# Properties

 Label 2240.2.a.j.1.1 Level $2240$ Weight $2$ Character 2240.1 Self dual yes Analytic conductor $17.886$ 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] = [2240,2,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.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: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2240.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} -5.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} +3.00000 q^{17} -6.00000 q^{19} -1.00000 q^{21} +6.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} +9.00000 q^{29} +5.00000 q^{33} +1.00000 q^{35} -6.00000 q^{37} +1.00000 q^{39} +8.00000 q^{41} +6.00000 q^{43} -2.00000 q^{45} -3.00000 q^{47} +1.00000 q^{49} -3.00000 q^{51} +12.0000 q^{53} -5.00000 q^{55} +6.00000 q^{57} +8.00000 q^{59} +4.00000 q^{61} -2.00000 q^{63} -1.00000 q^{65} -4.00000 q^{67} -6.00000 q^{69} -8.00000 q^{71} +10.0000 q^{73} -1.00000 q^{75} -5.00000 q^{77} +3.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +3.00000 q^{85} -9.00000 q^{87} -16.0000 q^{89} -1.00000 q^{91} -6.00000 q^{95} +7.00000 q^{97} +10.0000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 5.00000 0.870388
$$34$$ 0 0
$$35$$ 1.00000 0.169031
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ 0 0
$$45$$ −2.00000 −0.298142
$$46$$ 0 0
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −3.00000 −0.420084
$$52$$ 0 0
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ 0 0
$$55$$ −5.00000 −0.674200
$$56$$ 0 0
$$57$$ 6.00000 0.794719
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −5.00000 −0.569803
$$78$$ 0 0
$$79$$ 3.00000 0.337526 0.168763 0.985657i $$-0.446023\pi$$
0.168763 + 0.985657i $$0.446023\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 3.00000 0.325396
$$86$$ 0 0
$$87$$ −9.00000 −0.964901
$$88$$ 0 0
$$89$$ −16.0000 −1.69600 −0.847998 0.529999i $$-0.822192\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −6.00000 −0.615587
$$96$$ 0 0
$$97$$ 7.00000 0.710742 0.355371 0.934725i $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ 0 0
$$99$$ 10.0000 1.00504
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ 9.00000 0.886796 0.443398 0.896325i $$-0.353773\pi$$
0.443398 + 0.896325i $$0.353773\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −0.0975900
$$106$$ 0 0
$$107$$ 2.00000 0.193347 0.0966736 0.995316i $$-0.469180\pi$$
0.0966736 + 0.995316i $$0.469180\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 6.00000 0.559503
$$116$$ 0 0
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ −8.00000 −0.721336
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ −6.00000 −0.528271
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 0 0
$$133$$ −6.00000 −0.520266
$$134$$ 0 0
$$135$$ 5.00000 0.430331
$$136$$ 0 0
$$137$$ −16.0000 −1.36697 −0.683486 0.729964i $$-0.739537\pi$$
−0.683486 + 0.729964i $$0.739537\pi$$
$$138$$ 0 0
$$139$$ 18.0000 1.52674 0.763370 0.645961i $$-0.223543\pi$$
0.763370 + 0.645961i $$0.223543\pi$$
$$140$$ 0 0
$$141$$ 3.00000 0.252646
$$142$$ 0 0
$$143$$ 5.00000 0.418121
$$144$$ 0 0
$$145$$ 9.00000 0.747409
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ 19.0000 1.54620 0.773099 0.634285i $$-0.218706\pi$$
0.773099 + 0.634285i $$0.218706\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 6.00000 0.472866
