# Properties

 Label 14.2.a.a.1.1 Level $14$ Weight $2$ Character 14.1 Self dual yes Analytic conductor $0.112$ 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] = [14,2,Mod(1,14)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(14, 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("14.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 14.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: $$0.111790562830$$ 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 $$\chi$$ $$=$$ 14.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -2.00000 q^{21} +2.00000 q^{24} -5.00000 q^{25} +4.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} +8.00000 q^{39} +6.00000 q^{41} +2.00000 q^{42} +8.00000 q^{43} -12.0000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +5.00000 q^{50} -12.0000 q^{51} -4.00000 q^{52} +6.00000 q^{53} -4.00000 q^{54} -1.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} -6.00000 q^{59} +8.00000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} +6.00000 q^{68} -1.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} +10.0000 q^{75} +2.00000 q^{76} -8.00000 q^{78} +8.00000 q^{79} -11.0000 q^{81} -6.00000 q^{82} -6.00000 q^{83} -2.00000 q^{84} -8.00000 q^{86} +12.0000 q^{87} -6.00000 q^{89} -4.00000 q^{91} +8.00000 q^{93} +12.0000 q^{94} +2.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 2.00000 0.816497
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ −2.00000 −0.577350
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 2.00000 0.408248
$$25$$ −5.00000 −1.00000
$$26$$ 4.00000 0.784465
$$27$$ 4.00000 0.769800
$$28$$ 1.00000 0.188982
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 8.00000 1.28103
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 2.00000 0.308607
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ −2.00000 −0.288675
$$49$$ 1.00000 0.142857
$$50$$ 5.00000 0.707107
$$51$$ −12.0000 −1.68034
$$52$$ −4.00000 −0.554700
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ −4.00000 −0.529813
$$58$$ 6.00000 0.787839
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 10.0000 1.15470
$$76$$ 2.00000 0.229416
$$77$$ 0 0
$$78$$ −8.00000 −0.905822
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ −6.00000 −0.662589
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 12.0000 1.28654
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ 2.00000 0.204124
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ −5.00000 −0.500000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 12.0000 1.18818
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 4.00000 0.384900
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 1.00000 0.0944911
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ −4.00000 −0.369800
$$118$$ 6.00000 0.552345
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ −8.00000 −0.724286
$$123$$ −12.0000 −1.08200
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ −1.00000 −0.0890871
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −16.0000 −1.40872
$$130$$ 0 0
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ 0 0
$$133$$ 2.00000 0.173422
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 0 0
$$141$$ 24.0000 2.02116
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ −2.00000 −0.164957
$$148$$ 2.00000 0.164399
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ −10.0000 −0.816497
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 8.00000 0.640513
$$157$$ −4.00000 −0.319235 −0.159617 0.987179i $$-0.551026\pi$$
−0.159617 + 0.987179i $$0.551026\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 11.0000 0.864242
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 8.00000 0.609994
$$173$$ −12.0000 −0.912343 −0.456172 0.889892i $$-0.650780\pi$$
−0.456172 + 0.889892i $$0.650780\pi$$
$$174$$ −12.0000 −0.909718
$$175$$ −5.00000 −0.377964
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 6.00000 0.449719
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ 4.00000 0.296500
$$183$$ −16.0000 −1.18275
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ −12.0000 −0.875190
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ −2.00000 −0.144338
