# Properties

 Label 8092.2.a.l.1.2 Level $8092$ Weight $2$ Character 8092.1 Self dual yes Analytic conductor $64.615$ Analytic rank $1$ 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] = [8092,2,Mod(1,8092)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8092, 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("8092.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8092 = 2^{2} \cdot 7 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8092.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: $$64.6149453156$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 476) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 8092.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.30278 q^{3} +1.30278 q^{5} +1.00000 q^{7} +2.30278 q^{9} +O(q^{10})$$ $$q+2.30278 q^{3} +1.30278 q^{5} +1.00000 q^{7} +2.30278 q^{9} -4.00000 q^{11} -4.60555 q^{13} +3.00000 q^{15} +8.60555 q^{19} +2.30278 q^{21} -4.00000 q^{23} -3.30278 q^{25} -1.60555 q^{27} -9.21110 q^{29} -7.30278 q^{31} -9.21110 q^{33} +1.30278 q^{35} -9.81665 q^{37} -10.6056 q^{39} +11.5139 q^{41} -4.30278 q^{43} +3.00000 q^{45} -2.60555 q^{47} +1.00000 q^{49} +0.697224 q^{53} -5.21110 q^{55} +19.8167 q^{57} -8.00000 q^{59} -15.5139 q^{61} +2.30278 q^{63} -6.00000 q^{65} +2.69722 q^{67} -9.21110 q^{69} +3.39445 q^{71} -7.51388 q^{73} -7.60555 q^{75} -4.00000 q^{77} +2.60555 q^{79} -10.6056 q^{81} +3.21110 q^{83} -21.2111 q^{87} +7.81665 q^{89} -4.60555 q^{91} -16.8167 q^{93} +11.2111 q^{95} +13.3028 q^{97} -9.21110 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{5} + 2 q^{7} + q^{9}+O(q^{10})$$ 2 * q + q^3 - q^5 + 2 * q^7 + q^9 $$2 q + q^{3} - q^{5} + 2 q^{7} + q^{9} - 8 q^{11} - 2 q^{13} + 6 q^{15} + 10 q^{19} + q^{21} - 8 q^{23} - 3 q^{25} + 4 q^{27} - 4 q^{29} - 11 q^{31} - 4 q^{33} - q^{35} + 2 q^{37} - 14 q^{39} + 5 q^{41} - 5 q^{43} + 6 q^{45} + 2 q^{47} + 2 q^{49} + 5 q^{53} + 4 q^{55} + 18 q^{57} - 16 q^{59} - 13 q^{61} + q^{63} - 12 q^{65} + 9 q^{67} - 4 q^{69} + 14 q^{71} + 3 q^{73} - 8 q^{75} - 8 q^{77} - 2 q^{79} - 14 q^{81} - 8 q^{83} - 28 q^{87} - 6 q^{89} - 2 q^{91} - 12 q^{93} + 8 q^{95} + 23 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q + q^3 - q^5 + 2 * q^7 + q^9 - 8 * q^11 - 2 * q^13 + 6 * q^15 + 10 * q^19 + q^21 - 8 * q^23 - 3 * q^25 + 4 * q^27 - 4 * q^29 - 11 * q^31 - 4 * q^33 - q^35 + 2 * q^37 - 14 * q^39 + 5 * q^41 - 5 * q^43 + 6 * q^45 + 2 * q^47 + 2 * q^49 + 5 * q^53 + 4 * q^55 + 18 * q^57 - 16 * q^59 - 13 * q^61 + q^63 - 12 * q^65 + 9 * q^67 - 4 * q^69 + 14 * q^71 + 3 * q^73 - 8 * q^75 - 8 * q^77 - 2 * q^79 - 14 * q^81 - 8 * q^83 - 28 * q^87 - 6 * q^89 - 2 * q^91 - 12 * q^93 + 8 * q^95 + 23 * q^97 - 4 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.30278 1.32951 0.664754 0.747062i $$-0.268536\pi$$
0.664754 + 0.747062i $$0.268536\pi$$
$$4$$ 0 0
$$5$$ 1.30278 0.582619 0.291309 0.956629i $$-0.405909\pi$$
0.291309 + 0.956629i $$0.405909\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 2.30278 0.767592
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ −4.60555 −1.27735 −0.638675 0.769477i $$-0.720517\pi$$
−0.638675 + 0.769477i $$0.720517\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 0 0
$$17$$ 0 0
$$18$$ 0 0
$$19$$ 8.60555 1.97425 0.987124 0.159954i $$-0.0511347\pi$$
0.987124 + 0.159954i $$0.0511347\pi$$
$$20$$ 0 0
$$21$$ 2.30278 0.502507
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ −3.30278 −0.660555
$$26$$ 0 0
$$27$$ −1.60555 −0.308988
$$28$$ 0 0
$$29$$ −9.21110 −1.71046 −0.855229 0.518250i $$-0.826584\pi$$
−0.855229 + 0.518250i $$0.826584\pi$$
$$30$$ 0 0
$$31$$ −7.30278 −1.31162 −0.655809 0.754927i $$-0.727672\pi$$
−0.655809 + 0.754927i $$0.727672\pi$$
$$32$$ 0 0
$$33$$ −9.21110 −1.60345
$$34$$ 0 0
$$35$$ 1.30278 0.220209
$$36$$ 0 0
$$37$$ −9.81665 −1.61385 −0.806924 0.590655i $$-0.798869\pi$$
−0.806924 + 0.590655i $$0.798869\pi$$
$$38$$ 0 0
$$39$$ −10.6056 −1.69825
$$40$$ 0 0
$$41$$ 11.5139 1.79817 0.899083 0.437779i $$-0.144235\pi$$
0.899083 + 0.437779i $$0.144235\pi$$
$$42$$ 0 0
$$43$$ −4.30278 −0.656167 −0.328084 0.944649i $$-0.606403\pi$$
−0.328084 + 0.944649i $$0.606403\pi$$
$$44$$ 0 0
$$45$$ 3.00000 0.447214
$$46$$ 0 0
