# Properties

 Label 1840.2.a.n.1.1 Level $1840$ Weight $2$ Character 1840.1 Self dual yes Analytic conductor $14.692$ 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] = [1840,2,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.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: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.79129$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.79129 q^{3} -1.00000 q^{5} -2.79129 q^{7} +0.208712 q^{9} +O(q^{10})$$ $$q-1.79129 q^{3} -1.00000 q^{5} -2.79129 q^{7} +0.208712 q^{9} -3.79129 q^{11} +1.20871 q^{13} +1.79129 q^{15} -3.79129 q^{17} -1.20871 q^{19} +5.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} -1.58258 q^{29} -10.3739 q^{31} +6.79129 q^{33} +2.79129 q^{35} -4.00000 q^{37} -2.16515 q^{39} -2.20871 q^{41} +7.16515 q^{43} -0.208712 q^{45} +13.5826 q^{47} +0.791288 q^{49} +6.79129 q^{51} +6.00000 q^{53} +3.79129 q^{55} +2.16515 q^{57} +4.41742 q^{59} -3.37386 q^{61} -0.582576 q^{63} -1.20871 q^{65} +7.16515 q^{67} +1.79129 q^{69} +5.37386 q^{71} -14.7477 q^{73} -1.79129 q^{75} +10.5826 q^{77} -8.00000 q^{79} -9.58258 q^{81} +6.00000 q^{83} +3.79129 q^{85} +2.83485 q^{87} -3.16515 q^{89} -3.37386 q^{91} +18.5826 q^{93} +1.20871 q^{95} +14.9564 q^{97} -0.791288 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 2 q^{5} - q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q + q^3 - 2 * q^5 - q^7 + 5 * q^9 $$2 q + q^{3} - 2 q^{5} - q^{7} + 5 q^{9} - 3 q^{11} + 7 q^{13} - q^{15} - 3 q^{17} - 7 q^{19} + 10 q^{21} - 2 q^{23} + 2 q^{25} + 10 q^{27} + 6 q^{29} - 7 q^{31} + 9 q^{33} + q^{35} - 8 q^{37} + 14 q^{39} - 9 q^{41} - 4 q^{43} - 5 q^{45} + 18 q^{47} - 3 q^{49} + 9 q^{51} + 12 q^{53} + 3 q^{55} - 14 q^{57} + 18 q^{59} + 7 q^{61} + 8 q^{63} - 7 q^{65} - 4 q^{67} - q^{69} - 3 q^{71} - 2 q^{73} + q^{75} + 12 q^{77} - 16 q^{79} - 10 q^{81} + 12 q^{83} + 3 q^{85} + 24 q^{87} + 12 q^{89} + 7 q^{91} + 28 q^{93} + 7 q^{95} + 7 q^{97} + 3 q^{99}+O(q^{100})$$ 2 * q + q^3 - 2 * q^5 - q^7 + 5 * q^9 - 3 * q^11 + 7 * q^13 - q^15 - 3 * q^17 - 7 * q^19 + 10 * q^21 - 2 * q^23 + 2 * q^25 + 10 * q^27 + 6 * q^29 - 7 * q^31 + 9 * q^33 + q^35 - 8 * q^37 + 14 * q^39 - 9 * q^41 - 4 * q^43 - 5 * q^45 + 18 * q^47 - 3 * q^49 + 9 * q^51 + 12 * q^53 + 3 * q^55 - 14 * q^57 + 18 * q^59 + 7 * q^61 + 8 * q^63 - 7 * q^65 - 4 * q^67 - q^69 - 3 * q^71 - 2 * q^73 + q^75 + 12 * q^77 - 16 * q^79 - 10 * q^81 + 12 * q^83 + 3 * q^85 + 24 * q^87 + 12 * q^89 + 7 * q^91 + 28 * q^93 + 7 * q^95 + 7 * q^97 + 3 * 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
$$3$$ −1.79129 −1.03420 −0.517100 0.855925i $$-0.672989\pi$$
−0.517100 + 0.855925i $$0.672989\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −2.79129 −1.05501 −0.527504 0.849553i $$-0.676872\pi$$
−0.527504 + 0.849553i $$0.676872\pi$$
$$8$$ 0 0
$$9$$ 0.208712 0.0695707
$$10$$ 0 0
$$11$$ −3.79129 −1.14312 −0.571558 0.820562i $$-0.693661\pi$$
−0.571558 + 0.820562i $$0.693661\pi$$
$$12$$ 0 0
$$13$$ 1.20871 0.335236 0.167618 0.985852i $$-0.446392\pi$$
0.167618 + 0.985852i $$0.446392\pi$$
$$14$$ 0 0
$$15$$ 1.79129 0.462509
$$16$$ 0 0
$$17$$ −3.79129 −0.919522 −0.459761 0.888043i $$-0.652065\pi$$
−0.459761 + 0.888043i $$0.652065\pi$$
$$18$$ 0 0
$$19$$ −1.20871 −0.277298 −0.138649 0.990342i $$-0.544276\pi$$
−0.138649 + 0.990342i $$0.544276\pi$$
$$20$$ 0 0
$$21$$ 5.00000 1.09109
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ −1.58258 −0.293877 −0.146938 0.989146i $$-0.546942\pi$$
−0.146938 + 0.989146i $$0.546942\pi$$
$$30$$ 0 0
$$31$$ −10.3739 −1.86320 −0.931600 0.363484i $$-0.881587\pi$$
−0.931600 + 0.363484i $$0.881587\pi$$
$$32$$ 0 0
$$33$$ 6.79129 1.18221
$$34$$ 0 0
$$35$$ 2.79129 0.471814
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ −2.16515 −0.346702
$$40$$ 0 0
$$41$$ −2.20871 −0.344943 −0.172471 0.985015i $$-0.555175\pi$$
−0.172471 + 0.985015i $$0.555175\pi$$
$$42$$ 0 0
$$43$$ 7.16515 1.09268 0.546338 0.837565i $$-0.316022\pi$$
0.546338 + 0.837565i $$0.316022\pi$$
$$44$$ 0 0
$$45$$ −0.208712 −0.0311130
$$46$$ 0 0
$$47$$ 13.5826 1.98122 0.990611 0.136710i $$-0.0436528\pi$$
0.990611 + 0.136710i $$0.0436528\pi$$
$$48$$ 0 0
$$49$$ 0.791288 0.113041
$$50$$ 0 0
$$51$$ 6.79129 0.950971
