Properties

 Label 1734.2.b.d.577.2 Level $1734$ Weight $2$ Character 1734.577 Analytic conductor $13.846$ Analytic rank $0$ Dimension $2$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1734,2,Mod(577,1734)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1734, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1734.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1734 = 2 \cdot 3 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1734.b (of order $$2$$, degree $$1$$, not minimal)

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: no Analytic conductor: $$13.8460597105$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 577.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1734.577 Dual form 1734.2.b.d.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +4.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +4.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} +1.00000 q^{8} -1.00000 q^{9} +4.00000i q^{10} +1.00000i q^{12} -6.00000 q^{13} -2.00000i q^{14} -4.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.00000 q^{19} +4.00000i q^{20} +2.00000 q^{21} +6.00000i q^{23} +1.00000i q^{24} -11.0000 q^{25} -6.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} +4.00000i q^{29} -4.00000 q^{30} +6.00000i q^{31} +1.00000 q^{32} +8.00000 q^{35} -1.00000 q^{36} +4.00000i q^{37} -4.00000 q^{38} -6.00000i q^{39} +4.00000i q^{40} -10.0000i q^{41} +2.00000 q^{42} +4.00000 q^{43} -4.00000i q^{45} +6.00000i q^{46} +4.00000 q^{47} +1.00000i q^{48} +3.00000 q^{49} -11.0000 q^{50} -6.00000 q^{52} +2.00000 q^{53} -1.00000i q^{54} -2.00000i q^{56} -4.00000i q^{57} +4.00000i q^{58} -12.0000 q^{59} -4.00000 q^{60} -4.00000i q^{61} +6.00000i q^{62} +2.00000i q^{63} +1.00000 q^{64} -24.0000i q^{65} -12.0000 q^{67} -6.00000 q^{69} +8.00000 q^{70} +6.00000i q^{71} -1.00000 q^{72} -2.00000i q^{73} +4.00000i q^{74} -11.0000i q^{75} -4.00000 q^{76} -6.00000i q^{78} +10.0000i q^{79} +4.00000i q^{80} +1.00000 q^{81} -10.0000i q^{82} +12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} -4.00000 q^{87} -2.00000 q^{89} -4.00000i q^{90} +12.0000i q^{91} +6.00000i q^{92} -6.00000 q^{93} +4.00000 q^{94} -16.0000i q^{95} +1.00000i q^{96} -6.00000i q^{97} +3.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 12 q^{13} - 8 q^{15} + 2 q^{16} - 2 q^{18} - 8 q^{19} + 4 q^{21} - 22 q^{25} - 12 q^{26} - 8 q^{30} + 2 q^{32} + 16 q^{35} - 2 q^{36} - 8 q^{38} + 4 q^{42} + 8 q^{43} + 8 q^{47} + 6 q^{49} - 22 q^{50} - 12 q^{52} + 4 q^{53} - 24 q^{59} - 8 q^{60} + 2 q^{64} - 24 q^{67} - 12 q^{69} + 16 q^{70} - 2 q^{72} - 8 q^{76} + 2 q^{81} + 24 q^{83} + 4 q^{84} + 8 q^{86} - 8 q^{87} - 4 q^{89} - 12 q^{93} + 8 q^{94} + 6 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 12 * q^13 - 8 * q^15 + 2 * q^16 - 2 * q^18 - 8 * q^19 + 4 * q^21 - 22 * q^25 - 12 * q^26 - 8 * q^30 + 2 * q^32 + 16 * q^35 - 2 * q^36 - 8 * q^38 + 4 * q^42 + 8 * q^43 + 8 * q^47 + 6 * q^49 - 22 * q^50 - 12 * q^52 + 4 * q^53 - 24 * q^59 - 8 * q^60 + 2 * q^64 - 24 * q^67 - 12 * q^69 + 16 * q^70 - 2 * q^72 - 8 * q^76 + 2 * q^81 + 24 * q^83 + 4 * q^84 + 8 * q^86 - 8 * q^87 - 4 * q^89 - 12 * q^93 + 8 * q^94 + 6 * q^98

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times$$.

 $$n$$ $$1157$$ $$1159$$ $$\chi(n)$$ $$1$$ $$-1$$

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
$$3$$ 1.00000i 0.577350i
$$4$$ 1.00000 0.500000
$$5$$ 4.00000i 1.78885i 0.447214 + 0.894427i $$0.352416\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 1.00000i 0.408248i
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −1.00000 −0.333333
$$10$$ 4.00000i 1.26491i
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ − 2.00000i − 0.534522i
$$15$$ −4.00000 −1.03280
$$16$$ 1.00000 0.250000
$$17$$ 0 0
$$18$$ −1.00000 −0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 4.00000i 0.894427i
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −11.0000 −2.20000
$$26$$ −6.00000 −1.17670
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ 4.00000i 0.742781i 0.928477 + 0.371391i $$0.121119\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ −4.00000 −0.730297
$$31$$ 6.00000i 1.07763i 0.842424 + 0.538816i $$0.181128\pi$$
−0.842424 + 0.538816i $$0.818872\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 8.00000 1.35225
$$36$$ −1.00000 −0.166667
$$37$$ 4.00000i 0.657596i 0.944400 + 0.328798i $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ − 6.00000i − 0.960769i
$$40$$ 4.00000i 0.632456i
$$41$$ − 10.0000i − 1.56174i −0.624695 0.780869i $$-0.714777\pi$$
0.624695 0.780869i $$-0.285223\pi$$
$$42$$ 2.00000 0.308607
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ − 4.00000i − 0.596285i
