# Properties

 Label 1521.4.a.bf.1.3 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $9$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.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: $$89.7419051187$$ Analytic rank: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27$$ x^9 - 56*x^7 - 27*x^6 + 945*x^5 + 763*x^4 - 4139*x^3 - 2478*x^2 + 63*x + 27 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$13^{2}$$ Twist minimal: no (minimal twist has level 507) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-4.83218$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.03025 q^{2} +8.24289 q^{4} -8.08864 q^{5} +5.95078 q^{7} -0.978887 q^{8} +O(q^{10})$$ $$q-4.03025 q^{2} +8.24289 q^{4} -8.08864 q^{5} +5.95078 q^{7} -0.978887 q^{8} +32.5992 q^{10} -17.2359 q^{11} -23.9831 q^{14} -61.9979 q^{16} -92.9299 q^{17} +13.3832 q^{19} -66.6738 q^{20} +69.4648 q^{22} -219.710 q^{23} -59.5738 q^{25} +49.0516 q^{28} +199.485 q^{29} +307.777 q^{31} +257.698 q^{32} +374.531 q^{34} -48.1337 q^{35} -333.777 q^{37} -53.9376 q^{38} +7.91787 q^{40} +200.689 q^{41} +116.806 q^{43} -142.073 q^{44} +885.487 q^{46} -338.610 q^{47} -307.588 q^{49} +240.097 q^{50} +26.6215 q^{53} +139.415 q^{55} -5.82514 q^{56} -803.976 q^{58} +280.058 q^{59} -207.084 q^{61} -1240.42 q^{62} -542.603 q^{64} -285.981 q^{67} -766.011 q^{68} +193.991 q^{70} +317.673 q^{71} -63.0668 q^{73} +1345.20 q^{74} +110.316 q^{76} -102.567 q^{77} -623.835 q^{79} +501.479 q^{80} -808.824 q^{82} -659.874 q^{83} +751.677 q^{85} -470.755 q^{86} +16.8720 q^{88} -1273.19 q^{89} -1811.05 q^{92} +1364.68 q^{94} -108.252 q^{95} +603.746 q^{97} +1239.66 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10})$$ 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8 $$9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29} + 717 q^{31} - 707 q^{32} + 632 q^{34} + 418 q^{35} + 1108 q^{37} + 660 q^{38} - 1506 q^{40} - 334 q^{41} + 242 q^{43} + 307 q^{44} + 979 q^{46} + 184 q^{47} - 38 q^{49} + 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} + 1161 q^{58} - 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} + 2308 q^{67} - 2785 q^{68} - 1420 q^{70} - 96 q^{71} + 2505 q^{73} + 1191 q^{74} + 2409 q^{76} + 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 1517 q^{82} - 1539 q^{83} + 4296 q^{85} + 3763 q^{86} - 3716 q^{88} + 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} + 1445 q^{97} - 1457 q^{98}+O(q^{100})$$ 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8 - 54 * q^10 - 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 + 352 * q^19 - 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 + 940 * q^28 + 97 * q^29 + 717 * q^31 - 707 * q^32 + 632 * q^34 + 418 * q^35 + 1108 * q^37 + 660 * q^38 - 1506 * q^40 - 334 * q^41 + 242 * q^43 + 307 * q^44 + 979 * q^46 + 184 * q^47 - 38 * q^49 + 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 + 1161 * q^58 - 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 + 2308 * q^67 - 2785 * q^68 - 1420 * q^70 - 96 * q^71 + 2505 * q^73 + 1191 * q^74 + 2409 * q^76 + 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 1517 * q^82 - 1539 * q^83 + 4296 * q^85 + 3763 * q^86 - 3716 * q^88 + 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 + 1445 * q^97 - 1457 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.03025 −1.42491 −0.712454 0.701719i $$-0.752416\pi$$
−0.712454 + 0.701719i $$0.752416\pi$$
$$3$$ 0 0
$$4$$ 8.24289 1.03036
$$5$$ −8.08864 −0.723470 −0.361735 0.932281i $$-0.617816\pi$$
−0.361735 + 0.932281i $$0.617816\pi$$
$$6$$ 0 0
$$7$$ 5.95078 0.321312 0.160656 0.987010i $$-0.448639\pi$$
0.160656 + 0.987010i $$0.448639\pi$$
$$8$$ −0.978887 −0.0432611
$$9$$ 0 0
$$10$$ 32.5992 1.03088
$$11$$ −17.2359 −0.472437 −0.236219 0.971700i $$-0.575908\pi$$
−0.236219 + 0.971700i $$0.575908\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −23.9831 −0.457840
$$15$$ 0 0
$$16$$ −61.9979 −0.968718
$$17$$ −92.9299 −1.32581 −0.662907 0.748702i $$-0.730677\pi$$
−0.662907 + 0.748702i $$0.730677\pi$$
$$18$$ 0 0
$$19$$ 13.3832 0.161596 0.0807979 0.996731i $$-0.474253\pi$$
0.0807979 + 0.996731i $$0.474253\pi$$
$$20$$ −66.6738 −0.745435
$$21$$ 0 0
$$22$$ 69.4648 0.673179
$$23$$ −219.710 −1.99186 −0.995930 0.0901293i $$-0.971272\pi$$
−0.995930 + 0.0901293i $$0.971272\pi$$
$$24$$ 0 0
$$25$$ −59.5738 −0.476591
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 49.0516 0.331067
$$29$$ 199.485 1.27736 0.638681 0.769471i $$-0.279480\pi$$
0.638681 + 0.769471i $$0.279480\pi$$
$$30$$ 0 0
$$31$$ 307.777 1.78317 0.891586 0.452851i $$-0.149593\pi$$
0.891586 + 0.452851i $$0.149593\pi$$
$$32$$ 257.698 1.42359
$$33$$ 0 0
$$34$$ 374.531 1.88916
$$35$$ −48.1337 −0.232460
$$36$$ 0 0
