# Properties

 Label 8379.2.a.e.1.1 Level $8379$ Weight $2$ Character 8379.1 Self dual yes Analytic conductor $66.907$ Analytic rank $2$ 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] = [8379,2,Mod(1,8379)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8379, 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("8379.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8379 = 3^{2} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8379.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: $$66.9066518536$$ Analytic rank: $$2$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 8379.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{4} -2.00000 q^{5} +3.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{4} -2.00000 q^{5} +3.00000 q^{8} +2.00000 q^{10} -6.00000 q^{13} -1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{19} +2.00000 q^{20} -4.00000 q^{23} -1.00000 q^{25} +6.00000 q^{26} -2.00000 q^{29} -8.00000 q^{31} -5.00000 q^{32} +6.00000 q^{34} -10.0000 q^{37} -1.00000 q^{38} -6.00000 q^{40} -2.00000 q^{41} -4.00000 q^{43} +4.00000 q^{46} +12.0000 q^{47} +1.00000 q^{50} +6.00000 q^{52} +6.00000 q^{53} +2.00000 q^{58} -12.0000 q^{59} +2.00000 q^{61} +8.00000 q^{62} +7.00000 q^{64} +12.0000 q^{65} -4.00000 q^{67} +6.00000 q^{68} -10.0000 q^{73} +10.0000 q^{74} -1.00000 q^{76} +2.00000 q^{80} +2.00000 q^{82} +16.0000 q^{83} +12.0000 q^{85} +4.00000 q^{86} -2.00000 q^{89} +4.00000 q^{92} -12.0000 q^{94} -2.00000 q^{95} -10.0000 q^{97} +O(q^{100})$$

## 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 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 3.00000 1.06066
$$9$$ 0 0
$$10$$ 2.00000 0.632456
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 6.00000 1.17670
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ −6.00000 −0.948683
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 6.00000 0.832050
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 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 $$0.459132\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 12.0000 1.48842
$$66$$ 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 $$0.5\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 0 0
$$82$$ 2.00000 0.220863
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ 0 0
$$85$$ 12.0000 1.30158
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ −12.0000 −1.23771
$$95$$ −2.00000 −0.205196
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ −18.0000 −1.76505
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 8.00000 0.746004
$$116$$ 2.00000 0.185695
$$117$$ 0 0
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 3.00000 0.265165
$$129$$ 0 0
$$130$$ −12.0000 −1.05247
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −18.0000 −1.54349
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ 10.0000 0.827606
$$147$$ 0 0
$$148$$ 10.0000 0.821995
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 3.00000 0.243332
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 16.0000 1.28515
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 10.0000 0.790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −16.0000 −1.24184
$$167$$ 24.0000 1.85718 0.928588 0.371113i $$-0.121024\pi$$
0.928588 + 0.371113i $$0.121024\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ −12.0000 −0.920358
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 2.00000 0.149906
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −12.0000 −0.884652
$$185$$ 20.0000 1.47043
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −12.0000 −0.875190
$$189$$ 0 0
$$190$$ 2.00000 0.145095
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ −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
$$196$$ 0 0
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 0 0
$$202$$ 10.0000 0.703598
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 4.00000 0.279372
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ 6.00000 0.416025
$$209$$ 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
$$214$$ 4.00000 0.273434
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 10.0000 0.677285
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 36.0000 2.42162
