# Properties

 Label 5202.2.a.g.1.1 Level $5202$ Weight $2$ Character 5202.1 Self dual yes Analytic conductor $41.538$ Analytic rank $1$ 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] = [5202,2,Mod(1,5202)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5202, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5202.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5202 = 2 \cdot 3^{2} \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5202.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: $$41.5381791315$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5202.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} +2.00000 q^{7} +1.00000 q^{8} -4.00000 q^{10} -6.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +4.00000 q^{19} -4.00000 q^{20} +6.00000 q^{23} +11.0000 q^{25} -6.00000 q^{26} +2.00000 q^{28} -4.00000 q^{29} +6.00000 q^{31} +1.00000 q^{32} -8.00000 q^{35} +4.00000 q^{37} +4.00000 q^{38} -4.00000 q^{40} -10.0000 q^{41} -4.00000 q^{43} +6.00000 q^{46} -4.00000 q^{47} -3.00000 q^{49} +11.0000 q^{50} -6.00000 q^{52} +2.00000 q^{53} +2.00000 q^{56} -4.00000 q^{58} -12.0000 q^{59} +4.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +24.0000 q^{65} -12.0000 q^{67} -8.00000 q^{70} -6.00000 q^{71} -2.00000 q^{73} +4.00000 q^{74} +4.00000 q^{76} -10.0000 q^{79} -4.00000 q^{80} -10.0000 q^{82} +12.0000 q^{83} -4.00000 q^{86} +2.00000 q^{89} -12.0000 q^{91} +6.00000 q^{92} -4.00000 q^{94} -16.0000 q^{95} -6.00000 q^{97} -3.00000 q^{98} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −4.00000 −1.78885 −0.894427 0.447214i $$-0.852416\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ −4.00000 −1.26491
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ −4.00000 −0.894427
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ −6.00000 −1.17670
$$27$$ 0 0
$$28$$ 2.00000 0.377964
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −8.00000 −1.35225
$$36$$ 0 0
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 0 0
$$40$$ −4.00000 −0.632456
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 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
$$46$$ 6.00000 0.884652
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 11.0000 1.55563
$$51$$ 0 0
$$52$$ −6.00000 −0.832050
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ 0 0
$$58$$ −4.00000 −0.525226
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 24.0000 2.97683
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ −8.00000 −0.956183
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 0 0
$$82$$ −10.0000 −1.10432
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ −12.0000 −1.25794
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ −4.00000 −0.412568
$$95$$ −16.0000 −1.64157
$$96$$ 0 0
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 0 0
$$100$$ 11.0000 1.10000
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 2.00000 0.188982
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ −24.0000 −2.23801
$$116$$ −4.00000 −0.371391
$$117$$ 0 0
$$118$$ −12.0000 −1.10469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 4.00000 0.362143
$$123$$ 0 0
$$124$$ 6.00000 0.538816
$$125$$ −24.0000 −2.14663
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 24.0000 2.10494
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ 0 0
$$133$$ 8.00000 0.693688
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ −8.00000 −0.676123
$$141$$ 0 0
$$142$$ −6.00000 −0.503509
