# Properties

 Label 9025.2.a.j.1.1 Level $9025$ Weight $2$ Character 9025.1 Self dual yes Analytic conductor $72.065$ 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] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.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: $$72.0649878242$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9025.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +2.00000 q^{4} +4.00000 q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} +2.00000 q^{4} +4.00000 q^{7} -3.00000 q^{9} -1.00000 q^{11} -2.00000 q^{13} +8.00000 q^{14} -4.00000 q^{16} +2.00000 q^{17} -6.00000 q^{18} -2.00000 q^{22} -6.00000 q^{23} -4.00000 q^{26} +8.00000 q^{28} -9.00000 q^{29} +7.00000 q^{31} -8.00000 q^{32} +4.00000 q^{34} -6.00000 q^{36} +2.00000 q^{37} -2.00000 q^{41} -2.00000 q^{43} -2.00000 q^{44} -12.0000 q^{46} -6.00000 q^{47} +9.00000 q^{49} -4.00000 q^{52} +4.00000 q^{53} -18.0000 q^{58} -9.00000 q^{59} -7.00000 q^{61} +14.0000 q^{62} -12.0000 q^{63} -8.00000 q^{64} -10.0000 q^{67} +4.00000 q^{68} -1.00000 q^{71} -10.0000 q^{73} +4.00000 q^{74} -4.00000 q^{77} -1.00000 q^{79} +9.00000 q^{81} -4.00000 q^{82} +6.00000 q^{83} -4.00000 q^{86} +11.0000 q^{89} -8.00000 q^{91} -12.0000 q^{92} -12.0000 q^{94} -6.00000 q^{97} +18.0000 q^{98} +3.00000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ 0 0
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 8.00000 2.13809
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ −6.00000 −1.41421
$$19$$ 0 0
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −2.00000 −0.426401
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ 0 0
$$28$$ 8.00000 1.51186
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ −8.00000 −1.41421
$$33$$ 0 0
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ −6.00000 −1.00000
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ −12.0000 −1.76930
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −4.00000 −0.554700
$$53$$ 4.00000 0.549442 0.274721 0.961524i $$-0.411414\pi$$
0.274721 + 0.961524i $$0.411414\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −18.0000 −2.36352
$$59$$ −9.00000 −1.17170 −0.585850 0.810419i $$-0.699239\pi$$
−0.585850 + 0.810419i $$0.699239\pi$$
$$60$$ 0 0
$$61$$ −7.00000 −0.896258 −0.448129 0.893969i $$-0.647910\pi$$
−0.448129 + 0.893969i $$0.647910\pi$$
$$62$$ 14.0000 1.77800
$$63$$ −12.0000 −1.51186
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −10.0000 −1.22169 −0.610847 0.791748i $$-0.709171\pi$$
−0.610847 + 0.791748i $$0.709171\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −1.00000 −0.118678 −0.0593391 0.998238i $$-0.518899\pi$$
−0.0593391 + 0.998238i $$0.518899\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ −4.00000 −0.441726
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 11.0000 1.16600 0.582999 0.812473i $$-0.301879\pi$$
0.582999 + 0.812473i $$0.301879\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ −12.0000 −1.25109
$$93$$ 0 0
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ 18.0000 1.81827
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ 15.0000 1.49256 0.746278 0.665635i $$-0.231839\pi$$
0.746278 + 0.665635i $$0.231839\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 8.00000 0.777029
$$107$$ −10.0000 −0.966736 −0.483368 0.875417i $$-0.660587\pi$$
−0.483368 + 0.875417i $$0.660587\pi$$
$$108$$ 0 0
$$109$$ −15.0000 −1.43674 −0.718370 0.695662i $$-0.755111\pi$$
−0.718370 + 0.695662i $$0.755111\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −16.0000 −1.51186
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −18.0000 −1.67126
$$117$$ 6.00000 0.554700
$$118$$ −18.0000 −1.65703
