# Properties

 Label 4840.2.a.ba.1.6 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.25903625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5$$ x^6 - 3*x^5 - 7*x^4 + 17*x^3 + 16*x^2 - 20*x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$-1.71280$$ of defining polynomial Character $$\chi$$ $$=$$ 4840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.71280 q^{3} -1.00000 q^{5} -3.70505 q^{7} -0.0663151 q^{9} +O(q^{10})$$ $$q+1.71280 q^{3} -1.00000 q^{5} -3.70505 q^{7} -0.0663151 q^{9} +1.17750 q^{13} -1.71280 q^{15} +5.40027 q^{17} +3.84752 q^{19} -6.34602 q^{21} -8.68237 q^{23} +1.00000 q^{25} -5.25199 q^{27} +6.87597 q^{29} +8.75541 q^{31} +3.70505 q^{35} +2.25468 q^{37} +2.01682 q^{39} -1.59733 q^{41} -4.11979 q^{43} +0.0663151 q^{45} -12.6800 q^{47} +6.72743 q^{49} +9.24959 q^{51} -12.3795 q^{53} +6.59004 q^{57} -0.393999 q^{59} -1.80085 q^{61} +0.245701 q^{63} -1.17750 q^{65} -14.3809 q^{67} -14.8712 q^{69} -6.85527 q^{71} -10.0551 q^{73} +1.71280 q^{75} +1.11383 q^{79} -8.79666 q^{81} -0.0173212 q^{83} -5.40027 q^{85} +11.7772 q^{87} -8.49434 q^{89} -4.36270 q^{91} +14.9963 q^{93} -3.84752 q^{95} +10.5155 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9}+O(q^{10})$$ 6 * q - 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 $$6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} + q^{13} + 3 q^{15} + 6 q^{17} - 7 q^{19} + 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} + 10 q^{29} + q^{31} + 7 q^{35} + 3 q^{37} - q^{39} - 6 q^{41} + 18 q^{43} - 5 q^{45} - 3 q^{47} + 17 q^{49} + 15 q^{51} - 23 q^{53} + 9 q^{57} - 2 q^{59} + 6 q^{61} - 49 q^{63} - q^{65} - 22 q^{67} + 2 q^{69} - 13 q^{71} + 10 q^{73} - 3 q^{75} - 22 q^{79} + 10 q^{81} + 10 q^{83} - 6 q^{85} - 3 q^{87} - 25 q^{89} + 12 q^{91} + 19 q^{93} + 7 q^{95} - 33 q^{97}+O(q^{100})$$ 6 * q - 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 + q^13 + 3 * q^15 + 6 * q^17 - 7 * q^19 + 14 * q^21 - 9 * q^23 + 6 * q^25 - 21 * q^27 + 10 * q^29 + q^31 + 7 * q^35 + 3 * q^37 - q^39 - 6 * q^41 + 18 * q^43 - 5 * q^45 - 3 * q^47 + 17 * q^49 + 15 * q^51 - 23 * q^53 + 9 * q^57 - 2 * q^59 + 6 * q^61 - 49 * q^63 - q^65 - 22 * q^67 + 2 * q^69 - 13 * q^71 + 10 * q^73 - 3 * q^75 - 22 * q^79 + 10 * q^81 + 10 * q^83 - 6 * q^85 - 3 * q^87 - 25 * q^89 + 12 * q^91 + 19 * q^93 + 7 * q^95 - 33 * q^97

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.71280 0.988886 0.494443 0.869210i $$-0.335372\pi$$
0.494443 + 0.869210i $$0.335372\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −3.70505 −1.40038 −0.700189 0.713957i $$-0.746901\pi$$
−0.700189 + 0.713957i $$0.746901\pi$$
$$8$$ 0 0
$$9$$ −0.0663151 −0.0221050
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 1.17750 0.326580 0.163290 0.986578i $$-0.447789\pi$$
0.163290 + 0.986578i $$0.447789\pi$$
$$14$$ 0 0
$$15$$ −1.71280 −0.442243
$$16$$ 0 0
$$17$$ 5.40027 1.30976 0.654879 0.755734i $$-0.272720\pi$$
0.654879 + 0.755734i $$0.272720\pi$$
$$18$$ 0 0
$$19$$ 3.84752 0.882682 0.441341 0.897339i $$-0.354503\pi$$
0.441341 + 0.897339i $$0.354503\pi$$
$$20$$ 0 0
$$21$$ −6.34602 −1.38481
$$22$$ 0 0
$$23$$ −8.68237 −1.81040 −0.905200 0.424986i $$-0.860279\pi$$
−0.905200 + 0.424986i $$0.860279\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.25199 −1.01075
$$28$$ 0 0
$$29$$ 6.87597 1.27684 0.638418 0.769690i $$-0.279589\pi$$
0.638418 + 0.769690i $$0.279589\pi$$
$$30$$ 0 0
$$31$$ 8.75541 1.57252 0.786259 0.617898i $$-0.212015\pi$$
0.786259 + 0.617898i $$0.212015\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 3.70505 0.626268
$$36$$ 0 0
$$37$$ 2.25468 0.370667 0.185334 0.982676i $$-0.440663\pi$$
0.185334 + 0.982676i $$0.440663\pi$$
$$38$$ 0 0
$$39$$ 2.01682 0.322950
$$40$$ 0 0
$$41$$ −1.59733 −0.249461 −0.124730 0.992191i $$-0.539807\pi$$
−0.124730 + 0.992191i $$0.539807\pi$$
$$42$$ 0 0
$$43$$ −4.11979 −0.628262 −0.314131 0.949380i $$-0.601713\pi$$
−0.314131 + 0.949380i $$0.601713\pi$$
$$44$$ 0 0
$$45$$ 0.0663151 0.00988567
$$46$$ 0 0
$$47$$ −12.6800 −1.84956 −0.924782 0.380497i $$-0.875753\pi$$
−0.924782 + 0.380497i $$0.875753\pi$$
$$48$$ 0 0
$$49$$ 6.72743 0.961061
$$50$$ 0 0
$$51$$ 9.24959 1.29520
$$52$$ 0 0
$$53$$ −12.3795 −1.70046 −0.850229 0.526413i $$-0.823536\pi$$