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 3.79129 0.511217
$$56$$ 0 0
$$57$$ 2.16515 0.286781
$$58$$ 0 0
$$59$$ 4.41742 0.575100 0.287550 0.957766i $$-0.407159\pi$$
0.287550 + 0.957766i $$0.407159\pi$$
$$60$$ 0 0
$$61$$ −3.37386 −0.431979 −0.215989 0.976396i $$-0.569298\pi$$
−0.215989 + 0.976396i $$0.569298\pi$$
$$62$$ 0 0
$$63$$ −0.582576 −0.0733976
$$64$$ 0 0
$$65$$ −1.20871 −0.149922
$$66$$ 0 0
$$67$$ 7.16515 0.875363 0.437681 0.899130i $$-0.355800\pi$$
0.437681 + 0.899130i $$0.355800\pi$$
$$68$$ 0 0
$$69$$ 1.79129 0.215646
$$70$$ 0 0
$$71$$ 5.37386 0.637760 0.318880 0.947795i $$-0.396693\pi$$
0.318880 + 0.947795i $$0.396693\pi$$
$$72$$ 0 0
$$73$$ −14.7477 −1.72609 −0.863045 0.505126i $$-0.831446\pi$$
−0.863045 + 0.505126i $$0.831446\pi$$
$$74$$ 0 0
$$75$$ −1.79129 −0.206840
$$76$$ 0 0
$$77$$ 10.5826 1.20600
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −9.58258 −1.06473
$$82$$ 0 0
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 3.79129 0.411223
$$86$$ 0 0
$$87$$ 2.83485 0.303928
$$88$$ 0 0
$$89$$ −3.16515 −0.335505 −0.167753 0.985829i $$-0.553651\pi$$
−0.167753 + 0.985829i $$0.553651\pi$$
$$90$$ 0 0
$$91$$ −3.37386 −0.353677
$$92$$ 0 0
$$93$$ 18.5826 1.92692
$$94$$ 0 0
$$95$$ 1.20871 0.124011
$$96$$ 0 0
$$97$$ 14.9564 1.51860 0.759298 0.650743i $$-0.225542\pi$$
0.759298 + 0.650743i $$0.225542\pi$$
$$98$$ 0 0
$$99$$ −0.791288 −0.0795274
$$100$$ 0 0
$$101$$ 13.5826 1.35152 0.675758 0.737123i $$-0.263816\pi$$
0.675758 + 0.737123i $$0.263816\pi$$
$$102$$ 0 0
$$103$$ −7.37386 −0.726568 −0.363284 0.931678i $$-0.618345\pi$$
−0.363284 + 0.931678i $$0.618345\pi$$
$$104$$ 0 0
$$105$$ −5.00000 −0.487950
$$106$$ 0 0
$$107$$ −13.5826 −1.31308 −0.656539 0.754292i $$-0.727980\pi$$
−0.656539 + 0.754292i $$0.727980\pi$$
$$108$$ 0 0
$$109$$ 10.3739 0.993636 0.496818 0.867855i $$-0.334502\pi$$
0.496818 + 0.867855i $$0.334502\pi$$
$$110$$ 0 0
$$111$$ 7.16515 0.680086
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 0.252273 0.0233226
$$118$$ 0 0
$$119$$ 10.5826 0.970103
$$120$$ 0 0
$$121$$ 3.37386 0.306715
$$122$$ 0 0
$$123$$ 3.95644 0.356740
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 14.7477 1.30865 0.654325 0.756214i $$-0.272953\pi$$
0.654325 + 0.756214i $$0.272953\pi$$
$$128$$ 0 0
$$129$$ −12.8348 −1.13005
$$130$$ 0 0
$$131$$ −9.16515 −0.800763 −0.400381 0.916349i $$-0.631122\pi$$
−0.400381 + 0.916349i $$0.631122\pi$$
$$132$$ 0 0
$$133$$ 3.37386 0.292551
$$134$$ 0 0
$$135$$ −5.00000 −0.430331
$$136$$ 0 0
$$137$$ −0.791288 −0.0676043 −0.0338021 0.999429i $$-0.510762\pi$$
−0.0338021 + 0.999429i $$0.510762\pi$$
$$138$$ 0 0
$$139$$ 14.7477 1.25089 0.625443 0.780270i $$-0.284918\pi$$
0.625443 + 0.780270i $$0.284918\pi$$
$$140$$ 0 0
$$141$$ −24.3303 −2.04898
$$142$$ 0 0
$$143$$ −4.58258 −0.383214
$$144$$ 0 0
$$145$$ 1.58258 0.131426
$$146$$ 0 0
$$147$$ −1.41742 −0.116907
$$148$$ 0 0
$$149$$ 12.7913 1.04790 0.523952 0.851748i $$-0.324457\pi$$
0.523952 + 0.851748i $$0.324457\pi$$
$$150$$ 0 0
$$151$$ 6.20871 0.505258 0.252629 0.967563i $$-0.418705\pi$$
0.252629 + 0.967563i $$0.418705\pi$$
$$152$$ 0 0
$$153$$ −0.791288 −0.0639718
$$154$$ 0 0
$$155$$ 10.3739 0.833249
$$156$$ 0 0
$$157$$ 12.7477 1.01738 0.508690 0.860950i $$-0.330130\pi$$
0.508690 + 0.860950i $$0.330130\pi$$
$$158$$ 0 0
$$159$$ −10.7477 −0.852350
$$160$$ 0 0
$$161$$ 2.79129 0.219984
$$162$$ 0 0
$$163$$ −22.3739 −1.75246 −0.876228 0.481897i $$-0.839948\pi$$
−0.876228 + 0.481897i $$0.839948\pi$$
$$164$$ 0 0
$$165$$ −6.79129 −0.528701
$$166$$ 0 0
$$167$$ 18.3303 1.41844 0.709221 0.704987i $$-0.249047\pi$$
0.709221 + 0.704987i $$0.249047\pi$$
$$168$$ 0 0
$$169$$ −11.5390 −0.887617
$$170$$ 0 0
$$171$$ −0.252273 −0.0192918
$$172$$ 0 0
$$173$$ −14.2087 −1.08027 −0.540134 0.841579i $$-0.681627\pi$$
−0.540134 + 0.841579i $$0.681627\pi$$
$$174$$ 0 0
$$175$$ −2.79129 −0.211002
$$176$$ 0 0
$$177$$ −7.91288 −0.594768
$$178$$ 0 0
$$179$$ 16.7477 1.25178 0.625892 0.779910i $$-0.284735\pi$$
0.625892 + 0.779910i $$0.284735\pi$$
$$180$$ 0 0
$$181$$ 13.5390 1.00635 0.503174 0.864185i $$-0.332166\pi$$
0.503174 + 0.864185i $$0.332166\pi$$
$$182$$ 0 0
$$183$$ 6.04356 0.446753
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ 14.3739 1.05112
