# Properties

 Label 507.4.a.d.1.1 Level $507$ Weight $4$ Character 507.1 Self dual yes Analytic conductor $29.914$ 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] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, 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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.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: $$29.9139683729$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 507.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} -7.00000 q^{5} +3.00000 q^{6} +10.0000 q^{7} -15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} -7.00000 q^{5} +3.00000 q^{6} +10.0000 q^{7} -15.0000 q^{8} +9.00000 q^{9} -7.00000 q^{10} +22.0000 q^{11} -21.0000 q^{12} +10.0000 q^{14} -21.0000 q^{15} +41.0000 q^{16} +37.0000 q^{17} +9.00000 q^{18} -30.0000 q^{19} +49.0000 q^{20} +30.0000 q^{21} +22.0000 q^{22} -162.000 q^{23} -45.0000 q^{24} -76.0000 q^{25} +27.0000 q^{27} -70.0000 q^{28} -113.000 q^{29} -21.0000 q^{30} -196.000 q^{31} +161.000 q^{32} +66.0000 q^{33} +37.0000 q^{34} -70.0000 q^{35} -63.0000 q^{36} -13.0000 q^{37} -30.0000 q^{38} +105.000 q^{40} -285.000 q^{41} +30.0000 q^{42} -246.000 q^{43} -154.000 q^{44} -63.0000 q^{45} -162.000 q^{46} +462.000 q^{47} +123.000 q^{48} -243.000 q^{49} -76.0000 q^{50} +111.000 q^{51} -537.000 q^{53} +27.0000 q^{54} -154.000 q^{55} -150.000 q^{56} -90.0000 q^{57} -113.000 q^{58} -576.000 q^{59} +147.000 q^{60} -635.000 q^{61} -196.000 q^{62} +90.0000 q^{63} -167.000 q^{64} +66.0000 q^{66} -202.000 q^{67} -259.000 q^{68} -486.000 q^{69} -70.0000 q^{70} +1086.00 q^{71} -135.000 q^{72} +805.000 q^{73} -13.0000 q^{74} -228.000 q^{75} +210.000 q^{76} +220.000 q^{77} +884.000 q^{79} -287.000 q^{80} +81.0000 q^{81} -285.000 q^{82} -518.000 q^{83} -210.000 q^{84} -259.000 q^{85} -246.000 q^{86} -339.000 q^{87} -330.000 q^{88} -194.000 q^{89} -63.0000 q^{90} +1134.00 q^{92} -588.000 q^{93} +462.000 q^{94} +210.000 q^{95} +483.000 q^{96} +1202.00 q^{97} -243.000 q^{98} +198.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.353553 0.176777 0.984251i $$-0.443433\pi$$
0.176777 + 0.984251i $$0.443433\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −7.00000 −0.875000
$$5$$ −7.00000 −0.626099 −0.313050 0.949737i $$-0.601351\pi$$
−0.313050 + 0.949737i $$0.601351\pi$$
$$6$$ 3.00000 0.204124
$$7$$ 10.0000 0.539949 0.269975 0.962867i $$-0.412985\pi$$
0.269975 + 0.962867i $$0.412985\pi$$
$$8$$ −15.0000 −0.662913
$$9$$ 9.00000 0.333333
$$10$$ −7.00000 −0.221359
$$11$$ 22.0000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ −21.0000 −0.505181
$$13$$ 0 0
$$14$$ 10.0000 0.190901
$$15$$ −21.0000 −0.361478
$$16$$ 41.0000 0.640625
$$17$$ 37.0000 0.527872 0.263936 0.964540i $$-0.414979\pi$$
0.263936 + 0.964540i $$0.414979\pi$$
$$18$$ 9.00000 0.117851
$$19$$ −30.0000 −0.362235 −0.181118 0.983461i $$-0.557971\pi$$
−0.181118 + 0.983461i $$0.557971\pi$$
$$20$$ 49.0000 0.547837
$$21$$ 30.0000 0.311740
$$22$$ 22.0000 0.213201
$$23$$ −162.000 −1.46867 −0.734333 0.678789i $$-0.762505\pi$$
−0.734333 + 0.678789i $$0.762505\pi$$
$$24$$ −45.0000 −0.382733
$$25$$ −76.0000 −0.608000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ −70.0000 −0.472456
$$29$$ −113.000 −0.723571 −0.361786 0.932261i $$-0.617833\pi$$
−0.361786 + 0.932261i $$0.617833\pi$$
$$30$$ −21.0000 −0.127802
$$31$$ −196.000 −1.13557 −0.567785 0.823177i $$-0.692199\pi$$
−0.567785 + 0.823177i $$0.692199\pi$$
$$32$$ 161.000 0.889408
$$33$$ 66.0000 0.348155
$$34$$ 37.0000 0.186631
$$35$$ −70.0000 −0.338062
$$36$$ −63.0000 −0.291667
$$37$$ −13.0000 −0.0577618 −0.0288809 0.999583i $$-0.509194\pi$$
−0.0288809 + 0.999583i $$0.509194\pi$$
$$38$$ −30.0000 −0.128070
$$39$$ 0 0
$$40$$ 105.000 0.415049
$$41$$ −285.000 −1.08560 −0.542799 0.839863i $$-0.682635\pi$$
−0.542799 + 0.839863i $$0.682635\pi$$
$$42$$ 30.0000 0.110217
$$43$$ −246.000 −0.872434 −0.436217 0.899842i $$-0.643682\pi$$
−0.436217 + 0.899842i $$0.643682\pi$$
$$44$$ −154.000 −0.527645
$$45$$ −63.0000 −0.208700
$$46$$ −162.000 −0.519252
$$47$$ 462.000 1.43382 0.716911 0.697165i $$-0.245555\pi$$
0.716911 + 0.697165i $$0.245555\pi$$
$$48$$ 123.000 0.369865
$$49$$ −243.000 −0.708455
$$50$$ −76.0000 −0.214960
$$51$$ 111.000 0.304767
$$52$$ 0 0
$$53$$ −537.000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$54$$ 27.0000 0.0680414
$$55$$ −154.000 −0.377552
$$56$$ −150.000 −0.357939
$$57$$ −90.0000 −0.209137
$$58$$ −113.000 −0.255821
$$59$$ −576.000 −1.27100 −0.635498 0.772102i $$-0.719205\pi$$
−0.635498 + 0.772102i $$0.719205\pi$$
$$60$$ 147.000 0.316294
$$61$$ −635.000 −1.33284 −0.666421 0.745575i $$-0.732175\pi$$
−0.666421 + 0.745575i $$0.732175\pi$$
$$62$$ −196.000 −0.401484
$$63$$ 90.0000 0.179983
$$64$$ −167.000 −0.326172
$$65$$ 0 0
$$66$$ 66.0000 0.123091
$$67$$ −202.000 −0.368332 −0.184166 0.982895i $$-0.558958\pi$$
−0.184166 + 0.982895i $$0.558958\pi$$
$$68$$ −259.000 −0.461888
$$69$$ −486.000 −0.847935
$$70$$ −70.0000 −0.119523
$$71$$ 1086.00 1.81527 0.907637 0.419755i $$-0.137884\pi$$
