# Properties

 Label 1530.2.a.d.1.1 Level $1530$ Weight $2$ Character 1530.1 Self dual yes Analytic conductor $12.217$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1530,2,Mod(1,1530)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1530, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1530.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1530.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: $$12.2171115093$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 510) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1530.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +4.00000 q^{11} -2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -4.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{34} -4.00000 q^{35} -6.00000 q^{37} +4.00000 q^{38} -1.00000 q^{40} -2.00000 q^{41} -12.0000 q^{43} +4.00000 q^{44} -4.00000 q^{46} -8.00000 q^{47} +9.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} +2.00000 q^{53} +4.00000 q^{55} +4.00000 q^{56} +2.00000 q^{58} -12.0000 q^{59} +2.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{67} -1.00000 q^{68} +4.00000 q^{70} +4.00000 q^{71} -14.0000 q^{73} +6.00000 q^{74} -4.00000 q^{76} -16.0000 q^{77} -12.0000 q^{79} +1.00000 q^{80} +2.00000 q^{82} +4.00000 q^{83} -1.00000 q^{85} +12.0000 q^{86} -4.00000 q^{88} -10.0000 q^{89} +8.00000 q^{91} +4.00000 q^{92} +8.00000 q^{94} -4.00000 q^{95} +18.0000 q^{97} -9.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −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
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ −4.00000 −0.755929
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 1.00000 0.171499
$$35$$ −4.00000 −0.676123
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 4.00000 0.534522
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ 0 0
$$70$$ 4.00000 0.478091
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ 0 0
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ −16.0000 −1.82337
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ 2.00000 0.220863
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ −1.00000 −0.108465
$$86$$ 12.0000 1.29399
$$87$$ 0 0
$$88$$ −4.00000 −0.426401
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ 18.0000 1.82762 0.913812 0.406138i $$-0.133125\pi$$
0.913812 + 0.406138i $$0.133125\pi$$
$$98$$ −9.00000 −0.909137
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ 0 0
$$112$$ −4.00000 −0.377964
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ 12.0000 1.10469
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 2.00000 0.175412
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ 0 0
$$133$$ 16.0000 1.38738
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ 0 0
$$142$$ −4.00000 −0.335673
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ −2.00000 −0.166091
$$146$$ 14.0000 1.15865
$$147$$ 0 0
$$148$$ −6.00000 −0.493197
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 0 0
$$154$$ 16.0000 1.28932
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ 12.0000 0.954669
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ −16.0000 −1.26098
$$162$$ 0 0
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 1.00000 0.0766965
$$171$$ 0 0
$$172$$ −12.0000 −0.914991
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ 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 $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ −8.00000 −0.592999
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ 18.0000 1.29567 0.647834 0.761781i $$-0.275675\pi$$
0.647834 + 0.761781i $$0.275675\pi$$
$$194$$ −18.0000 −1.29232
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 14.0000 0.997459 0.498729 0.866758i $$-0.333800\pi$$
0.498729 + 0.866758i $$0.333800\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −10.0000 −0.703598
$$203$$ 8.00000 0.561490
