# Properties

 Label 338.2.a.d.1.1 Level $338$ Weight $2$ Character 338.1 Self dual yes Analytic conductor $2.699$ 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] = [338,2,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 338.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: $$2.69894358832$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 338.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -3.00000 q^{10} -1.00000 q^{12} -3.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} -6.00000 q^{19} -3.00000 q^{20} +3.00000 q^{21} +6.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +5.00000 q^{27} -3.00000 q^{28} +3.00000 q^{30} +1.00000 q^{32} -3.00000 q^{34} +9.00000 q^{35} -2.00000 q^{36} -3.00000 q^{37} -6.00000 q^{38} -3.00000 q^{40} +3.00000 q^{42} +1.00000 q^{43} +6.00000 q^{45} +6.00000 q^{46} -3.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +4.00000 q^{50} +3.00000 q^{51} -6.00000 q^{53} +5.00000 q^{54} -3.00000 q^{56} +6.00000 q^{57} +6.00000 q^{59} +3.00000 q^{60} -8.00000 q^{61} +6.00000 q^{63} +1.00000 q^{64} -12.0000 q^{67} -3.00000 q^{68} -6.00000 q^{69} +9.00000 q^{70} +15.0000 q^{71} -2.00000 q^{72} -6.00000 q^{73} -3.00000 q^{74} -4.00000 q^{75} -6.00000 q^{76} +10.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +6.00000 q^{83} +3.00000 q^{84} +9.00000 q^{85} +1.00000 q^{86} +6.00000 q^{89} +6.00000 q^{90} +6.00000 q^{92} -3.00000 q^{94} +18.0000 q^{95} -1.00000 q^{96} -12.0000 q^{97} +2.00000 q^{98} +O(q^{100})$$

## 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 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −2.00000 −0.666667
$$10$$ −3.00000 −0.948683
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −3.00000 −0.801784
$$15$$ 3.00000 0.774597
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ −2.00000 −0.471405
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ −3.00000 −0.670820
$$21$$ 3.00000 0.654654
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ −3.00000 −0.566947
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 3.00000 0.547723
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −3.00000 −0.514496
$$35$$ 9.00000 1.52128
$$36$$ −2.00000 −0.333333
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ −6.00000 −0.973329
$$39$$ 0 0
$$40$$ −3.00000 −0.474342
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 3.00000 0.462910
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 0 0
$$45$$ 6.00000 0.894427
$$46$$ 6.00000 0.884652
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 2.00000 0.285714
$$50$$ 4.00000 0.565685
$$51$$ 3.00000 0.420084
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 5.00000 0.680414
$$55$$ 0 0
$$56$$ −3.00000 −0.400892
$$57$$ 6.00000 0.794719
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 3.00000 0.387298
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 0 0
$$63$$ 6.00000 0.755929
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ −6.00000 −0.722315
$$70$$ 9.00000 1.07571
$$71$$ 15.0000 1.78017 0.890086 0.455792i $$-0.150644\pi$$
0.890086 + 0.455792i $$0.150644\pi$$
$$72$$ −2.00000 −0.235702
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ −3.00000 −0.348743
$$75$$ −4.00000 −0.461880
$$76$$ −6.00000 −0.688247
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 3.00000 0.327327
$$85$$ 9.00000 0.976187
$$86$$ 1.00000 0.107833
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 6.00000 0.632456
$$91$$ 0 0
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ −3.00000 −0.309426
$$95$$ 18.0000 1.84676
$$96$$ −1.00000 −0.102062
$$97$$ −12.0000 −1.21842 −0.609208 0.793011i $$-0.708512\pi$$
−0.609208 + 0.793011i $$0.708512\pi$$
$$98$$ 2.00000 0.202031
$$99$$ 0 0
$$100$$ 4.00000 0.400000
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 3.00000 0.297044
$$103$$ −14.0000 −1.37946 −0.689730 0.724066i $$-0.742271\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ 0 0
$$105$$ −9.00000 −0.878310
$$106$$ −6.00000 −0.582772
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 5.00000 0.481125
$$109$$ 9.00000 0.862044 0.431022 0.902342i $$-0.358153\pi$$
0.431022 + 0.902342i $$0.358153\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ −3.00000 −0.283473
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 6.00000 0.561951
