# Properties

 Label 23.4.a.a.1.1 Level $23$ Weight $4$ Character 23.1 Self dual yes Analytic conductor $1.357$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,4,Mod(1,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("23.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 23.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: $$1.35704393013$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 23.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -5.00000 q^{3} -4.00000 q^{4} -6.00000 q^{5} +10.0000 q^{6} -8.00000 q^{7} +24.0000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} -5.00000 q^{3} -4.00000 q^{4} -6.00000 q^{5} +10.0000 q^{6} -8.00000 q^{7} +24.0000 q^{8} -2.00000 q^{9} +12.0000 q^{10} +34.0000 q^{11} +20.0000 q^{12} -57.0000 q^{13} +16.0000 q^{14} +30.0000 q^{15} -16.0000 q^{16} -80.0000 q^{17} +4.00000 q^{18} -70.0000 q^{19} +24.0000 q^{20} +40.0000 q^{21} -68.0000 q^{22} +23.0000 q^{23} -120.000 q^{24} -89.0000 q^{25} +114.000 q^{26} +145.000 q^{27} +32.0000 q^{28} +245.000 q^{29} -60.0000 q^{30} +103.000 q^{31} -160.000 q^{32} -170.000 q^{33} +160.000 q^{34} +48.0000 q^{35} +8.00000 q^{36} -298.000 q^{37} +140.000 q^{38} +285.000 q^{39} -144.000 q^{40} +95.0000 q^{41} -80.0000 q^{42} +88.0000 q^{43} -136.000 q^{44} +12.0000 q^{45} -46.0000 q^{46} -357.000 q^{47} +80.0000 q^{48} -279.000 q^{49} +178.000 q^{50} +400.000 q^{51} +228.000 q^{52} -414.000 q^{53} -290.000 q^{54} -204.000 q^{55} -192.000 q^{56} +350.000 q^{57} -490.000 q^{58} -408.000 q^{59} -120.000 q^{60} +822.000 q^{61} -206.000 q^{62} +16.0000 q^{63} +448.000 q^{64} +342.000 q^{65} +340.000 q^{66} +926.000 q^{67} +320.000 q^{68} -115.000 q^{69} -96.0000 q^{70} +335.000 q^{71} -48.0000 q^{72} -899.000 q^{73} +596.000 q^{74} +445.000 q^{75} +280.000 q^{76} -272.000 q^{77} -570.000 q^{78} -1322.00 q^{79} +96.0000 q^{80} -671.000 q^{81} -190.000 q^{82} -36.0000 q^{83} -160.000 q^{84} +480.000 q^{85} -176.000 q^{86} -1225.00 q^{87} +816.000 q^{88} -460.000 q^{89} -24.0000 q^{90} +456.000 q^{91} -92.0000 q^{92} -515.000 q^{93} +714.000 q^{94} +420.000 q^{95} +800.000 q^{96} -964.000 q^{97} +558.000 q^{98} -68.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −0.707107 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ −5.00000 −0.962250 −0.481125 0.876652i $$-0.659772\pi$$
−0.481125 + 0.876652i $$0.659772\pi$$
$$4$$ −4.00000 −0.500000
$$5$$ −6.00000 −0.536656 −0.268328 0.963328i $$-0.586471\pi$$
−0.268328 + 0.963328i $$0.586471\pi$$
$$6$$ 10.0000 0.680414
$$7$$ −8.00000 −0.431959 −0.215980 0.976398i $$-0.569295\pi$$
−0.215980 + 0.976398i $$0.569295\pi$$
$$8$$ 24.0000 1.06066
$$9$$ −2.00000 −0.0740741
$$10$$ 12.0000 0.379473
$$11$$ 34.0000 0.931944 0.465972 0.884799i $$-0.345705\pi$$
0.465972 + 0.884799i $$0.345705\pi$$
$$12$$ 20.0000 0.481125
$$13$$ −57.0000 −1.21607 −0.608037 0.793909i $$-0.708043\pi$$
−0.608037 + 0.793909i $$0.708043\pi$$
$$14$$ 16.0000 0.305441
$$15$$ 30.0000 0.516398
$$16$$ −16.0000 −0.250000
$$17$$ −80.0000 −1.14134 −0.570672 0.821178i $$-0.693317\pi$$
−0.570672 + 0.821178i $$0.693317\pi$$
$$18$$ 4.00000 0.0523783
$$19$$ −70.0000 −0.845216 −0.422608 0.906313i $$-0.638885\pi$$
−0.422608 + 0.906313i $$0.638885\pi$$
$$20$$ 24.0000 0.268328
$$21$$ 40.0000 0.415653
$$22$$ −68.0000 −0.658984
$$23$$ 23.0000 0.208514
$$24$$ −120.000 −1.02062
$$25$$ −89.0000 −0.712000
$$26$$ 114.000 0.859894
$$27$$ 145.000 1.03353
$$28$$ 32.0000 0.215980
$$29$$ 245.000 1.56881 0.784403 0.620252i $$-0.212970\pi$$
0.784403 + 0.620252i $$0.212970\pi$$
$$30$$ −60.0000 −0.365148
$$31$$ 103.000 0.596753 0.298377 0.954448i $$-0.403555\pi$$
0.298377 + 0.954448i $$0.403555\pi$$
$$32$$ −160.000 −0.883883
$$33$$ −170.000 −0.896764
$$34$$ 160.000 0.807052
$$35$$ 48.0000 0.231814
$$36$$ 8.00000 0.0370370
$$37$$ −298.000 −1.32408 −0.662039 0.749469i $$-0.730309\pi$$
−0.662039 + 0.749469i $$0.730309\pi$$
$$38$$ 140.000 0.597658
$$39$$ 285.000 1.17017
$$40$$ −144.000 −0.569210
$$41$$ 95.0000 0.361866 0.180933 0.983495i $$-0.442088\pi$$
0.180933 + 0.983495i $$0.442088\pi$$
$$42$$ −80.0000 −0.293911
$$43$$ 88.0000 0.312090 0.156045 0.987750i $$-0.450125\pi$$
0.156045 + 0.987750i $$0.450125\pi$$
$$44$$ −136.000 −0.465972
$$45$$ 12.0000 0.0397523
$$46$$ −46.0000 −0.147442
$$47$$ −357.000 −1.10795 −0.553977 0.832532i $$-0.686890\pi$$
−0.553977 + 0.832532i $$0.686890\pi$$
$$48$$ 80.0000 0.240563
$$49$$ −279.000 −0.813411
$$50$$ 178.000 0.503460
$$51$$ 400.000 1.09826
$$52$$ 228.000 0.608037
$$53$$ −414.000 −1.07297 −0.536484 0.843911i $$-0.680248\pi$$
−0.536484 + 0.843911i $$0.680248\pi$$
$$54$$ −290.000 −0.730815
$$55$$ −204.000 −0.500134
$$56$$ −192.000 −0.458162
$$57$$ 350.000 0.813309
$$58$$ −490.000 −1.10931
$$59$$ −408.000 −0.900289 −0.450145 0.892956i $$-0.648628\pi$$
−0.450145 + 0.892956i $$0.648628\pi$$
$$60$$ −120.000 −0.258199
$$61$$ 822.000 1.72535 0.862675 0.505759i $$-0.168788\pi$$
0.862675 + 0.505759i $$0.168788\pi$$
$$62$$ −206.000 −0.421968
$$63$$ 16.0000 0.0319970
$$64$$ 448.000 0.875000
$$65$$ 342.000 0.652614
$$66$$ 340.000 0.634108
$$67$$ 926.000 1.68849 0.844246 0.535957i $$-0.180049\pi$$
0.844246 + 0.535957i $$0.180049\pi$$
