# Properties

 Label 690.2.a.g.1.1 Level $690$ Weight $2$ Character 690.1 Self dual yes Analytic conductor $5.510$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,2,Mod(1,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, 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("690.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.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: $$5.50967773947$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 690.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} +2.00000 q^{21} -2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} +2.00000 q^{33} -4.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -8.00000 q^{37} +6.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +2.00000 q^{42} -4.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} -6.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} -2.00000 q^{56} +2.00000 q^{58} +1.00000 q^{60} -8.00000 q^{61} -2.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} +2.00000 q^{66} -4.00000 q^{67} -4.00000 q^{68} -1.00000 q^{69} +2.00000 q^{70} +16.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -8.00000 q^{74} -1.00000 q^{75} +4.00000 q^{77} +6.00000 q^{78} +14.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +14.0000 q^{83} +2.00000 q^{84} +4.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} -2.00000 q^{88} -8.00000 q^{89} -1.00000 q^{90} +12.0000 q^{91} +1.00000 q^{92} -1.00000 q^{96} -6.00000 q^{97} -3.00000 q^{98} -2.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 2.00000 0.436436
$$22$$ −2.00000 −0.426401
$$23$$ 1.00000 0.208514
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ −6.00000 −1.17670
$$27$$ −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 $$0.440546\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 2.00000 0.348155
$$34$$ −4.00000 −0.685994
$$35$$ 2.00000 0.338062
$$36$$ 1.00000 0.166667
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 0 0
$$39$$ 6.00000 0.960769
$$40$$ −1.00000 −0.158114
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 2.00000 0.308607
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ −1.00000 −0.149071
$$46$$ 1.00000 0.147442
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 1.00000 0.141421
$$51$$ 4.00000 0.560112
$$52$$ −6.00000 −0.832050
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 2.00000 0.269680
$$56$$ −2.00000 −0.267261
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 1.00000 0.129099
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 6.00000 0.744208
$$66$$ 2.00000 0.246183
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ −1.00000 −0.120386
$$70$$ 2.00000 0.239046
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 4.00000 0.455842
$$78$$ 6.00000 0.679366
$$79$$ 14.0000 1.57512 0.787562 0.616236i $$-0.211343\pi$$
0.787562 + 0.616236i $$0.211343\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 14.0000 1.53670 0.768350 0.640030i $$-0.221078\pi$$
0.768350 + 0.640030i $$0.221078\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 4.00000 0.433861
$$86$$ −4.00000 −0.431331
$$87$$ −2.00000 −0.214423
$$88$$ −2.00000 −0.213201
$$89$$ −8.00000 −0.847998 −0.423999 0.905663i $$-0.639374\pi$$
−0.423999 + 0.905663i $$0.639374\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 12.0000 1.25794
$$92$$ 1.00000 0.104257
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ −2.00000 −0.201008
$$100$$ 1.00000 0.100000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 4.00000 0.396059
$$103$$ −2.00000 −0.197066 −0.0985329 0.995134i $$-0.531415\pi$$
−0.0985329 + 0.995134i $$0.531415\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ −2.00000 −0.195180
$$106$$ 6.00000 0.582772
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 2.00000 0.190693
$$111$$ 8.00000 0.759326
