Properties

 Label 4840.2.a.p.1.2 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 4840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.41421 q^{3} +1.00000 q^{5} +2.41421 q^{7} +2.82843 q^{9} +O(q^{10})$$ $$q+2.41421 q^{3} +1.00000 q^{5} +2.41421 q^{7} +2.82843 q^{9} +2.00000 q^{13} +2.41421 q^{15} +5.65685 q^{17} +4.82843 q^{19} +5.82843 q^{21} +3.65685 q^{23} +1.00000 q^{25} -0.414214 q^{27} +2.00000 q^{29} -5.65685 q^{31} +2.41421 q^{35} -4.00000 q^{37} +4.82843 q^{39} -9.48528 q^{41} +3.58579 q^{43} +2.82843 q^{45} +7.58579 q^{47} -1.17157 q^{49} +13.6569 q^{51} -7.65685 q^{53} +11.6569 q^{57} -11.3137 q^{59} +1.00000 q^{61} +6.82843 q^{63} +2.00000 q^{65} -6.41421 q^{67} +8.82843 q^{69} -4.00000 q^{71} +4.00000 q^{73} +2.41421 q^{75} -14.4853 q^{79} -9.48528 q^{81} -13.3137 q^{83} +5.65685 q^{85} +4.82843 q^{87} +2.65685 q^{89} +4.82843 q^{91} -13.6569 q^{93} +4.82843 q^{95} -17.3137 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 4 q^{13} + 2 q^{15} + 4 q^{19} + 6 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 2 q^{35} - 8 q^{37} + 4 q^{39} - 2 q^{41} + 10 q^{43} + 18 q^{47} - 8 q^{49} + 16 q^{51} - 4 q^{53} + 12 q^{57} + 2 q^{61} + 8 q^{63} + 4 q^{65} - 10 q^{67} + 12 q^{69} - 8 q^{71} + 8 q^{73} + 2 q^{75} - 12 q^{79} - 2 q^{81} - 4 q^{83} + 4 q^{87} - 6 q^{89} + 4 q^{91} - 16 q^{93} + 4 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 4 * q^13 + 2 * q^15 + 4 * q^19 + 6 * q^21 - 4 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 2 * q^35 - 8 * q^37 + 4 * q^39 - 2 * q^41 + 10 * q^43 + 18 * q^47 - 8 * q^49 + 16 * q^51 - 4 * q^53 + 12 * q^57 + 2 * q^61 + 8 * q^63 + 4 * q^65 - 10 * q^67 + 12 * q^69 - 8 * q^71 + 8 * q^73 + 2 * q^75 - 12 * q^79 - 2 * q^81 - 4 * q^83 + 4 * q^87 - 6 * q^89 + 4 * q^91 - 16 * q^93 + 4 * q^95 - 12 * q^97

Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.41421 1.39385 0.696923 0.717146i $$-0.254552\pi$$
0.696923 + 0.717146i $$0.254552\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.41421 0.912487 0.456243 0.889855i $$-0.349195\pi$$
0.456243 + 0.889855i $$0.349195\pi$$
$$8$$ 0 0
$$9$$ 2.82843 0.942809
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 2.41421 0.623347
$$16$$ 0 0
$$17$$ 5.65685 1.37199 0.685994 0.727607i $$-0.259367\pi$$
0.685994 + 0.727607i $$0.259367\pi$$
$$18$$ 0 0
$$19$$ 4.82843 1.10772 0.553859 0.832611i $$-0.313155\pi$$
0.553859 + 0.832611i $$0.313155\pi$$
$$20$$ 0 0
$$21$$ 5.82843 1.27187
$$22$$ 0 0
$$23$$ 3.65685 0.762507 0.381253 0.924471i $$-0.375493\pi$$
0.381253 + 0.924471i $$0.375493\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −0.414214 −0.0797154
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −5.65685 −1.01600 −0.508001 0.861357i $$-0.669615\pi$$
−0.508001 + 0.861357i $$0.669615\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.41421 0.408077
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ 4.82843 0.773167
$$40$$ 0 0
$$41$$ −9.48528 −1.48135 −0.740676 0.671862i $$-0.765495\pi$$
−0.740676 + 0.671862i $$0.765495\pi$$
$$42$$ 0 0
$$43$$ 3.58579 0.546827 0.273414 0.961897i $$-0.411847\pi$$
0.273414 + 0.961897i $$0.411847\pi$$
$$44$$ 0 0
$$45$$ 2.82843 0.421637
$$46$$ 0 0
$$47$$ 7.58579 1.10650 0.553250 0.833015i $$-0.313387\pi$$
0.553250 + 0.833015i $$0.313387\pi$$
$$48$$ 0 0
$$49$$ −1.17157 −0.167368
$$50$$ 0 0
$$51$$ 13.6569 1.91234
$$52$$ 0 0
$$53$$ −7.65685 −1.05175 −0.525875 0.850562i $$-0.676262\pi$$
−0.525875 + 0.850562i $$0.676262\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 11.6569 1.54399
$$58$$ 0 0
$$59$$ −11.3137 −1.47292 −0.736460 0.676481i $$-0.763504\pi$$
−0.736460 + 0.676481i $$0.763504\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 0 0
$$63$$ 6.82843 0.860301
$$64$$ 0 0
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ −6.41421 −0.783621 −0.391810 0.920046i $$-0.628151\pi$$
−0.391810 + 0.920046i $$0.628151\pi$$
$$68$$ 0 0
$$69$$ 8.82843 1.06282
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 0 0
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 0 0
