# Properties

 Label 575.4.a.j.1.2 Level $575$ Weight $4$ Character 575.1 Self dual yes Analytic conductor $33.926$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,4,Mod(1,575)]

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([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("575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$575 = 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 575.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: $$33.9260982533$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92$$ x^5 - x^4 - 27*x^3 + 7*x^2 + 168*x + 92 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 115) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.41740$$ of defining polynomial Character $$\chi$$ $$=$$ 575.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.41740 q^{2} -7.84147 q^{3} +11.5134 q^{4} +34.6389 q^{6} +8.97260 q^{7} -15.5200 q^{8} +34.4886 q^{9} +O(q^{10})$$ $$q-4.41740 q^{2} -7.84147 q^{3} +11.5134 q^{4} +34.6389 q^{6} +8.97260 q^{7} -15.5200 q^{8} +34.4886 q^{9} -28.9011 q^{11} -90.2818 q^{12} -16.0879 q^{13} -39.6355 q^{14} -23.5490 q^{16} -25.1771 q^{17} -152.350 q^{18} -35.6298 q^{19} -70.3583 q^{21} +127.668 q^{22} +23.0000 q^{23} +121.700 q^{24} +71.0668 q^{26} -58.7218 q^{27} +103.305 q^{28} +138.272 q^{29} +40.1277 q^{31} +228.186 q^{32} +226.627 q^{33} +111.217 q^{34} +397.081 q^{36} -379.745 q^{37} +157.391 q^{38} +126.153 q^{39} +412.514 q^{41} +310.801 q^{42} +402.095 q^{43} -332.750 q^{44} -101.600 q^{46} -110.070 q^{47} +184.659 q^{48} -262.492 q^{49} +197.426 q^{51} -185.227 q^{52} +421.300 q^{53} +259.397 q^{54} -139.255 q^{56} +279.390 q^{57} -610.802 q^{58} +755.913 q^{59} -307.032 q^{61} -177.260 q^{62} +309.453 q^{63} -819.594 q^{64} -1001.10 q^{66} -319.974 q^{67} -289.874 q^{68} -180.354 q^{69} -554.138 q^{71} -535.264 q^{72} +705.131 q^{73} +1677.48 q^{74} -410.219 q^{76} -259.318 q^{77} -557.268 q^{78} +1170.51 q^{79} -470.728 q^{81} -1822.24 q^{82} +455.978 q^{83} -810.063 q^{84} -1776.21 q^{86} -1084.26 q^{87} +448.546 q^{88} -1495.57 q^{89} -144.351 q^{91} +264.808 q^{92} -314.660 q^{93} +486.222 q^{94} -1789.31 q^{96} -1041.24 q^{97} +1159.53 q^{98} -996.760 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 6 q^{2} - 4 q^{3} + 22 q^{4} + 19 q^{6} + 3 q^{7} - 138 q^{8} + 77 q^{9}+O(q^{10})$$ 5 * q - 6 * q^2 - 4 * q^3 + 22 * q^4 + 19 * q^6 + 3 * q^7 - 138 * q^8 + 77 * q^9 $$5 q - 6 q^{2} - 4 q^{3} + 22 q^{4} + 19 q^{6} + 3 q^{7} - 138 q^{8} + 77 q^{9} + 23 q^{11} - 47 q^{12} - 132 q^{13} + 93 q^{14} + 282 q^{16} - 23 q^{17} + 15 q^{18} - 161 q^{19} - 60 q^{21} - 193 q^{22} + 115 q^{23} + 105 q^{24} - 257 q^{26} - 577 q^{27} - 17 q^{28} + 401 q^{29} + 32 q^{31} - 670 q^{32} - 189 q^{33} - 663 q^{34} - 659 q^{36} + 38 q^{37} + 875 q^{38} + 335 q^{39} - 12 q^{41} + 798 q^{42} + 566 q^{43} + 47 q^{44} - 138 q^{46} - 919 q^{47} + 773 q^{48} - 738 q^{49} - 993 q^{51} + 305 q^{52} - 1156 q^{53} - 8 q^{54} + 343 q^{56} - 114 q^{57} + 1042 q^{58} + 1324 q^{59} - 1673 q^{61} - 565 q^{62} - 270 q^{63} + 2466 q^{64} - 2781 q^{66} - 558 q^{67} + 2267 q^{68} - 92 q^{69} - 108 q^{71} + 789 q^{72} - 1173 q^{73} + 1458 q^{74} - 3477 q^{76} - 2608 q^{77} - 704 q^{78} + 656 q^{79} - 319 q^{81} - 3505 q^{82} + 82 q^{83} - 718 q^{84} + 112 q^{86} - 2389 q^{87} - 2397 q^{88} + 570 q^{89} - 1589 q^{91} + 506 q^{92} - 911 q^{93} - 948 q^{94} - 5991 q^{96} - 633 q^{97} + 2555 q^{98} + 2021 q^{99}+O(q^{100})$$ 5 * q - 6 * q^2 - 4 * q^3 + 22 * q^4 + 19 * q^6 + 3 * q^7 - 138 * q^8 + 77 * q^9 + 23 * q^11 - 47 * q^12 - 132 * q^13 + 93 * q^14 + 282 * q^16 - 23 * q^17 + 15 * q^18 - 161 * q^19 - 60 * q^21 - 193 * q^22 + 115 * q^23 + 105 * q^24 - 257 * q^26 - 577 * q^27 - 17 * q^28 + 401 * q^29 + 32 * q^31 - 670 * q^32 - 189 * q^33 - 663 * q^34 - 659 * q^36 + 38 * q^37 + 875 * q^38 + 335 * q^39 - 12 * q^41 + 798 * q^42 + 566 * q^43 + 47 * q^44 - 138 * q^46 - 919 * q^47 + 773 * q^48 - 738 * q^49 - 993 * q^51 + 305 * q^52 - 1156 * q^53 - 8 * q^54 + 343 * q^56 - 114 * q^57 + 1042 * q^58 + 1324 * q^59 - 1673 * q^61 - 565 * q^62 - 270 * q^63 + 2466 * q^64 - 2781 * q^66 - 558 * q^67 + 2267 * q^68 - 92 * q^69 - 108 * q^71 + 789 * q^72 - 1173 * q^73 + 1458 * q^74 - 3477 * q^76 - 2608 * q^77 - 704 * q^78 + 656 * q^79 - 319 * q^81 - 3505 * q^82 + 82 * q^83 - 718 * q^84 + 112 * q^86 - 2389 * q^87 - 2397 * q^88 + 570 * q^89 - 1589 * q^91 + 506 * q^92 - 911 * q^93 - 948 * q^94 - 5991 * q^96 - 633 * q^97 + 2555 * q^98 + 2021 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.41740 −1.56179 −0.780893 0.624665i $$-0.785235\pi$$
−0.780893 + 0.624665i $$0.785235\pi$$
$$3$$ −7.84147 −1.50909 −0.754546 0.656248i $$-0.772143\pi$$
−0.754546 + 0.656248i $$0.772143\pi$$
$$4$$ 11.5134 1.43917
$$5$$ 0 0
$$6$$ 34.6389 2.35688
$$7$$ 8.97260 0.484475 0.242237 0.970217i $$-0.422119\pi$$
0.242237 + 0.970217i $$0.422119\pi$$
$$8$$ −15.5200 −0.685894
$$9$$ 34.4886 1.27736
$$10$$ 0 0
$$11$$ −28.9011 −0.792183 −0.396092 0.918211i $$-0.629634\pi$$
−0.396092 + 0.918211i $$0.629634\pi$$