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 5.00000 0.353553
$$201$$ 8.00000 0.564276
$$202$$ 0 0
$$203$$ −6.00000 −0.421117
$$204$$ −12.0000 −0.840168
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ −4.00000 −0.277350
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ −4.00000 −0.272166
$$217$$ −4.00000 −0.271538
$$218$$ −2.00000 −0.135457
$$219$$ −4.00000 −0.270295
$$220$$ 0 0
$$221$$ −24.0000 −1.61441
$$222$$ 4.00000 0.268462
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −5.00000 −0.333333
$$226$$ −6.00000 −0.399114
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −4.00000 −0.264327 −0.132164 0.991228i $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ 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 $$0.562969\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ −16.0000 −1.03931
$$238$$ −6.00000 −0.388922
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 11.0000 0.707107
$$243$$ 10.0000 0.641500
$$244$$ 8.00000 0.512148
$$245$$ 0 0
$$246$$ 12.0000 0.765092
$$247$$ −8.00000 −0.509028
$$248$$ 4.00000 0.254000
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 0 0
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 16.0000 0.996116
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ −18.0000 −1.11204
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.00000 −0.122628
$$267$$ 12.0000 0.734388
$$268$$ −4.00000 −0.244339
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 8.00000 0.484182
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ −14.0000 −0.839664
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ −24.0000 −1.42918
$$283$$ −22.0000 −1.30776 −0.653882 0.756596i $$-0.726861\pi$$
−0.653882 + 0.756596i $$0.726861\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ −1.00000 −0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 20.0000 1.17242
$$292$$ 2.00000 0.117041
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 18.0000 1.04271
$$299$$ 0 0
$$300$$ 10.0000 0.577350
$$301$$ 8.00000 0.461112
$$302$$ −8.00000 −0.460348
$$303$$ 0 0
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ −8.00000 −0.452911
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 12.0000 0.672927
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −24.0000 −1.33955
$$322$$ 0 0
$$323$$ 12.0000 0.667698
$$324$$ −11.0000 −0.611111
$$325$$ 20.0000 1.10940
$$326$$ 16.0000 0.886158
$$327$$ −4.00000 −0.221201
$$328$$ −6.00000 −0.331295
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ 2.00000 0.109599
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −2.00000 −0.108148
$$343$$ 1.00000 0.0539949
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ 12.0000 0.645124
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ 12.0000 0.643268
$$349$$ −28.0000 −1.49881 −0.749403 0.662114i $$-0.769659\pi$$
−0.749403 + 0.662114i $$0.769659\pi$$
$$350$$ 5.00000 0.267261
$$351$$ −16.0000 −0.854017
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ −12.0000 −0.635107
$$358$$ 12.0000 0.634220
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −20.0000 −1.05118
$$363$$ 22.0000 1.15470
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 16.0000 0.836333
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ 8.00000 0.414781
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 12.0000 0.618853
$$377$$ 24.0000 1.23606
$$378$$ −4.00000 −0.205738
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 32.0000 1.63941
$$382$$ −24.0000 −1.22795
$$383$$ 36.0000 1.83951 0.919757 0.392488i $$-0.128386\pi$$
0.919757 + 0.392488i $$0.128386\pi$$
$$384$$ 2.00000 0.102062
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 8.00000 0.406663
$$388$$ −10.0000 −0.507673
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −1.00000 −0.0505076
$$393$$ −36.0000 −1.81596
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 20.0000 1.00377 0.501886 0.864934i $$-0.332640\pi$$
0.501886 + 0.864934i $$0.332640\pi$$
$$398$$ −20.0000 −1.00251
$$399$$ −4.00000 −0.200250
$$400$$ −5.00000 −0.250000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 16.0000 0.797017
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ 0 0
$$408$$ 12.0000 0.594089
