00 −0.0902449
$$502$$ 0 0
$$503$$ −18192.4 −1.61264 −0.806320 0.591479i $$-0.798544\pi$$
−0.806320 + 0.591479i $$0.798544\pi$$
$$504$$ 0 0
$$505$$ −9183.65 −0.809242
$$506$$ 0 0
$$507$$ −6437.39 −0.563895
$$508$$ 0 0
$$509$$ −5645.44 −0.491610 −0.245805 0.969319i $$-0.579052\pi$$
−0.245805 + 0.969319i $$0.579052\pi$$
$$510$$ 0 0
$$511$$ −22921.5 −1.98432
$$512$$ 0 0
$$513$$ −3393.60 −0.292068
$$514$$ 0 0
$$515$$ 3625.88 0.310243
$$516$$ 0 0
$$517$$ −16853.5 −1.43369
$$518$$ 0 0
$$519$$ 7944.87 0.671948
$$520$$ 0 0
$$521$$ 12338.7 1.03756 0.518780 0.854908i $$-0.326386\pi$$
0.518780 + 0.854908i $$0.326386\pi$$
$$522$$ 0 0
$$523$$ 10609.8 0.887062 0.443531 0.896259i $$-0.353726\pi$$
0.443531 + 0.896259i $$0.353726\pi$$
$$524$$ 0 0
$$525$$ −14420.5 −1.19878
$$526$$ 0 0
$$527$$ −5853.65 −0.483850
$$528$$ 0 0
$$529$$ −2335.17 −0.191926
$$530$$ 0 0
$$531$$ 2916.00 0.238312
$$532$$ 0 0
$$533$$ 3623.91 0.294501
$$534$$ 0 0
$$535$$ 6620.53 0.535010
$$536$$ 0 0
$$537$$ 8721.86 0.700887
$$538$$ 0 0
$$539$$ −8577.07 −0.685419
$$540$$ 0 0
$$541$$ 4035.42 0.320696 0.160348 0.987061i $$-0.448738\pi$$
0.160348 + 0.987061i $$0.448738\pi$$
$$542$$ 0 0
$$543$$ −11048.4 −0.873171
$$544$$ 0 0
$$545$$ −36286.8 −2.85203
$$546$$ 0 0
$$547$$ −7407.45 −0.579012 −0.289506 0.957176i $$-0.593491\pi$$
−0.289506 + 0.957176i $$0.593491\pi$$
$$548$$ 0 0
$$549$$ −2916.00 −0.226688
$$550$$ 0 0
$$551$$ 25866.7 1.99993
$$552$$ 0 0
$$553$$ −13518.6 −1.03954
$$554$$ 0 0
$$555$$ −6946.40 −0.531276
$$556$$ 0 0
$$557$$ −9500.77 −0.722730 −0.361365 0.932424i $$-0.617689\pi$$
−0.361365 + 0.932424i $$0.617689\pi$$
$$558$$ 0 0
$$559$$ −2960.19 −0.223976
$$560$$ 0 0
$$561$$ −6392.54 −0.481093
$$562$$ 0 0
$$563$$ −2700.26 −0.202136 −0.101068 0.994880i $$-0.532226\pi$$
−0.101068 + 0.994880i $$0.532226\pi$$
$$564$$ 0 0
$$565$$ −12778.4 −0.951492
$$566$$ 0 0
$$567$$ −1816.20 −0.134521
$$568$$ 0 0
$$569$$ 15904.9 1.17183 0.585913 0.810374i $$-0.300736\pi$$
0.585913 + 0.810374i $$0.300736\pi$$
$$570$$ 0 0
$$571$$ −18234.0 −1.33638 −0.668188 0.743992i $$-0.732930\pi$$
−0.668188 + 0.743992i $$0.732930\pi$$
$$572$$ 0 0
$$573$$ −3838.67 −0.279865
$$574$$ 0 0
$$575$$ −21256.7 −1.54168
$$576$$ 0 0
$$577$$ 4869.57 0.351339 0.175670 0.984449i $$-0.443791\pi$$
0.175670 + 0.984449i $$0.443791\pi$$
