# Properties

 Label 7800.2.a.be.1.1 Level $7800$ Weight $2$ Character 7800.1 Self dual yes Analytic conductor $62.283$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7800.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: $$62.2833135766$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1560) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 7800.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -2.70156 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -2.70156 q^{7} +1.00000 q^{9} +0.701562 q^{11} +1.00000 q^{13} +0.701562 q^{17} -2.00000 q^{19} -2.70156 q^{21} -6.70156 q^{23} +1.00000 q^{27} +9.40312 q^{29} -9.40312 q^{31} +0.701562 q^{33} +6.70156 q^{37} +1.00000 q^{39} +10.7016 q^{41} -4.00000 q^{43} +1.40312 q^{47} +0.298438 q^{49} +0.701562 q^{51} -10.7016 q^{53} -2.00000 q^{57} -14.8062 q^{59} +2.70156 q^{61} -2.70156 q^{63} +4.00000 q^{67} -6.70156 q^{69} +15.5078 q^{71} -13.4031 q^{73} -1.89531 q^{77} +4.70156 q^{79} +1.00000 q^{81} -4.00000 q^{83} +9.40312 q^{87} +8.10469 q^{89} -2.70156 q^{91} -9.40312 q^{93} -18.1047 q^{97} +0.701562 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + q^7 + 2 * q^9 $$2 q + 2 q^{3} + q^{7} + 2 q^{9} - 5 q^{11} + 2 q^{13} - 5 q^{17} - 4 q^{19} + q^{21} - 7 q^{23} + 2 q^{27} + 6 q^{29} - 6 q^{31} - 5 q^{33} + 7 q^{37} + 2 q^{39} + 15 q^{41} - 8 q^{43} - 10 q^{47} + 7 q^{49} - 5 q^{51} - 15 q^{53} - 4 q^{57} - 4 q^{59} - q^{61} + q^{63} + 8 q^{67} - 7 q^{69} - q^{71} - 14 q^{73} - 23 q^{77} + 3 q^{79} + 2 q^{81} - 8 q^{83} + 6 q^{87} - 3 q^{89} + q^{91} - 6 q^{93} - 17 q^{97} - 5 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + q^7 + 2 * q^9 - 5 * q^11 + 2 * q^13 - 5 * q^17 - 4 * q^19 + q^21 - 7 * q^23 + 2 * q^27 + 6 * q^29 - 6 * q^31 - 5 * q^33 + 7 * q^37 + 2 * q^39 + 15 * q^41 - 8 * q^43 - 10 * q^47 + 7 * q^49 - 5 * q^51 - 15 * q^53 - 4 * q^57 - 4 * q^59 - q^61 + q^63 + 8 * q^67 - 7 * q^69 - q^71 - 14 * q^73 - 23 * q^77 + 3 * q^79 + 2 * q^81 - 8 * q^83 + 6 * q^87 - 3 * q^89 + q^91 - 6 * q^93 - 17 * q^97 - 5 * q^99

## 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
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.70156 −1.02109 −0.510547 0.859850i $$-0.670557\pi$$
−0.510547 + 0.859850i $$0.670557\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0.701562 0.211529 0.105764 0.994391i $$-0.466271\pi$$
0.105764 + 0.994391i $$0.466271\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0.701562 0.170154 0.0850769 0.996374i $$-0.472886\pi$$
0.0850769 + 0.996374i $$0.472886\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −2.70156 −0.589529
$$22$$ 0 0
$$23$$ −6.70156 −1.39737 −0.698686 0.715428i $$-0.746232\pi$$
−0.698686 + 0.715428i $$0.746232\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 9.40312 1.74612 0.873058 0.487616i $$-0.162133\pi$$
0.873058 + 0.487616i $$0.162133\pi$$
$$30$$ 0 0
$$31$$ −9.40312 −1.68885 −0.844425 0.535673i $$-0.820058\pi$$
−0.844425 + 0.535673i $$0.820058\pi$$
$$32$$ 0 0
$$33$$ 0.701562 0.122126
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.70156 1.10173 0.550865 0.834594i $$-0.314298\pi$$
0.550865 + 0.834594i $$0.314298\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 10.7016 1.67130 0.835652 0.549260i $$-0.185090\pi$$
0.835652 + 0.549260i $$0.185090\pi$$
$$42$$ 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
$$46$$ 0 0
$$47$$ 1.40312 0.204667 0.102333 0.994750i $$-0.467369\pi$$
0.102333 + 0.994750i $$0.467369\pi$$
$$48$$ 0 0
$$49$$ 0.298438 0.0426340
$$50$$ 0 0
$$51$$ 0.701562 0.0982383
$$52$$ 0 0
$$53$$ −10.7016 −1.46997 −0.734986 0.678082i $$-0.762811\pi$$
−0.734986 + 0.678082i $$0.762811\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ 0 0
$$59$$ −14.8062 −1.92761 −0.963805 0.266609i $$-0.914097\pi$$
−0.963805 + 0.266609i $$0.914097\pi$$