$$47$$ −2.60555 −0.380059 −0.190029 0.981778i $$-0.560858\pi$$
−0.190029 + 0.981778i $$0.560858\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0.697224 0.0957711 0.0478856 0.998853i $$-0.484752\pi$$
0.0478856 + 0.998853i $$0.484752\pi$$
$$54$$ 0 0
$$55$$ −5.21110 −0.702665
$$56$$ 0 0
$$57$$ 19.8167 2.62478
$$58$$ 0 0
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ −15.5139 −1.98635 −0.993174 0.116640i $$-0.962788\pi$$
−0.993174 + 0.116640i $$0.962788\pi$$
$$62$$ 0 0
$$63$$ 2.30278 0.290122
$$64$$ 0 0
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ 2.69722 0.329518 0.164759 0.986334i $$-0.447315\pi$$
0.164759 + 0.986334i $$0.447315\pi$$
$$68$$ 0 0
$$69$$ −9.21110 −1.10889
$$70$$ 0 0
$$71$$ 3.39445 0.402847 0.201423 0.979504i $$-0.435443\pi$$
0.201423 + 0.979504i $$0.435443\pi$$
$$72$$ 0 0
$$73$$ −7.51388 −0.879433 −0.439716 0.898137i $$-0.644921\pi$$
−0.439716 + 0.898137i $$0.644921\pi$$
$$74$$ 0 0
$$75$$ −7.60555 −0.878213
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ 2.60555 0.293147 0.146574 0.989200i $$-0.453175\pi$$
0.146574 + 0.989200i $$0.453175\pi$$
$$80$$ 0 0
$$81$$ −10.6056 −1.17839
$$82$$ 0 0
$$83$$ 3.21110 0.352464 0.176232 0.984349i $$-0.443609\pi$$
0.176232 + 0.984349i $$0.443609\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −21.2111 −2.27407
$$88$$ 0 0
$$89$$ 7.81665 0.828564 0.414282 0.910149i $$-0.364033\pi$$
0.414282 + 0.910149i $$0.364033\pi$$
$$90$$ 0 0
$$91$$ −4.60555 −0.482793
$$92$$ 0 0
$$93$$ −16.8167 −1.74381
$$94$$ 0 0
$$95$$ 11.2111 1.15023
$$96$$ 0 0
$$97$$ 13.3028 1.35069 0.675346 0.737501i $$-0.263994\pi$$
0.675346 + 0.737501i $$0.263994\pi$$
$$98$$ 0 0
$$99$$ −9.21110 −0.925751
$$100$$ 0 0
$$101$$ −12.6056 −1.25430 −0.627150 0.778899i $$-0.715779\pi$$
−0.627150 + 0.778899i $$0.715779\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 0 0
$$105$$ 3.00000 0.292770
$$106$$ 0 0
$$107$$ 4.60555 0.445235 0.222618 0.974906i $$-0.428540\pi$$
0.222618 + 0.974906i $$0.428540\pi$$
$$108$$ 0 0
$$109$$ −4.78890 −0.458693 −0.229347 0.973345i $$-0.573659\pi$$
−0.229347 + 0.973345i $$0.573659\pi$$
$$110$$ 0 0
$$111$$ −22.6056 −2.14562
$$112$$ 0 0
$$113$$ 9.39445 0.883755 0.441878 0.897075i $$-0.354313\pi$$
0.441878 + 0.897075i $$0.354313\pi$$
$$114$$ 0 0
$$115$$ −5.21110 −0.485938
$$116$$ 0 0
$$117$$ −10.6056 −0.980484
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 26.5139 2.39068
$$124$$ 0 0
$$125$$ −10.8167 −0.967471
$$126$$ 0 0
$$127$$ 14.5139 1.28790 0.643949 0.765068i $$-0.277295\pi$$
0.643949 + 0.765068i $$0.277295\pi$$
$$128$$ 0 0
$$129$$ −9.90833 −0.872380
$$130$$ 0 0
$$131$$ 9.21110 0.804778 0.402389 0.915469i $$-0.368180\pi$$
0.402389 + 0.915469i $$0.368180\pi$$
$$132$$ 0 0
$$133$$ 8.60555 0.746196
$$134$$ 0 0
$$135$$ −2.09167 −0.180023
$$136$$ 0 0
$$137$$ 18.1194 1.54805 0.774024 0.633157i $$-0.218241\pi$$
0.774024 + 0.633157i $$0.218241\pi$$
$$138$$ 0 0
$$139$$ −8.51388 −0.722138 −0.361069 0.932539i $$-0.617588\pi$$
−0.361069 + 0.932539i $$0.617588\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ 18.4222 1.54054
$$144$$ 0 0
$$145$$ −12.0000 −0.996546
$$146$$ 0 0
$$147$$ 2.30278 0.189930
$$148$$ 0 0
$$149$$ −4.30278 −0.352497 −0.176249 0.984346i $$-0.556396\pi$$
−0.176249 + 0.984346i $$0.556396\pi$$
$$150$$ 0 0
$$151$$ 11.6972 0.951907 0.475953 0.879471i $$-0.342103\pi$$
0.475953 + 0.879471i $$0.342103\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −9.51388 −0.764173
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ 1.60555 0.127328
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ 0 0
$$163$$ −1.39445 −0.109222 −0.0546108 0.998508i $$-0.517392\pi$$
−0.0546108 + 0.998508i $$0.517392\pi$$
$$164$$ 0 0
$$165$$ −12.0000 −0.934199
$$166$$ 0 0
$$167$$ −12.1194 −0.937830 −0.468915 0.883243i $$-0.655355\pi$$
−0.468915 + 0.883243i $$0.655355\pi$$
$$168$$ 0 0
$$169$$ 8.21110 0.631623
$$170$$ 0 0
$$171$$ 19.8167 1.51542
$$172$$ 0 0
$$173$$ −20.7250 −1.57569 −0.787846 0.615873i $$-0.788803\pi$$
−0.787846 + 0.615873i $$0.788803\pi$$
$$174$$ 0 0
$$175$$ −3.30278 −0.249666
$$176$$ 0 0
$$177$$ −18.4222 −1.38470
$$178$$ 0 0
$$179$$ 9.51388 0.711101 0.355550 0.934657i $$-0.384293\pi$$
