# Properties

 Label 15.2.a.a.1.1 Level $15$ Weight $2$ Character 15.1 Self dual yes Analytic conductor $0.120$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,2,Mod(1,15)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, 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("15.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 15.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: $$0.119775603032$$ 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 $$\chi$$ $$=$$ 15.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} -3.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +1.00000 q^{30} -5.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} -1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} +2.00000 q^{39} +3.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{55} -4.00000 q^{57} +2.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} -2.00000 q^{61} +7.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} -8.00000 q^{71} +3.00000 q^{72} +10.0000 q^{73} +10.0000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +12.0000 q^{83} +2.00000 q^{85} -4.00000 q^{86} +2.00000 q^{87} -12.0000 q^{88} -6.00000 q^{89} -1.00000 q^{90} -8.00000 q^{94} +4.00000 q^{95} +5.00000 q^{96} +2.00000 q^{97} +7.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000 0.447214
$$6$$ 1.00000 0.408248
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 3.00000 1.06066
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 4.00000 0.696311
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 2.00000 0.320256
$$40$$ 3.00000 0.474342
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −7.00000 −1.00000
$$50$$ −1.00000 −0.141421
$$51$$ −2.00000 −0.280056
$$52$$ 2.00000 0.277350
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 2.00000 0.262613
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 1.00000 0.129099
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ −2.00000 −0.248069
$$66$$ −4.00000 −0.492366
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 3.00000 0.353553
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 10.0000 1.16248
$$75$$ −1.00000 −0.115470
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ −2.00000 −0.226455
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −10.0000 −1.10432
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ −4.00000 −0.431331
$$87$$ 2.00000 0.214423
$$88$$ −12.0000 −1.27920
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ 4.00000 0.410391
$$96$$ 5.00000 0.510310
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 7.00000 0.707107
$$99$$ −4.00000 −0.402015
$$100$$ −1.00000 −0.100000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 2.00000 0.198030
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 10.0000 0.949158
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ −2.00000 −0.184900
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ −3.00000 −0.273861
$$121$$ 5.00000 0.454545
$$122$$ 2.00000 0.181071
$$123$$ −10.0000 −0.901670
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 3.00000 0.265165
$$129$$ −4.00000 −0.352180
$$130$$ 2.00000 0.175412
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ −4.00000 −0.348155
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ −1.00000 −0.0860663
$$136$$ 6.00000 0.514496
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 8.00000 0.671345
$$143$$ 8.00000 0.668994
$$144$$ −1.00000 −0.0833333
$$145$$ −2.00000 −0.166091
$$146$$ −10.0000 −0.827606
$$147$$ 7.00000 0.577350
$$148$$ 10.0000 0.821995
$$149$$ 22.0000 1.80231 0.901155 0.433497i $$-0.142720\pi$$
0.901155 + 0.433497i $$0.142720\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 12.0000 0.973329
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 10.0000 0.793052