$$37$$ −333.777 −1.48304 −0.741521 0.670930i $$-0.765895\pi$$
−0.741521 + 0.670930i $$0.765895\pi$$
$$38$$ −53.9376 −0.230259
$$39$$ 0 0
$$40$$ 7.91787 0.0312981
$$41$$ 200.689 0.764446 0.382223 0.924070i $$-0.375159\pi$$
0.382223 + 0.924070i $$0.375159\pi$$
$$42$$ 0 0
$$43$$ 116.806 0.414248 0.207124 0.978315i $$-0.433590\pi$$
0.207124 + 0.978315i $$0.433590\pi$$
$$44$$ −142.073 −0.486781
$$45$$ 0 0
$$46$$ 885.487 2.83822
$$47$$ −338.610 −1.05088 −0.525440 0.850831i $$-0.676099\pi$$
−0.525440 + 0.850831i $$0.676099\pi$$
$$48$$ 0 0
$$49$$ −307.588 −0.896759
$$50$$ 240.097 0.679097
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 26.6215 0.0689951 0.0344975 0.999405i $$-0.489017\pi$$
0.0344975 + 0.999405i $$0.489017\pi$$
$$54$$ 0 0
$$55$$ 139.415 0.341794
$$56$$ −5.82514 −0.0139003
$$57$$ 0 0
$$58$$ −803.976 −1.82012
$$59$$ 280.058 0.617973 0.308987 0.951066i $$-0.400010\pi$$
0.308987 + 0.951066i $$0.400010\pi$$
$$60$$ 0 0
$$61$$ −207.084 −0.434663 −0.217332 0.976098i $$-0.569735\pi$$
−0.217332 + 0.976098i $$0.569735\pi$$
$$62$$ −1240.42 −2.54086
$$63$$ 0 0
$$64$$ −542.603 −1.05977
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −285.981 −0.521465 −0.260732 0.965411i $$-0.583964\pi$$
−0.260732 + 0.965411i $$0.583964\pi$$
$$68$$ −766.011 −1.36607
$$69$$ 0 0
$$70$$ 193.991 0.331233
$$71$$ 317.673 0.530998 0.265499 0.964111i $$-0.414463\pi$$
0.265499 + 0.964111i $$0.414463\pi$$
$$72$$ 0 0
$$73$$ −63.0668 −0.101115 −0.0505576 0.998721i $$-0.516100\pi$$
−0.0505576 + 0.998721i $$0.516100\pi$$
$$74$$ 1345.20 2.11320
$$75$$ 0 0
$$76$$ 110.316 0.166502
$$77$$ −102.567 −0.151800
$$78$$ 0 0
$$79$$ −623.835 −0.888443 −0.444221 0.895917i $$-0.646520\pi$$
−0.444221 + 0.895917i $$0.646520\pi$$
$$80$$ 501.479 0.700838
$$81$$ 0 0
$$82$$ −808.824 −1.08926
$$83$$ −659.874 −0.872658 −0.436329 0.899787i $$-0.643722\pi$$
−0.436329 + 0.899787i $$0.643722\pi$$
$$84$$ 0 0
$$85$$ 751.677 0.959186
$$86$$ −470.755 −0.590265
$$87$$ 0 0
$$88$$ 16.8720 0.0204382
$$89$$ −1273.19 −1.51638 −0.758191 0.652032i $$-0.773917\pi$$
−0.758191 + 0.652032i $$0.773917\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −1811.05 −2.05233
$$93$$ 0 0
$$94$$ 1364.68 1.49741
$$95$$ −108.252 −0.116910
$$96$$ 0 0
$$97$$ 603.746 0.631970 0.315985 0.948764i $$-0.397665\pi$$
0.315985 + 0.948764i $$0.397665\pi$$
$$98$$ 1239.66 1.27780
$$99$$ 0 0
$$100$$ −491.060 −0.491060
$$101$$ 740.588 0.729616 0.364808 0.931083i $$-0.381135\pi$$
0.364808 + 0.931083i $$0.381135\pi$$
$$102$$ 0 0
$$103$$ −1888.57 −1.80666 −0.903332 0.428942i $$-0.858887\pi$$
−0.903332 + 0.428942i $$0.858887\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −107.291 −0.0983116
$$107$$ 919.463 0.830728 0.415364 0.909655i $$-0.363654\pi$$
0.415364 + 0.909655i $$0.363654\pi$$
$$108$$ 0 0
$$109$$ 1570.40 1.37998 0.689988 0.723821i $$-0.257616\pi$$
0.689988 + 0.723821i $$0.257616\pi$$
$$110$$ −561.876 −0.487025
$$111$$ 0 0
$$112$$ −368.936 −0.311260
$$113$$ 1324.66 1.10277 0.551386 0.834250i $$-0.314099\pi$$
0.551386 + 0.834250i $$0.314099\pi$$
$$114$$ 0 0
$$115$$ 1777.16 1.44105
$$116$$ 1644.34 1.31614
$$117$$ 0 0
$$118$$ −1128.70 −0.880555
$$119$$ −553.006 −0.426000
$$120$$ 0 0
$$121$$ −1033.92 −0.776803
$$122$$ 834.601 0.619354
$$123$$ 0 0
$$124$$ 2536.97 1.83731
$$125$$ 1492.95 1.06827
$$126$$ 0 0
$$127$$ −2350.39 −1.64223 −0.821115 0.570763i $$-0.806648\pi$$
−0.821115 + 0.570763i $$0.806648\pi$$
$$128$$ 125.240 0.0864824
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2308.69 −1.53978 −0.769890 0.638177i $$-0.779689\pi$$
−0.769890 + 0.638177i $$0.779689\pi$$
$$132$$ 0 0
$$133$$ 79.6405 0.0519226
$$134$$ 1152.57 0.743039
$$135$$ 0 0
$$136$$ 90.9680 0.0573562
$$137$$ 374.912 0.233802 0.116901 0.993144i $$-0.462704\pi$$
0.116901 + 0.993144i $$0.462704\pi$$
$$138$$ 0 0
$$139$$ 487.711 0.297605 0.148802 0.988867i $$-0.452458\pi$$
0.148802 + 0.988867i $$0.452458\pi$$
$$140$$ −396.761 −0.239517
$$141$$ 0 0
$$142$$ −1280.30 −0.756623
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −1613.57 −0.924134
$$146$$ 254.175 0.144080
$$147$$ 0 0
$$148$$ −2751.28 −1.52807
$$149$$ −1055.91 −0.580558 −0.290279 0.956942i $$-0.593748\pi$$
−0.290279 + 0.956942i $$0.593748\pi$$
$$150$$ 0 0
$$151$$ −888.560 −0.478874 −0.239437 0.970912i $$-0.576963\pi$$
−0.239437 + 0.970912i $$0.576963\pi$$
$$152$$ −13.1007 −0.00699081
$$153$$ 0 0
$$154$$ 413.370 0.216301
$$155$$ −2489.50 −1.29007
$$156$$ 0 0
$$157$$ −3648.56 −1.85469 −0.927346 0.374206i $$-0.877915\pi$$
−0.927346 + 0.374206i $$0.877915\pi$$
$$158$$ 2514.21 1.26595
$$159$$ 0 0
$$160$$ −2084.43 −1.02993
$$161$$ −1307.45 −0.640008
$$162$$ 0 0
$$163$$ 1386.49 0.666249 0.333125 0.942883i $$-0.391897\pi$$
0.333125 + 0.942883i $$0.391897\pi$$
$$164$$ 1654.25 0.787655