0.355550 + 0.934657i $$0.384293\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 0 0
$$183$$ −35.7250 −2.64087
$$184$$ 0 0
$$185$$ −12.7889 −0.940258
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −1.60555 −0.116787
$$190$$ 0 0
$$191$$ −11.7250 −0.848390 −0.424195 0.905571i $$-0.639443\pi$$
−0.424195 + 0.905571i $$0.639443\pi$$
$$192$$ 0 0
$$193$$ 15.0278 1.08172 0.540861 0.841112i $$-0.318099\pi$$
0.540861 + 0.841112i $$0.318099\pi$$
$$194$$ 0 0
$$195$$ −13.8167 −0.989431
$$196$$ 0 0
$$197$$ 6.60555 0.470626 0.235313 0.971920i $$-0.424388\pi$$
0.235313 + 0.971920i $$0.424388\pi$$
$$198$$ 0 0
$$199$$ −9.69722 −0.687418 −0.343709 0.939076i $$-0.611683\pi$$
−0.343709 + 0.939076i $$0.611683\pi$$
$$200$$ 0 0
$$201$$ 6.21110 0.438097
$$202$$ 0 0
$$203$$ −9.21110 −0.646493
$$204$$ 0 0
$$205$$ 15.0000 1.04765
$$206$$ 0 0
$$207$$ −9.21110 −0.640216
$$208$$ 0 0
$$209$$ −34.4222 −2.38103
$$210$$ 0 0
$$211$$ −6.18335 −0.425679 −0.212840 0.977087i $$-0.568271\pi$$
−0.212840 + 0.977087i $$0.568271\pi$$
$$212$$ 0 0
$$213$$ 7.81665 0.535588
$$214$$ 0 0
$$215$$ −5.60555 −0.382295
$$216$$ 0 0
$$217$$ −7.30278 −0.495745
$$218$$ 0 0
$$219$$ −17.3028 −1.16921
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 13.8167 0.925232 0.462616 0.886559i $$-0.346911\pi$$
0.462616 + 0.886559i $$0.346911\pi$$
$$224$$ 0 0
$$225$$ −7.60555 −0.507037
$$226$$ 0 0
$$227$$ −2.09167 −0.138829 −0.0694146 0.997588i $$-0.522113\pi$$
−0.0694146 + 0.997588i $$0.522113\pi$$
$$228$$ 0 0
$$229$$ −1.21110 −0.0800319 −0.0400160 0.999199i $$-0.512741\pi$$
−0.0400160 + 0.999199i $$0.512741\pi$$
$$230$$ 0 0
$$231$$ −9.21110 −0.606046
$$232$$ 0 0
$$233$$ −26.6056 −1.74299 −0.871494 0.490407i $$-0.836848\pi$$
−0.871494 + 0.490407i $$0.836848\pi$$
$$234$$ 0 0
$$235$$ −3.39445 −0.221429
$$236$$ 0 0
$$237$$ 6.00000 0.389742
$$238$$ 0 0
$$239$$ 18.1194 1.17205 0.586024 0.810294i $$-0.300692\pi$$
0.586024 + 0.810294i $$0.300692\pi$$
$$240$$ 0 0
$$241$$ 26.5139 1.70791 0.853955 0.520348i $$-0.174198\pi$$
0.853955 + 0.520348i $$0.174198\pi$$
$$242$$ 0 0
$$243$$ −19.6056 −1.25770
$$244$$ 0 0
$$245$$ 1.30278 0.0832313
$$246$$ 0 0
$$247$$ −39.6333 −2.52181
$$248$$ 0 0
$$249$$ 7.39445 0.468604
$$250$$ 0 0
$$251$$ −2.18335 −0.137812 −0.0689058 0.997623i $$-0.521951\pi$$
−0.0689058 + 0.997623i $$0.521951\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −4.42221 −0.275850 −0.137925 0.990443i $$-0.544043\pi$$
−0.137925 + 0.990443i $$0.544043\pi$$
$$258$$ 0 0
$$259$$ −9.81665 −0.609977
$$260$$ 0 0
$$261$$ −21.2111 −1.31293
$$262$$ 0 0
$$263$$ −18.4222 −1.13596 −0.567981 0.823042i $$-0.692275\pi$$
−0.567981 + 0.823042i $$0.692275\pi$$
$$264$$ 0 0
$$265$$ 0.908327 0.0557981
$$266$$ 0 0
$$267$$ 18.0000 1.10158
$$268$$ 0 0
$$269$$ −12.7889 −0.779753 −0.389876 0.920867i $$-0.627482\pi$$
−0.389876 + 0.920867i $$0.627482\pi$$
$$270$$ 0 0
$$271$$ −10.4222 −0.633104 −0.316552 0.948575i $$-0.602525\pi$$
−0.316552 + 0.948575i $$0.602525\pi$$
$$272$$ 0 0
$$273$$ −10.6056 −0.641877
$$274$$ 0 0
$$275$$ 13.2111 0.796659
$$276$$ 0 0
$$277$$ −1.57779 −0.0948005 −0.0474003 0.998876i $$-0.515094\pi$$
−0.0474003 + 0.998876i $$0.515094\pi$$
$$278$$ 0 0
$$279$$ −16.8167 −1.00679
$$280$$ 0 0
$$281$$ −19.1194 −1.14057 −0.570285 0.821447i $$-0.693167\pi$$
−0.570285 + 0.821447i $$0.693167\pi$$
$$282$$ 0 0
$$283$$ −20.3305 −1.20852 −0.604262 0.796785i $$-0.706532\pi$$
−0.604262 + 0.796785i $$0.706532\pi$$
$$284$$ 0 0
$$285$$ 25.8167 1.52925
$$286$$ 0 0
$$287$$ 11.5139 0.679643
$$288$$ 0 0
$$289$$ 0 0
$$290$$ 0 0
$$291$$ 30.6333 1.79576
$$292$$ 0 0
$$293$$ −11.2111 −0.654960 −0.327480 0.944858i $$-0.606199\pi$$
−0.327480 + 0.944858i $$0.606199\pi$$
$$294$$ 0 0
$$295$$ −10.4222 −0.606804
$$296$$ 0 0
$$297$$ 6.42221 0.372654
$$298$$ 0 0
$$299$$ 18.4222 1.06538
$$300$$ 0 0
$$301$$ −4.30278 −0.248008
$$302$$ 0 0
$$303$$ −29.0278 −1.66760
$$304$$ 0 0
$$305$$ −20.2111 −1.15728
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ 0 0
$$309$$ 32.2389 1.83400
$$310$$ 0 0
$$311$$ 17.7250 1.00509 0.502546 0.864551i $$-0.332397\pi$$
0.502546 + 0.864551i $$0.332397\pi$$
$$312$$ 0 0
$$313$$ −3.90833 −0.220912 −0.110456 0.993881i $$-0.535231\pi$$
−0.110456 + 0.993881i $$0.535231\pi$$