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 16.0000 1.32873
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ 4.00000 0.328798
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −24.0000 −1.95309 −0.976546 0.215308i $$-0.930924\pi$$
−0.976546 + 0.215308i $$0.930924\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −24.0000 −1.92773
$$156$$ 0 0
$$157$$ 6.00000 0.478852 0.239426 0.970915i $$-0.423041\pi$$
0.239426 + 0.970915i $$0.423041\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ 0 0
$$160$$ −4.00000 −0.316228
$$161$$ 12.0000 0.945732
$$162$$ 0 0
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ −4.00000 −0.304114 −0.152057 0.988372i $$-0.548590\pi$$
−0.152057 + 0.988372i $$0.548590\pi$$
$$174$$ 0 0
$$175$$ 22.0000 1.66304
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 2.00000 0.149906
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ −12.0000 −0.889499
$$183$$ 0 0
$$184$$ 6.00000 0.442326
$$185$$ −16.0000 −1.17634
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −4.00000 −0.291730
$$189$$ 0 0
$$190$$ −16.0000 −1.16076
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 8.00000 0.569976 0.284988 0.958531i $$-0.408010\pi$$
0.284988 + 0.958531i $$0.408010\pi$$
$$198$$ 0 0
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 11.0000 0.777817
$$201$$ 0 0
$$202$$ −14.0000 −0.985037
$$203$$ −8.00000 −0.561490
$$204$$ 0 0
$$205$$ 40.0000 2.79372
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ −6.00000 −0.416025
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 16.0000 1.09119
$$216$$ 0 0
$$217$$ 12.0000 0.814613
$$218$$ −16.0000 −1.08366
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ −24.0000 −1.58251
$$231$$ 0 0
$$232$$ −4.00000 −0.262613
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 16.0000 1.04372
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ 0 0
$$244$$ 4.00000 0.256074
$$245$$ 12.0000 0.766652
$$246$$ 0 0
$$247$$ −24.0000 −1.52708
$$248$$ 6.00000 0.381000
$$249$$ 0 0
$$250$$ −24.0000 −1.51789
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 8.00000 0.497096
$$260$$ 24.0000 1.48842
$$261$$ 0 0
$$262$$ −16.0000 −0.988483
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ −8.00000 −0.491436
$$266$$ 8.00000 0.490511
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 8.00000 0.480673 0.240337 0.970690i $$-0.422742\pi$$
0.240337 + 0.970690i $$0.422742\pi$$
$$278$$ −8.00000 −0.479808
$$279$$ 0 0
$$280$$ −8.00000 −0.478091
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ −32.0000 −1.90220 −0.951101 0.308879i $$-0.900046\pi$$
−0.951101 + 0.308879i $$0.900046\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −20.0000 −1.18056
$$288$$ 0 0
$$289$$ 0 0
$$290$$ 16.0000 0.939552
$$291$$ 0 0
$$292$$ −2.00000 −0.117041
$$293$$ −2.00000 −0.116841 −0.0584206 0.998292i $$-0.518606\pi$$
−0.0584206 + 0.998292i $$0.518606\pi$$
$$294$$ 0 0
$$295$$ 48.0000 2.79467
$$296$$ 4.00000 0.232495
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ −36.0000 −2.08193
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ −24.0000 −1.38104
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ −16.0000 −0.916157
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −24.0000 −1.36311
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 0 0
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ −16.0000 −0.898650 −0.449325 0.893368i $$-0.648335\pi$$
−0.449325 + 0.893368i $$0.648335\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −4.00000 −0.223607
$$321$$ 0 0