$$165$$ 0 0
$$166$$ 2659.46 1.24346
$$167$$ 3376.19 1.56442 0.782209 0.623017i $$-0.214093\pi$$
0.782209 + 0.623017i $$0.214093\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −3029.44 −1.36675
$$171$$ 0 0
$$172$$ 962.815 0.426825
$$173$$ −341.817 −0.150219 −0.0751095 0.997175i $$-0.523931\pi$$
−0.0751095 + 0.997175i $$0.523931\pi$$
$$174$$ 0 0
$$175$$ −354.511 −0.153134
$$176$$ 1068.59 0.457658
$$177$$ 0 0
$$178$$ 5131.28 2.16070
$$179$$ 2885.55 1.20489 0.602446 0.798159i $$-0.294193\pi$$
0.602446 + 0.798159i $$0.294193\pi$$
$$180$$ 0 0
$$181$$ 3795.62 1.55871 0.779354 0.626584i $$-0.215547\pi$$
0.779354 + 0.626584i $$0.215547\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 215.072 0.0861701
$$185$$ 2699.80 1.07294
$$186$$ 0 0
$$187$$ 1601.73 0.626363
$$188$$ −2791.12 −1.08278
$$189$$ 0 0
$$190$$ 436.282 0.166586
$$191$$ 2805.90 1.06297 0.531486 0.847067i $$-0.321634\pi$$
0.531486 + 0.847067i $$0.321634\pi$$
$$192$$ 0 0
$$193$$ −2485.64 −0.927048 −0.463524 0.886084i $$-0.653415\pi$$
−0.463524 + 0.886084i $$0.653415\pi$$
$$194$$ −2433.24 −0.900499
$$195$$ 0 0
$$196$$ −2535.41 −0.923985
$$197$$ −2750.70 −0.994818 −0.497409 0.867516i $$-0.665715\pi$$
−0.497409 + 0.867516i $$0.665715\pi$$
$$198$$ 0 0
$$199$$ 3998.53 1.42436 0.712182 0.701994i $$-0.247707\pi$$
0.712182 + 0.701994i $$0.247707\pi$$
$$200$$ 58.3161 0.0206178
$$201$$ 0 0
$$202$$ −2984.75 −1.03964
$$203$$ 1187.09 0.410432
$$204$$ 0 0
$$205$$ −1623.30 −0.553054
$$206$$ 7611.41 2.57433
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −230.671 −0.0763438
$$210$$ 0 0
$$211$$ −1375.80 −0.448882 −0.224441 0.974488i $$-0.572056\pi$$
−0.224441 + 0.974488i $$0.572056\pi$$
$$212$$ 219.438 0.0710898
$$213$$ 0 0
$$214$$ −3705.66 −1.18371
$$215$$ −944.799 −0.299696
$$216$$ 0 0
$$217$$ 1831.51 0.572955
$$218$$ −6329.12 −1.96634
$$219$$ 0 0
$$220$$ 1149.18 0.352171
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 1694.37 0.508805 0.254403 0.967098i $$-0.418121\pi$$
0.254403 + 0.967098i $$0.418121\pi$$
$$224$$ 1533.50 0.457418
$$225$$ 0 0
$$226$$ −5338.70 −1.57135
$$227$$ 405.400 0.118534 0.0592672 0.998242i $$-0.481124\pi$$
0.0592672 + 0.998242i $$0.481124\pi$$
$$228$$ 0 0
$$229$$ 2359.54 0.680884 0.340442 0.940265i $$-0.389423\pi$$
0.340442 + 0.940265i $$0.389423\pi$$
$$230$$ −7162.39 −2.05337
$$231$$ 0 0
$$232$$ −195.274 −0.0552601
$$233$$ −938.507 −0.263878 −0.131939 0.991258i $$-0.542120\pi$$
−0.131939 + 0.991258i $$0.542120\pi$$
$$234$$ 0 0
$$235$$ 2738.90 0.760280
$$236$$ 2308.49 0.636736
$$237$$ 0 0
$$238$$ 2228.75 0.607010
$$239$$ −4664.25 −1.26237 −0.631183 0.775634i $$-0.717430\pi$$
−0.631183 + 0.775634i $$0.717430\pi$$
$$240$$ 0 0
$$241$$ −3774.94 −1.00898 −0.504492 0.863416i $$-0.668320\pi$$
−0.504492 + 0.863416i $$0.668320\pi$$
$$242$$ 4166.97 1.10687
$$243$$ 0 0
$$244$$ −1706.97 −0.447860
$$245$$ 2487.97 0.648778
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −301.279 −0.0771421
$$249$$ 0 0
$$250$$ −6016.96 −1.52219
$$251$$ 960.695 0.241588 0.120794 0.992678i $$-0.461456\pi$$
0.120794 + 0.992678i $$0.461456\pi$$
$$252$$ 0 0
$$253$$ 3786.90 0.941029
$$254$$ 9472.64 2.34003
$$255$$ 0 0
$$256$$ 3836.08 0.936542
$$257$$ 16.7302 0.00406070 0.00203035 0.999998i $$-0.499354\pi$$
0.00203035 + 0.999998i $$0.499354\pi$$
$$258$$ 0 0
$$259$$ −1986.23 −0.476519
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 9304.58 2.19404
$$263$$ 400.736 0.0939560 0.0469780 0.998896i $$-0.485041\pi$$
0.0469780 + 0.998896i $$0.485041\pi$$
$$264$$ 0 0
$$265$$ −215.332 −0.0499159
$$266$$ −320.971 −0.0739849
$$267$$ 0 0
$$268$$ −2357.31 −0.537297
$$269$$ −2236.97 −0.507027 −0.253514 0.967332i $$-0.581586\pi$$
−0.253514 + 0.967332i $$0.581586\pi$$
$$270$$ 0 0
$$271$$ −4018.89 −0.900850 −0.450425 0.892814i $$-0.648727\pi$$
−0.450425 + 0.892814i $$0.648727\pi$$
$$272$$ 5761.46 1.28434
$$273$$ 0 0
$$274$$ −1510.99 −0.333147
$$275$$ 1026.81 0.225159
$$276$$ 0 0
$$277$$ 6792.95 1.47346 0.736731 0.676186i $$-0.236368\pi$$
0.736731 + 0.676186i $$0.236368\pi$$
$$278$$ −1965.59 −0.424059
$$279$$ 0 0
$$280$$ 47.1175 0.0100565
$$281$$ 7286.80 1.54695 0.773477 0.633824i $$-0.218516\pi$$
0.773477 + 0.633824i $$0.218516\pi$$
$$282$$ 0 0
$$283$$ −2429.77 −0.510369 −0.255185 0.966892i $$-0.582136\pi$$
−0.255185 + 0.966892i $$0.582136\pi$$
$$284$$ 2618.54 0.547120
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1194.25 0.245626
$$288$$ 0 0
$$289$$ 3722.97 0.757780
$$290$$ 6503.07 1.31681
$$291$$ 0 0
$$292$$ −519.853 −0.104185
$$293$$ −4211.25 −0.839672 −0.419836 0.907600i $$-0.637912\pi$$
−0.419836 + 0.907600i $$0.637912\pi$$
$$294$$ 0 0
$$295$$ −2265.29 −0.447085
$$296$$ 326.730 0.0641580
$$297$$ 0 0
$$298$$ 4255.56 0.827242
