# Properties

 Label 9280.2.a.w.1.1 Level $9280$ Weight $2$ Character 9280.1 Self dual yes Analytic conductor $74.101$ 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] = [9280,2,Mod(1,9280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9280, 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("9280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9280 = 2^{6} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9280.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: $$74.1011730757$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 9280.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} -1.00000 q^{5} -0.828427 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} -1.00000 q^{5} -0.828427 q^{7} +1.00000 q^{9} -4.82843 q^{11} +2.00000 q^{13} +2.00000 q^{15} -2.82843 q^{17} +0.828427 q^{19} +1.65685 q^{21} +8.82843 q^{23} +1.00000 q^{25} +4.00000 q^{27} -1.00000 q^{29} +10.4853 q^{31} +9.65685 q^{33} +0.828427 q^{35} -8.48528 q^{37} -4.00000 q^{39} -6.00000 q^{41} -6.00000 q^{43} -1.00000 q^{45} +0.343146 q^{47} -6.31371 q^{49} +5.65685 q^{51} -7.65685 q^{53} +4.82843 q^{55} -1.65685 q^{57} -7.65685 q^{61} -0.828427 q^{63} -2.00000 q^{65} -10.4853 q^{67} -17.6569 q^{69} -7.31371 q^{71} -8.48528 q^{73} -2.00000 q^{75} +4.00000 q^{77} -14.4853 q^{79} -11.0000 q^{81} +12.8284 q^{83} +2.82843 q^{85} +2.00000 q^{87} +3.65685 q^{89} -1.65685 q^{91} -20.9706 q^{93} -0.828427 q^{95} +4.48528 q^{97} -4.82843 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^9 $$2 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{15} - 4 q^{19} - 8 q^{21} + 12 q^{23} + 2 q^{25} + 8 q^{27} - 2 q^{29} + 4 q^{31} + 8 q^{33} - 4 q^{35} - 8 q^{39} - 12 q^{41} - 12 q^{43} - 2 q^{45} + 12 q^{47} + 10 q^{49} - 4 q^{53} + 4 q^{55} + 8 q^{57} - 4 q^{61} + 4 q^{63} - 4 q^{65} - 4 q^{67} - 24 q^{69} + 8 q^{71} - 4 q^{75} + 8 q^{77} - 12 q^{79} - 22 q^{81} + 20 q^{83} + 4 q^{87} - 4 q^{89} + 8 q^{91} - 8 q^{93} + 4 q^{95} - 8 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^13 + 4 * q^15 - 4 * q^19 - 8 * q^21 + 12 * q^23 + 2 * q^25 + 8 * q^27 - 2 * q^29 + 4 * q^31 + 8 * q^33 - 4 * q^35 - 8 * q^39 - 12 * q^41 - 12 * q^43 - 2 * q^45 + 12 * q^47 + 10 * q^49 - 4 * q^53 + 4 * q^55 + 8 * q^57 - 4 * q^61 + 4 * q^63 - 4 * q^65 - 4 * q^67 - 24 * q^69 + 8 * q^71 - 4 * q^75 + 8 * q^77 - 12 * q^79 - 22 * q^81 + 20 * q^83 + 4 * q^87 - 4 * q^89 + 8 * q^91 - 8 * q^93 + 4 * q^95 - 8 * q^97 - 4 * q^99

## 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
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −0.828427 −0.313116 −0.156558 0.987669i $$-0.550040\pi$$
−0.156558 + 0.987669i $$0.550040\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −4.82843 −1.45583 −0.727913 0.685670i $$-0.759509\pi$$
−0.727913 + 0.685670i $$0.759509\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 0 0
$$17$$ −2.82843 −0.685994 −0.342997 0.939336i $$-0.611442\pi$$
−0.342997 + 0.939336i $$0.611442\pi$$
$$18$$ 0 0
$$19$$ 0.828427 0.190054 0.0950271 0.995475i $$-0.469706\pi$$
0.0950271 + 0.995475i $$0.469706\pi$$
$$20$$ 0 0
$$21$$ 1.65685 0.361555
$$22$$ 0 0
$$23$$ 8.82843 1.84085 0.920427 0.390914i $$-0.127841\pi$$
0.920427 + 0.390914i $$0.127841\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 10.4853 1.88321 0.941606 0.336717i $$-0.109316\pi$$
0.941606 + 0.336717i $$0.109316\pi$$
$$32$$ 0 0
$$33$$ 9.65685 1.68104
$$34$$ 0 0
$$35$$ 0.828427 0.140030
$$36$$ 0 0
$$37$$ −8.48528 −1.39497 −0.697486 0.716599i $$-0.745698\pi$$
−0.697486 + 0.716599i $$0.745698\pi$$
$$38$$ 0 0
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 0.343146 0.0500530 0.0250265 0.999687i $$-0.492033\pi$$
0.0250265 + 0.999687i $$0.492033\pi$$
$$48$$ 0 0
$$49$$ −6.31371 −0.901958
$$50$$ 0 0
$$51$$ 5.65685 0.792118
$$52$$ 0 0
$$53$$ −7.65685 −1.05175 −0.525875 0.850562i $$-0.676262\pi$$
−0.525875 + 0.850562i $$0.676262\pi$$
$$54$$ 0 0
$$55$$ 4.82843 0.651065
$$56$$ 0 0
$$57$$ −1.65685 −0.219456
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −7.65685 −0.980360 −0.490180 0.871621i $$-0.663069\pi$$
−0.490180 + 0.871621i $$0.663069\pi$$
$$62$$ 0 0
