Properties

 Label 70.2.a.a.1.1 Level $70$ Weight $2$ Character 70.1 Self dual yes Analytic conductor $0.559$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [70,2,Mod(1,70)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(70, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("70.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$70 = 2 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 70.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$0.558952814149$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 70.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} -6.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} -1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{28} +6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{35} -3.00000 q^{36} -10.0000 q^{37} -1.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} +4.00000 q^{44} +3.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} -2.00000 q^{53} -4.00000 q^{55} -1.00000 q^{56} +6.00000 q^{58} -8.00000 q^{59} -14.0000 q^{61} +8.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -12.0000 q^{67} +2.00000 q^{68} +1.00000 q^{70} -16.0000 q^{71} -3.00000 q^{72} +2.00000 q^{73} -10.0000 q^{74} -4.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} +2.00000 q^{82} +8.00000 q^{83} -2.00000 q^{85} +4.00000 q^{86} +4.00000 q^{88} +10.0000 q^{89} +3.00000 q^{90} +6.00000 q^{91} +8.00000 q^{94} +2.00000 q^{97} +1.00000 q^{98} -12.0000 q^{99} +O(q^{100})$$

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
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 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
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ −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
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −6.00000 −1.17670
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ 1.00000 0.169031
$$36$$ −3.00000 −0.500000
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 3.00000 0.447214
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ −6.00000 −0.832050
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ 6.00000 0.787839
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 3.00000 0.377964
$$64$$ 1.00000 0.125000
$$65$$ 6.00000 0.744208
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ 1.00000 0.119523
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 9.00000 1.00000
$$82$$ 2.00000 0.220863
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 3.00000 0.316228
$$91$$ 6.00000 0.628971
$$92$$ 0 0
$$93$$ 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 $$0.467625\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −12.0000 −1.20605
$$100$$ 1.00000 0.100000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 18.0000 1.66410
$$118$$ −8.00000 −0.736460
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −14.0000 −1.26750
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ −1.00000 −0.0894427
$$126$$ 3.00000 0.267261
$$127$$ −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
$$130$$ 6.00000 0.526235
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 0 0
$$142$$ −16.0000 −1.34269
$$143$$ −24.0000 −2.00698
$$144$$ −3.00000 −0.250000
$$145$$ −6.00000 −0.498273
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ −10.0000 −0.821995
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ −4.00000 −0.322329
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ 9.00000 0.707107
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 8.00000 0.620920
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ −2.00000 −0.153393
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ 0 0
$$175$$ −1.00000 −0.0755929
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 3.00000 0.223607
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 6.00000 0.444750
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 10.0000 0.735215
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 14.0000 0.997459 0.498729 0.866758i $$-0.333800\pi$$
0.498729 + 0.866758i $$0.333800\pi$$