$$75$$ 2.41421 0.278769
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −14.4853 −1.62972 −0.814861 0.579657i $$-0.803187\pi$$
−0.814861 + 0.579657i $$0.803187\pi$$
$$80$$ 0 0
$$81$$ −9.48528 −1.05392
$$82$$ 0 0
$$83$$ −13.3137 −1.46137 −0.730685 0.682715i $$-0.760799\pi$$
−0.730685 + 0.682715i $$0.760799\pi$$
$$84$$ 0 0
$$85$$ 5.65685 0.613572
$$86$$ 0 0
$$87$$ 4.82843 0.517662
$$88$$ 0 0
$$89$$ 2.65685 0.281626 0.140813 0.990036i $$-0.455028\pi$$
0.140813 + 0.990036i $$0.455028\pi$$
$$90$$ 0 0
$$91$$ 4.82843 0.506157
$$92$$ 0 0
$$93$$ −13.6569 −1.41615
$$94$$ 0 0
$$95$$ 4.82843 0.495386
$$96$$ 0 0
$$97$$ −17.3137 −1.75794 −0.878970 0.476876i $$-0.841769\pi$$
−0.878970 + 0.476876i $$0.841769\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 10.1716 1.01211 0.506055 0.862501i $$-0.331103\pi$$
0.506055 + 0.862501i $$0.331103\pi$$
$$102$$ 0 0
$$103$$ 13.3137 1.31184 0.655919 0.754831i $$-0.272281\pi$$
0.655919 + 0.754831i $$0.272281\pi$$
$$104$$ 0 0
$$105$$ 5.82843 0.568796
$$106$$ 0 0
$$107$$ 15.2426 1.47356 0.736781 0.676132i $$-0.236345\pi$$
0.736781 + 0.676132i $$0.236345\pi$$
$$108$$ 0 0
$$109$$ 16.6569 1.59544 0.797719 0.603030i $$-0.206040\pi$$
0.797719 + 0.603030i $$0.206040\pi$$
$$110$$ 0 0
$$111$$ −9.65685 −0.916588
$$112$$ 0 0
$$113$$ 17.3137 1.62874 0.814368 0.580348i $$-0.197084\pi$$
0.814368 + 0.580348i $$0.197084\pi$$
$$114$$ 0 0
$$115$$ 3.65685 0.341003
$$116$$ 0 0
$$117$$ 5.65685 0.522976
$$118$$ 0 0
$$119$$ 13.6569 1.25192
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −22.8995 −2.06478
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −4.41421 −0.391698 −0.195849 0.980634i $$-0.562746\pi$$
−0.195849 + 0.980634i $$0.562746\pi$$
$$128$$ 0 0
$$129$$ 8.65685 0.762194
$$130$$ 0 0
$$131$$ −2.48528 −0.217140 −0.108570 0.994089i $$-0.534627\pi$$
−0.108570 + 0.994089i $$0.534627\pi$$
$$132$$ 0 0
$$133$$ 11.6569 1.01078
$$134$$ 0 0
$$135$$ −0.414214 −0.0356498
$$136$$ 0 0
$$137$$ 13.3137 1.13747 0.568733 0.822522i $$-0.307434\pi$$
0.568733 + 0.822522i $$0.307434\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 18.3137 1.54229
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 2.00000 0.166091
$$146$$ 0 0
$$147$$ −2.82843 −0.233285
$$148$$ 0 0
$$149$$ −17.0000 −1.39269 −0.696347 0.717705i $$-0.745193\pi$$
−0.696347 + 0.717705i $$0.745193\pi$$
$$150$$ 0 0
$$151$$ 5.65685 0.460348 0.230174 0.973149i $$-0.426070\pi$$
0.230174 + 0.973149i $$0.426070\pi$$
$$152$$ 0 0
$$153$$ 16.0000 1.29352
$$154$$ 0 0
$$155$$ −5.65685 −0.454369
$$156$$ 0 0
$$157$$ 17.6569 1.40917 0.704585 0.709619i $$-0.251133\pi$$
0.704585 + 0.709619i $$0.251133\pi$$
$$158$$ 0 0
$$159$$ −18.4853 −1.46598
$$160$$ 0 0
$$161$$ 8.82843 0.695778
$$162$$ 0 0
$$163$$ 7.58579 0.594165 0.297082 0.954852i $$-0.403986\pi$$
0.297082 + 0.954852i $$0.403986\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.41421 0.496347 0.248173 0.968716i $$-0.420170\pi$$
0.248173 + 0.968716i $$0.420170\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 13.6569 1.04437
$$172$$ 0 0
$$173$$ −3.65685 −0.278025 −0.139013 0.990291i $$-0.544393\pi$$
−0.139013 + 0.990291i $$0.544393\pi$$
$$174$$ 0 0
$$175$$ 2.41421 0.182497
$$176$$ 0 0
$$177$$ −27.3137 −2.05302
$$178$$ 0 0
$$179$$ −17.7990 −1.33036 −0.665179 0.746684i $$-0.731645\pi$$
−0.665179 + 0.746684i $$0.731645\pi$$
$$180$$ 0 0
$$181$$ −25.8284 −1.91981 −0.959906 0.280322i $$-0.909559\pi$$
−0.959906 + 0.280322i $$0.909559\pi$$
$$182$$ 0 0
$$183$$ 2.41421 0.178464
$$184$$ 0 0
$$185$$ −4.00000 −0.294086
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ 16.8284 1.21766 0.608831 0.793300i $$-0.291639\pi$$
0.608831 + 0.793300i $$0.291639\pi$$
$$192$$ 0 0
$$193$$ 1.31371 0.0945628 0.0472814 0.998882i $$-0.484944\pi$$
0.0472814 + 0.998882i $$0.484944\pi$$
$$194$$ 0 0
$$195$$ 4.82843 0.345771
$$196$$ 0 0
$$197$$ −6.97056 −0.496632 −0.248316 0.968679i $$-0.579877\pi$$
−0.248316 + 0.968679i $$0.579877\pi$$
$$198$$ 0 0
$$199$$ −7.17157 −0.508379 −0.254190 0.967154i $$-0.581809\pi$$