$$162$$ 0 0
$$163$$ 6.00000 0.469956 0.234978 0.972001i $$-0.424498\pi$$
0.234978 + 0.972001i $$0.424498\pi$$
$$164$$ 0 0
$$165$$ 5.00000 0.389249
$$166$$ 0 0
$$167$$ 9.00000 0.696441 0.348220 0.937413i $$-0.386786\pi$$
0.348220 + 0.937413i $$0.386786\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 12.0000 0.917663
$$172$$ 0 0
$$173$$ −19.0000 −1.44454 −0.722272 0.691609i $$-0.756902\pi$$
−0.722272 + 0.691609i $$0.756902\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ −8.00000 −0.601317
$$178$$ 0 0
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ 0 0
$$183$$ −4.00000 −0.295689
$$184$$ 0 0
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ −15.0000 −1.09691
$$188$$ 0 0
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ −11.0000 −0.795932 −0.397966 0.917400i $$-0.630284\pi$$
−0.397966 + 0.917400i $$0.630284\pi$$
$$192$$ 0 0
$$193$$ 8.00000 0.575853 0.287926 0.957653i $$-0.407034\pi$$
0.287926 + 0.957653i $$0.407034\pi$$
$$194$$ 0 0
$$195$$ 1.00000 0.0716115
$$196$$ 0 0
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 9.00000 0.631676
$$204$$ 0 0
$$205$$ 8.00000 0.558744
$$206$$ 0 0
$$207$$ −12.0000 −0.834058
$$208$$ 0 0
$$209$$ 30.0000 2.07514
$$210$$ 0 0
$$211$$ 13.0000 0.894957 0.447478 0.894295i $$-0.352322\pi$$
0.447478 + 0.894295i $$0.352322\pi$$
$$212$$ 0 0
$$213$$ 8.00000 0.548151
$$214$$ 0 0
$$215$$ 6.00000 0.409197
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ −3.00000 −0.201802
$$222$$ 0 0
$$223$$ 25.0000 1.67412 0.837062 0.547108i $$-0.184271\pi$$
0.837062 + 0.547108i $$0.184271\pi$$
$$224$$ 0 0
$$225$$ −2.00000 −0.133333
$$226$$ 0 0
$$227$$ −13.0000 −0.862840 −0.431420 0.902151i $$-0.641987\pi$$
−0.431420 + 0.902151i $$0.641987\pi$$
$$228$$ 0 0
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ 5.00000 0.328976
$$232$$ 0 0
$$233$$ 8.00000 0.524097 0.262049 0.965055i $$-0.415602\pi$$
0.262049 + 0.965055i $$0.415602\pi$$
$$234$$ 0 0
$$235$$ −3.00000 −0.195698
$$236$$ 0 0
$$237$$ −3.00000 −0.194871
$$238$$ 0 0
$$239$$ 7.00000 0.452792 0.226396 0.974035i $$-0.427306\pi$$
0.226396 + 0.974035i $$0.427306\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 0 0
$$243$$ −16.0000 −1.02640
$$244$$ 0 0
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ 6.00000 0.381771
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 14.0000 0.883672 0.441836 0.897096i $$-0.354327\pi$$
0.441836 + 0.897096i $$0.354327\pi$$
$$252$$ 0 0
$$253$$ −30.0000 −1.88608
$$254$$ 0 0
$$255$$ −3.00000 −0.187867
$$256$$ 0 0
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ 0 0
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 0 0
$$265$$ 12.0000 0.737154
$$266$$ 0 0
$$267$$ 16.0000 0.979184
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ 0 0
$$273$$ 1.00000 0.0605228
$$274$$ 0 0
$$275$$ −5.00000 −0.301511
$$276$$ 0 0
$$277$$ −14.0000 −0.841178 −0.420589 0.907251i $$-0.638177\pi$$
−0.420589 + 0.907251i $$0.638177\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −13.0000 −0.775515 −0.387757 0.921761i $$-0.626750\pi$$
−0.387757 + 0.921761i $$0.626750\pi$$
$$282$$ 0 0
$$283$$ 29.0000 1.72387 0.861936 0.507018i $$-0.169252\pi$$
0.861936 + 0.507018i $$0.169252\pi$$
$$284$$ 0 0
$$285$$ 6.00000 0.355409
$$286$$ 0 0
$$287$$ 8.00000 0.472225