$$12$$ −90.2818 −2.17184
$$13$$ −16.0879 −0.343230 −0.171615 0.985164i $$-0.554898\pi$$
−0.171615 + 0.985164i $$0.554898\pi$$
$$14$$ −39.6355 −0.756646
$$15$$ 0 0
$$16$$ −23.5490 −0.367954
$$17$$ −25.1771 −0.359197 −0.179599 0.983740i $$-0.557480\pi$$
−0.179599 + 0.983740i $$0.557480\pi$$
$$18$$ −152.350 −1.99496
$$19$$ −35.6298 −0.430212 −0.215106 0.976591i $$-0.569010\pi$$
−0.215106 + 0.976591i $$0.569010\pi$$
$$20$$ 0 0
$$21$$ −70.3583 −0.731117
$$22$$ 127.668 1.23722
$$23$$ 23.0000 0.208514
$$24$$ 121.700 1.03508
$$25$$ 0 0
$$26$$ 71.0668 0.536051
$$27$$ −58.7218 −0.418556
$$28$$ 103.305 0.697243
$$29$$ 138.272 0.885395 0.442698 0.896671i $$-0.354022\pi$$
0.442698 + 0.896671i $$0.354022\pi$$
$$30$$ 0 0
$$31$$ 40.1277 0.232489 0.116244 0.993221i $$-0.462914\pi$$
0.116244 + 0.993221i $$0.462914\pi$$
$$32$$ 228.186 1.26056
$$33$$ 226.627 1.19548
$$34$$ 111.217 0.560989
$$35$$ 0 0
$$36$$ 397.081 1.83834
$$37$$ −379.745 −1.68729 −0.843645 0.536902i $$-0.819595\pi$$
−0.843645 + 0.536902i $$0.819595\pi$$
$$38$$ 157.391 0.671899
$$39$$ 126.153 0.517965
$$40$$ 0 0
$$41$$ 412.514 1.57132 0.785658 0.618662i $$-0.212325\pi$$
0.785658 + 0.618662i $$0.212325\pi$$
$$42$$ 310.801 1.14185
$$43$$ 402.095 1.42602 0.713011 0.701153i $$-0.247331\pi$$
0.713011 + 0.701153i $$0.247331\pi$$
$$44$$ −332.750 −1.14009
$$45$$ 0 0
$$46$$ −101.600 −0.325655
$$47$$ −110.070 −0.341603 −0.170801 0.985305i $$-0.554636\pi$$
−0.170801 + 0.985305i $$0.554636\pi$$
$$48$$ 184.659 0.555276
$$49$$ −262.492 −0.765284
$$50$$ 0 0
$$51$$ 197.426 0.542061
$$52$$ −185.227 −0.493967
$$53$$ 421.300 1.09189 0.545943 0.837822i $$-0.316171\pi$$
0.545943 + 0.837822i $$0.316171\pi$$
$$54$$ 259.397 0.653695
$$55$$ 0 0
$$56$$ −139.255 −0.332298
$$57$$ 279.390 0.649229
$$58$$ −610.802 −1.38280
$$59$$ 755.913 1.66799 0.833996 0.551770i $$-0.186048\pi$$
0.833996 + 0.551770i $$0.186048\pi$$
$$60$$ 0 0
$$61$$ −307.032 −0.644450 −0.322225 0.946663i $$-0.604431\pi$$
−0.322225 + 0.946663i $$0.604431\pi$$
$$62$$ −177.260 −0.363098
$$63$$ 309.453 0.618847
$$64$$ −819.594 −1.60077
$$65$$ 0 0
$$66$$ −1001.10 −1.86708
$$67$$ −319.974 −0.583448 −0.291724 0.956502i $$-0.594229\pi$$
−0.291724 + 0.956502i $$0.594229\pi$$
$$68$$ −289.874 −0.516947
$$69$$ −180.354 −0.314667
$$70$$ 0 0
$$71$$ −554.138 −0.926254 −0.463127 0.886292i $$-0.653273\pi$$
−0.463127 + 0.886292i $$0.653273\pi$$
$$72$$ −535.264 −0.876131
$$73$$ 705.131 1.13054 0.565269 0.824907i $$-0.308772\pi$$
0.565269 + 0.824907i $$0.308772\pi$$
$$74$$ 1677.48 2.63518
$$75$$ 0 0
$$76$$ −410.219 −0.619149
$$77$$ −259.318 −0.383793
$$78$$ −557.268 −0.808950
$$79$$ 1170.51 1.66699 0.833495 0.552526i $$-0.186336\pi$$
0.833495 + 0.552526i $$0.186336\pi$$
$$80$$ 0 0
$$81$$ −470.728 −0.645717
$$82$$ −1822.24 −2.45406
$$83$$ 455.978 0.603014 0.301507 0.953464i $$-0.402510\pi$$
0.301507 + 0.953464i $$0.402510\pi$$
$$84$$ −810.063 −1.05220
$$85$$ 0 0
$$86$$ −1776.21 −2.22714
$$87$$ −1084.26 −1.33614
$$88$$ 448.546 0.543354
$$89$$ −1495.57 −1.78124 −0.890621 0.454746i $$-0.849730\pi$$
−0.890621 + 0.454746i $$0.849730\pi$$
$$90$$ 0 0
$$91$$ −144.351 −0.166286
$$92$$ 264.808 0.300088
$$93$$ −314.660 −0.350847
$$94$$ 486.222 0.533510
$$95$$ 0 0
$$96$$ −1789.31 −1.90230
$$97$$ −1041.24 −1.08992 −0.544958 0.838463i $$-0.683455\pi$$
−0.544958 + 0.838463i $$0.683455\pi$$
$$98$$ 1159.53 1.19521
$$99$$ −996.760 −1.01190
$$100$$ 0 0
$$101$$ 1450.22 1.42873 0.714367 0.699771i $$-0.246715\pi$$
0.714367 + 0.699771i $$0.246715\pi$$
$$102$$ −872.108 −0.846584
$$103$$ −220.283 −0.210729 −0.105365 0.994434i $$-0.533601\pi$$
−0.105365 + 0.994434i $$0.533601\pi$$
$$104$$ 249.685 0.235419
$$105$$ 0 0
$$106$$ −1861.05 −1.70529
$$107$$ −59.2679 −0.0535481 −0.0267741 0.999642i $$-0.508523\pi$$
−0.0267741 + 0.999642i $$0.508523\pi$$
$$108$$ −676.087 −0.602375
$$109$$ 1966.29 1.72786 0.863930 0.503612i $$-0.167996\pi$$
0.863930 + 0.503612i $$0.167996\pi$$
$$110$$ 0 0
$$111$$ 2977.76 2.54627
$$112$$ −211.296 −0.178264
$$113$$ −1819.68 −1.51488 −0.757439 0.652905i $$-0.773550\pi$$
−0.757439 + 0.652905i $$0.773550\pi$$
$$114$$ −1234.17 −1.01396
$$115$$ 0 0
$$116$$ 1591.98 1.27424
$$117$$ −554.851 −0.438427
$$118$$ −3339.17 −2.60505
$$119$$ −225.904 −0.174022
$$120$$ 0 0
$$121$$ −495.725 −0.372445
$$122$$ 1356.28 1.00649
$$123$$ −3234.72 −2.37126
$$124$$ 462.006 0.334592
$$125$$ 0 0
$$126$$ −1366.97 −0.966506
$$127$$ −834.517 −0.583082 −0.291541 0.956558i $$-0.594168\pi$$
−0.291541 + 0.956558i $$0.594168\pi$$
$$128$$ 1794.98 1.23950
$$129$$ −3153.02 −2.15200
$$130$$ 0 0
$$131$$ 510.293 0.340340 0.170170 0.985415i $$-0.445568\pi$$
0.170170 + 0.985415i $$0.445568\pi$$
$$132$$ 2609.25 1.72050
$$133$$ −319.692 −0.208427
$$134$$ 1413.45 0.911221
$$135$$ 0 0
$$136$$ 390.750 0.246371
$$137$$ −11.2339 −0.00700567 −0.00350284 0.999994i $$-0.501115\pi$$
−0.00350284 + 0.999994i $$0.501115\pi$$
$$138$$ 796.694 0.491443
$$139$$ −1656.81 −1.01100 −0.505501 0.862826i $$-0.668692\pi$$
−0.505501 + 0.862826i $$0.668692\pi$$
$$140$$ 0 0
$$141$$ 863.109 0.515510
$$142$$ 2447.85 1.44661