$$63$$ −0.828427 −0.104372
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ −10.4853 −1.28098 −0.640490 0.767966i $$-0.721269\pi$$
−0.640490 + 0.767966i $$0.721269\pi$$
$$68$$ 0 0
$$69$$ −17.6569 −2.12564
$$70$$ 0 0
$$71$$ −7.31371 −0.867978 −0.433989 0.900918i $$-0.642894\pi$$
−0.433989 + 0.900918i $$0.642894\pi$$
$$72$$ 0 0
$$73$$ −8.48528 −0.993127 −0.496564 0.868000i $$-0.665405\pi$$
−0.496564 + 0.868000i $$0.665405\pi$$
$$74$$ 0 0
$$75$$ −2.00000 −0.230940
$$76$$ 0 0
$$77$$ 4.00000 0.455842
$$78$$ 0 0
$$79$$ −14.4853 −1.62972 −0.814861 0.579657i $$-0.803187\pi$$
−0.814861 + 0.579657i $$0.803187\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 12.8284 1.40810 0.704051 0.710149i $$-0.251372\pi$$
0.704051 + 0.710149i $$0.251372\pi$$
$$84$$ 0 0
$$85$$ 2.82843 0.306786
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ 3.65685 0.387626 0.193813 0.981039i $$-0.437915\pi$$
0.193813 + 0.981039i $$0.437915\pi$$
$$90$$ 0 0
$$91$$ −1.65685 −0.173686
$$92$$ 0 0
$$93$$ −20.9706 −2.17455
$$94$$ 0 0
$$95$$ −0.828427 −0.0849948
$$96$$ 0 0
$$97$$ 4.48528 0.455411 0.227706 0.973730i $$-0.426878\pi$$
0.227706 + 0.973730i $$0.426878\pi$$
$$98$$ 0 0
$$99$$ −4.82843 −0.485275
$$100$$ 0 0
$$101$$ −4.34315 −0.432159 −0.216080 0.976376i $$-0.569327\pi$$
−0.216080 + 0.976376i $$0.569327\pi$$
$$102$$ 0 0
$$103$$ 12.1421 1.19640 0.598200 0.801347i $$-0.295883\pi$$
0.598200 + 0.801347i $$0.295883\pi$$
$$104$$ 0 0
$$105$$ −1.65685 −0.161692
$$106$$ 0 0
$$107$$ −8.14214 −0.787130 −0.393565 0.919297i $$-0.628758\pi$$
−0.393565 + 0.919297i $$0.628758\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 16.9706 1.61077
$$112$$ 0 0
$$113$$ 2.82843 0.266076 0.133038 0.991111i $$-0.457527\pi$$
0.133038 + 0.991111i $$0.457527\pi$$
$$114$$ 0 0
$$115$$ −8.82843 −0.823255
$$116$$ 0 0
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ 2.34315 0.214796
$$120$$ 0 0
$$121$$ 12.3137 1.11943
$$122$$ 0 0
$$123$$ 12.0000 1.08200
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −6.00000 −0.532414 −0.266207 0.963916i $$-0.585770\pi$$
−0.266207 + 0.963916i $$0.585770\pi$$
$$128$$ 0 0
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ 16.1421 1.41034 0.705172 0.709036i $$-0.250870\pi$$
0.705172 + 0.709036i $$0.250870\pi$$
$$132$$ 0 0
$$133$$ −0.686292 −0.0595090
$$134$$ 0 0
$$135$$ −4.00000 −0.344265
$$136$$ 0 0
$$137$$ −10.8284 −0.925135 −0.462567 0.886584i $$-0.653072\pi$$
−0.462567 + 0.886584i $$0.653072\pi$$
$$138$$ 0 0
$$139$$ 10.3431 0.877294 0.438647 0.898659i $$-0.355458\pi$$
0.438647 + 0.898659i $$0.355458\pi$$
$$140$$ 0 0
$$141$$ −0.686292 −0.0577962
$$142$$ 0 0
$$143$$ −9.65685 −0.807547
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ 0 0
$$147$$ 12.6274 1.04149
$$148$$ 0 0
$$149$$ 13.3137 1.09070 0.545351 0.838208i $$-0.316396\pi$$
0.545351 + 0.838208i $$0.316396\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ −2.82843 −0.228665
$$154$$ 0 0
$$155$$ −10.4853 −0.842198
$$156$$ 0 0
$$157$$ 16.4853 1.31567 0.657834 0.753163i $$-0.271473\pi$$
0.657834 + 0.753163i $$0.271473\pi$$
$$158$$ 0 0
$$159$$ 15.3137 1.21446
$$160$$ 0 0
$$161$$ −7.31371 −0.576401
$$162$$ 0 0
$$163$$ −19.6569 −1.53964 −0.769822 0.638259i $$-0.779655\pi$$
−0.769822 + 0.638259i $$0.779655\pi$$
$$164$$ 0 0
$$165$$ −9.65685 −0.751785
$$166$$ 0 0
$$167$$ −14.4853 −1.12090 −0.560452 0.828187i $$-0.689373\pi$$
−0.560452 + 0.828187i $$0.689373\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0.828427 0.0633514
$$172$$ 0 0
$$173$$ 5.31371 0.403994 0.201997 0.979386i $$-0.435257\pi$$
0.201997 + 0.979386i $$0.435257\pi$$
$$174$$ 0 0
$$175$$ −0.828427 −0.0626232
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −0.686292 −0.0512958 −0.0256479 0.999671i $$-0.508165\pi$$
−0.0256479 + 0.999671i $$0.508165\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 0 0
$$183$$ 15.3137 1.13202
$$184$$ 0 0
$$185$$ 8.48528 0.623850
$$186$$ 0 0
$$187$$ 13.6569 0.998688
$$188$$ 0 0
$$189$$ −3.31371 −0.241037
$$190$$ 0 0
$$191$$ 15.1716 1.09778 0.548888 0.835896i $$-0.315051\pi$$
0.548888 + 0.835896i $$0.315051\pi$$
$$192$$ 0 0
