# Properties

 Label 575.4.b.i.24.9 Level $575$ Weight $4$ Character 575.24 Analytic conductor $33.926$ Analytic rank $0$ Dimension $10$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,4,Mod(24,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("575.24");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$575 = 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 575.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$33.9260982533$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} + 55x^{8} + 1079x^{6} + 8937x^{4} + 26936x^{2} + 8464$$ x^10 + 55*x^8 + 1079*x^6 + 8937*x^4 + 26936*x^2 + 8464 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 115) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 24.9 Root $$3.41740i$$ of defining polynomial Character $$\chi$$ $$=$$ 575.24 Dual form 575.4.b.i.24.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.41740i q^{2} -7.84147i q^{3} -11.5134 q^{4} +34.6389 q^{6} -8.97260i q^{7} -15.5200i q^{8} -34.4886 q^{9} +O(q^{10})$$ $$q+4.41740i q^{2} -7.84147i q^{3} -11.5134 q^{4} +34.6389 q^{6} -8.97260i q^{7} -15.5200i q^{8} -34.4886 q^{9} -28.9011 q^{11} +90.2818i q^{12} -16.0879i q^{13} +39.6355 q^{14} -23.5490 q^{16} +25.1771i q^{17} -152.350i q^{18} +35.6298 q^{19} -70.3583 q^{21} -127.668i q^{22} +23.0000i q^{23} -121.700 q^{24} +71.0668 q^{26} +58.7218i q^{27} +103.305i q^{28} -138.272 q^{29} +40.1277 q^{31} -228.186i q^{32} +226.627i q^{33} -111.217 q^{34} +397.081 q^{36} +379.745i q^{37} +157.391i q^{38} -126.153 q^{39} +412.514 q^{41} -310.801i q^{42} +402.095i q^{43} +332.750 q^{44} -101.600 q^{46} +110.070i q^{47} +184.659i q^{48} +262.492 q^{49} +197.426 q^{51} +185.227i q^{52} +421.300i q^{53} -259.397 q^{54} -139.255 q^{56} -279.390i q^{57} -610.802i q^{58} -755.913 q^{59} -307.032 q^{61} +177.260i q^{62} +309.453i q^{63} +819.594 q^{64} -1001.10 q^{66} +319.974i q^{67} -289.874i q^{68} +180.354 q^{69} -554.138 q^{71} +535.264i q^{72} +705.131i q^{73} -1677.48 q^{74} -410.219 q^{76} +259.318i q^{77} -557.268i q^{78} -1170.51 q^{79} -470.728 q^{81} +1822.24i q^{82} +455.978i q^{83} +810.063 q^{84} -1776.21 q^{86} +1084.26i q^{87} +448.546i q^{88} +1495.57 q^{89} -144.351 q^{91} -264.808i q^{92} -314.660i q^{93} -486.222 q^{94} -1789.31 q^{96} +1041.24i q^{97} +1159.53i q^{98} +996.760 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 44 q^{4} + 38 q^{6} - 154 q^{9}+O(q^{10})$$ 10 * q - 44 * q^4 + 38 * q^6 - 154 * q^9 $$10 q - 44 q^{4} + 38 q^{6} - 154 q^{9} + 46 q^{11} - 186 q^{14} + 564 q^{16} + 322 q^{19} - 120 q^{21} - 210 q^{24} - 514 q^{26} - 802 q^{29} + 64 q^{31} + 1326 q^{34} - 1318 q^{36} - 670 q^{39} - 24 q^{41} - 94 q^{44} - 276 q^{46} + 1476 q^{49} - 1986 q^{51} + 16 q^{54} + 686 q^{56} - 2648 q^{59} - 3346 q^{61} - 4932 q^{64} - 5562 q^{66} + 184 q^{69} - 216 q^{71} - 2916 q^{74} - 6954 q^{76} - 1312 q^{79} - 638 q^{81} + 1436 q^{84} + 224 q^{86} - 1140 q^{89} - 3178 q^{91} + 1896 q^{94} - 11982 q^{96} - 4042 q^{99}+O(q^{100})$$ 10 * q - 44 * q^4 + 38 * q^6 - 154 * q^9 + 46 * q^11 - 186 * q^14 + 564 * q^16 + 322 * q^19 - 120 * q^21 - 210 * q^24 - 514 * q^26 - 802 * q^29 + 64 * q^31 + 1326 * q^34 - 1318 * q^36 - 670 * q^39 - 24 * q^41 - 94 * q^44 - 276 * q^46 + 1476 * q^49 - 1986 * q^51 + 16 * q^54 + 686 * q^56 - 2648 * q^59 - 3346 * q^61 - 4932 * q^64 - 5562 * q^66 + 184 * q^69 - 216 * q^71 - 2916 * q^74 - 6954 * q^76 - 1312 * q^79 - 638 * q^81 + 1436 * q^84 + 224 * q^86 - 1140 * q^89 - 3178 * q^91 + 1896 * q^94 - 11982 * q^96 - 4042 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.41740i 1.56179i 0.624665 + 0.780893i $$0.285235\pi$$
−0.624665 + 0.780893i $$0.714765\pi$$
$$3$$ − 7.84147i − 1.50909i −0.656248 0.754546i $$-0.727857\pi$$
0.656248 0.754546i $$-0.272143\pi$$
$$4$$ −11.5134 −1.43917
$$5$$ 0 0
$$6$$ 34.6389 2.35688
$$7$$ − 8.97260i − 0.484475i −0.970217 0.242237i $$-0.922119\pi$$
0.970217 0.242237i $$-0.0778813\pi$$
$$8$$ − 15.5200i − 0.685894i
$$9$$ −34.4886 −1.27736
$$10$$ 0 0
$$11$$ −28.9011 −0.792183 −0.396092 0.918211i $$-0.629634\pi$$
−0.396092 + 0.918211i $$0.629634\pi$$
$$12$$ 90.2818i 2.17184i
$$13$$ − 16.0879i − 0.343230i −0.985164 0.171615i $$-0.945102\pi$$
0.985164 0.171615i $$-0.0548985\pi$$
$$14$$ 39.6355 0.756646
$$15$$ 0 0
$$16$$ −23.5490 −0.367954
$$17$$ 25.1771i 0.359197i 0.983740 + 0.179599i $$0.0574799\pi$$
−0.983740 + 0.179599i $$0.942520\pi$$
$$18$$ − 152.350i − 1.99496i
$$19$$ 35.6298 0.430212 0.215106 0.976591i $$-0.430990\pi$$
0.215106 + 0.976591i $$0.430990\pi$$
$$20$$ 0 0
$$21$$ −70.3583 −0.731117
$$22$$ − 127.668i − 1.23722i
$$23$$ 23.0000i 0.208514i
$$24$$ −121.700 −1.03508
$$25$$ 0 0
$$26$$ 71.0668 0.536051
$$27$$ 58.7218i 0.418556i
$$28$$ 103.305i 0.697243i
$$29$$ −138.272 −0.885395 −0.442698 0.896671i $$-0.645978\pi$$
−0.442698 + 0.896671i $$0.645978\pi$$
$$30$$ 0 0
$$31$$ 40.1277 0.232489 0.116244 0.993221i $$-0.462914\pi$$
0.116244 + 0.993221i $$0.462914\pi$$
$$32$$ − 228.186i − 1.26056i
$$33$$ 226.627i 1.19548i
$$34$$ −111.217 −0.560989
$$35$$ 0 0
$$36$$ 397.081 1.83834
$$37$$ 379.745i 1.68729i 0.536902 + 0.843645i $$0.319595\pi$$