$$143$$ 464.959 0.271901
$$144$$ −812.174 −0.470008
$$145$$ 0 0
$$146$$ −3114.84 −1.76566
$$147$$ 2058.33 1.15488
$$148$$ −4372.15 −2.42830
$$149$$ −510.206 −0.280522 −0.140261 0.990115i $$-0.544794\pi$$
−0.140261 + 0.990115i $$0.544794\pi$$
$$150$$ 0 0
$$151$$ −2337.38 −1.25969 −0.629846 0.776720i $$-0.716882\pi$$
−0.629846 + 0.776720i $$0.716882\pi$$
$$152$$ 552.974 0.295080
$$153$$ −868.325 −0.458823
$$154$$ 1145.51 0.599402
$$155$$ 0 0
$$156$$ 1452.45 0.745442
$$157$$ −146.592 −0.0745178 −0.0372589 0.999306i $$-0.511863\pi$$
−0.0372589 + 0.999306i $$0.511863\pi$$
$$158$$ −5170.59 −2.60348
$$159$$ −3303.61 −1.64776
$$160$$ 0 0
$$161$$ 206.370 0.101020
$$162$$ 2079.39 1.00847
$$163$$ 3278.19 1.57526 0.787630 0.616149i $$-0.211308\pi$$
0.787630 + 0.616149i $$0.211308\pi$$
$$164$$ 4749.44 2.26139
$$165$$ 0 0
$$166$$ −2014.24 −0.941778
$$167$$ 1555.42 0.720732 0.360366 0.932811i $$-0.382652\pi$$
0.360366 + 0.932811i $$0.382652\pi$$
$$168$$ 1091.96 0.501469
$$169$$ −1938.18 −0.882193
$$170$$ 0 0
$$171$$ −1228.82 −0.549534
$$172$$ 4629.48 2.05229
$$173$$ 472.392 0.207603 0.103801 0.994598i $$-0.466899\pi$$
0.103801 + 0.994598i $$0.466899\pi$$
$$174$$ 4789.58 2.08677
$$175$$ 0 0
$$176$$ 680.594 0.291487
$$177$$ −5927.47 −2.51715
$$178$$ 6606.54 2.78192
$$179$$ 2429.45 1.01444 0.507222 0.861815i $$-0.330672\pi$$
0.507222 + 0.861815i $$0.330672\pi$$
$$180$$ 0 0
$$181$$ −982.359 −0.403415 −0.201708 0.979446i $$-0.564649\pi$$
−0.201708 + 0.979446i $$0.564649\pi$$
$$182$$ 637.653 0.259703
$$183$$ 2407.58 0.972533
$$184$$ −356.960 −0.143019
$$185$$ 0 0
$$186$$ 1389.98 0.547947
$$187$$ 727.648 0.284550
$$188$$ −1267.28 −0.491626
$$189$$ −526.887 −0.202780
$$190$$ 0 0
$$191$$ 1361.14 0.515649 0.257824 0.966192i $$-0.416994\pi$$
0.257824 + 0.966192i $$0.416994\pi$$
$$192$$ 6426.82 2.41571
$$193$$ 1456.29 0.543142 0.271571 0.962418i $$-0.412457\pi$$
0.271571 + 0.962418i $$0.412457\pi$$
$$194$$ 4599.57 1.70222
$$195$$ 0 0
$$196$$ −3022.18 −1.10138
$$197$$ −1071.99 −0.387695 −0.193847 0.981032i $$-0.562097\pi$$
−0.193847 + 0.981032i $$0.562097\pi$$
$$198$$ 4403.08 1.58037
$$199$$ −2879.31 −1.02567 −0.512836 0.858486i $$-0.671405\pi$$
−0.512836 + 0.858486i $$0.671405\pi$$
$$200$$ 0 0
$$201$$ 2509.07 0.880477
$$202$$ −6406.19 −2.23138
$$203$$ 1240.66 0.428952
$$204$$ 2273.04 0.780120
$$205$$ 0 0
$$206$$ 973.077 0.329114
$$207$$ 793.238 0.266347
$$208$$ 378.855 0.126293
$$209$$ 1029.74 0.340807
$$210$$ 0 0
$$211$$ −4746.47 −1.54863 −0.774313 0.632802i $$-0.781905\pi$$
−0.774313 + 0.632802i $$0.781905\pi$$
$$212$$ 4850.59 1.57141
$$213$$ 4345.25 1.39780
$$214$$ 261.810 0.0836306
$$215$$ 0 0
$$216$$ 911.363 0.287085
$$217$$ 360.050 0.112635
$$218$$ −8685.90 −2.69855
$$219$$ −5529.26 −1.70609
$$220$$ 0 0
$$221$$ 405.048 0.123287
$$222$$ −13153.9 −3.97673
$$223$$ −4874.04 −1.46363 −0.731815 0.681503i $$-0.761327\pi$$
−0.731815 + 0.681503i $$0.761327\pi$$
$$224$$ 2047.42 0.610709
$$225$$ 0 0
$$226$$ 8038.26 2.36592
$$227$$ 2742.09 0.801757 0.400879 0.916131i $$-0.368705\pi$$
0.400879 + 0.916131i $$0.368705\pi$$
$$228$$ 3216.72 0.934353
$$229$$ 1528.38 0.441041 0.220520 0.975382i $$-0.429224\pi$$
0.220520 + 0.975382i $$0.429224\pi$$
$$230$$ 0 0
$$231$$ 2033.44 0.579179
$$232$$ −2145.98 −0.607287
$$233$$ −5552.78 −1.56126 −0.780632 0.624991i $$-0.785103\pi$$
−0.780632 + 0.624991i $$0.785103\pi$$
$$234$$ 2450.99 0.684729
$$235$$ 0 0
$$236$$ 8703.12 2.40053
$$237$$ −9178.49 −2.51564
$$238$$ 997.909 0.271785
$$239$$ 2779.63 0.752299 0.376149 0.926559i $$-0.377248\pi$$
0.376149 + 0.926559i $$0.377248\pi$$
$$240$$ 0 0
$$241$$ −5568.82 −1.48846 −0.744231 0.667922i $$-0.767184\pi$$
−0.744231 + 0.667922i $$0.767184\pi$$
$$242$$ 2189.81 0.581680
$$243$$ 5276.68 1.39300
$$244$$ −3534.98 −0.927475
$$245$$ 0 0
$$246$$ 14289.0 3.70340
$$247$$ 573.209 0.147662
$$248$$ −622.783 −0.159463
$$249$$ −3575.54 −0.910003
$$250$$ 0 0
$$251$$ −387.065 −0.0973360 −0.0486680 0.998815i $$-0.515498\pi$$
−0.0486680 + 0.998815i $$0.515498\pi$$
$$252$$ 3562.85 0.890628
$$253$$ −664.726 −0.165182
$$254$$ 3686.39 0.910649
$$255$$ 0 0
$$256$$ −1372.41 −0.335061
$$257$$ −1476.07 −0.358267 −0.179133 0.983825i $$-0.557329\pi$$
−0.179133 + 0.983825i $$0.557329\pi$$
$$258$$ 13928.1 3.36096
$$259$$ −3407.30 −0.817449
$$260$$ 0 0
$$261$$ 4768.81 1.13097
$$262$$ −2254.17 −0.531537
$$263$$ −203.984 −0.0478259 −0.0239130 0.999714i $$-0.507612\pi$$
−0.0239130 + 0.999714i $$0.507612\pi$$
$$264$$ −3517.26 −0.819971
$$265$$ 0 0
$$266$$ 1412.20 0.325518
$$267$$ 11727.5 2.68806
$$268$$ −3683.98 −0.839683
$$269$$ −6973.71 −1.58065 −0.790324 0.612689i $$-0.790088\pi$$
−0.790324 + 0.612689i $$0.790088\pi$$
$$270$$ 0 0
$$271$$ −2164.97 −0.485287 −0.242643 0.970116i $$-0.578015\pi$$
−0.242643 + 0.970116i $$0.578015\pi$$
$$272$$ 592.898 0.132168
$$273$$ 1131.92 0.250941
$$274$$ 49.6246 0.0109414
$$275$$ 0 0
$$276$$ −2076.48 −0.452861
$$277$$ −1194.00 −0.258992 −0.129496 0.991580i $$-0.541336\pi$$
−0.129496 + 0.991580i $$0.541336\pi$$
$$278$$ 7318.81 1.57897
$$279$$ 1383.95 0.296971
$$280$$ 0 0