$$314$$ 0 0
$$315$$ 3.00000 0.169031
$$316$$ 0 0
$$317$$ 33.6333 1.88903 0.944517 0.328461i $$-0.106530\pi$$
0.944517 + 0.328461i $$0.106530\pi$$
$$318$$ 0 0
$$319$$ 36.8444 2.06289
$$320$$ 0 0
$$321$$ 10.6056 0.591944
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 15.2111 0.843760
$$326$$ 0 0
$$327$$ −11.0278 −0.609836
$$328$$ 0 0
$$329$$ −2.60555 −0.143649
$$330$$ 0 0
$$331$$ 3.72498 0.204743 0.102372 0.994746i $$-0.467357\pi$$
0.102372 + 0.994746i $$0.467357\pi$$
$$332$$ 0 0
$$333$$ −22.6056 −1.23878
$$334$$ 0 0
$$335$$ 3.51388 0.191984
$$336$$ 0 0
$$337$$ 14.4222 0.785628 0.392814 0.919618i $$-0.371502\pi$$
0.392814 + 0.919618i $$0.371502\pi$$
$$338$$ 0 0
$$339$$ 21.6333 1.17496
$$340$$ 0 0
$$341$$ 29.2111 1.58187
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −12.0000 −0.646058
$$346$$ 0 0
$$347$$ −26.8444 −1.44108 −0.720542 0.693412i $$-0.756107\pi$$
−0.720542 + 0.693412i $$0.756107\pi$$
$$348$$ 0 0
$$349$$ 18.4222 0.986118 0.493059 0.869996i $$-0.335879\pi$$
0.493059 + 0.869996i $$0.335879\pi$$
$$350$$ 0 0
$$351$$ 7.39445 0.394686
$$352$$ 0 0
$$353$$ −16.0000 −0.851594 −0.425797 0.904819i $$-0.640006\pi$$
−0.425797 + 0.904819i $$0.640006\pi$$
$$354$$ 0 0
$$355$$ 4.42221 0.234706
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1.48612 −0.0784345 −0.0392173 0.999231i $$-0.512486\pi$$
−0.0392173 + 0.999231i $$0.512486\pi$$
$$360$$ 0 0
$$361$$ 55.0555 2.89766
$$362$$ 0 0
$$363$$ 11.5139 0.604322
$$364$$ 0 0
$$365$$ −9.78890 −0.512374
$$366$$ 0 0
$$367$$ −6.90833 −0.360612 −0.180306 0.983611i $$-0.557709\pi$$
−0.180306 + 0.983611i $$0.557709\pi$$
$$368$$ 0 0
$$369$$ 26.5139 1.38026
$$370$$ 0 0
$$371$$ 0.697224 0.0361981
$$372$$ 0 0
$$373$$ −15.3305 −0.793785 −0.396892 0.917865i $$-0.629911\pi$$
−0.396892 + 0.917865i $$0.629911\pi$$
$$374$$ 0 0
$$375$$ −24.9083 −1.28626
$$376$$ 0 0
$$377$$ 42.4222 2.18485
$$378$$ 0 0
$$379$$ −9.57779 −0.491978 −0.245989 0.969273i $$-0.579113\pi$$
−0.245989 + 0.969273i $$0.579113\pi$$
$$380$$ 0 0
$$381$$ 33.4222 1.71227
$$382$$ 0 0
$$383$$ −1.57779 −0.0806216 −0.0403108 0.999187i $$-0.512835\pi$$
−0.0403108 + 0.999187i $$0.512835\pi$$
$$384$$ 0 0
$$385$$ −5.21110 −0.265582
$$386$$ 0 0
$$387$$ −9.90833 −0.503669
$$388$$ 0 0
$$389$$ −29.7250 −1.50712 −0.753558 0.657381i $$-0.771664\pi$$
−0.753558 + 0.657381i $$0.771664\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 21.2111 1.06996
$$394$$ 0 0
$$395$$ 3.39445 0.170793
$$396$$ 0 0
$$397$$ 12.0917 0.606864 0.303432 0.952853i $$-0.401868\pi$$
0.303432 + 0.952853i $$0.401868\pi$$
$$398$$ 0 0
$$399$$ 19.8167 0.992074
$$400$$ 0 0
$$401$$ −2.60555 −0.130115 −0.0650575 0.997882i $$-0.520723\pi$$
−0.0650575 + 0.997882i $$0.520723\pi$$
$$402$$ 0 0
$$403$$ 33.6333 1.67539
$$404$$ 0 0
$$405$$ −13.8167 −0.686555
$$406$$ 0 0
$$407$$ 39.2666 1.94637
$$408$$ 0 0
$$409$$ −5.81665 −0.287615 −0.143808 0.989606i $$-0.545935\pi$$
−0.143808 + 0.989606i $$0.545935\pi$$
$$410$$ 0 0
$$411$$ 41.7250 2.05814
$$412$$ 0 0
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ 4.18335 0.205352
$$416$$ 0 0
$$417$$ −19.6056 −0.960088
$$418$$ 0 0
$$419$$ 8.51388 0.415930 0.207965 0.978136i $$-0.433316\pi$$
0.207965 + 0.978136i $$0.433316\pi$$
$$420$$ 0 0
$$421$$ 11.0917 0.540575 0.270288 0.962780i $$-0.412881\pi$$
0.270288 + 0.962780i $$0.412881\pi$$
$$422$$ 0 0
$$423$$ −6.00000 −0.291730
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −15.5139 −0.750769
$$428$$ 0 0
$$429$$ 42.4222 2.04816
$$430$$ 0 0
$$431$$ 28.8444 1.38939 0.694693 0.719306i $$-0.255540\pi$$
0.694693 + 0.719306i $$0.255540\pi$$
$$432$$ 0 0
$$433$$ −7.39445 −0.355355 −0.177677 0.984089i $$-0.556858\pi$$
−0.177677 + 0.984089i $$0.556858\pi$$
$$434$$ 0 0
$$435$$ −27.6333 −1.32492
$$436$$ 0 0
$$437$$ −34.4222 −1.64664
$$438$$ 0 0
$$439$$ −21.0917 −1.00665 −0.503325 0.864097i $$-0.667890\pi$$
−0.503325 + 0.864097i $$0.667890\pi$$
$$440$$ 0 0
$$441$$ 2.30278 0.109656
$$442$$ 0 0
$$443$$ −33.2111 −1.57791 −0.788954 0.614453i $$-0.789377\pi$$
−0.788954 + 0.614453i $$0.789377\pi$$
$$444$$ 0 0
$$445$$ 10.1833 0.482737
$$446$$ 0 0
$$447$$ −9.90833 −0.468648
$$448$$ 0 0
$$449$$ 5.02776 0.237274 0.118637 0.992938i $$-0.462147\pi$$
