# Properties

 Label 3332.2.a.u.1.5 Level $3332$ Weight $2$ Character 3332.1 Self dual yes Analytic conductor $26.606$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3332.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: $$26.6061539535$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} - 8x^{6} + 36x^{5} + 17x^{4} - 76x^{3} - 20x^{2} + 44x + 17$$ x^8 - 4*x^7 - 8*x^6 + 36*x^5 + 17*x^4 - 76*x^3 - 20*x^2 + 44*x + 17 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$1.26400$$ of defining polynomial Character $$\chi$$ $$=$$ 3332.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.26400 q^{3} -1.77431 q^{5} -1.40230 q^{9} +O(q^{10})$$ $$q+1.26400 q^{3} -1.77431 q^{5} -1.40230 q^{9} +0.318204 q^{11} +0.205935 q^{13} -2.24272 q^{15} -1.00000 q^{17} +5.45902 q^{19} +9.43694 q^{23} -1.85184 q^{25} -5.56451 q^{27} -8.61859 q^{29} +4.94814 q^{31} +0.402209 q^{33} +7.64938 q^{37} +0.260301 q^{39} +1.44153 q^{41} -1.98578 q^{43} +2.48812 q^{45} -1.56955 q^{47} -1.26400 q^{51} -13.5386 q^{53} -0.564590 q^{55} +6.90020 q^{57} +13.2075 q^{59} +3.37365 q^{61} -0.365391 q^{65} +13.3478 q^{67} +11.9283 q^{69} -2.62870 q^{71} +3.86935 q^{73} -2.34072 q^{75} +12.2324 q^{79} -2.82663 q^{81} +10.4915 q^{83} +1.77431 q^{85} -10.8939 q^{87} +0.00620724 q^{89} +6.25445 q^{93} -9.68597 q^{95} +4.93669 q^{97} -0.446218 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{3} + 4 q^{5} + 8 q^{9}+O(q^{10})$$ 8 * q + 4 * q^3 + 4 * q^5 + 8 * q^9 $$8 q + 4 q^{3} + 4 q^{5} + 8 q^{9} - 4 q^{11} + 20 q^{13} + 12 q^{15} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 8 q^{25} + 28 q^{27} - 16 q^{29} - 8 q^{31} + 16 q^{33} + 8 q^{37} + 20 q^{39} + 12 q^{41} - 4 q^{43} + 20 q^{45} + 4 q^{47} - 4 q^{51} + 12 q^{55} - 16 q^{59} + 32 q^{61} + 44 q^{69} + 24 q^{73} + 24 q^{75} + 4 q^{79} + 36 q^{81} + 28 q^{83} - 4 q^{85} - 40 q^{87} + 20 q^{89} - 16 q^{93} - 20 q^{95} + 56 q^{97} - 8 q^{99}+O(q^{100})$$ 8 * q + 4 * q^3 + 4 * q^5 + 8 * q^9 - 4 * q^11 + 20 * q^13 + 12 * q^15 - 8 * q^17 + 8 * q^19 + 4 * q^23 + 8 * q^25 + 28 * q^27 - 16 * q^29 - 8 * q^31 + 16 * q^33 + 8 * q^37 + 20 * q^39 + 12 * q^41 - 4 * q^43 + 20 * q^45 + 4 * q^47 - 4 * q^51 + 12 * q^55 - 16 * q^59 + 32 * q^61 + 44 * q^69 + 24 * q^73 + 24 * q^75 + 4 * q^79 + 36 * q^81 + 28 * q^83 - 4 * q^85 - 40 * q^87 + 20 * q^89 - 16 * q^93 - 20 * q^95 + 56 * q^97 - 8 * 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
$$3$$ 1.26400 0.729771 0.364885 0.931052i $$-0.381108\pi$$
0.364885 + 0.931052i $$0.381108\pi$$
$$4$$ 0 0
$$5$$ −1.77431 −0.793494 −0.396747 0.917928i $$-0.629861\pi$$
−0.396747 + 0.917928i $$0.629861\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −1.40230 −0.467435
$$10$$ 0 0
$$11$$ 0.318204 0.0959420 0.0479710 0.998849i $$-0.484724\pi$$
0.0479710 + 0.998849i $$0.484724\pi$$
$$12$$ 0 0
$$13$$ 0.205935 0.0571160 0.0285580 0.999592i $$-0.490908\pi$$
0.0285580 + 0.999592i $$0.490908\pi$$
$$14$$ 0 0
$$15$$ −2.24272 −0.579069
$$16$$ 0 0
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ 5.45902 1.25238 0.626192 0.779669i $$-0.284612\pi$$
0.626192 + 0.779669i $$0.284612\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 9.43694 1.96774 0.983870 0.178888i $$-0.0572499\pi$$
0.983870 + 0.178888i $$0.0572499\pi$$
$$24$$ 0 0
$$25$$ −1.85184 −0.370368
$$26$$ 0 0
$$27$$ −5.56451 −1.07089
$$28$$ 0 0
$$29$$ −8.61859 −1.60043 −0.800216 0.599712i $$-0.795282\pi$$
−0.800216 + 0.599712i $$0.795282\pi$$
$$30$$ 0 0
$$31$$ 4.94814 0.888712 0.444356 0.895850i $$-0.353432\pi$$
0.444356 + 0.895850i $$0.353432\pi$$
$$32$$ 0 0
$$33$$ 0.402209 0.0700157
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.64938 1.25755 0.628775 0.777587i $$-0.283557\pi$$
0.628775 + 0.777587i $$0.283557\pi$$
$$38$$ 0 0
$$39$$ 0.260301 0.0416816
$$40$$ 0 0
$$41$$ 1.44153 0.225129 0.112565 0.993644i $$-0.464093\pi$$
0.112565 + 0.993644i $$0.464093\pi$$
$$42$$ 0 0
$$43$$ −1.98578 −0.302829 −0.151414 0.988470i $$-0.548383\pi$$
−0.151414 + 0.988470i $$0.548383\pi$$
$$44$$ 0 0
$$45$$ 2.48812 0.370906
$$46$$ 0 0
$$47$$ −1.56955 −0.228942 −0.114471 0.993427i $$-0.536517\pi$$
−0.114471 + 0.993427i $$0.536517\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −1.26400 −0.176995