−0.254190 + 0.967154i $$0.581809\pi$$
$$200$$ 0 0
$$201$$ −15.4853 −1.09225
$$202$$ 0 0
$$203$$ 4.82843 0.338889
$$204$$ 0 0
$$205$$ −9.48528 −0.662481
$$206$$ 0 0
$$207$$ 10.3431 0.718898
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 0 0
$$213$$ −9.65685 −0.661677
$$214$$ 0 0
$$215$$ 3.58579 0.244549
$$216$$ 0 0
$$217$$ −13.6569 −0.927088
$$218$$ 0 0
$$219$$ 9.65685 0.652550
$$220$$ 0 0
$$221$$ 11.3137 0.761042
$$222$$ 0 0
$$223$$ 7.92893 0.530961 0.265480 0.964116i $$-0.414469\pi$$
0.265480 + 0.964116i $$0.414469\pi$$
$$224$$ 0 0
$$225$$ 2.82843 0.188562
$$226$$ 0 0
$$227$$ 26.0711 1.73040 0.865199 0.501429i $$-0.167192\pi$$
0.865199 + 0.501429i $$0.167192\pi$$
$$228$$ 0 0
$$229$$ 11.4853 0.758969 0.379484 0.925198i $$-0.376101\pi$$
0.379484 + 0.925198i $$0.376101\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 4.34315 0.284529 0.142264 0.989829i $$-0.454562\pi$$
0.142264 + 0.989829i $$0.454562\pi$$
$$234$$ 0 0
$$235$$ 7.58579 0.494842
$$236$$ 0 0
$$237$$ −34.9706 −2.27158
$$238$$ 0 0
$$239$$ −24.1421 −1.56162 −0.780812 0.624765i $$-0.785195\pi$$
−0.780812 + 0.624765i $$0.785195\pi$$
$$240$$ 0 0
$$241$$ 16.1716 1.04170 0.520851 0.853647i $$-0.325615\pi$$
0.520851 + 0.853647i $$0.325615\pi$$
$$242$$ 0 0
$$243$$ −21.6569 −1.38929
$$244$$ 0 0
$$245$$ −1.17157 −0.0748490
$$246$$ 0 0
$$247$$ 9.65685 0.614451
$$248$$ 0 0
$$249$$ −32.1421 −2.03693
$$250$$ 0 0
$$251$$ 0.686292 0.0433183 0.0216592 0.999765i $$-0.493105\pi$$
0.0216592 + 0.999765i $$0.493105\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 13.6569 0.855225
$$256$$ 0 0
$$257$$ 12.9706 0.809081 0.404541 0.914520i $$-0.367431\pi$$
0.404541 + 0.914520i $$0.367431\pi$$
$$258$$ 0 0
$$259$$ −9.65685 −0.600048
$$260$$ 0 0
$$261$$ 5.65685 0.350150
$$262$$ 0 0
$$263$$ 24.6274 1.51859 0.759296 0.650746i $$-0.225544\pi$$
0.759296 + 0.650746i $$0.225544\pi$$
$$264$$ 0 0
$$265$$ −7.65685 −0.470357
$$266$$ 0 0
$$267$$ 6.41421 0.392543
$$268$$ 0 0
$$269$$ −16.3137 −0.994664 −0.497332 0.867560i $$-0.665687\pi$$
−0.497332 + 0.867560i $$0.665687\pi$$
$$270$$ 0 0
$$271$$ −11.3137 −0.687259 −0.343629 0.939105i $$-0.611656\pi$$
−0.343629 + 0.939105i $$0.611656\pi$$
$$272$$ 0 0
$$273$$ 11.6569 0.705505
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 28.6274 1.72005 0.860027 0.510248i $$-0.170446\pi$$
0.860027 + 0.510248i $$0.170446\pi$$
$$278$$ 0 0
$$279$$ −16.0000 −0.957895
$$280$$ 0 0
$$281$$ −5.31371 −0.316989 −0.158495 0.987360i $$-0.550664\pi$$
−0.158495 + 0.987360i $$0.550664\pi$$
$$282$$ 0 0
$$283$$ −20.2132 −1.20155 −0.600775 0.799418i $$-0.705141\pi$$
−0.600775 + 0.799418i $$0.705141\pi$$
$$284$$ 0 0
$$285$$ 11.6569 0.690492
$$286$$ 0 0
$$287$$ −22.8995 −1.35171
$$288$$ 0 0
$$289$$ 15.0000 0.882353
$$290$$ 0 0
$$291$$ −41.7990 −2.45030
$$292$$ 0 0
$$293$$ −19.3137 −1.12832 −0.564159 0.825666i $$-0.690800\pi$$
−0.564159 + 0.825666i $$0.690800\pi$$
$$294$$ 0 0
$$295$$ −11.3137 −0.658710
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 7.31371 0.422963
$$300$$ 0 0
$$301$$ 8.65685 0.498973
$$302$$ 0 0
$$303$$ 24.5563 1.41073
$$304$$ 0 0
$$305$$ 1.00000 0.0572598
$$306$$ 0 0
$$307$$ −25.3137 −1.44473 −0.722365 0.691512i $$-0.756945\pi$$
−0.722365 + 0.691512i $$0.756945\pi$$
$$308$$ 0 0
$$309$$ 32.1421 1.82850
$$310$$ 0 0
$$311$$ 12.8284 0.727433 0.363717 0.931510i $$-0.381508\pi$$
0.363717 + 0.931510i $$0.381508\pi$$
$$312$$ 0 0
$$313$$ −21.3137 −1.20472 −0.602361 0.798224i $$-0.705773\pi$$
−0.602361 + 0.798224i $$0.705773\pi$$
$$314$$ 0 0
$$315$$ 6.82843 0.384738
$$316$$ 0 0
$$317$$ −2.68629 −0.150877 −0.0754386 0.997150i $$-0.524036\pi$$
−0.0754386 + 0.997150i $$0.524036\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 36.7990 2.05392
$$322$$ 0 0
$$323$$ 27.3137 1.51978
$$324$$ 0 0
$$325$$ 2.00000 0.110940
$$326$$ 0 0
$$327$$ 40.2132 2.22380
$$328$$ 0 0
$$329$$ 18.3137 1.00967
$$330$$ 0 0
$$331$$ 4.82843 0.265394 0.132697 0.991157i $$-0.457636\pi$$
0.132697 + 0.991157i $$0.457636\pi$$
$$332$$ 0 0
$$333$$ −11.3137 −0.619987
$$334$$ 0 0
