Properties

 Label 2496.4.a.bi.1.1 Level $2496$ Weight $4$ Character 2496.1 Self dual yes Analytic conductor $147.269$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2496.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: $$147.268767374$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{55})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 55$$ x^2 - 55 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 312) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-7.41620$$ of defining polynomial Character $$\chi$$ $$=$$ 2496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -12.8324 q^{5} +24.8324 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -12.8324 q^{5} +24.8324 q^{7} +9.00000 q^{9} -19.6648 q^{11} +13.0000 q^{13} -38.4972 q^{15} -63.6648 q^{17} +0.832397 q^{19} +74.4972 q^{21} +119.330 q^{23} +39.6704 q^{25} +27.0000 q^{27} +6.00000 q^{29} +185.503 q^{31} -58.9944 q^{33} -318.659 q^{35} -143.665 q^{37} +39.0000 q^{39} -117.156 q^{41} -67.6760 q^{43} -115.492 q^{45} +476.659 q^{47} +273.648 q^{49} -190.994 q^{51} -59.3184 q^{53} +252.346 q^{55} +2.49719 q^{57} -78.0000 q^{59} -609.955 q^{61} +223.492 q^{63} -166.821 q^{65} +654.452 q^{67} +357.989 q^{69} -390.994 q^{71} -84.3127 q^{73} +119.011 q^{75} -488.324 q^{77} +1176.65 q^{79} +81.0000 q^{81} +430.648 q^{83} +816.972 q^{85} +18.0000 q^{87} -1341.41 q^{89} +322.821 q^{91} +556.508 q^{93} -10.6816 q^{95} +802.949 q^{97} -176.983 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 4 q^{5} + 20 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 4 * q^5 + 20 * q^7 + 18 * q^9 $$2 q + 6 q^{3} + 4 q^{5} + 20 q^{7} + 18 q^{9} + 20 q^{11} + 26 q^{13} + 12 q^{15} - 68 q^{17} - 28 q^{19} + 60 q^{21} + 120 q^{23} + 198 q^{25} + 54 q^{27} + 12 q^{29} + 460 q^{31} + 60 q^{33} - 400 q^{35} - 228 q^{37} + 78 q^{39} + 92 q^{41} - 432 q^{43} + 36 q^{45} + 716 q^{47} - 46 q^{49} - 204 q^{51} + 356 q^{53} + 920 q^{55} - 84 q^{57} - 156 q^{59} + 204 q^{61} + 180 q^{63} + 52 q^{65} - 204 q^{67} + 360 q^{69} - 604 q^{71} + 484 q^{73} + 594 q^{75} - 680 q^{77} + 1760 q^{79} + 162 q^{81} + 268 q^{83} + 744 q^{85} + 36 q^{87} + 76 q^{89} + 260 q^{91} + 1380 q^{93} - 496 q^{95} + 4 q^{97} + 180 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 4 * q^5 + 20 * q^7 + 18 * q^9 + 20 * q^11 + 26 * q^13 + 12 * q^15 - 68 * q^17 - 28 * q^19 + 60 * q^21 + 120 * q^23 + 198 * q^25 + 54 * q^27 + 12 * q^29 + 460 * q^31 + 60 * q^33 - 400 * q^35 - 228 * q^37 + 78 * q^39 + 92 * q^41 - 432 * q^43 + 36 * q^45 + 716 * q^47 - 46 * q^49 - 204 * q^51 + 356 * q^53 + 920 * q^55 - 84 * q^57 - 156 * q^59 + 204 * q^61 + 180 * q^63 + 52 * q^65 - 204 * q^67 + 360 * q^69 - 604 * q^71 + 484 * q^73 + 594 * q^75 - 680 * q^77 + 1760 * q^79 + 162 * q^81 + 268 * q^83 + 744 * q^85 + 36 * q^87 + 76 * q^89 + 260 * q^91 + 1380 * q^93 - 496 * q^95 + 4 * q^97 + 180 * q^99

Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −12.8324 −1.14776 −0.573882 0.818938i $$-0.694563\pi$$
−0.573882 + 0.818938i $$0.694563\pi$$
$$6$$ 0 0
$$7$$ 24.8324 1.34082 0.670412 0.741989i $$-0.266118\pi$$
0.670412 + 0.741989i $$0.266118\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −19.6648 −0.539014 −0.269507 0.962998i $$-0.586861\pi$$
−0.269507 + 0.962998i $$0.586861\pi$$
$$12$$ 0 0
$$13$$ 13.0000 0.277350
$$14$$ 0 0
$$15$$ −38.4972 −0.662662
$$16$$ 0 0
$$17$$ −63.6648 −0.908293 −0.454146 0.890927i $$-0.650056\pi$$
−0.454146 + 0.890927i $$0.650056\pi$$
$$18$$ 0 0
$$19$$ 0.832397 0.0100508 0.00502539 0.999987i $$-0.498400\pi$$
0.00502539 + 0.999987i $$0.498400\pi$$
$$20$$ 0 0
$$21$$ 74.4972 0.774125
$$22$$ 0 0
$$23$$ 119.330 1.08182 0.540912 0.841079i $$-0.318079\pi$$
0.540912 + 0.841079i $$0.318079\pi$$
$$24$$ 0 0
$$25$$ 39.6704 0.317363
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 6.00000 0.0384197 0.0192099 0.999815i $$-0.493885\pi$$
0.0192099 + 0.999815i $$0.493885\pi$$
$$30$$ 0 0
$$31$$ 185.503 1.07475 0.537376 0.843343i $$-0.319416\pi$$
0.537376 + 0.843343i $$0.319416\pi$$
$$32$$ 0 0
$$33$$ −58.9944 −0.311200
$$34$$ 0 0
$$35$$ −318.659 −1.53895
$$36$$ 0 0
$$37$$ −143.665 −0.638334 −0.319167 0.947699i $$-0.603403\pi$$
−0.319167 + 0.947699i $$0.603403\pi$$
$$38$$ 0 0
$$39$$ 39.0000 0.160128
$$40$$ 0 0
$$41$$ −117.156 −0.446262 −0.223131 0.974788i $$-0.571628\pi$$
−0.223131 + 0.974788i $$0.571628\pi$$