$$46$$ 6.00000i 0.884652i
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 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 $$0.456138\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 0 0
$$56$$ − 2.00000i − 0.267261i
$$57$$ − 4.00000i − 0.529813i
$$58$$ 4.00000i 0.525226i
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ −4.00000 −0.516398
$$61$$ − 4.00000i − 0.512148i −0.966657 0.256074i $$-0.917571\pi$$
0.966657 0.256074i $$-0.0824290\pi$$
$$62$$ 6.00000i 0.762001i
$$63$$ 2.00000i 0.251976i
$$64$$ 1.00000 0.125000
$$65$$ − 24.0000i − 2.97683i
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 8.00000 0.956183
$$71$$ 6.00000i 0.712069i 0.934473 + 0.356034i $$0.115871\pi$$
−0.934473 + 0.356034i $$0.884129\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ 4.00000i 0.464991i
$$75$$ − 11.0000i − 1.27017i
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ − 6.00000i − 0.679366i
$$79$$ 10.0000i 1.12509i 0.826767 + 0.562544i $$0.190177\pi$$
−0.826767 + 0.562544i $$0.809823\pi$$
$$80$$ 4.00000i 0.447214i
$$81$$ 1.00000 0.111111
$$82$$ − 10.0000i − 1.10432i
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ −4.00000 −0.428845
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ − 4.00000i − 0.421637i
$$91$$ 12.0000i 1.25794i
$$92$$ 6.00000i 0.625543i
$$93$$ −6.00000 −0.622171
$$94$$ 4.00000 0.412568
$$95$$ − 16.0000i − 1.64157i
$$96$$ 1.00000i 0.102062i
$$97$$ − 6.00000i − 0.609208i −0.952479 0.304604i $$-0.901476\pi$$
0.952479 0.304604i $$-0.0985241\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 0 0
$$100$$ −11.0000 −1.10000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 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
$$105$$ 8.00000i 0.780720i
$$106$$ 2.00000 0.194257
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 16.0000i 1.53252i 0.642529 + 0.766261i $$0.277885\pi$$
−0.642529 + 0.766261i $$0.722115\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ − 2.00000i − 0.188982i
$$113$$ 2.00000i 0.188144i 0.995565 + 0.0940721i $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ − 4.00000i − 0.374634i
$$115$$ −24.0000 −2.23801
$$116$$ 4.00000i 0.371391i
$$117$$ 6.00000 0.554700
$$118$$ −12.0000 −1.10469
$$119$$ 0 0
$$120$$ −4.00000 −0.365148
$$121$$ 11.0000 1.00000
$$122$$ − 4.00000i − 0.362143i
$$123$$ 10.0000 0.901670
$$124$$ 6.00000i 0.538816i
$$125$$ − 24.0000i − 2.14663i
$$126$$ 2.00000i 0.178174i
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.00000i 0.352180i
$$130$$ − 24.0000i − 2.10494i
$$131$$ 16.0000i 1.39793i 0.715158 + 0.698963i $$0.246355\pi$$
−0.715158 + 0.698963i $$0.753645\pi$$
$$132$$ 0 0
$$133$$ 8.00000i 0.693688i
$$134$$ −12.0000 −1.03664
$$135$$ 4.00000 0.344265
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ −6.00000 −0.510754
$$139$$ − 8.00000i − 0.678551i −0.940687 0.339276i $$-0.889818\pi$$
0.940687 0.339276i $$-0.110182\pi$$
$$140$$ 8.00000 0.676123
$$141$$ 4.00000i 0.336861i
$$142$$ 6.00000i 0.503509i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ −16.0000 −1.32873
$$146$$ − 2.00000i − 0.165521i
$$147$$ 3.00000i 0.247436i
$$148$$ 4.00000i 0.328798i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ − 11.0000i − 0.898146i
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −24.0000 −1.92773
$$156$$ − 6.00000i − 0.480384i
$$157$$ 6.00000 0.478852 0.239426 0.970915i $$-0.423041\pi$$
0.239426 + 0.970915i $$0.423041\pi$$
$$158$$ 10.0000i 0.795557i
$$159$$ 2.00000i 0.158610i
$$160$$ 4.00000i 0.316228i
$$161$$ 12.0000 0.945732
$$162$$ 1.00000 0.0785674
$$163$$ 12.0000i 0.939913i 0.882690 + 0.469956i $$0.155730\pi$$
−0.882690 + 0.469956i $$0.844270\pi$$
$$164$$ − 10.0000i − 0.780869i
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 2.00000i 0.154765i 0.997001 + 0.0773823i $$0.0246562\pi$$
−0.997001 + 0.0773823i $$0.975344\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 4.00000 0.304997
$$173$$ 4.00000i 0.304114i 0.988372 + 0.152057i $$0.0485898\pi$$
−0.988372 + 0.152057i $$0.951410\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 22.0000i 1.66304i
$$176$$ 0 0
$$177$$ − 12.0000i − 0.901975i
$$178$$ −2.00000 −0.149906
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ − 4.00000i − 0.298142i
$$181$$ − 20.0000i − 1.48659i −0.668965 0.743294i $$-0.733262\pi$$
0.668965 0.743294i $$-0.266738\pi$$
$$182$$ 12.0000i 0.889499i
$$183$$ 4.00000 0.295689
$$184$$ 6.00000i 0.442326i
$$185$$ −16.0000 −1.17634
$$186$$ −6.00000 −0.439941
$$187$$ 0 0
$$188$$ 4.00000 0.291730
$$189$$ −2.00000 −0.145479
$$190$$ − 16.0000i − 1.16076i
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 6.00000i 0.431889i 0.976406 + 0.215945i $$0.0692831\pi$$
