Properties

 Label 1014.4.b.g.337.2 Level $1014$ Weight $4$ Character 1014.337 Analytic conductor $59.828$ 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] = [1014,4,Mod(337,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1014.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1014.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: $$59.8279367458$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 78) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 337.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1014.337 Dual form 1014.4.b.g.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} +4.00000i q^{5} +6.00000i q^{6} -4.00000i q^{7} -8.00000i q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} +4.00000i q^{5} +6.00000i q^{6} -4.00000i q^{7} -8.00000i q^{8} +9.00000 q^{9} -8.00000 q^{10} -2.00000i q^{11} -12.0000 q^{12} +8.00000 q^{14} +12.0000i q^{15} +16.0000 q^{16} +6.00000 q^{17} +18.0000i q^{18} -36.0000i q^{19} -16.0000i q^{20} -12.0000i q^{21} +4.00000 q^{22} +20.0000 q^{23} -24.0000i q^{24} +109.000 q^{25} +27.0000 q^{27} +16.0000i q^{28} -14.0000 q^{29} -24.0000 q^{30} -152.000i q^{31} +32.0000i q^{32} -6.00000i q^{33} +12.0000i q^{34} +16.0000 q^{35} -36.0000 q^{36} +258.000i q^{37} +72.0000 q^{38} +32.0000 q^{40} +84.0000i q^{41} +24.0000 q^{42} +188.000 q^{43} +8.00000i q^{44} +36.0000i q^{45} +40.0000i q^{46} -254.000i q^{47} +48.0000 q^{48} +327.000 q^{49} +218.000i q^{50} +18.0000 q^{51} +366.000 q^{53} +54.0000i q^{54} +8.00000 q^{55} -32.0000 q^{56} -108.000i q^{57} -28.0000i q^{58} -550.000i q^{59} -48.0000i q^{60} -14.0000 q^{61} +304.000 q^{62} -36.0000i q^{63} -64.0000 q^{64} +12.0000 q^{66} +448.000i q^{67} -24.0000 q^{68} +60.0000 q^{69} +32.0000i q^{70} +926.000i q^{71} -72.0000i q^{72} -254.000i q^{73} -516.000 q^{74} +327.000 q^{75} +144.000i q^{76} -8.00000 q^{77} +1328.00 q^{79} +64.0000i q^{80} +81.0000 q^{81} -168.000 q^{82} +186.000i q^{83} +48.0000i q^{84} +24.0000i q^{85} +376.000i q^{86} -42.0000 q^{87} -16.0000 q^{88} +336.000i q^{89} -72.0000 q^{90} -80.0000 q^{92} -456.000i q^{93} +508.000 q^{94} +144.000 q^{95} +96.0000i q^{96} +614.000i q^{97} +654.000i q^{98} -18.0000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 8 q^{4} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 8 * q^4 + 18 * q^9 $$2 q + 6 q^{3} - 8 q^{4} + 18 q^{9} - 16 q^{10} - 24 q^{12} + 16 q^{14} + 32 q^{16} + 12 q^{17} + 8 q^{22} + 40 q^{23} + 218 q^{25} + 54 q^{27} - 28 q^{29} - 48 q^{30} + 32 q^{35} - 72 q^{36} + 144 q^{38} + 64 q^{40} + 48 q^{42} + 376 q^{43} + 96 q^{48} + 654 q^{49} + 36 q^{51} + 732 q^{53} + 16 q^{55} - 64 q^{56} - 28 q^{61} + 608 q^{62} - 128 q^{64} + 24 q^{66} - 48 q^{68} + 120 q^{69} - 1032 q^{74} + 654 q^{75} - 16 q^{77} + 2656 q^{79} + 162 q^{81} - 336 q^{82} - 84 q^{87} - 32 q^{88} - 144 q^{90} - 160 q^{92} + 1016 q^{94} + 288 q^{95}+O(q^{100})$$ 2 * q + 6 * q^3 - 8 * q^4 + 18 * q^9 - 16 * q^10 - 24 * q^12 + 16 * q^14 + 32 * q^16 + 12 * q^17 + 8 * q^22 + 40 * q^23 + 218 * q^25 + 54 * q^27 - 28 * q^29 - 48 * q^30 + 32 * q^35 - 72 * q^36 + 144 * q^38 + 64 * q^40 + 48 * q^42 + 376 * q^43 + 96 * q^48 + 654 * q^49 + 36 * q^51 + 732 * q^53 + 16 * q^55 - 64 * q^56 - 28 * q^61 + 608 * q^62 - 128 * q^64 + 24 * q^66 - 48 * q^68 + 120 * q^69 - 1032 * q^74 + 654 * q^75 - 16 * q^77 + 2656 * q^79 + 162 * q^81 - 336 * q^82 - 84 * q^87 - 32 * q^88 - 144 * q^90 - 160 * q^92 + 1016 * q^94 + 288 * q^95

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000i 0.707107i
$$3$$ 3.00000 0.577350
$$4$$ −4.00000 −0.500000
$$5$$ 4.00000i 0.357771i 0.983870 + 0.178885i $$0.0572491\pi$$
−0.983870 + 0.178885i $$0.942751\pi$$
$$6$$ 6.00000i 0.408248i
$$7$$ − 4.00000i − 0.215980i −0.994152 0.107990i $$-0.965559\pi$$
0.994152 0.107990i $$-0.0344414\pi$$
$$8$$ − 8.00000i − 0.353553i
$$9$$ 9.00000 0.333333
$$10$$ −8.00000 −0.252982
$$11$$ − 2.00000i − 0.0548202i −0.999624 0.0274101i $$-0.991274\pi$$
0.999624 0.0274101i $$-0.00872601\pi$$
$$12$$ −12.0000 −0.288675
$$13$$ 0 0
$$14$$ 8.00000 0.152721
$$15$$ 12.0000i 0.206559i
$$16$$ 16.0000 0.250000
$$17$$ 6.00000 0.0856008 0.0428004 0.999084i $$-0.486372\pi$$
0.0428004 + 0.999084i $$0.486372\pi$$
$$18$$ 18.0000i 0.235702i
$$19$$ − 36.0000i − 0.434682i −0.976096 0.217341i $$-0.930262\pi$$
0.976096 0.217341i $$-0.0697384\pi$$
$$20$$ − 16.0000i − 0.178885i
$$21$$ − 12.0000i − 0.124696i
$$22$$ 4.00000 0.0387638
$$23$$ 20.0000 0.181317 0.0906584 0.995882i $$-0.471103\pi$$
0.0906584 + 0.995882i $$0.471103\pi$$
$$24$$ − 24.0000i − 0.204124i
$$25$$ 109.000 0.872000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 16.0000i 0.107990i
$$29$$ −14.0000 −0.0896460 −0.0448230 0.998995i $$-0.514272\pi$$
−0.0448230 + 0.998995i $$0.514272\pi$$
$$30$$ −24.0000 −0.146059
$$31$$ − 152.000i − 0.880645i −0.897840 0.440323i $$-0.854864\pi$$
0.897840 0.440323i $$-0.145136\pi$$
$$32$$ 32.0000i 0.176777i
$$33$$ − 6.00000i − 0.0316505i
$$34$$ 12.0000i 0.0605289i
$$35$$ 16.0000 0.0772712
$$36$$ −36.0000 −0.166667
$$37$$ 258.000i 1.14635i 0.819433 + 0.573175i $$0.194288\pi$$
−0.819433 + 0.573175i $$0.805712\pi$$
$$38$$ 72.0000 0.307367
$$39$$ 0 0
$$40$$ 32.0000 0.126491
$$41$$ 84.0000i 0.319966i 0.987120 + 0.159983i $$0.0511439\pi$$
−0.987120 + 0.159983i $$0.948856\pi$$
$$42$$ 24.0000 0.0881733
$$43$$ 188.000 0.666738 0.333369 0.942796i $$-0.391815\pi$$
0.333369 + 0.942796i $$0.391815\pi$$
