# Properties

 Label 768.4.a.e.1.1 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 1.1 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 $$-0.808187\pi$$
−0.823866 + 0.566785i $$0.808187\pi$$
$$6$$ 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 $$-0.236802\pi$$
0.735809 + 0.677190i $$0.236802\pi$$
$$12$$ 0 0
$$13$$ 7.15559 0.152662 0.0763309 0.997083i $$-0.475679\pi$$
0.0763309 + 0.997083i $$0.475679\pi$$
$$14$$ 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 $$-0.225778\pi$$
0.758816 + 0.651306i $$0.225778\pi$$
$$20$$ 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 $$-0.271023\pi$$
0.658898 + 0.752232i $$0.271023\pi$$
$$30$$ 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 $$-0.409920\pi$$
0.279231 + 0.960224i $$0.409920\pi$$
$$38$$ 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 $$-0.762147\pi$$
−0.733569 + 0.679615i $$0.762147\pi$$
$$44$$ 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 $$-0.518297\pi$$
−0.0574492 + 0.998348i $$0.518297\pi$$
$$54$$ 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 $$-0.616360\pi$$
−0.357468 + 0.933925i $$0.616360\pi$$
$$60$$ 0 0
$$61$$ 324.000 0.680065 0.340032 0.940414i $$-0.389562\pi$$
0.340032 + 0.940414i $$0.389562\pi$$
$$62$$ 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 $$-0.639120\pi$$
−0.423277 + 0.906000i $$0.639120\pi$$
$$68$$ 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 $$-0.503330\pi$$
−0.0104618 + 0.999945i $$0.503330\pi$$
$$84$$ 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 $$-0.421027\pi$$
0.245562 + 0.969381i $$0.421027\pi$$
$$102$$ 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 $$-0.551907\pi$$
−0.162347 + 0.986734i $$0.551907\pi$$
$$108$$ 0 0
$$109$$ 1969.73 1.73088 0.865441 0.501011i $$-0.167038\pi$$
0.865441 + 0.501011i $$0.167038\pi$$
$$110$$ 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 $$-0.565827\pi$$
−0.205332 + 0.978692i $$0.565827\pi$$
$$132$$ 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 $$-0.303256\pi$$
0.579480 + 0.814987i $$0.303256\pi$$
$$140$$ 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 $$-0.586500\pi$$
−0.268416 + 0.963303i $$0.586500\pi$$
$$150$$ 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 $$-0.541468\pi$$
−0.129907 + 0.991526i $$0.541468\pi$$
$$158$$ 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 $$-0.698070\pi$$
−0.582869 + 0.812566i $$0.698070\pi$$
$$164$$ 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 $$-0.697700\pi$$
−0.581924 + 0.813243i $$0.697700\pi$$
$$174$$ 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 $$-0.707621\pi$$
−0.606986 + 0.794713i $$0.707621\pi$$
$$180$$ 0 0
$$181$$ 3682.80 1.51238 0.756188 0.654354i $$-0.227060\pi$$
0.756188 + 0.654354i $$0.227060\pi$$
$$182$$ 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 $$-0.673642\pi$$
−0.518857 + 0.854861i $$0.673642\pi$$
$$198$$ 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 $$-0.527947\pi$$
−0.0876866 + 0.996148i $$0.527947\pi$$
$$212$$ 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 $$-0.648166\pi$$
−0.448850 + 0.893607i $$0.648166\pi$$
$$228$$ 0 0
$$229$$ −205.110 −0.0591881 −0.0295940 0.999562i $$-0.509421\pi$$
−0.0295940 + 0.999562i $$0.509421\pi$$
$$230$$ 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 $$-0.290157\pi$$
0.612518 + 0.790456i $$0.290157\pi$$
$$252$$ 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 $$-0.712089\pi$$
−0.618079 + 0.786116i $$0.712089\pi$$
$$270$$ 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 $$-0.592748\pi$$
−0.287271 + 0.957849i $$0.592748\pi$$