$$198$$ −12.0000 −0.852803
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ −6.00000 −0.422159
$$203$$ −6.00000 −0.421117
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ 16.0000 1.11477
$$207$$ 0 0
$$208$$ −6.00000 −0.416025
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ −8.00000 −0.543075
$$218$$ 6.00000 0.406371
$$219$$ 0 0
$$220$$ −4.00000 −0.269680
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −3.00000 −0.200000
$$226$$ 2.00000 0.133038
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 18.0000 1.17670
$$235$$ −8.00000 −0.521862
$$236$$ −8.00000 −0.520756
$$237$$ 0 0
$$238$$ −2.00000 −0.129641
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ −14.0000 −0.896258
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 8.00000 0.508001
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 3.00000 0.188982
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ 0 0
$$259$$ 10.0000 0.621370
$$260$$ 6.00000 0.372104
$$261$$ −18.0000 −1.11417
$$262$$ −16.0000 −0.988483
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 0 0
$$265$$ 2.00000 0.122859
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 4.00000 0.241209
$$276$$ 0 0
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ 16.0000 0.959616
$$279$$ −24.0000 −1.43684
$$280$$ 1.00000 0.0597614
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ 32.0000 1.90220 0.951101 0.308879i $$-0.0999539\pi$$
0.951101 + 0.308879i $$0.0999539\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ 0 0
$$286$$ −24.0000 −1.41915
$$287$$ −2.00000 −0.118056
$$288$$ −3.00000 −0.176777
$$289$$ −13.0000 −0.764706
$$290$$ −6.00000 −0.352332
$$291$$ 0 0
$$292$$ 2.00000 0.117041
$$293$$ 10.0000 0.584206 0.292103 0.956387i $$-0.405645\pi$$
0.292103 + 0.956387i $$0.405645\pi$$
$$294$$ 0 0
$$295$$ 8.00000 0.465778
$$296$$ −10.0000 −0.581238
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ 8.00000 0.460348
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 14.0000 0.801638
$$306$$ −6.00000 −0.342997
$$307$$ −8.00000 −0.456584 −0.228292 0.973593i $$-0.573314\pi$$
−0.228292 + 0.973593i $$0.573314\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ 0 0
$$310$$ −8.00000 −0.454369
$$311$$ −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 $$0.713581\pi$$
$$314$$ 10.0000 0.564333
$$315$$ −3.00000 −0.169031
$$316$$ −8.00000 −0.450035
$$317$$ 22.0000 1.23564 0.617822 0.786318i $$-0.288015\pi$$
0.617822 + 0.786318i $$0.288015\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 9.00000 0.500000
$$325$$ −6.00000 −0.332820
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ 2.00000 0.110432
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 8.00000 0.439057
$$333$$ 30.0000 1.64399
$$334$$ 0 0
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 23.0000 1.25104
$$339$$ 0 0
$$340$$ −2.00000 −0.108465
$$341$$ 32.0000 1.73290
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ −22.0000 −1.18273
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 3.00000 0.158114
$$361$$ −19.0000 −1.00000
$$362$$ −14.0000 −0.735824
$$363$$ 0 0
$$364$$ 6.00000 0.314485
$$365$$ −2.00000 −0.104685
$$366$$ 0 0
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 10.0000 0.519875
$$371$$ 2.00000 0.103835
$$372$$ 0 0
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ −36.0000 −1.85409
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 24.0000 1.22795
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 4.00000 0.203859
$$386$$ 2.00000 0.101797
$$387$$ −12.0000 −0.609994
$$388$$ 2.00000 0.101535
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ 14.0000 0.705310
$$395$$ 8.00000 0.402524
$$396$$ −12.0000 −0.603023
$$397$$ 10.0000 0.501886 0.250943 0.968002i $$-0.419259\pi$$
0.250943 + 0.968002i $$0.419259\pi$$
$$398$$ −16.0000 −0.802008
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 0 0
$$403$$ −48.0000 −2.39105
$$404$$ −6.00000 −0.298511
$$405$$ −9.00000 −0.447214
$$406$$ −6.00000 −0.297775
$$407$$ −40.0000 −1.98273
$$408$$ 0 0
$$409$$ −30.0000 −1.48340 −0.741702 0.670729i $$-0.765981\pi$$
