# Properties

 Label 57.2.a.b.1.1 Level $57$ Weight $2$ Character 57.1 Self dual yes Analytic conductor $0.455$ 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] = [57,2,Mod(1,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, 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("57.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$57 = 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 57.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: $$0.455147291521$$ 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 $$\chi$$ $$=$$ 57.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} -3.00000 q^{11} +2.00000 q^{12} -6.00000 q^{13} -6.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} -1.00000 q^{19} +2.00000 q^{20} +3.00000 q^{21} +6.00000 q^{22} +4.00000 q^{23} -4.00000 q^{25} +12.0000 q^{26} +1.00000 q^{27} +6.00000 q^{28} -10.0000 q^{29} -2.00000 q^{30} +2.00000 q^{31} +8.00000 q^{32} -3.00000 q^{33} -6.00000 q^{34} +3.00000 q^{35} +2.00000 q^{36} +8.00000 q^{37} +2.00000 q^{38} -6.00000 q^{39} -8.00000 q^{41} -6.00000 q^{42} -1.00000 q^{43} -6.00000 q^{44} +1.00000 q^{45} -8.00000 q^{46} +3.00000 q^{47} -4.00000 q^{48} +2.00000 q^{49} +8.00000 q^{50} +3.00000 q^{51} -12.0000 q^{52} -6.00000 q^{53} -2.00000 q^{54} -3.00000 q^{55} -1.00000 q^{57} +20.0000 q^{58} +2.00000 q^{60} +7.00000 q^{61} -4.00000 q^{62} +3.00000 q^{63} -8.00000 q^{64} -6.00000 q^{65} +6.00000 q^{66} +8.00000 q^{67} +6.00000 q^{68} +4.00000 q^{69} -6.00000 q^{70} +12.0000 q^{71} -11.0000 q^{73} -16.0000 q^{74} -4.00000 q^{75} -2.00000 q^{76} -9.00000 q^{77} +12.0000 q^{78} -4.00000 q^{80} +1.00000 q^{81} +16.0000 q^{82} +4.00000 q^{83} +6.00000 q^{84} +3.00000 q^{85} +2.00000 q^{86} -10.0000 q^{87} +10.0000 q^{89} -2.00000 q^{90} -18.0000 q^{91} +8.00000 q^{92} +2.00000 q^{93} -6.00000 q^{94} -1.00000 q^{95} +8.00000 q^{96} -2.00000 q^{97} -4.00000 q^{98} -3.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ −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
$$10$$ −2.00000 −0.632456
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 2.00000 0.577350
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ −6.00000 −1.60357
$$15$$ 1.00000 0.258199
$$16$$ −4.00000 −1.00000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ −2.00000 −0.471405
$$19$$ −1.00000 −0.229416
$$20$$ 2.00000 0.447214
$$21$$ 3.00000 0.654654
$$22$$ 6.00000 1.27920
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 12.0000 2.35339
$$27$$ 1.00000 0.192450
$$28$$ 6.00000 1.13389
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 8.00000 1.41421
$$33$$ −3.00000 −0.522233
$$34$$ −6.00000 −1.02899
$$35$$ 3.00000 0.507093
$$36$$ 2.00000 0.333333
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 2.00000 0.324443
$$39$$ −6.00000 −0.960769
$$40$$ 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 $$0.524294\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 1.00000 0.149071
$$46$$ −8.00000 −1.17954
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ −4.00000 −0.577350
$$49$$ 2.00000 0.285714
$$50$$ 8.00000 1.13137
$$51$$ 3.00000 0.420084
$$52$$ −12.0000 −1.66410
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −2.00000 −0.272166
$$55$$ −3.00000 −0.404520
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 20.0000 2.62613
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 2.00000 0.258199
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 3.00000 0.377964
$$64$$ −8.00000 −1.00000
$$65$$ −6.00000 −0.744208
$$66$$ 6.00000 0.738549
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 4.00000 0.481543
$$70$$ −6.00000 −0.717137
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ −11.0000 −1.28745 −0.643726 0.765256i $$-0.722612\pi$$
−0.643726 + 0.765256i $$0.722612\pi$$
$$74$$ −16.0000 −1.85996
$$75$$ −4.00000 −0.461880
$$76$$ −2.00000 −0.229416
$$77$$ −9.00000 −1.02565
$$78$$ 12.0000 1.35873
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 1.00000 0.111111
$$82$$ 16.0000 1.76690
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 6.00000 0.654654
$$85$$ 3.00000 0.325396
$$86$$ 2.00000 0.215666
$$87$$ −10.0000 −1.07211
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ −2.00000 −0.210819
$$91$$ −18.0000 −1.88691
$$92$$ 8.00000 0.834058
$$93$$ 2.00000 0.207390
$$94$$ −6.00000 −0.618853
$$95$$ −1.00000 −0.102598
$$96$$ 8.00000 0.816497