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ −24.0000 −1.56559
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 11.0000 0.707107
$$243$$ 0 0
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.00000 −0.381771
$$248$$ −24.0000 −1.52400
$$249$$ 0 0
$$250$$ −12.0000 −0.758947
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −12.0000 −0.744208
$$261$$ 0 0
$$262$$ −8.00000 −0.494242
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 4.00000 0.244339
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 20.0000 1.18888 0.594438 0.804141i $$-0.297374\pi$$
0.594438 + 0.804141i $$0.297374\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ −4.00000 −0.234888
$$291$$ 0 0
$$292$$ 10.0000 0.585206
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 0 0
$$295$$ 24.0000 1.39733
$$296$$ −30.0000 −1.74371
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 8.00000 0.460348
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ −4.00000 −0.229039
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −16.0000 −0.908739
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −14.0000 −0.782624
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −6.00000 −0.333849
$$324$$ 0 0
$$325$$ 6.00000 0.332820
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −6.00000 −0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ −16.0000 −0.878114
$$333$$ 0 0
$$334$$ −24.0000 −1.31322
$$335$$ 8.00000 0.437087
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ 0 0
$$340$$ −12.0000 −0.650791
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ 22.0000 1.18273
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −22.0000 −1.17094 −0.585471 0.810693i $$-0.699090\pi$$
−0.585471 + 0.810693i $$0.699090\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ −4.00000 −0.211407
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 14.0000 0.735824
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 20.0000 1.04685
$$366$$ 0 0
$$367$$ −32.0000 −1.67039 −0.835193 0.549957i $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ −20.0000 −1.03975
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 36.0000 1.85656
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 12.0000 0.616399 0.308199 0.951322i $$-0.400274\pi$$
0.308199 + 0.951322i $$0.400274\pi$$
$$380$$ 2.00000 0.102598
$$381$$ 0 0
$$382$$ −12.0000 −0.613973
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 0 0
$$388$$ 10.0000 0.507673
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −38.0000 −1.89763 −0.948815 0.315833i $$-0.897716\pi$$
−0.948815 + 0.315833i $$0.897716\pi$$
$$402$$ 0 0
$$403$$ 48.0000 2.39105
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ −4.00000 −0.197546
$$411$$ 0 0
$$412$$ 8.00000 0.394132
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −32.0000 −1.57082
$$416$$ 30.0000 1.47087
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 8.00000 0.390826 0.195413 0.980721i $$-0.437395\pi$$
0.195413 + 0.980721i $$0.437395\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 0 0
$$424$$ 18.0000 0.874157
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ −8.00000 −0.385794
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −36.0000 −1.71235
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 4.00000 0.189618
$$446$$ 16.0000 0.757622
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ 6.00000 0.280362
$$459$$ 0 0
$$460$$ −8.00000 −0.373002
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ −32.0000 −1.48078 −0.740392 0.672176i $$-0.765360\pi$$
−0.740392 + 0.672176i $$0.765360\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 24.0000 1.10704
$$471$$ 0 0
$$472$$ −36.0000 −1.65703
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −12.0000 −0.548867
$$479$$ 20.0000 0.913823 0.456912 0.889512i $$-0.348956\pi$$
0.456912 + 0.889512i $$0.348956\pi$$
$$480$$ 0 0
$$481$$ 60.0000 2.73576
$$482$$ −6.00000 −0.273293
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ 20.0000 0.908153
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 6.00000 0.271607
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 32.0000 1.44414 0.722070 0.691820i $$-0.243191\pi$$
0.722070 + 0.691820i $$0.243191\pi$$
$$492$$ 0 0
$$493$$ 12.0000 0.540453