$$115$$ −18.0000 −1.67851
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 6.00000 0.552345
$$119$$ 9.00000 0.825029
$$120$$ 3.00000 0.273861
$$121$$ −11.0000 −1.00000
$$122$$ −8.00000 −0.724286
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 3.00000 0.268328
$$126$$ 6.00000 0.534522
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −1.00000 −0.0880451
$$130$$ 0 0
$$131$$ −3.00000 −0.262111 −0.131056 0.991375i $$-0.541837\pi$$
−0.131056 + 0.991375i $$0.541837\pi$$
$$132$$ 0 0
$$133$$ 18.0000 1.56080
$$134$$ −12.0000 −1.03664
$$135$$ −15.0000 −1.29099
$$136$$ −3.00000 −0.257248
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ −6.00000 −0.510754
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 9.00000 0.760639
$$141$$ 3.00000 0.252646
$$142$$ 15.0000 1.25877
$$143$$ 0 0
$$144$$ −2.00000 −0.166667
$$145$$ 0 0
$$146$$ −6.00000 −0.496564
$$147$$ −2.00000 −0.164957
$$148$$ −3.00000 −0.246598
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ −4.00000 −0.326599
$$151$$ 15.0000 1.22068 0.610341 0.792139i $$-0.291032\pi$$
0.610341 + 0.792139i $$0.291032\pi$$
$$152$$ −6.00000 −0.486664
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −22.0000 −1.75579 −0.877896 0.478852i $$-0.841053\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ 10.0000 0.795557
$$159$$ 6.00000 0.475831
$$160$$ −3.00000 −0.237171
$$161$$ −18.0000 −1.41860
$$162$$ 1.00000 0.0785674
$$163$$ −6.00000 −0.469956 −0.234978 0.972001i $$-0.575502\pi$$
−0.234978 + 0.972001i $$0.575502\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 3.00000 0.231455
$$169$$ 0 0
$$170$$ 9.00000 0.690268
$$171$$ 12.0000 0.917663
$$172$$ 1.00000 0.0762493
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ −12.0000 −0.907115
$$176$$ 0 0
$$177$$ −6.00000 −0.450988
$$178$$ 6.00000 0.449719
$$179$$ −15.0000 −1.12115 −0.560576 0.828103i $$-0.689420\pi$$
−0.560576 + 0.828103i $$0.689420\pi$$
$$180$$ 6.00000 0.447214
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 8.00000 0.591377
$$184$$ 6.00000 0.442326
$$185$$ 9.00000 0.661693
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −3.00000 −0.218797
$$189$$ −15.0000 −1.09109
$$190$$ 18.0000 1.30586
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ 3.00000 0.213741 0.106871 0.994273i $$-0.465917\pi$$
0.106871 + 0.994273i $$0.465917\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 4.00000 0.282843
$$201$$ 12.0000 0.846415
$$202$$ −12.0000 −0.844317
$$203$$ 0 0
$$204$$ 3.00000 0.210042
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ −12.0000 −0.834058
$$208$$ 0 0
$$209$$ 0 0
$$210$$ −9.00000 −0.621059
$$211$$ −23.0000 −1.58339 −0.791693 0.610920i $$-0.790800\pi$$
−0.791693 + 0.610920i $$0.790800\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −15.0000 −1.02778
$$214$$ −12.0000 −0.820303
$$215$$ −3.00000 −0.204598
$$216$$ 5.00000 0.340207
$$217$$ 0 0
$$218$$ 9.00000 0.609557
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 3.00000 0.201347
$$223$$ −9.00000 −0.602685 −0.301342 0.953516i $$-0.597435\pi$$
−0.301342 + 0.953516i $$0.597435\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ −8.00000 −0.533333
$$226$$ −6.00000 −0.399114
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 6.00000 0.397360
$$229$$ −9.00000 −0.594737 −0.297368 0.954763i $$-0.596109\pi$$
−0.297368 + 0.954763i $$0.596109\pi$$
$$230$$ −18.0000 −1.18688
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 21.0000 1.37576 0.687878 0.725826i $$-0.258542\pi$$
0.687878 + 0.725826i $$0.258542\pi$$
$$234$$ 0 0
$$235$$ 9.00000 0.587095
$$236$$ 6.00000 0.390567
$$237$$ −10.0000 −0.649570
$$238$$ 9.00000 0.583383
$$239$$ 9.00000 0.582162 0.291081 0.956698i $$-0.405985\pi$$
0.291081 + 0.956698i $$0.405985\pi$$
$$240$$ 3.00000 0.193649
$$241$$ 30.0000 1.93247 0.966235 0.257663i $$-0.0829523\pi$$
0.966235 + 0.257663i $$0.0829523\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ −16.0000 −1.02640
$$244$$ −8.00000 −0.512148
$$245$$ −6.00000 −0.383326
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 3.00000 0.189737
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 6.00000 0.377964
$$253$$ 0 0
$$254$$ 2.00000 0.125491
$$255$$ −9.00000 −0.563602
$$256$$ 1.00000 0.0625000
$$257$$ −3.00000 −0.187135 −0.0935674 0.995613i $$-0.529827\pi$$
−0.0935674 + 0.995613i $$0.529827\pi$$