0.907637 + 0.419755i $$0.137884\pi$$
$$72$$ −135.000 −0.220971
$$73$$ 805.000 1.29066 0.645330 0.763904i $$-0.276720\pi$$
0.645330 + 0.763904i $$0.276720\pi$$
$$74$$ −13.0000 −0.0204219
$$75$$ −228.000 −0.351029
$$76$$ 210.000 0.316956
$$77$$ 220.000 0.325602
$$78$$ 0 0
$$79$$ 884.000 1.25896 0.629480 0.777017i $$-0.283268\pi$$
0.629480 + 0.777017i $$0.283268\pi$$
$$80$$ −287.000 −0.401095
$$81$$ 81.0000 0.111111
$$82$$ −285.000 −0.383817
$$83$$ −518.000 −0.685035 −0.342517 0.939511i $$-0.611280\pi$$
−0.342517 + 0.939511i $$0.611280\pi$$
$$84$$ −210.000 −0.272772
$$85$$ −259.000 −0.330500
$$86$$ −246.000 −0.308452
$$87$$ −339.000 −0.417754
$$88$$ −330.000 −0.399751
$$89$$ −194.000 −0.231056 −0.115528 0.993304i $$-0.536856\pi$$
−0.115528 + 0.993304i $$0.536856\pi$$
$$90$$ −63.0000 −0.0737865
$$91$$ 0 0
$$92$$ 1134.00 1.28508
$$93$$ −588.000 −0.655621
$$94$$ 462.000 0.506933
$$95$$ 210.000 0.226795
$$96$$ 483.000 0.513500
$$97$$ 1202.00 1.25819 0.629096 0.777328i $$-0.283425\pi$$
0.629096 + 0.777328i $$0.283425\pi$$
$$98$$ −243.000 −0.250477
$$99$$ 198.000 0.201008
$$100$$ 532.000 0.532000
$$101$$ −429.000 −0.422645 −0.211322 0.977416i $$-0.567777\pi$$
−0.211322 + 0.977416i $$0.567777\pi$$
$$102$$ 111.000 0.107751
$$103$$ −1302.00 −1.24553 −0.622766 0.782408i $$-0.713991\pi$$
−0.622766 + 0.782408i $$0.713991\pi$$
$$104$$ 0 0
$$105$$ −210.000 −0.195180
$$106$$ −537.000 −0.492057
$$107$$ −1338.00 −1.20887 −0.604436 0.796654i $$-0.706602\pi$$
−0.604436 + 0.796654i $$0.706602\pi$$
$$108$$ −189.000 −0.168394
$$109$$ 1034.00 0.908617 0.454308 0.890844i $$-0.349886\pi$$
0.454308 + 0.890844i $$0.349886\pi$$
$$110$$ −154.000 −0.133485
$$111$$ −39.0000 −0.0333488
$$112$$ 410.000 0.345905
$$113$$ 1077.00 0.896599 0.448299 0.893884i $$-0.352030\pi$$
0.448299 + 0.893884i $$0.352030\pi$$
$$114$$ −90.0000 −0.0739410
$$115$$ 1134.00 0.919531
$$116$$ 791.000 0.633125
$$117$$ 0 0
$$118$$ −576.000 −0.449365
$$119$$ 370.000 0.285024
$$120$$ 315.000 0.239629
$$121$$ −847.000 −0.636364
$$122$$ −635.000 −0.471231
$$123$$ −855.000 −0.626770
$$124$$ 1372.00 0.993623
$$125$$ 1407.00 1.00677
$$126$$ 90.0000 0.0636336
$$127$$ −988.000 −0.690321 −0.345161 0.938544i $$-0.612176\pi$$
−0.345161 + 0.938544i $$0.612176\pi$$
$$128$$ −1455.00 −1.00473
$$129$$ −738.000 −0.503700
$$130$$ 0 0
$$131$$ 560.000 0.373492 0.186746 0.982408i $$-0.440206\pi$$
0.186746 + 0.982408i $$0.440206\pi$$
$$132$$ −462.000 −0.304636
$$133$$ −300.000 −0.195589
$$134$$ −202.000 −0.130225
$$135$$ −189.000 −0.120493
$$136$$ −555.000 −0.349933
$$137$$ 519.000 0.323658 0.161829 0.986819i $$-0.448261\pi$$
0.161829 + 0.986819i $$0.448261\pi$$
$$138$$ −486.000 −0.299790
$$139$$ −348.000 −0.212352 −0.106176 0.994347i $$-0.533861\pi$$
−0.106176 + 0.994347i $$0.533861\pi$$
$$140$$ 490.000 0.295804
$$141$$ 1386.00 0.827817
$$142$$ 1086.00 0.641796
$$143$$ 0 0
$$144$$ 369.000 0.213542
$$145$$ 791.000 0.453027
$$146$$ 805.000 0.456317
$$147$$ −729.000 −0.409027
$$148$$ 91.0000 0.0505416
$$149$$ 645.000 0.354634 0.177317 0.984154i $$-0.443258\pi$$
0.177317 + 0.984154i $$0.443258\pi$$
$$150$$ −228.000 −0.124107
$$151$$ −2914.00 −1.57045 −0.785225 0.619211i $$-0.787453\pi$$
−0.785225 + 0.619211i $$0.787453\pi$$
$$152$$ 450.000 0.240130
$$153$$ 333.000 0.175957
$$154$$ 220.000 0.115118
$$155$$ 1372.00 0.710979
$$156$$ 0 0
$$157$$ −2079.00 −1.05683 −0.528415 0.848986i $$-0.677213\pi$$
−0.528415 + 0.848986i $$0.677213\pi$$
$$158$$ 884.000 0.445109
$$159$$ −1611.00 −0.803526
$$160$$ −1127.00 −0.556857
$$161$$ −1620.00 −0.793006
$$162$$ 81.0000 0.0392837
$$163$$ −1700.00 −0.816897 −0.408449 0.912781i $$-0.633930\pi$$
−0.408449 + 0.912781i $$0.633930\pi$$
$$164$$ 1995.00 0.949898
$$165$$ −462.000 −0.217980
$$166$$ −518.000 −0.242196
$$167$$ −3680.00 −1.70519 −0.852596 0.522571i $$-0.824973\pi$$
−0.852596 + 0.522571i $$0.824973\pi$$
$$168$$ −450.000 −0.206656
$$169$$ 0 0
$$170$$ −259.000 −0.116849
$$171$$ −270.000 −0.120745
$$172$$ 1722.00 0.763379
$$173$$ 4146.00 1.82205 0.911025 0.412352i $$-0.135293\pi$$
0.911025 + 0.412352i $$0.135293\pi$$
$$174$$ −339.000 −0.147698
$$175$$ −760.000 −0.328289
$$176$$ 902.000 0.386311
$$177$$ −1728.00 −0.733810
$$178$$ −194.000 −0.0816905
$$179$$ 3674.00 1.53412 0.767060 0.641575i $$-0.221719\pi$$
0.767060 + 0.641575i $$0.221719\pi$$
$$180$$ 441.000 0.182612
$$181$$ −3283.00 −1.34820 −0.674098 0.738642i $$-0.735467\pi$$
−0.674098 + 0.738642i $$0.735467\pi$$
$$182$$ 0 0
$$183$$ −1905.00 −0.769517
$$184$$ 2430.00 0.973598
$$185$$ 91.0000 0.0361646
$$186$$ −588.000 −0.231797
$$187$$ 814.000 0.318319
$$188$$ −3234.00 −1.25459
$$189$$ 270.000 0.103913
$$190$$ 210.000 0.0801842
$$191$$ −596.000 −0.225786 −0.112893 0.993607i $$-0.536012\pi$$
−0.112893 + 0.993607i $$0.536012\pi$$
$$192$$ −501.000 −0.188315
$$193$$ 393.000 0.146574 0.0732869 0.997311i $$-0.476651\pi$$
0.0732869 + 0.997311i $$0.476651\pi$$
$$194$$ 1202.00 0.444838
$$195$$ 0 0
$$196$$ 1701.00 0.619898
$$197$$ 3522.00 1.27377 0.636884 0.770960i $$-0.280223\pi$$
0.636884 + 0.770960i $$0.280223\pi$$
$$198$$ 198.000 0.0710669
