# Properties

 Label 75.2.a.a.1.1 Level $75$ Weight $2$ Character 75.1 Self dual yes Analytic conductor $0.599$ Analytic rank $0$ 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] = [75,2,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 75.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: $$0.598878015160$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 75.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +2.00000 q^{12} -1.00000 q^{13} -6.00000 q^{14} -4.00000 q^{16} -2.00000 q^{17} -2.00000 q^{18} -5.00000 q^{19} +3.00000 q^{21} -4.00000 q^{22} -6.00000 q^{23} +2.00000 q^{26} +1.00000 q^{27} +6.00000 q^{28} +10.0000 q^{29} -3.00000 q^{31} +8.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} +2.00000 q^{36} -2.00000 q^{37} +10.0000 q^{38} -1.00000 q^{39} -8.00000 q^{41} -6.00000 q^{42} -1.00000 q^{43} +4.00000 q^{44} +12.0000 q^{46} -2.00000 q^{47} -4.00000 q^{48} +2.00000 q^{49} -2.00000 q^{51} -2.00000 q^{52} +4.00000 q^{53} -2.00000 q^{54} -5.00000 q^{57} -20.0000 q^{58} -10.0000 q^{59} +7.00000 q^{61} +6.00000 q^{62} +3.00000 q^{63} -8.00000 q^{64} -4.00000 q^{66} +3.00000 q^{67} -4.00000 q^{68} -6.00000 q^{69} -8.00000 q^{71} +14.0000 q^{73} +4.00000 q^{74} -10.0000 q^{76} +6.00000 q^{77} +2.00000 q^{78} +1.00000 q^{81} +16.0000 q^{82} -6.00000 q^{83} +6.00000 q^{84} +2.00000 q^{86} +10.0000 q^{87} -3.00000 q^{91} -12.0000 q^{92} -3.00000 q^{93} +4.00000 q^{94} +8.00000 q^{96} -17.0000 q^{97} -4.00000 q^{98} +2.00000 q^{99} +O(q^{100})$$

## 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 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ −2.00000 −0.816497
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 2.00000 0.577350
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ −6.00000 −1.60357
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ −2.00000 −0.471405
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ −4.00000 −0.852803
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 1.00000 0.192450
$$28$$ 6.00000 1.13389
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 8.00000 1.41421
$$33$$ 2.00000 0.348155
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 10.0000 1.62221
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ −6.00000 −0.925820
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 12.0000 1.76930
$$47$$ −2.00000 −0.291730 −0.145865 0.989305i $$-0.546597\pi$$
−0.145865 + 0.989305i $$0.546597\pi$$
$$48$$ −4.00000 −0.577350
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ −2.00000 −0.277350
$$53$$ 4.00000 0.549442 0.274721 0.961524i $$-0.411414\pi$$
0.274721 + 0.961524i $$0.411414\pi$$
$$54$$ −2.00000 −0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −5.00000 −0.662266
$$58$$ −20.0000 −2.62613
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 3.00000 0.377964
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ 3.00000 0.366508 0.183254 0.983066i $$-0.441337\pi$$
0.183254 + 0.983066i $$0.441337\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ −10.0000 −1.14708
$$77$$ 6.00000 0.683763
$$78$$ 2.00000 0.226455
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 16.0000 1.76690
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 6.00000 0.654654
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 10.0000 1.07211
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ −12.0000 −1.25109
$$93$$ −3.00000 −0.311086
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 8.00000 0.816497
$$97$$ −17.0000 −1.72609 −0.863044 0.505128i $$-0.831445\pi$$
−0.863044 + 0.505128i $$0.831445\pi$$
$$98$$ −4.00000 −0.404061
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 4.00000 0.396059
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −8.00000 −0.777029
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 2.00000 0.192450
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ −12.0000 −1.13389
$$113$$ 4.00000 0.376288 0.188144 0.982141i $$-0.439753\pi$$