$$322$$ 12.0000 0.668734
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −66.0000 −3.66102
$$326$$ −12.0000 −0.664619
$$327$$ 0 0
$$328$$ −10.0000 −0.552158
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 0 0
$$334$$ −2.00000 −0.109435
$$335$$ 48.0000 2.62252
$$336$$ 0 0
$$337$$ 6.00000 0.326841 0.163420 0.986557i $$-0.447747\pi$$
0.163420 + 0.986557i $$0.447747\pi$$
$$338$$ 23.0000 1.25104
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −4.00000 −0.215041
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ 0 0
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 22.0000 1.17595
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ 24.0000 1.27379
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 20.0000 1.05118
$$363$$ 0 0
$$364$$ −12.0000 −0.628971
$$365$$ 8.00000 0.418739
$$366$$ 0 0
$$367$$ −10.0000 −0.521996 −0.260998 0.965339i $$-0.584052\pi$$
−0.260998 + 0.965339i $$0.584052\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 0 0
$$370$$ −16.0000 −0.831800
$$371$$ 4.00000 0.207670
$$372$$ 0 0
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ 24.0000 1.23606
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ −16.0000 −0.820783
$$381$$ 0 0
$$382$$ 4.00000 0.204658
$$383$$ −28.0000 −1.43073 −0.715367 0.698749i $$-0.753740\pi$$
−0.715367 + 0.698749i $$0.753740\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ 0 0
$$388$$ −6.00000 −0.304604
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −3.00000 −0.151523
$$393$$ 0 0
$$394$$ 8.00000 0.403034
$$395$$ 40.0000 2.01262
$$396$$ 0 0
$$397$$ −20.0000 −1.00377 −0.501886 0.864934i $$-0.667360\pi$$
−0.501886 + 0.864934i $$0.667360\pi$$
$$398$$ −14.0000 −0.701757
$$399$$ 0 0
$$400$$ 11.0000 0.550000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ −36.0000 −1.79329
$$404$$ −14.0000 −0.696526
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 40.0000 1.97546
$$411$$ 0 0
$$412$$ 4.00000 0.197066
$$413$$ −24.0000 −1.18096
$$414$$ 0 0
$$415$$ −48.0000 −2.35623
$$416$$ −6.00000 −0.294174
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ −8.00000 −0.389434
$$423$$ 0 0
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 8.00000 0.387147
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 16.0000 0.771589
$$431$$ 14.0000 0.674356 0.337178 0.941441i $$-0.390528\pi$$
0.337178 + 0.941441i $$0.390528\pi$$
$$432$$ 0 0
$$433$$ −18.0000 −0.865025 −0.432512 0.901628i $$-0.642373\pi$$
−0.432512 + 0.901628i $$0.642373\pi$$
$$434$$ 12.0000 0.576018
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ 24.0000 1.14808
$$438$$ 0 0
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ −8.00000 −0.379236
$$446$$ −4.00000 −0.189405
$$447$$ 0 0
$$448$$ 2.00000 0.0944911
$$449$$ −26.0000 −1.22702 −0.613508 0.789689i $$-0.710242\pi$$
−0.613508 + 0.789689i $$0.710242\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 2.00000 0.0940721
$$453$$ 0 0
$$454$$ 4.00000 0.187729
$$455$$ 48.0000 2.25027
$$456$$ 0 0
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ 0 0
$$460$$ −24.0000 −1.11901
$$461$$ 10.0000 0.465746 0.232873 0.972507i $$-0.425187\pi$$
0.232873 + 0.972507i $$0.425187\pi$$
$$462$$ 0 0
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ −24.0000 −1.10822
$$470$$ 16.0000 0.738025
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 44.0000 2.01886
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 20.0000 0.914779
$$479$$ 10.0000 0.456912 0.228456 0.973554i $$-0.426632\pi$$
0.228456 + 0.973554i $$0.426632\pi$$
