# Properties

 Label 9280.2.a.br.1.3 Level $9280$ Weight $2$ Character 9280.1 Self dual yes Analytic conductor $74.101$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Learn more

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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 1.3 Root $$0.311108$$ 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)

 $$f(q)$$ $$=$$ $$q+2.90321 q^{3} -1.00000 q^{5} +0.903212 q^{7} +5.42864 q^{9} +O(q^{10})$$ $$q+2.90321 q^{3} -1.00000 q^{5} +0.903212 q^{7} +5.42864 q^{9} -1.52543 q^{11} +0.622216 q^{13} -2.90321 q^{15} -7.95407 q^{17} -1.09679 q^{19} +2.62222 q^{21} -7.52543 q^{23} +1.00000 q^{25} +7.05086 q^{27} +1.00000 q^{29} +6.90321 q^{31} -4.42864 q^{33} -0.903212 q^{35} -3.95407 q^{37} +1.80642 q^{39} +3.67307 q^{41} -10.5161 q^{43} -5.42864 q^{45} -6.90321 q^{47} -6.18421 q^{49} -23.0923 q^{51} -6.42864 q^{53} +1.52543 q^{55} -3.18421 q^{57} -1.67307 q^{59} +1.86665 q^{61} +4.90321 q^{63} -0.622216 q^{65} +11.5254 q^{67} -21.8479 q^{69} -13.6731 q^{71} +10.1891 q^{73} +2.90321 q^{75} -1.37778 q^{77} -9.13828 q^{79} +4.18421 q^{81} +10.7096 q^{83} +7.95407 q^{85} +2.90321 q^{87} -7.80642 q^{89} +0.561993 q^{91} +20.0415 q^{93} +1.09679 q^{95} -4.08742 q^{97} -8.28100 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 - 3 * q^5 - 4 * q^7 + 3 * q^9 $$3 q + 2 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} - 4 q^{17} - 10 q^{19} + 8 q^{21} - 16 q^{23} + 3 q^{25} + 8 q^{27} + 3 q^{29} + 14 q^{31} + 4 q^{35} + 8 q^{37} - 8 q^{39} - 2 q^{41} + 2 q^{43} - 3 q^{45} - 14 q^{47} - 5 q^{49} - 16 q^{51} - 6 q^{53} - 2 q^{55} + 4 q^{57} + 8 q^{59} + 6 q^{61} + 8 q^{63} - 2 q^{65} + 28 q^{67} - 12 q^{69} - 28 q^{71} - 16 q^{73} + 2 q^{75} - 4 q^{77} + 6 q^{79} - q^{81} + 12 q^{83} + 4 q^{85} + 2 q^{87} - 10 q^{89} - 12 q^{91} + 20 q^{93} + 10 q^{95} + 8 q^{97} - 18 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 - 3 * q^5 - 4 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - 2 * q^15 - 4 * q^17 - 10 * q^19 + 8 * q^21 - 16 * q^23 + 3 * q^25 + 8 * q^27 + 3 * q^29 + 14 * q^31 + 4 * q^35 + 8 * q^37 - 8 * q^39 - 2 * q^41 + 2 * q^43 - 3 * q^45 - 14 * q^47 - 5 * q^49 - 16 * q^51 - 6 * q^53 - 2 * q^55 + 4 * q^57 + 8 * q^59 + 6 * q^61 + 8 * q^63 - 2 * q^65 + 28 * q^67 - 12 * q^69 - 28 * q^71 - 16 * q^73 + 2 * q^75 - 4 * q^77 + 6 * q^79 - q^81 + 12 * q^83 + 4 * q^85 + 2 * q^87 - 10 * q^89 - 12 * q^91 + 20 * q^93 + 10 * q^95 + 8 * q^97 - 18 * 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
$$3$$ 2.90321 1.67617 0.838085 0.545540i $$-0.183675\pi$$
0.838085 + 0.545540i $$0.183675\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0.903212 0.341382 0.170691 0.985325i $$-0.445400\pi$$
0.170691 + 0.985325i $$0.445400\pi$$
$$8$$ 0 0
$$9$$ 5.42864 1.80955
$$10$$ 0 0
$$11$$ −1.52543 −0.459934 −0.229967 0.973198i $$-0.573862\pi$$
−0.229967 + 0.973198i $$0.573862\pi$$
$$12$$ 0 0
$$13$$ 0.622216 0.172572 0.0862858 0.996270i $$-0.472500\pi$$
0.0862858 + 0.996270i $$0.472500\pi$$
$$14$$ 0 0
$$15$$ −2.90321 −0.749606
$$16$$ 0 0
$$17$$ −7.95407 −1.92914 −0.964572 0.263819i $$-0.915018\pi$$
−0.964572 + 0.263819i $$0.915018\pi$$
$$18$$ 0 0
$$19$$ −1.09679 −0.251620 −0.125810 0.992054i $$-0.540153\pi$$
−0.125810 + 0.992054i $$0.540153\pi$$
$$20$$ 0 0
$$21$$ 2.62222 0.572214
$$22$$ 0 0
$$23$$ −7.52543 −1.56916 −0.784580 0.620028i $$-0.787121\pi$$
−0.784580 + 0.620028i $$0.787121\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 7.05086 1.35694
$$28$$ 0 0
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 6.90321 1.23985 0.619927 0.784660i $$-0.287162\pi$$
0.619927 + 0.784660i $$0.287162\pi$$
$$32$$ 0 0
$$33$$ −4.42864 −0.770927
$$34$$ 0 0
$$35$$ −0.903212 −0.152671
$$36$$ 0 0
$$37$$ −3.95407 −0.650045 −0.325022 0.945706i $$-0.605372\pi$$
−0.325022 + 0.945706i $$0.605372\pi$$
$$38$$ 0 0
$$39$$ 1.80642 0.289259
$$40$$ 0 0
$$41$$ 3.67307 0.573637 0.286819 0.957985i $$-0.407402\pi$$
0.286819 + 0.957985i $$0.407402\pi$$
$$42$$ 0 0
$$43$$ −10.5161 −1.60368 −0.801842 0.597536i $$-0.796146\pi$$
−0.801842 + 0.597536i $$0.796146\pi$$
$$44$$ 0 0
$$45$$ −5.42864 −0.809254
$$46$$ 0 0
$$47$$ −6.90321 −1.00694 −0.503468 0.864014i $$-0.667943\pi$$
−0.503468 + 0.864014i $$0.667943\pi$$
$$48$$ 0 0
$$49$$ −6.18421 −0.883458
$$50$$ 0 0
$$51$$ −23.0923 −3.23357
$$52$$ 0 0
