Properties

 Label 384.4.d.e.193.3 Level $384$ Weight $4$ Character 384.193 Analytic conductor $22.657$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(193,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

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: no Analytic conductor: $$22.6567334422$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 193.3 Root $$-2.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.193 Dual form 384.4.d.e.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000i q^{3} -18.4222i q^{5} +22.4222 q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q+3.00000i q^{3} -18.4222i q^{5} +22.4222 q^{7} -9.00000 q^{9} +53.6888i q^{11} -7.15559i q^{13} +55.2666 q^{15} +39.6888 q^{17} -125.689i q^{19} +67.2666i q^{21} +99.1556 q^{23} -214.378 q^{25} -27.0000i q^{27} -205.800i q^{29} -147.489 q^{31} -161.066 q^{33} -413.066i q^{35} +125.689i q^{37} +21.4668 q^{39} +506.444 q^{41} -413.689i q^{43} +165.800i q^{45} +313.911 q^{47} +159.755 q^{49} +119.066i q^{51} -44.3331i q^{53} +989.066 q^{55} +377.066 q^{57} -324.000i q^{59} -324.000i q^{61} -201.800 q^{63} -131.822 q^{65} +464.266i q^{67} +297.467i q^{69} +1052.84 q^{71} -1022.27 q^{73} -643.133i q^{75} +1203.82i q^{77} +602.910 q^{79} +81.0000 q^{81} +15.8217i q^{83} -731.156i q^{85} +617.400 q^{87} -381.378 q^{89} -160.444i q^{91} -442.466i q^{93} -2315.47 q^{95} +659.154 q^{97} -483.199i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 32 q^{7} - 36 q^{9}+O(q^{10})$$ 4 * q + 32 * q^7 - 36 * q^9 $$4 q + 32 q^{7} - 36 q^{9} + 48 q^{15} - 72 q^{17} + 512 q^{23} - 396 q^{25} + 160 q^{31} + 48 q^{33} + 432 q^{39} + 872 q^{41} + 448 q^{47} - 284 q^{49} + 3264 q^{55} + 816 q^{57} - 288 q^{63} + 1088 q^{65} + 4096 q^{71} - 1320 q^{73} - 992 q^{79} + 324 q^{81} + 912 q^{87} - 1064 q^{89} - 4416 q^{95} - 2440 q^{97}+O(q^{100})$$ 4 * q + 32 * q^7 - 36 * q^9 + 48 * q^15 - 72 * q^17 + 512 * q^23 - 396 * q^25 + 160 * q^31 + 48 * q^33 + 432 * q^39 + 872 * q^41 + 448 * q^47 - 284 * q^49 + 3264 * q^55 + 816 * q^57 - 288 * q^63 + 1088 * q^65 + 4096 * q^71 - 1320 * q^73 - 992 * q^79 + 324 * q^81 + 912 * q^87 - 1064 * q^89 - 4416 * q^95 - 2440 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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
$$3$$ 3.00000i 0.577350i
$$4$$ 0 0
$$5$$ − 18.4222i − 1.64773i −0.566785 0.823866i $$-0.691813\pi$$
0.566785 0.823866i $$-0.308187\pi$$
$$6$$ 0 0
$$7$$ 22.4222 1.21069 0.605343 0.795965i $$-0.293036\pi$$
0.605343 + 0.795965i $$0.293036\pi$$
$$8$$ 0 0
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 53.6888i 1.47162i 0.677190 + 0.735809i $$0.263198\pi$$
−0.677190 + 0.735809i $$0.736802\pi$$
$$12$$ 0 0
$$13$$ − 7.15559i − 0.152662i −0.997083 0.0763309i $$-0.975679\pi$$
0.997083 0.0763309i $$-0.0243205\pi$$
$$14$$ 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
$$19$$ − 125.689i − 1.51763i −0.651306 0.758816i $$-0.725778\pi$$
0.651306 0.758816i $$-0.274222\pi$$
$$20$$ 0 0
$$21$$ 67.2666i 0.698989i
$$22$$ 0 0
$$23$$ 99.1556 0.898929 0.449465 0.893298i $$-0.351615\pi$$
0.449465 + 0.893298i $$0.351615\pi$$
$$24$$ 0 0
$$25$$ −214.378 −1.71502
$$26$$ 0 0
$$27$$ − 27.0000i − 0.192450i
$$28$$ 0 0
$$29$$ − 205.800i − 1.31780i −0.752232 0.658898i $$-0.771023\pi$$
0.752232 0.658898i $$-0.228977\pi$$
$$30$$ 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
$$35$$ − 413.066i − 1.99489i
$$36$$ 0 0
$$37$$ 125.689i 0.558463i 0.960224 + 0.279231i $$0.0900796\pi$$
−0.960224 + 0.279231i $$0.909920\pi$$
$$38$$ 0 0
$$39$$ 21.4668 0.0881393
$$40$$ 0 0
$$41$$ 506.444 1.92910 0.964552 0.263892i $$-0.0850063\pi$$
0.964552 + 0.263892i $$0.0850063\pi$$
$$42$$ 0 0
$$43$$ − 413.689i − 1.46714i −0.679615 0.733569i $$-0.737853\pi$$
0.679615 0.733569i $$-0.262147\pi$$
$$44$$ 0 0
$$45$$ 165.800i 0.549244i
$$46$$ 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
$$51$$ 119.066i 0.326914i
$$52$$ 0 0