$$44$$ 8.00000i 0.0274101i
$$45$$ 36.0000i 0.119257i
$$46$$ 40.0000i 0.128210i
$$47$$ − 254.000i − 0.788292i −0.919048 0.394146i $$-0.871040\pi$$
0.919048 0.394146i $$-0.128960\pi$$
$$48$$ 48.0000 0.144338
$$49$$ 327.000 0.953353
$$50$$ 218.000i 0.616597i
$$51$$ 18.0000 0.0494217
$$52$$ 0 0
$$53$$ 366.000 0.948565 0.474283 0.880373i $$-0.342707\pi$$
0.474283 + 0.880373i $$0.342707\pi$$
$$54$$ 54.0000i 0.136083i
$$55$$ 8.00000 0.0196131
$$56$$ −32.0000 −0.0763604
$$57$$ − 108.000i − 0.250964i
$$58$$ − 28.0000i − 0.0633893i
$$59$$ − 550.000i − 1.21363i −0.794845 0.606813i $$-0.792448\pi$$
0.794845 0.606813i $$-0.207552\pi$$
$$60$$ − 48.0000i − 0.103280i
$$61$$ −14.0000 −0.0293855 −0.0146928 0.999892i $$-0.504677\pi$$
−0.0146928 + 0.999892i $$0.504677\pi$$
$$62$$ 304.000 0.622710
$$63$$ − 36.0000i − 0.0719932i
$$64$$ −64.0000 −0.125000
$$65$$ 0 0
$$66$$ 12.0000 0.0223803
$$67$$ 448.000i 0.816894i 0.912782 + 0.408447i $$0.133930\pi$$
−0.912782 + 0.408447i $$0.866070\pi$$
$$68$$ −24.0000 −0.0428004
$$69$$ 60.0000 0.104683
$$70$$ 32.0000i 0.0546390i
$$71$$ 926.000i 1.54783i 0.633289 + 0.773915i $$0.281704\pi$$
−0.633289 + 0.773915i $$0.718296\pi$$
$$72$$ − 72.0000i − 0.117851i
$$73$$ − 254.000i − 0.407239i −0.979050 0.203620i $$-0.934729\pi$$
0.979050 0.203620i $$-0.0652706\pi$$
$$74$$ −516.000 −0.810592
$$75$$ 327.000 0.503449
$$76$$ 144.000i 0.217341i
$$77$$ −8.00000 −0.0118401
$$78$$ 0 0
$$79$$ 1328.00 1.89129 0.945644 0.325205i $$-0.105433\pi$$
0.945644 + 0.325205i $$0.105433\pi$$
$$80$$ 64.0000i 0.0894427i
$$81$$ 81.0000 0.111111
$$82$$ −168.000 −0.226250
$$83$$ 186.000i 0.245978i 0.992408 + 0.122989i $$0.0392479\pi$$
−0.992408 + 0.122989i $$0.960752\pi$$
$$84$$ 48.0000i 0.0623480i
$$85$$ 24.0000i 0.0306255i
$$86$$ 376.000i 0.471455i
$$87$$ −42.0000 −0.0517572
$$88$$ −16.0000 −0.0193819
$$89$$ 336.000i 0.400179i 0.979778 + 0.200089i $$0.0641233\pi$$
−0.979778 + 0.200089i $$0.935877\pi$$
$$90$$ −72.0000 −0.0843274
$$91$$ 0 0
$$92$$ −80.0000 −0.0906584
$$93$$ − 456.000i − 0.508441i
$$94$$ 508.000 0.557406
$$95$$ 144.000 0.155517
$$96$$ 96.0000i 0.102062i
$$97$$ 614.000i 0.642704i 0.946960 + 0.321352i $$0.104137\pi$$
−0.946960 + 0.321352i $$0.895863\pi$$
$$98$$ 654.000i 0.674122i
$$99$$ − 18.0000i − 0.0182734i
$$100$$ −436.000 −0.436000
$$101$$ 1606.00 1.58221 0.791104 0.611682i $$-0.209507\pi$$
0.791104 + 0.611682i $$0.209507\pi$$
$$102$$ 36.0000i 0.0349464i
$$103$$ −208.000 −0.198979 −0.0994896 0.995039i $$-0.531721\pi$$
−0.0994896 + 0.995039i $$0.531721\pi$$
$$104$$ 0 0
$$105$$ 48.0000 0.0446126
$$106$$ 732.000i 0.670737i
$$107$$ −248.000 −0.224066 −0.112033 0.993704i $$-0.535736\pi$$
−0.112033 + 0.993704i $$0.535736\pi$$
$$108$$ −108.000 −0.0962250
$$109$$ − 542.000i − 0.476277i −0.971231 0.238138i $$-0.923463\pi$$
0.971231 0.238138i $$-0.0765372\pi$$
$$110$$ 16.0000i 0.0138685i
$$111$$ 774.000i 0.661845i
$$112$$ − 64.0000i − 0.0539949i
$$113$$ −2042.00 −1.69996 −0.849979 0.526817i $$-0.823385\pi$$
−0.849979 + 0.526817i $$0.823385\pi$$
$$114$$ 216.000 0.177458
$$115$$ 80.0000i 0.0648699i
$$116$$ 56.0000 0.0448230
$$117$$ 0 0
$$118$$ 1100.00 0.858163
$$119$$ − 24.0000i − 0.0184880i
$$120$$ 96.0000 0.0730297
$$121$$ 1327.00 0.996995
$$122$$ − 28.0000i − 0.0207787i
$$123$$ 252.000i 0.184732i
$$124$$ 608.000i 0.440323i
$$125$$ 936.000i 0.669747i
$$126$$ 72.0000 0.0509069
$$127$$ 488.000 0.340968 0.170484 0.985360i $$-0.445467\pi$$
0.170484 + 0.985360i $$0.445467\pi$$
$$128$$ − 128.000i − 0.0883883i
$$129$$ 564.000 0.384941
$$130$$ 0 0
$$131$$ 1744.00 1.16316 0.581580 0.813489i $$-0.302435\pi$$
0.581580 + 0.813489i $$0.302435\pi$$
$$132$$ 24.0000i 0.0158252i
$$133$$ −144.000 −0.0938826
$$134$$ −896.000 −0.577631
$$135$$ 108.000i 0.0688530i
$$136$$ − 48.0000i − 0.0302645i
$$137$$ 828.000i 0.516356i 0.966097 + 0.258178i $$0.0831222\pi$$
−0.966097 + 0.258178i $$0.916878\pi$$
$$138$$ 120.000i 0.0740223i
$$139$$ −404.000 −0.246524 −0.123262 0.992374i $$-0.539336\pi$$
−0.123262 + 0.992374i $$0.539336\pi$$
$$140$$ −64.0000 −0.0386356
$$141$$ − 762.000i − 0.455120i
$$142$$ −1852.00 −1.09448
$$143$$ 0 0
$$144$$ 144.000 0.0833333
$$145$$ − 56.0000i − 0.0320727i
$$146$$ 508.000 0.287962
$$147$$ 981.000 0.550418
$$148$$ − 1032.00i − 0.573175i
$$149$$ 2928.00i 1.60987i 0.593361 + 0.804937i $$0.297801\pi$$
−0.593361 + 0.804937i $$0.702199\pi$$
$$150$$ 654.000i 0.355993i
$$151$$ − 1944.00i − 1.04769i −0.851815 0.523843i $$-0.824498\pi$$
0.851815 0.523843i $$-0.175502\pi$$
$$152$$ −288.000 −0.153683
$$153$$ 54.0000 0.0285336
$$154$$ − 16.0000i − 0.00837219i
$$155$$ 608.000 0.315069
$$156$$ 0 0
$$157$$ 3590.00 1.82492 0.912462 0.409161i $$-0.134178\pi$$
0.912462 + 0.409161i $$0.134178\pi$$
$$158$$ 2656.00i 1.33734i
$$159$$ 1098.00 0.547654
$$160$$ −128.000 −0.0632456
$$161$$ − 80.0000i − 0.0391608i
$$162$$ 162.000i 0.0785674i
$$163$$ 2284.00i 1.09753i 0.835978 + 0.548763i $$0.184901\pi$$
−0.835978 + 0.548763i $$0.815099\pi$$
$$164$$ − 336.000i − 0.159983i
$$165$$ 24.0000 0.0113236
$$166$$ −372.000 −0.173933
$$167$$ − 3174.00i − 1.47073i −0.677673 0.735364i $$-0.737011\pi$$
0.677673 0.735364i $$-0.262989\pi$$
$$168$$ −96.0000 −0.0440867
$$169$$ 0 0
$$170$$ −48.0000 −0.0216555
$$171$$ − 324.000i − 0.144894i
$$172$$ −752.000 −0.333369
$$173$$ 1358.00 0.596802 0.298401 0.954441i $$-0.403547\pi$$
0.298401 + 0.954441i $$0.403547\pi$$
$$174$$ − 84.0000i − 0.0365978i
$$175$$ − 436.000i − 0.188334i