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ −16.0000 −1.10674
$$210$$ 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
$$215$$ −12.0000 −0.818393
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ −10.0000 −0.677285
$$219$$ 0 0
$$220$$ 4.00000 0.269680
$$221$$ 2.00000 0.134535
$$222$$ 0 0
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ −4.00000 −0.259281
$$239$$ 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 $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 9.00000 0.574989
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ −4.00000 −0.254000
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ 28.0000 1.76734 0.883672 0.468106i $$-0.155064\pi$$
0.883672 + 0.468106i $$0.155064\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ 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 $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 24.0000 1.49129
$$260$$ −2.00000 −0.124035
$$261$$ 0 0
$$262$$ 20.0000 1.23560
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 2.00000 0.122859
$$266$$ −16.0000 −0.981023
$$267$$ 0 0
$$268$$ 4.00000 0.244339
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 4.00000 0.241209
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 20.0000 1.19952
$$279$$ 0 0
$$280$$ 4.00000 0.239046
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 4.00000 0.237356
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 8.00000 0.472225
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 2.00000 0.117444
$$291$$ 0 0
$$292$$ −14.0000 −0.819288
$$293$$ 26.0000 1.51894 0.759468 0.650545i $$-0.225459\pi$$
0.759468 + 0.650545i $$0.225459\pi$$
$$294$$ 0 0
$$295$$ −12.0000 −0.698667
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ −2.00000 −0.115857
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 48.0000 2.76667
$$302$$ 8.00000 0.460348
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 2.00000 0.114520
$$306$$ 0 0
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ −16.0000 −0.911685
$$309$$ 0 0
$$310$$ −4.00000 −0.227185
$$311$$ −20.0000 −1.13410 −0.567048 0.823685i $$-0.691915\pi$$
−0.567048 + 0.823685i $$0.691915\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ −10.0000 −0.561656 −0.280828 0.959758i $$-0.590609\pi$$
−0.280828 + 0.959758i $$0.590609\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ 16.0000 0.891645
$$323$$ 4.00000 0.222566
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 20.0000 1.10770
$$327$$ 0 0
$$328$$ 2.00000 0.110432
$$329$$ 32.0000 1.76422
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 4.00000 0.219529
$$333$$ 0 0
$$334$$ 12.0000 0.656611
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ −1.00000 −0.0542326
$$341$$ 16.0000 0.866449
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ −14.0000 −0.752645
$$347$$ −20.0000 −1.07366 −0.536828 0.843692i $$-0.680378\pi$$
−0.536828 + 0.843692i $$0.680378\pi$$
$$348$$ 0 0
$$349$$ 22.0000 1.17763 0.588817 0.808267i $$-0.299594\pi$$
0.588817 + 0.808267i $$0.299594\pi$$
$$350$$ 4.00000 0.213809
$$351$$ 0 0
$$352$$ −4.00000 −0.213201
$$353$$ −2.00000 −0.106449 −0.0532246 0.998583i $$-0.516950\pi$$
−0.0532246 + 0.998583i $$0.516950\pi$$
$$354$$ 0 0
$$355$$ 4.00000 0.212298
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 22.0000 1.15629
$$363$$ 0 0
$$364$$ 8.00000 0.419314
$$365$$ −14.0000 −0.732793
$$366$$ 0 0
$$367$$ 28.0000 1.46159 0.730794 0.682598i $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ 6.00000 0.311925
$$371$$ −8.00000 −0.415339
$$372$$ 0 0
$$373$$ 38.0000 1.96757 0.983783 0.179364i $$-0.0574041\pi$$
0.983783 + 0.179364i $$0.0574041\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 0 0
$$382$$ −8.00000 −0.409316
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 0 0
$$385$$ −16.0000 −0.815436
$$386$$ −18.0000 −0.916176