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ −14.0000 −1.26750
$$123$$ 0 0
$$124$$ 14.0000 1.25724
$$125$$ 0 0
$$126$$ −24.0000 −2.13809
$$127$$ −6.00000 −0.532414 −0.266207 0.963916i $$-0.585770\pi$$
−0.266207 + 0.963916i $$0.585770\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −20.0000 −1.72774
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −2.00000 −0.167836
$$143$$ 2.00000 0.167248
$$144$$ 12.0000 1.00000
$$145$$ 0 0
$$146$$ −20.0000 −1.65521
$$147$$ 0 0
$$148$$ 4.00000 0.328798
$$149$$ −1.00000 −0.0819232 −0.0409616 0.999161i $$-0.513042\pi$$
−0.0409616 + 0.999161i $$0.513042\pi$$
$$150$$ 0 0
$$151$$ −9.00000 −0.732410 −0.366205 0.930534i $$-0.619343\pi$$
−0.366205 + 0.930534i $$0.619343\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ −8.00000 −0.644658
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −4.00000 −0.319235 −0.159617 0.987179i $$-0.551026\pi$$
−0.159617 + 0.987179i $$0.551026\pi$$
$$158$$ −2.00000 −0.159111
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −24.0000 −1.89146
$$162$$ 18.0000 1.41421
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −4.00000 −0.312348
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ −24.0000 −1.82469 −0.912343 0.409426i $$-0.865729\pi$$
−0.912343 + 0.409426i $$0.865729\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ 22.0000 1.64897
$$179$$ −15.0000 −1.12115 −0.560576 0.828103i $$-0.689420\pi$$
−0.560576 + 0.828103i $$0.689420\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ −16.0000 −1.18600
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −2.00000 −0.146254
$$188$$ −12.0000 −0.875190
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3.00000 −0.217072 −0.108536 0.994092i $$-0.534616\pi$$
−0.108536 + 0.994092i $$0.534616\pi$$
$$192$$ 0 0
$$193$$ 16.0000 1.15171 0.575853 0.817554i $$-0.304670\pi$$
0.575853 + 0.817554i $$0.304670\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 0 0
$$196$$ 18.0000 1.28571
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 6.00000 0.426401
$$199$$ 13.0000 0.921546 0.460773 0.887518i $$-0.347572\pi$$
0.460773 + 0.887518i $$0.347572\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 30.0000 2.11079
$$203$$ −36.0000 −2.52670
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 32.0000 2.22955
$$207$$ 18.0000 1.25109
$$208$$ 8.00000 0.554700
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −5.00000 −0.344214 −0.172107 0.985078i $$-0.555058\pi$$
−0.172107 + 0.985078i $$0.555058\pi$$
$$212$$ 8.00000 0.549442
$$213$$ 0 0
$$214$$ −20.0000 −1.36717
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 28.0000 1.90076
$$218$$ −30.0000 −2.03186
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ −32.0000 −2.13809
$$225$$ 0 0
$$226$$ −24.0000 −1.59646
$$227$$ 14.0000 0.929213 0.464606 0.885517i $$-0.346196\pi$$
0.464606 + 0.885517i $$0.346196\pi$$
$$228$$ 0 0
$$229$$ 17.0000 1.12339 0.561696 0.827344i $$-0.310149\pi$$
0.561696 + 0.827344i $$0.310149\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 8.00000 0.524097 0.262049 0.965055i $$-0.415602\pi$$
0.262049 + 0.965055i $$0.415602\pi$$
$$234$$ 12.0000 0.784465
$$235$$ 0 0
$$236$$ −18.0000 −1.17170
$$237$$ 0 0
$$238$$ 16.0000 1.03713
$$239$$ −19.0000 −1.22901 −0.614504 0.788914i $$-0.710644\pi$$
−0.614504 + 0.788914i $$0.710644\pi$$
$$240$$ 0 0
$$241$$ −1.00000 −0.0644157 −0.0322078 0.999481i $$-0.510254\pi$$
−0.0322078 + 0.999481i $$0.510254\pi$$
$$242$$ −20.0000 −1.28565
$$243$$ 0 0
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 13.0000 0.820553 0.410276 0.911961i $$-0.365432\pi$$
0.410276 + 0.911961i $$0.365432\pi$$
$$252$$ −24.0000 −1.51186
$$253$$ 6.00000 0.377217
$$254$$ −12.0000 −0.752947