$$112$$ −2.00000 −0.188982
$$113$$ 12.0000 1.12887 0.564433 0.825479i $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 2.00000 0.185695
$$117$$ −6.00000 −0.554700
$$118$$ 0 0
$$119$$ 8.00000 0.733359
$$120$$ 1.00000 0.0912871
$$121$$ −7.00000 −0.636364
$$122$$ −8.00000 −0.724286
$$123$$ 6.00000 0.541002
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ −2.00000 −0.178174
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 6.00000 0.526235
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 2.00000 0.174078
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 1.00000 0.0860663
$$136$$ −4.00000 −0.342997
$$137$$ −8.00000 −0.683486 −0.341743 0.939793i $$-0.611017\pi$$
−0.341743 + 0.939793i $$0.611017\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 0 0
$$142$$ 16.0000 1.34269
$$143$$ 12.0000 1.00349
$$144$$ 1.00000 0.0833333
$$145$$ −2.00000 −0.166091
$$146$$ 6.00000 0.496564
$$147$$ 3.00000 0.247436
$$148$$ −8.00000 −0.657596
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ −4.00000 −0.323381
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ −16.0000 −1.27694 −0.638470 0.769647i $$-0.720432\pi$$
−0.638470 + 0.769647i $$0.720432\pi$$
$$158$$ 14.0000 1.11378
$$159$$ −6.00000 −0.475831
$$160$$ −1.00000 −0.0790569
$$161$$ −2.00000 −0.157622
$$162$$ 1.00000 0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ −2.00000 −0.155700
$$166$$ 14.0000 1.08661
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 23.0000 1.76923
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ −2.00000 −0.151186
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ −8.00000 −0.599625
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −12.0000 −0.891953 −0.445976 0.895045i $$-0.647144\pi$$
−0.445976 + 0.895045i $$0.647144\pi$$
$$182$$ 12.0000 0.889499
$$183$$ 8.00000 0.591377
$$184$$ 1.00000 0.0737210
$$185$$ 8.00000 0.588172
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ −6.00000 −0.429669
$$196$$ −3.00000 −0.214286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 4.00000 0.282138
$$202$$ 6.00000 0.422159
$$203$$ −4.00000 −0.280745
$$204$$ 4.00000 0.280056
$$205$$ 6.00000 0.419058
$$206$$ −2.00000 −0.139347
$$207$$ 1.00000 0.0695048
$$208$$ −6.00000 −0.416025
$$209$$ 0 0
$$210$$ −2.00000 −0.138013
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 6.00000 0.412082
$$213$$ −16.0000 −1.09630
$$214$$ −18.0000 −1.23045
$$215$$ 4.00000 0.272798
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ −4.00000 −0.270914
$$219$$ −6.00000 −0.405442
$$220$$ 2.00000 0.134840
$$221$$ 24.0000 1.61441
$$222$$ 8.00000 0.536925
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 1.00000 0.0666667
$$226$$ 12.0000 0.798228
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 0 0
$$229$$ −20.0000 −1.32164 −0.660819 0.750546i $$-0.729791\pi$$
−0.660819 + 0.750546i $$0.729791\pi$$
$$230$$ −1.00000 −0.0659380
$$231$$ −4.00000 −0.263181
$$232$$ 2.00000 0.131306
$$233$$ −26.0000 −1.70332 −0.851658 0.524097i $$-0.824403\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −14.0000 −0.909398
$$238$$ 8.00000 0.518563
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ −1.00000 −0.0641500
$$244$$ −8.00000 −0.512148
$$245$$ 3.00000 0.191663
$$246$$ 6.00000 0.382546
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −14.0000 −0.887214
$$250$$ −1.00000 −0.0632456
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ −2.00000 −0.125739
$$254$$ 0 0
$$255$$ −4.00000 −0.250490
$$256$$ 1.00000 0.0625000
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 16.0000 0.994192
$$260$$ 6.00000 0.372104
$$261$$ 2.00000 0.123797
$$262$$ 12.0000 0.741362
$$263$$ 4.00000 0.246651 0.123325 0.992366i $$-0.460644\pi$$
0.123325 + 0.992366i $$0.460644\pi$$
$$264$$ 2.00000 0.123091
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ 8.00000 0.489592