−0.536902 + 0.843645i $$0.680405\pi$$
$$38$$ 157.391i 0.671899i
$$39$$ −126.153 −0.517965
$$40$$ 0 0
$$41$$ 412.514 1.57132 0.785658 0.618662i $$-0.212325\pi$$
0.785658 + 0.618662i $$0.212325\pi$$
$$42$$ − 310.801i − 1.14185i
$$43$$ 402.095i 1.42602i 0.701153 + 0.713011i $$0.252669\pi$$
−0.701153 + 0.713011i $$0.747331\pi$$
$$44$$ 332.750 1.14009
$$45$$ 0 0
$$46$$ −101.600 −0.325655
$$47$$ 110.070i 0.341603i 0.985305 + 0.170801i $$0.0546357\pi$$
−0.985305 + 0.170801i $$0.945364\pi$$
$$48$$ 184.659i 0.555276i
$$49$$ 262.492 0.765284
$$50$$ 0 0
$$51$$ 197.426 0.542061
$$52$$ 185.227i 0.493967i
$$53$$ 421.300i 1.09189i 0.837822 + 0.545943i $$0.183829\pi$$
−0.837822 + 0.545943i $$0.816171\pi$$
$$54$$ −259.397 −0.653695
$$55$$ 0 0
$$56$$ −139.255 −0.332298
$$57$$ − 279.390i − 0.649229i
$$58$$ − 610.802i − 1.38280i
$$59$$ −755.913 −1.66799 −0.833996 0.551770i $$-0.813952\pi$$
−0.833996 + 0.551770i $$0.813952\pi$$
$$60$$ 0 0
$$61$$ −307.032 −0.644450 −0.322225 0.946663i $$-0.604431\pi$$
−0.322225 + 0.946663i $$0.604431\pi$$
$$62$$ 177.260i 0.363098i
$$63$$ 309.453i 0.618847i
$$64$$ 819.594 1.60077
$$65$$ 0 0
$$66$$ −1001.10 −1.86708
$$67$$ 319.974i 0.583448i 0.956502 + 0.291724i $$0.0942290\pi$$
−0.956502 + 0.291724i $$0.905771\pi$$
$$68$$ − 289.874i − 0.516947i
$$69$$ 180.354 0.314667
$$70$$ 0 0
$$71$$ −554.138 −0.926254 −0.463127 0.886292i $$-0.653273\pi$$
−0.463127 + 0.886292i $$0.653273\pi$$
$$72$$ 535.264i 0.876131i
$$73$$ 705.131i 1.13054i 0.824907 + 0.565269i $$0.191228\pi$$
−0.824907 + 0.565269i $$0.808772\pi$$
$$74$$ −1677.48 −2.63518
$$75$$ 0 0
$$76$$ −410.219 −0.619149
$$77$$ 259.318i 0.383793i
$$78$$ − 557.268i − 0.808950i
$$79$$ −1170.51 −1.66699 −0.833495 0.552526i $$-0.813664\pi$$
−0.833495 + 0.552526i $$0.813664\pi$$
$$80$$ 0 0
$$81$$ −470.728 −0.645717
$$82$$ 1822.24i 2.45406i
$$83$$ 455.978i 0.603014i 0.953464 + 0.301507i $$0.0974896\pi$$
−0.953464 + 0.301507i $$0.902510\pi$$
$$84$$ 810.063 1.05220
$$85$$ 0 0
$$86$$ −1776.21 −2.22714
$$87$$ 1084.26i 1.33614i
$$88$$ 448.546i 0.543354i
$$89$$ 1495.57 1.78124 0.890621 0.454746i $$-0.150270\pi$$
0.890621 + 0.454746i $$0.150270\pi$$
$$90$$ 0 0
$$91$$ −144.351 −0.166286
$$92$$ − 264.808i − 0.300088i
$$93$$ − 314.660i − 0.350847i
$$94$$ −486.222 −0.533510
$$95$$ 0 0
$$96$$ −1789.31 −1.90230
$$97$$ 1041.24i 1.08992i 0.838463 + 0.544958i $$0.183455\pi$$
−0.838463 + 0.544958i $$0.816545\pi$$
$$98$$ 1159.53i 1.19521i
$$99$$ 996.760 1.01190
$$100$$ 0 0
$$101$$ 1450.22 1.42873 0.714367 0.699771i $$-0.246715\pi$$
0.714367 + 0.699771i $$0.246715\pi$$
$$102$$ 872.108i 0.846584i
$$103$$ − 220.283i − 0.210729i −0.994434 0.105365i $$-0.966399\pi$$
0.994434 0.105365i $$-0.0336010\pi$$
$$104$$ −249.685 −0.235419
$$105$$ 0 0
$$106$$ −1861.05 −1.70529
$$107$$ 59.2679i 0.0535481i 0.999642 + 0.0267741i $$0.00852346\pi$$
−0.999642 + 0.0267741i $$0.991477\pi$$
$$108$$ − 676.087i − 0.602375i
$$109$$ −1966.29 −1.72786 −0.863930 0.503612i $$-0.832004\pi$$
−0.863930 + 0.503612i $$0.832004\pi$$
$$110$$ 0 0
$$111$$ 2977.76 2.54627
$$112$$ 211.296i 0.178264i
$$113$$ − 1819.68i − 1.51488i −0.652905 0.757439i $$-0.726450\pi$$
0.652905 0.757439i $$-0.273550\pi$$
$$114$$ 1234.17 1.01396
$$115$$ 0 0
$$116$$ 1591.98 1.27424
$$117$$ 554.851i 0.438427i
$$118$$ − 3339.17i − 2.60505i
$$119$$ 225.904 0.174022
$$120$$ 0 0
$$121$$ −495.725 −0.372445
$$122$$ − 1356.28i − 1.00649i
$$123$$ − 3234.72i − 2.37126i
$$124$$ −462.006 −0.334592
$$125$$ 0 0
$$126$$ −1366.97 −0.966506
$$127$$ 834.517i 0.583082i 0.956558 + 0.291541i $$0.0941680\pi$$
−0.956558 + 0.291541i $$0.905832\pi$$
$$128$$ 1794.98i 1.23950i
$$129$$ 3153.02 2.15200
$$130$$ 0 0
$$131$$ 510.293 0.340340 0.170170 0.985415i $$-0.445568\pi$$
0.170170 + 0.985415i $$0.445568\pi$$
$$132$$ − 2609.25i − 1.72050i
$$133$$ − 319.692i − 0.208427i
$$134$$ −1413.45 −0.911221
$$135$$ 0 0
$$136$$ 390.750 0.246371
$$137$$ 11.2339i 0.00700567i 0.999994 + 0.00350284i $$0.00111499\pi$$
−0.999994 + 0.00350284i $$0.998885\pi$$
$$138$$ 796.694i 0.491443i
$$139$$ 1656.81 1.01100 0.505501 0.862826i $$-0.331308\pi$$
0.505501 + 0.862826i $$0.331308\pi$$
$$140$$ 0 0
$$141$$ 863.109 0.515510
$$142$$ − 2447.85i − 1.44661i
$$143$$ 464.959i 0.271901i
$$144$$ 812.174 0.470008
$$145$$ 0 0
$$146$$ −3114.84 −1.76566
$$147$$ − 2058.33i − 1.15488i
$$148$$ − 4372.15i − 2.42830i
$$149$$ 510.206 0.280522 0.140261 0.990115i $$-0.455206\pi$$
0.140261 + 0.990115i $$0.455206\pi$$
$$150$$ 0 0
$$151$$ −2337.38 −1.25969 −0.629846 0.776720i $$-0.716882\pi$$
−0.629846 + 0.776720i $$0.716882\pi$$
$$152$$ − 552.974i − 0.295080i
$$153$$ − 868.325i − 0.458823i
$$154$$ −1145.51 −0.599402
$$155$$ 0 0
$$156$$ 1452.45 0.745442
$$157$$ 146.592i 0.0745178i 0.999306 + 0.0372589i $$0.0118626\pi$$
−0.999306 + 0.0372589i $$0.988137\pi$$
$$158$$ − 5170.59i − 2.60348i
$$159$$ 3303.61 1.64776
$$160$$ 0 0
$$161$$ 206.370 0.101020
$$162$$ − 2079.39i − 1.00847i
$$163$$ 3278.19i 1.57526i 0.616149 + 0.787630i $$0.288692\pi$$
−0.616149 + 0.787630i $$0.711308\pi$$
$$164$$ −4749.44 −2.26139
$$165$$ 0 0
$$166$$ −2014.24 −0.941778
