# Properties

 Label 207.4.a.a.1.1 Level $207$ Weight $4$ Character 207.1 Self dual yes Analytic conductor $12.213$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,4,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$207 = 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 207.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: $$12.2133953712$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 23) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 207.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -4.00000 q^{4} +6.00000 q^{5} -8.00000 q^{7} -24.0000 q^{8} +O(q^{10})$$ $$q+2.00000 q^{2} -4.00000 q^{4} +6.00000 q^{5} -8.00000 q^{7} -24.0000 q^{8} +12.0000 q^{10} -34.0000 q^{11} -57.0000 q^{13} -16.0000 q^{14} -16.0000 q^{16} +80.0000 q^{17} -70.0000 q^{19} -24.0000 q^{20} -68.0000 q^{22} -23.0000 q^{23} -89.0000 q^{25} -114.000 q^{26} +32.0000 q^{28} -245.000 q^{29} +103.000 q^{31} +160.000 q^{32} +160.000 q^{34} -48.0000 q^{35} -298.000 q^{37} -140.000 q^{38} -144.000 q^{40} -95.0000 q^{41} +88.0000 q^{43} +136.000 q^{44} -46.0000 q^{46} +357.000 q^{47} -279.000 q^{49} -178.000 q^{50} +228.000 q^{52} +414.000 q^{53} -204.000 q^{55} +192.000 q^{56} -490.000 q^{58} +408.000 q^{59} +822.000 q^{61} +206.000 q^{62} +448.000 q^{64} -342.000 q^{65} +926.000 q^{67} -320.000 q^{68} -96.0000 q^{70} -335.000 q^{71} -899.000 q^{73} -596.000 q^{74} +280.000 q^{76} +272.000 q^{77} -1322.00 q^{79} -96.0000 q^{80} -190.000 q^{82} +36.0000 q^{83} +480.000 q^{85} +176.000 q^{86} +816.000 q^{88} +460.000 q^{89} +456.000 q^{91} +92.0000 q^{92} +714.000 q^{94} -420.000 q^{95} -964.000 q^{97} -558.000 q^{98} +O(q^{100})$$

## 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 $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 0 0
$$4$$ −4.00000 −0.500000
$$5$$ 6.00000 0.536656 0.268328 0.963328i $$-0.413529\pi$$
0.268328 + 0.963328i $$0.413529\pi$$
$$6$$ 0 0
$$7$$ −8.00000 −0.431959 −0.215980 0.976398i $$-0.569295\pi$$
−0.215980 + 0.976398i $$0.569295\pi$$
$$8$$ −24.0000 −1.06066
$$9$$ 0 0
$$10$$ 12.0000 0.379473
$$11$$ −34.0000 −0.931944 −0.465972 0.884799i $$-0.654295\pi$$
−0.465972 + 0.884799i $$0.654295\pi$$
$$12$$ 0 0
$$13$$ −57.0000 −1.21607 −0.608037 0.793909i $$-0.708043\pi$$
−0.608037 + 0.793909i $$0.708043\pi$$
$$14$$ −16.0000 −0.305441
$$15$$ 0 0
$$16$$ −16.0000 −0.250000
$$17$$ 80.0000 1.14134 0.570672 0.821178i $$-0.306683\pi$$
0.570672 + 0.821178i $$0.306683\pi$$
$$18$$ 0 0
$$19$$ −70.0000 −0.845216 −0.422608 0.906313i $$-0.638885\pi$$
−0.422608 + 0.906313i $$0.638885\pi$$
$$20$$ −24.0000 −0.268328
$$21$$ 0 0
$$22$$ −68.0000 −0.658984
$$23$$ −23.0000 −0.208514
$$24$$ 0 0
$$25$$ −89.0000 −0.712000
$$26$$ −114.000 −0.859894
$$27$$ 0 0
$$28$$ 32.0000 0.215980
$$29$$ −245.000 −1.56881 −0.784403 0.620252i $$-0.787030\pi$$
−0.784403 + 0.620252i $$0.787030\pi$$
$$30$$ 0 0
$$31$$ 103.000 0.596753 0.298377 0.954448i $$-0.403555\pi$$
0.298377 + 0.954448i $$0.403555\pi$$
$$32$$ 160.000 0.883883
$$33$$ 0 0
$$34$$ 160.000 0.807052
$$35$$ −48.0000 −0.231814
$$36$$ 0 0
$$37$$ −298.000 −1.32408 −0.662039 0.749469i $$-0.730309\pi$$
−0.662039 + 0.749469i $$0.730309\pi$$
$$38$$ −140.000 −0.597658
$$39$$ 0 0
$$40$$ −144.000 −0.569210
$$41$$ −95.0000 −0.361866 −0.180933 0.983495i $$-0.557912\pi$$
−0.180933 + 0.983495i $$0.557912\pi$$
$$42$$ 0 0
$$43$$ 88.0000 0.312090 0.156045 0.987750i $$-0.450125\pi$$
0.156045 + 0.987750i $$0.450125\pi$$
$$44$$ 136.000 0.465972
$$45$$ 0 0
$$46$$ −46.0000 −0.147442
$$47$$ 357.000 1.10795 0.553977 0.832532i $$-0.313110\pi$$
0.553977 + 0.832532i $$0.313110\pi$$
$$48$$ 0 0
$$49$$ −279.000 −0.813411
$$50$$ −178.000 −0.503460
$$51$$ 0 0
$$52$$ 228.000 0.608037
$$53$$ 414.000 1.07297 0.536484 0.843911i $$-0.319752\pi$$
0.536484 + 0.843911i $$0.319752\pi$$
$$54$$ 0 0
$$55$$ −204.000 −0.500134
$$56$$ 192.000 0.458162
$$57$$ 0 0
$$58$$ −490.000 −1.10931
$$59$$ 408.000 0.900289 0.450145 0.892956i $$-0.351372\pi$$
0.450145 + 0.892956i $$0.351372\pi$$
$$60$$ 0 0
$$61$$ 822.000 1.72535 0.862675 0.505759i $$-0.168788\pi$$
0.862675 + 0.505759i $$0.168788\pi$$
$$62$$ 206.000 0.421968
$$63$$ 0 0
$$64$$ 448.000 0.875000
$$65$$ −342.000 −0.652614
$$66$$ 0 0
$$67$$ 926.000 1.68849 0.844246 0.535957i $$-0.180049\pi$$
0.844246 + 0.535957i $$0.180049\pi$$
$$68$$ −320.000 −0.570672
$$69$$ 0 0
$$70$$ −96.0000 −0.163917
$$71$$ −335.000 −0.559960 −0.279980 0.960006i $$-0.590328\pi$$
−0.279980 + 0.960006i $$0.590328\pi$$
$$72$$ 0 0
$$73$$ −899.000 −1.44137 −0.720685 0.693263i $$-0.756173\pi$$