$$68$$ 320.000 0.570672
$$69$$ −115.000 −0.200643
$$70$$ −96.0000 −0.163917
$$71$$ 335.000 0.559960 0.279980 0.960006i $$-0.409672\pi$$
0.279980 + 0.960006i $$0.409672\pi$$
$$72$$ −48.0000 −0.0785674
$$73$$ −899.000 −1.44137 −0.720685 0.693263i $$-0.756173\pi$$
−0.720685 + 0.693263i $$0.756173\pi$$
$$74$$ 596.000 0.936265
$$75$$ 445.000 0.685122
$$76$$ 280.000 0.422608
$$77$$ −272.000 −0.402562
$$78$$ −570.000 −0.827433
$$79$$ −1322.00 −1.88274 −0.941371 0.337373i $$-0.890462\pi$$
−0.941371 + 0.337373i $$0.890462\pi$$
$$80$$ 96.0000 0.134164
$$81$$ −671.000 −0.920439
$$82$$ −190.000 −0.255878
$$83$$ −36.0000 −0.0476086 −0.0238043 0.999717i $$-0.507578\pi$$
−0.0238043 + 0.999717i $$0.507578\pi$$
$$84$$ −160.000 −0.207827
$$85$$ 480.000 0.612510
$$86$$ −176.000 −0.220681
$$87$$ −1225.00 −1.50958
$$88$$ 816.000 0.988476
$$89$$ −460.000 −0.547864 −0.273932 0.961749i $$-0.588324\pi$$
−0.273932 + 0.961749i $$0.588324\pi$$
$$90$$ −24.0000 −0.0281091
$$91$$ 456.000 0.525294
$$92$$ −92.0000 −0.104257
$$93$$ −515.000 −0.574226
$$94$$ 714.000 0.783441
$$95$$ 420.000 0.453590
$$96$$ 800.000 0.850517
$$97$$ −964.000 −1.00907 −0.504533 0.863393i $$-0.668335\pi$$
−0.504533 + 0.863393i $$0.668335\pi$$
$$98$$ 558.000 0.575168
$$99$$ −68.0000 −0.0690329
$$100$$ 356.000 0.356000
$$101$$ −310.000 −0.305407 −0.152704 0.988272i $$-0.548798\pi$$
−0.152704 + 0.988272i $$0.548798\pi$$
$$102$$ −800.000 −0.776586
$$103$$ 1044.00 0.998722 0.499361 0.866394i $$-0.333568\pi$$
0.499361 + 0.866394i $$0.333568\pi$$
$$104$$ −1368.00 −1.28984
$$105$$ −240.000 −0.223063
$$106$$ 828.000 0.758703
$$107$$ 414.000 0.374046 0.187023 0.982356i $$-0.440116\pi$$
0.187023 + 0.982356i $$0.440116\pi$$
$$108$$ −580.000 −0.516764
$$109$$ 704.000 0.618633 0.309316 0.950959i $$-0.399900\pi$$
0.309316 + 0.950959i $$0.399900\pi$$
$$110$$ 408.000 0.353648
$$111$$ 1490.00 1.27409
$$112$$ 128.000 0.107990
$$113$$ 952.000 0.792537 0.396268 0.918135i $$-0.370305\pi$$
0.396268 + 0.918135i $$0.370305\pi$$
$$114$$ −700.000 −0.575097
$$115$$ −138.000 −0.111901
$$116$$ −980.000 −0.784403
$$117$$ 114.000 0.0900795
$$118$$ 816.000 0.636601
$$119$$ 640.000 0.493014
$$120$$ 720.000 0.547723
$$121$$ −175.000 −0.131480
$$122$$ −1644.00 −1.22001
$$123$$ −475.000 −0.348206
$$124$$ −412.000 −0.298377
$$125$$ 1284.00 0.918756
$$126$$ −32.0000 −0.0226253
$$127$$ 261.000 0.182362 0.0911811 0.995834i $$-0.470936\pi$$
0.0911811 + 0.995834i $$0.470936\pi$$
$$128$$ 384.000 0.265165
$$129$$ −440.000 −0.300309
$$130$$ −684.000 −0.461467
$$131$$ −1441.00 −0.961074 −0.480537 0.876974i $$-0.659558\pi$$
−0.480537 + 0.876974i $$0.659558\pi$$
$$132$$ 680.000 0.448382
$$133$$ 560.000 0.365099
$$134$$ −1852.00 −1.19394
$$135$$ −870.000 −0.554649
$$136$$ −1920.00 −1.21058
$$137$$ 1556.00 0.970351 0.485175 0.874417i $$-0.338756\pi$$
0.485175 + 0.874417i $$0.338756\pi$$
$$138$$ 230.000 0.141876
$$139$$ 25.0000 0.0152552 0.00762760 0.999971i $$-0.497572\pi$$
0.00762760 + 0.999971i $$0.497572\pi$$
$$140$$ −192.000 −0.115907
$$141$$ 1785.00 1.06613
$$142$$ −670.000 −0.395952
$$143$$ −1938.00 −1.13331
$$144$$ 32.0000 0.0185185
$$145$$ −1470.00 −0.841909
$$146$$ 1798.00 1.01920
$$147$$ 1395.00 0.782705
$$148$$ 1192.00 0.662039
$$149$$ 822.000 0.451952 0.225976 0.974133i $$-0.427443\pi$$
0.225976 + 0.974133i $$0.427443\pi$$
$$150$$ −890.000 −0.484455
$$151$$ −1489.00 −0.802471 −0.401235 0.915975i $$-0.631419\pi$$
−0.401235 + 0.915975i $$0.631419\pi$$
$$152$$ −1680.00 −0.896487
$$153$$ 160.000 0.0845440
$$154$$ 544.000 0.284654
$$155$$ −618.000 −0.320251
$$156$$ −1140.00 −0.585084
$$157$$ −632.000 −0.321268 −0.160634 0.987014i $$-0.551354\pi$$
−0.160634 + 0.987014i $$0.551354\pi$$
$$158$$ 2644.00 1.33130
$$159$$ 2070.00 1.03246
$$160$$ 960.000 0.474342
$$161$$ −184.000 −0.0900698
$$162$$ 1342.00 0.650849
$$163$$ −3043.00 −1.46225 −0.731123 0.682245i $$-0.761004\pi$$
−0.731123 + 0.682245i $$0.761004\pi$$
$$164$$ −380.000 −0.180933
$$165$$ 1020.00 0.481254
$$166$$ 72.0000 0.0336644
$$167$$ −2224.00 −1.03053 −0.515264 0.857031i $$-0.672306\pi$$
−0.515264 + 0.857031i $$0.672306\pi$$
$$168$$ 960.000 0.440867
$$169$$ 1052.00 0.478835
$$170$$ −960.000 −0.433110
$$171$$ 140.000 0.0626086
$$172$$ −352.000 −0.156045
$$173$$ 3230.00 1.41949 0.709747 0.704457i $$-0.248809\pi$$
0.709747 + 0.704457i $$0.248809\pi$$
$$174$$ 2450.00 1.06744
$$175$$ 712.000 0.307555
$$176$$ −544.000 −0.232986
$$177$$ 2040.00 0.866304
$$178$$ 920.000 0.387398
$$179$$ 369.000 0.154080 0.0770401 0.997028i $$-0.475453\pi$$
0.0770401 + 0.997028i $$0.475453\pi$$
$$180$$ −48.0000 −0.0198762
$$181$$ −1370.00 −0.562604 −0.281302 0.959619i $$-0.590766\pi$$
−0.281302 + 0.959619i $$0.590766\pi$$
$$182$$ −912.000 −0.371439
$$183$$ −4110.00 −1.66022
$$184$$ 552.000 0.221163
$$185$$ 1788.00 0.710575
$$186$$ 1030.00 0.406039
$$187$$ −2720.00 −1.06367
$$188$$ 1428.00 0.553977
$$189$$ −1160.00 −0.446442
$$190$$ −840.000 −0.320737
$$191$$ 4410.00 1.67066 0.835331 0.549747i $$-0.185276\pi$$
0.835331 + 0.549747i $$0.185276\pi$$
$$192$$ −2240.00 −0.841969
$$193$$ −135.000 −0.0503498 −0.0251749 0.999683i $$-0.508014\pi$$
−0.0251749 + 0.999683i $$0.508014\pi$$
$$194$$ 1928.00 0.713517