$$167$$ − 1555.42i − 0.720732i −0.932811 0.360366i $$-0.882652\pi$$
0.932811 0.360366i $$-0.117348\pi$$
$$168$$ 1091.96i 0.501469i
$$169$$ 1938.18 0.882193
$$170$$ 0 0
$$171$$ −1228.82 −0.549534
$$172$$ − 4629.48i − 2.05229i
$$173$$ 472.392i 0.207603i 0.994598 + 0.103801i $$0.0331006\pi$$
−0.994598 + 0.103801i $$0.966899\pi$$
$$174$$ −4789.58 −2.08677
$$175$$ 0 0
$$176$$ 680.594 0.291487
$$177$$ 5927.47i 2.51715i
$$178$$ 6606.54i 2.78192i
$$179$$ −2429.45 −1.01444 −0.507222 0.861815i $$-0.669328\pi$$
−0.507222 + 0.861815i $$0.669328\pi$$
$$180$$ 0 0
$$181$$ −982.359 −0.403415 −0.201708 0.979446i $$-0.564649\pi$$
−0.201708 + 0.979446i $$0.564649\pi$$
$$182$$ − 637.653i − 0.259703i
$$183$$ 2407.58i 0.972533i
$$184$$ 356.960 0.143019
$$185$$ 0 0
$$186$$ 1389.98 0.547947
$$187$$ − 727.648i − 0.284550i
$$188$$ − 1267.28i − 0.491626i
$$189$$ 526.887 0.202780
$$190$$ 0 0
$$191$$ 1361.14 0.515649 0.257824 0.966192i $$-0.416994\pi$$
0.257824 + 0.966192i $$0.416994\pi$$
$$192$$ − 6426.82i − 2.41571i
$$193$$ 1456.29i 0.543142i 0.962418 + 0.271571i $$0.0875432\pi$$
−0.962418 + 0.271571i $$0.912457\pi$$
$$194$$ −4599.57 −1.70222
$$195$$ 0 0
$$196$$ −3022.18 −1.10138
$$197$$ 1071.99i 0.387695i 0.981032 + 0.193847i $$0.0620966\pi$$
−0.981032 + 0.193847i $$0.937903\pi$$
$$198$$ 4403.08i 1.58037i
$$199$$ 2879.31 1.02567 0.512836 0.858486i $$-0.328595\pi$$
0.512836 + 0.858486i $$0.328595\pi$$
$$200$$ 0 0
$$201$$ 2509.07 0.880477
$$202$$ 6406.19i 2.23138i
$$203$$ 1240.66i 0.428952i
$$204$$ −2273.04 −0.780120
$$205$$ 0 0
$$206$$ 973.077 0.329114
$$207$$ − 793.238i − 0.266347i
$$208$$ 378.855i 0.126293i
$$209$$ −1029.74 −0.340807
$$210$$ 0 0
$$211$$ −4746.47 −1.54863 −0.774313 0.632802i $$-0.781905\pi$$
−0.774313 + 0.632802i $$0.781905\pi$$
$$212$$ − 4850.59i − 1.57141i
$$213$$ 4345.25i 1.39780i
$$214$$ −261.810 −0.0836306
$$215$$ 0 0
$$216$$ 911.363 0.287085
$$217$$ − 360.050i − 0.112635i
$$218$$ − 8685.90i − 2.69855i
$$219$$ 5529.26 1.70609
$$220$$ 0 0
$$221$$ 405.048 0.123287
$$222$$ 13153.9i 3.97673i
$$223$$ − 4874.04i − 1.46363i −0.681503 0.731815i $$-0.738673\pi$$
0.681503 0.731815i $$-0.261327\pi$$
$$224$$ −2047.42 −0.610709
$$225$$ 0 0
$$226$$ 8038.26 2.36592
$$227$$ − 2742.09i − 0.801757i −0.916131 0.400879i $$-0.868705\pi$$
0.916131 0.400879i $$-0.131295\pi$$
$$228$$ 3216.72i 0.934353i
$$229$$ −1528.38 −0.441041 −0.220520 0.975382i $$-0.570776\pi$$
−0.220520 + 0.975382i $$0.570776\pi$$
$$230$$ 0 0
$$231$$ 2033.44 0.579179
$$232$$ 2145.98i 0.607287i
$$233$$ − 5552.78i − 1.56126i −0.624991 0.780632i $$-0.714897\pi$$
0.624991 0.780632i $$-0.285103\pi$$
$$234$$ −2450.99 −0.684729
$$235$$ 0 0
$$236$$ 8703.12 2.40053
$$237$$ 9178.49i 2.51564i
$$238$$ 997.909i 0.271785i
$$239$$ −2779.63 −0.752299 −0.376149 0.926559i $$-0.622752\pi$$
−0.376149 + 0.926559i $$0.622752\pi$$
$$240$$ 0 0
$$241$$ −5568.82 −1.48846 −0.744231 0.667922i $$-0.767184\pi$$
−0.744231 + 0.667922i $$0.767184\pi$$
$$242$$ − 2189.81i − 0.581680i
$$243$$ 5276.68i 1.39300i
$$244$$ 3534.98 0.927475
$$245$$ 0 0
$$246$$ 14289.0 3.70340
$$247$$ − 573.209i − 0.147662i
$$248$$ − 622.783i − 0.159463i
$$249$$ 3575.54 0.910003
$$250$$ 0 0
$$251$$ −387.065 −0.0973360 −0.0486680 0.998815i $$-0.515498\pi$$
−0.0486680 + 0.998815i $$0.515498\pi$$
$$252$$ − 3562.85i − 0.890628i
$$253$$ − 664.726i − 0.165182i
$$254$$ −3686.39 −0.910649
$$255$$ 0 0
$$256$$ −1372.41 −0.335061
$$257$$ 1476.07i 0.358267i 0.983825 + 0.179133i $$0.0573293\pi$$
−0.983825 + 0.179133i $$0.942671\pi$$
$$258$$ 13928.1i 3.36096i
$$259$$ 3407.30 0.817449
$$260$$ 0 0
$$261$$ 4768.81 1.13097
$$262$$ 2254.17i 0.531537i
$$263$$ − 203.984i − 0.0478259i −0.999714 0.0239130i $$-0.992388\pi$$
0.999714 0.0239130i $$-0.00761246\pi$$
$$264$$ 3517.26 0.819971
$$265$$ 0 0
$$266$$ 1412.20 0.325518
$$267$$ − 11727.5i − 2.68806i
$$268$$ − 3683.98i − 0.839683i
$$269$$ 6973.71 1.58065 0.790324 0.612689i $$-0.209912\pi$$
0.790324 + 0.612689i $$0.209912\pi$$
$$270$$ 0 0
$$271$$ −2164.97 −0.485287 −0.242643 0.970116i $$-0.578015\pi$$
−0.242643 + 0.970116i $$0.578015\pi$$
$$272$$ − 592.898i − 0.132168i
$$273$$ 1131.92i 0.250941i
$$274$$ −49.6246 −0.0109414
$$275$$ 0 0
$$276$$ −2076.48 −0.452861
$$277$$ 1194.00i 0.258992i 0.991580 + 0.129496i $$0.0413359\pi$$
−0.991580 + 0.129496i $$0.958664\pi$$
$$278$$ 7318.81i 1.57897i
$$279$$ −1383.95 −0.296971
$$280$$ 0 0
$$281$$ 6485.98 1.37694 0.688472 0.725263i $$-0.258282\pi$$
0.688472 + 0.725263i $$0.258282\pi$$
$$282$$ 3812.69i 0.805116i
$$283$$ 3214.41i 0.675182i 0.941293 + 0.337591i $$0.109612\pi$$
−0.941293 + 0.337591i $$0.890388\pi$$
$$284$$ 6380.00 1.33304
$$285$$ 0 0
$$286$$ −2053.91 −0.424651
$$287$$ − 3701.33i − 0.761263i
$$288$$ 7869.81i 1.61018i
$$289$$ 4279.11 0.870977
$$290$$ 0 0
$$291$$ 8164.85 1.64478
$$292$$ − 8118.44i − 1.62704i
$$293$$ − 5585.47i − 1.11367i −0.830622 0.556837i $$-0.812015\pi$$
0.830622 0.556837i $$-0.187985\pi$$
$$294$$ 9092.44 1.80368
$$295$$ 0 0
$$296$$ 5893.65 1.15730
$$297$$ − 1697.13i − 0.331573i
$$298$$ 2253.78i 0.438115i
$$299$$ 370.022 0.0715684
$$300$$ 0 0
$$301$$ 3607.84 0.690872