$$52$$ 0 0
$$53$$ −13.5386 −1.85968 −0.929838 0.367969i $$-0.880053\pi$$
−0.929838 + 0.367969i $$0.880053\pi$$
$$54$$ 0 0
$$55$$ −0.564590 −0.0761293
$$56$$ 0 0
$$57$$ 6.90020 0.913954
$$58$$ 0 0
$$59$$ 13.2075 1.71947 0.859733 0.510743i $$-0.170630\pi$$
0.859733 + 0.510743i $$0.170630\pi$$
$$60$$ 0 0
$$61$$ 3.37365 0.431951 0.215976 0.976399i $$-0.430707\pi$$
0.215976 + 0.976399i $$0.430707\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −0.365391 −0.0453212
$$66$$ 0 0
$$67$$ 13.3478 1.63070 0.815348 0.578971i $$-0.196545\pi$$
0.815348 + 0.578971i $$0.196545\pi$$
$$68$$ 0 0
$$69$$ 11.9283 1.43600
$$70$$ 0 0
$$71$$ −2.62870 −0.311969 −0.155985 0.987759i $$-0.549855\pi$$
−0.155985 + 0.987759i $$0.549855\pi$$
$$72$$ 0 0
$$73$$ 3.86935 0.452873 0.226436 0.974026i $$-0.427292\pi$$
0.226436 + 0.974026i $$0.427292\pi$$
$$74$$ 0 0
$$75$$ −2.34072 −0.270284
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 12.2324 1.37625 0.688124 0.725594i $$-0.258435\pi$$
0.688124 + 0.725594i $$0.258435\pi$$
$$80$$ 0 0
$$81$$ −2.82663 −0.314070
$$82$$ 0 0
$$83$$ 10.4915 1.15159 0.575797 0.817592i $$-0.304692\pi$$
0.575797 + 0.817592i $$0.304692\pi$$
$$84$$ 0 0
$$85$$ 1.77431 0.192450
$$86$$ 0 0
$$87$$ −10.8939 −1.16795
$$88$$ 0 0
$$89$$ 0.00620724 0.000657966 0 0.000328983 1.00000i $$-0.499895\pi$$
0.000328983 1.00000i $$0.499895\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 6.25445 0.648556
$$94$$ 0 0
$$95$$ −9.68597 −0.993759
$$96$$ 0 0
$$97$$ 4.93669 0.501245 0.250622 0.968085i $$-0.419365\pi$$
0.250622 + 0.968085i $$0.419365\pi$$
$$98$$ 0 0
$$99$$ −0.446218 −0.0448466
$$100$$ 0 0
$$101$$ 8.69105 0.864792 0.432396 0.901684i $$-0.357668\pi$$
0.432396 + 0.901684i $$0.357668\pi$$
$$102$$ 0 0
$$103$$ 4.08166 0.402178 0.201089 0.979573i $$-0.435552\pi$$
0.201089 + 0.979573i $$0.435552\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 14.4600 1.39790 0.698950 0.715171i $$-0.253651\pi$$
0.698950 + 0.715171i $$0.253651\pi$$
$$108$$ 0 0
$$109$$ 1.57368 0.150731 0.0753655 0.997156i $$-0.475988\pi$$
0.0753655 + 0.997156i $$0.475988\pi$$
$$110$$ 0 0
$$111$$ 9.66882 0.917723
$$112$$ 0 0
$$113$$ −0.830486 −0.0781256 −0.0390628 0.999237i $$-0.512437\pi$$
−0.0390628 + 0.999237i $$0.512437\pi$$
$$114$$ 0 0
$$115$$ −16.7440 −1.56139
$$116$$ 0 0
$$117$$ −0.288783 −0.0266980
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.8987 −0.990795
$$122$$ 0 0
$$123$$ 1.82210 0.164293
$$124$$ 0 0
$$125$$ 12.1573 1.08738
$$126$$ 0 0
$$127$$ −4.93809 −0.438185 −0.219092 0.975704i $$-0.570310\pi$$
−0.219092 + 0.975704i $$0.570310\pi$$
$$128$$ 0 0
$$129$$ −2.51003 −0.220995
$$130$$ 0 0
$$131$$ 18.2925 1.59823 0.799113 0.601181i $$-0.205303\pi$$
0.799113 + 0.601181i $$0.205303\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 9.87315 0.849745
$$136$$ 0 0
$$137$$ 9.89826 0.845665 0.422833 0.906208i $$-0.361036\pi$$
0.422833 + 0.906208i $$0.361036\pi$$
$$138$$ 0 0
$$139$$ −12.6946 −1.07675 −0.538373 0.842707i $$-0.680961\pi$$
−0.538373 + 0.842707i $$0.680961\pi$$
$$140$$ 0 0
$$141$$ −1.98391 −0.167075
$$142$$ 0 0
$$143$$ 0.0655291 0.00547982
$$144$$ 0 0
$$145$$ 15.2920 1.26993
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1.30758 −0.107121 −0.0535607 0.998565i $$-0.517057\pi$$
−0.0535607 + 0.998565i $$0.517057\pi$$
$$150$$ 0 0
$$151$$ 3.68720 0.300060 0.150030 0.988681i $$-0.452063\pi$$
0.150030 + 0.988681i $$0.452063\pi$$
$$152$$ 0 0
$$153$$ 1.40230 0.113370
$$154$$ 0 0
$$155$$ −8.77952 −0.705188
$$156$$ 0 0
$$157$$ 15.0157 1.19838 0.599190 0.800607i $$-0.295489\pi$$
0.599190 + 0.800607i $$0.295489\pi$$
$$158$$ 0 0
$$159$$ −17.1128 −1.35714
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 9.11421 0.713880 0.356940 0.934127i $$-0.383820\pi$$
0.356940 + 0.934127i $$0.383820\pi$$
$$164$$ 0 0
$$165$$ −0.713642 −0.0555570
$$166$$ 0 0
$$167$$ 9.18362 0.710650 0.355325 0.934743i $$-0.384370\pi$$
0.355325 + 0.934743i $$0.384370\pi$$
$$168$$ 0 0
$$169$$ −12.9576 −0.996738
$$170$$ 0 0
$$171$$ −7.65520 −0.585408
$$172$$ 0 0
$$173$$ −15.5531 −1.18248 −0.591241 0.806495i $$-0.701362\pi$$
−0.591241 + 0.806495i $$0.701362\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 16.6942 1.25482
$$178$$ 0 0