$$268$$ −4.00000 −0.244339
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ −12.0000 −0.726273
$$274$$ −8.00000 −0.483298
$$275$$ −2.00000 −0.120605
$$276$$ −1.00000 −0.0601929
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 0 0
$$280$$ 2.00000 0.119523
$$281$$ 8.00000 0.477240 0.238620 0.971113i $$-0.423305\pi$$
0.238620 + 0.971113i $$0.423305\pi$$
$$282$$ 0 0
$$283$$ −16.0000 −0.951101 −0.475551 0.879688i $$-0.657751\pi$$
−0.475551 + 0.879688i $$0.657751\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ 12.0000 0.708338
$$288$$ 1.00000 0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ −2.00000 −0.117444
$$291$$ 6.00000 0.351726
$$292$$ 6.00000 0.351123
$$293$$ −10.0000 −0.584206 −0.292103 0.956387i $$-0.594355\pi$$
−0.292103 + 0.956387i $$0.594355\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ −8.00000 −0.464991
$$297$$ 2.00000 0.116052
$$298$$ 10.0000 0.579284
$$299$$ −6.00000 −0.346989
$$300$$ −1.00000 −0.0577350
$$301$$ 8.00000 0.461112
$$302$$ 12.0000 0.690522
$$303$$ −6.00000 −0.344691
$$304$$ 0 0
$$305$$ 8.00000 0.458079
$$306$$ −4.00000 −0.228665
$$307$$ 28.0000 1.59804 0.799022 0.601302i $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ 4.00000 0.227921
$$309$$ 2.00000 0.113776
$$310$$ 0 0
$$311$$ −32.0000 −1.81455 −0.907277 0.420534i $$-0.861843\pi$$
−0.907277 + 0.420534i $$0.861843\pi$$
$$312$$ 6.00000 0.339683
$$313$$ −34.0000 −1.92179 −0.960897 0.276907i $$-0.910691\pi$$
−0.960897 + 0.276907i $$0.910691\pi$$
$$314$$ −16.0000 −0.902932
$$315$$ 2.00000 0.112687
$$316$$ 14.0000 0.787562
$$317$$ −22.0000 −1.23564 −0.617822 0.786318i $$-0.711985\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ −4.00000 −0.223957
$$320$$ −1.00000 −0.0559017
$$321$$ 18.0000 1.00466
$$322$$ −2.00000 −0.111456
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ −6.00000 −0.332820
$$326$$ −4.00000 −0.221540
$$327$$ 4.00000 0.221201
$$328$$ −6.00000 −0.331295
$$329$$ 0 0
$$330$$ −2.00000 −0.110096
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 14.0000 0.768350
$$333$$ −8.00000 −0.438397
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 2.00000 0.109109
$$337$$ −34.0000 −1.85210 −0.926049 0.377403i $$-0.876817\pi$$
−0.926049 + 0.377403i $$0.876817\pi$$
$$338$$ 23.0000 1.25104
$$339$$ −12.0000 −0.651751
$$340$$ 4.00000 0.216930
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ −4.00000 −0.215666
$$345$$ 1.00000 0.0538382
$$346$$ 18.0000 0.967686
$$347$$ −8.00000 −0.429463 −0.214731 0.976673i $$-0.568888\pi$$
−0.214731 + 0.976673i $$0.568888\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ −6.00000 −0.321173 −0.160586 0.987022i $$-0.551338\pi$$
−0.160586 + 0.987022i $$0.551338\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 6.00000 0.320256
$$352$$ −2.00000 −0.106600
$$353$$ 26.0000 1.38384 0.691920 0.721974i $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ 0 0
$$355$$ −16.0000 −0.849192
$$356$$ −8.00000 −0.423999
$$357$$ −8.00000 −0.423405
$$358$$ −20.0000 −1.05703
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −19.0000 −1.00000
$$362$$ −12.0000 −0.630706
$$363$$ 7.00000 0.367405
$$364$$ 12.0000 0.628971
$$365$$ −6.00000 −0.314054
$$366$$ 8.00000 0.418167
$$367$$ 10.0000 0.521996 0.260998 0.965339i $$-0.415948\pi$$
0.260998 + 0.965339i $$0.415948\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ −6.00000 −0.312348
$$370$$ 8.00000 0.415900
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ 32.0000 1.65690 0.828449 0.560065i $$-0.189224\pi$$
0.828449 + 0.560065i $$0.189224\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 2.00000 0.102869
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −4.00000 −0.203859
$$386$$ −2.00000 −0.101797
$$387$$ −4.00000 −0.203331
$$388$$ −6.00000 −0.304604