$$199$$ 2018.00 0.718855 0.359428 0.933173i $$-0.382972\pi$$
0.359428 + 0.933173i $$0.382972\pi$$
$$200$$ 1140.00 0.403051
$$201$$ −606.000 −0.212656
$$202$$ −429.000 −0.149427
$$203$$ −1130.00 −0.390692
$$204$$ −777.000 −0.266671
$$205$$ 1995.00 0.679692
$$206$$ −1302.00 −0.440362
$$207$$ −1458.00 −0.489556
$$208$$ 0 0
$$209$$ −660.000 −0.218436
$$210$$ −210.000 −0.0690066
$$211$$ 160.000 0.0522031 0.0261016 0.999659i $$-0.491691\pi$$
0.0261016 + 0.999659i $$0.491691\pi$$
$$212$$ 3759.00 1.21778
$$213$$ 3258.00 1.04805
$$214$$ −1338.00 −0.427401
$$215$$ 1722.00 0.546230
$$216$$ −405.000 −0.127578
$$217$$ −1960.00 −0.613150
$$218$$ 1034.00 0.321245
$$219$$ 2415.00 0.745162
$$220$$ 1078.00 0.330358
$$221$$ 0 0
$$222$$ −39.0000 −0.0117906
$$223$$ −4072.00 −1.22279 −0.611393 0.791327i $$-0.709391\pi$$
−0.611393 + 0.791327i $$0.709391\pi$$
$$224$$ 1610.00 0.480235
$$225$$ −684.000 −0.202667
$$226$$ 1077.00 0.316995
$$227$$ 5794.00 1.69410 0.847051 0.531511i $$-0.178376\pi$$
0.847051 + 0.531511i $$0.178376\pi$$
$$228$$ 630.000 0.182995
$$229$$ −6482.00 −1.87049 −0.935246 0.353999i $$-0.884822\pi$$
−0.935246 + 0.353999i $$0.884822\pi$$
$$230$$ 1134.00 0.325103
$$231$$ 660.000 0.187986
$$232$$ 1695.00 0.479665
$$233$$ 6890.00 1.93725 0.968624 0.248530i $$-0.0799474\pi$$
0.968624 + 0.248530i $$0.0799474\pi$$
$$234$$ 0 0
$$235$$ −3234.00 −0.897714
$$236$$ 4032.00 1.11212
$$237$$ 2652.00 0.726860
$$238$$ 370.000 0.100771
$$239$$ −2466.00 −0.667415 −0.333708 0.942677i $$-0.608300\pi$$
−0.333708 + 0.942677i $$0.608300\pi$$
$$240$$ −861.000 −0.231572
$$241$$ 3617.00 0.966770 0.483385 0.875408i $$-0.339407\pi$$
0.483385 + 0.875408i $$0.339407\pi$$
$$242$$ −847.000 −0.224989
$$243$$ 243.000 0.0641500
$$244$$ 4445.00 1.16624
$$245$$ 1701.00 0.443563
$$246$$ −855.000 −0.221597
$$247$$ 0 0
$$248$$ 2940.00 0.752783
$$249$$ −1554.00 −0.395505
$$250$$ 1407.00 0.355946
$$251$$ 4860.00 1.22215 0.611077 0.791571i $$-0.290737\pi$$
0.611077 + 0.791571i $$0.290737\pi$$
$$252$$ −630.000 −0.157485
$$253$$ −3564.00 −0.885639
$$254$$ −988.000 −0.244065
$$255$$ −777.000 −0.190814
$$256$$ −119.000 −0.0290527
$$257$$ 565.000 0.137135 0.0685676 0.997646i $$-0.478157\pi$$
0.0685676 + 0.997646i $$0.478157\pi$$
$$258$$ −738.000 −0.178085
$$259$$ −130.000 −0.0311884
$$260$$ 0 0
$$261$$ −1017.00 −0.241190
$$262$$ 560.000 0.132049
$$263$$ −498.000 −0.116760 −0.0583802 0.998294i $$-0.518594\pi$$
−0.0583802 + 0.998294i $$0.518594\pi$$
$$264$$ −990.000 −0.230797
$$265$$ 3759.00 0.871372
$$266$$ −300.000 −0.0691511
$$267$$ −582.000 −0.133400
$$268$$ 1414.00 0.322290
$$269$$ 5546.00 1.25705 0.628523 0.777791i $$-0.283660\pi$$
0.628523 + 0.777791i $$0.283660\pi$$
$$270$$ −189.000 −0.0426006
$$271$$ 2256.00 0.505691 0.252845 0.967507i $$-0.418634\pi$$
0.252845 + 0.967507i $$0.418634\pi$$
$$272$$ 1517.00 0.338168
$$273$$ 0 0
$$274$$ 519.000 0.114430
$$275$$ −1672.00 −0.366638
$$276$$ 3402.00 0.741943
$$277$$ 2309.00 0.500846 0.250423 0.968137i $$-0.419430\pi$$
0.250423 + 0.968137i $$0.419430\pi$$
$$278$$ −348.000 −0.0750779
$$279$$ −1764.00 −0.378523
$$280$$ 1050.00 0.224105
$$281$$ −5833.00 −1.23832 −0.619159 0.785265i $$-0.712527\pi$$
−0.619159 + 0.785265i $$0.712527\pi$$
$$282$$ 1386.00 0.292678
$$283$$ 1650.00 0.346581 0.173290 0.984871i $$-0.444560\pi$$
0.173290 + 0.984871i $$0.444560\pi$$
$$284$$ −7602.00 −1.58837
$$285$$ 630.000 0.130940
$$286$$ 0 0
$$287$$ −2850.00 −0.586168
$$288$$ 1449.00 0.296469
$$289$$ −3544.00 −0.721352
$$290$$ 791.000 0.160169
$$291$$ 3606.00 0.726417
$$292$$ −5635.00 −1.12933
$$293$$ −2991.00 −0.596369 −0.298184 0.954508i $$-0.596381\pi$$
−0.298184 + 0.954508i $$0.596381\pi$$
$$294$$ −729.000 −0.144613
$$295$$ 4032.00 0.795770
$$296$$ 195.000 0.0382910
$$297$$ 594.000 0.116052
$$298$$ 645.000 0.125382
$$299$$ 0 0
$$300$$ 1596.00 0.307150
$$301$$ −2460.00 −0.471070
$$302$$ −2914.00 −0.555238
$$303$$ −1287.00 −0.244014
$$304$$ −1230.00 −0.232057
$$305$$ 4445.00 0.834492
$$306$$ 333.000 0.0622103
$$307$$ 2422.00 0.450263 0.225132 0.974328i $$-0.427719\pi$$
0.225132 + 0.974328i $$0.427719\pi$$
$$308$$ −1540.00 −0.284901
$$309$$ −3906.00 −0.719109
$$310$$ 1372.00 0.251369
$$311$$ −3402.00 −0.620288 −0.310144 0.950690i $$-0.600377\pi$$
−0.310144 + 0.950690i $$0.600377\pi$$
$$312$$ 0 0
$$313$$ 2310.00 0.417153 0.208577 0.978006i $$-0.433117\pi$$
0.208577 + 0.978006i $$0.433117\pi$$
$$314$$ −2079.00 −0.373646
$$315$$ −630.000 −0.112687
$$316$$ −6188.00 −1.10159
$$317$$ 257.000 0.0455349 0.0227674 0.999741i $$-0.492752\pi$$
0.0227674 + 0.999741i $$0.492752\pi$$
$$318$$ −1611.00 −0.284089
$$319$$ −2486.00 −0.436330
$$320$$ 1169.00 0.204216
$$321$$ −4014.00 −0.697943
$$322$$ −1620.00 −0.280370
$$323$$ −1110.00 −0.191214
$$324$$ −567.000 −0.0972222
$$325$$ 0 0
$$326$$ −1700.00 −0.288817
$$327$$ 3102.00 0.524590
$$328$$ 4275.00 0.719657
$$329$$ 4620.00 0.774191
$$330$$ −462.000 −0.0770675
$$331$$ −1028.00 −0.170707 −0.0853535 0.996351i $$-0.527202\pi$$
−0.0853535 + 0.996351i $$0.527202\pi$$
$$332$$ 3626.00 0.599405