$$281$$ 6485.98 1.37694 0.688472 0.725263i $$-0.258282\pi$$
0.688472 + 0.725263i $$0.258282\pi$$
$$282$$ −3812.69 −0.805116
$$283$$ 3214.41 0.675182 0.337591 0.941293i $$-0.390388\pi$$
0.337591 + 0.941293i $$0.390388\pi$$
$$284$$ −6380.00 −1.33304
$$285$$ 0 0
$$286$$ −2053.91 −0.424651
$$287$$ 3701.33 0.761263
$$288$$ 7869.81 1.61018
$$289$$ −4279.11 −0.870977
$$290$$ 0 0
$$291$$ 8164.85 1.64478
$$292$$ 8118.44 1.62704
$$293$$ −5585.47 −1.11367 −0.556837 0.830622i $$-0.687985\pi$$
−0.556837 + 0.830622i $$0.687985\pi$$
$$294$$ −9092.44 −1.80368
$$295$$ 0 0
$$296$$ 5893.65 1.15730
$$297$$ 1697.13 0.331573
$$298$$ 2253.78 0.438115
$$299$$ −370.022 −0.0715684
$$300$$ 0 0
$$301$$ 3607.84 0.690872
$$302$$ 10325.1 1.96737
$$303$$ −11371.8 −2.15609
$$304$$ 839.047 0.158298
$$305$$ 0 0
$$306$$ 3835.73 0.716583
$$307$$ −2849.51 −0.529740 −0.264870 0.964284i $$-0.585329\pi$$
−0.264870 + 0.964284i $$0.585329\pi$$
$$308$$ −2985.63 −0.552344
$$309$$ 1727.34 0.318010
$$310$$ 0 0
$$311$$ −6617.84 −1.20664 −0.603318 0.797501i $$-0.706155\pi$$
−0.603318 + 0.797501i $$0.706155\pi$$
$$312$$ −1957.90 −0.355269
$$313$$ 6271.26 1.13250 0.566250 0.824233i $$-0.308394\pi$$
0.566250 + 0.824233i $$0.308394\pi$$
$$314$$ 647.554 0.116381
$$315$$ 0 0
$$316$$ 13476.5 2.39909
$$317$$ −5650.73 −1.00119 −0.500594 0.865682i $$-0.666885\pi$$
−0.500594 + 0.865682i $$0.666885\pi$$
$$318$$ 14593.3 2.57344
$$319$$ −3996.22 −0.701395
$$320$$ 0 0
$$321$$ 464.748 0.0808090
$$322$$ −911.617 −0.157772
$$323$$ 897.055 0.154531
$$324$$ −5419.67 −0.929299
$$325$$ 0 0
$$326$$ −14481.0 −2.46022
$$327$$ −15418.6 −2.60750
$$328$$ −6402.23 −1.07776
$$329$$ −987.612 −0.165498
$$330$$ 0 0
$$331$$ −6391.40 −1.06134 −0.530669 0.847579i $$-0.678059\pi$$
−0.530669 + 0.847579i $$0.678059\pi$$
$$332$$ 5249.86 0.867841
$$333$$ −13096.9 −2.15527
$$334$$ −6870.92 −1.12563
$$335$$ 0 0
$$336$$ 1656.87 0.269017
$$337$$ −4153.87 −0.671441 −0.335721 0.941962i $$-0.608980\pi$$
−0.335721 + 0.941962i $$0.608980\pi$$
$$338$$ 8561.70 1.37780
$$339$$ 14269.0 2.28609
$$340$$ 0 0
$$341$$ −1159.74 −0.184174
$$342$$ 5428.19 0.858254
$$343$$ −5432.84 −0.855236
$$344$$ −6240.52 −0.978100
$$345$$ 0 0
$$346$$ −2086.74 −0.324231
$$347$$ 3071.86 0.475234 0.237617 0.971359i $$-0.423634\pi$$
0.237617 + 0.971359i $$0.423634\pi$$
$$348$$ −12483.4 −1.92294
$$349$$ −8358.91 −1.28207 −0.641035 0.767512i $$-0.721495\pi$$
−0.641035 + 0.767512i $$0.721495\pi$$
$$350$$ 0 0
$$351$$ 944.712 0.143661
$$352$$ −6594.82 −0.998594
$$353$$ −8018.12 −1.20896 −0.604478 0.796622i $$-0.706618\pi$$
−0.604478 + 0.796622i $$0.706618\pi$$
$$354$$ 26184.0 3.93125
$$355$$ 0 0
$$356$$ −17219.1 −2.56352
$$357$$ 1771.42 0.262615
$$358$$ −10731.8 −1.58434
$$359$$ 2138.46 0.314384 0.157192 0.987568i $$-0.449756\pi$$
0.157192 + 0.987568i $$0.449756\pi$$
$$360$$ 0 0
$$361$$ −5589.52 −0.814918
$$362$$ 4339.47 0.630048
$$363$$ 3887.21 0.562054
$$364$$ −1661.96 −0.239315
$$365$$ 0 0
$$366$$ −10635.2 −1.51889
$$367$$ 4509.96 0.641467 0.320733 0.947170i $$-0.396071\pi$$
0.320733 + 0.947170i $$0.396071\pi$$
$$368$$ −541.628 −0.0767237
$$369$$ 14227.1 2.00713
$$370$$ 0 0
$$371$$ 3780.15 0.528991
$$372$$ −3622.80 −0.504929
$$373$$ 3362.09 0.466709 0.233355 0.972392i $$-0.425030\pi$$
0.233355 + 0.972392i $$0.425030\pi$$
$$374$$ −3214.31 −0.444406
$$375$$ 0 0
$$376$$ 1708.29 0.234303
$$377$$ −2224.51 −0.303894
$$378$$ 2327.47 0.316699
$$379$$ 2107.16 0.285587 0.142793 0.989753i $$-0.454392\pi$$
0.142793 + 0.989753i $$0.454392\pi$$
$$380$$ 0 0
$$381$$ 6543.84 0.879924
$$382$$ −6012.71 −0.805333
$$383$$ −1373.05 −0.183184 −0.0915919 0.995797i $$-0.529196\pi$$
−0.0915919 + 0.995797i $$0.529196\pi$$
$$384$$ −14075.3 −1.87052
$$385$$ 0 0
$$386$$ −6433.03 −0.848271
$$387$$ 13867.7 1.82154
$$388$$ −11988.2 −1.56858
$$389$$ 7008.05 0.913425 0.456713 0.889614i $$-0.349027\pi$$
0.456713 + 0.889614i $$0.349027\pi$$
$$390$$ 0 0
$$391$$ −579.074 −0.0748978
$$392$$ 4073.89 0.524904
$$393$$ −4001.44 −0.513604
$$394$$ 4735.39 0.605496
$$395$$ 0 0
$$396$$ −11476.1 −1.45630
$$397$$ −2402.53 −0.303727 −0.151864 0.988401i $$-0.548527\pi$$
−0.151864 + 0.988401i $$0.548527\pi$$
$$398$$ 12719.0 1.60188
$$399$$ 2506.85 0.314535
$$400$$ 0 0
$$401$$ −11902.0 −1.48218 −0.741091 0.671404i $$-0.765691\pi$$
−0.741091 + 0.671404i $$0.765691\pi$$
$$402$$ −11083.5 −1.37512
$$403$$ −645.572 −0.0797971
$$404$$ 16696.9 2.05620
$$405$$ 0 0
$$406$$ −5480.48 −0.669930
$$407$$ 10975.1 1.33664
$$408$$ −3064.05 −0.371797
$$409$$ 5277.96 0.638089 0.319044 0.947740i $$-0.396638\pi$$
0.319044 + 0.947740i $$0.396638\pi$$
$$410$$ 0 0
$$411$$ 88.0903 0.0105722
$$412$$ −2536.20 −0.303276
$$413$$ 6782.51 0.808100
$$414$$ −3504.05 −0.415977
$$415$$ 0 0
$$416$$ −3671.03 −0.432662
$$417$$ 12991.9 1.52569
$$418$$ −4548.77 −0.532267
$$419$$ −11196.4 −1.30545 −0.652723 0.757597i $$-0.726373\pi$$
−0.652723 + 0.757597i $$0.726373\pi$$
$$420$$ 0 0
$$421$$ 5176.82 0.599293 0.299647 0.954050i $$-0.403131\pi$$
0.299647 + 0.954050i $$0.403131\pi$$
$$422$$ 20967.0 2.41862
$$423$$ −3796.16 −0.436349