$$53$$ − 44.3331i − 0.114898i −0.998348 0.0574492i $$-0.981703\pi$$
0.998348 0.0574492i $$-0.0182967\pi$$
$$54$$ 0 0
$$55$$ 989.066 2.42483
$$56$$ 0 0
$$57$$ 377.066 0.876205
$$58$$ 0 0
$$59$$ − 324.000i − 0.714936i −0.933925 0.357468i $$-0.883640\pi$$
0.933925 0.357468i $$-0.116360\pi$$
$$60$$ 0 0
$$61$$ − 324.000i − 0.680065i −0.940414 0.340032i $$-0.889562\pi$$
0.940414 0.340032i $$-0.110438\pi$$
$$62$$ 0 0
$$63$$ −201.800 −0.403562
$$64$$ 0 0
$$65$$ −131.822 −0.251546
$$66$$ 0 0
$$67$$ 464.266i 0.846554i 0.906000 + 0.423277i $$0.139120\pi$$
−0.906000 + 0.423277i $$0.860880\pi$$
$$68$$ 0 0
$$69$$ 297.467i 0.518997i
$$70$$ 0 0
$$71$$ 1052.84 1.75985 0.879927 0.475109i $$-0.157591\pi$$
0.879927 + 0.475109i $$0.157591\pi$$
$$72$$ 0 0
$$73$$ −1022.27 −1.63900 −0.819501 0.573078i $$-0.805749\pi$$
−0.819501 + 0.573078i $$0.805749\pi$$
$$74$$ 0 0
$$75$$ − 643.133i − 0.990168i
$$76$$ 0 0
$$77$$ 1203.82i 1.78167i
$$78$$ 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
$$83$$ 15.8217i 0.0209236i 0.999945 + 0.0104618i $$0.00333016\pi$$
−0.999945 + 0.0104618i $$0.996670\pi$$
$$84$$ 0 0
$$85$$ − 731.156i − 0.933000i
$$86$$ 0 0
$$87$$ 617.400 0.760830
$$88$$ 0 0
$$89$$ −381.378 −0.454224 −0.227112 0.973869i $$-0.572928\pi$$
−0.227112 + 0.973869i $$0.572928\pi$$
$$90$$ 0 0
$$91$$ − 160.444i − 0.184825i
$$92$$ 0 0
$$93$$ − 442.466i − 0.493350i
$$94$$ 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
$$99$$ − 483.199i − 0.490539i
$$100$$ 0 0
$$101$$ 498.510i 0.491125i 0.969381 + 0.245562i $$0.0789726\pi$$
−0.969381 + 0.245562i $$0.921027\pi$$
$$102$$ 0 0
$$103$$ −196.821 −0.188285 −0.0941425 0.995559i $$-0.530011\pi$$
−0.0941425 + 0.995559i $$0.530011\pi$$
$$104$$ 0 0
$$105$$ 1239.20 1.15175
$$106$$ 0 0
$$107$$ − 359.378i − 0.324695i −0.986734 0.162347i $$-0.948093\pi$$
0.986734 0.162347i $$-0.0519065\pi$$
$$108$$ 0 0
$$109$$ − 1969.73i − 1.73088i −0.501011 0.865441i $$-0.667038\pi$$
0.501011 0.865441i $$-0.332962\pi$$
$$110$$ 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
$$115$$ − 1826.66i − 1.48119i
$$116$$ 0 0
$$117$$ 64.4003i 0.0508873i
$$118$$ 0 0
$$119$$ 889.911 0.685529
$$120$$ 0 0
$$121$$ −1551.49 −1.16566
$$122$$ 0 0
$$123$$ 1519.33i 1.11377i
$$124$$ 0 0
$$125$$ 1646.53i 1.17816i
$$126$$ 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
$$131$$ 615.734i 0.410664i 0.978692 + 0.205332i $$0.0658274\pi$$
−0.978692 + 0.205332i $$0.934173\pi$$
$$132$$ 0 0
$$133$$ − 2818.22i − 1.83737i
$$134$$ 0 0
$$135$$ −497.400 −0.317106
$$136$$ 0 0
$$137$$ 613.290 0.382459 0.191230 0.981545i $$-0.438753\pi$$
0.191230 + 0.981545i $$0.438753\pi$$
$$138$$ 0 0
$$139$$ 1899.29i 1.15896i 0.814987 + 0.579480i $$0.196744\pi$$
−0.814987 + 0.579480i $$0.803256\pi$$
$$140$$ 0 0
$$141$$ 941.733i 0.562469i
$$142$$ 0 0
$$143$$ 384.175 0.224660
$$144$$ 0 0
$$145$$ −3791.29 −2.17137
$$146$$ 0 0
$$147$$ 479.266i 0.268906i
$$148$$ 0 0
$$149$$ − 976.377i − 0.536832i −0.963303 0.268416i $$-0.913500\pi$$
0.963303 0.268416i $$-0.0865001\pi$$
$$150$$ 0 0
$$151$$ 683.132 0.368162 0.184081 0.982911i $$-0.441069\pi$$
0.184081 + 0.982911i $$0.441069\pi$$
$$152$$ 0 0
$$153$$ −357.199 −0.188744
$$154$$ 0 0
$$155$$ 2717.07i 1.40800i
$$156$$ 0 0
$$157$$ 511.109i 0.259815i 0.991526 + 0.129907i $$0.0414680\pi$$
−0.991526 + 0.129907i $$0.958532\pi$$
$$158$$ 0 0
$$159$$ 132.999 0.0663366
$$160$$ 0 0
$$161$$ 2223.29 1.08832
$$162$$ 0 0
$$163$$ 2425.95i 1.16574i 0.812566 + 0.582869i $$0.198070\pi$$
−0.812566 + 0.582869i $$0.801930\pi$$
$$164$$ 0 0
$$165$$ 2967.20i 1.39998i
$$166$$ 0 0
$$167$$ 337.332 0.156309 0.0781544 0.996941i $$-0.475097\pi$$
0.0781544 + 0.996941i $$0.475097\pi$$
$$168$$ 0 0
$$169$$ 2145.80 0.976694
$$170$$ 0 0
$$171$$ 1131.20i 0.505877i
$$172$$ 0 0
$$173$$ 2648.29i 1.16385i 0.813243 + 0.581924i $$0.197700\pi$$
−0.813243 + 0.581924i $$0.802300\pi$$
$$174$$ 0 0
$$175$$ −4806.82 −2.07635
$$176$$ 0 0
$$177$$ 972.000 0.412768
$$178$$ 0 0