$$480$$ 0 0
$$481$$ −24.0000 −1.09431
$$482$$ 18.0000 0.819878
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 24.0000 1.08978
$$486$$ 0 0
$$487$$ 38.0000 1.72194 0.860972 0.508652i $$-0.169856\pi$$
0.860972 + 0.508652i $$0.169856\pi$$
$$488$$ 4.00000 0.181071
$$489$$ 0 0
$$490$$ 12.0000 0.542105
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −24.0000 −1.07981
$$495$$ 0 0
$$496$$ 6.00000 0.269408
$$497$$ −12.0000 −0.538274
$$498$$ 0 0
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ −24.0000 −1.07331
$$501$$ 0 0
$$502$$ −12.0000 −0.535586
$$503$$ −26.0000 −1.15928 −0.579641 0.814872i $$-0.696807\pi$$
−0.579641 + 0.814872i $$0.696807\pi$$
$$504$$ 0 0
$$505$$ 56.0000 2.49197
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ −26.0000 −1.15243 −0.576215 0.817298i $$-0.695471\pi$$
−0.576215 + 0.817298i $$0.695471\pi$$
$$510$$ 0 0
$$511$$ −4.00000 −0.176950
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −18.0000 −0.793946
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 8.00000 0.351500
$$519$$ 0 0
$$520$$ 24.0000 1.05247
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ −16.0000 −0.698963
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ −8.00000 −0.347498
$$531$$ 0 0
$$532$$ 8.00000 0.346844
$$533$$ 60.0000 2.59889
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ −12.0000 −0.517357
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 12.0000 0.515920 0.257960 0.966156i $$-0.416950\pi$$
0.257960 + 0.966156i $$0.416950\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 64.0000 2.74146
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −16.0000 −0.681623
$$552$$ 0 0
$$553$$ −20.0000 −0.850487
$$554$$ 8.00000 0.339887
$$555$$ 0 0
$$556$$ −8.00000 −0.339276
$$557$$ −14.0000 −0.593199 −0.296600 0.955002i $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ −8.00000 −0.338062
$$561$$ 0 0
$$562$$ 18.0000 0.759284
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 0 0
$$565$$ −8.00000 −0.336563
$$566$$ −32.0000 −1.34506
$$567$$ 0 0
$$568$$ −6.00000 −0.251754
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −20.0000 −0.834784
$$575$$ 66.0000 2.75239
$$576$$ 0 0
$$577$$ −30.0000 −1.24892 −0.624458 0.781058i $$-0.714680\pi$$
−0.624458 + 0.781058i $$0.714680\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 16.0000 0.664364
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −2.00000 −0.0826192
$$587$$ −36.0000 −1.48588 −0.742940 0.669359i $$-0.766569\pi$$
−0.742940 + 0.669359i $$0.766569\pi$$
$$588$$ 0 0
$$589$$ 24.0000 0.988903
$$590$$ 48.0000 1.97613
$$591$$ 0 0
$$592$$ 4.00000 0.164399
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ −36.0000 −1.47215
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ 0 0
$$604$$ −24.0000 −0.976546
$$605$$ 44.0000 1.78885
$$606$$ 0 0
$$607$$ 38.0000 1.54237 0.771186 0.636610i $$-0.219664\pi$$
0.771186 + 0.636610i $$0.219664\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ −16.0000 −0.647821
$$611$$ 24.0000 0.970936
$$612$$ 0 0
$$613$$ 30.0000 1.21169 0.605844 0.795583i $$-0.292835\pi$$
0.605844 + 0.795583i $$0.292835\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ −24.0000 −0.963863
$$621$$ 0 0
$$622$$ 30.0000 1.20289
$$623$$ 4.00000 0.160257
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ 6.00000 0.239426
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ −10.0000 −0.397779
$$633$$ 0 0
$$634$$ −16.0000 −0.635441
$$635$$ −32.0000 −1.26988
$$636$$ 0 0
$$637$$ 18.0000 0.713186