$$424$$ −6538.58 −0.748918
$$425$$ 0 0
$$426$$ −19194.7 −2.18307
$$427$$ −2754.88 −0.312220
$$428$$ −682.375 −0.0770650
$$429$$ −3645.96 −0.410324
$$430$$ 0 0
$$431$$ 9348.93 1.04483 0.522415 0.852691i $$-0.325031\pi$$
0.522415 + 0.852691i $$0.325031\pi$$
$$432$$ 1382.84 0.154009
$$433$$ −4320.91 −0.479560 −0.239780 0.970827i $$-0.577075\pi$$
−0.239780 + 0.970827i $$0.577075\pi$$
$$434$$ −1590.48 −0.175912
$$435$$ 0 0
$$436$$ 22638.7 2.48669
$$437$$ −819.484 −0.0897054
$$438$$ 24424.9 2.66454
$$439$$ −7016.03 −0.762772 −0.381386 0.924416i $$-0.624553\pi$$
−0.381386 + 0.924416i $$0.624553\pi$$
$$440$$ 0 0
$$441$$ −9053.00 −0.977541
$$442$$ −1789.26 −0.192548
$$443$$ −12343.6 −1.32384 −0.661921 0.749574i $$-0.730259\pi$$
−0.661921 + 0.749574i $$0.730259\pi$$
$$444$$ 34284.1 3.66453
$$445$$ 0 0
$$446$$ 21530.5 2.28588
$$447$$ 4000.77 0.423333
$$448$$ −7353.88 −0.775532
$$449$$ 18146.9 1.90737 0.953683 0.300815i $$-0.0972587\pi$$
0.953683 + 0.300815i $$0.0972587\pi$$
$$450$$ 0 0
$$451$$ −11922.1 −1.24477
$$452$$ −20950.7 −2.18017
$$453$$ 18328.5 1.90099
$$454$$ −12112.9 −1.25217
$$455$$ 0 0
$$456$$ −4336.13 −0.445302
$$457$$ −5233.82 −0.535728 −0.267864 0.963457i $$-0.586318\pi$$
−0.267864 + 0.963457i $$0.586318\pi$$
$$458$$ −6751.47 −0.688811
$$459$$ 1478.45 0.150344
$$460$$ 0 0
$$461$$ −2335.01 −0.235905 −0.117952 0.993019i $$-0.537633\pi$$
−0.117952 + 0.993019i $$0.537633\pi$$
$$462$$ −8982.49 −0.904553
$$463$$ −13644.1 −1.36954 −0.684769 0.728760i $$-0.740097\pi$$
−0.684769 + 0.728760i $$0.740097\pi$$
$$464$$ −3256.17 −0.325784
$$465$$ 0 0
$$466$$ 24528.8 2.43836
$$467$$ 5246.31 0.519851 0.259925 0.965629i $$-0.416302\pi$$
0.259925 + 0.965629i $$0.416302\pi$$
$$468$$ −6388.21 −0.630972
$$469$$ −2871.00 −0.282666
$$470$$ 0 0
$$471$$ 1149.50 0.112454
$$472$$ −11731.8 −1.14407
$$473$$ −11621.0 −1.12967
$$474$$ 40545.0 3.92889
$$475$$ 0 0
$$476$$ −2600.92 −0.250448
$$477$$ 14530.0 1.39473
$$478$$ −12278.7 −1.17493
$$479$$ 12278.3 1.17121 0.585603 0.810598i $$-0.300858\pi$$
0.585603 + 0.810598i $$0.300858\pi$$
$$480$$ 0 0
$$481$$ 6109.31 0.579128
$$482$$ 24599.7 2.32466
$$483$$ −1618.24 −0.152448
$$484$$ −5707.47 −0.536013
$$485$$ 0 0
$$486$$ −23309.2 −2.17557
$$487$$ 8945.24 0.832336 0.416168 0.909288i $$-0.363373\pi$$
0.416168 + 0.909288i $$0.363373\pi$$
$$488$$ 4765.14 0.442024
$$489$$ −25705.8 −2.37721
$$490$$ 0 0
$$491$$ −7792.94 −0.716274 −0.358137 0.933669i $$-0.616588\pi$$
−0.358137 + 0.933669i $$0.616588\pi$$
$$492$$ −37242.6 −3.41265
$$493$$ −3481.29 −0.318031
$$494$$ −2532.09 −0.230616
$$495$$ 0 0
$$496$$ −944.970 −0.0855451
$$497$$ −4972.06 −0.448747
$$498$$ 15794.6 1.42123
$$499$$ −8577.71 −0.769521 −0.384761 0.923016i $$-0.625716\pi$$
−0.384761 + 0.923016i $$0.625716\pi$$
$$500$$ 0 0
$$501$$ −12196.8 −1.08765
$$502$$ 1709.82 0.152018
$$503$$ −7784.99 −0.690091 −0.345045 0.938586i $$-0.612136\pi$$
−0.345045 + 0.938586i $$0.612136\pi$$
$$504$$ −4802.71 −0.424464
$$505$$ 0 0
$$506$$ 2936.36 0.257978
$$507$$ 15198.2 1.33131
$$508$$ −9608.12 −0.839156
$$509$$ −11390.4 −0.991891 −0.495946 0.868354i $$-0.665178\pi$$
−0.495946 + 0.868354i $$0.665178\pi$$
$$510$$ 0 0
$$511$$ 6326.85 0.547717
$$512$$ −8297.40 −0.716205
$$513$$ 2092.24 0.180068
$$514$$ 6520.37 0.559535
$$515$$ 0 0
$$516$$ −36301.9 −3.09710
$$517$$ 3181.14 0.270612
$$518$$ 15051.4 1.27668
$$519$$ −3704.24 −0.313292
$$520$$ 0 0
$$521$$ −12824.5 −1.07841 −0.539206 0.842174i $$-0.681276\pi$$
−0.539206 + 0.842174i $$0.681276\pi$$
$$522$$ −21065.7 −1.76632
$$523$$ −13087.4 −1.09421 −0.547107 0.837063i $$-0.684271\pi$$
−0.547107 + 0.837063i $$0.684271\pi$$
$$524$$ 5875.20 0.489808
$$525$$ 0 0
$$526$$ 901.080 0.0746938
$$527$$ −1010.30 −0.0835093
$$528$$ −5336.86 −0.439880
$$529$$ 529.000 0.0434783
$$530$$ 0 0
$$531$$ 26070.4 2.13062
$$532$$ −3680.73 −0.299962
$$533$$ −6636.50 −0.539322
$$534$$ −51805.0 −4.19817
$$535$$ 0 0
$$536$$ 4966.00 0.400184
$$537$$ −19050.4 −1.53089
$$538$$ 30805.6 2.46863
$$539$$ 7586.33 0.606245
$$540$$ 0 0
$$541$$ 15464.3 1.22895 0.614473 0.788938i $$-0.289369\pi$$
0.614473 + 0.788938i $$0.289369\pi$$
$$542$$ 9563.54 0.757914
$$543$$ 7703.14 0.608791
$$544$$ −5745.06 −0.452789
$$545$$ 0 0
$$546$$ −5000.14 −0.391916
$$547$$ −6981.32 −0.545703 −0.272852 0.962056i $$-0.587967\pi$$
−0.272852 + 0.962056i $$0.587967\pi$$
$$548$$ −129.340 −0.0100824
$$549$$ −10589.1 −0.823192
$$550$$ 0 0
$$551$$ −4926.60 −0.380908
$$552$$ 2799.09 0.215828
$$553$$ 10502.5 0.807615
$$554$$ 5274.38 0.404489
$$555$$ 0 0
$$556$$ −19075.5 −1.45501
$$557$$ −24964.8 −1.89909 −0.949543 0.313637i $$-0.898453\pi$$
−0.949543 + 0.313637i $$0.898453\pi$$
$$558$$ −6113.45 −0.463805
$$559$$ −6468.88 −0.489454
$$560$$ 0 0
$$561$$ −5705.83 −0.429412
$$562$$ −28651.2 −2.15049
$$563$$ 10820.3 0.809986 0.404993 0.914320i $$-0.367274\pi$$
0.404993 + 0.914320i $$0.367274\pi$$
$$564$$ 9937.31 0.741908
$$565$$ 0 0
$$566$$ −14199.3 −1.05449
$$567$$ −4223.65 −0.312834
$$568$$ 8600.23 0.635312
$$569$$ 8438.73 0.621740 0.310870 0.950453i $$-0.399380\pi$$
0.310870 + 0.950453i $$0.399380\pi$$