$$389$$ 10.0000 0.507020 0.253510 0.967333i $$-0.418415\pi$$
0.253510 + 0.967333i $$0.418415\pi$$
$$390$$ −6.00000 −0.303822
$$391$$ −4.00000 −0.202289
$$392$$ −3.00000 −0.151523
$$393$$ −12.0000 −0.605320
$$394$$ 6.00000 0.302276
$$395$$ −14.0000 −0.704416
$$396$$ −2.00000 −0.100504
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ −2.00000 −0.100251
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −24.0000 −1.19850 −0.599251 0.800561i $$-0.704535\pi$$
−0.599251 + 0.800561i $$0.704535\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ −1.00000 −0.0496904
$$406$$ −4.00000 −0.198517
$$407$$ 16.0000 0.793091
$$408$$ 4.00000 0.198030
$$409$$ 38.0000 1.87898 0.939490 0.342578i $$-0.111300\pi$$
0.939490 + 0.342578i $$0.111300\pi$$
$$410$$ 6.00000 0.296319
$$411$$ 8.00000 0.394611
$$412$$ −2.00000 −0.0985329
$$413$$ 0 0
$$414$$ 1.00000 0.0491473
$$415$$ −14.0000 −0.687233
$$416$$ −6.00000 −0.294174
$$417$$ 4.00000 0.195881
$$418$$ 0 0
$$419$$ −14.0000 −0.683945 −0.341972 0.939710i $$-0.611095\pi$$
−0.341972 + 0.939710i $$0.611095\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ −4.00000 −0.194029
$$426$$ −16.0000 −0.775203
$$427$$ 16.0000 0.774294
$$428$$ −18.0000 −0.870063
$$429$$ −12.0000 −0.579365
$$430$$ 4.00000 0.192897
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 30.0000 1.44171 0.720854 0.693087i $$-0.243750\pi$$
0.720854 + 0.693087i $$0.243750\pi$$
$$434$$ 0 0
$$435$$ 2.00000 0.0958927
$$436$$ −4.00000 −0.191565
$$437$$ 0 0
$$438$$ −6.00000 −0.286691
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ −3.00000 −0.142857
$$442$$ 24.0000 1.14156
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 8.00000 0.379236
$$446$$ −16.0000 −0.757622
$$447$$ −10.0000 −0.472984
$$448$$ −2.00000 −0.0944911
$$449$$ −22.0000 −1.03824 −0.519122 0.854700i $$-0.673741\pi$$
−0.519122 + 0.854700i $$0.673741\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 12.0000 0.565058
$$452$$ 12.0000 0.564433
$$453$$ −12.0000 −0.563809
$$454$$ −18.0000 −0.844782
$$455$$ −12.0000 −0.562569
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ −20.0000 −0.934539
$$459$$ 4.00000 0.186704
$$460$$ −1.00000 −0.0466252
$$461$$ −10.0000 −0.465746 −0.232873 0.972507i $$-0.574813\pi$$
−0.232873 + 0.972507i $$0.574813\pi$$
$$462$$ −4.00000 −0.186097
$$463$$ −36.0000 −1.67306 −0.836531 0.547920i $$-0.815420\pi$$
−0.836531 + 0.547920i $$0.815420\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −26.0000 −1.20443
$$467$$ −30.0000 −1.38823 −0.694117 0.719862i $$-0.744205\pi$$
−0.694117 + 0.719862i $$0.744205\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 16.0000 0.737241
$$472$$ 0 0
$$473$$ 8.00000 0.367840
$$474$$ −14.0000 −0.643041
$$475$$ 0 0
$$476$$ 8.00000 0.366679
$$477$$ 6.00000 0.274721
$$478$$ 16.0000 0.731823
$$479$$ 8.00000 0.365529 0.182765 0.983157i $$-0.441495\pi$$
0.182765 + 0.983157i $$0.441495\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 48.0000 2.18861
$$482$$ −14.0000 −0.637683
$$483$$ 2.00000 0.0910032
$$484$$ −7.00000 −0.318182
$$485$$ 6.00000 0.272446
$$486$$ −1.00000 −0.0453609
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ 4.00000 0.180886
$$490$$ 3.00000 0.135526
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 6.00000 0.270501
$$493$$ −8.00000 −0.360302
$$494$$ 0 0
$$495$$ 2.00000 0.0898933
$$496$$ 0 0
$$497$$ −32.0000 −1.43540
$$498$$ −14.0000 −0.627355
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 6.00000 0.267793
$$503$$ 4.00000 0.178351 0.0891756 0.996016i $$-0.471577\pi$$
0.0891756 + 0.996016i $$0.471577\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ −6.00000 −0.266996
$$506$$ −2.00000 −0.0889108
$$507$$ −23.0000 −1.02147
$$508$$ 0 0
$$509$$ −14.0000 −0.620539 −0.310270 0.950649i $$-0.600419\pi$$
−0.310270 + 0.950649i $$0.600419\pi$$
$$510$$ −4.00000 −0.177123