$$195$$ −1710.00 −0.627978
$$196$$ 1116.00 0.406706
$$197$$ 1221.00 0.441587 0.220794 0.975321i $$-0.429135\pi$$
0.220794 + 0.975321i $$0.429135\pi$$
$$198$$ 136.000 0.0488136
$$199$$ −1098.00 −0.391131 −0.195566 0.980691i $$-0.562654\pi$$
−0.195566 + 0.980691i $$0.562654\pi$$
$$200$$ −2136.00 −0.755190
$$201$$ −4630.00 −1.62475
$$202$$ 620.000 0.215956
$$203$$ −1960.00 −0.677660
$$204$$ −1600.00 −0.549129
$$205$$ −570.000 −0.194198
$$206$$ −2088.00 −0.706203
$$207$$ −46.0000 −0.0154455
$$208$$ 912.000 0.304018
$$209$$ −2380.00 −0.787694
$$210$$ 480.000 0.157729
$$211$$ −3676.00 −1.19937 −0.599683 0.800238i $$-0.704707\pi$$
−0.599683 + 0.800238i $$0.704707\pi$$
$$212$$ 1656.00 0.536484
$$213$$ −1675.00 −0.538822
$$214$$ −828.000 −0.264490
$$215$$ −528.000 −0.167485
$$216$$ 3480.00 1.09622
$$217$$ −824.000 −0.257773
$$218$$ −1408.00 −0.437439
$$219$$ 4495.00 1.38696
$$220$$ 816.000 0.250067
$$221$$ 4560.00 1.38796
$$222$$ −2980.00 −0.900921
$$223$$ 1656.00 0.497282 0.248641 0.968596i $$-0.420016\pi$$
0.248641 + 0.968596i $$0.420016\pi$$
$$224$$ 1280.00 0.381802
$$225$$ 178.000 0.0527407
$$226$$ −1904.00 −0.560408
$$227$$ 2940.00 0.859624 0.429812 0.902918i $$-0.358580\pi$$
0.429812 + 0.902918i $$0.358580\pi$$
$$228$$ −1400.00 −0.406655
$$229$$ 3612.00 1.04230 0.521152 0.853464i $$-0.325502\pi$$
0.521152 + 0.853464i $$0.325502\pi$$
$$230$$ 276.000 0.0791257
$$231$$ 1360.00 0.387366
$$232$$ 5880.00 1.66397
$$233$$ −4325.00 −1.21605 −0.608026 0.793917i $$-0.708038\pi$$
−0.608026 + 0.793917i $$0.708038\pi$$
$$234$$ −228.000 −0.0636958
$$235$$ 2142.00 0.594590
$$236$$ 1632.00 0.450145
$$237$$ 6610.00 1.81167
$$238$$ −1280.00 −0.348614
$$239$$ 2735.00 0.740219 0.370110 0.928988i $$-0.379320\pi$$
0.370110 + 0.928988i $$0.379320\pi$$
$$240$$ −480.000 −0.129099
$$241$$ −6710.00 −1.79348 −0.896741 0.442556i $$-0.854072\pi$$
−0.896741 + 0.442556i $$0.854072\pi$$
$$242$$ 350.000 0.0929705
$$243$$ −560.000 −0.147835
$$244$$ −3288.00 −0.862675
$$245$$ 1674.00 0.436522
$$246$$ 950.000 0.246219
$$247$$ 3990.00 1.02784
$$248$$ 2472.00 0.632952
$$249$$ 180.000 0.0458114
$$250$$ −2568.00 −0.649658
$$251$$ −6948.00 −1.74723 −0.873613 0.486621i $$-0.838229\pi$$
−0.873613 + 0.486621i $$0.838229\pi$$
$$252$$ −64.0000 −0.0159985
$$253$$ 782.000 0.194324
$$254$$ −522.000 −0.128950
$$255$$ −2400.00 −0.589388
$$256$$ −4352.00 −1.06250
$$257$$ −4929.00 −1.19635 −0.598176 0.801365i $$-0.704108\pi$$
−0.598176 + 0.801365i $$0.704108\pi$$
$$258$$ 880.000 0.212350
$$259$$ 2384.00 0.571948
$$260$$ −1368.00 −0.326307
$$261$$ −490.000 −0.116208
$$262$$ 2882.00 0.679582
$$263$$ 6138.00 1.43911 0.719554 0.694437i $$-0.244346\pi$$
0.719554 + 0.694437i $$0.244346\pi$$
$$264$$ −4080.00 −0.951162
$$265$$ 2484.00 0.575815
$$266$$ −1120.00 −0.258164
$$267$$ 2300.00 0.527182
$$268$$ −3704.00 −0.844246
$$269$$ −2063.00 −0.467596 −0.233798 0.972285i $$-0.575115\pi$$
−0.233798 + 0.972285i $$0.575115\pi$$
$$270$$ 1740.00 0.392196
$$271$$ −1064.00 −0.238500 −0.119250 0.992864i $$-0.538049\pi$$
−0.119250 + 0.992864i $$0.538049\pi$$
$$272$$ 1280.00 0.285336
$$273$$ −2280.00 −0.505465
$$274$$ −3112.00 −0.686142
$$275$$ −3026.00 −0.663544
$$276$$ 460.000 0.100322
$$277$$ 5729.00 1.24268 0.621340 0.783541i $$-0.286589\pi$$
0.621340 + 0.783541i $$0.286589\pi$$
$$278$$ −50.0000 −0.0107871
$$279$$ −206.000 −0.0442039
$$280$$ 1152.00 0.245876
$$281$$ −960.000 −0.203804 −0.101902 0.994794i $$-0.532493\pi$$
−0.101902 + 0.994794i $$0.532493\pi$$
$$282$$ −3570.00 −0.753867
$$283$$ −114.000 −0.0239456 −0.0119728 0.999928i $$-0.503811\pi$$
−0.0119728 + 0.999928i $$0.503811\pi$$
$$284$$ −1340.00 −0.279980
$$285$$ −2100.00 −0.436468
$$286$$ 3876.00 0.801373
$$287$$ −760.000 −0.156311
$$288$$ 320.000 0.0654729
$$289$$ 1487.00 0.302666
$$290$$ 2940.00 0.595320
$$291$$ 4820.00 0.970974
$$292$$ 3596.00 0.720685
$$293$$ −7048.00 −1.40529 −0.702643 0.711543i $$-0.747997\pi$$
−0.702643 + 0.711543i $$0.747997\pi$$
$$294$$ −2790.00 −0.553456
$$295$$ 2448.00 0.483146
$$296$$ −7152.00 −1.40440
$$297$$ 4930.00 0.963191
$$298$$ −1644.00 −0.319578
$$299$$ −1311.00 −0.253569
$$300$$ −1780.00 −0.342561
$$301$$ −704.000 −0.134810
$$302$$ 2978.00 0.567433
$$303$$ 1550.00 0.293878
$$304$$ 1120.00 0.211304
$$305$$ −4932.00 −0.925920
$$306$$ −320.000 −0.0597816
$$307$$ 3872.00 0.719826 0.359913 0.932986i $$-0.382806\pi$$
0.359913 + 0.932986i $$0.382806\pi$$
$$308$$ 1088.00 0.201281
$$309$$ −5220.00 −0.961021
$$310$$ 1236.00 0.226452
$$311$$ −4977.00 −0.907459 −0.453730 0.891139i $$-0.649907\pi$$
−0.453730 + 0.891139i $$0.649907\pi$$
$$312$$ 6840.00 1.24115
$$313$$ −2536.00 −0.457965 −0.228983 0.973430i $$-0.573540\pi$$
−0.228983 + 0.973430i $$0.573540\pi$$
$$314$$ 1264.00 0.227171
$$315$$ −96.0000 −0.0171714
$$316$$ 5288.00 0.941371
$$317$$ 1434.00 0.254074 0.127037 0.991898i $$-0.459453\pi$$
0.127037 + 0.991898i $$0.459453\pi$$
$$318$$ −4140.00 −0.730062
$$319$$ 8330.00 1.46204
$$320$$ −2688.00 −0.469574
$$321$$ −2070.00 −0.359926
$$322$$ 368.000 0.0636889
$$323$$ 5600.00 0.964682
$$324$$ 2684.00 0.460219
$$325$$ 5073.00 0.865844
$$326$$ 6086.00 1.03396
$$327$$ −3520.00 −0.595280
$$328$$ 2280.00 0.383817
$$329$$ 2856.00 0.478591