$$578$$ 0 0
$$579$$ −14510.5 −1.04151
$$580$$ 0 0
$$581$$ −354.758 −0.0253319
$$582$$ 0 0
$$583$$ −2380.19 −0.169086
$$584$$ 0 0
$$585$$ −1186.40 −0.0838486
$$586$$ 0 0
$$587$$ 1616.99 0.113697 0.0568485 0.998383i $$-0.481895\pi$$
0.0568485 + 0.998383i $$0.481895\pi$$
$$588$$ 0 0
$$589$$ 18537.7 1.29683
$$590$$ 0 0
$$591$$ 8607.93 0.599125
$$592$$ 0 0
$$593$$ 8117.01 0.562101 0.281050 0.959693i $$-0.409317\pi$$
0.281050 + 0.959693i $$0.409317\pi$$
$$594$$ 0 0
$$595$$ −16394.1 −1.12957
$$596$$ 0 0
$$597$$ −1956.73 −0.134143
$$598$$ 0 0
$$599$$ −9536.40 −0.650495 −0.325248 0.945629i $$-0.605448\pi$$
−0.325248 + 0.945629i $$0.605448\pi$$
$$600$$ 0 0
$$601$$ −16247.8 −1.10276 −0.551381 0.834253i $$-0.685899\pi$$
−0.551381 + 0.834253i $$0.685899\pi$$
$$602$$ 0 0
$$603$$ 4178.39 0.282185
$$604$$ 0 0
$$605$$ 28581.9 1.92069
$$606$$ 0 0
$$607$$ −27725.7 −1.85396 −0.926980 0.375111i $$-0.877605\pi$$
−0.926980 + 0.375111i $$0.877605\pi$$
$$608$$ 0 0
$$609$$ 13843.5 0.921125
$$610$$ 0 0
$$611$$ −2246.22 −0.148727
$$612$$ 0 0
$$613$$ 927.190 0.0610911 0.0305456 0.999533i $$-0.490276\pi$$
0.0305456 + 0.999533i $$0.490276\pi$$
$$614$$ 0 0
$$615$$ −27989.5 −1.83519
$$616$$ 0 0
$$617$$ −18727.8 −1.22196 −0.610982 0.791644i $$-0.709225\pi$$
−0.610982 + 0.791644i $$0.709225\pi$$
$$618$$ 0 0
$$619$$ −3210.22 −0.208449 −0.104224 0.994554i $$-0.533236\pi$$
−0.104224 + 0.994554i $$0.533236\pi$$
$$620$$ 0 0
$$621$$ −2677.20 −0.172999
$$622$$ 0 0
$$623$$ −8551.33 −0.549922
$$624$$ 0 0
$$625$$ 3535.57 0.226276
$$626$$ 0 0
$$627$$ 20244.3 1.28944
$$628$$ 0 0
$$629$$ −4988.44 −0.316220
$$630$$ 0 0
$$631$$ 11911.2 0.751468 0.375734 0.926728i $$-0.377391\pi$$
0.375734 + 0.926728i $$0.377391\pi$$
$$632$$ 0 0
$$633$$ 1612.53 0.101252
$$634$$ 0 0
$$635$$ −48943.7 −3.05869
$$636$$ 0 0
$$637$$ −1143.14 −0.0711036
$$638$$ 0 0
$$639$$ −9475.60 −0.586618
$$640$$ 0 0
$$641$$ 17232.6 1.06185 0.530924 0.847419i $$-0.321845\pi$$
0.530924 + 0.847419i $$0.321845\pi$$
$$642$$ 0 0
$$643$$ 12754.6 0.782262 0.391131 0.920335i $$-0.372084\pi$$
0.391131 + 0.920335i $$0.372084\pi$$
$$644$$ 0 0
$$645$$ 22863.2 1.39572
$$646$$ 0 0
$$647$$ −9441.73 −0.573714 −0.286857 0.957973i $$-0.592610\pi$$
−0.286857 + 0.957973i $$0.592610\pi$$
$$648$$ 0 0
$$649$$ −17395.2 −1.05211
$$650$$ 0 0