$$511$$ −12.0000 −0.530849
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 14.0000 0.617514
$$515$$ 2.00000 0.0881305
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 16.0000 0.703000
$$519$$ −18.0000 −0.790112
$$520$$ 6.00000 0.263117
$$521$$ −4.00000 −0.175243 −0.0876216 0.996154i $$-0.527927\pi$$
−0.0876216 + 0.996154i $$0.527927\pi$$
$$522$$ 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
$$525$$ 2.00000 0.0872872
$$526$$ 4.00000 0.174408
$$527$$ 0 0
$$528$$ 2.00000 0.0870388
$$529$$ 1.00000 0.0434783
$$530$$ −6.00000 −0.260623
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 36.0000 1.55933
$$534$$ 8.00000 0.346194
$$535$$ 18.0000 0.778208
$$536$$ −4.00000 −0.172774
$$537$$ 20.0000 0.863064
$$538$$ −18.0000 −0.776035
$$539$$ 6.00000 0.258438
$$540$$ 1.00000 0.0430331
$$541$$ 26.0000 1.11783 0.558914 0.829226i $$-0.311218\pi$$
0.558914 + 0.829226i $$0.311218\pi$$
$$542$$ 20.0000 0.859074
$$543$$ 12.0000 0.514969
$$544$$ −4.00000 −0.171499
$$545$$ 4.00000 0.171341
$$546$$ −12.0000 −0.513553
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ −8.00000 −0.341743
$$549$$ −8.00000 −0.341432
$$550$$ −2.00000 −0.0852803
$$551$$ 0 0
$$552$$ −1.00000 −0.0425628
$$553$$ −28.0000 −1.19068
$$554$$ −10.0000 −0.424859
$$555$$ −8.00000 −0.339581
$$556$$ −4.00000 −0.169638
$$557$$ −22.0000 −0.932170 −0.466085 0.884740i $$-0.654336\pi$$
−0.466085 + 0.884740i $$0.654336\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ 2.00000 0.0845154
$$561$$ −8.00000 −0.337760
$$562$$ 8.00000 0.337460
$$563$$ −10.0000 −0.421450 −0.210725 0.977545i $$-0.567582\pi$$
−0.210725 + 0.977545i $$0.567582\pi$$
$$564$$ 0 0
$$565$$ −12.0000 −0.504844
$$566$$ −16.0000 −0.672530
$$567$$ −2.00000 −0.0839921
$$568$$ 16.0000 0.671345
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 12.0000 0.501745
$$573$$ 0 0
$$574$$ 12.0000 0.500870
$$575$$ 1.00000 0.0417029
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 2.00000 0.0831172
$$580$$ −2.00000 −0.0830455
$$581$$ −28.0000 −1.16164
$$582$$ 6.00000 0.248708
$$583$$ −12.0000 −0.496989
$$584$$ 6.00000 0.248282
$$585$$ 6.00000 0.248069
$$586$$ −10.0000 −0.413096
$$587$$ 8.00000 0.330195 0.165098 0.986277i $$-0.447206\pi$$
0.165098 + 0.986277i $$0.447206\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ −8.00000 −0.328798
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ −8.00000 −0.327968
$$596$$ 10.0000 0.409616
$$597$$ 2.00000 0.0818546
$$598$$ −6.00000 −0.245358
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 8.00000 0.326056
$$603$$ −4.00000 −0.162893
$$604$$ 12.0000 0.488273
$$605$$ 7.00000 0.284590
$$606$$ −6.00000 −0.243733
$$607$$ 40.0000 1.62355 0.811775 0.583970i $$-0.198502\pi$$
0.811775 + 0.583970i $$0.198502\pi$$
$$608$$ 0 0
$$609$$ 4.00000 0.162088
$$610$$ 8.00000 0.323911
$$611$$ 0 0
$$612$$ −4.00000 −0.161690
$$613$$ −4.00000 −0.161558 −0.0807792 0.996732i $$-0.525741\pi$$
−0.0807792 + 0.996732i $$0.525741\pi$$
$$614$$ 28.0000 1.12999
$$615$$ −6.00000 −0.241943
$$616$$ 4.00000 0.161165
$$617$$ 28.0000 1.12724 0.563619 0.826035i $$-0.309409\pi$$
0.563619 + 0.826035i $$0.309409\pi$$
$$618$$ 2.00000 0.0804518
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ −32.0000 −1.28308
$$623$$ 16.0000 0.641026
$$624$$ 6.00000 0.240192
$$625$$ 1.00000 0.0400000
$$626$$ −34.0000 −1.35891
$$627$$ 0 0
$$628$$ −16.0000 −0.638470
$$629$$ 32.0000 1.27592
$$630$$ 2.00000 0.0796819
$$631$$ −22.0000 −0.875806 −0.437903 0.899022i $$-0.644279\pi$$
−0.437903 + 0.899022i $$0.644279\pi$$
$$632$$ 14.0000 0.556890
$$633$$ 20.0000 0.794929
$$634$$ −22.0000 −0.873732
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 18.0000 0.713186
$$638$$ −4.00000 −0.158362
$$639$$ 16.0000 0.632950
$$640$$ −1.00000 −0.0395285
$$641$$ 16.0000 0.631962 0.315981 0.948766i $$-0.397666\pi$$