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ −4.00000 −0.404061
$$99$$ −3.00000 −0.301511
$$100$$ −8.00000 −0.800000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ −6.00000 −0.594089
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 0 0
$$105$$ 3.00000 0.292770
$$106$$ 12.0000 1.16554
$$107$$ −2.00000 −0.193347 −0.0966736 0.995316i $$-0.530820\pi$$
−0.0966736 + 0.995316i $$0.530820\pi$$
$$108$$ 2.00000 0.192450
$$109$$ 20.0000 1.91565 0.957826 0.287348i $$-0.0927736\pi$$
0.957826 + 0.287348i $$0.0927736\pi$$
$$110$$ 6.00000 0.572078
$$111$$ 8.00000 0.759326
$$112$$ −12.0000 −1.13389
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 4.00000 0.373002
$$116$$ −20.0000 −1.85695
$$117$$ −6.00000 −0.554700
$$118$$ 0 0
$$119$$ 9.00000 0.825029
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −14.0000 −1.26750
$$123$$ −8.00000 −0.721336
$$124$$ 4.00000 0.359211
$$125$$ −9.00000 −0.804984
$$126$$ −6.00000 −0.534522
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 0 0
$$129$$ −1.00000 −0.0880451
$$130$$ 12.0000 1.05247
$$131$$ −13.0000 −1.13582 −0.567908 0.823092i $$-0.692247\pi$$
−0.567908 + 0.823092i $$0.692247\pi$$
$$132$$ −6.00000 −0.522233
$$133$$ −3.00000 −0.260133
$$134$$ −16.0000 −1.38219
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 3.00000 0.256307 0.128154 0.991754i $$-0.459095\pi$$
0.128154 + 0.991754i $$0.459095\pi$$
$$138$$ −8.00000 −0.681005
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 6.00000 0.507093
$$141$$ 3.00000 0.252646
$$142$$ −24.0000 −2.01404
$$143$$ 18.0000 1.50524
$$144$$ −4.00000 −0.333333
$$145$$ −10.0000 −0.830455
$$146$$ 22.0000 1.82073
$$147$$ 2.00000 0.164957
$$148$$ 16.0000 1.31519
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 8.00000 0.653197
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 3.00000 0.242536
$$154$$ 18.0000 1.45048
$$155$$ 2.00000 0.160644
$$156$$ −12.0000 −0.960769
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 8.00000 0.632456
$$161$$ 12.0000 0.945732
$$162$$ −2.00000 −0.157135
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ −16.0000 −1.24939
$$165$$ −3.00000 −0.233550
$$166$$ −8.00000 −0.620920
$$167$$ 18.0000 1.39288 0.696441 0.717614i $$-0.254766\pi$$
0.696441 + 0.717614i $$0.254766\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ −6.00000 −0.460179
$$171$$ −1.00000 −0.0764719
$$172$$ −2.00000 −0.152499
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 20.0000 1.51620
$$175$$ −12.0000 −0.907115
$$176$$ 12.0000 0.904534
$$177$$ 0 0
$$178$$ −20.0000 −1.49906
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 36.0000 2.66850
$$183$$ 7.00000 0.517455
$$184$$ 0 0
$$185$$ 8.00000 0.588172
$$186$$ −4.00000 −0.293294
$$187$$ −9.00000 −0.658145
$$188$$ 6.00000 0.437595
$$189$$ 3.00000 0.218218
$$190$$ 2.00000 0.145095
$$191$$ −3.00000 −0.217072 −0.108536 0.994092i $$-0.534616\pi$$
−0.108536 + 0.994092i $$0.534616\pi$$
$$192$$ −8.00000 −0.577350
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 4.00000 0.287183
$$195$$ −6.00000 −0.429669
$$196$$ 4.00000 0.285714
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 6.00000 0.426401
$$199$$ −5.00000 −0.354441 −0.177220 0.984171i $$-0.556711\pi$$
−0.177220 + 0.984171i $$0.556711\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ −4.00000 −0.281439
$$203$$ −30.0000 −2.10559
$$204$$ 6.00000 0.420084
$$205$$ −8.00000 −0.558744
$$206$$ −28.0000 −1.95085
$$207$$ 4.00000 0.278019
$$208$$ 24.0000 1.66410
$$209$$ 3.00000 0.207514
$$210$$ −6.00000 −0.414039
$$211$$ −28.0000 −1.92760 −0.963800 0.266627i $$-0.914091\pi$$
−0.963800 + 0.266627i $$0.914091\pi$$
$$212$$ −12.0000 −0.824163
$$213$$ 12.0000 0.822226
$$214$$ 4.00000 0.273434
$$215$$ −1.00000 −0.0681994
$$216$$ 0 0
$$217$$ 6.00000 0.407307
$$218$$ −40.0000 −2.70914
$$219$$ −11.0000 −0.743311
$$220$$ −6.00000 −0.404520
$$221$$ −18.0000 −1.21081
$$222$$ −16.0000 −1.07385
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 24.0000 1.60357
$$225$$ −4.00000 −0.266667
$$226$$ 12.0000 0.798228
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ −15.0000 −0.991228 −0.495614 0.868543i $$-0.665057\pi$$
−0.495614 + 0.868543i $$0.665057\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ −9.00000 −0.592157
$$232$$ 0 0
$$233$$ −11.0000 −0.720634 −0.360317 0.932830i $$-0.617331\pi$$
−0.360317 + 0.932830i $$0.617331\pi$$
$$234$$ 12.0000 0.784465
$$235$$ 3.00000 0.195698
$$236$$ 0 0