$$651$$ 9921.06 0.597292
$$652$$ 0 0
$$653$$ −5198.99 −0.311565 −0.155783 0.987791i $$-0.549790\pi$$
−0.155783 + 0.987791i $$0.549790\pi$$
$$654$$ 0 0
$$655$$ 11343.2 0.676664
$$656$$ 0 0
$$657$$ 9200.39 0.546334
$$658$$ 0 0
$$659$$ −6508.01 −0.384698 −0.192349 0.981327i $$-0.561611\pi$$
−0.192349 + 0.981327i $$0.561611\pi$$
$$660$$ 0 0
$$661$$ 25280.2 1.48757 0.743785 0.668419i $$-0.233028\pi$$
0.743785 + 0.668419i $$0.233028\pi$$
$$662$$ 0 0
$$663$$ −851.991 −0.0499074
$$664$$ 0 0
$$665$$ 51917.8 3.02750
$$666$$ 0 0
$$667$$ 20406.2 1.18460
$$668$$ 0 0
$$669$$ 12125.9 0.700770
$$670$$ 0 0
$$671$$ 17395.2 1.00079
$$672$$ 0 0
$$673$$ −5525.89 −0.316504 −0.158252 0.987399i $$-0.550586\pi$$
−0.158252 + 0.987399i $$0.550586\pi$$
$$674$$ 0 0
$$675$$ 5788.20 0.330056
$$676$$ 0 0
$$677$$ −6293.21 −0.357264 −0.178632 0.983916i $$-0.557167\pi$$
−0.178632 + 0.983916i $$0.557167\pi$$
$$678$$ 0 0
$$679$$ −14779.7 −0.835335
$$680$$ 0 0
$$681$$ 9210.66 0.518287
$$682$$ 0 0
$$683$$ 5675.91 0.317984 0.158992 0.987280i $$-0.449176\pi$$
0.158992 + 0.987280i $$0.449176\pi$$
$$684$$ 0 0
$$685$$ −11298.2 −0.630190
$$686$$ 0 0
$$687$$ 615.331 0.0341722
$$688$$ 0 0
$$689$$ −317.229 −0.0175406
$$690$$ 0 0
$$691$$ 3617.79 0.199171 0.0995854 0.995029i $$-0.468248\pi$$
0.0995854 + 0.995029i $$0.468248\pi$$
$$692$$ 0 0
$$693$$ 10834.4 0.593888
$$694$$ 0 0
$$695$$ −34989.1 −1.90966
$$696$$ 0 0
$$697$$ −20100.2 −1.09232
$$698$$ 0 0
$$699$$ 40.1329 0.00217163
$$700$$ 0 0
$$701$$ −7938.43 −0.427718 −0.213859 0.976865i $$-0.568603\pi$$
−0.213859 + 0.976865i $$0.568603\pi$$
$$702$$ 0 0
$$703$$ 15797.7 0.847540
$$704$$ 0 0
$$705$$ 17348.8 0.926799
$$706$$ 0 0
$$707$$ 11177.7 0.594597
$$708$$ 0 0
$$709$$ −25691.6 −1.36088 −0.680442 0.732802i $$-0.738212\pi$$
−0.680442 + 0.732802i $$0.738212\pi$$
$$710$$ 0 0
$$711$$ 5426.19 0.286214
$$712$$ 0 0
$$713$$ 14624.3 0.768142
$$714$$ 0 0
$$715$$ 7077.35 0.370179
$$716$$ 0 0
$$717$$ −12984.0 −0.676284
$$718$$ 0 0
$$719$$ −36803.3 −1.90895 −0.954473 0.298298i $$-0.903581\pi$$
−0.954473 + 0.298298i $$0.903581\pi$$
$$720$$ 0 0
$$721$$ −4413.16 −0.227954
$$722$$ 0 0
$$723$$ −4524.94 −0.232759
$$724$$ 0 0
$$725$$ −44118.9 −2.26005
$$726$$ 0 0
$$727$$ −28333.2 −1.44542 −0.722709 0.691152i $$-0.757103\pi$$