$$570$$ 0 0
$$571$$ 23107.6 1.69356 0.846781 0.531942i $$-0.178538\pi$$
0.846781 + 0.531942i $$0.178538\pi$$
$$572$$ 5353.26 0.391313
$$573$$ −10673.4 −0.778161
$$574$$ −16350.2 −1.18893
$$575$$ 0 0
$$576$$ −28266.7 −2.04475
$$577$$ −24848.7 −1.79283 −0.896416 0.443215i $$-0.853838\pi$$
−0.896416 + 0.443215i $$0.853838\pi$$
$$578$$ 18902.5 1.36028
$$579$$ −11419.5 −0.819651
$$580$$ 0 0
$$581$$ 4091.31 0.292145
$$582$$ −36067.4 −2.56880
$$583$$ −12176.0 −0.864974
$$584$$ −10943.6 −0.775430
$$585$$ 0 0
$$586$$ 24673.2 1.73932
$$587$$ −8663.94 −0.609198 −0.304599 0.952481i $$-0.598522\pi$$
−0.304599 + 0.952481i $$0.598522\pi$$
$$588$$ 23698.3 1.66208
$$589$$ −1429.74 −0.100019
$$590$$ 0 0
$$591$$ 8405.94 0.585066
$$592$$ 8942.64 0.620845
$$593$$ 24678.9 1.70901 0.854503 0.519446i $$-0.173862\pi$$
0.854503 + 0.519446i $$0.173862\pi$$
$$594$$ −7496.88 −0.517846
$$595$$ 0 0
$$596$$ −5874.20 −0.403719
$$597$$ 22578.0 1.54783
$$598$$ 1634.54 0.111774
$$599$$ 19698.4 1.34367 0.671833 0.740702i $$-0.265507\pi$$
0.671833 + 0.740702i $$0.265507\pi$$
$$600$$ 0 0
$$601$$ −18449.1 −1.25217 −0.626086 0.779754i $$-0.715344\pi$$
−0.626086 + 0.779754i $$0.715344\pi$$
$$602$$ −15937.3 −1.07899
$$603$$ −11035.5 −0.745272
$$604$$ −26911.2 −1.81291
$$605$$ 0 0
$$606$$ 50233.9 3.36735
$$607$$ 11321.0 0.757012 0.378506 0.925599i $$-0.376438\pi$$
0.378506 + 0.925599i $$0.376438\pi$$
$$608$$ −8130.20 −0.542308
$$609$$ −9728.59 −0.647327
$$610$$ 0 0
$$611$$ 1770.80 0.117248
$$612$$ −9997.36 −0.660326
$$613$$ −712.219 −0.0469270 −0.0234635 0.999725i $$-0.507469\pi$$
−0.0234635 + 0.999725i $$0.507469\pi$$
$$614$$ 12587.4 0.827341
$$615$$ 0 0
$$616$$ 4024.62 0.263241
$$617$$ −3234.25 −0.211031 −0.105515 0.994418i $$-0.533649\pi$$
−0.105515 + 0.994418i $$0.533649\pi$$
$$618$$ −7630.36 −0.496663
$$619$$ 26905.1 1.74703 0.873513 0.486801i $$-0.161836\pi$$
0.873513 + 0.486801i $$0.161836\pi$$
$$620$$ 0 0
$$621$$ −1350.60 −0.0872750
$$622$$ 29233.6 1.88451
$$623$$ −13419.2 −0.862967
$$624$$ −2970.78 −0.190587
$$625$$ 0 0
$$626$$ −27702.6 −1.76872
$$627$$ −8074.68 −0.514309
$$628$$ −1687.77 −0.107244
$$629$$ 9560.90 0.606070
$$630$$ 0 0
$$631$$ 16199.5 1.02202 0.511008 0.859576i $$-0.329272\pi$$
0.511008 + 0.859576i $$0.329272\pi$$
$$632$$ −18166.3 −1.14338
$$633$$ 37219.3 2.33702
$$634$$ 24961.5 1.56364
$$635$$ 0 0
$$636$$ −38035.7 −2.37141
$$637$$ 4222.96 0.262668
$$638$$ 17652.9 1.09543
$$639$$ −19111.5 −1.18316
$$640$$ 0 0
$$641$$ 19943.5 1.22889 0.614446 0.788959i $$-0.289380\pi$$
0.614446 + 0.788959i $$0.289380\pi$$
$$642$$ −2052.97 −0.126206
$$643$$ 27691.0 1.69833 0.849164 0.528129i $$-0.177106\pi$$
0.849164 + 0.528129i $$0.177106\pi$$
$$644$$ 2376.01 0.145385
$$645$$ 0 0
$$646$$ −3962.65 −0.241344
$$647$$ 23560.3 1.43161 0.715805 0.698300i $$-0.246060\pi$$
0.715805 + 0.698300i $$0.246060\pi$$
$$648$$ 7305.70 0.442893
$$649$$ −21846.7 −1.32136
$$650$$ 0 0
$$651$$ −2823.32 −0.169976
$$652$$ 37743.0 2.26707
$$653$$ −5571.58 −0.333894 −0.166947 0.985966i $$-0.553391\pi$$
−0.166947 + 0.985966i $$0.553391\pi$$
$$654$$ 68110.2 4.07235
$$655$$ 0 0
$$656$$ −9714.32 −0.578171
$$657$$ 24319.0 1.44410
$$658$$ 4362.68 0.258472
$$659$$ 10177.4 0.601602 0.300801 0.953687i $$-0.402746\pi$$
0.300801 + 0.953687i $$0.402746\pi$$
$$660$$ 0 0
$$661$$ −27413.8 −1.61312 −0.806561 0.591151i $$-0.798674\pi$$
−0.806561 + 0.591151i $$0.798674\pi$$
$$662$$ 28233.3 1.65758
$$663$$ −3176.17 −0.186052
$$664$$ −7076.79 −0.413604
$$665$$ 0 0
$$666$$ 57854.1 3.36607
$$667$$ 3180.25 0.184618
$$668$$ 17908.2 1.03726
$$669$$ 38219.6 2.20875
$$670$$ 0 0
$$671$$ 8873.57 0.510522
$$672$$ −16054.8 −0.921616
$$673$$ −12767.5 −0.731277 −0.365639 0.930757i $$-0.619149\pi$$
−0.365639 + 0.930757i $$0.619149\pi$$
$$674$$ 18349.3 1.04865
$$675$$ 0 0
$$676$$ −22315.0 −1.26963
$$677$$ −17036.9 −0.967180 −0.483590 0.875295i $$-0.660667\pi$$
−0.483590 + 0.875295i $$0.660667\pi$$
$$678$$ −63031.7 −3.57038
$$679$$ −9342.63 −0.528037
$$680$$ 0 0
$$681$$ −21502.0 −1.20993
$$682$$ 5123.02 0.287640
$$683$$ 510.213 0.0285838 0.0142919 0.999898i $$-0.495451\pi$$
0.0142919 + 0.999898i $$0.495451\pi$$
$$684$$ −14147.9 −0.790875
$$685$$ 0 0
$$686$$ 23999.0 1.33569
$$687$$ −11984.8 −0.665570
$$688$$ −9468.96 −0.524710
$$689$$ −6777.84 −0.374768
$$690$$ 0 0
$$691$$ 22793.1 1.25483 0.627416 0.778684i $$-0.284112\pi$$
0.627416 + 0.778684i $$0.284112\pi$$
$$692$$ 5438.83 0.298776
$$693$$ −8943.53 −0.490240
$$694$$ −13569.6 −0.742213
$$695$$ 0 0
$$696$$ 16827.7 0.916452
$$697$$ −10385.9 −0.564412
$$698$$ 36924.6 2.00232
$$699$$ 43541.9 2.35609
$$700$$ 0 0
$$701$$ 26324.3 1.41834 0.709168 0.705039i $$-0.249071\pi$$
0.709168 + 0.705039i $$0.249071\pi$$
$$702$$ −4173.17 −0.224368
$$703$$ 13530.2 0.725892
$$704$$ 23687.2 1.26810
$$705$$ 0 0
$$706$$ 35419.2 1.88813
$$707$$ 13012.2 0.692185
$$708$$ −68245.2 −3.62262
$$709$$ −10961.3 −0.580620 −0.290310 0.956933i $$-0.593758\pi$$
−0.290310 + 0.956933i $$0.593758\pi$$
$$710$$ 0 0
$$711$$ 40369.2 2.12934
$$712$$ 23211.3 1.22174
$$713$$ 922.938 0.0484773