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ −36.0000 −1.77575
$$412$$ −4.00000 −0.197066
$$413$$ −6.00000 −0.295241
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ −28.0000 −1.37117
$$418$$ 0 0
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ 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
$$423$$ −12.0000 −0.583460
$$424$$ −6.00000 −0.291386
$$425$$ −30.0000 −1.45521
$$426$$ 0 0
$$427$$ 8.00000 0.387147
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 4.00000 0.192450
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 0 0
$$438$$ 4.00000 0.191127
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 24.0000 1.14156
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ 36.0000 1.70274
$$448$$ 1.00000 0.0472456
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 5.00000 0.235702
$$451$$ 0 0
$$452$$ 6.00000 0.282216
$$453$$ −16.0000 −0.751746
$$454$$ −18.0000 −0.844782
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 4.00000 0.186908
$$459$$ 24.0000 1.12022
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ −6.00000 −0.277647 −0.138823 0.990317i $$-0.544332\pi$$
−0.138823 + 0.990317i $$0.544332\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 8.00000 0.368621
$$472$$ 6.00000 0.276172
$$473$$ 0 0
$$474$$ 16.0000 0.734904
$$475$$ −10.0000 −0.458831
$$476$$ 6.00000 0.275010
$$477$$ 6.00000 0.274721
$$478$$ −24.0000 −1.09773
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ −10.0000 −0.453609
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ 32.0000 1.44709
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ −12.0000 −0.541002
$$493$$ −36.0000 −1.62136
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 24.0000 1.07224
$$502$$ 18.0000 0.803379
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −6.00000 −0.266469
$$508$$ −16.0000 −0.709885
$$509$$ 36.0000 1.59567 0.797836 0.602875i $$-0.205978\pi$$
0.797836 + 0.602875i $$0.205978\pi$$
$$510$$ 0 0
$$511$$ 2.00000 0.0884748
$$512$$ −1.00000 −0.0441942
$$513$$ 8.00000 0.353209
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ −16.0000 −0.704361
$$517$$ 0 0
$$518$$ −2.00000 −0.0878750
$$519$$ 24.0000 1.05348
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 6.00000 0.262613
$$523$$ 2.00000 0.0874539 0.0437269 0.999044i $$-0.486077\pi$$
0.0437269 + 0.999044i $$0.486077\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 10.0000 0.436436
$$526$$ 0 0
$$527$$ −24.0000 −1.04546
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 2.00000 0.0867110
$$533$$ −24.0000 −1.03956
$$534$$ −12.0000 −0.519291
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 24.0000 1.03568
$$538$$ 12.0000 0.517357
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ 16.0000 0.687259
$$543$$ −40.0000 −1.71656
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 18.0000 0.768922
$$549$$ 8.00000 0.341432
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ 14.0000 0.593732
$$557$$ 6.00000 0.254228 0.127114 0.991888i $$-0.459429\pi$$
0.127114 + 0.991888i $$0.459429\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −32.0000 −1.35346
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 30.0000 1.26435 0.632175 0.774826i $$-0.282163\pi$$
0.632175 + 0.774826i $$0.282163\pi$$
$$564$$ 24.0000 1.01058
$$565$$ 0 0
$$566$$ 22.0000 0.924729
$$567$$ −11.0000 −0.461957
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 0 0
$$573$$ −48.0000 −2.00523
$$574$$ −6.00000 −0.250435
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ −28.0000 −1.16364
$$580$$ 0 0
$$581$$ −6.00000 −0.248922
$$582$$ −20.0000 −0.829027
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −24.0000 −0.991431
$$587$$ −42.0000 −1.73353 −0.866763 0.498721i $$-0.833803\pi$$
−0.866763 + 0.498721i $$0.833803\pi$$
$$588$$ −2.00000 −0.0824786
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 36.0000 1.48084
$$592$$ 2.00000 0.0821995
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ −40.0000 −1.63709
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ −10.0000 −0.408248
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ −4.00000 −0.162893
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 12.0000 0.486265
