# Properties

 Label 338.2.a.e.1.1 Level $338$ Weight $2$ Character 338.1 Self dual yes Analytic conductor $2.699$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} +4.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -3.00000 q^{18} -1.00000 q^{20} +4.00000 q^{22} -4.00000 q^{23} -4.00000 q^{25} +4.00000 q^{28} -1.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -4.00000 q^{35} -3.00000 q^{36} +3.00000 q^{37} -1.00000 q^{40} -9.00000 q^{41} -8.00000 q^{43} +4.00000 q^{44} +3.00000 q^{45} -4.00000 q^{46} -8.00000 q^{47} +9.00000 q^{49} -4.00000 q^{50} -9.00000 q^{53} -4.00000 q^{55} +4.00000 q^{56} -1.00000 q^{58} -4.00000 q^{59} +7.00000 q^{61} +4.00000 q^{62} -12.0000 q^{63} +1.00000 q^{64} +4.00000 q^{67} +3.00000 q^{68} -4.00000 q^{70} -8.00000 q^{71} -3.00000 q^{72} +11.0000 q^{73} +3.00000 q^{74} +16.0000 q^{77} -4.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} -9.00000 q^{82} -3.00000 q^{85} -8.00000 q^{86} +4.00000 q^{88} -6.00000 q^{89} +3.00000 q^{90} -4.00000 q^{92} -8.00000 q^{94} +2.00000 q^{97} +9.00000 q^{98} -12.0000 q^{99} +O(q^{100})$$

## 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
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 0 0
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −3.00000 −1.00000
$$10$$ −1.00000 −0.316228
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ −3.00000 −0.707107
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 4.00000 0.755929
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 3.00000 0.514496
$$35$$ −4.00000 −0.676123
$$36$$ −3.00000 −0.500000
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 3.00000 0.447214
$$46$$ −4.00000 −0.589768
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ −4.00000 −0.565685
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 4.00000 0.534522
$$57$$ 0 0
$$58$$ −1.00000 −0.131306
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −12.0000 −1.51186
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 3.00000 0.363803
$$69$$ 0 0
$$70$$ −4.00000 −0.478091
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 3.00000 0.348743
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 16.0000 1.82337
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 9.00000 1.00000
$$82$$ −9.00000 −0.993884
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ −3.00000 −0.325396
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 3.00000 0.316228
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 9.00000 0.909137
$$99$$ −12.0000 −1.20605
$$100$$ −4.00000 −0.400000
$$101$$ 7.00000 0.696526 0.348263 0.937397i $$-0.386772\pi$$
0.348263 + 0.937397i $$0.386772\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −9.00000 −0.874157
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ 0 0
$$112$$ 4.00000 0.377964
$$113$$ −1.00000 −0.0940721 −0.0470360 0.998893i $$-0.514978\pi$$
−0.0470360 + 0.998893i $$0.514978\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ −1.00000 −0.0928477
$$117$$ 0 0
$$118$$ −4.00000 −0.368230
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 7.00000 0.633750
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 9.00000 0.804984
$$126$$ −12.0000 −1.06904
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ −9.00000 −0.768922 −0.384461 0.923141i $$-0.625613\pi$$
−0.384461 + 0.923141i $$0.625613\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ 0 0
$$142$$ −8.00000 −0.671345
$$143$$ 0 0
$$144$$ −3.00000 −0.250000
$$145$$ 1.00000 0.0830455
$$146$$ 11.0000 0.910366
$$147$$ 0 0
$$148$$ 3.00000 0.246598
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ −9.00000 −0.727607
$$154$$ 16.0000 1.28932