$$160$$ −5.00000 −0.395285
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 4.00000 0.311400
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −2.00000 −0.153393
$$171$$ 4.00000 0.305888
$$172$$ −4.00000 −0.304997
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 4.00000 0.300658
$$178$$ 6.00000 0.449719
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ −10.0000 −0.735215
$$186$$ 0 0
$$187$$ −8.00000 −0.585018
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ −7.00000 −0.505181
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 2.00000 0.143223
$$196$$ 7.00000 0.500000
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 4.00000 0.284268
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 3.00000 0.212132
$$201$$ −12.0000 −0.846415
$$202$$ −6.00000 −0.422159
$$203$$ 0 0
$$204$$ 2.00000 0.140028
$$205$$ 10.0000 0.698430
$$206$$ 16.0000 1.11477
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 8.00000 0.548151
$$214$$ 12.0000 0.820303
$$215$$ 4.00000 0.272798
$$216$$ −3.00000 −0.204124
$$217$$ 0 0
$$218$$ −14.0000 −0.948200
$$219$$ −10.0000 −0.675737
$$220$$ 4.00000 0.269680
$$221$$ −4.00000 −0.269069
$$222$$ −10.0000 −0.671156
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ −2.00000 −0.133038
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 4.00000 0.264906
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 8.00000 0.521862
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ 2.00000 0.128037
$$245$$ −7.00000 −0.447214
$$246$$ 10.0000 0.637577
$$247$$ −8.00000 −0.509028
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ −1.00000 −0.0632456
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ −2.00000 −0.125245
$$256$$ −17.0000 −1.06250
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 0 0
$$260$$ 2.00000 0.124035
$$261$$ −2.00000 −0.123797
$$262$$ 12.0000 0.741362
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 12.0000 0.738549
$$265$$ −10.0000 −0.614295
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ −12.0000 −0.733017
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ 6.00000 0.360505 0.180253 0.983620i $$-0.442309\pi$$
0.180253 + 0.983620i $$0.442309\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 8.00000 0.476393
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ 8.00000 0.474713
$$285$$ −4.00000 −0.236940
$$286$$ −8.00000 −0.473050
$$287$$ 0 0
$$288$$ −5.00000 −0.294628
$$289$$ −13.0000 −0.764706
$$290$$ 2.00000 0.117444
$$291$$ −2.00000 −0.117242
$$292$$ −10.0000 −0.585206
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ −7.00000 −0.408248
$$295$$ −4.00000 −0.232889
$$296$$ −30.0000 −1.74371
$$297$$ 4.00000 0.232104
$$298$$ −22.0000 −1.27443
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 0 0
$$302$$ 8.00000 0.460348
$$303$$ −6.00000 −0.344691
$$304$$ −4.00000 −0.229416
$$305$$ −2.00000 −0.114520
$$306$$ −2.00000 −0.114332
$$307$$ 28.0000 1.59804 0.799022 0.601302i $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 6.00000 0.339683
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ −10.0000 −0.560772
$$319$$ 8.00000 0.447914
$$320$$ 7.00000 0.391312
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 8.00000 0.445132
$$324$$ −1.00000 −0.0555556
$$325$$ −2.00000 −0.110940
$$326$$ 4.00000 0.221540
$$327$$ −14.0000 −0.774202
$$328$$ 30.0000 1.65647
$$329$$ 0 0
$$330$$ −4.00000 −0.220193
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ −10.0000 −0.547997
$$334$$ 0 0
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −2.00000 −0.108625
$$340$$ −2.00000 −0.108465
$$341$$ 0 0
$$342$$ −4.00000 −0.216295
$$343$$ 0 0
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ −28.0000 −1.50312 −0.751559 0.659665i $$-0.770698\pi$$
−0.751559 + 0.659665i $$0.770698\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 20.0000 1.06600