$$237$$ 0 0
$$238$$ −18.0000 −1.16677
$$239$$ −15.0000 −0.970269 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ −4.00000 −0.258199
$$241$$ 12.0000 0.772988 0.386494 0.922292i $$-0.373686\pi$$
0.386494 + 0.922292i $$0.373686\pi$$
$$242$$ 4.00000 0.257130
$$243$$ 1.00000 0.0641500
$$244$$ 14.0000 0.896258
$$245$$ 2.00000 0.127775
$$246$$ 16.0000 1.02012
$$247$$ 6.00000 0.381771
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 18.0000 1.13842
$$251$$ 27.0000 1.70422 0.852112 0.523359i $$-0.175321\pi$$
0.852112 + 0.523359i $$0.175321\pi$$
$$252$$ 6.00000 0.377964
$$253$$ −12.0000 −0.754434
$$254$$ 4.00000 0.250982
$$255$$ 3.00000 0.187867
$$256$$ 16.0000 1.00000
$$257$$ 8.00000 0.499026 0.249513 0.968371i $$-0.419729\pi$$
0.249513 + 0.968371i $$0.419729\pi$$
$$258$$ 2.00000 0.124515
$$259$$ 24.0000 1.49129
$$260$$ −12.0000 −0.744208
$$261$$ −10.0000 −0.618984
$$262$$ 26.0000 1.60629
$$263$$ −21.0000 −1.29492 −0.647458 0.762101i $$-0.724168\pi$$
−0.647458 + 0.762101i $$0.724168\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 6.00000 0.367884
$$267$$ 10.0000 0.611990
$$268$$ 16.0000 0.977356
$$269$$ −30.0000 −1.82913 −0.914566 0.404436i $$-0.867468\pi$$
−0.914566 + 0.404436i $$0.867468\pi$$
$$270$$ −2.00000 −0.121716
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ −12.0000 −0.727607
$$273$$ −18.0000 −1.08941
$$274$$ −6.00000 −0.362473
$$275$$ 12.0000 0.723627
$$276$$ 8.00000 0.481543
$$277$$ 13.0000 0.781094 0.390547 0.920583i $$-0.372286\pi$$
0.390547 + 0.920583i $$0.372286\pi$$
$$278$$ 10.0000 0.599760
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ −6.00000 −0.357295
$$283$$ 19.0000 1.12943 0.564716 0.825285i $$-0.308986\pi$$
0.564716 + 0.825285i $$0.308986\pi$$
$$284$$ 24.0000 1.42414
$$285$$ −1.00000 −0.0592349
$$286$$ −36.0000 −2.12872
$$287$$ −24.0000 −1.41668
$$288$$ 8.00000 0.471405
$$289$$ −8.00000 −0.470588
$$290$$ 20.0000 1.17444
$$291$$ −2.00000 −0.117242
$$292$$ −22.0000 −1.28745
$$293$$ 4.00000 0.233682 0.116841 0.993151i $$-0.462723\pi$$
0.116841 + 0.993151i $$0.462723\pi$$
$$294$$ −4.00000 −0.233285
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −3.00000 −0.174078
$$298$$ −30.0000 −1.73785
$$299$$ −24.0000 −1.38796
$$300$$ −8.00000 −0.461880
$$301$$ −3.00000 −0.172917
$$302$$ 16.0000 0.920697
$$303$$ 2.00000 0.114897
$$304$$ 4.00000 0.229416
$$305$$ 7.00000 0.400819
$$306$$ −6.00000 −0.342997
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ −18.0000 −1.02565
$$309$$ 14.0000 0.796432
$$310$$ −4.00000 −0.227185
$$311$$ 7.00000 0.396934 0.198467 0.980108i $$-0.436404\pi$$
0.198467 + 0.980108i $$0.436404\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 3.00000 0.169031
$$316$$ 0 0
$$317$$ −12.0000 −0.673987 −0.336994 0.941507i $$-0.609410\pi$$
−0.336994 + 0.941507i $$0.609410\pi$$
$$318$$ 12.0000 0.672927
$$319$$ 30.0000 1.67968
$$320$$ −8.00000 −0.447214
$$321$$ −2.00000 −0.111629
$$322$$ −24.0000 −1.33747
$$323$$ −3.00000 −0.166924
$$324$$ 2.00000 0.111111
$$325$$ 24.0000 1.33128
$$326$$ 32.0000 1.77232
$$327$$ 20.0000 1.10600
$$328$$ 0 0
$$329$$ 9.00000 0.496186
$$330$$ 6.00000 0.330289
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 8.00000 0.439057
$$333$$ 8.00000 0.438397
$$334$$ −36.0000 −1.96983
$$335$$ 8.00000 0.437087
$$336$$ −12.0000 −0.654654
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ −46.0000 −2.50207
$$339$$ −6.00000 −0.325875
$$340$$ 6.00000 0.325396
$$341$$ −6.00000 −0.324918
$$342$$ 2.00000 0.108148
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 4.00000 0.215353
$$346$$ −28.0000 −1.50529
$$347$$ 3.00000 0.161048 0.0805242 0.996753i $$-0.474341\pi$$
0.0805242 + 0.996753i $$0.474341\pi$$
$$348$$ −20.0000 −1.07211
$$349$$ 25.0000 1.33822 0.669110 0.743164i $$-0.266676\pi$$
0.669110 + 0.743164i $$0.266676\pi$$
$$350$$ 24.0000 1.28285
$$351$$ −6.00000 −0.320256
$$352$$ −24.0000 −1.27920
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ 12.0000 0.636894
$$356$$ 20.0000 1.06000
$$357$$ 9.00000 0.476331
$$358$$ 20.0000 1.05703
$$359$$ 25.0000 1.31945 0.659725 0.751507i $$-0.270673\pi$$
0.659725 + 0.751507i $$0.270673\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −4.00000 −0.210235
$$363$$ −2.00000 −0.104973
$$364$$ −36.0000 −1.88691
$$365$$ −11.0000 −0.575766
$$366$$ −14.0000 −0.731792
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ −16.0000 −0.834058
$$369$$ −8.00000 −0.416463