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −4.00000 −0.158114
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −66.0000 −2.58873
$$651$$ 0 0
$$652$$ −12.0000 −0.469956
$$653$$ 20.0000 0.782660 0.391330 0.920250i $$-0.372015\pi$$
0.391330 + 0.920250i $$0.372015\pi$$
$$654$$ 0 0
$$655$$ 64.0000 2.50069
$$656$$ −10.0000 −0.390434
$$657$$ 0 0
$$658$$ −8.00000 −0.311872
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ −14.0000 −0.544537 −0.272268 0.962221i $$-0.587774\pi$$
−0.272268 + 0.962221i $$0.587774\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ −32.0000 −1.24091
$$666$$ 0 0
$$667$$ −24.0000 −0.929284
$$668$$ −2.00000 −0.0773823
$$669$$ 0 0
$$670$$ 48.0000 1.85440
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ 6.00000 0.231111
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 8.00000 0.307465 0.153732 0.988113i $$-0.450871\pi$$
0.153732 + 0.988113i $$0.450871\pi$$
$$678$$ 0 0
$$679$$ −12.0000 −0.460518
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ −24.0000 −0.916993
$$686$$ −20.0000 −0.763604
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −16.0000 −0.608669 −0.304334 0.952565i $$-0.598434\pi$$
−0.304334 + 0.952565i $$0.598434\pi$$
$$692$$ −4.00000 −0.152057
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 32.0000 1.21383
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −30.0000 −1.13552
$$699$$ 0 0
$$700$$ 22.0000 0.831522
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ 16.0000 0.603451
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −14.0000 −0.526897
$$707$$ −28.0000 −1.05305
$$708$$ 0 0
$$709$$ 16.0000 0.600893 0.300446 0.953799i $$-0.402864\pi$$
0.300446 + 0.953799i $$0.402864\pi$$
$$710$$ 24.0000 0.900704
$$711$$ 0 0
$$712$$ 2.00000 0.0749532
$$713$$ 36.0000 1.34821
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 42.0000 1.56634 0.783168 0.621810i $$-0.213603\pi$$
0.783168 + 0.621810i $$0.213603\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ −3.00000 −0.111648
$$723$$ 0 0
$$724$$ 20.0000 0.743294
$$725$$ −44.0000 −1.63412
$$726$$ 0 0
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ −12.0000 −0.444750
$$729$$ 0 0
$$730$$ 8.00000 0.296093
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 18.0000 0.664845 0.332423 0.943131i $$-0.392134\pi$$
0.332423 + 0.943131i $$0.392134\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ −16.0000 −0.588172
$$741$$ 0 0
$$742$$ 4.00000 0.146845
$$743$$ 38.0000 1.39408 0.697042 0.717030i $$-0.254499\pi$$
0.697042 + 0.717030i $$0.254499\pi$$
$$744$$ 0 0
$$745$$ −24.0000 −0.879292
$$746$$ −14.0000 −0.512576
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 34.0000 1.24068 0.620339 0.784334i $$-0.286995\pi$$
0.620339 + 0.784334i $$0.286995\pi$$
$$752$$ −4.00000 −0.145865
$$753$$ 0 0
$$754$$ 24.0000 0.874028
$$755$$ 96.0000 3.49380
$$756$$ 0 0
$$757$$ 14.0000 0.508839 0.254419 0.967094i $$-0.418116\pi$$
0.254419 + 0.967094i $$0.418116\pi$$
$$758$$ 4.00000 0.145287
$$759$$ 0 0
$$760$$ −16.0000 −0.580381
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ −32.0000 −1.15848
$$764$$ 4.00000 0.144715
$$765$$ 0 0
$$766$$ −28.0000 −1.01168
$$767$$ 72.0000 2.59977
$$768$$ 0 0
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −6.00000 −0.215945
$$773$$ −42.0000 −1.51064 −0.755318 0.655359i $$-0.772517\pi$$
−0.755318 + 0.655359i $$0.772517\pi$$
$$774$$ 0 0
$$775$$ 66.0000 2.37079
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ 14.0000 0.501924