$$335$$ −6.41421 −0.350446
$$336$$ 0 0
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ 0 0
$$339$$ 41.7990 2.27021
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −19.7279 −1.06521
$$344$$ 0 0
$$345$$ 8.82843 0.475307
$$346$$ 0 0
$$347$$ 1.38478 0.0743387 0.0371693 0.999309i $$-0.488166\pi$$
0.0371693 + 0.999309i $$0.488166\pi$$
$$348$$ 0 0
$$349$$ −5.31371 −0.284436 −0.142218 0.989835i $$-0.545423\pi$$
−0.142218 + 0.989835i $$0.545423\pi$$
$$350$$ 0 0
$$351$$ −0.828427 −0.0442182
$$352$$ 0 0
$$353$$ −34.6274 −1.84303 −0.921516 0.388341i $$-0.873048\pi$$
−0.921516 + 0.388341i $$0.873048\pi$$
$$354$$ 0 0
$$355$$ −4.00000 −0.212298
$$356$$ 0 0
$$357$$ 32.9706 1.74499
$$358$$ 0 0
$$359$$ −3.17157 −0.167389 −0.0836946 0.996491i $$-0.526672\pi$$
−0.0836946 + 0.996491i $$0.526672\pi$$
$$360$$ 0 0
$$361$$ 4.31371 0.227037
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 4.00000 0.209370
$$366$$ 0 0
$$367$$ 5.24264 0.273664 0.136832 0.990594i $$-0.456308\pi$$
0.136832 + 0.990594i $$0.456308\pi$$
$$368$$ 0 0
$$369$$ −26.8284 −1.39663
$$370$$ 0 0
$$371$$ −18.4853 −0.959708
$$372$$ 0 0
$$373$$ −20.9706 −1.08581 −0.542907 0.839793i $$-0.682677\pi$$
−0.542907 + 0.839793i $$0.682677\pi$$
$$374$$ 0 0
$$375$$ 2.41421 0.124669
$$376$$ 0 0
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ −20.1421 −1.03463 −0.517316 0.855794i $$-0.673069\pi$$
−0.517316 + 0.855794i $$0.673069\pi$$
$$380$$ 0 0
$$381$$ −10.6569 −0.545967
$$382$$ 0 0
$$383$$ 30.9706 1.58252 0.791261 0.611479i $$-0.209425\pi$$
0.791261 + 0.611479i $$0.209425\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 10.1421 0.515554
$$388$$ 0 0
$$389$$ 26.3137 1.33416 0.667079 0.744987i $$-0.267544\pi$$
0.667079 + 0.744987i $$0.267544\pi$$
$$390$$ 0 0
$$391$$ 20.6863 1.04615
$$392$$ 0 0
$$393$$ −6.00000 −0.302660
$$394$$ 0 0
$$395$$ −14.4853 −0.728834
$$396$$ 0 0
$$397$$ −24.0000 −1.20453 −0.602263 0.798298i $$-0.705734\pi$$
−0.602263 + 0.798298i $$0.705734\pi$$
$$398$$ 0 0
$$399$$ 28.1421 1.40887
$$400$$ 0 0
$$401$$ 6.31371 0.315292 0.157646 0.987496i $$-0.449610\pi$$
0.157646 + 0.987496i $$0.449610\pi$$
$$402$$ 0 0
$$403$$ −11.3137 −0.563576
$$404$$ 0 0
$$405$$ −9.48528 −0.471327
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −39.7696 −1.96648 −0.983239 0.182322i $$-0.941639\pi$$
−0.983239 + 0.182322i $$0.941639\pi$$
$$410$$ 0 0
$$411$$ 32.1421 1.58545
$$412$$ 0 0
$$413$$ −27.3137 −1.34402
$$414$$ 0 0
$$415$$ −13.3137 −0.653544
$$416$$ 0 0
$$417$$ 28.9706 1.41869
$$418$$ 0 0
$$419$$ −17.5147 −0.855650 −0.427825 0.903862i $$-0.640720\pi$$
−0.427825 + 0.903862i $$0.640720\pi$$
$$420$$ 0 0
$$421$$ −7.68629 −0.374607 −0.187303 0.982302i $$-0.559975\pi$$
−0.187303 + 0.982302i $$0.559975\pi$$
$$422$$ 0 0
$$423$$ 21.4558 1.04322
$$424$$ 0 0
$$425$$ 5.65685 0.274398
$$426$$ 0 0
$$427$$ 2.41421 0.116832
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 17.5147 0.843654 0.421827 0.906676i $$-0.361389\pi$$
0.421827 + 0.906676i $$0.361389\pi$$
$$432$$ 0 0
$$433$$ −28.9706 −1.39224 −0.696118 0.717927i $$-0.745091\pi$$
−0.696118 + 0.717927i $$0.745091\pi$$
$$434$$ 0 0
$$435$$ 4.82843 0.231505
$$436$$ 0 0
$$437$$ 17.6569 0.844642
$$438$$ 0 0
$$439$$ −24.2843 −1.15903 −0.579513 0.814963i $$-0.696757\pi$$
−0.579513 + 0.814963i $$0.696757\pi$$
$$440$$ 0 0
$$441$$ −3.31371 −0.157796
$$442$$ 0 0
$$443$$ 9.24264 0.439131 0.219566 0.975598i $$-0.429536\pi$$
0.219566 + 0.975598i $$0.429536\pi$$
$$444$$ 0 0
$$445$$ 2.65685 0.125947
$$446$$ 0 0
$$447$$ −41.0416 −1.94120
$$448$$ 0 0
$$449$$ −40.4558 −1.90923 −0.954615 0.297844i $$-0.903733\pi$$
−0.954615 + 0.297844i $$0.903733\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 13.6569 0.641655
$$454$$ 0 0
$$455$$ 4.82843 0.226360
$$456$$ 0 0
$$457$$ 3.31371 0.155009 0.0775044 0.996992i $$-0.475305\pi$$
0.0775044 + 0.996992i $$0.475305\pi$$
$$458$$ 0 0
$$459$$ −2.34315 −0.109369
$$460$$ 0 0
$$461$$ −9.14214 −0.425792 −0.212896 0.977075i $$-0.568290\pi$$
−0.212896 + 0.977075i $$0.568290\pi$$
$$462$$ 0 0