$$258$$ −1.00000 −0.0622573
$$259$$ 9.00000 0.559233
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −3.00000 −0.185341
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ 18.0000 1.10573
$$266$$ 18.0000 1.10365
$$267$$ −6.00000 −0.367194
$$268$$ −12.0000 −0.733017
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ −15.0000 −0.912871
$$271$$ −15.0000 −0.911185 −0.455593 0.890188i $$-0.650573\pi$$
−0.455593 + 0.890188i $$0.650573\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ 5.00000 0.299880
$$279$$ 0 0
$$280$$ 9.00000 0.537853
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 3.00000 0.178647
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 15.0000 0.890086
$$285$$ −18.0000 −1.06623
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −2.00000 −0.117851
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 12.0000 0.703452
$$292$$ −6.00000 −0.351123
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ −18.0000 −1.04800
$$296$$ −3.00000 −0.174371
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ −4.00000 −0.230940
$$301$$ −3.00000 −0.172917
$$302$$ 15.0000 0.863153
$$303$$ 12.0000 0.689382
$$304$$ −6.00000 −0.344124
$$305$$ 24.0000 1.37424
$$306$$ 6.00000 0.342997
$$307$$ −18.0000 −1.02731 −0.513657 0.857996i $$-0.671710\pi$$
−0.513657 + 0.857996i $$0.671710\pi$$
$$308$$ 0 0
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ 19.0000 1.07394 0.536972 0.843600i $$-0.319568\pi$$
0.536972 + 0.843600i $$0.319568\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ −18.0000 −1.01419
$$316$$ 10.0000 0.562544
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 0 0
$$320$$ −3.00000 −0.167705
$$321$$ 12.0000 0.669775
$$322$$ −18.0000 −1.00310
$$323$$ 18.0000 1.00155
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −6.00000 −0.332309
$$327$$ −9.00000 −0.497701
$$328$$ 0 0
$$329$$ 9.00000 0.496186
$$330$$ 0 0
$$331$$ 30.0000 1.64895 0.824475 0.565899i $$-0.191471\pi$$
0.824475 + 0.565899i $$0.191471\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 6.00000 0.328798
$$334$$ 12.0000 0.656611
$$335$$ 36.0000 1.96689
$$336$$ 3.00000 0.163663
$$337$$ −13.0000 −0.708155 −0.354078 0.935216i $$-0.615205\pi$$
−0.354078 + 0.935216i $$0.615205\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 9.00000 0.488094
$$341$$ 0 0
$$342$$ 12.0000 0.648886
$$343$$ 15.0000 0.809924
$$344$$ 1.00000 0.0539164
$$345$$ 18.0000 0.969087
$$346$$ 6.00000 0.322562
$$347$$ 33.0000 1.77153 0.885766 0.464131i $$-0.153633\pi$$
0.885766 + 0.464131i $$0.153633\pi$$
$$348$$ 0 0
$$349$$ 21.0000 1.12410 0.562052 0.827102i $$-0.310012\pi$$
0.562052 + 0.827102i $$0.310012\pi$$
$$350$$ −12.0000 −0.641427
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ −6.00000 −0.318896
$$355$$ −45.0000 −2.38835
$$356$$ 6.00000 0.317999
$$357$$ −9.00000 −0.476331
$$358$$ −15.0000 −0.792775
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 6.00000 0.316228
$$361$$ 17.0000 0.894737
$$362$$ −2.00000 −0.105118
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ 18.0000 0.942163
$$366$$ 8.00000 0.418167
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 0 0
$$370$$ 9.00000 0.467888
$$371$$ 18.0000 0.934513
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 0 0
$$375$$ −3.00000 −0.154919
$$376$$ −3.00000 −0.154713
$$377$$ 0 0
$$378$$ −15.0000 −0.771517
$$379$$ −6.00000 −0.308199 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$380$$ 18.0000 0.923381
$$381$$ −2.00000 −0.102463
$$382$$ 12.0000 0.613973
$$383$$ −9.00000 −0.459879 −0.229939 0.973205i $$-0.573853\pi$$
−0.229939 + 0.973205i $$0.573853\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ −2.00000 −0.101666
$$388$$ −12.0000 −0.609208
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −18.0000 −0.910299
$$392$$ 2.00000 0.101015
$$393$$ 3.00000 0.151330
$$394$$ 3.00000 0.151138
$$395$$ −30.0000 −1.50946
$$396$$ 0 0
$$397$$ −18.0000 −0.903394 −0.451697 0.892171i $$-0.649181\pi$$
−0.451697 + 0.892171i $$0.649181\pi$$
$$398$$ 20.0000 1.00251
$$399$$ −18.0000 −0.901127
$$400$$ 4.00000 0.200000
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 12.0000 0.598506
$$403$$ 0 0
$$404$$ −12.0000 −0.597022
$$405$$ −3.00000 −0.149071
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 3.00000 0.148522