−0.722709 + 0.691152i $$0.757103\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 16418.8 0.830742
$$732$$ 0 0
$$733$$ −10767.9 −0.542592 −0.271296 0.962496i $$-0.587452\pi$$
−0.271296 + 0.962496i $$0.587452\pi$$
$$734$$ 0 0
$$735$$ 8829.13 0.443085
$$736$$ 0 0
$$737$$ −24925.9 −1.24580
$$738$$ 0 0
$$739$$ 17301.4 0.861221 0.430610 0.902538i $$-0.358298\pi$$
0.430610 + 0.902538i $$0.358298\pi$$
$$740$$ 0 0
$$741$$ 2698.13 0.133763
$$742$$ 0 0
$$743$$ −24110.8 −1.19050 −0.595248 0.803542i $$-0.702946\pi$$
−0.595248 + 0.803542i $$0.702946\pi$$
$$744$$ 0 0
$$745$$ 17987.0 0.884555
$$746$$ 0 0
$$747$$ 142.396 0.00697455
$$748$$ 0 0
$$749$$ −8058.04 −0.393103
$$750$$ 0 0
$$751$$ −30052.8 −1.46024 −0.730121 0.683318i $$-0.760536\pi$$
−0.730121 + 0.683318i $$0.760536\pi$$
$$752$$ 0 0
$$753$$ −14614.4 −0.707275
$$754$$ 0 0
$$755$$ −12584.8 −0.606633
$$756$$ 0 0
$$757$$ 25599.4 1.22910 0.614549 0.788879i $$-0.289338\pi$$
0.614549 + 0.788879i $$0.289338\pi$$
$$758$$ 0 0
$$759$$ 15970.6 0.763765
$$760$$ 0 0
$$761$$ −28904.5 −1.37685 −0.688427 0.725306i $$-0.741698\pi$$
−0.688427 + 0.725306i $$0.741698\pi$$
$$762$$ 0 0
$$763$$ 44165.7 2.09555
$$764$$ 0 0
$$765$$ 6580.40 0.311000
$$766$$ 0 0
$$767$$ −2318.41 −0.109143
$$768$$ 0 0
$$769$$ −8756.13 −0.410603 −0.205302 0.978699i $$-0.565818\pi$$
−0.205302 + 0.978699i $$0.565818\pi$$
$$770$$ 0 0
$$771$$ −4996.66 −0.233399
$$772$$ 0 0
$$773$$ 28289.8 1.31632 0.658159 0.752879i $$-0.271335\pi$$
0.658159 + 0.752879i $$0.271335\pi$$
$$774$$ 0 0
$$775$$ −31618.3 −1.46550
$$776$$ 0 0
$$777$$ 8454.66 0.390359
$$778$$ 0 0
$$779$$ 63654.4 2.92767
$$780$$ 0 0
$$781$$ 56526.0 2.58983
$$782$$ 0 0
$$783$$ −5556.60 −0.253610
$$784$$ 0 0
$$785$$ 9415.75 0.428105
$$786$$ 0 0
$$787$$ −4859.74 −0.220116 −0.110058 0.993925i $$-0.535104\pi$$
−0.110058 + 0.993925i $$0.535104\pi$$
$$788$$ 0 0
$$789$$ −21502.9 −0.970246
$$790$$ 0 0
$$791$$ 15553.0 0.699117
$$792$$ 0 0
$$793$$ 2318.41 0.103820
$$794$$ 0 0
$$795$$ 2450.14 0.109305
$$796$$ 0 0
$$797$$ 17361.7 0.771623 0.385811 0.922578i $$-0.373922\pi$$
0.385811 + 0.922578i $$0.373922\pi$$
$$798$$ 0 0
$$799$$ 12458.8 0.551638
$$800$$ 0 0
$$801$$ 3432.40 0.151408
$$802$$ 0 0
$$803$$ −54884.2 −2.41198
$$804$$ 0 0
$$805$$ 40957.8 1.79326
$$806$$ 0 0
$$807$$ 16361.5 0.713696
$$808$$ 0 0