$$463$$ 15.2426 0.708386 0.354193 0.935172i $$-0.384756\pi$$
0.354193 + 0.935172i $$0.384756\pi$$
$$464$$ 0 0
$$465$$ −13.6569 −0.633321
$$466$$ 0 0
$$467$$ 42.5563 1.96927 0.984636 0.174617i $$-0.0558686\pi$$
0.984636 + 0.174617i $$0.0558686\pi$$
$$468$$ 0 0
$$469$$ −15.4853 −0.715044
$$470$$ 0 0
$$471$$ 42.6274 1.96417
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 4.82843 0.221543
$$476$$ 0 0
$$477$$ −21.6569 −0.991599
$$478$$ 0 0
$$479$$ −1.79899 −0.0821979 −0.0410990 0.999155i $$-0.513086\pi$$
−0.0410990 + 0.999155i $$0.513086\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 0 0
$$483$$ 21.3137 0.969807
$$484$$ 0 0
$$485$$ −17.3137 −0.786175
$$486$$ 0 0
$$487$$ 18.6863 0.846757 0.423378 0.905953i $$-0.360844\pi$$
0.423378 + 0.905953i $$0.360844\pi$$
$$488$$ 0 0
$$489$$ 18.3137 0.828175
$$490$$ 0 0
$$491$$ 17.5147 0.790428 0.395214 0.918589i $$-0.370670\pi$$
0.395214 + 0.918589i $$0.370670\pi$$
$$492$$ 0 0
$$493$$ 11.3137 0.509544
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −9.65685 −0.433169
$$498$$ 0 0
$$499$$ −37.7990 −1.69212 −0.846058 0.533092i $$-0.821030\pi$$
−0.846058 + 0.533092i $$0.821030\pi$$
$$500$$ 0 0
$$501$$ 15.4853 0.691831
$$502$$ 0 0
$$503$$ 9.38478 0.418446 0.209223 0.977868i $$-0.432906\pi$$
0.209223 + 0.977868i $$0.432906\pi$$
$$504$$ 0 0
$$505$$ 10.1716 0.452629
$$506$$ 0 0
$$507$$ −21.7279 −0.964971
$$508$$ 0 0
$$509$$ 11.4853 0.509076 0.254538 0.967063i $$-0.418077\pi$$
0.254538 + 0.967063i $$0.418077\pi$$
$$510$$ 0 0
$$511$$ 9.65685 0.427194
$$512$$ 0 0
$$513$$ −2.00000 −0.0883022
$$514$$ 0 0
$$515$$ 13.3137 0.586672
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −8.82843 −0.387525
$$520$$ 0 0
$$521$$ 6.02944 0.264154 0.132077 0.991239i $$-0.457835\pi$$
0.132077 + 0.991239i $$0.457835\pi$$
$$522$$ 0 0
$$523$$ −13.3137 −0.582168 −0.291084 0.956698i $$-0.594016\pi$$
−0.291084 + 0.956698i $$0.594016\pi$$
$$524$$ 0 0
$$525$$ 5.82843 0.254373
$$526$$ 0 0
$$527$$ −32.0000 −1.39394
$$528$$ 0 0
$$529$$ −9.62742 −0.418583
$$530$$ 0 0
$$531$$ −32.0000 −1.38868
$$532$$ 0 0
$$533$$ −18.9706 −0.821706
$$534$$ 0 0
$$535$$ 15.2426 0.658997
$$536$$ 0 0
$$537$$ −42.9706 −1.85432
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −5.14214 −0.221078 −0.110539 0.993872i $$-0.535258\pi$$
−0.110539 + 0.993872i $$0.535258\pi$$
$$542$$ 0 0
$$543$$ −62.3553 −2.67592
$$544$$ 0 0
$$545$$ 16.6569 0.713501
$$546$$ 0 0
$$547$$ 26.0000 1.11168 0.555840 0.831289i $$-0.312397\pi$$
0.555840 + 0.831289i $$0.312397\pi$$
$$548$$ 0 0
$$549$$ 2.82843 0.120714
$$550$$ 0 0
$$551$$ 9.65685 0.411396
$$552$$ 0 0
$$553$$ −34.9706 −1.48710
$$554$$ 0 0
$$555$$ −9.65685 −0.409911
$$556$$ 0 0
$$557$$ 22.6274 0.958754 0.479377 0.877609i $$-0.340863\pi$$
0.479377 + 0.877609i $$0.340863\pi$$
$$558$$ 0 0
$$559$$ 7.17157 0.303325
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 2.89949 0.122199 0.0610996 0.998132i $$-0.480539\pi$$
0.0610996 + 0.998132i $$0.480539\pi$$
$$564$$ 0 0
$$565$$ 17.3137 0.728393
$$566$$ 0 0
$$567$$ −22.8995 −0.961688
$$568$$ 0 0
$$569$$ −16.4558 −0.689865 −0.344932 0.938628i $$-0.612098\pi$$
−0.344932 + 0.938628i $$0.612098\pi$$
$$570$$ 0 0
$$571$$ 23.4558 0.981597 0.490798 0.871273i $$-0.336705\pi$$
0.490798 + 0.871273i $$0.336705\pi$$
$$572$$ 0 0
$$573$$ 40.6274 1.69723
$$574$$ 0 0
$$575$$ 3.65685 0.152501
$$576$$ 0 0
$$577$$ −33.9411 −1.41299 −0.706494 0.707719i $$-0.749724\pi$$
−0.706494 + 0.707719i $$0.749724\pi$$
$$578$$ 0 0
$$579$$ 3.17157 0.131806
$$580$$ 0 0
$$581$$ −32.1421 −1.33348
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 5.65685 0.233882
$$586$$ 0 0
$$587$$ 36.6985 1.51471 0.757354 0.653004i $$-0.226492\pi$$
0.757354 + 0.653004i $$0.226492\pi$$
$$588$$ 0 0
$$589$$ −27.3137 −1.12544
$$590$$ 0 0
$$591$$ −16.8284 −0.692229
$$592$$ 0 0
$$593$$ −23.6569 −0.971471 −0.485735 0.874106i $$-0.661448\pi$$
−0.485735 + 0.874106i $$0.661448\pi$$
$$594$$ 0 0
$$595$$ 13.6569 0.559876
$$596$$ 0 0
$$597$$ −17.3137 −0.708603
$$598$$ 0 0
$$599$$ 15.8579 0.647935 0.323967 0.946068i $$-0.394983\pi$$