$$193$$ −12.4853 −0.898710 −0.449355 0.893353i $$-0.648346\pi$$
−0.449355 + 0.893353i $$0.648346\pi$$
$$194$$ 0 0
$$195$$ 4.00000 0.286446
$$196$$ 0 0
$$197$$ 8.34315 0.594425 0.297212 0.954811i $$-0.403943\pi$$
0.297212 + 0.954811i $$0.403943\pi$$
$$198$$ 0 0
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ 0 0
$$201$$ 20.9706 1.47915
$$202$$ 0 0
$$203$$ 0.828427 0.0581442
$$204$$ 0 0
$$205$$ 6.00000 0.419058
$$206$$ 0 0
$$207$$ 8.82843 0.613618
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 4.82843 0.332403 0.166201 0.986092i $$-0.446850\pi$$
0.166201 + 0.986092i $$0.446850\pi$$
$$212$$ 0 0
$$213$$ 14.6274 1.00225
$$214$$ 0 0
$$215$$ 6.00000 0.409197
$$216$$ 0 0
$$217$$ −8.68629 −0.589664
$$218$$ 0 0
$$219$$ 16.9706 1.14676
$$220$$ 0 0
$$221$$ −5.65685 −0.380521
$$222$$ 0 0
$$223$$ −21.7990 −1.45977 −0.729884 0.683571i $$-0.760426\pi$$
−0.729884 + 0.683571i $$0.760426\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −8.14214 −0.540413 −0.270206 0.962802i $$-0.587092\pi$$
−0.270206 + 0.962802i $$0.587092\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ −0.343146 −0.0223844
$$236$$ 0 0
$$237$$ 28.9706 1.88184
$$238$$ 0 0
$$239$$ 23.3137 1.50804 0.754019 0.656852i $$-0.228113\pi$$
0.754019 + 0.656852i $$0.228113\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 0 0
$$245$$ 6.31371 0.403368
$$246$$ 0 0
$$247$$ 1.65685 0.105423
$$248$$ 0 0
$$249$$ −25.6569 −1.62594
$$250$$ 0 0
$$251$$ 3.17157 0.200188 0.100094 0.994978i $$-0.468086\pi$$
0.100094 + 0.994978i $$0.468086\pi$$
$$252$$ 0 0
$$253$$ −42.6274 −2.67996
$$254$$ 0 0
$$255$$ −5.65685 −0.354246
$$256$$ 0 0
$$257$$ 29.3137 1.82854 0.914269 0.405107i $$-0.132766\pi$$
0.914269 + 0.405107i $$0.132766\pi$$
$$258$$ 0 0
$$259$$ 7.02944 0.436788
$$260$$ 0 0
$$261$$ −1.00000 −0.0618984
$$262$$ 0 0
$$263$$ 8.34315 0.514460 0.257230 0.966350i $$-0.417190\pi$$
0.257230 + 0.966350i $$0.417190\pi$$
$$264$$ 0 0
$$265$$ 7.65685 0.470357
$$266$$ 0 0
$$267$$ −7.31371 −0.447592
$$268$$ 0 0
$$269$$ −1.31371 −0.0800982 −0.0400491 0.999198i $$-0.512751\pi$$
−0.0400491 + 0.999198i $$0.512751\pi$$
$$270$$ 0 0
$$271$$ −29.7990 −1.81016 −0.905080 0.425242i $$-0.860189\pi$$
−0.905080 + 0.425242i $$0.860189\pi$$
$$272$$ 0 0
$$273$$ 3.31371 0.200555
$$274$$ 0 0
$$275$$ −4.82843 −0.291165
$$276$$ 0 0
$$277$$ −7.65685 −0.460056 −0.230028 0.973184i $$-0.573882\pi$$
−0.230028 + 0.973184i $$0.573882\pi$$
$$278$$ 0 0
$$279$$ 10.4853 0.627737
$$280$$ 0 0
$$281$$ −6.68629 −0.398871 −0.199435 0.979911i $$-0.563911\pi$$
−0.199435 + 0.979911i $$0.563911\pi$$
$$282$$ 0 0
$$283$$ −0.828427 −0.0492449 −0.0246224 0.999697i $$-0.507838\pi$$
−0.0246224 + 0.999697i $$0.507838\pi$$
$$284$$ 0 0
$$285$$ 1.65685 0.0981436
$$286$$ 0 0
$$287$$ 4.97056 0.293403
$$288$$ 0 0
$$289$$ −9.00000 −0.529412
$$290$$ 0 0
$$291$$ −8.97056 −0.525864
$$292$$ 0 0
$$293$$ 8.48528 0.495715 0.247858 0.968796i $$-0.420273\pi$$
0.247858 + 0.968796i $$0.420273\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −19.3137 −1.12070
$$298$$ 0 0
$$299$$ 17.6569 1.02112
$$300$$ 0 0
$$301$$ 4.97056 0.286498
$$302$$ 0 0
$$303$$ 8.68629 0.499014
$$304$$ 0 0
$$305$$ 7.65685 0.438430
$$306$$ 0 0
$$307$$ 10.9706 0.626123 0.313062 0.949733i $$-0.398645\pi$$
0.313062 + 0.949733i $$0.398645\pi$$
$$308$$ 0 0
$$309$$ −24.2843 −1.38148
$$310$$ 0 0
$$311$$ 2.48528 0.140927 0.0704637 0.997514i $$-0.477552\pi$$
0.0704637 + 0.997514i $$0.477552\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 0 0
$$315$$ 0.828427 0.0466766
$$316$$ 0 0
$$317$$ 2.82843 0.158860 0.0794301 0.996840i $$-0.474690\pi$$
0.0794301 + 0.996840i $$0.474690\pi$$
$$318$$ 0 0
$$319$$ 4.82843 0.270340
$$320$$ 0 0
$$321$$ 16.2843 0.908899
$$322$$ 0 0
$$323$$ −2.34315 −0.130376
$$324$$ 0 0
$$325$$ 2.00000 0.110940
$$326$$ 0 0
$$327$$ 4.00000 0.221201
$$328$$ 0 0
$$329$$ −0.284271 −0.0156724
$$330$$ 0 0
$$331$$ −17.7990 −0.978321 −0.489160 0.872194i $$-0.662697\pi$$
−0.489160 + 0.872194i $$0.662697\pi$$
$$332$$ 0 0
$$333$$ −8.48528 −0.464991
$$334$$ 0 0
$$335$$ 10.4853 0.572872
$$336$$ 0 0
$$337$$ −6.82843 −0.371968 −0.185984 0.982553i $$-0.559547\pi$$