$$60$$ 0 0
$$61$$ 2.70156 0.345900 0.172950 0.984931i $$-0.444670\pi$$
0.172950 + 0.984931i $$0.444670\pi$$
$$62$$ 0 0
$$63$$ −2.70156 −0.340365
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ −6.70156 −0.806773
$$70$$ 0 0
$$71$$ 15.5078 1.84044 0.920219 0.391403i $$-0.128010\pi$$
0.920219 + 0.391403i $$0.128010\pi$$
$$72$$ 0 0
$$73$$ −13.4031 −1.56872 −0.784359 0.620308i $$-0.787008\pi$$
−0.784359 + 0.620308i $$0.787008\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1.89531 −0.215991
$$78$$ 0 0
$$79$$ 4.70156 0.528967 0.264484 0.964390i $$-0.414799\pi$$
0.264484 + 0.964390i $$0.414799\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 9.40312 1.00812
$$88$$ 0 0
$$89$$ 8.10469 0.859095 0.429548 0.903044i $$-0.358673\pi$$
0.429548 + 0.903044i $$0.358673\pi$$
$$90$$ 0 0
$$91$$ −2.70156 −0.283201
$$92$$ 0 0
$$93$$ −9.40312 −0.975059
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −18.1047 −1.83825 −0.919126 0.393963i $$-0.871104\pi$$
−0.919126 + 0.393963i $$0.871104\pi$$
$$98$$ 0 0
$$99$$ 0.701562 0.0705096
$$100$$ 0 0
$$101$$ −6.80625 −0.677247 −0.338624 0.940922i $$-0.609961\pi$$
−0.338624 + 0.940922i $$0.609961\pi$$
$$102$$ 0 0
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −18.1047 −1.75025 −0.875123 0.483900i $$-0.839220\pi$$
−0.875123 + 0.483900i $$0.839220\pi$$
$$108$$ 0 0
$$109$$ 10.8062 1.03505 0.517525 0.855668i $$-0.326853\pi$$
0.517525 + 0.855668i $$0.326853\pi$$
$$110$$ 0 0
$$111$$ 6.70156 0.636084
$$112$$ 0 0
$$113$$ −6.59688 −0.620582 −0.310291 0.950642i $$-0.600426\pi$$
−0.310291 + 0.950642i $$0.600426\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ −1.89531 −0.173743
$$120$$ 0 0
$$121$$ −10.5078 −0.955256
$$122$$ 0 0
$$123$$ 10.7016 0.964927
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 13.4031 1.18933 0.594667 0.803972i $$-0.297284\pi$$
0.594667 + 0.803972i $$0.297284\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −12.8062 −1.11889 −0.559444 0.828868i $$-0.688985\pi$$
−0.559444 + 0.828868i $$0.688985\pi$$
$$132$$ 0 0
$$133$$ 5.40312 0.468510
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ −4.70156 −0.398781 −0.199391 0.979920i $$-0.563896\pi$$
−0.199391 + 0.979920i $$0.563896\pi$$
$$140$$ 0 0
$$141$$ 1.40312 0.118164
$$142$$ 0 0
$$143$$ 0.701562 0.0586676
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0.298438 0.0246147
$$148$$ 0 0
$$149$$ 1.29844 0.106372 0.0531861 0.998585i $$-0.483062\pi$$
0.0531861 + 0.998585i $$0.483062\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0.701562 0.0567179
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −8.80625 −0.702815 −0.351408 0.936223i $$-0.614297\pi$$
−0.351408 + 0.936223i $$0.614297\pi$$
$$158$$ 0 0
$$159$$ −10.7016 −0.848689
$$160$$ 0 0
$$161$$ 18.1047 1.42685
$$162$$ 0 0
$$163$$ −11.2984 −0.884962 −0.442481 0.896778i $$-0.645902\pi$$
−0.442481 + 0.896778i $$0.645902\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 22.8062 1.76480 0.882400 0.470500i $$-0.155926\pi$$
0.882400 + 0.470500i $$0.155926\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −14.8062 −1.11291
$$178$$ 0 0
$$179$$ −22.0000 −1.64436 −0.822179 0.569230i $$-0.807242\pi$$
−0.822179 + 0.569230i $$0.807242\pi$$
$$180$$ 0 0
$$181$$ −1.29844 −0.0965121 −0.0482561 0.998835i $$-0.515366\pi$$
−0.0482561 + 0.998835i $$0.515366\pi$$
$$182$$ 0 0
$$183$$ 2.70156 0.199705
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.492189 0.0359925
$$188$$ 0 0
$$189$$ −2.70156 −0.196510
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 22.1047 1.59113 0.795565 0.605868i $$-0.207174\pi$$
0.795565 + 0.605868i $$0.207174\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 14.2094 1.01238 0.506188 0.862423i $$-0.331054\pi$$
0.506188 + 0.862423i $$0.331054\pi$$
$$198$$ 0 0
$$199$$ 2.80625 0.198930 0.0994648 0.995041i $$-0.468287\pi$$