0.315981 + 0.948766i $$0.397666\pi$$
$$642$$ 18.0000 0.710403
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ −2.00000 −0.0788110
$$645$$ −4.00000 −0.157500
$$646$$ 0 0
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ −6.00000 −0.235339
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 22.0000 0.860927 0.430463 0.902608i $$-0.358350\pi$$
0.430463 + 0.902608i $$0.358350\pi$$
$$654$$ 4.00000 0.156412
$$655$$ −12.0000 −0.468879
$$656$$ −6.00000 −0.234261
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ −2.00000 −0.0778499
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ 20.0000 0.777322
$$663$$ −24.0000 −0.932083
$$664$$ 14.0000 0.543305
$$665$$ 0 0
$$666$$ −8.00000 −0.309994
$$667$$ 2.00000 0.0774403
$$668$$ 0 0
$$669$$ 16.0000 0.618596
$$670$$ 4.00000 0.154533
$$671$$ 16.0000 0.617673
$$672$$ 2.00000 0.0771517
$$673$$ 42.0000 1.61898 0.809491 0.587133i $$-0.199743\pi$$
0.809491 + 0.587133i $$0.199743\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ −1.00000 −0.0384900
$$676$$ 23.0000 0.884615
$$677$$ 2.00000 0.0768662 0.0384331 0.999261i $$-0.487763\pi$$
0.0384331 + 0.999261i $$0.487763\pi$$
$$678$$ −12.0000 −0.460857
$$679$$ 12.0000 0.460518
$$680$$ 4.00000 0.153393
$$681$$ 18.0000 0.689761
$$682$$ 0 0
$$683$$ 48.0000 1.83667 0.918334 0.395805i $$-0.129534\pi$$
0.918334 + 0.395805i $$0.129534\pi$$
$$684$$ 0 0
$$685$$ 8.00000 0.305664
$$686$$ 20.0000 0.763604
$$687$$ 20.0000 0.763048
$$688$$ −4.00000 −0.152499
$$689$$ −36.0000 −1.37149
$$690$$ 1.00000 0.0380693
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 4.00000 0.151947
$$694$$ −8.00000 −0.303676
$$695$$ 4.00000 0.151729
$$696$$ −2.00000 −0.0758098
$$697$$ 24.0000 0.909065
$$698$$ −6.00000 −0.227103
$$699$$ 26.0000 0.983410
$$700$$ −2.00000 −0.0755929
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 6.00000 0.226455
$$703$$ 0 0
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ 26.0000 0.978523
$$707$$ −12.0000 −0.451306
$$708$$ 0 0
$$709$$ 16.0000 0.600893 0.300446 0.953799i $$-0.402864\pi$$
0.300446 + 0.953799i $$0.402864\pi$$
$$710$$ −16.0000 −0.600469
$$711$$ 14.0000 0.525041
$$712$$ −8.00000 −0.299813
$$713$$ 0 0
$$714$$ −8.00000 −0.299392
$$715$$ −12.0000 −0.448775
$$716$$ −20.0000 −0.747435
$$717$$ −16.0000 −0.597531
$$718$$ 24.0000 0.895672
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 4.00000 0.148968
$$722$$ −19.0000 −0.707107
$$723$$ 14.0000 0.520666
$$724$$ −12.0000 −0.445976
$$725$$ 2.00000 0.0742781
$$726$$ 7.00000 0.259794
$$727$$ 38.0000 1.40934 0.704671 0.709534i $$-0.251095\pi$$
0.704671 + 0.709534i $$0.251095\pi$$
$$728$$ 12.0000 0.444750
$$729$$ 1.00000 0.0370370
$$730$$ −6.00000 −0.222070
$$731$$ 16.0000 0.591781
$$732$$ 8.00000 0.295689
$$733$$ 36.0000 1.32969 0.664845 0.746981i $$-0.268498\pi$$
0.664845 + 0.746981i $$0.268498\pi$$
$$734$$ 10.0000 0.369107
$$735$$ −3.00000 −0.110657
$$736$$ 1.00000 0.0368605
$$737$$ 8.00000 0.294684
$$738$$ −6.00000 −0.220863
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 8.00000 0.294086
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ 4.00000 0.146746 0.0733729 0.997305i $$-0.476624\pi$$
0.0733729 + 0.997305i $$0.476624\pi$$
$$744$$ 0 0
$$745$$ −10.0000 −0.366372
$$746$$ 32.0000 1.17160
$$747$$ 14.0000 0.512233
$$748$$ 8.00000 0.292509
$$749$$ 36.0000 1.31541
$$750$$ 1.00000 0.0365148
$$751$$ −50.0000 −1.82453 −0.912263 0.409605i $$-0.865667\pi$$
−0.912263 + 0.409605i $$0.865667\pi$$
$$752$$ 0 0
$$753$$ −6.00000 −0.218652
$$754$$ −12.0000 −0.437014
$$755$$ −12.0000 −0.436725
$$756$$ 2.00000 0.0727393
$$757$$ 40.0000 1.45382 0.726912 0.686730i $$-0.240955\pi$$
0.726912 + 0.686730i $$0.240955\pi$$
$$758$$ 16.0000 0.581146
$$759$$ 2.00000 0.0725954
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ 8.00000 0.289619