−0.185984 + 0.982553i $$0.559547\pi$$
$$338$$ 0 0
$$339$$ −5.65685 −0.307238
$$340$$ 0 0
$$341$$ −50.6274 −2.74163
$$342$$ 0 0
$$343$$ 11.0294 0.595534
$$344$$ 0 0
$$345$$ 17.6569 0.950613
$$346$$ 0 0
$$347$$ 20.1421 1.08129 0.540643 0.841252i $$-0.318181\pi$$
0.540643 + 0.841252i $$0.318181\pi$$
$$348$$ 0 0
$$349$$ 24.6274 1.31828 0.659138 0.752022i $$-0.270921\pi$$
0.659138 + 0.752022i $$0.270921\pi$$
$$350$$ 0 0
$$351$$ 8.00000 0.427008
$$352$$ 0 0
$$353$$ −15.6569 −0.833330 −0.416665 0.909060i $$-0.636801\pi$$
−0.416665 + 0.909060i $$0.636801\pi$$
$$354$$ 0 0
$$355$$ 7.31371 0.388171
$$356$$ 0 0
$$357$$ −4.68629 −0.248025
$$358$$ 0 0
$$359$$ 32.1421 1.69640 0.848199 0.529678i $$-0.177687\pi$$
0.848199 + 0.529678i $$0.177687\pi$$
$$360$$ 0 0
$$361$$ −18.3137 −0.963879
$$362$$ 0 0
$$363$$ −24.6274 −1.29260
$$364$$ 0 0
$$365$$ 8.48528 0.444140
$$366$$ 0 0
$$367$$ −18.0000 −0.939592 −0.469796 0.882775i $$-0.655673\pi$$
−0.469796 + 0.882775i $$0.655673\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 6.34315 0.329320
$$372$$ 0 0
$$373$$ −26.9706 −1.39648 −0.698241 0.715862i $$-0.746034\pi$$
−0.698241 + 0.715862i $$0.746034\pi$$
$$374$$ 0 0
$$375$$ 2.00000 0.103280
$$376$$ 0 0
$$377$$ −2.00000 −0.103005
$$378$$ 0 0
$$379$$ 5.51472 0.283272 0.141636 0.989919i $$-0.454764\pi$$
0.141636 + 0.989919i $$0.454764\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ 0 0
$$383$$ 14.4853 0.740163 0.370082 0.928999i $$-0.379330\pi$$
0.370082 + 0.928999i $$0.379330\pi$$
$$384$$ 0 0
$$385$$ −4.00000 −0.203859
$$386$$ 0 0
$$387$$ −6.00000 −0.304997
$$388$$ 0 0
$$389$$ 6.68629 0.339008 0.169504 0.985529i $$-0.445783\pi$$
0.169504 + 0.985529i $$0.445783\pi$$
$$390$$ 0 0
$$391$$ −24.9706 −1.26282
$$392$$ 0 0
$$393$$ −32.2843 −1.62853
$$394$$ 0 0
$$395$$ 14.4853 0.728834
$$396$$ 0 0
$$397$$ 8.34315 0.418730 0.209365 0.977838i $$-0.432860\pi$$
0.209365 + 0.977838i $$0.432860\pi$$
$$398$$ 0 0
$$399$$ 1.37258 0.0687151
$$400$$ 0 0
$$401$$ −29.3137 −1.46386 −0.731928 0.681382i $$-0.761379\pi$$
−0.731928 + 0.681382i $$0.761379\pi$$
$$402$$ 0 0
$$403$$ 20.9706 1.04462
$$404$$ 0 0
$$405$$ 11.0000 0.546594
$$406$$ 0 0
$$407$$ 40.9706 2.03084
$$408$$ 0 0
$$409$$ 30.9706 1.53140 0.765698 0.643200i $$-0.222394\pi$$
0.765698 + 0.643200i $$0.222394\pi$$
$$410$$ 0 0
$$411$$ 21.6569 1.06825
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −12.8284 −0.629723
$$416$$ 0 0
$$417$$ −20.6863 −1.01301
$$418$$ 0 0
$$419$$ −4.97056 −0.242828 −0.121414 0.992602i $$-0.538743\pi$$
−0.121414 + 0.992602i $$0.538743\pi$$
$$420$$ 0 0
$$421$$ 14.9706 0.729621 0.364810 0.931082i $$-0.381134\pi$$
0.364810 + 0.931082i $$0.381134\pi$$
$$422$$ 0 0
$$423$$ 0.343146 0.0166843
$$424$$ 0 0
$$425$$ −2.82843 −0.137199
$$426$$ 0 0
$$427$$ 6.34315 0.306966
$$428$$ 0 0
$$429$$ 19.3137 0.932475
$$430$$ 0 0
$$431$$ 19.3137 0.930309 0.465154 0.885230i $$-0.345999\pi$$
0.465154 + 0.885230i $$0.345999\pi$$
$$432$$ 0 0
$$433$$ −34.8284 −1.67375 −0.836874 0.547396i $$-0.815619\pi$$
−0.836874 + 0.547396i $$0.815619\pi$$
$$434$$ 0 0
$$435$$ −2.00000 −0.0958927
$$436$$ 0 0
$$437$$ 7.31371 0.349862
$$438$$ 0 0
$$439$$ 21.6569 1.03363 0.516813 0.856099i $$-0.327118\pi$$
0.516813 + 0.856099i $$0.327118\pi$$
$$440$$ 0 0
$$441$$ −6.31371 −0.300653
$$442$$ 0 0
$$443$$ 3.65685 0.173742 0.0868712 0.996220i $$-0.472313\pi$$
0.0868712 + 0.996220i $$0.472313\pi$$
$$444$$ 0 0
$$445$$ −3.65685 −0.173352
$$446$$ 0 0
$$447$$ −26.6274 −1.25943
$$448$$ 0 0
$$449$$ 0.343146 0.0161940 0.00809702 0.999967i $$-0.497423\pi$$
0.00809702 + 0.999967i $$0.497423\pi$$
$$450$$ 0 0
$$451$$ 28.9706 1.36417
$$452$$ 0 0
$$453$$ −24.0000 −1.12762
$$454$$ 0 0
$$455$$ 1.65685 0.0776745
$$456$$ 0 0
$$457$$ 8.34315 0.390276 0.195138 0.980776i $$-0.437485\pi$$
0.195138 + 0.980776i $$0.437485\pi$$
$$458$$ 0 0
$$459$$ −11.3137 −0.528079
$$460$$ 0 0
$$461$$ 24.3431 1.13377 0.566887 0.823796i $$-0.308148\pi$$
0.566887 + 0.823796i $$0.308148\pi$$
$$462$$ 0 0
$$463$$ 17.7990 0.827189 0.413595 0.910461i $$-0.364273\pi$$
0.413595 + 0.910461i $$0.364273\pi$$
$$464$$ 0 0
$$465$$ 20.9706 0.972487
$$466$$ 0 0