$$333$$ −117.000 −0.0192539
$$334$$ −3680.00 −0.602876
$$335$$ 1414.00 0.230612
$$336$$ 1230.00 0.199708
$$337$$ 2487.00 0.402005 0.201002 0.979591i $$-0.435580\pi$$
0.201002 + 0.979591i $$0.435580\pi$$
$$338$$ 0 0
$$339$$ 3231.00 0.517651
$$340$$ 1813.00 0.289187
$$341$$ −4312.00 −0.684774
$$342$$ −270.000 −0.0426898
$$343$$ −5860.00 −0.922479
$$344$$ 3690.00 0.578347
$$345$$ 3402.00 0.530891
$$346$$ 4146.00 0.644192
$$347$$ −2850.00 −0.440911 −0.220455 0.975397i $$-0.570754\pi$$
−0.220455 + 0.975397i $$0.570754\pi$$
$$348$$ 2373.00 0.365535
$$349$$ 2018.00 0.309516 0.154758 0.987952i $$-0.450540\pi$$
0.154758 + 0.987952i $$0.450540\pi$$
$$350$$ −760.000 −0.116068
$$351$$ 0 0
$$352$$ 3542.00 0.536333
$$353$$ 5287.00 0.797163 0.398582 0.917133i $$-0.369503\pi$$
0.398582 + 0.917133i $$0.369503\pi$$
$$354$$ −1728.00 −0.259441
$$355$$ −7602.00 −1.13654
$$356$$ 1358.00 0.202174
$$357$$ 1110.00 0.164559
$$358$$ 3674.00 0.542394
$$359$$ 7278.00 1.06997 0.534983 0.844863i $$-0.320318\pi$$
0.534983 + 0.844863i $$0.320318\pi$$
$$360$$ 945.000 0.138350
$$361$$ −5959.00 −0.868786
$$362$$ −3283.00 −0.476659
$$363$$ −2541.00 −0.367405
$$364$$ 0 0
$$365$$ −5635.00 −0.808080
$$366$$ −1905.00 −0.272065
$$367$$ −4202.00 −0.597664 −0.298832 0.954306i $$-0.596597\pi$$
−0.298832 + 0.954306i $$0.596597\pi$$
$$368$$ −6642.00 −0.940865
$$369$$ −2565.00 −0.361866
$$370$$ 91.0000 0.0127861
$$371$$ −5370.00 −0.751473
$$372$$ 4116.00 0.573668
$$373$$ −1583.00 −0.219744 −0.109872 0.993946i $$-0.535044\pi$$
−0.109872 + 0.993946i $$0.535044\pi$$
$$374$$ 814.000 0.112543
$$375$$ 4221.00 0.581257
$$376$$ −6930.00 −0.950499
$$377$$ 0 0
$$378$$ 270.000 0.0367389
$$379$$ 2052.00 0.278111 0.139056 0.990285i $$-0.455593\pi$$
0.139056 + 0.990285i $$0.455593\pi$$
$$380$$ −1470.00 −0.198446
$$381$$ −2964.00 −0.398557
$$382$$ −596.000 −0.0798273
$$383$$ 6872.00 0.916822 0.458411 0.888740i $$-0.348419\pi$$
0.458411 + 0.888740i $$0.348419\pi$$
$$384$$ −4365.00 −0.580079
$$385$$ −1540.00 −0.203859
$$386$$ 393.000 0.0518217
$$387$$ −2214.00 −0.290811
$$388$$ −8414.00 −1.10092
$$389$$ −11653.0 −1.51884 −0.759422 0.650598i $$-0.774518\pi$$
−0.759422 + 0.650598i $$0.774518\pi$$
$$390$$ 0 0
$$391$$ −5994.00 −0.775268
$$392$$ 3645.00 0.469644
$$393$$ 1680.00 0.215636
$$394$$ 3522.00 0.450345
$$395$$ −6188.00 −0.788233
$$396$$ −1386.00 −0.175882
$$397$$ −6134.00 −0.775458 −0.387729 0.921774i $$-0.626740\pi$$
−0.387729 + 0.921774i $$0.626740\pi$$
$$398$$ 2018.00 0.254154
$$399$$ −900.000 −0.112923
$$400$$ −3116.00 −0.389500
$$401$$ 10795.0 1.34433 0.672165 0.740401i $$-0.265364\pi$$
0.672165 + 0.740401i $$0.265364\pi$$
$$402$$ −606.000 −0.0751854
$$403$$ 0 0
$$404$$ 3003.00 0.369814
$$405$$ −567.000 −0.0695666
$$406$$ −1130.00 −0.138130
$$407$$ −286.000 −0.0348317
$$408$$ −1665.00 −0.202034
$$409$$ 8489.00 1.02629 0.513147 0.858301i $$-0.328480\pi$$
0.513147 + 0.858301i $$0.328480\pi$$
$$410$$ 1995.00 0.240307
$$411$$ 1557.00 0.186864
$$412$$ 9114.00 1.08984
$$413$$ −5760.00 −0.686274
$$414$$ −1458.00 −0.173084
$$415$$ 3626.00 0.428900
$$416$$ 0 0
$$417$$ −1044.00 −0.122602
$$418$$ −660.000 −0.0772288
$$419$$ 1496.00 0.174426 0.0872129 0.996190i $$-0.472204\pi$$
0.0872129 + 0.996190i $$0.472204\pi$$
$$420$$ 1470.00 0.170783
$$421$$ 11695.0 1.35387 0.676935 0.736043i $$-0.263308\pi$$
0.676935 + 0.736043i $$0.263308\pi$$
$$422$$ 160.000 0.0184566
$$423$$ 4158.00 0.477941
$$424$$ 8055.00 0.922607
$$425$$ −2812.00 −0.320946
$$426$$ 3258.00 0.370541
$$427$$ −6350.00 −0.719668
$$428$$ 9366.00 1.05776
$$429$$ 0 0
$$430$$ 1722.00 0.193121
$$431$$ −10590.0 −1.18353 −0.591766 0.806110i $$-0.701569\pi$$
−0.591766 + 0.806110i $$0.701569\pi$$
$$432$$ 1107.00 0.123288
$$433$$ −13949.0 −1.54814 −0.774072 0.633098i $$-0.781783\pi$$
−0.774072 + 0.633098i $$0.781783\pi$$
$$434$$ −1960.00 −0.216781
$$435$$ 2373.00 0.261555
$$436$$ −7238.00 −0.795040
$$437$$ 4860.00 0.532003
$$438$$ 2415.00 0.263455
$$439$$ −10726.0 −1.16611 −0.583057 0.812431i $$-0.698144\pi$$
−0.583057 + 0.812431i $$0.698144\pi$$
$$440$$ 2310.00 0.250284
$$441$$ −2187.00 −0.236152
$$442$$ 0 0
$$443$$ 16228.0 1.74044 0.870221 0.492662i $$-0.163976\pi$$
0.870221 + 0.492662i $$0.163976\pi$$
$$444$$ 273.000 0.0291802
$$445$$ 1358.00 0.144664
$$446$$ −4072.00 −0.432320
$$447$$ 1935.00 0.204748
$$448$$ −1670.00 −0.176116
$$449$$ −7538.00 −0.792294 −0.396147 0.918187i $$-0.629653\pi$$
−0.396147 + 0.918187i $$0.629653\pi$$
$$450$$ −684.000 −0.0716535
$$451$$ −6270.00 −0.654640
$$452$$ −7539.00 −0.784524
$$453$$ −8742.00 −0.906700
$$454$$ 5794.00 0.598956
$$455$$ 0 0
$$456$$ 1350.00 0.138639
$$457$$ −15539.0 −1.59056 −0.795278 0.606245i $$-0.792675\pi$$
−0.795278 + 0.606245i $$0.792675\pi$$
$$458$$ −6482.00 −0.661319
$$459$$ 999.000 0.101589
$$460$$ −7938.00 −0.804589
$$461$$ −4811.00 −0.486053 −0.243027 0.970020i $$-0.578140\pi$$
−0.243027 + 0.970020i $$0.578140\pi$$
$$462$$ 660.000 0.0664632
$$463$$ −562.000 −0.0564111 −0.0282056 0.999602i $$-0.508979\pi$$
−0.0282056 + 0.999602i $$0.508979\pi$$
$$464$$ −4633.00 −0.463538