$$278$$ 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 $$-0.298585\pi$$
0.591376 + 0.806396i $$0.298585\pi$$
$$284$$ 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 $$-0.471135\pi$$
0.0905593 + 0.995891i $$0.471135\pi$$
$$294$$ 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 $$-0.487728\pi$$
0.0385448 + 0.999257i $$0.487728\pi$$
$$308$$ 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 $$-0.704900\pi$$
−0.600169 + 0.799873i $$0.704900\pi$$
$$318$$ 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 $$-0.780509\pi$$
−0.771532 + 0.636191i $$0.780509\pi$$
$$332$$ 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 $$-0.403204\pi$$
0.299427 + 0.954119i $$0.403204\pi$$
$$348$$ 0 0
$$349$$ 3474.57 0.532922 0.266461 0.963846i $$-0.414146\pi$$
0.266461 + 0.963846i $$0.414146\pi$$
$$350$$ 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 $$-0.213785\pi$$
0.782812 + 0.622258i $$0.213785\pi$$
$$374$$ 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 $$-0.515314\pi$$
−0.0480923 + 0.998843i $$0.515314\pi$$
$$380$$ 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 $$-0.646836\pi$$
−0.445113 + 0.895474i $$0.646836\pi$$
$$390$$ 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 $$-0.752945\pi$$
−0.713618 + 0.700535i $$0.752945\pi$$
$$398$$ 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 $$-0.854693\pi$$
−0.897602 + 0.440806i $$0.854693\pi$$
$$420$$ 0 0
$$421$$ 1034.45 0.119753 0.0598766 0.998206i $$-0.480929\pi$$
0.0598766 + 0.998206i $$0.480929\pi$$
$$422$$ 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 $$-0.573928\pi$$
−0.230171 + 0.973150i $$0.573928\pi$$
$$444$$ 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 $$-0.338622\pi$$
0.485542 + 0.874213i $$0.338622\pi$$
$$462$$ 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 $$-0.570469\pi$$
−0.219581 + 0.975594i $$0.570469\pi$$
$$468$$ 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 $$-0.535529\pi$$
−0.111386 + 0.993777i $$0.535529\pi$$
$$492$$ 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 $$-0.488406\pi$$
0.0364148 + 0.999337i $$0.488406\pi$$
$$500$$ 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 $$-0.420948\pi$$
0.245805 + 0.969319i $$0.420948\pi$$
$$510$$ 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 $$-0.646274\pi$$
−0.443531 + 0.896259i $$0.646274\pi$$
$$524$$ 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 $$-0.551262\pi$$
−0.160348 + 0.987061i $$0.551262\pi$$
$$542$$ 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 $$-0.406509\pi$$
0.289506 + 0.957176i $$0.406509\pi$$
$$548$$ 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 $$-0.382311\pi$$
0.361365 + 0.932424i $$0.382311\pi$$
$$558$$ 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 $$-0.467774\pi$$
0.101068 + 0.994880i $$0.467774\pi$$
$$564$$ 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 $$-0.267070\pi$$
0.668188 + 0.743992i $$0.267070\pi$$
$$572$$ 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 $$-0.518105\pi$$
−0.0568485 + 0.998383i $$0.518105\pi$$
$$588$$ 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 $$-0.509724\pi$$
−0.0305456 + 0.999533i $$0.509724\pi$$
$$614$$ 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 $$-0.466764\pi$$
0.104224 + 0.994554i $$0.466764\pi$$
$$620$$ 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 $$-0.627916\pi$$
−0.391131 + 0.920335i $$0.627916\pi$$
$$644$$ 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 $$-0.450210\pi$$
0.155783 + 0.987791i $$0.450210\pi$$
$$654$$ 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 $$-0.438389\pi$$