0.188144 + 0.982141i $$0.439753\pi$$
$$114$$ 10.0000 0.936586
$$115$$ 0 0
$$116$$ 20.0000 1.85695
$$117$$ −1.00000 −0.0924500
$$118$$ 20.0000 1.84115
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −14.0000 −1.26750
$$123$$ −8.00000 −0.721336
$$124$$ −6.00000 −0.538816
$$125$$ 0 0
$$126$$ −6.00000 −0.534522
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ −1.00000 −0.0880451
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 4.00000 0.348155
$$133$$ −15.0000 −1.30066
$$134$$ −6.00000 −0.518321
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 12.0000 1.02151
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ −2.00000 −0.168430
$$142$$ 16.0000 1.34269
$$143$$ −2.00000 −0.167248
$$144$$ −4.00000 −0.333333
$$145$$ 0 0
$$146$$ −28.0000 −2.31730
$$147$$ 2.00000 0.164957
$$148$$ −4.00000 −0.328798
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 7.00000 0.569652 0.284826 0.958579i $$-0.408064\pi$$
0.284826 + 0.958579i $$0.408064\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ −12.0000 −0.966988
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ 13.0000 1.03751 0.518756 0.854922i $$-0.326395\pi$$
0.518756 + 0.854922i $$0.326395\pi$$
$$158$$ 0 0
$$159$$ 4.00000 0.317221
$$160$$ 0 0
$$161$$ −18.0000 −1.41860
$$162$$ −2.00000 −0.157135
$$163$$ −11.0000 −0.861586 −0.430793 0.902451i $$-0.641766\pi$$
−0.430793 + 0.902451i $$0.641766\pi$$
$$164$$ −16.0000 −1.24939
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −5.00000 −0.382360
$$172$$ −2.00000 −0.152499
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ −20.0000 −1.51620
$$175$$ 0 0
$$176$$ −8.00000 −0.603023
$$177$$ −10.0000 −0.751646
$$178$$ 0 0
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ 17.0000 1.26360 0.631800 0.775131i $$-0.282316\pi$$
0.631800 + 0.775131i $$0.282316\pi$$
$$182$$ 6.00000 0.444750
$$183$$ 7.00000 0.517455
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 6.00000 0.439941
$$187$$ −4.00000 −0.292509
$$188$$ −4.00000 −0.291730
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ 22.0000 1.59186 0.795932 0.605386i $$-0.206981\pi$$
0.795932 + 0.605386i $$0.206981\pi$$
$$192$$ −8.00000 −0.577350
$$193$$ −11.0000 −0.791797 −0.395899 0.918294i $$-0.629567\pi$$
−0.395899 + 0.918294i $$0.629567\pi$$
$$194$$ 34.0000 2.44106
$$195$$ 0 0
$$196$$ 4.00000 0.285714
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ −5.00000 −0.354441 −0.177220 0.984171i $$-0.556711\pi$$
−0.177220 + 0.984171i $$0.556711\pi$$
$$200$$ 0 0
$$201$$ 3.00000 0.211604
$$202$$ −24.0000 −1.68863
$$203$$ 30.0000 2.10559
$$204$$ −4.00000 −0.280056
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ −6.00000 −0.417029
$$208$$ 4.00000 0.277350
$$209$$ −10.0000 −0.691714
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 8.00000 0.549442
$$213$$ −8.00000 −0.548151
$$214$$ 24.0000 1.64061
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −9.00000 −0.610960
$$218$$ −10.0000 −0.677285
$$219$$ 14.0000 0.946032
$$220$$ 0 0
$$221$$ 2.00000 0.134535
$$222$$ 4.00000 0.268462
$$223$$ 19.0000 1.27233 0.636167 0.771551i $$-0.280519\pi$$
0.636167 + 0.771551i $$0.280519\pi$$
$$224$$ 24.0000 1.60357
$$225$$ 0 0
$$226$$ −8.00000 −0.532152
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ −10.0000 −0.662266
$$229$$ −15.0000 −0.991228 −0.495614 0.868543i $$-0.665057\pi$$
−0.495614 + 0.868543i $$0.665057\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 0 0
$$233$$ 24.0000 1.57229 0.786146 0.618041i $$-0.212073\pi$$
0.786146 + 0.618041i $$0.212073\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ −20.0000 −1.30189
$$237$$ 0 0
$$238$$ 12.0000 0.777844
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ 0 0
$$241$$ −23.0000 −1.48156 −0.740780 0.671748i $$-0.765544\pi$$
−0.740780 + 0.671748i $$0.765544\pi$$
$$242$$ 14.0000 0.899954
$$243$$ 1.00000 0.0641500
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 16.0000 1.02012
$$247$$ 5.00000 0.318142