$$42$$ 0 0
$$43$$ −67.6760 −0.240012 −0.120006 0.992773i $$-0.538291\pi$$
−0.120006 + 0.992773i $$0.538291\pi$$
$$44$$ 0 0
$$45$$ −115.492 −0.382588
$$46$$ 0 0
$$47$$ 476.659 1.47932 0.739658 0.672983i $$-0.234987\pi$$
0.739658 + 0.672983i $$0.234987\pi$$
$$48$$ 0 0
$$49$$ 273.648 0.797807
$$50$$ 0 0
$$51$$ −190.994 −0.524403
$$52$$ 0 0
$$53$$ −59.3184 −0.153736 −0.0768679 0.997041i $$-0.524492\pi$$
−0.0768679 + 0.997041i $$0.524492\pi$$
$$54$$ 0 0
$$55$$ 252.346 0.618662
$$56$$ 0 0
$$57$$ 2.49719 0.00580282
$$58$$ 0 0
$$59$$ −78.0000 −0.172114 −0.0860571 0.996290i $$-0.527427\pi$$
−0.0860571 + 0.996290i $$0.527427\pi$$
$$60$$ 0 0
$$61$$ −609.955 −1.28027 −0.640137 0.768261i $$-0.721123\pi$$
−0.640137 + 0.768261i $$0.721123\pi$$
$$62$$ 0 0
$$63$$ 223.492 0.446941
$$64$$ 0 0
$$65$$ −166.821 −0.318333
$$66$$ 0 0
$$67$$ 654.452 1.19334 0.596672 0.802485i $$-0.296489\pi$$
0.596672 + 0.802485i $$0.296489\pi$$
$$68$$ 0 0
$$69$$ 357.989 0.624591
$$70$$ 0 0
$$71$$ −390.994 −0.653556 −0.326778 0.945101i $$-0.605963\pi$$
−0.326778 + 0.945101i $$0.605963\pi$$
$$72$$ 0 0
$$73$$ −84.3127 −0.135179 −0.0675894 0.997713i $$-0.521531\pi$$
−0.0675894 + 0.997713i $$0.521531\pi$$
$$74$$ 0 0
$$75$$ 119.011 0.183230
$$76$$ 0 0
$$77$$ −488.324 −0.722723
$$78$$ 0 0
$$79$$ 1176.65 1.67574 0.837869 0.545872i $$-0.183802\pi$$
0.837869 + 0.545872i $$0.183802\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 430.648 0.569515 0.284758 0.958600i $$-0.408087\pi$$
0.284758 + 0.958600i $$0.408087\pi$$
$$84$$ 0 0
$$85$$ 816.972 1.04251
$$86$$ 0 0
$$87$$ 18.0000 0.0221816
$$88$$ 0 0
$$89$$ −1341.41 −1.59763 −0.798817 0.601574i $$-0.794541\pi$$
−0.798817 + 0.601574i $$0.794541\pi$$
$$90$$ 0 0
$$91$$ 322.821 0.371878
$$92$$ 0 0
$$93$$ 556.508 0.620508
$$94$$ 0 0
$$95$$ −10.6816 −0.0115359
$$96$$ 0 0
$$97$$ 802.949 0.840486 0.420243 0.907412i $$-0.361945\pi$$
0.420243 + 0.907412i $$0.361945\pi$$
$$98$$ 0 0
$$99$$ −176.983 −0.179671
$$100$$ 0 0
$$101$$ 502.369 0.494926 0.247463 0.968897i $$-0.420403\pi$$
0.247463 + 0.968897i $$0.420403\pi$$
$$102$$ 0 0
$$103$$ 1928.93 1.84527 0.922635 0.385674i $$-0.126031\pi$$
0.922635 + 0.385674i $$0.126031\pi$$
$$104$$ 0 0
$$105$$ −955.978 −0.888513
$$106$$ 0 0
$$107$$ 1759.96 1.59011 0.795053 0.606540i $$-0.207443\pi$$
0.795053 + 0.606540i $$0.207443\pi$$
$$108$$ 0 0
$$109$$ 53.3071 0.0468431 0.0234215 0.999726i $$-0.492544\pi$$
0.0234215 + 0.999726i $$0.492544\pi$$
$$110$$ 0 0
$$111$$ −430.994 −0.368542
$$112$$ 0 0
$$113$$ 254.648 0.211993 0.105997 0.994366i $$-0.466197\pi$$
0.105997 + 0.994366i $$0.466197\pi$$
$$114$$ 0 0
$$115$$ −1531.28 −1.24168
$$116$$ 0 0
$$117$$ 117.000 0.0924500
$$118$$ 0 0
$$119$$ −1580.95 −1.21786
$$120$$ 0 0
$$121$$ −944.296 −0.709463
$$122$$ 0 0
$$123$$ −351.469 −0.257650
$$124$$ 0 0
$$125$$ 1094.98 0.783506
$$126$$ 0 0
$$127$$ 805.597 0.562876 0.281438 0.959579i $$-0.409189\pi$$
0.281438 + 0.959579i $$0.409189\pi$$
$$128$$ 0 0
$$129$$ −203.028 −0.138571
$$130$$ 0 0
$$131$$ −365.966 −0.244081 −0.122041 0.992525i $$-0.538944\pi$$
−0.122041 + 0.992525i $$0.538944\pi$$
$$132$$ 0 0
$$133$$ 20.6704 0.0134763
$$134$$ 0 0
$$135$$ −346.475 −0.220887
$$136$$ 0 0
$$137$$ −2319.79 −1.44667 −0.723333 0.690499i $$-0.757391\pi$$
−0.723333 + 0.690499i $$0.757391\pi$$
$$138$$ 0 0
$$139$$ −1816.67 −1.10855 −0.554273 0.832335i $$-0.687004\pi$$
−0.554273 + 0.832335i $$0.687004\pi$$
$$140$$ 0 0
$$141$$ 1429.98 0.854084
$$142$$ 0 0
$$143$$ −255.642 −0.149496
$$144$$ 0 0
$$145$$ −76.9944 −0.0440968
$$146$$ 0 0
$$147$$ 820.944 0.460614
$$148$$ 0 0
$$149$$ −1752.37 −0.963490 −0.481745 0.876311i $$-0.659997\pi$$
−0.481745 + 0.876311i $$0.659997\pi$$
$$150$$ 0 0
$$151$$ −590.128 −0.318039 −0.159020 0.987275i $$-0.550833\pi$$
−0.159020 + 0.987275i $$0.550833\pi$$
$$152$$ 0 0
$$153$$ −572.983 −0.302764
$$154$$ 0 0
$$155$$ −2380.45 −1.23356
$$156$$ 0 0
$$157$$ 2338.56 1.18877 0.594386 0.804180i $$-0.297395\pi$$
0.594386 + 0.804180i $$0.297395\pi$$
$$158$$ 0 0
$$159$$ −177.955 −0.0887595
$$160$$ 0 0
$$161$$ 2963.24 1.45053
$$162$$ 0 0
$$163$$ −12.8549 −0.00617712 −0.00308856 0.999995i $$-0.500983\pi$$
−0.00308856 + 0.999995i $$0.500983\pi$$
$$164$$ 0 0
$$165$$ 757.039 0.357184
$$166$$ 0 0
$$167$$ −3236.50 −1.49969 −0.749844 0.661614i $$-0.769872\pi$$
