# Properties

 Label 9360.2.a.ct.1.1 Level $9360$ Weight $2$ Character 9360.1 Self dual yes Analytic conductor $74.740$ 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] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, 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("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9360.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: $$74.7399762919$$ 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 $$\chi$$ $$=$$ 9360.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} -2.70156 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} -2.70156 q^{7} +0.701562 q^{11} -1.00000 q^{13} +0.701562 q^{17} +2.00000 q^{19} +6.70156 q^{23} +1.00000 q^{25} -9.40312 q^{29} +9.40312 q^{31} -2.70156 q^{35} -6.70156 q^{37} -10.7016 q^{41} -4.00000 q^{43} -1.40312 q^{47} +0.298438 q^{49} -10.7016 q^{53} +0.701562 q^{55} -14.8062 q^{59} +2.70156 q^{61} -1.00000 q^{65} +4.00000 q^{67} +15.5078 q^{71} +13.4031 q^{73} -1.89531 q^{77} -4.70156 q^{79} +4.00000 q^{83} +0.701562 q^{85} -8.10469 q^{89} +2.70156 q^{91} +2.00000 q^{95} +18.1047 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + q^7 $$2 q + 2 q^{5} + q^{7} - 5 q^{11} - 2 q^{13} - 5 q^{17} + 4 q^{19} + 7 q^{23} + 2 q^{25} - 6 q^{29} + 6 q^{31} + q^{35} - 7 q^{37} - 15 q^{41} - 8 q^{43} + 10 q^{47} + 7 q^{49} - 15 q^{53} - 5 q^{55} - 4 q^{59} - q^{61} - 2 q^{65} + 8 q^{67} - q^{71} + 14 q^{73} - 23 q^{77} - 3 q^{79} + 8 q^{83} - 5 q^{85} + 3 q^{89} - q^{91} + 4 q^{95} + 17 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + q^7 - 5 * q^11 - 2 * q^13 - 5 * q^17 + 4 * q^19 + 7 * q^23 + 2 * q^25 - 6 * q^29 + 6 * q^31 + q^35 - 7 * q^37 - 15 * q^41 - 8 * q^43 + 10 * q^47 + 7 * q^49 - 15 * q^53 - 5 * q^55 - 4 * q^59 - q^61 - 2 * q^65 + 8 * q^67 - q^71 + 14 * q^73 - 23 * q^77 - 3 * q^79 + 8 * q^83 - 5 * q^85 + 3 * q^89 - q^91 + 4 * q^95 + 17 * q^97

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.70156 −1.02109 −0.510547 0.859850i $$-0.670557\pi$$
−0.510547 + 0.859850i $$0.670557\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 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 $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.70156 1.39737 0.698686 0.715428i $$-0.253768\pi$$
0.698686 + 0.715428i $$0.253768\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −9.40312 −1.74612 −0.873058 0.487616i $$-0.837867\pi$$
−0.873058 + 0.487616i $$0.837867\pi$$
$$30$$ 0 0
$$31$$ 9.40312 1.68885 0.844425 0.535673i $$-0.179942\pi$$
0.844425 + 0.535673i $$0.179942\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.70156 −0.456647
$$36$$ 0 0
$$37$$ −6.70156 −1.10173 −0.550865 0.834594i $$-0.685702\pi$$
−0.550865 + 0.834594i $$0.685702\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −10.7016 −1.67130 −0.835652 0.549260i $$-0.814910\pi$$
−0.835652 + 0.549260i $$0.814910\pi$$
$$42$$ 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 $$-0.532631\pi$$
−0.102333 + 0.994750i $$0.532631\pi$$
$$48$$ 0 0
$$49$$ 0.298438 0.0426340
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −10.7016 −1.46997 −0.734986 0.678082i $$-0.762811\pi$$
−0.734986 + 0.678082i $$0.762811\pi$$
$$54$$ 0 0
$$55$$ 0.701562 0.0945986
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 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
$$63$$ 0 0
$$64$$ 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
$$69$$ 0 0
$$70$$ 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 $$-0.212992\pi$$
0.784359 + 0.620308i $$0.212992\pi$$
$$74$$ 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 $$-0.585201\pi$$
−0.264484 + 0.964390i $$0.585201\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 0.701562 0.0760951
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −8.10469 −0.859095 −0.429548 0.903044i $$-0.641327\pi$$
−0.429548 + 0.903044i $$0.641327\pi$$
$$90$$ 0 0
$$91$$ 2.70156 0.283201
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ 18.1047 1.83825 0.919126 0.393963i $$-0.128896\pi$$