$$494$$ 6.00000 0.269953
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 0 0
$$502$$ 24.0000 1.07117
$$503$$ 12.0000 0.535054 0.267527 0.963550i $$-0.413794\pi$$
0.267527 + 0.963550i $$0.413794\pi$$
$$504$$ 0 0
$$505$$ 20.0000 0.889988
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ −22.0000 −0.975133 −0.487566 0.873086i $$-0.662115\pi$$
−0.487566 + 0.873086i $$0.662115\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 11.0000 0.486136
$$513$$ 0 0
$$514$$ −14.0000 −0.617514
$$515$$ 16.0000 0.705044
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 36.0000 1.57870
$$521$$ 14.0000 0.613351 0.306676 0.951814i $$-0.400783\pi$$
0.306676 + 0.951814i $$0.400783\pi$$
$$522$$ 0 0
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ 48.0000 2.09091
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 12.0000 0.521247
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 12.0000 0.519778
$$534$$ 0 0
$$535$$ 8.00000 0.345870
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ 6.00000 0.258678
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 30.0000 1.28624
$$545$$ 20.0000 0.856706
$$546$$ 0 0
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ 18.0000 0.768922
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −2.00000 −0.0852029
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −10.0000 −0.421825
$$563$$ 20.0000 0.842900 0.421450 0.906852i $$-0.361521\pi$$
0.421450 + 0.906852i $$0.361521\pi$$
$$564$$ 0 0
$$565$$ 12.0000 0.504844
$$566$$ −20.0000 −0.840663
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ −18.0000 −0.749350 −0.374675 0.927156i $$-0.622246\pi$$
−0.374675 + 0.927156i $$0.622246\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 0 0
$$580$$ −4.00000 −0.166091
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −30.0000 −1.24141
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 8.00000 0.330195 0.165098 0.986277i $$-0.447206\pi$$
0.165098 + 0.986277i $$0.447206\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ −24.0000 −0.988064
$$591$$ 0 0
$$592$$ 10.0000 0.410997
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ −24.0000 −0.981433
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 8.00000 0.325515
$$605$$ 22.0000 0.894427
$$606$$ 0 0
$$607$$ −24.0000 −0.974130 −0.487065 0.873366i $$-0.661933\pi$$
−0.487065 + 0.873366i $$0.661933\pi$$
$$608$$ −5.00000 −0.202777
$$609$$ 0 0
$$610$$ 4.00000 0.161955
$$611$$ −72.0000 −2.91281
$$612$$ 0 0
$$613$$ 6.00000 0.242338 0.121169 0.992632i $$-0.461336\pi$$
0.121169 + 0.992632i $$0.461336\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2.00000 −0.0805170 −0.0402585 0.999189i $$-0.512818\pi$$
−0.0402585 + 0.999189i $$0.512818\pi$$
$$618$$ 0 0
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ −16.0000 −0.642575
$$621$$ 0 0
$$622$$ −4.00000 −0.160385
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ −2.00000 −0.0798087
$$629$$ 60.0000 2.39236
$$630$$ 0 0
$$631$$ −32.0000 −1.27390 −0.636950 0.770905i $$-0.719804\pi$$
−0.636950 + 0.770905i $$0.719804\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −6.00000 −0.238290
$$635$$ 16.0000 0.634941
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −6.00000 −0.237171
$$641$$ −38.0000 −1.50091 −0.750455 0.660922i $$-0.770166\pi$$
−0.750455 + 0.660922i $$0.770166\pi$$
$$642$$ 0 0
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −6.00000 −0.235339
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ 0 0
$$655$$ −16.0000 −0.625172
$$656$$ 2.00000 0.0780869
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 44.0000 1.71400 0.856998 0.515319i $$-0.172327\pi$$
0.856998 + 0.515319i $$0.172327\pi$$
$$660$$ 0 0
$$661$$ −6.00000 −0.233373 −0.116686 0.993169i $$-0.537227\pi$$
−0.116686 + 0.993169i $$0.537227\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ 0 0
$$664$$ 48.0000 1.86276
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 8.00000 0.309761
$$668$$ −24.0000 −0.928588
$$669$$ 0 0
$$670$$ −8.00000 −0.309067
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −46.0000 −1.77317 −0.886585 0.462566i $$-0.846929\pi$$
−0.886585 + 0.462566i $$0.846929\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ −23.0000 −0.884615