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ −8.00000 −0.424596
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ −20.0000 −1.05703
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 3.00000 0.158114
$$361$$ −3.00000 −0.157895
$$362$$ 10.0000 0.525588
$$363$$ −5.00000 −0.262432
$$364$$ 0 0
$$365$$ 10.0000 0.523424
$$366$$ −2.00000 −0.104542
$$367$$ −24.0000 −1.25279 −0.626395 0.779506i $$-0.715470\pi$$
−0.626395 + 0.779506i $$0.715470\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 10.0000 0.519875
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −26.0000 −1.34623 −0.673114 0.739538i $$-0.735044\pi$$
−0.673114 + 0.739538i $$0.735044\pi$$
$$374$$ 8.00000 0.413670
$$375$$ −1.00000 −0.0516398
$$376$$ 24.0000 1.23771
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 8.00000 0.409852
$$382$$ −16.0000 −0.818631
$$383$$ −24.0000 −1.22634 −0.613171 0.789950i $$-0.710106\pi$$
−0.613171 + 0.789950i $$0.710106\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ 4.00000 0.203331
$$388$$ −2.00000 −0.101535
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ −2.00000 −0.101274
$$391$$ 0 0
$$392$$ −21.0000 −1.06066
$$393$$ 12.0000 0.605320
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 8.00000 0.401004
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 12.0000 0.598506
$$403$$ 0 0
$$404$$ −6.00000 −0.298511
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 40.0000 1.98273
$$408$$ −6.00000 −0.297044
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ −10.0000 −0.493865
$$411$$ 6.00000 0.295958
$$412$$ 16.0000 0.788263
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 10.0000 0.490290
$$417$$ 4.00000 0.195881
$$418$$ 16.0000 0.782586
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 8.00000 0.388973
$$424$$ −30.0000 −1.45693
$$425$$ 2.00000 0.0970143
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ −8.00000 −0.386244
$$430$$ −4.00000 −0.192897
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ 2.00000 0.0958927
$$436$$ −14.0000 −0.670478
$$437$$ 0 0
$$438$$ 10.0000 0.477818
$$439$$ 40.0000 1.90910 0.954548 0.298057i $$-0.0963387\pi$$
0.954548 + 0.298057i $$0.0963387\pi$$
$$440$$ −12.0000 −0.572078
$$441$$ −7.00000 −0.333333
$$442$$ 4.00000 0.190261
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ −6.00000 −0.284427
$$446$$ −8.00000 −0.378811
$$447$$ −22.0000 −1.04056
$$448$$ 0 0
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −40.0000 −1.88353
$$452$$ −2.00000 −0.0940721
$$453$$ 8.00000 0.375873
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ −12.0000 −0.561951
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 0 0
$$470$$ −8.00000 −0.369012
$$471$$ −14.0000 −0.645086
$$472$$ −12.0000 −0.552345
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ −10.0000 −0.457869
$$478$$ 16.0000 0.731823
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 5.00000 0.228218
$$481$$ 20.0000 0.911922
$$482$$ 14.0000 0.637683
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 2.00000 0.0908153
$$486$$ 1.00000 0.0453609
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ 4.00000 0.180886
$$490$$ 7.00000 0.316228
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 10.0000 0.450835
$$493$$ −4.00000 −0.180151
$$494$$ 8.00000 0.359937
$$495$$ −4.00000 −0.179787
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ −12.0000 −0.535586
$$503$$ −32.0000 −1.42681 −0.713405 0.700752i $$-0.752848\pi$$
−0.713405 + 0.700752i $$0.752848\pi$$
$$504$$ 0 0
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ 8.00000 0.354943
$$509$$ −34.0000 −1.50702 −0.753512 0.657434i $$-0.771642\pi$$
−0.753512 + 0.657434i $$0.771642\pi$$
$$510$$ 2.00000 0.0885615
$$511$$ 0 0
$$512$$ 11.0000 0.486136
$$513$$ −4.00000 −0.176604