0.323967 + 0.946068i $$0.394983\pi$$
$$600$$ 0 0
$$601$$ 39.9411 1.62923 0.814616 0.580000i $$-0.196948\pi$$
0.814616 + 0.580000i $$0.196948\pi$$
$$602$$ 0 0
$$603$$ −18.1421 −0.738805
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 33.5980 1.36370 0.681850 0.731492i $$-0.261176\pi$$
0.681850 + 0.731492i $$0.261176\pi$$
$$608$$ 0 0
$$609$$ 11.6569 0.472360
$$610$$ 0 0
$$611$$ 15.1716 0.613776
$$612$$ 0 0
$$613$$ 14.9706 0.604655 0.302328 0.953204i $$-0.402236\pi$$
0.302328 + 0.953204i $$0.402236\pi$$
$$614$$ 0 0
$$615$$ −22.8995 −0.923397
$$616$$ 0 0
$$617$$ −40.0000 −1.61034 −0.805170 0.593045i $$-0.797926\pi$$
−0.805170 + 0.593045i $$0.797926\pi$$
$$618$$ 0 0
$$619$$ 7.31371 0.293963 0.146981 0.989139i $$-0.453044\pi$$
0.146981 + 0.989139i $$0.453044\pi$$
$$620$$ 0 0
$$621$$ −1.51472 −0.0607836
$$622$$ 0 0
$$623$$ 6.41421 0.256980
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −22.6274 −0.902214
$$630$$ 0 0
$$631$$ 34.3431 1.36718 0.683590 0.729867i $$-0.260418\pi$$
0.683590 + 0.729867i $$0.260418\pi$$
$$632$$ 0 0
$$633$$ −19.3137 −0.767651
$$634$$ 0 0
$$635$$ −4.41421 −0.175173
$$636$$ 0 0
$$637$$ −2.34315 −0.0928388
$$638$$ 0 0
$$639$$ −11.3137 −0.447563
$$640$$ 0 0
$$641$$ −48.6274 −1.92067 −0.960334 0.278853i $$-0.910046\pi$$
−0.960334 + 0.278853i $$0.910046\pi$$
$$642$$ 0 0
$$643$$ 0.272078 0.0107297 0.00536485 0.999986i $$-0.498292\pi$$
0.00536485 + 0.999986i $$0.498292\pi$$
$$644$$ 0 0
$$645$$ 8.65685 0.340863
$$646$$ 0 0
$$647$$ −29.8701 −1.17431 −0.587157 0.809473i $$-0.699753\pi$$
−0.587157 + 0.809473i $$0.699753\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −32.9706 −1.29222
$$652$$ 0 0
$$653$$ −7.65685 −0.299636 −0.149818 0.988714i $$-0.547869\pi$$
−0.149818 + 0.988714i $$0.547869\pi$$
$$654$$ 0 0
$$655$$ −2.48528 −0.0971080
$$656$$ 0 0
$$657$$ 11.3137 0.441390
$$658$$ 0 0
$$659$$ −23.1716 −0.902636 −0.451318 0.892363i $$-0.649046\pi$$
−0.451318 + 0.892363i $$0.649046\pi$$
$$660$$ 0 0
$$661$$ 35.3431 1.37469 0.687345 0.726332i $$-0.258776\pi$$
0.687345 + 0.726332i $$0.258776\pi$$
$$662$$ 0 0
$$663$$ 27.3137 1.06078
$$664$$ 0 0
$$665$$ 11.6569 0.452033
$$666$$ 0 0
$$667$$ 7.31371 0.283188
$$668$$ 0 0
$$669$$ 19.1421 0.740078
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 16.9706 0.654167 0.327084 0.944995i $$-0.393934\pi$$
0.327084 + 0.944995i $$0.393934\pi$$
$$674$$ 0 0
$$675$$ −0.414214 −0.0159431
$$676$$ 0 0
$$677$$ −8.68629 −0.333841 −0.166921 0.985970i $$-0.553382\pi$$
−0.166921 + 0.985970i $$0.553382\pi$$
$$678$$ 0 0
$$679$$ −41.7990 −1.60410
$$680$$ 0 0
$$681$$ 62.9411 2.41191
$$682$$ 0 0
$$683$$ 19.8701 0.760307 0.380153 0.924923i $$-0.375871\pi$$
0.380153 + 0.924923i $$0.375871\pi$$
$$684$$ 0 0
$$685$$ 13.3137 0.508691
$$686$$ 0 0
$$687$$ 27.7279 1.05789
$$688$$ 0 0
$$689$$ −15.3137 −0.583406
$$690$$ 0 0
$$691$$ −14.4853 −0.551046 −0.275523 0.961294i $$-0.588851\pi$$
−0.275523 + 0.961294i $$0.588851\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ −53.6569 −2.03240
$$698$$ 0 0
$$699$$ 10.4853 0.396590
$$700$$ 0 0
$$701$$ 39.9411 1.50856 0.754278 0.656555i $$-0.227987\pi$$
0.754278 + 0.656555i $$0.227987\pi$$
$$702$$ 0 0
$$703$$ −19.3137 −0.728430
$$704$$ 0 0
$$705$$ 18.3137 0.689734
$$706$$ 0 0
$$707$$ 24.5563 0.923537
$$708$$ 0 0
$$709$$ −35.1421 −1.31979 −0.659895 0.751358i $$-0.729399\pi$$
−0.659895 + 0.751358i $$0.729399\pi$$
$$710$$ 0 0
$$711$$ −40.9706 −1.53652
$$712$$ 0 0
$$713$$ −20.6863 −0.774708
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −58.2843 −2.17667
$$718$$ 0 0
$$719$$ −31.1716 −1.16250 −0.581252 0.813724i $$-0.697437\pi$$
−0.581252 + 0.813724i $$0.697437\pi$$
$$720$$ 0 0
$$721$$ 32.1421 1.19704
$$722$$ 0 0
$$723$$ 39.0416 1.45197
$$724$$ 0 0
$$725$$ 2.00000 0.0742781
$$726$$ 0 0
$$727$$ 52.0711 1.93121 0.965605 0.260015i $$-0.0837276\pi$$
0.965605 + 0.260015i $$0.0837276\pi$$
$$728$$ 0 0
$$729$$ −23.8284 −0.882534
$$730$$ 0 0
$$731$$ 20.2843 0.750241
$$732$$ 0 0
$$733$$ 40.2843 1.48793 0.743967 0.668217i $$-0.232942\pi$$