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 11.0000 0.877896 0.438948 0.898513i $$-0.355351\pi$$
0.438948 + 0.898513i $$0.355351\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ −16.0000 −1.26098
$$162$$ 9.00000 0.707107
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ −9.00000 −0.702782
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −3.00000 −0.230089
$$171$$ 0 0
$$172$$ −8.00000 −0.609994
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 0 0
$$175$$ −16.0000 −1.20949
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 3.00000 0.223607
$$181$$ −21.0000 −1.56092 −0.780459 0.625207i $$-0.785014\pi$$
−0.780459 + 0.625207i $$0.785014\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ −3.00000 −0.220564
$$186$$ 0 0
$$187$$ 12.0000 0.877527
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −20.0000 −1.44715 −0.723575 0.690246i $$-0.757502\pi$$
−0.723575 + 0.690246i $$0.757502\pi$$
$$192$$ 0 0
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ −12.0000 −0.852803
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ −4.00000 −0.282843
$$201$$ 0 0
$$202$$ 7.00000 0.492518
$$203$$ −4.00000 −0.280745
$$204$$ 0 0
$$205$$ 9.00000 0.628587
$$206$$ −8.00000 −0.557386
$$207$$ 12.0000 0.834058
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ −9.00000 −0.618123
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ 16.0000 1.08615
$$218$$ −2.00000 −0.135457
$$219$$ 0 0
$$220$$ −4.00000 −0.269680
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 12.0000 0.803579 0.401790 0.915732i $$-0.368388\pi$$
0.401790 + 0.915732i $$0.368388\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 12.0000 0.800000
$$226$$ −1.00000 −0.0665190
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 4.00000 0.263752
$$231$$ 0 0
$$232$$ −1.00000 −0.0656532
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 12.0000 0.777844
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 3.00000 0.193247 0.0966235 0.995321i $$-0.469196\pi$$
0.0966235 + 0.995321i $$0.469196\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ 7.00000 0.448129
$$245$$ −9.00000 −0.574989
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 4.00000 0.254000
$$249$$ 0 0
$$250$$ 9.00000 0.569210
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ −12.0000 −0.755929
$$253$$ −16.0000 −1.00591
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 15.0000 0.935674 0.467837 0.883815i $$-0.345033\pi$$
0.467837 + 0.883815i $$0.345033\pi$$
$$258$$ 0 0
$$259$$ 12.0000 0.745644
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ 20.0000 1.23560
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ 9.00000 0.552866
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 4.00000 0.244339
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 3.00000 0.181902
$$273$$ 0 0
$$274$$ −9.00000 −0.543710
$$275$$ −16.0000 −0.964836
$$276$$ 0 0
$$277$$ −9.00000 −0.540758 −0.270379 0.962754i $$-0.587149\pi$$
−0.270379 + 0.962754i $$0.587149\pi$$
$$278$$ 16.0000 0.959616
$$279$$ −12.0000 −0.718421
$$280$$ −4.00000 −0.239046
$$281$$ −5.00000 −0.298275 −0.149137 0.988816i $$-0.547650\pi$$
−0.149137 + 0.988816i $$0.547650\pi$$
$$282$$ 0 0
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −36.0000 −2.12501
$$288$$ −3.00000 −0.176777
$$289$$ −8.00000 −0.470588
$$290$$ 1.00000 0.0587220
$$291$$ 0 0
$$292$$ 11.0000 0.643726
$$293$$ −5.00000 −0.292103 −0.146052 0.989277i $$-0.546657\pi$$
−0.146052 + 0.989277i $$0.546657\pi$$
$$294$$ 0 0
$$295$$ 4.00000 0.232889
$$296$$ 3.00000 0.174371
$$297$$ 0 0
$$298$$ 15.0000 0.868927
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −32.0000 −1.84445
$$302$$ 12.0000 0.690522
