# Properties

 Label 350.2.a.f.1.1 Level $350$ Weight $2$ Character 350.1 Self dual yes Analytic conductor $2.795$ 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] = [350,2,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, 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("350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 350.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: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 350.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## 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 $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 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 $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 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
$$25$$ 0 0
$$26$$ 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 $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 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 $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 2.00000 0.288675
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −12.0000 −1.68034
$$52$$ 4.00000 0.554700
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −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 $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −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 $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 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 $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ −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 $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 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 $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −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 $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 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 $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 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 $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 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 $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 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 $$-0.448974\pi$$
0.159617 + 0.987179i $$0.448974\pi$$
$$158$$ 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 $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ −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 $$-0.349220\pi$$
0.456172 + 0.889892i $$0.349220\pi$$
$$174$$ −12.0000 −0.909718
$$175$$ 0 0
$$176$$ 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 $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ 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 $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 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 $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 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 $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ −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 $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 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 $$-0.273139\pi$$
0.653882 + 0.756596i $$0.273139\pi$$
$$284$$ 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 $$-0.747284\pi$$
−0.701047 + 0.713115i $$0.747284\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 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 $$-0.518177\pi$$
−0.0570730 + 0.998370i $$0.518177\pi$$
$$308$$ 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 $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ −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
$$325$$ 0 0
$$326$$ 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 $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 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 $$-0.277193\pi$$
0.644194 + 0.764862i $$0.277193\pi$$
$$348$$ −12.0000 −0.643268
$$349$$ −28.0000 −1.49881 −0.749403 0.662114i $$-0.769659\pi$$
−0.749403 + 0.662114i $$0.769659\pi$$
$$350$$ 0 0
$$351$$ −16.0000 −0.854017
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ −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 $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 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 $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 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 $$-0.871614\pi$$
−0.919757 + 0.392488i $$0.871614\pi$$
$$384$$ 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 $$-0.667360\pi$$
−0.501886 + 0.864934i $$0.667360\pi$$
$$398$$ 20.0000 1.00251
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ −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
$$425$$ 0 0
$$426$$ 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 $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ 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 $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ −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 $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 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 $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −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 $$-0.766875\pi$$
−0.743583 + 0.668644i $$0.766875\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 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
$$475$$ 0 0
$$476$$ 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 $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ 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 $$-0.513923\pi$$
−0.0437269 + 0.999044i $$0.513923\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 0 0
$$526$$ 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 $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ −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 $$-0.540571\pi$$
−0.127114 + 0.991888i $$0.540571\pi$$
$$558$$ −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 $$-0.717837\pi$$
−0.632175 + 0.774826i $$0.717837\pi$$
$$564$$ 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 $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 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 $$-0.166197\pi$$
0.866763 + 0.498721i $$0.166197\pi$$
$$588$$ 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 $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 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 $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 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 $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 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 $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 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
$$625$$ 0 0
$$626$$ 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 $$-0.589027\pi$$