−0.720685 + 0.693263i $$0.756173\pi$$
$$74$$ −596.000 −0.936265
$$75$$ 0 0
$$76$$ 280.000 0.422608
$$77$$ 272.000 0.402562
$$78$$ 0 0
$$79$$ −1322.00 −1.88274 −0.941371 0.337373i $$-0.890462\pi$$
−0.941371 + 0.337373i $$0.890462\pi$$
$$80$$ −96.0000 −0.134164
$$81$$ 0 0
$$82$$ −190.000 −0.255878
$$83$$ 36.0000 0.0476086 0.0238043 0.999717i $$-0.492422\pi$$
0.0238043 + 0.999717i $$0.492422\pi$$
$$84$$ 0 0
$$85$$ 480.000 0.612510
$$86$$ 176.000 0.220681
$$87$$ 0 0
$$88$$ 816.000 0.988476
$$89$$ 460.000 0.547864 0.273932 0.961749i $$-0.411676\pi$$
0.273932 + 0.961749i $$0.411676\pi$$
$$90$$ 0 0
$$91$$ 456.000 0.525294
$$92$$ 92.0000 0.104257
$$93$$ 0 0
$$94$$ 714.000 0.783441
$$95$$ −420.000 −0.453590
$$96$$ 0 0
$$97$$ −964.000 −1.00907 −0.504533 0.863393i $$-0.668335\pi$$
−0.504533 + 0.863393i $$0.668335\pi$$
$$98$$ −558.000 −0.575168
$$99$$ 0 0
$$100$$ 356.000 0.356000
$$101$$ 310.000 0.305407 0.152704 0.988272i $$-0.451202\pi$$
0.152704 + 0.988272i $$0.451202\pi$$
$$102$$ 0 0
$$103$$ 1044.00 0.998722 0.499361 0.866394i $$-0.333568\pi$$
0.499361 + 0.866394i $$0.333568\pi$$
$$104$$ 1368.00 1.28984
$$105$$ 0 0
$$106$$ 828.000 0.758703
$$107$$ −414.000 −0.374046 −0.187023 0.982356i $$-0.559884\pi$$
−0.187023 + 0.982356i $$0.559884\pi$$
$$108$$ 0 0
$$109$$ 704.000 0.618633 0.309316 0.950959i $$-0.399900\pi$$
0.309316 + 0.950959i $$0.399900\pi$$
$$110$$ −408.000 −0.353648
$$111$$ 0 0
$$112$$ 128.000 0.107990
$$113$$ −952.000 −0.792537 −0.396268 0.918135i $$-0.629695\pi$$
−0.396268 + 0.918135i $$0.629695\pi$$
$$114$$ 0 0
$$115$$ −138.000 −0.111901
$$116$$ 980.000 0.784403
$$117$$ 0 0
$$118$$ 816.000 0.636601
$$119$$ −640.000 −0.493014
$$120$$ 0 0
$$121$$ −175.000 −0.131480
$$122$$ 1644.00 1.22001
$$123$$ 0 0
$$124$$ −412.000 −0.298377
$$125$$ −1284.00 −0.918756
$$126$$ 0 0
$$127$$ 261.000 0.182362 0.0911811 0.995834i $$-0.470936\pi$$
0.0911811 + 0.995834i $$0.470936\pi$$
$$128$$ −384.000 −0.265165
$$129$$ 0 0
$$130$$ −684.000 −0.461467
$$131$$ 1441.00 0.961074 0.480537 0.876974i $$-0.340442\pi$$
0.480537 + 0.876974i $$0.340442\pi$$
$$132$$ 0 0
$$133$$ 560.000 0.365099
$$134$$ 1852.00 1.19394
$$135$$ 0 0
$$136$$ −1920.00 −1.21058
$$137$$ −1556.00 −0.970351 −0.485175 0.874417i $$-0.661244\pi$$
−0.485175 + 0.874417i $$0.661244\pi$$
$$138$$ 0 0
$$139$$ 25.0000 0.0152552 0.00762760 0.999971i $$-0.497572\pi$$
0.00762760 + 0.999971i $$0.497572\pi$$
$$140$$ 192.000 0.115907
$$141$$ 0 0
$$142$$ −670.000 −0.395952
$$143$$ 1938.00 1.13331
$$144$$ 0 0
$$145$$ −1470.00 −0.841909
$$146$$ −1798.00 −1.01920
$$147$$ 0 0
$$148$$ 1192.00 0.662039
$$149$$ −822.000 −0.451952 −0.225976 0.974133i $$-0.572557\pi$$
−0.225976 + 0.974133i $$0.572557\pi$$
$$150$$ 0 0
$$151$$ −1489.00 −0.802471 −0.401235 0.915975i $$-0.631419\pi$$
−0.401235 + 0.915975i $$0.631419\pi$$
$$152$$ 1680.00 0.896487
$$153$$ 0 0
$$154$$ 544.000 0.284654
$$155$$ 618.000 0.320251
$$156$$ 0 0
$$157$$ −632.000 −0.321268 −0.160634 0.987014i $$-0.551354\pi$$
−0.160634 + 0.987014i $$0.551354\pi$$
$$158$$ −2644.00 −1.33130
$$159$$ 0 0
$$160$$ 960.000 0.474342
$$161$$ 184.000 0.0900698
$$162$$ 0 0
$$163$$ −3043.00 −1.46225 −0.731123 0.682245i $$-0.761004\pi$$
−0.731123 + 0.682245i $$0.761004\pi$$
$$164$$ 380.000 0.180933
$$165$$ 0 0
$$166$$ 72.0000 0.0336644
$$167$$ 2224.00 1.03053 0.515264 0.857031i $$-0.327694\pi$$
0.515264 + 0.857031i $$0.327694\pi$$
$$168$$ 0 0
$$169$$ 1052.00 0.478835
$$170$$ 960.000 0.433110
$$171$$ 0 0
$$172$$ −352.000 −0.156045
$$173$$ −3230.00 −1.41949 −0.709747 0.704457i $$-0.751191\pi$$
−0.709747 + 0.704457i $$0.751191\pi$$
$$174$$ 0 0
$$175$$ 712.000 0.307555
$$176$$ 544.000 0.232986
$$177$$ 0 0
$$178$$ 920.000 0.387398
$$179$$ −369.000 −0.154080 −0.0770401 0.997028i $$-0.524547\pi$$
−0.0770401 + 0.997028i $$0.524547\pi$$
$$180$$ 0 0
$$181$$ −1370.00 −0.562604 −0.281302 0.959619i $$-0.590766\pi$$
−0.281302 + 0.959619i $$0.590766\pi$$
$$182$$ 912.000 0.371439
$$183$$ 0 0
$$184$$ 552.000 0.221163
$$185$$ −1788.00 −0.710575
$$186$$ 0 0
$$187$$ −2720.00 −1.06367
$$188$$ −1428.00 −0.553977
$$189$$ 0 0
$$190$$ −840.000 −0.320737
$$191$$ −4410.00 −1.67066 −0.835331 0.549747i $$-0.814724\pi$$
−0.835331 + 0.549747i $$0.814724\pi$$
$$192$$ 0 0
$$193$$ −135.000 −0.0503498 −0.0251749 0.999683i $$-0.508014\pi$$
−0.0251749 + 0.999683i $$0.508014\pi$$
$$194$$ −1928.00 −0.713517
$$195$$ 0 0
$$196$$ 1116.00 0.406706
$$197$$ −1221.00 −0.441587 −0.220794 0.975321i $$-0.570865\pi$$
−0.220794 + 0.975321i $$0.570865\pi$$