$$387$$ 0 0
$$388$$ 18.0000 0.913812
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ −9.00000 −0.454569
$$393$$ 0 0
$$394$$ −14.0000 −0.705310
$$395$$ −12.0000 −0.603786
$$396$$ 0 0
$$397$$ −30.0000 −1.50566 −0.752828 0.658217i $$-0.771311\pi$$
−0.752828 + 0.658217i $$0.771311\pi$$
$$398$$ −4.00000 −0.200502
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 0 0
$$403$$ −8.00000 −0.398508
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 48.0000 2.36193
$$414$$ 0 0
$$415$$ 4.00000 0.196352
$$416$$ 2.00000 0.0980581
$$417$$ 0 0
$$418$$ 16.0000 0.782586
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −18.0000 −0.877266 −0.438633 0.898666i $$-0.644537\pi$$
−0.438633 + 0.898666i $$0.644537\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ 0 0
$$424$$ −2.00000 −0.0971286
$$425$$ −1.00000 −0.0485071
$$426$$ 0 0
$$427$$ −8.00000 −0.387147
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 12.0000 0.578691
$$431$$ 4.00000 0.192673 0.0963366 0.995349i $$-0.469287\pi$$
0.0963366 + 0.995349i $$0.469287\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ −16.0000 −0.765384
$$438$$ 0 0
$$439$$ −12.0000 −0.572729 −0.286364 0.958121i $$-0.592447\pi$$
−0.286364 + 0.958121i $$0.592447\pi$$
$$440$$ −4.00000 −0.190693
$$441$$ 0 0
$$442$$ −2.00000 −0.0951303
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 0 0
$$445$$ −10.0000 −0.474045
$$446$$ 8.00000 0.378811
$$447$$ 0 0
$$448$$ −4.00000 −0.188982
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ −20.0000 −0.938647
$$455$$ 8.00000 0.375046
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ 0 0
$$460$$ 4.00000 0.186501
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 0 0
$$463$$ 40.0000 1.85896 0.929479 0.368875i $$-0.120257\pi$$
0.929479 + 0.368875i $$0.120257\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ 12.0000 0.552345
$$473$$ −48.0000 −2.20704
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 4.00000 0.183340
$$477$$ 0 0
$$478$$ −24.0000 −1.09773
$$479$$ −20.0000 −0.913823 −0.456912 0.889512i $$-0.651044\pi$$
−0.456912 + 0.889512i $$0.651044\pi$$
$$480$$ 0 0
$$481$$ 12.0000 0.547153
$$482$$ −10.0000 −0.455488
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 18.0000 0.817338
$$486$$ 0 0
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ 0 0
$$490$$ −9.00000 −0.406579
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 2.00000 0.0900755
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ −16.0000 −0.717698
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ −28.0000 −1.24970
$$503$$ −44.0000 −1.96186 −0.980932 0.194354i $$-0.937739\pi$$
−0.980932 + 0.194354i $$0.937739\pi$$
$$504$$ 0 0
$$505$$ 10.0000 0.444994
$$506$$ −16.0000 −0.711287
$$507$$ 0 0
$$508$$ −16.0000 −0.709885
$$509$$ −38.0000 −1.68432 −0.842160 0.539227i $$-0.818716\pi$$
−0.842160 + 0.539227i $$0.818716\pi$$
$$510$$ 0 0
$$511$$ 56.0000 2.47729
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −32.0000 −1.40736
$$518$$ −24.0000 −1.05450
$$519$$ 0 0
$$520$$ 2.00000 0.0877058
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ −4.00000 −0.174243
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −2.00000 −0.0868744
$$531$$ 0 0
$$532$$ 16.0000 0.693688
$$533$$ 4.00000 0.173259
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ −4.00000 −0.172774
$$537$$ 0 0
$$538$$ −30.0000 −1.29339
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ 18.0000 0.773880 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 1.00000 0.0428746
$$545$$ 10.0000 0.428353
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 0 0
$$550$$ −4.00000 −0.170561
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 48.0000 2.04117
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ 26.0000 1.10166 0.550828 0.834619i $$-0.314312\pi$$