−0.976406 + 0.215945i $$0.930717\pi$$
$$194$$ − 6.00000i − 0.430775i
$$195$$ 24.0000 1.71868
$$196$$ 3.00000 0.214286
$$197$$ 8.00000i 0.569976i 0.958531 + 0.284988i $$0.0919897\pi$$
−0.958531 + 0.284988i $$0.908010\pi$$
$$198$$ 0 0
$$199$$ − 14.0000i − 0.992434i −0.868199 0.496217i $$-0.834722\pi$$
0.868199 0.496217i $$-0.165278\pi$$
$$200$$ −11.0000 −0.777817
$$201$$ − 12.0000i − 0.846415i
$$202$$ 14.0000 0.985037
$$203$$ 8.00000 0.561490
$$204$$ 0 0
$$205$$ 40.0000 2.79372
$$206$$ 4.00000 0.278693
$$207$$ − 6.00000i − 0.417029i
$$208$$ −6.00000 −0.416025
$$209$$ 0 0
$$210$$ 8.00000i 0.552052i
$$211$$ 8.00000i 0.550743i 0.961338 + 0.275371i $$0.0888008\pi$$
−0.961338 + 0.275371i $$0.911199\pi$$
$$212$$ 2.00000 0.137361
$$213$$ −6.00000 −0.411113
$$214$$ 0 0
$$215$$ 16.0000i 1.09119i
$$216$$ − 1.00000i − 0.0680414i
$$217$$ 12.0000 0.814613
$$218$$ 16.0000i 1.08366i
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −4.00000 −0.268462
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ − 2.00000i − 0.133631i
$$225$$ 11.0000 0.733333
$$226$$ 2.00000i 0.133038i
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ −24.0000 −1.58251
$$231$$ 0 0
$$232$$ 4.00000i 0.262613i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 16.0000i 1.04372i
$$236$$ −12.0000 −0.781133
$$237$$ −10.0000 −0.649570
$$238$$ 0 0
$$239$$ −20.0000 −1.29369 −0.646846 0.762620i $$-0.723912\pi$$
−0.646846 + 0.762620i $$0.723912\pi$$
$$240$$ −4.00000 −0.258199
$$241$$ 18.0000i 1.15948i 0.814801 + 0.579741i $$0.196846\pi$$
−0.814801 + 0.579741i $$0.803154\pi$$
$$242$$ 11.0000 0.707107
$$243$$ 1.00000i 0.0641500i
$$244$$ − 4.00000i − 0.256074i
$$245$$ 12.0000i 0.766652i
$$246$$ 10.0000 0.637577
$$247$$ 24.0000 1.52708
$$248$$ 6.00000i 0.381000i
$$249$$ 12.0000i 0.760469i
$$250$$ − 24.0000i − 1.51789i
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 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 $$0.689739\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 8.00000 0.497096
$$260$$ − 24.0000i − 1.48842i
$$261$$ − 4.00000i − 0.247594i
$$262$$ 16.0000i 0.988483i
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ 8.00000i 0.491436i
$$266$$ 8.00000i 0.490511i
$$267$$ − 2.00000i − 0.122398i
$$268$$ −12.0000 −0.733017
$$269$$ 12.0000i 0.731653i 0.930683 + 0.365826i $$0.119214\pi$$
−0.930683 + 0.365826i $$0.880786\pi$$
$$270$$ 4.00000 0.243432
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ −12.0000 −0.726273
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ 8.00000i 0.480673i 0.970690 + 0.240337i $$0.0772579\pi$$
−0.970690 + 0.240337i $$0.922742\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ − 6.00000i − 0.359211i
$$280$$ 8.00000 0.478091
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 4.00000i 0.238197i
$$283$$ 32.0000i 1.90220i 0.308879 + 0.951101i $$0.400046\pi$$
−0.308879 + 0.951101i $$0.599954\pi$$
$$284$$ 6.00000i 0.356034i
$$285$$ 16.0000 0.947758
$$286$$ 0 0
$$287$$ −20.0000 −1.18056
$$288$$ −1.00000 −0.0589256
$$289$$ 0 0
$$290$$ −16.0000 −0.939552
$$291$$ 6.00000 0.351726
$$292$$ − 2.00000i − 0.117041i
$$293$$ 2.00000 0.116841 0.0584206 0.998292i $$-0.481394\pi$$
0.0584206 + 0.998292i $$0.481394\pi$$
$$294$$ 3.00000i 0.174964i
$$295$$ − 48.0000i − 2.79467i
$$296$$ 4.00000i 0.232495i
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ − 36.0000i − 2.08193i
$$300$$ − 11.0000i − 0.635085i
$$301$$ − 8.00000i − 0.461112i
$$302$$ 24.0000 1.38104
$$303$$ 14.0000i 0.804279i
$$304$$ −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
$$309$$ 4.00000i 0.227552i
$$310$$ −24.0000 −1.36311
$$311$$ − 30.0000i − 1.70114i −0.525859 0.850572i $$-0.676256\pi$$
0.525859 0.850572i $$-0.323744\pi$$
$$312$$ − 6.00000i − 0.339683i
$$313$$ − 26.0000i − 1.46961i −0.678280 0.734803i $$-0.737274\pi$$
0.678280 0.734803i $$-0.262726\pi$$
$$314$$ 6.00000 0.338600
$$315$$ −8.00000 −0.450749
$$316$$ 10.0000i 0.562544i
$$317$$ − 16.0000i − 0.898650i −0.893368 0.449325i $$-0.851665\pi$$
0.893368 0.449325i $$-0.148335\pi$$
$$318$$ 2.00000i 0.112154i
$$319$$ 0 0
$$320$$ 4.00000i 0.223607i
$$321$$ 0 0
$$322$$ 12.0000 0.668734
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 66.0000 3.66102
$$326$$ 12.0000i 0.664619i
$$327$$ −16.0000 −0.884802
$$328$$ − 10.0000i − 0.552158i
$$329$$ − 8.00000i − 0.441054i
$$330$$ 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
$$333$$ − 4.00000i − 0.219199i
$$334$$ 2.00000i 0.109435i
$$335$$ − 48.0000i − 2.62252i
$$336$$ 2.00000 0.109109
$$337$$ 6.00000i 0.326841i 0.986557 + 0.163420i $$0.0522527\pi$$
−0.986557 + 0.163420i $$0.947747\pi$$