$$179$$ −8.10572 −0.605850 −0.302925 0.953014i $$-0.597963\pi$$
−0.302925 + 0.953014i $$0.597963\pi$$
$$180$$ 0 0
$$181$$ 14.8172 1.10135 0.550677 0.834718i $$-0.314370\pi$$
0.550677 + 0.834718i $$0.314370\pi$$
$$182$$ 0 0
$$183$$ 4.26429 0.315226
$$184$$ 0 0
$$185$$ −13.5723 −0.997858
$$186$$ 0 0
$$187$$ −0.318204 −0.0232693
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −10.5824 −0.765719 −0.382860 0.923807i $$-0.625061\pi$$
−0.382860 + 0.923807i $$0.625061\pi$$
$$192$$ 0 0
$$193$$ −20.4950 −1.47526 −0.737631 0.675204i $$-0.764055\pi$$
−0.737631 + 0.675204i $$0.764055\pi$$
$$194$$ 0 0
$$195$$ −0.461854 −0.0330741
$$196$$ 0 0
$$197$$ −12.3238 −0.878037 −0.439018 0.898478i $$-0.644674\pi$$
−0.439018 + 0.898478i $$0.644674\pi$$
$$198$$ 0 0
$$199$$ −1.24045 −0.0879335 −0.0439668 0.999033i $$-0.514000\pi$$
−0.0439668 + 0.999033i $$0.514000\pi$$
$$200$$ 0 0
$$201$$ 16.8717 1.19003
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −2.55772 −0.178639
$$206$$ 0 0
$$207$$ −13.2335 −0.919789
$$208$$ 0 0
$$209$$ 1.73708 0.120156
$$210$$ 0 0
$$211$$ 6.88477 0.473967 0.236984 0.971514i $$-0.423841\pi$$
0.236984 + 0.971514i $$0.423841\pi$$
$$212$$ 0 0
$$213$$ −3.32268 −0.227666
$$214$$ 0 0
$$215$$ 3.52338 0.240293
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 4.89086 0.330493
$$220$$ 0 0
$$221$$ −0.205935 −0.0138527
$$222$$ 0 0
$$223$$ 13.5034 0.904258 0.452129 0.891953i $$-0.350665\pi$$
0.452129 + 0.891953i $$0.350665\pi$$
$$224$$ 0 0
$$225$$ 2.59684 0.173123
$$226$$ 0 0
$$227$$ −0.645775 −0.0428616 −0.0214308 0.999770i $$-0.506822\pi$$
−0.0214308 + 0.999770i $$0.506822\pi$$
$$228$$ 0 0
$$229$$ 17.8239 1.17784 0.588919 0.808192i $$-0.299554\pi$$
0.588919 + 0.808192i $$0.299554\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −8.09800 −0.530518 −0.265259 0.964177i $$-0.585457\pi$$
−0.265259 + 0.964177i $$0.585457\pi$$
$$234$$ 0 0
$$235$$ 2.78485 0.181664
$$236$$ 0 0
$$237$$ 15.4617 1.00434
$$238$$ 0 0
$$239$$ −3.12106 −0.201885 −0.100942 0.994892i $$-0.532186\pi$$
−0.100942 + 0.994892i $$0.532186\pi$$
$$240$$ 0 0
$$241$$ 11.3366 0.730252 0.365126 0.930958i $$-0.381026\pi$$
0.365126 + 0.930958i $$0.381026\pi$$
$$242$$ 0 0
$$243$$ 13.1207 0.841692
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.12420 0.0715312
$$248$$ 0 0
$$249$$ 13.2613 0.840400
$$250$$ 0 0
$$251$$ 3.57825 0.225857 0.112929 0.993603i $$-0.463977\pi$$
0.112929 + 0.993603i $$0.463977\pi$$
$$252$$ 0 0
$$253$$ 3.00287 0.188789
$$254$$ 0 0
$$255$$ 2.24272 0.140445
$$256$$ 0 0
$$257$$ −9.06411 −0.565404 −0.282702 0.959208i $$-0.591231\pi$$
−0.282702 + 0.959208i $$0.591231\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 12.0859 0.748097
$$262$$ 0 0
$$263$$ 4.93268 0.304162 0.152081 0.988368i $$-0.451403\pi$$
0.152081 + 0.988368i $$0.451403\pi$$
$$264$$ 0 0
$$265$$ 24.0217 1.47564
$$266$$ 0 0
$$267$$ 0.00784595 0.000480165 0
$$268$$ 0 0
$$269$$ −17.2672 −1.05280 −0.526400 0.850237i $$-0.676458\pi$$
−0.526400 + 0.850237i $$0.676458\pi$$
$$270$$ 0 0
$$271$$ 29.4257 1.78748 0.893742 0.448581i $$-0.148070\pi$$
0.893742 + 0.448581i $$0.148070\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −0.589262 −0.0355338
$$276$$ 0 0
$$277$$ −29.4835 −1.77149 −0.885745 0.464172i $$-0.846352\pi$$
−0.885745 + 0.464172i $$0.846352\pi$$
$$278$$ 0 0
$$279$$ −6.93880 −0.415415
$$280$$ 0 0
$$281$$ 5.51539 0.329021 0.164510 0.986375i $$-0.447396\pi$$
0.164510 + 0.986375i $$0.447396\pi$$
$$282$$ 0 0
$$283$$ −7.46379 −0.443676 −0.221838 0.975084i $$-0.571206\pi$$
−0.221838 + 0.975084i $$0.571206\pi$$
$$284$$ 0 0
$$285$$ −12.2431 −0.725217
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 6.23997 0.365794
$$292$$ 0 0
$$293$$ −20.6687 −1.20748 −0.603740 0.797181i $$-0.706323\pi$$
−0.603740 + 0.797181i $$0.706323\pi$$
$$294$$ 0 0
$$295$$ −23.4341 −1.36439
$$296$$ 0 0
$$297$$ −1.77065 −0.102743
$$298$$ 0 0
$$299$$ 1.94339 0.112389
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 10.9855 0.631100
$$304$$ 0 0
$$305$$ −5.98588 −0.342751
$$306$$ 0 0
$$307$$ −14.0562 −0.802228 −0.401114 0.916028i $$-0.631377\pi$$
−0.401114 + 0.916028i $$0.631377\pi$$
$$308$$ 0 0
$$309$$ 5.15922 0.293498
$$310$$ 0 0
$$311$$ −4.76105 −0.269975 −0.134987 0.990847i $$-0.543099\pi$$