$$188$$ 0 0
$$189$$ −13.9564 −1.01518
$$190$$ 0 0
$$191$$ −16.4174 −1.18792 −0.593962 0.804493i $$-0.702437\pi$$
−0.593962 + 0.804493i $$0.702437\pi$$
$$192$$ 0 0
$$193$$ 6.74773 0.485712 0.242856 0.970062i $$-0.421916\pi$$
0.242856 + 0.970062i $$0.421916\pi$$
$$194$$ 0 0
$$195$$ 2.16515 0.155050
$$196$$ 0 0
$$197$$ 20.5390 1.46334 0.731672 0.681657i $$-0.238740\pi$$
0.731672 + 0.681657i $$0.238740\pi$$
$$198$$ 0 0
$$199$$ −20.3303 −1.44118 −0.720588 0.693363i $$-0.756128\pi$$
−0.720588 + 0.693363i $$0.756128\pi$$
$$200$$ 0 0
$$201$$ −12.8348 −0.905300
$$202$$ 0 0
$$203$$ 4.41742 0.310042
$$204$$ 0 0
$$205$$ 2.20871 0.154263
$$206$$ 0 0
$$207$$ −0.208712 −0.0145065
$$208$$ 0 0
$$209$$ 4.58258 0.316983
$$210$$ 0 0
$$211$$ 10.0000 0.688428 0.344214 0.938891i $$-0.388145\pi$$
0.344214 + 0.938891i $$0.388145\pi$$
$$212$$ 0 0
$$213$$ −9.62614 −0.659572
$$214$$ 0 0
$$215$$ −7.16515 −0.488659
$$216$$ 0 0
$$217$$ 28.9564 1.96569
$$218$$ 0 0
$$219$$ 26.4174 1.78512
$$220$$ 0 0
$$221$$ −4.58258 −0.308257
$$222$$ 0 0
$$223$$ −11.1652 −0.747674 −0.373837 0.927494i $$-0.621958\pi$$
−0.373837 + 0.927494i $$0.621958\pi$$
$$224$$ 0 0
$$225$$ 0.208712 0.0139141
$$226$$ 0 0
$$227$$ −4.74773 −0.315118 −0.157559 0.987510i $$-0.550362\pi$$
−0.157559 + 0.987510i $$0.550362\pi$$
$$228$$ 0 0
$$229$$ −16.3303 −1.07914 −0.539568 0.841942i $$-0.681413\pi$$
−0.539568 + 0.841942i $$0.681413\pi$$
$$230$$ 0 0
$$231$$ −18.9564 −1.24724
$$232$$ 0 0
$$233$$ 7.58258 0.496751 0.248376 0.968664i $$-0.420103\pi$$
0.248376 + 0.968664i $$0.420103\pi$$
$$234$$ 0 0
$$235$$ −13.5826 −0.886030
$$236$$ 0 0
$$237$$ 14.3303 0.930853
$$238$$ 0 0
$$239$$ −3.16515 −0.204737 −0.102368 0.994747i $$-0.532642\pi$$
−0.102368 + 0.994747i $$0.532642\pi$$
$$240$$ 0 0
$$241$$ −28.0000 −1.80364 −0.901819 0.432113i $$-0.857768\pi$$
−0.901819 + 0.432113i $$0.857768\pi$$
$$242$$ 0 0
$$243$$ 2.16515 0.138895
$$244$$ 0 0
$$245$$ −0.791288 −0.0505535
$$246$$ 0 0
$$247$$ −1.46099 −0.0929603
$$248$$ 0 0
$$249$$ −10.7477 −0.681110
$$250$$ 0 0
$$251$$ −30.7913 −1.94353 −0.971764 0.235953i $$-0.924179\pi$$
−0.971764 + 0.235953i $$0.924179\pi$$
$$252$$ 0 0
$$253$$ 3.79129 0.238356
$$254$$ 0 0
$$255$$ −6.79129 −0.425287
$$256$$ 0 0
$$257$$ 22.7477 1.41896 0.709482 0.704723i $$-0.248929\pi$$
0.709482 + 0.704723i $$0.248929\pi$$
$$258$$ 0 0
$$259$$ 11.1652 0.693769
$$260$$ 0 0
$$261$$ −0.330303 −0.0204452
$$262$$ 0 0
$$263$$ −15.7913 −0.973733 −0.486866 0.873477i $$-0.661860\pi$$
−0.486866 + 0.873477i $$0.661860\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ 5.66970 0.346980
$$268$$ 0 0
$$269$$ 16.7477 1.02113 0.510563 0.859840i $$-0.329437\pi$$
0.510563 + 0.859840i $$0.329437\pi$$
$$270$$ 0 0
$$271$$ 23.1216 1.40454 0.702268 0.711912i $$-0.252171\pi$$
0.702268 + 0.711912i $$0.252171\pi$$
$$272$$ 0 0
$$273$$ 6.04356 0.365773
$$274$$ 0 0
$$275$$ −3.79129 −0.228623
$$276$$ 0 0
$$277$$ −1.16515 −0.0700072 −0.0350036 0.999387i $$-0.511144\pi$$
−0.0350036 + 0.999387i $$0.511144\pi$$
$$278$$ 0 0
$$279$$ −2.16515 −0.129624
$$280$$ 0 0
$$281$$ −16.7477 −0.999086 −0.499543 0.866289i $$-0.666499\pi$$
−0.499543 + 0.866289i $$0.666499\pi$$
$$282$$ 0 0
$$283$$ 28.3303 1.68406 0.842031 0.539429i $$-0.181360\pi$$
0.842031 + 0.539429i $$0.181360\pi$$
$$284$$ 0 0
$$285$$ −2.16515 −0.128252
$$286$$ 0 0
$$287$$ 6.16515 0.363917
$$288$$ 0 0
$$289$$ −2.62614 −0.154479
$$290$$ 0 0
$$291$$ −26.7913 −1.57053
$$292$$ 0 0
$$293$$ −27.4955 −1.60630 −0.803151 0.595776i $$-0.796845\pi$$
−0.803151 + 0.595776i $$0.796845\pi$$
$$294$$ 0 0
$$295$$ −4.41742 −0.257192
$$296$$ 0 0
$$297$$ −18.9564 −1.09996
$$298$$ 0 0
$$299$$ −1.20871 −0.0699016
$$300$$ 0 0
$$301$$ −20.0000 −1.15278
$$302$$ 0 0
$$303$$ −24.3303 −1.39774
$$304$$ 0 0
$$305$$ 3.37386 0.193187
$$306$$ 0 0
$$307$$ −16.5390 −0.943931 −0.471966 0.881617i $$-0.656455\pi$$
−0.471966 + 0.881617i $$0.656455\pi$$
$$308$$ 0 0
$$309$$ 13.2087 0.751417
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ −18.3739 −1.03855 −0.519276 0.854607i $$-0.673798\pi$$
−0.519276 + 0.854607i $$0.673798\pi$$
$$314$$ 0 0
$$315$$ 0.582576 0.0328244
$$316$$ 0 0
$$317$$ 5.20871 0.292550 0.146275 0.989244i $$-0.453271\pi$$