−0.749844 + 0.661614i $$0.769872\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 7.49157 0.00335026
$$172$$ 0 0
$$173$$ −3704.85 −1.62818 −0.814088 0.580742i $$-0.802763\pi$$
−0.814088 + 0.580742i $$0.802763\pi$$
$$174$$ 0 0
$$175$$ 985.111 0.425528
$$176$$ 0 0
$$177$$ −234.000 −0.0993702
$$178$$ 0 0
$$179$$ 2487.37 1.03863 0.519316 0.854582i $$-0.326187\pi$$
0.519316 + 0.854582i $$0.326187\pi$$
$$180$$ 0 0
$$181$$ 4362.47 1.79149 0.895745 0.444568i $$-0.146643\pi$$
0.895745 + 0.444568i $$0.146643\pi$$
$$182$$ 0 0
$$183$$ −1829.87 −0.739167
$$184$$ 0 0
$$185$$ 1843.56 0.732657
$$186$$ 0 0
$$187$$ 1251.96 0.489583
$$188$$ 0 0
$$189$$ 670.475 0.258042
$$190$$ 0 0
$$191$$ −2816.51 −1.06699 −0.533497 0.845802i $$-0.679122\pi$$
−0.533497 + 0.845802i $$0.679122\pi$$
$$192$$ 0 0
$$193$$ 4253.33 1.58633 0.793164 0.609008i $$-0.208432\pi$$
0.793164 + 0.609008i $$0.208432\pi$$
$$194$$ 0 0
$$195$$ −500.463 −0.183789
$$196$$ 0 0
$$197$$ 4164.98 1.50631 0.753153 0.657845i $$-0.228532\pi$$
0.753153 + 0.657845i $$0.228532\pi$$
$$198$$ 0 0
$$199$$ 2252.44 0.802367 0.401183 0.915998i $$-0.368599\pi$$
0.401183 + 0.915998i $$0.368599\pi$$
$$200$$ 0 0
$$201$$ 1963.36 0.688978
$$202$$ 0 0
$$203$$ 148.994 0.0515141
$$204$$ 0 0
$$205$$ 1503.40 0.512204
$$206$$ 0 0
$$207$$ 1073.97 0.360608
$$208$$ 0 0
$$209$$ −16.3689 −0.00541752
$$210$$ 0 0
$$211$$ 4507.73 1.47073 0.735367 0.677669i $$-0.237010\pi$$
0.735367 + 0.677669i $$0.237010\pi$$
$$212$$ 0 0
$$213$$ −1172.98 −0.377331
$$214$$ 0 0
$$215$$ 868.446 0.275477
$$216$$ 0 0
$$217$$ 4606.48 1.44105
$$218$$ 0 0
$$219$$ −252.938 −0.0780456
$$220$$ 0 0
$$221$$ −827.642 −0.251915
$$222$$ 0 0
$$223$$ −863.280 −0.259235 −0.129618 0.991564i $$-0.541375\pi$$
−0.129618 + 0.991564i $$0.541375\pi$$
$$224$$ 0 0
$$225$$ 357.034 0.105788
$$226$$ 0 0
$$227$$ 4662.68 1.36332 0.681658 0.731671i $$-0.261259\pi$$
0.681658 + 0.731671i $$0.261259\pi$$
$$228$$ 0 0
$$229$$ 3661.26 1.05652 0.528260 0.849083i $$-0.322845\pi$$
0.528260 + 0.849083i $$0.322845\pi$$
$$230$$ 0 0
$$231$$ −1464.97 −0.417264
$$232$$ 0 0
$$233$$ 783.845 0.220392 0.110196 0.993910i $$-0.464852\pi$$
0.110196 + 0.993910i $$0.464852\pi$$
$$234$$ 0 0
$$235$$ −6116.68 −1.69791
$$236$$ 0 0
$$237$$ 3529.94 0.967487
$$238$$ 0 0
$$239$$ 2279.44 0.616924 0.308462 0.951237i $$-0.400186\pi$$
0.308462 + 0.951237i $$0.400186\pi$$
$$240$$ 0 0
$$241$$ 4352.87 1.16346 0.581728 0.813383i $$-0.302377\pi$$
0.581728 + 0.813383i $$0.302377\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −3511.56 −0.915695
$$246$$ 0 0
$$247$$ 10.8212 0.00278759
$$248$$ 0 0
$$249$$ 1291.94 0.328810
$$250$$ 0 0
$$251$$ −3851.24 −0.968478 −0.484239 0.874936i $$-0.660904\pi$$
−0.484239 + 0.874936i $$0.660904\pi$$
$$252$$ 0 0
$$253$$ −2346.59 −0.583118
$$254$$ 0 0
$$255$$ 2450.92 0.601891
$$256$$ 0 0
$$257$$ 3090.99 0.750237 0.375119 0.926977i $$-0.377602\pi$$
0.375119 + 0.926977i $$0.377602\pi$$
$$258$$ 0 0
$$259$$ −3567.54 −0.855893
$$260$$ 0 0
$$261$$ 54.0000 0.0128066
$$262$$ 0 0
$$263$$ −8.67041 −0.00203285 −0.00101643 0.999999i $$-0.500324\pi$$
−0.00101643 + 0.999999i $$0.500324\pi$$
$$264$$ 0 0
$$265$$ 761.197 0.176453
$$266$$ 0 0
$$267$$ −4024.24 −0.922395
$$268$$ 0 0
$$269$$ 7485.51 1.69665 0.848326 0.529474i $$-0.177611\pi$$
0.848326 + 0.529474i $$0.177611\pi$$
$$270$$ 0 0
$$271$$ 6357.84 1.42513 0.712566 0.701605i $$-0.247533\pi$$
0.712566 + 0.701605i $$0.247533\pi$$
$$272$$ 0 0
$$273$$ 968.463 0.214704
$$274$$ 0 0
$$275$$ −780.110 −0.171063
$$276$$ 0 0
$$277$$ −7232.46 −1.56880 −0.784398 0.620258i $$-0.787028\pi$$
−0.784398 + 0.620258i $$0.787028\pi$$
$$278$$ 0 0
$$279$$ 1669.53 0.358250
$$280$$ 0 0
$$281$$ 2886.87 0.612868 0.306434 0.951892i $$-0.400864\pi$$
0.306434 + 0.951892i $$0.400864\pi$$
$$282$$ 0 0
$$283$$ 6413.26 1.34710 0.673549 0.739143i $$-0.264769\pi$$
0.673549 + 0.739143i $$0.264769\pi$$
$$284$$ 0 0
$$285$$ −32.0449 −0.00666028
$$286$$ 0 0
$$287$$ −2909.27 −0.598359
$$288$$ 0 0
$$289$$ −859.794 −0.175004
$$290$$ 0 0
$$291$$ 2408.85 0.485255
$$292$$ 0 0
$$293$$ 1223.82 0.244016 0.122008 0.992529i $$-0.461067\pi$$
0.122008 + 0.992529i $$0.461067\pi$$
$$294$$ 0 0
$$295$$ 1000.93 0.197547
$$296$$ 0 0
$$297$$ −530.949 −0.103733
$$298$$ 0 0
$$299$$ 1551.28 0.300044
$$300$$ 0 0
$$301$$ −1680.56 −0.321813
$$302$$ 0 0
$$303$$ 1507.11 0.285746
$$304$$ 0 0
$$305$$ 7827.19 1.46945
$$306$$ 0 0