$$714$$ −7825.07 −0.410148
$$715$$ 0 0
$$716$$ 27971.2 1.45996
$$717$$ −21796.4 −1.13529
$$718$$ −9446.44 −0.491000
$$719$$ −1304.68 −0.0676723 −0.0338362 0.999427i $$-0.510772\pi$$
−0.0338362 + 0.999427i $$0.510772\pi$$
$$720$$ 0 0
$$721$$ −1976.51 −0.102093
$$722$$ 24691.1 1.27273
$$723$$ 43667.7 2.24622
$$724$$ −11310.3 −0.580585
$$725$$ 0 0
$$726$$ −17171.3 −0.877808
$$727$$ −1583.59 −0.0807869 −0.0403934 0.999184i $$-0.512861\pi$$
−0.0403934 + 0.999184i $$0.512861\pi$$
$$728$$ 2240.32 0.114055
$$729$$ −28667.3 −1.45645
$$730$$ 0 0
$$731$$ −10123.6 −0.512223
$$732$$ 27719.4 1.39964
$$733$$ 34351.2 1.73095 0.865477 0.500948i $$-0.167015\pi$$
0.865477 + 0.500948i $$0.167015\pi$$
$$734$$ −19922.3 −1.00183
$$735$$ 0 0
$$736$$ 5248.27 0.262845
$$737$$ 9247.61 0.462198
$$738$$ −62846.5 −3.13471
$$739$$ 14996.6 0.746494 0.373247 0.927732i $$-0.378244\pi$$
0.373247 + 0.927732i $$0.378244\pi$$
$$740$$ 0 0
$$741$$ −4494.80 −0.222835
$$742$$ −16698.4 −0.826171
$$743$$ 32781.7 1.61863 0.809317 0.587372i $$-0.199838\pi$$
0.809317 + 0.587372i $$0.199838\pi$$
$$744$$ 4883.53 0.240644
$$745$$ 0 0
$$746$$ −14851.7 −0.728899
$$747$$ 15726.1 0.770263
$$748$$ 8377.69 0.409517
$$749$$ −531.787 −0.0259427
$$750$$ 0 0
$$751$$ −24044.4 −1.16830 −0.584149 0.811647i $$-0.698571\pi$$
−0.584149 + 0.811647i $$0.698571\pi$$
$$752$$ 2592.04 0.125694
$$753$$ 3035.16 0.146889
$$754$$ 9826.54 0.474617
$$755$$ 0 0
$$756$$ −6066.26 −0.291835
$$757$$ 21680.9 1.04096 0.520479 0.853874i $$-0.325753\pi$$
0.520479 + 0.853874i $$0.325753\pi$$
$$758$$ −9308.15 −0.446025
$$759$$ 5212.43 0.249274
$$760$$ 0 0
$$761$$ −10299.1 −0.490594 −0.245297 0.969448i $$-0.578885\pi$$
−0.245297 + 0.969448i $$0.578885\pi$$
$$762$$ −28906.7 −1.37425
$$763$$ 17642.8 0.837105
$$764$$ 15671.4 0.742108
$$765$$ 0 0
$$766$$ 6065.29 0.286094
$$767$$ −12161.1 −0.572505
$$768$$ 10761.7 0.505637
$$769$$ 28377.9 1.33073 0.665365 0.746518i $$-0.268276\pi$$
0.665365 + 0.746518i $$0.268276\pi$$
$$770$$ 0 0
$$771$$ 11574.5 0.540657
$$772$$ 16766.9 0.781675
$$773$$ −35409.5 −1.64759 −0.823797 0.566885i $$-0.808148\pi$$
−0.823797 + 0.566885i $$0.808148\pi$$
$$774$$ −61259.2 −2.84485
$$775$$ 0 0
$$776$$ 16160.1 0.747568
$$777$$ 26718.2 1.23361
$$778$$ −30957.3 −1.42657
$$779$$ −14697.8 −0.675999
$$780$$ 0 0
$$781$$ 16015.2 0.733763
$$782$$ 2558.00 0.116974
$$783$$ −8119.58 −0.370588
$$784$$ 6181.45 0.281589
$$785$$ 0 0
$$786$$ 17676.0 0.802138
$$787$$ −6117.56 −0.277087 −0.138544 0.990356i $$-0.544242\pi$$
−0.138544 + 0.990356i $$0.544242\pi$$
$$788$$ −12342.2 −0.557960
$$789$$ 1599.54 0.0721737
$$790$$ 0 0
$$791$$ −16327.3 −0.733921
$$792$$ 15469.7 0.694057
$$793$$ 4939.51 0.221194
$$794$$ 10612.9 0.474357
$$795$$ 0 0
$$796$$ −33150.6 −1.47612
$$797$$ −4099.46 −0.182196 −0.0910980 0.995842i $$-0.529038\pi$$
−0.0910980 + 0.995842i $$0.529038\pi$$
$$798$$ −11073.8 −0.491236
$$799$$ 2771.24 0.122703
$$800$$ 0 0
$$801$$ −51580.3 −2.27528
$$802$$ 52575.6 2.31485
$$803$$ −20379.1 −0.895594
$$804$$ 28887.8 1.26716
$$805$$ 0 0
$$806$$ 2851.75 0.124626
$$807$$ 54684.1 2.38534
$$808$$ −22507.4 −0.979960
$$809$$ 21358.6 0.928216 0.464108 0.885779i $$-0.346375\pi$$
0.464108 + 0.885779i $$0.346375\pi$$
$$810$$ 0 0
$$811$$ 13967.7 0.604776 0.302388 0.953185i $$-0.402216\pi$$
0.302388 + 0.953185i $$0.402216\pi$$
$$812$$ 14284.2 0.617336
$$813$$ 16976.6 0.732342
$$814$$ −48481.2 −2.08755
$$815$$ 0 0
$$816$$ −4649.19 −0.199454
$$817$$ −14326.6 −0.613492
$$818$$ −23314.8 −0.996557
$$819$$ −4978.45 −0.212407
$$820$$ 0 0
$$821$$ −22387.6 −0.951684 −0.475842 0.879531i $$-0.657857\pi$$
−0.475842 + 0.879531i $$0.657857\pi$$
$$822$$ −389.130 −0.0165115
$$823$$ 22615.7 0.957877 0.478939 0.877848i $$-0.341022\pi$$
0.478939 + 0.877848i $$0.341022\pi$$
$$824$$ 3418.80 0.144538
$$825$$ 0 0
$$826$$ −29961.0 −1.26208
$$827$$ −10878.0 −0.457394 −0.228697 0.973498i $$-0.573446\pi$$
−0.228697 + 0.973498i $$0.573446\pi$$
$$828$$ 9132.86 0.383320
$$829$$ 27382.3 1.14720 0.573600 0.819136i $$-0.305547\pi$$
0.573600 + 0.819136i $$0.305547\pi$$
$$830$$ 0 0
$$831$$ 9362.74 0.390842
$$832$$ 13185.6 0.549432
$$833$$ 6608.81 0.274888
$$834$$ −57390.2 −2.38281
$$835$$ 0 0
$$836$$ 11855.8 0.490480
$$837$$ −2356.37 −0.0973096
$$838$$ 49459.1 2.03883
$$839$$ −31799.7 −1.30852 −0.654260 0.756270i $$-0.727020\pi$$
−0.654260 + 0.756270i $$0.727020\pi$$
$$840$$ 0 0
$$841$$ −5269.87 −0.216076
$$842$$ −22868.0 −0.935968
$$843$$ −50859.6 −2.07793
$$844$$ −54647.9 −2.22874
$$845$$ 0 0
$$846$$ 16769.1 0.681483
$$847$$ −4447.94 −0.180440
$$848$$ −9921.21 −0.401764
$$849$$ −25205.7 −1.01891
$$850$$ 0 0
$$851$$ −8734.14 −0.351824
$$852$$ 50028.6 2.01168
$$853$$ −32016.4 −1.28514 −0.642568 0.766229i $$-0.722131\pi$$
−0.642568 + 0.766229i $$0.722131\pi$$
$$854$$ 12169.4 0.487620
$$855$$ 0 0
$$856$$ 919.839 0.0367283
$$857$$ −25280.1 −1.00764 −0.503822 0.863807i $$-0.668073\pi$$
−0.503822 + 0.863807i $$0.668073\pi$$
$$858$$ 16105.7 0.640837
$$859$$ −22313.5 −0.886296 −0.443148 0.896448i $$-0.646138\pi$$
−0.443148 + 0.896448i $$0.646138\pi$$