$$179$$ 2907.29i 1.21397i 0.794713 + 0.606986i $$0.207621\pi$$
−0.794713 + 0.606986i $$0.792379\pi$$
$$180$$ 0 0
$$181$$ 3682.80i 1.51238i 0.654354 + 0.756188i $$0.272940\pi$$
−0.654354 + 0.756188i $$0.727060\pi$$
$$182$$ 0 0
$$183$$ 972.000 0.392636
$$184$$ 0 0
$$185$$ 2315.47 0.920197
$$186$$ 0 0
$$187$$ 2130.85i 0.833277i
$$188$$ 0 0
$$189$$ − 605.400i − 0.232996i
$$190$$ 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
$$195$$ − 395.465i − 0.145230i
$$196$$ 0 0
$$197$$ − 2869.31i − 1.03771i −0.854861 0.518857i $$-0.826358\pi$$
0.854861 0.518857i $$-0.173642\pi$$
$$198$$ 0 0
$$199$$ 652.242 0.232343 0.116171 0.993229i $$-0.462938\pi$$
0.116171 + 0.993229i $$0.462938\pi$$
$$200$$ 0 0
$$201$$ −1392.80 −0.488758
$$202$$ 0 0
$$203$$ − 4614.49i − 1.59544i
$$204$$ 0 0
$$205$$ − 9329.82i − 3.17865i
$$206$$ 0 0
$$207$$ −892.400 −0.299643
$$208$$ 0 0
$$209$$ 6748.08 2.23337
$$210$$ 0 0
$$211$$ 537.511i 0.175373i 0.996148 + 0.0876866i $$0.0279474\pi$$
−0.996148 + 0.0876866i $$0.972053\pi$$
$$212$$ 0 0
$$213$$ 3158.53i 1.01605i
$$214$$ 0 0
$$215$$ −7621.06 −2.41745
$$216$$ 0 0
$$217$$ −3307.02 −1.03454
$$218$$ 0 0
$$219$$ − 3066.80i − 0.946278i
$$220$$ 0 0
$$221$$ − 283.997i − 0.0864421i
$$222$$ 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
$$227$$ 3070.22i 0.897699i 0.893607 + 0.448850i $$0.148166\pi$$
−0.893607 + 0.448850i $$0.851834\pi$$
$$228$$ 0 0
$$229$$ − 205.110i − 0.0591881i −0.999562 0.0295940i $$-0.990579\pi$$
0.999562 0.0295940i $$-0.00942145\pi$$
$$230$$ 0 0
$$231$$ −3611.47 −1.02864
$$232$$ 0 0
$$233$$ −13.3776 −0.00376137 −0.00188068 0.999998i $$-0.500599\pi$$
−0.00188068 + 0.999998i $$0.500599\pi$$
$$234$$ 0 0
$$235$$ − 5782.93i − 1.60526i
$$236$$ 0 0
$$237$$ 1808.73i 0.495737i
$$238$$ 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
$$243$$ 243.000i 0.0641500i
$$244$$ 0 0
$$245$$ − 2943.04i − 0.767446i
$$246$$ 0 0
$$247$$ −899.378 −0.231684
$$248$$ 0 0
$$249$$ −47.4652 −0.0120803
$$250$$ 0 0
$$251$$ 4871.47i 1.22504i 0.790456 + 0.612518i $$0.209843\pi$$
−0.790456 + 0.612518i $$0.790157\pi$$
$$252$$ 0 0
$$253$$ 5323.55i 1.32288i
$$254$$ 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
$$259$$ 2818.22i 0.676122i
$$260$$ 0 0
$$261$$ 1852.20i 0.439265i
$$262$$ 0 0
$$263$$ 7167.64 1.68052 0.840258 0.542188i $$-0.182404\pi$$
0.840258 + 0.542188i $$0.182404\pi$$
$$264$$ 0 0
$$265$$ −816.713 −0.189322
$$266$$ 0 0
$$267$$ − 1144.13i − 0.262246i
$$268$$ 0 0
$$269$$ 5453.84i 1.23616i 0.786116 + 0.618079i $$0.212089\pi$$
−0.786116 + 0.618079i $$0.787911\pi$$
$$270$$ 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
$$275$$ − 11509.7i − 2.52385i
$$276$$ 0 0
$$277$$ − 2648.75i − 0.574542i −0.957849 0.287271i $$-0.907252\pi$$
0.957849 0.287271i $$-0.0927479\pi$$
$$278$$ 0 0
$$279$$ 1327.40 0.284836
$$280$$ 0 0
$$281$$ −6664.57 −1.41486 −0.707429 0.706784i $$-0.750145\pi$$
−0.707429 + 0.706784i $$0.750145\pi$$
$$282$$ 0 0
$$283$$ 5630.84i 1.18275i 0.806396 + 0.591376i $$0.201415\pi$$
−0.806396 + 0.591376i $$0.798585\pi$$
$$284$$ 0 0
$$285$$ − 6946.40i − 1.44375i
$$286$$ 0 0
$$287$$ 11355.6 2.33554
$$288$$ 0 0
$$289$$ −3337.80 −0.679381
$$290$$ 0 0
$$291$$ 1977.46i 0.398354i
$$292$$ 0 0
$$293$$ 908.374i 0.181119i 0.995891 + 0.0905593i $$0.0288655\pi$$
−0.995891 + 0.0905593i $$0.971135\pi$$
$$294$$ 0 0
$$295$$ −5968.79 −1.17802
$$296$$ 0 0
$$297$$ 1449.60 0.283213
$$298$$ 0 0
$$299$$ − 709.517i − 0.137232i
$$300$$ 0 0
$$301$$ − 9275.82i − 1.77624i
$$302$$ 0 0
$$303$$ −1495.53 −0.283551
$$304$$ 0 0
$$305$$ −5968.79 −1.12056
$$306$$ 0 0
$$307$$ − 414.671i − 0.0770896i −0.999257 0.0385448i $$-0.987728\pi$$
0.999257 0.0385448i $$-0.0122722\pi$$
$$308$$ 0 0
$$309$$ − 590.463i − 0.108706i
$$310$$ 0 0
$$311$$ 1615.91 0.294629 0.147315 0.989090i $$-0.452937\pi$$
0.147315 + 0.989090i $$0.452937\pi$$
$$312$$ 0 0
$$313$$ 8479.33 1.53125 0.765623 0.643289i $$-0.222431\pi$$
0.765623 + 0.643289i $$0.222431\pi$$