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ −14.0000 −0.689730
$$413$$ −18.0000 −0.885722
$$414$$ −12.0000 −0.589768
$$415$$ −18.0000 −0.883585
$$416$$ 0 0
$$417$$ −5.00000 −0.244851
$$418$$ 0 0
$$419$$ 15.0000 0.732798 0.366399 0.930458i $$-0.380591\pi$$
0.366399 + 0.930458i $$0.380591\pi$$
$$420$$ −9.00000 −0.439155
$$421$$ −15.0000 −0.731055 −0.365528 0.930800i $$-0.619111\pi$$
−0.365528 + 0.930800i $$0.619111\pi$$
$$422$$ −23.0000 −1.11962
$$423$$ 6.00000 0.291730
$$424$$ −6.00000 −0.291386
$$425$$ −12.0000 −0.582086
$$426$$ −15.0000 −0.726752
$$427$$ 24.0000 1.16144
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ −3.00000 −0.144673
$$431$$ −15.0000 −0.722525 −0.361262 0.932464i $$-0.617654\pi$$
−0.361262 + 0.932464i $$0.617654\pi$$
$$432$$ 5.00000 0.240563
$$433$$ 11.0000 0.528626 0.264313 0.964437i $$-0.414855\pi$$
0.264313 + 0.964437i $$0.414855\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 9.00000 0.431022
$$437$$ −36.0000 −1.72211
$$438$$ 6.00000 0.286691
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ −4.00000 −0.190476
$$442$$ 0 0
$$443$$ −21.0000 −0.997740 −0.498870 0.866677i $$-0.666252\pi$$
−0.498870 + 0.866677i $$0.666252\pi$$
$$444$$ 3.00000 0.142374
$$445$$ −18.0000 −0.853282
$$446$$ −9.00000 −0.426162
$$447$$ 6.00000 0.283790
$$448$$ −3.00000 −0.141737
$$449$$ −24.0000 −1.13263 −0.566315 0.824189i $$-0.691631\pi$$
−0.566315 + 0.824189i $$0.691631\pi$$
$$450$$ −8.00000 −0.377124
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ −15.0000 −0.704761
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 6.00000 0.280976
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ −9.00000 −0.420542
$$459$$ −15.0000 −0.700140
$$460$$ −18.0000 −0.839254
$$461$$ −15.0000 −0.698620 −0.349310 0.937007i $$-0.613584\pi$$
−0.349310 + 0.937007i $$0.613584\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 21.0000 0.972806
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 36.0000 1.66233
$$470$$ 9.00000 0.415139
$$471$$ 22.0000 1.01371
$$472$$ 6.00000 0.276172
$$473$$ 0 0
$$474$$ −10.0000 −0.459315
$$475$$ −24.0000 −1.10120
$$476$$ 9.00000 0.412514
$$477$$ 12.0000 0.549442
$$478$$ 9.00000 0.411650
$$479$$ −39.0000 −1.78196 −0.890978 0.454047i $$-0.849980\pi$$
−0.890978 + 0.454047i $$0.849980\pi$$
$$480$$ 3.00000 0.136931
$$481$$ 0 0
$$482$$ 30.0000 1.36646
$$483$$ 18.0000 0.819028
$$484$$ −11.0000 −0.500000
$$485$$ 36.0000 1.63468
$$486$$ −16.0000 −0.725775
$$487$$ −12.0000 −0.543772 −0.271886 0.962329i $$-0.587647\pi$$
−0.271886 + 0.962329i $$0.587647\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ 6.00000 0.271329
$$490$$ −6.00000 −0.271052
$$491$$ −27.0000 −1.21849 −0.609246 0.792981i $$-0.708528\pi$$
−0.609246 + 0.792981i $$0.708528\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −45.0000 −2.01853
$$498$$ −6.00000 −0.268866
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ 3.00000 0.134164
$$501$$ −12.0000 −0.536120
$$502$$ −12.0000 −0.535586
$$503$$ −6.00000 −0.267527 −0.133763 0.991013i $$-0.542706\pi$$
−0.133763 + 0.991013i $$0.542706\pi$$
$$504$$ 6.00000 0.267261
$$505$$ 36.0000 1.60198
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 2.00000 0.0887357
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ −9.00000 −0.398527
$$511$$ 18.0000 0.796273
$$512$$ 1.00000 0.0441942
$$513$$ −30.0000 −1.32453
$$514$$ −3.00000 −0.132324
$$515$$ 42.0000 1.85074
$$516$$ −1.00000 −0.0440225
$$517$$ 0 0
$$518$$ 9.00000 0.395437
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ 27.0000 1.18289 0.591446 0.806345i $$-0.298557\pi$$
0.591446 + 0.806345i $$0.298557\pi$$
$$522$$ 0 0
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ −3.00000 −0.131056
$$525$$ 12.0000 0.523723
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 18.0000 0.781870
$$531$$ −12.0000 −0.520756
$$532$$ 18.0000 0.780399
$$533$$ 0 0
$$534$$ −6.00000 −0.259645
$$535$$ 36.0000 1.55642
$$536$$ −12.0000 −0.518321
$$537$$ 15.0000 0.647298
$$538$$ 0 0
$$539$$ 0 0
$$540$$ −15.0000 −0.645497
$$541$$ 15.0000 0.644900 0.322450 0.946586i $$-0.395494\pi$$
0.322450 + 0.946586i $$0.395494\pi$$
$$542$$ −15.0000 −0.644305
$$543$$ 2.00000 0.0858282
$$544$$ −3.00000 −0.128624
$$545$$ −27.0000 −1.15655
$$546$$ 0 0
$$547$$ −37.0000 −1.58201 −0.791003 0.611812i $$-0.790441\pi$$