$$330$$ −2040.00 −0.340298
$$331$$ 5469.00 0.908167 0.454084 0.890959i $$-0.349967\pi$$
0.454084 + 0.890959i $$0.349967\pi$$
$$332$$ 144.000 0.0238043
$$333$$ 596.000 0.0980799
$$334$$ 4448.00 0.728694
$$335$$ −5556.00 −0.906139
$$336$$ −640.000 −0.103913
$$337$$ −7796.00 −1.26016 −0.630082 0.776529i $$-0.716979\pi$$
−0.630082 + 0.776529i $$0.716979\pi$$
$$338$$ −2104.00 −0.338587
$$339$$ −4760.00 −0.762619
$$340$$ −1920.00 −0.306255
$$341$$ 3502.00 0.556141
$$342$$ −280.000 −0.0442710
$$343$$ 4976.00 0.783320
$$344$$ 2112.00 0.331022
$$345$$ 690.000 0.107676
$$346$$ −6460.00 −1.00373
$$347$$ −10068.0 −1.55758 −0.778788 0.627288i $$-0.784165\pi$$
−0.778788 + 0.627288i $$0.784165\pi$$
$$348$$ 4900.00 0.754792
$$349$$ −7495.00 −1.14956 −0.574782 0.818306i $$-0.694913\pi$$
−0.574782 + 0.818306i $$0.694913\pi$$
$$350$$ −1424.00 −0.217474
$$351$$ −8265.00 −1.25685
$$352$$ −5440.00 −0.823730
$$353$$ 10617.0 1.60081 0.800405 0.599460i $$-0.204618\pi$$
0.800405 + 0.599460i $$0.204618\pi$$
$$354$$ −4080.00 −0.612569
$$355$$ −2010.00 −0.300506
$$356$$ 1840.00 0.273932
$$357$$ −3200.00 −0.474403
$$358$$ −738.000 −0.108951
$$359$$ 2522.00 0.370769 0.185384 0.982666i $$-0.440647\pi$$
0.185384 + 0.982666i $$0.440647\pi$$
$$360$$ 288.000 0.0421637
$$361$$ −1959.00 −0.285610
$$362$$ 2740.00 0.397821
$$363$$ 875.000 0.126517
$$364$$ −1824.00 −0.262647
$$365$$ 5394.00 0.773520
$$366$$ 8220.00 1.17395
$$367$$ 7204.00 1.02465 0.512324 0.858792i $$-0.328785\pi$$
0.512324 + 0.858792i $$0.328785\pi$$
$$368$$ −368.000 −0.0521286
$$369$$ −190.000 −0.0268049
$$370$$ −3576.00 −0.502452
$$371$$ 3312.00 0.463478
$$372$$ 2060.00 0.287113
$$373$$ −13310.0 −1.84763 −0.923815 0.382840i $$-0.874946\pi$$
−0.923815 + 0.382840i $$0.874946\pi$$
$$374$$ 5440.00 0.752128
$$375$$ −6420.00 −0.884073
$$376$$ −8568.00 −1.17516
$$377$$ −13965.0 −1.90778
$$378$$ 2320.00 0.315682
$$379$$ 12952.0 1.75541 0.877704 0.479203i $$-0.159074\pi$$
0.877704 + 0.479203i $$0.159074\pi$$
$$380$$ −1680.00 −0.226795
$$381$$ −1305.00 −0.175478
$$382$$ −8820.00 −1.18134
$$383$$ −2812.00 −0.375161 −0.187580 0.982249i $$-0.560064\pi$$
−0.187580 + 0.982249i $$0.560064\pi$$
$$384$$ −1920.00 −0.255155
$$385$$ 1632.00 0.216037
$$386$$ 270.000 0.0356027
$$387$$ −176.000 −0.0231178
$$388$$ 3856.00 0.504533
$$389$$ 1264.00 0.164749 0.0823745 0.996601i $$-0.473750\pi$$
0.0823745 + 0.996601i $$0.473750\pi$$
$$390$$ 3420.00 0.444047
$$391$$ −1840.00 −0.237987
$$392$$ −6696.00 −0.862753
$$393$$ 7205.00 0.924794
$$394$$ −2442.00 −0.312249
$$395$$ 7932.00 1.01039
$$396$$ 272.000 0.0345165
$$397$$ 7119.00 0.899981 0.449990 0.893033i $$-0.351427\pi$$
0.449990 + 0.893033i $$0.351427\pi$$
$$398$$ 2196.00 0.276572
$$399$$ −2800.00 −0.351317
$$400$$ 1424.00 0.178000
$$401$$ 4262.00 0.530758 0.265379 0.964144i $$-0.414503\pi$$
0.265379 + 0.964144i $$0.414503\pi$$
$$402$$ 9260.00 1.14887
$$403$$ −5871.00 −0.725696
$$404$$ 1240.00 0.152704
$$405$$ 4026.00 0.493959
$$406$$ 3920.00 0.479178
$$407$$ −10132.0 −1.23397
$$408$$ 9600.00 1.16488
$$409$$ 229.000 0.0276854 0.0138427 0.999904i $$-0.495594\pi$$
0.0138427 + 0.999904i $$0.495594\pi$$
$$410$$ 1140.00 0.137319
$$411$$ −7780.00 −0.933720
$$412$$ −4176.00 −0.499361
$$413$$ 3264.00 0.388888
$$414$$ 92.0000 0.0109216
$$415$$ 216.000 0.0255495
$$416$$ 9120.00 1.07487
$$417$$ −125.000 −0.0146793
$$418$$ 4760.00 0.556984
$$419$$ 15776.0 1.83940 0.919699 0.392623i $$-0.128432\pi$$
0.919699 + 0.392623i $$0.128432\pi$$
$$420$$ 960.000 0.111531
$$421$$ −8728.00 −1.01040 −0.505198 0.863003i $$-0.668581\pi$$
−0.505198 + 0.863003i $$0.668581\pi$$
$$422$$ 7352.00 0.848080
$$423$$ 714.000 0.0820706
$$424$$ −9936.00 −1.13805
$$425$$ 7120.00 0.812637
$$426$$ 3350.00 0.381005
$$427$$ −6576.00 −0.745281
$$428$$ −1656.00 −0.187023
$$429$$ 9690.00 1.09053
$$430$$ 1056.00 0.118430
$$431$$ −2928.00 −0.327232 −0.163616 0.986524i $$-0.552316\pi$$
−0.163616 + 0.986524i $$0.552316\pi$$
$$432$$ −2320.00 −0.258382
$$433$$ −5314.00 −0.589780 −0.294890 0.955531i $$-0.595283\pi$$
−0.294890 + 0.955531i $$0.595283\pi$$
$$434$$ 1648.00 0.182273
$$435$$ 7350.00 0.810128
$$436$$ −2816.00 −0.309316
$$437$$ −1610.00 −0.176240
$$438$$ −8990.00 −0.980728
$$439$$ 2585.00 0.281037 0.140519 0.990078i $$-0.455123\pi$$
0.140519 + 0.990078i $$0.455123\pi$$
$$440$$ −4896.00 −0.530472
$$441$$ 558.000 0.0602527
$$442$$ −9120.00 −0.981435
$$443$$ −2997.00 −0.321426 −0.160713 0.987001i $$-0.551379\pi$$
−0.160713 + 0.987001i $$0.551379\pi$$
$$444$$ −5960.00 −0.637047
$$445$$ 2760.00 0.294015
$$446$$ −3312.00 −0.351632
$$447$$ −4110.00 −0.434891
$$448$$ −3584.00 −0.377964
$$449$$ −16562.0 −1.74078 −0.870389 0.492365i $$-0.836132\pi$$
−0.870389 + 0.492365i $$0.836132\pi$$
$$450$$ −356.000 −0.0372933
$$451$$ 3230.00 0.337239
$$452$$ −3808.00 −0.396268
$$453$$ 7445.00 0.772178
$$454$$ −5880.00 −0.607846
$$455$$ −2736.00 −0.281903
$$456$$ 8400.00 0.862645
$$457$$ 3924.00 0.401656 0.200828 0.979626i $$-0.435637\pi$$
0.200828 + 0.979626i $$0.435637\pi$$
$$458$$ −7224.00 −0.737020
$$459$$ −11600.0 −1.17961
$$460$$ 552.000 0.0559503
$$461$$ −4543.00 −0.458977 −0.229489 0.973311i $$-0.573705\pi$$
−0.229489 + 0.973311i $$0.573705\pi$$