−0.850229 + 0.526413i $$0.823536\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.59004 0.872872
$$58$$ 0 0
$$59$$ −0.393999 −0.0512944 −0.0256472 0.999671i $$-0.508165\pi$$
−0.0256472 + 0.999671i $$0.508165\pi$$
$$60$$ 0 0
$$61$$ −1.80085 −0.230575 −0.115288 0.993332i $$-0.536779\pi$$
−0.115288 + 0.993332i $$0.536779\pi$$
$$62$$ 0 0
$$63$$ 0.245701 0.0309554
$$64$$ 0 0
$$65$$ −1.17750 −0.146051
$$66$$ 0 0
$$67$$ −14.3809 −1.75691 −0.878455 0.477826i $$-0.841425\pi$$
−0.878455 + 0.477826i $$0.841425\pi$$
$$68$$ 0 0
$$69$$ −14.8712 −1.79028
$$70$$ 0 0
$$71$$ −6.85527 −0.813571 −0.406785 0.913524i $$-0.633350\pi$$
−0.406785 + 0.913524i $$0.633350\pi$$
$$72$$ 0 0
$$73$$ −10.0551 −1.17686 −0.588430 0.808548i $$-0.700254\pi$$
−0.588430 + 0.808548i $$0.700254\pi$$
$$74$$ 0 0
$$75$$ 1.71280 0.197777
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1.11383 0.125316 0.0626580 0.998035i $$-0.480042\pi$$
0.0626580 + 0.998035i $$0.480042\pi$$
$$80$$ 0 0
$$81$$ −8.79666 −0.977406
$$82$$ 0 0
$$83$$ −0.0173212 −0.00190125 −0.000950626 1.00000i $$-0.500303\pi$$
−0.000950626 1.00000i $$0.500303\pi$$
$$84$$ 0 0
$$85$$ −5.40027 −0.585742
$$86$$ 0 0
$$87$$ 11.7772 1.26265
$$88$$ 0 0
$$89$$ −8.49434 −0.900399 −0.450199 0.892928i $$-0.648647\pi$$
−0.450199 + 0.892928i $$0.648647\pi$$
$$90$$ 0 0
$$91$$ −4.36270 −0.457335
$$92$$ 0 0
$$93$$ 14.9963 1.55504
$$94$$ 0 0
$$95$$ −3.84752 −0.394747
$$96$$ 0 0
$$97$$ 10.5155 1.06768 0.533842 0.845584i $$-0.320748\pi$$
0.533842 + 0.845584i $$0.320748\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −7.17862 −0.714299 −0.357150 0.934047i $$-0.616251\pi$$
−0.357150 + 0.934047i $$0.616251\pi$$
$$102$$ 0 0
$$103$$ −3.75016 −0.369515 −0.184757 0.982784i $$-0.559150\pi$$
−0.184757 + 0.982784i $$0.559150\pi$$
$$104$$ 0 0
$$105$$ 6.34602 0.619308
$$106$$ 0 0
$$107$$ 6.35128 0.614001 0.307001 0.951709i $$-0.400675\pi$$
0.307001 + 0.951709i $$0.400675\pi$$
$$108$$ 0 0
$$109$$ −1.32407 −0.126823 −0.0634115 0.997987i $$-0.520198\pi$$
−0.0634115 + 0.997987i $$0.520198\pi$$
$$110$$ 0 0
$$111$$ 3.86182 0.366548
$$112$$ 0 0
$$113$$ 5.33169 0.501563 0.250782 0.968044i $$-0.419312\pi$$
0.250782 + 0.968044i $$0.419312\pi$$
$$114$$ 0 0
$$115$$ 8.68237 0.809636
$$116$$ 0 0
$$117$$ −0.0780860 −0.00721905
$$118$$ 0 0
$$119$$ −20.0083 −1.83416
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −2.73591 −0.246688
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −14.6406 −1.29914 −0.649571 0.760301i $$-0.725052\pi$$
−0.649571 + 0.760301i $$0.725052\pi$$
$$128$$ 0 0
$$129$$ −7.05637 −0.621279
$$130$$ 0 0
$$131$$ −5.93922 −0.518912 −0.259456 0.965755i $$-0.583543\pi$$
−0.259456 + 0.965755i $$0.583543\pi$$
$$132$$ 0 0
$$133$$ −14.2553 −1.23609
$$134$$ 0 0
$$135$$ 5.25199 0.452019
$$136$$ 0 0
$$137$$ 14.7698 1.26187 0.630934 0.775837i $$-0.282672\pi$$
0.630934 + 0.775837i $$0.282672\pi$$
$$138$$ 0 0
$$139$$ −1.29252 −0.109630 −0.0548149 0.998497i $$-0.517457\pi$$
−0.0548149 + 0.998497i $$0.517457\pi$$
$$140$$ 0 0
$$141$$ −21.7183 −1.82901
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −6.87597 −0.571018
$$146$$ 0 0
$$147$$ 11.5227 0.950379
$$148$$ 0 0
$$149$$ −23.9060 −1.95846 −0.979229 0.202757i $$-0.935010\pi$$
−0.979229 + 0.202757i $$0.935010\pi$$
$$150$$ 0 0
$$151$$ −8.28048 −0.673856 −0.336928 0.941530i $$-0.609388\pi$$
−0.336928 + 0.941530i $$0.609388\pi$$
$$152$$ 0 0
$$153$$ −0.358119 −0.0289522
$$154$$ 0 0
$$155$$ −8.75541 −0.703251
$$156$$ 0 0
$$157$$ −7.96444 −0.635631 −0.317816 0.948153i $$-0.602949\pi$$
−0.317816 + 0.948153i $$0.602949\pi$$
$$158$$ 0 0
$$159$$ −21.2036 −1.68156
$$160$$ 0 0
$$161$$ 32.1687 2.53525
$$162$$ 0 0
$$163$$ −4.08778 −0.320180 −0.160090 0.987102i $$-0.551178\pi$$
−0.160090 + 0.987102i $$0.551178\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3.98229 0.308159 0.154079 0.988058i $$-0.450759\pi$$
0.154079 + 0.988058i $$0.450759\pi$$
$$168$$ 0 0
$$169$$ −11.6135 −0.893346
$$170$$ 0 0
$$171$$ −0.255149 −0.0195117
$$172$$ 0 0
$$173$$ −17.6573 −1.34246 −0.671230 0.741250i $$-0.734234\pi$$
−0.671230 + 0.741250i $$0.734234\pi$$
$$174$$ 0 0
$$175$$ −3.70505 −0.280076
$$176$$ 0 0
$$177$$ −0.674842 −0.0507243
$$178$$ 0 0
$$179$$ 11.9417 0.892562 0.446281 0.894893i $$-0.352748\pi$$
0.446281 + 0.894893i $$0.352748\pi$$
$$180$$ 0 0