0.0994648 + 0.995041i $$0.468287\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ −25.4031 −1.78295
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −6.70156 −0.465791
$$208$$ 0 0
$$209$$ −1.40312 −0.0970561
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 0 0
$$213$$ 15.5078 1.06258
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 25.4031 1.72448
$$218$$ 0 0
$$219$$ −13.4031 −0.905699
$$220$$ 0 0
$$221$$ 0.701562 0.0471922
$$222$$ 0 0
$$223$$ −3.40312 −0.227890 −0.113945 0.993487i $$-0.536349\pi$$
−0.113945 + 0.993487i $$0.536349\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 10.5969 0.703339 0.351670 0.936124i $$-0.385614\pi$$
0.351670 + 0.936124i $$0.385614\pi$$
$$228$$ 0 0
$$229$$ −25.4031 −1.67869 −0.839343 0.543602i $$-0.817060\pi$$
−0.839343 + 0.543602i $$0.817060\pi$$
$$230$$ 0 0
$$231$$ −1.89531 −0.124702
$$232$$ 0 0
$$233$$ 10.1047 0.661980 0.330990 0.943634i $$-0.392617\pi$$
0.330990 + 0.943634i $$0.392617\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 4.70156 0.305399
$$238$$ 0 0
$$239$$ −6.10469 −0.394879 −0.197440 0.980315i $$-0.563263\pi$$
−0.197440 + 0.980315i $$0.563263\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ −18.2094 −1.14937 −0.574683 0.818376i $$-0.694875\pi$$
−0.574683 + 0.818376i $$0.694875\pi$$
$$252$$ 0 0
$$253$$ −4.70156 −0.295585
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −20.2094 −1.26063 −0.630313 0.776341i $$-0.717073\pi$$
−0.630313 + 0.776341i $$0.717073\pi$$
$$258$$ 0 0
$$259$$ −18.1047 −1.12497
$$260$$ 0 0
$$261$$ 9.40312 0.582039
$$262$$ 0 0
$$263$$ −4.59688 −0.283456 −0.141728 0.989906i $$-0.545266\pi$$
−0.141728 + 0.989906i $$0.545266\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 8.10469 0.495999
$$268$$ 0 0
$$269$$ −14.8062 −0.902753 −0.451376 0.892334i $$-0.649067\pi$$
−0.451376 + 0.892334i $$0.649067\pi$$
$$270$$ 0 0
$$271$$ −6.59688 −0.400732 −0.200366 0.979721i $$-0.564213\pi$$
−0.200366 + 0.979721i $$0.564213\pi$$
$$272$$ 0 0
$$273$$ −2.70156 −0.163506
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8.80625 −0.529116 −0.264558 0.964370i $$-0.585226\pi$$
−0.264558 + 0.964370i $$0.585226\pi$$
$$278$$ 0 0
$$279$$ −9.40312 −0.562950
$$280$$ 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 $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −28.9109 −1.70656
$$288$$ 0 0
$$289$$ −16.5078 −0.971048
$$290$$ 0 0
$$291$$ −18.1047 −1.06132
$$292$$ 0 0
$$293$$ −15.4031 −0.899860 −0.449930 0.893064i $$-0.648551\pi$$
−0.449930 + 0.893064i $$0.648551\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0.701562 0.0407088
$$298$$ 0 0
$$299$$ −6.70156 −0.387561
$$300$$ 0 0
$$301$$ 10.8062 0.622862
$$302$$ 0 0
$$303$$ −6.80625 −0.391009
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 13.8953 0.793047 0.396524 0.918024i $$-0.370216\pi$$
0.396524 + 0.918024i $$0.370216\pi$$
$$308$$ 0 0
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ −16.8062 −0.949945 −0.474973 0.880001i $$-0.657542\pi$$
−0.474973 + 0.880001i $$0.657542\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −14.2094 −0.798078 −0.399039 0.916934i $$-0.630656\pi$$
−0.399039 + 0.916934i $$0.630656\pi$$
$$318$$ 0 0
$$319$$ 6.59688 0.369354
$$320$$ 0 0
$$321$$ −18.1047 −1.01051
$$322$$ 0 0
$$323$$ −1.40312 −0.0780719
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 10.8062 0.597587
$$328$$ 0 0
$$329$$ −3.79063 −0.208984
$$330$$ 0 0
$$331$$ −14.2094 −0.781018 −0.390509 0.920599i $$-0.627701\pi$$
−0.390509 + 0.920599i $$0.627701\pi$$
$$332$$ 0 0
$$333$$ 6.70156 0.367243
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 6.20937 0.338246 0.169123 0.985595i $$-0.445906\pi$$
0.169123 + 0.985595i $$0.445906\pi$$
$$338$$ 0 0
$$339$$ −6.59688 −0.358293
$$340$$ 0 0
$$341$$ −6.59688 −0.357241