−0.741702 + 0.670729i $$0.765981\pi$$
$$410$$ −2.00000 −0.0987730
$$411$$ 0 0
$$412$$ 16.0000 0.788263
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ −8.00000 −0.392705
$$416$$ −6.00000 −0.294174
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 4.00000 0.194717
$$423$$ −24.0000 −1.16692
$$424$$ −2.00000 −0.0971286
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 14.0000 0.677507
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 0 0
$$436$$ 6.00000 0.287348
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ −4.00000 −0.190693
$$441$$ −3.00000 −0.142857
$$442$$ −12.0000 −0.570782
$$443$$ −20.0000 −0.950229 −0.475114 0.879924i $$-0.657593\pi$$
−0.475114 + 0.879924i $$0.657593\pi$$
$$444$$ 0 0
$$445$$ −10.0000 −0.474045
$$446$$ 16.0000 0.757622
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ −3.00000 −0.141421
$$451$$ 8.00000 0.376705
$$452$$ 2.00000 0.0940721
$$453$$ 0 0
$$454$$ 8.00000 0.375459
$$455$$ −6.00000 −0.281284
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10.0000 0.465746 0.232873 0.972507i $$-0.425187\pi$$
0.232873 + 0.972507i $$0.425187\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ −40.0000 −1.85098 −0.925490 0.378773i $$-0.876346\pi$$
−0.925490 + 0.378773i $$0.876346\pi$$
$$468$$ 18.0000 0.832050
$$469$$ 12.0000 0.554109
$$470$$ −8.00000 −0.369012
$$471$$ 0 0
$$472$$ −8.00000 −0.368230
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ 6.00000 0.274721
$$478$$ 16.0000 0.731823
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ 60.0000 2.73576
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ −2.00000 −0.0908153
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ −14.0000 −0.633750
$$489$$ 0 0
$$490$$ −1.00000 −0.0451754
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 12.0000 0.540453
$$494$$ 0 0
$$495$$ 12.0000 0.539360
$$496$$ 8.00000 0.359211
$$497$$ 16.0000 0.717698
$$498$$ 0 0
$$499$$ −12.0000 −0.537194 −0.268597 0.963253i $$-0.586560\pi$$
−0.268597 + 0.963253i $$0.586560\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 3.00000 0.133631
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ −2.00000 −0.0884748
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −22.0000 −0.970378
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ 32.0000 1.40736
$$518$$ 10.0000 0.439375
$$519$$ 0 0
$$520$$ 6.00000 0.263117
$$521$$ 2.00000 0.0876216 0.0438108 0.999040i $$-0.486050\pi$$
0.0438108 + 0.999040i $$0.486050\pi$$
$$522$$ −18.0000 −0.787839
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ −16.0000 −0.698963
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ 16.0000 0.696971
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 2.00000 0.0868744
$$531$$ 24.0000 1.04151
$$532$$ 0 0
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ −6.00000 −0.258678
$$539$$ 4.00000 0.172292
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 2.00000 0.0857493
$$545$$ −6.00000 −0.257012
$$546$$ 0 0
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 42.0000 1.79252
$$550$$ 4.00000 0.170561
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ −18.0000 −0.764747
$$555$$ 0 0
$$556$$ 16.0000 0.678551
$$557$$ 22.0000 0.932170 0.466085 0.884740i $$-0.345664\pi$$
0.466085 + 0.884740i $$0.345664\pi$$
$$558$$ −24.0000 −1.01600
$$559$$ −24.0000 −1.01509
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ 26.0000 1.09674
$$563$$ −16.0000 −0.674320 −0.337160 0.941447i $$-0.609466\pi$$
−0.337160 + 0.941447i $$0.609466\pi$$
$$564$$ 0 0
$$565$$ −2.00000 −0.0841406
$$566$$ 32.0000 1.34506
$$567$$ −9.00000 −0.377964
$$568$$ −16.0000 −0.671345
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ −24.0000 −1.00349
$$573$$ 0 0
$$574$$ −2.00000 −0.0834784
$$575$$ 0 0
$$576$$ −3.00000 −0.125000
$$577$$ 10.0000 0.416305 0.208153 0.978096i $$-0.433255\pi$$
0.208153 + 0.978096i $$0.433255\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ −6.00000 −0.249136
$$581$$ −8.00000 −0.331896
$$582$$ 0 0
$$583$$ −8.00000 −0.331326
$$584$$ 2.00000 0.0827606