$$514$$ −18.0000 −0.793946
$$515$$ −16.0000 −0.705044
$$516$$ 4.00000 0.176090
$$517$$ −32.0000 −1.40736
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ −6.00000 −0.263117
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ −4.00000 −0.174078
$$529$$ −23.0000 −1.00000
$$530$$ 10.0000 0.434372
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ −20.0000 −0.866296
$$534$$ −6.00000 −0.259645
$$535$$ −12.0000 −0.518805
$$536$$ 36.0000 1.55496
$$537$$ −20.0000 −0.863064
$$538$$ −14.0000 −0.603583
$$539$$ 28.0000 1.20605
$$540$$ 1.00000 0.0430331
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 10.0000 0.429141
$$544$$ −10.0000 −0.428746
$$545$$ 14.0000 0.599694
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −2.00000 −0.0853579
$$550$$ 4.00000 0.170561
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −6.00000 −0.254916
$$555$$ 10.0000 0.424476
$$556$$ 4.00000 0.169638
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 6.00000 0.253095
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 2.00000 0.0841406
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ −24.0000 −1.00702
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 4.00000 0.167542
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ −16.0000 −0.668410
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 13.0000 0.540729
$$579$$ −2.00000 −0.0831172
$$580$$ 2.00000 0.0830455
$$581$$ 0 0
$$582$$ 2.00000 0.0829027
$$583$$ 40.0000 1.65663
$$584$$ 30.0000 1.24141
$$585$$ −2.00000 −0.0826898
$$586$$ −6.00000 −0.247858
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −7.00000 −0.288675
$$589$$ 0 0
$$590$$ 4.00000 0.164677
$$591$$ −6.00000 −0.246807
$$592$$ 10.0000 0.410997
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −22.0000 −0.901155
$$597$$ 8.00000 0.327418
$$598$$ 0 0
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ 12.0000 0.488678
$$604$$ 8.00000 0.325515
$$605$$ 5.00000 0.203279
$$606$$ 6.00000 0.243733
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ −20.0000 −0.811107
$$609$$ 0 0
$$610$$ 2.00000 0.0809776
$$611$$ −16.0000 −0.647291
$$612$$ −2.00000 −0.0808452
$$613$$ 22.0000 0.888572 0.444286 0.895885i $$-0.353457\pi$$
0.444286 + 0.895885i $$0.353457\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ −10.0000 −0.403239
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ −16.0000 −0.643614
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 24.0000 0.962312
$$623$$ 0 0
$$624$$ −2.00000 −0.0800641
$$625$$ 1.00000 0.0400000
$$626$$ −26.0000 −1.03917
$$627$$ 16.0000 0.638978
$$628$$ −14.0000 −0.558661
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ −20.0000 −0.794929
$$634$$ 2.00000 0.0794301
$$635$$ −8.00000 −0.317470
$$636$$ −10.0000 −0.396526
$$637$$ 14.0000 0.554700
$$638$$ −8.00000 −0.316723
$$639$$ −8.00000 −0.316475
$$640$$ 3.00000 0.118585
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ −36.0000 −1.41970 −0.709851 0.704352i $$-0.751238\pi$$
−0.709851 + 0.704352i $$0.751238\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ −8.00000 −0.314756
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ 3.00000 0.117851
$$649$$ 16.0000 0.628055
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 46.0000 1.80012 0.900060 0.435767i $$-0.143523\pi$$
0.900060 + 0.435767i $$0.143523\pi$$
$$654$$ 14.0000 0.547443
$$655$$ −12.0000 −0.468879
$$656$$ −10.0000 −0.390434
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ −4.00000 −0.155700
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ 4.00000 0.155347
$$664$$ 36.0000 1.39707
$$665$$ 0 0
$$666$$ 10.0000 0.387492
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −8.00000 −0.309298
$$670$$ −12.0000 −0.463600
$$671$$ 8.00000 0.308837
$$672$$ 0 0
$$673$$ −30.0000 −1.15642 −0.578208 0.815890i $$-0.696248\pi$$