−0.276053 + 0.961142i $$0.589027\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −12.0000 −0.472134
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ −11.0000 −0.432121
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 8.00000 0.313545
$$652$$ 16.0000 0.626608
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 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 $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ −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 $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 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
$$700$$ 0 0
$$701$$ 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
$$725$$ 0 0
$$726$$ −22.0000 −0.816497
$$727$$ −44.0000 −1.63187 −0.815935 0.578144i $$-0.803777\pi$$
−0.815935 + 0.578144i $$0.803777\pi$$
$$728$$ −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 $$-0.235432\pi$$
0.738717 + 0.674016i $$0.235432\pi$$
$$734$$ −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 $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ −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 $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ −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 $$-0.642054\pi$$
−0.431610 + 0.902060i $$0.642054\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 0 0
$$776$$ 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 $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ 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 $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ −4.00000 −0.141598
$$799$$ −72.0000 −2.54718
$$800$$ 0 0
$$801$$ −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 $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 6.00000 0.208767
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 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
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −44.0000 −1.50653 −0.753266 0.657716i $$-0.771523\pi$$
−0.753266 + 0.657716i $$0.771523\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 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 $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ −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 $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 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 $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ −24.0000 −0.807207
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ −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
$$900$$ 0 0
$$901$$ 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 $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ −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
$$925$$ 0 0
$$926$$ −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 $$-0.510401\pi$$
−0.0326686 + 0.999466i $$0.510401\pi$$
$$938$$ −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 $$-0.627507\pi$$
−0.389948 + 0.920837i $$0.627507\pi$$
$$948$$ 16.0000 0.519656
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ −12.0000 −0.389127
$$952$$ 6.00000 0.194461
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ −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 $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ −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
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ 6.00000 0.191957 0.0959785 0.995383i $$-0.469402\pi$$
0.0959785 + 0.995383i $$0.469402\pi$$
$$978$$ 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 $$-0.305348\pi$$
0.574111 + 0.818778i $$0.305348\pi$$
$$984$$ 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 $$-0.540433\pi$$
−0.126681 + 0.991943i $$0.540433\pi$$
$$998$$ −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 350.2.a.f.1.1 1
3.2 odd 2 3150.2.a.i.1.1 1
4.3 odd 2 2800.2.a.g.1.1 1
5.2 odd 4 350.2.c.d.99.2 2
5.3 odd 4 350.2.c.d.99.1 2
5.4 even 2 14.2.a.a.1.1 1
7.6 odd 2 2450.2.a.t.1.1 1
15.2 even 4 3150.2.g.j.2899.1 2
15.8 even 4 3150.2.g.j.2899.2 2
15.14 odd 2 126.2.a.b.1.1 1
20.3 even 4 2800.2.g.h.449.1 2
20.7 even 4 2800.2.g.h.449.2 2
20.19 odd 2 112.2.a.c.1.1 1
35.4 even 6 98.2.c.b.79.1 2
35.9 even 6 98.2.c.b.67.1 2
35.13 even 4 2450.2.c.c.99.1 2
35.19 odd 6 98.2.c.a.67.1 2
35.24 odd 6 98.2.c.a.79.1 2
35.27 even 4 2450.2.c.c.99.2 2
35.34 odd 2 98.2.a.a.1.1 1
40.19 odd 2 448.2.a.a.1.1 1
40.29 even 2 448.2.a.g.1.1 1
45.4 even 6 1134.2.f.l.379.1 2
45.14 odd 6 1134.2.f.f.379.1 2
45.29 odd 6 1134.2.f.f.757.1 2
45.34 even 6 1134.2.f.l.757.1 2
55.54 odd 2 1694.2.a.e.1.1 1
60.59 even 2 1008.2.a.h.1.1 1
65.34 odd 4 2366.2.d.b.337.1 2
65.44 odd 4 2366.2.d.b.337.2 2
65.64 even 2 2366.2.a.j.1.1 1
80.19 odd 4 1792.2.b.g.897.2 2
80.29 even 4 1792.2.b.c.897.1 2
80.59 odd 4 1792.2.b.g.897.1 2
80.69 even 4 1792.2.b.c.897.2 2
85.84 even 2 4046.2.a.f.1.1 1
95.94 odd 2 5054.2.a.c.1.1 1
105.44 odd 6 882.2.g.c.361.1 2
105.59 even 6 882.2.g.d.667.1 2
105.74 odd 6 882.2.g.c.667.1 2
105.89 even 6 882.2.g.d.361.1 2
105.104 even 2 882.2.a.i.1.1 1
115.114 odd 2 7406.2.a.a.1.1 1
120.29 odd 2 4032.2.a.w.1.1 1
120.59 even 2 4032.2.a.r.1.1 1
140.19 even 6 784.2.i.i.753.1 2
140.39 odd 6 784.2.i.c.177.1 2
140.59 even 6 784.2.i.i.177.1 2
140.79 odd 6 784.2.i.c.753.1 2
140.139 even 2 784.2.a.b.1.1 1
280.69 odd 2 3136.2.a.e.1.1 1
280.139 even 2 3136.2.a.z.1.1 1
420.419 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 5.4 even 2
98.2.a.a.1.1 1 35.34 odd 2
98.2.c.a.67.1 2 35.19 odd 6
98.2.c.a.79.1 2 35.24 odd 6
98.2.c.b.67.1 2 35.9 even 6
98.2.c.b.79.1 2 35.4 even 6
112.2.a.c.1.1 1 20.19 odd 2
126.2.a.b.1.1 1 15.14 odd 2
350.2.a.f.1.1 1 1.1 even 1 trivial
350.2.c.d.99.1 2 5.3 odd 4
350.2.c.d.99.2 2 5.2 odd 4
448.2.a.a.1.1 1 40.19 odd 2
448.2.a.g.1.1 1 40.29 even 2
784.2.a.b.1.1 1 140.139 even 2
784.2.i.c.177.1 2 140.39 odd 6
784.2.i.c.753.1 2 140.79 odd 6
784.2.i.i.177.1 2 140.59 even 6
784.2.i.i.753.1 2 140.19 even 6
882.2.a.i.1.1 1 105.104 even 2
882.2.g.c.361.1 2 105.44 odd 6
882.2.g.c.667.1 2 105.74 odd 6
882.2.g.d.361.1 2 105.89 even 6
882.2.g.d.667.1 2 105.59 even 6
1008.2.a.h.1.1 1 60.59 even 2
1134.2.f.f.379.1 2 45.14 odd 6
1134.2.f.f.757.1 2 45.29 odd 6
1134.2.f.l.379.1 2 45.4 even 6
1134.2.f.l.757.1 2 45.34 even 6
1694.2.a.e.1.1 1 55.54 odd 2
1792.2.b.c.897.1 2 80.29 even 4
1792.2.b.c.897.2 2 80.69 even 4
1792.2.b.g.897.1 2 80.59 odd 4
1792.2.b.g.897.2 2 80.19 odd 4
2366.2.a.j.1.1 1 65.64 even 2
2366.2.d.b.337.1 2 65.34 odd 4
2366.2.d.b.337.2 2 65.44 odd 4
2450.2.a.t.1.1 1 7.6 odd 2
2450.2.c.c.99.1 2 35.13 even 4
2450.2.c.c.99.2 2 35.27 even 4
2800.2.a.g.1.1 1 4.3 odd 2
2800.2.g.h.449.1 2 20.3 even 4
2800.2.g.h.449.2 2 20.7 even 4
3136.2.a.e.1.1 1 280.69 odd 2
3136.2.a.z.1.1 1 280.139 even 2
3150.2.a.i.1.1 1 3.2 odd 2
3150.2.g.j.2899.1 2 15.2 even 4
3150.2.g.j.2899.2 2 15.8 even 4
4032.2.a.r.1.1 1 120.59 even 2
4032.2.a.w.1.1 1 120.29 odd 2
4046.2.a.f.1.1 1 85.84 even 2
5054.2.a.c.1.1 1 95.94 odd 2
7056.2.a.bd.1.1 1 420.419 odd 2
7406.2.a.a.1.1 1 115.114 odd 2