0.146275 + 0.989244i $$0.453271\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 24.3303 1.35799
$$322$$ 0 0
$$323$$ 4.58258 0.254981
$$324$$ 0 0
$$325$$ 1.20871 0.0670473
$$326$$ 0 0
$$327$$ −18.5826 −1.02762
$$328$$ 0 0
$$329$$ −37.9129 −2.09020
$$330$$ 0 0
$$331$$ −6.74773 −0.370889 −0.185444 0.982655i $$-0.559372\pi$$
−0.185444 + 0.982655i $$0.559372\pi$$
$$332$$ 0 0
$$333$$ −0.834849 −0.0457494
$$334$$ 0 0
$$335$$ −7.16515 −0.391474
$$336$$ 0 0
$$337$$ −16.7913 −0.914680 −0.457340 0.889292i $$-0.651198\pi$$
−0.457340 + 0.889292i $$0.651198\pi$$
$$338$$ 0 0
$$339$$ −10.7477 −0.583736
$$340$$ 0 0
$$341$$ 39.3303 2.12986
$$342$$ 0 0
$$343$$ 17.3303 0.935748
$$344$$ 0 0
$$345$$ −1.79129 −0.0964397
$$346$$ 0 0
$$347$$ −9.79129 −0.525624 −0.262812 0.964847i $$-0.584650\pi$$
−0.262812 + 0.964847i $$0.584650\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ 6.04356 0.322581
$$352$$ 0 0
$$353$$ −15.1652 −0.807160 −0.403580 0.914944i $$-0.632234\pi$$
−0.403580 + 0.914944i $$0.632234\pi$$
$$354$$ 0 0
$$355$$ −5.37386 −0.285215
$$356$$ 0 0
$$357$$ −18.9564 −1.00328
$$358$$ 0 0
$$359$$ 9.16515 0.483718 0.241859 0.970311i $$-0.422243\pi$$
0.241859 + 0.970311i $$0.422243\pi$$
$$360$$ 0 0
$$361$$ −17.5390 −0.923106
$$362$$ 0 0
$$363$$ −6.04356 −0.317205
$$364$$ 0 0
$$365$$ 14.7477 0.771931
$$366$$ 0 0
$$367$$ 0.834849 0.0435787 0.0217894 0.999763i $$-0.493064\pi$$
0.0217894 + 0.999763i $$0.493064\pi$$
$$368$$ 0 0
$$369$$ −0.460985 −0.0239979
$$370$$ 0 0
$$371$$ −16.7477 −0.869499
$$372$$ 0 0
$$373$$ −14.7477 −0.763608 −0.381804 0.924243i $$-0.624697\pi$$
−0.381804 + 0.924243i $$0.624697\pi$$
$$374$$ 0 0
$$375$$ 1.79129 0.0925017
$$376$$ 0 0
$$377$$ −1.91288 −0.0985183
$$378$$ 0 0
$$379$$ −7.37386 −0.378770 −0.189385 0.981903i $$-0.560649\pi$$
−0.189385 + 0.981903i $$0.560649\pi$$
$$380$$ 0 0
$$381$$ −26.4174 −1.35341
$$382$$ 0 0
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ −10.5826 −0.539338
$$386$$ 0 0
$$387$$ 1.49545 0.0760182
$$388$$ 0 0
$$389$$ 29.7042 1.50606 0.753031 0.657986i $$-0.228591\pi$$
0.753031 + 0.657986i $$0.228591\pi$$
$$390$$ 0 0
$$391$$ 3.79129 0.191734
$$392$$ 0 0
$$393$$ 16.4174 0.828150
$$394$$ 0 0
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ 16.5390 0.830069 0.415035 0.909806i $$-0.363769\pi$$
0.415035 + 0.909806i $$0.363769\pi$$
$$398$$ 0 0
$$399$$ −6.04356 −0.302556
$$400$$ 0 0
$$401$$ 22.7477 1.13597 0.567984 0.823040i $$-0.307724\pi$$
0.567984 + 0.823040i $$0.307724\pi$$
$$402$$ 0 0
$$403$$ −12.5390 −0.624613
$$404$$ 0 0
$$405$$ 9.58258 0.476162
$$406$$ 0 0
$$407$$ 15.1652 0.751709
$$408$$ 0 0
$$409$$ −22.7913 −1.12696 −0.563478 0.826131i $$-0.690537\pi$$
−0.563478 + 0.826131i $$0.690537\pi$$
$$410$$ 0 0
$$411$$ 1.41742 0.0699164
$$412$$ 0 0
$$413$$ −12.3303 −0.606735
$$414$$ 0 0
$$415$$ −6.00000 −0.294528
$$416$$ 0 0
$$417$$ −26.4174 −1.29367
$$418$$ 0 0
$$419$$ −39.1652 −1.91334 −0.956671 0.291170i $$-0.905956\pi$$
−0.956671 + 0.291170i $$0.905956\pi$$
$$420$$ 0 0
$$421$$ −23.1216 −1.12688 −0.563439 0.826158i $$-0.690522\pi$$
−0.563439 + 0.826158i $$0.690522\pi$$
$$422$$ 0 0
$$423$$ 2.83485 0.137835
$$424$$ 0 0
$$425$$ −3.79129 −0.183904
$$426$$ 0 0
$$427$$ 9.41742 0.455741
$$428$$ 0 0
$$429$$ 8.20871 0.396320
$$430$$ 0 0
$$431$$ 19.9129 0.959170 0.479585 0.877496i $$-0.340787\pi$$
0.479585 + 0.877496i $$0.340787\pi$$
$$432$$ 0 0
$$433$$ 1.53901 0.0739603 0.0369802 0.999316i $$-0.488226\pi$$
0.0369802 + 0.999316i $$0.488226\pi$$
$$434$$ 0 0
$$435$$ −2.83485 −0.135921
$$436$$ 0 0
$$437$$ 1.20871 0.0578205
$$438$$ 0 0
$$439$$ −25.5390 −1.21891 −0.609455 0.792820i $$-0.708612\pi$$
−0.609455 + 0.792820i $$0.708612\pi$$
$$440$$ 0 0
$$441$$ 0.165151 0.00786435
$$442$$ 0 0
$$443$$ 35.2087 1.67282 0.836408 0.548107i $$-0.184651\pi$$
0.836408 + 0.548107i $$0.184651\pi$$
$$444$$ 0 0
$$445$$ 3.16515 0.150043
$$446$$ 0 0
$$447$$ −22.9129 −1.08374
$$448$$ 0 0
$$449$$ 25.1216 1.18556 0.592781 0.805364i $$-0.298030\pi$$
0.592781 + 0.805364i $$0.298030\pi$$
$$450$$ 0 0
$$451$$ 8.37386 0.394310
$$452$$ 0 0
$$453$$ −11.1216 −0.522538
$$454$$ 0 0
$$455$$ 3.37386 0.158169
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 0 0
$$459$$ −18.9564 −0.884811
$$460$$ 0 0