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 6.00000 0.377964
$$253$$ −12.0000 −0.754434
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 2.00000 0.124515
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ −24.0000 −1.48272
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 30.0000 1.83942
$$267$$ 0 0
$$268$$ 6.00000 0.366508
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 8.00000 0.485071
$$273$$ −3.00000 −0.181568
$$274$$ −36.0000 −2.17484
$$275$$ 0 0
$$276$$ −12.0000 −0.722315
$$277$$ 3.00000 0.180253 0.0901263 0.995930i $$-0.471273\pi$$
0.0901263 + 0.995930i $$0.471273\pi$$
$$278$$ −40.0000 −2.39904
$$279$$ −3.00000 −0.179605
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 4.00000 0.238197
$$283$$ 9.00000 0.534994 0.267497 0.963559i $$-0.413803\pi$$
0.267497 + 0.963559i $$0.413803\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ −24.0000 −1.41668
$$288$$ 8.00000 0.471405
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −17.0000 −0.996558
$$292$$ 28.0000 1.63858
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ −4.00000 −0.233285
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 2.00000 0.116052
$$298$$ −20.0000 −1.15857
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ −14.0000 −0.805609
$$303$$ 12.0000 0.689382
$$304$$ 20.0000 1.14708
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ −7.00000 −0.399511 −0.199756 0.979846i $$-0.564015\pi$$
−0.199756 + 0.979846i $$0.564015\pi$$
$$308$$ 12.0000 0.683763
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ −11.0000 −0.621757 −0.310878 0.950450i $$-0.600623\pi$$
−0.310878 + 0.950450i $$0.600623\pi$$
$$314$$ −26.0000 −1.46726
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.00000 0.449325 0.224662 0.974437i $$-0.427872\pi$$
0.224662 + 0.974437i $$0.427872\pi$$
$$318$$ −8.00000 −0.448618
$$319$$ 20.0000 1.11979
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 36.0000 2.00620
$$323$$ 10.0000 0.556415
$$324$$ 2.00000 0.111111
$$325$$ 0 0
$$326$$ 22.0000 1.21847
$$327$$ 5.00000 0.276501
$$328$$ 0 0
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ −2.00000 −0.109599
$$334$$ 24.0000 1.31322
$$335$$ 0 0
$$336$$ −12.0000 −0.654654
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 24.0000 1.30543
$$339$$ 4.00000 0.217250
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 10.0000 0.540738
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 12.0000 0.645124
$$347$$ −2.00000 −0.107366 −0.0536828 0.998558i $$-0.517096\pi$$
−0.0536828 + 0.998558i $$0.517096\pi$$
$$348$$ 20.0000 1.07211
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 16.0000 0.852803
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 20.0000 1.06299
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −6.00000 −0.317554
$$358$$ 20.0000 1.05703
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ −34.0000 −1.78700
$$363$$ −7.00000 −0.367405
$$364$$ −6.00000 −0.314485
$$365$$ 0 0
$$366$$ −14.0000 −0.731792
$$367$$ −27.0000 −1.40939 −0.704694 0.709511i $$-0.748916\pi$$
−0.704694 + 0.709511i $$0.748916\pi$$
$$368$$ 24.0000 1.25109
$$369$$ −8.00000 −0.416463
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ −6.00000 −0.311086
$$373$$ 29.0000 1.50156 0.750782 0.660551i $$-0.229677\pi$$
0.750782 + 0.660551i $$0.229677\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −10.0000 −0.515026
$$378$$ −6.00000 −0.308607
$$379$$ 25.0000 1.28416 0.642082 0.766636i $$-0.278071\pi$$
0.642082 + 0.766636i $$0.278071\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ −44.0000 −2.25124
$$383$$ −36.0000 −1.83951 −0.919757 0.392488i $$-0.871614\pi$$
−0.919757 + 0.392488i $$0.871614\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ −1.00000 −0.0508329
$$388$$ −34.0000 −1.72609
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ −36.0000 −1.81365
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ −7.00000 −0.351320 −0.175660 0.984451i $$-0.556206\pi$$
−0.175660 + 0.984451i $$0.556206\pi$$