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 27.0000 1.67126
$$262$$ 24.0000 1.48272
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −20.0000 −1.22169
$$269$$ −3.00000 −0.182913 −0.0914566 0.995809i $$-0.529152\pi$$
−0.0914566 + 0.995809i $$0.529152\pi$$
$$270$$ 0 0
$$271$$ 3.00000 0.182237 0.0911185 0.995840i $$-0.470956\pi$$
0.0911185 + 0.995840i $$0.470956\pi$$
$$272$$ −8.00000 −0.485071
$$273$$ 0 0
$$274$$ −24.0000 −1.44989
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 28.0000 1.68236 0.841178 0.540758i $$-0.181862\pi$$
0.841178 + 0.540758i $$0.181862\pi$$
$$278$$ 40.0000 2.39904
$$279$$ −21.0000 −1.25724
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ −8.00000 −0.472225
$$288$$ 24.0000 1.41421
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −20.0000 −1.17041
$$293$$ 4.00000 0.233682 0.116841 0.993151i $$-0.462723\pi$$
0.116841 + 0.993151i $$0.462723\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ −2.00000 −0.115857
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ −18.0000 −1.03578
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −12.0000 −0.685994
$$307$$ 16.0000 0.913168 0.456584 0.889680i $$-0.349073\pi$$
0.456584 + 0.889680i $$0.349073\pi$$
$$308$$ −8.00000 −0.455842
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −30.0000 −1.69570 −0.847850 0.530236i $$-0.822103\pi$$
−0.847850 + 0.530236i $$0.822103\pi$$
$$314$$ −8.00000 −0.451466
$$315$$ 0 0
$$316$$ −2.00000 −0.112509
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 0 0
$$319$$ 9.00000 0.503903
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −48.0000 −2.67494
$$323$$ 0 0
$$324$$ 18.0000 1.00000
$$325$$ 0 0
$$326$$ −8.00000 −0.443079
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −6.00000 −0.328798
$$334$$ −24.0000 −1.31322
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ −18.0000 −0.979071
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −7.00000 −0.379071
$$342$$ 0 0
$$343$$ 8.00000 0.431959
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −48.0000 −2.58050
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 8.00000 0.426401
$$353$$ 8.00000 0.425797 0.212899 0.977074i $$-0.431710\pi$$
0.212899 + 0.977074i $$0.431710\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 22.0000 1.16600
$$357$$ 0 0
$$358$$ −30.0000 −1.58555
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ 12.0000 0.630706
$$363$$ 0 0
$$364$$ −16.0000 −0.838628
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ 24.0000 1.25109
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 16.0000 0.830679
$$372$$ 0 0
$$373$$ −12.0000 −0.621336 −0.310668 0.950518i $$-0.600553\pi$$
−0.310668 + 0.950518i $$0.600553\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 18.0000 0.927047
$$378$$ 0 0
$$379$$ 29.0000 1.48963 0.744815 0.667271i $$-0.232538\pi$$
0.744815 + 0.667271i $$0.232538\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −6.00000 −0.306987
$$383$$ 6.00000 0.306586 0.153293 0.988181i $$-0.451012\pi$$
0.153293 + 0.988181i $$0.451012\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 32.0000 1.62876
$$387$$ 6.00000 0.304997
$$388$$ −12.0000 −0.609208
$$389$$ −33.0000 −1.67317 −0.836583 0.547840i $$-0.815450\pi$$
−0.836583 + 0.547840i $$0.815450\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 36.0000 1.81365
$$395$$ 0 0
$$396$$ 6.00000 0.301511
$$397$$ 8.00000 0.401508 0.200754 0.979642i $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$398$$ 26.0000 1.30326
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −3.00000 −0.149813 −0.0749064 0.997191i $$-0.523866\pi$$
−0.0749064 + 0.997191i $$0.523866\pi$$
$$402$$ 0 0
$$403$$ −14.0000 −0.697390
$$404$$ 30.0000 1.49256
$$405$$ 0 0
$$406$$ −72.0000 −3.57330