$$764$$ 0 0
$$765$$ 4.00000 0.144620
$$766$$ −12.0000 −0.433578
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 6.00000 0.216366 0.108183 0.994131i $$-0.465497\pi$$
0.108183 + 0.994131i $$0.465497\pi$$
$$770$$ −4.00000 −0.144150
$$771$$ −14.0000 −0.504198
$$772$$ −2.00000 −0.0719816
$$773$$ −18.0000 −0.647415 −0.323708 0.946157i $$-0.604929\pi$$
−0.323708 + 0.946157i $$0.604929\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ −16.0000 −0.573997
$$778$$ 10.0000 0.358517
$$779$$ 0 0
$$780$$ −6.00000 −0.214834
$$781$$ −32.0000 −1.14505
$$782$$ −4.00000 −0.143040
$$783$$ −2.00000 −0.0714742
$$784$$ −3.00000 −0.107143
$$785$$ 16.0000 0.571064
$$786$$ −12.0000 −0.428026
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ 6.00000 0.213741
$$789$$ −4.00000 −0.142404
$$790$$ −14.0000 −0.498098
$$791$$ −24.0000 −0.853342
$$792$$ −2.00000 −0.0710669
$$793$$ 48.0000 1.70453
$$794$$ −22.0000 −0.780751
$$795$$ 6.00000 0.212798
$$796$$ −2.00000 −0.0708881
$$797$$ −46.0000 −1.62940 −0.814702 0.579880i $$-0.803099\pi$$
−0.814702 + 0.579880i $$0.803099\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 1.00000 0.0353553
$$801$$ −8.00000 −0.282666
$$802$$ −24.0000 −0.847469
$$803$$ −12.0000 −0.423471
$$804$$ 4.00000 0.141069
$$805$$ 2.00000 0.0704907
$$806$$ 0 0
$$807$$ 18.0000 0.633630
$$808$$ 6.00000 0.211079
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ −4.00000 −0.140372
$$813$$ −20.0000 −0.701431
$$814$$ 16.0000 0.560800
$$815$$ 4.00000 0.140114
$$816$$ 4.00000 0.140028
$$817$$ 0 0
$$818$$ 38.0000 1.32864
$$819$$ 12.0000 0.419314
$$820$$ 6.00000 0.209529
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ 8.00000 0.279032
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ −2.00000 −0.0696733
$$825$$ 2.00000 0.0696311
$$826$$ 0 0
$$827$$ −22.0000 −0.765015 −0.382507 0.923952i $$-0.624939\pi$$
−0.382507 + 0.923952i $$0.624939\pi$$
$$828$$ 1.00000 0.0347524
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ −14.0000 −0.485947
$$831$$ 10.0000 0.346896
$$832$$ −6.00000 −0.208013
$$833$$ 12.0000 0.415775
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −14.0000 −0.483622
$$839$$ −20.0000 −0.690477 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$840$$ −2.00000 −0.0690066
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −8.00000 −0.275535
$$844$$ −20.0000 −0.688428
$$845$$ −23.0000 −0.791224
$$846$$ 0 0
$$847$$ 14.0000 0.481046
$$848$$ 6.00000 0.206041
$$849$$ 16.0000 0.549119
$$850$$ −4.00000 −0.137199
$$851$$ −8.00000 −0.274236
$$852$$ −16.0000 −0.548151
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ 16.0000 0.547509
$$855$$ 0 0
$$856$$ −18.0000 −0.615227
$$857$$ 30.0000 1.02478 0.512390 0.858753i $$-0.328760\pi$$
0.512390 + 0.858753i $$0.328760\pi$$
$$858$$ −12.0000 −0.409673
$$859$$ −12.0000 −0.409435 −0.204717 0.978821i $$-0.565628\pi$$
−0.204717 + 0.978821i $$0.565628\pi$$
$$860$$ 4.00000 0.136399
$$861$$ −12.0000 −0.408959
$$862$$ −36.0000 −1.22616
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −18.0000 −0.612018
$$866$$ 30.0000 1.01944
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ −28.0000 −0.949835
$$870$$ 2.00000 0.0678064
$$871$$ 24.0000 0.813209
$$872$$ −4.00000 −0.135457
$$873$$ −6.00000 −0.203069
$$874$$ 0 0
$$875$$ 2.00000 0.0676123
$$876$$ −6.00000 −0.202721
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ 32.0000 1.07995
$$879$$ 10.0000 0.337292
$$880$$ 2.00000 0.0674200
$$881$$ −56.0000 −1.88669 −0.943344 0.331816i $$-0.892339\pi$$
−0.943344 + 0.331816i $$0.892339\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ −36.0000 −1.21150 −0.605748 0.795656i $$-0.707126\pi$$
−0.605748 + 0.795656i $$0.707126\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ 48.0000 1.61168 0.805841 0.592132i $$-0.201714\pi$$
0.805841 + 0.592132i $$0.201714\pi$$