$$302$$ − 10325.1i − 1.96737i
$$303$$ − 11371.8i − 2.15609i
$$304$$ −839.047 −0.158298
$$305$$ 0 0
$$306$$ 3835.73 0.716583
$$307$$ 2849.51i 0.529740i 0.964284 + 0.264870i $$0.0853291\pi$$
−0.964284 + 0.264870i $$0.914671\pi$$
$$308$$ − 2985.63i − 0.552344i
$$309$$ −1727.34 −0.318010
$$310$$ 0 0
$$311$$ −6617.84 −1.20664 −0.603318 0.797501i $$-0.706155\pi$$
−0.603318 + 0.797501i $$0.706155\pi$$
$$312$$ 1957.90i 0.355269i
$$313$$ 6271.26i 1.13250i 0.824233 + 0.566250i $$0.191606\pi$$
−0.824233 + 0.566250i $$0.808394\pi$$
$$314$$ −647.554 −0.116381
$$315$$ 0 0
$$316$$ 13476.5 2.39909
$$317$$ 5650.73i 1.00119i 0.865682 + 0.500594i $$0.166885\pi$$
−0.865682 + 0.500594i $$0.833115\pi$$
$$318$$ 14593.3i 2.57344i
$$319$$ 3996.22 0.701395
$$320$$ 0 0
$$321$$ 464.748 0.0808090
$$322$$ 911.617i 0.157772i
$$323$$ 897.055i 0.154531i
$$324$$ 5419.67 0.929299
$$325$$ 0 0
$$326$$ −14481.0 −2.46022
$$327$$ 15418.6i 2.60750i
$$328$$ − 6402.23i − 1.07776i
$$329$$ 987.612 0.165498
$$330$$ 0 0
$$331$$ −6391.40 −1.06134 −0.530669 0.847579i $$-0.678059\pi$$
−0.530669 + 0.847579i $$0.678059\pi$$
$$332$$ − 5249.86i − 0.867841i
$$333$$ − 13096.9i − 2.15527i
$$334$$ 6870.92 1.12563
$$335$$ 0 0
$$336$$ 1656.87 0.269017
$$337$$ 4153.87i 0.671441i 0.941962 + 0.335721i $$0.108980\pi$$
−0.941962 + 0.335721i $$0.891020\pi$$
$$338$$ 8561.70i 1.37780i
$$339$$ −14269.0 −2.28609
$$340$$ 0 0
$$341$$ −1159.74 −0.184174
$$342$$ − 5428.19i − 0.858254i
$$343$$ − 5432.84i − 0.855236i
$$344$$ 6240.52 0.978100
$$345$$ 0 0
$$346$$ −2086.74 −0.324231
$$347$$ − 3071.86i − 0.475234i −0.971359 0.237617i $$-0.923634\pi$$
0.971359 0.237617i $$-0.0763663\pi$$
$$348$$ − 12483.4i − 1.92294i
$$349$$ 8358.91 1.28207 0.641035 0.767512i $$-0.278505\pi$$
0.641035 + 0.767512i $$0.278505\pi$$
$$350$$ 0 0
$$351$$ 944.712 0.143661
$$352$$ 6594.82i 0.998594i
$$353$$ − 8018.12i − 1.20896i −0.796622 0.604478i $$-0.793382\pi$$
0.796622 0.604478i $$-0.206618\pi$$
$$354$$ −26184.0 −3.93125
$$355$$ 0 0
$$356$$ −17219.1 −2.56352
$$357$$ − 1771.42i − 0.262615i
$$358$$ − 10731.8i − 1.58434i
$$359$$ −2138.46 −0.314384 −0.157192 0.987568i $$-0.550244\pi$$
−0.157192 + 0.987568i $$0.550244\pi$$
$$360$$ 0 0
$$361$$ −5589.52 −0.814918
$$362$$ − 4339.47i − 0.630048i
$$363$$ 3887.21i 0.562054i
$$364$$ 1661.96 0.239315
$$365$$ 0 0
$$366$$ −10635.2 −1.51889
$$367$$ − 4509.96i − 0.641467i −0.947170 0.320733i $$-0.896071\pi$$
0.947170 0.320733i $$-0.103929\pi$$
$$368$$ − 541.628i − 0.0767237i
$$369$$ −14227.1 −2.00713
$$370$$ 0 0
$$371$$ 3780.15 0.528991
$$372$$ 3622.80i 0.504929i
$$373$$ 3362.09i 0.466709i 0.972392 + 0.233355i $$0.0749703\pi$$
−0.972392 + 0.233355i $$0.925030\pi$$
$$374$$ 3214.31 0.444406
$$375$$ 0 0
$$376$$ 1708.29 0.234303
$$377$$ 2224.51i 0.303894i
$$378$$ 2327.47i 0.316699i
$$379$$ −2107.16 −0.285587 −0.142793 0.989753i $$-0.545608\pi$$
−0.142793 + 0.989753i $$0.545608\pi$$
$$380$$ 0 0
$$381$$ 6543.84 0.879924
$$382$$ 6012.71i 0.805333i
$$383$$ − 1373.05i − 0.183184i −0.995797 0.0915919i $$-0.970804\pi$$
0.995797 0.0915919i $$-0.0291955\pi$$
$$384$$ 14075.3 1.87052
$$385$$ 0 0
$$386$$ −6433.03 −0.848271
$$387$$ − 13867.7i − 1.82154i
$$388$$ − 11988.2i − 1.56858i
$$389$$ −7008.05 −0.913425 −0.456713 0.889614i $$-0.650973\pi$$
−0.456713 + 0.889614i $$0.650973\pi$$
$$390$$ 0 0
$$391$$ −579.074 −0.0748978
$$392$$ − 4073.89i − 0.524904i
$$393$$ − 4001.44i − 0.513604i
$$394$$ −4735.39 −0.605496
$$395$$ 0 0
$$396$$ −11476.1 −1.45630
$$397$$ 2402.53i 0.303727i 0.988401 + 0.151864i $$0.0485275\pi$$
−0.988401 + 0.151864i $$0.951473\pi$$
$$398$$ 12719.0i 1.60188i
$$399$$ −2506.85 −0.314535
$$400$$ 0 0
$$401$$ −11902.0 −1.48218 −0.741091 0.671404i $$-0.765691\pi$$
−0.741091 + 0.671404i $$0.765691\pi$$
$$402$$ 11083.5i 1.37512i
$$403$$ − 645.572i − 0.0797971i
$$404$$ −16696.9 −2.05620
$$405$$ 0 0
$$406$$ −5480.48 −0.669930
$$407$$ − 10975.1i − 1.33664i
$$408$$ − 3064.05i − 0.371797i
$$409$$ −5277.96 −0.638089 −0.319044 0.947740i $$-0.603362\pi$$
−0.319044 + 0.947740i $$0.603362\pi$$
$$410$$ 0 0
$$411$$ 88.0903 0.0105722
$$412$$ 2536.20i 0.303276i
$$413$$ 6782.51i 0.808100i
$$414$$ 3504.05 0.415977
$$415$$ 0 0
$$416$$ −3671.03 −0.432662
$$417$$ − 12991.9i − 1.52569i
$$418$$ − 4548.77i − 0.532267i
$$419$$ 11196.4 1.30545 0.652723 0.757597i $$-0.273627\pi$$
0.652723 + 0.757597i $$0.273627\pi$$
$$420$$ 0 0
$$421$$ 5176.82 0.599293 0.299647 0.954050i $$-0.403131\pi$$
0.299647 + 0.954050i $$0.403131\pi$$
$$422$$ − 20967.0i − 2.41862i
$$423$$ − 3796.16i − 0.436349i
$$424$$ 6538.58 0.748918
$$425$$ 0 0
$$426$$ −19194.7 −2.18307
$$427$$ 2754.88i 0.312220i
$$428$$ − 682.375i − 0.0770650i
$$429$$ 3645.96 0.410324
$$430$$ 0 0
$$431$$ 9348.93 1.04483 0.522415 0.852691i $$-0.325031\pi$$
0.522415 + 0.852691i $$0.325031\pi$$
$$432$$ − 1382.84i − 0.154009i
$$433$$ − 4320.91i − 0.479560i −0.970827 0.239780i $$-0.922925\pi$$
0.970827 0.239780i $$-0.0770753\pi$$
$$434$$ 1590.48 0.175912
$$435$$ 0 0
$$436$$ 22638.7 2.48669
$$437$$ 819.484i 0.0897054i
$$438$$ 24424.9i 2.66454i
$$439$$ 7016.03 0.762772 0.381386 0.924416i $$-0.375447\pi$$