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 695.084 0.133103
$$302$$ 3581.12 0.682351
$$303$$ 0 0
$$304$$ −829.731 −0.156541
$$305$$ 1675.03 0.314466
$$306$$ 0 0
$$307$$ 8212.27 1.52671 0.763353 0.645982i $$-0.223552\pi$$
0.763353 + 0.645982i $$0.223552\pi$$
$$308$$ −845.447 −0.156408
$$309$$ 0 0
$$310$$ 10033.3 1.83823
$$311$$ 4238.55 0.772818 0.386409 0.922328i $$-0.373715\pi$$
0.386409 + 0.922328i $$0.373715\pi$$
$$312$$ 0 0
$$313$$ 3807.82 0.687638 0.343819 0.939036i $$-0.388279\pi$$
0.343819 + 0.939036i $$0.388279\pi$$
$$314$$ 14704.6 2.64276
$$315$$ 0 0
$$316$$ −5142.20 −0.915416
$$317$$ 10497.7 1.85996 0.929981 0.367608i $$-0.119823\pi$$
0.929981 + 0.367608i $$0.119823\pi$$
$$318$$ 0 0
$$319$$ −3438.31 −0.603474
$$320$$ 4388.92 0.766713
$$321$$ 0 0
$$322$$ 5269.34 0.911953
$$323$$ −1243.70 −0.214246
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −5587.91 −0.949343
$$327$$ 0 0
$$328$$ −196.452 −0.0330708
$$329$$ −2014.99 −0.337660
$$330$$ 0 0
$$331$$ 11905.2 1.97695 0.988474 0.151389i $$-0.0483746\pi$$
0.988474 + 0.151389i $$0.0483746\pi$$
$$332$$ −5439.27 −0.899152
$$333$$ 0 0
$$334$$ −13606.9 −2.22915
$$335$$ 2313.20 0.377264
$$336$$ 0 0
$$337$$ 8981.18 1.45174 0.725869 0.687833i $$-0.241438\pi$$
0.725869 + 0.687833i $$0.241438\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 6195.99 0.988308
$$341$$ −5304.80 −0.842437
$$342$$ 0 0
$$343$$ −3871.51 −0.609451
$$344$$ −114.339 −0.0179208
$$345$$ 0 0
$$346$$ 1377.61 0.214048
$$347$$ 5712.87 0.883813 0.441906 0.897061i $$-0.354302\pi$$
0.441906 + 0.897061i $$0.354302\pi$$
$$348$$ 0 0
$$349$$ 9507.28 1.45820 0.729102 0.684406i $$-0.239938\pi$$
0.729102 + 0.684406i $$0.239938\pi$$
$$350$$ 1428.77 0.218202
$$351$$ 0 0
$$352$$ −4441.65 −0.672559
$$353$$ 3972.96 0.599035 0.299518 0.954091i $$-0.403174\pi$$
0.299518 + 0.954091i $$0.403174\pi$$
$$354$$ 0 0
$$355$$ −2569.55 −0.384162
$$356$$ −10494.8 −1.56242
$$357$$ 0 0
$$358$$ −11629.5 −1.71686
$$359$$ 109.866 0.0161519 0.00807594 0.999967i $$-0.497429\pi$$
0.00807594 + 0.999967i $$0.497429\pi$$
$$360$$ 0 0
$$361$$ −6679.89 −0.973887
$$362$$ −15297.3 −2.22101
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 510.125 0.0731539
$$366$$ 0 0
$$367$$ 7020.02 0.998480 0.499240 0.866464i $$-0.333613\pi$$
0.499240 + 0.866464i $$0.333613\pi$$
$$368$$ 13621.6 1.92955
$$369$$ 0 0
$$370$$ −10880.9 −1.52884
$$371$$ 158.418 0.0221689
$$372$$ 0 0
$$373$$ 2227.18 0.309166 0.154583 0.987980i $$-0.450597\pi$$
0.154583 + 0.987980i $$0.450597\pi$$
$$374$$ −6455.36 −0.892510
$$375$$ 0 0
$$376$$ 331.461 0.0454622
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −11004.6 −1.49147 −0.745734 0.666244i $$-0.767901\pi$$
−0.745734 + 0.666244i $$0.767901\pi$$
$$380$$ −892.309 −0.120459
$$381$$ 0 0
$$382$$ −11308.5 −1.51464
$$383$$ 4436.85 0.591938 0.295969 0.955198i $$-0.404357\pi$$
0.295969 + 0.955198i $$0.404357\pi$$
$$384$$ 0 0
$$385$$ 829.627 0.109823
$$386$$ 10017.7 1.32096
$$387$$ 0 0
$$388$$ 4976.61 0.651157
$$389$$ 2561.58 0.333874 0.166937 0.985968i $$-0.446612\pi$$
0.166937 + 0.985968i $$0.446612\pi$$
$$390$$ 0 0
$$391$$ 20417.7 2.64083
$$392$$ 301.094 0.0387948
$$393$$ 0 0
$$394$$ 11086.0 1.41752
$$395$$ 5045.98 0.642762
$$396$$ 0 0
$$397$$ 2206.90 0.278996 0.139498 0.990222i $$-0.455451\pi$$
0.139498 + 0.990222i $$0.455451\pi$$
$$398$$ −16115.1 −2.02959
$$399$$ 0 0
$$400$$ 3693.45 0.461682
$$401$$ 8071.02 1.00511 0.502553 0.864546i $$-0.332394\pi$$
0.502553 + 0.864546i $$0.332394\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 6104.58 0.751768
$$405$$ 0 0
$$406$$ −4784.28 −0.584827
$$407$$ 5752.93 0.700644
$$408$$ 0 0
$$409$$ 6298.36 0.761452 0.380726 0.924688i $$-0.375674\pi$$
0.380726 + 0.924688i $$0.375674\pi$$
$$410$$ 6542.29 0.788051
$$411$$ 0 0
$$412$$ −15567.3 −1.86152
$$413$$ 1666.56 0.198562
$$414$$ 0 0
$$415$$ 5337.49 0.631342
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 929.662 0.108783
$$419$$ 1432.01 0.166965 0.0834824 0.996509i $$-0.473396\pi$$
0.0834824 + 0.996509i $$0.473396\pi$$
$$420$$ 0 0
$$421$$ 9641.43 1.11614 0.558069 0.829794i $$-0.311542\pi$$
0.558069 + 0.829794i $$0.311542\pi$$
$$422$$ 5544.82 0.639615
$$423$$ 0 0
$$424$$ −26.0594 −0.00298480
$$425$$ 5536.19 0.631870
$$426$$ 0 0
$$427$$ −1232.31 −0.139662
$$428$$ 7579.03 0.855949
$$429$$ 0 0
$$430$$ 3807.77 0.427040
$$431$$ −3370.61 −0.376698 −0.188349 0.982102i $$-0.560314\pi$$
−0.188349 + 0.982102i $$0.560314\pi$$
$$432$$ 0 0
$$433$$ 6248.91 0.693541 0.346770 0.937950i $$-0.387278\pi$$
0.346770 + 0.937950i $$0.387278\pi$$
$$434$$ −7381.45 −0.816407
$$435$$ 0 0
$$436$$ 12944.7 1.42187
$$437$$ −2940.43 −0.321876
$$438$$ 0 0
$$439$$ 2866.16 0.311604 0.155802 0.987788i $$-0.450204\pi$$
0.155802 + 0.987788i $$0.450204\pi$$