$$461$$ −1.25227 −0.0583242 −0.0291621 0.999575i $$-0.509284\pi$$
−0.0291621 + 0.999575i $$0.509284\pi$$
$$462$$ 0 0
$$463$$ 10.0000 0.464739 0.232370 0.972628i $$-0.425352\pi$$
0.232370 + 0.972628i $$0.425352\pi$$
$$464$$ 0 0
$$465$$ −18.5826 −0.861746
$$466$$ 0 0
$$467$$ −25.9129 −1.19911 −0.599553 0.800335i $$-0.704655\pi$$
−0.599553 + 0.800335i $$0.704655\pi$$
$$468$$ 0 0
$$469$$ −20.0000 −0.923514
$$470$$ 0 0
$$471$$ −22.8348 −1.05217
$$472$$ 0 0
$$473$$ −27.1652 −1.24905
$$474$$ 0 0
$$475$$ −1.20871 −0.0554595
$$476$$ 0 0
$$477$$ 1.25227 0.0573376
$$478$$ 0 0
$$479$$ 39.4955 1.80459 0.902297 0.431116i $$-0.141880\pi$$
0.902297 + 0.431116i $$0.141880\pi$$
$$480$$ 0 0
$$481$$ −4.83485 −0.220450
$$482$$ 0 0
$$483$$ −5.00000 −0.227508
$$484$$ 0 0
$$485$$ −14.9564 −0.679137
$$486$$ 0 0
$$487$$ −15.5826 −0.706114 −0.353057 0.935602i $$-0.614858\pi$$
−0.353057 + 0.935602i $$0.614858\pi$$
$$488$$ 0 0
$$489$$ 40.0780 1.81239
$$490$$ 0 0
$$491$$ 16.7477 0.755814 0.377907 0.925843i $$-0.376644\pi$$
0.377907 + 0.925843i $$0.376644\pi$$
$$492$$ 0 0
$$493$$ 6.00000 0.270226
$$494$$ 0 0
$$495$$ 0.791288 0.0355657
$$496$$ 0 0
$$497$$ −15.0000 −0.672842
$$498$$ 0 0
$$499$$ −23.1652 −1.03701 −0.518507 0.855073i $$-0.673512\pi$$
−0.518507 + 0.855073i $$0.673512\pi$$
$$500$$ 0 0
$$501$$ −32.8348 −1.46695
$$502$$ 0 0
$$503$$ −18.7913 −0.837862 −0.418931 0.908018i $$-0.637595\pi$$
−0.418931 + 0.908018i $$0.637595\pi$$
$$504$$ 0 0
$$505$$ −13.5826 −0.604417
$$506$$ 0 0
$$507$$ 20.6697 0.917973
$$508$$ 0 0
$$509$$ 7.25227 0.321451 0.160726 0.986999i $$-0.448617\pi$$
0.160726 + 0.986999i $$0.448617\pi$$
$$510$$ 0 0
$$511$$ 41.1652 1.82104
$$512$$ 0 0
$$513$$ −6.04356 −0.266830
$$514$$ 0 0
$$515$$ 7.37386 0.324931
$$516$$ 0 0
$$517$$ −51.4955 −2.26477
$$518$$ 0 0
$$519$$ 25.4519 1.11721
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ 1.16515 0.0509485 0.0254743 0.999675i $$-0.491890\pi$$
0.0254743 + 0.999675i $$0.491890\pi$$
$$524$$ 0 0
$$525$$ 5.00000 0.218218
$$526$$ 0 0
$$527$$ 39.3303 1.71325
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 0.921970 0.0400101
$$532$$ 0 0
$$533$$ −2.66970 −0.115637
$$534$$ 0 0
$$535$$ 13.5826 0.587226
$$536$$ 0 0
$$537$$ −30.0000 −1.29460
$$538$$ 0 0
$$539$$ −3.00000 −0.129219
$$540$$ 0 0
$$541$$ 38.3303 1.64795 0.823974 0.566627i $$-0.191752\pi$$
0.823974 + 0.566627i $$0.191752\pi$$
$$542$$ 0 0
$$543$$ −24.2523 −1.04076
$$544$$ 0 0
$$545$$ −10.3739 −0.444367
$$546$$ 0 0
$$547$$ −15.1216 −0.646553 −0.323276 0.946305i $$-0.604784\pi$$
−0.323276 + 0.946305i $$0.604784\pi$$
$$548$$ 0 0
$$549$$ −0.704166 −0.0300531
$$550$$ 0 0
$$551$$ 1.91288 0.0814914
$$552$$ 0 0
$$553$$ 22.3303 0.949581
$$554$$ 0 0
$$555$$ −7.16515 −0.304144
$$556$$ 0 0
$$557$$ 30.3303 1.28514 0.642568 0.766229i $$-0.277869\pi$$
0.642568 + 0.766229i $$0.277869\pi$$
$$558$$ 0 0
$$559$$ 8.66061 0.366305
$$560$$ 0 0
$$561$$ −25.7477 −1.08707
$$562$$ 0 0
$$563$$ −3.16515 −0.133395 −0.0666976 0.997773i $$-0.521246\pi$$
−0.0666976 + 0.997773i $$0.521246\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 26.7477 1.12330
$$568$$ 0 0
$$569$$ 15.4955 0.649603 0.324802 0.945782i $$-0.394702\pi$$
0.324802 + 0.945782i $$0.394702\pi$$
$$570$$ 0 0
$$571$$ −30.1216 −1.26055 −0.630275 0.776372i $$-0.717058\pi$$
−0.630275 + 0.776372i $$0.717058\pi$$
$$572$$ 0 0
$$573$$ 29.4083 1.22855
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ 22.8348 0.950627 0.475314 0.879816i $$-0.342335\pi$$
0.475314 + 0.879816i $$0.342335\pi$$
$$578$$ 0 0
$$579$$ −12.0871 −0.502324
$$580$$ 0 0
$$581$$ −16.7477 −0.694813
$$582$$ 0 0
$$583$$ −22.7477 −0.942115
$$584$$ 0 0
$$585$$ −0.252273 −0.0104302
$$586$$ 0 0
$$587$$ 26.2087 1.08175 0.540875 0.841103i $$-0.318093\pi$$
0.540875 + 0.841103i $$0.318093\pi$$
$$588$$ 0 0
$$589$$ 12.5390 0.516661
$$590$$ 0 0
$$591$$ −36.7913 −1.51339
$$592$$ 0 0
$$593$$ 13.9129 0.571333 0.285667 0.958329i $$-0.407785\pi$$
0.285667 + 0.958329i $$0.407785\pi$$
$$594$$ 0 0
$$595$$ −10.5826 −0.433843
$$596$$ 0 0
$$597$$ 36.4174 1.49047
$$598$$ 0 0
$$599$$ 40.1216 1.63932 0.819662 0.572848i $$-0.194161\pi$$
0.819662 + 0.572848i $$0.194161\pi$$
$$600$$ 0 0
$$601$$ −22.7913 −0.929676 −0.464838 0.885396i $$-0.653887\pi$$
