# Properties

 Label 8092.2.a.l.1.1 Level $8092$ Weight $2$ Character 8092.1 Self dual yes Analytic conductor $64.615$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

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 1.1 Root $$-1.30278$$ 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)

 $$f(q)$$ $$=$$ $$q-1.30278 q^{3} -2.30278 q^{5} +1.00000 q^{7} -1.30278 q^{9} +O(q^{10})$$ $$q-1.30278 q^{3} -2.30278 q^{5} +1.00000 q^{7} -1.30278 q^{9} -4.00000 q^{11} +2.60555 q^{13} +3.00000 q^{15} +1.39445 q^{19} -1.30278 q^{21} -4.00000 q^{23} +0.302776 q^{25} +5.60555 q^{27} +5.21110 q^{29} -3.69722 q^{31} +5.21110 q^{33} -2.30278 q^{35} +11.8167 q^{37} -3.39445 q^{39} -6.51388 q^{41} -0.697224 q^{43} +3.00000 q^{45} +4.60555 q^{47} +1.00000 q^{49} +4.30278 q^{53} +9.21110 q^{55} -1.81665 q^{57} -8.00000 q^{59} +2.51388 q^{61} -1.30278 q^{63} -6.00000 q^{65} +6.30278 q^{67} +5.21110 q^{69} +10.6056 q^{71} +10.5139 q^{73} -0.394449 q^{75} -4.00000 q^{77} -4.60555 q^{79} -3.39445 q^{81} -11.2111 q^{83} -6.78890 q^{87} -13.8167 q^{89} +2.60555 q^{91} +4.81665 q^{93} -3.21110 q^{95} +9.69722 q^{97} +5.21110 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{5} + 2 q^{7} + q^{9}+O(q^{10})$$ 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
$$3$$ −1.30278 −0.752158 −0.376079 0.926588i $$-0.622728\pi$$
−0.376079 + 0.926588i $$0.622728\pi$$
$$4$$ 0 0
$$5$$ −2.30278 −1.02983 −0.514916 0.857240i $$-0.672177\pi$$
−0.514916 + 0.857240i $$0.672177\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ −1.30278 −0.434259
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 2.60555 0.722650 0.361325 0.932440i $$-0.382325\pi$$
0.361325 + 0.932440i $$0.382325\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 0 0
$$17$$ 0 0
$$18$$ 0 0
$$19$$ 1.39445 0.319908 0.159954 0.987124i $$-0.448865\pi$$
0.159954 + 0.987124i $$0.448865\pi$$
$$20$$ 0 0
$$21$$ −1.30278 −0.284289
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 0.302776 0.0605551
$$26$$ 0 0
$$27$$ 5.60555 1.07879
$$28$$ 0 0
$$29$$ 5.21110 0.967677 0.483839 0.875157i $$-0.339242\pi$$
0.483839 + 0.875157i $$0.339242\pi$$
$$30$$ 0 0
$$31$$ −3.69722 −0.664041 −0.332021 0.943272i $$-0.607730\pi$$
−0.332021 + 0.943272i $$0.607730\pi$$
$$32$$ 0 0
$$33$$ 5.21110 0.907137
$$34$$ 0 0
$$35$$ −2.30278 −0.389240
$$36$$ 0 0
$$37$$ 11.8167 1.94265 0.971323 0.237764i $$-0.0764145\pi$$
0.971323 + 0.237764i $$0.0764145\pi$$
$$38$$ 0 0
$$39$$ −3.39445 −0.543547
$$40$$ 0 0
$$41$$ −6.51388 −1.01730 −0.508648 0.860974i $$-0.669855\pi$$
−0.508648 + 0.860974i $$0.669855\pi$$
$$42$$ 0 0
$$43$$ −0.697224 −0.106326 −0.0531629 0.998586i $$-0.516930\pi$$
−0.0531629 + 0.998586i $$0.516930\pi$$
$$44$$ 0 0
$$45$$ 3.00000 0.447214
$$46$$ 0 0
$$47$$ 4.60555 0.671789 0.335894 0.941900i $$-0.390961\pi$$
0.335894 + 0.941900i $$0.390961\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 4.30278 0.591032 0.295516 0.955338i $$-0.404508\pi$$
0.295516 + 0.955338i $$0.404508\pi$$
$$54$$ 0 0
$$55$$ 9.21110 1.24202
$$56$$ 0 0
$$57$$ −1.81665 −0.240622
$$58$$ 0 0
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 2.51388 0.321869 0.160935 0.986965i $$-0.448549\pi$$
0.160935 + 0.986965i $$0.448549\pi$$
$$62$$ 0 0
$$63$$ −1.30278 −0.164134
$$64$$ 0 0
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ 6.30278 0.770007 0.385003 0.922915i $$-0.374200\pi$$
0.385003 + 0.922915i $$0.374200\pi$$
$$68$$ 0 0
$$69$$ 5.21110 0.627343
$$70$$ 0 0
$$71$$ 10.6056 1.25865 0.629324 0.777143i $$-0.283332\pi$$
0.629324 + 0.777143i $$0.283332\pi$$
$$72$$ 0 0
$$73$$ 10.5139 1.23056 0.615278 0.788310i $$-0.289044\pi$$
0.615278 + 0.788310i $$0.289044\pi$$
$$74$$ 0 0
$$75$$ −0.394449 −0.0455470
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ −4.60555 −0.518165 −0.259083 0.965855i $$-0.583420\pi$$
−0.259083 + 0.965855i $$0.583420\pi$$
$$80$$ 0 0
$$81$$ −3.39445 −0.377161
$$82$$ 0 0
$$83$$ −11.2111 −1.23058 −0.615289 0.788301i $$-0.710961\pi$$