$$860$$ 0 0
$$861$$ −29023.8 −1.14881
$$862$$ −41297.9 −1.63180
$$863$$ 1478.28 0.0583096 0.0291548 0.999575i $$-0.490718\pi$$
0.0291548 + 0.999575i $$0.490718\pi$$
$$864$$ −13399.5 −0.527615
$$865$$ 0 0
$$866$$ 19087.2 0.748970
$$867$$ 33554.5 1.31438
$$868$$ 4145.39 0.162101
$$869$$ −33829.0 −1.32056
$$870$$ 0 0
$$871$$ 5147.72 0.200257
$$872$$ −30516.9 −1.18513
$$873$$ −35910.9 −1.39221
$$874$$ 3619.99 0.140101
$$875$$ 0 0
$$876$$ −63660.5 −2.45535
$$877$$ −32974.6 −1.26964 −0.634819 0.772661i $$-0.718925\pi$$
−0.634819 + 0.772661i $$0.718925\pi$$
$$878$$ 30992.6 1.19129
$$879$$ 43798.3 1.68064
$$880$$ 0 0
$$881$$ −32000.7 −1.22376 −0.611879 0.790951i $$-0.709586\pi$$
−0.611879 + 0.790951i $$0.709586\pi$$
$$882$$ 39990.7 1.52671
$$883$$ 44218.6 1.68525 0.842623 0.538503i $$-0.181010\pi$$
0.842623 + 0.538503i $$0.181010\pi$$
$$884$$ 4663.47 0.177432
$$885$$ 0 0
$$886$$ 54526.6 2.06756
$$887$$ −27374.8 −1.03625 −0.518126 0.855304i $$-0.673370\pi$$
−0.518126 + 0.855304i $$0.673370\pi$$
$$888$$ −46214.9 −1.74647
$$889$$ −7487.79 −0.282489
$$890$$ 0 0
$$891$$ 13604.6 0.511526
$$892$$ −56116.7 −2.10642
$$893$$ 3921.76 0.146962
$$894$$ −17673.0 −0.661155
$$895$$ 0 0
$$896$$ 16105.7 0.600505
$$897$$ 2901.52 0.108003
$$898$$ −80162.2 −2.97889
$$899$$ 5548.54 0.205844
$$900$$ 0 0
$$901$$ −10607.1 −0.392203
$$902$$ 52664.8 1.94406
$$903$$ −28290.8 −1.04259
$$904$$ 28241.5 1.03905
$$905$$ 0 0
$$906$$ −80964.3 −2.96894
$$907$$ −7835.16 −0.286838 −0.143419 0.989662i $$-0.545810\pi$$
−0.143419 + 0.989662i $$0.545810\pi$$
$$908$$ 31570.7 1.15387
$$909$$ 50016.0 1.82500
$$910$$ 0 0
$$911$$ −36528.6 −1.32848 −0.664241 0.747519i $$-0.731245\pi$$
−0.664241 + 0.747519i $$0.731245\pi$$
$$912$$ −6579.36 −0.238886
$$913$$ −13178.3 −0.477698
$$914$$ 23119.8 0.836692
$$915$$ 0 0
$$916$$ 17596.8 0.634734
$$917$$ 4578.65 0.164886
$$918$$ −6530.89 −0.234805
$$919$$ −15741.5 −0.565030 −0.282515 0.959263i $$-0.591169\pi$$
−0.282515 + 0.959263i $$0.591169\pi$$
$$920$$ 0 0
$$921$$ 22344.4 0.799427
$$922$$ 10314.7 0.368433
$$923$$ 8914.93 0.317918
$$924$$ 23411.7 0.833538
$$925$$ 0 0
$$926$$ 60271.5 2.13892
$$927$$ −7597.26 −0.269177
$$928$$ 31551.7 1.11609
$$929$$ 47428.9 1.67502 0.837510 0.546422i $$-0.184011\pi$$
0.837510 + 0.546422i $$0.184011\pi$$
$$930$$ 0 0
$$931$$ 9352.54 0.329234
$$932$$ −63931.3 −2.24693
$$933$$ 51893.6 1.82092
$$934$$ −23175.0 −0.811895
$$935$$ 0 0
$$936$$ 8611.29 0.300714
$$937$$ 34978.0 1.21951 0.609756 0.792589i $$-0.291267\pi$$
0.609756 + 0.792589i $$0.291267\pi$$
$$938$$ 12682.3 0.441464
$$939$$ −49175.9 −1.70905
$$940$$ 0 0
$$941$$ 45144.0 1.56392 0.781962 0.623326i $$-0.214219\pi$$
0.781962 + 0.623326i $$0.214219\pi$$
$$942$$ −5077.78 −0.175629
$$943$$ 9487.83 0.327642
$$944$$ −17801.0 −0.613744
$$945$$ 0 0
$$946$$ 51334.6 1.76430
$$947$$ −26123.6 −0.896413 −0.448207 0.893930i $$-0.647937\pi$$
−0.448207 + 0.893930i $$0.647937\pi$$
$$948$$ −105675. −3.62044
$$949$$ −11344.1 −0.388035
$$950$$ 0 0
$$951$$ 44310.0 1.51088
$$952$$ 3506.04 0.119361
$$953$$ 22143.5 0.752673 0.376336 0.926483i $$-0.377184\pi$$
0.376336 + 0.926483i $$0.377184\pi$$
$$954$$ −64185.0 −2.17827
$$955$$ 0 0
$$956$$ 32003.0 1.08269
$$957$$ 31336.2 1.05847
$$958$$ −54237.9 −1.82917
$$959$$ −100.797 −0.00339407
$$960$$ 0 0
$$961$$ −28180.8 −0.945949
$$962$$ −26987.3 −0.904474
$$963$$ −2044.07 −0.0684000
$$964$$ −64116.0 −2.14215
$$965$$ 0 0
$$966$$ 7148.42 0.238092
$$967$$ −44869.8 −1.49216 −0.746078 0.665858i $$-0.768066\pi$$
−0.746078 + 0.665858i $$0.768066\pi$$
$$968$$ 7693.65 0.255458
$$969$$ −7034.23 −0.233201
$$970$$ 0 0
$$971$$ −25048.3 −0.827845 −0.413923 0.910312i $$-0.635842\pi$$
−0.413923 + 0.910312i $$0.635842\pi$$
$$972$$ 60752.5 2.00477
$$973$$ −14865.9 −0.489805
$$974$$ −39514.7 −1.29993
$$975$$ 0 0
$$976$$ 7230.31 0.237128
$$977$$ −37320.7 −1.22210 −0.611052 0.791590i $$-0.709253\pi$$
−0.611052 + 0.791590i $$0.709253\pi$$
$$978$$ 113553. 3.71269
$$979$$ 43223.8 1.41107
$$980$$ 0 0
$$981$$ 67814.7 2.20709
$$982$$ 34424.5 1.11867
$$983$$ 17189.4 0.557737 0.278869 0.960329i $$-0.410041\pi$$
0.278869 + 0.960329i $$0.410041\pi$$
$$984$$ 50202.9 1.62643
$$985$$ 0 0
$$986$$ 15378.2 0.496697
$$987$$ 7744.33 0.249752
$$988$$ 6599.58 0.212511
$$989$$ 9248.19 0.297346
$$990$$ 0 0
$$991$$ −57797.1 −1.85266 −0.926330 0.376712i $$-0.877055\pi$$
−0.926330 + 0.376712i $$0.877055\pi$$
$$992$$ 9156.57 0.293066
$$993$$ 50117.9 1.60166
$$994$$ 21963.5 0.700846
$$995$$ 0 0
$$996$$ −41166.6 −1.30965
$$997$$ −46801.0 −1.48666 −0.743331 0.668923i $$-0.766755\pi$$
−0.743331 + 0.668923i $$0.766755\pi$$
$$998$$ 37891.2 1.20183
$$999$$ 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.a.j.1.2 5
5.2 odd 4 575.4.b.i.24.2 10
5.3 odd 4 575.4.b.i.24.9 10
5.4 even 2 115.4.a.e.1.4 5
15.14 odd 2 1035.4.a.k.1.2 5
20.19 odd 2 1840.4.a.n.1.2 5

By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.4 5 5.4 even 2
575.4.a.j.1.2 5 1.1 even 1 trivial
575.4.b.i.24.2 10 5.2 odd 4
575.4.b.i.24.9 10 5.3 odd 4
1035.4.a.k.1.2 5 15.14 odd 2
1840.4.a.n.1.2 5 20.19 odd 2