# Properties

 Label 384.4.d.c.193.1 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 193.1 Root $$-2.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.193 Dual form 384.4.d.c.193.4

## $q$-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 $$-0.706964\pi$$
−0.605343 + 0.795965i $$0.706964\pi$$
$$8$$ 0 0
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ − 53.6888i − 1.47162i −0.677190 0.735809i $$-0.736802\pi$$
0.677190 0.735809i $$-0.263198\pi$$
$$12$$ 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 $$0.274222\pi$$
−0.651306 + 0.758816i $$0.725778\pi$$
$$20$$ 0 0
$$21$$ 67.2666i 0.698989i
$$22$$ 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
$$27$$ 27.0000i 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 $$-0.359481\pi$$
0.427254 + 0.904132i $$0.359481\pi$$
$$32$$ 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 $$0.262147\pi$$
−0.679615 + 0.733569i $$0.737853\pi$$
$$44$$ 0 0
$$45$$ 165.800i 0.549244i
$$46$$ 0 0
$$47$$ −313.911 −0.974226 −0.487113 0.873339i $$-0.661950\pi$$
−0.487113 + 0.873339i $$0.661950\pi$$
$$48$$ 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 $$0.116360\pi$$
−0.933925 + 0.357468i $$0.883640\pi$$
$$60$$ 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 $$-0.860880\pi$$
0.906000 0.423277i $$-0.139120\pi$$
$$68$$ 0 0
$$69$$ 297.467i 0.518997i
$$70$$ 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 $$-0.805749\pi$$
−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 $$-0.641247\pi$$
−0.429321 + 0.903152i $$0.641247\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ − 15.8217i − 0.0209236i −0.999945 0.0104618i $$-0.996670\pi$$
0.999945 0.0104618i $$-0.00333016\pi$$
$$84$$ 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 $$-0.469989\pi$$
0.0941425 + 0.995559i $$0.469989\pi$$
$$104$$ 0 0
$$105$$ 1239.20 1.15175
$$106$$ 0 0
$$107$$ 359.378i 0.324695i 0.986734 + 0.162347i $$0.0519065\pi$$
−0.986734 + 0.162347i $$0.948093\pi$$
$$108$$ 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 $$-0.121396\pi$$
0.928153 + 0.372199i $$0.121396\pi$$
$$128$$ 0 0
$$129$$ 1241.07 0.847053
$$130$$ 0 0
$$131$$ − 615.734i − 0.410664i −0.978692 0.205332i $$-0.934173\pi$$
0.978692 0.205332i $$-0.0658274\pi$$
$$132$$ 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 $$-0.803256\pi$$
0.814987 0.579480i $$-0.196744\pi$$
$$140$$ 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 $$-0.558931\pi$$
−0.184081 + 0.982911i $$0.558931\pi$$
$$152$$ 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 $$-0.801930\pi$$
0.812566 0.582869i $$-0.198070\pi$$
$$164$$ 0 0
$$165$$ 2967.20i 1.39998i
$$166$$ 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
$$171$$ − 1131.20i − 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 $$-0.792379\pi$$
0.794713 0.606986i $$-0.207621\pi$$
$$180$$ 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 $$-0.422075\pi$$
0.242370 + 0.970184i $$0.422075\pi$$
$$192$$ 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 $$-0.537062\pi$$
−0.116171 + 0.993229i $$0.537062\pi$$
$$200$$ 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 $$-0.972053\pi$$
0.996148 0.0876866i $$-0.0279474\pi$$
$$212$$ 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 $$-0.707581\pi$$
−0.606885 + 0.794790i $$0.707581\pi$$
$$224$$ 0 0
$$225$$ 1929.40 0.571674
$$226$$ 0 0
$$227$$ − 3070.22i − 0.897699i −0.893607 0.448850i $$-0.851834\pi$$
0.893607 0.448850i $$-0.148166\pi$$
$$228$$ 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 $$-0.300828\pi$$
0.585679 + 0.810543i $$0.300828\pi$$
$$240$$ 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 $$-0.790157\pi$$
0.790456 0.612518i $$-0.209843\pi$$
$$252$$ 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 $$-0.817596\pi$$
−0.840258 + 0.542188i $$0.817596\pi$$