$$338$$ 23.0000 1.25104
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000 0.216295
$$343$$ − 20.0000i − 1.07990i
$$344$$ 4.00000 0.215666
$$345$$ − 24.0000i − 1.29212i
$$346$$ 4.00000i 0.215041i
$$347$$ − 4.00000i − 0.214731i −0.994220 0.107366i $$-0.965758\pi$$
0.994220 0.107366i $$-0.0342415\pi$$
$$348$$ −4.00000 −0.214423
$$349$$ 30.0000 1.60586 0.802932 0.596071i $$-0.203272\pi$$
0.802932 + 0.596071i $$0.203272\pi$$
$$350$$ 22.0000i 1.17595i
$$351$$ 6.00000i 0.320256i
$$352$$ 0 0
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ − 12.0000i − 0.637793i
$$355$$ −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 $$0.5\pi$$
$$360$$ − 4.00000i − 0.210819i
$$361$$ −3.00000 −0.157895
$$362$$ − 20.0000i − 1.05118i
$$363$$ 11.0000i 0.577350i
$$364$$ 12.0000i 0.628971i
$$365$$ 8.00000 0.418739
$$366$$ 4.00000 0.209083
$$367$$ 10.0000i 0.521996i 0.965339 + 0.260998i $$0.0840516\pi$$
−0.965339 + 0.260998i $$0.915948\pi$$
$$368$$ 6.00000i 0.312772i
$$369$$ 10.0000i 0.520579i
$$370$$ −16.0000 −0.831800
$$371$$ − 4.00000i − 0.207670i
$$372$$ −6.00000 −0.311086
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 0 0
$$375$$ 24.0000 1.23935
$$376$$ 4.00000 0.206284
$$377$$ − 24.0000i − 1.23606i
$$378$$ −2.00000 −0.102869
$$379$$ 4.00000i 0.205466i 0.994709 + 0.102733i $$0.0327588\pi$$
−0.994709 + 0.102733i $$0.967241\pi$$
$$380$$ − 16.0000i − 0.820783i
$$381$$ − 8.00000i − 0.409852i
$$382$$ −4.00000 −0.204658
$$383$$ −28.0000 −1.43073 −0.715367 0.698749i $$-0.753740\pi$$
−0.715367 + 0.698749i $$0.753740\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 0 0
$$386$$ 6.00000i 0.305392i
$$387$$ −4.00000 −0.203331
$$388$$ − 6.00000i − 0.304604i
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 24.0000 1.21529
$$391$$ 0 0
$$392$$ 3.00000 0.151523
$$393$$ −16.0000 −0.807093
$$394$$ 8.00000i 0.403034i
$$395$$ −40.0000 −2.01262
$$396$$ 0 0
$$397$$ 20.0000i 1.00377i 0.864934 + 0.501886i $$0.167360\pi$$
−0.864934 + 0.501886i $$0.832640\pi$$
$$398$$ − 14.0000i − 0.701757i
$$399$$ −8.00000 −0.400501
$$400$$ −11.0000 −0.550000
$$401$$ 30.0000i 1.49813i 0.662497 + 0.749064i $$0.269497\pi$$
−0.662497 + 0.749064i $$0.730503\pi$$
$$402$$ − 12.0000i − 0.598506i
$$403$$ − 36.0000i − 1.79329i
$$404$$ 14.0000 0.696526
$$405$$ 4.00000i 0.198762i
$$406$$ 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
$$411$$ − 6.00000i − 0.295958i
$$412$$ 4.00000 0.197066
$$413$$ 24.0000i 1.18096i
$$414$$ − 6.00000i − 0.294884i
$$415$$ 48.0000i 2.35623i
$$416$$ −6.00000 −0.294174
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ − 12.0000i − 0.586238i −0.956076 0.293119i $$-0.905307\pi$$
0.956076 0.293119i $$-0.0946933\pi$$
$$420$$ 8.00000i 0.390360i
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ 8.00000i 0.389434i
$$423$$ −4.00000 −0.194487
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ −6.00000 −0.290701
$$427$$ −8.00000 −0.387147
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 16.0000i 0.771589i
$$431$$ 14.0000i 0.674356i 0.941441 + 0.337178i $$0.109472\pi$$
−0.941441 + 0.337178i $$0.890528\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 18.0000 0.865025 0.432512 0.901628i $$-0.357627\pi$$
0.432512 + 0.901628i $$0.357627\pi$$
$$434$$ 12.0000 0.576018
$$435$$ − 16.0000i − 0.767141i
$$436$$ 16.0000i 0.766261i
$$437$$ − 24.0000i − 1.14808i
$$438$$ 2.00000 0.0955637
$$439$$ 10.0000i 0.477274i 0.971109 + 0.238637i $$0.0767006\pi$$
−0.971109 + 0.238637i $$0.923299\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ − 8.00000i − 0.379236i
$$446$$ 4.00000 0.189405
$$447$$ − 6.00000i − 0.283790i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ − 26.0000i − 1.22702i −0.789689 0.613508i $$-0.789758\pi$$
0.789689 0.613508i $$-0.210242\pi$$
$$450$$ 11.0000 0.518545
$$451$$ 0 0
$$452$$ 2.00000i 0.0940721i
$$453$$ 24.0000i 1.12762i
$$454$$ 4.00000i 0.187729i
$$455$$ −48.0000 −2.25027
$$456$$ − 4.00000i − 0.187317i
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 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 $$0.529629\pi$$
$$464$$ 4.00000i 0.185695i
$$465$$ − 24.0000i − 1.11297i
$$466$$ 6.00000i 0.277945i
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 6.00000 0.277350
$$469$$ 24.0000i 1.10822i
$$470$$ 16.0000i 0.738025i
$$471$$ 6.00000i 0.276465i
$$472$$ −12.0000 −0.552345
$$473$$ 0 0
$$474$$ −10.0000 −0.459315
$$475$$ 44.0000 2.01886
$$476$$ 0 0
$$477$$ −2.00000 −0.0915737
$$478$$ −20.0000 −0.914779
$$479$$ − 10.0000i − 0.456912i −0.973554 0.228456i $$-0.926632\pi$$
0.973554 0.228456i $$-0.0733677\pi$$
$$480$$ −4.00000 −0.182574
$$481$$ − 24.0000i − 1.09431i
$$482$$ 18.0000i 0.819878i