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 36.0000 1.38054
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 0 0
$$685$$ 36.0000 1.37549
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ −36.0000 −1.37149
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 22.0000 0.836315
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 8.00000 0.303457
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ −2.00000 −0.0757011
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ −10.0000 −0.377157
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 22.0000 0.827981
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −6.00000 −0.224860
$$713$$ 32.0000 1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ 0 0
$$718$$ −20.0000 −0.746393
$$719$$ −20.0000 −0.745874 −0.372937 0.927857i $$-0.621649\pi$$
−0.372937 + 0.927857i $$0.621649\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −1.00000 −0.0372161
$$723$$ 0 0
$$724$$ 14.0000 0.520306
$$725$$ 2.00000 0.0742781
$$726$$ 0 0
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −20.0000 −0.740233
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ −46.0000 −1.69905 −0.849524 0.527549i $$-0.823111\pi$$
−0.849524 + 0.527549i $$0.823111\pi$$
$$734$$ 32.0000 1.18114
$$735$$ 0 0
$$736$$ 20.0000 0.737210
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ −20.0000 −0.735215
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 12.0000 0.439646
$$746$$ 10.0000 0.366126
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ 0 0
$$754$$ −12.0000 −0.437014
$$755$$ 16.0000 0.582300
$$756$$ 0 0
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ −12.0000 −0.435860
$$759$$ 0 0
$$760$$ −6.00000 −0.217643
$$761$$ 50.0000 1.81250 0.906249 0.422744i $$-0.138933\pi$$
0.906249 + 0.422744i $$0.138933\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 72.0000 2.59977
$$768$$ 0 0
$$769$$ −18.0000 −0.649097 −0.324548 0.945869i $$-0.605212\pi$$
−0.324548 + 0.945869i $$0.605212\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 14.0000 0.503871
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ 0 0
$$775$$ 8.00000 0.287368
$$776$$ −30.0000 −1.07694
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ −2.00000 −0.0716574
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −24.0000 −0.858238
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −4.00000 −0.142766
$$786$$ 0 0
$$787$$ −44.0000 −1.56843 −0.784215 0.620489i $$-0.786934\pi$$
−0.784215 + 0.620489i $$0.786934\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −12.0000 −0.426132
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ −8.00000 −0.283552
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 0 0
$$799$$ −72.0000 −2.54718
$$800$$ 5.00000 0.176777
$$801$$ 0 0
$$802$$ 38.0000 1.34183
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −48.0000 −1.69073
$$807$$ 0 0
$$808$$ −30.0000 −1.05540
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ −4.00000 −0.139942
$$818$$ −14.0000 −0.489499
$$819$$ 0 0
$$820$$ −4.00000 −0.139686
$$821$$ 42.0000 1.46581 0.732905 0.680331i $$-0.238164\pi$$
0.732905 + 0.680331i $$0.238164\pi$$
$$822$$ 0 0
$$823$$ −32.0000 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$824$$ −24.0000 −0.836080
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.0000 0.973655 0.486828 0.873498i $$-0.338154\pi$$
0.486828 + 0.873498i $$0.338154\pi$$
$$828$$ 0 0
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 32.0000 1.11074
$$831$$ 0 0
$$832$$ −42.0000 −1.45609
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −48.0000 −1.66111
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −8.00000 −0.276355
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −14.0000 −0.482472
$$843$$ 0 0
$$844$$ 4.00000 0.137686
$$845$$ −46.0000 −1.58245
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ 0 0
$$850$$ −6.00000 −0.205798
$$851$$ 40.0000 1.37118
$$852$$ 0 0
$$853$$ −22.0000 −0.753266 −0.376633 0.926363i $$-0.622918\pi$$
−0.376633 + 0.926363i $$0.622918\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 30.0000 1.02478 0.512390 0.858753i $$-0.328760\pi$$
0.512390 + 0.858753i $$0.328760\pi$$
$$858$$ 0 0
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ 40.0000 1.36162 0.680808 0.732462i $$-0.261629\pi$$