$$467$$ −22.9706 −1.06295 −0.531475 0.847074i $$-0.678362\pi$$
−0.531475 + 0.847074i $$0.678362\pi$$
$$468$$ 0 0
$$469$$ 8.68629 0.401096
$$470$$ 0 0
$$471$$ −32.9706 −1.51920
$$472$$ 0 0
$$473$$ 28.9706 1.33207
$$474$$ 0 0
$$475$$ 0.828427 0.0380108
$$476$$ 0 0
$$477$$ −7.65685 −0.350583
$$478$$ 0 0
$$479$$ −12.8284 −0.586146 −0.293073 0.956090i $$-0.594678\pi$$
−0.293073 + 0.956090i $$0.594678\pi$$
$$480$$ 0 0
$$481$$ −16.9706 −0.773791
$$482$$ 0 0
$$483$$ 14.6274 0.665571
$$484$$ 0 0
$$485$$ −4.48528 −0.203666
$$486$$ 0 0
$$487$$ 29.7990 1.35032 0.675161 0.737671i $$-0.264074\pi$$
0.675161 + 0.737671i $$0.264074\pi$$
$$488$$ 0 0
$$489$$ 39.3137 1.77783
$$490$$ 0 0
$$491$$ 43.4558 1.96113 0.980567 0.196183i $$-0.0628545\pi$$
0.980567 + 0.196183i $$0.0628545\pi$$
$$492$$ 0 0
$$493$$ 2.82843 0.127386
$$494$$ 0 0
$$495$$ 4.82843 0.217022
$$496$$ 0 0
$$497$$ 6.05887 0.271778
$$498$$ 0 0
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 0 0
$$501$$ 28.9706 1.29431
$$502$$ 0 0
$$503$$ 30.0000 1.33763 0.668817 0.743427i $$-0.266801\pi$$
0.668817 + 0.743427i $$0.266801\pi$$
$$504$$ 0 0
$$505$$ 4.34315 0.193267
$$506$$ 0 0
$$507$$ 18.0000 0.799408
$$508$$ 0 0
$$509$$ 44.6274 1.97808 0.989038 0.147663i $$-0.0471751\pi$$
0.989038 + 0.147663i $$0.0471751\pi$$
$$510$$ 0 0
$$511$$ 7.02944 0.310964
$$512$$ 0 0
$$513$$ 3.31371 0.146304
$$514$$ 0 0
$$515$$ −12.1421 −0.535046
$$516$$ 0 0
$$517$$ −1.65685 −0.0728684
$$518$$ 0 0
$$519$$ −10.6274 −0.466492
$$520$$ 0 0
$$521$$ 1.31371 0.0575546 0.0287773 0.999586i $$-0.490839\pi$$
0.0287773 + 0.999586i $$0.490839\pi$$
$$522$$ 0 0
$$523$$ −14.4853 −0.633397 −0.316699 0.948526i $$-0.602574\pi$$
−0.316699 + 0.948526i $$0.602574\pi$$
$$524$$ 0 0
$$525$$ 1.65685 0.0723110
$$526$$ 0 0
$$527$$ −29.6569 −1.29187
$$528$$ 0 0
$$529$$ 54.9411 2.38874
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ 8.14214 0.352015
$$536$$ 0 0
$$537$$ 1.37258 0.0592313
$$538$$ 0 0
$$539$$ 30.4853 1.31309
$$540$$ 0 0
$$541$$ 38.9706 1.67548 0.837738 0.546073i $$-0.183878\pi$$
0.837738 + 0.546073i $$0.183878\pi$$
$$542$$ 0 0
$$543$$ −12.0000 −0.514969
$$544$$ 0 0
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ 14.4853 0.619346 0.309673 0.950843i $$-0.399780\pi$$
0.309673 + 0.950843i $$0.399780\pi$$
$$548$$ 0 0
$$549$$ −7.65685 −0.326787
$$550$$ 0 0
$$551$$ −0.828427 −0.0352922
$$552$$ 0 0
$$553$$ 12.0000 0.510292
$$554$$ 0 0
$$555$$ −16.9706 −0.720360
$$556$$ 0 0
$$557$$ −39.9411 −1.69236 −0.846180 0.532897i $$-0.821103\pi$$
−0.846180 + 0.532897i $$0.821103\pi$$
$$558$$ 0 0
$$559$$ −12.0000 −0.507546
$$560$$ 0 0
$$561$$ −27.3137 −1.15319
$$562$$ 0 0
$$563$$ −3.65685 −0.154118 −0.0770590 0.997027i $$-0.524553\pi$$
−0.0770590 + 0.997027i $$0.524553\pi$$
$$564$$ 0 0
$$565$$ −2.82843 −0.118993
$$566$$ 0 0
$$567$$ 9.11270 0.382697
$$568$$ 0 0
$$569$$ 16.3431 0.685140 0.342570 0.939492i $$-0.388703\pi$$
0.342570 + 0.939492i $$0.388703\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ 0 0
$$573$$ −30.3431 −1.26760
$$574$$ 0 0
$$575$$ 8.82843 0.368171
$$576$$ 0 0
$$577$$ −15.7990 −0.657721 −0.328860 0.944379i $$-0.606665\pi$$
−0.328860 + 0.944379i $$0.606665\pi$$
$$578$$ 0 0
$$579$$ 24.9706 1.03774
$$580$$ 0 0
$$581$$ −10.6274 −0.440900
$$582$$ 0 0
$$583$$ 36.9706 1.53116
$$584$$ 0 0
$$585$$ −2.00000 −0.0826898
$$586$$ 0 0
$$587$$ 9.79899 0.404448 0.202224 0.979339i $$-0.435183\pi$$
0.202224 + 0.979339i $$0.435183\pi$$
$$588$$ 0 0
$$589$$ 8.68629 0.357912
$$590$$ 0 0
$$591$$ −16.6863 −0.686382
$$592$$ 0 0
$$593$$ 3.65685 0.150169 0.0750845 0.997177i $$-0.476077\pi$$
0.0750845 + 0.997177i $$0.476077\pi$$
$$594$$ 0 0
$$595$$ −2.34315 −0.0960596
$$596$$ 0 0
$$597$$ 24.0000 0.982255
$$598$$ 0 0
$$599$$ −1.79899 −0.0735047 −0.0367524 0.999324i $$-0.511701\pi$$
−0.0367524 + 0.999324i $$0.511701\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ −10.4853 −0.426994
$$604$$ 0 0
$$605$$ −12.3137 −0.500623
$$606$$ 0 0
$$607$$ −42.9706 −1.74412 −0.872061 0.489398i $$-0.837217\pi$$
−0.872061 + 0.489398i $$0.837217\pi$$
$$608$$ 0 0
$$609$$ −1.65685 −0.0671391
$$610$$ 0 0
$$611$$ 0.686292 0.0277644
$$612$$ 0 0