$$465$$ 4116.00 0.410484
$$466$$ 6890.00 0.684921
$$467$$ 4914.00 0.486922 0.243461 0.969911i $$-0.421717\pi$$
0.243461 + 0.969911i $$0.421717\pi$$
$$468$$ 0 0
$$469$$ −2020.00 −0.198880
$$470$$ −3234.00 −0.317390
$$471$$ −6237.00 −0.610161
$$472$$ 8640.00 0.842560
$$473$$ −5412.00 −0.526097
$$474$$ 2652.00 0.256984
$$475$$ 2280.00 0.220239
$$476$$ −2590.00 −0.249396
$$477$$ −4833.00 −0.463916
$$478$$ −2466.00 −0.235967
$$479$$ 3600.00 0.343399 0.171700 0.985149i $$-0.445074\pi$$
0.171700 + 0.985149i $$0.445074\pi$$
$$480$$ −3381.00 −0.321502
$$481$$ 0 0
$$482$$ 3617.00 0.341805
$$483$$ −4860.00 −0.457842
$$484$$ 5929.00 0.556818
$$485$$ −8414.00 −0.787753
$$486$$ 243.000 0.0226805
$$487$$ 17130.0 1.59391 0.796955 0.604038i $$-0.206443\pi$$
0.796955 + 0.604038i $$0.206443\pi$$
$$488$$ 9525.00 0.883558
$$489$$ −5100.00 −0.471636
$$490$$ 1701.00 0.156823
$$491$$ −11838.0 −1.08807 −0.544034 0.839063i $$-0.683104\pi$$
−0.544034 + 0.839063i $$0.683104\pi$$
$$492$$ 5985.00 0.548424
$$493$$ −4181.00 −0.381953
$$494$$ 0 0
$$495$$ −1386.00 −0.125851
$$496$$ −8036.00 −0.727474
$$497$$ 10860.0 0.980156
$$498$$ −1554.00 −0.139832
$$499$$ −8976.00 −0.805252 −0.402626 0.915364i $$-0.631903\pi$$
−0.402626 + 0.915364i $$0.631903\pi$$
$$500$$ −9849.00 −0.880921
$$501$$ −11040.0 −0.984493
$$502$$ 4860.00 0.432096
$$503$$ 1682.00 0.149099 0.0745494 0.997217i $$-0.476248\pi$$
0.0745494 + 0.997217i $$0.476248\pi$$
$$504$$ −1350.00 −0.119313
$$505$$ 3003.00 0.264617
$$506$$ −3564.00 −0.313121
$$507$$ 0 0
$$508$$ 6916.00 0.604031
$$509$$ −15167.0 −1.32076 −0.660379 0.750933i $$-0.729604\pi$$
−0.660379 + 0.750933i $$0.729604\pi$$
$$510$$ −777.000 −0.0674630
$$511$$ 8050.00 0.696890
$$512$$ 11521.0 0.994455
$$513$$ −810.000 −0.0697122
$$514$$ 565.000 0.0484846
$$515$$ 9114.00 0.779827
$$516$$ 5166.00 0.440737
$$517$$ 10164.0 0.864627
$$518$$ −130.000 −0.0110268
$$519$$ 12438.0 1.05196
$$520$$ 0 0
$$521$$ −6783.00 −0.570381 −0.285191 0.958471i $$-0.592057\pi$$
−0.285191 + 0.958471i $$0.592057\pi$$
$$522$$ −1017.00 −0.0852737
$$523$$ −13918.0 −1.16366 −0.581828 0.813312i $$-0.697662\pi$$
−0.581828 + 0.813312i $$0.697662\pi$$
$$524$$ −3920.00 −0.326805
$$525$$ −2280.00 −0.189538
$$526$$ −498.000 −0.0412810
$$527$$ −7252.00 −0.599435
$$528$$ 2706.00 0.223037
$$529$$ 14077.0 1.15698
$$530$$ 3759.00 0.308076
$$531$$ −5184.00 −0.423666
$$532$$ 2100.00 0.171140
$$533$$ 0 0
$$534$$ −582.000 −0.0471641
$$535$$ 9366.00 0.756874
$$536$$ 3030.00 0.244172
$$537$$ 11022.0 0.885725
$$538$$ 5546.00 0.444433
$$539$$ −5346.00 −0.427214
$$540$$ 1323.00 0.105431
$$541$$ 1335.00 0.106093 0.0530463 0.998592i $$-0.483107\pi$$
0.0530463 + 0.998592i $$0.483107\pi$$
$$542$$ 2256.00 0.178789
$$543$$ −9849.00 −0.778381
$$544$$ 5957.00 0.469493
$$545$$ −7238.00 −0.568884
$$546$$ 0 0
$$547$$ −3806.00 −0.297501 −0.148750 0.988875i $$-0.547525\pi$$
−0.148750 + 0.988875i $$0.547525\pi$$
$$548$$ −3633.00 −0.283201
$$549$$ −5715.00 −0.444281
$$550$$ −1672.00 −0.129626
$$551$$ 3390.00 0.262103
$$552$$ 7290.00 0.562107
$$553$$ 8840.00 0.679774
$$554$$ 2309.00 0.177076
$$555$$ 273.000 0.0208796
$$556$$ 2436.00 0.185808
$$557$$ 1905.00 0.144915 0.0724573 0.997372i $$-0.476916\pi$$
0.0724573 + 0.997372i $$0.476916\pi$$
$$558$$ −1764.00 −0.133828
$$559$$ 0 0
$$560$$ −2870.00 −0.216571
$$561$$ 2442.00 0.183781
$$562$$ −5833.00 −0.437812
$$563$$ −4800.00 −0.359318 −0.179659 0.983729i $$-0.557499\pi$$
−0.179659 + 0.983729i $$0.557499\pi$$
$$564$$ −9702.00 −0.724340
$$565$$ −7539.00 −0.561359
$$566$$ 1650.00 0.122535
$$567$$ 810.000 0.0599944
$$568$$ −16290.0 −1.20337
$$569$$ 14678.0 1.08143 0.540715 0.841206i $$-0.318154\pi$$
0.540715 + 0.841206i $$0.318154\pi$$
$$570$$ 630.000 0.0462944
$$571$$ −586.000 −0.0429481 −0.0214740 0.999769i $$-0.506836\pi$$
−0.0214740 + 0.999769i $$0.506836\pi$$
$$572$$ 0 0
$$573$$ −1788.00 −0.130357
$$574$$ −2850.00 −0.207242
$$575$$ 12312.0 0.892949
$$576$$ −1503.00 −0.108724
$$577$$ −8939.00 −0.644949 −0.322474 0.946578i $$-0.604515\pi$$
−0.322474 + 0.946578i $$0.604515\pi$$
$$578$$ −3544.00 −0.255036
$$579$$ 1179.00 0.0846245
$$580$$ −5537.00 −0.396399
$$581$$ −5180.00 −0.369884
$$582$$ 3606.00 0.256827
$$583$$ −11814.0 −0.839255
$$584$$ −12075.0 −0.855594
$$585$$ 0 0
$$586$$ −2991.00 −0.210848
$$587$$ 13792.0 0.969773 0.484887 0.874577i $$-0.338861\pi$$
0.484887 + 0.874577i $$0.338861\pi$$
$$588$$ 5103.00 0.357898
$$589$$ 5880.00 0.411343
$$590$$ 4032.00 0.281347
$$591$$ 10566.0 0.735410
$$592$$ −533.000 −0.0370037
$$593$$ −9569.00 −0.662650 −0.331325 0.943517i $$-0.607496\pi$$
−0.331325 + 0.943517i $$0.607496\pi$$
$$594$$ 594.000 0.0410305
$$595$$ −2590.00 −0.178453
$$596$$ −4515.00 −0.310305
$$597$$ 6054.00 0.415031
$$598$$ 0 0
$$599$$ −5192.00 −0.354156 −0.177078 0.984197i $$-0.556664\pi$$
−0.177078 + 0.984197i $$0.556664\pi$$
$$600$$ 3420.00 0.232702
$$601$$ −3677.00 −0.249564 −0.124782 0.992184i $$-0.539823\pi$$
−0.124782 + 0.992184i $$0.539823\pi$$
$$602$$ −2460.00 −0.166548
$$603$$ −1818.00 −0.122777
$$604$$ 20398.0 1.37414
$$605$$ 5929.00 0.398427