$$440$$ −136.471 −0.0147864
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −10776.3 −1.15575 −0.577877 0.816124i $$-0.696119\pi$$
−0.577877 + 0.816124i $$0.696119\pi$$
$$444$$ 0 0
$$445$$ 10298.4 1.09706
$$446$$ −6828.74 −0.725000
$$447$$ 0 0
$$448$$ −3228.91 −0.340517
$$449$$ −13834.9 −1.45414 −0.727069 0.686564i $$-0.759118\pi$$
−0.727069 + 0.686564i $$0.759118\pi$$
$$450$$ 0 0
$$451$$ −3459.04 −0.361153
$$452$$ 10919.0 1.13625
$$453$$ 0 0
$$454$$ −1633.86 −0.168901
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17233.0 1.76395 0.881976 0.471295i $$-0.156213\pi$$
0.881976 + 0.471295i $$0.156213\pi$$
$$458$$ −9509.51 −0.970197
$$459$$ 0 0
$$460$$ 14648.9 1.48480
$$461$$ −11485.0 −1.16033 −0.580164 0.814500i $$-0.697012\pi$$
−0.580164 + 0.814500i $$0.697012\pi$$
$$462$$ 0 0
$$463$$ 1613.61 0.161967 0.0809837 0.996715i $$-0.474194\pi$$
0.0809837 + 0.996715i $$0.474194\pi$$
$$464$$ −12367.7 −1.23740
$$465$$ 0 0
$$466$$ 3782.41 0.376002
$$467$$ 8149.20 0.807495 0.403747 0.914871i $$-0.367707\pi$$
0.403747 + 0.914871i $$0.367707\pi$$
$$468$$ 0 0
$$469$$ −1701.81 −0.167553
$$470$$ −11038.4 −1.08333
$$471$$ 0 0
$$472$$ −274.145 −0.0267342
$$473$$ −2013.24 −0.195706
$$474$$ 0 0
$$475$$ −797.289 −0.0770150
$$476$$ −4558.36 −0.438933
$$477$$ 0 0
$$478$$ 18798.1 1.79875
$$479$$ −1827.62 −0.174334 −0.0871670 0.996194i $$-0.527781\pi$$
−0.0871670 + 0.996194i $$0.527781\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 15213.9 1.43771
$$483$$ 0 0
$$484$$ −8522.52 −0.800387
$$485$$ −4883.49 −0.457212
$$486$$ 0 0
$$487$$ −7838.95 −0.729398 −0.364699 0.931125i $$-0.618828\pi$$
−0.364699 + 0.931125i $$0.618828\pi$$
$$488$$ 202.712 0.0188040
$$489$$ 0 0
$$490$$ −10027.1 −0.924449
$$491$$ −17196.1 −1.58055 −0.790273 0.612755i $$-0.790061\pi$$
−0.790273 + 0.612755i $$0.790061\pi$$
$$492$$ 0 0
$$493$$ −18538.2 −1.69354
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −19081.5 −1.72739
$$497$$ 1890.40 0.170616
$$498$$ 0 0
$$499$$ 5355.28 0.480431 0.240216 0.970720i $$-0.422782\pi$$
0.240216 + 0.970720i $$0.422782\pi$$
$$500$$ 12306.2 1.10070
$$501$$ 0 0
$$502$$ −3871.84 −0.344240
$$503$$ −2979.80 −0.264141 −0.132071 0.991240i $$-0.542163\pi$$
−0.132071 + 0.991240i $$0.542163\pi$$
$$504$$ 0 0
$$505$$ −5990.35 −0.527856
$$506$$ −15262.1 −1.34088
$$507$$ 0 0
$$508$$ −19374.0 −1.69209
$$509$$ −15353.7 −1.33701 −0.668507 0.743706i $$-0.733066\pi$$
−0.668507 + 0.743706i $$0.733066\pi$$
$$510$$ 0 0
$$511$$ −375.297 −0.0324895
$$512$$ −16462.3 −1.42097
$$513$$ 0 0
$$514$$ −67.4267 −0.00578611
$$515$$ 15276.0 1.30707
$$516$$ 0 0
$$517$$ 5836.24 0.496475
$$518$$ 8005.00 0.678995
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −433.801 −0.0364783 −0.0182391 0.999834i $$-0.505806\pi$$
−0.0182391 + 0.999834i $$0.505806\pi$$
$$522$$ 0 0
$$523$$ 18900.3 1.58021 0.790106 0.612970i $$-0.210026\pi$$
0.790106 + 0.612970i $$0.210026\pi$$
$$524$$ −19030.3 −1.58653
$$525$$ 0 0
$$526$$ −1615.06 −0.133879
$$527$$ −28601.7 −2.36415
$$528$$ 0 0
$$529$$ 36105.7 2.96751
$$530$$ 867.839 0.0711255
$$531$$ 0 0
$$532$$ 656.468 0.0534990
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −7437.21 −0.601007
$$536$$ 279.943 0.0225592
$$537$$ 0 0
$$538$$ 9015.53 0.722467
$$539$$ 5301.55 0.423662
$$540$$ 0 0
$$541$$ 13247.5 1.05278 0.526391 0.850243i $$-0.323545\pi$$
0.526391 + 0.850243i $$0.323545\pi$$
$$542$$ 16197.1 1.28363
$$543$$ 0 0
$$544$$ −23947.9 −1.88742
$$545$$ −12702.4 −0.998372
$$546$$ 0 0
$$547$$ −5543.52 −0.433316 −0.216658 0.976248i $$-0.569516\pi$$
−0.216658 + 0.976248i $$0.569516\pi$$
$$548$$ 3090.36 0.240901
$$549$$ 0 0
$$550$$ −4138.28 −0.320831
$$551$$ 2669.76 0.206416
$$552$$ 0 0
$$553$$ −3712.31 −0.285467
$$554$$ −27377.3 −2.09955
$$555$$ 0 0
$$556$$ 4020.14 0.306640
$$557$$ −6733.19 −0.512198 −0.256099 0.966651i $$-0.582437\pi$$
−0.256099 + 0.966651i $$0.582437\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 2984.19 0.225188
$$561$$ 0 0
$$562$$ −29367.6 −2.20427
$$563$$ −24282.4 −1.81773 −0.908863 0.417096i $$-0.863048\pi$$
−0.908863 + 0.417096i $$0.863048\pi$$
$$564$$ 0 0
$$565$$ −10714.7 −0.797823
$$566$$ 9792.55 0.727229
$$567$$ 0 0
$$568$$ −310.966 −0.0229716
$$569$$ 19876.2 1.46441 0.732207 0.681082i $$-0.238490\pi$$
0.732207 + 0.681082i $$0.238490\pi$$
$$570$$ 0 0
$$571$$ 225.014 0.0164913 0.00824567 0.999966i $$-0.497375\pi$$
0.00824567 + 0.999966i $$0.497375\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −4813.14 −0.349994
$$575$$ 13089.0 0.949302
$$576$$ 0 0
$$577$$ −12258.1 −0.884424 −0.442212 0.896910i $$-0.645806\pi$$
−0.442212 + 0.896910i $$0.645806\pi$$
$$578$$ −15004.5 −1.07977
$$579$$ 0 0
$$580$$ −13300.4 −0.952191
$$581$$ −3926.77 −0.280395
$$582$$ 0 0
$$583$$ −458.844 −0.0325958