$$264$$ 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 $$-0.707640\pi$$
−0.607031 + 0.794678i $$0.707640\pi$$
$$272$$ 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 $$-0.798585\pi$$
0.806396 0.591376i $$-0.201415\pi$$
$$284$$ 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 $$0.0122722\pi$$
−0.999257 + 0.0385448i $$0.987728\pi$$
$$308$$ 0 0
$$309$$ − 590.463i − 0.108706i
$$310$$ 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 $$-0.222431\pi$$
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 $$0.280509\pi$$
−0.636191 + 0.771532i $$0.719491\pi$$
$$332$$ 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 $$-0.903204\pi$$
0.954119 0.299427i $$-0.0967956\pi$$
$$348$$ 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 $$-0.295205\pi$$
0.599904 + 0.800072i $$0.295205\pi$$
$$360$$ 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 $$-0.398036\pi$$
0.314880 + 0.949132i $$0.398036\pi$$
$$368$$ 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 $$0.0153142\pi$$
−0.998843 + 0.0480923i $$0.984686\pi$$
$$380$$ 0 0
$$381$$ − 7970.33i − 1.07174i
$$382$$ 0 0
$$383$$ −1233.69 −0.164592 −0.0822962 0.996608i $$-0.526225\pi$$
−0.0822962 + 0.996608i $$0.526225\pi$$
$$384$$ 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 $$-0.645307\pi$$
0.440806 0.897602i $$-0.354693\pi$$
$$420$$ 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 $$-0.589096\pi$$
−0.276261 + 0.961083i $$0.589096\pi$$
$$432$$ 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 $$-0.578554\pi$$
−0.244286 + 0.969703i $$0.578554\pi$$
$$440$$ 0 0
$$441$$ −1437.80 −0.155253
$$442$$ 0 0
$$443$$ 4292.26i 0.460341i 0.973150 + 0.230171i $$0.0739285\pi$$
−0.973150 + 0.230171i $$0.926072\pi$$
$$444$$ 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 $$-0.731586\pi$$
−0.665041 + 0.746807i $$0.731586\pi$$
$$464$$ 0 0
$$465$$ −8151.20 −0.812909
$$466$$ 0 0
$$467$$ − 4432.00i − 0.439161i −0.975594 0.219581i $$-0.929531\pi$$
0.975594 0.219581i $$-0.0704689\pi$$
$$468$$ 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 $$-0.357491\pi$$
0.432897 + 0.901443i $$0.357491\pi$$
$$480$$ 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 $$-0.450278\pi$$
0.155571 + 0.987825i $$0.450278\pi$$
$$488$$ 0 0
$$489$$ −7277.86 −0.673040
$$490$$ 0 0
$$491$$ 2423.73i 0.222773i 0.993777 + 0.111386i $$0.0355291\pi$$
−0.993777 + 0.111386i $$0.964471\pi$$
$$492$$ 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 $$0.0115938\pi$$
−0.999337 + 0.0364148i $$0.988406\pi$$
$$500$$ 0 0
$$501$$ 1012.00i 0.0902449i
$$502$$ 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
$$507$$ − 6437.39i − 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 $$0.146274\pi$$
−0.896259 + 0.443531i $$0.853726\pi$$
$$524$$ 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 $$0.0934911\pi$$
−0.957176 + 0.289506i $$0.906509\pi$$
$$548$$ 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 $$0.0322259\pi$$
−0.994880 + 0.101068i $$0.967774\pi$$
$$564$$ 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 $$-0.767070\pi$$
0.743992 0.668188i $$-0.232930\pi$$
$$572$$ 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 $$0.0181052\pi$$
−0.998383 + 0.0568485i $$0.981895\pi$$
$$588$$ 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 $$-0.605448\pi$$
−0.325248 + 0.945629i $$0.605448\pi$$
$$600$$ 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 $$-0.122395\pi$$
0.926980 + 0.375111i $$0.122395\pi$$
$$608$$ 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 $$-0.966764\pi$$
0.994554 0.104224i $$-0.0332360\pi$$
$$620$$ 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 $$-0.377391\pi$$
0.375734 + 0.926728i $$0.377391\pi$$
$$632$$ 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 $$-0.872084\pi$$
0.920335 0.391131i $$-0.127916\pi$$
$$644$$ 0 0
$$645$$ − 22863.2i − 1.39572i