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ −7.00000 −0.410347
$$292$$ 0 0
$$293$$ 1.00000 0.0584206 0.0292103 0.999573i $$-0.490701\pi$$
0.0292103 + 0.999573i $$0.490701\pi$$
$$294$$ 0 0
$$295$$ 8.00000 0.465778
$$296$$ 0 0
$$297$$ −25.0000 −1.45065
$$298$$ 0 0
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 6.00000 0.345834
$$302$$ 0 0
$$303$$ −14.0000 −0.804279
$$304$$ 0 0
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ −27.0000 −1.54097 −0.770486 0.637457i $$-0.779986\pi$$
−0.770486 + 0.637457i $$0.779986\pi$$
$$308$$ 0 0
$$309$$ −9.00000 −0.511992
$$310$$ 0 0
$$311$$ 14.0000 0.793867 0.396934 0.917847i $$-0.370074\pi$$
0.396934 + 0.917847i $$0.370074\pi$$
$$312$$ 0 0
$$313$$ 29.0000 1.63918 0.819588 0.572953i $$-0.194202\pi$$
0.819588 + 0.572953i $$0.194202\pi$$
$$314$$ 0 0
$$315$$ −2.00000 −0.112687
$$316$$ 0 0
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ 0 0
$$319$$ −45.0000 −2.51952
$$320$$ 0 0
$$321$$ −2.00000 −0.111629
$$322$$ 0 0
$$323$$ −18.0000 −1.00155
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ −11.0000 −0.608301
$$328$$ 0 0
$$329$$ −3.00000 −0.165395
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ 12.0000 0.657596
$$334$$ 0 0
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −6.00000 −0.323029
$$346$$ 0 0
$$347$$ −10.0000 −0.536828 −0.268414 0.963304i $$-0.586500\pi$$
−0.268414 + 0.963304i $$0.586500\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ −5.00000 −0.266880
$$352$$ 0 0
$$353$$ −33.0000 −1.75641 −0.878206 0.478282i $$-0.841260\pi$$
−0.878206 + 0.478282i $$0.841260\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ 0 0
$$357$$ −3.00000 −0.158777
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ −14.0000 −0.734809
$$364$$ 0 0
$$365$$ 10.0000 0.523424
$$366$$ 0 0
$$367$$ 29.0000 1.51379 0.756894 0.653538i $$-0.226716\pi$$
0.756894 + 0.653538i $$0.226716\pi$$
$$368$$ 0 0
$$369$$ −16.0000 −0.832927
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ −9.00000 −0.463524
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ 0 0
$$385$$ −5.00000 −0.254824
$$386$$ 0 0
$$387$$ −12.0000 −0.609994
$$388$$ 0 0
$$389$$ −25.0000 −1.26755 −0.633775 0.773517i $$-0.718496\pi$$
−0.633775 + 0.773517i $$0.718496\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ 0 0
$$393$$ 6.00000 0.302660
$$394$$ 0 0
$$395$$ 3.00000 0.150946
$$396$$ 0 0
$$397$$ 29.0000 1.45547 0.727734 0.685859i $$-0.240573\pi$$
0.727734 + 0.685859i $$0.240573\pi$$
$$398$$ 0 0
$$399$$ 6.00000 0.300376
$$400$$ 0 0
$$401$$ 9.00000 0.449439 0.224719 0.974424i $$-0.427853\pi$$
0.224719 + 0.974424i $$0.427853\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 30.0000 1.48704
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 16.0000 0.789222
$$412$$ 0 0
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ −18.0000 −0.881464
$$418$$ 0 0
$$419$$ 36.0000 1.75872 0.879358 0.476162i $$-0.157972\pi$$
0.879358 + 0.476162i $$0.157972\pi$$
$$420$$ 0 0
$$421$$ 19.0000 0.926003 0.463002 0.886357i $$-0.346772\pi$$
0.463002 + 0.886357i $$0.346772\pi$$
$$422$$ 0 0
$$423$$ 6.00000 0.291730
$$424$$ 0 0
$$425$$ 3.00000 0.145521
$$426$$ 0 0
$$427$$ 4.00000 0.193574
$$428$$ 0 0
$$429$$ −5.00000 −0.241402
$$430$$ 0 0
$$431$$ −23.0000 −1.10787 −0.553936 0.832560i $$-0.686875\pi$$
−0.553936 + 0.832560i $$0.686875\pi$$