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ 0 0
$$615$$ −12.0000 −0.483887
$$616$$ 0 0
$$617$$ −14.8284 −0.596970 −0.298485 0.954414i $$-0.596481\pi$$
−0.298485 + 0.954414i $$0.596481\pi$$
$$618$$ 0 0
$$619$$ 29.7990 1.19772 0.598861 0.800853i $$-0.295620\pi$$
0.598861 + 0.800853i $$0.295620\pi$$
$$620$$ 0 0
$$621$$ 35.3137 1.41709
$$622$$ 0 0
$$623$$ −3.02944 −0.121372
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 8.00000 0.319489
$$628$$ 0 0
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ −3.02944 −0.120600 −0.0603000 0.998180i $$-0.519206\pi$$
−0.0603000 + 0.998180i $$0.519206\pi$$
$$632$$ 0 0
$$633$$ −9.65685 −0.383825
$$634$$ 0 0
$$635$$ 6.00000 0.238103
$$636$$ 0 0
$$637$$ −12.6274 −0.500316
$$638$$ 0 0
$$639$$ −7.31371 −0.289326
$$640$$ 0 0
$$641$$ −44.6274 −1.76268 −0.881338 0.472485i $$-0.843357\pi$$
−0.881338 + 0.472485i $$0.843357\pi$$
$$642$$ 0 0
$$643$$ −31.4558 −1.24050 −0.620249 0.784405i $$-0.712968\pi$$
−0.620249 + 0.784405i $$0.712968\pi$$
$$644$$ 0 0
$$645$$ −12.0000 −0.472500
$$646$$ 0 0
$$647$$ −21.1127 −0.830026 −0.415013 0.909816i $$-0.636223\pi$$
−0.415013 + 0.909816i $$0.636223\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 17.3726 0.680885
$$652$$ 0 0
$$653$$ 22.8284 0.893345 0.446673 0.894697i $$-0.352609\pi$$
0.446673 + 0.894697i $$0.352609\pi$$
$$654$$ 0 0
$$655$$ −16.1421 −0.630725
$$656$$ 0 0
$$657$$ −8.48528 −0.331042
$$658$$ 0 0
$$659$$ −37.7990 −1.47244 −0.736220 0.676743i $$-0.763391\pi$$
−0.736220 + 0.676743i $$0.763391\pi$$
$$660$$ 0 0
$$661$$ −26.0000 −1.01128 −0.505641 0.862744i $$-0.668744\pi$$
−0.505641 + 0.862744i $$0.668744\pi$$
$$662$$ 0 0
$$663$$ 11.3137 0.439388
$$664$$ 0 0
$$665$$ 0.686292 0.0266132
$$666$$ 0 0
$$667$$ −8.82843 −0.341838
$$668$$ 0 0
$$669$$ 43.5980 1.68560
$$670$$ 0 0
$$671$$ 36.9706 1.42723
$$672$$ 0 0
$$673$$ −10.9706 −0.422884 −0.211442 0.977391i $$-0.567816\pi$$
−0.211442 + 0.977391i $$0.567816\pi$$
$$674$$ 0 0
$$675$$ 4.00000 0.153960
$$676$$ 0 0
$$677$$ −36.7696 −1.41317 −0.706584 0.707629i $$-0.749765\pi$$
−0.706584 + 0.707629i $$0.749765\pi$$
$$678$$ 0 0
$$679$$ −3.71573 −0.142597
$$680$$ 0 0
$$681$$ 16.2843 0.624015
$$682$$ 0 0
$$683$$ 40.1421 1.53600 0.767998 0.640452i $$-0.221253\pi$$
0.767998 + 0.640452i $$0.221253\pi$$
$$684$$ 0 0
$$685$$ 10.8284 0.413733
$$686$$ 0 0
$$687$$ −4.00000 −0.152610
$$688$$ 0 0
$$689$$ −15.3137 −0.583406
$$690$$ 0 0
$$691$$ −11.0294 −0.419580 −0.209790 0.977747i $$-0.567278\pi$$
−0.209790 + 0.977747i $$0.567278\pi$$
$$692$$ 0 0
$$693$$ 4.00000 0.151947
$$694$$ 0 0
$$695$$ −10.3431 −0.392338
$$696$$ 0 0
$$697$$ 16.9706 0.642806
$$698$$ 0 0
$$699$$ −36.0000 −1.36165
$$700$$ 0 0
$$701$$ −29.3137 −1.10716 −0.553582 0.832795i $$-0.686739\pi$$
−0.553582 + 0.832795i $$0.686739\pi$$
$$702$$ 0 0
$$703$$ −7.02944 −0.265120
$$704$$ 0 0
$$705$$ 0.686292 0.0258472
$$706$$ 0 0
$$707$$ 3.59798 0.135316
$$708$$ 0 0
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 0 0
$$711$$ −14.4853 −0.543240
$$712$$ 0 0
$$713$$ 92.5685 3.46672
$$714$$ 0 0
$$715$$ 9.65685 0.361146
$$716$$ 0 0
$$717$$ −46.6274 −1.74133
$$718$$ 0 0
$$719$$ −10.6274 −0.396336 −0.198168 0.980168i $$-0.563499\pi$$
−0.198168 + 0.980168i $$0.563499\pi$$
$$720$$ 0 0
$$721$$ −10.0589 −0.374612
$$722$$ 0 0
$$723$$ −20.0000 −0.743808
$$724$$ 0 0
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ −43.9411 −1.62969 −0.814843 0.579682i $$-0.803177\pi$$
−0.814843 + 0.579682i $$0.803177\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 16.9706 0.627679
$$732$$ 0 0
$$733$$ 17.1716 0.634247 0.317123 0.948384i $$-0.397283\pi$$
0.317123 + 0.948384i $$0.397283\pi$$
$$734$$ 0 0
$$735$$ −12.6274 −0.465769
$$736$$ 0 0
$$737$$ 50.6274 1.86488
$$738$$ 0 0
$$739$$ −2.48528 −0.0914226 −0.0457113 0.998955i $$-0.514555\pi$$
−0.0457113 + 0.998955i $$0.514555\pi$$
$$740$$ 0 0
$$741$$ −3.31371 −0.121732
$$742$$ 0 0
$$743$$ 7.37258 0.270474 0.135237 0.990813i $$-0.456820\pi$$
0.135237 + 0.990813i $$0.456820\pi$$
$$744$$ 0 0
$$745$$ −13.3137 −0.487777
$$746$$ 0 0
$$747$$ 12.8284 0.469368
$$748$$ 0 0
$$749$$ 6.74517 0.246463
$$750$$ 0 0