$$342$$ 0 0
$$343$$ 18.1047 0.977561
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −6.10469 −0.327717 −0.163858 0.986484i $$-0.552394\pi$$
−0.163858 + 0.986484i $$0.552394\pi$$
$$348$$ 0 0
$$349$$ −34.8062 −1.86314 −0.931568 0.363567i $$-0.881559\pi$$
−0.931568 + 0.363567i $$0.881559\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ −12.8062 −0.681608 −0.340804 0.940134i $$-0.610699\pi$$
−0.340804 + 0.940134i $$0.610699\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −1.89531 −0.100311
$$358$$ 0 0
$$359$$ −21.6125 −1.14066 −0.570332 0.821414i $$-0.693185\pi$$
−0.570332 + 0.821414i $$0.693185\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ −10.5078 −0.551517
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −1.19375 −0.0623133 −0.0311567 0.999515i $$-0.509919\pi$$
−0.0311567 + 0.999515i $$0.509919\pi$$
$$368$$ 0 0
$$369$$ 10.7016 0.557101
$$370$$ 0 0
$$371$$ 28.9109 1.50098
$$372$$ 0 0
$$373$$ 8.59688 0.445129 0.222565 0.974918i $$-0.428557\pi$$
0.222565 + 0.974918i $$0.428557\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 9.40312 0.484286
$$378$$ 0 0
$$379$$ 15.6125 0.801960 0.400980 0.916087i $$-0.368670\pi$$
0.400980 + 0.916087i $$0.368670\pi$$
$$380$$ 0 0
$$381$$ 13.4031 0.686663
$$382$$ 0 0
$$383$$ 5.40312 0.276087 0.138043 0.990426i $$-0.455919\pi$$
0.138043 + 0.990426i $$0.455919\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ 25.6125 1.29861 0.649303 0.760530i $$-0.275061\pi$$
0.649303 + 0.760530i $$0.275061\pi$$
$$390$$ 0 0
$$391$$ −4.70156 −0.237768
$$392$$ 0 0
$$393$$ −12.8062 −0.645990
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 22.7016 1.13936 0.569679 0.821867i $$-0.307067\pi$$
0.569679 + 0.821867i $$0.307067\pi$$
$$398$$ 0 0
$$399$$ 5.40312 0.270495
$$400$$ 0 0
$$401$$ 0.806248 0.0402621 0.0201311 0.999797i $$-0.493592\pi$$
0.0201311 + 0.999797i $$0.493592\pi$$
$$402$$ 0 0
$$403$$ −9.40312 −0.468403
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.70156 0.233048
$$408$$ 0 0
$$409$$ −4.80625 −0.237654 −0.118827 0.992915i $$-0.537913\pi$$
−0.118827 + 0.992915i $$0.537913\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ 0 0
$$413$$ 40.0000 1.96827
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −4.70156 −0.230236
$$418$$ 0 0
$$419$$ 27.6125 1.34896 0.674479 0.738294i $$-0.264368\pi$$
0.674479 + 0.738294i $$0.264368\pi$$
$$420$$ 0 0
$$421$$ −25.6125 −1.24828 −0.624138 0.781314i $$-0.714550\pi$$
−0.624138 + 0.781314i $$0.714550\pi$$
$$422$$ 0 0
$$423$$ 1.40312 0.0682222
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −7.29844 −0.353196
$$428$$ 0 0
$$429$$ 0.701562 0.0338717
$$430$$ 0 0
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ 0 0
$$433$$ −12.5969 −0.605367 −0.302684 0.953091i $$-0.597883\pi$$
−0.302684 + 0.953091i $$0.597883\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 13.4031 0.641158
$$438$$ 0 0
$$439$$ −1.89531 −0.0904584 −0.0452292 0.998977i $$-0.514402\pi$$
−0.0452292 + 0.998977i $$0.514402\pi$$
$$440$$ 0 0
$$441$$ 0.298438 0.0142113
$$442$$ 0 0
$$443$$ 26.3141 1.25022 0.625109 0.780537i $$-0.285054\pi$$
0.625109 + 0.780537i $$0.285054\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 1.29844 0.0614140
$$448$$ 0 0
$$449$$ 3.89531 0.183831 0.0919156 0.995767i $$-0.470701\pi$$
0.0919156 + 0.995767i $$0.470701\pi$$
$$450$$ 0 0
$$451$$ 7.50781 0.353529
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −7.29844 −0.341407 −0.170703 0.985322i $$-0.554604\pi$$
−0.170703 + 0.985322i $$0.554604\pi$$
$$458$$ 0 0
$$459$$ 0.701562 0.0327461
$$460$$ 0 0
$$461$$ 19.8953 0.926617 0.463309 0.886197i $$-0.346662\pi$$
0.463309 + 0.886197i $$0.346662\pi$$
$$462$$ 0 0
$$463$$ 4.10469 0.190761 0.0953805 0.995441i $$-0.469593\pi$$