$$370$$ −16.0000 −0.831800
$$371$$ −18.0000 −0.934513
$$372$$ 4.00000 0.207390
$$373$$ −16.0000 −0.828449 −0.414224 0.910175i $$-0.635947\pi$$
−0.414224 + 0.910175i $$0.635947\pi$$
$$374$$ 18.0000 0.930758
$$375$$ −9.00000 −0.464758
$$376$$ 0 0
$$377$$ 60.0000 3.09016
$$378$$ −6.00000 −0.308607
$$379$$ −30.0000 −1.54100 −0.770498 0.637442i $$-0.779993\pi$$
−0.770498 + 0.637442i $$0.779993\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ −2.00000 −0.102463
$$382$$ 6.00000 0.306987
$$383$$ 14.0000 0.715367 0.357683 0.933843i $$-0.383567\pi$$
0.357683 + 0.933843i $$0.383567\pi$$
$$384$$ 0 0
$$385$$ −9.00000 −0.458682
$$386$$ −8.00000 −0.407189
$$387$$ −1.00000 −0.0508329
$$388$$ −4.00000 −0.203069
$$389$$ −15.0000 −0.760530 −0.380265 0.924878i $$-0.624167\pi$$
−0.380265 + 0.924878i $$0.624167\pi$$
$$390$$ 12.0000 0.607644
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ −13.0000 −0.655763
$$394$$ 4.00000 0.201517
$$395$$ 0 0
$$396$$ −6.00000 −0.301511
$$397$$ −7.00000 −0.351320 −0.175660 0.984451i $$-0.556206\pi$$
−0.175660 + 0.984451i $$0.556206\pi$$
$$398$$ 10.0000 0.501255
$$399$$ −3.00000 −0.150188
$$400$$ 16.0000 0.800000
$$401$$ −28.0000 −1.39825 −0.699127 0.714998i $$-0.746428\pi$$
−0.699127 + 0.714998i $$0.746428\pi$$
$$402$$ −16.0000 −0.798007
$$403$$ −12.0000 −0.597763
$$404$$ 4.00000 0.199007
$$405$$ 1.00000 0.0496904
$$406$$ 60.0000 2.97775
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 16.0000 0.790184
$$411$$ 3.00000 0.147979
$$412$$ 28.0000 1.37946
$$413$$ 0 0
$$414$$ −8.00000 −0.393179
$$415$$ 4.00000 0.196352
$$416$$ −48.0000 −2.35339
$$417$$ −5.00000 −0.244851
$$418$$ −6.00000 −0.293470
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 6.00000 0.292770
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 56.0000 2.72604
$$423$$ 3.00000 0.145865
$$424$$ 0 0
$$425$$ −12.0000 −0.582086
$$426$$ −24.0000 −1.16280
$$427$$ 21.0000 1.01626
$$428$$ −4.00000 −0.193347
$$429$$ 18.0000 0.869048
$$430$$ 2.00000 0.0964486
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ −26.0000 −1.24948 −0.624740 0.780833i $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ −12.0000 −0.576018
$$435$$ −10.0000 −0.479463
$$436$$ 40.0000 1.91565
$$437$$ −4.00000 −0.191346
$$438$$ 22.0000 1.05120
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 36.0000 1.71235
$$443$$ 39.0000 1.85295 0.926473 0.376361i $$-0.122825\pi$$
0.926473 + 0.376361i $$0.122825\pi$$
$$444$$ 16.0000 0.759326
$$445$$ 10.0000 0.474045
$$446$$ −8.00000 −0.378811
$$447$$ 15.0000 0.709476
$$448$$ −24.0000 −1.13389
$$449$$ −20.0000 −0.943858 −0.471929 0.881636i $$-0.656442\pi$$
−0.471929 + 0.881636i $$0.656442\pi$$
$$450$$ 8.00000 0.377124
$$451$$ 24.0000 1.13012
$$452$$ −12.0000 −0.564433
$$453$$ −8.00000 −0.375873
$$454$$ −36.0000 −1.68956
$$455$$ −18.0000 −0.843853
$$456$$ 0 0
$$457$$ 3.00000 0.140334 0.0701670 0.997535i $$-0.477647\pi$$
0.0701670 + 0.997535i $$0.477647\pi$$
$$458$$ 30.0000 1.40181
$$459$$ 3.00000 0.140028
$$460$$ 8.00000 0.373002
$$461$$ −33.0000 −1.53696 −0.768482 0.639872i $$-0.778987\pi$$
−0.768482 + 0.639872i $$0.778987\pi$$
$$462$$ 18.0000 0.837436
$$463$$ −31.0000 −1.44069 −0.720346 0.693615i $$-0.756017\pi$$
−0.720346 + 0.693615i $$0.756017\pi$$
$$464$$ 40.0000 1.85695
$$465$$ 2.00000 0.0927478
$$466$$ 22.0000 1.01913
$$467$$ −17.0000 −0.786666 −0.393333 0.919396i $$-0.628678\pi$$
−0.393333 + 0.919396i $$0.628678\pi$$
$$468$$ −12.0000 −0.554700
$$469$$ 24.0000 1.10822
$$470$$ −6.00000 −0.276759
$$471$$ −2.00000 −0.0921551
$$472$$ 0 0
$$473$$ 3.00000 0.137940
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 18.0000 0.825029
$$477$$ −6.00000 −0.274721
$$478$$ 30.0000 1.37217
$$479$$ −40.0000 −1.82765 −0.913823 0.406112i $$-0.866884\pi$$
−0.913823 + 0.406112i $$0.866884\pi$$
$$480$$ 8.00000 0.365148
$$481$$ −48.0000 −2.18861
$$482$$ −24.0000 −1.09317
$$483$$ 12.0000 0.546019
$$484$$ −4.00000 −0.181818
$$485$$ −2.00000 −0.0908153
$$486$$ −2.00000 −0.0907218
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 0 0
$$489$$ −16.0000 −0.723545
$$490$$ −4.00000 −0.180702
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ −16.0000 −0.721336
$$493$$ −30.0000 −1.35113
$$494$$ −12.0000 −0.539906
$$495$$ −3.00000 −0.134840
$$496$$ −8.00000 −0.359211
$$497$$ 36.0000 1.61482
$$498$$ −8.00000 −0.358489
$$499$$ −35.0000 −1.56682 −0.783408 0.621508i $$-0.786520\pi$$