$$483$$ 12.0000i 0.546019i
$$484$$ 11.0000 0.500000
$$485$$ 24.0000 1.08978
$$486$$ 1.00000i 0.0453609i
$$487$$ − 38.0000i − 1.72194i −0.508652 0.860972i $$-0.669856\pi$$
0.508652 0.860972i $$-0.330144\pi$$
$$488$$ − 4.00000i − 0.181071i
$$489$$ −12.0000 −0.542659
$$490$$ 12.0000i 0.542105i
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 10.0000 0.450835
$$493$$ 0 0
$$494$$ 24.0000 1.07981
$$495$$ 0 0
$$496$$ 6.00000i 0.269408i
$$497$$ 12.0000 0.538274
$$498$$ 12.0000i 0.537733i
$$499$$ 32.0000i 1.43252i 0.697835 + 0.716258i $$0.254147\pi$$
−0.697835 + 0.716258i $$0.745853\pi$$
$$500$$ − 24.0000i − 1.07331i
$$501$$ −2.00000 −0.0893534
$$502$$ 12.0000 0.535586
$$503$$ − 26.0000i − 1.15928i −0.814872 0.579641i $$-0.803193\pi$$
0.814872 0.579641i $$-0.196807\pi$$
$$504$$ 2.00000i 0.0890871i
$$505$$ 56.0000i 2.49197i
$$506$$ 0 0
$$507$$ 23.0000i 1.02147i
$$508$$ −8.00000 −0.354943
$$509$$ 26.0000 1.15243 0.576215 0.817298i $$-0.304529\pi$$
0.576215 + 0.817298i $$0.304529\pi$$
$$510$$ 0 0
$$511$$ −4.00000 −0.176950
$$512$$ 1.00000 0.0441942
$$513$$ 4.00000i 0.176604i
$$514$$ −18.0000 −0.793946
$$515$$ 16.0000i 0.705044i
$$516$$ 4.00000i 0.176090i
$$517$$ 0 0
$$518$$ 8.00000 0.351500
$$519$$ −4.00000 −0.175581
$$520$$ − 24.0000i − 1.05247i
$$521$$ − 10.0000i − 0.438108i −0.975713 0.219054i $$-0.929703\pi$$
0.975713 0.219054i $$-0.0702971\pi$$
$$522$$ − 4.00000i − 0.175075i
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 16.0000i 0.698963i
$$525$$ −22.0000 −0.960159
$$526$$ 12.0000 0.523225
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 8.00000i 0.347498i
$$531$$ 12.0000 0.520756
$$532$$ 8.00000i 0.346844i
$$533$$ 60.0000i 2.59889i
$$534$$ − 2.00000i − 0.0865485i
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 12.0000i 0.517838i
$$538$$ 12.0000i 0.517357i
$$539$$ 0 0
$$540$$ 4.00000 0.172133
$$541$$ 12.0000i 0.515920i 0.966156 + 0.257960i $$0.0830503\pi$$
−0.966156 + 0.257960i $$0.916950\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 20.0000 0.858282
$$544$$ 0 0
$$545$$ −64.0000 −2.74146
$$546$$ −12.0000 −0.513553
$$547$$ 8.00000i 0.342055i 0.985266 + 0.171028i $$0.0547087\pi$$
−0.985266 + 0.171028i $$0.945291\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 4.00000i 0.170716i
$$550$$ 0 0
$$551$$ − 16.0000i − 0.681623i
$$552$$ −6.00000 −0.255377
$$553$$ 20.0000 0.850487
$$554$$ 8.00000i 0.339887i
$$555$$ − 16.0000i − 0.679162i
$$556$$ − 8.00000i − 0.339276i
$$557$$ 14.0000 0.593199 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$558$$ − 6.00000i − 0.254000i
$$559$$ −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 $$0.473138\pi$$
$$564$$ 4.00000i 0.168430i
$$565$$ −8.00000 −0.336563
$$566$$ 32.0000i 1.34506i
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 6.00000i 0.251754i
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 16.0000 0.670166
$$571$$ 44.0000i 1.84134i 0.390339 + 0.920671i $$0.372358\pi$$
−0.390339 + 0.920671i $$0.627642\pi$$
$$572$$ 0 0
$$573$$ − 4.00000i − 0.167102i
$$574$$ −20.0000 −0.834784
$$575$$ − 66.0000i − 2.75239i
$$576$$ −1.00000 −0.0416667
$$577$$ −30.0000 −1.24892 −0.624458 0.781058i $$-0.714680\pi$$
−0.624458 + 0.781058i $$0.714680\pi$$
$$578$$ 0 0
$$579$$ −6.00000 −0.249351
$$580$$ −16.0000 −0.664364
$$581$$ − 24.0000i − 0.995688i
$$582$$ 6.00000 0.248708
$$583$$ 0 0
$$584$$ − 2.00000i − 0.0827606i
$$585$$ 24.0000i 0.992278i
$$586$$ 2.00000 0.0826192
$$587$$ −36.0000 −1.48588 −0.742940 0.669359i $$-0.766569\pi$$
−0.742940 + 0.669359i $$0.766569\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ − 24.0000i − 0.988903i
$$590$$ − 48.0000i − 1.97613i
$$591$$ −8.00000 −0.329076
$$592$$ 4.00000i 0.164399i
$$593$$ 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
$$597$$ 14.0000 0.572982
$$598$$ − 36.0000i − 1.47215i
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ − 11.0000i − 0.449073i
$$601$$ − 38.0000i − 1.55005i −0.631929 0.775026i $$-0.717737\pi$$
0.631929 0.775026i $$-0.282263\pi$$
$$602$$ − 8.00000i − 0.326056i
$$603$$ 12.0000 0.488678
$$604$$ 24.0000 0.976546
$$605$$ 44.0000i 1.78885i
$$606$$ 14.0000i 0.568711i
$$607$$ 38.0000i 1.54237i 0.636610 + 0.771186i $$0.280336\pi$$
−0.636610 + 0.771186i $$0.719664\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 8.00000i 0.324176i
$$610$$ 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
$$615$$ 40.0000i 1.61296i
$$616$$ 0 0
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ 4.00000i 0.160904i
$$619$$ − 20.0000i − 0.803868i −0.915669 0.401934i $$-0.868338\pi$$
0.915669 0.401934i $$-0.131662\pi$$
$$620$$ −24.0000 −0.963863
$$621$$ 6.00000 0.240772