$$462$$ −2720.00 −0.273909
$$463$$ 9616.00 0.965213 0.482606 0.875837i $$-0.339690\pi$$
0.482606 + 0.875837i $$0.339690\pi$$
$$464$$ −3920.00 −0.392201
$$465$$ 3090.00 0.308162
$$466$$ 8650.00 0.859879
$$467$$ 7826.00 0.775469 0.387735 0.921771i $$-0.373258\pi$$
0.387735 + 0.921771i $$0.373258\pi$$
$$468$$ −456.000 −0.0450398
$$469$$ −7408.00 −0.729360
$$470$$ −4284.00 −0.420439
$$471$$ 3160.00 0.309140
$$472$$ −9792.00 −0.954901
$$473$$ 2992.00 0.290851
$$474$$ −13220.0 −1.28104
$$475$$ 6230.00 0.601794
$$476$$ −2560.00 −0.246507
$$477$$ 828.000 0.0794791
$$478$$ −5470.00 −0.523414
$$479$$ 11404.0 1.08781 0.543906 0.839146i $$-0.316945\pi$$
0.543906 + 0.839146i $$0.316945\pi$$
$$480$$ −4800.00 −0.456435
$$481$$ 16986.0 1.61018
$$482$$ 13420.0 1.26818
$$483$$ 920.000 0.0866697
$$484$$ 700.000 0.0657400
$$485$$ 5784.00 0.541521
$$486$$ 1120.00 0.104535
$$487$$ −9267.00 −0.862275 −0.431137 0.902286i $$-0.641888\pi$$
−0.431137 + 0.902286i $$0.641888\pi$$
$$488$$ 19728.0 1.83001
$$489$$ 15215.0 1.40705
$$490$$ −3348.00 −0.308668
$$491$$ −18191.0 −1.67199 −0.835996 0.548735i $$-0.815110\pi$$
−0.835996 + 0.548735i $$0.815110\pi$$
$$492$$ 1900.00 0.174103
$$493$$ −19600.0 −1.79055
$$494$$ −7980.00 −0.726796
$$495$$ 408.000 0.0370469
$$496$$ −1648.00 −0.149188
$$497$$ −2680.00 −0.241880
$$498$$ −360.000 −0.0323935
$$499$$ 19315.0 1.73278 0.866391 0.499366i $$-0.166434\pi$$
0.866391 + 0.499366i $$0.166434\pi$$
$$500$$ −5136.00 −0.459378
$$501$$ 11120.0 0.991627
$$502$$ 13896.0 1.23548
$$503$$ 8422.00 0.746557 0.373279 0.927719i $$-0.378234\pi$$
0.373279 + 0.927719i $$0.378234\pi$$
$$504$$ 384.000 0.0339379
$$505$$ 1860.00 0.163899
$$506$$ −1564.00 −0.137408
$$507$$ −5260.00 −0.460759
$$508$$ −1044.00 −0.0911811
$$509$$ −863.000 −0.0751509 −0.0375754 0.999294i $$-0.511963\pi$$
−0.0375754 + 0.999294i $$0.511963\pi$$
$$510$$ 4800.00 0.416760
$$511$$ 7192.00 0.622613
$$512$$ 5632.00 0.486136
$$513$$ −10150.0 −0.873554
$$514$$ 9858.00 0.845949
$$515$$ −6264.00 −0.535971
$$516$$ 1760.00 0.150154
$$517$$ −12138.0 −1.03255
$$518$$ −4768.00 −0.404428
$$519$$ −16150.0 −1.36591
$$520$$ 8208.00 0.692201
$$521$$ 19260.0 1.61957 0.809785 0.586727i $$-0.199584\pi$$
0.809785 + 0.586727i $$0.199584\pi$$
$$522$$ 980.000 0.0821713
$$523$$ −11740.0 −0.981557 −0.490779 0.871284i $$-0.663288\pi$$
−0.490779 + 0.871284i $$0.663288\pi$$
$$524$$ 5764.00 0.480537
$$525$$ −3560.00 −0.295945
$$526$$ −12276.0 −1.01760
$$527$$ −8240.00 −0.681101
$$528$$ 2720.00 0.224191
$$529$$ 529.000 0.0434783
$$530$$ −4968.00 −0.407163
$$531$$ 816.000 0.0666881
$$532$$ −2240.00 −0.182549
$$533$$ −5415.00 −0.440056
$$534$$ −4600.00 −0.372774
$$535$$ −2484.00 −0.200734
$$536$$ 22224.0 1.79092
$$537$$ −1845.00 −0.148264
$$538$$ 4126.00 0.330640
$$539$$ −9486.00 −0.758054
$$540$$ 3480.00 0.277325
$$541$$ 17741.0 1.40988 0.704940 0.709267i $$-0.250974\pi$$
0.704940 + 0.709267i $$0.250974\pi$$
$$542$$ 2128.00 0.168645
$$543$$ 6850.00 0.541366
$$544$$ 12800.0 1.00882
$$545$$ −4224.00 −0.331993
$$546$$ 4560.00 0.357418
$$547$$ −6571.00 −0.513630 −0.256815 0.966461i $$-0.582673\pi$$
−0.256815 + 0.966461i $$0.582673\pi$$
$$548$$ −6224.00 −0.485175
$$549$$ −1644.00 −0.127804
$$550$$ 6052.00 0.469197
$$551$$ −17150.0 −1.32598
$$552$$ −2760.00 −0.212814
$$553$$ 10576.0 0.813268
$$554$$ −11458.0 −0.878707
$$555$$ −8940.00 −0.683751
$$556$$ −100.000 −0.00762760
$$557$$ −1372.00 −0.104369 −0.0521845 0.998637i $$-0.516618\pi$$
−0.0521845 + 0.998637i $$0.516618\pi$$
$$558$$ 412.000 0.0312569
$$559$$ −5016.00 −0.379524
$$560$$ −768.000 −0.0579534
$$561$$ 13600.0 1.02352
$$562$$ 1920.00 0.144111
$$563$$ 4332.00 0.324284 0.162142 0.986767i $$-0.448160\pi$$
0.162142 + 0.986767i $$0.448160\pi$$
$$564$$ −7140.00 −0.533064
$$565$$ −5712.00 −0.425320
$$566$$ 228.000 0.0169321
$$567$$ 5368.00 0.397592
$$568$$ 8040.00 0.593928
$$569$$ −3546.00 −0.261258 −0.130629 0.991431i $$-0.541700\pi$$
−0.130629 + 0.991431i $$0.541700\pi$$
$$570$$ 4200.00 0.308629
$$571$$ −6160.00 −0.451468 −0.225734 0.974189i $$-0.572478\pi$$
−0.225734 + 0.974189i $$0.572478\pi$$
$$572$$ 7752.00 0.566656
$$573$$ −22050.0 −1.60760
$$574$$ 1520.00 0.110529
$$575$$ −2047.00 −0.148462
$$576$$ −896.000 −0.0648148
$$577$$ 2953.00 0.213059 0.106529 0.994310i $$-0.466026\pi$$
0.106529 + 0.994310i $$0.466026\pi$$
$$578$$ −2974.00 −0.214017
$$579$$ 675.000 0.0484491
$$580$$ 5880.00 0.420955
$$581$$ 288.000 0.0205650
$$582$$ −9640.00 −0.686582
$$583$$ −14076.0 −0.999946
$$584$$ −21576.0 −1.52880
$$585$$ −684.000 −0.0483417
$$586$$ 14096.0 0.993687
$$587$$ −2949.00 −0.207356 −0.103678 0.994611i $$-0.533061\pi$$
−0.103678 + 0.994611i $$0.533061\pi$$
$$588$$ −5580.00 −0.391353
$$589$$ −7210.00 −0.504385
$$590$$ −4896.00 −0.341636
$$591$$ −6105.00 −0.424917
$$592$$ 4768.00 0.331020
$$593$$ 16390.0 1.13500 0.567501 0.823372i $$-0.307910\pi$$
0.567501 + 0.823372i $$0.307910\pi$$
$$594$$ −9860.00 −0.681079
$$595$$ −3840.00 −0.264579
$$596$$ −3288.00 −0.225976
$$597$$ 5490.00 0.376366
$$598$$ 2622.00 0.179300
$$599$$ −12920.0 −0.881297 −0.440648 0.897680i $$-0.645251\pi$$
−0.440648 + 0.897680i $$0.645251\pi$$
$$600$$ 10680.0 0.726682