$$888$$ 8.00000 0.268462
$$889$$ 0 0
$$890$$ 8.00000 0.268161
$$891$$ −2.00000 −0.0670025
$$892$$ −16.0000 −0.535720
$$893$$ 0 0
$$894$$ −10.0000 −0.334450
$$895$$ 20.0000 0.668526
$$896$$ −2.00000 −0.0668153
$$897$$ 6.00000 0.200334
$$898$$ −22.0000 −0.734150
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ −24.0000 −0.799556
$$902$$ 12.0000 0.399556
$$903$$ −8.00000 −0.266223
$$904$$ 12.0000 0.399114
$$905$$ 12.0000 0.398893
$$906$$ −12.0000 −0.398673
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ −18.0000 −0.597351
$$909$$ 6.00000 0.199007
$$910$$ −12.0000 −0.397796
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ 0 0
$$913$$ −28.0000 −0.926665
$$914$$ 18.0000 0.595387
$$915$$ −8.00000 −0.264472
$$916$$ −20.0000 −0.660819
$$917$$ −24.0000 −0.792550
$$918$$ 4.00000 0.132020
$$919$$ −2.00000 −0.0659739 −0.0329870 0.999456i $$-0.510502\pi$$
−0.0329870 + 0.999456i $$0.510502\pi$$
$$920$$ −1.00000 −0.0329690
$$921$$ −28.0000 −0.922631
$$922$$ −10.0000 −0.329332
$$923$$ −96.0000 −3.15988
$$924$$ −4.00000 −0.131590
$$925$$ −8.00000 −0.263038
$$926$$ −36.0000 −1.18303
$$927$$ −2.00000 −0.0656886
$$928$$ 2.00000 0.0656532
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −26.0000 −0.851658
$$933$$ 32.0000 1.04763
$$934$$ −30.0000 −0.981630
$$935$$ −8.00000 −0.261628
$$936$$ −6.00000 −0.196116
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ 8.00000 0.261209
$$939$$ 34.0000 1.10955
$$940$$ 0 0
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ 16.0000 0.521308
$$943$$ −6.00000 −0.195387
$$944$$ 0 0
$$945$$ −2.00000 −0.0650600
$$946$$ 8.00000 0.260102
$$947$$ 8.00000 0.259965 0.129983 0.991516i $$-0.458508\pi$$
0.129983 + 0.991516i $$0.458508\pi$$
$$948$$ −14.0000 −0.454699
$$949$$ −36.0000 −1.16861
$$950$$ 0 0
$$951$$ 22.0000 0.713399
$$952$$ 8.00000 0.259281
$$953$$ 44.0000 1.42530 0.712650 0.701520i $$-0.247495\pi$$
0.712650 + 0.701520i $$0.247495\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ 4.00000 0.129302
$$958$$ 8.00000 0.258468
$$959$$ 16.0000 0.516667
$$960$$ 1.00000 0.0322749
$$961$$ −31.0000 −1.00000
$$962$$ 48.0000 1.54758
$$963$$ −18.0000 −0.580042
$$964$$ −14.0000 −0.450910
$$965$$ 2.00000 0.0643823
$$966$$ 2.00000 0.0643489
$$967$$ 52.0000 1.67221 0.836104 0.548572i $$-0.184828\pi$$
0.836104 + 0.548572i $$0.184828\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ 0 0
$$970$$ 6.00000 0.192648
$$971$$ −46.0000 −1.47621 −0.738105 0.674686i $$-0.764279\pi$$
−0.738105 + 0.674686i $$0.764279\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 8.00000 0.256468
$$974$$ 12.0000 0.384505
$$975$$ 6.00000 0.192154
$$976$$ −8.00000 −0.256074
$$977$$ 24.0000 0.767828 0.383914 0.923369i $$-0.374576\pi$$
0.383914 + 0.923369i $$0.374576\pi$$
$$978$$ 4.00000 0.127906
$$979$$ 16.0000 0.511362
$$980$$ 3.00000 0.0958315
$$981$$ −4.00000 −0.127710
$$982$$ −12.0000 −0.382935
$$983$$ −12.0000 −0.382741 −0.191370 0.981518i $$-0.561293\pi$$
−0.191370 + 0.981518i $$0.561293\pi$$
$$984$$ 6.00000 0.191273
$$985$$ −6.00000 −0.191176
$$986$$ −8.00000 −0.254772
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −4.00000 −0.127193
$$990$$ 2.00000 0.0635642
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ −20.0000 −0.634681
$$994$$ −32.0000 −1.01498
$$995$$ 2.00000 0.0634043
$$996$$ −14.0000 −0.443607
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ 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 690.2.a.g.1.1 1
3.2 odd 2 2070.2.a.e.1.1 1
4.3 odd 2 5520.2.a.z.1.1 1
5.2 odd 4 3450.2.d.d.2899.2 2
5.3 odd 4 3450.2.d.d.2899.1 2
5.4 even 2 3450.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.g.1.1 1 1.1 even 1 trivial
2070.2.a.e.1.1 1 3.2 odd 2
3450.2.a.l.1.1 1 5.4 even 2
3450.2.d.d.2899.1 2 5.3 odd 4
3450.2.d.d.2899.2 2 5.2 odd 4
5520.2.a.z.1.1 1 4.3 odd 2