$$610$$ 0 0
$$611$$ 48.0000 1.94187
$$612$$ 6.00000 0.242536
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ 26.0000 1.04503 0.522514 0.852631i $$-0.324994\pi$$
0.522514 + 0.852631i $$0.324994\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 24.0000 0.962312
$$623$$ −6.00000 −0.240385
$$624$$ 8.00000 0.320256
$$625$$ 25.0000 1.00000
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ −4.00000 −0.159617
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 8.00000 0.317971
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ −12.0000 −0.475831
$$637$$ −4.00000 −0.158486
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 24.0000 0.947204
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −12.0000 −0.472134
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ 11.0000 0.432121
$$649$$ 0 0
$$650$$ −20.0000 −0.784465
$$651$$ 8.00000 0.313545
$$652$$ −16.0000 −0.626608
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 4.00000 0.156412
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 2.00000 0.0780274
$$658$$ 12.0000 0.467809
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ −40.0000 −1.55582 −0.777910 0.628376i $$-0.783720\pi$$
−0.777910 + 0.628376i $$0.783720\pi$$
$$662$$ −8.00000 −0.310929
$$663$$ 48.0000 1.86417
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ −12.0000 −0.464294
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 2.00000 0.0771517
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ −20.0000 −0.769800
$$676$$ 3.00000 0.115385
$$677$$ −12.0000 −0.461197 −0.230599 0.973049i $$-0.574068\pi$$
−0.230599 + 0.973049i $$0.574068\pi$$
$$678$$ 12.0000 0.460857
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ −36.0000 −1.37952
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ 8.00000 0.305219
$$688$$ 8.00000 0.304997
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ −46.0000 −1.74992 −0.874961 0.484193i $$-0.839113\pi$$
−0.874961 + 0.484193i $$0.839113\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 0 0
$$694$$ 24.0000 0.911028
$$695$$ 0 0
$$696$$ −12.0000 −0.454859
$$697$$ 36.0000 1.36360
$$698$$ 28.0000 1.05982
$$699$$ 12.0000 0.453882
$$700$$ −5.00000 −0.188982
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 16.0000 0.603881
$$703$$ 4.00000 0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 0 0
$$708$$ 12.0000 0.450988
$$709$$ −46.0000 −1.72757 −0.863783 0.503864i $$-0.831911\pi$$
−0.863783 + 0.503864i $$0.831911\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ 12.0000 0.449089
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ −48.0000 −1.79259
$$718$$ 24.0000 0.895672
$$719$$ 12.0000 0.447524 0.223762 0.974644i $$-0.428166\pi$$
0.223762 + 0.974644i $$0.428166\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ 15.0000 0.558242
$$723$$ 20.0000 0.743808
$$724$$ 20.0000 0.743294
$$725$$ 30.0000 1.11417
$$726$$ −22.0000 −0.816497
$$727$$ 44.0000 1.63187 0.815935 0.578144i $$-0.196223\pi$$
0.815935 + 0.578144i $$0.196223\pi$$
$$728$$ 4.00000 0.148250
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 48.0000 1.77534
$$732$$ −16.0000 −0.591377
$$733$$ −40.0000 −1.47743 −0.738717 0.674016i $$-0.764568\pi$$
−0.738717 + 0.674016i $$0.764568\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ −6.00000 −0.220863
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 0 0
$$741$$ 16.0000 0.587775
$$742$$ −6.00000 −0.220267
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ −6.00000 −0.219529
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ 36.0000 1.31191
$$754$$ −24.0000 −0.874028
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 16.0000 0.581146
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ −32.0000 −1.15924
$$763$$ 2.00000 0.0724049
$$764$$ 24.0000 0.868290
$$765$$ 0 0
$$766$$ −36.0000 −1.30073
$$767$$ 24.0000 0.866590
$$768$$ −2.00000 −0.0721688
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ −36.0000 −1.29651
$$772$$ 14.0000 0.503871
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 20.0000 0.718421
$$776$$ 10.0000 0.358979
$$777$$ −4.00000 −0.143499
$$778$$ −18.0000 −0.645331
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −24.0000 −0.857690