$$307$$ 3853.21 0.716333 0.358167 0.933658i $$-0.383402\pi$$
0.358167 + 0.933658i $$0.383402\pi$$
$$308$$ 0 0
$$309$$ 5786.78 1.06537
$$310$$ 0 0
$$311$$ −7694.88 −1.40301 −0.701506 0.712664i $$-0.747489\pi$$
−0.701506 + 0.712664i $$0.747489\pi$$
$$312$$ 0 0
$$313$$ 2743.30 0.495400 0.247700 0.968837i $$-0.420325\pi$$
0.247700 + 0.968837i $$0.420325\pi$$
$$314$$ 0 0
$$315$$ −2867.93 −0.512983
$$316$$ 0 0
$$317$$ −4706.66 −0.833920 −0.416960 0.908925i $$-0.636904\pi$$
−0.416960 + 0.908925i $$0.636904\pi$$
$$318$$ 0 0
$$319$$ −117.989 −0.0207088
$$320$$ 0 0
$$321$$ 5279.87 0.918048
$$322$$ 0 0
$$323$$ −52.9944 −0.00912906
$$324$$ 0 0
$$325$$ 515.715 0.0880207
$$326$$ 0 0
$$327$$ 159.921 0.0270449
$$328$$ 0 0
$$329$$ 11836.6 1.98350
$$330$$ 0 0
$$331$$ 9270.83 1.53949 0.769745 0.638352i $$-0.220384\pi$$
0.769745 + 0.638352i $$0.220384\pi$$
$$332$$ 0 0
$$333$$ −1292.98 −0.212778
$$334$$ 0 0
$$335$$ −8398.19 −1.36968
$$336$$ 0 0
$$337$$ −6888.86 −1.11353 −0.556766 0.830670i $$-0.687958\pi$$
−0.556766 + 0.830670i $$0.687958\pi$$
$$338$$ 0 0
$$339$$ 763.944 0.122394
$$340$$ 0 0
$$341$$ −3647.87 −0.579306
$$342$$ 0 0
$$343$$ −1722.18 −0.271105
$$344$$ 0 0
$$345$$ −4593.85 −0.716883
$$346$$ 0 0
$$347$$ 2864.49 0.443152 0.221576 0.975143i $$-0.428880\pi$$
0.221576 + 0.975143i $$0.428880\pi$$
$$348$$ 0 0
$$349$$ −7865.20 −1.20635 −0.603173 0.797610i $$-0.706097\pi$$
−0.603173 + 0.797610i $$0.706097\pi$$
$$350$$ 0 0
$$351$$ 351.000 0.0533761
$$352$$ 0 0
$$353$$ 6071.98 0.915521 0.457761 0.889075i $$-0.348652\pi$$
0.457761 + 0.889075i $$0.348652\pi$$
$$354$$ 0 0
$$355$$ 5017.40 0.750129
$$356$$ 0 0
$$357$$ −4742.85 −0.703132
$$358$$ 0 0
$$359$$ 4951.16 0.727889 0.363945 0.931421i $$-0.381430\pi$$
0.363945 + 0.931421i $$0.381430\pi$$
$$360$$ 0 0
$$361$$ −6858.31 −0.999899
$$362$$ 0 0
$$363$$ −2832.89 −0.409609
$$364$$ 0 0
$$365$$ 1081.93 0.155154
$$366$$ 0 0
$$367$$ 2475.62 0.352115 0.176058 0.984380i $$-0.443666\pi$$
0.176058 + 0.984380i $$0.443666\pi$$
$$368$$ 0 0
$$369$$ −1054.41 −0.148754
$$370$$ 0 0
$$371$$ −1473.02 −0.206133
$$372$$ 0 0
$$373$$ 7342.00 1.01918 0.509590 0.860417i $$-0.329797\pi$$
0.509590 + 0.860417i $$0.329797\pi$$
$$374$$ 0 0
$$375$$ 3284.95 0.452357
$$376$$ 0 0
$$377$$ 78.0000 0.0106557
$$378$$ 0 0
$$379$$ 6188.34 0.838717 0.419359 0.907821i $$-0.362255\pi$$
0.419359 + 0.907821i $$0.362255\pi$$
$$380$$ 0 0
$$381$$ 2416.79 0.324976
$$382$$ 0 0
$$383$$ −3583.67 −0.478113 −0.239056 0.971006i $$-0.576838\pi$$
−0.239056 + 0.971006i $$0.576838\pi$$
$$384$$ 0 0
$$385$$ 6266.37 0.829516
$$386$$ 0 0
$$387$$ −609.084 −0.0800039
$$388$$ 0 0
$$389$$ 2377.34 0.309861 0.154931 0.987925i $$-0.450485\pi$$
0.154931 + 0.987925i $$0.450485\pi$$
$$390$$ 0 0
$$391$$ −7597.09 −0.982613
$$392$$ 0 0
$$393$$ −1097.90 −0.140920
$$394$$ 0 0
$$395$$ −15099.2 −1.92335
$$396$$ 0 0
$$397$$ 8775.17 1.10935 0.554677 0.832066i $$-0.312842\pi$$
0.554677 + 0.832066i $$0.312842\pi$$
$$398$$ 0 0
$$399$$ 62.0112 0.00778056
$$400$$ 0 0
$$401$$ 11512.5 1.43368 0.716842 0.697236i $$-0.245587\pi$$
0.716842 + 0.697236i $$0.245587\pi$$
$$402$$ 0 0
$$403$$ 2411.54 0.298082
$$404$$ 0 0
$$405$$ −1039.42 −0.127529
$$406$$ 0 0
$$407$$ 2825.14 0.344071
$$408$$ 0 0
$$409$$ −13259.2 −1.60300 −0.801499 0.597996i $$-0.795964\pi$$
−0.801499 + 0.597996i $$0.795964\pi$$
$$410$$ 0 0
$$411$$ −6959.38 −0.835233
$$412$$ 0 0
$$413$$ −1936.93 −0.230775
$$414$$ 0 0
$$415$$ −5526.25 −0.653669
$$416$$ 0 0
$$417$$ −5450.01 −0.640020
$$418$$ 0 0
$$419$$ 2101.74 0.245052 0.122526 0.992465i $$-0.460901\pi$$
0.122526 + 0.992465i $$0.460901\pi$$
$$420$$ 0 0
$$421$$ 15777.9 1.82653 0.913264 0.407369i $$-0.133554\pi$$
0.913264 + 0.407369i $$0.133554\pi$$
$$422$$ 0 0
$$423$$ 4289.93 0.493106
$$424$$ 0 0
$$425$$ −2525.61 −0.288259
$$426$$ 0 0
$$427$$ −15146.6 −1.71662
$$428$$ 0 0
$$429$$ −766.927 −0.0863114
$$430$$ 0 0
$$431$$ −4091.03 −0.457211 −0.228606 0.973519i $$-0.573417\pi$$
−0.228606 + 0.973519i $$0.573417\pi$$
$$432$$ 0 0
$$433$$ −419.071 −0.0465110 −0.0232555 0.999730i $$-0.507403\pi$$
−0.0232555 + 0.999730i $$0.507403\pi$$
$$434$$ 0 0
$$435$$ −230.983 −0.0254593
$$436$$ 0 0
$$437$$ 99.3296 0.0108732
$$438$$ 0 0
$$439$$ −15041.1 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$440$$ 0 0
$$441$$ 2462.83 0.265936
$$442$$ 0 0
