# Properties

 Label 195.2.a.c.1.1 Level $195$ Weight $2$ Character 195.1 Self dual yes Analytic conductor $1.557$ 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] = [195,2,Mod(1,195)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(195, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("195.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 195.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: $$1.55708283941$$ 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 $$\chi$$ $$=$$ 195.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} +5.00000 q^{11} +2.00000 q^{12} -1.00000 q^{13} -2.00000 q^{14} -1.00000 q^{15} -4.00000 q^{16} -7.00000 q^{17} +2.00000 q^{18} -6.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} +10.0000 q^{22} +3.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} +2.00000 q^{29} -2.00000 q^{30} +2.00000 q^{31} -8.00000 q^{32} +5.00000 q^{33} -14.0000 q^{34} +1.00000 q^{35} +2.00000 q^{36} +7.00000 q^{37} -12.0000 q^{38} -1.00000 q^{39} +9.00000 q^{41} -2.00000 q^{42} -8.00000 q^{43} +10.0000 q^{44} -1.00000 q^{45} +6.00000 q^{46} +10.0000 q^{47} -4.00000 q^{48} -6.00000 q^{49} +2.00000 q^{50} -7.00000 q^{51} -2.00000 q^{52} +5.00000 q^{53} +2.00000 q^{54} -5.00000 q^{55} -6.00000 q^{57} +4.00000 q^{58} -2.00000 q^{60} +5.00000 q^{61} +4.00000 q^{62} -1.00000 q^{63} -8.00000 q^{64} +1.00000 q^{65} +10.0000 q^{66} -4.00000 q^{67} -14.0000 q^{68} +3.00000 q^{69} +2.00000 q^{70} +9.00000 q^{71} -6.00000 q^{73} +14.0000 q^{74} +1.00000 q^{75} -12.0000 q^{76} -5.00000 q^{77} -2.00000 q^{78} -3.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +18.0000 q^{82} -4.00000 q^{83} -2.00000 q^{84} +7.00000 q^{85} -16.0000 q^{86} +2.00000 q^{87} +11.0000 q^{89} -2.00000 q^{90} +1.00000 q^{91} +6.00000 q^{92} +2.00000 q^{93} +20.0000 q^{94} +6.00000 q^{95} -8.00000 q^{96} -11.0000 q^{97} -12.0000 q^{98} +5.00000 q^{99} +O(q^{100})$$

## 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 $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 2.00000 1.00000
$$5$$ −1.00000 −0.447214
$$6$$ 2.00000 0.816497
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ −2.00000 −0.632456
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 2.00000 0.577350
$$13$$ −1.00000 −0.277350
$$14$$ −2.00000 −0.534522
$$15$$ −1.00000 −0.258199
$$16$$ −4.00000 −1.00000
$$17$$ −7.00000 −1.69775 −0.848875 0.528594i $$-0.822719\pi$$
−0.848875 + 0.528594i $$0.822719\pi$$
$$18$$ 2.00000 0.471405
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ −1.00000 −0.218218
$$22$$ 10.0000 2.13201
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ −2.00000 −0.377964
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ −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
$$33$$ 5.00000 0.870388
$$34$$ −14.0000 −2.40098
$$35$$ 1.00000 0.169031
$$36$$ 2.00000 0.333333
$$37$$ 7.00000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ −12.0000 −1.94666
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 9.00000 1.40556 0.702782 0.711405i $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 10.0000 1.50756
$$45$$ −1.00000 −0.149071
$$46$$ 6.00000 0.884652
$$47$$ 10.0000 1.45865 0.729325 0.684167i $$-0.239834\pi$$
0.729325 + 0.684167i $$0.239834\pi$$
$$48$$ −4.00000 −0.577350
$$49$$ −6.00000 −0.857143
$$50$$ 2.00000 0.282843
$$51$$ −7.00000 −0.980196
$$52$$ −2.00000 −0.277350
$$53$$ 5.00000 0.686803 0.343401 0.939189i $$-0.388421\pi$$
0.343401 + 0.939189i $$0.388421\pi$$
$$54$$ 2.00000 0.272166
$$55$$ −5.00000 −0.674200
$$56$$ 0 0
$$57$$ −6.00000 −0.794719
$$58$$ 4.00000 0.525226
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −1.00000 −0.125988
$$64$$ −8.00000 −1.00000
$$65$$ 1.00000 0.124035
$$66$$ 10.0000 1.23091
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −14.0000 −1.69775
$$69$$ 3.00000 0.361158
$$70$$ 2.00000 0.239046
$$71$$ 9.00000 1.06810 0.534052 0.845452i $$-0.320669\pi$$
0.534052 + 0.845452i $$0.320669\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ 14.0000 1.62747
$$75$$ 1.00000 0.115470
$$76$$ −12.0000 −1.37649
$$77$$ −5.00000 −0.569803
$$78$$ −2.00000 −0.226455
$$79$$ −3.00000 −0.337526 −0.168763 0.985657i $$-0.553977\pi$$
−0.168763 + 0.985657i $$0.553977\pi$$
$$80$$ 4.00000 0.447214
$$81$$ 1.00000 0.111111
$$82$$ 18.0000 1.98777
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 7.00000 0.759257
$$86$$ −16.0000 −1.72532
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ 11.0000 1.16600 0.582999 0.812473i $$-0.301879\pi$$