−0.464838 + 0.885396i $$0.653887\pi$$
$$602$$ 0 0
$$603$$ 1.49545 0.0608996
$$604$$ 0 0
$$605$$ −3.37386 −0.137167
$$606$$ 0 0
$$607$$ 28.0000 1.13648 0.568242 0.822861i $$-0.307624\pi$$
0.568242 + 0.822861i $$0.307624\pi$$
$$608$$ 0 0
$$609$$ −7.91288 −0.320646
$$610$$ 0 0
$$611$$ 16.4174 0.664178
$$612$$ 0 0
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ 0 0
$$615$$ −3.95644 −0.159539
$$616$$ 0 0
$$617$$ 44.8693 1.80637 0.903185 0.429251i $$-0.141222\pi$$
0.903185 + 0.429251i $$0.141222\pi$$
$$618$$ 0 0
$$619$$ −2.79129 −0.112191 −0.0560957 0.998425i $$-0.517865\pi$$
−0.0560957 + 0.998425i $$0.517865\pi$$
$$620$$ 0 0
$$621$$ −5.00000 −0.200643
$$622$$ 0 0
$$623$$ 8.83485 0.353961
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −8.20871 −0.327824
$$628$$ 0 0
$$629$$ 15.1652 0.604674
$$630$$ 0 0
$$631$$ 17.9129 0.713100 0.356550 0.934276i $$-0.383953\pi$$
0.356550 + 0.934276i $$0.383953\pi$$
$$632$$ 0 0
$$633$$ −17.9129 −0.711973
$$634$$ 0 0
$$635$$ −14.7477 −0.585246
$$636$$ 0 0
$$637$$ 0.956439 0.0378955
$$638$$ 0 0
$$639$$ 1.12159 0.0443694
$$640$$ 0 0
$$641$$ −3.16515 −0.125016 −0.0625080 0.998044i $$-0.519910\pi$$
−0.0625080 + 0.998044i $$0.519910\pi$$
$$642$$ 0 0
$$643$$ 20.7477 0.818210 0.409105 0.912487i $$-0.365841\pi$$
0.409105 + 0.912487i $$0.365841\pi$$
$$644$$ 0 0
$$645$$ 12.8348 0.505372
$$646$$ 0 0
$$647$$ −2.83485 −0.111449 −0.0557247 0.998446i $$-0.517747\pi$$
−0.0557247 + 0.998446i $$0.517747\pi$$
$$648$$ 0 0
$$649$$ −16.7477 −0.657406
$$650$$ 0 0
$$651$$ −51.8693 −2.03292
$$652$$ 0 0
$$653$$ 35.5390 1.39075 0.695375 0.718647i $$-0.255238\pi$$
0.695375 + 0.718647i $$0.255238\pi$$
$$654$$ 0 0
$$655$$ 9.16515 0.358112
$$656$$ 0 0
$$657$$ −3.07803 −0.120085
$$658$$ 0 0
$$659$$ 27.1652 1.05820 0.529102 0.848558i $$-0.322529\pi$$
0.529102 + 0.848558i $$0.322529\pi$$
$$660$$ 0 0
$$661$$ −39.3739 −1.53147 −0.765733 0.643159i $$-0.777624\pi$$
−0.765733 + 0.643159i $$0.777624\pi$$
$$662$$ 0 0
$$663$$ 8.20871 0.318800
$$664$$ 0 0
$$665$$ −3.37386 −0.130833
$$666$$ 0 0
$$667$$ 1.58258 0.0612776
$$668$$ 0 0
$$669$$ 20.0000 0.773245
$$670$$ 0 0
$$671$$ 12.7913 0.493802
$$672$$ 0 0
$$673$$ 38.0000 1.46479 0.732396 0.680879i $$-0.238402\pi$$
0.732396 + 0.680879i $$0.238402\pi$$
$$674$$ 0 0
$$675$$ 5.00000 0.192450
$$676$$ 0 0
$$677$$ −30.6606 −1.17838 −0.589191 0.807993i $$-0.700554\pi$$
−0.589191 + 0.807993i $$0.700554\pi$$
$$678$$ 0 0
$$679$$ −41.7477 −1.60213
$$680$$ 0 0
$$681$$ 8.50455 0.325895
$$682$$ 0 0
$$683$$ −2.37386 −0.0908334 −0.0454167 0.998968i $$-0.514462\pi$$
−0.0454167 + 0.998968i $$0.514462\pi$$
$$684$$ 0 0
$$685$$ 0.791288 0.0302336
$$686$$ 0 0
$$687$$ 29.2523 1.11604
$$688$$ 0 0
$$689$$ 7.25227 0.276290
$$690$$ 0 0
$$691$$ −15.2523 −0.580224 −0.290112 0.956993i $$-0.593693\pi$$
−0.290112 + 0.956993i $$0.593693\pi$$
$$692$$ 0 0
$$693$$ 2.20871 0.0839020
$$694$$ 0 0
$$695$$ −14.7477 −0.559413
$$696$$ 0 0
$$697$$ 8.37386 0.317183
$$698$$ 0 0
$$699$$ −13.5826 −0.513740
$$700$$ 0 0
$$701$$ 9.62614 0.363574 0.181787 0.983338i $$-0.441812\pi$$
0.181787 + 0.983338i $$0.441812\pi$$
$$702$$ 0 0
$$703$$ 4.83485 0.182350
$$704$$ 0 0
$$705$$ 24.3303 0.916332
$$706$$ 0 0
$$707$$ −37.9129 −1.42586
$$708$$ 0 0
$$709$$ 34.5390 1.29714 0.648570 0.761155i $$-0.275367\pi$$
0.648570 + 0.761155i $$0.275367\pi$$
$$710$$ 0 0
$$711$$ −1.66970 −0.0626185
$$712$$ 0 0
$$713$$ 10.3739 0.388504
$$714$$ 0 0
$$715$$ 4.58258 0.171379
$$716$$ 0 0
$$717$$ 5.66970 0.211739
$$718$$ 0 0
$$719$$ 29.5390 1.10162 0.550810 0.834631i $$-0.314319\pi$$
0.550810 + 0.834631i $$0.314319\pi$$
$$720$$ 0 0
$$721$$ 20.5826 0.766535
$$722$$ 0 0
$$723$$ 50.1561 1.86532
$$724$$ 0 0
$$725$$ −1.58258 −0.0587754
$$726$$ 0 0
$$727$$ 2.12159 0.0786854 0.0393427 0.999226i $$-0.487474\pi$$
0.0393427 + 0.999226i $$0.487474\pi$$
$$728$$ 0 0
$$729$$ 24.8693 0.921086
$$730$$ 0 0
$$731$$ −27.1652 −1.00474
$$732$$ 0 0
$$733$$ 26.0000 0.960332 0.480166 0.877178i $$-0.340576\pi$$
0.480166 + 0.877178i $$0.340576\pi$$
$$734$$ 0 0
$$735$$ 1.41742 0.0522825
$$736$$ 0 0
$$737$$ −27.1652 −1.00064
$$738$$ 0 0
$$739$$ −8.00000 −0.294285 −0.147142 0.989115i $$-0.547008\pi$$
−0.147142 + 0.989115i $$0.547008\pi$$
$$740$$ 0 0