$$53$$ −6.42864 −0.883042 −0.441521 0.897251i $$-0.645561\pi$$
−0.441521 + 0.897251i $$0.645561\pi$$
$$54$$ 0 0
$$55$$ 1.52543 0.205689
$$56$$ 0 0
$$57$$ −3.18421 −0.421759
$$58$$ 0 0
$$59$$ −1.67307 −0.217815 −0.108908 0.994052i $$-0.534735\pi$$
−0.108908 + 0.994052i $$0.534735\pi$$
$$60$$ 0 0
$$61$$ 1.86665 0.239000 0.119500 0.992834i $$-0.461871\pi$$
0.119500 + 0.992834i $$0.461871\pi$$
$$62$$ 0 0
$$63$$ 4.90321 0.617747
$$64$$ 0 0
$$65$$ −0.622216 −0.0771764
$$66$$ 0 0
$$67$$ 11.5254 1.40806 0.704028 0.710173i $$-0.251383\pi$$
0.704028 + 0.710173i $$0.251383\pi$$
$$68$$ 0 0
$$69$$ −21.8479 −2.63018
$$70$$ 0 0
$$71$$ −13.6731 −1.62269 −0.811347 0.584564i $$-0.801266\pi$$
−0.811347 + 0.584564i $$0.801266\pi$$
$$72$$ 0 0
$$73$$ 10.1891 1.19255 0.596274 0.802781i $$-0.296647\pi$$
0.596274 + 0.802781i $$0.296647\pi$$
$$74$$ 0 0
$$75$$ 2.90321 0.335234
$$76$$ 0 0
$$77$$ −1.37778 −0.157013
$$78$$ 0 0
$$79$$ −9.13828 −1.02814 −0.514068 0.857749i $$-0.671862\pi$$
−0.514068 + 0.857749i $$0.671862\pi$$
$$80$$ 0 0
$$81$$ 4.18421 0.464912
$$82$$ 0 0
$$83$$ 10.7096 1.17554 0.587768 0.809030i $$-0.300007\pi$$
0.587768 + 0.809030i $$0.300007\pi$$
$$84$$ 0 0
$$85$$ 7.95407 0.862740
$$86$$ 0 0
$$87$$ 2.90321 0.311257
$$88$$ 0 0
$$89$$ −7.80642 −0.827479 −0.413740 0.910395i $$-0.635778\pi$$
−0.413740 + 0.910395i $$0.635778\pi$$
$$90$$ 0 0
$$91$$ 0.561993 0.0589128
$$92$$ 0 0
$$93$$ 20.0415 2.07821
$$94$$ 0 0
$$95$$ 1.09679 0.112528
$$96$$ 0 0
$$97$$ −4.08742 −0.415015 −0.207507 0.978233i $$-0.566535\pi$$
−0.207507 + 0.978233i $$0.566535\pi$$
$$98$$ 0 0
$$99$$ −8.28100 −0.832271
$$100$$ 0 0
$$101$$ −13.9081 −1.38391 −0.691956 0.721940i $$-0.743251\pi$$
−0.691956 + 0.721940i $$0.743251\pi$$
$$102$$ 0 0
$$103$$ 12.9447 1.27548 0.637740 0.770252i $$-0.279870\pi$$
0.637740 + 0.770252i $$0.279870\pi$$
$$104$$ 0 0
$$105$$ −2.62222 −0.255902
$$106$$ 0 0
$$107$$ 11.0049 1.06389 0.531943 0.846780i $$-0.321462\pi$$
0.531943 + 0.846780i $$0.321462\pi$$
$$108$$ 0 0
$$109$$ 18.0415 1.72806 0.864031 0.503439i $$-0.167932\pi$$
0.864031 + 0.503439i $$0.167932\pi$$
$$110$$ 0 0
$$111$$ −11.4795 −1.08959
$$112$$ 0 0
$$113$$ −10.2810 −0.967155 −0.483577 0.875302i $$-0.660663\pi$$
−0.483577 + 0.875302i $$0.660663\pi$$
$$114$$ 0 0
$$115$$ 7.52543 0.701750
$$116$$ 0 0
$$117$$ 3.37778 0.312276
$$118$$ 0 0
$$119$$ −7.18421 −0.658575
$$120$$ 0 0
$$121$$ −8.67307 −0.788461
$$122$$ 0 0
$$123$$ 10.6637 0.961514
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −6.22077 −0.552004 −0.276002 0.961157i $$-0.589010\pi$$
−0.276002 + 0.961157i $$0.589010\pi$$
$$128$$ 0 0
$$129$$ −30.5303 −2.68805
$$130$$ 0 0
$$131$$ −11.7605 −1.02752 −0.513759 0.857934i $$-0.671748\pi$$
−0.513759 + 0.857934i $$0.671748\pi$$
$$132$$ 0 0
$$133$$ −0.990632 −0.0858987
$$134$$ 0 0
$$135$$ −7.05086 −0.606841
$$136$$ 0 0
$$137$$ −3.56691 −0.304742 −0.152371 0.988323i $$-0.548691\pi$$
−0.152371 + 0.988323i $$0.548691\pi$$
$$138$$ 0 0
$$139$$ −8.56199 −0.726219 −0.363109 0.931747i $$-0.618285\pi$$
−0.363109 + 0.931747i $$0.618285\pi$$
$$140$$ 0 0
$$141$$ −20.0415 −1.68780
$$142$$ 0 0
$$143$$ −0.949145 −0.0793715
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ 0 0
$$147$$ −17.9541 −1.48083
$$148$$ 0 0
$$149$$ 5.61285 0.459822 0.229911 0.973212i $$-0.426156\pi$$
0.229911 + 0.973212i $$0.426156\pi$$
$$150$$ 0 0
$$151$$ −10.7971 −0.878652 −0.439326 0.898328i $$-0.644783\pi$$
−0.439326 + 0.898328i $$0.644783\pi$$
$$152$$ 0 0
$$153$$ −43.1798 −3.49088
$$154$$ 0 0
$$155$$ −6.90321 −0.554479
$$156$$ 0 0
$$157$$ −2.28100 −0.182043 −0.0910217 0.995849i $$-0.529013\pi$$
−0.0910217 + 0.995849i $$0.529013\pi$$
$$158$$ 0 0
$$159$$ −18.6637 −1.48013
$$160$$ 0 0
$$161$$ −6.79706 −0.535683
$$162$$ 0 0
$$163$$ 16.3225 1.27848 0.639238 0.769009i $$-0.279250\pi$$
0.639238 + 0.769009i $$0.279250\pi$$
$$164$$ 0 0
$$165$$ 4.42864 0.344769
$$166$$ 0 0
$$167$$ 4.76986 0.369103 0.184551 0.982823i $$-0.440917\pi$$
0.184551 + 0.982823i $$0.440917\pi$$
$$168$$ 0 0
$$169$$ −12.6128 −0.970219
$$170$$ 0 0
$$171$$ −5.95407 −0.455319
$$172$$ 0 0
$$173$$ −4.23506 −0.321986 −0.160993 0.986956i $$-0.551470\pi$$
−0.160993 + 0.986956i $$0.551470\pi$$
$$174$$ 0 0
$$175$$ 0.903212 0.0682764
$$176$$ 0 0
$$177$$ −4.85728 −0.365095
$$178$$ 0 0
$$179$$ 9.71456 0.726100 0.363050 0.931770i $$-0.381735\pi$$