$$751$$ 12.1421 0.443073 0.221536 0.975152i $$-0.428893\pi$$
0.221536 + 0.975152i $$0.428893\pi$$
$$752$$ 0 0
$$753$$ −6.34315 −0.231157
$$754$$ 0 0
$$755$$ −12.0000 −0.436725
$$756$$ 0 0
$$757$$ −36.4853 −1.32608 −0.663040 0.748584i $$-0.730734\pi$$
−0.663040 + 0.748584i $$0.730734\pi$$
$$758$$ 0 0
$$759$$ 85.2548 3.09455
$$760$$ 0 0
$$761$$ −36.6274 −1.32774 −0.663871 0.747847i $$-0.731088\pi$$
−0.663871 + 0.747847i $$0.731088\pi$$
$$762$$ 0 0
$$763$$ 1.65685 0.0599822
$$764$$ 0 0
$$765$$ 2.82843 0.102262
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −4.34315 −0.156618 −0.0783089 0.996929i $$-0.524952\pi$$
−0.0783089 + 0.996929i $$0.524952\pi$$
$$770$$ 0 0
$$771$$ −58.6274 −2.11141
$$772$$ 0 0
$$773$$ −8.48528 −0.305194 −0.152597 0.988288i $$-0.548764\pi$$
−0.152597 + 0.988288i $$0.548764\pi$$
$$774$$ 0 0
$$775$$ 10.4853 0.376642
$$776$$ 0 0
$$777$$ −14.0589 −0.504359
$$778$$ 0 0
$$779$$ −4.97056 −0.178089
$$780$$ 0 0
$$781$$ 35.3137 1.26362
$$782$$ 0 0
$$783$$ −4.00000 −0.142948
$$784$$ 0 0
$$785$$ −16.4853 −0.588385
$$786$$ 0 0
$$787$$ −21.7990 −0.777050 −0.388525 0.921438i $$-0.627015\pi$$
−0.388525 + 0.921438i $$0.627015\pi$$
$$788$$ 0 0
$$789$$ −16.6863 −0.594048
$$790$$ 0 0
$$791$$ −2.34315 −0.0833127
$$792$$ 0 0
$$793$$ −15.3137 −0.543806
$$794$$ 0 0
$$795$$ −15.3137 −0.543121
$$796$$ 0 0
$$797$$ 34.1421 1.20938 0.604688 0.796462i $$-0.293298\pi$$
0.604688 + 0.796462i $$0.293298\pi$$
$$798$$ 0 0
$$799$$ −0.970563 −0.0343360
$$800$$ 0 0
$$801$$ 3.65685 0.129209
$$802$$ 0 0
$$803$$ 40.9706 1.44582
$$804$$ 0 0
$$805$$ 7.31371 0.257774
$$806$$ 0 0
$$807$$ 2.62742 0.0924895
$$808$$ 0 0
$$809$$ −14.2843 −0.502208 −0.251104 0.967960i $$-0.580794\pi$$
−0.251104 + 0.967960i $$0.580794\pi$$
$$810$$ 0 0
$$811$$ 26.3431 0.925033 0.462516 0.886611i $$-0.346947\pi$$
0.462516 + 0.886611i $$0.346947\pi$$
$$812$$ 0 0
$$813$$ 59.5980 2.09019
$$814$$ 0 0
$$815$$ 19.6569 0.688550
$$816$$ 0 0
$$817$$ −4.97056 −0.173898
$$818$$ 0 0
$$819$$ −1.65685 −0.0578952
$$820$$ 0 0
$$821$$ 45.3137 1.58146 0.790730 0.612165i $$-0.209701\pi$$
0.790730 + 0.612165i $$0.209701\pi$$
$$822$$ 0 0
$$823$$ −2.97056 −0.103547 −0.0517737 0.998659i $$-0.516487\pi$$
−0.0517737 + 0.998659i $$0.516487\pi$$
$$824$$ 0 0
$$825$$ 9.65685 0.336209
$$826$$ 0 0
$$827$$ −5.31371 −0.184776 −0.0923879 0.995723i $$-0.529450\pi$$
−0.0923879 + 0.995723i $$0.529450\pi$$
$$828$$ 0 0
$$829$$ 24.6274 0.855346 0.427673 0.903934i $$-0.359334\pi$$
0.427673 + 0.903934i $$0.359334\pi$$
$$830$$ 0 0
$$831$$ 15.3137 0.531227
$$832$$ 0 0
$$833$$ 17.8579 0.618738
$$834$$ 0 0
$$835$$ 14.4853 0.501284
$$836$$ 0 0
$$837$$ 41.9411 1.44970
$$838$$ 0 0
$$839$$ 14.4853 0.500087 0.250044 0.968235i $$-0.419555\pi$$
0.250044 + 0.968235i $$0.419555\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ 13.3726 0.460576
$$844$$ 0 0
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ −10.2010 −0.350511
$$848$$ 0 0
$$849$$ 1.65685 0.0568631
$$850$$ 0 0
$$851$$ −74.9117 −2.56794
$$852$$ 0 0
$$853$$ 11.1127 0.380492 0.190246 0.981736i $$-0.439072\pi$$
0.190246 + 0.981736i $$0.439072\pi$$
$$854$$ 0 0
$$855$$ −0.828427 −0.0283316
$$856$$ 0 0
$$857$$ 48.6274 1.66108 0.830540 0.556958i $$-0.188032\pi$$
0.830540 + 0.556958i $$0.188032\pi$$
$$858$$ 0 0
$$859$$ 28.4264 0.969896 0.484948 0.874543i $$-0.338838\pi$$
0.484948 + 0.874543i $$0.338838\pi$$
$$860$$ 0 0
$$861$$ −9.94113 −0.338793
$$862$$ 0 0
$$863$$ 7.85786 0.267485 0.133742 0.991016i $$-0.457301\pi$$
0.133742 + 0.991016i $$0.457301\pi$$
$$864$$ 0 0
$$865$$ −5.31371 −0.180672
$$866$$ 0 0
$$867$$ 18.0000 0.611312
$$868$$ 0 0
$$869$$ 69.9411 2.37259
$$870$$ 0 0
$$871$$ −20.9706 −0.710560
$$872$$ 0 0
$$873$$ 4.48528 0.151804
$$874$$ 0 0
$$875$$ 0.828427 0.0280059
$$876$$ 0 0
$$877$$ 18.2843 0.617416 0.308708 0.951157i $$-0.400103\pi$$
0.308708 + 0.951157i $$0.400103\pi$$
$$878$$ 0 0
$$879$$ −16.9706 −0.572403
$$880$$ 0 0
$$881$$ 6.68629 0.225267 0.112633 0.993637i $$-0.464071\pi$$
0.112633 + 0.993637i $$0.464071\pi$$
$$882$$ 0 0
$$883$$ 2.48528 0.0836364 0.0418182 0.999125i $$-0.486685\pi$$