$$198$$ 0 0
$$199$$ −1098.00 −0.391131 −0.195566 0.980691i $$-0.562654\pi$$
−0.195566 + 0.980691i $$0.562654\pi$$
$$200$$ 2136.00 0.755190
$$201$$ 0 0
$$202$$ 620.000 0.215956
$$203$$ 1960.00 0.677660
$$204$$ 0 0
$$205$$ −570.000 −0.194198
$$206$$ 2088.00 0.706203
$$207$$ 0 0
$$208$$ 912.000 0.304018
$$209$$ 2380.00 0.787694
$$210$$ 0 0
$$211$$ −3676.00 −1.19937 −0.599683 0.800238i $$-0.704707\pi$$
−0.599683 + 0.800238i $$0.704707\pi$$
$$212$$ −1656.00 −0.536484
$$213$$ 0 0
$$214$$ −828.000 −0.264490
$$215$$ 528.000 0.167485
$$216$$ 0 0
$$217$$ −824.000 −0.257773
$$218$$ 1408.00 0.437439
$$219$$ 0 0
$$220$$ 816.000 0.250067
$$221$$ −4560.00 −1.38796
$$222$$ 0 0
$$223$$ 1656.00 0.497282 0.248641 0.968596i $$-0.420016\pi$$
0.248641 + 0.968596i $$0.420016\pi$$
$$224$$ −1280.00 −0.381802
$$225$$ 0 0
$$226$$ −1904.00 −0.560408
$$227$$ −2940.00 −0.859624 −0.429812 0.902918i $$-0.641420\pi$$
−0.429812 + 0.902918i $$0.641420\pi$$
$$228$$ 0 0
$$229$$ 3612.00 1.04230 0.521152 0.853464i $$-0.325502\pi$$
0.521152 + 0.853464i $$0.325502\pi$$
$$230$$ −276.000 −0.0791257
$$231$$ 0 0
$$232$$ 5880.00 1.66397
$$233$$ 4325.00 1.21605 0.608026 0.793917i $$-0.291962\pi$$
0.608026 + 0.793917i $$0.291962\pi$$
$$234$$ 0 0
$$235$$ 2142.00 0.594590
$$236$$ −1632.00 −0.450145
$$237$$ 0 0
$$238$$ −1280.00 −0.348614
$$239$$ −2735.00 −0.740219 −0.370110 0.928988i $$-0.620680\pi$$
−0.370110 + 0.928988i $$0.620680\pi$$
$$240$$ 0 0
$$241$$ −6710.00 −1.79348 −0.896741 0.442556i $$-0.854072\pi$$
−0.896741 + 0.442556i $$0.854072\pi$$
$$242$$ −350.000 −0.0929705
$$243$$ 0 0
$$244$$ −3288.00 −0.862675
$$245$$ −1674.00 −0.436522
$$246$$ 0 0
$$247$$ 3990.00 1.02784
$$248$$ −2472.00 −0.632952
$$249$$ 0 0
$$250$$ −2568.00 −0.649658
$$251$$ 6948.00 1.74723 0.873613 0.486621i $$-0.161771\pi$$
0.873613 + 0.486621i $$0.161771\pi$$
$$252$$ 0 0
$$253$$ 782.000 0.194324
$$254$$ 522.000 0.128950
$$255$$ 0 0
$$256$$ −4352.00 −1.06250
$$257$$ 4929.00 1.19635 0.598176 0.801365i $$-0.295892\pi$$
0.598176 + 0.801365i $$0.295892\pi$$
$$258$$ 0 0
$$259$$ 2384.00 0.571948
$$260$$ 1368.00 0.326307
$$261$$ 0 0
$$262$$ 2882.00 0.679582
$$263$$ −6138.00 −1.43911 −0.719554 0.694437i $$-0.755654\pi$$
−0.719554 + 0.694437i $$0.755654\pi$$
$$264$$ 0 0
$$265$$ 2484.00 0.575815
$$266$$ 1120.00 0.258164
$$267$$ 0 0
$$268$$ −3704.00 −0.844246
$$269$$ 2063.00 0.467596 0.233798 0.972285i $$-0.424885\pi$$
0.233798 + 0.972285i $$0.424885\pi$$
$$270$$ 0 0
$$271$$ −1064.00 −0.238500 −0.119250 0.992864i $$-0.538049\pi$$
−0.119250 + 0.992864i $$0.538049\pi$$
$$272$$ −1280.00 −0.285336
$$273$$ 0 0
$$274$$ −3112.00 −0.686142
$$275$$ 3026.00 0.663544
$$276$$ 0 0
$$277$$ 5729.00 1.24268 0.621340 0.783541i $$-0.286589\pi$$
0.621340 + 0.783541i $$0.286589\pi$$
$$278$$ 50.0000 0.0107871
$$279$$ 0 0
$$280$$ 1152.00 0.245876
$$281$$ 960.000 0.203804 0.101902 0.994794i $$-0.467507\pi$$
0.101902 + 0.994794i $$0.467507\pi$$
$$282$$ 0 0
$$283$$ −114.000 −0.0239456 −0.0119728 0.999928i $$-0.503811\pi$$
−0.0119728 + 0.999928i $$0.503811\pi$$
$$284$$ 1340.00 0.279980
$$285$$ 0 0
$$286$$ 3876.00 0.801373
$$287$$ 760.000 0.156311
$$288$$ 0 0
$$289$$ 1487.00 0.302666
$$290$$ −2940.00 −0.595320
$$291$$ 0 0
$$292$$ 3596.00 0.720685
$$293$$ 7048.00 1.40529 0.702643 0.711543i $$-0.252003\pi$$
0.702643 + 0.711543i $$0.252003\pi$$
$$294$$ 0 0
$$295$$ 2448.00 0.483146
$$296$$ 7152.00 1.40440
$$297$$ 0 0
$$298$$ −1644.00 −0.319578
$$299$$ 1311.00 0.253569
$$300$$ 0 0
$$301$$ −704.000 −0.134810
$$302$$ −2978.00 −0.567433
$$303$$ 0 0
$$304$$ 1120.00 0.211304
$$305$$ 4932.00 0.925920
$$306$$ 0 0
$$307$$ 3872.00 0.719826 0.359913 0.932986i $$-0.382806\pi$$
0.359913 + 0.932986i $$0.382806\pi$$
$$308$$ −1088.00 −0.201281
$$309$$ 0 0
$$310$$ 1236.00 0.226452
$$311$$ 4977.00 0.907459 0.453730 0.891139i $$-0.350093\pi$$
0.453730 + 0.891139i $$0.350093\pi$$
$$312$$ 0 0
$$313$$ −2536.00 −0.457965 −0.228983 0.973430i $$-0.573540\pi$$
−0.228983 + 0.973430i $$0.573540\pi$$
$$314$$ −1264.00 −0.227171
$$315$$ 0 0
$$316$$ 5288.00 0.941371
$$317$$ −1434.00 −0.254074 −0.127037 0.991898i $$-0.540547\pi$$
−0.127037 + 0.991898i $$0.540547\pi$$
$$318$$ 0 0
$$319$$ 8330.00 1.46204
$$320$$ 2688.00 0.469574
$$321$$ 0 0
$$322$$ 368.000 0.0636889
$$323$$ −5600.00 −0.964682
$$324$$ 0 0
$$325$$ 5073.00 0.865844
$$326$$ −6086.00 −1.03396
$$327$$ 0 0
$$328$$ 2280.00 0.383817
$$329$$ −2856.00 −0.478591
$$330$$ 0 0
$$331$$ 5469.00 0.908167 0.454084 0.890959i $$-0.349967\pi$$
0.454084 + 0.890959i $$0.349967\pi$$