$$601$$ −13835.0 −0.939004 −0.469502 0.882931i $$-0.655567\pi$$
−0.469502 + 0.882931i $$0.655567\pi$$
$$602$$ 1408.00 0.0953252
$$603$$ −1852.00 −0.125073
$$604$$ 5956.00 0.401235
$$605$$ 1050.00 0.0705596
$$606$$ −3100.00 −0.207803
$$607$$ 6004.00 0.401474 0.200737 0.979645i $$-0.435666\pi$$
0.200737 + 0.979645i $$0.435666\pi$$
$$608$$ 11200.0 0.747072
$$609$$ 9800.00 0.652079
$$610$$ 9864.00 0.654724
$$611$$ 20349.0 1.34735
$$612$$ −640.000 −0.0422720
$$613$$ −16416.0 −1.08162 −0.540812 0.841143i $$-0.681883\pi$$
−0.540812 + 0.841143i $$0.681883\pi$$
$$614$$ −7744.00 −0.508994
$$615$$ 2850.00 0.186867
$$616$$ −6528.00 −0.426982
$$617$$ 3786.00 0.247032 0.123516 0.992343i $$-0.460583\pi$$
0.123516 + 0.992343i $$0.460583\pi$$
$$618$$ 10440.0 0.679544
$$619$$ 15824.0 1.02750 0.513748 0.857941i $$-0.328257\pi$$
0.513748 + 0.857941i $$0.328257\pi$$
$$620$$ 2472.00 0.160126
$$621$$ 3335.00 0.215506
$$622$$ 9954.00 0.641670
$$623$$ 3680.00 0.236655
$$624$$ −4560.00 −0.292542
$$625$$ 3421.00 0.218944
$$626$$ 5072.00 0.323830
$$627$$ 11900.0 0.757959
$$628$$ 2528.00 0.160634
$$629$$ 23840.0 1.51123
$$630$$ 192.000 0.0121420
$$631$$ 17852.0 1.12627 0.563135 0.826365i $$-0.309595\pi$$
0.563135 + 0.826365i $$0.309595\pi$$
$$632$$ −31728.0 −1.99695
$$633$$ 18380.0 1.15409
$$634$$ −2868.00 −0.179657
$$635$$ −1566.00 −0.0978658
$$636$$ −8280.00 −0.516232
$$637$$ 15903.0 0.989168
$$638$$ −16660.0 −1.03382
$$639$$ −670.000 −0.0414785
$$640$$ −2304.00 −0.142302
$$641$$ 10324.0 0.636152 0.318076 0.948065i $$-0.396963\pi$$
0.318076 + 0.948065i $$0.396963\pi$$
$$642$$ 4140.00 0.254506
$$643$$ −14702.0 −0.901696 −0.450848 0.892601i $$-0.648878\pi$$
−0.450848 + 0.892601i $$0.648878\pi$$
$$644$$ 736.000 0.0450349
$$645$$ 2640.00 0.161163
$$646$$ −11200.0 −0.682133
$$647$$ 11939.0 0.725457 0.362728 0.931895i $$-0.381845\pi$$
0.362728 + 0.931895i $$0.381845\pi$$
$$648$$ −16104.0 −0.976273
$$649$$ −13872.0 −0.839019
$$650$$ −10146.0 −0.612244
$$651$$ 4120.00 0.248042
$$652$$ 12172.0 0.731123
$$653$$ 6159.00 0.369097 0.184548 0.982823i $$-0.440918\pi$$
0.184548 + 0.982823i $$0.440918\pi$$
$$654$$ 7040.00 0.420926
$$655$$ 8646.00 0.515767
$$656$$ −1520.00 −0.0904665
$$657$$ 1798.00 0.106768
$$658$$ −5712.00 −0.338415
$$659$$ −21692.0 −1.28225 −0.641123 0.767438i $$-0.721531\pi$$
−0.641123 + 0.767438i $$0.721531\pi$$
$$660$$ −4080.00 −0.240627
$$661$$ 16502.0 0.971034 0.485517 0.874227i $$-0.338631\pi$$
0.485517 + 0.874227i $$0.338631\pi$$
$$662$$ −10938.0 −0.642171
$$663$$ −22800.0 −1.33556
$$664$$ −864.000 −0.0504965
$$665$$ −3360.00 −0.195933
$$666$$ −1192.00 −0.0693529
$$667$$ 5635.00 0.327119
$$668$$ 8896.00 0.515264
$$669$$ −8280.00 −0.478510
$$670$$ 11112.0 0.640737
$$671$$ 27948.0 1.60793
$$672$$ −6400.00 −0.367389
$$673$$ −27733.0 −1.58845 −0.794226 0.607622i $$-0.792124\pi$$
−0.794226 + 0.607622i $$0.792124\pi$$
$$674$$ 15592.0 0.891070
$$675$$ −12905.0 −0.735872
$$676$$ −4208.00 −0.239417
$$677$$ −8814.00 −0.500369 −0.250184 0.968198i $$-0.580491\pi$$
−0.250184 + 0.968198i $$0.580491\pi$$
$$678$$ 9520.00 0.539253
$$679$$ 7712.00 0.435875
$$680$$ 11520.0 0.649664
$$681$$ −14700.0 −0.827174
$$682$$ −7004.00 −0.393251
$$683$$ −22999.0 −1.28848 −0.644240 0.764823i $$-0.722826\pi$$
−0.644240 + 0.764823i $$0.722826\pi$$
$$684$$ −560.000 −0.0313043
$$685$$ −9336.00 −0.520745
$$686$$ −9952.00 −0.553891
$$687$$ −18060.0 −1.00296
$$688$$ −1408.00 −0.0780225
$$689$$ 23598.0 1.30481
$$690$$ −1380.00 −0.0761387
$$691$$ −12140.0 −0.668346 −0.334173 0.942512i $$-0.608457\pi$$
−0.334173 + 0.942512i $$0.608457\pi$$
$$692$$ −12920.0 −0.709747
$$693$$ 544.000 0.0298194
$$694$$ 20136.0 1.10137
$$695$$ −150.000 −0.00818680
$$696$$ −29400.0 −1.60116
$$697$$ −7600.00 −0.413014
$$698$$ 14990.0 0.812865
$$699$$ 21625.0 1.17015
$$700$$ −2848.00 −0.153778
$$701$$ −20024.0 −1.07888 −0.539441 0.842024i $$-0.681364\pi$$
−0.539441 + 0.842024i $$0.681364\pi$$
$$702$$ 16530.0 0.888725
$$703$$ 20860.0 1.11913
$$704$$ 15232.0 0.815451
$$705$$ −10710.0 −0.572145
$$706$$ −21234.0 −1.13194
$$707$$ 2480.00 0.131924
$$708$$ −8160.00 −0.433152
$$709$$ −4956.00 −0.262520 −0.131260 0.991348i $$-0.541902\pi$$
−0.131260 + 0.991348i $$0.541902\pi$$
$$710$$ 4020.00 0.212490
$$711$$ 2644.00 0.139462
$$712$$ −11040.0 −0.581098
$$713$$ 2369.00 0.124432
$$714$$ 6400.00 0.335454
$$715$$ 11628.0 0.608199
$$716$$ −1476.00 −0.0770401
$$717$$ −13675.0 −0.712276
$$718$$ −5044.00 −0.262173
$$719$$ 2760.00 0.143158 0.0715790 0.997435i $$-0.477196\pi$$
0.0715790 + 0.997435i $$0.477196\pi$$
$$720$$ −192.000 −0.00993808
$$721$$ −8352.00 −0.431407
$$722$$ 3918.00 0.201957
$$723$$ 33550.0 1.72578
$$724$$ 5480.00 0.281302
$$725$$ −21805.0 −1.11699
$$726$$ −1750.00 −0.0894609
$$727$$ 7746.00 0.395163 0.197581 0.980287i $$-0.436691\pi$$
0.197581 + 0.980287i $$0.436691\pi$$
$$728$$ 10944.0 0.557159
$$729$$ 20917.0 1.06269
$$730$$ −10788.0 −0.546961
$$731$$ −7040.00 −0.356202
$$732$$ 16440.0 0.830109
$$733$$ −11976.0 −0.603470 −0.301735 0.953392i $$-0.597566\pi$$
−0.301735 + 0.953392i $$0.597566\pi$$
$$734$$ −14408.0 −0.724535
$$735$$ −8370.00 −0.420044
$$736$$ −3680.00 −0.184302
$$737$$ 31484.0 1.57358
$$738$$ 380.000 0.0189539