$$176$$ − 32.0000i − 0.0137051i
$$177$$ − 1650.00i − 0.700687i
$$178$$ −672.000 −0.282969
$$179$$ −708.000 −0.295634 −0.147817 0.989015i $$-0.547225\pi$$
−0.147817 + 0.989015i $$0.547225\pi$$
$$180$$ − 144.000i − 0.0596285i
$$181$$ 546.000 0.224220 0.112110 0.993696i $$-0.464239\pi$$
0.112110 + 0.993696i $$0.464239\pi$$
$$182$$ 0 0
$$183$$ −42.0000 −0.0169657
$$184$$ − 160.000i − 0.0641052i
$$185$$ −1032.00 −0.410131
$$186$$ 912.000 0.359522
$$187$$ − 12.0000i − 0.00469266i
$$188$$ 1016.00i 0.394146i
$$189$$ − 108.000i − 0.0415653i
$$190$$ 288.000i 0.109967i
$$191$$ −3472.00 −1.31531 −0.657657 0.753317i $$-0.728453\pi$$
−0.657657 + 0.753317i $$0.728453\pi$$
$$192$$ −192.000 −0.0721688
$$193$$ 310.000i 0.115618i 0.998328 + 0.0578090i $$0.0184115\pi$$
−0.998328 + 0.0578090i $$0.981589\pi$$
$$194$$ −1228.00 −0.454460
$$195$$ 0 0
$$196$$ −1308.00 −0.476676
$$197$$ 1020.00i 0.368893i 0.982843 + 0.184447i $$0.0590493\pi$$
−0.982843 + 0.184447i $$0.940951\pi$$
$$198$$ 36.0000 0.0129213
$$199$$ 3256.00 1.15986 0.579929 0.814667i $$-0.303080\pi$$
0.579929 + 0.814667i $$0.303080\pi$$
$$200$$ − 872.000i − 0.308299i
$$201$$ 1344.00i 0.471634i
$$202$$ 3212.00i 1.11879i
$$203$$ 56.0000i 0.0193617i
$$204$$ −72.0000 −0.0247108
$$205$$ −336.000 −0.114474
$$206$$ − 416.000i − 0.140699i
$$207$$ 180.000 0.0604390
$$208$$ 0 0
$$209$$ −72.0000 −0.0238294
$$210$$ 96.0000i 0.0315459i
$$211$$ −4564.00 −1.48909 −0.744547 0.667570i $$-0.767334\pi$$
−0.744547 + 0.667570i $$0.767334\pi$$
$$212$$ −1464.00 −0.474283
$$213$$ 2778.00i 0.893640i
$$214$$ − 496.000i − 0.158439i
$$215$$ 752.000i 0.238539i
$$216$$ − 216.000i − 0.0680414i
$$217$$ −608.000 −0.190202
$$218$$ 1084.00 0.336779
$$219$$ − 762.000i − 0.235120i
$$220$$ −32.0000 −0.00980654
$$221$$ 0 0
$$222$$ −1548.00 −0.467995
$$223$$ − 72.0000i − 0.0216210i −0.999942 0.0108105i $$-0.996559\pi$$
0.999942 0.0108105i $$-0.00344115\pi$$
$$224$$ 128.000 0.0381802
$$225$$ 981.000 0.290667
$$226$$ − 4084.00i − 1.20205i
$$227$$ 2694.00i 0.787696i 0.919176 + 0.393848i $$0.128856\pi$$
−0.919176 + 0.393848i $$0.871144\pi$$
$$228$$ 432.000i 0.125482i
$$229$$ − 5922.00i − 1.70889i −0.519538 0.854447i $$-0.673896\pi$$
0.519538 0.854447i $$-0.326104\pi$$
$$230$$ −160.000 −0.0458699
$$231$$ −24.0000 −0.00683586
$$232$$ 112.000i 0.0316947i
$$233$$ 5122.00 1.44014 0.720072 0.693900i $$-0.244109\pi$$
0.720072 + 0.693900i $$0.244109\pi$$
$$234$$ 0 0
$$235$$ 1016.00 0.282028
$$236$$ 2200.00i 0.606813i
$$237$$ 3984.00 1.09194
$$238$$ 48.0000 0.0130730
$$239$$ 5022.00i 1.35919i 0.733588 + 0.679595i $$0.237844\pi$$
−0.733588 + 0.679595i $$0.762156\pi$$
$$240$$ 192.000i 0.0516398i
$$241$$ 1218.00i 0.325553i 0.986663 + 0.162777i $$0.0520450\pi$$
−0.986663 + 0.162777i $$0.947955\pi$$
$$242$$ 2654.00i 0.704982i
$$243$$ 243.000 0.0641500
$$244$$ 56.0000 0.0146928
$$245$$ 1308.00i 0.341082i
$$246$$ −504.000 −0.130625
$$247$$ 0 0
$$248$$ −1216.00 −0.311355
$$249$$ 558.000i 0.142015i
$$250$$ −1872.00 −0.473583
$$251$$ 2112.00 0.531109 0.265554 0.964096i $$-0.414445\pi$$
0.265554 + 0.964096i $$0.414445\pi$$
$$252$$ 144.000i 0.0359966i
$$253$$ − 40.0000i − 0.00993984i
$$254$$ 976.000i 0.241101i
$$255$$ 72.0000i 0.0176816i
$$256$$ 256.000 0.0625000
$$257$$ −2814.00 −0.683006 −0.341503 0.939881i $$-0.610936\pi$$
−0.341503 + 0.939881i $$0.610936\pi$$
$$258$$ 1128.00i 0.272195i
$$259$$ 1032.00 0.247588
$$260$$ 0 0
$$261$$ −126.000 −0.0298820
$$262$$ 3488.00i 0.822478i
$$263$$ −4044.00 −0.948151 −0.474076 0.880484i $$-0.657218\pi$$
−0.474076 + 0.880484i $$0.657218\pi$$
$$264$$ −48.0000 −0.0111901
$$265$$ 1464.00i 0.339369i
$$266$$ − 288.000i − 0.0663850i
$$267$$ 1008.00i 0.231043i
$$268$$ − 1792.00i − 0.408447i
$$269$$ −1470.00 −0.333188 −0.166594 0.986026i $$-0.553277\pi$$
−0.166594 + 0.986026i $$0.553277\pi$$
$$270$$ −216.000 −0.0486864
$$271$$ 1844.00i 0.413340i 0.978411 + 0.206670i $$0.0662626\pi$$
−0.978411 + 0.206670i $$0.933737\pi$$
$$272$$ 96.0000 0.0214002
$$273$$ 0 0
$$274$$ −1656.00 −0.365119
$$275$$ − 218.000i − 0.0478033i
$$276$$ −240.000 −0.0523417
$$277$$ −5766.00 −1.25071 −0.625353 0.780342i $$-0.715045\pi$$
−0.625353 + 0.780342i $$0.715045\pi$$
$$278$$ − 808.000i − 0.174319i
$$279$$ − 1368.00i − 0.293548i
$$280$$ − 128.000i − 0.0273195i
$$281$$ 7468.00i 1.58542i 0.609598 + 0.792711i $$0.291331\pi$$
−0.609598 + 0.792711i $$0.708669\pi$$
$$282$$ 1524.00 0.321819
$$283$$ −1228.00 −0.257940 −0.128970 0.991648i $$-0.541167\pi$$
−0.128970 + 0.991648i $$0.541167\pi$$
$$284$$ − 3704.00i − 0.773915i
$$285$$ 432.000 0.0897876
$$286$$ 0 0
$$287$$ 336.000 0.0691061
$$288$$ 288.000i 0.0589256i
$$289$$ −4877.00 −0.992673
$$290$$ 112.000 0.0226788
$$291$$ 1842.00i 0.371065i
$$292$$ 1016.00i 0.203620i
$$293$$ − 6608.00i − 1.31755i −0.752338 0.658777i $$-0.771074\pi$$
0.752338 0.658777i $$-0.228926\pi$$
$$294$$ 1962.00i 0.389205i
$$295$$ 2200.00 0.434200
$$296$$ 2064.00 0.405296
$$297$$ − 54.0000i − 0.0105502i
$$298$$ −5856.00 −1.13835
$$299$$ 0 0
$$300$$ −1308.00 −0.251725
$$301$$ − 752.000i − 0.144002i
$$302$$ 3888.00 0.740825
$$303$$ 4818.00 0.913488
$$304$$ − 576.000i − 0.108671i
$$305$$ − 56.0000i − 0.0105133i
$$306$$ 108.000i 0.0201763i
$$307$$ − 7664.00i − 1.42478i −0.701784 0.712390i $$-0.747613\pi$$
0.701784 0.712390i $$-0.252387\pi$$
$$308$$ 32.0000 0.00592003
$$309$$ −624.000 −0.114881
$$310$$ 1216.00i 0.222788i
$$311$$ 2340.00 0.426653 0.213327 0.976981i $$-0.431570\pi$$