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 36.0000 1.28408
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ −32.0000 −1.13635
$$794$$ −20.0000 −0.709773
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ −12.0000 −0.425062 −0.212531 0.977154i $$-0.568171\pi$$
−0.212531 + 0.977154i $$0.568171\pi$$
$$798$$ 4.00000 0.141598
$$799$$ −72.0000 −2.54718
$$800$$ 5.00000 0.176777
$$801$$ −6.00000 −0.212000
$$802$$ 18.0000 0.635602
$$803$$ 0 0
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ 24.0000 0.844840
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ 32.0000 1.12229
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −12.0000 −0.420084
$$817$$ 16.0000 0.559769
$$818$$ −14.0000 −0.489499
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 36.0000 1.25564
$$823$$ −40.0000 −1.39431 −0.697156 0.716919i $$-0.745552\pi$$
−0.697156 + 0.716919i $$0.745552\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 6.00000 0.208767
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 0 0
$$829$$ 56.0000 1.94496 0.972480 0.232986i $$-0.0748495\pi$$
0.972480 + 0.232986i $$0.0748495\pi$$
$$830$$ 0 0
$$831$$ 20.0000 0.693792
$$832$$ −4.00000 −0.138675
$$833$$ 6.00000 0.207888
$$834$$ 28.0000 0.969561
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −16.0000 −0.553041
$$838$$ −6.00000 −0.207267
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000 0.344623
$$843$$ 12.0000 0.413302
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ −11.0000 −0.377964
$$848$$ 6.00000 0.206041
$$849$$ 44.0000 1.51008
$$850$$ 30.0000 1.02899
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 44.0000 1.50653 0.753266 0.657716i $$-0.228477\pi$$
0.753266 + 0.657716i $$0.228477\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 0 0
$$856$$ −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
$$859$$ 14.0000 0.477674 0.238837 0.971060i $$-0.423234\pi$$
0.238837 + 0.971060i $$0.423234\pi$$
$$860$$ 0 0
$$861$$ −12.0000 −0.408959
$$862$$ −24.0000 −0.817443
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ −38.0000 −1.29055
$$868$$ −4.00000 −0.135769
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ −2.00000 −0.0677285
$$873$$ −10.0000 −0.338449
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ −48.0000 −1.61900
$$880$$ 0 0
$$881$$ −54.0000 −1.81931 −0.909653 0.415369i $$-0.863653\pi$$
−0.909653 + 0.415369i $$0.863653\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ −24.0000 −0.807207
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 4.00000 0.134231
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 8.00000 0.267860
$$893$$ −24.0000 −0.803129
$$894$$ −36.0000 −1.20402
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ 24.0000 0.800445
$$900$$ −5.00000 −0.166667
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ −16.0000 −0.532447
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 16.0000 0.531564
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 18.0000 0.597351
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 0 0
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −4.00000 −0.132164
$$917$$ 18.0000 0.594412
$$918$$ −24.0000 −0.792118
$$919$$ 56.0000 1.84727 0.923635 0.383274i $$-0.125203\pi$$
0.923635 + 0.383274i $$0.125203\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ −12.0000 −0.395199
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ −32.0000 −1.05159
$$927$$ −4.00000 −0.131377
$$928$$ 6.00000 0.196960
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ −6.00000 −0.196537
$$933$$ 48.0000 1.57145
$$934$$ 6.00000 0.196326
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 4.00000 0.130605
$$939$$ 20.0000 0.652675
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ −8.00000 −0.260654
$$943$$ 0 0
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 24.0000 0.779895 0.389948 0.920837i $$-0.372493\pi$$
0.389948 + 0.920837i $$0.372493\pi$$
$$948$$ −16.0000 −0.519656
$$949$$ −8.00000 −0.259691
$$950$$ 10.0000 0.324443
$$951$$ −12.0000 −0.389127
$$952$$ −6.00000 −0.194461
$$953$$ −54.0000 −1.74923 −0.874616 0.484817i $$-0.838886\pi$$
−0.874616 + 0.484817i $$0.838886\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ 36.0000 1.16311