$$606$$ −1287.00 −0.0862719
$$607$$ −10960.0 −0.732871 −0.366435 0.930443i $$-0.619422\pi$$
−0.366435 + 0.930443i $$0.619422\pi$$
$$608$$ −4830.00 −0.322175
$$609$$ −3390.00 −0.225566
$$610$$ 4445.00 0.295037
$$611$$ 0 0
$$612$$ −2331.00 −0.153963
$$613$$ 26027.0 1.71488 0.857439 0.514585i $$-0.172054\pi$$
0.857439 + 0.514585i $$0.172054\pi$$
$$614$$ 2422.00 0.159192
$$615$$ 5985.00 0.392420
$$616$$ −3300.00 −0.215845
$$617$$ −17681.0 −1.15366 −0.576832 0.816863i $$-0.695711\pi$$
−0.576832 + 0.816863i $$0.695711\pi$$
$$618$$ −3906.00 −0.254243
$$619$$ −3192.00 −0.207265 −0.103633 0.994616i $$-0.533047\pi$$
−0.103633 + 0.994616i $$0.533047\pi$$
$$620$$ −9604.00 −0.622106
$$621$$ −4374.00 −0.282645
$$622$$ −3402.00 −0.219305
$$623$$ −1940.00 −0.124758
$$624$$ 0 0
$$625$$ −349.000 −0.0223360
$$626$$ 2310.00 0.147486
$$627$$ −1980.00 −0.126114
$$628$$ 14553.0 0.924726
$$629$$ −481.000 −0.0304908
$$630$$ −630.000 −0.0398410
$$631$$ −7580.00 −0.478217 −0.239109 0.970993i $$-0.576855\pi$$
−0.239109 + 0.970993i $$0.576855\pi$$
$$632$$ −13260.0 −0.834580
$$633$$ 480.000 0.0301395
$$634$$ 257.000 0.0160990
$$635$$ 6916.00 0.432210
$$636$$ 11277.0 0.703085
$$637$$ 0 0
$$638$$ −2486.00 −0.154266
$$639$$ 9774.00 0.605091
$$640$$ 10185.0 0.629059
$$641$$ −27707.0 −1.70727 −0.853635 0.520871i $$-0.825607\pi$$
−0.853635 + 0.520871i $$0.825607\pi$$
$$642$$ −4014.00 −0.246760
$$643$$ −11216.0 −0.687894 −0.343947 0.938989i $$-0.611764\pi$$
−0.343947 + 0.938989i $$0.611764\pi$$
$$644$$ 11340.0 0.693880
$$645$$ 5166.00 0.315366
$$646$$ −1110.00 −0.0676043
$$647$$ −2536.00 −0.154097 −0.0770483 0.997027i $$-0.524550\pi$$
−0.0770483 + 0.997027i $$0.524550\pi$$
$$648$$ −1215.00 −0.0736570
$$649$$ −12672.0 −0.766440
$$650$$ 0 0
$$651$$ −5880.00 −0.354002
$$652$$ 11900.0 0.714785
$$653$$ 17730.0 1.06252 0.531262 0.847207i $$-0.321718\pi$$
0.531262 + 0.847207i $$0.321718\pi$$
$$654$$ 3102.00 0.185471
$$655$$ −3920.00 −0.233843
$$656$$ −11685.0 −0.695461
$$657$$ 7245.00 0.430220
$$658$$ 4620.00 0.273718
$$659$$ 18920.0 1.11839 0.559195 0.829036i $$-0.311110\pi$$
0.559195 + 0.829036i $$0.311110\pi$$
$$660$$ 3234.00 0.190732
$$661$$ −5241.00 −0.308398 −0.154199 0.988040i $$-0.549280\pi$$
−0.154199 + 0.988040i $$0.549280\pi$$
$$662$$ −1028.00 −0.0603540
$$663$$ 0 0
$$664$$ 7770.00 0.454118
$$665$$ 2100.00 0.122458
$$666$$ −117.000 −0.00680729
$$667$$ 18306.0 1.06269
$$668$$ 25760.0 1.49204
$$669$$ −12216.0 −0.705976
$$670$$ 1414.00 0.0815337
$$671$$ −13970.0 −0.803735
$$672$$ 4830.00 0.277264
$$673$$ 20467.0 1.17228 0.586140 0.810210i $$-0.300647\pi$$
0.586140 + 0.810210i $$0.300647\pi$$
$$674$$ 2487.00 0.142130
$$675$$ −2052.00 −0.117010
$$676$$ 0 0
$$677$$ −70.0000 −0.00397388 −0.00198694 0.999998i $$-0.500632\pi$$
−0.00198694 + 0.999998i $$0.500632\pi$$
$$678$$ 3231.00 0.183017
$$679$$ 12020.0 0.679360
$$680$$ 3885.00 0.219093
$$681$$ 17382.0 0.978091
$$682$$ −4312.00 −0.242104
$$683$$ −6432.00 −0.360342 −0.180171 0.983635i $$-0.557665\pi$$
−0.180171 + 0.983635i $$0.557665\pi$$
$$684$$ 1890.00 0.105652
$$685$$ −3633.00 −0.202642
$$686$$ −5860.00 −0.326146
$$687$$ −19446.0 −1.07993
$$688$$ −10086.0 −0.558903
$$689$$ 0 0
$$690$$ 3402.00 0.187698
$$691$$ 6666.00 0.366985 0.183492 0.983021i $$-0.441260\pi$$
0.183492 + 0.983021i $$0.441260\pi$$
$$692$$ −29022.0 −1.59429
$$693$$ 1980.00 0.108534
$$694$$ −2850.00 −0.155885
$$695$$ 2436.00 0.132954
$$696$$ 5085.00 0.276935
$$697$$ −10545.0 −0.573056
$$698$$ 2018.00 0.109430
$$699$$ 20670.0 1.11847
$$700$$ 5320.00 0.287253
$$701$$ −14054.0 −0.757221 −0.378611 0.925556i $$-0.623598\pi$$
−0.378611 + 0.925556i $$0.623598\pi$$
$$702$$ 0 0
$$703$$ 390.000 0.0209234
$$704$$ −3674.00 −0.196689
$$705$$ −9702.00 −0.518296
$$706$$ 5287.00 0.281840
$$707$$ −4290.00 −0.228207
$$708$$ 12096.0 0.642084
$$709$$ 71.0000 0.00376088 0.00188044 0.999998i $$-0.499401\pi$$
0.00188044 + 0.999998i $$0.499401\pi$$
$$710$$ −7602.00 −0.401828
$$711$$ 7956.00 0.419653
$$712$$ 2910.00 0.153170
$$713$$ 31752.0 1.66777
$$714$$ 1110.00 0.0581803
$$715$$ 0 0
$$716$$ −25718.0 −1.34236
$$717$$ −7398.00 −0.385332
$$718$$ 7278.00 0.378290
$$719$$ 3936.00 0.204156 0.102078 0.994776i $$-0.467451\pi$$
0.102078 + 0.994776i $$0.467451\pi$$
$$720$$ −2583.00 −0.133698
$$721$$ −13020.0 −0.672524
$$722$$ −5959.00 −0.307162
$$723$$ 10851.0 0.558165
$$724$$ 22981.0 1.17967
$$725$$ 8588.00 0.439931
$$726$$ −2541.00 −0.129897
$$727$$ 34202.0 1.74482 0.872409 0.488777i $$-0.162557\pi$$
0.872409 + 0.488777i $$0.162557\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ −5635.00 −0.285700
$$731$$ −9102.00 −0.460533
$$732$$ 13335.0 0.673328
$$733$$ 27363.0 1.37882 0.689410 0.724371i $$-0.257870\pi$$
0.689410 + 0.724371i $$0.257870\pi$$
$$734$$ −4202.00 −0.211306
$$735$$ 5103.00 0.256091
$$736$$ −26082.0 −1.30624
$$737$$ −4444.00 −0.222112
$$738$$ −2565.00 −0.127939
$$739$$ −21776.0 −1.08396 −0.541978 0.840393i $$-0.682324\pi$$
−0.541978 + 0.840393i $$0.682324\pi$$
$$740$$ −637.000 −0.0316440
$$741$$ 0 0
$$742$$ −5370.00 −0.265686