$$407$$ −2.00000 −0.0991363
$$408$$ 0 0
$$409$$ 5.00000 0.247234 0.123617 0.992330i $$-0.460551\pi$$
0.123617 + 0.992330i $$0.460551\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 32.0000 1.57653
$$413$$ −36.0000 −1.77144
$$414$$ 36.0000 1.76930
$$415$$ 0 0
$$416$$ 16.0000 0.784465
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 9.00000 0.439679 0.219839 0.975536i $$-0.429447\pi$$
0.219839 + 0.975536i $$0.429447\pi$$
$$420$$ 0 0
$$421$$ 15.0000 0.731055 0.365528 0.930800i $$-0.380889\pi$$
0.365528 + 0.930800i $$0.380889\pi$$
$$422$$ −10.0000 −0.486792
$$423$$ 18.0000 0.875190
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −28.0000 −1.35501
$$428$$ −20.0000 −0.966736
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −21.0000 −1.01153 −0.505767 0.862670i $$-0.668791\pi$$
−0.505767 + 0.862670i $$0.668791\pi$$
$$432$$ 0 0
$$433$$ −4.00000 −0.192228 −0.0961139 0.995370i $$-0.530641\pi$$
−0.0961139 + 0.995370i $$0.530641\pi$$
$$434$$ 56.0000 2.68809
$$435$$ 0 0
$$436$$ −30.0000 −1.43674
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 13.0000 0.620456 0.310228 0.950662i $$-0.399595\pi$$
0.310228 + 0.950662i $$0.399595\pi$$
$$440$$ 0 0
$$441$$ −27.0000 −1.28571
$$442$$ −8.00000 −0.380521
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −4.00000 −0.189405
$$447$$ 0 0
$$448$$ −32.0000 −1.51186
$$449$$ −31.0000 −1.46298 −0.731490 0.681852i $$-0.761175\pi$$
−0.731490 + 0.681852i $$0.761175\pi$$
$$450$$ 0 0
$$451$$ 2.00000 0.0941763
$$452$$ −24.0000 −1.12887
$$453$$ 0 0
$$454$$ 28.0000 1.31411
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −34.0000 −1.59045 −0.795226 0.606313i $$-0.792648\pi$$
−0.795226 + 0.606313i $$0.792648\pi$$
$$458$$ 34.0000 1.58872
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −25.0000 −1.16437 −0.582183 0.813058i $$-0.697801\pi$$
−0.582183 + 0.813058i $$0.697801\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 36.0000 1.67126
$$465$$ 0 0
$$466$$ 16.0000 0.741186
$$467$$ −10.0000 −0.462745 −0.231372 0.972865i $$-0.574322\pi$$
−0.231372 + 0.972865i $$0.574322\pi$$
$$468$$ 12.0000 0.554700
$$469$$ −40.0000 −1.84703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 2.00000 0.0919601
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 16.0000 0.733359
$$477$$ −12.0000 −0.549442
$$478$$ −38.0000 −1.73808
$$479$$ 15.0000 0.685367 0.342684 0.939451i $$-0.388664\pi$$
0.342684 + 0.939451i $$0.388664\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ −2.00000 −0.0910975
$$483$$ 0 0
$$484$$ −20.0000 −0.909091
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −38.0000 −1.72194 −0.860972 0.508652i $$-0.830144\pi$$
−0.860972 + 0.508652i $$0.830144\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 13.0000 0.586682 0.293341 0.956008i $$-0.405233\pi$$
0.293341 + 0.956008i $$0.405233\pi$$
$$492$$ 0 0
$$493$$ −18.0000 −0.810679
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −28.0000 −1.25724
$$497$$ −4.00000 −0.179425
$$498$$ 0 0
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 26.0000 1.16044
$$503$$ 26.0000 1.15928 0.579641 0.814872i $$-0.303193\pi$$
0.579641 + 0.814872i $$0.303193\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 12.0000 0.533465
$$507$$ 0 0
$$508$$ −12.0000 −0.532414
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ −40.0000 −1.76950
$$512$$ 32.0000 1.41421
$$513$$ 0 0
$$514$$ −12.0000 −0.529297
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 6.00000 0.263880
$$518$$ 16.0000 0.703000
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −15.0000 −0.657162 −0.328581 0.944476i $$-0.606570\pi$$
−0.328581 + 0.944476i $$0.606570\pi$$
$$522$$ 54.0000 2.36352
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ 24.0000 1.04844