$$585$$ −18.0000 −0.744208
$$586$$ 10.0000 0.413096
$$587$$ 8.00000 0.330195 0.165098 0.986277i $$-0.447206\pi$$
0.165098 + 0.986277i $$0.447206\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 8.00000 0.329355
$$591$$ 0 0
$$592$$ −10.0000 −0.410997
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 2.00000 0.0819920
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ −4.00000 −0.163028
$$603$$ 36.0000 1.46603
$$604$$ 8.00000 0.325515
$$605$$ −5.00000 −0.203279
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 14.0000 0.566843
$$611$$ −48.0000 −1.94187
$$612$$ −6.00000 −0.242536
$$613$$ 6.00000 0.242338 0.121169 0.992632i $$-0.461336\pi$$
0.121169 + 0.992632i $$0.461336\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 0 0
$$616$$ −4.00000 −0.161165
$$617$$ −22.0000 −0.885687 −0.442843 0.896599i $$-0.646030\pi$$
−0.442843 + 0.896599i $$0.646030\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ −8.00000 −0.321288
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ −10.0000 −0.400642
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ 10.0000 0.399043
$$629$$ −20.0000 −0.797452
$$630$$ −3.00000 −0.119523
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 0 0
$$634$$ 22.0000 0.873732
$$635$$ 8.00000 0.317470
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 24.0000 0.950169
$$639$$ 48.0000 1.89885
$$640$$ −1.00000 −0.0395285
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ −32.0000 −1.26196 −0.630978 0.775800i $$-0.717346\pi$$
−0.630978 + 0.775800i $$0.717346\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 16.0000 0.629025 0.314512 0.949253i $$-0.398159\pi$$
0.314512 + 0.949253i $$0.398159\pi$$
$$648$$ 9.00000 0.353553
$$649$$ −32.0000 −1.25611
$$650$$ −6.00000 −0.235339
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ 0 0
$$655$$ 16.0000 0.625172
$$656$$ 2.00000 0.0780869
$$657$$ −6.00000 −0.234082
$$658$$ −8.00000 −0.311872
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ 8.00000 0.310460
$$665$$ 0 0
$$666$$ 30.0000 1.16248
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 12.0000 0.463600
$$671$$ −56.0000 −2.16186
$$672$$ 0 0
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 0 0
$$679$$ −2.00000 −0.0767530
$$680$$ −2.00000 −0.0766965
$$681$$ 0 0
$$682$$ 32.0000 1.22534
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ 6.00000 0.229248
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −32.0000 −1.21734 −0.608669 0.793424i $$-0.708296\pi$$
−0.608669 + 0.793424i $$0.708296\pi$$
$$692$$ −22.0000 −0.836315
$$693$$ 12.0000 0.455842
$$694$$ 4.00000 0.151838
$$695$$ −16.0000 −0.606915
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ 10.0000 0.378506
$$699$$ 0 0
$$700$$ −1.00000 −0.0377964
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 6.00000 0.225653
$$708$$ 0 0
$$709$$ −2.00000 −0.0751116 −0.0375558 0.999295i $$-0.511957\pi$$
−0.0375558 + 0.999295i $$0.511957\pi$$
$$710$$ 16.0000 0.600469
$$711$$ 24.0000 0.900070
$$712$$ 10.0000 0.374766
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 24.0000 0.897549
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 8.00000 0.298557
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 3.00000 0.111803
$$721$$ −16.0000 −0.595871
$$722$$ −19.0000 −0.707107
$$723$$ 0 0
$$724$$ −14.0000 −0.520306
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 6.00000 0.222375
$$729$$ −27.0000 −1.00000
$$730$$ −2.00000 −0.0740233
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −48.0000 −1.76810
$$738$$ −6.00000 −0.220863
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 10.0000 0.367607
$$741$$ 0 0
$$742$$ 2.00000 0.0734223
$$743$$ 48.0000 1.76095 0.880475 0.474093i $$-0.157224\pi$$
0.880475 + 0.474093i $$0.157224\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 14.0000 0.512576
$$747$$ −24.0000 −0.878114
$$748$$ 8.00000 0.292509
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 0 0
$$754$$ −36.0000 −1.31104
$$755$$ −8.00000 −0.291150
$$756$$ 0 0