0.582999 + 0.812473i $$0.301879\pi$$
$$90$$ −2.00000 −0.210819
$$91$$ 1.00000 0.104828
$$92$$ 6.00000 0.625543
$$93$$ 2.00000 0.207390
$$94$$ 20.0000 2.06284
$$95$$ 6.00000 0.615587
$$96$$ −8.00000 −0.816497
$$97$$ −11.0000 −1.11688 −0.558440 0.829545i $$-0.688600\pi$$
−0.558440 + 0.829545i $$0.688600\pi$$
$$98$$ −12.0000 −1.21218
$$99$$ 5.00000 0.502519
$$100$$ 2.00000 0.200000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ −14.0000 −1.38621
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ 1.00000 0.0975900
$$106$$ 10.0000 0.971286
$$107$$ −17.0000 −1.64345 −0.821726 0.569883i $$-0.806989\pi$$
−0.821726 + 0.569883i $$0.806989\pi$$
$$108$$ 2.00000 0.192450
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ −10.0000 −0.953463
$$111$$ 7.00000 0.664411
$$112$$ 4.00000 0.377964
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ −12.0000 −1.12390
$$115$$ −3.00000 −0.279751
$$116$$ 4.00000 0.371391
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ 7.00000 0.641689
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 10.0000 0.905357
$$123$$ 9.00000 0.811503
$$124$$ 4.00000 0.359211
$$125$$ −1.00000 −0.0894427
$$126$$ −2.00000 −0.178174
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 2.00000 0.175412
$$131$$ −22.0000 −1.92215 −0.961074 0.276289i $$-0.910895\pi$$
−0.961074 + 0.276289i $$0.910895\pi$$
$$132$$ 10.0000 0.870388
$$133$$ 6.00000 0.520266
$$134$$ −8.00000 −0.691095
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 6.00000 0.510754
$$139$$ 15.0000 1.27228 0.636142 0.771572i $$-0.280529\pi$$
0.636142 + 0.771572i $$0.280529\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 10.0000 0.842152
$$142$$ 18.0000 1.51053
$$143$$ −5.00000 −0.418121
$$144$$ −4.00000 −0.333333
$$145$$ −2.00000 −0.166091
$$146$$ −12.0000 −0.993127
$$147$$ −6.00000 −0.494872
$$148$$ 14.0000 1.15079
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 2.00000 0.163299
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ −7.00000 −0.565916
$$154$$ −10.0000 −0.805823
$$155$$ −2.00000 −0.160644
$$156$$ −2.00000 −0.160128
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ −6.00000 −0.477334
$$159$$ 5.00000 0.396526
$$160$$ 8.00000 0.632456
$$161$$ −3.00000 −0.236433
$$162$$ 2.00000 0.157135
$$163$$ 15.0000 1.17489 0.587445 0.809264i $$-0.300134\pi$$
0.587445 + 0.809264i $$0.300134\pi$$
$$164$$ 18.0000 1.40556
$$165$$ −5.00000 −0.389249
$$166$$ −8.00000 −0.620920
$$167$$ −24.0000 −1.85718 −0.928588 0.371113i $$-0.878976\pi$$
−0.928588 + 0.371113i $$0.878976\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 14.0000 1.07375
$$171$$ −6.00000 −0.458831
$$172$$ −16.0000 −1.21999
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 4.00000 0.303239
$$175$$ −1.00000 −0.0755929
$$176$$ −20.0000 −1.50756
$$177$$ 0 0
$$178$$ 22.0000 1.64897
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ −2.00000 −0.149071
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ 2.00000 0.148250
$$183$$ 5.00000 0.369611
$$184$$ 0 0
$$185$$ −7.00000 −0.514650
$$186$$ 4.00000 0.293294
$$187$$ −35.0000 −2.55945
$$188$$ 20.0000 1.45865
$$189$$ −1.00000 −0.0727393
$$190$$ 12.0000 0.870572
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ −8.00000 −0.577350
$$193$$ −17.0000 −1.22369 −0.611843 0.790979i $$-0.709572\pi$$
−0.611843 + 0.790979i $$0.709572\pi$$
$$194$$ −22.0000 −1.57951
$$195$$ 1.00000 0.0716115
$$196$$ −12.0000 −0.857143
$$197$$ 24.0000 1.70993 0.854965 0.518686i $$-0.173579\pi$$
0.854965 + 0.518686i $$0.173579\pi$$
$$198$$ 10.0000 0.710669
$$199$$ −28.0000 −1.98487 −0.992434 0.122782i $$-0.960818\pi$$
−0.992434 + 0.122782i $$0.960818\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ −2.00000 −0.140372
$$204$$ −14.0000 −0.980196
$$205$$ −9.00000 −0.628587
$$206$$ −8.00000 −0.557386
$$207$$ 3.00000 0.208514
$$208$$ 4.00000 0.277350
$$209$$ −30.0000 −2.07514
$$210$$ 2.00000 0.138013
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 9.00000 0.616670
$$214$$ −34.0000 −2.32419
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ −2.00000 −0.135769
$$218$$ 8.00000 0.541828
$$219$$ −6.00000 −0.405442
$$220$$ −10.0000 −0.674200
$$221$$ 7.00000 0.470871
$$222$$ 14.0000 0.939618
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 8.00000 0.534522
$$225$$ 1.00000 0.0666667
$$226$$ 20.0000 1.33038
$$227$$ −2.00000 −0.132745 −0.0663723 0.997795i $$-0.521143\pi$$
−0.0663723 + 0.997795i $$0.521143\pi$$
$$228$$ −12.0000 −0.794719
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ −6.00000 −0.395628