0.363050 + 0.931770i $$0.381735\pi$$
$$180$$ 0 0
$$181$$ −0.326929 −0.0243005 −0.0121502 0.999926i $$-0.503868\pi$$
−0.0121502 + 0.999926i $$0.503868\pi$$
$$182$$ 0 0
$$183$$ 5.41927 0.400604
$$184$$ 0 0
$$185$$ 3.95407 0.290709
$$186$$ 0 0
$$187$$ 12.1334 0.887279
$$188$$ 0 0
$$189$$ 6.36842 0.463234
$$190$$ 0 0
$$191$$ 14.9447 1.08136 0.540680 0.841228i $$-0.318167\pi$$
0.540680 + 0.841228i $$0.318167\pi$$
$$192$$ 0 0
$$193$$ −14.1476 −1.01837 −0.509185 0.860657i $$-0.670053\pi$$
−0.509185 + 0.860657i $$0.670053\pi$$
$$194$$ 0 0
$$195$$ −1.80642 −0.129361
$$196$$ 0 0
$$197$$ 5.70471 0.406444 0.203222 0.979133i $$-0.434859\pi$$
0.203222 + 0.979133i $$0.434859\pi$$
$$198$$ 0 0
$$199$$ 22.1432 1.56969 0.784845 0.619692i $$-0.212743\pi$$
0.784845 + 0.619692i $$0.212743\pi$$
$$200$$ 0 0
$$201$$ 33.4608 2.36014
$$202$$ 0 0
$$203$$ 0.903212 0.0633930
$$204$$ 0 0
$$205$$ −3.67307 −0.256538
$$206$$ 0 0
$$207$$ −40.8528 −2.83947
$$208$$ 0 0
$$209$$ 1.67307 0.115729
$$210$$ 0 0
$$211$$ −20.8430 −1.43489 −0.717445 0.696615i $$-0.754689\pi$$
−0.717445 + 0.696615i $$0.754689\pi$$
$$212$$ 0 0
$$213$$ −39.6958 −2.71991
$$214$$ 0 0
$$215$$ 10.5161 0.717189
$$216$$ 0 0
$$217$$ 6.23506 0.423264
$$218$$ 0 0
$$219$$ 29.5812 1.99891
$$220$$ 0 0
$$221$$ −4.94914 −0.332916
$$222$$ 0 0
$$223$$ −9.03657 −0.605133 −0.302567 0.953128i $$-0.597843\pi$$
−0.302567 + 0.953128i $$0.597843\pi$$
$$224$$ 0 0
$$225$$ 5.42864 0.361909
$$226$$ 0 0
$$227$$ 19.4050 1.28795 0.643977 0.765045i $$-0.277283\pi$$
0.643977 + 0.765045i $$0.277283\pi$$
$$228$$ 0 0
$$229$$ 25.6128 1.69254 0.846272 0.532751i $$-0.178842\pi$$
0.846272 + 0.532751i $$0.178842\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 0 0
$$233$$ −3.12399 −0.204659 −0.102330 0.994751i $$-0.532630\pi$$
−0.102330 + 0.994751i $$0.532630\pi$$
$$234$$ 0 0
$$235$$ 6.90321 0.450316
$$236$$ 0 0
$$237$$ −26.5303 −1.72333
$$238$$ 0 0
$$239$$ −13.9398 −0.901689 −0.450845 0.892602i $$-0.648877\pi$$
−0.450845 + 0.892602i $$0.648877\pi$$
$$240$$ 0 0
$$241$$ −18.4701 −1.18977 −0.594883 0.803813i $$-0.702802\pi$$
−0.594883 + 0.803813i $$0.702802\pi$$
$$242$$ 0 0
$$243$$ −9.00492 −0.577666
$$244$$ 0 0
$$245$$ 6.18421 0.395095
$$246$$ 0 0
$$247$$ −0.682439 −0.0434225
$$248$$ 0 0
$$249$$ 31.0923 1.97040
$$250$$ 0 0
$$251$$ −13.7921 −0.870552 −0.435276 0.900297i $$-0.643349\pi$$
−0.435276 + 0.900297i $$0.643349\pi$$
$$252$$ 0 0
$$253$$ 11.4795 0.721710
$$254$$ 0 0
$$255$$ 23.0923 1.44610
$$256$$ 0 0
$$257$$ −1.47949 −0.0922883 −0.0461442 0.998935i $$-0.514693\pi$$
−0.0461442 + 0.998935i $$0.514693\pi$$
$$258$$ 0 0
$$259$$ −3.57136 −0.221914
$$260$$ 0 0
$$261$$ 5.42864 0.336024
$$262$$ 0 0
$$263$$ 0.442930 0.0273122 0.0136561 0.999907i $$-0.495653\pi$$
0.0136561 + 0.999907i $$0.495653\pi$$
$$264$$ 0 0
$$265$$ 6.42864 0.394908
$$266$$ 0 0
$$267$$ −22.6637 −1.38700
$$268$$ 0 0
$$269$$ −3.93978 −0.240212 −0.120106 0.992761i $$-0.538324\pi$$
−0.120106 + 0.992761i $$0.538324\pi$$
$$270$$ 0 0
$$271$$ −6.20787 −0.377101 −0.188551 0.982063i $$-0.560379\pi$$
−0.188551 + 0.982063i $$0.560379\pi$$
$$272$$ 0 0
$$273$$ 1.63158 0.0987479
$$274$$ 0 0
$$275$$ −1.52543 −0.0919867
$$276$$ 0 0
$$277$$ −5.57136 −0.334751 −0.167375 0.985893i $$-0.553529\pi$$
−0.167375 + 0.985893i $$0.553529\pi$$
$$278$$ 0 0
$$279$$ 37.4750 2.24357
$$280$$ 0 0
$$281$$ 6.69535 0.399411 0.199705 0.979856i $$-0.436001\pi$$
0.199705 + 0.979856i $$0.436001\pi$$
$$282$$ 0 0
$$283$$ 25.8020 1.53377 0.766884 0.641785i $$-0.221806\pi$$
0.766884 + 0.641785i $$0.221806\pi$$
$$284$$ 0 0
$$285$$ 3.18421 0.188616
$$286$$ 0 0
$$287$$ 3.31756 0.195829
$$288$$ 0 0
$$289$$ 46.2672 2.72160
$$290$$ 0 0
$$291$$ −11.8666 −0.695635
$$292$$ 0 0
$$293$$ 18.8430 1.10082 0.550410 0.834895i $$-0.314472\pi$$
0.550410 + 0.834895i $$0.314472\pi$$
$$294$$ 0 0
$$295$$ 1.67307 0.0974099
$$296$$ 0 0
$$297$$ −10.7556 −0.624101
$$298$$ 0 0
$$299$$ −4.68244 −0.270792
$$300$$ 0 0
$$301$$ −9.49823 −0.547469
$$302$$ 0 0
$$303$$ −40.3783 −2.31967
$$304$$ 0 0
$$305$$ −1.86665 −0.106884
$$306$$ 0 0
$$307$$ −1.65878 −0.0946716 −0.0473358 0.998879i $$-0.515073\pi$$
−0.0473358 + 0.998879i $$0.515073\pi$$
$$308$$ 0 0
$$309$$ 37.5812 2.13792
$$310$$ 0 0
$$311$$ −21.3002 −1.20782 −0.603912 0.797051i $$-0.706392\pi$$
−0.603912 + 0.797051i $$0.706392\pi$$
$$312$$ 0 0
$$313$$ 8.62222 0.487356 0.243678 0.969856i $$-0.421646\pi$$