$$584$$ 61.7353 0.00437436
$$585$$ 0 0
$$586$$ 16972.4 1.19645
$$587$$ −10282.3 −0.722992 −0.361496 0.932374i $$-0.617734\pi$$
−0.361496 + 0.932374i $$0.617734\pi$$
$$588$$ 0 0
$$589$$ 4119.04 0.288153
$$590$$ 9129.67 0.637055
$$591$$ 0 0
$$592$$ 20693.5 1.43665
$$593$$ −12355.1 −0.855586 −0.427793 0.903877i $$-0.640709\pi$$
−0.427793 + 0.903877i $$0.640709\pi$$
$$594$$ 0 0
$$595$$ 4473.07 0.308198
$$596$$ −8703.71 −0.598184
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −13035.2 −0.889157 −0.444578 0.895740i $$-0.646646\pi$$
−0.444578 + 0.895740i $$0.646646\pi$$
$$600$$ 0 0
$$601$$ −5153.36 −0.349767 −0.174884 0.984589i $$-0.555955\pi$$
−0.174884 + 0.984589i $$0.555955\pi$$
$$602$$ −2801.36 −0.189659
$$603$$ 0 0
$$604$$ −7324.30 −0.493413
$$605$$ 8363.05 0.561994
$$606$$ 0 0
$$607$$ −1406.55 −0.0940530 −0.0470265 0.998894i $$-0.514975\pi$$
−0.0470265 + 0.998894i $$0.514975\pi$$
$$608$$ 3448.83 0.230047
$$609$$ 0 0
$$610$$ −6750.79 −0.448085
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −15984.3 −1.05318 −0.526590 0.850120i $$-0.676530\pi$$
−0.526590 + 0.850120i $$0.676530\pi$$
$$614$$ −33097.5 −2.17541
$$615$$ 0 0
$$616$$ 100.401 0.00656703
$$617$$ 10324.4 0.673652 0.336826 0.941567i $$-0.390647\pi$$
0.336826 + 0.941567i $$0.390647\pi$$
$$618$$ 0 0
$$619$$ −7423.00 −0.481996 −0.240998 0.970526i $$-0.577475\pi$$
−0.240998 + 0.970526i $$0.577475\pi$$
$$620$$ −20520.6 −1.32924
$$621$$ 0 0
$$622$$ −17082.4 −1.10119
$$623$$ −7576.48 −0.487232
$$624$$ 0 0
$$625$$ −4629.23 −0.296271
$$626$$ −15346.4 −0.979820
$$627$$ 0 0
$$628$$ −30074.6 −1.91100
$$629$$ 31017.8 1.96624
$$630$$ 0 0
$$631$$ 1190.30 0.0750954 0.0375477 0.999295i $$-0.488045\pi$$
0.0375477 + 0.999295i $$0.488045\pi$$
$$632$$ 610.665 0.0384350
$$633$$ 0 0
$$634$$ −42308.2 −2.65027
$$635$$ 19011.5 1.18810
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 13857.2 0.859894
$$639$$ 0 0
$$640$$ −1013.02 −0.0625674
$$641$$ −705.064 −0.0434451 −0.0217226 0.999764i $$-0.506915\pi$$
−0.0217226 + 0.999764i $$0.506915\pi$$
$$642$$ 0 0
$$643$$ 14641.4 0.897976 0.448988 0.893538i $$-0.351785\pi$$
0.448988 + 0.893538i $$0.351785\pi$$
$$644$$ −10777.1 −0.659439
$$645$$ 0 0
$$646$$ 5012.42 0.305280
$$647$$ −1520.83 −0.0924111 −0.0462056 0.998932i $$-0.514713\pi$$
−0.0462056 + 0.998932i $$0.514713\pi$$
$$648$$ 0 0
$$649$$ −4827.04 −0.291954
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 11428.7 0.686477
$$653$$ 22054.4 1.32168 0.660838 0.750529i $$-0.270201\pi$$
0.660838 + 0.750529i $$0.270201\pi$$
$$654$$ 0 0
$$655$$ 18674.2 1.11398
$$656$$ −12442.3 −0.740532
$$657$$ 0 0
$$658$$ 8120.92 0.481134
$$659$$ 12652.3 0.747895 0.373947 0.927450i $$-0.378004\pi$$
0.373947 + 0.927450i $$0.378004\pi$$
$$660$$ 0 0
$$661$$ −10893.2 −0.640994 −0.320497 0.947250i $$-0.603850\pi$$
−0.320497 + 0.947250i $$0.603850\pi$$
$$662$$ −47981.0 −2.81697
$$663$$ 0 0
$$664$$ 645.942 0.0377522
$$665$$ −644.184 −0.0375645
$$666$$ 0 0
$$667$$ −43829.0 −2.54433
$$668$$ 27829.6 1.61191
$$669$$ 0 0
$$670$$ −9322.76 −0.537567
$$671$$ 3569.28 0.205351
$$672$$ 0 0
$$673$$ −9019.75 −0.516621 −0.258310 0.966062i $$-0.583166\pi$$
−0.258310 + 0.966062i $$0.583166\pi$$
$$674$$ −36196.4 −2.06859
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 24923.6 1.41491 0.707454 0.706760i $$-0.249844\pi$$
0.707454 + 0.706760i $$0.249844\pi$$
$$678$$ 0 0
$$679$$ 3592.76 0.203060
$$680$$ −735.807 −0.0414955
$$681$$ 0 0
$$682$$ 21379.7 1.20039
$$683$$ 24634.2 1.38009 0.690044 0.723767i $$-0.257591\pi$$
0.690044 + 0.723767i $$0.257591\pi$$
$$684$$ 0 0
$$685$$ −3032.53 −0.169149
$$686$$ 15603.1 0.868411
$$687$$ 0 0
$$688$$ −7241.70 −0.401290
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −10340.2 −0.569264 −0.284632 0.958637i $$-0.591871\pi$$
−0.284632 + 0.958637i $$0.591871\pi$$
$$692$$ −2817.56 −0.154780
$$693$$ 0 0
$$694$$ −23024.3 −1.25935
$$695$$ −3944.92 −0.215308
$$696$$ 0 0
$$697$$ −18650.0 −1.01351
$$698$$ −38316.7 −2.07780
$$699$$ 0 0
$$700$$ −2922.19 −0.157784
$$701$$ 20833.9 1.12252 0.561258 0.827641i $$-0.310317\pi$$
0.561258 + 0.827641i $$0.310317\pi$$
$$702$$ 0 0
$$703$$ −4467.00 −0.239653
$$704$$ 9352.23 0.500676
$$705$$ 0 0
$$706$$ −16012.0 −0.853569
$$707$$ 4407.07 0.234434
$$708$$ 0 0
$$709$$ 9189.56 0.486772 0.243386 0.969930i $$-0.421742\pi$$
0.243386 + 0.969930i $$0.421742\pi$$
$$710$$ 10355.9 0.547395
$$711$$ 0 0
$$712$$ 1246.31 0.0656004
$$713$$ −67621.8 −3.55183
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 23785.2 1.24147
$$717$$ 0 0
$$718$$ −442.788 −0.0230149
$$719$$ −7503.22 −0.389183 −0.194592 0.980884i $$-0.562338\pi$$
−0.194592 + 0.980884i $$0.562338\pi$$
$$720$$ 0 0
$$721$$ −11238.5 −0.580503
$$722$$ 26921.6 1.38770
$$723$$ 0 0
$$724$$ 31286.8 1.60603