$$231$$ −5.00000 −0.328976
$$232$$ 0 0
$$233$$ 19.0000 1.24473 0.622366 0.782727i $$-0.286172\pi$$
0.622366 + 0.782727i $$0.286172\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ −10.0000 −0.652328
$$236$$ 0 0
$$237$$ −3.00000 −0.194871
$$238$$ 14.0000 0.907485
$$239$$ 9.00000 0.582162 0.291081 0.956698i $$-0.405985\pi$$
0.291081 + 0.956698i $$0.405985\pi$$
$$240$$ 4.00000 0.258199
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ 28.0000 1.79991
$$243$$ 1.00000 0.0641500
$$244$$ 10.0000 0.640184
$$245$$ 6.00000 0.383326
$$246$$ 18.0000 1.14764
$$247$$ 6.00000 0.381771
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ −2.00000 −0.126491
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 15.0000 0.943042
$$254$$ −4.00000 −0.250982
$$255$$ 7.00000 0.438357
$$256$$ 16.0000 1.00000
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ −16.0000 −0.996116
$$259$$ −7.00000 −0.434959
$$260$$ 2.00000 0.124035
$$261$$ 2.00000 0.123797
$$262$$ −44.0000 −2.71833
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ −5.00000 −0.307148
$$266$$ 12.0000 0.735767
$$267$$ 11.0000 0.673189
$$268$$ −8.00000 −0.488678
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ −2.00000 −0.121716
$$271$$ −22.0000 −1.33640 −0.668202 0.743980i $$-0.732936\pi$$
−0.668202 + 0.743980i $$0.732936\pi$$
$$272$$ 28.0000 1.69775
$$273$$ 1.00000 0.0605228
$$274$$ −28.0000 −1.69154
$$275$$ 5.00000 0.301511
$$276$$ 6.00000 0.361158
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 30.0000 1.79928
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 20.0000 1.19098
$$283$$ 20.0000 1.18888 0.594438 0.804141i $$-0.297374\pi$$
0.594438 + 0.804141i $$0.297374\pi$$
$$284$$ 18.0000 1.06810
$$285$$ 6.00000 0.355409
$$286$$ −10.0000 −0.591312
$$287$$ −9.00000 −0.531253
$$288$$ −8.00000 −0.471405
$$289$$ 32.0000 1.88235
$$290$$ −4.00000 −0.234888
$$291$$ −11.0000 −0.644831
$$292$$ −12.0000 −0.702247
$$293$$ 4.00000 0.233682 0.116841 0.993151i $$-0.462723\pi$$
0.116841 + 0.993151i $$0.462723\pi$$
$$294$$ −12.0000 −0.699854
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5.00000 0.290129
$$298$$ 30.0000 1.73785
$$299$$ −3.00000 −0.173494
$$300$$ 2.00000 0.115470
$$301$$ 8.00000 0.461112
$$302$$ −16.0000 −0.920697
$$303$$ 0 0
$$304$$ 24.0000 1.37649
$$305$$ −5.00000 −0.286299
$$306$$ −14.0000 −0.800327
$$307$$ 23.0000 1.31268 0.656340 0.754466i $$-0.272104\pi$$
0.656340 + 0.754466i $$0.272104\pi$$
$$308$$ −10.0000 −0.569803
$$309$$ −4.00000 −0.227552
$$310$$ −4.00000 −0.227185
$$311$$ 20.0000 1.13410 0.567048 0.823685i $$-0.308085\pi$$
0.567048 + 0.823685i $$0.308085\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ 36.0000 2.03160
$$315$$ 1.00000 0.0563436
$$316$$ −6.00000 −0.337526
$$317$$ −24.0000 −1.34797 −0.673987 0.738743i $$-0.735420\pi$$
−0.673987 + 0.738743i $$0.735420\pi$$
$$318$$ 10.0000 0.560772
$$319$$ 10.0000 0.559893
$$320$$ 8.00000 0.447214
$$321$$ −17.0000 −0.948847
$$322$$ −6.00000 −0.334367
$$323$$ 42.0000 2.33694
$$324$$ 2.00000 0.111111
$$325$$ −1.00000 −0.0554700
$$326$$ 30.0000 1.66155
$$327$$ 4.00000 0.221201
$$328$$ 0 0
$$329$$ −10.0000 −0.551318
$$330$$ −10.0000 −0.550482
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ −8.00000 −0.439057
$$333$$ 7.00000 0.383598
$$334$$ −48.0000 −2.62644
$$335$$ 4.00000 0.218543
$$336$$ 4.00000 0.218218
$$337$$ 8.00000 0.435788 0.217894 0.975972i $$-0.430081\pi$$
0.217894 + 0.975972i $$0.430081\pi$$
$$338$$ 2.00000 0.108786
$$339$$ 10.0000 0.543125
$$340$$ 14.0000 0.759257
$$341$$ 10.0000 0.541530
$$342$$ −12.0000 −0.648886
$$343$$ 13.0000 0.701934
$$344$$ 0 0
$$345$$ −3.00000 −0.161515
$$346$$ −36.0000 −1.93537
$$347$$ 11.0000 0.590511 0.295255 0.955418i $$-0.404595\pi$$
0.295255 + 0.955418i $$0.404595\pi$$
$$348$$ 4.00000 0.214423
$$349$$ 24.0000 1.28469 0.642345 0.766415i $$-0.277962\pi$$
0.642345 + 0.766415i $$0.277962\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ −1.00000 −0.0533761
$$352$$ −40.0000 −2.13201
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ −9.00000 −0.477670
$$356$$ 22.0000 1.16600
$$357$$ 7.00000 0.370479
$$358$$ 12.0000 0.634220
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ −14.0000 −0.735824
$$363$$ 14.0000 0.734809
$$364$$ 2.00000 0.104828
$$365$$ 6.00000 0.314054
$$366$$ 10.0000 0.522708
$$367$$ 4.00000 0.208798 0.104399 0.994535i $$-0.466708\pi$$
0.104399 + 0.994535i $$0.466708\pi$$
$$368$$ −12.0000 −0.625543
$$369$$ 9.00000 0.468521
