# Properties

 Label 2240.2.a.v.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 $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 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 $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 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 $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 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 $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ 0 0
$$45$$ −2.00000 −0.298142
$$46$$ 0 0
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ 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 $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 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 $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 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 $$-0.553977\pi$$
−0.168763 + 0.985657i $$0.553977\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 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 $$-0.646227\pi$$
−0.443398 + 0.896325i $$0.646227\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −0.0975900
$$106$$ 0 0
$$107$$ −2.00000 −0.193347 −0.0966736 0.995316i $$-0.530820\pi$$
−0.0966736 + 0.995316i $$0.530820\pi$$
$$108$$ 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 $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ −6.00000 −0.528271
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 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 $$-0.776457\pi$$
−0.763370 + 0.645961i $$0.776457\pi$$
$$140$$ 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 $$-0.781294\pi$$
−0.773099 + 0.634285i $$0.781294\pi$$
$$152$$ 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 $$-0.575502\pi$$
−0.234978 + 0.972001i $$0.575502\pi$$
$$164$$ 0 0
$$165$$ 5.00000 0.389249
$$166$$ 0 0
$$167$$ −9.00000 −0.696441 −0.348220 0.937413i $$-0.613214\pi$$
−0.348220 + 0.937413i $$0.613214\pi$$
$$168$$ 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 $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 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 $$-0.369716\pi$$
0.397966 + 0.917400i $$0.369716\pi$$
$$192$$ 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 $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 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 $$-0.815729\pi$$
−0.837062 + 0.547108i $$0.815729\pi$$
$$224$$ 0 0
$$225$$ −2.00000 −0.133333
$$226$$ 0 0
$$227$$ 13.0000 0.862840 0.431420 0.902151i $$-0.358013\pi$$
0.431420 + 0.902151i $$0.358013\pi$$
$$228$$ 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 $$-0.572694\pi$$
−0.226396 + 0.974035i $$0.572694\pi$$
$$240$$ 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 $$-0.645673\pi$$
−0.441836 + 0.897096i $$0.645673\pi$$
$$252$$ 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 $$-0.312732\pi$$
0.554964 + 0.831875i $$0.312732\pi$$
$$264$$ 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 $$-0.461232\pi$$
0.121491 + 0.992592i $$0.461232\pi$$
$$272$$ 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 $$-0.830748\pi$$
−0.861936 + 0.507018i $$0.830748\pi$$
$$284$$ 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 $$-0.220014\pi$$
0.770486 + 0.637457i $$0.220014\pi$$
$$308$$ 0 0
$$309$$ −9.00000 −0.511992
$$310$$ 0 0
$$311$$ −14.0000 −0.793867 −0.396934 0.917847i $$-0.629926\pi$$
−0.396934 + 0.917847i $$0.629926\pi$$
$$312$$ 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 $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 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 $$-0.413500\pi$$
0.268414 + 0.963304i $$0.413500\pi$$
$$348$$ 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 $$-0.773284\pi$$
−0.756894 + 0.653538i $$0.773284\pi$$
$$368$$ 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 $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 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 $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 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 $$-0.313125\pi$$
0.553936 + 0.832560i $$0.313125\pi$$
$$432$$ 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 $$-0.801275\pi$$
−0.811366 + 0.584539i $$0.801275\pi$$
$$440$$ 0 0
$$441$$ −2.00000 −0.0952381
$$442$$ 0 0
$$443$$ −30.0000 −1.42534 −0.712672 0.701498i $$-0.752515\pi$$
−0.712672 + 0.701498i $$0.752515\pi$$
$$444$$ 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 $$-0.346148\pi$$
0.464739 + 0.885448i $$0.346148\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −33.0000 −1.52706 −0.763529 0.645774i $$-0.776535\pi$$
−0.763529 + 0.645774i $$0.776535\pi$$
$$468$$ 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 $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 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 $$-0.299488\pi$$
0.589086 + 0.808070i $$0.299488\pi$$
$$488$$ 0 0
$$489$$ −6.00000 −0.271329
$$490$$ 0 0
$$491$$ −33.0000 −1.48927 −0.744635 0.667472i $$-0.767376\pi$$
−0.744635 + 0.667472i $$0.767376\pi$$
$$492$$ 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 $$-0.310964\pi$$
0.559577 + 0.828778i $$0.310964\pi$$
$$500$$ 0 0
$$501$$ −9.00000 −0.402090
$$502$$ 0 0
$$503$$ −31.0000 −1.38222 −0.691111 0.722749i $$-0.742878\pi$$