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ −12.0000 −0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ 0 0
$$763$$ −6.00000 −0.217215
$$764$$ 24.0000 0.868290
$$765$$ 6.00000 0.216930
$$766$$ 24.0000 0.867155
$$767$$ 48.0000 1.73318
$$768$$ 0 0
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 4.00000 0.144150
$$771$$ 0 0
$$772$$ 2.00000 0.0719816
$$773$$ 42.0000 1.51064 0.755318 0.655359i $$-0.227483\pi$$
0.755318 + 0.655359i $$0.227483\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 8.00000 0.287368
$$776$$ 2.00000 0.0717958
$$777$$ 0 0
$$778$$ 14.0000 0.501924
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −64.0000 −2.29010
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ −10.0000 −0.356915
$$786$$ 0 0
$$787$$ −8.00000 −0.285169 −0.142585 0.989783i $$-0.545541\pi$$
−0.142585 + 0.989783i $$0.545541\pi$$
$$788$$ 14.0000 0.498729
$$789$$ 0 0
$$790$$ 8.00000 0.284627
$$791$$ −2.00000 −0.0711118
$$792$$ −12.0000 −0.426401
$$793$$ 84.0000 2.98293
$$794$$ 10.0000 0.354887
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 1.00000 0.0353553
$$801$$ −30.0000 −1.06000
$$802$$ −14.0000 −0.494357
$$803$$ 8.00000 0.282314
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −48.0000 −1.69073
$$807$$ 0 0
$$808$$ −6.00000 −0.211079
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ −9.00000 −0.316228
$$811$$ 8.00000 0.280918 0.140459 0.990086i $$-0.455142\pi$$
0.140459 + 0.990086i $$0.455142\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ 0 0
$$814$$ −40.0000 −1.40200
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −30.0000 −1.04893
$$819$$ −18.0000 −0.628971
$$820$$ −2.00000 −0.0698430
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ −8.00000 −0.278862 −0.139431 0.990232i $$-0.544527\pi$$
−0.139431 + 0.990232i $$0.544527\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ 44.0000 1.53003 0.765015 0.644013i $$-0.222732\pi$$
0.765015 + 0.644013i $$0.222732\pi$$
$$828$$ 0 0
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ −8.00000 −0.277684
$$831$$ 0 0
$$832$$ −6.00000 −0.208013
$$833$$ 2.00000 0.0692959
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 24.0000 0.829066
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −10.0000 −0.344623
$$843$$ 0 0
$$844$$ 4.00000 0.137686
$$845$$ −23.0000 −0.791224
$$846$$ −24.0000 −0.825137
$$847$$ −5.00000 −0.171802
$$848$$ −2.00000 −0.0686803
$$849$$ 0 0
$$850$$ 2.00000 0.0685994
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −46.0000 −1.57501 −0.787505 0.616308i $$-0.788628\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ 14.0000 0.479070
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ −54.0000 −1.84460 −0.922302 0.386469i $$-0.873695\pi$$
−0.922302 + 0.386469i $$0.873695\pi$$
$$858$$ 0 0
$$859$$ −48.0000 −1.63774 −0.818869 0.573980i $$-0.805399\pi$$
−0.818869 + 0.573980i $$0.805399\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 0 0
$$865$$ 22.0000 0.748022
$$866$$ 2.00000 0.0679628
$$867$$ 0 0
$$868$$ −8.00000 −0.271538
$$869$$ −32.0000 −1.08553
$$870$$ 0 0
$$871$$ 72.0000 2.43963
$$872$$ 6.00000 0.203186
$$873$$ −6.00000 −0.203069
$$874$$ 0 0
$$875$$ 1.00000 0.0338062
$$876$$ 0 0
$$877$$ 14.0000 0.472746 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ 0 0
$$880$$ −4.00000 −0.134840
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ −12.0000 −0.403832 −0.201916 0.979403i $$-0.564717\pi$$
−0.201916 + 0.979403i $$0.564717\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ −20.0000 −0.671913
$$887$$ −16.0000 −0.537227 −0.268614 0.963248i $$-0.586566\pi$$
−0.268614 + 0.963248i $$0.586566\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ −10.0000 −0.335201
$$891$$ 36.0000 1.20605
$$892$$ 16.0000 0.535720
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 12.0000 0.401116
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ 48.0000 1.60089
$$900$$ −3.00000 −0.100000
$$901$$ −4.00000 −0.133259
$$902$$ 8.00000 0.266371
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ 14.0000 0.465376
$$906$$ 0 0