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −7.00000 −0.400819
$$306$$ −9.00000 −0.514496
$$307$$ −28.0000 −1.59804 −0.799022 0.601302i $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ 16.0000 0.911685
$$309$$ 0 0
$$310$$ −4.00000 −0.227185
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ 11.0000 0.620766
$$315$$ 12.0000 0.676123
$$316$$ −4.00000 −0.225018
$$317$$ 3.00000 0.168497 0.0842484 0.996445i $$-0.473151\pi$$
0.0842484 + 0.996445i $$0.473151\pi$$
$$318$$ 0 0
$$319$$ −4.00000 −0.223957
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ −16.0000 −0.891645
$$323$$ 0 0
$$324$$ 9.00000 0.500000
$$325$$ 0 0
$$326$$ 8.00000 0.443079
$$327$$ 0 0
$$328$$ −9.00000 −0.496942
$$329$$ −32.0000 −1.76422
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ 0 0
$$333$$ −9.00000 −0.493197
$$334$$ 12.0000 0.656611
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −3.00000 −0.162698
$$341$$ 16.0000 0.866449
$$342$$ 0 0
$$343$$ 8.00000 0.431959
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ −16.0000 −0.855236
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ 7.00000 0.372572 0.186286 0.982496i $$-0.440355\pi$$
0.186286 + 0.982496i $$0.440355\pi$$
$$354$$ 0 0
$$355$$ 8.00000 0.424596
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ −24.0000 −1.26844
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 3.00000 0.158114
$$361$$ −19.0000 −1.00000
$$362$$ −21.0000 −1.10374
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −11.0000 −0.575766
$$366$$ 0 0
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 27.0000 1.40556
$$370$$ −3.00000 −0.155963
$$371$$ −36.0000 −1.86903
$$372$$ 0 0
$$373$$ −13.0000 −0.673114 −0.336557 0.941663i $$-0.609263\pi$$
−0.336557 + 0.941663i $$0.609263\pi$$
$$374$$ 12.0000 0.620505
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −20.0000 −1.02329
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ −16.0000 −0.815436
$$386$$ 11.0000 0.559885
$$387$$ 24.0000 1.21999
$$388$$ 2.00000 0.101535
$$389$$ −9.00000 −0.456318 −0.228159 0.973624i $$-0.573271\pi$$
−0.228159 + 0.973624i $$0.573271\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ 9.00000 0.454569
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 4.00000 0.201262
$$396$$ −12.0000 −0.603023
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ −4.00000 −0.200502
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ 3.00000 0.149813 0.0749064 0.997191i $$-0.476134\pi$$
0.0749064 + 0.997191i $$0.476134\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 7.00000 0.348263
$$405$$ −9.00000 −0.447214
$$406$$ −4.00000 −0.198517
$$407$$ 12.0000 0.594818
$$408$$ 0 0
$$409$$ 31.0000 1.53285 0.766426 0.642333i $$-0.222033\pi$$
0.766426 + 0.642333i $$0.222033\pi$$
$$410$$ 9.00000 0.444478
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ −16.0000 −0.787309
$$414$$ 12.0000 0.589768
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −5.00000 −0.243685 −0.121843 0.992549i $$-0.538880\pi$$
−0.121843 + 0.992549i $$0.538880\pi$$
$$422$$ 20.0000 0.973585
$$423$$ 24.0000 1.16692
$$424$$ −9.00000 −0.437079
$$425$$ −12.0000 −0.582086
$$426$$ 0 0
$$427$$ 28.0000 1.35501
$$428$$ −4.00000 −0.193347
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ −5.00000 −0.240285 −0.120142 0.992757i $$-0.538335\pi$$
−0.120142 + 0.992757i $$0.538335\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ −4.00000 −0.190693
$$441$$ −27.0000 −1.28571
$$442$$ 0 0
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ 12.0000 0.568216
$$447$$ 0 0
$$448$$ 4.00000 0.188982
$$449$$ 34.0000 1.60456 0.802280 0.596948i $$-0.203620\pi$$
0.802280 + 0.596948i $$0.203620\pi$$
$$450$$ 12.0000 0.565685