$$432$$ 0 0
$$433$$ −30.0000 −1.44171 −0.720854 0.693087i $$-0.756250\pi$$
−0.720854 + 0.693087i $$0.756250\pi$$
$$434$$ 0 0
$$435$$ −9.00000 −0.431517
$$436$$ 0 0
$$437$$ −36.0000 −1.72211
$$438$$ 0 0
$$439$$ 34.0000 1.62273 0.811366 0.584539i $$-0.198725\pi$$
0.811366 + 0.584539i $$0.198725\pi$$
$$440$$ 0 0
$$441$$ −2.00000 −0.0952381
$$442$$ 0 0
$$443$$ 30.0000 1.42534 0.712672 0.701498i $$-0.247485\pi$$
0.712672 + 0.701498i $$0.247485\pi$$
$$444$$ 0 0
$$445$$ −16.0000 −0.758473
$$446$$ 0 0
$$447$$ −14.0000 −0.662177
$$448$$ 0 0
$$449$$ −33.0000 −1.55737 −0.778683 0.627417i $$-0.784112\pi$$
−0.778683 + 0.627417i $$0.784112\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ 0 0
$$453$$ −19.0000 −0.892698
$$454$$ 0 0
$$455$$ −1.00000 −0.0468807
$$456$$ 0 0
$$457$$ −28.0000 −1.30978 −0.654892 0.755722i $$-0.727286\pi$$
−0.654892 + 0.755722i $$0.727286\pi$$
$$458$$ 0 0
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ 24.0000 1.11779 0.558896 0.829238i $$-0.311225\pi$$
0.558896 + 0.829238i $$0.311225\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 33.0000 1.52706 0.763529 0.645774i $$-0.223465\pi$$
0.763529 + 0.645774i $$0.223465\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 0 0
$$473$$ −30.0000 −1.37940
$$474$$ 0 0
$$475$$ −6.00000 −0.275299
$$476$$ 0 0
$$477$$ −24.0000 −1.09888
$$478$$ 0 0
$$479$$ 6.00000 0.274147 0.137073 0.990561i $$-0.456230\pi$$
0.137073 + 0.990561i $$0.456230\pi$$
$$480$$ 0 0
$$481$$ 6.00000 0.273576
$$482$$ 0 0
$$483$$ −6.00000 −0.273009
$$484$$ 0 0
$$485$$ 7.00000 0.317854
$$486$$ 0 0
$$487$$ −26.0000 −1.17817 −0.589086 0.808070i $$-0.700512\pi$$
−0.589086 + 0.808070i $$0.700512\pi$$
$$488$$ 0 0
$$489$$ −6.00000 −0.271329
$$490$$ 0 0
$$491$$ 33.0000 1.48927 0.744635 0.667472i $$-0.232624\pi$$
0.744635 + 0.667472i $$0.232624\pi$$
$$492$$ 0 0
$$493$$ 27.0000 1.21602
$$494$$ 0 0
$$495$$ 10.0000 0.449467
$$496$$ 0 0
$$497$$ −8.00000 −0.358849
$$498$$ 0 0
$$499$$ −25.0000 −1.11915 −0.559577 0.828778i $$-0.689036\pi$$
−0.559577 + 0.828778i $$0.689036\pi$$
$$500$$ 0 0
$$501$$ −9.00000 −0.402090
$$502$$ 0 0
$$503$$ 31.0000 1.38222 0.691111 0.722749i $$-0.257122\pi$$
0.691111 + 0.722749i $$0.257122\pi$$
$$504$$ 0 0
$$505$$ 14.0000 0.622992
$$506$$ 0 0
$$507$$ 12.0000 0.532939
$$508$$ 0 0
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ 10.0000 0.442374
$$512$$ 0 0
$$513$$ −30.0000 −1.32453
$$514$$ 0 0
$$515$$ 9.00000 0.396587
$$516$$ 0 0
$$517$$ 15.0000 0.659699
$$518$$ 0 0
$$519$$ 19.0000 0.834007
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 0 0
$$525$$ −1.00000 −0.0436436
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −16.0000 −0.694341
$$532$$ 0 0
$$533$$ −8.00000 −0.346518
$$534$$ 0 0
$$535$$ 2.00000 0.0864675
$$536$$ 0 0
$$537$$ −4.00000 −0.172613
$$538$$ 0 0
$$539$$ −5.00000 −0.215365
$$540$$ 0 0
$$541$$ 9.00000 0.386940 0.193470 0.981106i $$-0.438026\pi$$
0.193470 + 0.981106i $$0.438026\pi$$
$$542$$ 0 0
$$543$$ −20.0000 −0.858282
$$544$$ 0 0
$$545$$ 11.0000 0.471188
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 0 0
$$549$$ −8.00000 −0.341432
$$550$$ 0 0
$$551$$ −54.0000 −2.30048
$$552$$ 0 0
$$553$$ 3.00000 0.127573
$$554$$ 0 0
$$555$$ 6.00000 0.254686
$$556$$ 0 0
$$557$$ −24.0000 −1.01691 −0.508456 0.861088i $$-0.669784\pi$$