0.381386 + 0.924416i $$0.375447\pi$$
$$440$$ 0 0
$$441$$ −9053.00 −0.977541
$$442$$ 1789.26i 0.192548i
$$443$$ − 12343.6i − 1.32384i −0.749574 0.661921i $$-0.769741\pi$$
0.749574 0.661921i $$-0.230259\pi$$
$$444$$ −34284.1 −3.66453
$$445$$ 0 0
$$446$$ 21530.5 2.28588
$$447$$ − 4000.77i − 0.423333i
$$448$$ − 7353.88i − 0.775532i
$$449$$ −18146.9 −1.90737 −0.953683 0.300815i $$-0.902741\pi$$
−0.953683 + 0.300815i $$0.902741\pi$$
$$450$$ 0 0
$$451$$ −11922.1 −1.24477
$$452$$ 20950.7i 2.18017i
$$453$$ 18328.5i 1.90099i
$$454$$ 12112.9 1.25217
$$455$$ 0 0
$$456$$ −4336.13 −0.445302
$$457$$ 5233.82i 0.535728i 0.963457 + 0.267864i $$0.0863177\pi$$
−0.963457 + 0.267864i $$0.913682\pi$$
$$458$$ − 6751.47i − 0.688811i
$$459$$ −1478.45 −0.150344
$$460$$ 0 0
$$461$$ −2335.01 −0.235905 −0.117952 0.993019i $$-0.537633\pi$$
−0.117952 + 0.993019i $$0.537633\pi$$
$$462$$ 8982.49i 0.904553i
$$463$$ − 13644.1i − 1.36954i −0.728760 0.684769i $$-0.759903\pi$$
0.728760 0.684769i $$-0.240097\pi$$
$$464$$ 3256.17 0.325784
$$465$$ 0 0
$$466$$ 24528.8 2.43836
$$467$$ − 5246.31i − 0.519851i −0.965629 0.259925i $$-0.916302\pi$$
0.965629 0.259925i $$-0.0836979\pi$$
$$468$$ − 6388.21i − 0.630972i
$$469$$ 2871.00 0.282666
$$470$$ 0 0
$$471$$ 1149.50 0.112454
$$472$$ 11731.8i 1.14407i
$$473$$ − 11621.0i − 1.12967i
$$474$$ −40545.0 −3.92889
$$475$$ 0 0
$$476$$ −2600.92 −0.250448
$$477$$ − 14530.0i − 1.39473i
$$478$$ − 12278.7i − 1.17493i
$$479$$ −12278.3 −1.17121 −0.585603 0.810598i $$-0.699142\pi$$
−0.585603 + 0.810598i $$0.699142\pi$$
$$480$$ 0 0
$$481$$ 6109.31 0.579128
$$482$$ − 24599.7i − 2.32466i
$$483$$ − 1618.24i − 0.152448i
$$484$$ 5707.47 0.536013
$$485$$ 0 0
$$486$$ −23309.2 −2.17557
$$487$$ − 8945.24i − 0.832336i −0.909288 0.416168i $$-0.863373\pi$$
0.909288 0.416168i $$-0.136627\pi$$
$$488$$ 4765.14i 0.442024i
$$489$$ 25705.8 2.37721
$$490$$ 0 0
$$491$$ −7792.94 −0.716274 −0.358137 0.933669i $$-0.616588\pi$$
−0.358137 + 0.933669i $$0.616588\pi$$
$$492$$ 37242.6i 3.41265i
$$493$$ − 3481.29i − 0.318031i
$$494$$ 2532.09 0.230616
$$495$$ 0 0
$$496$$ −944.970 −0.0855451
$$497$$ 4972.06i 0.448747i
$$498$$ 15794.6i 1.42123i
$$499$$ 8577.71 0.769521 0.384761 0.923016i $$-0.374284\pi$$
0.384761 + 0.923016i $$0.374284\pi$$
$$500$$ 0 0
$$501$$ −12196.8 −1.08765
$$502$$ − 1709.82i − 0.152018i
$$503$$ − 7784.99i − 0.690091i −0.938586 0.345045i $$-0.887864\pi$$
0.938586 0.345045i $$-0.112136\pi$$
$$504$$ 4802.71 0.424464
$$505$$ 0 0
$$506$$ 2936.36 0.257978
$$507$$ − 15198.2i − 1.33131i
$$508$$ − 9608.12i − 0.839156i
$$509$$ 11390.4 0.991891 0.495946 0.868354i $$-0.334822\pi$$
0.495946 + 0.868354i $$0.334822\pi$$
$$510$$ 0 0
$$511$$ 6326.85 0.547717
$$512$$ 8297.40i 0.716205i
$$513$$ 2092.24i 0.180068i
$$514$$ −6520.37 −0.559535
$$515$$ 0 0
$$516$$ −36301.9 −3.09710
$$517$$ − 3181.14i − 0.270612i
$$518$$ 15051.4i 1.27668i
$$519$$ 3704.24 0.313292
$$520$$ 0 0
$$521$$ −12824.5 −1.07841 −0.539206 0.842174i $$-0.681276\pi$$
−0.539206 + 0.842174i $$0.681276\pi$$
$$522$$ 21065.7i 1.76632i
$$523$$ − 13087.4i − 1.09421i −0.837063 0.547107i $$-0.815729\pi$$
0.837063 0.547107i $$-0.184271\pi$$
$$524$$ −5875.20 −0.489808
$$525$$ 0 0
$$526$$ 901.080 0.0746938
$$527$$ 1010.30i 0.0835093i
$$528$$ − 5336.86i − 0.439880i
$$529$$ −529.000 −0.0434783
$$530$$ 0 0
$$531$$ 26070.4 2.13062
$$532$$ 3680.73i 0.299962i
$$533$$ − 6636.50i − 0.539322i
$$534$$ 51805.0 4.19817
$$535$$ 0 0
$$536$$ 4966.00 0.400184
$$537$$ 19050.4i 1.53089i
$$538$$ 30805.6i 2.46863i
$$539$$ −7586.33 −0.606245
$$540$$ 0 0
$$541$$ 15464.3 1.22895 0.614473 0.788938i $$-0.289369\pi$$
0.614473 + 0.788938i $$0.289369\pi$$
$$542$$ − 9563.54i − 0.757914i
$$543$$ 7703.14i 0.608791i
$$544$$ 5745.06 0.452789
$$545$$ 0 0
$$546$$ −5000.14 −0.391916
$$547$$ 6981.32i 0.545703i 0.962056 + 0.272852i $$0.0879668\pi$$
−0.962056 + 0.272852i $$0.912033\pi$$
$$548$$ − 129.340i − 0.0100824i
$$549$$ 10589.1 0.823192
$$550$$ 0 0
$$551$$ −4926.60 −0.380908
$$552$$ − 2799.09i − 0.215828i
$$553$$ 10502.5i 0.807615i
$$554$$ −5274.38 −0.404489
$$555$$ 0 0
$$556$$ −19075.5 −1.45501
$$557$$ 24964.8i 1.89909i 0.313637 + 0.949543i $$0.398453\pi$$
−0.313637 + 0.949543i $$0.601547\pi$$
$$558$$ − 6113.45i − 0.463805i
$$559$$ 6468.88 0.489454
$$560$$ 0 0
$$561$$ −5705.83 −0.429412
$$562$$ 28651.2i 2.15049i
$$563$$ 10820.3i 0.809986i 0.914320 + 0.404993i $$0.132726\pi$$
−0.914320 + 0.404993i $$0.867274\pi$$
$$564$$ −9937.31 −0.741908
$$565$$ 0 0
$$566$$ −14199.3 −1.05449
$$567$$ 4223.65i 0.312834i
$$568$$ 8600.23i 0.635312i
$$569$$ −8438.73 −0.621740 −0.310870 0.950453i $$-0.600620\pi$$
−0.310870 + 0.950453i $$0.600620\pi$$
$$570$$ 0 0
$$571$$ 23107.6 1.69356 0.846781 0.531942i $$-0.178538\pi$$
0.846781 + 0.531942i $$0.178538\pi$$
$$572$$ − 5353.26i − 0.391313i
$$573$$ − 10673.4i − 0.778161i
$$574$$ 16350.2 1.18893
$$575$$ 0 0
$$576$$ −28266.7 −2.04475
$$577$$ 24848.7i 1.79283i 0.443215 + 0.896416i $$0.353838\pi$$
−0.443215 + 0.896416i $$0.646162\pi$$
$$578$$ 18902.5i 1.36028i
$$579$$ 11419.5 0.819651
$$580$$ 0 0
$$581$$ 4091.31 0.292145
$$582$$ 36067.4i 2.56880i