$$370$$ −14.0000 −0.727825
$$371$$ −5.00000 −0.259587
$$372$$ 4.00000 0.207390
$$373$$ −32.0000 −1.65690 −0.828449 0.560065i $$-0.810776\pi$$
−0.828449 + 0.560065i $$0.810776\pi$$
$$374$$ −70.0000 −3.61961
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ −2.00000 −0.103005
$$378$$ −2.00000 −0.102869
$$379$$ 10.0000 0.513665 0.256833 0.966456i $$-0.417321\pi$$
0.256833 + 0.966456i $$0.417321\pi$$
$$380$$ 12.0000 0.615587
$$381$$ −2.00000 −0.102463
$$382$$ −24.0000 −1.22795
$$383$$ −6.00000 −0.306586 −0.153293 0.988181i $$-0.548988\pi$$
−0.153293 + 0.988181i $$0.548988\pi$$
$$384$$ 0 0
$$385$$ 5.00000 0.254824
$$386$$ −34.0000 −1.73055
$$387$$ −8.00000 −0.406663
$$388$$ −22.0000 −1.11688
$$389$$ −4.00000 −0.202808 −0.101404 0.994845i $$-0.532333\pi$$
−0.101404 + 0.994845i $$0.532333\pi$$
$$390$$ 2.00000 0.101274
$$391$$ −21.0000 −1.06202
$$392$$ 0 0
$$393$$ −22.0000 −1.10975
$$394$$ 48.0000 2.41821
$$395$$ 3.00000 0.150946
$$396$$ 10.0000 0.502519
$$397$$ 15.0000 0.752828 0.376414 0.926451i $$-0.377157\pi$$
0.376414 + 0.926451i $$0.377157\pi$$
$$398$$ −56.0000 −2.80703
$$399$$ 6.00000 0.300376
$$400$$ −4.00000 −0.200000
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ −2.00000 −0.0996271
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ −4.00000 −0.198517
$$407$$ 35.0000 1.73489
$$408$$ 0 0
$$409$$ 30.0000 1.48340 0.741702 0.670729i $$-0.234019\pi$$
0.741702 + 0.670729i $$0.234019\pi$$
$$410$$ −18.0000 −0.888957
$$411$$ −14.0000 −0.690569
$$412$$ −8.00000 −0.394132
$$413$$ 0 0
$$414$$ 6.00000 0.294884
$$415$$ 4.00000 0.196352
$$416$$ 8.00000 0.392232
$$417$$ 15.0000 0.734553
$$418$$ −60.0000 −2.93470
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ −20.0000 −0.974740 −0.487370 0.873195i $$-0.662044\pi$$
−0.487370 + 0.873195i $$0.662044\pi$$
$$422$$ −24.0000 −1.16830
$$423$$ 10.0000 0.486217
$$424$$ 0 0
$$425$$ −7.00000 −0.339550
$$426$$ 18.0000 0.872103
$$427$$ −5.00000 −0.241967
$$428$$ −34.0000 −1.64345
$$429$$ −5.00000 −0.241402
$$430$$ 16.0000 0.771589
$$431$$ −40.0000 −1.92673 −0.963366 0.268190i $$-0.913575\pi$$
−0.963366 + 0.268190i $$0.913575\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ −20.0000 −0.961139 −0.480569 0.876957i $$-0.659570\pi$$
−0.480569 + 0.876957i $$0.659570\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ −2.00000 −0.0958927
$$436$$ 8.00000 0.383131
$$437$$ −18.0000 −0.861057
$$438$$ −12.0000 −0.573382
$$439$$ 15.0000 0.715911 0.357955 0.933739i $$-0.383474\pi$$
0.357955 + 0.933739i $$0.383474\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 14.0000 0.665912
$$443$$ −1.00000 −0.0475114 −0.0237557 0.999718i $$-0.507562\pi$$
−0.0237557 + 0.999718i $$0.507562\pi$$
$$444$$ 14.0000 0.664411
$$445$$ −11.0000 −0.521450
$$446$$ −16.0000 −0.757622
$$447$$ 15.0000 0.709476
$$448$$ 8.00000 0.377964
$$449$$ 9.00000 0.424736 0.212368 0.977190i $$-0.431882\pi$$
0.212368 + 0.977190i $$0.431882\pi$$
$$450$$ 2.00000 0.0942809
$$451$$ 45.0000 2.11897
$$452$$ 20.0000 0.940721
$$453$$ −8.00000 −0.375873
$$454$$ −4.00000 −0.187729
$$455$$ −1.00000 −0.0468807
$$456$$ 0 0
$$457$$ −7.00000 −0.327446 −0.163723 0.986506i $$-0.552350\pi$$
−0.163723 + 0.986506i $$0.552350\pi$$
$$458$$ 28.0000 1.30835
$$459$$ −7.00000 −0.326732
$$460$$ −6.00000 −0.279751
$$461$$ 37.0000 1.72326 0.861631 0.507535i $$-0.169443\pi$$
0.861631 + 0.507535i $$0.169443\pi$$
$$462$$ −10.0000 −0.465242
$$463$$ 15.0000 0.697109 0.348555 0.937288i $$-0.386673\pi$$
0.348555 + 0.937288i $$0.386673\pi$$
$$464$$ −8.00000 −0.371391
$$465$$ −2.00000 −0.0927478
$$466$$ 38.0000 1.76032
$$467$$ −1.00000 −0.0462745 −0.0231372 0.999732i $$-0.507365\pi$$
−0.0231372 + 0.999732i $$0.507365\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 4.00000 0.184703
$$470$$ −20.0000 −0.922531
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ −40.0000 −1.83920
$$474$$ −6.00000 −0.275589
$$475$$ −6.00000 −0.275299
$$476$$ 14.0000 0.641689
$$477$$ 5.00000 0.228934
$$478$$ 18.0000 0.823301
$$479$$ 3.00000 0.137073 0.0685367 0.997649i $$-0.478167\pi$$
0.0685367 + 0.997649i $$0.478167\pi$$
$$480$$ 8.00000 0.365148
$$481$$ −7.00000 −0.319173
$$482$$ 44.0000 2.00415
$$483$$ −3.00000 −0.136505
$$484$$ 28.0000 1.27273
$$485$$ 11.0000 0.499484
$$486$$ 2.00000 0.0907218
$$487$$ 5.00000 0.226572 0.113286 0.993562i $$-0.463862\pi$$
0.113286 + 0.993562i $$0.463862\pi$$
$$488$$ 0 0
$$489$$ 15.0000 0.678323
$$490$$ 12.0000 0.542105
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 18.0000 0.811503