$$809$$ −24475.3 −1.06367 −0.531833 0.846849i $$-0.678497\pi$$
−0.531833 + 0.846849i $$0.678497\pi$$
$$810$$ 0 0
$$811$$ −19875.4 −0.860566 −0.430283 0.902694i $$-0.641586\pi$$
−0.430283 + 0.902694i $$0.641586\pi$$
$$812$$ 0 0
$$813$$ 16248.6 0.700939
$$814$$ 0 0
$$815$$ 44691.4 1.92083
$$816$$ 0 0
$$817$$ −51996.1 −2.22658
$$818$$ 0 0
$$819$$ 1444.00 0.0616085
$$820$$ 0 0
$$821$$ 21682.1 0.921693 0.460846 0.887480i $$-0.347546\pi$$
0.460846 + 0.887480i $$0.347546\pi$$
$$822$$ 0 0
$$823$$ −6698.17 −0.283698 −0.141849 0.989888i $$-0.545305\pi$$
−0.141849 + 0.989888i $$0.545305\pi$$
$$824$$ 0 0
$$825$$ −34529.0 −1.45715
$$826$$ 0 0
$$827$$ 3390.85 0.142577 0.0712886 0.997456i $$-0.477289\pi$$
0.0712886 + 0.997456i $$0.477289\pi$$
$$828$$ 0 0
$$829$$ −40093.4 −1.67974 −0.839869 0.542789i $$-0.817368\pi$$
−0.839869 + 0.542789i $$0.817368\pi$$
$$830$$ 0 0
$$831$$ 7946.25 0.331712
$$832$$ 0 0
$$833$$ 6340.50 0.263728
$$834$$ 0 0
$$835$$ −6214.40 −0.257555
$$836$$ 0 0
$$837$$ −3982.19 −0.164450
$$838$$ 0 0
$$839$$ 6172.12 0.253975 0.126988 0.991904i $$-0.459469\pi$$
0.126988 + 0.991904i $$0.459469\pi$$
$$840$$ 0 0
$$841$$ 17964.6 0.736585
$$842$$ 0 0
$$843$$ 19993.7 0.816869
$$844$$ 0 0
$$845$$ −39530.3 −1.60933
$$846$$ 0 0
$$847$$ −34787.8 −1.41124
$$848$$ 0 0
$$849$$ −16892.5 −0.682862
$$850$$ 0 0
$$851$$ 12462.7 0.502018
$$852$$ 0 0
$$853$$ −276.632 −0.0111040 −0.00555198 0.999985i $$-0.501767\pi$$
−0.00555198 + 0.999985i $$0.501767\pi$$
$$854$$ 0 0
$$855$$ −20839.2 −0.833550
$$856$$ 0 0
$$857$$ 3704.41 0.147655 0.0738274 0.997271i $$-0.476479\pi$$
0.0738274 + 0.997271i $$0.476479\pi$$
$$858$$ 0 0
$$859$$ −26915.5 −1.06909 −0.534544 0.845141i $$-0.679516\pi$$
−0.534544 + 0.845141i $$0.679516\pi$$
$$860$$ 0 0
$$861$$ 34066.8 1.34842
$$862$$ 0 0
$$863$$ 23623.9 0.931828 0.465914 0.884830i $$-0.345726\pi$$
0.465914 + 0.884830i $$0.345726\pi$$
$$864$$ 0 0
$$865$$ 48787.3 1.91771
$$866$$ 0 0
$$867$$ −10013.4 −0.392241
$$868$$ 0 0
$$869$$ −32369.5 −1.26359
$$870$$ 0 0
$$871$$ −3322.10 −0.129236
$$872$$ 0 0
$$873$$ 5932.39 0.229990
$$874$$ 0 0
$$875$$ −36918.9 −1.42638
$$876$$ 0 0
$$877$$ 14094.0 0.542667 0.271333 0.962485i $$-0.412535\pi$$
0.271333 + 0.962485i $$0.412535\pi$$
$$878$$ 0 0
$$879$$ −2725.12 −0.104569
$$880$$ 0 0
$$881$$ 18967.3 0.725341 0.362671 0.931917i $$-0.381865\pi$$