0.243678 + 0.969856i $$0.421646\pi$$
$$314$$ 0 0
$$315$$ −4.90321 −0.276265
$$316$$ 0 0
$$317$$ 27.5955 1.54992 0.774959 0.632012i $$-0.217771\pi$$
0.774959 + 0.632012i $$0.217771\pi$$
$$318$$ 0 0
$$319$$ −1.52543 −0.0854075
$$320$$ 0 0
$$321$$ 31.9496 1.78325
$$322$$ 0 0
$$323$$ 8.72393 0.485412
$$324$$ 0 0
$$325$$ 0.622216 0.0345143
$$326$$ 0 0
$$327$$ 52.3783 2.89652
$$328$$ 0 0
$$329$$ −6.23506 −0.343750
$$330$$ 0 0
$$331$$ 16.9131 0.929626 0.464813 0.885409i $$-0.346122\pi$$
0.464813 + 0.885409i $$0.346122\pi$$
$$332$$ 0 0
$$333$$ −21.4652 −1.17629
$$334$$ 0 0
$$335$$ −11.5254 −0.629701
$$336$$ 0 0
$$337$$ 11.9956 0.653439 0.326720 0.945121i $$-0.394057\pi$$
0.326720 + 0.945121i $$0.394057\pi$$
$$338$$ 0 0
$$339$$ −29.8479 −1.62112
$$340$$ 0 0
$$341$$ −10.5303 −0.570250
$$342$$ 0 0
$$343$$ −11.9081 −0.642979
$$344$$ 0 0
$$345$$ 21.8479 1.17625
$$346$$ 0 0
$$347$$ 6.14764 0.330023 0.165011 0.986292i $$-0.447234\pi$$
0.165011 + 0.986292i $$0.447234\pi$$
$$348$$ 0 0
$$349$$ 7.12399 0.381338 0.190669 0.981654i $$-0.438934\pi$$
0.190669 + 0.981654i $$0.438934\pi$$
$$350$$ 0 0
$$351$$ 4.38715 0.234169
$$352$$ 0 0
$$353$$ −16.9175 −0.900428 −0.450214 0.892921i $$-0.648652\pi$$
−0.450214 + 0.892921i $$0.648652\pi$$
$$354$$ 0 0
$$355$$ 13.6731 0.725691
$$356$$ 0 0
$$357$$ −20.8573 −1.10388
$$358$$ 0 0
$$359$$ −36.7096 −1.93746 −0.968730 0.248116i $$-0.920188\pi$$
−0.968730 + 0.248116i $$0.920188\pi$$
$$360$$ 0 0
$$361$$ −17.7971 −0.936687
$$362$$ 0 0
$$363$$ −25.1798 −1.32159
$$364$$ 0 0
$$365$$ −10.1891 −0.533323
$$366$$ 0 0
$$367$$ −8.41435 −0.439225 −0.219613 0.975587i $$-0.570479\pi$$
−0.219613 + 0.975587i $$0.570479\pi$$
$$368$$ 0 0
$$369$$ 19.9398 1.03802
$$370$$ 0 0
$$371$$ −5.80642 −0.301455
$$372$$ 0 0
$$373$$ 8.66370 0.448590 0.224295 0.974521i $$-0.427992\pi$$
0.224295 + 0.974521i $$0.427992\pi$$
$$374$$ 0 0
$$375$$ −2.90321 −0.149921
$$376$$ 0 0
$$377$$ 0.622216 0.0320457
$$378$$ 0 0
$$379$$ −2.76986 −0.142278 −0.0711390 0.997466i $$-0.522663\pi$$
−0.0711390 + 0.997466i $$0.522663\pi$$
$$380$$ 0 0
$$381$$ −18.0602 −0.925253
$$382$$ 0 0
$$383$$ −1.67752 −0.0857171 −0.0428585 0.999081i $$-0.513646\pi$$
−0.0428585 + 0.999081i $$0.513646\pi$$
$$384$$ 0 0
$$385$$ 1.37778 0.0702184
$$386$$ 0 0
$$387$$ −57.0879 −2.90194
$$388$$ 0 0
$$389$$ 5.77478 0.292793 0.146397 0.989226i $$-0.453232\pi$$
0.146397 + 0.989226i $$0.453232\pi$$
$$390$$ 0 0
$$391$$ 59.8578 3.02714
$$392$$ 0 0
$$393$$ −34.1432 −1.72230
$$394$$ 0 0
$$395$$ 9.13828 0.459797
$$396$$ 0 0
$$397$$ −29.9081 −1.50105 −0.750523 0.660844i $$-0.770198\pi$$
−0.750523 + 0.660844i $$0.770198\pi$$
$$398$$ 0 0
$$399$$ −2.87601 −0.143981
$$400$$ 0 0
$$401$$ 8.53035 0.425985 0.212993 0.977054i $$-0.431679\pi$$
0.212993 + 0.977054i $$0.431679\pi$$
$$402$$ 0 0
$$403$$ 4.29529 0.213963
$$404$$ 0 0
$$405$$ −4.18421 −0.207915
$$406$$ 0 0
$$407$$ 6.03164 0.298977
$$408$$ 0 0
$$409$$ 5.09234 0.251800 0.125900 0.992043i $$-0.459818\pi$$
0.125900 + 0.992043i $$0.459818\pi$$
$$410$$ 0 0
$$411$$ −10.3555 −0.510800
$$412$$ 0 0
$$413$$ −1.51114 −0.0743582
$$414$$ 0 0
$$415$$ −10.7096 −0.525715
$$416$$ 0 0
$$417$$ −24.8573 −1.21727
$$418$$ 0 0
$$419$$ −24.3368 −1.18893 −0.594465 0.804122i $$-0.702636\pi$$
−0.594465 + 0.804122i $$0.702636\pi$$
$$420$$ 0 0
$$421$$ −24.5018 −1.19414 −0.597072 0.802188i $$-0.703669\pi$$
−0.597072 + 0.802188i $$0.703669\pi$$
$$422$$ 0 0
$$423$$ −37.4750 −1.82210
$$424$$ 0 0
$$425$$ −7.95407 −0.385829
$$426$$ 0 0
$$427$$ 1.68598 0.0815902
$$428$$ 0 0
$$429$$ −2.75557 −0.133040
$$430$$ 0 0
$$431$$ −4.26671 −0.205520 −0.102760 0.994706i $$-0.532767\pi$$
−0.102760 + 0.994706i $$0.532767\pi$$
$$432$$ 0 0
$$433$$ 27.0049 1.29777 0.648887 0.760885i $$-0.275235\pi$$
0.648887 + 0.760885i $$0.275235\pi$$
$$434$$ 0 0
$$435$$ −2.90321 −0.139198
$$436$$ 0 0
$$437$$ 8.25380 0.394833
$$438$$ 0 0
$$439$$ 2.03164 0.0969650 0.0484825 0.998824i $$-0.484561\pi$$
0.0484825 + 0.998824i $$0.484561\pi$$
$$440$$ 0 0
$$441$$ −33.5718 −1.59866
$$442$$ 0 0
$$443$$ −3.46520 −0.164637 −0.0823184 0.996606i $$-0.526232\pi$$
−0.0823184 + 0.996606i $$0.526232\pi$$
$$444$$ 0 0
$$445$$ 7.80642 0.370060
$$446$$ 0 0
$$447$$ 16.2953 0.770741
$$448$$ 0 0
$$449$$ 37.3590 1.76308 0.881541 0.472107i $$-0.156506\pi$$
0.881541 + 0.472107i $$0.156506\pi$$
$$450$$ 0 0