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 8.00000 0.257930
$$963$$ 12.0000 0.386695
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 11.0000 0.353553
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ 10.0000 0.320750
$$973$$ 14.0000 0.448819
$$974$$ 16.0000 0.512673
$$975$$ −40.0000 −1.28103
$$976$$ 8.00000 0.256074
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ −32.0000 −1.02325
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 12.0000 0.382935
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ 12.0000 0.382546
$$985$$ 0 0
$$986$$ 36.0000 1.14647
$$987$$ 24.0000 0.763928
$$988$$ −8.00000 −0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −16.0000 −0.507745
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ 8.00000 0.253363 0.126681 0.991943i $$-0.459567\pi$$
0.126681 + 0.991943i $$0.459567\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 8.00000 0.253109
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 14.2.a.a.1.1 1
3.2 odd 2 126.2.a.b.1.1 1
4.3 odd 2 112.2.a.c.1.1 1
5.2 odd 4 350.2.c.d.99.1 2
5.3 odd 4 350.2.c.d.99.2 2
5.4 even 2 350.2.a.f.1.1 1
7.2 even 3 98.2.c.b.67.1 2
7.3 odd 6 98.2.c.a.79.1 2
7.4 even 3 98.2.c.b.79.1 2
7.5 odd 6 98.2.c.a.67.1 2
7.6 odd 2 98.2.a.a.1.1 1
8.3 odd 2 448.2.a.a.1.1 1
8.5 even 2 448.2.a.g.1.1 1
9.2 odd 6 1134.2.f.f.757.1 2
9.4 even 3 1134.2.f.l.379.1 2
9.5 odd 6 1134.2.f.f.379.1 2
9.7 even 3 1134.2.f.l.757.1 2
11.10 odd 2 1694.2.a.e.1.1 1
12.11 even 2 1008.2.a.h.1.1 1
13.5 odd 4 2366.2.d.b.337.2 2
13.8 odd 4 2366.2.d.b.337.1 2
13.12 even 2 2366.2.a.j.1.1 1
15.2 even 4 3150.2.g.j.2899.2 2
15.8 even 4 3150.2.g.j.2899.1 2
15.14 odd 2 3150.2.a.i.1.1 1
16.3 odd 4 1792.2.b.g.897.2 2
16.5 even 4 1792.2.b.c.897.2 2
16.11 odd 4 1792.2.b.g.897.1 2
16.13 even 4 1792.2.b.c.897.1 2
17.16 even 2 4046.2.a.f.1.1 1
19.18 odd 2 5054.2.a.c.1.1 1
20.3 even 4 2800.2.g.h.449.2 2
20.7 even 4 2800.2.g.h.449.1 2
20.19 odd 2 2800.2.a.g.1.1 1
21.2 odd 6 882.2.g.c.361.1 2
21.5 even 6 882.2.g.d.361.1 2
21.11 odd 6 882.2.g.c.667.1 2
21.17 even 6 882.2.g.d.667.1 2
21.20 even 2 882.2.a.i.1.1 1
23.22 odd 2 7406.2.a.a.1.1 1
24.5 odd 2 4032.2.a.w.1.1 1
24.11 even 2 4032.2.a.r.1.1 1
28.3 even 6 784.2.i.i.177.1 2
28.11 odd 6 784.2.i.c.177.1 2
28.19 even 6 784.2.i.i.753.1 2
28.23 odd 6 784.2.i.c.753.1 2
28.27 even 2 784.2.a.b.1.1 1
35.13 even 4 2450.2.c.c.99.2 2
35.27 even 4 2450.2.c.c.99.1 2
35.34 odd 2 2450.2.a.t.1.1 1
56.13 odd 2 3136.2.a.e.1.1 1
56.27 even 2 3136.2.a.z.1.1 1
84.83 odd 2 7056.2.a.bd.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 1 1.1 even 1 trivial
98.2.a.a.1.1 1 7.6 odd 2
98.2.c.a.67.1 2 7.5 odd 6
98.2.c.a.79.1 2 7.3 odd 6
98.2.c.b.67.1 2 7.2 even 3
98.2.c.b.79.1 2 7.4 even 3
112.2.a.c.1.1 1 4.3 odd 2
126.2.a.b.1.1 1 3.2 odd 2
350.2.a.f.1.1 1 5.4 even 2
350.2.c.d.99.1 2 5.2 odd 4
350.2.c.d.99.2 2 5.3 odd 4
448.2.a.a.1.1 1 8.3 odd 2
448.2.a.g.1.1 1 8.5 even 2
784.2.a.b.1.1 1 28.27 even 2
784.2.i.c.177.1 2 28.11 odd 6
784.2.i.c.753.1 2 28.23 odd 6
784.2.i.i.177.1 2 28.3 even 6
784.2.i.i.753.1 2 28.19 even 6
882.2.a.i.1.1 1 21.20 even 2
882.2.g.c.361.1 2 21.2 odd 6
882.2.g.c.667.1 2 21.11 odd 6
882.2.g.d.361.1 2 21.5 even 6
882.2.g.d.667.1 2 21.17 even 6
1008.2.a.h.1.1 1 12.11 even 2
1134.2.f.f.379.1 2 9.5 odd 6
1134.2.f.f.757.1 2 9.2 odd 6
1134.2.f.l.379.1 2 9.4 even 3
1134.2.f.l.757.1 2 9.7 even 3
1694.2.a.e.1.1 1 11.10 odd 2
1792.2.b.c.897.1 2 16.13 even 4
1792.2.b.c.897.2 2 16.5 even 4
1792.2.b.g.897.1 2 16.11 odd 4
1792.2.b.g.897.2 2 16.3 odd 4
2366.2.a.j.1.1 1 13.12 even 2
2366.2.d.b.337.1 2 13.8 odd 4
2366.2.d.b.337.2 2 13.5 odd 4
2450.2.a.t.1.1 1 35.34 odd 2
2450.2.c.c.99.1 2 35.27 even 4
2450.2.c.c.99.2 2 35.13 even 4
2800.2.a.g.1.1 1 20.19 odd 2
2800.2.g.h.449.1 2 20.7 even 4
2800.2.g.h.449.2 2 20.3 even 4
3136.2.a.e.1.1 1 56.13 odd 2
3136.2.a.z.1.1 1 56.27 even 2
3150.2.a.i.1.1 1 15.14 odd 2
3150.2.g.j.2899.1 2 15.8 even 4
3150.2.g.j.2899.2 2 15.2 even 4
4032.2.a.r.1.1 1 24.11 even 2
4032.2.a.w.1.1 1 24.5 odd 2
4046.2.a.f.1.1 1 17.16 even 2
5054.2.a.c.1.1 1 19.18 odd 2
7056.2.a.bd.1.1 1 84.83 odd 2
7406.2.a.a.1.1 1 23.22 odd 2