$$451$$ −36.0000 −1.69517
$$452$$ −1.00000 −0.0470360
$$453$$ 0 0
$$454$$ 24.0000 1.12638
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 31.0000 1.45012 0.725059 0.688686i $$-0.241812\pi$$
0.725059 + 0.688686i $$0.241812\pi$$
$$458$$ 6.00000 0.280362
$$459$$ 0 0
$$460$$ 4.00000 0.186501
$$461$$ −33.0000 −1.53696 −0.768482 0.639872i $$-0.778987\pi$$
−0.768482 + 0.639872i $$0.778987\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ 20.0000 0.925490 0.462745 0.886492i $$-0.346865\pi$$
0.462745 + 0.886492i $$0.346865\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ −4.00000 −0.184115
$$473$$ −32.0000 −1.47136
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ 27.0000 1.23625
$$478$$ −12.0000 −0.548867
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 3.00000 0.136646
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ −2.00000 −0.0908153
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 7.00000 0.316875
$$489$$ 0 0
$$490$$ −9.00000 −0.406579
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ 0 0
$$493$$ −3.00000 −0.135113
$$494$$ 0 0
$$495$$ 12.0000 0.539360
$$496$$ 4.00000 0.179605
$$497$$ −32.0000 −1.43540
$$498$$ 0 0
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 9.00000 0.402492
$$501$$ 0 0
$$502$$ −12.0000 −0.535586
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ −12.0000 −0.534522
$$505$$ −7.00000 −0.311496
$$506$$ −16.0000 −0.711287
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ 43.0000 1.90594 0.952971 0.303062i $$-0.0980090\pi$$
0.952971 + 0.303062i $$0.0980090\pi$$
$$510$$ 0 0
$$511$$ 44.0000 1.94645
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 15.0000 0.661622
$$515$$ 8.00000 0.352522
$$516$$ 0 0
$$517$$ −32.0000 −1.40736
$$518$$ 12.0000 0.527250
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 39.0000 1.70862 0.854311 0.519763i $$-0.173980\pi$$
0.854311 + 0.519763i $$0.173980\pi$$
$$522$$ 3.00000 0.131306
$$523$$ 36.0000 1.57417 0.787085 0.616844i $$-0.211589\pi$$
0.787085 + 0.616844i $$0.211589\pi$$
$$524$$ 20.0000 0.873704
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 9.00000 0.390935
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 4.00000 0.172935
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ −18.0000 −0.776035
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ −25.0000 −1.07483 −0.537417 0.843317i $$-0.680600\pi$$
−0.537417 + 0.843317i $$0.680600\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ 0 0
$$544$$ 3.00000 0.128624
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ 16.0000 0.684111 0.342055 0.939680i $$-0.388877\pi$$
0.342055 + 0.939680i $$0.388877\pi$$
$$548$$ −9.00000 −0.384461
$$549$$ −21.0000 −0.896258
$$550$$ −16.0000 −0.682242
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ −9.00000 −0.382373
$$555$$ 0 0
$$556$$ 16.0000 0.678551
$$557$$ 39.0000 1.65248 0.826242 0.563316i $$-0.190475\pi$$
0.826242 + 0.563316i $$0.190475\pi$$
$$558$$ −12.0000 −0.508001
$$559$$ 0 0
$$560$$ −4.00000 −0.169031
$$561$$ 0 0
$$562$$ −5.00000 −0.210912
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ 1.00000 0.0420703
$$566$$ 28.0000 1.17693
$$567$$ 36.0000 1.51186
$$568$$ −8.00000 −0.335673
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −36.0000 −1.50261
$$575$$ 16.0000 0.667246
$$576$$ −3.00000 −0.125000
$$577$$ 39.0000 1.62359 0.811796 0.583942i $$-0.198490\pi$$
0.811796 + 0.583942i $$0.198490\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 0 0
$$580$$ 1.00000 0.0415227
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −36.0000 −1.49097
$$584$$ 11.0000 0.455183
$$585$$ 0 0
$$586$$ −5.00000 −0.206548