$$332$$ −144.000 −0.0238043
$$333$$ 0 0
$$334$$ 4448.00 0.728694
$$335$$ 5556.00 0.906139
$$336$$ 0 0
$$337$$ −7796.00 −1.26016 −0.630082 0.776529i $$-0.716979\pi$$
−0.630082 + 0.776529i $$0.716979\pi$$
$$338$$ 2104.00 0.338587
$$339$$ 0 0
$$340$$ −1920.00 −0.306255
$$341$$ −3502.00 −0.556141
$$342$$ 0 0
$$343$$ 4976.00 0.783320
$$344$$ −2112.00 −0.331022
$$345$$ 0 0
$$346$$ −6460.00 −1.00373
$$347$$ 10068.0 1.55758 0.778788 0.627288i $$-0.215835\pi$$
0.778788 + 0.627288i $$0.215835\pi$$
$$348$$ 0 0
$$349$$ −7495.00 −1.14956 −0.574782 0.818306i $$-0.694913\pi$$
−0.574782 + 0.818306i $$0.694913\pi$$
$$350$$ 1424.00 0.217474
$$351$$ 0 0
$$352$$ −5440.00 −0.823730
$$353$$ −10617.0 −1.60081 −0.800405 0.599460i $$-0.795382\pi$$
−0.800405 + 0.599460i $$0.795382\pi$$
$$354$$ 0 0
$$355$$ −2010.00 −0.300506
$$356$$ −1840.00 −0.273932
$$357$$ 0 0
$$358$$ −738.000 −0.108951
$$359$$ −2522.00 −0.370769 −0.185384 0.982666i $$-0.559353\pi$$
−0.185384 + 0.982666i $$0.559353\pi$$
$$360$$ 0 0
$$361$$ −1959.00 −0.285610
$$362$$ −2740.00 −0.397821
$$363$$ 0 0
$$364$$ −1824.00 −0.262647
$$365$$ −5394.00 −0.773520
$$366$$ 0 0
$$367$$ 7204.00 1.02465 0.512324 0.858792i $$-0.328785\pi$$
0.512324 + 0.858792i $$0.328785\pi$$
$$368$$ 368.000 0.0521286
$$369$$ 0 0
$$370$$ −3576.00 −0.502452
$$371$$ −3312.00 −0.463478
$$372$$ 0 0
$$373$$ −13310.0 −1.84763 −0.923815 0.382840i $$-0.874946\pi$$
−0.923815 + 0.382840i $$0.874946\pi$$
$$374$$ −5440.00 −0.752128
$$375$$ 0 0
$$376$$ −8568.00 −1.17516
$$377$$ 13965.0 1.90778
$$378$$ 0 0
$$379$$ 12952.0 1.75541 0.877704 0.479203i $$-0.159074\pi$$
0.877704 + 0.479203i $$0.159074\pi$$
$$380$$ 1680.00 0.226795
$$381$$ 0 0
$$382$$ −8820.00 −1.18134
$$383$$ 2812.00 0.375161 0.187580 0.982249i $$-0.439936\pi$$
0.187580 + 0.982249i $$0.439936\pi$$
$$384$$ 0 0
$$385$$ 1632.00 0.216037
$$386$$ −270.000 −0.0356027
$$387$$ 0 0
$$388$$ 3856.00 0.504533
$$389$$ −1264.00 −0.164749 −0.0823745 0.996601i $$-0.526250\pi$$
−0.0823745 + 0.996601i $$0.526250\pi$$
$$390$$ 0 0
$$391$$ −1840.00 −0.237987
$$392$$ 6696.00 0.862753
$$393$$ 0 0
$$394$$ −2442.00 −0.312249
$$395$$ −7932.00 −1.01039
$$396$$ 0 0
$$397$$ 7119.00 0.899981 0.449990 0.893033i $$-0.351427\pi$$
0.449990 + 0.893033i $$0.351427\pi$$
$$398$$ −2196.00 −0.276572
$$399$$ 0 0
$$400$$ 1424.00 0.178000
$$401$$ −4262.00 −0.530758 −0.265379 0.964144i $$-0.585497\pi$$
−0.265379 + 0.964144i $$0.585497\pi$$
$$402$$ 0 0
$$403$$ −5871.00 −0.725696
$$404$$ −1240.00 −0.152704
$$405$$ 0 0
$$406$$ 3920.00 0.479178
$$407$$ 10132.0 1.23397
$$408$$ 0 0
$$409$$ 229.000 0.0276854 0.0138427 0.999904i $$-0.495594\pi$$
0.0138427 + 0.999904i $$0.495594\pi$$
$$410$$ −1140.00 −0.137319
$$411$$ 0 0
$$412$$ −4176.00 −0.499361
$$413$$ −3264.00 −0.388888
$$414$$ 0 0
$$415$$ 216.000 0.0255495
$$416$$ −9120.00 −1.07487
$$417$$ 0 0
$$418$$ 4760.00 0.556984
$$419$$ −15776.0 −1.83940 −0.919699 0.392623i $$-0.871568\pi$$
−0.919699 + 0.392623i $$0.871568\pi$$
$$420$$ 0 0
$$421$$ −8728.00 −1.01040 −0.505198 0.863003i $$-0.668581\pi$$
−0.505198 + 0.863003i $$0.668581\pi$$
$$422$$ −7352.00 −0.848080
$$423$$ 0 0
$$424$$ −9936.00 −1.13805
$$425$$ −7120.00 −0.812637
$$426$$ 0 0
$$427$$ −6576.00 −0.745281
$$428$$ 1656.00 0.187023
$$429$$ 0 0
$$430$$ 1056.00 0.118430
$$431$$ 2928.00 0.327232 0.163616 0.986524i $$-0.447684\pi$$
0.163616 + 0.986524i $$0.447684\pi$$
$$432$$ 0 0
$$433$$ −5314.00 −0.589780 −0.294890 0.955531i $$-0.595283\pi$$
−0.294890 + 0.955531i $$0.595283\pi$$
$$434$$ −1648.00 −0.182273
$$435$$ 0 0
$$436$$ −2816.00 −0.309316
$$437$$ 1610.00 0.176240
$$438$$ 0 0
$$439$$ 2585.00 0.281037 0.140519 0.990078i $$-0.455123\pi$$
0.140519 + 0.990078i $$0.455123\pi$$
$$440$$ 4896.00 0.530472
$$441$$ 0 0
$$442$$ −9120.00 −0.981435
$$443$$ 2997.00 0.321426 0.160713 0.987001i $$-0.448621\pi$$
0.160713 + 0.987001i $$0.448621\pi$$
$$444$$ 0 0
$$445$$ 2760.00 0.294015
$$446$$ 3312.00 0.351632
$$447$$ 0 0
$$448$$ −3584.00 −0.377964
$$449$$ 16562.0 1.74078 0.870389 0.492365i $$-0.163868\pi$$
0.870389 + 0.492365i $$0.163868\pi$$
$$450$$ 0 0
$$451$$ 3230.00 0.337239
$$452$$ 3808.00 0.396268
$$453$$ 0 0
$$454$$ −5880.00 −0.607846
$$455$$ 2736.00 0.281903
$$456$$ 0 0
$$457$$ 3924.00 0.401656 0.200828 0.979626i $$-0.435637\pi$$
0.200828 + 0.979626i $$0.435637\pi$$
$$458$$ 7224.00 0.737020
$$459$$ 0 0
$$460$$ 552.000 0.0559503
$$461$$ 4543.00 0.458977 0.229489 0.973311i $$-0.426295\pi$$
0.229489 + 0.973311i $$0.426295\pi$$
$$462$$ 0 0
$$463$$ 9616.00 0.965213 0.482606 0.875837i $$-0.339690\pi$$