−0.134987 + 0.990847i $$0.543099\pi$$
$$312$$ 0 0
$$313$$ −26.9097 −1.52103 −0.760513 0.649323i $$-0.775052\pi$$
−0.760513 + 0.649323i $$0.775052\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −20.3180 −1.14117 −0.570587 0.821237i $$-0.693284\pi$$
−0.570587 + 0.821237i $$0.693284\pi$$
$$318$$ 0 0
$$319$$ −2.74246 −0.153549
$$320$$ 0 0
$$321$$ 18.2774 1.02015
$$322$$ 0 0
$$323$$ −5.45902 −0.303748
$$324$$ 0 0
$$325$$ −0.381358 −0.0211539
$$326$$ 0 0
$$327$$ 1.98913 0.109999
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1.29216 0.0710238 0.0355119 0.999369i $$-0.488694\pi$$
0.0355119 + 0.999369i $$0.488694\pi$$
$$332$$ 0 0
$$333$$ −10.7268 −0.587822
$$334$$ 0 0
$$335$$ −23.6831 −1.29395
$$336$$ 0 0
$$337$$ 4.35840 0.237417 0.118708 0.992929i $$-0.462125\pi$$
0.118708 + 0.992929i $$0.462125\pi$$
$$338$$ 0 0
$$339$$ −1.04973 −0.0570138
$$340$$ 0 0
$$341$$ 1.57452 0.0852648
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −21.1645 −1.13946
$$346$$ 0 0
$$347$$ 5.83404 0.313188 0.156594 0.987663i $$-0.449949\pi$$
0.156594 + 0.987663i $$0.449949\pi$$
$$348$$ 0 0
$$349$$ −35.2609 −1.88747 −0.943736 0.330700i $$-0.892715\pi$$
−0.943736 + 0.330700i $$0.892715\pi$$
$$350$$ 0 0
$$351$$ −1.14593 −0.0611650
$$352$$ 0 0
$$353$$ 21.7586 1.15809 0.579046 0.815295i $$-0.303425\pi$$
0.579046 + 0.815295i $$0.303425\pi$$
$$354$$ 0 0
$$355$$ 4.66412 0.247546
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 3.68306 0.194385 0.0971923 0.995266i $$-0.469014\pi$$
0.0971923 + 0.995266i $$0.469014\pi$$
$$360$$ 0 0
$$361$$ 10.8009 0.568468
$$362$$ 0 0
$$363$$ −13.7760 −0.723053
$$364$$ 0 0
$$365$$ −6.86541 −0.359352
$$366$$ 0 0
$$367$$ 8.43369 0.440235 0.220117 0.975473i $$-0.429356\pi$$
0.220117 + 0.975473i $$0.429356\pi$$
$$368$$ 0 0
$$369$$ −2.02146 −0.105233
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −31.3060 −1.62097 −0.810483 0.585762i $$-0.800795\pi$$
−0.810483 + 0.585762i $$0.800795\pi$$
$$374$$ 0 0
$$375$$ 15.3668 0.793537
$$376$$ 0 0
$$377$$ −1.77486 −0.0914102
$$378$$ 0 0
$$379$$ −27.3855 −1.40670 −0.703349 0.710844i $$-0.748313\pi$$
−0.703349 + 0.710844i $$0.748313\pi$$
$$380$$ 0 0
$$381$$ −6.24175 −0.319774
$$382$$ 0 0
$$383$$ −22.2937 −1.13916 −0.569579 0.821937i $$-0.692894\pi$$
−0.569579 + 0.821937i $$0.692894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2.78467 0.141553
$$388$$ 0 0
$$389$$ −26.3008 −1.33351 −0.666753 0.745279i $$-0.732316\pi$$
−0.666753 + 0.745279i $$0.732316\pi$$
$$390$$ 0 0
$$391$$ −9.43694 −0.477247
$$392$$ 0 0
$$393$$ 23.1218 1.16634
$$394$$ 0 0
$$395$$ −21.7039 −1.09204
$$396$$ 0 0
$$397$$ 38.8034 1.94749 0.973745 0.227644i $$-0.0731021\pi$$
0.973745 + 0.227644i $$0.0731021\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 14.9758 0.747855 0.373928 0.927458i $$-0.378011\pi$$
0.373928 + 0.927458i $$0.378011\pi$$
$$402$$ 0 0
$$403$$ 1.01899 0.0507597
$$404$$ 0 0
$$405$$ 5.01531 0.249213
$$406$$ 0 0
$$407$$ 2.43406 0.120652
$$408$$ 0 0
$$409$$ 38.3051 1.89407 0.947034 0.321134i $$-0.104064\pi$$
0.947034 + 0.321134i $$0.104064\pi$$
$$410$$ 0 0
$$411$$ 12.5114 0.617142
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −18.6152 −0.913783
$$416$$ 0 0
$$417$$ −16.0460 −0.785778
$$418$$ 0 0
$$419$$ 23.3744 1.14192 0.570958 0.820979i $$-0.306572\pi$$
0.570958 + 0.820979i $$0.306572\pi$$
$$420$$ 0 0
$$421$$ 8.17711 0.398528 0.199264 0.979946i $$-0.436145\pi$$
0.199264 + 0.979946i $$0.436145\pi$$
$$422$$ 0 0
$$423$$ 2.20098 0.107015
$$424$$ 0 0
$$425$$ 1.85184 0.0898274
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0.0828288 0.00399901
$$430$$ 0 0
$$431$$ −20.1422 −0.970217 −0.485108 0.874454i $$-0.661220\pi$$
−0.485108 + 0.874454i $$0.661220\pi$$
$$432$$ 0 0
$$433$$ 30.8004 1.48017 0.740085 0.672513i $$-0.234785\pi$$
0.740085 + 0.672513i $$0.234785\pi$$
$$434$$ 0 0
$$435$$ 19.3291 0.926759
$$436$$ 0 0
$$437$$ 51.5165 2.46437
$$438$$ 0 0
$$439$$ −37.7595 −1.80216 −0.901082 0.433649i $$-0.857226\pi$$
−0.901082 + 0.433649i $$0.857226\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0.877302 0.0416819 0.0208409 0.999783i $$-0.493366\pi$$
0.0208409 + 0.999783i $$0.493366\pi$$
$$444$$ 0 0
$$445$$ −0.0110135 −0.000522092 0