$$587$$ −16.0000 −0.660391 −0.330195 0.943913i $$-0.607115\pi$$
−0.330195 + 0.943913i $$0.607115\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 4.00000 0.164677
$$591$$ 0 0
$$592$$ 3.00000 0.123299
$$593$$ −1.00000 −0.0410651 −0.0205325 0.999789i $$-0.506536\pi$$
−0.0205325 + 0.999789i $$0.506536\pi$$
$$594$$ 0 0
$$595$$ −12.0000 −0.491952
$$596$$ 15.0000 0.614424
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −44.0000 −1.79779 −0.898896 0.438163i $$-0.855629\pi$$
−0.898896 + 0.438163i $$0.855629\pi$$
$$600$$ 0 0
$$601$$ 19.0000 0.775026 0.387513 0.921864i $$-0.373334\pi$$
0.387513 + 0.921864i $$0.373334\pi$$
$$602$$ −32.0000 −1.30422
$$603$$ −12.0000 −0.488678
$$604$$ 12.0000 0.488273
$$605$$ −5.00000 −0.203279
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ −7.00000 −0.283422
$$611$$ 0 0
$$612$$ −9.00000 −0.363803
$$613$$ 11.0000 0.444286 0.222143 0.975014i $$-0.428695\pi$$
0.222143 + 0.975014i $$0.428695\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ 16.0000 0.644658
$$617$$ −29.0000 −1.16750 −0.583748 0.811935i $$-0.698414\pi$$
−0.583748 + 0.811935i $$0.698414\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ −24.0000 −0.961540
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ 11.0000 0.438948
$$629$$ 9.00000 0.358854
$$630$$ 12.0000 0.478091
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ −4.00000 −0.159111
$$633$$ 0 0
$$634$$ 3.00000 0.119145
$$635$$ 8.00000 0.317470
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −4.00000 −0.158362
$$639$$ 24.0000 0.949425
$$640$$ −1.00000 −0.0395285
$$641$$ 19.0000 0.750455 0.375227 0.926933i $$-0.377565\pi$$
0.375227 + 0.926933i $$0.377565\pi$$
$$642$$ 0 0
$$643$$ 44.0000 1.73519 0.867595 0.497271i $$-0.165665\pi$$
0.867595 + 0.497271i $$0.165665\pi$$
$$644$$ −16.0000 −0.630488
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 28.0000 1.10079 0.550397 0.834903i $$-0.314476\pi$$
0.550397 + 0.834903i $$0.314476\pi$$
$$648$$ 9.00000 0.353553
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 8.00000 0.313304
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 0 0
$$655$$ −20.0000 −0.781465
$$656$$ −9.00000 −0.351391
$$657$$ −33.0000 −1.28745
$$658$$ −32.0000 −1.24749
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ −49.0000 −1.90588 −0.952940 0.303160i $$-0.901958\pi$$
−0.952940 + 0.303160i $$0.901958\pi$$
$$662$$ 8.00000 0.310929
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −9.00000 −0.348743
$$667$$ 4.00000 0.154881
$$668$$ 12.0000 0.464294
$$669$$ 0 0
$$670$$ −4.00000 −0.154533
$$671$$ 28.0000 1.08093
$$672$$ 0 0
$$673$$ −29.0000 −1.11787 −0.558934 0.829212i $$-0.688789\pi$$
−0.558934 + 0.829212i $$0.688789\pi$$
$$674$$ 23.0000 0.885927
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 0 0
$$679$$ 8.00000 0.307012
$$680$$ −3.00000 −0.115045
$$681$$ 0 0
$$682$$ 16.0000 0.612672
$$683$$ −44.0000 −1.68361 −0.841807 0.539779i $$-0.818508\pi$$
−0.841807 + 0.539779i $$0.818508\pi$$
$$684$$ 0 0
$$685$$ 9.00000 0.343872
$$686$$ 8.00000 0.305441
$$687$$ 0 0
$$688$$ −8.00000 −0.304997
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 14.0000 0.532200
$$693$$ −48.0000 −1.82337
$$694$$ 12.0000 0.455514
$$695$$ −16.0000 −0.606915
$$696$$ 0 0
$$697$$ −27.0000 −1.02270
$$698$$ −2.00000 −0.0757011
$$699$$ 0 0
$$700$$ −16.0000 −0.604743
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 7.00000 0.263448
$$707$$ 28.0000 1.05305
$$708$$ 0 0
$$709$$ −13.0000 −0.488225 −0.244113 0.969747i $$-0.578497\pi$$
−0.244113 + 0.969747i $$0.578497\pi$$
$$710$$ 8.00000 0.300235
$$711$$ 12.0000 0.450035
$$712$$ −6.00000 −0.224860
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −24.0000 −0.896922
$$717$$ 0 0