$$741$$ 2.61704 0.0961395
$$742$$ 0 0
$$743$$ −9.95644 −0.365266 −0.182633 0.983181i $$-0.558462\pi$$
−0.182633 + 0.983181i $$0.558462\pi$$
$$744$$ 0 0
$$745$$ −12.7913 −0.468637
$$746$$ 0 0
$$747$$ 1.25227 0.0458183
$$748$$ 0 0
$$749$$ 37.9129 1.38531
$$750$$ 0 0
$$751$$ −18.7477 −0.684114 −0.342057 0.939679i $$-0.611124\pi$$
−0.342057 + 0.939679i $$0.611124\pi$$
$$752$$ 0 0
$$753$$ 55.1561 2.01000
$$754$$ 0 0
$$755$$ −6.20871 −0.225958
$$756$$ 0 0
$$757$$ 26.3303 0.956991 0.478496 0.878090i $$-0.341182\pi$$
0.478496 + 0.878090i $$0.341182\pi$$
$$758$$ 0 0
$$759$$ −6.79129 −0.246508
$$760$$ 0 0
$$761$$ −33.9564 −1.23092 −0.615460 0.788168i $$-0.711030\pi$$
−0.615460 + 0.788168i $$0.711030\pi$$
$$762$$ 0 0
$$763$$ −28.9564 −1.04829
$$764$$ 0 0
$$765$$ 0.791288 0.0286091
$$766$$ 0 0
$$767$$ 5.33939 0.192794
$$768$$ 0 0
$$769$$ −3.66970 −0.132333 −0.0661663 0.997809i $$-0.521077\pi$$
−0.0661663 + 0.997809i $$0.521077\pi$$
$$770$$ 0 0
$$771$$ −40.7477 −1.46749
$$772$$ 0 0
$$773$$ −21.4955 −0.773138 −0.386569 0.922261i $$-0.626340\pi$$
−0.386569 + 0.922261i $$0.626340\pi$$
$$774$$ 0 0
$$775$$ −10.3739 −0.372640
$$776$$ 0 0
$$777$$ −20.0000 −0.717496
$$778$$ 0 0
$$779$$ 2.66970 0.0956518
$$780$$ 0 0
$$781$$ −20.3739 −0.729034
$$782$$ 0 0
$$783$$ −7.91288 −0.282783
$$784$$ 0 0
$$785$$ −12.7477 −0.454986
$$786$$ 0 0
$$787$$ 8.41742 0.300049 0.150024 0.988682i $$-0.452065\pi$$
0.150024 + 0.988682i $$0.452065\pi$$
$$788$$ 0 0
$$789$$ 28.2867 1.00703
$$790$$ 0 0
$$791$$ −16.7477 −0.595481
$$792$$ 0 0
$$793$$ −4.07803 −0.144815
$$794$$ 0 0
$$795$$ 10.7477 0.381183
$$796$$ 0 0
$$797$$ −49.9129 −1.76800 −0.884002 0.467482i $$-0.845161\pi$$
−0.884002 + 0.467482i $$0.845161\pi$$
$$798$$ 0 0
$$799$$ −51.4955 −1.82178
$$800$$ 0 0
$$801$$ −0.660606 −0.0233413
$$802$$ 0 0
$$803$$ 55.9129 1.97312
$$804$$ 0 0
$$805$$ −2.79129 −0.0983800
$$806$$ 0 0
$$807$$ −30.0000 −1.05605
$$808$$ 0 0
$$809$$ 11.0436 0.388271 0.194135 0.980975i $$-0.437810\pi$$
0.194135 + 0.980975i $$0.437810\pi$$
$$810$$ 0 0
$$811$$ 47.9129 1.68245 0.841224 0.540686i $$-0.181835\pi$$
0.841224 + 0.540686i $$0.181835\pi$$
$$812$$ 0 0
$$813$$ −41.4174 −1.45257
$$814$$ 0 0
$$815$$ 22.3739 0.783722
$$816$$ 0 0
$$817$$ −8.66061 −0.302996
$$818$$ 0 0
$$819$$ −0.704166 −0.0246056
$$820$$ 0 0
$$821$$ 2.83485 0.0989369 0.0494684 0.998776i $$-0.484247\pi$$
0.0494684 + 0.998776i $$0.484247\pi$$
$$822$$ 0 0
$$823$$ −41.1652 −1.43493 −0.717463 0.696596i $$-0.754697\pi$$
−0.717463 + 0.696596i $$0.754697\pi$$
$$824$$ 0 0
$$825$$ 6.79129 0.236442
$$826$$ 0 0
$$827$$ −41.0780 −1.42842 −0.714212 0.699930i $$-0.753215\pi$$
−0.714212 + 0.699930i $$0.753215\pi$$
$$828$$ 0 0
$$829$$ −31.4955 −1.09388 −0.546941 0.837171i $$-0.684208\pi$$
−0.546941 + 0.837171i $$0.684208\pi$$
$$830$$ 0 0
$$831$$ 2.08712 0.0724014
$$832$$ 0 0
$$833$$ −3.00000 −0.103944
$$834$$ 0 0
$$835$$ −18.3303 −0.634346
$$836$$ 0 0
$$837$$ −51.8693 −1.79287
$$838$$ 0 0
$$839$$ 22.4174 0.773935 0.386968 0.922093i $$-0.373522\pi$$
0.386968 + 0.922093i $$0.373522\pi$$
$$840$$ 0 0
$$841$$ −26.4955 −0.913636
$$842$$ 0 0
$$843$$ 30.0000 1.03325
$$844$$ 0 0
$$845$$ 11.5390 0.396954
$$846$$ 0 0
$$847$$ −9.41742 −0.323587
$$848$$ 0 0
$$849$$ −50.7477 −1.74166
$$850$$ 0 0
$$851$$ 4.00000 0.137118
$$852$$ 0 0
$$853$$ 8.46099 0.289699 0.144849 0.989454i $$-0.453730\pi$$
0.144849 + 0.989454i $$0.453730\pi$$
$$854$$ 0 0
$$855$$ 0.252273 0.00862755
$$856$$ 0 0
$$857$$ 9.16515 0.313076 0.156538 0.987672i $$-0.449967\pi$$
0.156538 + 0.987672i $$0.449967\pi$$
$$858$$ 0 0
$$859$$ −0.747727 −0.0255121 −0.0127561 0.999919i $$-0.504060\pi$$
−0.0127561 + 0.999919i $$0.504060\pi$$
$$860$$ 0 0
$$861$$ −11.0436 −0.376364
$$862$$ 0 0
$$863$$ 31.5826 1.07508 0.537542 0.843237i $$-0.319353\pi$$
0.537542 + 0.843237i $$0.319353\pi$$
$$864$$ 0 0
$$865$$ 14.2087 0.483111
$$866$$ 0 0
$$867$$ 4.70417 0.159762
$$868$$ 0 0
$$869$$ 30.3303 1.02889
$$870$$ 0 0
$$871$$ 8.66061 0.293453
$$872$$ 0 0
$$873$$ 3.12159 0.105650
$$874$$ 0 0
$$875$$ 2.79129 0.0943628
$$876$$ 0 0
$$877$$ 7.70417 0.260151 0.130076 0.991504i $$-0.458478\pi$$
0.130076 + 0.991504i $$0.458478\pi$$
$$878$$ 0 0
$$879$$ 49.2523 1.66124
$$880$$ 0 0