$$725$$ −11884.1 −0.608779
$$726$$ 0 0
$$727$$ −20727.2 −1.05740 −0.528700 0.848809i $$-0.677320\pi$$
−0.528700 + 0.848809i $$0.677320\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −2055.93 −0.104237
$$731$$ −10854.7 −0.549216
$$732$$ 0 0
$$733$$ 17361.3 0.874836 0.437418 0.899258i $$-0.355893\pi$$
0.437418 + 0.899258i $$0.355893\pi$$
$$734$$ −28292.4 −1.42274
$$735$$ 0 0
$$736$$ −56618.9 −2.83560
$$737$$ 4929.13 0.246359
$$738$$ 0 0
$$739$$ 18093.8 0.900663 0.450331 0.892861i $$-0.351306\pi$$
0.450331 + 0.892861i $$0.351306\pi$$
$$740$$ 22254.1 1.10551
$$741$$ 0 0
$$742$$ −638.466 −0.0315887
$$743$$ 14875.3 0.734482 0.367241 0.930126i $$-0.380302\pi$$
0.367241 + 0.930126i $$0.380302\pi$$
$$744$$ 0 0
$$745$$ 8540.85 0.420017
$$746$$ −8976.07 −0.440533
$$747$$ 0 0
$$748$$ 13202.9 0.645380
$$749$$ 5471.52 0.266923
$$750$$ 0 0
$$751$$ −17750.9 −0.862502 −0.431251 0.902232i $$-0.641928\pi$$
−0.431251 + 0.902232i $$0.641928\pi$$
$$752$$ 20993.1 1.01801
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 7187.25 0.346451
$$756$$ 0 0
$$757$$ −10220.3 −0.490702 −0.245351 0.969434i $$-0.578903\pi$$
−0.245351 + 0.969434i $$0.578903\pi$$
$$758$$ 44351.1 2.12520
$$759$$ 0 0
$$760$$ 105.967 0.00505765
$$761$$ 14112.0 0.672221 0.336110 0.941823i $$-0.390888\pi$$
0.336110 + 0.941823i $$0.390888\pi$$
$$762$$ 0 0
$$763$$ 9345.13 0.443403
$$764$$ 23128.7 1.09525
$$765$$ 0 0
$$766$$ −17881.6 −0.843457
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −22736.7 −1.06620 −0.533100 0.846052i $$-0.678973\pi$$
−0.533100 + 0.846052i $$0.678973\pi$$
$$770$$ −3343.60 −0.156487
$$771$$ 0 0
$$772$$ −20488.8 −0.955194
$$773$$ 343.173 0.0159678 0.00798388 0.999968i $$-0.497459\pi$$
0.00798388 + 0.999968i $$0.497459\pi$$
$$774$$ 0 0
$$775$$ −18335.4 −0.849844
$$776$$ −590.999 −0.0273397
$$777$$ 0 0
$$778$$ −10323.8 −0.475740
$$779$$ 2685.86 0.123531
$$780$$ 0 0
$$781$$ −5475.37 −0.250863
$$782$$ −82288.3 −3.76294
$$783$$ 0 0
$$784$$ 19069.8 0.868706
$$785$$ 29511.9 1.34181
$$786$$ 0 0
$$787$$ −19086.4 −0.864496 −0.432248 0.901755i $$-0.642279\pi$$
−0.432248 + 0.901755i $$0.642279\pi$$
$$788$$ −22673.7 −1.02502
$$789$$ 0 0
$$790$$ −20336.6 −0.915876
$$791$$ 7882.74 0.354334
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −8894.36 −0.397543
$$795$$ 0 0
$$796$$ 32959.5 1.46761
$$797$$ −12031.1 −0.534709 −0.267354 0.963598i $$-0.586149\pi$$
−0.267354 + 0.963598i $$0.586149\pi$$
$$798$$ 0 0
$$799$$ 31467.0 1.39327
$$800$$ −15352.1 −0.678471
$$801$$ 0 0
$$802$$ −32528.2 −1.43218
$$803$$ 1087.01 0.0477706
$$804$$ 0 0
$$805$$ 10575.5 0.463027
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −724.952 −0.0315640
$$809$$ −10175.4 −0.442210 −0.221105 0.975250i $$-0.570966\pi$$
−0.221105 + 0.975250i $$0.570966\pi$$
$$810$$ 0 0
$$811$$ 26754.1 1.15840 0.579201 0.815185i $$-0.303365\pi$$
0.579201 + 0.815185i $$0.303365\pi$$
$$812$$ 9785.08 0.422893
$$813$$ 0 0
$$814$$ −23185.7 −0.998353
$$815$$ −11214.9 −0.482011
$$816$$ 0 0
$$817$$ 1563.23 0.0669408
$$818$$ −25383.9 −1.08500
$$819$$ 0 0
$$820$$ −13380.7 −0.569845
$$821$$ 15777.5 0.670693 0.335347 0.942095i $$-0.391147\pi$$
0.335347 + 0.942095i $$0.391147\pi$$
$$822$$ 0 0
$$823$$ −13863.9 −0.587197 −0.293599 0.955929i $$-0.594853\pi$$
−0.293599 + 0.955929i $$0.594853\pi$$
$$824$$ 1848.70 0.0781583
$$825$$ 0 0
$$826$$ −6716.66 −0.282933
$$827$$ 26835.0 1.12835 0.564175 0.825655i $$-0.309194\pi$$
0.564175 + 0.825655i $$0.309194\pi$$
$$828$$ 0 0
$$829$$ 625.251 0.0261953 0.0130976 0.999914i $$-0.495831\pi$$
0.0130976 + 0.999914i $$0.495831\pi$$
$$830$$ −21511.4 −0.899604
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 28584.2 1.18893
$$834$$ 0 0
$$835$$ −27308.8 −1.13181
$$836$$ −1901.40 −0.0786617
$$837$$ 0 0
$$838$$ −5771.35 −0.237909
$$839$$ 27307.4 1.12367 0.561833 0.827251i $$-0.310096\pi$$
0.561833 + 0.827251i $$0.310096\pi$$
$$840$$ 0 0
$$841$$ 15405.4 0.631656
$$842$$ −38857.3 −1.59039
$$843$$ 0 0
$$844$$ −11340.6 −0.462510
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −6152.66 −0.249596
$$848$$ −1650.48 −0.0668368
$$849$$ 0 0
$$850$$ −22312.2 −0.900356
$$851$$ 73334.2 2.95401
$$852$$ 0 0
$$853$$ 44801.9 1.79834 0.899172 0.437595i $$-0.144170\pi$$
0.899172 + 0.437595i $$0.144170\pi$$
$$854$$ 4966.53 0.199006
$$855$$ 0 0
$$856$$ −900.051 −0.0359382
$$857$$ 25167.1 1.00314 0.501571 0.865116i $$-0.332756\pi$$
0.501571 + 0.865116i $$0.332756\pi$$
$$858$$ 0 0
$$859$$ 4059.10 0.161228 0.0806138 0.996745i $$-0.474312\pi$$
0.0806138 + 0.996745i $$0.474312\pi$$
$$860$$ −7787.87 −0.308795
$$861$$ 0 0
$$862$$ 13584.4 0.536759
$$863$$ −818.924 −0.0323019 −0.0161509 0.999870i $$-0.505141\pi$$
−0.0161509 + 0.999870i $$0.505141\pi$$
$$864$$ 0 0
$$865$$ 2764.84 0.108679