$$181$$ −1.03171 −0.0766863 −0.0383431 0.999265i $$-0.512208\pi$$
−0.0383431 + 0.999265i $$0.512208\pi$$
$$182$$ 0 0
$$183$$ −3.08450 −0.228013
$$184$$ 0 0
$$185$$ −2.25468 −0.165767
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 19.4589 1.41543
$$190$$ 0 0
$$191$$ 19.0153 1.37590 0.687950 0.725758i $$-0.258511\pi$$
0.687950 + 0.725758i $$0.258511\pi$$
$$192$$ 0 0
$$193$$ −5.40874 −0.389329 −0.194665 0.980870i $$-0.562362\pi$$
−0.194665 + 0.980870i $$0.562362\pi$$
$$194$$ 0 0
$$195$$ −2.01682 −0.144428
$$196$$ 0 0
$$197$$ −4.07618 −0.290416 −0.145208 0.989401i $$-0.546385\pi$$
−0.145208 + 0.989401i $$0.546385\pi$$
$$198$$ 0 0
$$199$$ −12.0711 −0.855700 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$200$$ 0 0
$$201$$ −24.6316 −1.73738
$$202$$ 0 0
$$203$$ −25.4759 −1.78805
$$204$$ 0 0
$$205$$ 1.59733 0.111562
$$206$$ 0 0
$$207$$ 0.575772 0.0400190
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 9.31289 0.641126 0.320563 0.947227i $$-0.396128\pi$$
0.320563 + 0.947227i $$0.396128\pi$$
$$212$$ 0 0
$$213$$ −11.7417 −0.804528
$$214$$ 0 0
$$215$$ 4.11979 0.280967
$$216$$ 0 0
$$217$$ −32.4392 −2.20212
$$218$$ 0 0
$$219$$ −17.2224 −1.16378
$$220$$ 0 0
$$221$$ 6.35881 0.427740
$$222$$ 0 0
$$223$$ −4.74364 −0.317658 −0.158829 0.987306i $$-0.550772\pi$$
−0.158829 + 0.987306i $$0.550772\pi$$
$$224$$ 0 0
$$225$$ −0.0663151 −0.00442101
$$226$$ 0 0
$$227$$ −3.51523 −0.233314 −0.116657 0.993172i $$-0.537218\pi$$
−0.116657 + 0.993172i $$0.537218\pi$$
$$228$$ 0 0
$$229$$ −9.12613 −0.603072 −0.301536 0.953455i $$-0.597499\pi$$
−0.301536 + 0.953455i $$0.597499\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 21.5251 1.41016 0.705079 0.709128i $$-0.250911\pi$$
0.705079 + 0.709128i $$0.250911\pi$$
$$234$$ 0 0
$$235$$ 12.6800 0.827150
$$236$$ 0 0
$$237$$ 1.90777 0.123923
$$238$$ 0 0
$$239$$ −11.3074 −0.731417 −0.365709 0.930729i $$-0.619173\pi$$
−0.365709 + 0.930729i $$0.619173\pi$$
$$240$$ 0 0
$$241$$ 26.1144 1.68218 0.841089 0.540897i $$-0.181915\pi$$
0.841089 + 0.540897i $$0.181915\pi$$
$$242$$ 0 0
$$243$$ 0.689039 0.0442019
$$244$$ 0 0
$$245$$ −6.72743 −0.429799
$$246$$ 0 0
$$247$$ 4.53045 0.288266
$$248$$ 0 0
$$249$$ −0.0296678 −0.00188012
$$250$$ 0 0
$$251$$ −3.71812 −0.234685 −0.117343 0.993091i $$-0.537438\pi$$
−0.117343 + 0.993091i $$0.537438\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −9.24959 −0.579231
$$256$$ 0 0
$$257$$ −7.09278 −0.442436 −0.221218 0.975224i $$-0.571003\pi$$
−0.221218 + 0.975224i $$0.571003\pi$$
$$258$$ 0 0
$$259$$ −8.35372 −0.519075
$$260$$ 0 0
$$261$$ −0.455981 −0.0282245
$$262$$ 0 0
$$263$$ −23.3578 −1.44030 −0.720151 0.693817i $$-0.755927\pi$$
−0.720151 + 0.693817i $$0.755927\pi$$
$$264$$ 0 0
$$265$$ 12.3795 0.760468
$$266$$ 0 0
$$267$$ −14.5491 −0.890391
$$268$$ 0 0
$$269$$ −17.0818 −1.04149 −0.520746 0.853711i $$-0.674346\pi$$
−0.520746 + 0.853711i $$0.674346\pi$$
$$270$$ 0 0
$$271$$ 27.4343 1.66652 0.833258 0.552885i $$-0.186473\pi$$
0.833258 + 0.552885i $$0.186473\pi$$
$$272$$ 0 0
$$273$$ −7.47243 −0.452252
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 24.2250 1.45554 0.727771 0.685820i $$-0.240556\pi$$
0.727771 + 0.685820i $$0.240556\pi$$
$$278$$ 0 0
$$279$$ −0.580616 −0.0347605
$$280$$ 0 0
$$281$$ 17.1634 1.02388 0.511941 0.859020i $$-0.328927\pi$$
0.511941 + 0.859020i $$0.328927\pi$$
$$282$$ 0 0
$$283$$ −4.62908 −0.275170 −0.137585 0.990490i $$-0.543934\pi$$
−0.137585 + 0.990490i $$0.543934\pi$$
$$284$$ 0 0
$$285$$ −6.59004 −0.390360
$$286$$ 0 0
$$287$$ 5.91819 0.349340
$$288$$ 0 0
$$289$$ 12.1629 0.715466
$$290$$ 0 0
$$291$$ 18.0109 1.05582
$$292$$ 0 0
$$293$$ 18.5006 1.08082 0.540409 0.841402i $$-0.318270\pi$$
0.540409 + 0.841402i $$0.318270\pi$$
$$294$$ 0 0
$$295$$ 0.393999 0.0229395
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −10.2235 −0.591240
$$300$$ 0 0
$$301$$ 15.2640 0.879804
$$302$$ 0 0
$$303$$ −12.2955 −0.706360
$$304$$ 0 0
$$305$$ 1.80085 0.103116
$$306$$ 0 0
$$307$$ 27.2021 1.55250 0.776252 0.630422i $$-0.217118\pi$$
0.776252 + 0.630422i $$0.217118\pi$$
$$308$$ 0 0
$$309$$ −6.42328 −0.365408
$$310$$ 0 0
$$311$$ 32.2598 1.82929 0.914644 0.404260i $$-0.132471\pi$$
0.914644 + 0.404260i $$0.132471\pi$$
$$312$$ 0 0
$$313$$ −0.290748 −0.0164341 −0.00821704 0.999966i $$-0.502616\pi$$