−0.691111 + 0.722749i $$0.742878\pi$$
$$504$$ 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 $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 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 $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 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 $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 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 $$-0.228472\pi$$
0.753277 + 0.657704i $$0.228472\pi$$
$$572$$ 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 $$-0.552794\pi$$
−0.165098 + 0.986277i $$0.552794\pi$$
$$588$$ 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 $$-0.735499\pi$$
−0.674172 + 0.738575i $$0.735499\pi$$
$$600$$ 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 $$-0.387877\pi$$
0.345004 + 0.938601i $$0.387877\pi$$
$$608$$ 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 $$-0.512797\pi$$
−0.0401934 + 0.999192i $$0.512797\pi$$
$$620$$ 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 $$-0.557332\pi$$
−0.179142 + 0.983823i $$0.557332\pi$$
$$632$$ 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 $$-0.877409\pi$$
−0.926750 + 0.375680i $$0.877409\pi$$
$$644$$ 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 $$-0.338116\pi$$
0.486931 + 0.873441i $$0.338116\pi$$
$$660$$ 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 $$-0.375015\pi$$
0.382639 + 0.923898i $$0.375015\pi$$
$$684$$ 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 $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 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 $$-0.117761\pi$$
0.932343 + 0.361576i $$0.117761\pi$$
$$720$$ 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 $$-0.595887\pi$$
−0.296704 + 0.954970i $$0.595887\pi$$
$$728$$ 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 $$-0.261748\pi$$
0.680534 + 0.732717i $$0.261748\pi$$
$$740$$ 0 0
$$741$$ −6.00000 −0.220416
$$742$$ 0 0
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 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 $$-0.720480\pi$$
−0.638584 + 0.769552i $$0.720480\pi$$
$$752$$ 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 $$-0.562813\pi$$
−0.196054 + 0.980593i $$0.562813\pi$$
$$788$$ 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 $$-0.732499\pi$$
−0.667180 + 0.744896i $$0.732499\pi$$
$$812$$ 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 $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ 0 0
$$825$$ 5.00000 0.174078
$$826$$ 0 0
$$827$$ −2.00000 −0.0695468 −0.0347734 0.999395i $$-0.511071\pi$$
−0.0347734 + 0.999395i $$0.511071\pi$$
$$828$$ 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 $$-0.648152\pi$$
−0.448810 + 0.893627i $$0.648152\pi$$
$$840$$ 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 $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ 20.0000 0.680808 0.340404 0.940279i $$-0.389436\pi$$
0.340404 + 0.940279i $$0.389436\pi$$
$$864$$ 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 $$-0.413234\pi$$
0.269221 + 0.963078i $$0.413234\pi$$
$$884$$ 0 0
$$885$$ −8.00000 −0.268917
$$886$$ 0 0
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ 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 $$-0.334043\pi$$
0.498067 + 0.867139i $$0.334043\pi$$
$$908$$ 0 0
$$909$$ −28.0000 −0.928701
$$910$$ 0 0
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ 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 $$-0.364713\pi$$
0.412337 + 0.911031i $$0.364713\pi$$
$$920$$ 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 $$-0.179668\pi$$
0.844886 + 0.534946i $$0.179668\pi$$
$$948$$ 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 $$-0.384911\pi$$
0.353736 + 0.935345i $$0.384911\pi$$
$$968$$ 0 0
$$969$$ 18.0000 0.578243
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 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 $$-0.454155\pi$$
0.143528 + 0.989646i $$0.454155\pi$$
$$984$$ 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 $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 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.v.1.1 1
4.3 odd 2 2240.2.a.j.1.1 1
8.3 odd 2 560.2.a.e.1.1 1
8.5 even 2 280.2.a.b.1.1 1
24.5 odd 2 2520.2.a.p.1.1 1
24.11 even 2 5040.2.a.be.1.1 1
40.3 even 4 2800.2.g.m.449.2 2
40.13 odd 4 1400.2.g.e.449.1 2
40.19 odd 2 2800.2.a.i.1.1 1
40.27 even 4 2800.2.g.m.449.1 2
40.29 even 2 1400.2.a.k.1.1 1
40.37 odd 4 1400.2.g.e.449.2 2
56.5 odd 6 1960.2.q.e.361.1 2
56.13 odd 2 1960.2.a.k.1.1 1
56.27 even 2 3920.2.a.r.1.1 1
56.37 even 6 1960.2.q.m.361.1 2
56.45 odd 6 1960.2.q.e.961.1 2
56.53 even 6 1960.2.q.m.961.1 2
280.69 odd 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 8.5 even 2
560.2.a.e.1.1 1 8.3 odd 2
1400.2.a.k.1.1 1 40.29 even 2
1400.2.g.e.449.1 2 40.13 odd 4
1400.2.g.e.449.2 2 40.37 odd 4
1960.2.a.k.1.1 1 56.13 odd 2
1960.2.q.e.361.1 2 56.5 odd 6
1960.2.q.e.961.1 2 56.45 odd 6
1960.2.q.m.361.1 2 56.37 even 6
1960.2.q.m.961.1 2 56.53 even 6
2240.2.a.j.1.1 1 4.3 odd 2
2240.2.a.v.1.1 1 1.1 even 1 trivial
2520.2.a.p.1.1 1 24.5 odd 2
2800.2.a.i.1.1 1 40.19 odd 2
2800.2.g.m.449.1 2 40.27 even 4
2800.2.g.m.449.2 2 40.3 even 4
3920.2.a.r.1.1 1 56.27 even 2
5040.2.a.be.1.1 1 24.11 even 2
9800.2.a.n.1.1 1 280.69 odd 2