$$451$$ −5.60300 −0.263835
$$452$$ 0 0
$$453$$ −31.3461 −1.47277
$$454$$ 0 0
$$455$$ −0.561993 −0.0263466
$$456$$ 0 0
$$457$$ 13.4509 0.629207 0.314604 0.949223i $$-0.398128\pi$$
0.314604 + 0.949223i $$0.398128\pi$$
$$458$$ 0 0
$$459$$ −56.0830 −2.61773
$$460$$ 0 0
$$461$$ 16.2766 0.758075 0.379037 0.925381i $$-0.376255\pi$$
0.379037 + 0.925381i $$0.376255\pi$$
$$462$$ 0 0
$$463$$ 30.3926 1.41246 0.706231 0.707982i $$-0.250394\pi$$
0.706231 + 0.707982i $$0.250394\pi$$
$$464$$ 0 0
$$465$$ −20.0415 −0.929402
$$466$$ 0 0
$$467$$ −1.18865 −0.0550043 −0.0275022 0.999622i $$-0.508755\pi$$
−0.0275022 + 0.999622i $$0.508755\pi$$
$$468$$ 0 0
$$469$$ 10.4099 0.480685
$$470$$ 0 0
$$471$$ −6.62222 −0.305136
$$472$$ 0 0
$$473$$ 16.0415 0.737588
$$474$$ 0 0
$$475$$ −1.09679 −0.0503241
$$476$$ 0 0
$$477$$ −34.8988 −1.59790
$$478$$ 0 0
$$479$$ −41.0464 −1.87546 −0.937729 0.347367i $$-0.887076\pi$$
−0.937729 + 0.347367i $$0.887076\pi$$
$$480$$ 0 0
$$481$$ −2.46028 −0.112179
$$482$$ 0 0
$$483$$ −19.7333 −0.897896
$$484$$ 0 0
$$485$$ 4.08742 0.185600
$$486$$ 0 0
$$487$$ 10.1476 0.459834 0.229917 0.973210i $$-0.426155\pi$$
0.229917 + 0.973210i $$0.426155\pi$$
$$488$$ 0 0
$$489$$ 47.3876 2.14294
$$490$$ 0 0
$$491$$ −29.2083 −1.31815 −0.659077 0.752075i $$-0.729053\pi$$
−0.659077 + 0.752075i $$0.729053\pi$$
$$492$$ 0 0
$$493$$ −7.95407 −0.358233
$$494$$ 0 0
$$495$$ 8.28100 0.372203
$$496$$ 0 0
$$497$$ −12.3497 −0.553959
$$498$$ 0 0
$$499$$ 21.9813 0.984017 0.492008 0.870591i $$-0.336263\pi$$
0.492008 + 0.870591i $$0.336263\pi$$
$$500$$ 0 0
$$501$$ 13.8479 0.618679
$$502$$ 0 0
$$503$$ −5.77923 −0.257683 −0.128841 0.991665i $$-0.541126\pi$$
−0.128841 + 0.991665i $$0.541126\pi$$
$$504$$ 0 0
$$505$$ 13.9081 0.618904
$$506$$ 0 0
$$507$$ −36.6178 −1.62625
$$508$$ 0 0
$$509$$ −13.6543 −0.605218 −0.302609 0.953115i $$-0.597858\pi$$
−0.302609 + 0.953115i $$0.597858\pi$$
$$510$$ 0 0
$$511$$ 9.20294 0.407114
$$512$$ 0 0
$$513$$ −7.73329 −0.341433
$$514$$ 0 0
$$515$$ −12.9447 −0.570412
$$516$$ 0 0
$$517$$ 10.5303 0.463124
$$518$$ 0 0
$$519$$ −12.2953 −0.539703
$$520$$ 0 0
$$521$$ −19.6731 −0.861893 −0.430946 0.902378i $$-0.641820\pi$$
−0.430946 + 0.902378i $$0.641820\pi$$
$$522$$ 0 0
$$523$$ −15.1383 −0.661951 −0.330975 0.943639i $$-0.607378\pi$$
−0.330975 + 0.943639i $$0.607378\pi$$
$$524$$ 0 0
$$525$$ 2.62222 0.114443
$$526$$ 0 0
$$527$$ −54.9086 −2.39186
$$528$$ 0 0
$$529$$ 33.6321 1.46226
$$530$$ 0 0
$$531$$ −9.08250 −0.394147
$$532$$ 0 0
$$533$$ 2.28544 0.0989935
$$534$$ 0 0
$$535$$ −11.0049 −0.475784
$$536$$ 0 0
$$537$$ 28.2034 1.21707
$$538$$ 0 0
$$539$$ 9.43356 0.406332
$$540$$ 0 0
$$541$$ −2.68244 −0.115327 −0.0576635 0.998336i $$-0.518365\pi$$
−0.0576635 + 0.998336i $$0.518365\pi$$
$$542$$ 0 0
$$543$$ −0.949145 −0.0407317
$$544$$ 0 0
$$545$$ −18.0415 −0.772812
$$546$$ 0 0
$$547$$ −15.3635 −0.656896 −0.328448 0.944522i $$-0.606525\pi$$
−0.328448 + 0.944522i $$0.606525\pi$$
$$548$$ 0 0
$$549$$ 10.1334 0.432481
$$550$$ 0 0
$$551$$ −1.09679 −0.0467247
$$552$$ 0 0
$$553$$ −8.25380 −0.350987
$$554$$ 0 0
$$555$$ 11.4795 0.487277
$$556$$ 0 0
$$557$$ 9.87955 0.418610 0.209305 0.977850i $$-0.432880\pi$$
0.209305 + 0.977850i $$0.432880\pi$$
$$558$$ 0 0
$$559$$ −6.54326 −0.276750
$$560$$ 0 0
$$561$$ 35.2257 1.48723
$$562$$ 0 0
$$563$$ −27.4938 −1.15872 −0.579362 0.815070i $$-0.696698\pi$$
−0.579362 + 0.815070i $$0.696698\pi$$
$$564$$ 0 0
$$565$$ 10.2810 0.432525
$$566$$ 0 0
$$567$$ 3.77923 0.158713
$$568$$ 0 0
$$569$$ 17.3590 0.727729 0.363865 0.931452i $$-0.381457\pi$$
0.363865 + 0.931452i $$0.381457\pi$$
$$570$$ 0 0
$$571$$ −25.4479 −1.06496 −0.532480 0.846443i $$-0.678740\pi$$
−0.532480 + 0.846443i $$0.678740\pi$$
$$572$$ 0 0
$$573$$ 43.3876 1.81254
$$574$$ 0 0
$$575$$ −7.52543 −0.313832
$$576$$ 0 0
$$577$$ −10.6178 −0.442024 −0.221012 0.975271i $$-0.570936\pi$$
−0.221012 + 0.975271i $$0.570936\pi$$
$$578$$ 0 0
$$579$$ −41.0736 −1.70696
$$580$$ 0 0
$$581$$ 9.67307 0.401307
$$582$$ 0 0
$$583$$ 9.80642 0.406141
$$584$$ 0 0
$$585$$ −3.37778 −0.139654
$$586$$ 0 0
$$587$$ −8.94470 −0.369187 −0.184594 0.982815i $$-0.559097\pi$$
−0.184594 + 0.982815i $$0.559097\pi$$
$$588$$ 0 0
$$589$$ −7.57136 −0.311972
$$590$$ 0 0
$$591$$ 16.5620 0.681269
$$592$$ 0 0
$$593$$ 14.1619 0.581561 0.290780 0.956790i $$-0.406085\pi$$
0.290780 + 0.956790i $$0.406085\pi$$