−0.00821704 + 0.999966i $$0.502616\pi$$
$$314$$ 0 0
$$315$$ −0.245701 −0.0138437
$$316$$ 0 0
$$317$$ 22.8346 1.28252 0.641258 0.767325i $$-0.278413\pi$$
0.641258 + 0.767325i $$0.278413\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 10.8785 0.607177
$$322$$ 0 0
$$323$$ 20.7777 1.15610
$$324$$ 0 0
$$325$$ 1.17750 0.0653159
$$326$$ 0 0
$$327$$ −2.26787 −0.125413
$$328$$ 0 0
$$329$$ 46.9800 2.59009
$$330$$ 0 0
$$331$$ −14.5262 −0.798433 −0.399217 0.916857i $$-0.630718\pi$$
−0.399217 + 0.916857i $$0.630718\pi$$
$$332$$ 0 0
$$333$$ −0.149519 −0.00819361
$$334$$ 0 0
$$335$$ 14.3809 0.785714
$$336$$ 0 0
$$337$$ −14.0343 −0.764498 −0.382249 0.924059i $$-0.624850\pi$$
−0.382249 + 0.924059i $$0.624850\pi$$
$$338$$ 0 0
$$339$$ 9.13212 0.495989
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00990 0.0545297
$$344$$ 0 0
$$345$$ 14.8712 0.800637
$$346$$ 0 0
$$347$$ 27.2726 1.46407 0.732034 0.681268i $$-0.238571\pi$$
0.732034 + 0.681268i $$0.238571\pi$$
$$348$$ 0 0
$$349$$ 24.9117 1.33349 0.666747 0.745284i $$-0.267686\pi$$
0.666747 + 0.745284i $$0.267686\pi$$
$$350$$ 0 0
$$351$$ −6.18421 −0.330089
$$352$$ 0 0
$$353$$ −22.2916 −1.18646 −0.593232 0.805031i $$-0.702148\pi$$
−0.593232 + 0.805031i $$0.702148\pi$$
$$354$$ 0 0
$$355$$ 6.85527 0.363840
$$356$$ 0 0
$$357$$ −34.2702 −1.81377
$$358$$ 0 0
$$359$$ −1.86792 −0.0985850 −0.0492925 0.998784i $$-0.515697\pi$$
−0.0492925 + 0.998784i $$0.515697\pi$$
$$360$$ 0 0
$$361$$ −4.19658 −0.220872
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 10.0551 0.526308
$$366$$ 0 0
$$367$$ 2.57693 0.134515 0.0672574 0.997736i $$-0.478575\pi$$
0.0672574 + 0.997736i $$0.478575\pi$$
$$368$$ 0 0
$$369$$ 0.105927 0.00551434
$$370$$ 0 0
$$371$$ 45.8668 2.38128
$$372$$ 0 0
$$373$$ −7.12031 −0.368675 −0.184338 0.982863i $$-0.559014\pi$$
−0.184338 + 0.982863i $$0.559014\pi$$
$$374$$ 0 0
$$375$$ −1.71280 −0.0884486
$$376$$ 0 0
$$377$$ 8.09645 0.416989
$$378$$ 0 0
$$379$$ −28.5424 −1.46612 −0.733061 0.680163i $$-0.761909\pi$$
−0.733061 + 0.680163i $$0.761909\pi$$
$$380$$ 0 0
$$381$$ −25.0764 −1.28470
$$382$$ 0 0
$$383$$ −9.33065 −0.476774 −0.238387 0.971170i $$-0.576619\pi$$
−0.238387 + 0.971170i $$0.576619\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0.273204 0.0138877
$$388$$ 0 0
$$389$$ 0.444940 0.0225594 0.0112797 0.999936i $$-0.496409\pi$$
0.0112797 + 0.999936i $$0.496409\pi$$
$$390$$ 0 0
$$391$$ −46.8872 −2.37119
$$392$$ 0 0
$$393$$ −10.1727 −0.513145
$$394$$ 0 0
$$395$$ −1.11383 −0.0560431
$$396$$ 0 0
$$397$$ 1.66941 0.0837850 0.0418925 0.999122i $$-0.486661\pi$$
0.0418925 + 0.999122i $$0.486661\pi$$
$$398$$ 0 0
$$399$$ −24.4164 −1.22235
$$400$$ 0 0
$$401$$ 33.8566 1.69072 0.845360 0.534197i $$-0.179386\pi$$
0.845360 + 0.534197i $$0.179386\pi$$
$$402$$ 0 0
$$403$$ 10.3095 0.513552
$$404$$ 0 0
$$405$$ 8.79666 0.437109
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 21.7117 1.07357 0.536787 0.843718i $$-0.319638\pi$$
0.536787 + 0.843718i $$0.319638\pi$$
$$410$$ 0 0
$$411$$ 25.2977 1.24784
$$412$$ 0 0
$$413$$ 1.45979 0.0718315
$$414$$ 0 0
$$415$$ 0.0173212 0.000850265 0
$$416$$ 0 0
$$417$$ −2.21382 −0.108411
$$418$$ 0 0
$$419$$ −26.5502 −1.29706 −0.648532 0.761187i $$-0.724617\pi$$
−0.648532 + 0.761187i $$0.724617\pi$$
$$420$$ 0 0
$$421$$ −10.4845 −0.510984 −0.255492 0.966811i $$-0.582237\pi$$
−0.255492 + 0.966811i $$0.582237\pi$$
$$422$$ 0 0
$$423$$ 0.840874 0.0408847
$$424$$ 0 0
$$425$$ 5.40027 0.261952
$$426$$ 0 0
$$427$$ 6.67225 0.322893
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −5.41218 −0.260696 −0.130348 0.991468i $$-0.541609\pi$$
−0.130348 + 0.991468i $$0.541609\pi$$
$$432$$ 0 0
$$433$$ 9.08123 0.436416 0.218208 0.975902i $$-0.429979\pi$$
0.218208 + 0.975902i $$0.429979\pi$$
$$434$$ 0 0
$$435$$ −11.7772 −0.564672
$$436$$ 0 0
$$437$$ −33.4056 −1.59801
$$438$$ 0 0
$$439$$ −17.3683 −0.828945 −0.414473 0.910062i $$-0.636034\pi$$
−0.414473 + 0.910062i $$0.636034\pi$$
$$440$$ 0 0
$$441$$ −0.446130 −0.0212443
$$442$$ 0 0
$$443$$ −13.6634 −0.649170 −0.324585 0.945857i $$-0.605225\pi$$
−0.324585 + 0.945857i $$0.605225\pi$$
$$444$$ 0 0
$$445$$ 8.49434 0.402671
$$446$$ 0 0
$$447$$ −40.9463 −1.93669
$$448$$ 0 0
$$449$$ −15.0222 −0.708942 −0.354471 0.935067i $$-0.615339\pi$$