−0.578208 + 0.815890i $$0.696248\pi$$
$$674$$ 14.0000 0.539260
$$675$$ −1.00000 −0.0384900
$$676$$ 9.00000 0.346154
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 2.00000 0.0768095
$$679$$ 0 0
$$680$$ 6.00000 0.230089
$$681$$ 20.0000 0.766402
$$682$$ 0 0
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ −6.00000 −0.229248
$$686$$ 0 0
$$687$$ −6.00000 −0.228914
$$688$$ −4.00000 −0.152499
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 0 0
$$694$$ 28.0000 1.06287
$$695$$ −4.00000 −0.151729
$$696$$ 6.00000 0.227429
$$697$$ 20.0000 0.757554
$$698$$ 2.00000 0.0757011
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ −40.0000 −1.50863
$$704$$ −28.0000 −1.05529
$$705$$ −8.00000 −0.301297
$$706$$ −18.0000 −0.677439
$$707$$ 0 0
$$708$$ −4.00000 −0.150329
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 8.00000 0.300235
$$711$$ 0 0
$$712$$ −18.0000 −0.674579
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 8.00000 0.299183
$$716$$ −20.0000 −0.747435
$$717$$ 16.0000 0.597531
$$718$$ 24.0000 0.895672
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ 14.0000 0.520666
$$724$$ 10.0000 0.371647
$$725$$ −2.00000 −0.0742781
$$726$$ 5.00000 0.185567
$$727$$ −16.0000 −0.593407 −0.296704 0.954970i $$-0.595887\pi$$
−0.296704 + 0.954970i $$0.595887\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −10.0000 −0.370117
$$731$$ 8.00000 0.295891
$$732$$ −2.00000 −0.0739221
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 24.0000 0.885856
$$735$$ 7.00000 0.258199
$$736$$ 0 0
$$737$$ −48.0000 −1.76810
$$738$$ −10.0000 −0.368105
$$739$$ −44.0000 −1.61857 −0.809283 0.587419i $$-0.800144\pi$$
−0.809283 + 0.587419i $$0.800144\pi$$
$$740$$ 10.0000 0.367607
$$741$$ 8.00000 0.293887
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 0 0
$$745$$ 22.0000 0.806018
$$746$$ 26.0000 0.951928
$$747$$ 12.0000 0.439057
$$748$$ 8.00000 0.292509
$$749$$ 0 0
$$750$$ 1.00000 0.0365148
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −12.0000 −0.437304
$$754$$ −4.00000 −0.145671
$$755$$ −8.00000 −0.291150
$$756$$ 0 0
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ 12.0000 0.435286
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ 0 0
$$764$$ −16.0000 −0.578860
$$765$$ 2.00000 0.0723102
$$766$$ 24.0000 0.867155
$$767$$ 8.00000 0.288863
$$768$$ 17.0000 0.613435
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ −2.00000 −0.0719816
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 0 0
$$778$$ −6.00000 −0.215110
$$779$$ 40.0000 1.43315
$$780$$ −2.00000 −0.0716115
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 7.00000 0.250000
$$785$$ 14.0000 0.499681
$$786$$ −12.0000 −0.428026
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −12.0000 −0.426401
$$793$$ 4.00000 0.142044
$$794$$ 2.00000 0.0709773
$$795$$ 10.0000 0.354663
$$796$$ 8.00000 0.283552
$$797$$ −2.00000 −0.0708436 −0.0354218 0.999372i $$-0.511277\pi$$
−0.0354218 + 0.999372i $$0.511277\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ −5.00000 −0.176777
$$801$$ −6.00000 −0.212000
$$802$$ −18.0000 −0.635602
$$803$$ −40.0000 −1.41157
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −14.0000 −0.492823
$$808$$ 18.0000 0.633238
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ −16.0000 −0.561144
$$814$$ −40.0000 −1.40200
$$815$$ −4.00000 −0.140114
$$816$$ 2.00000 0.0700140
$$817$$ 16.0000 0.559769
$$818$$ −26.0000 −0.909069
$$819$$ 0 0
$$820$$ −10.0000 −0.349215
$$821$$ 54.0000 1.88461 0.942306 0.334751i $$-0.108652\pi$$
0.942306 + 0.334751i $$0.108652\pi$$
$$822$$ −6.00000 −0.209274
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ −48.0000 −1.67216
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ 0 0
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ −12.0000 −0.416526