$$622$$ − 30.0000i − 1.20289i
$$623$$ 4.00000i 0.160257i
$$624$$ − 6.00000i − 0.240192i
$$625$$ 41.0000 1.64000
$$626$$ − 26.0000i − 1.03917i
$$627$$ 0 0
$$628$$ 6.00000 0.239426
$$629$$ 0 0
$$630$$ −8.00000 −0.318728
$$631$$ −20.0000 −0.796187 −0.398094 0.917345i $$-0.630328\pi$$
−0.398094 + 0.917345i $$0.630328\pi$$
$$632$$ 10.0000i 0.397779i
$$633$$ −8.00000 −0.317971
$$634$$ − 16.0000i − 0.635441i
$$635$$ − 32.0000i − 1.26988i
$$636$$ 2.00000i 0.0793052i
$$637$$ −18.0000 −0.713186
$$638$$ 0 0
$$639$$ − 6.00000i − 0.237356i
$$640$$ 4.00000i 0.158114i
$$641$$ 2.00000i 0.0789953i 0.999220 + 0.0394976i $$0.0125758\pi$$
−0.999220 + 0.0394976i $$0.987424\pi$$
$$642$$ 0 0
$$643$$ − 4.00000i − 0.157745i −0.996885 0.0788723i $$-0.974868\pi$$
0.996885 0.0788723i $$-0.0251319\pi$$
$$644$$ 12.0000 0.472866
$$645$$ −16.0000 −0.629999
$$646$$ 0 0
$$647$$ 28.0000 1.10079 0.550397 0.834903i $$-0.314476\pi$$
0.550397 + 0.834903i $$0.314476\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 66.0000 2.58873
$$651$$ 12.0000i 0.470317i
$$652$$ 12.0000i 0.469956i
$$653$$ 20.0000i 0.782660i 0.920250 + 0.391330i $$0.127985\pi$$
−0.920250 + 0.391330i $$0.872015\pi$$
$$654$$ −16.0000 −0.625650
$$655$$ −64.0000 −2.50069
$$656$$ − 10.0000i − 0.390434i
$$657$$ 2.00000i 0.0780274i
$$658$$ − 8.00000i − 0.311872i
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ −32.0000 −1.24091
$$666$$ − 4.00000i − 0.154997i
$$667$$ −24.0000 −0.929284
$$668$$ 2.00000i 0.0773823i
$$669$$ 4.00000i 0.154649i
$$670$$ − 48.0000i − 1.85440i
$$671$$ 0 0
$$672$$ 2.00000 0.0771517
$$673$$ − 26.0000i − 1.00223i −0.865382 0.501113i $$-0.832924\pi$$
0.865382 0.501113i $$-0.167076\pi$$
$$674$$ 6.00000i 0.231111i
$$675$$ 11.0000i 0.423390i
$$676$$ 23.0000 0.884615
$$677$$ − 8.00000i − 0.307465i −0.988113 0.153732i $$-0.950871\pi$$
0.988113 0.153732i $$-0.0491294\pi$$
$$678$$ −2.00000 −0.0768095
$$679$$ −12.0000 −0.460518
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 4.00000 0.152944
$$685$$ − 24.0000i − 0.916993i
$$686$$ − 20.0000i − 0.763604i
$$687$$ 2.00000i 0.0763048i
$$688$$ 4.00000 0.152499
$$689$$ −12.0000 −0.457164
$$690$$ − 24.0000i − 0.913664i
$$691$$ 16.0000i 0.608669i 0.952565 + 0.304334i $$0.0984340\pi$$
−0.952565 + 0.304334i $$0.901566\pi$$
$$692$$ 4.00000i 0.152057i
$$693$$ 0 0
$$694$$ − 4.00000i − 0.151838i
$$695$$ 32.0000 1.21383
$$696$$ −4.00000 −0.151620
$$697$$ 0 0
$$698$$ 30.0000 1.13552
$$699$$ −6.00000 −0.226941
$$700$$ 22.0000i 0.831522i
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 6.00000i 0.226455i
$$703$$ − 16.0000i − 0.603451i
$$704$$ 0 0
$$705$$ −16.0000 −0.602595
$$706$$ 14.0000 0.526897
$$707$$ − 28.0000i − 1.05305i
$$708$$ − 12.0000i − 0.450988i
$$709$$ 16.0000i 0.600893i 0.953799 + 0.300446i $$0.0971356\pi$$
−0.953799 + 0.300446i $$0.902864\pi$$
$$710$$ −24.0000 −0.900704
$$711$$ − 10.0000i − 0.375029i
$$712$$ −2.00000 −0.0749532
$$713$$ −36.0000 −1.34821
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ − 20.0000i − 0.746914i
$$718$$ 0 0
$$719$$ − 42.0000i − 1.56634i −0.621810 0.783168i $$-0.713603\pi$$
0.621810 0.783168i $$-0.286397\pi$$
$$720$$ − 4.00000i − 0.149071i
$$721$$ − 8.00000i − 0.297936i
$$722$$ −3.00000 −0.111648
$$723$$ −18.0000 −0.669427
$$724$$ − 20.0000i − 0.743294i
$$725$$ − 44.0000i − 1.63412i
$$726$$ 11.0000i 0.408248i
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 12.0000i 0.444750i
$$729$$ −1.00000 −0.0370370
$$730$$ 8.00000 0.296093
$$731$$ 0 0
$$732$$ 4.00000 0.147844
$$733$$ −18.0000 −0.664845 −0.332423 0.943131i $$-0.607866\pi$$
−0.332423 + 0.943131i $$0.607866\pi$$
$$734$$ 10.0000i 0.369107i
$$735$$ −12.0000 −0.442627
$$736$$ 6.00000i 0.221163i
$$737$$ 0 0
$$738$$ 10.0000i 0.368105i
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ −16.0000 −0.588172
$$741$$ 24.0000i 0.881662i
$$742$$ − 4.00000i − 0.146845i
$$743$$ − 38.0000i − 1.39408i −0.717030 0.697042i $$-0.754499\pi$$
0.717030 0.697042i $$-0.245501\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ − 24.0000i − 0.879292i
$$746$$ −14.0000 −0.512576
$$747$$ −12.0000 −0.439057
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 24.0000 0.876356
$$751$$ 34.0000i 1.24068i 0.784334 + 0.620339i $$0.213005\pi$$
−0.784334 + 0.620339i $$0.786995\pi$$
$$752$$ 4.00000 0.145865
$$753$$ 12.0000i 0.437304i
$$754$$ − 24.0000i − 0.874028i
$$755$$ 96.0000i 3.49380i
$$756$$ −2.00000 −0.0727393
$$757$$ −14.0000 −0.508839 −0.254419 0.967094i $$-0.581884\pi$$
−0.254419 + 0.967094i $$0.581884\pi$$
$$758$$ 4.00000i 0.145287i
$$759$$ 0 0
$$760$$ − 16.0000i − 0.580381i
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ − 8.00000i − 0.289809i