0.919126 + 0.393963i $$0.128896\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.80625 0.677247 0.338624 0.940922i $$-0.390039\pi$$
0.338624 + 0.940922i $$0.390039\pi$$
$$102$$ 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 $$-0.160780\pi$$
0.875123 + 0.483900i $$0.160780\pi$$
$$108$$ 0 0
$$109$$ 10.8062 1.03505 0.517525 0.855668i $$-0.326853\pi$$
0.517525 + 0.855668i $$0.326853\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.59688 −0.620582 −0.310291 0.950642i $$-0.600426\pi$$
−0.310291 + 0.950642i $$0.600426\pi$$
$$114$$ 0 0
$$115$$ 6.70156 0.624924
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −1.89531 −0.173743
$$120$$ 0 0
$$121$$ −10.5078 −0.955256
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 13.4031 1.18933 0.594667 0.803972i $$-0.297284\pi$$
0.594667 + 0.803972i $$0.297284\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 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 $$-0.436104\pi$$
0.199391 + 0.979920i $$0.436104\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −0.701562 −0.0586676
$$144$$ 0 0
$$145$$ −9.40312 −0.780887
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1.29844 −0.106372 −0.0531861 0.998585i $$-0.516938\pi$$
−0.0531861 + 0.998585i $$0.516938\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 9.40312 0.755277
$$156$$ 0 0
$$157$$ 8.80625 0.702815 0.351408 0.936223i $$-0.385703\pi$$
0.351408 + 0.936223i $$0.385703\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 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 $$-0.844074\pi$$
−0.882400 + 0.470500i $$0.844074\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ −2.70156 −0.204219
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 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
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −6.70156 −0.492709
$$186$$ 0 0
$$187$$ 0.492189 0.0359925
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 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 $$-0.792826\pi$$
−0.795565 + 0.605868i $$0.792826\pi$$
$$194$$ 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 $$-0.531713\pi$$
−0.0994648 + 0.995041i $$0.531713\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 25.4031 1.78295
$$204$$ 0 0
$$205$$ −10.7016 −0.747430
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1.40312 0.0970561
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ −25.4031 −1.72448
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 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 $$-0.614386\pi$$
−0.351670 + 0.936124i $$0.614386\pi$$
$$228$$ 0 0
$$229$$ −25.4031 −1.67869 −0.839343 0.543602i $$-0.817060\pi$$
−0.839343 + 0.543602i $$0.817060\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 10.1047 0.661980 0.330990 0.943634i $$-0.392617\pi$$
0.330990 + 0.943634i $$0.392617\pi$$
$$234$$ 0 0
$$235$$ −1.40312 −0.0915297
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 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
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0.298438 0.0190665
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 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
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 4.59688 0.283456 0.141728 0.989906i $$-0.454734\pi$$
0.141728 + 0.989906i $$0.454734\pi$$
$$264$$ 0 0
$$265$$ −10.7016 −0.657392
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 14.8062 0.902753 0.451376 0.892334i $$-0.350933\pi$$
0.451376 + 0.892334i $$0.350933\pi$$
$$270$$ 0 0
$$271$$ 6.59688 0.400732 0.200366 0.979721i $$-0.435787\pi$$
0.200366 + 0.979721i $$0.435787\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0.701562 0.0423058
$$276$$ 0 0
$$277$$ 8.80625 0.529116 0.264558 0.964370i $$-0.414774\pi$$
0.264558 + 0.964370i $$0.414774\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 28.9109 1.70656
$$288$$ 0 0
$$289$$ −16.5078 −0.971048
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −15.4031 −0.899860 −0.449930 0.893064i $$-0.648551\pi$$
−0.449930 + 0.893064i $$0.648551\pi$$
$$294$$ 0 0
$$295$$ −14.8062 −0.862053