$$314$$ 0 0
$$315$$ 3717.60i 0.664962i
$$316$$ 0 0
$$317$$ 6774.73i 1.20034i 0.799873 + 0.600169i $$0.204900\pi$$
−0.799873 + 0.600169i $$0.795100\pi$$
$$318$$ 0 0
$$319$$ 11049.2 1.93929
$$320$$ 0 0
$$321$$ 1078.13 0.187463
$$322$$ 0 0
$$323$$ − 4988.44i − 0.859332i
$$324$$ 0 0
$$325$$ 1534.00i 0.261818i
$$326$$ 0 0
$$327$$ 5909.20 0.999325
$$328$$ 0 0
$$329$$ 7038.57 1.17948
$$330$$ 0 0
$$331$$ − 9292.36i − 1.54306i −0.636191 0.771532i $$-0.719491\pi$$
0.636191 0.771532i $$-0.280509\pi$$
$$332$$ 0 0
$$333$$ − 1131.20i − 0.186154i
$$334$$ 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
$$339$$ − 2080.93i − 0.333394i
$$340$$ 0 0
$$341$$ − 7918.49i − 1.25751i
$$342$$ 0 0
$$343$$ −4108.75 −0.646798
$$344$$ 0 0
$$345$$ 5479.99 0.855168
$$346$$ 0 0
$$347$$ 3870.93i 0.598855i 0.954119 + 0.299427i $$0.0967956\pi$$
−0.954119 + 0.299427i $$0.903204\pi$$
$$348$$ 0 0
$$349$$ − 3474.57i − 0.532922i −0.963846 0.266461i $$-0.914146\pi$$
0.963846 0.266461i $$-0.0858543\pi$$
$$350$$ 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
$$355$$ − 19395.7i − 2.89977i
$$356$$ 0 0
$$357$$ 2669.73i 0.395791i
$$358$$ 0 0
$$359$$ −8161.19 −1.19981 −0.599904 0.800072i $$-0.704795\pi$$
−0.599904 + 0.800072i $$0.704795\pi$$
$$360$$ 0 0
$$361$$ −8938.68 −1.30320
$$362$$ 0 0
$$363$$ − 4654.47i − 0.672992i
$$364$$ 0 0
$$365$$ 18832.4i 2.70064i
$$366$$ 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
$$371$$ − 994.045i − 0.139106i
$$372$$ 0 0
$$373$$ 11278.5i 1.56562i 0.622258 + 0.782812i $$0.286215\pi$$
−0.622258 + 0.782812i $$0.713785\pi$$
$$374$$ 0 0
$$375$$ −4939.60 −0.680213
$$376$$ 0 0
$$377$$ −1472.62 −0.201177
$$378$$ 0 0
$$379$$ − 709.683i − 0.0961846i −0.998843 0.0480923i $$-0.984686\pi$$
0.998843 0.0480923i $$-0.0153142\pi$$
$$380$$ 0 0
$$381$$ − 7970.33i − 1.07174i
$$382$$ 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
$$387$$ 3723.20i 0.489046i
$$388$$ 0 0
$$389$$ − 6830.06i − 0.890226i −0.895474 0.445113i $$-0.853164\pi$$
0.895474 0.445113i $$-0.146836\pi$$
$$390$$ 0 0
$$391$$ 3935.37 0.509003
$$392$$ 0 0
$$393$$ −1847.20 −0.237097
$$394$$ 0 0
$$395$$ − 11106.9i − 1.41481i
$$396$$ 0 0
$$397$$ 11289.7i 1.42724i 0.700535 + 0.713618i $$0.252945\pi$$
−0.700535 + 0.713618i $$0.747055\pi$$
$$398$$ 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
$$403$$ 1055.37i 0.130451i
$$404$$ 0 0
$$405$$ − 1492.20i − 0.183081i
$$406$$ 0 0
$$407$$ −6748.08 −0.821843
$$408$$ 0 0
$$409$$ −4089.01 −0.494349 −0.247175 0.968971i $$-0.579502\pi$$
−0.247175 + 0.968971i $$0.579502\pi$$
$$410$$ 0 0
$$411$$ 1839.87i 0.220813i
$$412$$ 0 0
$$413$$ − 7264.79i − 0.865562i
$$414$$ 0 0
$$415$$ 291.471 0.0344765
$$416$$ 0 0
$$417$$ −5697.86 −0.669126
$$418$$ 0 0
$$419$$ 15397.0i 1.79520i 0.440806 + 0.897602i $$0.354693\pi$$
−0.440806 + 0.897602i $$0.645307\pi$$
$$420$$ 0 0
$$421$$ 1034.45i 0.119753i 0.998206 + 0.0598766i $$0.0190707\pi$$
−0.998206 + 0.0598766i $$0.980929\pi$$
$$422$$ 0 0
$$423$$ −2825.20 −0.324742
$$424$$ 0 0
$$425$$ −8508.40 −0.971101
$$426$$ 0 0
$$427$$ − 7264.79i − 0.823344i
$$428$$ 0 0
$$429$$ 1152.53i 0.129707i
$$430$$ 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
$$435$$ − 11373.9i − 1.25364i
$$436$$ 0 0
$$437$$ − 12462.7i − 1.36424i
$$438$$ 0 0
$$439$$ 4493.93 0.488573 0.244286 0.969703i $$-0.421446\pi$$
0.244286 + 0.969703i $$0.421446\pi$$
$$440$$ 0 0
$$441$$ −1437.80 −0.155253
$$442$$ 0 0
$$443$$ − 4292.26i − 0.460341i −0.973150 0.230171i $$-0.926072\pi$$
0.973150 0.230171i $$-0.0739285\pi$$
$$444$$ 0 0
$$445$$ 7025.82i 0.748440i
$$446$$ 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
$$451$$ 27190.4i 2.83890i
$$452$$ 0 0
$$453$$ 2049.40i 0.212559i
$$454$$ 0 0
$$455$$ −2955.73 −0.304543
$$456$$ 0 0
$$457$$ −301.643 −0.0308759 −0.0154380 0.999881i $$-0.504914\pi$$
−0.0154380 + 0.999881i $$0.504914\pi$$
$$458$$ 0 0
$$459$$ − 1071.60i − 0.108971i
$$460$$ 0 0