$$763$$ 32.0000 1.15848
$$764$$ −4.00000 −0.144715
$$765$$ 0 0
$$766$$ −28.0000 −1.01168
$$767$$ 72.0000 2.59977
$$768$$ 1.00000i 0.0360844i
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ − 18.0000i − 0.648254i
$$772$$ 6.00000i 0.215945i
$$773$$ −42.0000 −1.51064 −0.755318 0.655359i $$-0.772517\pi$$
−0.755318 + 0.655359i $$0.772517\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ − 66.0000i − 2.37079i
$$776$$ − 6.00000i − 0.215387i
$$777$$ 8.00000i 0.286998i
$$778$$ 14.0000 0.501924
$$779$$ 40.0000i 1.43315i
$$780$$ 24.0000 0.859338
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 4.00000 0.142948
$$784$$ 3.00000 0.107143
$$785$$ 24.0000i 0.856597i
$$786$$ −16.0000 −0.570701
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ 8.00000i 0.284988i
$$789$$ 12.0000i 0.427211i
$$790$$ −40.0000 −1.42314
$$791$$ 4.00000 0.142224
$$792$$ 0 0
$$793$$ 24.0000i 0.852265i
$$794$$ 20.0000i 0.709773i
$$795$$ −8.00000 −0.283731
$$796$$ − 14.0000i − 0.496217i
$$797$$ −46.0000 −1.62940 −0.814702 0.579880i $$-0.803099\pi$$
−0.814702 + 0.579880i $$0.803099\pi$$
$$798$$ −8.00000 −0.283197
$$799$$ 0 0
$$800$$ −11.0000 −0.388909
$$801$$ 2.00000 0.0706665
$$802$$ 30.0000i 1.05934i
$$803$$ 0 0
$$804$$ − 12.0000i − 0.423207i
$$805$$ 48.0000i 1.69178i
$$806$$ − 36.0000i − 1.26805i
$$807$$ −12.0000 −0.422420
$$808$$ 14.0000 0.492518
$$809$$ − 26.0000i − 0.914111i −0.889438 0.457056i $$-0.848904\pi$$
0.889438 0.457056i $$-0.151096\pi$$
$$810$$ 4.00000i 0.140546i
$$811$$ 12.0000i 0.421377i 0.977553 + 0.210688i $$0.0675706\pi$$
−0.977553 + 0.210688i $$0.932429\pi$$
$$812$$ 8.00000 0.280745
$$813$$ − 16.0000i − 0.561144i
$$814$$ 0 0
$$815$$ −48.0000 −1.68137
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ −26.0000 −0.909069
$$819$$ − 12.0000i − 0.419314i
$$820$$ 40.0000 1.39686
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ − 6.00000i − 0.209274i
$$823$$ 18.0000i 0.627441i 0.949515 + 0.313720i $$0.101575\pi$$
−0.949515 + 0.313720i $$0.898425\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 24.0000i 0.835067i
$$827$$ 16.0000i 0.556375i 0.960527 + 0.278187i $$0.0897336\pi$$
−0.960527 + 0.278187i $$0.910266\pi$$
$$828$$ − 6.00000i − 0.208514i
$$829$$ 6.00000 0.208389 0.104194 0.994557i $$-0.466774\pi$$
0.104194 + 0.994557i $$0.466774\pi$$
$$830$$ 48.0000i 1.66610i
$$831$$ −8.00000 −0.277517
$$832$$ −6.00000 −0.208013
$$833$$ 0 0
$$834$$ 8.00000 0.277017
$$835$$ −8.00000 −0.276851
$$836$$ 0 0
$$837$$ 6.00000 0.207390
$$838$$ − 12.0000i − 0.414533i
$$839$$ − 30.0000i − 1.03572i −0.855467 0.517858i $$-0.826730\pi$$
0.855467 0.517858i $$-0.173270\pi$$
$$840$$ 8.00000i 0.276026i
$$841$$ 13.0000 0.448276
$$842$$ 34.0000 1.17172
$$843$$ 18.0000i 0.619953i
$$844$$ 8.00000i 0.275371i
$$845$$ 92.0000i 3.16490i
$$846$$ −4.00000 −0.137523
$$847$$ − 22.0000i − 0.755929i
$$848$$ 2.00000 0.0686803
$$849$$ −32.0000 −1.09824
$$850$$ 0 0
$$851$$ −24.0000 −0.822709
$$852$$ −6.00000 −0.205557
$$853$$ 16.0000i 0.547830i 0.961754 + 0.273915i $$0.0883186\pi$$
−0.961754 + 0.273915i $$0.911681\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 16.0000i 0.547188i
$$856$$ 0 0
$$857$$ − 10.0000i − 0.341593i −0.985306 0.170797i $$-0.945366\pi$$
0.985306 0.170797i $$-0.0546341\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 16.0000i 0.545595i
$$861$$ − 20.0000i − 0.681598i
$$862$$ 14.0000i 0.476842i
$$863$$ −48.0000 −1.63394 −0.816970 0.576681i $$-0.804348\pi$$
−0.816970 + 0.576681i $$0.804348\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ −16.0000 −0.544016
$$866$$ 18.0000 0.611665
$$867$$ 0 0
$$868$$ 12.0000 0.407307
$$869$$ 0 0
$$870$$ − 16.0000i − 0.542451i
$$871$$ 72.0000 2.43963
$$872$$ 16.0000i 0.541828i
$$873$$ 6.00000i 0.203069i
$$874$$ − 24.0000i − 0.811812i
$$875$$ −48.0000 −1.62270
$$876$$ 2.00000 0.0675737
$$877$$ − 24.0000i − 0.810422i −0.914223 0.405211i $$-0.867198\pi$$
0.914223 0.405211i $$-0.132802\pi$$
$$878$$ 10.0000i 0.337484i
$$879$$ 2.00000i 0.0674583i
$$880$$ 0 0
$$881$$ − 50.0000i − 1.68454i −0.539054 0.842271i $$-0.681218\pi$$
0.539054 0.842271i $$-0.318782\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ −52.0000 −1.74994 −0.874970 0.484178i $$-0.839119\pi$$
−0.874970 + 0.484178i $$0.839119\pi$$
$$884$$ 0 0
$$885$$ 48.0000 1.61350
$$886$$ 12.0000 0.403148
$$887$$ − 18.0000i − 0.604381i −0.953248 0.302190i $$-0.902282\pi$$
0.953248 0.302190i $$-0.0977178\pi$$
$$888$$ −4.00000 −0.134231
$$889$$ 16.0000i 0.536623i
$$890$$ − 8.00000i − 0.268161i
$$891$$ 0 0
$$892$$ 4.00000 0.133930
$$893$$ −16.0000 −0.535420
$$894$$ − 6.00000i − 0.200670i
$$895$$ 48.0000i 1.60446i
$$896$$ − 2.00000i − 0.0668153i