−0.791003 + 0.611812i $$0.790441\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 16.0000 0.682863
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −6.00000 −0.255377
$$553$$ −30.0000 −1.27573
$$554$$ −8.00000 −0.339887
$$555$$ −9.00000 −0.382029
$$556$$ 5.00000 0.212047
$$557$$ 27.0000 1.14403 0.572013 0.820244i $$-0.306163\pi$$
0.572013 + 0.820244i $$0.306163\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 9.00000 0.380319
$$561$$ 0 0
$$562$$ 30.0000 1.26547
$$563$$ −39.0000 −1.64365 −0.821827 0.569737i $$-0.807045\pi$$
−0.821827 + 0.569737i $$0.807045\pi$$
$$564$$ 3.00000 0.126323
$$565$$ 18.0000 0.757266
$$566$$ −4.00000 −0.168133
$$567$$ −3.00000 −0.125988
$$568$$ 15.0000 0.629386
$$569$$ −45.0000 −1.88650 −0.943249 0.332086i $$-0.892248\pi$$
−0.943249 + 0.332086i $$0.892248\pi$$
$$570$$ −18.0000 −0.753937
$$571$$ 23.0000 0.962520 0.481260 0.876578i $$-0.340179\pi$$
0.481260 + 0.876578i $$0.340179\pi$$
$$572$$ 0 0
$$573$$ −12.0000 −0.501307
$$574$$ 0 0
$$575$$ 24.0000 1.00087
$$576$$ −2.00000 −0.0833333
$$577$$ −42.0000 −1.74848 −0.874241 0.485491i $$-0.838641\pi$$
−0.874241 + 0.485491i $$0.838641\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 6.00000 0.249351
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ 12.0000 0.497416
$$583$$ 0 0
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 9.00000 0.371787
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ −2.00000 −0.0824786
$$589$$ 0 0
$$590$$ −18.0000 −0.741048
$$591$$ −3.00000 −0.123404
$$592$$ −3.00000 −0.123299
$$593$$ −36.0000 −1.47834 −0.739171 0.673517i $$-0.764783\pi$$
−0.739171 + 0.673517i $$0.764783\pi$$
$$594$$ 0 0
$$595$$ −27.0000 −1.10689
$$596$$ −6.00000 −0.245770
$$597$$ −20.0000 −0.818546
$$598$$ 0 0
$$599$$ −30.0000 −1.22577 −0.612883 0.790173i $$-0.709990\pi$$
−0.612883 + 0.790173i $$0.709990\pi$$
$$600$$ −4.00000 −0.163299
$$601$$ 37.0000 1.50926 0.754631 0.656150i $$-0.227816\pi$$
0.754631 + 0.656150i $$0.227816\pi$$
$$602$$ −3.00000 −0.122271
$$603$$ 24.0000 0.977356
$$604$$ 15.0000 0.610341
$$605$$ 33.0000 1.34164
$$606$$ 12.0000 0.487467
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ −6.00000 −0.243332
$$609$$ 0 0
$$610$$ 24.0000 0.971732
$$611$$ 0 0
$$612$$ 6.00000 0.242536
$$613$$ 6.00000 0.242338 0.121169 0.992632i $$-0.461336\pi$$
0.121169 + 0.992632i $$0.461336\pi$$
$$614$$ −18.0000 −0.726421
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.0000 −0.483102 −0.241551 0.970388i $$-0.577656\pi$$
−0.241551 + 0.970388i $$0.577656\pi$$
$$618$$ 14.0000 0.563163
$$619$$ −24.0000 −0.964641 −0.482321 0.875995i $$-0.660206\pi$$
−0.482321 + 0.875995i $$0.660206\pi$$
$$620$$ 0 0
$$621$$ 30.0000 1.20386
$$622$$ 18.0000 0.721734
$$623$$ −18.0000 −0.721155
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 19.0000 0.759393
$$627$$ 0 0
$$628$$ −22.0000 −0.877896
$$629$$ 9.00000 0.358854
$$630$$ −18.0000 −0.717137
$$631$$ 15.0000 0.597141 0.298570 0.954388i $$-0.403490\pi$$
0.298570 + 0.954388i $$0.403490\pi$$
$$632$$ 10.0000 0.397779
$$633$$ 23.0000 0.914168
$$634$$ 18.0000 0.714871
$$635$$ −6.00000 −0.238103
$$636$$ 6.00000 0.237915
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −30.0000 −1.18678
$$640$$ −3.00000 −0.118585
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 36.0000 1.41970 0.709851 0.704352i $$-0.248762\pi$$
0.709851 + 0.704352i $$0.248762\pi$$
$$644$$ −18.0000 −0.709299
$$645$$ 3.00000 0.118125
$$646$$ 18.0000 0.708201
$$647$$ 42.0000 1.65119 0.825595 0.564263i $$-0.190840\pi$$
0.825595 + 0.564263i $$0.190840\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −6.00000 −0.234978
$$653$$ −36.0000 −1.40879 −0.704394 0.709809i $$-0.748781\pi$$
−0.704394 + 0.709809i $$0.748781\pi$$
$$654$$ −9.00000 −0.351928
$$655$$ 9.00000 0.351659
$$656$$ 0 0
$$657$$ 12.0000 0.468165
$$658$$ 9.00000 0.350857
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ 30.0000 1.16598
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ −54.0000 −2.09403
$$666$$ 6.00000 0.232495
$$667$$ 0 0
$$668$$ 12.0000 0.464294
$$669$$ 9.00000 0.347960
$$670$$ 36.0000 1.39080
$$671$$ 0 0
$$672$$ 3.00000 0.115728
$$673$$ 1.00000 0.0385472 0.0192736 0.999814i $$-0.493865\pi$$
0.0192736 + 0.999814i $$0.493865\pi$$
$$674$$ −13.0000 −0.500741
$$675$$ 20.0000 0.769800