0.482606 + 0.875837i $$0.339690\pi$$
$$464$$ 3920.00 0.392201
$$465$$ 0 0
$$466$$ 8650.00 0.859879
$$467$$ −7826.00 −0.775469 −0.387735 0.921771i $$-0.626742\pi$$
−0.387735 + 0.921771i $$0.626742\pi$$
$$468$$ 0 0
$$469$$ −7408.00 −0.729360
$$470$$ 4284.00 0.420439
$$471$$ 0 0
$$472$$ −9792.00 −0.954901
$$473$$ −2992.00 −0.290851
$$474$$ 0 0
$$475$$ 6230.00 0.601794
$$476$$ 2560.00 0.246507
$$477$$ 0 0
$$478$$ −5470.00 −0.523414
$$479$$ −11404.0 −1.08781 −0.543906 0.839146i $$-0.683055\pi$$
−0.543906 + 0.839146i $$0.683055\pi$$
$$480$$ 0 0
$$481$$ 16986.0 1.61018
$$482$$ −13420.0 −1.26818
$$483$$ 0 0
$$484$$ 700.000 0.0657400
$$485$$ −5784.00 −0.541521
$$486$$ 0 0
$$487$$ −9267.00 −0.862275 −0.431137 0.902286i $$-0.641888\pi$$
−0.431137 + 0.902286i $$0.641888\pi$$
$$488$$ −19728.0 −1.83001
$$489$$ 0 0
$$490$$ −3348.00 −0.308668
$$491$$ 18191.0 1.67199 0.835996 0.548735i $$-0.184890\pi$$
0.835996 + 0.548735i $$0.184890\pi$$
$$492$$ 0 0
$$493$$ −19600.0 −1.79055
$$494$$ 7980.00 0.726796
$$495$$ 0 0
$$496$$ −1648.00 −0.149188
$$497$$ 2680.00 0.241880
$$498$$ 0 0
$$499$$ 19315.0 1.73278 0.866391 0.499366i $$-0.166434\pi$$
0.866391 + 0.499366i $$0.166434\pi$$
$$500$$ 5136.00 0.459378
$$501$$ 0 0
$$502$$ 13896.0 1.23548
$$503$$ −8422.00 −0.746557 −0.373279 0.927719i $$-0.621766\pi$$
−0.373279 + 0.927719i $$0.621766\pi$$
$$504$$ 0 0
$$505$$ 1860.00 0.163899
$$506$$ 1564.00 0.137408
$$507$$ 0 0
$$508$$ −1044.00 −0.0911811
$$509$$ 863.000 0.0751509 0.0375754 0.999294i $$-0.488037\pi$$
0.0375754 + 0.999294i $$0.488037\pi$$
$$510$$ 0 0
$$511$$ 7192.00 0.622613
$$512$$ −5632.00 −0.486136
$$513$$ 0 0
$$514$$ 9858.00 0.845949
$$515$$ 6264.00 0.535971
$$516$$ 0 0
$$517$$ −12138.0 −1.03255
$$518$$ 4768.00 0.404428
$$519$$ 0 0
$$520$$ 8208.00 0.692201
$$521$$ −19260.0 −1.61957 −0.809785 0.586727i $$-0.800416\pi$$
−0.809785 + 0.586727i $$0.800416\pi$$
$$522$$ 0 0
$$523$$ −11740.0 −0.981557 −0.490779 0.871284i $$-0.663288\pi$$
−0.490779 + 0.871284i $$0.663288\pi$$
$$524$$ −5764.00 −0.480537
$$525$$ 0 0
$$526$$ −12276.0 −1.01760
$$527$$ 8240.00 0.681101
$$528$$ 0 0
$$529$$ 529.000 0.0434783
$$530$$ 4968.00 0.407163
$$531$$ 0 0
$$532$$ −2240.00 −0.182549
$$533$$ 5415.00 0.440056
$$534$$ 0 0
$$535$$ −2484.00 −0.200734
$$536$$ −22224.0 −1.79092
$$537$$ 0 0
$$538$$ 4126.00 0.330640
$$539$$ 9486.00 0.758054
$$540$$ 0 0
$$541$$ 17741.0 1.40988 0.704940 0.709267i $$-0.250974\pi$$
0.704940 + 0.709267i $$0.250974\pi$$
$$542$$ −2128.00 −0.168645
$$543$$ 0 0
$$544$$ 12800.0 1.00882
$$545$$ 4224.00 0.331993
$$546$$ 0 0
$$547$$ −6571.00 −0.513630 −0.256815 0.966461i $$-0.582673\pi$$
−0.256815 + 0.966461i $$0.582673\pi$$
$$548$$ 6224.00 0.485175
$$549$$ 0 0
$$550$$ 6052.00 0.469197
$$551$$ 17150.0 1.32598
$$552$$ 0 0
$$553$$ 10576.0 0.813268
$$554$$ 11458.0 0.878707
$$555$$ 0 0
$$556$$ −100.000 −0.00762760
$$557$$ 1372.00 0.104369 0.0521845 0.998637i $$-0.483382\pi$$
0.0521845 + 0.998637i $$0.483382\pi$$
$$558$$ 0 0
$$559$$ −5016.00 −0.379524
$$560$$ 768.000 0.0579534
$$561$$ 0 0
$$562$$ 1920.00 0.144111
$$563$$ −4332.00 −0.324284 −0.162142 0.986767i $$-0.551840\pi$$
−0.162142 + 0.986767i $$0.551840\pi$$
$$564$$ 0 0
$$565$$ −5712.00 −0.425320
$$566$$ −228.000 −0.0169321
$$567$$ 0 0
$$568$$ 8040.00 0.593928
$$569$$ 3546.00 0.261258 0.130629 0.991431i $$-0.458300\pi$$
0.130629 + 0.991431i $$0.458300\pi$$
$$570$$ 0 0
$$571$$ −6160.00 −0.451468 −0.225734 0.974189i $$-0.572478\pi$$
−0.225734 + 0.974189i $$0.572478\pi$$
$$572$$ −7752.00 −0.566656
$$573$$ 0 0
$$574$$ 1520.00 0.110529
$$575$$ 2047.00 0.148462
$$576$$ 0 0
$$577$$ 2953.00 0.213059 0.106529 0.994310i $$-0.466026\pi$$
0.106529 + 0.994310i $$0.466026\pi$$
$$578$$ 2974.00 0.214017
$$579$$ 0 0
$$580$$ 5880.00 0.420955
$$581$$ −288.000 −0.0205650
$$582$$ 0 0
$$583$$ −14076.0 −0.999946
$$584$$ 21576.0 1.52880
$$585$$ 0 0
$$586$$ 14096.0 0.993687
$$587$$ 2949.00 0.207356 0.103678 0.994611i $$-0.466939\pi$$
0.103678 + 0.994611i $$0.466939\pi$$
$$588$$ 0 0
$$589$$ −7210.00 −0.504385
$$590$$ 4896.00 0.341636
$$591$$ 0 0
$$592$$ 4768.00 0.331020
$$593$$ −16390.0 −1.13500 −0.567501 0.823372i $$-0.692090\pi$$
−0.567501 + 0.823372i $$0.692090\pi$$
$$594$$ 0 0
$$595$$ −3840.00 −0.264579
$$596$$ 3288.00 0.225976
$$597$$ 0 0
$$598$$ 2622.00 0.179300
$$599$$ 12920.0 0.881297 0.440648 0.897680i $$-0.354749\pi$$
0.440648 + 0.897680i $$0.354749\pi$$
$$600$$ 0 0
$$601$$ −13835.0 −0.939004 −0.469502 0.882931i $$-0.655567\pi$$
−0.469502 + 0.882931i $$0.655567\pi$$