0.0953805 + 0.995441i $$0.469593\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −18.1047 −0.837785 −0.418892 0.908036i $$-0.637582\pi$$
−0.418892 + 0.908036i $$0.637582\pi$$
$$468$$ 0 0
$$469$$ −10.8062 −0.498986
$$470$$ 0 0
$$471$$ −8.80625 −0.405771
$$472$$ 0 0
$$473$$ −2.80625 −0.129031
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −10.7016 −0.489991
$$478$$ 0 0
$$479$$ −28.7016 −1.31141 −0.655704 0.755018i $$-0.727628\pi$$
−0.655704 + 0.755018i $$0.727628\pi$$
$$480$$ 0 0
$$481$$ 6.70156 0.305565
$$482$$ 0 0
$$483$$ 18.1047 0.823792
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 36.3141 1.64555 0.822774 0.568369i $$-0.192426\pi$$
0.822774 + 0.568369i $$0.192426\pi$$
$$488$$ 0 0
$$489$$ −11.2984 −0.510933
$$490$$ 0 0
$$491$$ −6.20937 −0.280225 −0.140113 0.990136i $$-0.544746\pi$$
−0.140113 + 0.990136i $$0.544746\pi$$
$$492$$ 0 0
$$493$$ 6.59688 0.297108
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −41.8953 −1.87926
$$498$$ 0 0
$$499$$ 15.1938 0.680166 0.340083 0.940395i $$-0.389545\pi$$
0.340083 + 0.940395i $$0.389545\pi$$
$$500$$ 0 0
$$501$$ 22.8062 1.01891
$$502$$ 0 0
$$503$$ −31.4031 −1.40020 −0.700098 0.714047i $$-0.746860\pi$$
−0.700098 + 0.714047i $$0.746860\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ −32.3141 −1.43230 −0.716148 0.697949i $$-0.754096\pi$$
−0.716148 + 0.697949i $$0.754096\pi$$
$$510$$ 0 0
$$511$$ 36.2094 1.60181
$$512$$ 0 0
$$513$$ −2.00000 −0.0883022
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0.984379 0.0432929
$$518$$ 0 0
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −6.59688 −0.287364
$$528$$ 0 0
$$529$$ 21.9109 0.952649
$$530$$ 0 0
$$531$$ −14.8062 −0.642536
$$532$$ 0 0
$$533$$ 10.7016 0.463536
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −22.0000 −0.949370
$$538$$ 0 0
$$539$$ 0.209373 0.00901832
$$540$$ 0 0
$$541$$ 40.2094 1.72874 0.864368 0.502860i $$-0.167719\pi$$
0.864368 + 0.502860i $$0.167719\pi$$
$$542$$ 0 0
$$543$$ −1.29844 −0.0557213
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 25.6125 1.09511 0.547556 0.836769i $$-0.315558\pi$$
0.547556 + 0.836769i $$0.315558\pi$$
$$548$$ 0 0
$$549$$ 2.70156 0.115300
$$550$$ 0 0
$$551$$ −18.8062 −0.801173
$$552$$ 0 0
$$553$$ −12.7016 −0.540125
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 0.492189 0.0207803
$$562$$ 0 0
$$563$$ −22.1047 −0.931601 −0.465801 0.884890i $$-0.654234\pi$$
−0.465801 + 0.884890i $$0.654234\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −2.70156 −0.113455
$$568$$ 0 0
$$569$$ −27.4031 −1.14880 −0.574399 0.818575i $$-0.694764\pi$$
−0.574399 + 0.818575i $$0.694764\pi$$
$$570$$ 0 0
$$571$$ 26.3141 1.10121 0.550605 0.834766i $$-0.314397\pi$$
0.550605 + 0.834766i $$0.314397\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 31.5078 1.31169 0.655844 0.754897i $$-0.272313\pi$$
0.655844 + 0.754897i $$0.272313\pi$$
$$578$$ 0 0
$$579$$ 22.1047 0.918639
$$580$$ 0 0
$$581$$ 10.8062 0.448319
$$582$$ 0 0
$$583$$ −7.50781 −0.310942
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 24.2094 0.999228 0.499614 0.866248i $$-0.333475\pi$$
0.499614 + 0.866248i $$0.333475\pi$$
$$588$$ 0 0
$$589$$ 18.8062 0.774898
$$590$$ 0 0
$$591$$ 14.2094 0.584495
$$592$$ 0 0
$$593$$ 3.40312 0.139750 0.0698748 0.997556i $$-0.477740\pi$$
0.0698748 + 0.997556i $$0.477740\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 2.80625 0.114852
$$598$$ 0 0
$$599$$ −42.8062 −1.74902 −0.874508 0.485011i $$-0.838816\pi$$
−0.874508 + 0.485011i $$0.838816\pi$$
$$600$$ 0 0
$$601$$ 9.29844 0.379291 0.189646 0.981853i $$-0.439266\pi$$
0.189646 + 0.981853i $$0.439266\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 41.6125 1.68900 0.844500 0.535556i $$-0.179898\pi$$
0.844500 + 0.535556i $$0.179898\pi$$
$$608$$ 0 0
$$609$$ −25.4031 −1.02939
$$610$$ 0 0
$$611$$ 1.40312 0.0567643
$$612$$ 0 0