$$646$$ 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
$$651$$ 9921.06i 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 $$0.0616105\pi$$
−0.981327 + 0.192349i $$0.938389\pi$$
$$660$$ 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 $$0.0508243\pi$$
−0.987280 + 0.158992i $$0.949176\pi$$
$$684$$ 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 $$-0.968248\pi$$
0.995029 0.0995854i $$-0.0317517\pi$$
$$692$$ 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 $$-0.0964190\pi$$
0.954473 + 0.298298i $$0.0964190\pi$$
$$720$$ 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 $$-0.757103\pi$$
−0.722709 + 0.691152i $$0.757103\pi$$
$$728$$ 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 $$-0.858298\pi$$
0.902538 0.430610i $$-0.141702\pi$$
$$740$$ 0 0
$$741$$ − 2698.13i − 0.133763i
$$742$$ 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
$$747$$ 142.396i 0.00697455i
$$748$$ 0 0
$$749$$ − 8058.04i − 0.393103i
$$750$$ 0 0
$$751$$ 30052.8 1.46024 0.730121 0.683318i $$-0.239464\pi$$
0.730121 + 0.683318i $$0.239464\pi$$
$$752$$ 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 $$0.0351036\pi$$
−0.993925 + 0.110058i $$0.964896\pi$$
$$788$$ 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 $$-0.858414\pi$$
0.902694 0.430283i $$-0.141586\pi$$
$$812$$ 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 $$-0.545305\pi$$
−0.141849 + 0.989888i $$0.545305\pi$$
$$824$$ 0 0
$$825$$ 34529.0 1.45715
$$826$$ 0 0
$$827$$ 3390.85i 0.142577i 0.997456 + 0.0712886i $$0.0227111\pi$$
−0.997456 + 0.0712886i $$0.977289\pi$$
$$828$$ 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 $$-0.459469\pi$$
0.126988 + 0.991904i $$0.459469\pi$$
$$840$$ 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 $$-0.820484\pi$$
0.845141 0.534544i $$-0.179516\pi$$
$$860$$ 0 0
$$861$$ 34066.8i 1.34842i
$$862$$ 0 0
$$863$$ −23623.9 −0.931828 −0.465914 0.884830i $$-0.654274\pi$$
−0.465914 + 0.884830i $$0.654274\pi$$
$$864$$ 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 $$0.215583\pi$$
−0.779283 + 0.626672i $$0.784417\pi$$
$$884$$ 0 0
$$885$$ − 17906.4i − 0.680132i
$$886$$ 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
$$891$$ − 4348.79i − 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 $$0.0323631\pi$$
−0.994836 + 0.101497i $$0.967637\pi$$
$$908$$ 0 0
$$909$$ − 4486.59i − 0.163708i
$$910$$ 0 0
$$911$$ 19638.9 0.714233 0.357116 0.934060i $$-0.383760\pi$$
0.357116 + 0.934060i $$0.383760\pi$$
$$912$$ 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 $$-0.630000\pi$$
−0.397149 + 0.917754i $$0.630000\pi$$
$$920$$ 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 $$-0.800148\pi$$
0.809289 0.587410i $$-0.199852\pi$$
$$948$$ 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 $$-0.146599\pi$$
0.895806 + 0.444445i $$0.146599\pi$$
$$968$$ 0 0
$$969$$ 14965.3 0.496136
$$970$$ 0 0
$$971$$ − 42901.5i − 1.41789i −0.705262 0.708947i $$-0.749171\pi$$
0.705262 0.708947i $$-0.250829\pi$$
$$972$$ 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 $$-0.365390\pi$$
0.410396 + 0.911907i $$0.365390\pi$$
$$984$$ 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 $$-0.445632\pi$$
0.169972 + 0.985449i $$0.445632\pi$$
$$992$$ 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.c.193.1 4
3.2 odd 2 1152.4.d.i.577.4 4
4.3 odd 2 384.4.d.e.193.3 yes 4
8.3 odd 2 384.4.d.e.193.2 yes 4
8.5 even 2 inner 384.4.d.c.193.4 yes 4
12.11 even 2 1152.4.d.o.577.4 4
16.3 odd 4 768.4.a.e.1.1 2
16.5 even 4 768.4.a.j.1.2 2
16.11 odd 4 768.4.a.p.1.2 2
16.13 even 4 768.4.a.k.1.1 2
24.5 odd 2 1152.4.d.i.577.1 4
24.11 even 2 1152.4.d.o.577.1 4
48.5 odd 4 2304.4.a.t.1.1 2
48.11 even 4 2304.4.a.s.1.1 2
48.29 odd 4 2304.4.a.bq.1.2 2
48.35 even 4 2304.4.a.bp.1.2 2

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