$$594$$ 0 0
$$595$$ 7.18421 0.294524
$$596$$ 0 0
$$597$$ 64.2864 2.63107
$$598$$ 0 0
$$599$$ 22.5575 0.921676 0.460838 0.887484i $$-0.347549\pi$$
0.460838 + 0.887484i $$0.347549\pi$$
$$600$$ 0 0
$$601$$ −40.6133 −1.65665 −0.828326 0.560246i $$-0.810706\pi$$
−0.828326 + 0.560246i $$0.810706\pi$$
$$602$$ 0 0
$$603$$ 62.5674 2.54794
$$604$$ 0 0
$$605$$ 8.67307 0.352610
$$606$$ 0 0
$$607$$ −13.5955 −0.551824 −0.275912 0.961183i $$-0.588980\pi$$
−0.275912 + 0.961183i $$0.588980\pi$$
$$608$$ 0 0
$$609$$ 2.62222 0.106258
$$610$$ 0 0
$$611$$ −4.29529 −0.173769
$$612$$ 0 0
$$613$$ 42.0830 1.69972 0.849858 0.527012i $$-0.176688\pi$$
0.849858 + 0.527012i $$0.176688\pi$$
$$614$$ 0 0
$$615$$ −10.6637 −0.430002
$$616$$ 0 0
$$617$$ 33.5067 1.34893 0.674464 0.738307i $$-0.264375\pi$$
0.674464 + 0.738307i $$0.264375\pi$$
$$618$$ 0 0
$$619$$ −14.6780 −0.589958 −0.294979 0.955504i $$-0.595313\pi$$
−0.294979 + 0.955504i $$0.595313\pi$$
$$620$$ 0 0
$$621$$ −53.0607 −2.12925
$$622$$ 0 0
$$623$$ −7.05086 −0.282487
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 4.85728 0.193981
$$628$$ 0 0
$$629$$ 31.4509 1.25403
$$630$$ 0 0
$$631$$ −11.3176 −0.450545 −0.225273 0.974296i $$-0.572327\pi$$
−0.225273 + 0.974296i $$0.572327\pi$$
$$632$$ 0 0
$$633$$ −60.5116 −2.40512
$$634$$ 0 0
$$635$$ 6.22077 0.246864
$$636$$ 0 0
$$637$$ −3.84791 −0.152460
$$638$$ 0 0
$$639$$ −74.2262 −2.93634
$$640$$ 0 0
$$641$$ 34.8988 1.37842 0.689209 0.724562i $$-0.257958\pi$$
0.689209 + 0.724562i $$0.257958\pi$$
$$642$$ 0 0
$$643$$ −41.9768 −1.65540 −0.827702 0.561168i $$-0.810352\pi$$
−0.827702 + 0.561168i $$0.810352\pi$$
$$644$$ 0 0
$$645$$ 30.5303 1.20213
$$646$$ 0 0
$$647$$ −5.46520 −0.214859 −0.107430 0.994213i $$-0.534262\pi$$
−0.107430 + 0.994213i $$0.534262\pi$$
$$648$$ 0 0
$$649$$ 2.55215 0.100181
$$650$$ 0 0
$$651$$ 18.1017 0.709462
$$652$$ 0 0
$$653$$ 8.76986 0.343191 0.171596 0.985167i $$-0.445108\pi$$
0.171596 + 0.985167i $$0.445108\pi$$
$$654$$ 0 0
$$655$$ 11.7605 0.459520
$$656$$ 0 0
$$657$$ 55.3131 2.15797
$$658$$ 0 0
$$659$$ 3.29036 0.128174 0.0640872 0.997944i $$-0.479586\pi$$
0.0640872 + 0.997944i $$0.479586\pi$$
$$660$$ 0 0
$$661$$ −19.7560 −0.768421 −0.384211 0.923246i $$-0.625526\pi$$
−0.384211 + 0.923246i $$0.625526\pi$$
$$662$$ 0 0
$$663$$ −14.3684 −0.558023
$$664$$ 0 0
$$665$$ 0.990632 0.0384151
$$666$$ 0 0
$$667$$ −7.52543 −0.291386
$$668$$ 0 0
$$669$$ −26.2351 −1.01431
$$670$$ 0 0
$$671$$ −2.84743 −0.109924
$$672$$ 0 0
$$673$$ 44.3970 1.71138 0.855689 0.517490i $$-0.173134\pi$$
0.855689 + 0.517490i $$0.173134\pi$$
$$674$$ 0 0
$$675$$ 7.05086 0.271388
$$676$$ 0 0
$$677$$ −6.09726 −0.234337 −0.117168 0.993112i $$-0.537382\pi$$
−0.117168 + 0.993112i $$0.537382\pi$$
$$678$$ 0 0
$$679$$ −3.69181 −0.141679
$$680$$ 0 0
$$681$$ 56.3368 2.15883
$$682$$ 0 0
$$683$$ 37.9224 1.45106 0.725531 0.688190i $$-0.241594\pi$$
0.725531 + 0.688190i $$0.241594\pi$$
$$684$$ 0 0
$$685$$ 3.56691 0.136285
$$686$$ 0 0
$$687$$ 74.3595 2.83699
$$688$$ 0 0
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ 13.3145 0.506507 0.253254 0.967400i $$-0.418499\pi$$
0.253254 + 0.967400i $$0.418499\pi$$
$$692$$ 0 0
$$693$$ −7.47949 −0.284123
$$694$$ 0 0
$$695$$ 8.56199 0.324775
$$696$$ 0 0
$$697$$ −29.2159 −1.10663
$$698$$ 0 0
$$699$$ −9.06959 −0.343043
$$700$$ 0 0
$$701$$ 23.4893 0.887180 0.443590 0.896230i $$-0.353705\pi$$
0.443590 + 0.896230i $$0.353705\pi$$
$$702$$ 0 0
$$703$$ 4.33677 0.163565
$$704$$ 0 0
$$705$$ 20.0415 0.754806
$$706$$ 0 0
$$707$$ −12.5620 −0.472442
$$708$$ 0 0
$$709$$ −11.6731 −0.438391 −0.219196 0.975681i $$-0.570343\pi$$
−0.219196 + 0.975681i $$0.570343\pi$$
$$710$$ 0 0
$$711$$ −49.6084 −1.86046
$$712$$ 0 0
$$713$$ −51.9496 −1.94553
$$714$$ 0 0
$$715$$ 0.949145 0.0354960
$$716$$ 0 0
$$717$$ −40.4701 −1.51138
$$718$$ 0 0
$$719$$ −29.5526 −1.10213 −0.551063 0.834463i $$-0.685778\pi$$
−0.551063 + 0.834463i $$0.685778\pi$$
$$720$$ 0 0
$$721$$ 11.6918 0.435426
$$722$$ 0 0
$$723$$ −53.6227 −1.99425
$$724$$ 0 0
$$725$$ 1.00000 0.0371391
$$726$$ 0 0
$$727$$ −3.88094 −0.143936 −0.0719680 0.997407i $$-0.522928\pi$$
−0.0719680 + 0.997407i $$0.522928\pi$$
$$728$$ 0 0
$$729$$ −38.6958 −1.43318
$$730$$ 0 0
$$731$$ 83.6454 3.09374
$$732$$ 0 0
$$733$$ −14.8845 −0.549771 −0.274885 0.961477i $$-0.588640\pi$$
−0.274885 + 0.961477i $$0.588640\pi$$