0.362671 + 0.931917i $$0.381865\pi$$
$$882$$ 0 0
$$883$$ −32886.0 −1.25334 −0.626672 0.779283i $$-0.715583\pi$$
−0.626672 + 0.779283i $$0.715583\pi$$
$$884$$ 0 0
$$885$$ 17906.4 0.680132
$$886$$ 0 0
$$887$$ 27945.6 1.05786 0.528929 0.848666i $$-0.322594\pi$$
0.528929 + 0.848666i $$0.322594\pi$$
$$888$$ 0 0
$$889$$ 59570.8 2.24740
$$890$$ 0 0
$$891$$ −4348.79 −0.163513
$$892$$ 0 0
$$893$$ −39455.1 −1.47852
$$894$$ 0 0
$$895$$ 53558.6 2.00030
$$896$$ 0 0
$$897$$ 2128.55 0.0792310
$$898$$ 0 0
$$899$$ 30353.1 1.12607
$$900$$ 0 0
$$901$$ 1759.53 0.0650592
$$902$$ 0 0
$$903$$ −27827.4 −1.02551
$$904$$ 0 0
$$905$$ −67845.2 −2.49199
$$906$$ 0 0
$$907$$ 5544.89 0.202993 0.101497 0.994836i $$-0.467637\pi$$
0.101497 + 0.994836i $$0.467637\pi$$
$$908$$ 0 0
$$909$$ −4486.59 −0.163708
$$910$$ 0 0
$$911$$ −19638.9 −0.714233 −0.357116 0.934060i $$-0.616240\pi$$
−0.357116 + 0.934060i $$0.616240\pi$$
$$912$$ 0 0
$$913$$ −849.451 −0.0307916
$$914$$ 0 0
$$915$$ −17906.4 −0.646958
$$916$$ 0 0
$$917$$ −13806.1 −0.497184
$$918$$ 0 0
$$919$$ −22128.7 −0.794297 −0.397149 0.917754i $$-0.630000\pi$$
−0.397149 + 0.917754i $$0.630000\pi$$
$$920$$ 0 0
$$921$$ −1244.01 −0.0445077
$$922$$ 0 0
$$923$$ 7533.72 0.268663
$$924$$ 0 0
$$925$$ −26944.9 −0.957775
$$926$$ 0 0
$$927$$ 1771.39 0.0627616
$$928$$ 0 0
$$929$$ 17599.3 0.621544 0.310772 0.950484i $$-0.399412\pi$$
0.310772 + 0.950484i $$0.399412\pi$$
$$930$$ 0 0
$$931$$ −20079.5 −0.706850
$$932$$ 0 0
$$933$$ −4847.72 −0.170104
$$934$$ 0 0
$$935$$ −39254.9 −1.37302
$$936$$ 0 0
$$937$$ 441.299 0.0153859 0.00769297 0.999970i $$-0.497551\pi$$
0.00769297 + 0.999970i $$0.497551\pi$$
$$938$$ 0 0
$$939$$ −25438.0 −0.884065
$$940$$ 0 0
$$941$$ 5408.01 0.187350 0.0936749 0.995603i $$-0.470139\pi$$
0.0936749 + 0.995603i $$0.470139\pi$$
$$942$$ 0 0
$$943$$ 50216.8 1.73413
$$944$$ 0 0
$$945$$ −11152.8 −0.383916
$$946$$ 0 0
$$947$$ 34237.1 1.17482 0.587410 0.809289i $$-0.300148\pi$$
0.587410 + 0.809289i $$0.300148\pi$$
$$948$$ 0 0
$$949$$ −7314.92 −0.250213
$$950$$ 0 0
$$951$$ 20324.2 0.693015
$$952$$ 0 0
$$953$$ 21410.7 0.727766 0.363883 0.931445i $$-0.381451\pi$$
0.363883 + 0.931445i $$0.381451\pi$$
$$954$$ 0 0
$$955$$ −23572.2 −0.798722
$$956$$ 0 0
$$957$$ 33147.5 1.11965
$$958$$ 0 0