−0.508456 + 0.861088i $$0.669784\pi$$
$$558$$ 0 0
$$559$$ −6.00000 −0.253773
$$560$$ 0 0
$$561$$ 15.0000 0.633300
$$562$$ 0 0
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 34.0000 1.42535 0.712677 0.701492i $$-0.247483\pi$$
0.712677 + 0.701492i $$0.247483\pi$$
$$570$$ 0 0
$$571$$ −36.0000 −1.50655 −0.753277 0.657704i $$-0.771528\pi$$
−0.753277 + 0.657704i $$0.771528\pi$$
$$572$$ 0 0
$$573$$ 11.0000 0.459532
$$574$$ 0 0
$$575$$ 6.00000 0.250217
$$576$$ 0 0
$$577$$ 17.0000 0.707719 0.353860 0.935299i $$-0.384869\pi$$
0.353860 + 0.935299i $$0.384869\pi$$
$$578$$ 0 0
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ −60.0000 −2.48495
$$584$$ 0 0
$$585$$ 2.00000 0.0826898
$$586$$ 0 0
$$587$$ 8.00000 0.330195 0.165098 0.986277i $$-0.447206\pi$$
0.165098 + 0.986277i $$0.447206\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 0 0
$$593$$ −7.00000 −0.287456 −0.143728 0.989617i $$-0.545909\pi$$
−0.143728 + 0.989617i $$0.545909\pi$$
$$594$$ 0 0
$$595$$ 3.00000 0.122988
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 33.0000 1.34834 0.674172 0.738575i $$-0.264501\pi$$
0.674172 + 0.738575i $$0.264501\pi$$
$$600$$ 0 0
$$601$$ −34.0000 −1.38689 −0.693444 0.720510i $$-0.743908\pi$$
−0.693444 + 0.720510i $$0.743908\pi$$
$$602$$ 0 0
$$603$$ 8.00000 0.325785
$$604$$ 0 0
$$605$$ 14.0000 0.569181
$$606$$ 0 0
$$607$$ −17.0000 −0.690009 −0.345004 0.938601i $$-0.612123\pi$$
−0.345004 + 0.938601i $$0.612123\pi$$
$$608$$ 0 0
$$609$$ −9.00000 −0.364698
$$610$$ 0 0
$$611$$ 3.00000 0.121367
$$612$$ 0 0
$$613$$ 22.0000 0.888572 0.444286 0.895885i $$-0.353457\pi$$
0.444286 + 0.895885i $$0.353457\pi$$
$$614$$ 0 0
$$615$$ −8.00000 −0.322591
$$616$$ 0 0
$$617$$ 10.0000 0.402585 0.201292 0.979531i $$-0.435486\pi$$
0.201292 + 0.979531i $$0.435486\pi$$
$$618$$ 0 0
$$619$$ 2.00000 0.0803868 0.0401934 0.999192i $$-0.487203\pi$$
0.0401934 + 0.999192i $$0.487203\pi$$
$$620$$ 0 0
$$621$$ 30.0000 1.20386
$$622$$ 0 0
$$623$$ −16.0000 −0.641026
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −30.0000 −1.19808
$$628$$ 0 0
$$629$$ −18.0000 −0.717707
$$630$$ 0 0
$$631$$ 9.00000 0.358284 0.179142 0.983823i $$-0.442668\pi$$
0.179142 + 0.983823i $$0.442668\pi$$
$$632$$ 0 0
$$633$$ −13.0000 −0.516704
$$634$$ 0 0
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 0 0
$$639$$ 16.0000 0.632950
$$640$$ 0 0
$$641$$ 34.0000 1.34292 0.671460 0.741041i $$-0.265668\pi$$
0.671460 + 0.741041i $$0.265668\pi$$
$$642$$ 0 0
$$643$$ 47.0000 1.85350 0.926750 0.375680i $$-0.122591\pi$$
0.926750 + 0.375680i $$0.122591\pi$$
$$644$$ 0 0
$$645$$ −6.00000 −0.236250
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ −40.0000 −1.57014
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 0 0
$$655$$ −6.00000 −0.234439
$$656$$ 0 0
$$657$$ −20.0000 −0.780274
$$658$$ 0 0
$$659$$ −25.0000 −0.973862 −0.486931 0.873441i $$-0.661884\pi$$
−0.486931 + 0.873441i $$0.661884\pi$$
$$660$$ 0 0
$$661$$ −8.00000 −0.311164 −0.155582 0.987823i $$-0.549725\pi$$
−0.155582 + 0.987823i $$0.549725\pi$$
$$662$$ 0 0
$$663$$ 3.00000 0.116510
$$664$$ 0 0
$$665$$ −6.00000 −0.232670
$$666$$ 0 0
$$667$$ 54.0000 2.09089
$$668$$ 0 0
$$669$$ −25.0000 −0.966556
$$670$$ 0 0
$$671$$ −20.0000 −0.772091
$$672$$ 0 0
$$673$$ 32.0000 1.23351 0.616755 0.787155i $$-0.288447\pi$$