$$881$$ −6.33030 −0.213273 −0.106637 0.994298i $$-0.534008\pi$$
−0.106637 + 0.994298i $$0.534008\pi$$
$$882$$ 0 0
$$883$$ 12.0436 0.405298 0.202649 0.979251i $$-0.435045\pi$$
0.202649 + 0.979251i $$0.435045\pi$$
$$884$$ 0 0
$$885$$ 7.91288 0.265989
$$886$$ 0 0
$$887$$ 3.16515 0.106275 0.0531377 0.998587i $$-0.483078\pi$$
0.0531377 + 0.998587i $$0.483078\pi$$
$$888$$ 0 0
$$889$$ −41.1652 −1.38063
$$890$$ 0 0
$$891$$ 36.3303 1.21711
$$892$$ 0 0
$$893$$ −16.4174 −0.549388
$$894$$ 0 0
$$895$$ −16.7477 −0.559815
$$896$$ 0 0
$$897$$ 2.16515 0.0722923
$$898$$ 0 0
$$899$$ 16.4174 0.547552
$$900$$ 0 0
$$901$$ −22.7477 −0.757837
$$902$$ 0 0
$$903$$ 35.8258 1.19221
$$904$$ 0 0
$$905$$ −13.5390 −0.450052
$$906$$ 0 0
$$907$$ 20.7477 0.688917 0.344458 0.938802i $$-0.388063\pi$$
0.344458 + 0.938802i $$0.388063\pi$$
$$908$$ 0 0
$$909$$ 2.83485 0.0940260
$$910$$ 0 0
$$911$$ −13.5826 −0.450011 −0.225005 0.974358i $$-0.572240\pi$$
−0.225005 + 0.974358i $$0.572240\pi$$
$$912$$ 0 0
$$913$$ −22.7477 −0.752840
$$914$$ 0 0
$$915$$ −6.04356 −0.199794
$$916$$ 0 0
$$917$$ 25.5826 0.844811
$$918$$ 0 0
$$919$$ 36.8348 1.21507 0.607535 0.794293i $$-0.292159\pi$$
0.607535 + 0.794293i $$0.292159\pi$$
$$920$$ 0 0
$$921$$ 29.6261 0.976214
$$922$$ 0 0
$$923$$ 6.49545 0.213800
$$924$$ 0 0
$$925$$ −4.00000 −0.131519
$$926$$ 0 0
$$927$$ −1.53901 −0.0505479
$$928$$ 0 0
$$929$$ −39.4955 −1.29580 −0.647902 0.761724i $$-0.724353\pi$$
−0.647902 + 0.761724i $$0.724353\pi$$
$$930$$ 0 0
$$931$$ −0.956439 −0.0313460
$$932$$ 0 0
$$933$$ 21.4955 0.703730
$$934$$ 0 0
$$935$$ −14.3739 −0.470076
$$936$$ 0 0
$$937$$ 58.3739 1.90699 0.953495 0.301407i $$-0.0974564\pi$$
0.953495 + 0.301407i $$0.0974564\pi$$
$$938$$ 0 0
$$939$$ 32.9129 1.07407
$$940$$ 0 0
$$941$$ 54.9564 1.79153 0.895764 0.444529i $$-0.146629\pi$$
0.895764 + 0.444529i $$0.146629\pi$$
$$942$$ 0 0
$$943$$ 2.20871 0.0719256
$$944$$ 0 0
$$945$$ 13.9564 0.454003
$$946$$ 0 0
$$947$$ 29.5390 0.959889 0.479945 0.877299i $$-0.340657\pi$$
0.479945 + 0.877299i $$0.340657\pi$$
$$948$$ 0 0
$$949$$ −17.8258 −0.578649
$$950$$ 0 0
$$951$$ −9.33030 −0.302556
$$952$$ 0 0
$$953$$ −26.5390 −0.859683 −0.429842 0.902904i $$-0.641431\pi$$
−0.429842 + 0.902904i $$0.641431\pi$$
$$954$$ 0 0
$$955$$ 16.4174 0.531255
$$956$$ 0 0
$$957$$ −10.7477 −0.347425
$$958$$ 0 0
$$959$$ 2.20871 0.0713230
$$960$$ 0 0
$$961$$ 76.6170 2.47152
$$962$$ 0 0
$$963$$ −2.83485 −0.0913517
$$964$$ 0 0
$$965$$ −6.74773 −0.217217
$$966$$ 0 0
$$967$$ 5.25227 0.168902 0.0844509 0.996428i $$-0.473086\pi$$
0.0844509 + 0.996428i $$0.473086\pi$$
$$968$$ 0 0
$$969$$ −8.20871 −0.263702
$$970$$ 0 0
$$971$$ 6.95644 0.223243 0.111621 0.993751i $$-0.464396\pi$$
0.111621 + 0.993751i $$0.464396\pi$$
$$972$$ 0 0
$$973$$ −41.1652 −1.31969
$$974$$ 0 0
$$975$$ −2.16515 −0.0693403
$$976$$ 0 0
$$977$$ 7.12159 0.227840 0.113920 0.993490i $$-0.463659\pi$$
0.113920 + 0.993490i $$0.463659\pi$$
$$978$$ 0 0
$$979$$ 12.0000 0.383522
$$980$$ 0 0
$$981$$ 2.16515 0.0691280
$$982$$ 0 0
$$983$$ 0.626136 0.0199707 0.00998533 0.999950i $$-0.496822\pi$$
0.00998533 + 0.999950i $$0.496822\pi$$
$$984$$ 0 0
$$985$$ −20.5390 −0.654427
$$986$$ 0 0
$$987$$ 67.9129 2.16169
$$988$$ 0 0
$$989$$ −7.16515 −0.227839
$$990$$ 0 0
$$991$$ 37.7913 1.20048 0.600240 0.799820i $$-0.295072\pi$$
0.600240 + 0.799820i $$0.295072\pi$$
$$992$$ 0 0
$$993$$ 12.0871 0.383573
$$994$$ 0 0
$$995$$ 20.3303 0.644514
$$996$$ 0 0
$$997$$ 11.4955 0.364065 0.182032 0.983293i $$-0.441732\pi$$
0.182032 + 0.983293i $$0.441732\pi$$
$$998$$ 0 0
$$999$$ −20.0000 −0.632772
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 1840.2.a.n.1.1 2
4.3 odd 2 230.2.a.a.1.2 2
5.4 even 2 9200.2.a.bs.1.2 2
8.3 odd 2 7360.2.a.bq.1.1 2
8.5 even 2 7360.2.a.bk.1.2 2
12.11 even 2 2070.2.a.x.1.2 2
20.3 even 4 1150.2.b.g.599.4 4
20.7 even 4 1150.2.b.g.599.1 4
20.19 odd 2 1150.2.a.o.1.1 2
92.91 even 2 5290.2.a.e.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.2 2 4.3 odd 2
1150.2.a.o.1.1 2 20.19 odd 2
1150.2.b.g.599.1 4 20.7 even 4
1150.2.b.g.599.4 4 20.3 even 4
1840.2.a.n.1.1 2 1.1 even 1 trivial
2070.2.a.x.1.2 2 12.11 even 2
5290.2.a.e.1.2 2 92.91 even 2
7360.2.a.bk.1.2 2 8.5 even 2
7360.2.a.bq.1.1 2 8.3 odd 2
9200.2.a.bs.1.2 2 5.4 even 2