0.550828 + 0.834619i $$0.314312\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ −4.00000 −0.169031
$$561$$ 0 0
$$562$$ 10.0000 0.421825
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 0 0
$$565$$ 6.00000 0.252422
$$566$$ 20.0000 0.840663
$$567$$ 0 0
$$568$$ −4.00000 −0.167836
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ 0 0
$$574$$ −8.00000 −0.333914
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 0 0
$$580$$ −2.00000 −0.0830455
$$581$$ −16.0000 −0.663792
$$582$$ 0 0
$$583$$ 8.00000 0.331326
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ −44.0000 −1.81607 −0.908037 0.418890i $$-0.862419\pi$$
−0.908037 + 0.418890i $$0.862419\pi$$
$$588$$ 0 0
$$589$$ −16.0000 −0.659269
$$590$$ 12.0000 0.494032
$$591$$ 0 0
$$592$$ −6.00000 −0.246598
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 0 0
$$595$$ 4.00000 0.163984
$$596$$ 2.00000 0.0819232
$$597$$ 0 0
$$598$$ 8.00000 0.327144
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ −48.0000 −1.95633
$$603$$ 0 0
$$604$$ −8.00000 −0.325515
$$605$$ 5.00000 0.203279
$$606$$ 0 0
$$607$$ 28.0000 1.13648 0.568242 0.822861i $$-0.307624\pi$$
0.568242 + 0.822861i $$0.307624\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ −2.00000 −0.0809776
$$611$$ 16.0000 0.647291
$$612$$ 0 0
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ 16.0000 0.644658
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 0 0
$$622$$ 20.0000 0.801927
$$623$$ 40.0000 1.60257
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −10.0000 −0.399680
$$627$$ 0 0
$$628$$ −10.0000 −0.399043
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 12.0000 0.477334
$$633$$ 0 0
$$634$$ 10.0000 0.397151
$$635$$ −16.0000 −0.634941
$$636$$ 0 0
$$637$$ −18.0000 −0.713186
$$638$$ 8.00000 0.316723
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 0 0
$$643$$ 28.0000 1.10421 0.552106 0.833774i $$-0.313824\pi$$
0.552106 + 0.833774i $$0.313824\pi$$
$$644$$ −16.0000 −0.630488
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ 16.0000 0.629025 0.314512 0.949253i $$-0.398159\pi$$
0.314512 + 0.949253i $$0.398159\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ −20.0000 −0.783260
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ 0 0
$$655$$ −20.0000 −0.781465
$$656$$ −2.00000 −0.0780869
$$657$$ 0 0
$$658$$ −32.0000 −1.24749
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ 0 0
$$664$$ −4.00000 −0.155230
$$665$$ 16.0000 0.620453
$$666$$ 0 0
$$667$$ −8.00000 −0.309761
$$668$$ −12.0000 −0.464294
$$669$$ 0 0
$$670$$ −4.00000 −0.154533
$$671$$ 8.00000 0.308837
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 0 0
$$679$$ −72.0000 −2.76311
$$680$$ 1.00000 0.0383482
$$681$$ 0 0
$$682$$ −16.0000 −0.612672
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 0 0
$$685$$ 6.00000 0.229248
$$686$$ 8.00000 0.305441
$$687$$ 0 0
$$688$$ −12.0000 −0.457496
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ 14.0000 0.532200
$$693$$ 0 0
$$694$$ 20.0000 0.759190
$$695$$ −20.0000 −0.758643
$$696$$ 0 0
$$697$$ 2.00000 0.0757554
$$698$$ −22.0000 −0.832712
$$699$$ 0 0
$$700$$ −4.00000 −0.151186
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ 24.0000 0.905177
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 2.00000 0.0752710
$$707$$ −40.0000 −1.50435
$$708$$ 0 0
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ −4.00000 −0.150117
$$711$$ 0 0
$$712$$ 10.0000 0.374766
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ −8.00000 −0.299183
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ −24.0000 −0.895672
$$719$$ 44.0000 1.64092 0.820462 0.571702i $$-0.193717\pi$$
0.820462 + 0.571702i $$0.193717\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ 0 0
$$724$$ −22.0000 −0.817624
$$725$$ −2.00000 −0.0742781
$$726$$ 0 0
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ −8.00000 −0.296500
$$729$$ 0 0
$$730$$ 14.0000 0.518163