$$602$$ −1408.00 −0.0953252
$$603$$ 0 0
$$604$$ 5956.00 0.401235
$$605$$ −1050.00 −0.0705596
$$606$$ 0 0
$$607$$ 6004.00 0.401474 0.200737 0.979645i $$-0.435666\pi$$
0.200737 + 0.979645i $$0.435666\pi$$
$$608$$ −11200.0 −0.747072
$$609$$ 0 0
$$610$$ 9864.00 0.654724
$$611$$ −20349.0 −1.34735
$$612$$ 0 0
$$613$$ −16416.0 −1.08162 −0.540812 0.841143i $$-0.681883\pi$$
−0.540812 + 0.841143i $$0.681883\pi$$
$$614$$ 7744.00 0.508994
$$615$$ 0 0
$$616$$ −6528.00 −0.426982
$$617$$ −3786.00 −0.247032 −0.123516 0.992343i $$-0.539417\pi$$
−0.123516 + 0.992343i $$0.539417\pi$$
$$618$$ 0 0
$$619$$ 15824.0 1.02750 0.513748 0.857941i $$-0.328257\pi$$
0.513748 + 0.857941i $$0.328257\pi$$
$$620$$ −2472.00 −0.160126
$$621$$ 0 0
$$622$$ 9954.00 0.641670
$$623$$ −3680.00 −0.236655
$$624$$ 0 0
$$625$$ 3421.00 0.218944
$$626$$ −5072.00 −0.323830
$$627$$ 0 0
$$628$$ 2528.00 0.160634
$$629$$ −23840.0 −1.51123
$$630$$ 0 0
$$631$$ 17852.0 1.12627 0.563135 0.826365i $$-0.309595\pi$$
0.563135 + 0.826365i $$0.309595\pi$$
$$632$$ 31728.0 1.99695
$$633$$ 0 0
$$634$$ −2868.00 −0.179657
$$635$$ 1566.00 0.0978658
$$636$$ 0 0
$$637$$ 15903.0 0.989168
$$638$$ 16660.0 1.03382
$$639$$ 0 0
$$640$$ −2304.00 −0.142302
$$641$$ −10324.0 −0.636152 −0.318076 0.948065i $$-0.603037\pi$$
−0.318076 + 0.948065i $$0.603037\pi$$
$$642$$ 0 0
$$643$$ −14702.0 −0.901696 −0.450848 0.892601i $$-0.648878\pi$$
−0.450848 + 0.892601i $$0.648878\pi$$
$$644$$ −736.000 −0.0450349
$$645$$ 0 0
$$646$$ −11200.0 −0.682133
$$647$$ −11939.0 −0.725457 −0.362728 0.931895i $$-0.618155\pi$$
−0.362728 + 0.931895i $$0.618155\pi$$
$$648$$ 0 0
$$649$$ −13872.0 −0.839019
$$650$$ 10146.0 0.612244
$$651$$ 0 0
$$652$$ 12172.0 0.731123
$$653$$ −6159.00 −0.369097 −0.184548 0.982823i $$-0.559082\pi$$
−0.184548 + 0.982823i $$0.559082\pi$$
$$654$$ 0 0
$$655$$ 8646.00 0.515767
$$656$$ 1520.00 0.0904665
$$657$$ 0 0
$$658$$ −5712.00 −0.338415
$$659$$ 21692.0 1.28225 0.641123 0.767438i $$-0.278469\pi$$
0.641123 + 0.767438i $$0.278469\pi$$
$$660$$ 0 0
$$661$$ 16502.0 0.971034 0.485517 0.874227i $$-0.338631\pi$$
0.485517 + 0.874227i $$0.338631\pi$$
$$662$$ 10938.0 0.642171
$$663$$ 0 0
$$664$$ −864.000 −0.0504965
$$665$$ 3360.00 0.195933
$$666$$ 0 0
$$667$$ 5635.00 0.327119
$$668$$ −8896.00 −0.515264
$$669$$ 0 0
$$670$$ 11112.0 0.640737
$$671$$ −27948.0 −1.60793
$$672$$ 0 0
$$673$$ −27733.0 −1.58845 −0.794226 0.607622i $$-0.792124\pi$$
−0.794226 + 0.607622i $$0.792124\pi$$
$$674$$ −15592.0 −0.891070
$$675$$ 0 0
$$676$$ −4208.00 −0.239417
$$677$$ 8814.00 0.500369 0.250184 0.968198i $$-0.419509\pi$$
0.250184 + 0.968198i $$0.419509\pi$$
$$678$$ 0 0
$$679$$ 7712.00 0.435875
$$680$$ −11520.0 −0.649664
$$681$$ 0 0
$$682$$ −7004.00 −0.393251
$$683$$ 22999.0 1.28848 0.644240 0.764823i $$-0.277174\pi$$
0.644240 + 0.764823i $$0.277174\pi$$
$$684$$ 0 0
$$685$$ −9336.00 −0.520745
$$686$$ 9952.00 0.553891
$$687$$ 0 0
$$688$$ −1408.00 −0.0780225
$$689$$ −23598.0 −1.30481
$$690$$ 0 0
$$691$$ −12140.0 −0.668346 −0.334173 0.942512i $$-0.608457\pi$$
−0.334173 + 0.942512i $$0.608457\pi$$
$$692$$ 12920.0 0.709747
$$693$$ 0 0
$$694$$ 20136.0 1.10137
$$695$$ 150.000 0.00818680
$$696$$ 0 0
$$697$$ −7600.00 −0.413014
$$698$$ −14990.0 −0.812865
$$699$$ 0 0
$$700$$ −2848.00 −0.153778
$$701$$ 20024.0 1.07888 0.539441 0.842024i $$-0.318636\pi$$
0.539441 + 0.842024i $$0.318636\pi$$
$$702$$ 0 0
$$703$$ 20860.0 1.11913
$$704$$ −15232.0 −0.815451
$$705$$ 0 0
$$706$$ −21234.0 −1.13194
$$707$$ −2480.00 −0.131924
$$708$$ 0 0
$$709$$ −4956.00 −0.262520 −0.131260 0.991348i $$-0.541902\pi$$
−0.131260 + 0.991348i $$0.541902\pi$$
$$710$$ −4020.00 −0.212490
$$711$$ 0 0
$$712$$ −11040.0 −0.581098
$$713$$ −2369.00 −0.124432
$$714$$ 0 0
$$715$$ 11628.0 0.608199
$$716$$ 1476.00 0.0770401
$$717$$ 0 0
$$718$$ −5044.00 −0.262173
$$719$$ −2760.00 −0.143158 −0.0715790 0.997435i $$-0.522804\pi$$
−0.0715790 + 0.997435i $$0.522804\pi$$
$$720$$ 0 0
$$721$$ −8352.00 −0.431407
$$722$$ −3918.00 −0.201957
$$723$$ 0 0
$$724$$ 5480.00 0.281302
$$725$$ 21805.0 1.11699
$$726$$ 0 0
$$727$$ 7746.00 0.395163 0.197581 0.980287i $$-0.436691\pi$$
0.197581 + 0.980287i $$0.436691\pi$$
$$728$$ −10944.0 −0.557159
$$729$$ 0 0
$$730$$ −10788.0 −0.546961
$$731$$ 7040.00 0.356202
$$732$$ 0 0
$$733$$ −11976.0 −0.603470 −0.301735 0.953392i $$-0.597566\pi$$
−0.301735 + 0.953392i $$0.597566\pi$$
$$734$$ 14408.0 0.724535
$$735$$ 0 0
$$736$$ −3680.00 −0.184302
$$737$$ −31484.0 −1.57358
$$738$$ 0 0
$$739$$ 15057.0 0.749500 0.374750 0.927126i $$-0.377728\pi$$