0.118637 + 0.992938i $$0.462147\pi$$
$$450$$ 0 0
$$451$$ −46.0555 −2.16867
$$452$$ 0 0
$$453$$ 26.9361 1.26557
$$454$$ 0 0
$$455$$ −6.00000 −0.281284
$$456$$ 0 0
$$457$$ −13.9083 −0.650604 −0.325302 0.945610i $$-0.605466\pi$$
−0.325302 + 0.945610i $$0.605466\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 23.3944 1.08959 0.544794 0.838570i $$-0.316608\pi$$
0.544794 + 0.838570i $$0.316608\pi$$
$$462$$ 0 0
$$463$$ −6.72498 −0.312536 −0.156268 0.987715i $$-0.549946\pi$$
−0.156268 + 0.987715i $$0.549946\pi$$
$$464$$ 0 0
$$465$$ −21.9083 −1.01597
$$466$$ 0 0
$$467$$ −16.6056 −0.768413 −0.384207 0.923247i $$-0.625525\pi$$
−0.384207 + 0.923247i $$0.625525\pi$$
$$468$$ 0 0
$$469$$ 2.69722 0.124546
$$470$$ 0 0
$$471$$ 4.60555 0.212213
$$472$$ 0 0
$$473$$ 17.2111 0.791367
$$474$$ 0 0
$$475$$ −28.4222 −1.30410
$$476$$ 0 0
$$477$$ 1.60555 0.0735131
$$478$$ 0 0
$$479$$ −17.0917 −0.780938 −0.390469 0.920616i $$-0.627687\pi$$
−0.390469 + 0.920616i $$0.627687\pi$$
$$480$$ 0 0
$$481$$ 45.2111 2.06145
$$482$$ 0 0
$$483$$ −9.21110 −0.419120
$$484$$ 0 0
$$485$$ 17.3305 0.786939
$$486$$ 0 0
$$487$$ 28.0000 1.26880 0.634401 0.773004i $$-0.281247\pi$$
0.634401 + 0.773004i $$0.281247\pi$$
$$488$$ 0 0
$$489$$ −3.21110 −0.145211
$$490$$ 0 0
$$491$$ −22.5139 −1.01604 −0.508019 0.861346i $$-0.669622\pi$$
−0.508019 + 0.861346i $$0.669622\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −12.0000 −0.539360
$$496$$ 0 0
$$497$$ 3.39445 0.152262
$$498$$ 0 0
$$499$$ 29.2111 1.30767 0.653834 0.756638i $$-0.273159\pi$$
0.653834 + 0.756638i $$0.273159\pi$$
$$500$$ 0 0
$$501$$ −27.9083 −1.24685
$$502$$ 0 0
$$503$$ 31.1194 1.38755 0.693773 0.720193i $$-0.255947\pi$$
0.693773 + 0.720193i $$0.255947\pi$$
$$504$$ 0 0
$$505$$ −16.4222 −0.730779
$$506$$ 0 0
$$507$$ 18.9083 0.839748
$$508$$ 0 0
$$509$$ 6.60555 0.292786 0.146393 0.989227i $$-0.453234\pi$$
0.146393 + 0.989227i $$0.453234\pi$$
$$510$$ 0 0
$$511$$ −7.51388 −0.332394
$$512$$ 0 0
$$513$$ −13.8167 −0.610020
$$514$$ 0 0
$$515$$ 18.2389 0.803700
$$516$$ 0 0
$$517$$ 10.4222 0.458368
$$518$$ 0 0
$$519$$ −47.7250 −2.09489
$$520$$ 0 0
$$521$$ 0.486122 0.0212974 0.0106487 0.999943i $$-0.496610\pi$$
0.0106487 + 0.999943i $$0.496610\pi$$
$$522$$ 0 0
$$523$$ 38.0555 1.66405 0.832026 0.554737i $$-0.187181\pi$$
0.832026 + 0.554737i $$0.187181\pi$$
$$524$$ 0 0
$$525$$ −7.60555 −0.331933
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −18.4222 −0.799456
$$532$$ 0 0
$$533$$ −53.0278 −2.29689
$$534$$ 0 0
$$535$$ 6.00000 0.259403
$$536$$ 0 0
$$537$$ 21.9083 0.945414
$$538$$ 0 0
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −4.00000 −0.171973 −0.0859867 0.996296i $$-0.527404\pi$$
−0.0859867 + 0.996296i $$0.527404\pi$$
$$542$$ 0 0
$$543$$ 13.8167 0.592929
$$544$$ 0 0
$$545$$ −6.23886 −0.267243
$$546$$ 0 0
$$547$$ −2.00000 −0.0855138 −0.0427569 0.999086i $$-0.513614\pi$$
−0.0427569 + 0.999086i $$0.513614\pi$$
$$548$$ 0 0
$$549$$ −35.7250 −1.52471
$$550$$ 0 0
$$551$$ −79.2666 −3.37687
$$552$$ 0 0
$$553$$ 2.60555 0.110799
$$554$$ 0 0
$$555$$ −29.4500 −1.25008
$$556$$ 0 0
$$557$$ −20.7889 −0.880854 −0.440427 0.897788i $$-0.645173\pi$$
−0.440427 + 0.897788i $$0.645173\pi$$
$$558$$ 0 0
$$559$$ 19.8167 0.838155
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 0 0
$$565$$ 12.2389 0.514893
$$566$$ 0 0
$$567$$ −10.6056 −0.445391
$$568$$ 0 0
$$569$$ 22.1194 0.927295 0.463647 0.886020i $$-0.346540\pi$$
0.463647 + 0.886020i $$0.346540\pi$$
$$570$$ 0 0
$$571$$ 45.4500 1.90202 0.951011 0.309158i $$-0.100047\pi$$
0.951011 + 0.309158i $$0.100047\pi$$
$$572$$ 0 0
$$573$$ −27.0000 −1.12794
$$574$$ 0 0
$$575$$ 13.2111 0.550941
$$576$$ 0 0
$$577$$ −8.18335 −0.340677 −0.170339 0.985386i $$-0.554486\pi$$
−0.170339 + 0.985386i $$0.554486\pi$$
$$578$$ 0 0
$$579$$ 34.6056 1.43816
$$580$$ 0 0
$$581$$ 3.21110 0.133219
$$582$$ 0 0
$$583$$ −2.78890 −0.115504
$$584$$ 0 0
$$585$$ −13.8167 −0.571248
$$586$$ 0 0
$$587$$ −11.3944 −0.470299 −0.235150 0.971959i $$-0.575558\pi$$
−0.235150 + 0.971959i $$0.575558\pi$$
$$588$$ 0 0
$$589$$ −62.8444 −2.58946
$$590$$ 0 0
$$591$$ 15.2111 0.625701
$$592$$ 0 0
$$593$$ 22.1833 0.910961 0.455480 0.890246i $$-0.349468\pi$$