$$743$$ −2484.00 −0.122650 −0.0613251 0.998118i $$-0.519533\pi$$
−0.0613251 + 0.998118i $$0.519533\pi$$
$$744$$ 8820.00 0.434619
$$745$$ −4515.00 −0.222036
$$746$$ −1583.00 −0.0776914
$$747$$ −4662.00 −0.228345
$$748$$ −5698.00 −0.278529
$$749$$ −13380.0 −0.652730
$$750$$ 4221.00 0.205506
$$751$$ 32906.0 1.59888 0.799439 0.600748i $$-0.205130\pi$$
0.799439 + 0.600748i $$0.205130\pi$$
$$752$$ 18942.0 0.918542
$$753$$ 14580.0 0.705611
$$754$$ 0 0
$$755$$ 20398.0 0.983257
$$756$$ −1890.00 −0.0909241
$$757$$ −3914.00 −0.187922 −0.0939609 0.995576i $$-0.529953\pi$$
−0.0939609 + 0.995576i $$0.529953\pi$$
$$758$$ 2052.00 0.0983272
$$759$$ −10692.0 −0.511324
$$760$$ −3150.00 −0.150345
$$761$$ 33038.0 1.57375 0.786877 0.617110i $$-0.211697\pi$$
0.786877 + 0.617110i $$0.211697\pi$$
$$762$$ −2964.00 −0.140911
$$763$$ 10340.0 0.490607
$$764$$ 4172.00 0.197562
$$765$$ −2331.00 −0.110167
$$766$$ 6872.00 0.324145
$$767$$ 0 0
$$768$$ −357.000 −0.0167736
$$769$$ 17586.0 0.824665 0.412332 0.911033i $$-0.364714\pi$$
0.412332 + 0.911033i $$0.364714\pi$$
$$770$$ −1540.00 −0.0720750
$$771$$ 1695.00 0.0791750
$$772$$ −2751.00 −0.128252
$$773$$ −18314.0 −0.852146 −0.426073 0.904689i $$-0.640103\pi$$
−0.426073 + 0.904689i $$0.640103\pi$$
$$774$$ −2214.00 −0.102817
$$775$$ 14896.0 0.690426
$$776$$ −18030.0 −0.834071
$$777$$ −390.000 −0.0180067
$$778$$ −11653.0 −0.536993
$$779$$ 8550.00 0.393242
$$780$$ 0 0
$$781$$ 23892.0 1.09465
$$782$$ −5994.00 −0.274098
$$783$$ −3051.00 −0.139251
$$784$$ −9963.00 −0.453854
$$785$$ 14553.0 0.661680
$$786$$ 1680.00 0.0762387
$$787$$ 42068.0 1.90542 0.952708 0.303888i $$-0.0982847\pi$$
0.952708 + 0.303888i $$0.0982847\pi$$
$$788$$ −24654.0 −1.11455
$$789$$ −1494.00 −0.0674117
$$790$$ −6188.00 −0.278682
$$791$$ 10770.0 0.484118
$$792$$ −2970.00 −0.133250
$$793$$ 0 0
$$794$$ −6134.00 −0.274166
$$795$$ 11277.0 0.503087
$$796$$ −14126.0 −0.628998
$$797$$ −4282.00 −0.190309 −0.0951545 0.995463i $$-0.530335\pi$$
−0.0951545 + 0.995463i $$0.530335\pi$$
$$798$$ −900.000 −0.0399244
$$799$$ 17094.0 0.756874
$$800$$ −12236.0 −0.540760
$$801$$ −1746.00 −0.0770186
$$802$$ 10795.0 0.475293
$$803$$ 17710.0 0.778297
$$804$$ 4242.00 0.186074
$$805$$ 11340.0 0.496500
$$806$$ 0 0
$$807$$ 16638.0 0.725756
$$808$$ 6435.00 0.280176
$$809$$ 40221.0 1.74795 0.873977 0.485967i $$-0.161532\pi$$
0.873977 + 0.485967i $$0.161532\pi$$
$$810$$ −567.000 −0.0245955
$$811$$ 7084.00 0.306724 0.153362 0.988170i $$-0.450990\pi$$
0.153362 + 0.988170i $$0.450990\pi$$
$$812$$ 7910.00 0.341855
$$813$$ 6768.00 0.291961
$$814$$ −286.000 −0.0123149
$$815$$ 11900.0 0.511459
$$816$$ 4551.00 0.195241
$$817$$ 7380.00 0.316026
$$818$$ 8489.00 0.362850
$$819$$ 0 0
$$820$$ −13965.0 −0.594730
$$821$$ −17338.0 −0.737028 −0.368514 0.929622i $$-0.620133\pi$$
−0.368514 + 0.929622i $$0.620133\pi$$
$$822$$ 1557.00 0.0660664
$$823$$ 35496.0 1.50342 0.751709 0.659495i $$-0.229230\pi$$
0.751709 + 0.659495i $$0.229230\pi$$
$$824$$ 19530.0 0.825679
$$825$$ −5016.00 −0.211678
$$826$$ −5760.00 −0.242634
$$827$$ 14992.0 0.630378 0.315189 0.949029i $$-0.397932\pi$$
0.315189 + 0.949029i $$0.397932\pi$$
$$828$$ 10206.0 0.428361
$$829$$ −20659.0 −0.865521 −0.432760 0.901509i $$-0.642460\pi$$
−0.432760 + 0.901509i $$0.642460\pi$$
$$830$$ 3626.00 0.151639
$$831$$ 6927.00 0.289164
$$832$$ 0 0
$$833$$ −8991.00 −0.373973
$$834$$ −1044.00 −0.0433462
$$835$$ 25760.0 1.06762
$$836$$ 4620.00 0.191132
$$837$$ −5292.00 −0.218540
$$838$$ 1496.00 0.0616688
$$839$$ −28716.0 −1.18163 −0.590814 0.806808i $$-0.701193\pi$$
−0.590814 + 0.806808i $$0.701193\pi$$
$$840$$ 3150.00 0.129387
$$841$$ −11620.0 −0.476444
$$842$$ 11695.0 0.478665
$$843$$ −17499.0 −0.714944
$$844$$ −1120.00 −0.0456777
$$845$$ 0 0
$$846$$ 4158.00 0.168978
$$847$$ −8470.00 −0.343604
$$848$$ −22017.0 −0.891588
$$849$$ 4950.00 0.200098
$$850$$ −2812.00 −0.113472
$$851$$ 2106.00 0.0848328
$$852$$ −22806.0 −0.917043
$$853$$ −13377.0 −0.536952 −0.268476 0.963286i $$-0.586520\pi$$
−0.268476 + 0.963286i $$0.586520\pi$$
$$854$$ −6350.00 −0.254441
$$855$$ 1890.00 0.0755984
$$856$$ 20070.0 0.801377
$$857$$ −27419.0 −1.09290 −0.546450 0.837492i $$-0.684021\pi$$
−0.546450 + 0.837492i $$0.684021\pi$$
$$858$$ 0 0
$$859$$ 2422.00 0.0962021 0.0481010 0.998842i $$-0.484683\pi$$
0.0481010 + 0.998842i $$0.484683\pi$$
$$860$$ −12054.0 −0.477951
$$861$$ −8550.00 −0.338424
$$862$$ −10590.0 −0.418442
$$863$$ 34522.0 1.36169 0.680847 0.732425i $$-0.261612\pi$$
0.680847 + 0.732425i $$0.261612\pi$$
$$864$$ 4347.00 0.171167
$$865$$ −29022.0 −1.14078
$$866$$ −13949.0 −0.547351
$$867$$ −10632.0 −0.416472
$$868$$ 13720.0 0.536506
$$869$$ 19448.0 0.759181
$$870$$ 2373.00 0.0924738
$$871$$ 0 0
$$872$$ −15510.0 −0.602334
$$873$$ 10818.0 0.419397
$$874$$ 4860.00 0.188091
$$875$$ 14070.0 0.543603
$$876$$ −16905.0 −0.652017
$$877$$ −13733.0 −0.528769 −0.264385 0.964417i $$-0.585169\pi$$
−0.264385 + 0.964417i $$0.585169\pi$$
$$878$$ −10726.0 −0.412284
$$879$$ −8973.00 −0.344314
$$880$$ −6314.00 −0.241869
$$881$$ −22759.0 −0.870341 −0.435170 0.900348i $$-0.643312\pi$$