−0.615289 + 0.788301i $$0.710961\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −6.78890 −0.727846
$$88$$ 0 0
$$89$$ −13.8167 −1.46456 −0.732281 0.681002i $$-0.761544\pi$$
−0.732281 + 0.681002i $$0.761544\pi$$
$$90$$ 0 0
$$91$$ 2.60555 0.273136
$$92$$ 0 0
$$93$$ 4.81665 0.499464
$$94$$ 0 0
$$95$$ −3.21110 −0.329452
$$96$$ 0 0
$$97$$ 9.69722 0.984604 0.492302 0.870424i $$-0.336156\pi$$
0.492302 + 0.870424i $$0.336156\pi$$
$$98$$ 0 0
$$99$$ 5.21110 0.523736
$$100$$ 0 0
$$101$$ −5.39445 −0.536768 −0.268384 0.963312i $$-0.586490\pi$$
−0.268384 + 0.963312i $$0.586490\pi$$
$$102$$ 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
$$107$$ −2.60555 −0.251888 −0.125944 0.992037i $$-0.540196\pi$$
−0.125944 + 0.992037i $$0.540196\pi$$
$$108$$ 0 0
$$109$$ −19.2111 −1.84009 −0.920045 0.391813i $$-0.871848\pi$$
−0.920045 + 0.391813i $$0.871848\pi$$
$$110$$ 0 0
$$111$$ −15.3944 −1.46118
$$112$$ 0 0
$$113$$ 16.6056 1.56212 0.781059 0.624457i $$-0.214680\pi$$
0.781059 + 0.624457i $$0.214680\pi$$
$$114$$ 0 0
$$115$$ 9.21110 0.858940
$$116$$ 0 0
$$117$$ −3.39445 −0.313817
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 8.48612 0.765168
$$124$$ 0 0
$$125$$ 10.8167 0.967471
$$126$$ 0 0
$$127$$ −3.51388 −0.311806 −0.155903 0.987772i $$-0.549829\pi$$
−0.155903 + 0.987772i $$0.549829\pi$$
$$128$$ 0 0
$$129$$ 0.908327 0.0799737
$$130$$ 0 0
$$131$$ −5.21110 −0.455296 −0.227648 0.973743i $$-0.573104\pi$$
−0.227648 + 0.973743i $$0.573104\pi$$
$$132$$ 0 0
$$133$$ 1.39445 0.120914
$$134$$ 0 0
$$135$$ −12.9083 −1.11097
$$136$$ 0 0
$$137$$ −7.11943 −0.608254 −0.304127 0.952632i $$-0.598365\pi$$
−0.304127 + 0.952632i $$0.598365\pi$$
$$138$$ 0 0
$$139$$ 9.51388 0.806957 0.403478 0.914989i $$-0.367801\pi$$
0.403478 + 0.914989i $$0.367801\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ −10.4222 −0.871549
$$144$$ 0 0
$$145$$ −12.0000 −0.996546
$$146$$ 0 0
$$147$$ −1.30278 −0.107451
$$148$$ 0 0
$$149$$ −0.697224 −0.0571188 −0.0285594 0.999592i $$-0.509092\pi$$
−0.0285594 + 0.999592i $$0.509092\pi$$
$$150$$ 0 0
$$151$$ 15.3028 1.24532 0.622661 0.782492i $$-0.286052\pi$$
0.622661 + 0.782492i $$0.286052\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 8.51388 0.683851
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ −5.60555 −0.444549
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ 0 0
$$163$$ −8.60555 −0.674039 −0.337019 0.941498i $$-0.609419\pi$$
−0.337019 + 0.941498i $$0.609419\pi$$
$$164$$ 0 0
$$165$$ −12.0000 −0.934199
$$166$$ 0 0
$$167$$ 13.1194 1.01521 0.507606 0.861589i $$-0.330531\pi$$
0.507606 + 0.861589i $$0.330531\pi$$
$$168$$ 0 0
$$169$$ −6.21110 −0.477777
$$170$$ 0 0
$$171$$ −1.81665 −0.138923
$$172$$ 0 0
$$173$$ 11.7250 0.891434 0.445717 0.895174i $$-0.352949\pi$$
0.445717 + 0.895174i $$0.352949\pi$$
$$174$$ 0 0
$$175$$ 0.302776 0.0228877
$$176$$ 0 0
$$177$$ 10.4222 0.783381
$$178$$ 0 0
$$179$$ −8.51388 −0.636357 −0.318179 0.948031i $$-0.603071\pi$$
−0.318179 + 0.948031i $$0.603071\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 0 0
$$183$$ −3.27502 −0.242096
$$184$$ 0 0
$$185$$ −27.2111 −2.00060
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 5.60555 0.407744
$$190$$ 0 0
$$191$$ 20.7250 1.49961 0.749803 0.661661i $$-0.230148\pi$$
0.749803 + 0.661661i $$0.230148\pi$$
$$192$$ 0 0
$$193$$ −21.0278 −1.51361 −0.756806 0.653640i $$-0.773241\pi$$
−0.756806 + 0.653640i $$0.773241\pi$$
$$194$$ 0 0
$$195$$ 7.81665 0.559762
$$196$$ 0 0
$$197$$ −0.605551 −0.0431437 −0.0215719 0.999767i $$-0.506867\pi$$
−0.0215719 + 0.999767i $$0.506867\pi$$
$$198$$ 0 0
$$199$$ −13.3028 −0.943009 −0.471504 0.881864i $$-0.656289\pi$$
−0.471504 + 0.881864i $$0.656289\pi$$
$$200$$ 0 0
$$201$$ −8.21110 −0.579167
$$202$$ 0 0
$$203$$ 5.21110 0.365748
$$204$$ 0 0
$$205$$ 15.0000 1.04765
$$206$$ 0 0
$$207$$ 5.21110 0.362197
$$208$$ 0 0
$$209$$ −5.57779 −0.385824
$$210$$ 0 0