$$583$$ − 12176.0i − 0.864974i
$$584$$ 10943.6 0.775430
$$585$$ 0 0
$$586$$ 24673.2 1.73932
$$587$$ 8663.94i 0.609198i 0.952481 + 0.304599i $$0.0985224\pi$$
−0.952481 + 0.304599i $$0.901478\pi$$
$$588$$ 23698.3i 1.66208i
$$589$$ 1429.74 0.100019
$$590$$ 0 0
$$591$$ 8405.94 0.585066
$$592$$ − 8942.64i − 0.620845i
$$593$$ 24678.9i 1.70901i 0.519446 + 0.854503i $$0.326138\pi$$
−0.519446 + 0.854503i $$0.673862\pi$$
$$594$$ 7496.88 0.517846
$$595$$ 0 0
$$596$$ −5874.20 −0.403719
$$597$$ − 22578.0i − 1.54783i
$$598$$ 1634.54i 0.111774i
$$599$$ −19698.4 −1.34367 −0.671833 0.740702i $$-0.734493\pi$$
−0.671833 + 0.740702i $$0.734493\pi$$
$$600$$ 0 0
$$601$$ −18449.1 −1.25217 −0.626086 0.779754i $$-0.715344\pi$$
−0.626086 + 0.779754i $$0.715344\pi$$
$$602$$ 15937.3i 1.07899i
$$603$$ − 11035.5i − 0.745272i
$$604$$ 26911.2 1.81291
$$605$$ 0 0
$$606$$ 50233.9 3.36735
$$607$$ − 11321.0i − 0.757012i −0.925599 0.378506i $$-0.876438\pi$$
0.925599 0.378506i $$-0.123562\pi$$
$$608$$ − 8130.20i − 0.542308i
$$609$$ 9728.59 0.647327
$$610$$ 0 0
$$611$$ 1770.80 0.117248
$$612$$ 9997.36i 0.660326i
$$613$$ − 712.219i − 0.0469270i −0.999725 0.0234635i $$-0.992531\pi$$
0.999725 0.0234635i $$-0.00746935\pi$$
$$614$$ −12587.4 −0.827341
$$615$$ 0 0
$$616$$ 4024.62 0.263241
$$617$$ 3234.25i 0.211031i 0.994418 + 0.105515i $$0.0336492\pi$$
−0.994418 + 0.105515i $$0.966351\pi$$
$$618$$ − 7630.36i − 0.496663i
$$619$$ −26905.1 −1.74703 −0.873513 0.486801i $$-0.838164\pi$$
−0.873513 + 0.486801i $$0.838164\pi$$
$$620$$ 0 0
$$621$$ −1350.60 −0.0872750
$$622$$ − 29233.6i − 1.88451i
$$623$$ − 13419.2i − 0.862967i
$$624$$ 2970.78 0.190587
$$625$$ 0 0
$$626$$ −27702.6 −1.76872
$$627$$ 8074.68i 0.514309i
$$628$$ − 1687.77i − 0.107244i
$$629$$ −9560.90 −0.606070
$$630$$ 0 0
$$631$$ 16199.5 1.02202 0.511008 0.859576i $$-0.329272\pi$$
0.511008 + 0.859576i $$0.329272\pi$$
$$632$$ 18166.3i 1.14338i
$$633$$ 37219.3i 2.33702i
$$634$$ −24961.5 −1.56364
$$635$$ 0 0
$$636$$ −38035.7 −2.37141
$$637$$ − 4222.96i − 0.262668i
$$638$$ 17652.9i 1.09543i
$$639$$ 19111.5 1.18316
$$640$$ 0 0
$$641$$ 19943.5 1.22889 0.614446 0.788959i $$-0.289380\pi$$
0.614446 + 0.788959i $$0.289380\pi$$
$$642$$ 2052.97i 0.126206i
$$643$$ 27691.0i 1.69833i 0.528129 + 0.849164i $$0.322894\pi$$
−0.528129 + 0.849164i $$0.677106\pi$$
$$644$$ −2376.01 −0.145385
$$645$$ 0 0
$$646$$ −3962.65 −0.241344
$$647$$ − 23560.3i − 1.43161i −0.698300 0.715805i $$-0.746060\pi$$
0.698300 0.715805i $$-0.253940\pi$$
$$648$$ 7305.70i 0.442893i
$$649$$ 21846.7 1.32136
$$650$$ 0 0
$$651$$ −2823.32 −0.169976
$$652$$ − 37743.0i − 2.26707i
$$653$$ − 5571.58i − 0.333894i −0.985966 0.166947i $$-0.946609\pi$$
0.985966 0.166947i $$-0.0533909\pi$$
$$654$$ −68110.2 −4.07235
$$655$$ 0 0
$$656$$ −9714.32 −0.578171
$$657$$ − 24319.0i − 1.44410i
$$658$$ 4362.68i 0.258472i
$$659$$ −10177.4 −0.601602 −0.300801 0.953687i $$-0.597254\pi$$
−0.300801 + 0.953687i $$0.597254\pi$$
$$660$$ 0 0
$$661$$ −27413.8 −1.61312 −0.806561 0.591151i $$-0.798674\pi$$
−0.806561 + 0.591151i $$0.798674\pi$$
$$662$$ − 28233.3i − 1.65758i
$$663$$ − 3176.17i − 0.186052i
$$664$$ 7076.79 0.413604
$$665$$ 0 0
$$666$$ 57854.1 3.36607
$$667$$ − 3180.25i − 0.184618i
$$668$$ 17908.2i 1.03726i
$$669$$ −38219.6 −2.20875
$$670$$ 0 0
$$671$$ 8873.57 0.510522
$$672$$ 16054.8i 0.921616i
$$673$$ − 12767.5i − 0.731277i −0.930757 0.365639i $$-0.880851\pi$$
0.930757 0.365639i $$-0.119149\pi$$
$$674$$ −18349.3 −1.04865
$$675$$ 0 0
$$676$$ −22315.0 −1.26963
$$677$$ 17036.9i 0.967180i 0.875295 + 0.483590i $$0.160667\pi$$
−0.875295 + 0.483590i $$0.839333\pi$$
$$678$$ − 63031.7i − 3.57038i
$$679$$ 9342.63 0.528037
$$680$$ 0 0
$$681$$ −21502.0 −1.20993
$$682$$ − 5123.02i − 0.287640i
$$683$$ 510.213i 0.0285838i 0.999898 + 0.0142919i $$0.00454941\pi$$
−0.999898 + 0.0142919i $$0.995451\pi$$
$$684$$ 14147.9 0.790875
$$685$$ 0 0
$$686$$ 23999.0 1.33569
$$687$$ 11984.8i 0.665570i
$$688$$ − 9468.96i − 0.524710i
$$689$$ 6777.84 0.374768
$$690$$ 0 0
$$691$$ 22793.1 1.25483 0.627416 0.778684i $$-0.284112\pi$$
0.627416 + 0.778684i $$0.284112\pi$$
$$692$$ − 5438.83i − 0.298776i
$$693$$ − 8943.53i − 0.490240i
$$694$$ 13569.6 0.742213
$$695$$ 0 0
$$696$$ 16827.7 0.916452
$$697$$ 10385.9i 0.564412i
$$698$$ 36924.6i 2.00232i
$$699$$ −43541.9 −2.35609
$$700$$ 0 0
$$701$$ 26324.3 1.41834 0.709168 0.705039i $$-0.249071\pi$$
0.709168 + 0.705039i $$0.249071\pi$$
$$702$$ 4173.17i 0.224368i
$$703$$ 13530.2i 0.725892i
$$704$$ −23687.2 −1.26810
$$705$$ 0 0
$$706$$ 35419.2 1.88813
$$707$$ − 13012.2i − 0.692185i
$$708$$ − 68245.2i − 3.62262i
$$709$$ 10961.3 0.580620 0.290310 0.956933i $$-0.406242\pi$$
0.290310 + 0.956933i $$0.406242\pi$$
$$710$$ 0 0
$$711$$ 40369.2 2.12934
$$712$$ − 23211.3i − 1.22174i
$$713$$ 922.938i 0.0484773i
$$714$$ 7825.07 0.410148
$$715$$ 0 0
$$716$$ 27971.2 1.45996
$$717$$ 21796.4i 1.13529i
$$718$$ − 9446.44i − 0.491000i
$$719$$ 1304.68 0.0676723 0.0338362 0.999427i $$-0.489228\pi$$
0.0338362 + 0.999427i $$0.489228\pi$$
$$720$$ 0 0
$$721$$ −1976.51 −0.102093
$$722$$ − 24691.1i − 1.27273i
$$723$$ 43667.7i 2.24622i