$$398$$ 10.0000 0.501255
$$399$$ −15.0000 −0.750939
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ −6.00000 −0.299253
$$403$$ 3.00000 0.149441
$$404$$ 24.0000 1.19404
$$405$$ 0 0
$$406$$ −60.0000 −2.97775
$$407$$ −4.00000 −0.198273
$$408$$ 0 0
$$409$$ 5.00000 0.247234 0.123617 0.992330i $$-0.460551\pi$$
0.123617 + 0.992330i $$0.460551\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ 8.00000 0.394132
$$413$$ −30.0000 −1.47620
$$414$$ 12.0000 0.589768
$$415$$ 0 0
$$416$$ −8.00000 −0.392232
$$417$$ 20.0000 0.979404
$$418$$ 20.0000 0.978232
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 26.0000 1.26566
$$423$$ −2.00000 −0.0972433
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 16.0000 0.775203
$$427$$ 21.0000 1.01626
$$428$$ −24.0000 −1.16008
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ 29.0000 1.39365 0.696826 0.717241i $$-0.254595\pi$$
0.696826 + 0.717241i $$0.254595\pi$$
$$434$$ 18.0000 0.864028
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ 30.0000 1.43509
$$438$$ −28.0000 −1.33789
$$439$$ −35.0000 −1.67046 −0.835229 0.549902i $$-0.814665\pi$$
−0.835229 + 0.549902i $$0.814665\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ −4.00000 −0.190261
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ −38.0000 −1.79935
$$447$$ 10.0000 0.472984
$$448$$ −24.0000 −1.13389
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ 0 0
$$451$$ −16.0000 −0.753411
$$452$$ 8.00000 0.376288
$$453$$ 7.00000 0.328889
$$454$$ −16.0000 −0.750917
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 30.0000 1.40181
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ −12.0000 −0.558291
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ −40.0000 −1.85695
$$465$$ 0 0
$$466$$ −48.0000 −2.22356
$$467$$ 38.0000 1.75843 0.879215 0.476425i $$-0.158068\pi$$
0.879215 + 0.476425i $$0.158068\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 9.00000 0.415581
$$470$$ 0 0
$$471$$ 13.0000 0.599008
$$472$$ 0 0
$$473$$ −2.00000 −0.0919601
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −12.0000 −0.550019
$$477$$ 4.00000 0.183147
$$478$$ −40.0000 −1.82956
$$479$$ 30.0000 1.37073 0.685367 0.728197i $$-0.259642\pi$$
0.685367 + 0.728197i $$0.259642\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 46.0000 2.09524
$$483$$ −18.0000 −0.819028
$$484$$ −14.0000 −0.636364
$$485$$ 0 0
$$486$$ −2.00000 −0.0907218
$$487$$ 13.0000 0.589086 0.294543 0.955638i $$-0.404833\pi$$
0.294543 + 0.955638i $$0.404833\pi$$
$$488$$ 0 0
$$489$$ −11.0000 −0.497437
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ −16.0000 −0.721336
$$493$$ −20.0000 −0.900755
$$494$$ −10.0000 −0.449921
$$495$$ 0 0
$$496$$ 12.0000 0.538816
$$497$$ −24.0000 −1.07655
$$498$$ 12.0000 0.537733
$$499$$ −5.00000 −0.223831 −0.111915 0.993718i $$-0.535699\pi$$
−0.111915 + 0.993718i $$0.535699\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ −24.0000 −1.07117
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 24.0000 1.06693
$$507$$ −12.0000 −0.532939
$$508$$ 16.0000 0.709885
$$509$$ −10.0000 −0.443242 −0.221621 0.975133i $$-0.571135\pi$$
−0.221621 + 0.975133i $$0.571135\pi$$
$$510$$ 0 0
$$511$$ 42.0000 1.85797
$$512$$ −32.0000 −1.41421
$$513$$ −5.00000 −0.220755
$$514$$ 24.0000 1.05859
$$515$$ 0 0
$$516$$ −2.00000 −0.0880451
$$517$$ −4.00000 −0.175920
$$518$$ 12.0000 0.527250
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ −20.0000 −0.875376
$$523$$ −31.0000 −1.35554 −0.677768 0.735276i $$-0.737052\pi$$
−0.677768 + 0.735276i $$0.737052\pi$$
$$524$$ 24.0000 1.04844
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ 6.00000 0.261364
$$528$$ −8.00000 −0.348155
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −10.0000 −0.433963
$$532$$ −30.0000 −1.30066
$$533$$ 8.00000 0.346518
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −10.0000 −0.431532
$$538$$ 20.0000 0.862261
$$539$$ 4.00000 0.172292
$$540$$ 0 0
$$541$$ −3.00000 −0.128980 −0.0644900 0.997918i $$-0.520542\pi$$