$$461$$ − 9611.88i − 0.971085i −0.874213 0.485542i $$-0.838622\pi$$
0.874213 0.485542i $$-0.161378\pi$$
$$462$$ 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
$$467$$ 4432.00i 0.439161i 0.975594 + 0.219581i $$0.0704689\pi$$
−0.975594 + 0.219581i $$0.929531\pi$$
$$468$$ 0 0
$$469$$ 10409.9i 1.02491i
$$470$$ 0 0
$$471$$ −1533.33 −0.150004
$$472$$ 0 0
$$473$$ 22210.5 2.15907
$$474$$ 0 0
$$475$$ 26944.9i 2.60277i
$$476$$ 0 0
$$477$$ 398.998i 0.0382995i
$$478$$ 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
$$483$$ 6669.86i 0.628342i
$$484$$ 0 0
$$485$$ − 12143.1i − 1.13688i
$$486$$ 0 0
$$487$$ −3343.89 −0.311142 −0.155571 0.987825i $$-0.549722\pi$$
−0.155571 + 0.987825i $$0.549722\pi$$
$$488$$ 0 0
$$489$$ −7277.86 −0.673040
$$490$$ 0 0
$$491$$ − 2423.73i − 0.222773i −0.993777 0.111386i $$-0.964471\pi$$
0.993777 0.111386i $$-0.0355291\pi$$
$$492$$ 0 0
$$493$$ − 8167.95i − 0.746179i
$$494$$ 0 0
$$495$$ −8901.60 −0.808277
$$496$$ 0 0
$$497$$ 23607.1 2.13063
$$498$$ 0 0
$$499$$ − 811.819i − 0.0728296i −0.999337 0.0364148i $$-0.988406\pi$$
0.999337 0.0364148i $$-0.0115938\pi$$
$$500$$ 0 0
$$501$$ 1012.00i 0.0902449i
$$502$$ 0 0
$$503$$ 18192.4 1.61264 0.806320 0.591479i $$-0.201456\pi$$
0.806320 + 0.591479i $$0.201456\pi$$
$$504$$ 0 0
$$505$$ 9183.65 0.809242
$$506$$ 0 0
$$507$$ 6437.39i 0.563895i
$$508$$ 0 0
$$509$$ − 5645.44i − 0.491610i −0.969319 0.245805i $$-0.920948\pi$$
0.969319 0.245805i $$-0.0790523\pi$$
$$510$$ 0 0
$$511$$ −22921.5 −1.98432
$$512$$ 0 0
$$513$$ −3393.60 −0.292068
$$514$$ 0 0
$$515$$ 3625.88i 0.310243i
$$516$$ 0 0
$$517$$ 16853.5i 1.43369i
$$518$$ 0 0
$$519$$ −7944.87 −0.671948
$$520$$ 0 0
$$521$$ −12338.7 −1.03756 −0.518780 0.854908i $$-0.673614\pi$$
−0.518780 + 0.854908i $$0.673614\pi$$
$$522$$ 0 0
$$523$$ − 10609.8i − 0.887062i −0.896259 0.443531i $$-0.853726\pi$$
0.896259 0.443531i $$-0.146274\pi$$
$$524$$ 0 0
$$525$$ − 14420.5i − 1.19878i
$$526$$ 0 0
$$527$$ −5853.65 −0.483850
$$528$$ 0 0
$$529$$ −2335.17 −0.191926
$$530$$ 0 0
$$531$$ 2916.00i 0.238312i
$$532$$ 0 0
$$533$$ − 3623.91i − 0.294501i
$$534$$ 0 0
$$535$$ −6620.53 −0.535010
$$536$$ 0 0
$$537$$ −8721.86 −0.700887
$$538$$ 0 0
$$539$$ 8577.07i 0.685419i
$$540$$ 0 0
$$541$$ 4035.42i 0.320696i 0.987061 + 0.160348i $$0.0512616\pi$$
−0.987061 + 0.160348i $$0.948738\pi$$
$$542$$ 0 0
$$543$$ −11048.4 −0.873171
$$544$$ 0 0
$$545$$ −36286.8 −2.85203
$$546$$ 0 0
$$547$$ − 7407.45i − 0.579012i −0.957176 0.289506i $$-0.906509\pi$$
0.957176 0.289506i $$-0.0934911\pi$$
$$548$$ 0 0
$$549$$ 2916.00i 0.226688i
$$550$$ 0 0
$$551$$ −25866.7 −1.99993
$$552$$ 0 0
$$553$$ 13518.6 1.03954
$$554$$ 0 0
$$555$$ 6946.40i 0.531276i
$$556$$ 0 0
$$557$$ − 9500.77i − 0.722730i −0.932424 0.361365i $$-0.882311\pi$$
0.932424 0.361365i $$-0.117689\pi$$
$$558$$ 0 0
$$559$$ −2960.19 −0.223976
$$560$$ 0 0
$$561$$ −6392.54 −0.481093
$$562$$ 0 0
$$563$$ − 2700.26i − 0.202136i −0.994880 0.101068i $$-0.967774\pi$$
0.994880 0.101068i $$-0.0322259\pi$$
$$564$$ 0 0
$$565$$ 12778.4i 0.951492i
$$566$$ 0 0
$$567$$ 1816.20 0.134521
$$568$$ 0 0
$$569$$ −15904.9 −1.17183 −0.585913 0.810374i $$-0.699264\pi$$
−0.585913 + 0.810374i $$0.699264\pi$$
$$570$$ 0 0
$$571$$ 18234.0i 1.33638i 0.743992 + 0.668188i $$0.232930\pi$$
−0.743992 + 0.668188i $$0.767070\pi$$
$$572$$ 0 0
$$573$$ − 3838.67i − 0.279865i
$$574$$ 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
$$579$$ − 14510.5i − 1.04151i
$$580$$ 0 0
$$581$$ 354.758i 0.0253319i
$$582$$ 0 0
$$583$$ 2380.19 0.169086
$$584$$ 0 0
$$585$$ 1186.40 0.0838486
$$586$$ 0 0
$$587$$ − 1616.99i − 0.113697i −0.998383 0.0568485i $$-0.981895\pi$$
0.998383 0.0568485i $$-0.0181052\pi$$
$$588$$ 0 0
$$589$$ 18537.7i 1.29683i
$$590$$ 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
$$595$$ − 16394.1i − 1.12957i
$$596$$ 0 0
$$597$$ 1956.73i 0.134143i
$$598$$ 0 0
$$599$$ 9536.40 0.650495 0.325248 0.945629i $$-0.394552\pi$$
0.325248 + 0.945629i $$0.394552\pi$$