$$724$$ 11310.3 0.580585
$$725$$ 0 0
$$726$$ −17171.3 −0.877808
$$727$$ 1583.59i 0.0807869i 0.999184 + 0.0403934i $$0.0128611\pi$$
−0.999184 + 0.0403934i $$0.987139\pi$$
$$728$$ 2240.32i 0.114055i
$$729$$ 28667.3 1.45645
$$730$$ 0 0
$$731$$ −10123.6 −0.512223
$$732$$ − 27719.4i − 1.39964i
$$733$$ 34351.2i 1.73095i 0.500948 + 0.865477i $$0.332985\pi$$
−0.500948 + 0.865477i $$0.667015\pi$$
$$734$$ 19922.3 1.00183
$$735$$ 0 0
$$736$$ 5248.27 0.262845
$$737$$ − 9247.61i − 0.462198i
$$738$$ − 62846.5i − 3.13471i
$$739$$ −14996.6 −0.746494 −0.373247 0.927732i $$-0.621756\pi$$
−0.373247 + 0.927732i $$0.621756\pi$$
$$740$$ 0 0
$$741$$ −4494.80 −0.222835
$$742$$ 16698.4i 0.826171i
$$743$$ 32781.7i 1.61863i 0.587372 + 0.809317i $$0.300162\pi$$
−0.587372 + 0.809317i $$0.699838\pi$$
$$744$$ −4883.53 −0.240644
$$745$$ 0 0
$$746$$ −14851.7 −0.728899
$$747$$ − 15726.1i − 0.770263i
$$748$$ 8377.69i 0.409517i
$$749$$ 531.787 0.0259427
$$750$$ 0 0
$$751$$ −24044.4 −1.16830 −0.584149 0.811647i $$-0.698571\pi$$
−0.584149 + 0.811647i $$0.698571\pi$$
$$752$$ − 2592.04i − 0.125694i
$$753$$ 3035.16i 0.146889i
$$754$$ −9826.54 −0.474617
$$755$$ 0 0
$$756$$ −6066.26 −0.291835
$$757$$ − 21680.9i − 1.04096i −0.853874 0.520479i $$-0.825753\pi$$
0.853874 0.520479i $$-0.174247\pi$$
$$758$$ − 9308.15i − 0.446025i
$$759$$ −5212.43 −0.249274
$$760$$ 0 0
$$761$$ −10299.1 −0.490594 −0.245297 0.969448i $$-0.578885\pi$$
−0.245297 + 0.969448i $$0.578885\pi$$
$$762$$ 28906.7i 1.37425i
$$763$$ 17642.8i 0.837105i
$$764$$ −15671.4 −0.742108
$$765$$ 0 0
$$766$$ 6065.29 0.286094
$$767$$ 12161.1i 0.572505i
$$768$$ 10761.7i 0.505637i
$$769$$ −28377.9 −1.33073 −0.665365 0.746518i $$-0.731724\pi$$
−0.665365 + 0.746518i $$0.731724\pi$$
$$770$$ 0 0
$$771$$ 11574.5 0.540657
$$772$$ − 16766.9i − 0.781675i
$$773$$ − 35409.5i − 1.64759i −0.566885 0.823797i $$-0.691852\pi$$
0.566885 0.823797i $$-0.308148\pi$$
$$774$$ 61259.2 2.84485
$$775$$ 0 0
$$776$$ 16160.1 0.747568
$$777$$ − 26718.2i − 1.23361i
$$778$$ − 30957.3i − 1.42657i
$$779$$ 14697.8 0.675999
$$780$$ 0 0
$$781$$ 16015.2 0.733763
$$782$$ − 2558.00i − 0.116974i
$$783$$ − 8119.58i − 0.370588i
$$784$$ −6181.45 −0.281589
$$785$$ 0 0
$$786$$ 17676.0 0.802138
$$787$$ 6117.56i 0.277087i 0.990356 + 0.138544i $$0.0442421\pi$$
−0.990356 + 0.138544i $$0.955758\pi$$
$$788$$ − 12342.2i − 0.557960i
$$789$$ −1599.54 −0.0721737
$$790$$ 0 0
$$791$$ −16327.3 −0.733921
$$792$$ − 15469.7i − 0.694057i
$$793$$ 4939.51i 0.221194i
$$794$$ −10612.9 −0.474357
$$795$$ 0 0
$$796$$ −33150.6 −1.47612
$$797$$ 4099.46i 0.182196i 0.995842 + 0.0910980i $$0.0290377\pi$$
−0.995842 + 0.0910980i $$0.970962\pi$$
$$798$$ − 11073.8i − 0.491236i
$$799$$ −2771.24 −0.122703
$$800$$ 0 0
$$801$$ −51580.3 −2.27528
$$802$$ − 52575.6i − 2.31485i
$$803$$ − 20379.1i − 0.895594i
$$804$$ −28887.8 −1.26716
$$805$$ 0 0
$$806$$ 2851.75 0.124626
$$807$$ − 54684.1i − 2.38534i
$$808$$ − 22507.4i − 0.979960i
$$809$$ −21358.6 −0.928216 −0.464108 0.885779i $$-0.653625\pi$$
−0.464108 + 0.885779i $$0.653625\pi$$
$$810$$ 0 0
$$811$$ 13967.7 0.604776 0.302388 0.953185i $$-0.402216\pi$$
0.302388 + 0.953185i $$0.402216\pi$$
$$812$$ − 14284.2i − 0.617336i
$$813$$ 16976.6i 0.732342i
$$814$$ 48481.2 2.08755
$$815$$ 0 0
$$816$$ −4649.19 −0.199454
$$817$$ 14326.6i 0.613492i
$$818$$ − 23314.8i − 0.996557i
$$819$$ 4978.45 0.212407
$$820$$ 0 0
$$821$$ −22387.6 −0.951684 −0.475842 0.879531i $$-0.657857\pi$$
−0.475842 + 0.879531i $$0.657857\pi$$
$$822$$ 389.130i 0.0165115i
$$823$$ 22615.7i 0.957877i 0.877848 + 0.478939i $$0.158978\pi$$
−0.877848 + 0.478939i $$0.841022\pi$$
$$824$$ −3418.80 −0.144538
$$825$$ 0 0
$$826$$ −29961.0 −1.26208
$$827$$ 10878.0i 0.457394i 0.973498 + 0.228697i $$0.0734464\pi$$
−0.973498 + 0.228697i $$0.926554\pi$$
$$828$$ 9132.86i 0.383320i
$$829$$ −27382.3 −1.14720 −0.573600 0.819136i $$-0.694453\pi$$
−0.573600 + 0.819136i $$0.694453\pi$$
$$830$$ 0 0
$$831$$ 9362.74 0.390842
$$832$$ − 13185.6i − 0.549432i
$$833$$ 6608.81i 0.274888i
$$834$$ 57390.2 2.38281
$$835$$ 0 0
$$836$$ 11855.8 0.490480
$$837$$ 2356.37i 0.0973096i
$$838$$ 49459.1i 2.03883i
$$839$$ 31799.7 1.30852 0.654260 0.756270i $$-0.272980\pi$$
0.654260 + 0.756270i $$0.272980\pi$$
$$840$$ 0 0
$$841$$ −5269.87 −0.216076
$$842$$ 22868.0i 0.935968i
$$843$$ − 50859.6i − 2.07793i
$$844$$ 54647.9 2.22874
$$845$$ 0 0
$$846$$ 16769.1 0.681483
$$847$$ 4447.94i 0.180440i
$$848$$ − 9921.21i − 0.401764i
$$849$$ 25205.7 1.01891
$$850$$ 0 0
$$851$$ −8734.14 −0.351824
$$852$$ − 50028.6i − 2.01168i
$$853$$ − 32016.4i − 1.28514i −0.766229 0.642568i $$-0.777869\pi$$
0.766229 0.642568i $$-0.222131\pi$$
$$854$$ −12169.4 −0.487620
$$855$$ 0 0
$$856$$ 919.839 0.0367283
$$857$$ 25280.1i 1.00764i 0.863807 + 0.503822i $$0.168073\pi$$
−0.863807 + 0.503822i $$0.831927\pi$$
$$858$$ 16105.7i 0.640837i
$$859$$ 22313.5 0.886296 0.443148 0.896448i $$-0.353862\pi$$
0.443148 + 0.896448i $$0.353862\pi$$
$$860$$ 0 0
$$861$$ −29023.8 −1.14881
$$862$$ 41297.9i 1.63180i
$$863$$ 1478.28i 0.0583096i 0.999575 + 0.0291548i $$0.00928158\pi$$
−0.999575 + 0.0291548i $$0.990718\pi$$