0.455480 + 0.890246i $$0.349468\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −22.3305 −0.913928
$$598$$ 0 0
$$599$$ 26.7250 1.09195 0.545977 0.837800i $$-0.316159\pi$$
0.545977 + 0.837800i $$0.316159\pi$$
$$600$$ 0 0
$$601$$ −45.2666 −1.84646 −0.923232 0.384243i $$-0.874462\pi$$
−0.923232 + 0.384243i $$0.874462\pi$$
$$602$$ 0 0
$$603$$ 6.21110 0.252936
$$604$$ 0 0
$$605$$ 6.51388 0.264827
$$606$$ 0 0
$$607$$ −0.330532 −0.0134159 −0.00670794 0.999978i $$-0.502135\pi$$
−0.00670794 + 0.999978i $$0.502135\pi$$
$$608$$ 0 0
$$609$$ −21.2111 −0.859517
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ −21.7250 −0.877464 −0.438732 0.898618i $$-0.644572\pi$$
−0.438732 + 0.898618i $$0.644572\pi$$
$$614$$ 0 0
$$615$$ 34.5416 1.39285
$$616$$ 0 0
$$617$$ −7.63331 −0.307305 −0.153653 0.988125i $$-0.549104\pi$$
−0.153653 + 0.988125i $$0.549104\pi$$
$$618$$ 0 0
$$619$$ −37.2111 −1.49564 −0.747820 0.663901i $$-0.768900\pi$$
−0.747820 + 0.663901i $$0.768900\pi$$
$$620$$ 0 0
$$621$$ 6.42221 0.257714
$$622$$ 0 0
$$623$$ 7.81665 0.313168
$$624$$ 0 0
$$625$$ 2.42221 0.0968882
$$626$$ 0 0
$$627$$ −79.2666 −3.16560
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 9.11943 0.363039 0.181519 0.983387i $$-0.441898\pi$$
0.181519 + 0.983387i $$0.441898\pi$$
$$632$$ 0 0
$$633$$ −14.2389 −0.565944
$$634$$ 0 0
$$635$$ 18.9083 0.750354
$$636$$ 0 0
$$637$$ −4.60555 −0.182479
$$638$$ 0 0
$$639$$ 7.81665 0.309222
$$640$$ 0 0
$$641$$ −43.6333 −1.72341 −0.861706 0.507408i $$-0.830604\pi$$
−0.861706 + 0.507408i $$0.830604\pi$$
$$642$$ 0 0
$$643$$ −7.88057 −0.310779 −0.155390 0.987853i $$-0.549663\pi$$
−0.155390 + 0.987853i $$0.549663\pi$$
$$644$$ 0 0
$$645$$ −12.9083 −0.508265
$$646$$ 0 0
$$647$$ −35.6333 −1.40089 −0.700445 0.713706i $$-0.747015\pi$$
−0.700445 + 0.713706i $$0.747015\pi$$
$$648$$ 0 0
$$649$$ 32.0000 1.25611
$$650$$ 0 0
$$651$$ −16.8167 −0.659097
$$652$$ 0 0
$$653$$ −40.4222 −1.58184 −0.790922 0.611918i $$-0.790398\pi$$
−0.790922 + 0.611918i $$0.790398\pi$$
$$654$$ 0 0
$$655$$ 12.0000 0.468879
$$656$$ 0 0
$$657$$ −17.3028 −0.675046
$$658$$ 0 0
$$659$$ 21.1194 0.822696 0.411348 0.911478i $$-0.365058\pi$$
0.411348 + 0.911478i $$0.365058\pi$$
$$660$$ 0 0
$$661$$ −5.02776 −0.195557 −0.0977785 0.995208i $$-0.531174\pi$$
−0.0977785 + 0.995208i $$0.531174\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 11.2111 0.434748
$$666$$ 0 0
$$667$$ 36.8444 1.42662
$$668$$ 0 0
$$669$$ 31.8167 1.23010
$$670$$ 0 0
$$671$$ 62.0555 2.39563
$$672$$ 0 0
$$673$$ 7.02776 0.270900 0.135450 0.990784i $$-0.456752\pi$$
0.135450 + 0.990784i $$0.456752\pi$$
$$674$$ 0 0
$$675$$ 5.30278 0.204104
$$676$$ 0 0
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 0 0
$$679$$ 13.3028 0.510514
$$680$$ 0 0
$$681$$ −4.81665 −0.184575
$$682$$ 0 0
$$683$$ −13.3944 −0.512524 −0.256262 0.966607i $$-0.582491\pi$$
−0.256262 + 0.966607i $$0.582491\pi$$
$$684$$ 0 0
$$685$$ 23.6056 0.901922
$$686$$ 0 0
$$687$$ −2.78890 −0.106403
$$688$$ 0 0
$$689$$ −3.21110 −0.122333
$$690$$ 0 0
$$691$$ 1.93608 0.0736521 0.0368260 0.999322i $$-0.488275\pi$$
0.0368260 + 0.999322i $$0.488275\pi$$
$$692$$ 0 0
$$693$$ −9.21110 −0.349901
$$694$$ 0 0
$$695$$ −11.0917 −0.420731
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ −61.2666 −2.31732
$$700$$ 0 0
$$701$$ −11.5778 −0.437287 −0.218644 0.975805i $$-0.570163\pi$$
−0.218644 + 0.975805i $$0.570163\pi$$
$$702$$ 0 0
$$703$$ −84.4777 −3.18614
$$704$$ 0 0
$$705$$ −7.81665 −0.294392
$$706$$ 0 0
$$707$$ −12.6056 −0.474081
$$708$$ 0 0
$$709$$ 4.23886 0.159194 0.0795968 0.996827i $$-0.474637\pi$$
0.0795968 + 0.996827i $$0.474637\pi$$
$$710$$ 0 0
$$711$$ 6.00000 0.225018
$$712$$ 0 0
$$713$$ 29.2111 1.09396
$$714$$ 0 0
$$715$$ 24.0000 0.897549
$$716$$ 0 0
$$717$$ 41.7250 1.55825
$$718$$ 0 0
$$719$$ −9.69722 −0.361645 −0.180823 0.983516i $$-0.557876\pi$$
−0.180823 + 0.983516i $$0.557876\pi$$
$$720$$ 0 0
$$721$$ 14.0000 0.521387
$$722$$ 0 0
$$723$$ 61.0555 2.27068
$$724$$ 0 0
$$725$$ 30.4222 1.12985
$$726$$ 0 0
$$727$$ −14.4222 −0.534890 −0.267445 0.963573i $$-0.586179\pi$$
−0.267445 + 0.963573i $$0.586179\pi$$
$$728$$ 0 0
$$729$$ −13.3305 −0.493723
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 16.2389 0.599796 0.299898 0.953971i $$-0.403047\pi$$