$$493$$ −14.0000 −0.630528
$$494$$ 12.0000 0.539906
$$495$$ −5.00000 −0.224733
$$496$$ −8.00000 −0.359211
$$497$$ −9.00000 −0.403705
$$498$$ −8.00000 −0.358489
$$499$$ −14.0000 −0.626726 −0.313363 0.949633i $$-0.601456\pi$$
−0.313363 + 0.949633i $$0.601456\pi$$
$$500$$ −2.00000 −0.0894427
$$501$$ −24.0000 −1.07224
$$502$$ 0 0
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 30.0000 1.33366
$$507$$ 1.00000 0.0444116
$$508$$ −4.00000 −0.177471
$$509$$ 27.0000 1.19675 0.598377 0.801215i $$-0.295813\pi$$
0.598377 + 0.801215i $$0.295813\pi$$
$$510$$ 14.0000 0.619930
$$511$$ 6.00000 0.265424
$$512$$ 32.0000 1.41421
$$513$$ −6.00000 −0.264906
$$514$$ 4.00000 0.176432
$$515$$ 4.00000 0.176261
$$516$$ −16.0000 −0.704361
$$517$$ 50.0000 2.19900
$$518$$ −14.0000 −0.615125
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 4.00000 0.175075
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ −44.0000 −1.92215
$$525$$ −1.00000 −0.0436436
$$526$$ −32.0000 −1.39527
$$527$$ −14.0000 −0.609850
$$528$$ −20.0000 −0.870388
$$529$$ −14.0000 −0.608696
$$530$$ −10.0000 −0.434372
$$531$$ 0 0
$$532$$ 12.0000 0.520266
$$533$$ −9.00000 −0.389833
$$534$$ 22.0000 0.952033
$$535$$ 17.0000 0.734974
$$536$$ 0 0
$$537$$ 6.00000 0.258919
$$538$$ −48.0000 −2.06943
$$539$$ −30.0000 −1.29219
$$540$$ −2.00000 −0.0860663
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ −44.0000 −1.88996
$$543$$ −7.00000 −0.300399
$$544$$ 56.0000 2.40098
$$545$$ −4.00000 −0.171341
$$546$$ 2.00000 0.0855921
$$547$$ 36.0000 1.53925 0.769624 0.638497i $$-0.220443\pi$$
0.769624 + 0.638497i $$0.220443\pi$$
$$548$$ −28.0000 −1.19610
$$549$$ 5.00000 0.213395
$$550$$ 10.0000 0.426401
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ 3.00000 0.127573
$$554$$ −4.00000 −0.169944
$$555$$ −7.00000 −0.297133
$$556$$ 30.0000 1.27228
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 8.00000 0.338364
$$560$$ −4.00000 −0.169031
$$561$$ −35.0000 −1.47770
$$562$$ 36.0000 1.51857
$$563$$ 11.0000 0.463595 0.231797 0.972764i $$-0.425539\pi$$
0.231797 + 0.972764i $$0.425539\pi$$
$$564$$ 20.0000 0.842152
$$565$$ −10.0000 −0.420703
$$566$$ 40.0000 1.68133
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −20.0000 −0.838444 −0.419222 0.907884i $$-0.637697\pi$$
−0.419222 + 0.907884i $$0.637697\pi$$
$$570$$ 12.0000 0.502625
$$571$$ −39.0000 −1.63210 −0.816050 0.577982i $$-0.803840\pi$$
−0.816050 + 0.577982i $$0.803840\pi$$
$$572$$ −10.0000 −0.418121
$$573$$ −12.0000 −0.501307
$$574$$ −18.0000 −0.751305
$$575$$ 3.00000 0.125109
$$576$$ −8.00000 −0.333333
$$577$$ −7.00000 −0.291414 −0.145707 0.989328i $$-0.546546\pi$$
−0.145707 + 0.989328i $$0.546546\pi$$
$$578$$ 64.0000 2.66205
$$579$$ −17.0000 −0.706496
$$580$$ −4.00000 −0.166091
$$581$$ 4.00000 0.165948
$$582$$ −22.0000 −0.911929
$$583$$ 25.0000 1.03539
$$584$$ 0 0
$$585$$ 1.00000 0.0413449
$$586$$ 8.00000 0.330477
$$587$$ −2.00000 −0.0825488 −0.0412744 0.999148i $$-0.513142\pi$$
−0.0412744 + 0.999148i $$0.513142\pi$$
$$588$$ −12.0000 −0.494872
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ 24.0000 0.987228
$$592$$ −28.0000 −1.15079
$$593$$ −16.0000 −0.657041 −0.328521 0.944497i $$-0.606550\pi$$
−0.328521 + 0.944497i $$0.606550\pi$$
$$594$$ 10.0000 0.410305
$$595$$ −7.00000 −0.286972
$$596$$ 30.0000 1.22885
$$597$$ −28.0000 −1.14596
$$598$$ −6.00000 −0.245358
$$599$$ 20.0000 0.817178 0.408589 0.912719i $$-0.366021\pi$$
0.408589 + 0.912719i $$0.366021\pi$$
$$600$$ 0 0
$$601$$ −5.00000 −0.203954 −0.101977 0.994787i $$-0.532517\pi$$
−0.101977 + 0.994787i $$0.532517\pi$$
$$602$$ 16.0000 0.652111
$$603$$ −4.00000 −0.162893
$$604$$ −16.0000 −0.651031
$$605$$ −14.0000 −0.569181
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 48.0000 1.94666
$$609$$ −2.00000 −0.0810441
$$610$$ −10.0000 −0.404888
$$611$$ −10.0000 −0.404557
$$612$$ −14.0000 −0.565916
$$613$$ 9.00000 0.363507 0.181753 0.983344i $$-0.441823\pi$$
0.181753 + 0.983344i $$0.441823\pi$$
$$614$$ 46.0000 1.85641
$$615$$ −9.00000 −0.362915
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ −38.0000 −1.52735 −0.763674 0.645601i $$-0.776607\pi$$
−0.763674 + 0.645601i $$0.776607\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 3.00000 0.120386
$$622$$ 40.0000 1.60385
$$623$$ −11.0000 −0.440706
$$624$$ 4.00000 0.160128
$$625$$ 1.00000 0.0400000
$$626$$ −44.0000 −1.75859
$$627$$ −30.0000 −1.19808
$$628$$ 36.0000 1.43656
$$629$$ −49.0000 −1.95376
$$630$$ 2.00000 0.0796819
$$631$$ −20.0000 −0.796187 −0.398094 0.917345i $$-0.630328\pi$$