0.213327 + 0.976981i $$0.431570\pi$$
$$312$$ 0 0
$$313$$ 6710.00 1.21173 0.605865 0.795567i $$-0.292827\pi$$
0.605865 + 0.795567i $$0.292827\pi$$
$$314$$ 7180.00i 1.29042i
$$315$$ 144.000 0.0257571
$$316$$ −5312.00 −0.945644
$$317$$ 4164.00i 0.737771i 0.929475 + 0.368886i $$0.120261\pi$$
−0.929475 + 0.368886i $$0.879739\pi$$
$$318$$ 2196.00i 0.387250i
$$319$$ 28.0000i 0.00491442i
$$320$$ − 256.000i − 0.0447214i
$$321$$ −744.000 −0.129365
$$322$$ 160.000 0.0276908
$$323$$ − 216.000i − 0.0372092i
$$324$$ −324.000 −0.0555556
$$325$$ 0 0
$$326$$ −4568.00 −0.776068
$$327$$ − 1626.00i − 0.274979i
$$328$$ 672.000 0.113125
$$329$$ −1016.00 −0.170255
$$330$$ 48.0000i 0.00800701i
$$331$$ − 10072.0i − 1.67253i −0.548326 0.836265i $$-0.684735\pi$$
0.548326 0.836265i $$-0.315265\pi$$
$$332$$ − 744.000i − 0.122989i
$$333$$ 2322.00i 0.382117i
$$334$$ 6348.00 1.03996
$$335$$ −1792.00 −0.292261
$$336$$ − 192.000i − 0.0311740i
$$337$$ −2990.00 −0.483311 −0.241655 0.970362i $$-0.577690\pi$$
−0.241655 + 0.970362i $$0.577690\pi$$
$$338$$ 0 0
$$339$$ −6126.00 −0.981471
$$340$$ − 96.0000i − 0.0153127i
$$341$$ −304.000 −0.0482772
$$342$$ 648.000 0.102456
$$343$$ − 2680.00i − 0.421885i
$$344$$ − 1504.00i − 0.235727i
$$345$$ 240.000i 0.0374527i
$$346$$ 2716.00i 0.422003i
$$347$$ 6564.00 1.01549 0.507743 0.861508i $$-0.330480\pi$$
0.507743 + 0.861508i $$0.330480\pi$$
$$348$$ 168.000 0.0258786
$$349$$ 674.000i 0.103376i 0.998663 + 0.0516882i $$0.0164602\pi$$
−0.998663 + 0.0516882i $$0.983540\pi$$
$$350$$ 872.000 0.133172
$$351$$ 0 0
$$352$$ 64.0000 0.00969094
$$353$$ − 10732.0i − 1.61815i −0.587706 0.809075i $$-0.699969\pi$$
0.587706 0.809075i $$-0.300031\pi$$
$$354$$ 3300.00 0.495461
$$355$$ −3704.00 −0.553769
$$356$$ − 1344.00i − 0.200089i
$$357$$ − 72.0000i − 0.0106741i
$$358$$ − 1416.00i − 0.209044i
$$359$$ 4842.00i 0.711841i 0.934516 + 0.355921i $$0.115833\pi$$
−0.934516 + 0.355921i $$0.884167\pi$$
$$360$$ 288.000 0.0421637
$$361$$ 5563.00 0.811051
$$362$$ 1092.00i 0.158548i
$$363$$ 3981.00 0.575615
$$364$$ 0 0
$$365$$ 1016.00 0.145698
$$366$$ − 84.0000i − 0.0119966i
$$367$$ −6280.00 −0.893224 −0.446612 0.894728i $$-0.647370\pi$$
−0.446612 + 0.894728i $$0.647370\pi$$
$$368$$ 320.000 0.0453292
$$369$$ 756.000i 0.106655i
$$370$$ − 2064.00i − 0.290006i
$$371$$ − 1464.00i − 0.204871i
$$372$$ 1824.00i 0.254220i
$$373$$ 6434.00 0.893136 0.446568 0.894750i $$-0.352646\pi$$
0.446568 + 0.894750i $$0.352646\pi$$
$$374$$ 24.0000 0.00331821
$$375$$ 2808.00i 0.386679i
$$376$$ −2032.00 −0.278703
$$377$$ 0 0
$$378$$ 216.000 0.0293911
$$379$$ − 9068.00i − 1.22900i −0.788916 0.614501i $$-0.789357\pi$$
0.788916 0.614501i $$-0.210643\pi$$
$$380$$ −576.000 −0.0777584
$$381$$ 1464.00 0.196858
$$382$$ − 6944.00i − 0.930068i
$$383$$ 3162.00i 0.421855i 0.977502 + 0.210928i $$0.0676485\pi$$
−0.977502 + 0.210928i $$0.932352\pi$$
$$384$$ − 384.000i − 0.0510310i
$$385$$ − 32.0000i − 0.00423603i
$$386$$ −620.000 −0.0817543
$$387$$ 1692.00 0.222246
$$388$$ − 2456.00i − 0.321352i
$$389$$ 3666.00 0.477824 0.238912 0.971041i $$-0.423209\pi$$
0.238912 + 0.971041i $$0.423209\pi$$
$$390$$ 0 0
$$391$$ 120.000 0.0155209
$$392$$ − 2616.00i − 0.337061i
$$393$$ 5232.00 0.671551
$$394$$ −2040.00 −0.260847
$$395$$ 5312.00i 0.676647i
$$396$$ 72.0000i 0.00913671i
$$397$$ − 11054.0i − 1.39744i −0.715394 0.698721i $$-0.753753\pi$$
0.715394 0.698721i $$-0.246247\pi$$
$$398$$ 6512.00i 0.820143i
$$399$$ −432.000 −0.0542031
$$400$$ 1744.00 0.218000
$$401$$ 5328.00i 0.663510i 0.943366 + 0.331755i $$0.107641\pi$$
−0.943366 + 0.331755i $$0.892359\pi$$
$$402$$ −2688.00 −0.333496
$$403$$ 0 0
$$404$$ −6424.00 −0.791104
$$405$$ 324.000i 0.0397523i
$$406$$ −112.000 −0.0136908
$$407$$ 516.000 0.0628432
$$408$$ − 144.000i − 0.0174732i
$$409$$ − 12074.0i − 1.45971i −0.683603 0.729854i $$-0.739588\pi$$
0.683603 0.729854i $$-0.260412\pi$$
$$410$$ − 672.000i − 0.0809456i
$$411$$ 2484.00i 0.298118i
$$412$$ 832.000 0.0994896
$$413$$ −2200.00 −0.262118
$$414$$ 360.000i 0.0427368i
$$415$$ −744.000 −0.0880037
$$416$$ 0 0
$$417$$ −1212.00 −0.142331
$$418$$ − 144.000i − 0.0168499i
$$419$$ 13584.0 1.58382 0.791911 0.610636i $$-0.209086\pi$$
0.791911 + 0.610636i $$0.209086\pi$$
$$420$$ −192.000 −0.0223063
$$421$$ − 7406.00i − 0.857355i −0.903458 0.428677i $$-0.858980\pi$$
0.903458 0.428677i $$-0.141020\pi$$
$$422$$ − 9128.00i − 1.05295i
$$423$$ − 2286.00i − 0.262764i
$$424$$ − 2928.00i − 0.335369i
$$425$$ 654.000 0.0746439
$$426$$ −5556.00 −0.631899
$$427$$ 56.0000i 0.00634667i
$$428$$ 992.000 0.112033
$$429$$ 0 0
$$430$$ −1504.00 −0.168673
$$431$$ − 10134.0i − 1.13257i −0.824210 0.566285i $$-0.808380\pi$$
0.824210 0.566285i $$-0.191620\pi$$
$$432$$ 432.000 0.0481125
$$433$$ −9406.00 −1.04393 −0.521967 0.852966i $$-0.674802\pi$$
−0.521967 + 0.852966i $$0.674802\pi$$
$$434$$ − 1216.00i − 0.134493i
$$435$$ − 168.000i − 0.0185172i
$$436$$ 2168.00i 0.238138i
$$437$$ − 720.000i − 0.0788153i
$$438$$ 1524.00 0.166255
$$439$$ −4088.00 −0.444441 −0.222220 0.974996i $$-0.571330\pi$$
−0.222220 + 0.974996i $$0.571330\pi$$
$$440$$ − 64.0000i − 0.00693427i
$$441$$ 2943.00 0.317784
$$442$$ 0 0
$$443$$ −5328.00 −0.571424 −0.285712 0.958315i $$-0.592230\pi$$
−0.285712 + 0.958315i $$0.592230\pi$$
$$444$$ − 3096.00i − 0.330923i
$$445$$ −1344.00 −0.143172
$$446$$ 144.000 0.0152883
$$447$$ 8784.00i 0.929461i
$$448$$ 256.000i 0.0269975i