$$211$$ −27.8167 −1.91498 −0.957489 0.288471i $$-0.906853\pi$$
−0.957489 + 0.288471i $$0.906853\pi$$
$$212$$ 0 0
$$213$$ −13.8167 −0.946702
$$214$$ 0 0
$$215$$ 1.60555 0.109498
$$216$$ 0 0
$$217$$ −3.69722 −0.250984
$$218$$ 0 0
$$219$$ −13.6972 −0.925573
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −7.81665 −0.523442 −0.261721 0.965144i $$-0.584290\pi$$
−0.261721 + 0.965144i $$0.584290\pi$$
$$224$$ 0 0
$$225$$ −0.394449 −0.0262966
$$226$$ 0 0
$$227$$ −12.9083 −0.856756 −0.428378 0.903600i $$-0.640915\pi$$
−0.428378 + 0.903600i $$0.640915\pi$$
$$228$$ 0 0
$$229$$ 13.2111 0.873014 0.436507 0.899701i $$-0.356215\pi$$
0.436507 + 0.899701i $$0.356215\pi$$
$$230$$ 0 0
$$231$$ 5.21110 0.342865
$$232$$ 0 0
$$233$$ −19.3944 −1.27057 −0.635286 0.772277i $$-0.719118\pi$$
−0.635286 + 0.772277i $$0.719118\pi$$
$$234$$ 0 0
$$235$$ −10.6056 −0.691830
$$236$$ 0 0
$$237$$ 6.00000 0.389742
$$238$$ 0 0
$$239$$ −7.11943 −0.460518 −0.230259 0.973129i $$-0.573957\pi$$
−0.230259 + 0.973129i $$0.573957\pi$$
$$240$$ 0 0
$$241$$ 8.48612 0.546639 0.273320 0.961923i $$-0.411878\pi$$
0.273320 + 0.961923i $$0.411878\pi$$
$$242$$ 0 0
$$243$$ −12.3944 −0.795104
$$244$$ 0 0
$$245$$ −2.30278 −0.147119
$$246$$ 0 0
$$247$$ 3.63331 0.231182
$$248$$ 0 0
$$249$$ 14.6056 0.925589
$$250$$ 0 0
$$251$$ −23.8167 −1.50329 −0.751647 0.659566i $$-0.770740\pi$$
−0.751647 + 0.659566i $$0.770740\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 24.4222 1.52342 0.761708 0.647921i $$-0.224361\pi$$
0.761708 + 0.647921i $$0.224361\pi$$
$$258$$ 0 0
$$259$$ 11.8167 0.734251
$$260$$ 0 0
$$261$$ −6.78890 −0.420222
$$262$$ 0 0
$$263$$ 10.4222 0.642661 0.321330 0.946967i $$-0.395870\pi$$
0.321330 + 0.946967i $$0.395870\pi$$
$$264$$ 0 0
$$265$$ −9.90833 −0.608664
$$266$$ 0 0
$$267$$ 18.0000 1.10158
$$268$$ 0 0
$$269$$ −27.2111 −1.65909 −0.829545 0.558440i $$-0.811400\pi$$
−0.829545 + 0.558440i $$0.811400\pi$$
$$270$$ 0 0
$$271$$ 18.4222 1.11907 0.559535 0.828807i $$-0.310980\pi$$
0.559535 + 0.828807i $$0.310980\pi$$
$$272$$ 0 0
$$273$$ −3.39445 −0.205441
$$274$$ 0 0
$$275$$ −1.21110 −0.0730322
$$276$$ 0 0
$$277$$ −30.4222 −1.82789 −0.913947 0.405835i $$-0.866981\pi$$
−0.913947 + 0.405835i $$0.866981\pi$$
$$278$$ 0 0
$$279$$ 4.81665 0.288366
$$280$$ 0 0
$$281$$ 6.11943 0.365055 0.182527 0.983201i $$-0.441572\pi$$
0.182527 + 0.983201i $$0.441572\pi$$
$$282$$ 0 0
$$283$$ 19.3305 1.14908 0.574540 0.818476i $$-0.305181\pi$$
0.574540 + 0.818476i $$0.305181\pi$$
$$284$$ 0 0
$$285$$ 4.18335 0.247800
$$286$$ 0 0
$$287$$ −6.51388 −0.384502
$$288$$ 0 0
$$289$$ 0 0
$$290$$ 0 0
$$291$$ −12.6333 −0.740578
$$292$$ 0 0
$$293$$ 3.21110 0.187595 0.0937973 0.995591i $$-0.470099\pi$$
0.0937973 + 0.995591i $$0.470099\pi$$
$$294$$ 0 0
$$295$$ 18.4222 1.07258
$$296$$ 0 0
$$297$$ −22.4222 −1.30107
$$298$$ 0 0
$$299$$ −10.4222 −0.602732
$$300$$ 0 0
$$301$$ −0.697224 −0.0401873
$$302$$ 0 0
$$303$$ 7.02776 0.403734
$$304$$ 0 0
$$305$$ −5.78890 −0.331471
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ 0 0
$$309$$ −18.2389 −1.03757
$$310$$ 0 0
$$311$$ −14.7250 −0.834977 −0.417489 0.908682i $$-0.637090\pi$$
−0.417489 + 0.908682i $$0.637090\pi$$
$$312$$ 0 0
$$313$$ 6.90833 0.390482 0.195241 0.980755i $$-0.437451\pi$$
0.195241 + 0.980755i $$0.437451\pi$$
$$314$$ 0 0
$$315$$ 3.00000 0.169031
$$316$$ 0 0
$$317$$ −9.63331 −0.541060 −0.270530 0.962711i $$-0.587199\pi$$
−0.270530 + 0.962711i $$0.587199\pi$$
$$318$$ 0 0
$$319$$ −20.8444 −1.16706
$$320$$ 0 0
$$321$$ 3.39445 0.189460
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0.788897 0.0437602
$$326$$ 0 0
$$327$$ 25.0278 1.38404
$$328$$ 0 0
$$329$$ 4.60555 0.253912
$$330$$ 0 0
$$331$$ −28.7250 −1.57887 −0.789434 0.613836i $$-0.789626\pi$$
−0.789434 + 0.613836i $$0.789626\pi$$
$$332$$ 0 0
$$333$$ −15.3944 −0.843611
$$334$$ 0 0
$$335$$ −14.5139 −0.792978
$$336$$ 0 0
$$337$$ −14.4222 −0.785628 −0.392814 0.919618i $$-0.628498\pi$$
−0.392814 + 0.919618i $$0.628498\pi$$