−0.398094 + 0.917345i $$0.630328\pi$$
$$632$$ 0 0
$$633$$ −12.0000 −0.476957
$$634$$ −48.0000 −1.90632
$$635$$ 2.00000 0.0793676
$$636$$ 10.0000 0.396526
$$637$$ 6.00000 0.237729
$$638$$ 20.0000 0.791808
$$639$$ 9.00000 0.356034
$$640$$ 0 0
$$641$$ 20.0000 0.789953 0.394976 0.918691i $$-0.370753\pi$$
0.394976 + 0.918691i $$0.370753\pi$$
$$642$$ −34.0000 −1.34187
$$643$$ −37.0000 −1.45914 −0.729569 0.683907i $$-0.760279\pi$$
−0.729569 + 0.683907i $$0.760279\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ 8.00000 0.315000
$$646$$ 84.0000 3.30494
$$647$$ 17.0000 0.668339 0.334169 0.942513i $$-0.391544\pi$$
0.334169 + 0.942513i $$0.391544\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −2.00000 −0.0784465
$$651$$ −2.00000 −0.0783862
$$652$$ 30.0000 1.17489
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 8.00000 0.312825
$$655$$ 22.0000 0.859611
$$656$$ −36.0000 −1.40556
$$657$$ −6.00000 −0.234082
$$658$$ −20.0000 −0.779681
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ −10.0000 −0.389249
$$661$$ −40.0000 −1.55582 −0.777910 0.628376i $$-0.783720\pi$$
−0.777910 + 0.628376i $$0.783720\pi$$
$$662$$ 0 0
$$663$$ 7.00000 0.271857
$$664$$ 0 0
$$665$$ −6.00000 −0.232670
$$666$$ 14.0000 0.542489
$$667$$ 6.00000 0.232321
$$668$$ −48.0000 −1.85718
$$669$$ −8.00000 −0.309298
$$670$$ 8.00000 0.309067
$$671$$ 25.0000 0.965114
$$672$$ 8.00000 0.308607
$$673$$ 42.0000 1.61898 0.809491 0.587133i $$-0.199743\pi$$
0.809491 + 0.587133i $$0.199743\pi$$
$$674$$ 16.0000 0.616297
$$675$$ 1.00000 0.0384900
$$676$$ 2.00000 0.0769231
$$677$$ 21.0000 0.807096 0.403548 0.914959i $$-0.367777\pi$$
0.403548 + 0.914959i $$0.367777\pi$$
$$678$$ 20.0000 0.768095
$$679$$ 11.0000 0.422141
$$680$$ 0 0
$$681$$ −2.00000 −0.0766402
$$682$$ 20.0000 0.765840
$$683$$ 16.0000 0.612223 0.306111 0.951996i $$-0.400972\pi$$
0.306111 + 0.951996i $$0.400972\pi$$
$$684$$ −12.0000 −0.458831
$$685$$ 14.0000 0.534913
$$686$$ 26.0000 0.992685
$$687$$ 14.0000 0.534133
$$688$$ 32.0000 1.21999
$$689$$ −5.00000 −0.190485
$$690$$ −6.00000 −0.228416
$$691$$ 10.0000 0.380418 0.190209 0.981744i $$-0.439083\pi$$
0.190209 + 0.981744i $$0.439083\pi$$
$$692$$ −36.0000 −1.36851
$$693$$ −5.00000 −0.189934
$$694$$ 22.0000 0.835109
$$695$$ −15.0000 −0.568982
$$696$$ 0 0
$$697$$ −63.0000 −2.38630
$$698$$ 48.0000 1.81683
$$699$$ 19.0000 0.718646
$$700$$ −2.00000 −0.0755929
$$701$$ 28.0000 1.05755 0.528773 0.848763i $$-0.322652\pi$$
0.528773 + 0.848763i $$0.322652\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ −42.0000 −1.58406
$$704$$ −40.0000 −1.50756
$$705$$ −10.0000 −0.376622
$$706$$ −12.0000 −0.451626
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −44.0000 −1.65245 −0.826227 0.563337i $$-0.809517\pi$$
−0.826227 + 0.563337i $$0.809517\pi$$
$$710$$ −18.0000 −0.675528
$$711$$ −3.00000 −0.112509
$$712$$ 0 0
$$713$$ 6.00000 0.224702
$$714$$ 14.0000 0.523937
$$715$$ 5.00000 0.186989
$$716$$ 12.0000 0.448461
$$717$$ 9.00000 0.336111
$$718$$ −32.0000 −1.19423
$$719$$ 16.0000 0.596699 0.298350 0.954457i $$-0.403564\pi$$
0.298350 + 0.954457i $$0.403564\pi$$
$$720$$ 4.00000 0.149071
$$721$$ 4.00000 0.148968
$$722$$ 34.0000 1.26535
$$723$$ 22.0000 0.818189
$$724$$ −14.0000 −0.520306
$$725$$ 2.00000 0.0742781
$$726$$ 28.0000 1.03918
$$727$$ −18.0000 −0.667583 −0.333792 0.942647i $$-0.608328\pi$$
−0.333792 + 0.942647i $$0.608328\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 12.0000 0.444140
$$731$$ 56.0000 2.07123
$$732$$ 10.0000 0.369611
$$733$$ 43.0000 1.58824 0.794121 0.607760i $$-0.207932\pi$$
0.794121 + 0.607760i $$0.207932\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 6.00000 0.221313
$$736$$ −24.0000 −0.884652
$$737$$ −20.0000 −0.736709
$$738$$ 18.0000 0.662589
$$739$$ −6.00000 −0.220714 −0.110357 0.993892i $$-0.535199\pi$$
−0.110357 + 0.993892i $$0.535199\pi$$
$$740$$ −14.0000 −0.514650
$$741$$ 6.00000 0.220416
$$742$$ −10.0000 −0.367112
$$743$$ 30.0000 1.10059 0.550297 0.834969i $$-0.314515\pi$$
0.550297 + 0.834969i $$0.314515\pi$$
$$744$$ 0 0
$$745$$ −15.0000 −0.549557
$$746$$ −64.0000 −2.34321
$$747$$ −4.00000 −0.146352
$$748$$ −70.0000 −2.55945
$$749$$ 17.0000 0.621166
$$750$$ −2.00000 −0.0730297
$$751$$ −13.0000 −0.474377 −0.237188 0.971464i $$-0.576226\pi$$
−0.237188 + 0.971464i $$0.576226\pi$$
$$752$$ −40.0000 −1.45865
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ 8.00000 0.291150
$$756$$ −2.00000 −0.0727393
$$757$$ 20.0000 0.726912 0.363456 0.931611i $$-0.381597\pi$$