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ −24.0000 −0.856597
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 8.00000 0.284988
$$789$$ 0 0
$$790$$ 40.0000 1.42314
$$791$$ 4.00000 0.142224
$$792$$ 0 0
$$793$$ −24.0000 −0.852265
$$794$$ −20.0000 −0.709773
$$795$$ 0 0
$$796$$ −14.0000 −0.496217
$$797$$ −46.0000 −1.62940 −0.814702 0.579880i $$-0.803099\pi$$
−0.814702 + 0.579880i $$0.803099\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 11.0000 0.388909
$$801$$ 0 0
$$802$$ 30.0000 1.05934
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −48.0000 −1.69178
$$806$$ −36.0000 −1.26805
$$807$$ 0 0
$$808$$ −14.0000 −0.492518
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ −8.00000 −0.280745
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 48.0000 1.68137
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ −26.0000 −0.909069
$$819$$ 0 0
$$820$$ 40.0000 1.39686
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 0 0
$$823$$ −18.0000 −0.627441 −0.313720 0.949515i $$-0.601575\pi$$
−0.313720 + 0.949515i $$0.601575\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ −24.0000 −0.835067
$$827$$ 16.0000 0.556375 0.278187 0.960527i $$-0.410266\pi$$
0.278187 + 0.960527i $$0.410266\pi$$
$$828$$ 0 0
$$829$$ 6.00000 0.208389 0.104194 0.994557i $$-0.466774\pi$$
0.104194 + 0.994557i $$0.466774\pi$$
$$830$$ −48.0000 −1.66610
$$831$$ 0 0
$$832$$ −6.00000 −0.208013
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −12.0000 −0.414533
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 34.0000 1.17172
$$843$$ 0 0
$$844$$ −8.00000 −0.275371
$$845$$ −92.0000 −3.16490
$$846$$ 0 0
$$847$$ −22.0000 −0.755929
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ 0 0
$$853$$ 16.0000 0.547830 0.273915 0.961754i $$-0.411681\pi$$
0.273915 + 0.961754i $$0.411681\pi$$
$$854$$ 8.00000 0.273754
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 16.0000 0.545595
$$861$$ 0 0
$$862$$ 14.0000 0.476842
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ 0 0
$$865$$ 16.0000 0.544016
$$866$$ −18.0000 −0.611665
$$867$$ 0 0
$$868$$ 12.0000 0.407307
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 72.0000 2.43963
$$872$$ −16.0000 −0.541828
$$873$$ 0 0
$$874$$ 24.0000 0.811812
$$875$$ −48.0000 −1.62270
$$876$$ 0 0
$$877$$ 24.0000 0.810422 0.405211 0.914223i $$-0.367198\pi$$
0.405211 + 0.914223i $$0.367198\pi$$
$$878$$ 10.0000 0.337484
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 50.0000 1.68454 0.842271 0.539054i $$-0.181218\pi$$
0.842271 + 0.539054i $$0.181218\pi$$
$$882$$ 0 0
$$883$$ −52.0000 −1.74994 −0.874970 0.484178i $$-0.839119\pi$$
−0.874970 + 0.484178i $$0.839119\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 18.0000 0.604381 0.302190 0.953248i $$-0.402282\pi$$
0.302190 + 0.953248i $$0.402282\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ −8.00000 −0.268161
$$891$$ 0 0
$$892$$ −4.00000 −0.133930
$$893$$ −16.0000 −0.535420
$$894$$ 0 0
$$895$$ −48.0000 −1.60446
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ −26.0000 −0.867631
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ −80.0000 −2.65929
$$906$$ 0 0
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ 4.00000 0.132745
$$909$$ 0 0
$$910$$ 48.0000 1.59118
$$911$$ 26.0000 0.861418 0.430709 0.902491i $$-0.358263\pi$$
0.430709 + 0.902491i $$0.358263\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 22.0000 0.727695
$$915$$ 0 0
$$916$$ −2.00000 −0.0660819
$$917$$ −32.0000 −1.05673
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ −24.0000 −0.791257