$$338$$ 0 0
$$339$$ −21.6333 −1.17496
$$340$$ 0 0
$$341$$ 14.7889 0.800864
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −12.0000 −0.646058
$$346$$ 0 0
$$347$$ 30.8444 1.65581 0.827907 0.560865i $$-0.189531\pi$$
0.827907 + 0.560865i $$0.189531\pi$$
$$348$$ 0 0
$$349$$ −10.4222 −0.557888 −0.278944 0.960307i $$-0.589984\pi$$
−0.278944 + 0.960307i $$0.589984\pi$$
$$350$$ 0 0
$$351$$ 14.6056 0.779587
$$352$$ 0 0
$$353$$ −16.0000 −0.851594 −0.425797 0.904819i $$-0.640006\pi$$
−0.425797 + 0.904819i $$0.640006\pi$$
$$354$$ 0 0
$$355$$ −24.4222 −1.29620
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −19.5139 −1.02990 −0.514952 0.857219i $$-0.672190\pi$$
−0.514952 + 0.857219i $$0.672190\pi$$
$$360$$ 0 0
$$361$$ −17.0555 −0.897659
$$362$$ 0 0
$$363$$ −6.51388 −0.341890
$$364$$ 0 0
$$365$$ −24.2111 −1.26727
$$366$$ 0 0
$$367$$ 3.90833 0.204013 0.102007 0.994784i $$-0.467474\pi$$
0.102007 + 0.994784i $$0.467474\pi$$
$$368$$ 0 0
$$369$$ 8.48612 0.441770
$$370$$ 0 0
$$371$$ 4.30278 0.223389
$$372$$ 0 0
$$373$$ 24.3305 1.25979 0.629894 0.776681i $$-0.283098\pi$$
0.629894 + 0.776681i $$0.283098\pi$$
$$374$$ 0 0
$$375$$ −14.0917 −0.727691
$$376$$ 0 0
$$377$$ 13.5778 0.699292
$$378$$ 0 0
$$379$$ −38.4222 −1.97362 −0.986808 0.161895i $$-0.948240\pi$$
−0.986808 + 0.161895i $$0.948240\pi$$
$$380$$ 0 0
$$381$$ 4.57779 0.234528
$$382$$ 0 0
$$383$$ −30.4222 −1.55450 −0.777251 0.629191i $$-0.783386\pi$$
−0.777251 + 0.629191i $$0.783386\pi$$
$$384$$ 0 0
$$385$$ 9.21110 0.469441
$$386$$ 0 0
$$387$$ 0.908327 0.0461729
$$388$$ 0 0
$$389$$ 2.72498 0.138162 0.0690810 0.997611i $$-0.477993\pi$$
0.0690810 + 0.997611i $$0.477993\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 6.78890 0.342455
$$394$$ 0 0
$$395$$ 10.6056 0.533623
$$396$$ 0 0
$$397$$ 22.9083 1.14974 0.574868 0.818246i $$-0.305053\pi$$
0.574868 + 0.818246i $$0.305053\pi$$
$$398$$ 0 0
$$399$$ −1.81665 −0.0909464
$$400$$ 0 0
$$401$$ 4.60555 0.229990 0.114995 0.993366i $$-0.463315\pi$$
0.114995 + 0.993366i $$0.463315\pi$$
$$402$$ 0 0
$$403$$ −9.63331 −0.479869
$$404$$ 0 0
$$405$$ 7.81665 0.388413
$$406$$ 0 0
$$407$$ −47.2666 −2.34292
$$408$$ 0 0
$$409$$ 15.8167 0.782083 0.391042 0.920373i $$-0.372115\pi$$
0.391042 + 0.920373i $$0.372115\pi$$
$$410$$ 0 0
$$411$$ 9.27502 0.457503
$$412$$ 0 0
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ 25.8167 1.26729
$$416$$ 0 0
$$417$$ −12.3944 −0.606959
$$418$$ 0 0
$$419$$ −9.51388 −0.464783 −0.232392 0.972622i $$-0.574655\pi$$
−0.232392 + 0.972622i $$0.574655\pi$$
$$420$$ 0 0
$$421$$ 21.9083 1.06775 0.533873 0.845565i $$-0.320736\pi$$
0.533873 + 0.845565i $$0.320736\pi$$
$$422$$ 0 0
$$423$$ −6.00000 −0.291730
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 2.51388 0.121655
$$428$$ 0 0
$$429$$ 13.5778 0.655542
$$430$$ 0 0
$$431$$ −28.8444 −1.38939 −0.694693 0.719306i $$-0.744460\pi$$
−0.694693 + 0.719306i $$0.744460\pi$$
$$432$$ 0 0
$$433$$ −14.6056 −0.701898 −0.350949 0.936395i $$-0.614141\pi$$
−0.350949 + 0.936395i $$0.614141\pi$$
$$434$$ 0 0
$$435$$ 15.6333 0.749560
$$436$$ 0 0
$$437$$ −5.57779 −0.266822
$$438$$ 0 0
$$439$$ −31.9083 −1.52290 −0.761451 0.648223i $$-0.775513\pi$$
−0.761451 + 0.648223i $$0.775513\pi$$
$$440$$ 0 0
$$441$$ −1.30278 −0.0620369
$$442$$ 0 0
$$443$$ −18.7889 −0.892687 −0.446344 0.894862i $$-0.647274\pi$$
−0.446344 + 0.894862i $$0.647274\pi$$
$$444$$ 0 0
$$445$$ 31.8167 1.50825
$$446$$ 0 0
$$447$$ 0.908327 0.0429624
$$448$$ 0 0
$$449$$ −31.0278 −1.46429 −0.732145 0.681149i $$-0.761481\pi$$
−0.732145 + 0.681149i $$0.761481\pi$$
$$450$$ 0 0
$$451$$ 26.0555 1.22691
$$452$$ 0 0
$$453$$ −19.9361 −0.936679
$$454$$ 0 0
$$455$$ −6.00000 −0.281284
$$456$$ 0 0
$$457$$ −3.09167 −0.144622 −0.0723112 0.997382i $$-0.523037\pi$$
−0.0723112 + 0.997382i $$0.523037\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 30.6056 1.42544 0.712721 0.701447i $$-0.247462\pi$$
0.712721 + 0.701447i $$0.247462\pi$$