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ 8.00000 0.265489
$$909$$ 18.0000 0.597022
$$910$$ −6.00000 −0.198898
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 32.0000 1.05905
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 16.0000 0.528367
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 10.0000 0.329332
$$923$$ 96.0000 3.15988
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ −16.0000 −0.525793
$$927$$ −48.0000 −1.57653
$$928$$ 6.00000 0.196960
$$929$$ 58.0000 1.90292 0.951459 0.307775i $$-0.0995844\pi$$
0.951459 + 0.307775i $$0.0995844\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −6.00000 −0.196537
$$933$$ 0 0
$$934$$ −40.0000 −1.30884
$$935$$ −8.00000 −0.261628
$$936$$ 18.0000 0.588348
$$937$$ 50.0000 1.63343 0.816714 0.577042i $$-0.195793\pi$$
0.816714 + 0.577042i $$0.195793\pi$$
$$938$$ 12.0000 0.391814
$$939$$ 0 0
$$940$$ −8.00000 −0.260931
$$941$$ 2.00000 0.0651981 0.0325991 0.999469i $$-0.489622\pi$$
0.0325991 + 0.999469i $$0.489622\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 44.0000 1.42981 0.714904 0.699223i $$-0.246470\pi$$
0.714904 + 0.699223i $$0.246470\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −2.00000 −0.0648204
$$953$$ −54.0000 −1.74923 −0.874616 0.484817i $$-0.838886\pi$$
−0.874616 + 0.484817i $$0.838886\pi$$
$$954$$ 6.00000 0.194257
$$955$$ −24.0000 −0.776622
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 60.0000 1.93448
$$963$$ −36.0000 −1.16008
$$964$$ 10.0000 0.322078
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ 16.0000 0.514525 0.257263 0.966342i $$-0.417179\pi$$
0.257263 + 0.966342i $$0.417179\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ −2.00000 −0.0642161
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ 0 0
$$973$$ −16.0000 −0.512936
$$974$$ 32.0000 1.02535
$$975$$ 0 0
$$976$$ −14.0000 −0.448129
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 0 0
$$979$$ 40.0000 1.27841
$$980$$ −1.00000 −0.0319438
$$981$$ −18.0000 −0.574696
$$982$$ 12.0000 0.382935
$$983$$ −16.0000 −0.510321 −0.255160 0.966899i $$-0.582128\pi$$
−0.255160 + 0.966899i $$0.582128\pi$$
$$984$$ 0 0
$$985$$ −14.0000 −0.446077
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 12.0000 0.381385
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 8.00000 0.254000
$$993$$ 0 0
$$994$$ 16.0000 0.507489
$$995$$ 16.0000 0.507234
$$996$$ 0 0
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ −12.0000 −0.379853
$$999$$ 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 70.2.a.a.1.1 1
3.2 odd 2 630.2.a.d.1.1 1
4.3 odd 2 560.2.a.d.1.1 1
5.2 odd 4 350.2.c.b.99.2 2
5.3 odd 4 350.2.c.b.99.1 2
5.4 even 2 350.2.a.b.1.1 1
7.2 even 3 490.2.e.d.361.1 2
7.3 odd 6 490.2.e.c.471.1 2
7.4 even 3 490.2.e.d.471.1 2
7.5 odd 6 490.2.e.c.361.1 2
7.6 odd 2 490.2.a.h.1.1 1
8.3 odd 2 2240.2.a.q.1.1 1
8.5 even 2 2240.2.a.n.1.1 1
11.10 odd 2 8470.2.a.j.1.1 1
12.11 even 2 5040.2.a.bm.1.1 1
15.2 even 4 3150.2.g.c.2899.1 2
15.8 even 4 3150.2.g.c.2899.2 2
15.14 odd 2 3150.2.a.bj.1.1 1
20.3 even 4 2800.2.g.n.449.1 2
20.7 even 4 2800.2.g.n.449.2 2
20.19 odd 2 2800.2.a.m.1.1 1
21.20 even 2 4410.2.a.b.1.1 1
28.27 even 2 3920.2.a.t.1.1 1
35.13 even 4 2450.2.c.k.99.1 2
35.27 even 4 2450.2.c.k.99.2 2
35.34 odd 2 2450.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.a.a.1.1 1 1.1 even 1 trivial
350.2.a.b.1.1 1 5.4 even 2
350.2.c.b.99.1 2 5.3 odd 4
350.2.c.b.99.2 2 5.2 odd 4
490.2.a.h.1.1 1 7.6 odd 2
490.2.e.c.361.1 2 7.5 odd 6
490.2.e.c.471.1 2 7.3 odd 6
490.2.e.d.361.1 2 7.2 even 3
490.2.e.d.471.1 2 7.4 even 3
560.2.a.d.1.1 1 4.3 odd 2
630.2.a.d.1.1 1 3.2 odd 2
2240.2.a.n.1.1 1 8.5 even 2
2240.2.a.q.1.1 1 8.3 odd 2
2450.2.a.l.1.1 1 35.34 odd 2
2450.2.c.k.99.1 2 35.13 even 4
2450.2.c.k.99.2 2 35.27 even 4
2800.2.a.m.1.1 1 20.19 odd 2
2800.2.g.n.449.1 2 20.3 even 4
2800.2.g.n.449.2 2 20.7 even 4
3150.2.a.bj.1.1 1 15.14 odd 2
3150.2.g.c.2899.1 2 15.2 even 4
3150.2.g.c.2899.2 2 15.8 even 4
3920.2.a.t.1.1 1 28.27 even 2
4410.2.a.b.1.1 1 21.20 even 2
5040.2.a.bm.1.1 1 12.11 even 2
8470.2.a.j.1.1 1 11.10 odd 2