$$613$$ 45.7172 1.84650 0.923250 0.384200i $$-0.125523\pi$$
0.923250 + 0.384200i $$0.125523\pi$$
$$614$$ 0 0
$$615$$ 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
$$619$$ 39.6125 1.59216 0.796080 0.605191i $$-0.206903\pi$$
0.796080 + 0.605191i $$0.206903\pi$$
$$620$$ 0 0
$$621$$ −6.70156 −0.268924
$$622$$ 0 0
$$623$$ −21.8953 −0.877217
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −1.40312 −0.0560354
$$628$$ 0 0
$$629$$ 4.70156 0.187464
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.298438 0.0118245
$$638$$ 0 0
$$639$$ 15.5078 0.613480
$$640$$ 0 0
$$641$$ −10.2094 −0.403246 −0.201623 0.979463i $$-0.564622\pi$$
−0.201623 + 0.979463i $$0.564622\pi$$
$$642$$ 0 0
$$643$$ −31.2984 −1.23429 −0.617145 0.786849i $$-0.711711\pi$$
−0.617145 + 0.786849i $$0.711711\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −20.1047 −0.790397 −0.395198 0.918596i $$-0.629324\pi$$
−0.395198 + 0.918596i $$0.629324\pi$$
$$648$$ 0 0
$$649$$ −10.3875 −0.407745
$$650$$ 0 0
$$651$$ 25.4031 0.995627
$$652$$ 0 0
$$653$$ −15.6125 −0.610964 −0.305482 0.952198i $$-0.598818\pi$$
−0.305482 + 0.952198i $$0.598818\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −13.4031 −0.522906
$$658$$ 0 0
$$659$$ −27.4031 −1.06747 −0.533737 0.845650i $$-0.679213\pi$$
−0.533737 + 0.845650i $$0.679213\pi$$
$$660$$ 0 0
$$661$$ −12.0000 −0.466746 −0.233373 0.972387i $$-0.574976\pi$$
−0.233373 + 0.972387i $$0.574976\pi$$
$$662$$ 0 0
$$663$$ 0.701562 0.0272464
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −63.0156 −2.43997
$$668$$ 0 0
$$669$$ −3.40312 −0.131572
$$670$$ 0 0
$$671$$ 1.89531 0.0731678
$$672$$ 0 0
$$673$$ 3.19375 0.123110 0.0615550 0.998104i $$-0.480394\pi$$
0.0615550 + 0.998104i $$0.480394\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 4.10469 0.157756 0.0788780 0.996884i $$-0.474866\pi$$
0.0788780 + 0.996884i $$0.474866\pi$$
$$678$$ 0 0
$$679$$ 48.9109 1.87703
$$680$$ 0 0
$$681$$ 10.5969 0.406073
$$682$$ 0 0
$$683$$ −28.0000 −1.07139 −0.535695 0.844411i $$-0.679950\pi$$
−0.535695 + 0.844411i $$0.679950\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −25.4031 −0.969190
$$688$$ 0 0
$$689$$ −10.7016 −0.407697
$$690$$ 0 0
$$691$$ −10.0000 −0.380418 −0.190209 0.981744i $$-0.560917\pi$$
−0.190209 + 0.981744i $$0.560917\pi$$
$$692$$ 0 0
$$693$$ −1.89531 −0.0719970
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 7.50781 0.284379
$$698$$ 0 0
$$699$$ 10.1047 0.382194
$$700$$ 0 0
$$701$$ −25.6125 −0.967371 −0.483685 0.875242i $$-0.660702\pi$$
−0.483685 + 0.875242i $$0.660702\pi$$
$$702$$ 0 0
$$703$$ −13.4031 −0.505508
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 18.3875 0.691533
$$708$$ 0 0
$$709$$ 42.8062 1.60762 0.803811 0.594884i $$-0.202802\pi$$
0.803811 + 0.594884i $$0.202802\pi$$
$$710$$ 0 0
$$711$$ 4.70156 0.176322
$$712$$ 0 0
$$713$$ 63.0156 2.35995
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −6.10469 −0.227984
$$718$$ 0 0
$$719$$ 9.19375 0.342869 0.171435 0.985196i $$-0.445160\pi$$
0.171435 + 0.985196i $$0.445160\pi$$
$$720$$ 0 0
$$721$$ 32.4187 1.20734
$$722$$ 0 0
$$723$$ 2.00000 0.0743808
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 9.40312 0.348743 0.174371 0.984680i $$-0.444211\pi$$
0.174371 + 0.984680i $$0.444211\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −2.80625 −0.103793
$$732$$ 0 0
$$733$$ −41.7172 −1.54086 −0.770430 0.637525i $$-0.779958\pi$$
−0.770430 + 0.637525i $$0.779958\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 2.80625 0.103369
$$738$$ 0 0
$$739$$ −8.80625 −0.323943 −0.161972 0.986795i $$-0.551785\pi$$
−0.161972 + 0.986795i $$0.551785\pi$$
$$740$$ 0 0
$$741$$ −2.00000 −0.0734718
$$742$$ 0 0
$$743$$ 10.5969 0.388762 0.194381 0.980926i $$-0.437730\pi$$
0.194381 + 0.980926i $$0.437730\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −4.00000 −0.146352