−0.783408 + 0.621508i $$0.786520\pi$$
$$500$$ −18.0000 −0.804984
$$501$$ 18.0000 0.804181
$$502$$ −54.0000 −2.41014
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 2.00000 0.0889988
$$506$$ 24.0000 1.06693
$$507$$ 23.0000 1.02147
$$508$$ −4.00000 −0.177471
$$509$$ −10.0000 −0.443242 −0.221621 0.975133i $$-0.571135\pi$$
−0.221621 + 0.975133i $$0.571135\pi$$
$$510$$ −6.00000 −0.265684
$$511$$ −33.0000 −1.45983
$$512$$ −32.0000 −1.41421
$$513$$ −1.00000 −0.0441511
$$514$$ −16.0000 −0.705730
$$515$$ 14.0000 0.616914
$$516$$ −2.00000 −0.0880451
$$517$$ −9.00000 −0.395820
$$518$$ −48.0000 −2.10900
$$519$$ 14.0000 0.614532
$$520$$ 0 0
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ 20.0000 0.875376
$$523$$ 14.0000 0.612177 0.306089 0.952003i $$-0.400980\pi$$
0.306089 + 0.952003i $$0.400980\pi$$
$$524$$ −26.0000 −1.13582
$$525$$ −12.0000 −0.523723
$$526$$ 42.0000 1.83129
$$527$$ 6.00000 0.261364
$$528$$ 12.0000 0.522233
$$529$$ −7.00000 −0.304348
$$530$$ 12.0000 0.521247
$$531$$ 0 0
$$532$$ −6.00000 −0.260133
$$533$$ 48.0000 2.07911
$$534$$ −20.0000 −0.865485
$$535$$ −2.00000 −0.0864675
$$536$$ 0 0
$$537$$ −10.0000 −0.431532
$$538$$ 60.0000 2.58678
$$539$$ −6.00000 −0.258438
$$540$$ 2.00000 0.0860663
$$541$$ −13.0000 −0.558914 −0.279457 0.960158i $$-0.590154\pi$$
−0.279457 + 0.960158i $$0.590154\pi$$
$$542$$ −24.0000 −1.03089
$$543$$ 2.00000 0.0858282
$$544$$ 24.0000 1.02899
$$545$$ 20.0000 0.856706
$$546$$ 36.0000 1.54066
$$547$$ −2.00000 −0.0855138 −0.0427569 0.999086i $$-0.513614\pi$$
−0.0427569 + 0.999086i $$0.513614\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 7.00000 0.298753
$$550$$ −24.0000 −1.02336
$$551$$ 10.0000 0.426014
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −26.0000 −1.10463
$$555$$ 8.00000 0.339581
$$556$$ −10.0000 −0.424094
$$557$$ 3.00000 0.127114 0.0635570 0.997978i $$-0.479756\pi$$
0.0635570 + 0.997978i $$0.479756\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ 6.00000 0.253773
$$560$$ −12.0000 −0.507093
$$561$$ −9.00000 −0.379980
$$562$$ −4.00000 −0.168730
$$563$$ 44.0000 1.85438 0.927189 0.374593i $$-0.122217\pi$$
0.927189 + 0.374593i $$0.122217\pi$$
$$564$$ 6.00000 0.252646
$$565$$ −6.00000 −0.252422
$$566$$ −38.0000 −1.59726
$$567$$ 3.00000 0.125988
$$568$$ 0 0
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 2.00000 0.0837708
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 36.0000 1.50524
$$573$$ −3.00000 −0.125327
$$574$$ 48.0000 2.00348
$$575$$ −16.0000 −0.667246
$$576$$ −8.00000 −0.333333
$$577$$ 3.00000 0.124892 0.0624458 0.998048i $$-0.480110\pi$$
0.0624458 + 0.998048i $$0.480110\pi$$
$$578$$ 16.0000 0.665512
$$579$$ 4.00000 0.166234
$$580$$ −20.0000 −0.830455
$$581$$ 12.0000 0.497844
$$582$$ 4.00000 0.165805
$$583$$ 18.0000 0.745484
$$584$$ 0 0
$$585$$ −6.00000 −0.248069
$$586$$ −8.00000 −0.330477
$$587$$ −37.0000 −1.52715 −0.763577 0.645717i $$-0.776559\pi$$
−0.763577 + 0.645717i $$0.776559\pi$$
$$588$$ 4.00000 0.164957
$$589$$ −2.00000 −0.0824086
$$590$$ 0 0
$$591$$ −2.00000 −0.0822690
$$592$$ −32.0000 −1.31519
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 6.00000 0.246183
$$595$$ 9.00000 0.368964
$$596$$ 30.0000 1.22885
$$597$$ −5.00000 −0.204636
$$598$$ 48.0000 1.96287
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ 6.00000 0.244542
$$603$$ 8.00000 0.325785
$$604$$ −16.0000 −0.651031
$$605$$ −2.00000 −0.0813116
$$606$$ −4.00000 −0.162489
$$607$$ 18.0000 0.730597 0.365299 0.930890i $$-0.380967\pi$$
0.365299 + 0.930890i $$0.380967\pi$$
$$608$$ −8.00000 −0.324443
$$609$$ −30.0000 −1.21566
$$610$$ −14.0000 −0.566843
$$611$$ −18.0000 −0.728202
$$612$$ 6.00000 0.242536
$$613$$ 9.00000 0.363507 0.181753 0.983344i $$-0.441823\pi$$
0.181753 + 0.983344i $$0.441823\pi$$
$$614$$ 24.0000 0.968561
$$615$$ −8.00000 −0.322591
$$616$$ 0 0
$$617$$ 23.0000 0.925945 0.462973 0.886373i $$-0.346783\pi$$
0.462973 + 0.886373i $$0.346783\pi$$
$$618$$ −28.0000 −1.12633
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 4.00000 0.160514
$$622$$ −14.0000 −0.561349
$$623$$ 30.0000 1.20192
$$624$$ 24.0000 0.960769
$$625$$ 11.0000 0.440000
$$626$$ −28.0000 −1.11911
$$627$$ 3.00000 0.119808
$$628$$ −4.00000 −0.159617
$$629$$ 24.0000 0.956943
$$630$$ −6.00000 −0.239046
$$631$$ 7.00000 0.278666 0.139333 0.990246i $$-0.455504\pi$$
0.139333 + 0.990246i $$0.455504\pi$$
$$632$$ 0 0
$$633$$ −28.0000 −1.11290