0.374750 + 0.927126i $$0.377728\pi$$
$$740$$ 7152.00 0.355287
$$741$$ 0 0
$$742$$ −6624.00 −0.327729
$$743$$ −18532.0 −0.915038 −0.457519 0.889200i $$-0.651262\pi$$
−0.457519 + 0.889200i $$0.651262\pi$$
$$744$$ 0 0
$$745$$ −4932.00 −0.242543
$$746$$ −26620.0 −1.30647
$$747$$ 0 0
$$748$$ 10880.0 0.531834
$$749$$ 3312.00 0.161573
$$750$$ 0 0
$$751$$ −192.000 −0.00932913 −0.00466457 0.999989i $$-0.501485\pi$$
−0.00466457 + 0.999989i $$0.501485\pi$$
$$752$$ −5712.00 −0.276988
$$753$$ 0 0
$$754$$ 27930.0 1.34901
$$755$$ −8934.00 −0.430651
$$756$$ 0 0
$$757$$ −9830.00 −0.471965 −0.235982 0.971757i $$-0.575831\pi$$
−0.235982 + 0.971757i $$0.575831\pi$$
$$758$$ 25904.0 1.24126
$$759$$ 0 0
$$760$$ 10080.0 0.481105
$$761$$ 30219.0 1.43947 0.719736 0.694248i $$-0.244263\pi$$
0.719736 + 0.694248i $$0.244263\pi$$
$$762$$ 0 0
$$763$$ −5632.00 −0.267224
$$764$$ 17640.0 0.835331
$$765$$ 0 0
$$766$$ 5624.00 0.265279
$$767$$ −23256.0 −1.09482
$$768$$ 0 0
$$769$$ 1122.00 0.0526142 0.0263071 0.999654i $$-0.491625\pi$$
0.0263071 + 0.999654i $$0.491625\pi$$
$$770$$ 3264.00 0.152762
$$771$$ 0 0
$$772$$ 540.000 0.0251749
$$773$$ −19300.0 −0.898024 −0.449012 0.893526i $$-0.648224\pi$$
−0.449012 + 0.893526i $$0.648224\pi$$
$$774$$ 0 0
$$775$$ −9167.00 −0.424888
$$776$$ 23136.0 1.07028
$$777$$ 0 0
$$778$$ −2528.00 −0.116495
$$779$$ 6650.00 0.305855
$$780$$ 0 0
$$781$$ 11390.0 0.521852
$$782$$ −3680.00 −0.168282
$$783$$ 0 0
$$784$$ 4464.00 0.203353
$$785$$ −3792.00 −0.172411
$$786$$ 0 0
$$787$$ −19396.0 −0.878517 −0.439258 0.898361i $$-0.644759\pi$$
−0.439258 + 0.898361i $$0.644759\pi$$
$$788$$ 4884.00 0.220794
$$789$$ 0 0
$$790$$ −15864.0 −0.714450
$$791$$ 7616.00 0.342344
$$792$$ 0 0
$$793$$ −46854.0 −2.09815
$$794$$ 14238.0 0.636383
$$795$$ 0 0
$$796$$ 4392.00 0.195566
$$797$$ 39034.0 1.73482 0.867412 0.497590i $$-0.165782\pi$$
0.867412 + 0.497590i $$0.165782\pi$$
$$798$$ 0 0
$$799$$ 28560.0 1.26456
$$800$$ −14240.0 −0.629325
$$801$$ 0 0
$$802$$ −8524.00 −0.375303
$$803$$ 30566.0 1.34328
$$804$$ 0 0
$$805$$ 1104.00 0.0483365
$$806$$ −11742.0 −0.513144
$$807$$ 0 0
$$808$$ −7440.00 −0.323934
$$809$$ 10310.0 0.448060 0.224030 0.974582i $$-0.428079\pi$$
0.224030 + 0.974582i $$0.428079\pi$$
$$810$$ 0 0
$$811$$ −40693.0 −1.76193 −0.880965 0.473182i $$-0.843105\pi$$
−0.880965 + 0.473182i $$0.843105\pi$$
$$812$$ −7840.00 −0.338830
$$813$$ 0 0
$$814$$ 20264.0 0.872546
$$815$$ −18258.0 −0.784724
$$816$$ 0 0
$$817$$ −6160.00 −0.263784
$$818$$ 458.000 0.0195765
$$819$$ 0 0
$$820$$ 2280.00 0.0970988
$$821$$ 13934.0 0.592326 0.296163 0.955137i $$-0.404293\pi$$
0.296163 + 0.955137i $$0.404293\pi$$
$$822$$ 0 0
$$823$$ 6175.00 0.261539 0.130770 0.991413i $$-0.458255\pi$$
0.130770 + 0.991413i $$0.458255\pi$$
$$824$$ −25056.0 −1.05930
$$825$$ 0 0
$$826$$ −6528.00 −0.274986
$$827$$ −28664.0 −1.20525 −0.602627 0.798023i $$-0.705879\pi$$
−0.602627 + 0.798023i $$0.705879\pi$$
$$828$$ 0 0
$$829$$ −39590.0 −1.65865 −0.829323 0.558770i $$-0.811274\pi$$
−0.829323 + 0.558770i $$0.811274\pi$$
$$830$$ 432.000 0.0180662
$$831$$ 0 0
$$832$$ −25536.0 −1.06406
$$833$$ −22320.0 −0.928382
$$834$$ 0 0
$$835$$ 13344.0 0.553040
$$836$$ −9520.00 −0.393847
$$837$$ 0 0
$$838$$ −31552.0 −1.30065
$$839$$ 14316.0 0.589086 0.294543 0.955638i $$-0.404833\pi$$
0.294543 + 0.955638i $$0.404833\pi$$
$$840$$ 0 0
$$841$$ 35636.0 1.46115
$$842$$ −17456.0 −0.714458
$$843$$ 0 0
$$844$$ 14704.0 0.599683
$$845$$ 6312.00 0.256970
$$846$$ 0 0
$$847$$ 1400.00 0.0567941
$$848$$ −6624.00 −0.268242
$$849$$ 0 0
$$850$$ −14240.0 −0.574621
$$851$$ 6854.00 0.276089
$$852$$ 0 0
$$853$$ 28366.0 1.13861 0.569304 0.822127i $$-0.307213\pi$$
0.569304 + 0.822127i $$0.307213\pi$$
$$854$$ −13152.0 −0.526993
$$855$$ 0 0
$$856$$ 9936.00 0.396735
$$857$$ −19283.0 −0.768605 −0.384303 0.923207i $$-0.625558\pi$$
−0.384303 + 0.923207i $$0.625558\pi$$
$$858$$ 0 0
$$859$$ −26101.0 −1.03673 −0.518367 0.855158i $$-0.673460\pi$$
−0.518367 + 0.855158i $$0.673460\pi$$
$$860$$ −2112.00 −0.0837426
$$861$$ 0 0
$$862$$ 5856.00 0.231388
$$863$$ −973.000 −0.0383793 −0.0191896 0.999816i $$-0.506109\pi$$
−0.0191896 + 0.999816i $$0.506109\pi$$
$$864$$ 0 0
$$865$$ −19380.0 −0.761780
$$866$$ −10628.0 −0.417037
$$867$$ 0 0
$$868$$ 3296.00 0.128887
$$869$$ 44948.0 1.75461
$$870$$ 0 0
$$871$$ −52782.0 −2.05333
$$872$$ −16896.0 −0.656159
$$873$$ 0 0
$$874$$ 3220.00 0.124620
$$875$$ 10272.0 0.396865
$$876$$ 0 0
$$877$$ 5694.00 0.219239 0.109620 0.993974i $$-0.465037\pi$$
0.109620 + 0.993974i $$0.465037\pi$$