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −6.70156 −0.387561
$$300$$ 0 0
$$301$$ 10.8062 0.622862
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 2.70156 0.154691
$$306$$ 0 0
$$307$$ 13.8953 0.793047 0.396524 0.918024i $$-0.370216\pi$$
0.396524 + 0.918024i $$0.370216\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 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 $$-0.342458\pi$$
0.474973 + 0.880001i $$0.342458\pi$$
$$314$$ 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
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 1.40312 0.0780719
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 3.79063 0.208984
$$330$$ 0 0
$$331$$ 14.2094 0.781018 0.390509 0.920599i $$-0.372299\pi$$
0.390509 + 0.920599i $$0.372299\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ −6.20937 −0.338246 −0.169123 0.985595i $$-0.554094\pi$$
−0.169123 + 0.985595i $$0.554094\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 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 $$-0.447606\pi$$
0.163858 + 0.986484i $$0.447606\pi$$
$$348$$ 0 0
$$349$$ −34.8062 −1.86314 −0.931568 0.363567i $$-0.881559\pi$$
−0.931568 + 0.363567i $$0.881559\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −12.8062 −0.681608 −0.340804 0.940134i $$-0.610699\pi$$
−0.340804 + 0.940134i $$0.610699\pi$$
$$354$$ 0 0
$$355$$ 15.5078 0.823069
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 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
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 13.4031 0.701552
$$366$$ 0 0
$$367$$ −1.19375 −0.0623133 −0.0311567 0.999515i $$-0.509919\pi$$
−0.0311567 + 0.999515i $$0.509919\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 28.9109 1.50098
$$372$$ 0 0
$$373$$ −8.59688 −0.445129 −0.222565 0.974918i $$-0.571443\pi$$
−0.222565 + 0.974918i $$0.571443\pi$$
$$374$$ 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 $$-0.631330\pi$$
−0.400980 + 0.916087i $$0.631330\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −5.40312 −0.276087 −0.138043 0.990426i $$-0.544081\pi$$
−0.138043 + 0.990426i $$0.544081\pi$$
$$384$$ 0 0
$$385$$ −1.89531 −0.0965941
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −25.6125 −1.29861 −0.649303 0.760530i $$-0.724939\pi$$
−0.649303 + 0.760530i $$0.724939\pi$$
$$390$$ 0 0
$$391$$ 4.70156 0.237768
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −4.70156 −0.236561
$$396$$ 0 0
$$397$$ −22.7016 −1.13936 −0.569679 0.821867i $$-0.692933\pi$$
−0.569679 + 0.821867i $$0.692933\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −0.806248 −0.0402621 −0.0201311 0.999797i $$-0.506408\pi$$
−0.0201311 + 0.999797i $$0.506408\pi$$
$$402$$ 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
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 40.0000 1.96827
$$414$$ 0 0
$$415$$ 4.00000 0.196352
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 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
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0.701562 0.0340308
$$426$$ 0 0
$$427$$ −7.29844 −0.353196
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 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 $$-0.402117\pi$$
0.302684 + 0.953091i $$0.402117\pi$$
$$434$$ 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 $$-0.485598\pi$$
0.0452292 + 0.998977i $$0.485598\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −26.3141 −1.25022 −0.625109 0.780537i $$-0.714946\pi$$
−0.625109 + 0.780537i $$0.714946\pi$$
$$444$$ 0 0
$$445$$ −8.10469 −0.384199
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −3.89531 −0.183831 −0.0919156 0.995767i $$-0.529299\pi$$
−0.0919156 + 0.995767i $$0.529299\pi$$
$$450$$ 0 0
$$451$$ −7.50781 −0.353529
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 2.70156 0.126651
$$456$$ 0 0
$$457$$ 7.29844 0.341407 0.170703 0.985322i $$-0.445396\pi$$
0.170703 + 0.985322i $$0.445396\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −19.8953 −0.926617 −0.463309 0.886197i $$-0.653338\pi$$
−0.463309 + 0.886197i $$0.653338\pi$$