0.743967 + 0.668217i $$0.232942\pi$$
$$734$$ 0 0
$$735$$ −2.82843 −0.104328
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −33.5147 −1.23286 −0.616429 0.787410i $$-0.711421\pi$$
−0.616429 + 0.787410i $$0.711421\pi$$
$$740$$ 0 0
$$741$$ 23.3137 0.856450
$$742$$ 0 0
$$743$$ 32.4142 1.18916 0.594581 0.804036i $$-0.297318\pi$$
0.594581 + 0.804036i $$0.297318\pi$$
$$744$$ 0 0
$$745$$ −17.0000 −0.622832
$$746$$ 0 0
$$747$$ −37.6569 −1.37779
$$748$$ 0 0
$$749$$ 36.7990 1.34461
$$750$$ 0 0
$$751$$ 4.82843 0.176192 0.0880959 0.996112i $$-0.471922\pi$$
0.0880959 + 0.996112i $$0.471922\pi$$
$$752$$ 0 0
$$753$$ 1.65685 0.0603791
$$754$$ 0 0
$$755$$ 5.65685 0.205874
$$756$$ 0 0
$$757$$ −30.9706 −1.12564 −0.562822 0.826578i $$-0.690284\pi$$
−0.562822 + 0.826578i $$0.690284\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ 40.2132 1.45582
$$764$$ 0 0
$$765$$ 16.0000 0.578481
$$766$$ 0 0
$$767$$ −22.6274 −0.817029
$$768$$ 0 0
$$769$$ −18.0000 −0.649097 −0.324548 0.945869i $$-0.605212\pi$$
−0.324548 + 0.945869i $$0.605212\pi$$
$$770$$ 0 0
$$771$$ 31.3137 1.12774
$$772$$ 0 0
$$773$$ −28.9706 −1.04200 −0.520999 0.853557i $$-0.674441\pi$$
−0.520999 + 0.853557i $$0.674441\pi$$
$$774$$ 0 0
$$775$$ −5.65685 −0.203200
$$776$$ 0 0
$$777$$ −23.3137 −0.836375
$$778$$ 0 0
$$779$$ −45.7990 −1.64092
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −0.828427 −0.0296056
$$784$$ 0 0
$$785$$ 17.6569 0.630200
$$786$$ 0 0
$$787$$ 0.213203 0.00759988 0.00379994 0.999993i $$-0.498790\pi$$
0.00379994 + 0.999993i $$0.498790\pi$$
$$788$$ 0 0
$$789$$ 59.4558 2.11668
$$790$$ 0 0
$$791$$ 41.7990 1.48620
$$792$$ 0 0
$$793$$ 2.00000 0.0710221
$$794$$ 0 0
$$795$$ −18.4853 −0.655605
$$796$$ 0 0
$$797$$ 34.3431 1.21650 0.608248 0.793747i $$-0.291872\pi$$
0.608248 + 0.793747i $$0.291872\pi$$
$$798$$ 0 0
$$799$$ 42.9117 1.51811
$$800$$ 0 0
$$801$$ 7.51472 0.265520
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 8.82843 0.311161
$$806$$ 0 0
$$807$$ −39.3848 −1.38641
$$808$$ 0 0
$$809$$ 24.6274 0.865854 0.432927 0.901429i $$-0.357481\pi$$
0.432927 + 0.901429i $$0.357481\pi$$
$$810$$ 0 0
$$811$$ −5.65685 −0.198639 −0.0993195 0.995056i $$-0.531667\pi$$
−0.0993195 + 0.995056i $$0.531667\pi$$
$$812$$ 0 0
$$813$$ −27.3137 −0.957934
$$814$$ 0 0
$$815$$ 7.58579 0.265719
$$816$$ 0 0
$$817$$ 17.3137 0.605730
$$818$$ 0 0
$$819$$ 13.6569 0.477209
$$820$$ 0 0
$$821$$ −3.97056 −0.138574 −0.0692868 0.997597i $$-0.522072\pi$$
−0.0692868 + 0.997597i $$0.522072\pi$$
$$822$$ 0 0
$$823$$ −1.92893 −0.0672383 −0.0336192 0.999435i $$-0.510703\pi$$
−0.0336192 + 0.999435i $$0.510703\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −32.3553 −1.12511 −0.562553 0.826761i $$-0.690181\pi$$
−0.562553 + 0.826761i $$0.690181\pi$$
$$828$$ 0 0
$$829$$ 32.9411 1.14409 0.572046 0.820221i $$-0.306150\pi$$
0.572046 + 0.820221i $$0.306150\pi$$
$$830$$ 0 0
$$831$$ 69.1127 2.39749
$$832$$ 0 0
$$833$$ −6.62742 −0.229626
$$834$$ 0 0
$$835$$ 6.41421 0.221973
$$836$$ 0 0
$$837$$ 2.34315 0.0809910
$$838$$ 0 0
$$839$$ 41.7990 1.44306 0.721531 0.692382i $$-0.243439\pi$$
0.721531 + 0.692382i $$0.243439\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −12.8284 −0.441834
$$844$$ 0 0
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −48.7990 −1.67478
$$850$$ 0 0
$$851$$ −14.6274 −0.501421
$$852$$ 0 0
$$853$$ −29.6569 −1.01543 −0.507716 0.861525i $$-0.669510\pi$$
−0.507716 + 0.861525i $$0.669510\pi$$
$$854$$ 0 0
$$855$$ 13.6569 0.467055
$$856$$ 0 0
$$857$$ −10.9706 −0.374747 −0.187374 0.982289i $$-0.559998\pi$$
−0.187374 + 0.982289i $$0.559998\pi$$
$$858$$ 0 0
$$859$$ −7.85786 −0.268107 −0.134053 0.990974i $$-0.542799\pi$$
−0.134053 + 0.990974i $$0.542799\pi$$
$$860$$ 0 0
$$861$$ −55.2843 −1.88408
$$862$$ 0 0
$$863$$ −7.87006 −0.267900 −0.133950 0.990988i $$-0.542766\pi$$
−0.133950 + 0.990988i $$0.542766\pi$$
$$864$$ 0 0
$$865$$ −3.65685 −0.124337
$$866$$ 0 0
$$867$$ 36.2132 1.22986
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −12.8284 −0.434675
$$872$$ 0 0