−0.0644900 + 0.997918i $$0.520542\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 17.0000 0.729540
$$544$$ −16.0000 −0.685994
$$545$$ 0 0
$$546$$ 6.00000 0.256776
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 36.0000 1.53784
$$549$$ 7.00000 0.298753
$$550$$ 0 0
$$551$$ −50.0000 −2.13007
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −6.00000 −0.254916
$$555$$ 0 0
$$556$$ 40.0000 1.69638
$$557$$ −42.0000 −1.77960 −0.889799 0.456354i $$-0.849155\pi$$
−0.889799 + 0.456354i $$0.849155\pi$$
$$558$$ 6.00000 0.254000
$$559$$ 1.00000 0.0422955
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 36.0000 1.51857
$$563$$ −6.00000 −0.252870 −0.126435 0.991975i $$-0.540353\pi$$
−0.126435 + 0.991975i $$0.540353\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ −18.0000 −0.756596
$$567$$ 3.00000 0.125988
$$568$$ 0 0
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ −13.0000 −0.544033 −0.272017 0.962293i $$-0.587691\pi$$
−0.272017 + 0.962293i $$0.587691\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ 22.0000 0.919063
$$574$$ 48.0000 2.00348
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ 13.0000 0.541197 0.270599 0.962692i $$-0.412778\pi$$
0.270599 + 0.962692i $$0.412778\pi$$
$$578$$ 26.0000 1.08146
$$579$$ −11.0000 −0.457144
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ 34.0000 1.40935
$$583$$ 8.00000 0.331326
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 12.0000 0.495715
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 4.00000 0.164957
$$589$$ 15.0000 0.618064
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 8.00000 0.328798
$$593$$ −16.0000 −0.657041 −0.328521 0.944497i $$-0.606550\pi$$
−0.328521 + 0.944497i $$0.606550\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ 20.0000 0.819232
$$597$$ −5.00000 −0.204636
$$598$$ −12.0000 −0.490716
$$599$$ −20.0000 −0.817178 −0.408589 0.912719i $$-0.633979\pi$$
−0.408589 + 0.912719i $$0.633979\pi$$
$$600$$ 0 0
$$601$$ −13.0000 −0.530281 −0.265141 0.964210i $$-0.585418\pi$$
−0.265141 + 0.964210i $$0.585418\pi$$
$$602$$ 6.00000 0.244542
$$603$$ 3.00000 0.122169
$$604$$ 14.0000 0.569652
$$605$$ 0 0
$$606$$ −24.0000 −0.974933
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ −40.0000 −1.62221
$$609$$ 30.0000 1.21566
$$610$$ 0 0
$$611$$ 2.00000 0.0809113
$$612$$ −4.00000 −0.161690
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ 14.0000 0.564994
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.0000 −0.483102 −0.241551 0.970388i $$-0.577656\pi$$
−0.241551 + 0.970388i $$0.577656\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ −25.0000 −1.00483 −0.502417 0.864625i $$-0.667556\pi$$
−0.502417 + 0.864625i $$0.667556\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 36.0000 1.44347
$$623$$ 0 0
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ 22.0000 0.879297
$$627$$ −10.0000 −0.399362
$$628$$ 26.0000 1.03751
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ −23.0000 −0.915616 −0.457808 0.889051i $$-0.651365\pi$$
−0.457808 + 0.889051i $$0.651365\pi$$
$$632$$ 0 0
$$633$$ −13.0000 −0.516704
$$634$$ −16.0000 −0.635441
$$635$$ 0 0
$$636$$ 8.00000 0.317221
$$637$$ −2.00000 −0.0792429
$$638$$ −40.0000 −1.58362
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ 12.0000 0.473972 0.236986 0.971513i $$-0.423841\pi$$
0.236986 + 0.971513i $$0.423841\pi$$
$$642$$ 24.0000 0.947204
$$643$$ −36.0000 −1.41970 −0.709851 0.704352i $$-0.751238\pi$$
−0.709851 + 0.704352i $$0.751238\pi$$
$$644$$ −36.0000 −1.41860
$$645$$ 0 0
$$646$$ −20.0000 −0.786889
$$647$$ 28.0000 1.10079 0.550397 0.834903i $$-0.314476\pi$$
0.550397 + 0.834903i $$0.314476\pi$$
$$648$$ 0 0
$$649$$ −20.0000 −0.785069
$$650$$ 0 0
$$651$$ −9.00000 −0.352738
$$652$$ −22.0000 −0.861586
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ 0 0
$$656$$ 32.0000 1.24939
$$657$$ 14.0000 0.546192
$$658$$ 12.0000 0.467809
$$659$$ −40.0000 −1.55818 −0.779089 0.626913i $$-0.784318\pi$$