$$676$$ 0 0
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 36.0000 1.38155
$$680$$ 9.00000 0.345134
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ −6.00000 −0.229584 −0.114792 0.993390i $$-0.536620\pi$$
−0.114792 + 0.993390i $$0.536620\pi$$
$$684$$ 12.0000 0.458831
$$685$$ 54.0000 2.06323
$$686$$ 15.0000 0.572703
$$687$$ 9.00000 0.343371
$$688$$ 1.00000 0.0381246
$$689$$ 0 0
$$690$$ 18.0000 0.685248
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ 33.0000 1.25266
$$695$$ −15.0000 −0.568982
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 21.0000 0.794862
$$699$$ −21.0000 −0.794293
$$700$$ −12.0000 −0.453557
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 0 0
$$703$$ 18.0000 0.678883
$$704$$ 0 0
$$705$$ −9.00000 −0.338960
$$706$$ 6.00000 0.225813
$$707$$ 36.0000 1.35392
$$708$$ −6.00000 −0.225494
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ −45.0000 −1.68882
$$711$$ −20.0000 −0.750059
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ −9.00000 −0.336817
$$715$$ 0 0
$$716$$ −15.0000 −0.560576
$$717$$ −9.00000 −0.336111
$$718$$ −24.0000 −0.895672
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 6.00000 0.223607
$$721$$ 42.0000 1.56416
$$722$$ 17.0000 0.632674
$$723$$ −30.0000 −1.11571
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ 11.0000 0.408248
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 18.0000 0.666210
$$731$$ −3.00000 −0.110959
$$732$$ 8.00000 0.295689
$$733$$ −9.00000 −0.332423 −0.166211 0.986090i $$-0.553153\pi$$
−0.166211 + 0.986090i $$0.553153\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 6.00000 0.221313
$$736$$ 6.00000 0.221163
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 36.0000 1.32428 0.662141 0.749380i $$-0.269648\pi$$
0.662141 + 0.749380i $$0.269648\pi$$
$$740$$ 9.00000 0.330847
$$741$$ 0 0
$$742$$ 18.0000 0.660801
$$743$$ −39.0000 −1.43077 −0.715386 0.698730i $$-0.753749\pi$$
−0.715386 + 0.698730i $$0.753749\pi$$
$$744$$ 0 0
$$745$$ 18.0000 0.659469
$$746$$ 4.00000 0.146450
$$747$$ −12.0000 −0.439057
$$748$$ 0 0
$$749$$ 36.0000 1.31541
$$750$$ −3.00000 −0.109545
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ −3.00000 −0.109399
$$753$$ 12.0000 0.437304
$$754$$ 0 0
$$755$$ −45.0000 −1.63772
$$756$$ −15.0000 −0.545545
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ −6.00000 −0.217930
$$759$$ 0 0
$$760$$ 18.0000 0.652929
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ −2.00000 −0.0724524
$$763$$ −27.0000 −0.977466
$$764$$ 12.0000 0.434145
$$765$$ −18.0000 −0.650791
$$766$$ −9.00000 −0.325183
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 24.0000 0.865462 0.432731 0.901523i $$-0.357550\pi$$
0.432731 + 0.901523i $$0.357550\pi$$
$$770$$ 0 0
$$771$$ 3.00000 0.108042
$$772$$ −6.00000 −0.215945
$$773$$ 21.0000 0.755318 0.377659 0.925945i $$-0.376729\pi$$
0.377659 + 0.925945i $$0.376729\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ 0 0
$$776$$ −12.0000 −0.430775
$$777$$ −9.00000 −0.322873
$$778$$ 30.0000 1.07555
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −18.0000 −0.643679
$$783$$ 0 0
$$784$$ 2.00000 0.0714286
$$785$$ 66.0000 2.35564
$$786$$ 3.00000 0.107006
$$787$$ 12.0000 0.427754 0.213877 0.976861i $$-0.431391\pi$$
0.213877 + 0.976861i $$0.431391\pi$$
$$788$$ 3.00000 0.106871
$$789$$ −24.0000 −0.854423
$$790$$ −30.0000 −1.06735
$$791$$ 18.0000 0.640006
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −18.0000 −0.638796
$$795$$ −18.0000 −0.638394
$$796$$ 20.0000 0.708881
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ −18.0000 −0.637193
$$799$$ 9.00000 0.318397
$$800$$ 4.00000 0.141421
$$801$$ −12.0000 −0.423999
$$802$$ −30.0000 −1.05934
$$803$$ 0 0
$$804$$ 12.0000 0.423207
$$805$$ 54.0000 1.90325
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −12.0000 −0.422159
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ −3.00000 −0.105409
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ 15.0000 0.526073
$$814$$ 0 0
$$815$$ 18.0000 0.630512
$$816$$ 3.00000 0.105021
$$817$$ −6.00000 −0.209913
$$818$$ −6.00000 −0.209785
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 45.0000 1.57051 0.785255 0.619172i $$-0.212532\pi$$
0.785255 + 0.619172i $$0.212532\pi$$
$$822$$ 18.0000 0.627822
$$823$$ −14.0000 −0.488009 −0.244005 0.969774i $$-0.578461\pi$$