0.616755 + 0.787155i $$0.288447\pi$$
$$674$$ 0 0
$$675$$ 5.00000 0.192450
$$676$$ 0 0
$$677$$ −33.0000 −1.26829 −0.634147 0.773213i $$-0.718648\pi$$
−0.634147 + 0.773213i $$0.718648\pi$$
$$678$$ 0 0
$$679$$ 7.00000 0.268635
$$680$$ 0 0
$$681$$ 13.0000 0.498161
$$682$$ 0 0
$$683$$ −20.0000 −0.765279 −0.382639 0.923898i $$-0.624985\pi$$
−0.382639 + 0.923898i $$0.624985\pi$$
$$684$$ 0 0
$$685$$ −16.0000 −0.611329
$$686$$ 0 0
$$687$$ 16.0000 0.610438
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 0 0
$$693$$ 10.0000 0.379869
$$694$$ 0 0
$$695$$ 18.0000 0.682779
$$696$$ 0 0
$$697$$ 24.0000 0.909065
$$698$$ 0 0
$$699$$ −8.00000 −0.302588
$$700$$ 0 0
$$701$$ 21.0000 0.793159 0.396580 0.918000i $$-0.370197\pi$$
0.396580 + 0.918000i $$0.370197\pi$$
$$702$$ 0 0
$$703$$ 36.0000 1.35777
$$704$$ 0 0
$$705$$ 3.00000 0.112987
$$706$$ 0 0
$$707$$ 14.0000 0.526524
$$708$$ 0 0
$$709$$ 41.0000 1.53979 0.769894 0.638172i $$-0.220309\pi$$
0.769894 + 0.638172i $$0.220309\pi$$
$$710$$ 0 0
$$711$$ −6.00000 −0.225018
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 5.00000 0.186989
$$716$$ 0 0
$$717$$ −7.00000 −0.261420
$$718$$ 0 0
$$719$$ −50.0000 −1.86469 −0.932343 0.361576i $$-0.882239\pi$$
−0.932343 + 0.361576i $$0.882239\pi$$
$$720$$ 0 0
$$721$$ 9.00000 0.335178
$$722$$ 0 0
$$723$$ −18.0000 −0.669427
$$724$$ 0 0
$$725$$ 9.00000 0.334252
$$726$$ 0 0
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 18.0000 0.665754
$$732$$ 0 0
$$733$$ −5.00000 −0.184679 −0.0923396 0.995728i $$-0.529435\pi$$
−0.0923396 + 0.995728i $$0.529435\pi$$
$$734$$ 0 0
$$735$$ −1.00000 −0.0368856
$$736$$ 0 0
$$737$$ 20.0000 0.736709
$$738$$ 0 0
$$739$$ −37.0000 −1.36107 −0.680534 0.732717i $$-0.738252\pi$$
−0.680534 + 0.732717i $$0.738252\pi$$
$$740$$ 0 0
$$741$$ −6.00000 −0.220416
$$742$$ 0 0
$$743$$ 36.0000 1.32071 0.660356 0.750953i $$-0.270405\pi$$
0.660356 + 0.750953i $$0.270405\pi$$
$$744$$ 0 0
$$745$$ 14.0000 0.512920
$$746$$ 0 0
$$747$$ 24.0000 0.878114
$$748$$ 0 0
$$749$$ 2.00000 0.0730784
$$750$$ 0 0
$$751$$ 35.0000 1.27717 0.638584 0.769552i $$-0.279520\pi$$
0.638584 + 0.769552i $$0.279520\pi$$
$$752$$ 0 0
$$753$$ −14.0000 −0.510188
$$754$$ 0 0
$$755$$ 19.0000 0.691481
$$756$$ 0 0
$$757$$ −8.00000 −0.290765 −0.145382 0.989376i $$-0.546441\pi$$
−0.145382 + 0.989376i $$0.546441\pi$$
$$758$$ 0 0
$$759$$ 30.0000 1.08893
$$760$$ 0 0
$$761$$ −46.0000 −1.66750 −0.833749 0.552143i $$-0.813810\pi$$
−0.833749 + 0.552143i $$0.813810\pi$$
$$762$$ 0 0
$$763$$ 11.0000 0.398227
$$764$$ 0 0
$$765$$ −6.00000 −0.216930
$$766$$ 0 0
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ 0 0
$$773$$ −1.00000 −0.0359675 −0.0179838 0.999838i $$-0.505725\pi$$
−0.0179838 + 0.999838i $$0.505725\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 6.00000 0.215249
$$778$$ 0 0
$$779$$ −48.0000 −1.71978
$$780$$ 0 0
$$781$$ 40.0000 1.43131
$$782$$ 0 0
$$783$$ 45.0000 1.60817
$$784$$ 0 0
$$785$$ 2.00000 0.0713831
$$786$$ 0 0
$$787$$ 11.0000 0.392108 0.196054 0.980593i $$-0.437187\pi$$
0.196054 + 0.980593i $$0.437187\pi$$
$$788$$ 0 0
$$789$$ 18.0000 0.640817
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 0 0
$$795$$ −12.0000 −0.425596
$$796$$ 0 0
$$797$$ −35.0000 −1.23976 −0.619882 0.784695i $$-0.712819\pi$$
−0.619882 + 0.784695i $$0.712819\pi$$
$$798$$ 0 0