$$600$$ 0 0
$$601$$ 16247.8 1.10276 0.551381 0.834253i $$-0.314101\pi$$
0.551381 + 0.834253i $$0.314101\pi$$
$$602$$ 0 0
$$603$$ − 4178.39i − 0.282185i
$$604$$ 0 0
$$605$$ 28581.9i 1.92069i
$$606$$ 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
$$611$$ − 2246.22i − 0.148727i
$$612$$ 0 0
$$613$$ − 927.190i − 0.0610911i −0.999533 0.0305456i $$-0.990276\pi$$
0.999533 0.0305456i $$-0.00972447\pi$$
$$614$$ 0 0
$$615$$ 27989.5 1.83519
$$616$$ 0 0
$$617$$ 18727.8 1.22196 0.610982 0.791644i $$-0.290775\pi$$
0.610982 + 0.791644i $$0.290775\pi$$
$$618$$ 0 0
$$619$$ 3210.22i 0.208449i 0.994554 + 0.104224i $$0.0332360\pi$$
−0.994554 + 0.104224i $$0.966764\pi$$
$$620$$ 0 0
$$621$$ − 2677.20i − 0.172999i
$$622$$ 0 0
$$623$$ −8551.33 −0.549922
$$624$$ 0 0
$$625$$ 3535.57 0.226276
$$626$$ 0 0
$$627$$ 20244.3i 1.28944i
$$628$$ 0 0
$$629$$ 4988.44i 0.316220i
$$630$$ 0 0
$$631$$ −11911.2 −0.751468 −0.375734 0.926728i $$-0.622609\pi$$
−0.375734 + 0.926728i $$0.622609\pi$$
$$632$$ 0 0
$$633$$ −1612.53 −0.101252
$$634$$ 0 0
$$635$$ 48943.7i 3.05869i
$$636$$ 0 0
$$637$$ − 1143.14i − 0.0711036i
$$638$$ 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
$$643$$ 12754.6i 0.782262i 0.920335 + 0.391131i $$0.127916\pi$$
−0.920335 + 0.391131i $$0.872084\pi$$
$$644$$ 0 0
$$645$$ − 22863.2i − 1.39572i
$$646$$ 0 0
$$647$$ 9441.73 0.573714 0.286857 0.957973i $$-0.407390\pi$$
0.286857 + 0.957973i $$0.407390\pi$$
$$648$$ 0 0
$$649$$ 17395.2 1.05211
$$650$$ 0 0
$$651$$ − 9921.06i − 0.597292i
$$652$$ 0 0
$$653$$ − 5198.99i − 0.311565i −0.987791 0.155783i $$-0.950210\pi$$
0.987791 0.155783i $$-0.0497900\pi$$
$$654$$ 0 0
$$655$$ 11343.2 0.676664
$$656$$ 0 0
$$657$$ 9200.39 0.546334
$$658$$ 0 0
$$659$$ − 6508.01i − 0.384698i −0.981327 0.192349i $$-0.938389\pi$$
0.981327 0.192349i $$-0.0616105\pi$$
$$660$$ 0 0
$$661$$ − 25280.2i − 1.48757i −0.668419 0.743785i $$-0.733028\pi$$
0.668419 0.743785i $$-0.266972\pi$$
$$662$$ 0 0
$$663$$ 851.991 0.0499074
$$664$$ 0 0
$$665$$ −51917.8 −3.02750
$$666$$ 0 0
$$667$$ − 20406.2i − 1.18460i
$$668$$ 0 0
$$669$$ 12125.9i 0.700770i
$$670$$ 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
$$675$$ 5788.20i 0.330056i
$$676$$ 0 0
$$677$$ 6293.21i 0.357264i 0.983916 + 0.178632i $$0.0571671\pi$$
−0.983916 + 0.178632i $$0.942833\pi$$
$$678$$ 0 0
$$679$$ 14779.7 0.835335
$$680$$ 0 0
$$681$$ −9210.66 −0.518287
$$682$$ 0 0
$$683$$ − 5675.91i − 0.317984i −0.987280 0.158992i $$-0.949176\pi$$
0.987280 0.158992i $$-0.0508243\pi$$
$$684$$ 0 0
$$685$$ − 11298.2i − 0.630190i
$$686$$ 0 0
$$687$$ 615.331 0.0341722
$$688$$ 0 0
$$689$$ −317.229 −0.0175406
$$690$$ 0 0
$$691$$ 3617.79i 0.199171i 0.995029 + 0.0995854i $$0.0317517\pi$$
−0.995029 + 0.0995854i $$0.968248\pi$$
$$692$$ 0 0
$$693$$ − 10834.4i − 0.593888i
$$694$$ 0 0
$$695$$ 34989.1 1.90966
$$696$$ 0 0
$$697$$ 20100.2 1.09232
$$698$$ 0 0
$$699$$ − 40.1329i − 0.00217163i
$$700$$ 0 0
$$701$$ − 7938.43i − 0.427718i −0.976865 0.213859i $$-0.931397\pi$$
0.976865 0.213859i $$-0.0686033\pi$$
$$702$$ 0 0
$$703$$ 15797.7 0.847540
$$704$$ 0 0
$$705$$ 17348.8 0.926799
$$706$$ 0 0
$$707$$ 11177.7i 0.594597i
$$708$$ 0 0
$$709$$ 25691.6i 1.36088i 0.732802 + 0.680442i $$0.238212\pi$$
−0.732802 + 0.680442i $$0.761788\pi$$
$$710$$ 0 0
$$711$$ −5426.19 −0.286214
$$712$$ 0 0
$$713$$ −14624.3 −0.768142
$$714$$ 0 0
$$715$$ − 7077.35i − 0.370179i
$$716$$ 0 0
$$717$$ − 12984.0i − 0.676284i
$$718$$ 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
$$723$$ − 4524.94i − 0.232759i
$$724$$ 0 0
$$725$$ 44118.9i 2.26005i
$$726$$ 0 0
$$727$$ 28333.2 1.44542 0.722709 0.691152i $$-0.242897\pi$$
0.722709 + 0.691152i $$0.242897\pi$$
$$728$$ 0 0
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ − 16418.8i − 0.830742i
$$732$$ 0 0
$$733$$ − 10767.9i − 0.542592i −0.962496 0.271296i $$-0.912548\pi$$
0.962496 0.271296i $$-0.0874522\pi$$
$$734$$ 0 0
$$735$$ 8829.13 0.443085
$$736$$ 0 0