$$443$$ 11829.0 1.26865 0.634326 0.773066i $$-0.281278\pi$$
0.634326 + 0.773066i $$0.281278\pi$$
$$444$$ 0 0
$$445$$ 17213.5 1.83371
$$446$$ 0 0
$$447$$ −5257.12 −0.556271
$$448$$ 0 0
$$449$$ −9655.47 −1.01485 −0.507427 0.861695i $$-0.669403\pi$$
−0.507427 + 0.861695i $$0.669403\pi$$
$$450$$ 0 0
$$451$$ 2303.86 0.240542
$$452$$ 0 0
$$453$$ −1770.38 −0.183620
$$454$$ 0 0
$$455$$ −4142.57 −0.426828
$$456$$ 0 0
$$457$$ −2304.65 −0.235901 −0.117951 0.993019i $$-0.537632\pi$$
−0.117951 + 0.993019i $$0.537632\pi$$
$$458$$ 0 0
$$459$$ −1718.95 −0.174801
$$460$$ 0 0
$$461$$ 16181.3 1.63479 0.817393 0.576080i $$-0.195418\pi$$
0.817393 + 0.576080i $$0.195418\pi$$
$$462$$ 0 0
$$463$$ 13304.0 1.33540 0.667699 0.744431i $$-0.267279\pi$$
0.667699 + 0.744431i $$0.267279\pi$$
$$464$$ 0 0
$$465$$ −7141.34 −0.712197
$$466$$ 0 0
$$467$$ −6280.78 −0.622355 −0.311178 0.950352i $$-0.600723\pi$$
−0.311178 + 0.950352i $$0.600723\pi$$
$$468$$ 0 0
$$469$$ 16251.6 1.60006
$$470$$ 0 0
$$471$$ 7015.67 0.686338
$$472$$ 0 0
$$473$$ 1330.84 0.129370
$$474$$ 0 0
$$475$$ 33.0215 0.00318975
$$476$$ 0 0
$$477$$ −533.865 −0.0512453
$$478$$ 0 0
$$479$$ −12938.7 −1.23420 −0.617102 0.786883i $$-0.711693\pi$$
−0.617102 + 0.786883i $$0.711693\pi$$
$$480$$ 0 0
$$481$$ −1867.64 −0.177042
$$482$$ 0 0
$$483$$ 8889.72 0.837466
$$484$$ 0 0
$$485$$ −10303.8 −0.964680
$$486$$ 0 0
$$487$$ 17162.2 1.59690 0.798452 0.602059i $$-0.205653\pi$$
0.798452 + 0.602059i $$0.205653\pi$$
$$488$$ 0 0
$$489$$ −38.5646 −0.00356636
$$490$$ 0 0
$$491$$ 1295.35 0.119060 0.0595300 0.998227i $$-0.481040\pi$$
0.0595300 + 0.998227i $$0.481040\pi$$
$$492$$ 0 0
$$493$$ −381.989 −0.0348964
$$494$$ 0 0
$$495$$ 2271.12 0.206221
$$496$$ 0 0
$$497$$ −9709.33 −0.876304
$$498$$ 0 0
$$499$$ −50.9972 −0.00457504 −0.00228752 0.999997i $$-0.500728\pi$$
−0.00228752 + 0.999997i $$0.500728\pi$$
$$500$$ 0 0
$$501$$ −9709.51 −0.865846
$$502$$ 0 0
$$503$$ 18949.3 1.67973 0.839866 0.542794i $$-0.182633\pi$$
0.839866 + 0.542794i $$0.182633\pi$$
$$504$$ 0 0
$$505$$ −6446.60 −0.568059
$$506$$ 0 0
$$507$$ 507.000 0.0444116
$$508$$ 0 0
$$509$$ 19440.3 1.69288 0.846439 0.532486i $$-0.178742\pi$$
0.846439 + 0.532486i $$0.178742\pi$$
$$510$$ 0 0
$$511$$ −2093.69 −0.181251
$$512$$ 0 0
$$513$$ 22.4747 0.00193427
$$514$$ 0 0
$$515$$ −24752.8 −2.11794
$$516$$ 0 0
$$517$$ −9373.40 −0.797373
$$518$$ 0 0
$$519$$ −11114.5 −0.940028
$$520$$ 0 0
$$521$$ 2480.08 0.208549 0.104275 0.994549i $$-0.466748\pi$$
0.104275 + 0.994549i $$0.466748\pi$$
$$522$$ 0 0
$$523$$ −14540.2 −1.21568 −0.607840 0.794060i $$-0.707964\pi$$
−0.607840 + 0.794060i $$0.707964\pi$$
$$524$$ 0 0
$$525$$ 2955.33 0.245679
$$526$$ 0 0
$$527$$ −11810.0 −0.976189
$$528$$ 0 0
$$529$$ 2072.55 0.170342
$$530$$ 0 0
$$531$$ −702.000 −0.0573714
$$532$$ 0 0
$$533$$ −1523.03 −0.123771
$$534$$ 0 0
$$535$$ −22584.4 −1.82507
$$536$$ 0 0
$$537$$ 7462.12 0.599654
$$538$$ 0 0
$$539$$ −5381.23 −0.430030
$$540$$ 0 0
$$541$$ −14166.6 −1.12582 −0.562909 0.826519i $$-0.690318\pi$$
−0.562909 + 0.826519i $$0.690318\pi$$
$$542$$ 0 0
$$543$$ 13087.4 1.03432
$$544$$ 0 0
$$545$$ −684.058 −0.0537648
$$546$$ 0 0
$$547$$ 3102.57 0.242516 0.121258 0.992621i $$-0.461307\pi$$
0.121258 + 0.992621i $$0.461307\pi$$
$$548$$ 0 0
$$549$$ −5489.60 −0.426758
$$550$$ 0 0
$$551$$ 4.99438 0.000386148 0
$$552$$ 0 0
$$553$$ 29219.0 2.24687
$$554$$ 0 0
$$555$$ 5530.69 0.423000
$$556$$ 0 0
$$557$$ −6829.69 −0.519539 −0.259770 0.965671i $$-0.583647\pi$$
−0.259770 + 0.965671i $$0.583647\pi$$
$$558$$ 0 0
$$559$$ −879.788 −0.0665672
$$560$$ 0 0
$$561$$ 3755.87 0.282661
$$562$$ 0 0
$$563$$ −15516.1 −1.16150 −0.580752 0.814080i $$-0.697242\pi$$
−0.580752 + 0.814080i $$0.697242\pi$$
$$564$$ 0 0
$$565$$ −3267.74 −0.243319
$$566$$ 0 0
$$567$$ 2011.42 0.148980
$$568$$ 0 0
$$569$$ −9960.00 −0.733822 −0.366911 0.930256i $$-0.619585\pi$$
−0.366911 + 0.930256i $$0.619585\pi$$
$$570$$ 0 0
$$571$$ −22117.9 −1.62102 −0.810512 0.585722i $$-0.800811\pi$$
−0.810512 + 0.585722i $$0.800811\pi$$
$$572$$ 0 0
$$573$$ −8449.54 −0.616029
$$574$$ 0 0
$$575$$ 4733.85 0.343331
$$576$$ 0 0
$$577$$ 8171.54 0.589576 0.294788 0.955563i $$-0.404751\pi$$
0.294788 + 0.955563i $$0.404751\pi$$
$$578$$ 0 0
$$579$$ 12760.0 0.915867
$$580$$ 0 0
$$581$$ 10694.0 0.763619
$$582$$ 0 0
$$583$$ 1166.48 0.0828659
$$584$$ 0 0
$$585$$ −1501.39 −0.106111
$$586$$ 0 0