−0.244005 + 0.969774i $$0.578461\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ 0 0
$$826$$ −18.0000 −0.626300
$$827$$ −18.0000 −0.625921 −0.312961 0.949766i $$-0.601321\pi$$
−0.312961 + 0.949766i $$0.601321\pi$$
$$828$$ −12.0000 −0.417029
$$829$$ 20.0000 0.694629 0.347314 0.937749i $$-0.387094\pi$$
0.347314 + 0.937749i $$0.387094\pi$$
$$830$$ −18.0000 −0.624789
$$831$$ 8.00000 0.277517
$$832$$ 0 0
$$833$$ −6.00000 −0.207888
$$834$$ −5.00000 −0.173136
$$835$$ −36.0000 −1.24583
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 15.0000 0.518166
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ −9.00000 −0.310530
$$841$$ −29.0000 −1.00000
$$842$$ −15.0000 −0.516934
$$843$$ −30.0000 −1.03325
$$844$$ −23.0000 −0.791693
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ 33.0000 1.13389
$$848$$ −6.00000 −0.206041
$$849$$ 4.00000 0.137280
$$850$$ −12.0000 −0.411597
$$851$$ −18.0000 −0.617032
$$852$$ −15.0000 −0.513892
$$853$$ 39.0000 1.33533 0.667667 0.744460i $$-0.267293\pi$$
0.667667 + 0.744460i $$0.267293\pi$$
$$854$$ 24.0000 0.821263
$$855$$ −36.0000 −1.23117
$$856$$ −12.0000 −0.410152
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ −3.00000 −0.102299
$$861$$ 0 0
$$862$$ −15.0000 −0.510902
$$863$$ −39.0000 −1.32758 −0.663788 0.747921i $$-0.731052\pi$$
−0.663788 + 0.747921i $$0.731052\pi$$
$$864$$ 5.00000 0.170103
$$865$$ −18.0000 −0.612018
$$866$$ 11.0000 0.373795
$$867$$ 8.00000 0.271694
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 9.00000 0.304778
$$873$$ 24.0000 0.812277
$$874$$ −36.0000 −1.21772
$$875$$ −9.00000 −0.304256
$$876$$ 6.00000 0.202721
$$877$$ 3.00000 0.101303 0.0506514 0.998716i $$-0.483870\pi$$
0.0506514 + 0.998716i $$0.483870\pi$$
$$878$$ 10.0000 0.337484
$$879$$ −9.00000 −0.303562
$$880$$ 0 0
$$881$$ 33.0000 1.11180 0.555899 0.831250i $$-0.312374\pi$$
0.555899 + 0.831250i $$0.312374\pi$$
$$882$$ −4.00000 −0.134687
$$883$$ 11.0000 0.370179 0.185090 0.982722i $$-0.440742\pi$$
0.185090 + 0.982722i $$0.440742\pi$$
$$884$$ 0 0
$$885$$ 18.0000 0.605063
$$886$$ −21.0000 −0.705509
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ 3.00000 0.100673
$$889$$ −6.00000 −0.201234
$$890$$ −18.0000 −0.603361
$$891$$ 0 0
$$892$$ −9.00000 −0.301342
$$893$$ 18.0000 0.602347
$$894$$ 6.00000 0.200670
$$895$$ 45.0000 1.50418
$$896$$ −3.00000 −0.100223
$$897$$ 0 0
$$898$$ −24.0000 −0.800890
$$899$$ 0 0
$$900$$ −8.00000 −0.266667
$$901$$ 18.0000 0.599667
$$902$$ 0 0
$$903$$ 3.00000 0.0998337
$$904$$ −6.00000 −0.199557
$$905$$ 6.00000 0.199447
$$906$$ −15.0000 −0.498342
$$907$$ 17.0000 0.564476 0.282238 0.959344i $$-0.408923\pi$$
0.282238 + 0.959344i $$0.408923\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 24.0000 0.796030
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 6.00000 0.198680
$$913$$ 0 0
$$914$$ 18.0000 0.595387
$$915$$ −24.0000 −0.793416
$$916$$ −9.00000 −0.297368
$$917$$ 9.00000 0.297206
$$918$$ −15.0000 −0.495074
$$919$$ −20.0000 −0.659739 −0.329870 0.944027i $$-0.607005\pi$$
−0.329870 + 0.944027i $$0.607005\pi$$
$$920$$ −18.0000 −0.593442
$$921$$ 18.0000 0.593120
$$922$$ −15.0000 −0.493999
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −12.0000 −0.394558
$$926$$ 24.0000 0.788689
$$927$$ 28.0000 0.919641
$$928$$ 0 0
$$929$$ −36.0000 −1.18112 −0.590561 0.806993i $$-0.701093\pi$$
−0.590561 + 0.806993i $$0.701093\pi$$
$$930$$ 0 0
$$931$$ −12.0000 −0.393284
$$932$$ 21.0000 0.687878
$$933$$ −18.0000 −0.589294
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −2.00000 −0.0653372 −0.0326686 0.999466i $$-0.510401\pi$$
−0.0326686 + 0.999466i $$0.510401\pi$$
$$938$$ 36.0000 1.17544
$$939$$ −19.0000 −0.620042
$$940$$ 9.00000 0.293548
$$941$$ −45.0000 −1.46696 −0.733479 0.679712i $$-0.762105\pi$$
−0.733479 + 0.679712i $$0.762105\pi$$
$$942$$ 22.0000 0.716799
$$943$$ 0 0
$$944$$ 6.00000 0.195283
$$945$$ 45.0000 1.46385
$$946$$ 0 0
$$947$$ −48.0000 −1.55979 −0.779895 0.625910i $$-0.784728\pi$$
−0.779895 + 0.625910i $$0.784728\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ 0 0
$$950$$ −24.0000 −0.778663
$$951$$ −18.0000 −0.583690
$$952$$ 9.00000 0.291692
$$953$$ −9.00000 −0.291539 −0.145769 0.989319i $$-0.546566\pi$$
−0.145769 + 0.989319i $$0.546566\pi$$
$$954$$ 12.0000 0.388514
$$955$$ −36.0000 −1.16493