$$449$$ − 13160.0i − 1.38320i −0.722279 0.691602i $$-0.756905\pi$$
0.722279 0.691602i $$-0.243095\pi$$
$$450$$ 1962.00i 0.205532i
$$451$$ 168.000 0.0175406
$$452$$ 8168.00 0.849979
$$453$$ − 5832.00i − 0.604881i
$$454$$ −5388.00 −0.556985
$$455$$ 0 0
$$456$$ −864.000 −0.0887292
$$457$$ − 9146.00i − 0.936175i −0.883682 0.468087i $$-0.844943\pi$$
0.883682 0.468087i $$-0.155057\pi$$
$$458$$ 11844.0 1.20837
$$459$$ 162.000 0.0164739
$$460$$ − 320.000i − 0.0324349i
$$461$$ 5580.00i 0.563745i 0.959452 + 0.281873i $$0.0909555\pi$$
−0.959452 + 0.281873i $$0.909044\pi$$
$$462$$ − 48.0000i − 0.00483368i
$$463$$ − 14788.0i − 1.48436i −0.670203 0.742178i $$-0.733793\pi$$
0.670203 0.742178i $$-0.266207\pi$$
$$464$$ −224.000 −0.0224115
$$465$$ 1824.00 0.181905
$$466$$ 10244.0i 1.01834i
$$467$$ −12376.0 −1.22632 −0.613162 0.789957i $$-0.710103\pi$$
−0.613162 + 0.789957i $$0.710103\pi$$
$$468$$ 0 0
$$469$$ 1792.00 0.176433
$$470$$ 2032.00i 0.199424i
$$471$$ 10770.0 1.05362
$$472$$ −4400.00 −0.429081
$$473$$ − 376.000i − 0.0365507i
$$474$$ 7968.00i 0.772115i
$$475$$ − 3924.00i − 0.379043i
$$476$$ 96.0000i 0.00924402i
$$477$$ 3294.00 0.316188
$$478$$ −10044.0 −0.961092
$$479$$ − 834.000i − 0.0795541i −0.999209 0.0397771i $$-0.987335\pi$$
0.999209 0.0397771i $$-0.0126648\pi$$
$$480$$ −384.000 −0.0365148
$$481$$ 0 0
$$482$$ −2436.00 −0.230201
$$483$$ − 240.000i − 0.0226095i
$$484$$ −5308.00 −0.498497
$$485$$ −2456.00 −0.229941
$$486$$ 486.000i 0.0453609i
$$487$$ − 13192.0i − 1.22749i −0.789505 0.613744i $$-0.789663\pi$$
0.789505 0.613744i $$-0.210337\pi$$
$$488$$ 112.000i 0.0103893i
$$489$$ 6852.00i 0.633657i
$$490$$ −2616.00 −0.241181
$$491$$ −16568.0 −1.52282 −0.761409 0.648272i $$-0.775492\pi$$
−0.761409 + 0.648272i $$0.775492\pi$$
$$492$$ − 1008.00i − 0.0923662i
$$493$$ −84.0000 −0.00767377
$$494$$ 0 0
$$495$$ 72.0000 0.00653770
$$496$$ − 2432.00i − 0.220161i
$$497$$ 3704.00 0.334300
$$498$$ −1116.00 −0.100420
$$499$$ − 10136.0i − 0.909318i −0.890666 0.454659i $$-0.849761\pi$$
0.890666 0.454659i $$-0.150239\pi$$
$$500$$ − 3744.00i − 0.334874i
$$501$$ − 9522.00i − 0.849125i
$$502$$ 4224.00i 0.375550i
$$503$$ 10412.0 0.922959 0.461479 0.887151i $$-0.347319\pi$$
0.461479 + 0.887151i $$0.347319\pi$$
$$504$$ −288.000 −0.0254535
$$505$$ 6424.00i 0.566068i
$$506$$ 80.0000 0.00702853
$$507$$ 0 0
$$508$$ −1952.00 −0.170484
$$509$$ − 4180.00i − 0.363999i −0.983299 0.181999i $$-0.941743\pi$$
0.983299 0.181999i $$-0.0582568\pi$$
$$510$$ −144.000 −0.0125028
$$511$$ −1016.00 −0.0879554
$$512$$ 512.000i 0.0441942i
$$513$$ − 972.000i − 0.0836547i
$$514$$ − 5628.00i − 0.482958i
$$515$$ − 832.000i − 0.0711889i
$$516$$ −2256.00 −0.192471
$$517$$ −508.000 −0.0432143
$$518$$ 2064.00i 0.175071i
$$519$$ 4074.00 0.344564
$$520$$ 0 0
$$521$$ −14610.0 −1.22855 −0.614276 0.789091i $$-0.710552\pi$$
−0.614276 + 0.789091i $$0.710552\pi$$
$$522$$ − 252.000i − 0.0211298i
$$523$$ −2172.00 −0.181596 −0.0907982 0.995869i $$-0.528942\pi$$
−0.0907982 + 0.995869i $$0.528942\pi$$
$$524$$ −6976.00 −0.581580
$$525$$ − 1308.00i − 0.108735i
$$526$$ − 8088.00i − 0.670444i
$$527$$ − 912.000i − 0.0753840i
$$528$$ − 96.0000i − 0.00791262i
$$529$$ −11767.0 −0.967124
$$530$$ −2928.00 −0.239970
$$531$$ − 4950.00i − 0.404542i
$$532$$ 576.000 0.0469413
$$533$$ 0 0
$$534$$ −2016.00 −0.163372
$$535$$ − 992.000i − 0.0801643i
$$536$$ 3584.00 0.288816
$$537$$ −2124.00 −0.170684
$$538$$ − 2940.00i − 0.235599i
$$539$$ − 654.000i − 0.0522630i
$$540$$ − 432.000i − 0.0344265i
$$541$$ 11758.0i 0.934410i 0.884149 + 0.467205i $$0.154739\pi$$
−0.884149 + 0.467205i $$0.845261\pi$$
$$542$$ −3688.00 −0.292275
$$543$$ 1638.00 0.129454
$$544$$ 192.000i 0.0151322i
$$545$$ 2168.00 0.170398
$$546$$ 0 0
$$547$$ 340.000 0.0265765 0.0132883 0.999912i $$-0.495770\pi$$
0.0132883 + 0.999912i $$0.495770\pi$$
$$548$$ − 3312.00i − 0.258178i
$$549$$ −126.000 −0.00979517
$$550$$ 436.000 0.0338020
$$551$$ 504.000i 0.0389676i
$$552$$ − 480.000i − 0.0370112i
$$553$$ − 5312.00i − 0.408480i
$$554$$ − 11532.0i − 0.884382i
$$555$$ −3096.00 −0.236789
$$556$$ 1616.00 0.123262
$$557$$ 3768.00i 0.286634i 0.989677 + 0.143317i $$0.0457769\pi$$
−0.989677 + 0.143317i $$0.954223\pi$$
$$558$$ 2736.00 0.207570
$$559$$ 0 0
$$560$$ 256.000 0.0193178
$$561$$ − 36.0000i − 0.00270931i
$$562$$ −14936.0 −1.12106
$$563$$ 10172.0 0.761454 0.380727 0.924687i $$-0.375674\pi$$
0.380727 + 0.924687i $$0.375674\pi$$
$$564$$ 3048.00i 0.227560i
$$565$$ − 8168.00i − 0.608195i
$$566$$ − 2456.00i − 0.182391i
$$567$$ − 324.000i − 0.0239977i
$$568$$ 7408.00 0.547241
$$569$$ 5506.00 0.405665 0.202833 0.979213i $$-0.434985\pi$$
0.202833 + 0.979213i $$0.434985\pi$$
$$570$$ 864.000i 0.0634894i
$$571$$ −2340.00 −0.171499 −0.0857495 0.996317i $$-0.527328\pi$$
−0.0857495 + 0.996317i $$0.527328\pi$$
$$572$$ 0 0
$$573$$ −10416.0 −0.759397
$$574$$ 672.000i 0.0488654i
$$575$$ 2180.00 0.158108
$$576$$ −576.000 −0.0416667
$$577$$ − 20094.0i − 1.44978i −0.688864 0.724891i $$-0.741890\pi$$
0.688864 0.724891i $$-0.258110\pi$$
$$578$$ − 9754.00i − 0.701925i
$$579$$ 930.000i 0.0667521i
$$580$$ 224.000i 0.0160364i
$$581$$ 744.000 0.0531262
$$582$$ −3684.00 −0.262383
$$583$$ − 732.000i − 0.0520006i
$$584$$ −2032.00 −0.143981
$$585$$ 0 0
$$586$$ 13216.0 0.931652
$$587$$ − 7118.00i − 0.500496i −0.968182 0.250248i $$-0.919488\pi$$
0.968182 0.250248i $$-0.0805122\pi$$
$$588$$ −3924.00 −0.275209
$$589$$ −5472.00 −0.382801
$$590$$ 4400.00i 0.307026i