0.363456 + 0.931611i $$0.381597\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 15.0000 0.544466
$$760$$ 0 0
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ −4.00000 −0.144905
$$763$$ −4.00000 −0.144810
$$764$$ −24.0000 −0.868290
$$765$$ 7.00000 0.253086
$$766$$ −12.0000 −0.433578
$$767$$ 0 0
$$768$$ 16.0000 0.577350
$$769$$ −28.0000 −1.00971 −0.504853 0.863205i $$-0.668453\pi$$
−0.504853 + 0.863205i $$0.668453\pi$$
$$770$$ 10.0000 0.360375
$$771$$ 2.00000 0.0720282
$$772$$ −34.0000 −1.22369
$$773$$ −20.0000 −0.719350 −0.359675 0.933078i $$-0.617112\pi$$
−0.359675 + 0.933078i $$0.617112\pi$$
$$774$$ −16.0000 −0.575108
$$775$$ 2.00000 0.0718421
$$776$$ 0 0
$$777$$ −7.00000 −0.251124
$$778$$ −8.00000 −0.286814
$$779$$ −54.0000 −1.93475
$$780$$ 2.00000 0.0716115
$$781$$ 45.0000 1.61023
$$782$$ −42.0000 −1.50192
$$783$$ 2.00000 0.0714742
$$784$$ 24.0000 0.857143
$$785$$ −18.0000 −0.642448
$$786$$ −44.0000 −1.56943
$$787$$ 44.0000 1.56843 0.784215 0.620489i $$-0.213066\pi$$
0.784215 + 0.620489i $$0.213066\pi$$
$$788$$ 48.0000 1.70993
$$789$$ −16.0000 −0.569615
$$790$$ 6.00000 0.213470
$$791$$ −10.0000 −0.355559
$$792$$ 0 0
$$793$$ −5.00000 −0.177555
$$794$$ 30.0000 1.06466
$$795$$ −5.00000 −0.177332
$$796$$ −56.0000 −1.98487
$$797$$ 39.0000 1.38145 0.690725 0.723117i $$-0.257291\pi$$
0.690725 + 0.723117i $$0.257291\pi$$
$$798$$ 12.0000 0.424795
$$799$$ −70.0000 −2.47642
$$800$$ −8.00000 −0.282843
$$801$$ 11.0000 0.388666
$$802$$ −28.0000 −0.988714
$$803$$ −30.0000 −1.05868
$$804$$ −8.00000 −0.282138
$$805$$ 3.00000 0.105736
$$806$$ −4.00000 −0.140894
$$807$$ −24.0000 −0.844840
$$808$$ 0 0
$$809$$ −50.0000 −1.75791 −0.878953 0.476908i $$-0.841757\pi$$
−0.878953 + 0.476908i $$0.841757\pi$$
$$810$$ −2.00000 −0.0702728
$$811$$ 24.0000 0.842754 0.421377 0.906886i $$-0.361547\pi$$
0.421377 + 0.906886i $$0.361547\pi$$
$$812$$ −4.00000 −0.140372
$$813$$ −22.0000 −0.771574
$$814$$ 70.0000 2.45350
$$815$$ −15.0000 −0.525427
$$816$$ 28.0000 0.980196
$$817$$ 48.0000 1.67931
$$818$$ 60.0000 2.09785
$$819$$ 1.00000 0.0349428
$$820$$ −18.0000 −0.628587
$$821$$ 17.0000 0.593304 0.296652 0.954986i $$-0.404130\pi$$
0.296652 + 0.954986i $$0.404130\pi$$
$$822$$ −28.0000 −0.976612
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 0 0
$$825$$ 5.00000 0.174078
$$826$$ 0 0
$$827$$ −26.0000 −0.904109 −0.452054 0.891990i $$-0.649309\pi$$
−0.452054 + 0.891990i $$0.649309\pi$$
$$828$$ 6.00000 0.208514
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 8.00000 0.277684
$$831$$ −2.00000 −0.0693792
$$832$$ 8.00000 0.277350
$$833$$ 42.0000 1.45521
$$834$$ 30.0000 1.03882
$$835$$ 24.0000 0.830554
$$836$$ −60.0000 −2.07514
$$837$$ 2.00000 0.0691301
$$838$$ 52.0000 1.79631
$$839$$ 17.0000 0.586905 0.293453 0.955974i $$-0.405196\pi$$
0.293453 + 0.955974i $$0.405196\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −40.0000 −1.37849
$$843$$ 18.0000 0.619953
$$844$$ −24.0000 −0.826114
$$845$$ −1.00000 −0.0344010
$$846$$ 20.0000 0.687614
$$847$$ −14.0000 −0.481046
$$848$$ −20.0000 −0.686803
$$849$$ 20.0000 0.686398
$$850$$ −14.0000 −0.480196
$$851$$ 21.0000 0.719871
$$852$$ 18.0000 0.616670
$$853$$ −39.0000 −1.33533 −0.667667 0.744460i $$-0.732707\pi$$
−0.667667 + 0.744460i $$0.732707\pi$$
$$854$$ −10.0000 −0.342193
$$855$$ 6.00000 0.205196
$$856$$ 0 0
$$857$$ −45.0000 −1.53717 −0.768585 0.639747i $$-0.779039\pi$$
−0.768585 + 0.639747i $$0.779039\pi$$
$$858$$ −10.0000 −0.341394
$$859$$ 19.0000 0.648272 0.324136 0.946011i $$-0.394927\pi$$
0.324136 + 0.946011i $$0.394927\pi$$
$$860$$ 16.0000 0.545595
$$861$$ −9.00000 −0.306719
$$862$$ −80.0000 −2.72481
$$863$$ 6.00000 0.204242 0.102121 0.994772i $$-0.467437\pi$$
0.102121 + 0.994772i $$0.467437\pi$$
$$864$$ −8.00000 −0.272166
$$865$$ 18.0000 0.612018
$$866$$ −40.0000 −1.35926
$$867$$ 32.0000 1.08678
$$868$$ −4.00000 −0.135769
$$869$$ −15.0000 −0.508840
$$870$$ −4.00000 −0.135613
$$871$$ 4.00000 0.135535
$$872$$ 0 0
$$873$$ −11.0000 −0.372294
$$874$$ −36.0000 −1.21772
$$875$$ 1.00000 0.0338062
$$876$$ −12.0000 −0.405442
$$877$$ 18.0000 0.607817 0.303908 0.952701i $$-0.401708\pi$$
0.303908 + 0.952701i $$0.401708\pi$$
$$878$$ 30.0000 1.01245
$$879$$ 4.00000 0.134917
$$880$$ 20.0000 0.674200
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ −12.0000 −0.404061
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ 14.0000 0.470871
$$885$$ 0 0
$$886$$ −2.00000 −0.0671913
$$887$$ 13.0000 0.436497 0.218249 0.975893i $$-0.429966\pi$$