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 14.0000 0.609850
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 27.0000 1.17170
$$532$$ 0 0
$$533$$ 4.00000 0.173259
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −6.00000 −0.258678
$$539$$ −9.00000 −0.387657
$$540$$ 0 0
$$541$$ 23.0000 0.988847 0.494424 0.869221i $$-0.335379\pi$$
0.494424 + 0.869221i $$0.335379\pi$$
$$542$$ 6.00000 0.257722
$$543$$ 0 0
$$544$$ −16.0000 −0.685994
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 22.0000 0.940652 0.470326 0.882493i $$-0.344136\pi$$
0.470326 + 0.882493i $$0.344136\pi$$
$$548$$ −24.0000 −1.02523
$$549$$ 21.0000 0.896258
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −4.00000 −0.170097
$$554$$ 56.0000 2.37921
$$555$$ 0 0
$$556$$ 40.0000 1.69638
$$557$$ −14.0000 −0.593199 −0.296600 0.955002i $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$558$$ −42.0000 −1.77800
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −20.0000 −0.843649
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −28.0000 −1.17693
$$567$$ 36.0000 1.51186
$$568$$ 0 0
$$569$$ −15.0000 −0.628833 −0.314416 0.949285i $$-0.601809\pi$$
−0.314416 + 0.949285i $$0.601809\pi$$
$$570$$ 0 0
$$571$$ 13.0000 0.544033 0.272017 0.962293i $$-0.412309\pi$$
0.272017 + 0.962293i $$0.412309\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ −16.0000 −0.667827
$$575$$ 0 0
$$576$$ 24.0000 1.00000
$$577$$ −6.00000 −0.249783 −0.124892 0.992170i $$-0.539858\pi$$
−0.124892 + 0.992170i $$0.539858\pi$$
$$578$$ −26.0000 −1.08146
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ −4.00000 −0.165663
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 8.00000 0.330477
$$587$$ 24.0000 0.990586 0.495293 0.868726i $$-0.335061\pi$$
0.495293 + 0.868726i $$0.335061\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −8.00000 −0.328798
$$593$$ 32.0000 1.31408 0.657041 0.753855i $$-0.271808\pi$$
0.657041 + 0.753855i $$0.271808\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −2.00000 −0.0819232
$$597$$ 0 0
$$598$$ 24.0000 0.981433
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ −17.0000 −0.693444 −0.346722 0.937968i $$-0.612705\pi$$
−0.346722 + 0.937968i $$0.612705\pi$$
$$602$$ −16.0000 −0.652111
$$603$$ 30.0000 1.22169
$$604$$ −18.0000 −0.732410
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −14.0000 −0.568242 −0.284121 0.958788i $$-0.591702\pi$$
−0.284121 + 0.958788i $$0.591702\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ −12.0000 −0.485071
$$613$$ −24.0000 −0.969351 −0.484675 0.874694i $$-0.661062\pi$$
−0.484675 + 0.874694i $$0.661062\pi$$
$$614$$ 32.0000 1.29141
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 44.0000 1.76282
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −60.0000 −2.39808
$$627$$ 0 0
$$628$$ −8.00000 −0.319235
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ 1.00000 0.0398094 0.0199047 0.999802i $$-0.493664\pi$$
0.0199047 + 0.999802i $$0.493664\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −4.00000 −0.158860
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −18.0000 −0.713186
$$638$$ 18.0000 0.712627
$$639$$ 3.00000 0.118678
$$640$$ 0 0
$$641$$ −21.0000 −0.829450 −0.414725 0.909947i $$-0.636122\pi$$
−0.414725 + 0.909947i $$0.636122\pi$$
$$642$$ 0 0
$$643$$ 46.0000 1.81406 0.907031 0.421063i $$-0.138343\pi$$
0.907031 + 0.421063i $$0.138343\pi$$
$$644$$ −48.0000 −1.89146
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ 0 0
$$649$$ 9.00000 0.353281
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −8.00000 −0.313304
$$653$$ −10.0000 −0.391330 −0.195665 0.980671i $$-0.562687\pi$$