$$748$$ 0 0
$$749$$ 48.9109 1.78717
$$750$$ 0 0
$$751$$ −42.3141 −1.54406 −0.772031 0.635585i $$-0.780759\pi$$
−0.772031 + 0.635585i $$0.780759\pi$$
$$752$$ 0 0
$$753$$ −18.2094 −0.663586
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 47.4031 1.72290 0.861448 0.507846i $$-0.169558\pi$$
0.861448 + 0.507846i $$0.169558\pi$$
$$758$$ 0 0
$$759$$ −4.70156 −0.170656
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ −29.1938 −1.05688
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −14.8062 −0.534623
$$768$$ 0 0
$$769$$ 51.4031 1.85364 0.926822 0.375501i $$-0.122529\pi$$
0.926822 + 0.375501i $$0.122529\pi$$
$$770$$ 0 0
$$771$$ −20.2094 −0.727823
$$772$$ 0 0
$$773$$ 32.5969 1.17243 0.586214 0.810156i $$-0.300618\pi$$
0.586214 + 0.810156i $$0.300618\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −18.1047 −0.649502
$$778$$ 0 0
$$779$$ −21.4031 −0.766847
$$780$$ 0 0
$$781$$ 10.8797 0.389306
$$782$$ 0 0
$$783$$ 9.40312 0.336040
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −5.19375 −0.185137 −0.0925686 0.995706i $$-0.529508\pi$$
−0.0925686 + 0.995706i $$0.529508\pi$$
$$788$$ 0 0
$$789$$ −4.59688 −0.163653
$$790$$ 0 0
$$791$$ 17.8219 0.633673
$$792$$ 0 0
$$793$$ 2.70156 0.0959353
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 14.7016 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$798$$ 0 0
$$799$$ 0.984379 0.0348248
$$800$$ 0 0
$$801$$ 8.10469 0.286365
$$802$$ 0 0
$$803$$ −9.40312 −0.331829
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −14.8062 −0.521205
$$808$$ 0 0
$$809$$ −0.806248 −0.0283462 −0.0141731 0.999900i $$-0.504512\pi$$
−0.0141731 + 0.999900i $$0.504512\pi$$
$$810$$ 0 0
$$811$$ −14.2094 −0.498959 −0.249479 0.968380i $$-0.580259\pi$$
−0.249479 + 0.968380i $$0.580259\pi$$
$$812$$ 0 0
$$813$$ −6.59688 −0.231363
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 8.00000 0.279885
$$818$$ 0 0
$$819$$ −2.70156 −0.0944002
$$820$$ 0 0
$$821$$ −2.49219 −0.0869780 −0.0434890 0.999054i $$-0.513847\pi$$
−0.0434890 + 0.999054i $$0.513847\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −34.5969 −1.20305 −0.601526 0.798854i $$-0.705440\pi$$
−0.601526 + 0.798854i $$0.705440\pi$$
$$828$$ 0 0
$$829$$ −47.6125 −1.65365 −0.826825 0.562459i $$-0.809855\pi$$
−0.826825 + 0.562459i $$0.809855\pi$$
$$830$$ 0 0
$$831$$ −8.80625 −0.305485
$$832$$ 0 0
$$833$$ 0.209373 0.00725433
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −9.40312 −0.325020
$$838$$ 0 0
$$839$$ 45.1203 1.55773 0.778863 0.627194i $$-0.215797\pi$$
0.778863 + 0.627194i $$0.215797\pi$$
$$840$$ 0 0
$$841$$ 59.4187 2.04892
$$842$$ 0 0
$$843$$ 10.0000 0.344418
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 28.3875 0.975406
$$848$$ 0 0
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ −44.9109 −1.53953
$$852$$ 0 0
$$853$$ 21.7172 0.743582 0.371791 0.928316i $$-0.378744\pi$$
0.371791 + 0.928316i $$0.378744\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −12.7016 −0.433877 −0.216939 0.976185i $$-0.569607\pi$$
−0.216939 + 0.976185i $$0.569607\pi$$
$$858$$ 0 0
$$859$$ 31.2984 1.06789 0.533944 0.845520i $$-0.320709\pi$$
0.533944 + 0.845520i $$0.320709\pi$$
$$860$$ 0 0
$$861$$ −28.9109 −0.985282
$$862$$ 0 0
$$863$$ −13.4031 −0.456248 −0.228124 0.973632i $$-0.573259\pi$$
−0.228124 + 0.973632i $$0.573259\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −16.5078 −0.560635
$$868$$ 0 0
$$869$$ 3.29844 0.111892
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 0 0
$$873$$ −18.1047 −0.612751
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 3.61250 0.121985 0.0609927 0.998138i $$-0.480573\pi$$
0.0609927 + 0.998138i $$0.480573\pi$$
$$878$$ 0 0
$$879$$ −15.4031 −0.519534
$$880$$ 0 0
$$881$$ −12.8062 −0.431453 −0.215727 0.976454i $$-0.569212\pi$$