$$634$$ 24.0000 0.953162
$$635$$ −2.00000 −0.0793676
$$636$$ −12.0000 −0.475831
$$637$$ −12.0000 −0.475457
$$638$$ −60.0000 −2.37542
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 4.00000 0.157867
$$643$$ −1.00000 −0.0394362 −0.0197181 0.999806i $$-0.506277\pi$$
−0.0197181 + 0.999806i $$0.506277\pi$$
$$644$$ 24.0000 0.945732
$$645$$ −1.00000 −0.0393750
$$646$$ 6.00000 0.236067
$$647$$ −27.0000 −1.06148 −0.530740 0.847535i $$-0.678086\pi$$
−0.530740 + 0.847535i $$0.678086\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −48.0000 −1.88271
$$651$$ 6.00000 0.235159
$$652$$ −32.0000 −1.25322
$$653$$ −1.00000 −0.0391330 −0.0195665 0.999809i $$-0.506229\pi$$
−0.0195665 + 0.999809i $$0.506229\pi$$
$$654$$ −40.0000 −1.56412
$$655$$ −13.0000 −0.507952
$$656$$ 32.0000 1.24939
$$657$$ −11.0000 −0.429151
$$658$$ −18.0000 −0.701713
$$659$$ 10.0000 0.389545 0.194772 0.980848i $$-0.437603\pi$$
0.194772 + 0.980848i $$0.437603\pi$$
$$660$$ −6.00000 −0.233550
$$661$$ 12.0000 0.466746 0.233373 0.972387i $$-0.425024\pi$$
0.233373 + 0.972387i $$0.425024\pi$$
$$662$$ −24.0000 −0.932786
$$663$$ −18.0000 −0.699062
$$664$$ 0 0
$$665$$ −3.00000 −0.116335
$$666$$ −16.0000 −0.619987
$$667$$ −40.0000 −1.54881
$$668$$ 36.0000 1.39288
$$669$$ 4.00000 0.154649
$$670$$ −16.0000 −0.618134
$$671$$ −21.0000 −0.810696
$$672$$ 24.0000 0.925820
$$673$$ −16.0000 −0.616755 −0.308377 0.951264i $$-0.599786\pi$$
−0.308377 + 0.951264i $$0.599786\pi$$
$$674$$ 44.0000 1.69482
$$675$$ −4.00000 −0.153960
$$676$$ 46.0000 1.76923
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ 12.0000 0.460857
$$679$$ −6.00000 −0.230259
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ 12.0000 0.459504
$$683$$ −6.00000 −0.229584 −0.114792 0.993390i $$-0.536620\pi$$
−0.114792 + 0.993390i $$0.536620\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 3.00000 0.114624
$$686$$ 30.0000 1.14541
$$687$$ −15.0000 −0.572286
$$688$$ 4.00000 0.152499
$$689$$ 36.0000 1.37149
$$690$$ −8.00000 −0.304555
$$691$$ 17.0000 0.646710 0.323355 0.946278i $$-0.395189\pi$$
0.323355 + 0.946278i $$0.395189\pi$$
$$692$$ 28.0000 1.06440
$$693$$ −9.00000 −0.341882
$$694$$ −6.00000 −0.227757
$$695$$ −5.00000 −0.189661
$$696$$ 0 0
$$697$$ −24.0000 −0.909065
$$698$$ −50.0000 −1.89253
$$699$$ −11.0000 −0.416058
$$700$$ −24.0000 −0.907115
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ 12.0000 0.452911
$$703$$ −8.00000 −0.301726
$$704$$ 24.0000 0.904534
$$705$$ 3.00000 0.112987
$$706$$ −28.0000 −1.05379
$$707$$ 6.00000 0.225653
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ −24.0000 −0.900704
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 8.00000 0.299602
$$714$$ −18.0000 −0.673633
$$715$$ 18.0000 0.673162
$$716$$ −20.0000 −0.747435
$$717$$ −15.0000 −0.560185
$$718$$ −50.0000 −1.86598
$$719$$ −35.0000 −1.30528 −0.652640 0.757668i $$-0.726339\pi$$
−0.652640 + 0.757668i $$0.726339\pi$$
$$720$$ −4.00000 −0.149071
$$721$$ 42.0000 1.56416
$$722$$ −2.00000 −0.0744323
$$723$$ 12.0000 0.446285
$$724$$ 4.00000 0.148659
$$725$$ 40.0000 1.48556
$$726$$ 4.00000 0.148454
$$727$$ −7.00000 −0.259616 −0.129808 0.991539i $$-0.541436\pi$$
−0.129808 + 0.991539i $$0.541436\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 22.0000 0.814257
$$731$$ −3.00000 −0.110959
$$732$$ 14.0000 0.517455
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 2.00000 0.0737711
$$736$$ 32.0000 1.17954
$$737$$ −24.0000 −0.884051
$$738$$ 16.0000 0.588968
$$739$$ −45.0000 −1.65535 −0.827676 0.561206i $$-0.810337\pi$$
−0.827676 + 0.561206i $$0.810337\pi$$
$$740$$ 16.0000 0.588172
$$741$$ 6.00000 0.220416
$$742$$ 36.0000 1.32160
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ 15.0000 0.549557
$$746$$ 32.0000 1.17160
$$747$$ 4.00000 0.146352
$$748$$ −18.0000 −0.658145
$$749$$ −6.00000 −0.219235
$$750$$ 18.0000 0.657267
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ 27.0000 0.983935
$$754$$ −120.000 −4.37014
$$755$$ −8.00000 −0.291150
$$756$$ 6.00000 0.218218
$$757$$ 23.0000 0.835949 0.417975 0.908459i $$-0.362740\pi$$
0.417975 + 0.908459i $$0.362740\pi$$
$$758$$ 60.0000 2.17930
$$759$$ −12.0000 −0.435572
$$760$$ 0 0
$$761$$ −13.0000 −0.471250 −0.235625 0.971844i $$-0.575714\pi$$
−0.235625 + 0.971844i $$0.575714\pi$$
$$762$$ 4.00000 0.144905
$$763$$ 60.0000 2.17215
$$764$$ −6.00000 −0.217072
$$765$$ 3.00000 0.108465
$$766$$ −28.0000 −1.01168
$$767$$ 0 0