0.0418182 + 0.999125i $$0.486685\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 29.3137 0.984258 0.492129 0.870522i $$-0.336219\pi$$
0.492129 + 0.870522i $$0.336219\pi$$
$$888$$ 0 0
$$889$$ 4.97056 0.166707
$$890$$ 0 0
$$891$$ 53.1127 1.77934
$$892$$ 0 0
$$893$$ 0.284271 0.00951277
$$894$$ 0 0
$$895$$ 0.686292 0.0229402
$$896$$ 0 0
$$897$$ −35.3137 −1.17909
$$898$$ 0 0
$$899$$ −10.4853 −0.349704
$$900$$ 0 0
$$901$$ 21.6569 0.721494
$$902$$ 0 0
$$903$$ −9.94113 −0.330820
$$904$$ 0 0
$$905$$ −6.00000 −0.199447
$$906$$ 0 0
$$907$$ −10.0000 −0.332045 −0.166022 0.986122i $$-0.553092\pi$$
−0.166022 + 0.986122i $$0.553092\pi$$
$$908$$ 0 0
$$909$$ −4.34315 −0.144053
$$910$$ 0 0
$$911$$ −3.85786 −0.127817 −0.0639084 0.997956i $$-0.520357\pi$$
−0.0639084 + 0.997956i $$0.520357\pi$$
$$912$$ 0 0
$$913$$ −61.9411 −2.04995
$$914$$ 0 0
$$915$$ −15.3137 −0.506256
$$916$$ 0 0
$$917$$ −13.3726 −0.441602
$$918$$ 0 0
$$919$$ 36.0000 1.18753 0.593765 0.804638i $$-0.297641\pi$$
0.593765 + 0.804638i $$0.297641\pi$$
$$920$$ 0 0
$$921$$ −21.9411 −0.722985
$$922$$ 0 0
$$923$$ −14.6274 −0.481467
$$924$$ 0 0
$$925$$ −8.48528 −0.278994
$$926$$ 0 0
$$927$$ 12.1421 0.398800
$$928$$ 0 0
$$929$$ 40.6274 1.33294 0.666471 0.745531i $$-0.267804\pi$$
0.666471 + 0.745531i $$0.267804\pi$$
$$930$$ 0 0
$$931$$ −5.23045 −0.171421
$$932$$ 0 0
$$933$$ −4.97056 −0.162729
$$934$$ 0 0
$$935$$ −13.6569 −0.446627
$$936$$ 0 0
$$937$$ 8.34315 0.272559 0.136279 0.990670i $$-0.456486\pi$$
0.136279 + 0.990670i $$0.456486\pi$$
$$938$$ 0 0
$$939$$ 12.0000 0.391605
$$940$$ 0 0
$$941$$ −39.9411 −1.30204 −0.651022 0.759059i $$-0.725659\pi$$
−0.651022 + 0.759059i $$0.725659\pi$$
$$942$$ 0 0
$$943$$ −52.9706 −1.72496
$$944$$ 0 0
$$945$$ 3.31371 0.107795
$$946$$ 0 0
$$947$$ −56.9117 −1.84938 −0.924691 0.380719i $$-0.875676\pi$$
−0.924691 + 0.380719i $$0.875676\pi$$
$$948$$ 0 0
$$949$$ −16.9706 −0.550888
$$950$$ 0 0
$$951$$ −5.65685 −0.183436
$$952$$ 0 0
$$953$$ 6.68629 0.216590 0.108295 0.994119i $$-0.465461\pi$$
0.108295 + 0.994119i $$0.465461\pi$$
$$954$$ 0 0
$$955$$ −15.1716 −0.490941
$$956$$ 0 0
$$957$$ −9.65685 −0.312162
$$958$$ 0 0
$$959$$ 8.97056 0.289675
$$960$$ 0 0
$$961$$ 78.9411 2.54649
$$962$$ 0 0
$$963$$ −8.14214 −0.262377
$$964$$ 0 0
$$965$$ 12.4853 0.401915
$$966$$ 0 0
$$967$$ 18.9706 0.610052 0.305026 0.952344i $$-0.401335\pi$$
0.305026 + 0.952344i $$0.401335\pi$$
$$968$$ 0 0
$$969$$ 4.68629 0.150545
$$970$$ 0 0
$$971$$ −0.142136 −0.00456135 −0.00228067 0.999997i $$-0.500726\pi$$
−0.00228067 + 0.999997i $$0.500726\pi$$
$$972$$ 0 0
$$973$$ −8.56854 −0.274695
$$974$$ 0 0
$$975$$ −4.00000 −0.128103
$$976$$ 0 0
$$977$$ −25.3137 −0.809857 −0.404929 0.914348i $$-0.632704\pi$$
−0.404929 + 0.914348i $$0.632704\pi$$
$$978$$ 0 0
$$979$$ −17.6569 −0.564316
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ 0 0
$$983$$ −13.3137 −0.424641 −0.212321 0.977200i $$-0.568102\pi$$
−0.212321 + 0.977200i $$0.568102\pi$$
$$984$$ 0 0
$$985$$ −8.34315 −0.265835
$$986$$ 0 0
$$987$$ 0.568542 0.0180969
$$988$$ 0 0
$$989$$ −52.9706 −1.68437
$$990$$ 0 0
$$991$$ −52.0000 −1.65183 −0.825917 0.563791i $$-0.809342\pi$$
−0.825917 + 0.563791i $$0.809342\pi$$
$$992$$ 0 0
$$993$$ 35.5980 1.12967
$$994$$ 0 0
$$995$$ 12.0000 0.380426
$$996$$ 0 0
$$997$$ −1.17157 −0.0371041 −0.0185520 0.999828i $$-0.505906\pi$$
−0.0185520 + 0.999828i $$0.505906\pi$$
$$998$$ 0 0
$$999$$ −33.9411 −1.07385
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 9280.2.a.w.1.1 2
4.3 odd 2 9280.2.a.be.1.2 2
8.3 odd 2 145.2.a.b.1.1 2
8.5 even 2 2320.2.a.k.1.1 2
24.11 even 2 1305.2.a.n.1.2 2
40.3 even 4 725.2.b.c.349.4 4
40.19 odd 2 725.2.a.c.1.2 2
40.27 even 4 725.2.b.c.349.1 4
56.27 even 2 7105.2.a.e.1.1 2
120.59 even 2 6525.2.a.p.1.1 2
232.115 odd 2 4205.2.a.d.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.1 2 8.3 odd 2
725.2.a.c.1.2 2 40.19 odd 2
725.2.b.c.349.1 4 40.27 even 4
725.2.b.c.349.4 4 40.3 even 4
1305.2.a.n.1.2 2 24.11 even 2
2320.2.a.k.1.1 2 8.5 even 2
4205.2.a.d.1.2 2 232.115 odd 2
6525.2.a.p.1.1 2 120.59 even 2
7105.2.a.e.1.1 2 56.27 even 2
9280.2.a.w.1.1 2 1.1 even 1 trivial
9280.2.a.be.1.2 2 4.3 odd 2