$$462$$ 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 $$-0.362418\pi$$
0.418892 + 0.908036i $$0.362418\pi$$
$$468$$ 0 0
$$469$$ −10.8062 −0.498986
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −2.80625 −0.129031
$$474$$ 0 0
$$475$$ 2.00000 0.0917663
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 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
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 18.1047 0.822091
$$486$$ 0 0
$$487$$ 36.3141 1.64555 0.822774 0.568369i $$-0.192426\pi$$
0.822774 + 0.568369i $$0.192426\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 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 $$-0.610455\pi$$
−0.340083 + 0.940395i $$0.610455\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 31.4031 1.40020 0.700098 0.714047i $$-0.253140\pi$$
0.700098 + 0.714047i $$0.253140\pi$$
$$504$$ 0 0
$$505$$ 6.80625 0.302874
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 32.3141 1.43230 0.716148 0.697949i $$-0.245904\pi$$
0.716148 + 0.697949i $$0.245904\pi$$
$$510$$ 0 0
$$511$$ −36.2094 −1.60181
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −12.0000 −0.528783
$$516$$ 0 0
$$517$$ −0.984379 −0.0432929
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 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
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 10.7016 0.463536
$$534$$ 0 0
$$535$$ 18.1047 0.782734
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 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
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 10.8062 0.462889
$$546$$ 0 0
$$547$$ 25.6125 1.09511 0.547556 0.836769i $$-0.315558\pi$$
0.547556 + 0.836769i $$0.315558\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 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 0
$$562$$ 0 0
$$563$$ 22.1047 0.931601 0.465801 0.884890i $$-0.345766\pi$$
0.465801 + 0.884890i $$0.345766\pi$$
$$564$$ 0 0
$$565$$ −6.59688 −0.277533
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 27.4031 1.14880 0.574399 0.818575i $$-0.305236\pi$$
0.574399 + 0.818575i $$0.305236\pi$$
$$570$$ 0 0
$$571$$ −26.3141 −1.10121 −0.550605 0.834766i $$-0.685603\pi$$
−0.550605 + 0.834766i $$0.685603\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 6.70156 0.279474
$$576$$ 0 0
$$577$$ −31.5078 −1.31169 −0.655844 0.754897i $$-0.727687\pi$$
−0.655844 + 0.754897i $$0.727687\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 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 $$-0.666525\pi$$
−0.499614 + 0.866248i $$0.666525\pi$$
$$588$$ 0 0
$$589$$ 18.8062 0.774898
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 3.40312 0.139750 0.0698748 0.997556i $$-0.477740\pi$$
0.0698748 + 0.997556i $$0.477740\pi$$
$$594$$ 0 0
$$595$$ −1.89531 −0.0777003
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 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
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −10.5078 −0.427203
$$606$$ 0 0
$$607$$ 41.6125 1.68900 0.844500 0.535556i $$-0.179898\pi$$
0.844500 + 0.535556i $$0.179898\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.40312 0.0567643
$$612$$ 0 0
$$613$$ −45.7172 −1.84650 −0.923250 0.384200i $$-0.874477\pi$$
−0.923250 + 0.384200i $$0.874477\pi$$
$$614$$ 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 $$-0.793097\pi$$
−0.796080 + 0.605191i $$0.793097\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 21.8953 0.877217
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −4.70156 −0.187464
$$630$$ 0 0
$$631$$ −20.0000 −0.796187 −0.398094 0.917345i $$-0.630328\pi$$
−0.398094 + 0.917345i $$0.630328\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 13.4031 0.531887
$$636$$ 0 0
$$637$$ −0.298438 −0.0118245
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 10.2094 0.403246 0.201623 0.979463i $$-0.435378\pi$$
0.201623 + 0.979463i $$0.435378\pi$$
$$642$$ 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 $$-0.370676\pi$$
0.395198 + 0.918596i $$0.370676\pi$$
$$648$$ 0 0
$$649$$ −10.3875 −0.407745