$$731$$ 12.0000 0.443836
$$732$$ 0 0
$$733$$ −18.0000 −0.664845 −0.332423 0.943131i $$-0.607866\pi$$
−0.332423 + 0.943131i $$0.607866\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ −6.00000 −0.220564
$$741$$ 0 0
$$742$$ 8.00000 0.293689
$$743$$ −28.0000 −1.02722 −0.513610 0.858024i $$-0.671692\pi$$
−0.513610 + 0.858024i $$0.671692\pi$$
$$744$$ 0 0
$$745$$ 2.00000 0.0732743
$$746$$ −38.0000 −1.39128
$$747$$ 0 0
$$748$$ −4.00000 −0.146254
$$749$$ 48.0000 1.75388
$$750$$ 0 0
$$751$$ 12.0000 0.437886 0.218943 0.975738i $$-0.429739\pi$$
0.218943 + 0.975738i $$0.429739\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ −8.00000 −0.291150
$$756$$ 0 0
$$757$$ −34.0000 −1.23575 −0.617876 0.786276i $$-0.712006\pi$$
−0.617876 + 0.786276i $$0.712006\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ 4.00000 0.145095
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ −40.0000 −1.44810
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ 24.0000 0.866590
$$768$$ 0 0
$$769$$ −46.0000 −1.65880 −0.829401 0.558653i $$-0.811318\pi$$
−0.829401 + 0.558653i $$0.811318\pi$$
$$770$$ 16.0000 0.576600
$$771$$ 0 0
$$772$$ 18.0000 0.647834
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ −18.0000 −0.646162
$$777$$ 0 0
$$778$$ −18.0000 −0.645331
$$779$$ 8.00000 0.286630
$$780$$ 0 0
$$781$$ 16.0000 0.572525
$$782$$ 4.00000 0.143040
$$783$$ 0 0
$$784$$ 9.00000 0.321429
$$785$$ −10.0000 −0.356915
$$786$$ 0 0
$$787$$ 36.0000 1.28326 0.641631 0.767014i $$-0.278258\pi$$
0.641631 + 0.767014i $$0.278258\pi$$
$$788$$ 14.0000 0.498729
$$789$$ 0 0
$$790$$ 12.0000 0.426941
$$791$$ −24.0000 −0.853342
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 30.0000 1.06466
$$795$$ 0 0
$$796$$ 4.00000 0.141776
$$797$$ 26.0000 0.920967 0.460484 0.887668i $$-0.347676\pi$$
0.460484 + 0.887668i $$0.347676\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ −6.00000 −0.211867
$$803$$ −56.0000 −1.97620
$$804$$ 0 0
$$805$$ −16.0000 −0.563926
$$806$$ 8.00000 0.281788
$$807$$ 0 0
$$808$$ −10.0000 −0.351799
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 8.00000 0.280745
$$813$$ 0 0
$$814$$ 24.0000 0.841200
$$815$$ −20.0000 −0.700569
$$816$$ 0 0
$$817$$ 48.0000 1.67931
$$818$$ −10.0000 −0.349642
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 0 0
$$823$$ −28.0000 −0.976019 −0.488009 0.872838i $$-0.662277\pi$$
−0.488009 + 0.872838i $$0.662277\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −48.0000 −1.67013
$$827$$ −44.0000 −1.53003 −0.765015 0.644013i $$-0.777268\pi$$
−0.765015 + 0.644013i $$0.777268\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ −4.00000 −0.138842
$$831$$ 0 0
$$832$$ −2.00000 −0.0693375
$$833$$ −9.00000 −0.311832
$$834$$ 0 0
$$835$$ −12.0000 −0.415277
$$836$$ −16.0000 −0.553372
$$837$$ 0 0
$$838$$ 12.0000 0.414533
$$839$$ 36.0000 1.24286 0.621429 0.783470i $$-0.286552\pi$$
0.621429 + 0.783470i $$0.286552\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 18.0000 0.620321
$$843$$ 0 0
$$844$$ 4.00000 0.137686
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ −20.0000 −0.687208
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ 1.00000 0.0342997
$$851$$ −24.0000 −0.822709
$$852$$ 0 0
$$853$$ −46.0000 −1.57501 −0.787505 0.616308i $$-0.788628\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ 8.00000 0.273754
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ 0 0
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ −12.0000 −0.409197
$$861$$ 0 0
$$862$$ −4.00000 −0.136241
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 0 0
$$865$$ 14.0000 0.476014
$$866$$ 14.0000 0.475739
$$867$$ 0 0
$$868$$ −16.0000 −0.543075
$$869$$ −48.0000 −1.62829
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ −10.0000 −0.338643
$$873$$ 0 0
$$874$$ 16.0000 0.541208
$$875$$ −4.00000 −0.135225
$$876$$ 0 0