0.192349 + 0.981327i $$0.438389\pi$$
$$660$$ 0 0
$$661$$ −25280.2 −1.48757 −0.743785 0.668419i $$-0.766972\pi$$
−0.743785 + 0.668419i $$0.766972\pi$$
$$662$$ 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 $$-0.442833\pi$$
0.178632 + 0.983916i $$0.442833\pi$$
$$678$$ 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 $$-0.550824\pi$$
−0.158992 + 0.987280i $$0.550824\pi$$
$$684$$ 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 $$-0.531752\pi$$
−0.0995854 + 0.995029i $$0.531752\pi$$
$$692$$ 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 $$-0.431397\pi$$
0.213859 + 0.976865i $$0.431397\pi$$
$$702$$ 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 $$-0.261788\pi$$
0.680442 + 0.732802i $$0.261788\pi$$
$$710$$ 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 $$-0.412548\pi$$
0.271296 + 0.962496i $$0.412548\pi$$
$$734$$ 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 $$-0.641702\pi$$
−0.430610 + 0.902538i $$0.641702\pi$$
$$740$$ 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 $$-0.710662\pi$$
−0.614549 + 0.788879i $$0.710662\pi$$
$$758$$ 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 $$-0.728665\pi$$
−0.658159 + 0.752879i $$0.728665\pi$$
$$774$$ 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 $$-0.464896\pi$$
0.110058 + 0.993925i $$0.464896\pi$$
$$788$$ 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 $$-0.626078\pi$$
−0.385811 + 0.922578i $$0.626078\pi$$
$$798$$ 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 $$-0.358414\pi$$
0.430283 + 0.902694i $$0.358414\pi$$
$$812$$ 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 $$-0.652454\pi$$
−0.460846 + 0.887480i $$0.652454\pi$$
$$822$$ 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 $$-0.522711\pi$$
−0.0712886 + 0.997456i $$0.522711\pi$$
$$828$$ 0 0
$$829$$ 40093.4 1.67974 0.839869 0.542789i $$-0.182632\pi$$
0.839869 + 0.542789i $$0.182632\pi$$
$$830$$ 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 $$-0.498233\pi$$
0.00555198 + 0.999985i $$0.498233\pi$$
$$854$$ 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 $$-0.320484\pi$$
0.534544 + 0.845141i $$0.320484\pi$$
$$860$$ 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 $$-0.587465\pi$$
−0.271333 + 0.962485i $$0.587465\pi$$
$$878$$ 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 $$-0.284417\pi$$
0.626672 + 0.779283i $$0.284417\pi$$
$$884$$ 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 $$-0.532363\pi$$
−0.101497 + 0.994836i $$0.532363\pi$$
$$908$$ 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 $$-0.529861\pi$$
−0.0936749 + 0.995603i $$0.529861\pi$$
$$942$$ 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 $$-0.699852\pi$$
−0.587410 + 0.809289i $$0.699852\pi$$
$$948$$ 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 $$-0.249171\pi$$
0.708947 + 0.705262i $$0.249171\pi$$
$$972$$ 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 $$-0.470787\pi$$
0.0916469 + 0.995792i $$0.470787\pi$$
$$998$$ 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.e.1.1 2
3.2 odd 2 2304.4.a.bp.1.2 2
4.3 odd 2 768.4.a.k.1.1 2
8.3 odd 2 768.4.a.j.1.2 2
8.5 even 2 768.4.a.p.1.2 2
12.11 even 2 2304.4.a.bq.1.2 2
16.3 odd 4 384.4.d.c.193.4 yes 4
16.5 even 4 384.4.d.e.193.3 yes 4
16.11 odd 4 384.4.d.c.193.1 4
16.13 even 4 384.4.d.e.193.2 yes 4
24.5 odd 2 2304.4.a.s.1.1 2
24.11 even 2 2304.4.a.t.1.1 2
48.5 odd 4 1152.4.d.o.577.4 4
48.11 even 4 1152.4.d.i.577.4 4
48.29 odd 4 1152.4.d.o.577.1 4
48.35 even 4 1152.4.d.i.577.1 4

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