$$768$$ 16.0000 0.577350
$$769$$ −45.0000 −1.62274 −0.811371 0.584532i $$-0.801278\pi$$
−0.811371 + 0.584532i $$0.801278\pi$$
$$770$$ 18.0000 0.648675
$$771$$ 8.00000 0.288113
$$772$$ 8.00000 0.287926
$$773$$ −36.0000 −1.29483 −0.647415 0.762138i $$-0.724150\pi$$
−0.647415 + 0.762138i $$0.724150\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ −8.00000 −0.287368
$$776$$ 0 0
$$777$$ 24.0000 0.860995
$$778$$ 30.0000 1.07555
$$779$$ 8.00000 0.286630
$$780$$ −12.0000 −0.429669
$$781$$ −36.0000 −1.28818
$$782$$ −24.0000 −0.858238
$$783$$ −10.0000 −0.357371
$$784$$ −8.00000 −0.285714
$$785$$ −2.00000 −0.0713831
$$786$$ 26.0000 0.927389
$$787$$ 8.00000 0.285169 0.142585 0.989783i $$-0.454459\pi$$
0.142585 + 0.989783i $$0.454459\pi$$
$$788$$ −4.00000 −0.142494
$$789$$ −21.0000 −0.747620
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ 0 0
$$793$$ −42.0000 −1.49146
$$794$$ 14.0000 0.496841
$$795$$ −6.00000 −0.212798
$$796$$ −10.0000 −0.354441
$$797$$ −12.0000 −0.425062 −0.212531 0.977154i $$-0.568171\pi$$
−0.212531 + 0.977154i $$0.568171\pi$$
$$798$$ 6.00000 0.212398
$$799$$ 9.00000 0.318397
$$800$$ −32.0000 −1.13137
$$801$$ 10.0000 0.353333
$$802$$ 56.0000 1.97743
$$803$$ 33.0000 1.16454
$$804$$ 16.0000 0.564276
$$805$$ 12.0000 0.422944
$$806$$ 24.0000 0.845364
$$807$$ −30.0000 −1.05605
$$808$$ 0 0
$$809$$ 5.00000 0.175791 0.0878953 0.996130i $$-0.471986\pi$$
0.0878953 + 0.996130i $$0.471986\pi$$
$$810$$ −2.00000 −0.0702728
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ −60.0000 −2.10559
$$813$$ 12.0000 0.420858
$$814$$ 48.0000 1.68240
$$815$$ −16.0000 −0.560456
$$816$$ −12.0000 −0.420084
$$817$$ 1.00000 0.0349856
$$818$$ −20.0000 −0.699284
$$819$$ −18.0000 −0.628971
$$820$$ −16.0000 −0.558744
$$821$$ −33.0000 −1.15171 −0.575854 0.817553i $$-0.695330\pi$$
−0.575854 + 0.817553i $$0.695330\pi$$
$$822$$ −6.00000 −0.209274
$$823$$ 19.0000 0.662298 0.331149 0.943578i $$-0.392564\pi$$
0.331149 + 0.943578i $$0.392564\pi$$
$$824$$ 0 0
$$825$$ 12.0000 0.417786
$$826$$ 0 0
$$827$$ −52.0000 −1.80822 −0.904109 0.427303i $$-0.859464\pi$$
−0.904109 + 0.427303i $$0.859464\pi$$
$$828$$ 8.00000 0.278019
$$829$$ 20.0000 0.694629 0.347314 0.937749i $$-0.387094\pi$$
0.347314 + 0.937749i $$0.387094\pi$$
$$830$$ −8.00000 −0.277684
$$831$$ 13.0000 0.450965
$$832$$ 48.0000 1.66410
$$833$$ 6.00000 0.207888
$$834$$ 10.0000 0.346272
$$835$$ 18.0000 0.622916
$$836$$ 6.00000 0.207514
$$837$$ 2.00000 0.0691301
$$838$$ −40.0000 −1.38178
$$839$$ 30.0000 1.03572 0.517858 0.855467i $$-0.326730\pi$$
0.517858 + 0.855467i $$0.326730\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ −4.00000 −0.137849
$$843$$ 2.00000 0.0688837
$$844$$ −56.0000 −1.92760
$$845$$ 23.0000 0.791224
$$846$$ −6.00000 −0.206284
$$847$$ −6.00000 −0.206162
$$848$$ 24.0000 0.824163
$$849$$ 19.0000 0.652078
$$850$$ 24.0000 0.823193
$$851$$ 32.0000 1.09695
$$852$$ 24.0000 0.822226
$$853$$ 34.0000 1.16414 0.582069 0.813139i $$-0.302243\pi$$
0.582069 + 0.813139i $$0.302243\pi$$
$$854$$ −42.0000 −1.43721
$$855$$ −1.00000 −0.0341993
$$856$$ 0 0
$$857$$ 48.0000 1.63965 0.819824 0.572615i $$-0.194071\pi$$
0.819824 + 0.572615i $$0.194071\pi$$
$$858$$ −36.0000 −1.22902
$$859$$ 35.0000 1.19418 0.597092 0.802173i $$-0.296323\pi$$
0.597092 + 0.802173i $$0.296323\pi$$
$$860$$ −2.00000 −0.0681994
$$861$$ −24.0000 −0.817918
$$862$$ 36.0000 1.22616
$$863$$ 44.0000 1.49778 0.748889 0.662696i $$-0.230588\pi$$
0.748889 + 0.662696i $$0.230588\pi$$
$$864$$ 8.00000 0.272166
$$865$$ 14.0000 0.476014
$$866$$ 52.0000 1.76703
$$867$$ −8.00000 −0.271694
$$868$$ 12.0000 0.407307
$$869$$ 0 0
$$870$$ 20.0000 0.678064
$$871$$ −48.0000 −1.62642
$$872$$ 0 0
$$873$$ −2.00000 −0.0676897
$$874$$ 8.00000 0.270604
$$875$$ −27.0000 −0.912767
$$876$$ −22.0000 −0.743311
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 4.00000 0.134917
$$880$$ 12.0000 0.404520
$$881$$ 7.00000 0.235836 0.117918 0.993023i $$-0.462378\pi$$
0.117918 + 0.993023i $$0.462378\pi$$
$$882$$ −4.00000 −0.134687
$$883$$ −21.0000 −0.706706 −0.353353 0.935490i $$-0.614959\pi$$
−0.353353 + 0.935490i $$0.614959\pi$$
$$884$$ −36.0000 −1.21081
$$885$$ 0 0
$$886$$ −78.0000 −2.62046
$$887$$ −52.0000 −1.74599 −0.872995 0.487730i $$-0.837825\pi$$
−0.872995 + 0.487730i $$0.837825\pi$$
$$888$$ 0 0
$$889$$ −6.00000 −0.201234
$$890$$ −20.0000 −0.670402
$$891$$ −3.00000 −0.100504
$$892$$ 8.00000 0.267860