0.218249 + 0.975893i $$0.429966\pi$$
$$888$$ 0 0
$$889$$ 2.00000 0.0670778
$$890$$ −22.0000 −0.737442
$$891$$ 5.00000 0.167506
$$892$$ −16.0000 −0.535720
$$893$$ −60.0000 −2.00782
$$894$$ 30.0000 1.00335
$$895$$ −6.00000 −0.200558
$$896$$ 0 0
$$897$$ −3.00000 −0.100167
$$898$$ 18.0000 0.600668
$$899$$ 4.00000 0.133407
$$900$$ 2.00000 0.0666667
$$901$$ −35.0000 −1.16602
$$902$$ 90.0000 2.99667
$$903$$ 8.00000 0.266223
$$904$$ 0 0
$$905$$ 7.00000 0.232688
$$906$$ −16.0000 −0.531564
$$907$$ 38.0000 1.26177 0.630885 0.775877i $$-0.282692\pi$$
0.630885 + 0.775877i $$0.282692\pi$$
$$908$$ −4.00000 −0.132745
$$909$$ 0 0
$$910$$ −2.00000 −0.0662994
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 24.0000 0.794719
$$913$$ −20.0000 −0.661903
$$914$$ −14.0000 −0.463079
$$915$$ −5.00000 −0.165295
$$916$$ 28.0000 0.925146
$$917$$ 22.0000 0.726504
$$918$$ −14.0000 −0.462069
$$919$$ −19.0000 −0.626752 −0.313376 0.949629i $$-0.601460\pi$$
−0.313376 + 0.949629i $$0.601460\pi$$
$$920$$ 0 0
$$921$$ 23.0000 0.757876
$$922$$ 74.0000 2.43706
$$923$$ −9.00000 −0.296239
$$924$$ −10.0000 −0.328976
$$925$$ 7.00000 0.230159
$$926$$ 30.0000 0.985861
$$927$$ −4.00000 −0.131377
$$928$$ −16.0000 −0.525226
$$929$$ −29.0000 −0.951459 −0.475730 0.879592i $$-0.657816\pi$$
−0.475730 + 0.879592i $$0.657816\pi$$
$$930$$ −4.00000 −0.131165
$$931$$ 36.0000 1.17985
$$932$$ 38.0000 1.24473
$$933$$ 20.0000 0.654771
$$934$$ −2.00000 −0.0654420
$$935$$ 35.0000 1.14462
$$936$$ 0 0
$$937$$ −6.00000 −0.196011 −0.0980057 0.995186i $$-0.531246\pi$$
−0.0980057 + 0.995186i $$0.531246\pi$$
$$938$$ 8.00000 0.261209
$$939$$ −22.0000 −0.717943
$$940$$ −20.0000 −0.652328
$$941$$ 15.0000 0.488986 0.244493 0.969651i $$-0.421378\pi$$
0.244493 + 0.969651i $$0.421378\pi$$
$$942$$ 36.0000 1.17294
$$943$$ 27.0000 0.879241
$$944$$ 0 0
$$945$$ 1.00000 0.0325300
$$946$$ −80.0000 −2.60102
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ −6.00000 −0.194871
$$949$$ 6.00000 0.194768
$$950$$ −12.0000 −0.389331
$$951$$ −24.0000 −0.778253
$$952$$ 0 0
$$953$$ −15.0000 −0.485898 −0.242949 0.970039i $$-0.578115\pi$$
−0.242949 + 0.970039i $$0.578115\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 12.0000 0.388311
$$956$$ 18.0000 0.582162
$$957$$ 10.0000 0.323254
$$958$$ 6.00000 0.193851
$$959$$ 14.0000 0.452084
$$960$$ 8.00000 0.258199
$$961$$ −27.0000 −0.870968
$$962$$ −14.0000 −0.451378
$$963$$ −17.0000 −0.547817
$$964$$ 44.0000 1.41714
$$965$$ 17.0000 0.547249
$$966$$ −6.00000 −0.193047
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 0 0
$$969$$ 42.0000 1.34923
$$970$$ 22.0000 0.706377
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 2.00000 0.0641500
$$973$$ −15.0000 −0.480878
$$974$$ 10.0000 0.320421
$$975$$ −1.00000 −0.0320256
$$976$$ −20.0000 −0.640184
$$977$$ −12.0000 −0.383914 −0.191957 0.981403i $$-0.561483\pi$$
−0.191957 + 0.981403i $$0.561483\pi$$
$$978$$ 30.0000 0.959294
$$979$$ 55.0000 1.75781
$$980$$ 12.0000 0.383326
$$981$$ 4.00000 0.127710
$$982$$ 0 0
$$983$$ −4.00000 −0.127580 −0.0637901 0.997963i $$-0.520319\pi$$
−0.0637901 + 0.997963i $$0.520319\pi$$
$$984$$ 0 0
$$985$$ −24.0000 −0.764704
$$986$$ −28.0000 −0.891702
$$987$$ −10.0000 −0.318304
$$988$$ 12.0000 0.381771
$$989$$ −24.0000 −0.763156
$$990$$ −10.0000 −0.317821
$$991$$ −57.0000 −1.81066 −0.905332 0.424704i $$-0.860378\pi$$
−0.905332 + 0.424704i $$0.860378\pi$$
$$992$$ −16.0000 −0.508001
$$993$$ 0 0
$$994$$ −18.0000 −0.570925
$$995$$ 28.0000 0.887660
$$996$$ −8.00000 −0.253490
$$997$$ 8.00000 0.253363 0.126681 0.991943i $$-0.459567\pi$$
0.126681 + 0.991943i $$0.459567\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ 7.00000 0.221470
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 195.2.a.c.1.1 1
3.2 odd 2 585.2.a.c.1.1 1
4.3 odd 2 3120.2.a.d.1.1 1
5.2 odd 4 975.2.c.c.274.2 2
5.3 odd 4 975.2.c.c.274.1 2
5.4 even 2 975.2.a.a.1.1 1
7.6 odd 2 9555.2.a.u.1.1 1
12.11 even 2 9360.2.a.bv.1.1 1
13.12 even 2 2535.2.a.d.1.1 1
15.2 even 4 2925.2.c.a.2224.1 2
15.8 even 4 2925.2.c.a.2224.2 2
15.14 odd 2 2925.2.a.s.1.1 1
39.38 odd 2 7605.2.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.c.1.1 1 1.1 even 1 trivial
585.2.a.c.1.1 1 3.2 odd 2
975.2.a.a.1.1 1 5.4 even 2
975.2.c.c.274.1 2 5.3 odd 4
975.2.c.c.274.2 2 5.2 odd 4
2535.2.a.d.1.1 1 13.12 even 2
2925.2.a.s.1.1 1 15.14 odd 2
2925.2.c.a.2224.1 2 15.2 even 4
2925.2.c.a.2224.2 2 15.8 even 4
3120.2.a.d.1.1 1 4.3 odd 2
7605.2.a.t.1.1 1 39.38 odd 2
9360.2.a.bv.1.1 1 12.11 even 2
9555.2.a.u.1.1 1 7.6 odd 2