−0.435170 + 0.900348i $$0.643312\pi$$
$$882$$ −2187.00 −0.0834922
$$883$$ −2168.00 −0.0826263 −0.0413131 0.999146i $$-0.513154\pi$$
−0.0413131 + 0.999146i $$0.513154\pi$$
$$884$$ 0 0
$$885$$ 12096.0 0.459438
$$886$$ 16228.0 0.615339
$$887$$ 15888.0 0.601428 0.300714 0.953714i $$-0.402775\pi$$
0.300714 + 0.953714i $$0.402775\pi$$
$$888$$ 585.000 0.0221073
$$889$$ −9880.00 −0.372739
$$890$$ 1358.00 0.0511464
$$891$$ 1782.00 0.0670025
$$892$$ 28504.0 1.06994
$$893$$ −13860.0 −0.519381
$$894$$ 1935.00 0.0723894
$$895$$ −25718.0 −0.960512
$$896$$ −14550.0 −0.542502
$$897$$ 0 0
$$898$$ −7538.00 −0.280118
$$899$$ 22148.0 0.821665
$$900$$ 4788.00 0.177333
$$901$$ −19869.0 −0.734664
$$902$$ −6270.00 −0.231450
$$903$$ −7380.00 −0.271972
$$904$$ −16155.0 −0.594366
$$905$$ 22981.0 0.844104
$$906$$ −8742.00 −0.320567
$$907$$ −11628.0 −0.425691 −0.212845 0.977086i $$-0.568273\pi$$
−0.212845 + 0.977086i $$0.568273\pi$$
$$908$$ −40558.0 −1.48234
$$909$$ −3861.00 −0.140882
$$910$$ 0 0
$$911$$ −12584.0 −0.457658 −0.228829 0.973467i $$-0.573490\pi$$
−0.228829 + 0.973467i $$0.573490\pi$$
$$912$$ −3690.00 −0.133978
$$913$$ −11396.0 −0.413092
$$914$$ −15539.0 −0.562346
$$915$$ 13335.0 0.481794
$$916$$ 45374.0 1.63668
$$917$$ 5600.00 0.201667
$$918$$ 999.000 0.0359171
$$919$$ 17184.0 0.616809 0.308405 0.951255i $$-0.400205\pi$$
0.308405 + 0.951255i $$0.400205\pi$$
$$920$$ −17010.0 −0.609569
$$921$$ 7266.00 0.259960
$$922$$ −4811.00 −0.171846
$$923$$ 0 0
$$924$$ −4620.00 −0.164488
$$925$$ 988.000 0.0351192
$$926$$ −562.000 −0.0199443
$$927$$ −11718.0 −0.415178
$$928$$ −18193.0 −0.643550
$$929$$ −12777.0 −0.451238 −0.225619 0.974216i $$-0.572440\pi$$
−0.225619 + 0.974216i $$0.572440\pi$$
$$930$$ 4116.00 0.145128
$$931$$ 7290.00 0.256627
$$932$$ −48230.0 −1.69509
$$933$$ −10206.0 −0.358124
$$934$$ 4914.00 0.172153
$$935$$ −5698.00 −0.199299
$$936$$ 0 0
$$937$$ 9191.00 0.320445 0.160222 0.987081i $$-0.448779\pi$$
0.160222 + 0.987081i $$0.448779\pi$$
$$938$$ −2020.00 −0.0703149
$$939$$ 6930.00 0.240843
$$940$$ 22638.0 0.785500
$$941$$ −50498.0 −1.74940 −0.874701 0.484662i $$-0.838942\pi$$
−0.874701 + 0.484662i $$0.838942\pi$$
$$942$$ −6237.00 −0.215724
$$943$$ 46170.0 1.59438
$$944$$ −23616.0 −0.814232
$$945$$ −1890.00 −0.0650600
$$946$$ −5412.00 −0.186003
$$947$$ −1560.00 −0.0535303 −0.0267651 0.999642i $$-0.508521\pi$$
−0.0267651 + 0.999642i $$0.508521\pi$$
$$948$$ −18564.0 −0.636003
$$949$$ 0 0
$$950$$ 2280.00 0.0778663
$$951$$ 771.000 0.0262896
$$952$$ −5550.00 −0.188946
$$953$$ −21498.0 −0.730733 −0.365366 0.930864i $$-0.619056\pi$$
−0.365366 + 0.930864i $$0.619056\pi$$
$$954$$ −4833.00 −0.164019
$$955$$ 4172.00 0.141364
$$956$$ 17262.0 0.583988
$$957$$ −7458.00 −0.251915
$$958$$ 3600.00 0.121410
$$959$$ 5190.00 0.174759
$$960$$ 3507.00 0.117904
$$961$$ 8625.00 0.289517
$$962$$ 0 0
$$963$$ −12042.0 −0.402957
$$964$$ −25319.0 −0.845923
$$965$$ −2751.00 −0.0917698
$$966$$ −4860.00 −0.161872
$$967$$ 418.000 0.0139007 0.00695035 0.999976i $$-0.497788\pi$$
0.00695035 + 0.999976i $$0.497788\pi$$
$$968$$ 12705.0 0.421853
$$969$$ −3330.00 −0.110397
$$970$$ −8414.00 −0.278513
$$971$$ 18132.0 0.599262 0.299631 0.954055i $$-0.403136\pi$$
0.299631 + 0.954055i $$0.403136\pi$$
$$972$$ −1701.00 −0.0561313
$$973$$ −3480.00 −0.114659
$$974$$ 17130.0 0.563532
$$975$$ 0 0
$$976$$ −26035.0 −0.853853
$$977$$ −12501.0 −0.409358 −0.204679 0.978829i $$-0.565615\pi$$
−0.204679 + 0.978829i $$0.565615\pi$$
$$978$$ −5100.00 −0.166748
$$979$$ −4268.00 −0.139332
$$980$$ −11907.0 −0.388118
$$981$$ 9306.00 0.302872
$$982$$ −11838.0 −0.384690
$$983$$ 43708.0 1.41818 0.709089 0.705119i $$-0.249106\pi$$
0.709089 + 0.705119i $$0.249106\pi$$
$$984$$ 12825.0 0.415494
$$985$$ −24654.0 −0.797504
$$986$$ −4181.00 −0.135041
$$987$$ 13860.0 0.446979
$$988$$ 0 0
$$989$$ 39852.0 1.28131
$$990$$ −1386.00 −0.0444949
$$991$$ −39614.0 −1.26981 −0.634904 0.772591i $$-0.718960\pi$$
−0.634904 + 0.772591i $$0.718960\pi$$
$$992$$ −31556.0 −1.00998
$$993$$ −3084.00 −0.0985577
$$994$$ 10860.0 0.346538
$$995$$ −14126.0 −0.450075
$$996$$ 10878.0 0.346067
$$997$$ −36503.0 −1.15954 −0.579770 0.814780i $$-0.696858\pi$$
−0.579770 + 0.814780i $$0.696858\pi$$
$$998$$ −8976.00 −0.284700
$$999$$ −351.000 −0.0111163
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 507.4.a.d.1.1 1
3.2 odd 2 1521.4.a.e.1.1 1
13.4 even 6 39.4.e.b.16.1 2
13.5 odd 4 507.4.b.d.337.1 2
13.8 odd 4 507.4.b.d.337.2 2
13.10 even 6 39.4.e.b.22.1 yes 2
13.12 even 2 507.4.a.b.1.1 1
39.17 odd 6 117.4.g.a.55.1 2
39.23 odd 6 117.4.g.a.100.1 2
39.38 odd 2 1521.4.a.h.1.1 1
52.23 odd 6 624.4.q.c.529.1 2
52.43 odd 6 624.4.q.c.289.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.b.16.1 2 13.4 even 6
39.4.e.b.22.1 yes 2 13.10 even 6
117.4.g.a.55.1 2 39.17 odd 6
117.4.g.a.100.1 2 39.23 odd 6
507.4.a.b.1.1 1 13.12 even 2
507.4.a.d.1.1 1 1.1 even 1 trivial
507.4.b.d.337.1 2 13.5 odd 4
507.4.b.d.337.2 2 13.8 odd 4
624.4.q.c.289.1 2 52.43 odd 6
624.4.q.c.529.1 2 52.23 odd 6
1521.4.a.e.1.1 1 3.2 odd 2
1521.4.a.h.1.1 1 39.38 odd 2