$$462$$ 0 0
$$463$$ 25.7250 1.19554 0.597771 0.801667i $$-0.296053\pi$$
0.597771 + 0.801667i $$0.296053\pi$$
$$464$$ 0 0
$$465$$ −11.0917 −0.514364
$$466$$ 0 0
$$467$$ −9.39445 −0.434723 −0.217362 0.976091i $$-0.569745\pi$$
−0.217362 + 0.976091i $$0.569745\pi$$
$$468$$ 0 0
$$469$$ 6.30278 0.291035
$$470$$ 0 0
$$471$$ −2.60555 −0.120057
$$472$$ 0 0
$$473$$ 2.78890 0.128234
$$474$$ 0 0
$$475$$ 0.422205 0.0193721
$$476$$ 0 0
$$477$$ −5.60555 −0.256661
$$478$$ 0 0
$$479$$ −27.9083 −1.27516 −0.637582 0.770382i $$-0.720065\pi$$
−0.637582 + 0.770382i $$0.720065\pi$$
$$480$$ 0 0
$$481$$ 30.7889 1.40385
$$482$$ 0 0
$$483$$ 5.21110 0.237113
$$484$$ 0 0
$$485$$ −22.3305 −1.01398
$$486$$ 0 0
$$487$$ 28.0000 1.26880 0.634401 0.773004i $$-0.281247\pi$$
0.634401 + 0.773004i $$0.281247\pi$$
$$488$$ 0 0
$$489$$ 11.2111 0.506984
$$490$$ 0 0
$$491$$ −4.48612 −0.202456 −0.101228 0.994863i $$-0.532277\pi$$
−0.101228 + 0.994863i $$0.532277\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −12.0000 −0.539360
$$496$$ 0 0
$$497$$ 10.6056 0.475724
$$498$$ 0 0
$$499$$ 14.7889 0.662042 0.331021 0.943623i $$-0.392607\pi$$
0.331021 + 0.943623i $$0.392607\pi$$
$$500$$ 0 0
$$501$$ −17.0917 −0.763600
$$502$$ 0 0
$$503$$ 5.88057 0.262202 0.131101 0.991369i $$-0.458149\pi$$
0.131101 + 0.991369i $$0.458149\pi$$
$$504$$ 0 0
$$505$$ 12.4222 0.552781
$$506$$ 0 0
$$507$$ 8.09167 0.359364
$$508$$ 0 0
$$509$$ −0.605551 −0.0268406 −0.0134203 0.999910i $$-0.504272\pi$$
−0.0134203 + 0.999910i $$0.504272\pi$$
$$510$$ 0 0
$$511$$ 10.5139 0.465107
$$512$$ 0 0
$$513$$ 7.81665 0.345114
$$514$$ 0 0
$$515$$ −32.2389 −1.42061
$$516$$ 0 0
$$517$$ −18.4222 −0.810208
$$518$$ 0 0
$$519$$ −15.2750 −0.670499
$$520$$ 0 0
$$521$$ 18.5139 0.811108 0.405554 0.914071i $$-0.367079\pi$$
0.405554 + 0.914071i $$0.367079\pi$$
$$522$$ 0 0
$$523$$ −34.0555 −1.48914 −0.744572 0.667542i $$-0.767346\pi$$
−0.744572 + 0.667542i $$0.767346\pi$$
$$524$$ 0 0
$$525$$ −0.394449 −0.0172152
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 10.4222 0.452285
$$532$$ 0 0
$$533$$ −16.9722 −0.735149
$$534$$ 0 0
$$535$$ 6.00000 0.259403
$$536$$ 0 0
$$537$$ 11.0917 0.478641
$$538$$ 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
$$543$$ −7.81665 −0.335445
$$544$$ 0 0
$$545$$ 44.2389 1.89498
$$546$$ 0 0
$$547$$ −2.00000 −0.0855138 −0.0427569 0.999086i $$-0.513614\pi$$
−0.0427569 + 0.999086i $$0.513614\pi$$
$$548$$ 0 0
$$549$$ −3.27502 −0.139774
$$550$$ 0 0
$$551$$ 7.26662 0.309568
$$552$$ 0 0
$$553$$ −4.60555 −0.195848
$$554$$ 0 0
$$555$$ 35.4500 1.50477
$$556$$ 0 0
$$557$$ −35.2111 −1.49194 −0.745971 0.665978i $$-0.768014\pi$$
−0.745971 + 0.665978i $$0.768014\pi$$
$$558$$ 0 0
$$559$$ −1.81665 −0.0768363
$$560$$ 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
$$565$$ −38.2389 −1.60872
$$566$$ 0 0
$$567$$ −3.39445 −0.142553
$$568$$ 0 0
$$569$$ −3.11943 −0.130773 −0.0653866 0.997860i $$-0.520828\pi$$
−0.0653866 + 0.997860i $$0.520828\pi$$
$$570$$ 0 0
$$571$$ −19.4500 −0.813956 −0.406978 0.913438i $$-0.633417\pi$$
−0.406978 + 0.913438i $$0.633417\pi$$
$$572$$ 0 0
$$573$$ −27.0000 −1.12794
$$574$$ 0 0
$$575$$ −1.21110 −0.0505065
$$576$$ 0 0
$$577$$ −29.8167 −1.24128 −0.620642 0.784094i $$-0.713128\pi$$
−0.620642 + 0.784094i $$0.713128\pi$$
$$578$$ 0 0
$$579$$ 27.3944 1.13847
$$580$$ 0 0
$$581$$ −11.2111 −0.465115
$$582$$ 0 0
$$583$$ −17.2111 −0.712811
$$584$$ 0 0
$$585$$ 7.81665 0.323179
$$586$$ 0 0
$$587$$ −18.6056 −0.767933 −0.383967 0.923347i $$-0.625442\pi$$
−0.383967 + 0.923347i $$0.625442\pi$$
$$588$$ 0 0
$$589$$ −5.15559 −0.212432
$$590$$ 0 0
$$591$$ 0.788897 0.0324509
$$592$$ 0 0
$$593$$ 43.8167 1.79933 0.899667 0.436576i $$-0.143809\pi$$
0.899667 + 0.436576i $$0.143809\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 17.3305 0.709291
$$598$$ 0 0
$$599$$ −5.72498 −0.233916 −0.116958 0.993137i $$-0.537314\pi$$
−0.116958 + 0.993137i $$0.537314\pi$$
$$600$$ 0 0