−0.195665 + 0.980671i $$0.562687\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 8.00000 0.312348
$$657$$ 30.0000 1.17041
$$658$$ −48.0000 −1.87123
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ 15.0000 0.583432 0.291716 0.956505i $$-0.405774\pi$$
0.291716 + 0.956505i $$0.405774\pi$$
$$662$$ −40.0000 −1.55464
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −12.0000 −0.464991
$$667$$ 54.0000 2.09089
$$668$$ −24.0000 −0.928588
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 7.00000 0.270232
$$672$$ 0 0
$$673$$ −20.0000 −0.770943 −0.385472 0.922720i $$-0.625961\pi$$
−0.385472 + 0.922720i $$0.625961\pi$$
$$674$$ 68.0000 2.61926
$$675$$ 0 0
$$676$$ −18.0000 −0.692308
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ −24.0000 −0.921035
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −14.0000 −0.536088
$$683$$ 30.0000 1.14792 0.573959 0.818884i $$-0.305407\pi$$
0.573959 + 0.818884i $$0.305407\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 16.0000 0.610883
$$687$$ 0 0
$$688$$ 8.00000 0.304997
$$689$$ −8.00000 −0.304776
$$690$$ 0 0
$$691$$ 3.00000 0.114125 0.0570627 0.998371i $$-0.481827\pi$$
0.0570627 + 0.998371i $$0.481827\pi$$
$$692$$ −48.0000 −1.82469
$$693$$ 12.0000 0.455842
$$694$$ 24.0000 0.911028
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −4.00000 −0.151511
$$698$$ 28.0000 1.05982
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 8.00000 0.301511
$$705$$ 0 0
$$706$$ 16.0000 0.602168
$$707$$ 60.0000 2.25653
$$708$$ 0 0
$$709$$ 25.0000 0.938895 0.469447 0.882960i $$-0.344453\pi$$
0.469447 + 0.882960i $$0.344453\pi$$
$$710$$ 0 0
$$711$$ 3.00000 0.112509
$$712$$ 0 0
$$713$$ −42.0000 −1.57291
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −30.0000 −1.12115
$$717$$ 0 0
$$718$$ −40.0000 −1.49279
$$719$$ −21.0000 −0.783168 −0.391584 0.920142i $$-0.628073\pi$$
−0.391584 + 0.920142i $$0.628073\pi$$
$$720$$ 0 0
$$721$$ 64.0000 2.38348
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 12.0000 0.445976
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −4.00000 −0.147945
$$732$$ 0 0
$$733$$ 6.00000 0.221615 0.110808 0.993842i $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ 32.0000 1.18114
$$735$$ 0 0
$$736$$ 48.0000 1.76930
$$737$$ 10.0000 0.368355
$$738$$ 12.0000 0.441726
$$739$$ −29.0000 −1.06678 −0.533391 0.845869i $$-0.679083\pi$$
−0.533391 + 0.845869i $$0.679083\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 32.0000 1.17476
$$743$$ 50.0000 1.83432 0.917161 0.398517i $$-0.130475\pi$$
0.917161 + 0.398517i $$0.130475\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −24.0000 −0.878702
$$747$$ −18.0000 −0.658586
$$748$$ −4.00000 −0.146254
$$749$$ −40.0000 −1.46157
$$750$$ 0 0
$$751$$ −41.0000 −1.49611 −0.748056 0.663636i $$-0.769012\pi$$
−0.748056 + 0.663636i $$0.769012\pi$$
$$752$$ 24.0000 0.875190
$$753$$ 0 0
$$754$$ 36.0000 1.31104
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ 58.0000 2.10665
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 0 0
$$763$$ −60.0000 −2.17215
$$764$$ −6.00000 −0.217072
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ 18.0000 0.649942
$$768$$ 0 0
$$769$$ 5.00000 0.180305 0.0901523 0.995928i $$-0.471265\pi$$
0.0901523 + 0.995928i $$0.471265\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 32.0000 1.15171
$$773$$ 34.0000 1.22290 0.611448 0.791285i $$-0.290588\pi$$
0.611448 + 0.791285i $$0.290588\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −66.0000 −2.36621
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 1.00000 0.0357828
$$782$$ −24.0000 −0.858238
$$783$$ 0 0
$$784$$ −36.0000 −1.28571
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 22.0000 0.784215 0.392108 0.919919i $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ 36.0000 1.28245