$$591$$ 3060.00i 0.212981i
$$592$$ 4128.00i 0.286587i
$$593$$ 10328.0i 0.715211i 0.933873 + 0.357606i $$0.116407\pi$$
−0.933873 + 0.357606i $$0.883593\pi$$
$$594$$ 108.000 0.00746009
$$595$$ 96.0000 0.00661448
$$596$$ − 11712.0i − 0.804937i
$$597$$ 9768.00 0.669644
$$598$$ 0 0
$$599$$ −19732.0 −1.34596 −0.672978 0.739662i $$-0.734985\pi$$
−0.672978 + 0.739662i $$0.734985\pi$$
$$600$$ − 2616.00i − 0.177996i
$$601$$ −12026.0 −0.816224 −0.408112 0.912932i $$-0.633813\pi$$
−0.408112 + 0.912932i $$0.633813\pi$$
$$602$$ 1504.00 0.101825
$$603$$ 4032.00i 0.272298i
$$604$$ 7776.00i 0.523843i
$$605$$ 5308.00i 0.356696i
$$606$$ 9636.00i 0.645934i
$$607$$ 17016.0 1.13782 0.568911 0.822399i $$-0.307365\pi$$
0.568911 + 0.822399i $$0.307365\pi$$
$$608$$ 1152.00 0.0768417
$$609$$ 168.000i 0.0111785i
$$610$$ 112.000 0.00743401
$$611$$ 0 0
$$612$$ −216.000 −0.0142668
$$613$$ 11654.0i 0.767864i 0.923361 + 0.383932i $$0.125430\pi$$
−0.923361 + 0.383932i $$0.874570\pi$$
$$614$$ 15328.0 1.00747
$$615$$ −1008.00 −0.0660918
$$616$$ 64.0000i 0.00418609i
$$617$$ 11612.0i 0.757669i 0.925465 + 0.378834i $$0.123675\pi$$
−0.925465 + 0.378834i $$0.876325\pi$$
$$618$$ − 1248.00i − 0.0812329i
$$619$$ − 4024.00i − 0.261290i −0.991429 0.130645i $$-0.958295\pi$$
0.991429 0.130645i $$-0.0417047\pi$$
$$620$$ −2432.00 −0.157535
$$621$$ 540.000 0.0348945
$$622$$ 4680.00i 0.301690i
$$623$$ 1344.00 0.0864305
$$624$$ 0 0
$$625$$ 9881.00 0.632384
$$626$$ 13420.0i 0.856823i
$$627$$ −216.000 −0.0137579
$$628$$ −14360.0 −0.912462
$$629$$ 1548.00i 0.0981285i
$$630$$ 288.000i 0.0182130i
$$631$$ 1088.00i 0.0686412i 0.999411 + 0.0343206i $$0.0109267\pi$$
−0.999411 + 0.0343206i $$0.989073\pi$$
$$632$$ − 10624.0i − 0.668671i
$$633$$ −13692.0 −0.859729
$$634$$ −8328.00 −0.521683
$$635$$ 1952.00i 0.121989i
$$636$$ −4392.00 −0.273827
$$637$$ 0 0
$$638$$ −56.0000 −0.00347502
$$639$$ 8334.00i 0.515944i
$$640$$ 512.000 0.0316228
$$641$$ 7078.00 0.436138 0.218069 0.975933i $$-0.430024\pi$$
0.218069 + 0.975933i $$0.430024\pi$$
$$642$$ − 1488.00i − 0.0914746i
$$643$$ 8336.00i 0.511259i 0.966775 + 0.255630i $$0.0822828\pi$$
−0.966775 + 0.255630i $$0.917717\pi$$
$$644$$ 320.000i 0.0195804i
$$645$$ 2256.00i 0.137721i
$$646$$ 432.000 0.0263109
$$647$$ −32.0000 −0.00194444 −0.000972218 1.00000i $$-0.500309\pi$$
−0.000972218 1.00000i $$0.500309\pi$$
$$648$$ − 648.000i − 0.0392837i
$$649$$ −1100.00 −0.0665312
$$650$$ 0 0
$$651$$ −1824.00 −0.109813
$$652$$ − 9136.00i − 0.548763i
$$653$$ −15822.0 −0.948182 −0.474091 0.880476i $$-0.657223\pi$$
−0.474091 + 0.880476i $$0.657223\pi$$
$$654$$ 3252.00 0.194439
$$655$$ 6976.00i 0.416145i
$$656$$ 1344.00i 0.0799914i
$$657$$ − 2286.00i − 0.135746i
$$658$$ − 2032.00i − 0.120388i
$$659$$ 21540.0 1.27326 0.636631 0.771169i $$-0.280328\pi$$
0.636631 + 0.771169i $$0.280328\pi$$
$$660$$ −96.0000 −0.00566181
$$661$$ − 8270.00i − 0.486635i −0.969947 0.243317i $$-0.921764\pi$$
0.969947 0.243317i $$-0.0782357\pi$$
$$662$$ 20144.0 1.18266
$$663$$ 0 0
$$664$$ 1488.00 0.0869663
$$665$$ − 576.000i − 0.0335885i
$$666$$ −4644.00 −0.270197
$$667$$ −280.000 −0.0162543
$$668$$ 12696.0i 0.735364i
$$669$$ − 216.000i − 0.0124829i
$$670$$ − 3584.00i − 0.206660i
$$671$$ 28.0000i 0.00161092i
$$672$$ 384.000 0.0220433
$$673$$ −8482.00 −0.485820 −0.242910 0.970049i $$-0.578102\pi$$
−0.242910 + 0.970049i $$0.578102\pi$$
$$674$$ − 5980.00i − 0.341752i
$$675$$ 2943.00 0.167816
$$676$$ 0 0
$$677$$ 2550.00 0.144763 0.0723814 0.997377i $$-0.476940\pi$$
0.0723814 + 0.997377i $$0.476940\pi$$
$$678$$ − 12252.0i − 0.694005i
$$679$$ 2456.00 0.138811
$$680$$ 192.000 0.0108277
$$681$$ 8082.00i 0.454777i
$$682$$ − 608.000i − 0.0341371i
$$683$$ 31534.0i 1.76664i 0.468771 + 0.883320i $$0.344697\pi$$
−0.468771 + 0.883320i $$0.655303\pi$$
$$684$$ 1296.00i 0.0724471i
$$685$$ −3312.00 −0.184737
$$686$$ 5360.00 0.298317
$$687$$ − 17766.0i − 0.986631i
$$688$$ 3008.00 0.166684
$$689$$ 0 0
$$690$$ −480.000 −0.0264830
$$691$$ 33832.0i 1.86256i 0.364302 + 0.931281i $$0.381307\pi$$
−0.364302 + 0.931281i $$0.618693\pi$$
$$692$$ −5432.00 −0.298401
$$693$$ −72.0000 −0.00394669
$$694$$ 13128.0i 0.718058i
$$695$$ − 1616.00i − 0.0881991i
$$696$$ 336.000i 0.0182989i
$$697$$ 504.000i 0.0273893i
$$698$$ −1348.00 −0.0730982
$$699$$ 15366.0 0.831467
$$700$$ 1744.00i 0.0941671i
$$701$$ −19422.0 −1.04645 −0.523223 0.852196i $$-0.675271\pi$$
−0.523223 + 0.852196i $$0.675271\pi$$
$$702$$ 0 0
$$703$$ 9288.00 0.498298
$$704$$ 128.000i 0.00685253i
$$705$$ 3048.00 0.162829
$$706$$ 21464.0 1.14420
$$707$$ − 6424.00i − 0.341725i
$$708$$ 6600.00i 0.350343i
$$709$$ 1894.00i 0.100325i 0.998741 + 0.0501627i $$0.0159740\pi$$
−0.998741 + 0.0501627i $$0.984026\pi$$
$$710$$ − 7408.00i − 0.391574i
$$711$$ 11952.0 0.630429
$$712$$ 2688.00 0.141485
$$713$$ − 3040.00i − 0.159676i
$$714$$ 144.000 0.00754771
$$715$$ 0 0
$$716$$ 2832.00 0.147817
$$717$$ 15066.0i 0.784728i
$$718$$ −9684.00 −0.503348
$$719$$ 20156.0 1.04547 0.522734 0.852496i $$-0.324912\pi$$
0.522734 + 0.852496i $$0.324912\pi$$
$$720$$ 576.000i 0.0298142i
$$721$$ 832.000i 0.0429754i
$$722$$ 11126.0i 0.573500i
$$723$$ 3654.00i 0.187958i
$$724$$ −2184.00 −0.112110
$$725$$ −1526.00 −0.0781713
$$726$$ 7962.00i 0.407021i
$$727$$ −11128.0 −0.567696 −0.283848 0.958869i $$-0.591611\pi$$
−0.283848 + 0.958869i $$0.591611\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 2032.00i 0.103024i
$$731$$ 1128.00 0.0570733