$$601$$ 41.2666 1.68330 0.841650 0.540023i $$-0.181585\pi$$
0.841650 + 0.540023i $$0.181585\pi$$
$$602$$ 0 0
$$603$$ −8.21110 −0.334382
$$604$$ 0 0
$$605$$ −11.5139 −0.468106
$$606$$ 0 0
$$607$$ 39.3305 1.59638 0.798189 0.602408i $$-0.205792\pi$$
0.798189 + 0.602408i $$0.205792\pi$$
$$608$$ 0 0
$$609$$ −6.78890 −0.275100
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ 10.7250 0.433178 0.216589 0.976263i $$-0.430507\pi$$
0.216589 + 0.976263i $$0.430507\pi$$
$$614$$ 0 0
$$615$$ −19.5416 −0.787995
$$616$$ 0 0
$$617$$ 35.6333 1.43454 0.717271 0.696794i $$-0.245391\pi$$
0.717271 + 0.696794i $$0.245391\pi$$
$$618$$ 0 0
$$619$$ −22.7889 −0.915963 −0.457982 0.888962i $$-0.651427\pi$$
−0.457982 + 0.888962i $$0.651427\pi$$
$$620$$ 0 0
$$621$$ −22.4222 −0.899772
$$622$$ 0 0
$$623$$ −13.8167 −0.553553
$$624$$ 0 0
$$625$$ −26.4222 −1.05689
$$626$$ 0 0
$$627$$ 7.26662 0.290201
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −16.1194 −0.641704 −0.320852 0.947129i $$-0.603969\pi$$
−0.320852 + 0.947129i $$0.603969\pi$$
$$632$$ 0 0
$$633$$ 36.2389 1.44037
$$634$$ 0 0
$$635$$ 8.09167 0.321108
$$636$$ 0 0
$$637$$ 2.60555 0.103236
$$638$$ 0 0
$$639$$ −13.8167 −0.546578
$$640$$ 0 0
$$641$$ −0.366692 −0.0144835 −0.00724174 0.999974i $$-0.502305\pi$$
−0.00724174 + 0.999974i $$0.502305\pi$$
$$642$$ 0 0
$$643$$ −33.1194 −1.30610 −0.653051 0.757314i $$-0.726511\pi$$
−0.653051 + 0.757314i $$0.726511\pi$$
$$644$$ 0 0
$$645$$ −2.09167 −0.0823595
$$646$$ 0 0
$$647$$ 7.63331 0.300096 0.150048 0.988679i $$-0.452057\pi$$
0.150048 + 0.988679i $$0.452057\pi$$
$$648$$ 0 0
$$649$$ 32.0000 1.25611
$$650$$ 0 0
$$651$$ 4.81665 0.188780
$$652$$ 0 0
$$653$$ −11.5778 −0.453074 −0.226537 0.974003i $$-0.572740\pi$$
−0.226537 + 0.974003i $$0.572740\pi$$
$$654$$ 0 0
$$655$$ 12.0000 0.468879
$$656$$ 0 0
$$657$$ −13.6972 −0.534380
$$658$$ 0 0
$$659$$ −4.11943 −0.160470 −0.0802351 0.996776i $$-0.525567\pi$$
−0.0802351 + 0.996776i $$0.525567\pi$$
$$660$$ 0 0
$$661$$ 31.0278 1.20684 0.603420 0.797424i $$-0.293804\pi$$
0.603420 + 0.797424i $$0.293804\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −3.21110 −0.124521
$$666$$ 0 0
$$667$$ −20.8444 −0.807099
$$668$$ 0 0
$$669$$ 10.1833 0.393711
$$670$$ 0 0
$$671$$ −10.0555 −0.388189
$$672$$ 0 0
$$673$$ −29.0278 −1.11894 −0.559469 0.828851i $$-0.688995\pi$$
−0.559469 + 0.828851i $$0.688995\pi$$
$$674$$ 0 0
$$675$$ 1.69722 0.0653262
$$676$$ 0 0
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 0 0
$$679$$ 9.69722 0.372145
$$680$$ 0 0
$$681$$ 16.8167 0.644416
$$682$$ 0 0
$$683$$ −20.6056 −0.788450 −0.394225 0.919014i $$-0.628987\pi$$
−0.394225 + 0.919014i $$0.628987\pi$$
$$684$$ 0 0
$$685$$ 16.3944 0.626400
$$686$$ 0 0
$$687$$ −17.2111 −0.656645
$$688$$ 0 0
$$689$$ 11.2111 0.427109
$$690$$ 0 0
$$691$$ −44.9361 −1.70945 −0.854725 0.519082i $$-0.826274\pi$$
−0.854725 + 0.519082i $$0.826274\pi$$
$$692$$ 0 0
$$693$$ 5.21110 0.197953
$$694$$ 0 0
$$695$$ −21.9083 −0.831030
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 25.2666 0.955671
$$700$$ 0 0
$$701$$ −40.4222 −1.52673 −0.763363 0.645970i $$-0.776453\pi$$
−0.763363 + 0.645970i $$0.776453\pi$$
$$702$$ 0 0
$$703$$ 16.4777 0.621469
$$704$$ 0 0
$$705$$ 13.8167 0.520365
$$706$$ 0 0
$$707$$ −5.39445 −0.202879
$$708$$ 0 0
$$709$$ −46.2389 −1.73654 −0.868268 0.496095i $$-0.834767\pi$$
−0.868268 + 0.496095i $$0.834767\pi$$
$$710$$ 0 0
$$711$$ 6.00000 0.225018
$$712$$ 0 0
$$713$$ 14.7889 0.553849
$$714$$ 0 0
$$715$$ 24.0000 0.897549
$$716$$ 0 0
$$717$$ 9.27502 0.346382
$$718$$ 0 0
$$719$$ −13.3028 −0.496110 −0.248055 0.968746i $$-0.579791\pi$$
−0.248055 + 0.968746i $$0.579791\pi$$
$$720$$ 0 0
$$721$$ 14.0000 0.521387
$$722$$ 0 0
$$723$$ −11.0555 −0.411159
$$724$$ 0 0
$$725$$ 1.57779 0.0585978
$$726$$ 0 0
$$727$$ 14.4222 0.534890 0.267445 0.963573i $$-0.413821\pi$$
0.267445 + 0.963573i $$0.413821\pi$$
$$728$$ 0 0
$$729$$ 26.3305 0.975205