−0.779089 + 0.626913i $$0.784318\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ −24.0000 −0.932786
$$663$$ 2.00000 0.0776736
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ −60.0000 −2.32321
$$668$$ −24.0000 −0.928588
$$669$$ 19.0000 0.734582
$$670$$ 0 0
$$671$$ 14.0000 0.540464
$$672$$ 24.0000 0.925820
$$673$$ −6.00000 −0.231283 −0.115642 0.993291i $$-0.536892\pi$$
−0.115642 + 0.993291i $$0.536892\pi$$
$$674$$ −46.0000 −1.77185
$$675$$ 0 0
$$676$$ −24.0000 −0.923077
$$677$$ −42.0000 −1.61419 −0.807096 0.590421i $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ −8.00000 −0.307238
$$679$$ −51.0000 −1.95720
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ 12.0000 0.459504
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ −10.0000 −0.382360
$$685$$ 0 0
$$686$$ 30.0000 1.14541
$$687$$ −15.0000 −0.572286
$$688$$ 4.00000 0.152499
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 6.00000 0.227921
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 16.0000 0.606043
$$698$$ −20.0000 −0.757011
$$699$$ 24.0000 0.907763
$$700$$ 0 0
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ 10.0000 0.377157
$$704$$ −16.0000 −0.603023
$$705$$ 0 0
$$706$$ 12.0000 0.451626
$$707$$ 36.0000 1.35392
$$708$$ −20.0000 −0.751646
$$709$$ 25.0000 0.938895 0.469447 0.882960i $$-0.344453\pi$$
0.469447 + 0.882960i $$0.344453\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 18.0000 0.674105
$$714$$ 12.0000 0.449089
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ 20.0000 0.746914
$$718$$ 0 0
$$719$$ 30.0000 1.11881 0.559406 0.828894i $$-0.311029\pi$$
0.559406 + 0.828894i $$0.311029\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ −12.0000 −0.446594
$$723$$ −23.0000 −0.855379
$$724$$ 34.0000 1.26360
$$725$$ 0 0
$$726$$ 14.0000 0.519589
$$727$$ 43.0000 1.59478 0.797391 0.603463i $$-0.206213\pi$$
0.797391 + 0.603463i $$0.206213\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 2.00000 0.0739727
$$732$$ 14.0000 0.517455
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ 54.0000 1.99318
$$735$$ 0 0
$$736$$ −48.0000 −1.76930
$$737$$ 6.00000 0.221013
$$738$$ 16.0000 0.588968
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 5.00000 0.183680
$$742$$ −24.0000 −0.881068
$$743$$ 4.00000 0.146746 0.0733729 0.997305i $$-0.476624\pi$$
0.0733729 + 0.997305i $$0.476624\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −58.0000 −2.12353
$$747$$ −6.00000 −0.219529
$$748$$ −8.00000 −0.292509
$$749$$ −36.0000 −1.31541
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 12.0000 0.437304
$$754$$ 20.0000 0.728357
$$755$$ 0 0
$$756$$ 6.00000 0.218218
$$757$$ 23.0000 0.835949 0.417975 0.908459i $$-0.362740\pi$$
0.417975 + 0.908459i $$0.362740\pi$$
$$758$$ −50.0000 −1.81608
$$759$$ −12.0000 −0.435572
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ −16.0000 −0.579619
$$763$$ 15.0000 0.543036
$$764$$ 44.0000 1.59186
$$765$$ 0 0
$$766$$ 72.0000 2.60147
$$767$$ 10.0000 0.361079
$$768$$ 16.0000 0.577350
$$769$$ 35.0000 1.26213 0.631066 0.775729i $$-0.282618\pi$$
0.631066 + 0.775729i $$0.282618\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ −22.0000 −0.791797
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −6.00000 −0.215249
$$778$$ 0 0
$$779$$ 40.0000 1.43315
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ −24.0000 −0.858238
$$783$$ 10.0000 0.357371
$$784$$ −8.00000 −0.285714
$$785$$ 0 0
$$786$$ −24.0000 −0.856052
$$787$$ −7.00000 −0.249523 −0.124762 0.992187i $$-0.539817\pi$$
−0.124762 + 0.992187i $$0.539817\pi$$
$$788$$ 36.0000 1.28245
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 0 0
$$793$$ −7.00000 −0.248577
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ −52.0000 −1.84193 −0.920967 0.389640i $$-0.872599\pi$$
−0.920967 + 0.389640i $$0.872599\pi$$
$$798$$ 30.0000 1.06199
$$799$$ 4.00000 0.141510
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −24.0000 −0.847469
$$803$$ 28.0000 0.988099
$$804$$ 6.00000 0.211604