$$587$$ 6436.54 0.452580 0.226290 0.974060i $$-0.427340\pi$$
0.226290 + 0.974060i $$0.427340\pi$$
$$588$$ 0 0
$$589$$ 154.412 0.0108021
$$590$$ 0 0
$$591$$ 12494.9 0.869666
$$592$$ 0 0
$$593$$ 16421.7 1.13720 0.568599 0.822615i $$-0.307486\pi$$
0.568599 + 0.822615i $$0.307486\pi$$
$$594$$ 0 0
$$595$$ 20287.4 1.39782
$$596$$ 0 0
$$597$$ 6757.31 0.463247
$$598$$ 0 0
$$599$$ −914.614 −0.0623875 −0.0311938 0.999513i $$-0.509931\pi$$
−0.0311938 + 0.999513i $$0.509931\pi$$
$$600$$ 0 0
$$601$$ −28107.4 −1.90770 −0.953848 0.300289i $$-0.902917\pi$$
−0.953848 + 0.300289i $$0.902917\pi$$
$$602$$ 0 0
$$603$$ 5890.07 0.397781
$$604$$ 0 0
$$605$$ 12117.6 0.814297
$$606$$ 0 0
$$607$$ 9581.92 0.640722 0.320361 0.947296i $$-0.396196\pi$$
0.320361 + 0.947296i $$0.396196\pi$$
$$608$$ 0 0
$$609$$ 446.983 0.0297417
$$610$$ 0 0
$$611$$ 6196.57 0.410289
$$612$$ 0 0
$$613$$ 1118.28 0.0736814 0.0368407 0.999321i $$-0.488271\pi$$
0.0368407 + 0.999321i $$0.488271\pi$$
$$614$$ 0 0
$$615$$ 4510.19 0.295721
$$616$$ 0 0
$$617$$ 8230.82 0.537051 0.268525 0.963273i $$-0.413464\pi$$
0.268525 + 0.963273i $$0.413464\pi$$
$$618$$ 0 0
$$619$$ −5312.60 −0.344962 −0.172481 0.985013i $$-0.555178\pi$$
−0.172481 + 0.985013i $$0.555178\pi$$
$$620$$ 0 0
$$621$$ 3221.90 0.208197
$$622$$ 0 0
$$623$$ −33310.5 −2.14215
$$624$$ 0 0
$$625$$ −19010.1 −1.21664
$$626$$ 0 0
$$627$$ −49.1067 −0.00312781
$$628$$ 0 0
$$629$$ 9146.39 0.579794
$$630$$ 0 0
$$631$$ −6797.18 −0.428830 −0.214415 0.976743i $$-0.568784\pi$$
−0.214415 + 0.976743i $$0.568784\pi$$
$$632$$ 0 0
$$633$$ 13523.2 0.849129
$$634$$ 0 0
$$635$$ −10337.7 −0.646049
$$636$$ 0 0
$$637$$ 3557.42 0.221272
$$638$$ 0 0
$$639$$ −3518.95 −0.217852
$$640$$ 0 0
$$641$$ 7436.63 0.458236 0.229118 0.973399i $$-0.426416\pi$$
0.229118 + 0.973399i $$0.426416\pi$$
$$642$$ 0 0
$$643$$ −3976.51 −0.243886 −0.121943 0.992537i $$-0.538912\pi$$
−0.121943 + 0.992537i $$0.538912\pi$$
$$644$$ 0 0
$$645$$ 2605.34 0.159047
$$646$$ 0 0
$$647$$ 29726.1 1.80626 0.903132 0.429362i $$-0.141262\pi$$
0.903132 + 0.429362i $$0.141262\pi$$
$$648$$ 0 0
$$649$$ 1533.85 0.0927720
$$650$$ 0 0
$$651$$ 13819.4 0.831992
$$652$$ 0 0
$$653$$ 18774.1 1.12510 0.562548 0.826765i $$-0.309821\pi$$
0.562548 + 0.826765i $$0.309821\pi$$
$$654$$ 0 0
$$655$$ 4696.22 0.280148
$$656$$ 0 0
$$657$$ −758.815 −0.0450596
$$658$$ 0 0
$$659$$ −7601.47 −0.449334 −0.224667 0.974436i $$-0.572129\pi$$
−0.224667 + 0.974436i $$0.572129\pi$$
$$660$$ 0 0
$$661$$ −14701.5 −0.865089 −0.432544 0.901613i $$-0.642384\pi$$
−0.432544 + 0.901613i $$0.642384\pi$$
$$662$$ 0 0
$$663$$ −2482.93 −0.145443
$$664$$ 0 0
$$665$$ −265.251 −0.0154677
$$666$$ 0 0
$$667$$ 715.978 0.0415634
$$668$$ 0 0
$$669$$ −2589.84 −0.149670
$$670$$ 0 0
$$671$$ 11994.6 0.690086
$$672$$ 0 0
$$673$$ 2577.04 0.147604 0.0738020 0.997273i $$-0.476487\pi$$
0.0738020 + 0.997273i $$0.476487\pi$$
$$674$$ 0 0
$$675$$ 1071.10 0.0610766
$$676$$ 0 0
$$677$$ −10227.0 −0.580583 −0.290291 0.956938i $$-0.593752\pi$$
−0.290291 + 0.956938i $$0.593752\pi$$
$$678$$ 0 0
$$679$$ 19939.2 1.12694
$$680$$ 0 0
$$681$$ 13988.0 0.787111
$$682$$ 0 0
$$683$$ −11239.2 −0.629656 −0.314828 0.949149i $$-0.601947\pi$$
−0.314828 + 0.949149i $$0.601947\pi$$
$$684$$ 0 0
$$685$$ 29768.5 1.66043
$$686$$ 0 0
$$687$$ 10983.8 0.609982
$$688$$ 0 0
$$689$$ −771.139 −0.0426387
$$690$$ 0 0
$$691$$ 4388.42 0.241597 0.120798 0.992677i $$-0.461455\pi$$
0.120798 + 0.992677i $$0.461455\pi$$
$$692$$ 0 0
$$693$$ −4394.92 −0.240908
$$694$$ 0 0
$$695$$ 23312.2 1.27235
$$696$$ 0 0
$$697$$ 7458.74 0.405337
$$698$$ 0 0
$$699$$ 2351.53 0.127243
$$700$$ 0 0
$$701$$ −15247.0 −0.821500 −0.410750 0.911748i $$-0.634733\pi$$
−0.410750 + 0.911748i $$0.634733\pi$$
$$702$$ 0 0
$$703$$ −119.586 −0.00641576
$$704$$ 0 0
$$705$$ −18350.0 −0.980287
$$706$$ 0 0
$$707$$ 12475.0 0.663609
$$708$$ 0 0
$$709$$ −22073.9 −1.16926 −0.584628 0.811301i $$-0.698760\pi$$
−0.584628 + 0.811301i $$0.698760\pi$$
$$710$$ 0 0
$$711$$ 10589.8 0.558579
$$712$$ 0 0
$$713$$ 22136.0 1.16269
$$714$$ 0 0
$$715$$ 3280.50 0.171586
$$716$$ 0 0
$$717$$ 6838.32 0.356181
$$718$$ 0 0
$$719$$ 20988.2 1.08863 0.544317 0.838879i $$-0.316789\pi$$
0.544317 + 0.838879i $$0.316789\pi$$
$$720$$ 0 0
$$721$$ 47899.9 2.47418
$$722$$ 0 0
$$723$$ 13058.6 0.671722
$$724$$ 0 0
$$725$$ 238.022 0.0121930
$$726$$ 0 0