0.680808 + 0.732462i $$0.261629\pi$$
$$864$$ 0 0
$$865$$ 44.0000 1.49604
$$866$$ −14.0000 −0.475739
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ −30.0000 −1.01593
$$873$$ 0 0
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −34.0000 −1.14810 −0.574049 0.818821i $$-0.694628\pi$$
−0.574049 + 0.818821i $$0.694628\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −38.0000 −1.28025 −0.640126 0.768270i $$-0.721118\pi$$
−0.640126 + 0.768270i $$0.721118\pi$$
$$882$$ 0 0
$$883$$ −36.0000 −1.21150 −0.605748 0.795656i $$-0.707126\pi$$
−0.605748 + 0.795656i $$0.707126\pi$$
$$884$$ −36.0000 −1.21081
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −40.0000 −1.34307 −0.671534 0.740973i $$-0.734364\pi$$
−0.671534 + 0.740973i $$0.734364\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −4.00000 −0.134080
$$891$$ 0 0
$$892$$ 16.0000 0.535720
$$893$$ 12.0000 0.401565
$$894$$ 0 0
$$895$$ −8.00000 −0.267411
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −2.00000 −0.0667409
$$899$$ 16.0000 0.533630
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −18.0000 −0.598671
$$905$$ 28.0000 0.930751
$$906$$ 0 0
$$907$$ 4.00000 0.132818 0.0664089 0.997792i $$-0.478846\pi$$
0.0664089 + 0.997792i $$0.478846\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 6.00000 0.198462
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 24.0000 0.791257
$$921$$ 0 0
$$922$$ 18.0000 0.592798
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 10.0000 0.328798
$$926$$ −32.0000 −1.05159
$$927$$ 0 0
$$928$$ 10.0000 0.328266
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 10.0000 0.327561
$$933$$ 0 0
$$934$$ 32.0000 1.04707
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 24.0000 0.782794
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ 0 0
$$943$$ 8.00000 0.260516
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 8.00000 0.259965 0.129983 0.991516i $$-0.458508\pi$$
0.129983 + 0.991516i $$0.458508\pi$$
$$948$$ 0 0
$$949$$ 60.0000 1.94768
$$950$$ 1.00000 0.0324443
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −38.0000 −1.23094 −0.615470 0.788160i $$-0.711034\pi$$
−0.615470 + 0.788160i $$0.711034\pi$$
$$954$$ 0 0
$$955$$ −24.0000 −0.776622
$$956$$ −12.0000 −0.388108
$$957$$ 0 0
$$958$$ −20.0000 −0.646171
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −60.0000 −1.93448
$$963$$ 0 0
$$964$$ −6.00000 −0.193247
$$965$$ 28.0000 0.901352
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ −33.0000 −1.06066
$$969$$ 0 0
$$970$$ −20.0000 −0.642161
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −32.0000 −1.02116
$$983$$ −8.00000 −0.255160 −0.127580 0.991828i $$-0.540721\pi$$
−0.127580 + 0.991828i $$0.540721\pi$$
$$984$$ 0 0
$$985$$ −4.00000 −0.127451
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ 6.00000 0.190885
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 40.0000 1.27000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −16.0000 −0.507234
$$996$$ 0 0
$$997$$ 58.0000 1.83688 0.918439 0.395562i $$-0.129450\pi$$
0.918439 + 0.395562i $$0.129450\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ 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 8379.2.a.e.1.1 1
3.2 odd 2 2793.2.a.i.1.1 1
7.6 odd 2 171.2.a.a.1.1 1
21.20 even 2 57.2.a.c.1.1 1
28.27 even 2 2736.2.a.s.1.1 1
35.34 odd 2 4275.2.a.m.1.1 1
84.83 odd 2 912.2.a.b.1.1 1
105.62 odd 4 1425.2.c.g.799.2 2
105.83 odd 4 1425.2.c.g.799.1 2
105.104 even 2 1425.2.a.a.1.1 1
133.132 even 2 3249.2.a.g.1.1 1
168.83 odd 2 3648.2.a.bf.1.1 1
168.125 even 2 3648.2.a.o.1.1 1
231.230 odd 2 6897.2.a.a.1.1 1
273.272 even 2 9633.2.a.h.1.1 1
399.398 odd 2 1083.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.c.1.1 1 21.20 even 2
171.2.a.a.1.1 1 7.6 odd 2
912.2.a.b.1.1 1 84.83 odd 2
1083.2.a.a.1.1 1 399.398 odd 2
1425.2.a.a.1.1 1 105.104 even 2
1425.2.c.g.799.1 2 105.83 odd 4
1425.2.c.g.799.2 2 105.62 odd 4
2736.2.a.s.1.1 1 28.27 even 2
2793.2.a.i.1.1 1 3.2 odd 2
3249.2.a.g.1.1 1 133.132 even 2
3648.2.a.o.1.1 1 168.125 even 2
3648.2.a.bf.1.1 1 168.83 odd 2
4275.2.a.m.1.1 1 35.34 odd 2
6897.2.a.a.1.1 1 231.230 odd 2
8379.2.a.e.1.1 1 1.1 even 1 trivial
9633.2.a.h.1.1 1 273.272 even 2