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −15.6125 −0.610964 −0.305482 0.952198i $$-0.598818\pi$$
−0.305482 + 0.952198i $$0.598818\pi$$
$$654$$ 0 0
$$655$$ −12.8062 −0.500382
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 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 0
$$664$$ 0 0
$$665$$ −5.40312 −0.209524
$$666$$ 0 0
$$667$$ −63.0156 −2.43997
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 1.89531 0.0731678
$$672$$ 0 0
$$673$$ −3.19375 −0.123110 −0.0615550 0.998104i $$-0.519606\pi$$
−0.0615550 + 0.998104i $$0.519606\pi$$
$$674$$ 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
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 28.0000 1.07139 0.535695 0.844411i $$-0.320050\pi$$
0.535695 + 0.844411i $$0.320050\pi$$
$$684$$ 0 0
$$685$$ −18.0000 −0.687745
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 10.7016 0.407697
$$690$$ 0 0
$$691$$ 10.0000 0.380418 0.190209 0.981744i $$-0.439083\pi$$
0.190209 + 0.981744i $$0.439083\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4.70156 0.178340
$$696$$ 0 0
$$697$$ −7.50781 −0.284379
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 25.6125 0.967371 0.483685 0.875242i $$-0.339298\pi$$
0.483685 + 0.875242i $$0.339298\pi$$
$$702$$ 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
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 63.0156 2.35995
$$714$$ 0 0
$$715$$ −0.701562 −0.0262369
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 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
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −9.40312 −0.349223
$$726$$ 0 0
$$727$$ 9.40312 0.348743 0.174371 0.984680i $$-0.444211\pi$$
0.174371 + 0.984680i $$0.444211\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2.80625 −0.103793
$$732$$ 0 0
$$733$$ 41.7172 1.54086 0.770430 0.637525i $$-0.220042\pi$$
0.770430 + 0.637525i $$0.220042\pi$$
$$734$$ 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 $$-0.448215\pi$$
0.161972 + 0.986795i $$0.448215\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −10.5969 −0.388762 −0.194381 0.980926i $$-0.562270\pi$$
−0.194381 + 0.980926i $$0.562270\pi$$
$$744$$ 0 0
$$745$$ −1.29844 −0.0475711
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −48.9109 −1.78717
$$750$$ 0 0
$$751$$ 42.3141 1.54406 0.772031 0.635585i $$-0.219241\pi$$
0.772031 + 0.635585i $$0.219241\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −47.4031 −1.72290 −0.861448 0.507846i $$-0.830442\pi$$
−0.861448 + 0.507846i $$0.830442\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 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
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 32.5969 1.17243 0.586214 0.810156i $$-0.300618\pi$$
0.586214 + 0.810156i $$0.300618\pi$$
$$774$$ 0 0
$$775$$ 9.40312 0.337770
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −21.4031 −0.766847
$$780$$ 0 0
$$781$$ 10.8797 0.389306
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 8.80625 0.314308
$$786$$ 0 0
$$787$$ −5.19375 −0.185137 −0.0925686 0.995706i $$-0.529508\pi$$
−0.0925686 + 0.995706i $$0.529508\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 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
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 9.40312 0.331829
$$804$$ 0 0
$$805$$ −18.1047 −0.638106
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0.806248 0.0283462 0.0141731 0.999900i $$-0.495488\pi$$
0.0141731 + 0.999900i $$0.495488\pi$$
$$810$$ 0 0
$$811$$ 14.2094 0.498959 0.249479 0.968380i $$-0.419741\pi$$
0.249479 + 0.968380i $$0.419741\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −11.2984 −0.395767
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 2.49219 0.0869780 0.0434890 0.999054i $$-0.486153\pi$$
0.0434890 + 0.999054i $$0.486153\pi$$
$$822$$ 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 $$-0.294560\pi$$
0.601526 + 0.798854i $$0.294560\pi$$
$$828$$ 0 0
$$829$$ −47.6125 −1.65365 −0.826825 0.562459i $$-0.809855\pi$$
−0.826825 + 0.562459i $$0.809855\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0.209373 0.00725433