0.299898 + 0.953971i $$0.403047\pi$$
$$734$$ 0 0
$$735$$ 3.00000 0.110657
$$736$$ 0 0
$$737$$ −10.7889 −0.397414
$$738$$ 0 0
$$739$$ −23.3305 −0.858227 −0.429114 0.903250i $$-0.641174\pi$$
−0.429114 + 0.903250i $$0.641174\pi$$
$$740$$ 0 0
$$741$$ −91.2666 −3.35276
$$742$$ 0 0
$$743$$ 34.6056 1.26955 0.634777 0.772695i $$-0.281092\pi$$
0.634777 + 0.772695i $$0.281092\pi$$
$$744$$ 0 0
$$745$$ −5.60555 −0.205372
$$746$$ 0 0
$$747$$ 7.39445 0.270549
$$748$$ 0 0
$$749$$ 4.60555 0.168283
$$750$$ 0 0
$$751$$ −3.39445 −0.123865 −0.0619326 0.998080i $$-0.519726\pi$$
−0.0619326 + 0.998080i $$0.519726\pi$$
$$752$$ 0 0
$$753$$ −5.02776 −0.183222
$$754$$ 0 0
$$755$$ 15.2389 0.554599
$$756$$ 0 0
$$757$$ −7.69722 −0.279760 −0.139880 0.990168i $$-0.544672\pi$$
−0.139880 + 0.990168i $$0.544672\pi$$
$$758$$ 0 0
$$759$$ 36.8444 1.33737
$$760$$ 0 0
$$761$$ −50.2389 −1.82116 −0.910579 0.413336i $$-0.864364\pi$$
−0.910579 + 0.413336i $$0.864364\pi$$
$$762$$ 0 0
$$763$$ −4.78890 −0.173370
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 36.8444 1.33037
$$768$$ 0 0
$$769$$ −36.0555 −1.30020 −0.650098 0.759851i $$-0.725272\pi$$
−0.650098 + 0.759851i $$0.725272\pi$$
$$770$$ 0 0
$$771$$ −10.1833 −0.366744
$$772$$ 0 0
$$773$$ 17.0278 0.612446 0.306223 0.951960i $$-0.400935\pi$$
0.306223 + 0.951960i $$0.400935\pi$$
$$774$$ 0 0
$$775$$ 24.1194 0.866395
$$776$$ 0 0
$$777$$ −22.6056 −0.810970
$$778$$ 0 0
$$779$$ 99.0833 3.55003
$$780$$ 0 0
$$781$$ −13.5778 −0.485852
$$782$$ 0 0
$$783$$ 14.7889 0.528512
$$784$$ 0 0
$$785$$ 2.60555 0.0929961
$$786$$ 0 0
$$787$$ −50.4222 −1.79736 −0.898679 0.438607i $$-0.855472\pi$$
−0.898679 + 0.438607i $$0.855472\pi$$
$$788$$ 0 0
$$789$$ −42.4222 −1.51027
$$790$$ 0 0
$$791$$ 9.39445 0.334028
$$792$$ 0 0
$$793$$ 71.4500 2.53726
$$794$$ 0 0
$$795$$ 2.09167 0.0741840
$$796$$ 0 0
$$797$$ 48.0555 1.70221 0.851107 0.524993i $$-0.175932\pi$$
0.851107 + 0.524993i $$0.175932\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 18.0000 0.635999
$$802$$ 0 0
$$803$$ 30.0555 1.06064
$$804$$ 0 0
$$805$$ −5.21110 −0.183667
$$806$$ 0 0
$$807$$ −29.4500 −1.03669
$$808$$ 0 0
$$809$$ 19.8167 0.696716 0.348358 0.937361i $$-0.386739\pi$$
0.348358 + 0.937361i $$0.386739\pi$$
$$810$$ 0 0
$$811$$ 4.48612 0.157529 0.0787645 0.996893i $$-0.474902\pi$$
0.0787645 + 0.996893i $$0.474902\pi$$
$$812$$ 0 0
$$813$$ −24.0000 −0.841717
$$814$$ 0 0
$$815$$ −1.81665 −0.0636346
$$816$$ 0 0
$$817$$ −37.0278 −1.29544
$$818$$ 0 0
$$819$$ −10.6056 −0.370588
$$820$$ 0 0
$$821$$ 8.23886 0.287538 0.143769 0.989611i $$-0.454078\pi$$
0.143769 + 0.989611i $$0.454078\pi$$
$$822$$ 0 0
$$823$$ 38.6056 1.34570 0.672852 0.739777i $$-0.265069\pi$$
0.672852 + 0.739777i $$0.265069\pi$$
$$824$$ 0 0
$$825$$ 30.4222 1.05917
$$826$$ 0 0
$$827$$ −45.2111 −1.57214 −0.786072 0.618135i $$-0.787889\pi$$
−0.786072 + 0.618135i $$0.787889\pi$$
$$828$$ 0 0
$$829$$ −16.2389 −0.563999 −0.281999 0.959415i $$-0.590998\pi$$
−0.281999 + 0.959415i $$0.590998\pi$$
$$830$$ 0 0
$$831$$ −3.63331 −0.126038
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −15.7889 −0.546397
$$836$$ 0 0
$$837$$ 11.7250 0.405275
$$838$$ 0 0
$$839$$ −18.4222 −0.636005 −0.318003 0.948090i $$-0.603012\pi$$
−0.318003 + 0.948090i $$0.603012\pi$$
$$840$$ 0 0
$$841$$ 55.8444 1.92567
$$842$$ 0 0
$$843$$ −44.0278 −1.51640
$$844$$ 0 0
$$845$$ 10.6972 0.367996
$$846$$ 0 0
$$847$$ 5.00000 0.171802
$$848$$ 0 0
$$849$$ −46.8167 −1.60674
$$850$$ 0 0
$$851$$ 39.2666 1.34604
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 0 0
$$855$$ 25.8167 0.882911
$$856$$ 0 0
$$857$$ 45.9638 1.57009 0.785047 0.619436i $$-0.212639\pi$$
0.785047 + 0.619436i $$0.212639\pi$$
$$858$$ 0 0
$$859$$ −54.6056 −1.86312 −0.931559 0.363591i $$-0.881551\pi$$
−0.931559 + 0.363591i $$0.881551\pi$$
$$860$$ 0 0
$$861$$ 26.5139 0.903591
$$862$$ 0 0
$$863$$ 23.3583 0.795125 0.397563 0.917575i $$-0.369856\pi$$
0.397563 + 0.917575i $$0.369856\pi$$
$$864$$ 0 0
$$865$$ −27.0000 −0.918028
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −10.4222 −0.353549
$$870$$ 0 0
$$871$$ −12.4222 −0.420910
$$872$$ 0 0
$$873$$ 30.6333 1.03678
$$874$$ 0 0
$$875$$ −10.8167 −0.365670
$$876$$ 0 0