$$866$$ −25184.6 −0.988231
$$867$$ 0 0
$$868$$ 15096.9 0.590350
$$869$$ 10752.3 0.419733
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −1537.25 −0.0596993
$$873$$ 0 0
$$874$$ 11850.7 0.458644
$$875$$ 8884.23 0.343248
$$876$$ 0 0
$$877$$ 190.251 0.00732533 0.00366267 0.999993i $$-0.498834\pi$$
0.00366267 + 0.999993i $$0.498834\pi$$
$$878$$ −11551.3 −0.444007
$$879$$ 0 0
$$880$$ −8643.43 −0.331102
$$881$$ 19803.4 0.757315 0.378657 0.925537i $$-0.376386\pi$$
0.378657 + 0.925537i $$0.376386\pi$$
$$882$$ 0 0
$$883$$ 19652.1 0.748974 0.374487 0.927232i $$-0.377819\pi$$
0.374487 + 0.927232i $$0.377819\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 43431.2 1.64684
$$887$$ 26295.0 0.995379 0.497689 0.867355i $$-0.334182\pi$$
0.497689 + 0.867355i $$0.334182\pi$$
$$888$$ 0 0
$$889$$ −13986.6 −0.527668
$$890$$ −41505.1 −1.56321
$$891$$ 0 0
$$892$$ 13966.5 0.524253
$$893$$ −4531.69 −0.169818
$$894$$ 0 0
$$895$$ −23340.2 −0.871704
$$896$$ 745.275 0.0277878
$$897$$ 0 0
$$898$$ 55758.0 2.07201
$$899$$ 61397.0 2.27776
$$900$$ 0 0
$$901$$ −2473.93 −0.0914746
$$902$$ 13940.8 0.514609
$$903$$ 0 0
$$904$$ −1296.69 −0.0477072
$$905$$ −30701.4 −1.12768
$$906$$ 0 0
$$907$$ 42417.3 1.55286 0.776429 0.630204i $$-0.217029\pi$$
0.776429 + 0.630204i $$0.217029\pi$$
$$908$$ 3341.66 0.122133
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 872.245 0.0317220 0.0158610 0.999874i $$-0.494951\pi$$
0.0158610 + 0.999874i $$0.494951\pi$$
$$912$$ 0 0
$$913$$ 11373.5 0.412276
$$914$$ −69453.2 −2.51347
$$915$$ 0 0
$$916$$ 19449.4 0.701557
$$917$$ −13738.5 −0.494749
$$918$$ 0 0
$$919$$ −17181.4 −0.616716 −0.308358 0.951270i $$-0.599779\pi$$
−0.308358 + 0.951270i $$0.599779\pi$$
$$920$$ −1739.64 −0.0623415
$$921$$ 0 0
$$922$$ 46287.5 1.65336
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 19884.4 0.706804
$$926$$ −6503.25 −0.230788
$$927$$ 0 0
$$928$$ 51407.0 1.81845
$$929$$ 56042.9 1.97923 0.989617 0.143728i $$-0.0459091\pi$$
0.989617 + 0.143728i $$0.0459091\pi$$
$$930$$ 0 0
$$931$$ −4116.52 −0.144912
$$932$$ −7736.00 −0.271890
$$933$$ 0 0
$$934$$ −32843.3 −1.15060
$$935$$ −12955.8 −0.453155
$$936$$ 0 0
$$937$$ −36672.5 −1.27859 −0.639295 0.768961i $$-0.720774\pi$$
−0.639295 + 0.768961i $$0.720774\pi$$
$$938$$ 6858.71 0.238747
$$939$$ 0 0
$$940$$ 22576.4 0.783363
$$941$$ −21069.0 −0.729895 −0.364947 0.931028i $$-0.618913\pi$$
−0.364947 + 0.931028i $$0.618913\pi$$
$$942$$ 0 0
$$943$$ −44093.4 −1.52267
$$944$$ −17363.0 −0.598642
$$945$$ 0 0
$$946$$ 8113.87 0.278863
$$947$$ −1838.70 −0.0630938 −0.0315469 0.999502i $$-0.510043\pi$$
−0.0315469 + 0.999502i $$0.510043\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 3213.27 0.109739
$$951$$ 0 0
$$952$$ 541.330 0.0184292
$$953$$ −13599.8 −0.462266 −0.231133 0.972922i $$-0.574243\pi$$
−0.231133 + 0.972922i $$0.574243\pi$$
$$954$$ 0 0
$$955$$ −22695.9 −0.769029
$$956$$ −38446.9 −1.30069
$$957$$ 0 0
$$958$$ 7365.75 0.248410
$$959$$ 2231.02 0.0751235
$$960$$ 0 0
$$961$$ 64935.6 2.17971
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −31116.4 −1.03962
$$965$$ 20105.5 0.670692
$$966$$ 0 0
$$967$$ −2081.30 −0.0692141 −0.0346070 0.999401i $$-0.511018\pi$$
−0.0346070 + 0.999401i $$0.511018\pi$$
$$968$$ 1012.10 0.0336054
$$969$$ 0 0
$$970$$ 19681.6 0.651484
$$971$$ −1636.62 −0.0540904 −0.0270452 0.999634i $$-0.508610\pi$$
−0.0270452 + 0.999634i $$0.508610\pi$$
$$972$$ 0 0
$$973$$ 2902.26 0.0956240
$$974$$ 31592.9 1.03932
$$975$$ 0 0
$$976$$ 12838.8 0.421066
$$977$$ −29387.1 −0.962311 −0.481156 0.876635i $$-0.659783\pi$$
−0.481156 + 0.876635i $$0.659783\pi$$
$$978$$ 0 0
$$979$$ 21944.6 0.716396
$$980$$ 20508.1 0.668476
$$981$$ 0 0
$$982$$ 69304.4 2.25213
$$983$$ 24084.4 0.781457 0.390728 0.920506i $$-0.372223\pi$$
0.390728 + 0.920506i $$0.372223\pi$$
$$984$$ 0 0
$$985$$ 22249.4 0.719722
$$986$$ 74713.4 2.41314
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −25663.4 −0.825125
$$990$$ 0 0
$$991$$ 1413.43 0.0453068 0.0226534 0.999743i $$-0.492789\pi$$
0.0226534 + 0.999743i $$0.492789\pi$$
$$992$$ 79313.5 2.53851
$$993$$ 0 0
$$994$$ −7618.79 −0.243112
$$995$$ −32342.7 −1.03049
$$996$$ 0 0
$$997$$ −33357.4 −1.05962 −0.529809 0.848117i $$-0.677737\pi$$
−0.529809 + 0.848117i $$0.677737\pi$$
$$998$$ −21583.1 −0.684570
$$999$$ 0 0
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 1521.4.a.bf.1.3 9
3.2 odd 2 507.4.a.p.1.7 yes 9
13.12 even 2 1521.4.a.bi.1.7 9
39.5 even 4 507.4.b.k.337.4 18
39.8 even 4 507.4.b.k.337.15 18
39.38 odd 2 507.4.a.o.1.3 9

By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.3 9 39.38 odd 2
507.4.a.p.1.7 yes 9 3.2 odd 2
507.4.b.k.337.4 18 39.5 even 4
507.4.b.k.337.15 18 39.8 even 4
1521.4.a.bf.1.3 9 1.1 even 1 trivial
1521.4.a.bi.1.7 9 13.12 even 2