−0.354471 + 0.935067i $$0.615339\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −14.1828 −0.666367
$$454$$ 0 0
$$455$$ 4.36270 0.204526
$$456$$ 0 0
$$457$$ 2.62422 0.122756 0.0613779 0.998115i $$-0.480451\pi$$
0.0613779 + 0.998115i $$0.480451\pi$$
$$458$$ 0 0
$$459$$ −28.3621 −1.32383
$$460$$ 0 0
$$461$$ −39.2715 −1.82906 −0.914529 0.404521i $$-0.867438\pi$$
−0.914529 + 0.404521i $$0.867438\pi$$
$$462$$ 0 0
$$463$$ −3.26421 −0.151701 −0.0758505 0.997119i $$-0.524167\pi$$
−0.0758505 + 0.997119i $$0.524167\pi$$
$$464$$ 0 0
$$465$$ −14.9963 −0.695435
$$466$$ 0 0
$$467$$ −25.1521 −1.16390 −0.581949 0.813225i $$-0.697710\pi$$
−0.581949 + 0.813225i $$0.697710\pi$$
$$468$$ 0 0
$$469$$ 53.2821 2.46034
$$470$$ 0 0
$$471$$ −13.6415 −0.628567
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 3.84752 0.176536
$$476$$ 0 0
$$477$$ 0.820949 0.0375887
$$478$$ 0 0
$$479$$ −29.8987 −1.36611 −0.683053 0.730369i $$-0.739348\pi$$
−0.683053 + 0.730369i $$0.739348\pi$$
$$480$$ 0 0
$$481$$ 2.65489 0.121052
$$482$$ 0 0
$$483$$ 55.0985 2.50707
$$484$$ 0 0
$$485$$ −10.5155 −0.477483
$$486$$ 0 0
$$487$$ 31.2658 1.41679 0.708393 0.705818i $$-0.249420\pi$$
0.708393 + 0.705818i $$0.249420\pi$$
$$488$$ 0 0
$$489$$ −7.00156 −0.316621
$$490$$ 0 0
$$491$$ −8.15752 −0.368144 −0.184072 0.982913i $$-0.558928\pi$$
−0.184072 + 0.982913i $$0.558928\pi$$
$$492$$ 0 0
$$493$$ 37.1321 1.67235
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 25.3991 1.13931
$$498$$ 0 0
$$499$$ 8.52746 0.381742 0.190871 0.981615i $$-0.438869\pi$$
0.190871 + 0.981615i $$0.438869\pi$$
$$500$$ 0 0
$$501$$ 6.82087 0.304734
$$502$$ 0 0
$$503$$ −9.83853 −0.438678 −0.219339 0.975649i $$-0.570390\pi$$
−0.219339 + 0.975649i $$0.570390\pi$$
$$504$$ 0 0
$$505$$ 7.17862 0.319444
$$506$$ 0 0
$$507$$ −19.8916 −0.883417
$$508$$ 0 0
$$509$$ −0.488846 −0.0216677 −0.0108339 0.999941i $$-0.503449\pi$$
−0.0108339 + 0.999941i $$0.503449\pi$$
$$510$$ 0 0
$$511$$ 37.2547 1.64805
$$512$$ 0 0
$$513$$ −20.2071 −0.892167
$$514$$ 0 0
$$515$$ 3.75016 0.165252
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −30.2434 −1.32754
$$520$$ 0 0
$$521$$ 11.1536 0.488650 0.244325 0.969693i $$-0.421434\pi$$
0.244325 + 0.969693i $$0.421434\pi$$
$$522$$ 0 0
$$523$$ −3.36279 −0.147045 −0.0735223 0.997294i $$-0.523424\pi$$
−0.0735223 + 0.997294i $$0.523424\pi$$
$$524$$ 0 0
$$525$$ −6.34602 −0.276963
$$526$$ 0 0
$$527$$ 47.2816 2.05962
$$528$$ 0 0
$$529$$ 52.3836 2.27755
$$530$$ 0 0
$$531$$ 0.0261281 0.00113386
$$532$$ 0 0
$$533$$ −1.88085 −0.0814688
$$534$$ 0 0
$$535$$ −6.35128 −0.274590
$$536$$ 0 0
$$537$$ 20.4537 0.882642
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −2.63681 −0.113365 −0.0566827 0.998392i $$-0.518052\pi$$
−0.0566827 + 0.998392i $$0.518052\pi$$
$$542$$ 0 0
$$543$$ −1.76711 −0.0758340
$$544$$ 0 0
$$545$$ 1.32407 0.0567170
$$546$$ 0 0
$$547$$ −38.2751 −1.63652 −0.818262 0.574845i $$-0.805062\pi$$
−0.818262 + 0.574845i $$0.805062\pi$$
$$548$$ 0 0
$$549$$ 0.119424 0.00509687
$$550$$ 0 0
$$551$$ 26.4555 1.12704
$$552$$ 0 0
$$553$$ −4.12681 −0.175490
$$554$$ 0 0
$$555$$ −3.86182 −0.163925
$$556$$ 0 0
$$557$$ 18.1087 0.767289 0.383645 0.923481i $$-0.374669\pi$$
0.383645 + 0.923481i $$0.374669\pi$$
$$558$$ 0 0
$$559$$ −4.85105 −0.205177
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −3.56855 −0.150396 −0.0751982 0.997169i $$-0.523959\pi$$
−0.0751982 + 0.997169i $$0.523959\pi$$
$$564$$ 0 0
$$565$$ −5.33169 −0.224306
$$566$$ 0 0
$$567$$ 32.5921 1.36874
$$568$$ 0 0
$$569$$ 40.3751 1.69261 0.846307 0.532696i $$-0.178821\pi$$
0.846307 + 0.532696i $$0.178821\pi$$
$$570$$ 0 0
$$571$$ −36.3873 −1.52276 −0.761381 0.648305i $$-0.775478\pi$$
−0.761381 + 0.648305i $$0.775478\pi$$
$$572$$ 0 0
$$573$$ 32.5694 1.36061
$$574$$ 0 0
$$575$$ −8.68237 −0.362080
$$576$$ 0 0
$$577$$ −1.48422 −0.0617887 −0.0308944 0.999523i $$-0.509836\pi$$
−0.0308944 + 0.999523i $$0.509836\pi$$
$$578$$ 0 0
$$579$$ −9.26409 −0.385002
$$580$$ 0 0
$$581$$ 0.0641761 0.00266247
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0.0780860 0.00322846
$$586$$ 0 0
$$587$$ −39.3563 −1.62441 −0.812204 0.583374i $$-0.801732\pi$$
−0.812204 + 0.583374i $$0.801732\pi$$
$$588$$ 0 0
$$589$$ 33.6866 1.38803
$$590$$ 0 0
$$591$$ −6.98168 −0.287188
$$592$$ 0 0