$$805$$ 0 0
$$806$$ −6.00000 −0.211341
$$807$$ −10.0000 −0.352017
$$808$$ 0 0
$$809$$ −20.0000 −0.703163 −0.351581 0.936157i $$-0.614356\pi$$
−0.351581 + 0.936157i $$0.614356\pi$$
$$810$$ 0 0
$$811$$ 27.0000 0.948098 0.474049 0.880498i $$-0.342792\pi$$
0.474049 + 0.880498i $$0.342792\pi$$
$$812$$ 60.0000 2.10559
$$813$$ −8.00000 −0.280572
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ 8.00000 0.280056
$$817$$ 5.00000 0.174928
$$818$$ −10.0000 −0.349642
$$819$$ −3.00000 −0.104828
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ −36.0000 −1.25564
$$823$$ −41.0000 −1.42917 −0.714585 0.699549i $$-0.753384\pi$$
−0.714585 + 0.699549i $$0.753384\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 60.0000 2.08767
$$827$$ 28.0000 0.973655 0.486828 0.873498i $$-0.338154\pi$$
0.486828 + 0.873498i $$0.338154\pi$$
$$828$$ −12.0000 −0.417029
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ 3.00000 0.104069
$$832$$ 8.00000 0.277350
$$833$$ −4.00000 −0.138592
$$834$$ −40.0000 −1.38509
$$835$$ 0 0
$$836$$ −20.0000 −0.691714
$$837$$ −3.00000 −0.103695
$$838$$ 40.0000 1.38178
$$839$$ 10.0000 0.345238 0.172619 0.984989i $$-0.444777\pi$$
0.172619 + 0.984989i $$0.444777\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ −44.0000 −1.51634
$$843$$ −18.0000 −0.619953
$$844$$ −26.0000 −0.894957
$$845$$ 0 0
$$846$$ 4.00000 0.137523
$$847$$ −21.0000 −0.721569
$$848$$ −16.0000 −0.549442
$$849$$ 9.00000 0.308879
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ −16.0000 −0.548151
$$853$$ −51.0000 −1.74621 −0.873103 0.487535i $$-0.837896\pi$$
−0.873103 + 0.487535i $$0.837896\pi$$
$$854$$ −42.0000 −1.43721
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 28.0000 0.956462 0.478231 0.878234i $$-0.341278\pi$$
0.478231 + 0.878234i $$0.341278\pi$$
$$858$$ 4.00000 0.136558
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ 36.0000 1.22616
$$863$$ −16.0000 −0.544646 −0.272323 0.962206i $$-0.587792\pi$$
−0.272323 + 0.962206i $$0.587792\pi$$
$$864$$ 8.00000 0.272166
$$865$$ 0 0
$$866$$ −58.0000 −1.97092
$$867$$ −13.0000 −0.441503
$$868$$ −18.0000 −0.610960
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −3.00000 −0.101651
$$872$$ 0 0
$$873$$ −17.0000 −0.575363
$$874$$ −60.0000 −2.02953
$$875$$ 0 0
$$876$$ 28.0000 0.946032
$$877$$ −27.0000 −0.911725 −0.455863 0.890050i $$-0.650669\pi$$
−0.455863 + 0.890050i $$0.650669\pi$$
$$878$$ 70.0000 2.36239
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ 32.0000 1.07811 0.539054 0.842271i $$-0.318782\pi$$
0.539054 + 0.842271i $$0.318782\pi$$
$$882$$ −4.00000 −0.134687
$$883$$ −41.0000 −1.37976 −0.689880 0.723924i $$-0.742337\pi$$
−0.689880 + 0.723924i $$0.742337\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ −48.0000 −1.61259
$$887$$ 18.0000 0.604381 0.302190 0.953248i $$-0.402282\pi$$
0.302190 + 0.953248i $$0.402282\pi$$
$$888$$ 0 0
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 38.0000 1.27233
$$893$$ 10.0000 0.334637
$$894$$ −20.0000 −0.668900
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ −40.0000 −1.33482
$$899$$ −30.0000 −1.00056
$$900$$ 0 0
$$901$$ −8.00000 −0.266519
$$902$$ 32.0000 1.06548
$$903$$ −3.00000 −0.0998337
$$904$$ 0 0
$$905$$ 0 0
$$906$$ −14.0000 −0.465119
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ 16.0000 0.530979
$$909$$ 12.0000 0.398015
$$910$$ 0 0
$$911$$ −58.0000 −1.92163 −0.960813 0.277198i $$-0.910594\pi$$
−0.960813 + 0.277198i $$0.910594\pi$$
$$912$$ 20.0000 0.662266
$$913$$ −12.0000 −0.397142
$$914$$ 44.0000 1.45539
$$915$$ 0 0
$$916$$ −30.0000 −0.991228
$$917$$ 36.0000 1.18882
$$918$$ 4.00000 0.132020
$$919$$ 55.0000 1.81428 0.907141 0.420826i $$-0.138260\pi$$
0.907141 + 0.420826i $$0.138260\pi$$
$$920$$ 0 0
$$921$$ −7.00000 −0.230658
$$922$$ −24.0000 −0.790398
$$923$$ 8.00000 0.263323
$$924$$ 12.0000 0.394771
$$925$$ 0 0
$$926$$ −48.0000 −1.57738
$$927$$ 4.00000 0.131377
$$928$$ 80.0000 2.62613
$$929$$ −50.0000 −1.64045 −0.820223 0.572043i $$-0.806151\pi$$