$$446$$ 0 0
$$447$$ −1.65279 −0.0781741
$$448$$ 0 0
$$449$$ 33.0313 1.55884 0.779421 0.626500i $$-0.215513\pi$$
0.779421 + 0.626500i $$0.215513\pi$$
$$450$$ 0 0
$$451$$ 0.458700 0.0215993
$$452$$ 0 0
$$453$$ 4.66062 0.218975
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 31.6690 1.48141 0.740707 0.671828i $$-0.234491\pi$$
0.740707 + 0.671828i $$0.234491\pi$$
$$458$$ 0 0
$$459$$ 5.56451 0.259729
$$460$$ 0 0
$$461$$ −29.9124 −1.39316 −0.696580 0.717479i $$-0.745296\pi$$
−0.696580 + 0.717479i $$0.745296\pi$$
$$462$$ 0 0
$$463$$ −15.8809 −0.738050 −0.369025 0.929419i $$-0.620308\pi$$
−0.369025 + 0.929419i $$0.620308\pi$$
$$464$$ 0 0
$$465$$ −11.0973 −0.514625
$$466$$ 0 0
$$467$$ −13.0111 −0.602081 −0.301041 0.953611i $$-0.597334\pi$$
−0.301041 + 0.953611i $$0.597334\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 18.9798 0.874543
$$472$$ 0 0
$$473$$ −0.631882 −0.0290540
$$474$$ 0 0
$$475$$ −10.1092 −0.463843
$$476$$ 0 0
$$477$$ 18.9853 0.869277
$$478$$ 0 0
$$479$$ −33.6377 −1.53695 −0.768473 0.639882i $$-0.778983\pi$$
−0.768473 + 0.639882i $$0.778983\pi$$
$$480$$ 0 0
$$481$$ 1.57527 0.0718262
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −8.75919 −0.397734
$$486$$ 0 0
$$487$$ 33.3793 1.51256 0.756281 0.654247i $$-0.227014\pi$$
0.756281 + 0.654247i $$0.227014\pi$$
$$488$$ 0 0
$$489$$ 11.5204 0.520969
$$490$$ 0 0
$$491$$ 1.61369 0.0728249 0.0364125 0.999337i $$-0.488407\pi$$
0.0364125 + 0.999337i $$0.488407\pi$$
$$492$$ 0 0
$$493$$ 8.61859 0.388162
$$494$$ 0 0
$$495$$ 0.791727 0.0355855
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 7.08060 0.316971 0.158486 0.987361i $$-0.449339\pi$$
0.158486 + 0.987361i $$0.449339\pi$$
$$500$$ 0 0
$$501$$ 11.6081 0.518611
$$502$$ 0 0
$$503$$ −17.2097 −0.767342 −0.383671 0.923470i $$-0.625340\pi$$
−0.383671 + 0.923470i $$0.625340\pi$$
$$504$$ 0 0
$$505$$ −15.4206 −0.686207
$$506$$ 0 0
$$507$$ −16.3784 −0.727390
$$508$$ 0 0
$$509$$ 0.569329 0.0252351 0.0126175 0.999920i $$-0.495984\pi$$
0.0126175 + 0.999920i $$0.495984\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −30.3768 −1.34117
$$514$$ 0 0
$$515$$ −7.24211 −0.319125
$$516$$ 0 0
$$517$$ −0.499435 −0.0219651
$$518$$ 0 0
$$519$$ −19.6591 −0.862941
$$520$$ 0 0
$$521$$ −4.96590 −0.217560 −0.108780 0.994066i $$-0.534694\pi$$
−0.108780 + 0.994066i $$0.534694\pi$$
$$522$$ 0 0
$$523$$ 10.5968 0.463364 0.231682 0.972792i $$-0.425577\pi$$
0.231682 + 0.972792i $$0.425577\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −4.94814 −0.215544
$$528$$ 0 0
$$529$$ 66.0559 2.87200
$$530$$ 0 0
$$531$$ −18.5209 −0.803738
$$532$$ 0 0
$$533$$ 0.296861 0.0128585
$$534$$ 0 0
$$535$$ −25.6564 −1.10922
$$536$$ 0 0
$$537$$ −10.2456 −0.442132
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −8.64750 −0.371785 −0.185892 0.982570i $$-0.559518\pi$$
−0.185892 + 0.982570i $$0.559518\pi$$
$$542$$ 0 0
$$543$$ 18.7289 0.803736
$$544$$ 0 0
$$545$$ −2.79219 −0.119604
$$546$$ 0 0
$$547$$ 27.9601 1.19549 0.597744 0.801687i $$-0.296064\pi$$
0.597744 + 0.801687i $$0.296064\pi$$
$$548$$ 0 0
$$549$$ −4.73088 −0.201909
$$550$$ 0 0
$$551$$ −47.0490 −2.00436
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −17.1554 −0.728208
$$556$$ 0 0
$$557$$ 23.4737 0.994613 0.497306 0.867575i $$-0.334322\pi$$
0.497306 + 0.867575i $$0.334322\pi$$
$$558$$ 0 0
$$559$$ −0.408941 −0.0172963
$$560$$ 0 0
$$561$$ −0.402209 −0.0169813
$$562$$ 0 0
$$563$$ 26.6059 1.12130 0.560652 0.828052i $$-0.310551\pi$$
0.560652 + 0.828052i $$0.310551\pi$$
$$564$$ 0 0
$$565$$ 1.47354 0.0619921
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 28.0713 1.17681 0.588405 0.808566i $$-0.299756\pi$$
0.588405 + 0.808566i $$0.299756\pi$$
$$570$$ 0 0
$$571$$ 39.1974 1.64036 0.820180 0.572105i $$-0.193873\pi$$
0.820180 + 0.572105i $$0.193873\pi$$
$$572$$ 0 0
$$573$$ −13.3762 −0.558799
$$574$$ 0 0
$$575$$ −17.4757 −0.728787
$$576$$ 0 0
$$577$$ 6.80895 0.283460 0.141730 0.989905i $$-0.454733\pi$$
0.141730 + 0.989905i $$0.454733\pi$$
$$578$$ 0 0
$$579$$ −25.9057 −1.07660
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −4.30804 −0.178421
$$584$$ 0 0
$$585$$ 0.512389 0.0211847
$$586$$ 0 0
$$587$$ −23.8016 −0.982397 −0.491198 0.871048i $$-0.663441\pi$$