$$593$$ −27.2990 −1.12104 −0.560518 0.828142i $$-0.689398\pi$$
−0.560518 + 0.828142i $$0.689398\pi$$
$$594$$ 0 0
$$595$$ 20.0083 0.820260
$$596$$ 0 0
$$597$$ −20.6755 −0.846190
$$598$$ 0 0
$$599$$ 21.1311 0.863393 0.431696 0.902019i $$-0.357915\pi$$
0.431696 + 0.902019i $$0.357915\pi$$
$$600$$ 0 0
$$601$$ 2.86260 0.116768 0.0583840 0.998294i $$-0.481405\pi$$
0.0583840 + 0.998294i $$0.481405\pi$$
$$602$$ 0 0
$$603$$ 0.953672 0.0388365
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −31.9944 −1.29861 −0.649306 0.760527i $$-0.724941\pi$$
−0.649306 + 0.760527i $$0.724941\pi$$
$$608$$ 0 0
$$609$$ −43.6350 −1.76818
$$610$$ 0 0
$$611$$ −14.9307 −0.604030
$$612$$ 0 0
$$613$$ 35.6008 1.43790 0.718952 0.695060i $$-0.244622\pi$$
0.718952 + 0.695060i $$0.244622\pi$$
$$614$$ 0 0
$$615$$ 2.73591 0.110322
$$616$$ 0 0
$$617$$ −36.9894 −1.48914 −0.744568 0.667546i $$-0.767345\pi$$
−0.744568 + 0.667546i $$0.767345\pi$$
$$618$$ 0 0
$$619$$ 33.9911 1.36622 0.683109 0.730317i $$-0.260628\pi$$
0.683109 + 0.730317i $$0.260628\pi$$
$$620$$ 0 0
$$621$$ 45.5997 1.82985
$$622$$ 0 0
$$623$$ 31.4720 1.26090
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 12.1759 0.485485
$$630$$ 0 0
$$631$$ 2.88653 0.114911 0.0574555 0.998348i $$-0.481701\pi$$
0.0574555 + 0.998348i $$0.481701\pi$$
$$632$$ 0 0
$$633$$ 15.9511 0.634000
$$634$$ 0 0
$$635$$ 14.6406 0.580994
$$636$$ 0 0
$$637$$ 7.92154 0.313863
$$638$$ 0 0
$$639$$ 0.454608 0.0179840
$$640$$ 0 0
$$641$$ 11.6233 0.459092 0.229546 0.973298i $$-0.426276\pi$$
0.229546 + 0.973298i $$0.426276\pi$$
$$642$$ 0 0
$$643$$ −27.0864 −1.06818 −0.534091 0.845427i $$-0.679346\pi$$
−0.534091 + 0.845427i $$0.679346\pi$$
$$644$$ 0 0
$$645$$ 7.05637 0.277844
$$646$$ 0 0
$$647$$ 1.66070 0.0652888 0.0326444 0.999467i $$-0.489607\pi$$
0.0326444 + 0.999467i $$0.489607\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −55.5620 −2.17764
$$652$$ 0 0
$$653$$ −40.2905 −1.57669 −0.788345 0.615234i $$-0.789062\pi$$
−0.788345 + 0.615234i $$0.789062\pi$$
$$654$$ 0 0
$$655$$ 5.93922 0.232065
$$656$$ 0 0
$$657$$ 0.666804 0.0260145
$$658$$ 0 0
$$659$$ −33.7115 −1.31321 −0.656607 0.754233i $$-0.728009\pi$$
−0.656607 + 0.754233i $$0.728009\pi$$
$$660$$ 0 0
$$661$$ 42.4279 1.65026 0.825128 0.564946i $$-0.191103\pi$$
0.825128 + 0.564946i $$0.191103\pi$$
$$662$$ 0 0
$$663$$ 10.8914 0.422986
$$664$$ 0 0
$$665$$ 14.2553 0.552796
$$666$$ 0 0
$$667$$ −59.6998 −2.31158
$$668$$ 0 0
$$669$$ −8.12491 −0.314127
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 38.1273 1.46970 0.734850 0.678230i $$-0.237253\pi$$
0.734850 + 0.678230i $$0.237253\pi$$
$$674$$ 0 0
$$675$$ −5.25199 −0.202149
$$676$$ 0 0
$$677$$ 12.0095 0.461561 0.230781 0.973006i $$-0.425872\pi$$
0.230781 + 0.973006i $$0.425872\pi$$
$$678$$ 0 0
$$679$$ −38.9604 −1.49516
$$680$$ 0 0
$$681$$ −6.02089 −0.230721
$$682$$ 0 0
$$683$$ 37.6564 1.44088 0.720441 0.693516i $$-0.243939\pi$$
0.720441 + 0.693516i $$0.243939\pi$$
$$684$$ 0 0
$$685$$ −14.7698 −0.564324
$$686$$ 0 0
$$687$$ −15.6312 −0.596369
$$688$$ 0 0
$$689$$ −14.5769 −0.555335
$$690$$ 0 0
$$691$$ −9.93572 −0.377973 −0.188986 0.981980i $$-0.560520\pi$$
−0.188986 + 0.981980i $$0.560520\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 1.29252 0.0490280
$$696$$ 0 0
$$697$$ −8.62601 −0.326733
$$698$$ 0 0
$$699$$ 36.8683 1.39449
$$700$$ 0 0
$$701$$ −10.0853 −0.380915 −0.190458 0.981695i $$-0.560997\pi$$
−0.190458 + 0.981695i $$0.560997\pi$$
$$702$$ 0 0
$$703$$ 8.67494 0.327181
$$704$$ 0 0
$$705$$ 21.7183 0.817957
$$706$$ 0 0
$$707$$ 26.5972 1.00029
$$708$$ 0 0
$$709$$ 5.21637 0.195905 0.0979524 0.995191i $$-0.468771\pi$$
0.0979524 + 0.995191i $$0.468771\pi$$
$$710$$ 0 0
$$711$$ −0.0738640 −0.00277012
$$712$$ 0 0
$$713$$ −76.0177 −2.84689
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −19.3674 −0.723288
$$718$$ 0 0
$$719$$ −7.90947 −0.294973 −0.147487 0.989064i $$-0.547118\pi$$
−0.147487 + 0.989064i $$0.547118\pi$$
$$720$$ 0 0
$$721$$ 13.8946 0.517460
$$722$$ 0 0
$$723$$ 44.7288 1.66348
$$724$$ 0 0
$$725$$ 6.87597 0.255367
$$726$$ 0 0
$$727$$ 10.2281 0.379338 0.189669 0.981848i $$-0.439258\pi$$
0.189669 + 0.981848i $$0.439258\pi$$
$$728$$ 0 0
$$729$$ 27.5702 1.02112
$$730$$ 0 0
$$731$$ −22.2480 −0.822871
$$732$$ 0 0
$$733$$ 15.3672 0.567601 0.283801 0.958883i $$-0.408405\pi$$