$$718$$ 24.0000 0.895672
$$719$$ −28.0000 −1.04422 −0.522112 0.852877i $$-0.674856\pi$$
−0.522112 + 0.852877i $$0.674856\pi$$
$$720$$ 3.00000 0.111803
$$721$$ −32.0000 −1.19174
$$722$$ −19.0000 −0.707107
$$723$$ 0 0
$$724$$ −21.0000 −0.780459
$$725$$ 4.00000 0.148556
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ −11.0000 −0.407128
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ −1.00000 −0.0369358 −0.0184679 0.999829i $$-0.505879\pi$$
−0.0184679 + 0.999829i $$0.505879\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 16.0000 0.589368
$$738$$ 27.0000 0.993884
$$739$$ −24.0000 −0.882854 −0.441427 0.897297i $$-0.645528\pi$$
−0.441427 + 0.897297i $$0.645528\pi$$
$$740$$ −3.00000 −0.110282
$$741$$ 0 0
$$742$$ −36.0000 −1.32160
$$743$$ −8.00000 −0.293492 −0.146746 0.989174i $$-0.546880\pi$$
−0.146746 + 0.989174i $$0.546880\pi$$
$$744$$ 0 0
$$745$$ −15.0000 −0.549557
$$746$$ −13.0000 −0.475964
$$747$$ 0 0
$$748$$ 12.0000 0.438763
$$749$$ −16.0000 −0.584627
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −12.0000 −0.436725
$$756$$ 0 0
$$757$$ 6.00000 0.218074 0.109037 0.994038i $$-0.465223\pi$$
0.109037 + 0.994038i $$0.465223\pi$$
$$758$$ −8.00000 −0.290573
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 0 0
$$763$$ −8.00000 −0.289619
$$764$$ −20.0000 −0.723575
$$765$$ 9.00000 0.325396
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ −16.0000 −0.576600
$$771$$ 0 0
$$772$$ 11.0000 0.395899
$$773$$ 38.0000 1.36677 0.683383 0.730061i $$-0.260508\pi$$
0.683383 + 0.730061i $$0.260508\pi$$
$$774$$ 24.0000 0.862662
$$775$$ −16.0000 −0.574737
$$776$$ 2.00000 0.0717958
$$777$$ 0 0
$$778$$ −9.00000 −0.322666
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ −12.0000 −0.429119
$$783$$ 0 0
$$784$$ 9.00000 0.321429
$$785$$ −11.0000 −0.392607
$$786$$ 0 0
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ 4.00000 0.142314
$$791$$ −4.00000 −0.142224
$$792$$ −12.0000 −0.426401
$$793$$ 0 0
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ −4.00000 −0.141421
$$801$$ 18.0000 0.635999
$$802$$ 3.00000 0.105934
$$803$$ 44.0000 1.55273
$$804$$ 0 0
$$805$$ 16.0000 0.563926
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 7.00000 0.246259
$$809$$ 39.0000 1.37117 0.685583 0.727994i $$-0.259547\pi$$
0.685583 + 0.727994i $$0.259547\pi$$
$$810$$ −9.00000 −0.316228
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ −4.00000 −0.140372
$$813$$ 0 0
$$814$$ 12.0000 0.420600
$$815$$ −8.00000 −0.280228
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 31.0000 1.08389
$$819$$ 0 0
$$820$$ 9.00000 0.314294
$$821$$ 38.0000 1.32621 0.663105 0.748527i $$-0.269238\pi$$
0.663105 + 0.748527i $$0.269238\pi$$
$$822$$ 0 0
$$823$$ 12.0000 0.418294 0.209147 0.977884i $$-0.432931\pi$$
0.209147 + 0.977884i $$0.432931\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ −16.0000 −0.556711
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 12.0000 0.417029
$$829$$ 7.00000 0.243120 0.121560 0.992584i $$-0.461210\pi$$
0.121560 + 0.992584i $$0.461210\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 27.0000 0.935495
$$834$$ 0 0
$$835$$ −12.0000 −0.415277
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 12.0000 0.414533
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ −5.00000 −0.172311
$$843$$ 0 0
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 24.0000 0.825137
$$847$$ 20.0000 0.687208
$$848$$ −9.00000 −0.309061
$$849$$ 0 0
$$850$$ −12.0000 −0.411597
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ 7.00000 0.239675 0.119838 0.992793i $$-0.461763\pi$$
0.119838 + 0.992793i $$0.461763\pi$$
$$854$$ 28.0000 0.958140