$$831$$ −6.00000 −0.208138
$$832$$ −14.0000 −0.485363
$$833$$ −14.0000 −0.485071
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ −4.00000 −0.138178
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 26.0000 0.896019
$$843$$ 6.00000 0.206651
$$844$$ −20.0000 −0.688428
$$845$$ −9.00000 −0.309609
$$846$$ −8.00000 −0.275046
$$847$$ 0 0
$$848$$ 10.0000 0.343401
$$849$$ 12.0000 0.411839
$$850$$ −2.00000 −0.0685994
$$851$$ 0 0
$$852$$ −8.00000 −0.274075
$$853$$ 6.00000 0.205436 0.102718 0.994711i $$-0.467246\pi$$
0.102718 + 0.994711i $$0.467246\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ −36.0000 −1.23045
$$857$$ −22.0000 −0.751506 −0.375753 0.926720i $$-0.622616\pi$$
−0.375753 + 0.926720i $$0.622616\pi$$
$$858$$ 8.00000 0.273115
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −56.0000 −1.90626 −0.953131 0.302558i $$-0.902160\pi$$
−0.953131 + 0.302558i $$0.902160\pi$$
$$864$$ 5.00000 0.170103
$$865$$ −18.0000 −0.612018
$$866$$ 14.0000 0.475739
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −2.00000 −0.0678064
$$871$$ −24.0000 −0.813209
$$872$$ 42.0000 1.42230
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ 30.0000 1.01303 0.506514 0.862232i $$-0.330934\pi$$
0.506514 + 0.862232i $$0.330934\pi$$
$$878$$ −40.0000 −1.34993
$$879$$ −6.00000 −0.202375
$$880$$ 4.00000 0.134840
$$881$$ −46.0000 −1.54978 −0.774890 0.632096i $$-0.782195\pi$$
−0.774890 + 0.632096i $$0.782195\pi$$
$$882$$ 7.00000 0.235702
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 4.00000 0.134459
$$886$$ 12.0000 0.403148
$$887$$ 48.0000 1.61168 0.805841 0.592132i $$-0.201714\pi$$
0.805841 + 0.592132i $$0.201714\pi$$
$$888$$ 30.0000 1.00673
$$889$$ 0 0
$$890$$ 6.00000 0.201120
$$891$$ −4.00000 −0.134005
$$892$$ −8.00000 −0.267860
$$893$$ 32.0000 1.07084
$$894$$ 22.0000 0.735790
$$895$$ 20.0000 0.668526
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −2.00000 −0.0667409
$$899$$ 0 0
$$900$$ −1.00000 −0.0333333
$$901$$ −20.0000 −0.666297
$$902$$ 40.0000 1.33185
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ −10.0000 −0.332411
$$906$$ −8.00000 −0.265782
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ 20.0000 0.663723
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ 4.00000 0.132453
$$913$$ −48.0000 −1.58857
$$914$$ −10.0000 −0.330771
$$915$$ 2.00000 0.0661180
$$916$$ −6.00000 −0.198246
$$917$$ 0 0
$$918$$ 2.00000 0.0660098
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 18.0000 0.592798
$$923$$ 16.0000 0.526646
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ −24.0000 −0.788689
$$927$$ −16.0000 −0.525509
$$928$$ 10.0000 0.328266
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ 6.00000 0.196537
$$933$$ 24.0000 0.785725
$$934$$ −28.0000 −0.916188
$$935$$ −8.00000 −0.261628
$$936$$ −6.00000 −0.196116
$$937$$ −54.0000 −1.76410 −0.882052 0.471153i $$-0.843838\pi$$
−0.882052 + 0.471153i $$0.843838\pi$$
$$938$$ 0 0
$$939$$ −26.0000 −0.848478
$$940$$ −8.00000 −0.260931
$$941$$ −50.0000 −1.62995 −0.814977 0.579494i $$-0.803250\pi$$
−0.814977 + 0.579494i $$0.803250\pi$$
$$942$$ 14.0000 0.456145
$$943$$ 0 0
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ 0 0
$$949$$ −20.0000 −0.649227
$$950$$ −4.00000 −0.129777
$$951$$ 2.00000 0.0648544
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 16.0000 0.517748
$$956$$ 16.0000 0.517477
$$957$$ −8.00000 −0.258603
$$958$$ 0 0
$$959$$ 0 0
$$960$$ −7.00000 −0.225924
$$961$$ −31.0000 −1.00000
$$962$$ −20.0000 −0.644826
$$963$$ −12.0000 −0.386695
$$964$$ 14.0000 0.450910
$$965$$ 2.00000 0.0643823
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 15.0000 0.482118
$$969$$ −8.00000 −0.256997
$$970$$ −2.00000 −0.0642161
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ −32.0000 −1.02535
$$975$$ 2.00000 0.0640513