$$897$$ 36.0000 1.20201
$$898$$ − 26.0000i − 0.867631i
$$899$$ −24.0000 −0.800445
$$900$$ 11.0000 0.366667
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 8.00000 0.266223
$$904$$ 2.00000i 0.0665190i
$$905$$ 80.0000 2.65929
$$906$$ 24.0000i 0.797347i
$$907$$ 24.0000i 0.796907i 0.917189 + 0.398453i $$0.130453\pi$$
−0.917189 + 0.398453i $$0.869547\pi$$
$$908$$ 4.00000i 0.132745i
$$909$$ −14.0000 −0.464351
$$910$$ −48.0000 −1.59118
$$911$$ 26.0000i 0.861418i 0.902491 + 0.430709i $$0.141737\pi$$
−0.902491 + 0.430709i $$0.858263\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ 0 0
$$914$$ −22.0000 −0.727695
$$915$$ 16.0000i 0.528944i
$$916$$ 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
$$921$$ − 12.0000i − 0.395413i
$$922$$ 10.0000 0.329332
$$923$$ − 36.0000i − 1.18495i
$$924$$ 0 0
$$925$$ − 44.0000i − 1.44671i
$$926$$ −4.00000 −0.131448
$$927$$ −4.00000 −0.131377
$$928$$ 4.00000i 0.131306i
$$929$$ − 34.0000i − 1.11550i −0.830008 0.557752i $$-0.811664\pi$$
0.830008 0.557752i $$-0.188336\pi$$
$$930$$ − 24.0000i − 0.786991i
$$931$$ −12.0000 −0.393284
$$932$$ 6.00000i 0.196537i
$$933$$ 30.0000 0.982156
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 24.0000i 0.783628i
$$939$$ 26.0000 0.848478
$$940$$ 16.0000i 0.521862i
$$941$$ 48.0000i 1.56476i 0.622804 + 0.782378i $$0.285993\pi$$
−0.622804 + 0.782378i $$0.714007\pi$$
$$942$$ 6.00000i 0.195491i
$$943$$ 60.0000 1.95387
$$944$$ −12.0000 −0.390567
$$945$$ − 8.00000i − 0.260240i
$$946$$ 0 0
$$947$$ 24.0000i 0.779895i 0.920837 + 0.389948i $$0.127507\pi$$
−0.920837 + 0.389948i $$0.872493\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ 12.0000i 0.389536i
$$950$$ 44.0000 1.42755
$$951$$ 16.0000 0.518836
$$952$$ 0 0
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ − 16.0000i − 0.517748i
$$956$$ −20.0000 −0.646846
$$957$$ 0 0
$$958$$ − 10.0000i − 0.323085i
$$959$$ 12.0000i 0.387500i
$$960$$ −4.00000 −0.129099
$$961$$ −5.00000 −0.161290
$$962$$ − 24.0000i − 0.773791i
$$963$$ 0 0
$$964$$ 18.0000i 0.579741i
$$965$$ −24.0000 −0.772587
$$966$$ 12.0000i 0.386094i
$$967$$ 44.0000 1.41494 0.707472 0.706741i $$-0.249835\pi$$
0.707472 + 0.706741i $$0.249835\pi$$
$$968$$ 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 $$0.603990\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ −16.0000 −0.512936
$$974$$ − 38.0000i − 1.21760i
$$975$$ 66.0000i 2.11369i
$$976$$ − 4.00000i − 0.128037i
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ −12.0000 −0.383718
$$979$$ 0 0
$$980$$ 12.0000i 0.383326i
$$981$$ − 16.0000i − 0.510841i
$$982$$ 20.0000 0.638226
$$983$$ 38.0000i 1.21201i 0.795460 + 0.606006i $$0.207229\pi$$
−0.795460 + 0.606006i $$0.792771\pi$$
$$984$$ 10.0000 0.318788
$$985$$ −32.0000 −1.01960
$$986$$ 0 0
$$987$$ 8.00000 0.254643
$$988$$ 24.0000 0.763542
$$989$$ 24.0000i 0.763156i
$$990$$ 0 0
$$991$$ 34.0000i 1.08005i 0.841650 + 0.540023i $$0.181584\pi$$
−0.841650 + 0.540023i $$0.818416\pi$$
$$992$$ 6.00000i 0.190500i
$$993$$ − 20.0000i − 0.634681i
$$994$$ 12.0000 0.380617
$$995$$ 56.0000 1.77532
$$996$$ 12.0000i 0.380235i
$$997$$ − 20.0000i − 0.633406i −0.948525 0.316703i $$-0.897424\pi$$
0.948525 0.316703i $$-0.102576\pi$$
$$998$$ 32.0000i 1.01294i
$$999$$ 4.00000 0.126554
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 1734.2.b.d.577.2 2
17.2 even 8 1734.2.f.g.829.2 4
17.4 even 4 1734.2.a.h.1.1 1
17.8 even 8 1734.2.f.g.1483.2 4
17.9 even 8 1734.2.f.g.1483.1 4
17.13 even 4 102.2.a.a.1.1 1
17.15 even 8 1734.2.f.g.829.1 4
17.16 even 2 inner 1734.2.b.d.577.1 2
51.38 odd 4 5202.2.a.g.1.1 1
51.47 odd 4 306.2.a.d.1.1 1
68.47 odd 4 816.2.a.h.1.1 1
85.13 odd 4 2550.2.d.q.2449.2 2
85.47 odd 4 2550.2.d.q.2449.1 2
85.64 even 4 2550.2.a.be.1.1 1
119.13 odd 4 4998.2.a.x.1.1 1
136.13 even 4 3264.2.a.bf.1.1 1
136.115 odd 4 3264.2.a.p.1.1 1
204.47 even 4 2448.2.a.t.1.1 1
255.149 odd 4 7650.2.a.z.1.1 1
408.149 odd 4 9792.2.a.a.1.1 1
408.251 even 4 9792.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.a.1.1 1 17.13 even 4
306.2.a.d.1.1 1 51.47 odd 4
816.2.a.h.1.1 1 68.47 odd 4
1734.2.a.h.1.1 1 17.4 even 4
1734.2.b.d.577.1 2 17.16 even 2 inner
1734.2.b.d.577.2 2 1.1 even 1 trivial
1734.2.f.g.829.1 4 17.15 even 8
1734.2.f.g.829.2 4 17.2 even 8
1734.2.f.g.1483.1 4 17.9 even 8
1734.2.f.g.1483.2 4 17.8 even 8
2448.2.a.t.1.1 1 204.47 even 4
2550.2.a.be.1.1 1 85.64 even 4
2550.2.d.q.2449.1 2 85.47 odd 4
2550.2.d.q.2449.2 2 85.13 odd 4
3264.2.a.p.1.1 1 136.115 odd 4
3264.2.a.bf.1.1 1 136.13 even 4
4998.2.a.x.1.1 1 119.13 odd 4
5202.2.a.g.1.1 1 51.38 odd 4
7650.2.a.z.1.1 1 255.149 odd 4
9792.2.a.a.1.1 1 408.149 odd 4
9792.2.a.b.1.1 1 408.251 even 4