$$734$$ 0 0
$$735$$ 17.9541 0.662246
$$736$$ 0 0
$$737$$ −17.5812 −0.647612
$$738$$ 0 0
$$739$$ 2.24935 0.0827438 0.0413719 0.999144i $$-0.486827\pi$$
0.0413719 + 0.999144i $$0.486827\pi$$
$$740$$ 0 0
$$741$$ −1.98126 −0.0727836
$$742$$ 0 0
$$743$$ 3.46520 0.127126 0.0635630 0.997978i $$-0.479754\pi$$
0.0635630 + 0.997978i $$0.479754\pi$$
$$744$$ 0 0
$$745$$ −5.61285 −0.205639
$$746$$ 0 0
$$747$$ 58.1388 2.12719
$$748$$ 0 0
$$749$$ 9.93978 0.363192
$$750$$ 0 0
$$751$$ −3.16992 −0.115672 −0.0578360 0.998326i $$-0.518420\pi$$
−0.0578360 + 0.998326i $$0.518420\pi$$
$$752$$ 0 0
$$753$$ −40.0415 −1.45919
$$754$$ 0 0
$$755$$ 10.7971 0.392945
$$756$$ 0 0
$$757$$ 52.0785 1.89283 0.946413 0.322958i $$-0.104677\pi$$
0.946413 + 0.322958i $$0.104677\pi$$
$$758$$ 0 0
$$759$$ 33.3274 1.20971
$$760$$ 0 0
$$761$$ −14.9777 −0.542942 −0.271471 0.962447i $$-0.587510\pi$$
−0.271471 + 0.962447i $$0.587510\pi$$
$$762$$ 0 0
$$763$$ 16.2953 0.589929
$$764$$ 0 0
$$765$$ 43.1798 1.56117
$$766$$ 0 0
$$767$$ −1.04101 −0.0375887
$$768$$ 0 0
$$769$$ −1.90813 −0.0688091 −0.0344045 0.999408i $$-0.510953\pi$$
−0.0344045 + 0.999408i $$0.510953\pi$$
$$770$$ 0 0
$$771$$ −4.29529 −0.154691
$$772$$ 0 0
$$773$$ −21.7891 −0.783698 −0.391849 0.920029i $$-0.628165\pi$$
−0.391849 + 0.920029i $$0.628165\pi$$
$$774$$ 0 0
$$775$$ 6.90321 0.247971
$$776$$ 0 0
$$777$$ −10.3684 −0.371965
$$778$$ 0 0
$$779$$ −4.02858 −0.144339
$$780$$ 0 0
$$781$$ 20.8573 0.746332
$$782$$ 0 0
$$783$$ 7.05086 0.251977
$$784$$ 0 0
$$785$$ 2.28100 0.0814122
$$786$$ 0 0
$$787$$ −18.1388 −0.646577 −0.323288 0.946301i $$-0.604788\pi$$
−0.323288 + 0.946301i $$0.604788\pi$$
$$788$$ 0 0
$$789$$ 1.28592 0.0457799
$$790$$ 0 0
$$791$$ −9.28592 −0.330169
$$792$$ 0 0
$$793$$ 1.16146 0.0412445
$$794$$ 0 0
$$795$$ 18.6637 0.661933
$$796$$ 0 0
$$797$$ −2.96343 −0.104970 −0.0524851 0.998622i $$-0.516714\pi$$
−0.0524851 + 0.998622i $$0.516714\pi$$
$$798$$ 0 0
$$799$$ 54.9086 1.94253
$$800$$ 0 0
$$801$$ −42.3783 −1.49736
$$802$$ 0 0
$$803$$ −15.5428 −0.548493
$$804$$ 0 0
$$805$$ 6.79706 0.239565
$$806$$ 0 0
$$807$$ −11.4380 −0.402637
$$808$$ 0 0
$$809$$ −26.2953 −0.924493 −0.462247 0.886751i $$-0.652956\pi$$
−0.462247 + 0.886751i $$0.652956\pi$$
$$810$$ 0 0
$$811$$ 24.3783 0.856037 0.428018 0.903770i $$-0.359212\pi$$
0.428018 + 0.903770i $$0.359212\pi$$
$$812$$ 0 0
$$813$$ −18.0228 −0.632085
$$814$$ 0 0
$$815$$ −16.3225 −0.571752
$$816$$ 0 0
$$817$$ 11.5339 0.403520
$$818$$ 0 0
$$819$$ 3.05086 0.106606
$$820$$ 0 0
$$821$$ −1.52987 −0.0533929 −0.0266965 0.999644i $$-0.508499\pi$$
−0.0266965 + 0.999644i $$0.508499\pi$$
$$822$$ 0 0
$$823$$ −46.7195 −1.62854 −0.814269 0.580487i $$-0.802862\pi$$
−0.814269 + 0.580487i $$0.802862\pi$$
$$824$$ 0 0
$$825$$ −4.42864 −0.154185
$$826$$ 0 0
$$827$$ −29.6499 −1.03103 −0.515514 0.856881i $$-0.672399\pi$$
−0.515514 + 0.856881i $$0.672399\pi$$
$$828$$ 0 0
$$829$$ 8.79706 0.305534 0.152767 0.988262i $$-0.451182\pi$$
0.152767 + 0.988262i $$0.451182\pi$$
$$830$$ 0 0
$$831$$ −16.1748 −0.561099
$$832$$ 0 0
$$833$$ 49.1896 1.70432
$$834$$ 0 0
$$835$$ −4.76986 −0.165068
$$836$$ 0 0
$$837$$ 48.6735 1.68240
$$838$$ 0 0
$$839$$ −11.3319 −0.391219 −0.195609 0.980682i $$-0.562668\pi$$
−0.195609 + 0.980682i $$0.562668\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ 19.4380 0.669481
$$844$$ 0 0
$$845$$ 12.6128 0.433895
$$846$$ 0 0
$$847$$ −7.83362 −0.269166
$$848$$ 0 0
$$849$$ 74.9086 2.57086
$$850$$ 0 0
$$851$$ 29.7560 1.02002
$$852$$ 0 0
$$853$$ −54.8845 −1.87921 −0.939604 0.342263i $$-0.888807\pi$$
−0.939604 + 0.342263i $$0.888807\pi$$
$$854$$ 0 0
$$855$$ 5.95407 0.203625
$$856$$ 0 0
$$857$$ 36.4385 1.24471 0.622357 0.782733i $$-0.286175\pi$$
0.622357 + 0.782733i $$0.286175\pi$$
$$858$$ 0 0
$$859$$ 1.72885 0.0589875 0.0294938 0.999565i $$-0.490610\pi$$
0.0294938 + 0.999565i $$0.490610\pi$$
$$860$$ 0 0
$$861$$ 9.63158 0.328243
$$862$$ 0 0
$$863$$ 9.40192 0.320045 0.160023 0.987113i $$-0.448843\pi$$
0.160023 + 0.987113i $$0.448843\pi$$
$$864$$ 0 0
$$865$$ 4.23506 0.143996
$$866$$ 0 0
$$867$$ 134.323 4.56186
$$868$$ 0 0
$$869$$ 13.9398 0.472875
$$870$$ 0 0
$$871$$ 7.17130 0.242990
$$872$$ 0 0
$$873$$ −22.1891 −0.750988
$$874$$ 0 0
$$875$$ −0.903212 −0.0305341
$$876$$ 0 0
$$877$$ −8.91750 −0.301123 −0.150561 0.988601i $$-0.548108\pi$$
−0.150561 + 0.988601i $$0.548108\pi$$