$$727$$ 11692.8 0.596509 0.298254 0.954486i $$-0.403596\pi$$
0.298254 + 0.954486i $$0.403596\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 4308.58 0.218001
$$732$$ 0 0
$$733$$ −21190.2 −1.06777 −0.533887 0.845556i $$-0.679269\pi$$
−0.533887 + 0.845556i $$0.679269\pi$$
$$734$$ 0 0
$$735$$ −10534.7 −0.528677
$$736$$ 0 0
$$737$$ −12869.7 −0.643230
$$738$$ 0 0
$$739$$ −6905.92 −0.343759 −0.171880 0.985118i $$-0.554984\pi$$
−0.171880 + 0.985118i $$0.554984\pi$$
$$740$$ 0 0
$$741$$ 32.4635 0.00160941
$$742$$ 0 0
$$743$$ −2539.96 −0.125413 −0.0627067 0.998032i $$-0.519973\pi$$
−0.0627067 + 0.998032i $$0.519973\pi$$
$$744$$ 0 0
$$745$$ 22487.2 1.10586
$$746$$ 0 0
$$747$$ 3875.83 0.189838
$$748$$ 0 0
$$749$$ 43703.9 2.13205
$$750$$ 0 0
$$751$$ 3805.24 0.184894 0.0924468 0.995718i $$-0.470531\pi$$
0.0924468 + 0.995718i $$0.470531\pi$$
$$752$$ 0 0
$$753$$ −11553.7 −0.559151
$$754$$ 0 0
$$755$$ 7572.76 0.365034
$$756$$ 0 0
$$757$$ 20266.6 0.973052 0.486526 0.873666i $$-0.338264\pi$$
0.486526 + 0.873666i $$0.338264\pi$$
$$758$$ 0 0
$$759$$ −7039.78 −0.336664
$$760$$ 0 0
$$761$$ 28056.2 1.33645 0.668224 0.743960i $$-0.267055\pi$$
0.668224 + 0.743960i $$0.267055\pi$$
$$762$$ 0 0
$$763$$ 1323.74 0.0628083
$$764$$ 0 0
$$765$$ 7352.75 0.347502
$$766$$ 0 0
$$767$$ −1014.00 −0.0477359
$$768$$ 0 0
$$769$$ 16703.0 0.783260 0.391630 0.920123i $$-0.371911\pi$$
0.391630 + 0.920123i $$0.371911\pi$$
$$770$$ 0 0
$$771$$ 9272.98 0.433150
$$772$$ 0 0
$$773$$ −25155.9 −1.17050 −0.585250 0.810853i $$-0.699003\pi$$
−0.585250 + 0.810853i $$0.699003\pi$$
$$774$$ 0 0
$$775$$ 7358.97 0.341087
$$776$$ 0 0
$$777$$ −10702.6 −0.494150
$$778$$ 0 0
$$779$$ −97.5206 −0.00448529
$$780$$ 0 0
$$781$$ 7688.82 0.352276
$$782$$ 0 0
$$783$$ 162.000 0.00739388
$$784$$ 0 0
$$785$$ −30009.3 −1.36443
$$786$$ 0 0
$$787$$ 8287.15 0.375356 0.187678 0.982231i $$-0.439904\pi$$
0.187678 + 0.982231i $$0.439904\pi$$
$$788$$ 0 0
$$789$$ −26.0112 −0.00117367
$$790$$ 0 0
$$791$$ 6323.52 0.284246
$$792$$ 0 0
$$793$$ −7929.42 −0.355084
$$794$$ 0 0
$$795$$ 2283.59 0.101875
$$796$$ 0 0
$$797$$ −28506.5 −1.26694 −0.633470 0.773767i $$-0.718370\pi$$
−0.633470 + 0.773767i $$0.718370\pi$$
$$798$$ 0 0
$$799$$ −30346.4 −1.34365
$$800$$ 0 0
$$801$$ −12072.7 −0.532545
$$802$$ 0 0
$$803$$ 1657.99 0.0728634
$$804$$ 0 0
$$805$$ −38025.5 −1.66487
$$806$$ 0 0
$$807$$ 22456.5 0.979562
$$808$$ 0 0
$$809$$ −31026.7 −1.34838 −0.674191 0.738557i $$-0.735508\pi$$
−0.674191 + 0.738557i $$0.735508\pi$$
$$810$$ 0 0
$$811$$ −11304.9 −0.489480 −0.244740 0.969589i $$-0.578703\pi$$
−0.244740 + 0.969589i $$0.578703\pi$$
$$812$$ 0 0
$$813$$ 19073.5 0.822801
$$814$$ 0 0
$$815$$ 164.959 0.00708988
$$816$$ 0 0
$$817$$ −56.3333 −0.00241231
$$818$$ 0 0
$$819$$ 2905.39 0.123959
$$820$$ 0 0
$$821$$ 32811.2 1.39479 0.697393 0.716688i $$-0.254343\pi$$
0.697393 + 0.716688i $$0.254343\pi$$
$$822$$ 0 0
$$823$$ 39056.0 1.65420 0.827099 0.562056i $$-0.189990\pi$$
0.827099 + 0.562056i $$0.189990\pi$$
$$824$$ 0 0
$$825$$ −2340.33 −0.0987635
$$826$$ 0 0
$$827$$ −8221.77 −0.345706 −0.172853 0.984948i $$-0.555299\pi$$
−0.172853 + 0.984948i $$0.555299\pi$$
$$828$$ 0 0
$$829$$ −17201.9 −0.720686 −0.360343 0.932820i $$-0.617340\pi$$
−0.360343 + 0.932820i $$0.617340\pi$$
$$830$$ 0 0
$$831$$ −21697.4 −0.905744
$$832$$ 0 0
$$833$$ −17421.7 −0.724643
$$834$$ 0 0
$$835$$ 41532.1 1.72129
$$836$$ 0 0
$$837$$ 5008.58 0.206836
$$838$$ 0 0
$$839$$ −37656.5 −1.54952 −0.774759 0.632257i $$-0.782129\pi$$
−0.774759 + 0.632257i $$0.782129\pi$$
$$840$$ 0 0
$$841$$ −24353.0 −0.998524
$$842$$ 0 0
$$843$$ 8660.60 0.353840
$$844$$ 0 0
$$845$$ −2168.68 −0.0882896
$$846$$ 0 0
$$847$$ −23449.1 −0.951265
$$848$$ 0 0
$$849$$ 19239.8 0.777747
$$850$$ 0 0
$$851$$ −17143.5 −0.690564
$$852$$ 0 0
$$853$$ −19882.5 −0.798082 −0.399041 0.916933i $$-0.630657\pi$$
−0.399041 + 0.916933i $$0.630657\pi$$
$$854$$ 0 0
$$855$$ −96.1348 −0.00384531
$$856$$ 0 0
$$857$$ −45416.6 −1.81027 −0.905135 0.425125i $$-0.860230\pi$$
−0.905135 + 0.425125i $$0.860230\pi$$
$$858$$ 0 0
$$859$$ 18572.2 0.737689 0.368845 0.929491i $$-0.379753\pi$$
0.368845 + 0.929491i $$0.379753\pi$$
$$860$$ 0 0
$$861$$ −8727.82 −0.345463
$$862$$ 0 0
$$863$$ −43854.1 −1.72979 −0.864896 0.501952i $$-0.832616\pi$$
−0.864896 + 0.501952i $$0.832616\pi$$
$$864$$ 0 0
$$865$$ 47542.1 1.86876
$$866$$ 0 0
$$867$$ −2579.38 −0.101039