0.283801 + 0.958883i $$0.408405\pi$$
$$734$$ 0 0
$$735$$ −11.5227 −0.425023
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 28.2071 1.03762 0.518808 0.854891i $$-0.326376\pi$$
0.518808 + 0.854891i $$0.326376\pi$$
$$740$$ 0 0
$$741$$ 7.75976 0.285062
$$742$$ 0 0
$$743$$ −44.1059 −1.61809 −0.809044 0.587748i $$-0.800015\pi$$
−0.809044 + 0.587748i $$0.800015\pi$$
$$744$$ 0 0
$$745$$ 23.9060 0.875849
$$746$$ 0 0
$$747$$ 0.00114866 4.20272e−5 0
$$748$$ 0 0
$$749$$ −23.5318 −0.859834
$$750$$ 0 0
$$751$$ −43.1816 −1.57572 −0.787860 0.615854i $$-0.788811\pi$$
−0.787860 + 0.615854i $$0.788811\pi$$
$$752$$ 0 0
$$753$$ −6.36839 −0.232077
$$754$$ 0 0
$$755$$ 8.28048 0.301358
$$756$$ 0 0
$$757$$ 19.2800 0.700743 0.350371 0.936611i $$-0.386055\pi$$
0.350371 + 0.936611i $$0.386055\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −34.9095 −1.26547 −0.632733 0.774370i $$-0.718067\pi$$
−0.632733 + 0.774370i $$0.718067\pi$$
$$762$$ 0 0
$$763$$ 4.90576 0.177600
$$764$$ 0 0
$$765$$ 0.358119 0.0129478
$$766$$ 0 0
$$767$$ −0.463934 −0.0167517
$$768$$ 0 0
$$769$$ −10.7367 −0.387177 −0.193588 0.981083i $$-0.562013\pi$$
−0.193588 + 0.981083i $$0.562013\pi$$
$$770$$ 0 0
$$771$$ −12.1485 −0.437518
$$772$$ 0 0
$$773$$ 31.6402 1.13802 0.569010 0.822331i $$-0.307327\pi$$
0.569010 + 0.822331i $$0.307327\pi$$
$$774$$ 0 0
$$775$$ 8.75541 0.314503
$$776$$ 0 0
$$777$$ −14.3082 −0.513306
$$778$$ 0 0
$$779$$ −6.14576 −0.220195
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −36.1125 −1.29056
$$784$$ 0 0
$$785$$ 7.96444 0.284263
$$786$$ 0 0
$$787$$ −15.6444 −0.557662 −0.278831 0.960340i $$-0.589947\pi$$
−0.278831 + 0.960340i $$0.589947\pi$$
$$788$$ 0 0
$$789$$ −40.0072 −1.42429
$$790$$ 0 0
$$791$$ −19.7542 −0.702379
$$792$$ 0 0
$$793$$ −2.12050 −0.0753011
$$794$$ 0 0
$$795$$ 21.2036 0.752016
$$796$$ 0 0
$$797$$ −4.97978 −0.176393 −0.0881964 0.996103i $$-0.528110\pi$$
−0.0881964 + 0.996103i $$0.528110\pi$$
$$798$$ 0 0
$$799$$ −68.4753 −2.42248
$$800$$ 0 0
$$801$$ 0.563303 0.0199033
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −32.1687 −1.13380
$$806$$ 0 0
$$807$$ −29.2576 −1.02992
$$808$$ 0 0
$$809$$ −30.8902 −1.08604 −0.543021 0.839719i $$-0.682720\pi$$
−0.543021 + 0.839719i $$0.682720\pi$$
$$810$$ 0 0
$$811$$ −4.47943 −0.157294 −0.0786471 0.996903i $$-0.525060\pi$$
−0.0786471 + 0.996903i $$0.525060\pi$$
$$812$$ 0 0
$$813$$ 46.9895 1.64799
$$814$$ 0 0
$$815$$ 4.08778 0.143189
$$816$$ 0 0
$$817$$ −15.8510 −0.554555
$$818$$ 0 0
$$819$$ 0.289313 0.0101094
$$820$$ 0 0
$$821$$ −12.4808 −0.435583 −0.217791 0.975995i $$-0.569885\pi$$
−0.217791 + 0.975995i $$0.569885\pi$$
$$822$$ 0 0
$$823$$ 16.5038 0.575288 0.287644 0.957737i $$-0.407128\pi$$
0.287644 + 0.957737i $$0.407128\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 45.1967 1.57164 0.785822 0.618453i $$-0.212240\pi$$
0.785822 + 0.618453i $$0.212240\pi$$
$$828$$ 0 0
$$829$$ −11.3874 −0.395502 −0.197751 0.980252i $$-0.563364\pi$$
−0.197751 + 0.980252i $$0.563364\pi$$
$$830$$ 0 0
$$831$$ 41.4927 1.43936
$$832$$ 0 0
$$833$$ 36.3299 1.25876
$$834$$ 0 0
$$835$$ −3.98229 −0.137813
$$836$$ 0 0
$$837$$ −45.9833 −1.58941
$$838$$ 0 0
$$839$$ −14.1100 −0.487131 −0.243565 0.969884i $$-0.578317\pi$$
−0.243565 + 0.969884i $$0.578317\pi$$
$$840$$ 0 0
$$841$$ 18.2790 0.630310
$$842$$ 0 0
$$843$$ 29.3975 1.01250
$$844$$ 0 0
$$845$$ 11.6135 0.399516
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −7.92870 −0.272112
$$850$$ 0 0
$$851$$ −19.5760 −0.671056
$$852$$ 0 0
$$853$$ 20.9805 0.718360 0.359180 0.933268i $$-0.383056\pi$$
0.359180 + 0.933268i $$0.383056\pi$$
$$854$$ 0 0
$$855$$ 0.255149 0.00872590
$$856$$ 0 0
$$857$$ −25.0649 −0.856201 −0.428101 0.903731i $$-0.640817\pi$$
−0.428101 + 0.903731i $$0.640817\pi$$
$$858$$ 0 0
$$859$$ −38.0496 −1.29823 −0.649117 0.760689i $$-0.724861\pi$$
−0.649117 + 0.760689i $$0.724861\pi$$
$$860$$ 0 0
$$861$$ 10.1367 0.345457
$$862$$ 0 0
$$863$$ −8.58888 −0.292369 −0.146184 0.989257i $$-0.546699\pi$$
−0.146184 + 0.989257i $$0.546699\pi$$
$$864$$ 0 0
$$865$$ 17.6573 0.600366
$$866$$ 0 0
$$867$$ 20.8327 0.707514
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −16.9335 −0.573771
$$872$$ 0 0
$$873$$ −0.697334 −0.0236012
$$874$$ 0 0
$$875$$ 3.70505 0.125254
$$876$$ 0 0