$$873$$ −48.9706 −1.65740
$$874$$ 0 0
$$875$$ 2.41421 0.0816153
$$876$$ 0 0
$$877$$ −47.3137 −1.59767 −0.798835 0.601550i $$-0.794550\pi$$
−0.798835 + 0.601550i $$0.794550\pi$$
$$878$$ 0 0
$$879$$ −46.6274 −1.57270
$$880$$ 0 0
$$881$$ 2.51472 0.0847230 0.0423615 0.999102i $$-0.486512\pi$$
0.0423615 + 0.999102i $$0.486512\pi$$
$$882$$ 0 0
$$883$$ −32.3431 −1.08843 −0.544217 0.838945i $$-0.683173\pi$$
−0.544217 + 0.838945i $$0.683173\pi$$
$$884$$ 0 0
$$885$$ −27.3137 −0.918140
$$886$$ 0 0
$$887$$ −0.615224 −0.0206572 −0.0103286 0.999947i $$-0.503288\pi$$
−0.0103286 + 0.999947i $$0.503288\pi$$
$$888$$ 0 0
$$889$$ −10.6569 −0.357419
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 36.6274 1.22569
$$894$$ 0 0
$$895$$ −17.7990 −0.594955
$$896$$ 0 0
$$897$$ 17.6569 0.589545
$$898$$ 0 0
$$899$$ −11.3137 −0.377333
$$900$$ 0 0
$$901$$ −43.3137 −1.44299
$$902$$ 0 0
$$903$$ 20.8995 0.695492
$$904$$ 0 0
$$905$$ −25.8284 −0.858566
$$906$$ 0 0
$$907$$ 40.6985 1.35137 0.675686 0.737190i $$-0.263848\pi$$
0.675686 + 0.737190i $$0.263848\pi$$
$$908$$ 0 0
$$909$$ 28.7696 0.954226
$$910$$ 0 0
$$911$$ −27.1716 −0.900234 −0.450117 0.892969i $$-0.648618\pi$$
−0.450117 + 0.892969i $$0.648618\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 2.41421 0.0798114
$$916$$ 0 0
$$917$$ −6.00000 −0.198137
$$918$$ 0 0
$$919$$ 32.4264 1.06965 0.534824 0.844963i $$-0.320378\pi$$
0.534824 + 0.844963i $$0.320378\pi$$
$$920$$ 0 0
$$921$$ −61.1127 −2.01373
$$922$$ 0 0
$$923$$ −8.00000 −0.263323
$$924$$ 0 0
$$925$$ −4.00000 −0.131519
$$926$$ 0 0
$$927$$ 37.6569 1.23681
$$928$$ 0 0
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ −5.65685 −0.185396
$$932$$ 0 0
$$933$$ 30.9706 1.01393
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −30.0000 −0.980057 −0.490029 0.871706i $$-0.663014\pi$$
−0.490029 + 0.871706i $$0.663014\pi$$
$$938$$ 0 0
$$939$$ −51.4558 −1.67920
$$940$$ 0 0
$$941$$ 51.8284 1.68956 0.844779 0.535115i $$-0.179732\pi$$
0.844779 + 0.535115i $$0.179732\pi$$
$$942$$ 0 0
$$943$$ −34.6863 −1.12954
$$944$$ 0 0
$$945$$ −1.00000 −0.0325300
$$946$$ 0 0
$$947$$ 10.0000 0.324956 0.162478 0.986712i $$-0.448051\pi$$
0.162478 + 0.986712i $$0.448051\pi$$
$$948$$ 0 0
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ −6.48528 −0.210300
$$952$$ 0 0
$$953$$ 24.2843 0.786645 0.393322 0.919401i $$-0.371326\pi$$
0.393322 + 0.919401i $$0.371326\pi$$
$$954$$ 0 0
$$955$$ 16.8284 0.544555
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 32.1421 1.03792
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 0 0
$$963$$ 43.1127 1.38929
$$964$$ 0 0
$$965$$ 1.31371 0.0422898
$$966$$ 0 0
$$967$$ 41.5980 1.33770 0.668850 0.743397i $$-0.266787\pi$$
0.668850 + 0.743397i $$0.266787\pi$$
$$968$$ 0 0
$$969$$ 65.9411 2.11833
$$970$$ 0 0
$$971$$ −3.71573 −0.119243 −0.0596217 0.998221i $$-0.518989\pi$$
−0.0596217 + 0.998221i $$0.518989\pi$$
$$972$$ 0 0
$$973$$ 28.9706 0.928754
$$974$$ 0 0
$$975$$ 4.82843 0.154633
$$976$$ 0 0
$$977$$ 2.62742 0.0840585 0.0420293 0.999116i $$-0.486618\pi$$
0.0420293 + 0.999116i $$0.486618\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 47.1127 1.50419
$$982$$ 0 0
$$983$$ −5.92893 −0.189104 −0.0945518 0.995520i $$-0.530142\pi$$
−0.0945518 + 0.995520i $$0.530142\pi$$
$$984$$ 0 0
$$985$$ −6.97056 −0.222101
$$986$$ 0 0
$$987$$ 44.2132 1.40732
$$988$$ 0 0
$$989$$ 13.1127 0.416960
$$990$$ 0 0
$$991$$ 58.4853 1.85785 0.928923 0.370273i $$-0.120736\pi$$
0.928923 + 0.370273i $$0.120736\pi$$
$$992$$ 0 0
$$993$$ 11.6569 0.369919
$$994$$ 0 0
$$995$$ −7.17157 −0.227354
$$996$$ 0 0
$$997$$ −29.3137 −0.928374 −0.464187 0.885737i $$-0.653654\pi$$
−0.464187 + 0.885737i $$0.653654\pi$$
$$998$$ 0 0
$$999$$ 1.65685 0.0524205
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 4840.2.a.p.1.2 yes 2
4.3 odd 2 9680.2.a.bj.1.1 2
11.10 odd 2 4840.2.a.o.1.2 2
44.43 even 2 9680.2.a.bk.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.o.1.2 2 11.10 odd 2
4840.2.a.p.1.2 yes 2 1.1 even 1 trivial
9680.2.a.bj.1.1 2 4.3 odd 2
9680.2.a.bk.1.1 2 44.43 even 2