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −48.0000 −1.70668
$$792$$ 0 0
$$793$$ 14.0000 0.497155
$$794$$ 16.0000 0.567819
$$795$$ 0 0
$$796$$ 26.0000 0.921546
$$797$$ 24.0000 0.850124 0.425062 0.905164i $$-0.360252\pi$$
0.425062 + 0.905164i $$0.360252\pi$$
$$798$$ 0 0
$$799$$ −12.0000 −0.424529
$$800$$ 0 0
$$801$$ −33.0000 −1.16600
$$802$$ −6.00000 −0.211867
$$803$$ 10.0000 0.352892
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −28.0000 −0.986258
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −43.0000 −1.51180 −0.755900 0.654687i $$-0.772800\pi$$
−0.755900 + 0.654687i $$0.772800\pi$$
$$810$$ 0 0
$$811$$ −9.00000 −0.316033 −0.158016 0.987436i $$-0.550510\pi$$
−0.158016 + 0.987436i $$0.550510\pi$$
$$812$$ −72.0000 −2.52670
$$813$$ 0 0
$$814$$ −4.00000 −0.140200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 10.0000 0.349642
$$819$$ 24.0000 0.838628
$$820$$ 0 0
$$821$$ 21.0000 0.732905 0.366453 0.930437i $$-0.380572\pi$$
0.366453 + 0.930437i $$0.380572\pi$$
$$822$$ 0 0
$$823$$ 30.0000 1.04573 0.522867 0.852414i $$-0.324862\pi$$
0.522867 + 0.852414i $$0.324862\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −72.0000 −2.50520
$$827$$ 6.00000 0.208640 0.104320 0.994544i $$-0.466733\pi$$
0.104320 + 0.994544i $$0.466733\pi$$
$$828$$ 36.0000 1.25109
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 16.0000 0.554700
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 18.0000 0.621800
$$839$$ −48.0000 −1.65714 −0.828572 0.559883i $$-0.810846\pi$$
−0.828572 + 0.559883i $$0.810846\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 30.0000 1.03387
$$843$$ 0 0
$$844$$ −10.0000 −0.344214
$$845$$ 0 0
$$846$$ 36.0000 1.23771
$$847$$ −40.0000 −1.37442
$$848$$ −16.0000 −0.549442
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ 16.0000 0.547830 0.273915 0.961754i $$-0.411681\pi$$
0.273915 + 0.961754i $$0.411681\pi$$
$$854$$ −56.0000 −1.91628
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ 1.00000 0.0341196 0.0170598 0.999854i $$-0.494569\pi$$
0.0170598 + 0.999854i $$0.494569\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −42.0000 −1.43053
$$863$$ 20.0000 0.680808 0.340404 0.940279i $$-0.389436\pi$$
0.340404 + 0.940279i $$0.389436\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −8.00000 −0.271851
$$867$$ 0 0
$$868$$ 56.0000 1.90076
$$869$$ 1.00000 0.0339227
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 0 0
$$873$$ 18.0000 0.609208
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −58.0000 −1.95852 −0.979260 0.202606i $$-0.935059\pi$$
−0.979260 + 0.202606i $$0.935059\pi$$
$$878$$ 26.0000 0.877457
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 37.0000 1.24656 0.623281 0.781998i $$-0.285799\pi$$
0.623281 + 0.781998i $$0.285799\pi$$
$$882$$ −54.0000 −1.81827
$$883$$ 8.00000 0.269221 0.134611 0.990899i $$-0.457022\pi$$
0.134611 + 0.990899i $$0.457022\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 0 0
$$886$$ −8.00000 −0.268765
$$887$$ −16.0000 −0.537227 −0.268614 0.963248i $$-0.586566\pi$$
−0.268614 + 0.963248i $$0.586566\pi$$
$$888$$ 0 0
$$889$$ −24.0000 −0.804934
$$890$$ 0 0
$$891$$ −9.00000 −0.301511
$$892$$ −4.00000 −0.133930
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −62.0000 −2.06897
$$899$$ −63.0000 −2.10117
$$900$$ 0 0
$$901$$ 8.00000 0.266519
$$902$$ 4.00000 0.133185
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ 28.0000 0.929213
$$909$$ −45.0000 −1.49256
$$910$$ 0 0
$$911$$ −41.0000 −1.35839 −0.679195 0.733958i $$-0.737671\pi$$
−0.679195 + 0.733958i $$0.737671\pi$$
$$912$$ 0 0
$$913$$ −6.00000 −0.198571
$$914$$ −68.0000 −2.24924
$$915$$ 0 0
$$916$$ 34.0000 1.12339
$$917$$ 48.0000 1.58510