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ −17.0000 −0.580709 −0.290354 0.956919i $$-0.593773\pi$$
−0.290354 + 0.956919i $$0.593773\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 8.00000 0.272798
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ 52.0000 1.77010 0.885050 0.465495i $$-0.154124\pi$$
0.885050 + 0.465495i $$0.154124\pi$$
$$864$$ 0 0
$$865$$ −14.0000 −0.476014
$$866$$ −5.00000 −0.169907
$$867$$ 0 0
$$868$$ 16.0000 0.543075
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −2.00000 −0.0677285
$$873$$ −6.00000 −0.203069
$$874$$ 0 0
$$875$$ 36.0000 1.21702
$$876$$ 0 0
$$877$$ 15.0000 0.506514 0.253257 0.967399i $$-0.418498\pi$$
0.253257 + 0.967399i $$0.418498\pi$$
$$878$$ 8.00000 0.269987
$$879$$ 0 0
$$880$$ −4.00000 −0.134840
$$881$$ −9.00000 −0.303218 −0.151609 0.988441i $$-0.548445\pi$$
−0.151609 + 0.988441i $$0.548445\pi$$
$$882$$ −27.0000 −0.909137
$$883$$ 16.0000 0.538443 0.269221 0.963078i $$-0.413234\pi$$
0.269221 + 0.963078i $$0.413234\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ −20.0000 −0.671534 −0.335767 0.941945i $$-0.608996\pi$$
−0.335767 + 0.941945i $$0.608996\pi$$
$$888$$ 0 0
$$889$$ −32.0000 −1.07325
$$890$$ 6.00000 0.201120
$$891$$ 36.0000 1.20605
$$892$$ 12.0000 0.401790
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 4.00000 0.133631
$$897$$ 0 0
$$898$$ 34.0000 1.13459
$$899$$ −4.00000 −0.133407
$$900$$ 12.0000 0.400000
$$901$$ −27.0000 −0.899500
$$902$$ −36.0000 −1.19867
$$903$$ 0 0
$$904$$ −1.00000 −0.0332595
$$905$$ 21.0000 0.698064
$$906$$ 0 0
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ 24.0000 0.796468
$$909$$ −21.0000 −0.696526
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 31.0000 1.02539
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 80.0000 2.64183
$$918$$ 0 0
$$919$$ −48.0000 −1.58337 −0.791687 0.610927i $$-0.790797\pi$$
−0.791687 + 0.610927i $$0.790797\pi$$
$$920$$ 4.00000 0.131876
$$921$$ 0 0
$$922$$ −33.0000 −1.08680
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −12.0000 −0.394558
$$926$$ 16.0000 0.525793
$$927$$ 24.0000 0.788263
$$928$$ −1.00000 −0.0328266
$$929$$ −21.0000 −0.688988 −0.344494 0.938789i $$-0.611949\pi$$
−0.344494 + 0.938789i $$0.611949\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 10.0000 0.327561
$$933$$ 0 0
$$934$$ 20.0000 0.654420
$$935$$ −12.0000 −0.392442
$$936$$ 0 0
$$937$$ −21.0000 −0.686040 −0.343020 0.939328i $$-0.611450\pi$$
−0.343020 + 0.939328i $$0.611450\pi$$
$$938$$ 16.0000 0.522419
$$939$$ 0 0
$$940$$ 8.00000 0.260931
$$941$$ 46.0000 1.49956 0.749779 0.661689i $$-0.230160\pi$$
0.749779 + 0.661689i $$0.230160\pi$$
$$942$$ 0 0
$$943$$ 36.0000 1.17232
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ −32.0000 −1.04041
$$947$$ 24.0000 0.779895 0.389948 0.920837i $$-0.372493\pi$$
0.389948 + 0.920837i $$0.372493\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 12.0000 0.388922
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ 27.0000 0.874157
$$955$$ 20.0000 0.647185
$$956$$ −12.0000 −0.388108
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ 3.00000 0.0966235
$$965$$ −11.0000 −0.354103
$$966$$ 0 0
$$967$$ −40.0000 −1.28631 −0.643157 0.765735i $$-0.722376\pi$$
−0.643157 + 0.765735i $$0.722376\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ −2.00000 −0.0642161
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 64.0000 2.05175
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ 7.00000 0.224065
$$977$$ −45.0000 −1.43968 −0.719839 0.694141i $$-0.755784\pi$$
−0.719839 + 0.694141i $$0.755784\pi$$
$$978$$ 0 0
$$979$$ −24.0000 −0.767043
$$980$$ −9.00000 −0.287494
$$981$$ 6.00000 0.191565
$$982$$ −8.00000 −0.255290