$$732$$ 168.000 0.00848287
$$733$$ 16202.0i 0.816418i 0.912888 + 0.408209i $$0.133847\pi$$
−0.912888 + 0.408209i $$0.866153\pi$$
$$734$$ − 12560.0i − 0.631605i
$$735$$ 3924.00i 0.196924i
$$736$$ 640.000i 0.0320526i
$$737$$ 896.000 0.0447823
$$738$$ −1512.00 −0.0754167
$$739$$ 5328.00i 0.265215i 0.991169 + 0.132607i $$0.0423349\pi$$
−0.991169 + 0.132607i $$0.957665\pi$$
$$740$$ 4128.00 0.205065
$$741$$ 0 0
$$742$$ 2928.00 0.144866
$$743$$ − 20482.0i − 1.01132i −0.862732 0.505661i $$-0.831249\pi$$
0.862732 0.505661i $$-0.168751\pi$$
$$744$$ −3648.00 −0.179761
$$745$$ −11712.0 −0.575966
$$746$$ 12868.0i 0.631543i
$$747$$ 1674.00i 0.0819926i
$$748$$ 48.0000i 0.00234633i
$$749$$ 992.000i 0.0483937i
$$750$$ −5616.00 −0.273423
$$751$$ −8040.00 −0.390657 −0.195329 0.980738i $$-0.562577\pi$$
−0.195329 + 0.980738i $$0.562577\pi$$
$$752$$ − 4064.00i − 0.197073i
$$753$$ 6336.00 0.306636
$$754$$ 0 0
$$755$$ 7776.00 0.374831
$$756$$ 432.000i 0.0207827i
$$757$$ −15822.0 −0.759657 −0.379829 0.925057i $$-0.624017\pi$$
−0.379829 + 0.925057i $$0.624017\pi$$
$$758$$ 18136.0 0.869036
$$759$$ − 120.000i − 0.00573877i
$$760$$ − 1152.00i − 0.0549835i
$$761$$ 1452.00i 0.0691655i 0.999402 + 0.0345828i $$0.0110102\pi$$
−0.999402 + 0.0345828i $$0.988990\pi$$
$$762$$ 2928.00i 0.139200i
$$763$$ −2168.00 −0.102866
$$764$$ 13888.0 0.657657
$$765$$ 216.000i 0.0102085i
$$766$$ −6324.00 −0.298297
$$767$$ 0 0
$$768$$ 768.000 0.0360844
$$769$$ 32298.0i 1.51456i 0.653091 + 0.757279i $$0.273472\pi$$
−0.653091 + 0.757279i $$0.726528\pi$$
$$770$$ 64.0000 0.00299532
$$771$$ −8442.00 −0.394334
$$772$$ − 1240.00i − 0.0578090i
$$773$$ 18736.0i 0.871781i 0.900000 + 0.435891i $$0.143567\pi$$
−0.900000 + 0.435891i $$0.856433\pi$$
$$774$$ 3384.00i 0.157152i
$$775$$ − 16568.0i − 0.767923i
$$776$$ 4912.00 0.227230
$$777$$ 3096.00 0.142945
$$778$$ 7332.00i 0.337873i
$$779$$ 3024.00 0.139083
$$780$$ 0 0
$$781$$ 1852.00 0.0848525
$$782$$ 240.000i 0.0109749i
$$783$$ −378.000 −0.0172524
$$784$$ 5232.00 0.238338
$$785$$ 14360.0i 0.652905i
$$786$$ 10464.0i 0.474858i
$$787$$ 40816.0i 1.84871i 0.381536 + 0.924354i $$0.375395\pi$$
−0.381536 + 0.924354i $$0.624605\pi$$
$$788$$ − 4080.00i − 0.184447i
$$789$$ −12132.0 −0.547415
$$790$$ −10624.0 −0.478462
$$791$$ 8168.00i 0.367156i
$$792$$ −144.000 −0.00646063
$$793$$ 0 0
$$794$$ 22108.0 0.988141
$$795$$ 4392.00i 0.195935i
$$796$$ −13024.0 −0.579929
$$797$$ −4518.00 −0.200798 −0.100399 0.994947i $$-0.532012\pi$$
−0.100399 + 0.994947i $$0.532012\pi$$
$$798$$ − 864.000i − 0.0383274i
$$799$$ − 1524.00i − 0.0674784i
$$800$$ 3488.00i 0.154149i
$$801$$ 3024.00i 0.133393i
$$802$$ −10656.0 −0.469173
$$803$$ −508.000 −0.0223249
$$804$$ − 5376.00i − 0.235817i
$$805$$ 320.000 0.0140106
$$806$$ 0 0
$$807$$ −4410.00 −0.192366
$$808$$ − 12848.0i − 0.559395i
$$809$$ −5058.00 −0.219814 −0.109907 0.993942i $$-0.535055\pi$$
−0.109907 + 0.993942i $$0.535055\pi$$
$$810$$ −648.000 −0.0281091
$$811$$ − 22564.0i − 0.976978i −0.872570 0.488489i $$-0.837548\pi$$
0.872570 0.488489i $$-0.162452\pi$$
$$812$$ − 224.000i − 0.00968086i
$$813$$ 5532.00i 0.238642i
$$814$$ 1032.00i 0.0444368i
$$815$$ −9136.00 −0.392663
$$816$$ 288.000 0.0123554
$$817$$ − 6768.00i − 0.289819i
$$818$$ 24148.0 1.03217
$$819$$ 0 0
$$820$$ 1344.00 0.0572372
$$821$$ 32584.0i 1.38513i 0.721357 + 0.692564i $$0.243519\pi$$
−0.721357 + 0.692564i $$0.756481\pi$$
$$822$$ −4968.00 −0.210802
$$823$$ 9288.00 0.393389 0.196695 0.980465i $$-0.436979\pi$$
0.196695 + 0.980465i $$0.436979\pi$$
$$824$$ 1664.00i 0.0703497i
$$825$$ − 654.000i − 0.0275992i
$$826$$ − 4400.00i − 0.185346i
$$827$$ − 20586.0i − 0.865593i −0.901492 0.432796i $$-0.857527\pi$$
0.901492 0.432796i $$-0.142473\pi$$
$$828$$ −720.000 −0.0302195
$$829$$ 46118.0 1.93214 0.966070 0.258280i $$-0.0831556\pi$$
0.966070 + 0.258280i $$0.0831556\pi$$
$$830$$ − 1488.00i − 0.0622280i
$$831$$ −17298.0 −0.722095
$$832$$ 0 0
$$833$$ 1962.00 0.0816078
$$834$$ − 2424.00i − 0.100643i
$$835$$ 12696.0 0.526183
$$836$$ 288.000 0.0119147
$$837$$ − 4104.00i − 0.169480i
$$838$$ 27168.0i 1.11993i
$$839$$ 39230.0i 1.61427i 0.590369 + 0.807133i $$0.298982\pi$$
−0.590369 + 0.807133i $$0.701018\pi$$
$$840$$ − 384.000i − 0.0157729i
$$841$$ −24193.0 −0.991964
$$842$$ 14812.0 0.606241
$$843$$ 22404.0i 0.915344i
$$844$$ 18256.0 0.744547
$$845$$ 0 0
$$846$$ 4572.00 0.185802
$$847$$ − 5308.00i − 0.215331i
$$848$$ 5856.00 0.237141
$$849$$ −3684.00 −0.148922
$$850$$ 1308.00i 0.0527812i
$$851$$ 5160.00i 0.207853i
$$852$$ − 11112.0i − 0.446820i
$$853$$ 18674.0i 0.749573i 0.927111 + 0.374786i $$0.122284\pi$$
−0.927111 + 0.374786i $$0.877716\pi$$
$$854$$ −112.000 −0.00448778
$$855$$ 1296.00 0.0518389
$$856$$ 1984.00i 0.0792193i
$$857$$ −41678.0 −1.66125 −0.830626 0.556830i $$-0.812017\pi$$
−0.830626 + 0.556830i $$0.812017\pi$$
$$858$$ 0 0
$$859$$ −14740.0 −0.585474 −0.292737 0.956193i $$-0.594566\pi$$
−0.292737 + 0.956193i $$0.594566\pi$$
$$860$$ − 3008.00i − 0.119270i
$$861$$ 1008.00 0.0398984
$$862$$ 20268.0 0.800848
$$863$$ − 24982.0i − 0.985396i −0.870200 0.492698i $$-0.836011\pi$$
0.870200 0.492698i $$-0.163989\pi$$
$$864$$ 864.000i 0.0340207i
$$865$$ 5432.00i 0.213519i
$$866$$ − 18812.0i − 0.738173i
$$867$$ −14631.0 −0.573120
$$868$$ 2432.00 0.0951008
$$869$$ − 2656.00i − 0.103681i
$$870$$ 336.000 0.0130936
$$871$$ 0 0
$$872$$ −4336.00 −0.168389
$$873$$ 5526.00i 0.214235i
$$874$$ 1440.00 0.0557308