$$877$$ 1.39445 0.0470872 0.0235436 0.999723i $$-0.492505\pi$$
0.0235436 + 0.999723i $$0.492505\pi$$
$$878$$ 0 0
$$879$$ −25.8167 −0.870774
$$880$$ 0 0
$$881$$ 13.1194 0.442005 0.221002 0.975273i $$-0.429067\pi$$
0.221002 + 0.975273i $$0.429067\pi$$
$$882$$ 0 0
$$883$$ 12.0917 0.406917 0.203459 0.979084i $$-0.434782\pi$$
0.203459 + 0.979084i $$0.434782\pi$$
$$884$$ 0 0
$$885$$ −24.0000 −0.806751
$$886$$ 0 0
$$887$$ −55.7805 −1.87293 −0.936463 0.350767i $$-0.885921\pi$$
−0.936463 + 0.350767i $$0.885921\pi$$
$$888$$ 0 0
$$889$$ 14.5139 0.486780
$$890$$ 0 0
$$891$$ 42.4222 1.42120
$$892$$ 0 0
$$893$$ −22.4222 −0.750330
$$894$$ 0 0
$$895$$ 12.3944 0.414301
$$896$$ 0 0
$$897$$ 42.4222 1.41644
$$898$$ 0 0
$$899$$ 67.2666 2.24347
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ −9.90833 −0.329728
$$904$$ 0 0
$$905$$ 7.81665 0.259834
$$906$$ 0 0
$$907$$ −27.2111 −0.903530 −0.451765 0.892137i $$-0.649205\pi$$
−0.451765 + 0.892137i $$0.649205\pi$$
$$908$$ 0 0
$$909$$ −29.0278 −0.962790
$$910$$ 0 0
$$911$$ 15.8167 0.524029 0.262015 0.965064i $$-0.415613\pi$$
0.262015 + 0.965064i $$0.415613\pi$$
$$912$$ 0 0
$$913$$ −12.8444 −0.425088
$$914$$ 0 0
$$915$$ −46.5416 −1.53862
$$916$$ 0 0
$$917$$ 9.21110 0.304177
$$918$$ 0 0
$$919$$ 26.3583 0.869480 0.434740 0.900556i $$-0.356840\pi$$
0.434740 + 0.900556i $$0.356840\pi$$
$$920$$ 0 0
$$921$$ 9.21110 0.303516
$$922$$ 0 0
$$923$$ −15.6333 −0.514577
$$924$$ 0 0
$$925$$ 32.4222 1.06604
$$926$$ 0 0
$$927$$ 32.2389 1.05886
$$928$$ 0 0
$$929$$ −36.0917 −1.18413 −0.592065 0.805890i $$-0.701687\pi$$
−0.592065 + 0.805890i $$0.701687\pi$$
$$930$$ 0 0
$$931$$ 8.60555 0.282036
$$932$$ 0 0
$$933$$ 40.8167 1.33628
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −37.2111 −1.21563 −0.607817 0.794077i $$-0.707955\pi$$
−0.607817 + 0.794077i $$0.707955\pi$$
$$938$$ 0 0
$$939$$ −9.00000 −0.293704
$$940$$ 0 0
$$941$$ 33.6972 1.09850 0.549249 0.835659i $$-0.314914\pi$$
0.549249 + 0.835659i $$0.314914\pi$$
$$942$$ 0 0
$$943$$ −46.0555 −1.49977
$$944$$ 0 0
$$945$$ −2.09167 −0.0680421
$$946$$ 0 0
$$947$$ −24.8444 −0.807335 −0.403667 0.914906i $$-0.632265\pi$$
−0.403667 + 0.914906i $$0.632265\pi$$
$$948$$ 0 0
$$949$$ 34.6056 1.12334
$$950$$ 0 0
$$951$$ 77.4500 2.51149
$$952$$ 0 0
$$953$$ −21.3583 −0.691863 −0.345931 0.938260i $$-0.612437\pi$$
−0.345931 + 0.938260i $$0.612437\pi$$
$$954$$ 0 0
$$955$$ −15.2750 −0.494288
$$956$$ 0 0
$$957$$ 84.8444 2.74263
$$958$$ 0 0
$$959$$ 18.1194 0.585107
$$960$$ 0 0
$$961$$ 22.3305 0.720340
$$962$$ 0 0
$$963$$ 10.6056 0.341759
$$964$$ 0 0
$$965$$ 19.5778 0.630232
$$966$$ 0 0
$$967$$ −9.51388 −0.305946 −0.152973 0.988230i $$-0.548885\pi$$
−0.152973 + 0.988230i $$0.548885\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 2.78890 0.0895000 0.0447500 0.998998i $$-0.485751\pi$$
0.0447500 + 0.998998i $$0.485751\pi$$
$$972$$ 0 0
$$973$$ −8.51388 −0.272942
$$974$$ 0 0
$$975$$ 35.0278 1.12179
$$976$$ 0 0
$$977$$ −8.69722 −0.278249 −0.139124 0.990275i $$-0.544429\pi$$
−0.139124 + 0.990275i $$0.544429\pi$$
$$978$$ 0 0
$$979$$ −31.2666 −0.999285
$$980$$ 0 0
$$981$$ −11.0278 −0.352089
$$982$$ 0 0
$$983$$ −47.7805 −1.52396 −0.761981 0.647600i $$-0.775773\pi$$
−0.761981 + 0.647600i $$0.775773\pi$$
$$984$$ 0 0
$$985$$ 8.60555 0.274196
$$986$$ 0 0
$$987$$ −6.00000 −0.190982
$$988$$ 0 0
$$989$$ 17.2111 0.547281
$$990$$ 0 0
$$991$$ 11.2111 0.356132 0.178066 0.984019i $$-0.443016\pi$$
0.178066 + 0.984019i $$0.443016\pi$$
$$992$$ 0 0
$$993$$ 8.57779 0.272208
$$994$$ 0 0
$$995$$ −12.6333 −0.400503
$$996$$ 0 0
$$997$$ 45.6972 1.44725 0.723623 0.690196i $$-0.242476\pi$$
0.723623 + 0.690196i $$0.242476\pi$$
$$998$$ 0 0
$$999$$ 15.7611 0.498660
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 8092.2.a.l.1.2 2
17.16 even 2 476.2.a.c.1.1 2
51.50 odd 2 4284.2.a.l.1.2 2
68.67 odd 2 1904.2.a.k.1.2 2
119.118 odd 2 3332.2.a.k.1.2 2
136.67 odd 2 7616.2.a.o.1.1 2
136.101 even 2 7616.2.a.t.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.c.1.1 2 17.16 even 2
1904.2.a.k.1.2 2 68.67 odd 2
3332.2.a.k.1.2 2 119.118 odd 2
4284.2.a.l.1.2 2 51.50 odd 2
7616.2.a.o.1.1 2 136.67 odd 2
7616.2.a.t.1.2 2 136.101 even 2
8092.2.a.l.1.2 2 1.1 even 1 trivial