−0.491198 + 0.871048i $$0.663441\pi$$
$$588$$ 0 0
$$589$$ 27.0120 1.11301
$$590$$ 0 0
$$591$$ −15.5773 −0.640766
$$592$$ 0 0
$$593$$ −30.2650 −1.24284 −0.621418 0.783479i $$-0.713443\pi$$
−0.621418 + 0.783479i $$0.713443\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −1.56793 −0.0641713
$$598$$ 0 0
$$599$$ −20.3977 −0.833427 −0.416714 0.909038i $$-0.636818\pi$$
−0.416714 + 0.909038i $$0.636818\pi$$
$$600$$ 0 0
$$601$$ −19.6591 −0.801912 −0.400956 0.916097i $$-0.631322\pi$$
−0.400956 + 0.916097i $$0.631322\pi$$
$$602$$ 0 0
$$603$$ −18.7177 −0.762244
$$604$$ 0 0
$$605$$ 19.3377 0.786190
$$606$$ 0 0
$$607$$ 0.670435 0.0272121 0.0136061 0.999907i $$-0.495669\pi$$
0.0136061 + 0.999907i $$0.495669\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −0.323224 −0.0130762
$$612$$ 0 0
$$613$$ 20.1900 0.815466 0.407733 0.913101i $$-0.366319\pi$$
0.407733 + 0.913101i $$0.366319\pi$$
$$614$$ 0 0
$$615$$ −3.23295 −0.130365
$$616$$ 0 0
$$617$$ 15.1299 0.609107 0.304554 0.952495i $$-0.401493\pi$$
0.304554 + 0.952495i $$0.401493\pi$$
$$618$$ 0 0
$$619$$ −23.6956 −0.952407 −0.476204 0.879335i $$-0.657987\pi$$
−0.476204 + 0.879335i $$0.657987\pi$$
$$620$$ 0 0
$$621$$ −52.5120 −2.10723
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −12.3115 −0.492460
$$626$$ 0 0
$$627$$ 2.19567 0.0876865
$$628$$ 0 0
$$629$$ −7.64938 −0.305001
$$630$$ 0 0
$$631$$ −0.879472 −0.0350112 −0.0175056 0.999847i $$-0.505572\pi$$
−0.0175056 + 0.999847i $$0.505572\pi$$
$$632$$ 0 0
$$633$$ 8.70235 0.345887
$$634$$ 0 0
$$635$$ 8.76168 0.347697
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 3.68624 0.145825
$$640$$ 0 0
$$641$$ −32.1852 −1.27124 −0.635620 0.772002i $$-0.719255\pi$$
−0.635620 + 0.772002i $$0.719255\pi$$
$$642$$ 0 0
$$643$$ 18.7957 0.741231 0.370615 0.928786i $$-0.379147\pi$$
0.370615 + 0.928786i $$0.379147\pi$$
$$644$$ 0 0
$$645$$ 4.45355 0.175358
$$646$$ 0 0
$$647$$ −44.7477 −1.75921 −0.879607 0.475701i $$-0.842194\pi$$
−0.879607 + 0.475701i $$0.842194\pi$$
$$648$$ 0 0
$$649$$ 4.20266 0.164969
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −13.6090 −0.532560 −0.266280 0.963896i $$-0.585795\pi$$
−0.266280 + 0.963896i $$0.585795\pi$$
$$654$$ 0 0
$$655$$ −32.4565 −1.26818
$$656$$ 0 0
$$657$$ −5.42600 −0.211688
$$658$$ 0 0
$$659$$ 31.7616 1.23725 0.618627 0.785684i $$-0.287689\pi$$
0.618627 + 0.785684i $$0.287689\pi$$
$$660$$ 0 0
$$661$$ 20.4414 0.795078 0.397539 0.917585i $$-0.369864\pi$$
0.397539 + 0.917585i $$0.369864\pi$$
$$662$$ 0 0
$$663$$ −0.260301 −0.0101093
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −81.3331 −3.14923
$$668$$ 0 0
$$669$$ 17.0684 0.659901
$$670$$ 0 0
$$671$$ 1.07351 0.0414423
$$672$$ 0 0
$$673$$ −8.40756 −0.324087 −0.162044 0.986784i $$-0.551809\pi$$
−0.162044 + 0.986784i $$0.551809\pi$$
$$674$$ 0 0
$$675$$ 10.3046 0.396624
$$676$$ 0 0
$$677$$ 10.9126 0.419405 0.209703 0.977765i $$-0.432750\pi$$
0.209703 + 0.977765i $$0.432750\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −0.816259 −0.0312791
$$682$$ 0 0
$$683$$ 0.402969 0.0154192 0.00770959 0.999970i $$-0.497546\pi$$
0.00770959 + 0.999970i $$0.497546\pi$$
$$684$$ 0 0
$$685$$ −17.5625 −0.671030
$$686$$ 0 0
$$687$$ 22.5294 0.859552
$$688$$ 0 0
$$689$$ −2.78807 −0.106217
$$690$$ 0 0
$$691$$ 28.7597 1.09407 0.547036 0.837109i $$-0.315756\pi$$
0.547036 + 0.837109i $$0.315756\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 22.5242 0.854391
$$696$$ 0 0
$$697$$ −1.44153 −0.0546019
$$698$$ 0 0
$$699$$ −10.2359 −0.387157
$$700$$ 0 0
$$701$$ 30.4661 1.15069 0.575345 0.817911i $$-0.304868\pi$$
0.575345 + 0.817911i $$0.304868\pi$$
$$702$$ 0 0
$$703$$ 41.7581 1.57494
$$704$$ 0 0
$$705$$ 3.52005 0.132573
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −13.1020 −0.492057 −0.246029 0.969263i $$-0.579126\pi$$
−0.246029 + 0.969263i $$0.579126\pi$$
$$710$$ 0 0
$$711$$ −17.1535 −0.643305
$$712$$ 0 0
$$713$$ 46.6953 1.74875
$$714$$ 0 0
$$715$$ −0.116269 −0.00434820
$$716$$ 0 0
$$717$$ −3.94502 −0.147330
$$718$$ 0 0
$$719$$ 15.4948 0.577858 0.288929 0.957350i $$-0.406701\pi$$
0.288929 + 0.957350i $$0.406701\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 14.3294 0.532917
$$724$$ 0 0