$$875$$ 3744.00 0.144652
$$876$$ 3048.00i 0.117560i
$$877$$ 1134.00i 0.0436630i 0.999762 + 0.0218315i $$0.00694974\pi$$
−0.999762 + 0.0218315i $$0.993050\pi$$
$$878$$ − 8176.00i − 0.314267i
$$879$$ − 19824.0i − 0.760690i
$$880$$ 128.000 0.00490327
$$881$$ −34950.0 −1.33654 −0.668272 0.743917i $$-0.732966\pi$$
−0.668272 + 0.743917i $$0.732966\pi$$
$$882$$ 5886.00i 0.224707i
$$883$$ 3068.00 0.116927 0.0584634 0.998290i $$-0.481380\pi$$
0.0584634 + 0.998290i $$0.481380\pi$$
$$884$$ 0 0
$$885$$ 6600.00 0.250685
$$886$$ − 10656.0i − 0.404058i
$$887$$ −14080.0 −0.532988 −0.266494 0.963837i $$-0.585865\pi$$
−0.266494 + 0.963837i $$0.585865\pi$$
$$888$$ 6192.00 0.233998
$$889$$ − 1952.00i − 0.0736423i
$$890$$ − 2688.00i − 0.101238i
$$891$$ − 162.000i − 0.00609114i
$$892$$ 288.000i 0.0108105i
$$893$$ −9144.00 −0.342657
$$894$$ −17568.0 −0.657228
$$895$$ − 2832.00i − 0.105769i
$$896$$ −512.000 −0.0190901
$$897$$ 0 0
$$898$$ 26320.0 0.978073
$$899$$ 2128.00i 0.0789464i
$$900$$ −3924.00 −0.145333
$$901$$ 2196.00 0.0811980
$$902$$ 336.000i 0.0124031i
$$903$$ − 2256.00i − 0.0831395i
$$904$$ 16336.0i 0.601026i
$$905$$ 2184.00i 0.0802195i
$$906$$ 11664.0 0.427716
$$907$$ 24876.0 0.910688 0.455344 0.890316i $$-0.349516\pi$$
0.455344 + 0.890316i $$0.349516\pi$$
$$908$$ − 10776.0i − 0.393848i
$$909$$ 14454.0 0.527403
$$910$$ 0 0
$$911$$ 51456.0 1.87136 0.935682 0.352843i $$-0.114785\pi$$
0.935682 + 0.352843i $$0.114785\pi$$
$$912$$ − 1728.00i − 0.0627410i
$$913$$ 372.000 0.0134846
$$914$$ 18292.0 0.661975
$$915$$ − 168.000i − 0.00606985i
$$916$$ 23688.0i 0.854447i
$$917$$ − 6976.00i − 0.251219i
$$918$$ 324.000i 0.0116488i
$$919$$ −31032.0 −1.11388 −0.556938 0.830554i $$-0.688024\pi$$
−0.556938 + 0.830554i $$0.688024\pi$$
$$920$$ 640.000 0.0229350
$$921$$ − 22992.0i − 0.822597i
$$922$$ −11160.0 −0.398628
$$923$$ 0 0
$$924$$ 96.0000 0.00341793
$$925$$ 28122.0i 0.999617i
$$926$$ 29576.0 1.04960
$$927$$ −1872.00 −0.0663264
$$928$$ − 448.000i − 0.0158473i
$$929$$ 50820.0i 1.79478i 0.441239 + 0.897390i $$0.354539\pi$$
−0.441239 + 0.897390i $$0.645461\pi$$
$$930$$ 3648.00i 0.128626i
$$931$$ − 11772.0i − 0.414406i
$$932$$ −20488.0 −0.720072
$$933$$ 7020.00 0.246328
$$934$$ − 24752.0i − 0.867142i
$$935$$ 48.0000 0.00167890
$$936$$ 0 0
$$937$$ 5982.00 0.208563 0.104281 0.994548i $$-0.466746\pi$$
0.104281 + 0.994548i $$0.466746\pi$$
$$938$$ 3584.00i 0.124757i
$$939$$ 20130.0 0.699593
$$940$$ −4064.00 −0.141014
$$941$$ − 20224.0i − 0.700620i −0.936634 0.350310i $$-0.886076\pi$$
0.936634 0.350310i $$-0.113924\pi$$
$$942$$ 21540.0i 0.745022i
$$943$$ 1680.00i 0.0580152i
$$944$$ − 8800.00i − 0.303406i
$$945$$ 432.000 0.0148709
$$946$$ 752.000 0.0258453
$$947$$ − 8478.00i − 0.290917i −0.989364 0.145458i $$-0.953534\pi$$
0.989364 0.145458i $$-0.0464657\pi$$
$$948$$ −15936.0 −0.545968
$$949$$ 0 0
$$950$$ 7848.00 0.268024
$$951$$ 12492.0i 0.425953i
$$952$$ −192.000 −0.00653651
$$953$$ −40918.0 −1.39083 −0.695417 0.718607i $$-0.744780\pi$$
−0.695417 + 0.718607i $$0.744780\pi$$
$$954$$ 6588.00i 0.223579i
$$955$$ − 13888.0i − 0.470581i
$$956$$ − 20088.0i − 0.679595i
$$957$$ 84.0000i 0.00283734i
$$958$$ 1668.00 0.0562533
$$959$$ 3312.00 0.111522
$$960$$ − 768.000i − 0.0258199i
$$961$$ 6687.00 0.224464
$$962$$ 0 0
$$963$$ −2232.00 −0.0746887
$$964$$ − 4872.00i − 0.162777i
$$965$$ −1240.00 −0.0413648
$$966$$ 480.000 0.0159873
$$967$$ − 4624.00i − 0.153772i −0.997040 0.0768862i $$-0.975502\pi$$
0.997040 0.0768862i $$-0.0244978\pi$$
$$968$$ − 10616.0i − 0.352491i
$$969$$ − 648.000i − 0.0214827i
$$970$$ − 4912.00i − 0.162593i
$$971$$ 15300.0 0.505665 0.252832 0.967510i $$-0.418638\pi$$
0.252832 + 0.967510i $$0.418638\pi$$
$$972$$ −972.000 −0.0320750
$$973$$ 1616.00i 0.0532442i
$$974$$ 26384.0 0.867965
$$975$$ 0 0
$$976$$ −224.000 −0.00734638
$$977$$ 19584.0i 0.641298i 0.947198 + 0.320649i $$0.103901\pi$$
−0.947198 + 0.320649i $$0.896099\pi$$
$$978$$ −13704.0 −0.448063
$$979$$ 672.000 0.0219379
$$980$$ − 5232.00i − 0.170541i
$$981$$ − 4878.00i − 0.158759i
$$982$$ − 33136.0i − 1.07679i
$$983$$ 17582.0i 0.570477i 0.958457 + 0.285238i $$0.0920728\pi$$
−0.958457 + 0.285238i $$0.907927\pi$$
$$984$$ 2016.00 0.0653127
$$985$$ −4080.00 −0.131979
$$986$$ − 168.000i − 0.00542618i
$$987$$ −3048.00 −0.0982968
$$988$$ 0 0
$$989$$ 3760.00 0.120891
$$990$$ 144.000i 0.00462285i
$$991$$ 47904.0 1.53554 0.767770 0.640725i $$-0.221366\pi$$
0.767770 + 0.640725i $$0.221366\pi$$
$$992$$ 4864.00 0.155678
$$993$$ − 30216.0i − 0.965635i
$$994$$ 7408.00i 0.236386i
$$995$$ 13024.0i 0.414963i
$$996$$ − 2232.00i − 0.0710077i
$$997$$ −44578.0 −1.41605 −0.708024 0.706189i $$-0.750413\pi$$
−0.708024 + 0.706189i $$0.750413\pi$$
$$998$$ 20272.0 0.642985
$$999$$ 6966.00i 0.220615i
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 1014.4.b.g.337.2 2
13.5 odd 4 78.4.a.f.1.1 1
13.8 odd 4 1014.4.a.e.1.1 1
13.12 even 2 inner 1014.4.b.g.337.1 2
39.5 even 4 234.4.a.c.1.1 1
52.31 even 4 624.4.a.c.1.1 1
65.44 odd 4 1950.4.a.a.1.1 1
104.5 odd 4 2496.4.a.c.1.1 1
104.83 even 4 2496.4.a.l.1.1 1
156.83 odd 4 1872.4.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.f.1.1 1 13.5 odd 4
234.4.a.c.1.1 1 39.5 even 4
624.4.a.c.1.1 1 52.31 even 4
1014.4.a.e.1.1 1 13.8 odd 4
1014.4.b.g.337.1 2 13.12 even 2 inner
1014.4.b.g.337.2 2 1.1 even 1 trivial
1872.4.a.f.1.1 1 156.83 odd 4
1950.4.a.a.1.1 1 65.44 odd 4
2496.4.a.c.1.1 1 104.5 odd 4
2496.4.a.l.1.1 1 104.83 even 4