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −50.0000 −1.64666
$$923$$ 2.00000 0.0658308
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 8.00000 0.262896
$$927$$ −48.0000 −1.57653
$$928$$ 72.0000 2.36352
$$929$$ 51.0000 1.67326 0.836628 0.547772i $$-0.184524\pi$$
0.836628 + 0.547772i $$0.184524\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 16.0000 0.524097
$$933$$ 0 0
$$934$$ −20.0000 −0.654420
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 56.0000 1.82944 0.914720 0.404088i $$-0.132411\pi$$
0.914720 + 0.404088i $$0.132411\pi$$
$$938$$ −80.0000 −2.61209
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −35.0000 −1.14097 −0.570484 0.821309i $$-0.693244\pi$$
−0.570484 + 0.821309i $$0.693244\pi$$
$$942$$ 0 0
$$943$$ 12.0000 0.390774
$$944$$ 36.0000 1.17170
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ 42.0000 1.36482 0.682408 0.730971i $$-0.260933\pi$$
0.682408 + 0.730971i $$0.260933\pi$$
$$948$$ 0 0
$$949$$ 20.0000 0.649227
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −24.0000 −0.777436 −0.388718 0.921357i $$-0.627082\pi$$
−0.388718 + 0.921357i $$0.627082\pi$$
$$954$$ −24.0000 −0.777029
$$955$$ 0 0
$$956$$ −38.0000 −1.22901
$$957$$ 0 0
$$958$$ 30.0000 0.969256
$$959$$ −48.0000 −1.55000
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ −8.00000 −0.257930
$$963$$ 30.0000 0.966736
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −22.0000 −0.707472 −0.353736 0.935345i $$-0.615089\pi$$
−0.353736 + 0.935345i $$0.615089\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 0 0
$$973$$ 80.0000 2.56468
$$974$$ −76.0000 −2.43520
$$975$$ 0 0
$$976$$ 28.0000 0.896258
$$977$$ −44.0000 −1.40768 −0.703842 0.710356i $$-0.748534\pi$$
−0.703842 + 0.710356i $$0.748534\pi$$
$$978$$ 0 0
$$979$$ −11.0000 −0.351562
$$980$$ 0 0
$$981$$ 45.0000 1.43674
$$982$$ 26.0000 0.829693
$$983$$ 36.0000 1.14822 0.574111 0.818778i $$-0.305348\pi$$
0.574111 + 0.818778i $$0.305348\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −36.0000 −1.14647
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 12.0000 0.381578
$$990$$ 0 0
$$991$$ −44.0000 −1.39771 −0.698853 0.715265i $$-0.746306\pi$$
−0.698853 + 0.715265i $$0.746306\pi$$
$$992$$ −56.0000 −1.77800
$$993$$ 0 0
$$994$$ −8.00000 −0.253745
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 56.0000 1.77265
$$999$$ 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 9025.2.a.j.1.1 1
5.2 odd 4 1805.2.b.a.1084.2 2
5.3 odd 4 1805.2.b.a.1084.1 2
5.4 even 2 9025.2.a.a.1.1 1
19.8 odd 6 475.2.e.c.26.1 2
19.12 odd 6 475.2.e.c.201.1 2
19.18 odd 2 9025.2.a.b.1.1 1
95.8 even 12 95.2.i.a.64.2 yes 4
95.12 even 12 95.2.i.a.49.2 yes 4
95.18 even 4 1805.2.b.b.1084.2 2
95.27 even 12 95.2.i.a.64.1 yes 4
95.37 even 4 1805.2.b.b.1084.1 2
95.69 odd 6 475.2.e.a.201.1 2
95.84 odd 6 475.2.e.a.26.1 2
95.88 even 12 95.2.i.a.49.1 4
95.94 odd 2 9025.2.a.i.1.1 1
285.8 odd 12 855.2.be.a.64.1 4
285.107 odd 12 855.2.be.a.334.1 4
285.122 odd 12 855.2.be.a.64.2 4
285.278 odd 12 855.2.be.a.334.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.i.a.49.1 4 95.88 even 12
95.2.i.a.49.2 yes 4 95.12 even 12
95.2.i.a.64.1 yes 4 95.27 even 12
95.2.i.a.64.2 yes 4 95.8 even 12
475.2.e.a.26.1 2 95.84 odd 6
475.2.e.a.201.1 2 95.69 odd 6
475.2.e.c.26.1 2 19.8 odd 6
475.2.e.c.201.1 2 19.12 odd 6
855.2.be.a.64.1 4 285.8 odd 12
855.2.be.a.64.2 4 285.122 odd 12
855.2.be.a.334.1 4 285.107 odd 12
855.2.be.a.334.2 4 285.278 odd 12
1805.2.b.a.1084.1 2 5.3 odd 4
1805.2.b.a.1084.2 2 5.2 odd 4
1805.2.b.b.1084.1 2 95.37 even 4
1805.2.b.b.1084.2 2 95.18 even 4
9025.2.a.a.1.1 1 5.4 even 2
9025.2.a.b.1.1 1 19.18 odd 2
9025.2.a.i.1.1 1 95.94 odd 2
9025.2.a.j.1.1 1 1.1 even 1 trivial