$$959$$ 13751.3 0.463038
$$960$$ 0 0
$$961$$ −8038.09 −0.269816
$$962$$ 0 0
$$963$$ 3234.40 0.108232
$$964$$ 0 0
$$965$$ −89105.3 −2.97243
$$966$$ 0 0
$$967$$ 53874.6 1.79161 0.895806 0.444445i $$-0.146599\pi$$
0.895806 + 0.444445i $$0.146599\pi$$
$$968$$ 0 0
$$969$$ −14965.3 −0.496136
$$970$$ 0 0
$$971$$ −42901.5 −1.41789 −0.708947 0.705262i $$-0.750829\pi$$
−0.708947 + 0.705262i $$0.750829\pi$$
$$972$$ 0 0
$$973$$ 42586.2 1.40314
$$974$$ 0 0
$$975$$ −4602.00 −0.151161
$$976$$ 0 0
$$977$$ 58636.5 1.92011 0.960055 0.279812i $$-0.0902721\pi$$
0.960055 + 0.279812i $$0.0902721\pi$$
$$978$$ 0 0
$$979$$ −20475.7 −0.668444
$$980$$ 0 0
$$981$$ −17727.6 −0.576961
$$982$$ 0 0
$$983$$ 25296.7 0.820793 0.410396 0.911907i $$-0.365390\pi$$
0.410396 + 0.911907i $$0.365390\pi$$
$$984$$ 0 0
$$985$$ 52859.0 1.70988
$$986$$ 0 0
$$987$$ −21115.7 −0.680973
$$988$$ 0 0
$$989$$ −41019.6 −1.31885
$$990$$ 0 0
$$991$$ −10605.2 −0.339944 −0.169972 0.985449i $$-0.554368\pi$$
−0.169972 + 0.985449i $$0.554368\pi$$
$$992$$ 0 0
$$993$$ 27877.1 0.890888
$$994$$ 0 0
$$995$$ −12015.7 −0.382839
$$996$$ 0 0
$$997$$ −5770.19 −0.183294 −0.0916469 0.995792i $$-0.529213\pi$$
−0.0916469 + 0.995792i $$0.529213\pi$$
$$998$$ 0 0
$$999$$ −3393.60 −0.107476
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 768.4.a.p.1.2 2
3.2 odd 2 2304.4.a.s.1.1 2
4.3 odd 2 768.4.a.j.1.2 2
8.3 odd 2 768.4.a.k.1.1 2
8.5 even 2 768.4.a.e.1.1 2
12.11 even 2 2304.4.a.t.1.1 2
16.3 odd 4 384.4.d.c.193.1 4
16.5 even 4 384.4.d.e.193.2 yes 4
16.11 odd 4 384.4.d.c.193.4 yes 4
16.13 even 4 384.4.d.e.193.3 yes 4
24.5 odd 2 2304.4.a.bp.1.2 2
24.11 even 2 2304.4.a.bq.1.2 2
48.5 odd 4 1152.4.d.o.577.1 4
48.11 even 4 1152.4.d.i.577.1 4
48.29 odd 4 1152.4.d.o.577.4 4
48.35 even 4 1152.4.d.i.577.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.c.193.1 4 16.3 odd 4
384.4.d.c.193.4 yes 4 16.11 odd 4
384.4.d.e.193.2 yes 4 16.5 even 4
384.4.d.e.193.3 yes 4 16.13 even 4
768.4.a.e.1.1 2 8.5 even 2
768.4.a.j.1.2 2 4.3 odd 2
768.4.a.k.1.1 2 8.3 odd 2
768.4.a.p.1.2 2 1.1 even 1 trivial
1152.4.d.i.577.1 4 48.11 even 4
1152.4.d.i.577.4 4 48.35 even 4
1152.4.d.o.577.1 4 48.5 odd 4
1152.4.d.o.577.4 4 48.29 odd 4
2304.4.a.s.1.1 2 3.2 odd 2
2304.4.a.t.1.1 2 12.11 even 2
2304.4.a.bp.1.2 2 24.5 odd 2
2304.4.a.bq.1.2 2 24.11 even 2