$$834$$ 0 0
$$835$$ −22.8062 −0.789243
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 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
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ 28.3875 0.975406
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −44.9109 −1.53953
$$852$$ 0 0
$$853$$ −21.7172 −0.743582 −0.371791 0.928316i $$-0.621256\pi$$
−0.371791 + 0.928316i $$0.621256\pi$$
$$854$$ 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 $$-0.679291\pi$$
−0.533944 + 0.845520i $$0.679291\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 13.4031 0.456248 0.228124 0.973632i $$-0.426741\pi$$
0.228124 + 0.973632i $$0.426741\pi$$
$$864$$ 0 0
$$865$$ 2.00000 0.0680020
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −3.29844 −0.111892
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −2.70156 −0.0913295
$$876$$ 0 0
$$877$$ −3.61250 −0.121985 −0.0609927 0.998138i $$-0.519427\pi$$
−0.0609927 + 0.998138i $$0.519427\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 12.8062 0.431453 0.215727 0.976454i $$-0.430788\pi$$
0.215727 + 0.976454i $$0.430788\pi$$
$$882$$ 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 $$-0.399519\pi$$
0.310453 + 0.950589i $$0.399519\pi$$
$$888$$ 0 0
$$889$$ −36.2094 −1.21442
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −2.80625 −0.0939075
$$894$$ 0 0
$$895$$ −22.0000 −0.735379
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −88.4187 −2.94893
$$900$$ 0 0
$$901$$ −7.50781 −0.250121
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −1.29844 −0.0431615
$$906$$ 0 0
$$907$$ 16.2094 0.538223 0.269112 0.963109i $$-0.413270\pi$$
0.269112 + 0.963109i $$0.413270\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 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 $$-0.537303\pi$$
−0.116923 + 0.993141i $$0.537303\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −15.5078 −0.510446
$$924$$ 0 0
$$925$$ −6.70156 −0.220346
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −8.10469 −0.265906 −0.132953 0.991122i $$-0.542446\pi$$
−0.132953 + 0.991122i $$0.542446\pi$$
$$930$$ 0 0
$$931$$ 0.596876 0.0195618
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0.492189 0.0160963
$$936$$ 0 0
$$937$$ 52.8062 1.72510 0.862552 0.505968i $$-0.168864\pi$$
0.862552 + 0.505968i $$0.168864\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 54.7016 1.78322 0.891610 0.452804i $$-0.149576\pi$$
0.891610 + 0.452804i $$0.149576\pi$$
$$942$$ 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 $$-0.549919\pi$$
−0.156182 + 0.987728i $$0.549919\pi$$
$$948$$ 0 0
$$949$$ −13.4031 −0.435084
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 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
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 48.6281 1.57028
$$960$$ 0 0
$$961$$ 57.4187 1.85222
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −22.1047 −0.711575
$$966$$ 0 0
$$967$$ −39.8219 −1.28058 −0.640292 0.768131i $$-0.721187\pi$$
−0.640292 + 0.768131i $$0.721187\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 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
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −20.0000 −0.637901 −0.318950 0.947771i $$-0.603330\pi$$
−0.318950 + 0.947771i $$0.603330\pi$$
$$984$$ 0 0
$$985$$ 14.2094 0.452748
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −26.8062 −0.852389
$$990$$ 0 0
$$991$$ −27.7172 −0.880465 −0.440233 0.897884i $$-0.645104\pi$$
−0.440233 + 0.897884i $$0.645104\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −2.80625 −0.0889641
$$996$$ 0 0
$$997$$ −51.4031 −1.62795 −0.813977 0.580898i $$-0.802702\pi$$
−0.813977 + 0.580898i $$0.802702\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9360.2.a.ct.1.1 2
3.2 odd 2 3120.2.a.bf.1.1 2
4.3 odd 2 4680.2.a.bb.1.2 2
12.11 even 2 1560.2.a.m.1.2 2
60.59 even 2 7800.2.a.be.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.m.1.2 2 12.11 even 2
3120.2.a.bf.1.1 2 3.2 odd 2
4680.2.a.bb.1.2 2 4.3 odd 2
7800.2.a.be.1.1 2 60.59 even 2
9360.2.a.ct.1.1 2 1.1 even 1 trivial