$$725$$ 15.9602 0.592748
$$726$$ 0 0
$$727$$ −35.6680 −1.32285 −0.661427 0.750010i $$-0.730049\pi$$
−0.661427 + 0.750010i $$0.730049\pi$$
$$728$$ 0 0
$$729$$ 25.0644 0.928312
$$730$$ 0 0
$$731$$ 1.98578 0.0734467
$$732$$ 0 0
$$733$$ −12.5978 −0.465311 −0.232656 0.972559i $$-0.574742\pi$$
−0.232656 + 0.972559i $$0.574742\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4.24733 0.156452
$$738$$ 0 0
$$739$$ 32.5853 1.19867 0.599334 0.800499i $$-0.295432\pi$$
0.599334 + 0.800499i $$0.295432\pi$$
$$740$$ 0 0
$$741$$ 1.42099 0.0522014
$$742$$ 0 0
$$743$$ −1.53311 −0.0562444 −0.0281222 0.999604i $$-0.508953\pi$$
−0.0281222 + 0.999604i $$0.508953\pi$$
$$744$$ 0 0
$$745$$ 2.32005 0.0850001
$$746$$ 0 0
$$747$$ −14.7123 −0.538295
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −24.5508 −0.895871 −0.447935 0.894066i $$-0.647841\pi$$
−0.447935 + 0.894066i $$0.647841\pi$$
$$752$$ 0 0
$$753$$ 4.52291 0.164824
$$754$$ 0 0
$$755$$ −6.54221 −0.238095
$$756$$ 0 0
$$757$$ 30.8744 1.12215 0.561075 0.827765i $$-0.310388\pi$$
0.561075 + 0.827765i $$0.310388\pi$$
$$758$$ 0 0
$$759$$ 3.79563 0.137773
$$760$$ 0 0
$$761$$ −26.8426 −0.973045 −0.486522 0.873668i $$-0.661735\pi$$
−0.486522 + 0.873668i $$0.661735\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −2.48812 −0.0899580
$$766$$ 0 0
$$767$$ 2.71988 0.0982090
$$768$$ 0 0
$$769$$ −52.2087 −1.88269 −0.941347 0.337440i $$-0.890439\pi$$
−0.941347 + 0.337440i $$0.890439\pi$$
$$770$$ 0 0
$$771$$ −11.4570 −0.412615
$$772$$ 0 0
$$773$$ 15.3808 0.553207 0.276604 0.960984i $$-0.410791\pi$$
0.276604 + 0.960984i $$0.410791\pi$$
$$774$$ 0 0
$$775$$ −9.16316 −0.329150
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 7.86934 0.281949
$$780$$ 0 0
$$781$$ −0.836462 −0.0299309
$$782$$ 0 0
$$783$$ 47.9582 1.71389
$$784$$ 0 0
$$785$$ −26.6424 −0.950907
$$786$$ 0 0
$$787$$ 25.6237 0.913387 0.456693 0.889624i $$-0.349034\pi$$
0.456693 + 0.889624i $$0.349034\pi$$
$$788$$ 0 0
$$789$$ 6.23491 0.221969
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0.694751 0.0246713
$$794$$ 0 0
$$795$$ 30.3634 1.07688
$$796$$ 0 0
$$797$$ −15.4806 −0.548350 −0.274175 0.961680i $$-0.588405\pi$$
−0.274175 + 0.961680i $$0.588405\pi$$
$$798$$ 0 0
$$799$$ 1.56955 0.0555265
$$800$$ 0 0
$$801$$ −0.00870444 −0.000307556 0
$$802$$ 0 0
$$803$$ 1.23124 0.0434495
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −21.8257 −0.768302
$$808$$ 0 0
$$809$$ −38.5138 −1.35407 −0.677036 0.735950i $$-0.736736\pi$$
−0.677036 + 0.735950i $$0.736736\pi$$
$$810$$ 0 0
$$811$$ −41.6065 −1.46100 −0.730502 0.682911i $$-0.760714\pi$$
−0.730502 + 0.682911i $$0.760714\pi$$
$$812$$ 0 0
$$813$$ 37.1941 1.30445
$$814$$ 0 0
$$815$$ −16.1714 −0.566459
$$816$$ 0 0
$$817$$ −10.8404 −0.379258
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 20.4853 0.714943 0.357472 0.933924i $$-0.383639\pi$$
0.357472 + 0.933924i $$0.383639\pi$$
$$822$$ 0 0
$$823$$ 17.1271 0.597014 0.298507 0.954407i $$-0.403511\pi$$
0.298507 + 0.954407i $$0.403511\pi$$
$$824$$ 0 0
$$825$$ −0.744827 −0.0259315
$$826$$ 0 0
$$827$$ −50.9089 −1.77028 −0.885138 0.465329i $$-0.845936\pi$$
−0.885138 + 0.465329i $$0.845936\pi$$
$$828$$ 0 0
$$829$$ −37.5712 −1.30490 −0.652451 0.757831i $$-0.726259\pi$$
−0.652451 + 0.757831i $$0.726259\pi$$
$$830$$ 0 0
$$831$$ −37.2671 −1.29278
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −16.2945 −0.563896
$$836$$ 0 0
$$837$$ −27.5340 −0.951714
$$838$$ 0 0
$$839$$ 33.2696 1.14860 0.574298 0.818646i $$-0.305275\pi$$
0.574298 + 0.818646i $$0.305275\pi$$
$$840$$ 0 0
$$841$$ 45.2800 1.56138
$$842$$ 0 0
$$843$$ 6.97146 0.240110
$$844$$ 0 0
$$845$$ 22.9907 0.790905
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −9.43423 −0.323782
$$850$$ 0 0
$$851$$ 72.1868 2.47453
$$852$$ 0 0
$$853$$ −30.3159 −1.03800 −0.518998 0.854776i $$-0.673695\pi$$
−0.518998 + 0.854776i $$0.673695\pi$$
$$854$$ 0 0
$$855$$ 13.5827 0.464517
$$856$$ 0 0
$$857$$ −38.7778 −1.32463 −0.662313 0.749228i $$-0.730425\pi$$
−0.662313 + 0.749228i $$0.730425\pi$$
$$858$$ 0 0
$$859$$ 36.4346 1.24313 0.621566 0.783362i $$-0.286497\pi$$
0.621566 + 0.783362i $$0.286497\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −41.0825 −1.39847 −0.699233 0.714894i $$-0.746475\pi$$