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −34.2389 −1.26464 −0.632321 0.774707i $$-0.717897\pi$$
−0.632321 + 0.774707i $$0.717897\pi$$
$$734$$ 0 0
$$735$$ 3.00000 0.110657
$$736$$ 0 0
$$737$$ −25.2111 −0.928663
$$738$$ 0 0
$$739$$ 16.3305 0.600728 0.300364 0.953825i $$-0.402892\pi$$
0.300364 + 0.953825i $$0.402892\pi$$
$$740$$ 0 0
$$741$$ −4.73338 −0.173885
$$742$$ 0 0
$$743$$ 27.3944 1.00500 0.502502 0.864576i $$-0.332413\pi$$
0.502502 + 0.864576i $$0.332413\pi$$
$$744$$ 0 0
$$745$$ 1.60555 0.0588228
$$746$$ 0 0
$$747$$ 14.6056 0.534389
$$748$$ 0 0
$$749$$ −2.60555 −0.0952048
$$750$$ 0 0
$$751$$ −10.6056 −0.387002 −0.193501 0.981100i $$-0.561984\pi$$
−0.193501 + 0.981100i $$0.561984\pi$$
$$752$$ 0 0
$$753$$ 31.0278 1.13071
$$754$$ 0 0
$$755$$ −35.2389 −1.28247
$$756$$ 0 0
$$757$$ −11.3028 −0.410806 −0.205403 0.978677i $$-0.565851\pi$$
−0.205403 + 0.978677i $$0.565851\pi$$
$$758$$ 0 0
$$759$$ −20.8444 −0.756604
$$760$$ 0 0
$$761$$ 0.238859 0.00865863 0.00432931 0.999991i $$-0.498622\pi$$
0.00432931 + 0.999991i $$0.498622\pi$$
$$762$$ 0 0
$$763$$ −19.2111 −0.695489
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −20.8444 −0.752648
$$768$$ 0 0
$$769$$ 36.0555 1.30020 0.650098 0.759851i $$-0.274728\pi$$
0.650098 + 0.759851i $$0.274728\pi$$
$$770$$ 0 0
$$771$$ −31.8167 −1.14585
$$772$$ 0 0
$$773$$ −19.0278 −0.684381 −0.342190 0.939631i $$-0.611169\pi$$
−0.342190 + 0.939631i $$0.611169\pi$$
$$774$$ 0 0
$$775$$ −1.11943 −0.0402111
$$776$$ 0 0
$$777$$ −15.3944 −0.552273
$$778$$ 0 0
$$779$$ −9.08327 −0.325442
$$780$$ 0 0
$$781$$ −42.4222 −1.51799
$$782$$ 0 0
$$783$$ 29.2111 1.04392
$$784$$ 0 0
$$785$$ −4.60555 −0.164379
$$786$$ 0 0
$$787$$ −21.5778 −0.769165 −0.384583 0.923091i $$-0.625655\pi$$
−0.384583 + 0.923091i $$0.625655\pi$$
$$788$$ 0 0
$$789$$ −13.5778 −0.483382
$$790$$ 0 0
$$791$$ 16.6056 0.590425
$$792$$ 0 0
$$793$$ 6.55004 0.232599
$$794$$ 0 0
$$795$$ 12.9083 0.457811
$$796$$ 0 0
$$797$$ −24.0555 −0.852090 −0.426045 0.904702i $$-0.640093\pi$$
−0.426045 + 0.904702i $$0.640093\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 18.0000 0.635999
$$802$$ 0 0
$$803$$ −42.0555 −1.48411
$$804$$ 0 0
$$805$$ 9.21110 0.324649
$$806$$ 0 0
$$807$$ 35.4500 1.24790
$$808$$ 0 0
$$809$$ −1.81665 −0.0638701 −0.0319351 0.999490i $$-0.510167\pi$$
−0.0319351 + 0.999490i $$0.510167\pi$$
$$810$$ 0 0
$$811$$ 22.5139 0.790569 0.395285 0.918559i $$-0.370646\pi$$
0.395285 + 0.918559i $$0.370646\pi$$
$$812$$ 0 0
$$813$$ −24.0000 −0.841717
$$814$$ 0 0
$$815$$ 19.8167 0.694147
$$816$$ 0 0
$$817$$ −0.972244 −0.0340145
$$818$$ 0 0
$$819$$ −3.39445 −0.118612
$$820$$ 0 0
$$821$$ −42.2389 −1.47415 −0.737073 0.675813i $$-0.763793\pi$$
−0.737073 + 0.675813i $$0.763793\pi$$
$$822$$ 0 0
$$823$$ 31.3944 1.09434 0.547171 0.837021i $$-0.315705\pi$$
0.547171 + 0.837021i $$0.315705\pi$$
$$824$$ 0 0
$$825$$ 1.57779 0.0549318
$$826$$ 0 0
$$827$$ −30.7889 −1.07063 −0.535317 0.844651i $$-0.679808\pi$$
−0.535317 + 0.844651i $$0.679808\pi$$
$$828$$ 0 0
$$829$$ 34.2389 1.18916 0.594582 0.804035i $$-0.297317\pi$$
0.594582 + 0.804035i $$0.297317\pi$$
$$830$$ 0 0
$$831$$ 39.6333 1.37486
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −30.2111 −1.04550
$$836$$ 0 0
$$837$$ −20.7250 −0.716360
$$838$$ 0 0
$$839$$ 10.4222 0.359814 0.179907 0.983684i $$-0.442420\pi$$
0.179907 + 0.983684i $$0.442420\pi$$
$$840$$ 0 0
$$841$$ −1.84441 −0.0636004
$$842$$ 0 0
$$843$$ −7.97224 −0.274579
$$844$$ 0 0
$$845$$ 14.3028 0.492030
$$846$$ 0 0
$$847$$ 5.00000 0.171802
$$848$$ 0 0
$$849$$ −25.1833 −0.864290
$$850$$ 0 0
$$851$$ −47.2666 −1.62028
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 0 0
$$855$$ 4.18335 0.143067
$$856$$ 0 0
$$857$$ −36.9638 −1.26266 −0.631330 0.775514i $$-0.717491\pi$$
−0.631330 + 0.775514i $$0.717491\pi$$
$$858$$ 0 0
$$859$$ −47.3944 −1.61708 −0.808539 0.588443i $$-0.799741\pi$$
−0.808539 + 0.588443i $$0.799741\pi$$
$$860$$ 0 0
$$861$$ 8.48612 0.289206
$$862$$ 0 0