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ 12.0000 0.404980
$$879$$ 0 0
$$880$$ 4.00000 0.134840
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 0 0
$$883$$ −28.0000 −0.942275 −0.471138 0.882060i $$-0.656156\pi$$
−0.471138 + 0.882060i $$0.656156\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 0 0
$$889$$ 64.0000 2.14649
$$890$$ 10.0000 0.335201
$$891$$ 0 0
$$892$$ −8.00000 −0.267860
$$893$$ 32.0000 1.07084
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 4.00000 0.133631
$$897$$ 0 0
$$898$$ −6.00000 −0.200223
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ −2.00000 −0.0666297
$$902$$ 8.00000 0.266371
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ −22.0000 −0.731305
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 20.0000 0.663723
$$909$$ 0 0
$$910$$ −8.00000 −0.265197
$$911$$ −20.0000 −0.662630 −0.331315 0.943520i $$-0.607492\pi$$
−0.331315 + 0.943520i $$0.607492\pi$$
$$912$$ 0 0
$$913$$ 16.0000 0.529523
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 80.0000 2.64183
$$918$$ 0 0
$$919$$ −56.0000 −1.84727 −0.923635 0.383274i $$-0.874797\pi$$
−0.923635 + 0.383274i $$0.874797\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ 0 0
$$922$$ 14.0000 0.461065
$$923$$ −8.00000 −0.263323
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ −40.0000 −1.31448
$$927$$ 0 0
$$928$$ 2.00000 0.0656532
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −36.0000 −1.17985
$$932$$ −18.0000 −0.589610
$$933$$ 0 0
$$934$$ 36.0000 1.17796
$$935$$ −4.00000 −0.130814
$$936$$ 0 0
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 16.0000 0.522419
$$939$$ 0 0
$$940$$ −8.00000 −0.260931
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ 0 0
$$943$$ −8.00000 −0.260516
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 48.0000 1.56061
$$947$$ 28.0000 0.909878 0.454939 0.890523i $$-0.349661\pi$$
0.454939 + 0.890523i $$0.349661\pi$$
$$948$$ 0 0
$$949$$ 28.0000 0.908918
$$950$$ 4.00000 0.129777
$$951$$ 0 0
$$952$$ −4.00000 −0.129641
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 0 0
$$955$$ 8.00000 0.258874
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ 20.0000 0.646171
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −12.0000 −0.386896
$$963$$ 0 0
$$964$$ 10.0000 0.322078
$$965$$ 18.0000 0.579441
$$966$$ 0 0
$$967$$ −16.0000 −0.514525 −0.257263 0.966342i $$-0.582821\pi$$
−0.257263 + 0.966342i $$0.582821\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 0 0
$$970$$ −18.0000 −0.577945
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 0 0
$$973$$ 80.0000 2.56468
$$974$$ −20.0000 −0.640841
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ −40.0000 −1.27841
$$980$$ 9.00000 0.287494
$$981$$ 0 0
$$982$$ −12.0000 −0.382935
$$983$$ 28.0000 0.893061 0.446531 0.894768i $$-0.352659\pi$$
0.446531 + 0.894768i $$0.352659\pi$$
$$984$$ 0 0
$$985$$ 14.0000 0.446077
$$986$$ −2.00000 −0.0636930
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ −48.0000 −1.52631
$$990$$ 0 0
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ 16.0000 0.507489
$$995$$ 4.00000 0.126809
$$996$$ 0 0
$$997$$ −38.0000 −1.20347 −0.601736 0.798695i $$-0.705524\pi$$
−0.601736 + 0.798695i $$0.705524\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ 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 1530.2.a.d.1.1 1
3.2 odd 2 510.2.a.c.1.1 1
5.4 even 2 7650.2.a.cn.1.1 1
12.11 even 2 4080.2.a.x.1.1 1
15.2 even 4 2550.2.d.b.2449.2 2
15.8 even 4 2550.2.d.b.2449.1 2
15.14 odd 2 2550.2.a.n.1.1 1
51.50 odd 2 8670.2.a.bb.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.c.1.1 1 3.2 odd 2
1530.2.a.d.1.1 1 1.1 even 1 trivial
2550.2.a.n.1.1 1 15.14 odd 2
2550.2.d.b.2449.1 2 15.8 even 4
2550.2.d.b.2449.2 2 15.2 even 4
4080.2.a.x.1.1 1 12.11 even 2
7650.2.a.cn.1.1 1 5.4 even 2
8670.2.a.bb.1.1 1 51.50 odd 2