−0.699233 + 0.714894i $$0.746475\pi$$
$$864$$ 0 0
$$865$$ 27.5960 0.938292
$$866$$ 0 0
$$867$$ 1.26400 0.0429277
$$868$$ 0 0
$$869$$ 3.89238 0.132040
$$870$$ 0 0
$$871$$ 2.74878 0.0931388
$$872$$ 0 0
$$873$$ −6.92273 −0.234299
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 6.12907 0.206964 0.103482 0.994631i $$-0.467002\pi$$
0.103482 + 0.994631i $$0.467002\pi$$
$$878$$ 0 0
$$879$$ −26.1253 −0.881183
$$880$$ 0 0
$$881$$ −32.4381 −1.09287 −0.546434 0.837502i $$-0.684015\pi$$
−0.546434 + 0.837502i $$0.684015\pi$$
$$882$$ 0 0
$$883$$ −35.1411 −1.18259 −0.591296 0.806454i $$-0.701384\pi$$
−0.591296 + 0.806454i $$0.701384\pi$$
$$884$$ 0 0
$$885$$ −29.6207 −0.995689
$$886$$ 0 0
$$887$$ −26.9435 −0.904673 −0.452336 0.891847i $$-0.649409\pi$$
−0.452336 + 0.891847i $$0.649409\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −0.899445 −0.0301325
$$892$$ 0 0
$$893$$ −8.56818 −0.286723
$$894$$ 0 0
$$895$$ 14.3820 0.480738
$$896$$ 0 0
$$897$$ 2.45645 0.0820184
$$898$$ 0 0
$$899$$ −42.6460 −1.42232
$$900$$ 0 0
$$901$$ 13.5386 0.451038
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −26.2903 −0.873918
$$906$$ 0 0
$$907$$ −23.1647 −0.769170 −0.384585 0.923090i $$-0.625655\pi$$
−0.384585 + 0.923090i $$0.625655\pi$$
$$908$$ 0 0
$$909$$ −12.1875 −0.404234
$$910$$ 0 0
$$911$$ −28.5642 −0.946376 −0.473188 0.880962i $$-0.656897\pi$$
−0.473188 + 0.880962i $$0.656897\pi$$
$$912$$ 0 0
$$913$$ 3.33844 0.110486
$$914$$ 0 0
$$915$$ −7.56616 −0.250129
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −29.4563 −0.971673 −0.485836 0.874050i $$-0.661485\pi$$
−0.485836 + 0.874050i $$0.661485\pi$$
$$920$$ 0 0
$$921$$ −17.7670 −0.585442
$$922$$ 0 0
$$923$$ −0.541340 −0.0178184
$$924$$ 0 0
$$925$$ −14.1654 −0.465756
$$926$$ 0 0
$$927$$ −5.72372 −0.187992
$$928$$ 0 0
$$929$$ −6.91683 −0.226934 −0.113467 0.993542i $$-0.536196\pi$$
−0.113467 + 0.993542i $$0.536196\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −6.01797 −0.197020
$$934$$ 0 0
$$935$$ 0.564590 0.0184641
$$936$$ 0 0
$$937$$ 39.1000 1.27734 0.638671 0.769480i $$-0.279484\pi$$
0.638671 + 0.769480i $$0.279484\pi$$
$$938$$ 0 0
$$939$$ −34.0138 −1.11000
$$940$$ 0 0
$$941$$ 19.8018 0.645522 0.322761 0.946481i $$-0.395389\pi$$
0.322761 + 0.946481i $$0.395389\pi$$
$$942$$ 0 0
$$943$$ 13.6036 0.442996
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −6.95286 −0.225938 −0.112969 0.993599i $$-0.536036\pi$$
−0.112969 + 0.993599i $$0.536036\pi$$
$$948$$ 0 0
$$949$$ 0.796833 0.0258663
$$950$$ 0 0
$$951$$ −25.6820 −0.832795
$$952$$ 0 0
$$953$$ 44.2489 1.43336 0.716681 0.697402i $$-0.245661\pi$$
0.716681 + 0.697402i $$0.245661\pi$$
$$954$$ 0 0
$$955$$ 18.7765 0.607593
$$956$$ 0 0
$$957$$ −3.46648 −0.112055
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −6.51590 −0.210190
$$962$$ 0 0
$$963$$ −20.2773 −0.653427
$$964$$ 0 0
$$965$$ 36.3644 1.17061
$$966$$ 0 0
$$967$$ 3.70560 0.119164 0.0595821 0.998223i $$-0.481023\pi$$
0.0595821 + 0.998223i $$0.481023\pi$$
$$968$$ 0 0
$$969$$ −6.90020 −0.221666
$$970$$ 0 0
$$971$$ 7.59954 0.243881 0.121940 0.992537i $$-0.461088\pi$$
0.121940 + 0.992537i $$0.461088\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −0.482036 −0.0154375
$$976$$ 0 0
$$977$$ −37.4738 −1.19889 −0.599446 0.800415i $$-0.704612\pi$$
−0.599446 + 0.800415i $$0.704612\pi$$
$$978$$ 0 0
$$979$$ 0.00197517 6.31266e−5 0
$$980$$ 0 0
$$981$$ −2.20678 −0.0704569
$$982$$ 0 0
$$983$$ 20.6504 0.658647 0.329323 0.944217i $$-0.393179\pi$$
0.329323 + 0.944217i $$0.393179\pi$$
$$984$$ 0 0
$$985$$ 21.8663 0.696717
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −18.7397 −0.595888
$$990$$ 0 0
$$991$$ −17.2648 −0.548435 −0.274218 0.961668i $$-0.588419\pi$$
−0.274218 + 0.961668i $$0.588419\pi$$
$$992$$ 0 0
$$993$$ 1.63330 0.0518311
$$994$$ 0 0
$$995$$ 2.20095 0.0697747
$$996$$ 0 0
$$997$$ 13.6573 0.432530 0.216265 0.976335i $$-0.430613\pi$$
0.216265 + 0.976335i $$0.430613\pi$$
$$998$$ 0 0
$$999$$ −42.5651 −1.34670
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 3332.2.a.u.1.5 yes 8
7.6 odd 2 3332.2.a.t.1.4 8

By twisted newform
Twist Min Dim Char Parity Ord Type
3332.2.a.t.1.4 8 7.6 odd 2
3332.2.a.u.1.5 yes 8 1.1 even 1 trivial