−0.820223 + 0.572043i $$0.806151\pi$$
$$930$$ 0 0
$$931$$ −10.0000 −0.327737
$$932$$ 48.0000 1.57229
$$933$$ −18.0000 −0.589294
$$934$$ −76.0000 −2.48680
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 33.0000 1.07806 0.539032 0.842286i $$-0.318790\pi$$
0.539032 + 0.842286i $$0.318790\pi$$
$$938$$ −18.0000 −0.587721
$$939$$ −11.0000 −0.358971
$$940$$ 0 0
$$941$$ 22.0000 0.717180 0.358590 0.933495i $$-0.383258\pi$$
0.358590 + 0.933495i $$0.383258\pi$$
$$942$$ −26.0000 −0.847126
$$943$$ 48.0000 1.56310
$$944$$ 40.0000 1.30189
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ 18.0000 0.584921 0.292461 0.956278i $$-0.405526\pi$$
0.292461 + 0.956278i $$0.405526\pi$$
$$948$$ 0 0
$$949$$ −14.0000 −0.454459
$$950$$ 0 0
$$951$$ 8.00000 0.259418
$$952$$ 0 0
$$953$$ −56.0000 −1.81402 −0.907009 0.421111i $$-0.861640\pi$$
−0.907009 + 0.421111i $$0.861640\pi$$
$$954$$ −8.00000 −0.259010
$$955$$ 0 0
$$956$$ 40.0000 1.29369
$$957$$ 20.0000 0.646508
$$958$$ −60.0000 −1.93851
$$959$$ 54.0000 1.74375
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ −4.00000 −0.128965
$$963$$ −12.0000 −0.386695
$$964$$ −46.0000 −1.48156
$$965$$ 0 0
$$966$$ 36.0000 1.15828
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ 10.0000 0.321246
$$970$$ 0 0
$$971$$ 42.0000 1.34784 0.673922 0.738802i $$-0.264608\pi$$
0.673922 + 0.738802i $$0.264608\pi$$
$$972$$ 2.00000 0.0641500
$$973$$ 60.0000 1.92351
$$974$$ −26.0000 −0.833094
$$975$$ 0 0
$$976$$ −28.0000 −0.896258
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ 22.0000 0.703482
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 5.00000 0.159638
$$982$$ 16.0000 0.510581
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 40.0000 1.27386
$$987$$ −6.00000 −0.190982
$$988$$ 10.0000 0.318142
$$989$$ 6.00000 0.190789
$$990$$ 0 0
$$991$$ 17.0000 0.540023 0.270011 0.962857i $$-0.412973\pi$$
0.270011 + 0.962857i $$0.412973\pi$$
$$992$$ −24.0000 −0.762001
$$993$$ 12.0000 0.380808
$$994$$ 48.0000 1.52247
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ −42.0000 −1.33015 −0.665077 0.746775i $$-0.731601\pi$$
−0.665077 + 0.746775i $$0.731601\pi$$
$$998$$ 10.0000 0.316544
$$999$$ −2.00000 −0.0632772
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 75.2.a.a.1.1 1
3.2 odd 2 225.2.a.e.1.1 1
4.3 odd 2 1200.2.a.c.1.1 1
5.2 odd 4 75.2.b.a.49.1 2
5.3 odd 4 75.2.b.a.49.2 2
5.4 even 2 75.2.a.c.1.1 yes 1
7.6 odd 2 3675.2.a.b.1.1 1
8.3 odd 2 4800.2.a.br.1.1 1
8.5 even 2 4800.2.a.bb.1.1 1
11.10 odd 2 9075.2.a.s.1.1 1
12.11 even 2 3600.2.a.j.1.1 1
15.2 even 4 225.2.b.a.199.2 2
15.8 even 4 225.2.b.a.199.1 2
15.14 odd 2 225.2.a.a.1.1 1
20.3 even 4 1200.2.f.d.49.1 2
20.7 even 4 1200.2.f.d.49.2 2
20.19 odd 2 1200.2.a.p.1.1 1
35.34 odd 2 3675.2.a.q.1.1 1
40.3 even 4 4800.2.f.y.3649.2 2
40.13 odd 4 4800.2.f.l.3649.1 2
40.19 odd 2 4800.2.a.be.1.1 1
40.27 even 4 4800.2.f.y.3649.1 2
40.29 even 2 4800.2.a.bq.1.1 1
40.37 odd 4 4800.2.f.l.3649.2 2
55.54 odd 2 9075.2.a.a.1.1 1
60.23 odd 4 3600.2.f.p.2449.2 2
60.47 odd 4 3600.2.f.p.2449.1 2
60.59 even 2 3600.2.a.bk.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.a.a.1.1 1 1.1 even 1 trivial
75.2.a.c.1.1 yes 1 5.4 even 2
75.2.b.a.49.1 2 5.2 odd 4
75.2.b.a.49.2 2 5.3 odd 4
225.2.a.a.1.1 1 15.14 odd 2
225.2.a.e.1.1 1 3.2 odd 2
225.2.b.a.199.1 2 15.8 even 4
225.2.b.a.199.2 2 15.2 even 4
1200.2.a.c.1.1 1 4.3 odd 2
1200.2.a.p.1.1 1 20.19 odd 2
1200.2.f.d.49.1 2 20.3 even 4
1200.2.f.d.49.2 2 20.7 even 4
3600.2.a.j.1.1 1 12.11 even 2
3600.2.a.bk.1.1 1 60.59 even 2
3600.2.f.p.2449.1 2 60.47 odd 4
3600.2.f.p.2449.2 2 60.23 odd 4
3675.2.a.b.1.1 1 7.6 odd 2
3675.2.a.q.1.1 1 35.34 odd 2
4800.2.a.bb.1.1 1 8.5 even 2
4800.2.a.be.1.1 1 40.19 odd 2
4800.2.a.bq.1.1 1 40.29 even 2
4800.2.a.br.1.1 1 8.3 odd 2
4800.2.f.l.3649.1 2 40.13 odd 4
4800.2.f.l.3649.2 2 40.37 odd 4
4800.2.f.y.3649.1 2 40.27 even 4
4800.2.f.y.3649.2 2 40.3 even 4
9075.2.a.a.1.1 1 55.54 odd 2
9075.2.a.s.1.1 1 11.10 odd 2