$$877$$ 10.5663 0.356798 0.178399 0.983958i $$-0.442908\pi$$
0.178399 + 0.983958i $$0.442908\pi$$
$$878$$ 0 0
$$879$$ 31.6879 1.06881
$$880$$ 0 0
$$881$$ −35.5069 −1.19626 −0.598129 0.801400i $$-0.704089\pi$$
−0.598129 + 0.801400i $$0.704089\pi$$
$$882$$ 0 0
$$883$$ −34.0160 −1.14473 −0.572364 0.820000i $$-0.693974\pi$$
−0.572364 + 0.820000i $$0.693974\pi$$
$$884$$ 0 0
$$885$$ 0.674842 0.0226846
$$886$$ 0 0
$$887$$ 9.99766 0.335689 0.167844 0.985814i $$-0.446319\pi$$
0.167844 + 0.985814i $$0.446319\pi$$
$$888$$ 0 0
$$889$$ 54.2442 1.81929
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −48.7865 −1.63258
$$894$$ 0 0
$$895$$ −11.9417 −0.399166
$$896$$ 0 0
$$897$$ −17.5108 −0.584668
$$898$$ 0 0
$$899$$ 60.2019 2.00785
$$900$$ 0 0
$$901$$ −66.8528 −2.22719
$$902$$ 0 0
$$903$$ 26.1442 0.870026
$$904$$ 0 0
$$905$$ 1.03171 0.0342952
$$906$$ 0 0
$$907$$ 14.3972 0.478051 0.239025 0.971013i $$-0.423172\pi$$
0.239025 + 0.971013i $$0.423172\pi$$
$$908$$ 0 0
$$909$$ 0.476051 0.0157896
$$910$$ 0 0
$$911$$ −16.1765 −0.535951 −0.267975 0.963426i $$-0.586355\pi$$
−0.267975 + 0.963426i $$0.586355\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 3.08450 0.101970
$$916$$ 0 0
$$917$$ 22.0051 0.726674
$$918$$ 0 0
$$919$$ 18.1790 0.599671 0.299836 0.953991i $$-0.403068\pi$$
0.299836 + 0.953991i $$0.403068\pi$$
$$920$$ 0 0
$$921$$ 46.5917 1.53525
$$922$$ 0 0
$$923$$ −8.07207 −0.265696
$$924$$ 0 0
$$925$$ 2.25468 0.0741335
$$926$$ 0 0
$$927$$ 0.248692 0.00816813
$$928$$ 0 0
$$929$$ −24.3272 −0.798150 −0.399075 0.916918i $$-0.630669\pi$$
−0.399075 + 0.916918i $$0.630669\pi$$
$$930$$ 0 0
$$931$$ 25.8839 0.848311
$$932$$ 0 0
$$933$$ 55.2547 1.80896
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −24.1898 −0.790246 −0.395123 0.918628i $$-0.629298\pi$$
−0.395123 + 0.918628i $$0.629298\pi$$
$$938$$ 0 0
$$939$$ −0.497994 −0.0162514
$$940$$ 0 0
$$941$$ 6.12261 0.199591 0.0997956 0.995008i $$-0.468181\pi$$
0.0997956 + 0.995008i $$0.468181\pi$$
$$942$$ 0 0
$$943$$ 13.8686 0.451624
$$944$$ 0 0
$$945$$ −19.4589 −0.632998
$$946$$ 0 0
$$947$$ 30.4758 0.990330 0.495165 0.868799i $$-0.335108\pi$$
0.495165 + 0.868799i $$0.335108\pi$$
$$948$$ 0 0
$$949$$ −11.8399 −0.384338
$$950$$ 0 0
$$951$$ 39.1110 1.26826
$$952$$ 0 0
$$953$$ −12.3055 −0.398614 −0.199307 0.979937i $$-0.563869\pi$$
−0.199307 + 0.979937i $$0.563869\pi$$
$$954$$ 0 0
$$955$$ −19.0153 −0.615321
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −54.7228 −1.76709
$$960$$ 0 0
$$961$$ 45.6571 1.47281
$$962$$ 0 0
$$963$$ −0.421186 −0.0135725
$$964$$ 0 0
$$965$$ 5.40874 0.174113
$$966$$ 0 0
$$967$$ −22.5927 −0.726532 −0.363266 0.931685i $$-0.618338\pi$$
−0.363266 + 0.931685i $$0.618338\pi$$
$$968$$ 0 0
$$969$$ 35.5880 1.14325
$$970$$ 0 0
$$971$$ 43.2544 1.38810 0.694050 0.719927i $$-0.255825\pi$$
0.694050 + 0.719927i $$0.255825\pi$$
$$972$$ 0 0
$$973$$ 4.78884 0.153523
$$974$$ 0 0
$$975$$ 2.01682 0.0645900
$$976$$ 0 0
$$977$$ 15.0741 0.482263 0.241131 0.970492i $$-0.422482\pi$$
0.241131 + 0.970492i $$0.422482\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0.0878059 0.00280343
$$982$$ 0 0
$$983$$ −41.9188 −1.33700 −0.668502 0.743711i $$-0.733064\pi$$
−0.668502 + 0.743711i $$0.733064\pi$$
$$984$$ 0 0
$$985$$ 4.07618 0.129878
$$986$$ 0 0
$$987$$ 80.4673 2.56130
$$988$$ 0 0
$$989$$ 35.7695 1.13740
$$990$$ 0 0
$$991$$ 29.5068 0.937314 0.468657 0.883380i $$-0.344738\pi$$
0.468657 + 0.883380i $$0.344738\pi$$
$$992$$ 0 0
$$993$$ −24.8805 −0.789559
$$994$$ 0 0
$$995$$ 12.0711 0.382681
$$996$$ 0 0
$$997$$ −48.6155 −1.53967 −0.769835 0.638243i $$-0.779661\pi$$
−0.769835 + 0.638243i $$0.779661\pi$$
$$998$$ 0 0
$$999$$ −11.8416 −0.374650
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 4840.2.a.ba.1.6 6
4.3 odd 2 9680.2.a.dd.1.1 6
11.2 odd 10 440.2.y.c.81.3 12
11.6 odd 10 440.2.y.c.201.3 yes 12
11.10 odd 2 4840.2.a.bb.1.6 6
44.35 even 10 880.2.bo.i.81.1 12
44.39 even 10 880.2.bo.i.641.1 12
44.43 even 2 9680.2.a.dc.1.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.81.3 12 11.2 odd 10
440.2.y.c.201.3 yes 12 11.6 odd 10
880.2.bo.i.81.1 12 44.35 even 10
880.2.bo.i.641.1 12 44.39 even 10
4840.2.a.ba.1.6 6 1.1 even 1 trivial
4840.2.a.bb.1.6 6 11.10 odd 2
9680.2.a.dc.1.1 6 44.43 even 2
9680.2.a.dd.1.1 6 4.3 odd 2