$$983$$ −52.0000 −1.65854 −0.829271 0.558846i $$-0.811244\pi$$
−0.829271 + 0.558846i $$0.811244\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ −3.00000 −0.0955395
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 32.0000 1.01754
$$990$$ 12.0000 0.381385
$$991$$ −24.0000 −0.762385 −0.381193 0.924496i $$-0.624487\pi$$
−0.381193 + 0.924496i $$0.624487\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 0 0
$$994$$ −32.0000 −1.01498
$$995$$ 4.00000 0.126809
$$996$$ 0 0
$$997$$ −25.0000 −0.791758 −0.395879 0.918303i $$-0.629560\pi$$
−0.395879 + 0.918303i $$0.629560\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ 0 0
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.e.1.1 1
3.2 odd 2 3042.2.a.e.1.1 1
4.3 odd 2 2704.2.a.h.1.1 1
5.4 even 2 8450.2.a.f.1.1 1
13.2 odd 12 338.2.e.b.147.2 4
13.3 even 3 26.2.c.a.9.1 yes 2
13.4 even 6 338.2.c.e.315.1 2
13.5 odd 4 338.2.b.b.337.1 2
13.6 odd 12 338.2.e.b.23.1 4
13.7 odd 12 338.2.e.b.23.2 4
13.8 odd 4 338.2.b.b.337.2 2
13.9 even 3 26.2.c.a.3.1 2
13.10 even 6 338.2.c.e.191.1 2
13.11 odd 12 338.2.e.b.147.1 4
13.12 even 2 338.2.a.c.1.1 1
39.5 even 4 3042.2.b.e.1351.2 2
39.8 even 4 3042.2.b.e.1351.1 2
39.29 odd 6 234.2.h.c.217.1 2
39.35 odd 6 234.2.h.c.55.1 2
39.38 odd 2 3042.2.a.k.1.1 1
52.3 odd 6 208.2.i.b.113.1 2
52.31 even 4 2704.2.f.g.337.2 2
52.35 odd 6 208.2.i.b.81.1 2
52.47 even 4 2704.2.f.g.337.1 2
52.51 odd 2 2704.2.a.i.1.1 1
65.3 odd 12 650.2.o.c.399.2 4
65.9 even 6 650.2.e.c.601.1 2
65.22 odd 12 650.2.o.c.549.2 4
65.29 even 6 650.2.e.c.451.1 2
65.42 odd 12 650.2.o.c.399.1 4
65.48 odd 12 650.2.o.c.549.1 4
65.64 even 2 8450.2.a.s.1.1 1
91.3 odd 6 1274.2.h.a.373.1 2
91.9 even 3 1274.2.h.b.263.1 2
91.16 even 3 1274.2.e.n.165.1 2
91.48 odd 6 1274.2.g.a.393.1 2
91.55 odd 6 1274.2.g.a.295.1 2
91.61 odd 6 1274.2.h.a.263.1 2
91.68 odd 6 1274.2.e.m.165.1 2
91.74 even 3 1274.2.e.n.471.1 2
91.81 even 3 1274.2.h.b.373.1 2
91.87 odd 6 1274.2.e.m.471.1 2
104.3 odd 6 832.2.i.f.321.1 2
104.29 even 6 832.2.i.e.321.1 2
104.35 odd 6 832.2.i.f.705.1 2
104.61 even 6 832.2.i.e.705.1 2
156.35 even 6 1872.2.t.k.289.1 2
156.107 even 6 1872.2.t.k.1153.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.c.a.3.1 2 13.9 even 3
26.2.c.a.9.1 yes 2 13.3 even 3
208.2.i.b.81.1 2 52.35 odd 6
208.2.i.b.113.1 2 52.3 odd 6
234.2.h.c.55.1 2 39.35 odd 6
234.2.h.c.217.1 2 39.29 odd 6
338.2.a.c.1.1 1 13.12 even 2
338.2.a.e.1.1 1 1.1 even 1 trivial
338.2.b.b.337.1 2 13.5 odd 4
338.2.b.b.337.2 2 13.8 odd 4
338.2.c.e.191.1 2 13.10 even 6
338.2.c.e.315.1 2 13.4 even 6
338.2.e.b.23.1 4 13.6 odd 12
338.2.e.b.23.2 4 13.7 odd 12
338.2.e.b.147.1 4 13.11 odd 12
338.2.e.b.147.2 4 13.2 odd 12
650.2.e.c.451.1 2 65.29 even 6
650.2.e.c.601.1 2 65.9 even 6
650.2.o.c.399.1 4 65.42 odd 12
650.2.o.c.399.2 4 65.3 odd 12
650.2.o.c.549.1 4 65.48 odd 12
650.2.o.c.549.2 4 65.22 odd 12
832.2.i.e.321.1 2 104.29 even 6
832.2.i.e.705.1 2 104.61 even 6
832.2.i.f.321.1 2 104.3 odd 6
832.2.i.f.705.1 2 104.35 odd 6
1274.2.e.m.165.1 2 91.68 odd 6
1274.2.e.m.471.1 2 91.87 odd 6
1274.2.e.n.165.1 2 91.16 even 3
1274.2.e.n.471.1 2 91.74 even 3
1274.2.g.a.295.1 2 91.55 odd 6
1274.2.g.a.393.1 2 91.48 odd 6
1274.2.h.a.263.1 2 91.61 odd 6
1274.2.h.a.373.1 2 91.3 odd 6
1274.2.h.b.263.1 2 91.9 even 3
1274.2.h.b.373.1 2 91.81 even 3
1872.2.t.k.289.1 2 156.35 even 6
1872.2.t.k.1153.1 2 156.107 even 6
2704.2.a.h.1.1 1 4.3 odd 2
2704.2.a.i.1.1 1 52.51 odd 2
2704.2.f.g.337.1 2 52.47 even 4
2704.2.f.g.337.2 2 52.31 even 4
3042.2.a.e.1.1 1 3.2 odd 2
3042.2.a.k.1.1 1 39.38 odd 2
3042.2.b.e.1351.1 2 39.8 even 4
3042.2.b.e.1351.2 2 39.5 even 4
8450.2.a.f.1.1 1 5.4 even 2
8450.2.a.s.1.1 1 65.64 even 2