$$878$$ 5170.00 0.198723
$$879$$ 0 0
$$880$$ 3264.00 0.125033
$$881$$ −45960.0 −1.75758 −0.878792 0.477205i $$-0.841650\pi$$
−0.878792 + 0.477205i $$0.841650\pi$$
$$882$$ 0 0
$$883$$ 17188.0 0.655065 0.327532 0.944840i $$-0.393783\pi$$
0.327532 + 0.944840i $$0.393783\pi$$
$$884$$ 18240.0 0.693979
$$885$$ 0 0
$$886$$ 5994.00 0.227283
$$887$$ −8451.00 −0.319906 −0.159953 0.987125i $$-0.551134\pi$$
−0.159953 + 0.987125i $$0.551134\pi$$
$$888$$ 0 0
$$889$$ −2088.00 −0.0787731
$$890$$ 5520.00 0.207900
$$891$$ 0 0
$$892$$ −6624.00 −0.248641
$$893$$ −24990.0 −0.936460
$$894$$ 0 0
$$895$$ −2214.00 −0.0826881
$$896$$ 3072.00 0.114541
$$897$$ 0 0
$$898$$ 33124.0 1.23092
$$899$$ −25235.0 −0.936190
$$900$$ 0 0
$$901$$ 33120.0 1.22463
$$902$$ 6460.00 0.238464
$$903$$ 0 0
$$904$$ 22848.0 0.840612
$$905$$ −8220.00 −0.301925
$$906$$ 0 0
$$907$$ 32774.0 1.19983 0.599913 0.800065i $$-0.295202\pi$$
0.599913 + 0.800065i $$0.295202\pi$$
$$908$$ 11760.0 0.429812
$$909$$ 0 0
$$910$$ 5472.00 0.199335
$$911$$ 23690.0 0.861564 0.430782 0.902456i $$-0.358238\pi$$
0.430782 + 0.902456i $$0.358238\pi$$
$$912$$ 0 0
$$913$$ −1224.00 −0.0443686
$$914$$ 7848.00 0.284014
$$915$$ 0 0
$$916$$ −14448.0 −0.521152
$$917$$ −11528.0 −0.415145
$$918$$ 0 0
$$919$$ −30044.0 −1.07841 −0.539206 0.842174i $$-0.681275\pi$$
−0.539206 + 0.842174i $$0.681275\pi$$
$$920$$ 3312.00 0.118688
$$921$$ 0 0
$$922$$ 9086.00 0.324546
$$923$$ 19095.0 0.680953
$$924$$ 0 0
$$925$$ 26522.0 0.942744
$$926$$ 19232.0 0.682508
$$927$$ 0 0
$$928$$ −39200.0 −1.38664
$$929$$ 39705.0 1.40224 0.701119 0.713044i $$-0.252684\pi$$
0.701119 + 0.713044i $$0.252684\pi$$
$$930$$ 0 0
$$931$$ 19530.0 0.687508
$$932$$ −17300.0 −0.608026
$$933$$ 0 0
$$934$$ −15652.0 −0.548339
$$935$$ −16320.0 −0.570825
$$936$$ 0 0
$$937$$ 17422.0 0.607419 0.303710 0.952765i $$-0.401775\pi$$
0.303710 + 0.952765i $$0.401775\pi$$
$$938$$ −14816.0 −0.515735
$$939$$ 0 0
$$940$$ −8568.00 −0.297295
$$941$$ 25292.0 0.876191 0.438095 0.898928i $$-0.355653\pi$$
0.438095 + 0.898928i $$0.355653\pi$$
$$942$$ 0 0
$$943$$ 2185.00 0.0754543
$$944$$ −6528.00 −0.225072
$$945$$ 0 0
$$946$$ −5984.00 −0.205662
$$947$$ −33211.0 −1.13961 −0.569806 0.821779i $$-0.692982\pi$$
−0.569806 + 0.821779i $$0.692982\pi$$
$$948$$ 0 0
$$949$$ 51243.0 1.75281
$$950$$ 12460.0 0.425532
$$951$$ 0 0
$$952$$ 15360.0 0.522921
$$953$$ 14154.0 0.481105 0.240552 0.970636i $$-0.422671\pi$$
0.240552 + 0.970636i $$0.422671\pi$$
$$954$$ 0 0
$$955$$ −26460.0 −0.896571
$$956$$ 10940.0 0.370110
$$957$$ 0 0
$$958$$ −22808.0 −0.769199
$$959$$ 12448.0 0.419152
$$960$$ 0 0
$$961$$ −19182.0 −0.643886
$$962$$ 33972.0 1.13857
$$963$$ 0 0
$$964$$ 26840.0 0.896741
$$965$$ −810.000 −0.0270205
$$966$$ 0 0
$$967$$ −46343.0 −1.54115 −0.770574 0.637350i $$-0.780030\pi$$
−0.770574 + 0.637350i $$0.780030\pi$$
$$968$$ 4200.00 0.139456
$$969$$ 0 0
$$970$$ −11568.0 −0.382914
$$971$$ −11710.0 −0.387015 −0.193508 0.981099i $$-0.561986\pi$$
−0.193508 + 0.981099i $$0.561986\pi$$
$$972$$ 0 0
$$973$$ −200.000 −0.00658963
$$974$$ −18534.0 −0.609720
$$975$$ 0 0
$$976$$ −13152.0 −0.431337
$$977$$ −47854.0 −1.56703 −0.783513 0.621375i $$-0.786574\pi$$
−0.783513 + 0.621375i $$0.786574\pi$$
$$978$$ 0 0
$$979$$ −15640.0 −0.510579
$$980$$ 6696.00 0.218261
$$981$$ 0 0
$$982$$ 36382.0 1.18228
$$983$$ 22078.0 0.716357 0.358178 0.933653i $$-0.383398\pi$$
0.358178 + 0.933653i $$0.383398\pi$$
$$984$$ 0 0
$$985$$ −7326.00 −0.236980
$$986$$ −39200.0 −1.26611
$$987$$ 0 0
$$988$$ −15960.0 −0.513922
$$989$$ −2024.00 −0.0650753
$$990$$ 0 0
$$991$$ −4288.00 −0.137450 −0.0687249 0.997636i $$-0.521893\pi$$
−0.0687249 + 0.997636i $$0.521893\pi$$
$$992$$ 16480.0 0.527460
$$993$$ 0 0
$$994$$ 5360.00 0.171035
$$995$$ −6588.00 −0.209903
$$996$$ 0 0
$$997$$ 28966.0 0.920123 0.460061 0.887887i $$-0.347827\pi$$
0.460061 + 0.887887i $$0.347827\pi$$
$$998$$ 38630.0 1.22526
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 207.4.a.a.1.1 1
3.2 odd 2 23.4.a.a.1.1 1
12.11 even 2 368.4.a.d.1.1 1
15.2 even 4 575.4.b.b.24.1 2
15.8 even 4 575.4.b.b.24.2 2
15.14 odd 2 575.4.a.g.1.1 1
21.20 even 2 1127.4.a.a.1.1 1
24.5 odd 2 1472.4.a.h.1.1 1
24.11 even 2 1472.4.a.c.1.1 1
69.68 even 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 3.2 odd 2
207.4.a.a.1.1 1 1.1 even 1 trivial
368.4.a.d.1.1 1 12.11 even 2
529.4.a.a.1.1 1 69.68 even 2
575.4.a.g.1.1 1 15.14 odd 2
575.4.b.b.24.1 2 15.2 even 4
575.4.b.b.24.2 2 15.8 even 4
1127.4.a.a.1.1 1 21.20 even 2
1472.4.a.c.1.1 1 24.11 even 2
1472.4.a.h.1.1 1 24.5 odd 2