$$956$$ 9.00000 0.291081
$$957$$ 0 0
$$958$$ −39.0000 −1.26003
$$959$$ 54.0000 1.74375
$$960$$ 3.00000 0.0968246
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 24.0000 0.773389
$$964$$ 30.0000 0.966235
$$965$$ 18.0000 0.579441
$$966$$ 18.0000 0.579141
$$967$$ 3.00000 0.0964735 0.0482367 0.998836i $$-0.484640\pi$$
0.0482367 + 0.998836i $$0.484640\pi$$
$$968$$ −11.0000 −0.353553
$$969$$ −18.0000 −0.578243
$$970$$ 36.0000 1.15589
$$971$$ 27.0000 0.866471 0.433236 0.901281i $$-0.357372\pi$$
0.433236 + 0.901281i $$0.357372\pi$$
$$972$$ −16.0000 −0.513200
$$973$$ −15.0000 −0.480878
$$974$$ −12.0000 −0.384505
$$975$$ 0 0
$$976$$ −8.00000 −0.256074
$$977$$ −12.0000 −0.383914 −0.191957 0.981403i $$-0.561483\pi$$
−0.191957 + 0.981403i $$0.561483\pi$$
$$978$$ 6.00000 0.191859
$$979$$ 0 0
$$980$$ −6.00000 −0.191663
$$981$$ −18.0000 −0.574696
$$982$$ −27.0000 −0.861605
$$983$$ 9.00000 0.287055 0.143528 0.989646i $$-0.454155\pi$$
0.143528 + 0.989646i $$0.454155\pi$$
$$984$$ 0 0
$$985$$ −9.00000 −0.286764
$$986$$ 0 0
$$987$$ −9.00000 −0.286473
$$988$$ 0 0
$$989$$ 6.00000 0.190789
$$990$$ 0 0
$$991$$ 2.00000 0.0635321 0.0317660 0.999495i $$-0.489887\pi$$
0.0317660 + 0.999495i $$0.489887\pi$$
$$992$$ 0 0
$$993$$ −30.0000 −0.952021
$$994$$ −45.0000 −1.42731
$$995$$ −60.0000 −1.90213
$$996$$ −6.00000 −0.190117
$$997$$ 8.00000 0.253363 0.126681 0.991943i $$-0.459567\pi$$
0.126681 + 0.991943i $$0.459567\pi$$
$$998$$ −36.0000 −1.13956
$$999$$ −15.0000 −0.474579
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 338.2.a.d.1.1 1
3.2 odd 2 3042.2.a.g.1.1 1
4.3 odd 2 2704.2.a.j.1.1 1
5.4 even 2 8450.2.a.h.1.1 1
13.2 odd 12 338.2.e.c.147.2 4
13.3 even 3 338.2.c.b.191.1 2
13.4 even 6 338.2.c.f.315.1 2
13.5 odd 4 26.2.b.a.25.1 2
13.6 odd 12 338.2.e.c.23.1 4
13.7 odd 12 338.2.e.c.23.2 4
13.8 odd 4 26.2.b.a.25.2 yes 2
13.9 even 3 338.2.c.b.315.1 2
13.10 even 6 338.2.c.f.191.1 2
13.11 odd 12 338.2.e.c.147.1 4
13.12 even 2 338.2.a.b.1.1 1
39.5 even 4 234.2.b.b.181.2 2
39.8 even 4 234.2.b.b.181.1 2
39.38 odd 2 3042.2.a.j.1.1 1
52.31 even 4 208.2.f.a.129.2 2
52.47 even 4 208.2.f.a.129.1 2
52.51 odd 2 2704.2.a.k.1.1 1
65.8 even 4 650.2.c.d.649.1 2
65.18 even 4 650.2.c.a.649.1 2
65.34 odd 4 650.2.d.b.51.1 2
65.44 odd 4 650.2.d.b.51.2 2
65.47 even 4 650.2.c.a.649.2 2
65.57 even 4 650.2.c.d.649.2 2
65.64 even 2 8450.2.a.u.1.1 1
91.5 even 12 1274.2.n.c.753.1 4
91.18 odd 12 1274.2.n.d.961.2 4
91.31 even 12 1274.2.n.c.961.2 4
91.34 even 4 1274.2.d.c.883.2 2
91.44 odd 12 1274.2.n.d.753.1 4
91.47 even 12 1274.2.n.c.753.2 4
91.60 odd 12 1274.2.n.d.961.1 4
91.73 even 12 1274.2.n.c.961.1 4
91.83 even 4 1274.2.d.c.883.1 2
91.86 odd 12 1274.2.n.d.753.2 4
104.5 odd 4 832.2.f.d.129.1 2
104.21 odd 4 832.2.f.d.129.2 2
104.83 even 4 832.2.f.b.129.1 2
104.99 even 4 832.2.f.b.129.2 2
156.47 odd 4 1872.2.c.f.1585.2 2
156.83 odd 4 1872.2.c.f.1585.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.b.a.25.1 2 13.5 odd 4
26.2.b.a.25.2 yes 2 13.8 odd 4
208.2.f.a.129.1 2 52.47 even 4
208.2.f.a.129.2 2 52.31 even 4
234.2.b.b.181.1 2 39.8 even 4
234.2.b.b.181.2 2 39.5 even 4
338.2.a.b.1.1 1 13.12 even 2
338.2.a.d.1.1 1 1.1 even 1 trivial
338.2.c.b.191.1 2 13.3 even 3
338.2.c.b.315.1 2 13.9 even 3
338.2.c.f.191.1 2 13.10 even 6
338.2.c.f.315.1 2 13.4 even 6
338.2.e.c.23.1 4 13.6 odd 12
338.2.e.c.23.2 4 13.7 odd 12
338.2.e.c.147.1 4 13.11 odd 12
338.2.e.c.147.2 4 13.2 odd 12
650.2.c.a.649.1 2 65.18 even 4
650.2.c.a.649.2 2 65.47 even 4
650.2.c.d.649.1 2 65.8 even 4
650.2.c.d.649.2 2 65.57 even 4
650.2.d.b.51.1 2 65.34 odd 4
650.2.d.b.51.2 2 65.44 odd 4
832.2.f.b.129.1 2 104.83 even 4
832.2.f.b.129.2 2 104.99 even 4
832.2.f.d.129.1 2 104.5 odd 4
832.2.f.d.129.2 2 104.21 odd 4
1274.2.d.c.883.1 2 91.83 even 4
1274.2.d.c.883.2 2 91.34 even 4
1274.2.n.c.753.1 4 91.5 even 12
1274.2.n.c.753.2 4 91.47 even 12
1274.2.n.c.961.1 4 91.73 even 12
1274.2.n.c.961.2 4 91.31 even 12
1274.2.n.d.753.1 4 91.44 odd 12
1274.2.n.d.753.2 4 91.86 odd 12
1274.2.n.d.961.1 4 91.60 odd 12
1274.2.n.d.961.2 4 91.18 odd 12
1872.2.c.f.1585.1 2 156.83 odd 4
1872.2.c.f.1585.2 2 156.47 odd 4
2704.2.a.j.1.1 1 4.3 odd 2
2704.2.a.k.1.1 1 52.51 odd 2
3042.2.a.g.1.1 1 3.2 odd 2
3042.2.a.j.1.1 1 39.38 odd 2
8450.2.a.h.1.1 1 5.4 even 2
8450.2.a.u.1.1 1 65.64 even 2