$$737$$ −24925.9 −1.24580
$$738$$ 0 0
$$739$$ 17301.4i 0.861221i 0.902538 + 0.430610i $$0.141702\pi$$
−0.902538 + 0.430610i $$0.858298\pi$$
$$740$$ 0 0
$$741$$ − 2698.13i − 0.133763i
$$742$$ 0 0
$$743$$ 24110.8 1.19050 0.595248 0.803542i $$-0.297054\pi$$
0.595248 + 0.803542i $$0.297054\pi$$
$$744$$ 0 0
$$745$$ −17987.0 −0.884555
$$746$$ 0 0
$$747$$ − 142.396i − 0.00697455i
$$748$$ 0 0
$$749$$ − 8058.04i − 0.393103i
$$750$$ 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
$$755$$ − 12584.8i − 0.606633i
$$756$$ 0 0
$$757$$ − 25599.4i − 1.22910i −0.788879 0.614549i $$-0.789338\pi$$
0.788879 0.614549i $$-0.210662\pi$$
$$758$$ 0 0
$$759$$ −15970.6 −0.763765
$$760$$ 0 0
$$761$$ 28904.5 1.37685 0.688427 0.725306i $$-0.258302\pi$$
0.688427 + 0.725306i $$0.258302\pi$$
$$762$$ 0 0
$$763$$ − 44165.7i − 2.09555i
$$764$$ 0 0
$$765$$ 6580.40i 0.311000i
$$766$$ 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
$$771$$ − 4996.66i − 0.233399i
$$772$$ 0 0
$$773$$ − 28289.8i − 1.31632i −0.752879 0.658159i $$-0.771335\pi$$
0.752879 0.658159i $$-0.228665\pi$$
$$774$$ 0 0
$$775$$ 31618.3 1.46550
$$776$$ 0 0
$$777$$ −8454.66 −0.390359
$$778$$ 0 0
$$779$$ − 63654.4i − 2.92767i
$$780$$ 0 0
$$781$$ 56526.0i 2.58983i
$$782$$ 0 0
$$783$$ −5556.60 −0.253610
$$784$$ 0 0
$$785$$ 9415.75 0.428105
$$786$$ 0 0
$$787$$ − 4859.74i − 0.220116i −0.993925 0.110058i $$-0.964896\pi$$
0.993925 0.110058i $$-0.0351036\pi$$
$$788$$ 0 0
$$789$$ 21502.9i 0.970246i
$$790$$ 0 0
$$791$$ −15553.0 −0.699117
$$792$$ 0 0
$$793$$ −2318.41 −0.103820
$$794$$ 0 0
$$795$$ − 2450.14i − 0.109305i
$$796$$ 0 0
$$797$$ 17361.7i 0.771623i 0.922578 + 0.385811i $$0.126078\pi$$
−0.922578 + 0.385811i $$0.873922\pi$$
$$798$$ 0 0
$$799$$ 12458.8 0.551638
$$800$$ 0 0
$$801$$ 3432.40 0.151408
$$802$$ 0 0
$$803$$ − 54884.2i − 2.41198i
$$804$$ 0 0
$$805$$ − 40957.8i − 1.79326i
$$806$$ 0 0
$$807$$ −16361.5 −0.713696
$$808$$ 0 0
$$809$$ 24475.3 1.06367 0.531833 0.846849i $$-0.321503\pi$$
0.531833 + 0.846849i $$0.321503\pi$$
$$810$$ 0 0
$$811$$ 19875.4i 0.860566i 0.902694 + 0.430283i $$0.141586\pi$$
−0.902694 + 0.430283i $$0.858414\pi$$
$$812$$ 0 0
$$813$$ 16248.6i 0.700939i
$$814$$ 0 0
$$815$$ 44691.4 1.92083
$$816$$ 0 0
$$817$$ −51996.1 −2.22658
$$818$$ 0 0
$$819$$ 1444.00i 0.0616085i
$$820$$ 0 0
$$821$$ − 21682.1i − 0.921693i −0.887480 0.460846i $$-0.847546\pi$$
0.887480 0.460846i $$-0.152454\pi$$
$$822$$ 0 0
$$823$$ 6698.17 0.283698 0.141849 0.989888i $$-0.454695\pi$$
0.141849 + 0.989888i $$0.454695\pi$$
$$824$$ 0 0
$$825$$ 34529.0 1.45715
$$826$$ 0 0
$$827$$ − 3390.85i − 0.142577i −0.997456 0.0712886i $$-0.977289\pi$$
0.997456 0.0712886i $$-0.0227111\pi$$
$$828$$ 0 0
$$829$$ − 40093.4i − 1.67974i −0.542789 0.839869i $$-0.682632\pi$$
0.542789 0.839869i $$-0.317368\pi$$
$$830$$ 0 0
$$831$$ 7946.25 0.331712
$$832$$ 0 0
$$833$$ 6340.50 0.263728
$$834$$ 0 0
$$835$$ − 6214.40i − 0.257555i
$$836$$ 0 0
$$837$$ 3982.19i 0.164450i
$$838$$ 0 0
$$839$$ −6172.12 −0.253975 −0.126988 0.991904i $$-0.540531\pi$$
−0.126988 + 0.991904i $$0.540531\pi$$
$$840$$ 0 0
$$841$$ −17964.6 −0.736585
$$842$$ 0 0
$$843$$ − 19993.7i − 0.816869i
$$844$$ 0 0
$$845$$ − 39530.3i − 1.60933i
$$846$$ 0 0
$$847$$ −34787.8 −1.41124
$$848$$ 0 0
$$849$$ −16892.5 −0.682862
$$850$$ 0 0
$$851$$ 12462.7i 0.502018i
$$852$$ 0 0
$$853$$ 276.632i 0.0111040i 0.999985 + 0.00555198i $$0.00176726\pi$$
−0.999985 + 0.00555198i $$0.998233\pi$$
$$854$$ 0 0
$$855$$ 20839.2 0.833550
$$856$$ 0 0
$$857$$ −3704.41 −0.147655 −0.0738274 0.997271i $$-0.523521\pi$$
−0.0738274 + 0.997271i $$0.523521\pi$$
$$858$$ 0 0
$$859$$ 26915.5i 1.06909i 0.845141 + 0.534544i $$0.179516\pi$$
−0.845141 + 0.534544i $$0.820484\pi$$
$$860$$ 0 0
$$861$$ 34066.8i 1.34842i
$$862$$ 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
$$867$$ − 10013.4i − 0.392241i
$$868$$ 0 0
$$869$$ 32369.5i 1.26359i
$$870$$ 0 0
$$871$$ 3322.10 0.129236
$$872$$ 0 0
$$873$$ −5932.39 −0.229990