$$868$$ 0 0
$$869$$ −23138.5 −0.903246
$$870$$ 0 0
$$871$$ 8507.88 0.330974
$$872$$ 0 0
$$873$$ 7226.54 0.280162
$$874$$ 0 0
$$875$$ 27191.1 1.05054
$$876$$ 0 0
$$877$$ −8076.73 −0.310983 −0.155491 0.987837i $$-0.549696\pi$$
−0.155491 + 0.987837i $$0.549696\pi$$
$$878$$ 0 0
$$879$$ 3671.47 0.140883
$$880$$ 0 0
$$881$$ −28647.3 −1.09552 −0.547759 0.836636i $$-0.684519\pi$$
−0.547759 + 0.836636i $$0.684519\pi$$
$$882$$ 0 0
$$883$$ −25901.8 −0.987164 −0.493582 0.869699i $$-0.664313\pi$$
−0.493582 + 0.869699i $$0.664313\pi$$
$$884$$ 0 0
$$885$$ 3002.78 0.114054
$$886$$ 0 0
$$887$$ −9023.06 −0.341561 −0.170780 0.985309i $$-0.554629\pi$$
−0.170780 + 0.985309i $$0.554629\pi$$
$$888$$ 0 0
$$889$$ 20004.9 0.754717
$$890$$ 0 0
$$891$$ −1592.85 −0.0598905
$$892$$ 0 0
$$893$$ 396.770 0.0148683
$$894$$ 0 0
$$895$$ −31919.0 −1.19210
$$896$$ 0 0
$$897$$ 4653.85 0.173230
$$898$$ 0 0
$$899$$ 1113.02 0.0412916
$$900$$ 0 0
$$901$$ 3776.49 0.139637
$$902$$ 0 0
$$903$$ −5041.67 −0.185799
$$904$$ 0 0
$$905$$ −55980.9 −2.05621
$$906$$ 0 0
$$907$$ 9526.00 0.348738 0.174369 0.984680i $$-0.444211\pi$$
0.174369 + 0.984680i $$0.444211\pi$$
$$908$$ 0 0
$$909$$ 4521.32 0.164975
$$910$$ 0 0
$$911$$ −8695.33 −0.316234 −0.158117 0.987420i $$-0.550542\pi$$
−0.158117 + 0.987420i $$0.550542\pi$$
$$912$$ 0 0
$$913$$ −8468.60 −0.306977
$$914$$ 0 0
$$915$$ 23481.6 0.848389
$$916$$ 0 0
$$917$$ −9087.82 −0.327270
$$918$$ 0 0
$$919$$ −3367.18 −0.120863 −0.0604315 0.998172i $$-0.519248\pi$$
−0.0604315 + 0.998172i $$0.519248\pi$$
$$920$$ 0 0
$$921$$ 11559.6 0.413575
$$922$$ 0 0
$$923$$ −5082.93 −0.181264
$$924$$ 0 0
$$925$$ −5699.24 −0.202584
$$926$$ 0 0
$$927$$ 17360.3 0.615090
$$928$$ 0 0
$$929$$ −25124.3 −0.887300 −0.443650 0.896200i $$-0.646317\pi$$
−0.443650 + 0.896200i $$0.646317\pi$$
$$930$$ 0 0
$$931$$ 227.784 0.00801859
$$932$$ 0 0
$$933$$ −23084.6 −0.810029
$$934$$ 0 0
$$935$$ −16065.6 −0.561926
$$936$$ 0 0
$$937$$ 1387.27 0.0483674 0.0241837 0.999708i $$-0.492301\pi$$
0.0241837 + 0.999708i $$0.492301\pi$$
$$938$$ 0 0
$$939$$ 8229.89 0.286019
$$940$$ 0 0
$$941$$ −579.261 −0.0200674 −0.0100337 0.999950i $$-0.503194\pi$$
−0.0100337 + 0.999950i $$0.503194\pi$$
$$942$$ 0 0
$$943$$ −13980.2 −0.482777
$$944$$ 0 0
$$945$$ −8603.80 −0.296171
$$946$$ 0 0
$$947$$ 23162.3 0.794798 0.397399 0.917646i $$-0.369913\pi$$
0.397399 + 0.917646i $$0.369913\pi$$
$$948$$ 0 0
$$949$$ −1096.07 −0.0374919
$$950$$ 0 0
$$951$$ −14120.0 −0.481464
$$952$$ 0 0
$$953$$ 4896.52 0.166436 0.0832182 0.996531i $$-0.473480\pi$$
0.0832182 + 0.996531i $$0.473480\pi$$
$$954$$ 0 0
$$955$$ 36142.6 1.22466
$$956$$ 0 0
$$957$$ −353.966 −0.0119562
$$958$$ 0 0
$$959$$ −57606.0 −1.93972
$$960$$ 0 0
$$961$$ 4620.29 0.155090
$$962$$ 0 0
$$963$$ 15839.6 0.530035
$$964$$ 0 0
$$965$$ −54580.4 −1.82073
$$966$$ 0 0
$$967$$ 48707.2 1.61977 0.809885 0.586589i $$-0.199530\pi$$
0.809885 + 0.586589i $$0.199530\pi$$
$$968$$ 0 0
$$969$$ −158.983 −0.00527067
$$970$$ 0 0
$$971$$ −55808.4 −1.84446 −0.922232 0.386636i $$-0.873637\pi$$
−0.922232 + 0.386636i $$0.873637\pi$$
$$972$$ 0 0
$$973$$ −45112.3 −1.48637
$$974$$ 0 0
$$975$$ 1547.15 0.0508188
$$976$$ 0 0
$$977$$ 12136.2 0.397413 0.198707 0.980059i $$-0.436326\pi$$
0.198707 + 0.980059i $$0.436326\pi$$
$$978$$ 0 0
$$979$$ 26378.6 0.861148
$$980$$ 0 0
$$981$$ 479.764 0.0156144
$$982$$ 0 0
$$983$$ 43305.9 1.40513 0.702566 0.711619i $$-0.252038\pi$$
0.702566 + 0.711619i $$0.252038\pi$$
$$984$$ 0 0
$$985$$ −53446.6 −1.72888
$$986$$ 0 0
$$987$$ 35509.8 1.14518
$$988$$ 0 0
$$989$$ −8075.75 −0.259650
$$990$$ 0 0
$$991$$ −36327.3 −1.16446 −0.582228 0.813025i $$-0.697819\pi$$
−0.582228 + 0.813025i $$0.697819\pi$$
$$992$$ 0 0
$$993$$ 27812.5 0.888825
$$994$$ 0 0
$$995$$ −28904.2 −0.920928
$$996$$ 0 0
$$997$$ 57063.1 1.81264 0.906322 0.422588i $$-0.138878\pi$$
0.906322 + 0.422588i $$0.138878\pi$$
$$998$$ 0 0
$$999$$ −3878.95 −0.122847
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 2496.4.a.bi.1.1 2
4.3 odd 2 2496.4.a.x.1.1 2
8.3 odd 2 624.4.a.n.1.2 2
8.5 even 2 312.4.a.b.1.2 2
24.5 odd 2 936.4.a.h.1.1 2
24.11 even 2 1872.4.a.bd.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.b.1.2 2 8.5 even 2
624.4.a.n.1.2 2 8.3 odd 2
936.4.a.h.1.1 2 24.5 odd 2
1872.4.a.bd.1.1 2 24.11 even 2
2496.4.a.x.1.1 2 4.3 odd 2
2496.4.a.bi.1.1 2 1.1 even 1 trivial