$$893$$ −3.00000 −0.100391
$$894$$ −30.0000 −1.00335
$$895$$ −10.0000 −0.334263
$$896$$ 0 0
$$897$$ −24.0000 −0.801337
$$898$$ 40.0000 1.33482
$$899$$ −20.0000 −0.667037
$$900$$ −8.00000 −0.266667
$$901$$ −18.0000 −0.599667
$$902$$ −48.0000 −1.59823
$$903$$ −3.00000 −0.0998337
$$904$$ 0 0
$$905$$ 2.00000 0.0664822
$$906$$ 16.0000 0.531564
$$907$$ −2.00000 −0.0664089 −0.0332045 0.999449i $$-0.510571\pi$$
−0.0332045 + 0.999449i $$0.510571\pi$$
$$908$$ 36.0000 1.19470
$$909$$ 2.00000 0.0663358
$$910$$ 36.0000 1.19339
$$911$$ 2.00000 0.0662630 0.0331315 0.999451i $$-0.489452\pi$$
0.0331315 + 0.999451i $$0.489452\pi$$
$$912$$ 4.00000 0.132453
$$913$$ −12.0000 −0.397142
$$914$$ −6.00000 −0.198462
$$915$$ 7.00000 0.231413
$$916$$ −30.0000 −0.991228
$$917$$ −39.0000 −1.28789
$$918$$ −6.00000 −0.198030
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 66.0000 2.17359
$$923$$ −72.0000 −2.36991
$$924$$ −18.0000 −0.592157
$$925$$ −32.0000 −1.05215
$$926$$ 62.0000 2.03745
$$927$$ 14.0000 0.459820
$$928$$ −80.0000 −2.62613
$$929$$ −50.0000 −1.64045 −0.820223 0.572043i $$-0.806151\pi$$
−0.820223 + 0.572043i $$0.806151\pi$$
$$930$$ −4.00000 −0.131165
$$931$$ −2.00000 −0.0655474
$$932$$ −22.0000 −0.720634
$$933$$ 7.00000 0.229170
$$934$$ 34.0000 1.11251
$$935$$ −9.00000 −0.294331
$$936$$ 0 0
$$937$$ 53.0000 1.73143 0.865717 0.500533i $$-0.166863\pi$$
0.865717 + 0.500533i $$0.166863\pi$$
$$938$$ −48.0000 −1.56726
$$939$$ 14.0000 0.456873
$$940$$ 6.00000 0.195698
$$941$$ 2.00000 0.0651981 0.0325991 0.999469i $$-0.489622\pi$$
0.0325991 + 0.999469i $$0.489622\pi$$
$$942$$ 4.00000 0.130327
$$943$$ −32.0000 −1.04206
$$944$$ 0 0
$$945$$ 3.00000 0.0975900
$$946$$ −6.00000 −0.195077
$$947$$ 28.0000 0.909878 0.454939 0.890523i $$-0.349661\pi$$
0.454939 + 0.890523i $$0.349661\pi$$
$$948$$ 0 0
$$949$$ 66.0000 2.14245
$$950$$ −8.00000 −0.259554
$$951$$ −12.0000 −0.389127
$$952$$ 0 0
$$953$$ −16.0000 −0.518291 −0.259145 0.965838i $$-0.583441\pi$$
−0.259145 + 0.965838i $$0.583441\pi$$
$$954$$ 12.0000 0.388514
$$955$$ −3.00000 −0.0970777
$$956$$ −30.0000 −0.970269
$$957$$ 30.0000 0.969762
$$958$$ 80.0000 2.58468
$$959$$ 9.00000 0.290625
$$960$$ −8.00000 −0.258199
$$961$$ −27.0000 −0.870968
$$962$$ 96.0000 3.09516
$$963$$ −2.00000 −0.0644491
$$964$$ 24.0000 0.772988
$$965$$ 4.00000 0.128765
$$966$$ −24.0000 −0.772187
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ −3.00000 −0.0963739
$$970$$ 4.00000 0.128432
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 2.00000 0.0641500
$$973$$ −15.0000 −0.480878
$$974$$ −16.0000 −0.512673
$$975$$ 24.0000 0.768615
$$976$$ −28.0000 −0.896258
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 32.0000 1.02325
$$979$$ −30.0000 −0.958804
$$980$$ 4.00000 0.127775
$$981$$ 20.0000 0.638551
$$982$$ 16.0000 0.510581
$$983$$ 4.00000 0.127580 0.0637901 0.997963i $$-0.479681\pi$$
0.0637901 + 0.997963i $$0.479681\pi$$
$$984$$ 0 0
$$985$$ −2.00000 −0.0637253
$$986$$ 60.0000 1.91079
$$987$$ 9.00000 0.286473
$$988$$ 12.0000 0.381771
$$989$$ −4.00000 −0.127193
$$990$$ 6.00000 0.190693
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 16.0000 0.508001
$$993$$ 12.0000 0.380808
$$994$$ −72.0000 −2.28370
$$995$$ −5.00000 −0.158511
$$996$$ 8.00000 0.253490
$$997$$ −7.00000 −0.221692 −0.110846 0.993838i $$-0.535356\pi$$
−0.110846 + 0.993838i $$0.535356\pi$$
$$998$$ 70.0000 2.21581
$$999$$ 8.00000 0.253109
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 57.2.a.b.1.1 1
3.2 odd 2 171.2.a.c.1.1 1
4.3 odd 2 912.2.a.d.1.1 1
5.2 odd 4 1425.2.c.a.799.1 2
5.3 odd 4 1425.2.c.a.799.2 2
5.4 even 2 1425.2.a.i.1.1 1
7.6 odd 2 2793.2.a.a.1.1 1
8.3 odd 2 3648.2.a.y.1.1 1
8.5 even 2 3648.2.a.h.1.1 1
11.10 odd 2 6897.2.a.g.1.1 1
12.11 even 2 2736.2.a.h.1.1 1
13.12 even 2 9633.2.a.p.1.1 1
15.14 odd 2 4275.2.a.a.1.1 1
19.18 odd 2 1083.2.a.d.1.1 1
21.20 even 2 8379.2.a.q.1.1 1
57.56 even 2 3249.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.b.1.1 1 1.1 even 1 trivial
171.2.a.c.1.1 1 3.2 odd 2
912.2.a.d.1.1 1 4.3 odd 2
1083.2.a.d.1.1 1 19.18 odd 2
1425.2.a.i.1.1 1 5.4 even 2
1425.2.c.a.799.1 2 5.2 odd 4
1425.2.c.a.799.2 2 5.3 odd 4
2736.2.a.h.1.1 1 12.11 even 2
2793.2.a.a.1.1 1 7.6 odd 2
3249.2.a.a.1.1 1 57.56 even 2
3648.2.a.h.1.1 1 8.5 even 2
3648.2.a.y.1.1 1 8.3 odd 2
4275.2.a.a.1.1 1 15.14 odd 2
6897.2.a.g.1.1 1 11.10 odd 2
8379.2.a.q.1.1 1 21.20 even 2
9633.2.a.p.1.1 1 13.12 even 2