$$921$$ 0 0
$$922$$ 10.0000 0.329332
$$923$$ 36.0000 1.18495
$$924$$ 0 0
$$925$$ 44.0000 1.44671
$$926$$ −4.00000 −0.131448
$$927$$ 0 0
$$928$$ −4.00000 −0.131306
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ −12.0000 −0.393284
$$932$$ −6.00000 −0.196537
$$933$$ 0 0
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ −24.0000 −0.783628
$$939$$ 0 0
$$940$$ 16.0000 0.521862
$$941$$ 48.0000 1.56476 0.782378 0.622804i $$-0.214007\pi$$
0.782378 + 0.622804i $$0.214007\pi$$
$$942$$ 0 0
$$943$$ −60.0000 −1.95387
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −24.0000 −0.779895 −0.389948 0.920837i $$-0.627507\pi$$
−0.389948 + 0.920837i $$0.627507\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ 44.0000 1.42755
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ 0 0
$$955$$ −16.0000 −0.517748
$$956$$ 20.0000 0.646846
$$957$$ 0 0
$$958$$ 10.0000 0.323085
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ −24.0000 −0.773791
$$963$$ 0 0
$$964$$ 18.0000 0.579741
$$965$$ 24.0000 0.772587
$$966$$ 0 0
$$967$$ −44.0000 −1.41494 −0.707472 0.706741i $$-0.750165\pi$$
−0.707472 + 0.706741i $$0.750165\pi$$
$$968$$ −11.0000 −0.353553
$$969$$ 0 0
$$970$$ 24.0000 0.770594
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ 0 0
$$973$$ −16.0000 −0.512936
$$974$$ 38.0000 1.21760
$$975$$ 0 0
$$976$$ 4.00000 0.128037
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 12.0000 0.383326
$$981$$ 0 0
$$982$$ 20.0000 0.638226
$$983$$ −38.0000 −1.21201 −0.606006 0.795460i $$-0.707229\pi$$
−0.606006 + 0.795460i $$0.707229\pi$$
$$984$$ 0 0
$$985$$ −32.0000 −1.01960
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −24.0000 −0.763542
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 34.0000 1.08005 0.540023 0.841650i $$-0.318416\pi$$
0.540023 + 0.841650i $$0.318416\pi$$
$$992$$ 6.00000 0.190500
$$993$$ 0 0
$$994$$ −12.0000 −0.380617
$$995$$ 56.0000 1.77532
$$996$$ 0 0
$$997$$ 20.0000 0.633406 0.316703 0.948525i $$-0.397424\pi$$
0.316703 + 0.948525i $$0.397424\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ 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 5202.2.a.g.1.1 1
3.2 odd 2 1734.2.a.h.1.1 1
17.16 even 2 306.2.a.d.1.1 1
51.2 odd 8 1734.2.f.g.1483.2 4
51.8 odd 8 1734.2.f.g.829.1 4
51.26 odd 8 1734.2.f.g.829.2 4
51.32 odd 8 1734.2.f.g.1483.1 4
51.38 odd 4 1734.2.b.d.577.1 2
51.47 odd 4 1734.2.b.d.577.2 2
51.50 odd 2 102.2.a.a.1.1 1
68.67 odd 2 2448.2.a.t.1.1 1
85.84 even 2 7650.2.a.z.1.1 1
136.67 odd 2 9792.2.a.b.1.1 1
136.101 even 2 9792.2.a.a.1.1 1
204.203 even 2 816.2.a.h.1.1 1
255.152 even 4 2550.2.d.q.2449.1 2
255.203 even 4 2550.2.d.q.2449.2 2
255.254 odd 2 2550.2.a.be.1.1 1
357.356 even 2 4998.2.a.x.1.1 1
408.101 odd 2 3264.2.a.bf.1.1 1
408.203 even 2 3264.2.a.p.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.a.1.1 1 51.50 odd 2
306.2.a.d.1.1 1 17.16 even 2
816.2.a.h.1.1 1 204.203 even 2
1734.2.a.h.1.1 1 3.2 odd 2
1734.2.b.d.577.1 2 51.38 odd 4
1734.2.b.d.577.2 2 51.47 odd 4
1734.2.f.g.829.1 4 51.8 odd 8
1734.2.f.g.829.2 4 51.26 odd 8
1734.2.f.g.1483.1 4 51.32 odd 8
1734.2.f.g.1483.2 4 51.2 odd 8
2448.2.a.t.1.1 1 68.67 odd 2
2550.2.a.be.1.1 1 255.254 odd 2
2550.2.d.q.2449.1 2 255.152 even 4
2550.2.d.q.2449.2 2 255.203 even 4
3264.2.a.p.1.1 1 408.203 even 2
3264.2.a.bf.1.1 1 408.101 odd 2
4998.2.a.x.1.1 1 357.356 even 2
5202.2.a.g.1.1 1 1.1 even 1 trivial
7650.2.a.z.1.1 1 85.84 even 2
9792.2.a.a.1.1 1 136.101 even 2
9792.2.a.b.1.1 1 136.67 odd 2