−0.215727 + 0.976454i $$0.569212\pi$$
$$882$$ 0 0
$$883$$ 17.6125 0.592708 0.296354 0.955078i $$-0.404229\pi$$
0.296354 + 0.955078i $$0.404229\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −18.4922 −0.620907 −0.310453 0.950589i $$-0.600481\pi$$
−0.310453 + 0.950589i $$0.600481\pi$$
$$888$$ 0 0
$$889$$ −36.2094 −1.21442
$$890$$ 0 0
$$891$$ 0.701562 0.0235032
$$892$$ 0 0
$$893$$ −2.80625 −0.0939075
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −6.70156 −0.223759
$$898$$ 0 0
$$899$$ −88.4187 −2.94893
$$900$$ 0 0
$$901$$ −7.50781 −0.250121
$$902$$ 0 0
$$903$$ 10.8062 0.359609
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 16.2094 0.538223 0.269112 0.963109i $$-0.413270\pi$$
0.269112 + 0.963109i $$0.413270\pi$$
$$908$$ 0 0
$$909$$ −6.80625 −0.225749
$$910$$ 0 0
$$911$$ 6.80625 0.225501 0.112751 0.993623i $$-0.464034\pi$$
0.112751 + 0.993623i $$0.464034\pi$$
$$912$$ 0 0
$$913$$ −2.80625 −0.0928733
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 34.5969 1.14249
$$918$$ 0 0
$$919$$ 7.08907 0.233847 0.116923 0.993141i $$-0.462697\pi$$
0.116923 + 0.993141i $$0.462697\pi$$
$$920$$ 0 0
$$921$$ 13.8953 0.457866
$$922$$ 0 0
$$923$$ 15.5078 0.510446
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −12.0000 −0.394132
$$928$$ 0 0
$$929$$ 8.10469 0.265906 0.132953 0.991122i $$-0.457554\pi$$
0.132953 + 0.991122i $$0.457554\pi$$
$$930$$ 0 0
$$931$$ −0.596876 −0.0195618
$$932$$ 0 0
$$933$$ 8.00000 0.261908
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −52.8062 −1.72510 −0.862552 0.505968i $$-0.831136\pi$$
−0.862552 + 0.505968i $$0.831136\pi$$
$$938$$ 0 0
$$939$$ −16.8062 −0.548451
$$940$$ 0 0
$$941$$ −54.7016 −1.78322 −0.891610 0.452804i $$-0.850424\pi$$
−0.891610 + 0.452804i $$0.850424\pi$$
$$942$$ 0 0
$$943$$ −71.7172 −2.33543
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 9.61250 0.312364 0.156182 0.987728i $$-0.450081\pi$$
0.156182 + 0.987728i $$0.450081\pi$$
$$948$$ 0 0
$$949$$ −13.4031 −0.435084
$$950$$ 0 0
$$951$$ −14.2094 −0.460770
$$952$$ 0 0
$$953$$ −4.91093 −0.159081 −0.0795404 0.996832i $$-0.525345\pi$$
−0.0795404 + 0.996832i $$0.525345\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 6.59688 0.213247
$$958$$ 0 0
$$959$$ 48.6281 1.57028
$$960$$ 0 0
$$961$$ 57.4187 1.85222
$$962$$ 0 0
$$963$$ −18.1047 −0.583415
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −39.8219 −1.28058 −0.640292 0.768131i $$-0.721187\pi$$
−0.640292 + 0.768131i $$0.721187\pi$$
$$968$$ 0 0
$$969$$ −1.40312 −0.0450748
$$970$$ 0 0
$$971$$ 57.0156 1.82972 0.914859 0.403773i $$-0.132301\pi$$
0.914859 + 0.403773i $$0.132301\pi$$
$$972$$ 0 0
$$973$$ 12.7016 0.407193
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −26.2094 −0.838512 −0.419256 0.907868i $$-0.637709\pi$$
−0.419256 + 0.907868i $$0.637709\pi$$
$$978$$ 0 0
$$979$$ 5.68594 0.181723
$$980$$ 0 0
$$981$$ 10.8062 0.345017
$$982$$ 0 0
$$983$$ 20.0000 0.637901 0.318950 0.947771i $$-0.396670\pi$$
0.318950 + 0.947771i $$0.396670\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −3.79063 −0.120657
$$988$$ 0 0
$$989$$ 26.8062 0.852389
$$990$$ 0 0
$$991$$ 27.7172 0.880465 0.440233 0.897884i $$-0.354896\pi$$
0.440233 + 0.897884i $$0.354896\pi$$
$$992$$ 0 0
$$993$$ −14.2094 −0.450921
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 51.4031 1.62795 0.813977 0.580898i $$-0.197298\pi$$
0.813977 + 0.580898i $$0.197298\pi$$
$$998$$ 0 0
$$999$$ 6.70156 0.212028
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 7800.2.a.be.1.1 2
5.4 even 2 1560.2.a.m.1.2 2
15.14 odd 2 4680.2.a.bb.1.2 2
20.19 odd 2 3120.2.a.bf.1.1 2
60.59 even 2 9360.2.a.ct.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.m.1.2 2 5.4 even 2
3120.2.a.bf.1.1 2 20.19 odd 2
4680.2.a.bb.1.2 2 15.14 odd 2
7800.2.a.be.1.1 2 1.1 even 1 trivial
9360.2.a.ct.1.1 2 60.59 even 2