$$739$$ 15057.0 0.749500 0.374750 0.927126i $$-0.377728\pi$$
0.374750 + 0.927126i $$0.377728\pi$$
$$740$$ −7152.00 −0.355287
$$741$$ −19950.0 −0.989044
$$742$$ −6624.00 −0.327729
$$743$$ 18532.0 0.915038 0.457519 0.889200i $$-0.348738\pi$$
0.457519 + 0.889200i $$0.348738\pi$$
$$744$$ −12360.0 −0.609059
$$745$$ −4932.00 −0.242543
$$746$$ 26620.0 1.30647
$$747$$ 72.0000 0.00352656
$$748$$ 10880.0 0.531834
$$749$$ −3312.00 −0.161573
$$750$$ 12840.0 0.625134
$$751$$ −192.000 −0.00932913 −0.00466457 0.999989i $$-0.501485\pi$$
−0.00466457 + 0.999989i $$0.501485\pi$$
$$752$$ 5712.00 0.276988
$$753$$ 34740.0 1.68127
$$754$$ 27930.0 1.34901
$$755$$ 8934.00 0.430651
$$756$$ 4640.00 0.223221
$$757$$ −9830.00 −0.471965 −0.235982 0.971757i $$-0.575831\pi$$
−0.235982 + 0.971757i $$0.575831\pi$$
$$758$$ −25904.0 −1.24126
$$759$$ −3910.00 −0.186988
$$760$$ 10080.0 0.481105
$$761$$ −30219.0 −1.43947 −0.719736 0.694248i $$-0.755737\pi$$
−0.719736 + 0.694248i $$0.755737\pi$$
$$762$$ 2610.00 0.124082
$$763$$ −5632.00 −0.267224
$$764$$ −17640.0 −0.835331
$$765$$ −960.000 −0.0453711
$$766$$ 5624.00 0.265279
$$767$$ 23256.0 1.09482
$$768$$ 21760.0 1.02239
$$769$$ 1122.00 0.0526142 0.0263071 0.999654i $$-0.491625\pi$$
0.0263071 + 0.999654i $$0.491625\pi$$
$$770$$ −3264.00 −0.152762
$$771$$ 24645.0 1.15119
$$772$$ 540.000 0.0251749
$$773$$ 19300.0 0.898024 0.449012 0.893526i $$-0.351776\pi$$
0.449012 + 0.893526i $$0.351776\pi$$
$$774$$ 352.000 0.0163467
$$775$$ −9167.00 −0.424888
$$776$$ −23136.0 −1.07028
$$777$$ −11920.0 −0.550357
$$778$$ −2528.00 −0.116495
$$779$$ −6650.00 −0.305855
$$780$$ 6840.00 0.313989
$$781$$ 11390.0 0.521852
$$782$$ 3680.00 0.168282
$$783$$ 35525.0 1.62140
$$784$$ 4464.00 0.203353
$$785$$ 3792.00 0.172411
$$786$$ −14410.0 −0.653928
$$787$$ −19396.0 −0.878517 −0.439258 0.898361i $$-0.644759\pi$$
−0.439258 + 0.898361i $$0.644759\pi$$
$$788$$ −4884.00 −0.220794
$$789$$ −30690.0 −1.38478
$$790$$ −15864.0 −0.714450
$$791$$ −7616.00 −0.342344
$$792$$ −1632.00 −0.0732204
$$793$$ −46854.0 −2.09815
$$794$$ −14238.0 −0.636383
$$795$$ −12420.0 −0.554078
$$796$$ 4392.00 0.195566
$$797$$ −39034.0 −1.73482 −0.867412 0.497590i $$-0.834218\pi$$
−0.867412 + 0.497590i $$0.834218\pi$$
$$798$$ 5600.00 0.248418
$$799$$ 28560.0 1.26456
$$800$$ 14240.0 0.629325
$$801$$ 920.000 0.0405825
$$802$$ −8524.00 −0.375303
$$803$$ −30566.0 −1.34328
$$804$$ 18520.0 0.812376
$$805$$ 1104.00 0.0483365
$$806$$ 11742.0 0.513144
$$807$$ 10315.0 0.449944
$$808$$ −7440.00 −0.323934
$$809$$ −10310.0 −0.448060 −0.224030 0.974582i $$-0.571921\pi$$
−0.224030 + 0.974582i $$0.571921\pi$$
$$810$$ −8052.00 −0.349282
$$811$$ −40693.0 −1.76193 −0.880965 0.473182i $$-0.843105\pi$$
−0.880965 + 0.473182i $$0.843105\pi$$
$$812$$ 7840.00 0.338830
$$813$$ 5320.00 0.229496
$$814$$ 20264.0 0.872546
$$815$$ 18258.0 0.784724
$$816$$ −6400.00 −0.274565
$$817$$ −6160.00 −0.263784
$$818$$ −458.000 −0.0195765
$$819$$ −912.000 −0.0389107
$$820$$ 2280.00 0.0970988
$$821$$ −13934.0 −0.592326 −0.296163 0.955137i $$-0.595707\pi$$
−0.296163 + 0.955137i $$0.595707\pi$$
$$822$$ 15560.0 0.660240
$$823$$ 6175.00 0.261539 0.130770 0.991413i $$-0.458255\pi$$
0.130770 + 0.991413i $$0.458255\pi$$
$$824$$ 25056.0 1.05930
$$825$$ 15130.0 0.638496
$$826$$ −6528.00 −0.274986
$$827$$ 28664.0 1.20525 0.602627 0.798023i $$-0.294121\pi$$
0.602627 + 0.798023i $$0.294121\pi$$
$$828$$ 184.000 0.00772276
$$829$$ −39590.0 −1.65865 −0.829323 0.558770i $$-0.811274\pi$$
−0.829323 + 0.558770i $$0.811274\pi$$
$$830$$ −432.000 −0.0180662
$$831$$ −28645.0 −1.19577
$$832$$ −25536.0 −1.06406
$$833$$ 22320.0 0.928382
$$834$$ 250.000 0.0103798
$$835$$ 13344.0 0.553040
$$836$$ 9520.00 0.393847
$$837$$ 14935.0 0.616761
$$838$$ −31552.0 −1.30065
$$839$$ −14316.0 −0.589086 −0.294543 0.955638i $$-0.595167\pi$$
−0.294543 + 0.955638i $$0.595167\pi$$
$$840$$ −5760.00 −0.236594
$$841$$ 35636.0 1.46115
$$842$$ 17456.0 0.714458
$$843$$ 4800.00 0.196110
$$844$$ 14704.0 0.599683
$$845$$ −6312.00 −0.256970
$$846$$ −1428.00 −0.0580327
$$847$$ 1400.00 0.0567941
$$848$$ 6624.00 0.268242
$$849$$ 570.000 0.0230416
$$850$$ −14240.0 −0.574621
$$851$$ −6854.00 −0.276089
$$852$$ 6700.00 0.269411
$$853$$ 28366.0 1.13861 0.569304 0.822127i $$-0.307213\pi$$
0.569304 + 0.822127i $$0.307213\pi$$
$$854$$ 13152.0 0.526993
$$855$$ −840.000 −0.0335993
$$856$$ 9936.00 0.396735
$$857$$ 19283.0 0.768605 0.384303 0.923207i $$-0.374442\pi$$
0.384303 + 0.923207i $$0.374442\pi$$
$$858$$ −19380.0 −0.771122
$$859$$ −26101.0 −1.03673 −0.518367 0.855158i $$-0.673460\pi$$
−0.518367 + 0.855158i $$0.673460\pi$$
$$860$$ 2112.00 0.0837426
$$861$$ 3800.00 0.150411
$$862$$ 5856.00 0.231388
$$863$$ 973.000 0.0383793 0.0191896 0.999816i $$-0.493891\pi$$
0.0191896 + 0.999816i $$0.493891\pi$$
$$864$$ −23200.0 −0.913519
$$865$$ −19380.0 −0.761780
$$866$$ 10628.0 0.417037
$$867$$ −7435.00 −0.291241
$$868$$ 3296.00 0.128887
$$869$$ −44948.0 −1.75461
$$870$$ −14700.0 −0.572847
$$871$$ −52782.0 −2.05333
$$872$$ 16896.0 0.656159
$$873$$ 1928.00 0.0747456
$$874$$ 3220.00 0.124620
$$875$$ −10272.0 −0.396865
$$876$$ −17980.0 −0.693479
$$877$$ 5694.00 0.219239 0.109620 0.993974i $$-0.465037\pi$$