$$976$$ 2.00000 0.0640184
$$977$$ 2.00000 0.0639857 0.0319928 0.999488i $$-0.489815\pi$$
0.0319928 + 0.999488i $$0.489815\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ 24.0000 0.767043
$$980$$ 7.00000 0.223607
$$981$$ 14.0000 0.446986
$$982$$ −28.0000 −0.893516
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ −30.0000 −0.956365
$$985$$ 6.00000 0.191176
$$986$$ 4.00000 0.127386
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ 4.00000 0.127128
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 0 0
$$993$$ −12.0000 −0.380808
$$994$$ 0 0
$$995$$ −8.00000 −0.253617
$$996$$ 12.0000 0.380235
$$997$$ 54.0000 1.71020 0.855099 0.518465i $$-0.173497\pi$$
0.855099 + 0.518465i $$0.173497\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ 10.0000 0.316386
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 15.2.a.a.1.1 1
3.2 odd 2 45.2.a.a.1.1 1
4.3 odd 2 240.2.a.d.1.1 1
5.2 odd 4 75.2.b.b.49.1 2
5.3 odd 4 75.2.b.b.49.2 2
5.4 even 2 75.2.a.b.1.1 1
7.2 even 3 735.2.i.e.361.1 2
7.3 odd 6 735.2.i.d.226.1 2
7.4 even 3 735.2.i.e.226.1 2
7.5 odd 6 735.2.i.d.361.1 2
7.6 odd 2 735.2.a.c.1.1 1
8.3 odd 2 960.2.a.a.1.1 1
8.5 even 2 960.2.a.l.1.1 1
9.2 odd 6 405.2.e.c.271.1 2
9.4 even 3 405.2.e.f.136.1 2
9.5 odd 6 405.2.e.c.136.1 2
9.7 even 3 405.2.e.f.271.1 2
11.10 odd 2 1815.2.a.d.1.1 1
12.11 even 2 720.2.a.c.1.1 1
13.12 even 2 2535.2.a.j.1.1 1
15.2 even 4 225.2.b.b.199.2 2
15.8 even 4 225.2.b.b.199.1 2
15.14 odd 2 225.2.a.b.1.1 1
16.3 odd 4 3840.2.k.r.1921.2 2
16.5 even 4 3840.2.k.m.1921.2 2
16.11 odd 4 3840.2.k.r.1921.1 2
16.13 even 4 3840.2.k.m.1921.1 2
17.16 even 2 4335.2.a.c.1.1 1
19.18 odd 2 5415.2.a.j.1.1 1
20.3 even 4 1200.2.f.h.49.2 2
20.7 even 4 1200.2.f.h.49.1 2
20.19 odd 2 1200.2.a.e.1.1 1
21.20 even 2 2205.2.a.i.1.1 1
23.22 odd 2 7935.2.a.d.1.1 1
24.5 odd 2 2880.2.a.y.1.1 1
24.11 even 2 2880.2.a.bc.1.1 1
33.32 even 2 5445.2.a.c.1.1 1
35.34 odd 2 3675.2.a.j.1.1 1
39.38 odd 2 7605.2.a.g.1.1 1
40.3 even 4 4800.2.f.c.3649.1 2
40.13 odd 4 4800.2.f.bf.3649.2 2
40.19 odd 2 4800.2.a.bz.1.1 1
40.27 even 4 4800.2.f.c.3649.2 2
40.29 even 2 4800.2.a.t.1.1 1
40.37 odd 4 4800.2.f.bf.3649.1 2
55.54 odd 2 9075.2.a.g.1.1 1
60.23 odd 4 3600.2.f.e.2449.1 2
60.47 odd 4 3600.2.f.e.2449.2 2
60.59 even 2 3600.2.a.u.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.2.a.a.1.1 1 1.1 even 1 trivial
45.2.a.a.1.1 1 3.2 odd 2
75.2.a.b.1.1 1 5.4 even 2
75.2.b.b.49.1 2 5.2 odd 4
75.2.b.b.49.2 2 5.3 odd 4
225.2.a.b.1.1 1 15.14 odd 2
225.2.b.b.199.1 2 15.8 even 4
225.2.b.b.199.2 2 15.2 even 4
240.2.a.d.1.1 1 4.3 odd 2
405.2.e.c.136.1 2 9.5 odd 6
405.2.e.c.271.1 2 9.2 odd 6
405.2.e.f.136.1 2 9.4 even 3
405.2.e.f.271.1 2 9.7 even 3
720.2.a.c.1.1 1 12.11 even 2
735.2.a.c.1.1 1 7.6 odd 2
735.2.i.d.226.1 2 7.3 odd 6
735.2.i.d.361.1 2 7.5 odd 6
735.2.i.e.226.1 2 7.4 even 3
735.2.i.e.361.1 2 7.2 even 3
960.2.a.a.1.1 1 8.3 odd 2
960.2.a.l.1.1 1 8.5 even 2
1200.2.a.e.1.1 1 20.19 odd 2
1200.2.f.h.49.1 2 20.7 even 4
1200.2.f.h.49.2 2 20.3 even 4
1815.2.a.d.1.1 1 11.10 odd 2
2205.2.a.i.1.1 1 21.20 even 2
2535.2.a.j.1.1 1 13.12 even 2
2880.2.a.y.1.1 1 24.5 odd 2
2880.2.a.bc.1.1 1 24.11 even 2
3600.2.a.u.1.1 1 60.59 even 2
3600.2.f.e.2449.1 2 60.23 odd 4
3600.2.f.e.2449.2 2 60.47 odd 4
3675.2.a.j.1.1 1 35.34 odd 2
3840.2.k.m.1921.1 2 16.13 even 4
3840.2.k.m.1921.2 2 16.5 even 4
3840.2.k.r.1921.1 2 16.11 odd 4
3840.2.k.r.1921.2 2 16.3 odd 4
4335.2.a.c.1.1 1 17.16 even 2
4800.2.a.t.1.1 1 40.29 even 2
4800.2.a.bz.1.1 1 40.19 odd 2
4800.2.f.c.3649.1 2 40.3 even 4
4800.2.f.c.3649.2 2 40.27 even 4
4800.2.f.bf.3649.1 2 40.37 odd 4
4800.2.f.bf.3649.2 2 40.13 odd 4
5415.2.a.j.1.1 1 19.18 odd 2
5445.2.a.c.1.1 1 33.32 even 2
7605.2.a.g.1.1 1 39.38 odd 2
7935.2.a.d.1.1 1 23.22 odd 2
9075.2.a.g.1.1 1 55.54 odd 2