$$799$$ −9.00000 −0.318397
$$800$$ 0 0
$$801$$ 32.0000 1.13066
$$802$$ 0 0
$$803$$ −50.0000 −1.76446
$$804$$ 0 0
$$805$$ 6.00000 0.211472
$$806$$ 0 0
$$807$$ 10.0000 0.352017
$$808$$ 0 0
$$809$$ −23.0000 −0.808637 −0.404318 0.914618i $$-0.632491\pi$$
−0.404318 + 0.914618i $$0.632491\pi$$
$$810$$ 0 0
$$811$$ 38.0000 1.33436 0.667180 0.744896i $$-0.267501\pi$$
0.667180 + 0.744896i $$0.267501\pi$$
$$812$$ 0 0
$$813$$ 4.00000 0.140286
$$814$$ 0 0
$$815$$ 6.00000 0.210171
$$816$$ 0 0
$$817$$ −36.0000 −1.25948
$$818$$ 0 0
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ 7.00000 0.244302 0.122151 0.992512i $$-0.461021\pi$$
0.122151 + 0.992512i $$0.461021\pi$$
$$822$$ 0 0
$$823$$ −32.0000 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$824$$ 0 0
$$825$$ 5.00000 0.174078
$$826$$ 0 0
$$827$$ 2.00000 0.0695468 0.0347734 0.999395i $$-0.488929\pi$$
0.0347734 + 0.999395i $$0.488929\pi$$
$$828$$ 0 0
$$829$$ −16.0000 −0.555703 −0.277851 0.960624i $$-0.589622\pi$$
−0.277851 + 0.960624i $$0.589622\pi$$
$$830$$ 0 0
$$831$$ 14.0000 0.485655
$$832$$ 0 0
$$833$$ 3.00000 0.103944
$$834$$ 0 0
$$835$$ 9.00000 0.311458
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 26.0000 0.897620 0.448810 0.893627i $$-0.351848\pi$$
0.448810 + 0.893627i $$0.351848\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 13.0000 0.447744
$$844$$ 0 0
$$845$$ −12.0000 −0.412813
$$846$$ 0 0
$$847$$ 14.0000 0.481046
$$848$$ 0 0
$$849$$ −29.0000 −0.995277
$$850$$ 0 0
$$851$$ −36.0000 −1.23406
$$852$$ 0 0
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ 0 0
$$855$$ 12.0000 0.410391
$$856$$ 0 0
$$857$$ −14.0000 −0.478231 −0.239115 0.970991i $$-0.576857\pi$$
−0.239115 + 0.970991i $$0.576857\pi$$
$$858$$ 0 0
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ −20.0000 −0.680808 −0.340404 0.940279i $$-0.610564\pi$$
−0.340404 + 0.940279i $$0.610564\pi$$
$$864$$ 0 0
$$865$$ −19.0000 −0.646019
$$866$$ 0 0
$$867$$ 8.00000 0.271694
$$868$$ 0 0
$$869$$ −15.0000 −0.508840
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 0 0
$$873$$ −14.0000 −0.473828
$$874$$ 0 0
$$875$$ 1.00000 0.0338062
$$876$$ 0 0
$$877$$ 10.0000 0.337676 0.168838 0.985644i $$-0.445999\pi$$
0.168838 + 0.985644i $$0.445999\pi$$
$$878$$ 0 0
$$879$$ −1.00000 −0.0337292
$$880$$ 0 0
$$881$$ −16.0000 −0.539054 −0.269527 0.962993i $$-0.586867\pi$$
−0.269527 + 0.962993i $$0.586867\pi$$
$$882$$ 0 0
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ 0 0
$$885$$ −8.00000 −0.268917
$$886$$ 0 0
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ −5.00000 −0.167506
$$892$$ 0 0
$$893$$ 18.0000 0.602347
$$894$$ 0 0
$$895$$ 4.00000 0.133705
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ −6.00000 −0.199667
$$904$$ 0 0
$$905$$ 20.0000 0.664822
$$906$$ 0 0
$$907$$ −30.0000 −0.996134 −0.498067 0.867139i $$-0.665957\pi$$
−0.498067 + 0.867139i $$0.665957\pi$$
$$908$$ 0 0
$$909$$ −28.0000 −0.928701
$$910$$ 0 0
$$911$$ 40.0000 1.32526 0.662630 0.748947i $$-0.269440\pi$$
0.662630 + 0.748947i $$0.269440\pi$$
$$912$$ 0 0
$$913$$ 60.0000 1.98571
$$914$$ 0 0
$$915$$ −4.00000 −0.132236
$$916$$ 0 0
$$917$$ −6.00000 −0.198137
$$918$$ 0 0
$$919$$ −25.0000 −0.824674 −0.412337 0.911031i $$-0.635287\pi$$
−0.412337 + 0.911031i $$0.635287\pi$$
$$920$$ 0 0
$$921$$ 27.0000 0.889680
$$922$$ 0 0