0.109620 + 0.993974i $$0.465037\pi$$
$$878$$ −5170.00 −0.198723
$$879$$ 35240.0 1.35224
$$880$$ 3264.00 0.125033
$$881$$ 45960.0 1.75758 0.878792 0.477205i $$-0.158350\pi$$
0.878792 + 0.477205i $$0.158350\pi$$
$$882$$ −1116.00 −0.0426051
$$883$$ 17188.0 0.655065 0.327532 0.944840i $$-0.393783\pi$$
0.327532 + 0.944840i $$0.393783\pi$$
$$884$$ −18240.0 −0.693979
$$885$$ −12240.0 −0.464907
$$886$$ 5994.00 0.227283
$$887$$ 8451.00 0.319906 0.159953 0.987125i $$-0.448866\pi$$
0.159953 + 0.987125i $$0.448866\pi$$
$$888$$ 35760.0 1.35138
$$889$$ −2088.00 −0.0787731
$$890$$ −5520.00 −0.207900
$$891$$ −22814.0 −0.857798
$$892$$ −6624.00 −0.248641
$$893$$ 24990.0 0.936460
$$894$$ 8220.00 0.307514
$$895$$ −2214.00 −0.0826881
$$896$$ −3072.00 −0.114541
$$897$$ 6555.00 0.243997
$$898$$ 33124.0 1.23092
$$899$$ 25235.0 0.936190
$$900$$ −712.000 −0.0263704
$$901$$ 33120.0 1.22463
$$902$$ −6460.00 −0.238464
$$903$$ 3520.00 0.129721
$$904$$ 22848.0 0.840612
$$905$$ 8220.00 0.301925
$$906$$ −14890.0 −0.546012
$$907$$ 32774.0 1.19983 0.599913 0.800065i $$-0.295202\pi$$
0.599913 + 0.800065i $$0.295202\pi$$
$$908$$ −11760.0 −0.429812
$$909$$ 620.000 0.0226228
$$910$$ 5472.00 0.199335
$$911$$ −23690.0 −0.861564 −0.430782 0.902456i $$-0.641762\pi$$
−0.430782 + 0.902456i $$0.641762\pi$$
$$912$$ −5600.00 −0.203327
$$913$$ −1224.00 −0.0443686
$$914$$ −7848.00 −0.284014
$$915$$ 24660.0 0.890967
$$916$$ −14448.0 −0.521152
$$917$$ 11528.0 0.415145
$$918$$ 23200.0 0.834111
$$919$$ −30044.0 −1.07841 −0.539206 0.842174i $$-0.681275\pi$$
−0.539206 + 0.842174i $$0.681275\pi$$
$$920$$ −3312.00 −0.118688
$$921$$ −19360.0 −0.692653
$$922$$ 9086.00 0.324546
$$923$$ −19095.0 −0.680953
$$924$$ −5440.00 −0.193683
$$925$$ 26522.0 0.942744
$$926$$ −19232.0 −0.682508
$$927$$ −2088.00 −0.0739794
$$928$$ −39200.0 −1.38664
$$929$$ −39705.0 −1.40224 −0.701119 0.713044i $$-0.747316\pi$$
−0.701119 + 0.713044i $$0.747316\pi$$
$$930$$ −6180.00 −0.217903
$$931$$ 19530.0 0.687508
$$932$$ 17300.0 0.608026
$$933$$ 24885.0 0.873203
$$934$$ −15652.0 −0.548339
$$935$$ 16320.0 0.570825
$$936$$ 2736.00 0.0955438
$$937$$ 17422.0 0.607419 0.303710 0.952765i $$-0.401775\pi$$
0.303710 + 0.952765i $$0.401775\pi$$
$$938$$ 14816.0 0.515735
$$939$$ 12680.0 0.440677
$$940$$ −8568.00 −0.297295
$$941$$ −25292.0 −0.876191 −0.438095 0.898928i $$-0.644347\pi$$
−0.438095 + 0.898928i $$0.644347\pi$$
$$942$$ −6320.00 −0.218595
$$943$$ 2185.00 0.0754543
$$944$$ 6528.00 0.225072
$$945$$ 6960.00 0.239586
$$946$$ −5984.00 −0.205662
$$947$$ 33211.0 1.13961 0.569806 0.821779i $$-0.307018\pi$$
0.569806 + 0.821779i $$0.307018\pi$$
$$948$$ −26440.0 −0.905835
$$949$$ 51243.0 1.75281
$$950$$ −12460.0 −0.425532
$$951$$ −7170.00 −0.244483
$$952$$ 15360.0 0.522921
$$953$$ −14154.0 −0.481105 −0.240552 0.970636i $$-0.577329\pi$$
−0.240552 + 0.970636i $$0.577329\pi$$
$$954$$ −1656.00 −0.0562002
$$955$$ −26460.0 −0.896571
$$956$$ −10940.0 −0.370110
$$957$$ −41650.0 −1.40685
$$958$$ −22808.0 −0.769199
$$959$$ −12448.0 −0.419152
$$960$$ 13440.0 0.451848
$$961$$ −19182.0 −0.643886
$$962$$ −33972.0 −1.13857
$$963$$ −828.000 −0.0277071
$$964$$ 26840.0 0.896741
$$965$$ 810.000 0.0270205
$$966$$ −1840.00 −0.0612847
$$967$$ −46343.0 −1.54115 −0.770574 0.637350i $$-0.780030\pi$$
−0.770574 + 0.637350i $$0.780030\pi$$
$$968$$ −4200.00 −0.139456
$$969$$ −28000.0 −0.928266
$$970$$ −11568.0 −0.382914
$$971$$ 11710.0 0.387015 0.193508 0.981099i $$-0.438014\pi$$
0.193508 + 0.981099i $$0.438014\pi$$
$$972$$ 2240.00 0.0739177
$$973$$ −200.000 −0.00658963
$$974$$ 18534.0 0.609720
$$975$$ −25365.0 −0.833159
$$976$$ −13152.0 −0.431337
$$977$$ 47854.0 1.56703 0.783513 0.621375i $$-0.213426\pi$$
0.783513 + 0.621375i $$0.213426\pi$$
$$978$$ −30430.0 −0.994933
$$979$$ −15640.0 −0.510579
$$980$$ −6696.00 −0.218261
$$981$$ −1408.00 −0.0458246
$$982$$ 36382.0 1.18228
$$983$$ −22078.0 −0.716357 −0.358178 0.933653i $$-0.616602\pi$$
−0.358178 + 0.933653i $$0.616602\pi$$
$$984$$ −11400.0 −0.369328
$$985$$ −7326.00 −0.236980
$$986$$ 39200.0 1.26611
$$987$$ −14280.0 −0.460524
$$988$$ −15960.0 −0.513922
$$989$$ 2024.00 0.0650753
$$990$$ −816.000 −0.0261961
$$991$$ −4288.00 −0.137450 −0.0687249 0.997636i $$-0.521893\pi$$
−0.0687249 + 0.997636i $$0.521893\pi$$
$$992$$ −16480.0 −0.527460
$$993$$ −27345.0 −0.873885
$$994$$ 5360.00 0.171035
$$995$$ 6588.00 0.209903
$$996$$ −720.000 −0.0229057
$$997$$ 28966.0 0.920123 0.460061 0.887887i $$-0.347827\pi$$
0.460061 + 0.887887i $$0.347827\pi$$
$$998$$ −38630.0 −1.22526
$$999$$ −43210.0 −1.36847
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 23.4.a.a.1.1 1
3.2 odd 2 207.4.a.a.1.1 1
4.3 odd 2 368.4.a.d.1.1 1
5.2 odd 4 575.4.b.b.24.1 2
5.3 odd 4 575.4.b.b.24.2 2
5.4 even 2 575.4.a.g.1.1 1
7.6 odd 2 1127.4.a.a.1.1 1
8.3 odd 2 1472.4.a.c.1.1 1
8.5 even 2 1472.4.a.h.1.1 1
23.22 odd 2 529.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
23.4.a.a.1.1 1 1.1 even 1 trivial
207.4.a.a.1.1 1 3.2 odd 2
368.4.a.d.1.1 1 4.3 odd 2
529.4.a.a.1.1 1 23.22 odd 2
575.4.a.g.1.1 1 5.4 even 2
575.4.b.b.24.1 2 5.2 odd 4
575.4.b.b.24.2 2 5.3 odd 4
1127.4.a.a.1.1 1 7.6 odd 2
1472.4.a.c.1.1 1 8.3 odd 2
1472.4.a.h.1.1 1 8.5 even 2