$$864$$ 13399.5 0.527615
$$865$$ 0 0
$$866$$ 19087.2 0.748970
$$867$$ − 33554.5i − 1.31438i
$$868$$ 4145.39i 0.162101i
$$869$$ 33829.0 1.32056
$$870$$ 0 0
$$871$$ 5147.72 0.200257
$$872$$ 30516.9i 1.18513i
$$873$$ − 35910.9i − 1.39221i
$$874$$ −3619.99 −0.140101
$$875$$ 0 0
$$876$$ −63660.5 −2.45535
$$877$$ 32974.6i 1.26964i 0.772661 + 0.634819i $$0.218925\pi$$
−0.772661 + 0.634819i $$0.781075\pi$$
$$878$$ 30992.6i 1.19129i
$$879$$ −43798.3 −1.68064
$$880$$ 0 0
$$881$$ −32000.7 −1.22376 −0.611879 0.790951i $$-0.709586\pi$$
−0.611879 + 0.790951i $$0.709586\pi$$
$$882$$ − 39990.7i − 1.52671i
$$883$$ 44218.6i 1.68525i 0.538503 + 0.842623i $$0.318990\pi$$
−0.538503 + 0.842623i $$0.681010\pi$$
$$884$$ −4663.47 −0.177432
$$885$$ 0 0
$$886$$ 54526.6 2.06756
$$887$$ 27374.8i 1.03625i 0.855304 + 0.518126i $$0.173370\pi$$
−0.855304 + 0.518126i $$0.826630\pi$$
$$888$$ − 46214.9i − 1.74647i
$$889$$ 7487.79 0.282489
$$890$$ 0 0
$$891$$ 13604.6 0.511526
$$892$$ 56116.7i 2.10642i
$$893$$ 3921.76i 0.146962i
$$894$$ 17673.0 0.661155
$$895$$ 0 0
$$896$$ 16105.7 0.600505
$$897$$ − 2901.52i − 0.108003i
$$898$$ − 80162.2i − 2.97889i
$$899$$ −5548.54 −0.205844
$$900$$ 0 0
$$901$$ −10607.1 −0.392203
$$902$$ − 52664.8i − 1.94406i
$$903$$ − 28290.8i − 1.04259i
$$904$$ −28241.5 −1.03905
$$905$$ 0 0
$$906$$ −80964.3 −2.96894
$$907$$ 7835.16i 0.286838i 0.989662 + 0.143419i $$0.0458097\pi$$
−0.989662 + 0.143419i $$0.954190\pi$$
$$908$$ 31570.7i 1.15387i
$$909$$ −50016.0 −1.82500
$$910$$ 0 0
$$911$$ −36528.6 −1.32848 −0.664241 0.747519i $$-0.731245\pi$$
−0.664241 + 0.747519i $$0.731245\pi$$
$$912$$ 6579.36i 0.238886i
$$913$$ − 13178.3i − 0.477698i
$$914$$ −23119.8 −0.836692
$$915$$ 0 0
$$916$$ 17596.8 0.634734
$$917$$ − 4578.65i − 0.164886i
$$918$$ − 6530.89i − 0.234805i
$$919$$ 15741.5 0.565030 0.282515 0.959263i $$-0.408831\pi$$
0.282515 + 0.959263i $$0.408831\pi$$
$$920$$ 0 0
$$921$$ 22344.4 0.799427
$$922$$ − 10314.7i − 0.368433i
$$923$$ 8914.93i 0.317918i
$$924$$ −23411.7 −0.833538
$$925$$ 0 0
$$926$$ 60271.5 2.13892
$$927$$ 7597.26i 0.269177i
$$928$$ 31551.7i 1.11609i
$$929$$ −47428.9 −1.67502 −0.837510 0.546422i $$-0.815989\pi$$
−0.837510 + 0.546422i $$0.815989\pi$$
$$930$$ 0 0
$$931$$ 9352.54 0.329234
$$932$$ 63931.3i 2.24693i
$$933$$ 51893.6i 1.82092i
$$934$$ 23175.0 0.811895
$$935$$ 0 0
$$936$$ 8611.29 0.300714
$$937$$ − 34978.0i − 1.21951i −0.792589 0.609756i $$-0.791267\pi$$
0.792589 0.609756i $$-0.208733\pi$$
$$938$$ 12682.3i 0.441464i
$$939$$ 49175.9 1.70905
$$940$$ 0 0
$$941$$ 45144.0 1.56392 0.781962 0.623326i $$-0.214219\pi$$
0.781962 + 0.623326i $$0.214219\pi$$
$$942$$ 5077.78i 0.175629i
$$943$$ 9487.83i 0.327642i
$$944$$ 17801.0 0.613744
$$945$$ 0 0
$$946$$ 51334.6 1.76430
$$947$$ 26123.6i 0.896413i 0.893930 + 0.448207i $$0.147937\pi$$
−0.893930 + 0.448207i $$0.852063\pi$$
$$948$$ − 105675.i − 3.62044i
$$949$$ 11344.1 0.388035
$$950$$ 0 0
$$951$$ 44310.0 1.51088
$$952$$ − 3506.04i − 0.119361i
$$953$$ 22143.5i 0.752673i 0.926483 + 0.376336i $$0.122816\pi$$
−0.926483 + 0.376336i $$0.877184\pi$$
$$954$$ 64185.0 2.17827
$$955$$ 0 0
$$956$$ 32003.0 1.08269
$$957$$ − 31336.2i − 1.05847i
$$958$$ − 54237.9i − 1.82917i
$$959$$ 100.797 0.00339407
$$960$$ 0 0
$$961$$ −28180.8 −0.945949
$$962$$ 26987.3i 0.904474i
$$963$$ − 2044.07i − 0.0684000i
$$964$$ 64116.0 2.14215
$$965$$ 0 0
$$966$$ 7148.42 0.238092
$$967$$ 44869.8i 1.49216i 0.665858 + 0.746078i $$0.268066\pi$$
−0.665858 + 0.746078i $$0.731934\pi$$
$$968$$ 7693.65i 0.255458i
$$969$$ 7034.23 0.233201
$$970$$ 0 0
$$971$$ −25048.3 −0.827845 −0.413923 0.910312i $$-0.635842\pi$$
−0.413923 + 0.910312i $$0.635842\pi$$
$$972$$ − 60752.5i − 2.00477i
$$973$$ − 14865.9i − 0.489805i
$$974$$ 39514.7 1.29993
$$975$$ 0 0
$$976$$ 7230.31 0.237128
$$977$$ 37320.7i 1.22210i 0.791590 + 0.611052i $$0.209253\pi$$
−0.791590 + 0.611052i $$0.790747\pi$$
$$978$$ 113553.i 3.71269i
$$979$$ −43223.8 −1.41107
$$980$$ 0 0
$$981$$ 67814.7 2.20709
$$982$$ − 34424.5i − 1.11867i
$$983$$ 17189.4i 0.557737i 0.960329 + 0.278869i $$0.0899594\pi$$
−0.960329 + 0.278869i $$0.910041\pi$$
$$984$$ −50202.9 −1.62643
$$985$$ 0 0
$$986$$ 15378.2 0.496697
$$987$$ − 7744.33i − 0.249752i
$$988$$ 6599.58i 0.212511i
$$989$$ −9248.19 −0.297346
$$990$$ 0 0
$$991$$ −57797.1 −1.85266 −0.926330 0.376712i $$-0.877055\pi$$
−0.926330 + 0.376712i $$0.877055\pi$$
$$992$$ − 9156.57i − 0.293066i
$$993$$ 50117.9i 1.60166i
$$994$$ −21963.5 −0.700846
$$995$$ 0 0
$$996$$ −41166.6 −1.30965
$$997$$ 46801.0i 1.48666i 0.668923 + 0.743331i $$0.266755\pi$$
−0.668923 + 0.743331i $$0.733245\pi$$
$$998$$ 37891.2i 1.20183i
$$999$$ −22299.3 −0.706226
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 575.4.b.i.24.9 10
5.2 odd 4 575.4.a.j.1.2 5
5.3 odd 4 115.4.a.e.1.4 5
5.4 even 2 inner 575.4.b.i.24.2 10
15.8 even 4 1035.4.a.k.1.2 5
20.3 even 4 1840.4.a.n.1.2 5

By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.4 5 5.3 odd 4
575.4.a.j.1.2 5 5.2 odd 4
575.4.b.i.24.2 10 5.4 even 2 inner
575.4.b.i.24.9 10 1.1 even 1 trivial
1035.4.a.k.1.2 5 15.8 even 4
1840.4.a.n.1.2 5 20.3 even 4