$$878$$ 0 0
$$879$$ 54.7052 1.84516
$$880$$ 0 0
$$881$$ −42.1245 −1.41921 −0.709605 0.704600i $$-0.751126\pi$$
−0.709605 + 0.704600i $$0.751126\pi$$
$$882$$ 0 0
$$883$$ −38.4340 −1.29341 −0.646704 0.762741i $$-0.723853\pi$$
−0.646704 + 0.762741i $$0.723853\pi$$
$$884$$ 0 0
$$885$$ 4.85728 0.163276
$$886$$ 0 0
$$887$$ 38.6365 1.29729 0.648643 0.761092i $$-0.275337\pi$$
0.648643 + 0.761092i $$0.275337\pi$$
$$888$$ 0 0
$$889$$ −5.61868 −0.188444
$$890$$ 0 0
$$891$$ −6.38271 −0.213829
$$892$$ 0 0
$$893$$ 7.57136 0.253366
$$894$$ 0 0
$$895$$ −9.71456 −0.324722
$$896$$ 0 0
$$897$$ −13.5941 −0.453894
$$898$$ 0 0
$$899$$ 6.90321 0.230235
$$900$$ 0 0
$$901$$ 51.1338 1.70351
$$902$$ 0 0
$$903$$ −27.5754 −0.917651
$$904$$ 0 0
$$905$$ 0.326929 0.0108675
$$906$$ 0 0
$$907$$ 0.534795 0.0177576 0.00887880 0.999961i $$-0.497174\pi$$
0.00887880 + 0.999961i $$0.497174\pi$$
$$908$$ 0 0
$$909$$ −75.5022 −2.50425
$$910$$ 0 0
$$911$$ 23.6686 0.784177 0.392088 0.919928i $$-0.371753\pi$$
0.392088 + 0.919928i $$0.371753\pi$$
$$912$$ 0 0
$$913$$ −16.3368 −0.540668
$$914$$ 0 0
$$915$$ −5.41927 −0.179156
$$916$$ 0 0
$$917$$ −10.6222 −0.350776
$$918$$ 0 0
$$919$$ −35.7748 −1.18010 −0.590051 0.807366i $$-0.700892\pi$$
−0.590051 + 0.807366i $$0.700892\pi$$
$$920$$ 0 0
$$921$$ −4.81579 −0.158686
$$922$$ 0 0
$$923$$ −8.50760 −0.280031
$$924$$ 0 0
$$925$$ −3.95407 −0.130009
$$926$$ 0 0
$$927$$ 70.2721 2.30804
$$928$$ 0 0
$$929$$ −52.7753 −1.73150 −0.865750 0.500477i $$-0.833158\pi$$
−0.865750 + 0.500477i $$0.833158\pi$$
$$930$$ 0 0
$$931$$ 6.78277 0.222296
$$932$$ 0 0
$$933$$ −61.8390 −2.02452
$$934$$ 0 0
$$935$$ −12.1334 −0.396803
$$936$$ 0 0
$$937$$ 42.1245 1.37615 0.688073 0.725641i $$-0.258457\pi$$
0.688073 + 0.725641i $$0.258457\pi$$
$$938$$ 0 0
$$939$$ 25.0321 0.816892
$$940$$ 0 0
$$941$$ 3.89829 0.127081 0.0635403 0.997979i $$-0.479761\pi$$
0.0635403 + 0.997979i $$0.479761\pi$$
$$942$$ 0 0
$$943$$ −27.6414 −0.900129
$$944$$ 0 0
$$945$$ −6.36842 −0.207165
$$946$$ 0 0
$$947$$ 9.56691 0.310883 0.155441 0.987845i $$-0.450320\pi$$
0.155441 + 0.987845i $$0.450320\pi$$
$$948$$ 0 0
$$949$$ 6.33984 0.205800
$$950$$ 0 0
$$951$$ 80.1156 2.59793
$$952$$ 0 0
$$953$$ 27.2070 0.881320 0.440660 0.897674i $$-0.354744\pi$$
0.440660 + 0.897674i $$0.354744\pi$$
$$954$$ 0 0
$$955$$ −14.9447 −0.483599
$$956$$ 0 0
$$957$$ −4.42864 −0.143158
$$958$$ 0 0
$$959$$ −3.22168 −0.104033
$$960$$ 0 0
$$961$$ 16.6543 0.537237
$$962$$ 0 0
$$963$$ 59.7418 1.92515
$$964$$ 0 0
$$965$$ 14.1476 0.455429
$$966$$ 0 0
$$967$$ 16.8015 0.540300 0.270150 0.962818i $$-0.412927\pi$$
0.270150 + 0.962818i $$0.412927\pi$$
$$968$$ 0 0
$$969$$ 25.3274 0.813633
$$970$$ 0 0
$$971$$ −17.4465 −0.559884 −0.279942 0.960017i $$-0.590315\pi$$
−0.279942 + 0.960017i $$0.590315\pi$$
$$972$$ 0 0
$$973$$ −7.73329 −0.247918
$$974$$ 0 0
$$975$$ 1.80642 0.0578519
$$976$$ 0 0
$$977$$ 32.0513 1.02541 0.512706 0.858564i $$-0.328643\pi$$
0.512706 + 0.858564i $$0.328643\pi$$
$$978$$ 0 0
$$979$$ 11.9081 0.380586
$$980$$ 0 0
$$981$$ 97.9407 3.12701
$$982$$ 0 0
$$983$$ 16.5259 0.527094 0.263547 0.964646i $$-0.415108\pi$$
0.263547 + 0.964646i $$0.415108\pi$$
$$984$$ 0 0
$$985$$ −5.70471 −0.181767
$$986$$ 0 0
$$987$$ −18.1017 −0.576184
$$988$$ 0 0
$$989$$ 79.1378 2.51644
$$990$$ 0 0
$$991$$ 9.34920 0.296987 0.148494 0.988913i $$-0.452558\pi$$
0.148494 + 0.988913i $$0.452558\pi$$
$$992$$ 0 0
$$993$$ 49.1022 1.55821
$$994$$ 0 0
$$995$$ −22.1432 −0.701987
$$996$$ 0 0
$$997$$ 15.9956 0.506584 0.253292 0.967390i $$-0.418487\pi$$
0.253292 + 0.967390i $$0.418487\pi$$
$$998$$ 0 0
$$999$$ −27.8796 −0.882070
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.br.1.3 3
4.3 odd 2 9280.2.a.bj.1.1 3
8.3 odd 2 145.2.a.c.1.2 3
8.5 even 2 2320.2.a.n.1.1 3
24.11 even 2 1305.2.a.p.1.2 3
40.3 even 4 725.2.b.e.349.3 6
40.19 odd 2 725.2.a.e.1.2 3
40.27 even 4 725.2.b.e.349.4 6
56.27 even 2 7105.2.a.o.1.2 3
120.59 even 2 6525.2.a.be.1.2 3
232.115 odd 2 4205.2.a.f.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.2 3 8.3 odd 2
725.2.a.e.1.2 3 40.19 odd 2
725.2.b.e.349.3 6 40.3 even 4
725.2.b.e.349.4 6 40.27 even 4
1305.2.a.p.1.2 3 24.11 even 2
2320.2.a.n.1.1 3 8.5 even 2
4205.2.a.f.1.2 3 232.115 odd 2
6525.2.a.be.1.2 3 120.59 even 2
7105.2.a.o.1.2 3 56.27 even 2
9280.2.a.bj.1.1 3 4.3 odd 2
9280.2.a.br.1.3 3 1.1 even 1 trivial