$$923$$ 8.00000 0.263323
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ 0 0
$$927$$ −18.0000 −0.591198
$$928$$ 0 0
$$929$$ 36.0000 1.18112 0.590561 0.806993i $$-0.298907\pi$$
0.590561 + 0.806993i $$0.298907\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 0 0
$$933$$ −14.0000 −0.458339
$$934$$ 0 0
$$935$$ −15.0000 −0.490552
$$936$$ 0 0
$$937$$ 7.00000 0.228680 0.114340 0.993442i $$-0.463525\pi$$
0.114340 + 0.993442i $$0.463525\pi$$
$$938$$ 0 0
$$939$$ −29.0000 −0.946379
$$940$$ 0 0
$$941$$ −4.00000 −0.130396 −0.0651981 0.997872i $$-0.520768\pi$$
−0.0651981 + 0.997872i $$0.520768\pi$$
$$942$$ 0 0
$$943$$ 48.0000 1.56310
$$944$$ 0 0
$$945$$ 5.00000 0.162650
$$946$$ 0 0
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ 0 0
$$949$$ −10.0000 −0.324614
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ 0 0
$$953$$ 20.0000 0.647864 0.323932 0.946080i $$-0.394995\pi$$
0.323932 + 0.946080i $$0.394995\pi$$
$$954$$ 0 0
$$955$$ −11.0000 −0.355952
$$956$$ 0 0
$$957$$ 45.0000 1.45464
$$958$$ 0 0
$$959$$ −16.0000 −0.516667
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −4.00000 −0.128898
$$964$$ 0 0
$$965$$ 8.00000 0.257529
$$966$$ 0 0
$$967$$ −22.0000 −0.707472 −0.353736 0.935345i $$-0.615089\pi$$
−0.353736 + 0.935345i $$0.615089\pi$$
$$968$$ 0 0
$$969$$ 18.0000 0.578243
$$970$$ 0 0
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ 0 0
$$973$$ 18.0000 0.577054
$$974$$ 0 0
$$975$$ 1.00000 0.0320256
$$976$$ 0 0
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 0 0
$$979$$ 80.0000 2.55681
$$980$$ 0 0
$$981$$ −22.0000 −0.702406
$$982$$ 0 0
$$983$$ −9.00000 −0.287055 −0.143528 0.989646i $$-0.545845\pi$$
−0.143528 + 0.989646i $$0.545845\pi$$
$$984$$ 0 0
$$985$$ 12.0000 0.382352
$$986$$ 0 0
$$987$$ 3.00000 0.0954911
$$988$$ 0 0
$$989$$ 36.0000 1.14473
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ 20.0000 0.634681
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.00000 0.0316703 0.0158352 0.999875i $$-0.494959\pi$$
0.0158352 + 0.999875i $$0.494959\pi$$
$$998$$ 0 0
$$999$$ −30.0000 −0.949158
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 2240.2.a.j.1.1 1
4.3 odd 2 2240.2.a.v.1.1 1
8.3 odd 2 280.2.a.b.1.1 1
8.5 even 2 560.2.a.e.1.1 1
24.5 odd 2 5040.2.a.be.1.1 1
24.11 even 2 2520.2.a.p.1.1 1
40.3 even 4 1400.2.g.e.449.1 2
40.13 odd 4 2800.2.g.m.449.2 2
40.19 odd 2 1400.2.a.k.1.1 1
40.27 even 4 1400.2.g.e.449.2 2
40.29 even 2 2800.2.a.i.1.1 1
40.37 odd 4 2800.2.g.m.449.1 2
56.3 even 6 1960.2.q.e.961.1 2
56.11 odd 6 1960.2.q.m.961.1 2
56.13 odd 2 3920.2.a.r.1.1 1
56.19 even 6 1960.2.q.e.361.1 2
56.27 even 2 1960.2.a.k.1.1 1
56.51 odd 6 1960.2.q.m.361.1 2
280.139 even 2 9800.2.a.n.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.b.1.1 1 8.3 odd 2
560.2.a.e.1.1 1 8.5 even 2
1400.2.a.k.1.1 1 40.19 odd 2
1400.2.g.e.449.1 2 40.3 even 4
1400.2.g.e.449.2 2 40.27 even 4
1960.2.a.k.1.1 1 56.27 even 2
1960.2.q.e.361.1 2 56.19 even 6
1960.2.q.e.961.1 2 56.3 even 6
1960.2.q.m.361.1 2 56.51 odd 6
1960.2.q.m.961.1 2 56.11 odd 6
2240.2.a.j.1.1 1 1.1 even 1 trivial
2240.2.a.v.1.1 1 4.3 odd 2
2520.2.a.p.1.1 1 24.11 even 2
2800.2.a.i.1.1 1 40.29 even 2
2800.2.g.m.449.1 2 40.37 odd 4
2800.2.g.m.449.2 2 40.13 odd 4
3920.2.a.r.1.1 1 56.13 odd 2
5040.2.a.be.1.1 1 24.5 odd 2
9800.2.a.n.1.1 1 280.139 even 2