$$874$$ 0 0
$$875$$ 36918.9i 1.42638i
$$876$$ 0 0
$$877$$ 14094.0i 0.542667i 0.962485 + 0.271333i $$0.0874646\pi$$
−0.962485 + 0.271333i $$0.912535\pi$$
$$878$$ 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
$$883$$ − 32886.0i − 1.25334i −0.779283 0.626672i $$-0.784417\pi$$
0.779283 0.626672i $$-0.215583\pi$$
$$884$$ 0 0
$$885$$ − 17906.4i − 0.680132i
$$886$$ 0 0
$$887$$ −27945.6 −1.05786 −0.528929 0.848666i $$-0.677406\pi$$
−0.528929 + 0.848666i $$0.677406\pi$$
$$888$$ 0 0
$$889$$ −59570.8 −2.24740
$$890$$ 0 0
$$891$$ 4348.79i 0.163513i
$$892$$ 0 0
$$893$$ − 39455.1i − 1.47852i
$$894$$ 0 0
$$895$$ 53558.6 2.00030
$$896$$ 0 0
$$897$$ 2128.55 0.0792310
$$898$$ 0 0
$$899$$ 30353.1i 1.12607i
$$900$$ 0 0
$$901$$ − 1759.53i − 0.0650592i
$$902$$ 0 0
$$903$$ 27827.4 1.02551
$$904$$ 0 0
$$905$$ 67845.2 2.49199
$$906$$ 0 0
$$907$$ − 5544.89i − 0.202993i −0.994836 0.101497i $$-0.967637\pi$$
0.994836 0.101497i $$-0.0323631\pi$$
$$908$$ 0 0
$$909$$ − 4486.59i − 0.163708i
$$910$$ 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
$$915$$ − 17906.4i − 0.646958i
$$916$$ 0 0
$$917$$ 13806.1i 0.497184i
$$918$$ 0 0
$$919$$ 22128.7 0.794297 0.397149 0.917754i $$-0.370000\pi$$
0.397149 + 0.917754i $$0.370000\pi$$
$$920$$ 0 0
$$921$$ 1244.01 0.0445077
$$922$$ 0 0
$$923$$ − 7533.72i − 0.268663i
$$924$$ 0 0
$$925$$ − 26944.9i − 0.957775i
$$926$$ 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
$$931$$ − 20079.5i − 0.706850i
$$932$$ 0 0
$$933$$ 4847.72i 0.170104i
$$934$$ 0 0
$$935$$ 39254.9 1.37302
$$936$$ 0 0
$$937$$ −441.299 −0.0153859 −0.00769297 0.999970i $$-0.502449\pi$$
−0.00769297 + 0.999970i $$0.502449\pi$$
$$938$$ 0 0
$$939$$ 25438.0i 0.884065i
$$940$$ 0 0
$$941$$ 5408.01i 0.187350i 0.995603 + 0.0936749i $$0.0298614\pi$$
−0.995603 + 0.0936749i $$0.970139\pi$$
$$942$$ 0 0
$$943$$ 50216.8 1.73413
$$944$$ 0 0
$$945$$ −11152.8 −0.383916
$$946$$ 0 0
$$947$$ 34237.1i 1.17482i 0.809289 + 0.587410i $$0.199852\pi$$
−0.809289 + 0.587410i $$0.800148\pi$$
$$948$$ 0 0
$$949$$ 7314.92i 0.250213i
$$950$$ 0 0
$$951$$ −20324.2 −0.693015
$$952$$ 0 0
$$953$$ −21410.7 −0.727766 −0.363883 0.931445i $$-0.618549\pi$$
−0.363883 + 0.931445i $$0.618549\pi$$
$$954$$ 0 0
$$955$$ 23572.2i 0.798722i
$$956$$ 0 0
$$957$$ 33147.5i 1.11965i
$$958$$ 0 0
$$959$$ 13751.3 0.463038
$$960$$ 0 0
$$961$$ −8038.09 −0.269816
$$962$$ 0 0
$$963$$ 3234.40i 0.108232i
$$964$$ 0 0
$$965$$ 89105.3i 2.97243i
$$966$$ 0 0
$$967$$ −53874.6 −1.79161 −0.895806 0.444445i $$-0.853401\pi$$
−0.895806 + 0.444445i $$0.853401\pi$$
$$968$$ 0 0
$$969$$ 14965.3 0.496136
$$970$$ 0 0
$$971$$ 42901.5i 1.41789i 0.705262 + 0.708947i $$0.250829\pi$$
−0.705262 + 0.708947i $$0.749171\pi$$
$$972$$ 0 0
$$973$$ 42586.2i 1.40314i
$$974$$ 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
$$979$$ − 20475.7i − 0.668444i
$$980$$ 0 0
$$981$$ 17727.6i 0.576961i
$$982$$ 0 0
$$983$$ −25296.7 −0.820793 −0.410396 0.911907i $$-0.634610\pi$$
−0.410396 + 0.911907i $$0.634610\pi$$
$$984$$ 0 0
$$985$$ −52859.0 −1.70988
$$986$$ 0 0
$$987$$ 21115.7i 0.680973i
$$988$$ 0 0
$$989$$ − 41019.6i − 1.31885i
$$990$$ 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
$$995$$ − 12015.7i − 0.382839i
$$996$$ 0 0
$$997$$ 5770.19i 0.183294i 0.995792 + 0.0916469i $$0.0292131\pi$$
−0.995792 + 0.0916469i $$0.970787\pi$$
$$998$$ 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 384.4.d.e.193.3 yes 4
3.2 odd 2 1152.4.d.o.577.4 4
4.3 odd 2 384.4.d.c.193.1 4
8.3 odd 2 384.4.d.c.193.4 yes 4
8.5 even 2 inner 384.4.d.e.193.2 yes 4
12.11 even 2 1152.4.d.i.577.4 4
16.3 odd 4 768.4.a.k.1.1 2
16.5 even 4 768.4.a.p.1.2 2
16.11 odd 4 768.4.a.j.1.2 2
16.13 even 4 768.4.a.e.1.1 2
24.5 odd 2 1152.4.d.o.577.1 4
24.11 even 2 1152.4.d.i.577.1 4
48.5 odd 4 2304.4.a.s.1.1 2
48.11 even 4 2304.4.a.t.1.1 2
48.29 odd 4 2304.4.a.bp.1.2 2
48.35 even 4 2304.4.a.bq.1.2 2

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