$$863$$ −52.3583 −1.78230 −0.891148 0.453712i $$-0.850100\pi$$
−0.891148 + 0.453712i $$0.850100\pi$$
$$864$$ 0 0
$$865$$ −27.0000 −0.918028
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 18.4222 0.624931
$$870$$ 0 0
$$871$$ 16.4222 0.556445
$$872$$ 0 0
$$873$$ −12.6333 −0.427573
$$874$$ 0 0
$$875$$ 10.8167 0.365670
$$876$$ 0 0
$$877$$ 8.60555 0.290589 0.145294 0.989388i $$-0.453587\pi$$
0.145294 + 0.989388i $$0.453587\pi$$
$$878$$ 0 0
$$879$$ −4.18335 −0.141101
$$880$$ 0 0
$$881$$ −12.1194 −0.408314 −0.204157 0.978938i $$-0.565445\pi$$
−0.204157 + 0.978938i $$0.565445\pi$$
$$882$$ 0 0
$$883$$ 22.9083 0.770927 0.385463 0.922723i $$-0.374042\pi$$
0.385463 + 0.922723i $$0.374042\pi$$
$$884$$ 0 0
$$885$$ −24.0000 −0.806751
$$886$$ 0 0
$$887$$ 48.7805 1.63789 0.818944 0.573873i $$-0.194560\pi$$
0.818944 + 0.573873i $$0.194560\pi$$
$$888$$ 0 0
$$889$$ −3.51388 −0.117852
$$890$$ 0 0
$$891$$ 13.5778 0.454873
$$892$$ 0 0
$$893$$ 6.42221 0.214911
$$894$$ 0 0
$$895$$ 19.6056 0.655341
$$896$$ 0 0
$$897$$ 13.5778 0.453349
$$898$$ 0 0
$$899$$ −19.2666 −0.642578
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0.908327 0.0302272
$$904$$ 0 0
$$905$$ −13.8167 −0.459281
$$906$$ 0 0
$$907$$ −12.7889 −0.424648 −0.212324 0.977199i $$-0.568103\pi$$
−0.212324 + 0.977199i $$0.568103\pi$$
$$908$$ 0 0
$$909$$ 7.02776 0.233096
$$910$$ 0 0
$$911$$ −5.81665 −0.192714 −0.0963572 0.995347i $$-0.530719\pi$$
−0.0963572 + 0.995347i $$0.530719\pi$$
$$912$$ 0 0
$$913$$ 44.8444 1.48413
$$914$$ 0 0
$$915$$ 7.54163 0.249319
$$916$$ 0 0
$$917$$ −5.21110 −0.172086
$$918$$ 0 0
$$919$$ −49.3583 −1.62818 −0.814090 0.580739i $$-0.802764\pi$$
−0.814090 + 0.580739i $$0.802764\pi$$
$$920$$ 0 0
$$921$$ −5.21110 −0.171712
$$922$$ 0 0
$$923$$ 27.6333 0.909561
$$924$$ 0 0
$$925$$ 3.57779 0.117637
$$926$$ 0 0
$$927$$ −18.2389 −0.599043
$$928$$ 0 0
$$929$$ −46.9083 −1.53901 −0.769506 0.638639i $$-0.779498\pi$$
−0.769506 + 0.638639i $$0.779498\pi$$
$$930$$ 0 0
$$931$$ 1.39445 0.0457012
$$932$$ 0 0
$$933$$ 19.1833 0.628035
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −22.7889 −0.744481 −0.372240 0.928136i $$-0.621410\pi$$
−0.372240 + 0.928136i $$0.621410\pi$$
$$938$$ 0 0
$$939$$ −9.00000 −0.293704
$$940$$ 0 0
$$941$$ 37.3028 1.21604 0.608018 0.793923i $$-0.291965\pi$$
0.608018 + 0.793923i $$0.291965\pi$$
$$942$$ 0 0
$$943$$ 26.0555 0.848484
$$944$$ 0 0
$$945$$ −12.9083 −0.419908
$$946$$ 0 0
$$947$$ 32.8444 1.06730 0.533650 0.845705i $$-0.320820\pi$$
0.533650 + 0.845705i $$0.320820\pi$$
$$948$$ 0 0
$$949$$ 27.3944 0.889261
$$950$$ 0 0
$$951$$ 12.5500 0.406963
$$952$$ 0 0
$$953$$ 54.3583 1.76084 0.880419 0.474197i $$-0.157262\pi$$
0.880419 + 0.474197i $$0.157262\pi$$
$$954$$ 0 0
$$955$$ −47.7250 −1.54434
$$956$$ 0 0
$$957$$ 27.1556 0.877816
$$958$$ 0 0
$$959$$ −7.11943 −0.229898
$$960$$ 0 0
$$961$$ −17.3305 −0.559049
$$962$$ 0 0
$$963$$ 3.39445 0.109385
$$964$$ 0 0
$$965$$ 48.4222 1.55877
$$966$$ 0 0
$$967$$ 8.51388 0.273788 0.136894 0.990586i $$-0.456288\pi$$
0.136894 + 0.990586i $$0.456288\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 17.2111 0.552331 0.276165 0.961110i $$-0.410936\pi$$
0.276165 + 0.961110i $$0.410936\pi$$
$$972$$ 0 0
$$973$$ 9.51388 0.305001
$$974$$ 0 0
$$975$$ −1.02776 −0.0329145
$$976$$ 0 0
$$977$$ −12.3028 −0.393601 −0.196800 0.980444i $$-0.563055\pi$$
−0.196800 + 0.980444i $$0.563055\pi$$
$$978$$ 0 0
$$979$$ 55.2666 1.76633
$$980$$ 0 0
$$981$$ 25.0278 0.799075
$$982$$ 0 0
$$983$$ 56.7805 1.81102 0.905508 0.424329i $$-0.139490\pi$$
0.905508 + 0.424329i $$0.139490\pi$$
$$984$$ 0 0
$$985$$ 1.39445 0.0444308
$$986$$ 0 0
$$987$$ −6.00000 −0.190982
$$988$$ 0 0
$$989$$ 2.78890 0.0886818
$$990$$ 0 0
$$991$$ −3.21110 −0.102004 −0.0510020 0.998699i $$-0.516241\pi$$
−0.0510020 + 0.998699i $$0.516241\pi$$
$$992$$ 0 0
$$993$$ 37.4222 1.18756
$$994$$ 0 0
$$995$$ 30.6333 0.971141
$$996$$ 0 0
$$997$$ 49.3028 1.56143 0.780717 0.624884i $$-0.214854\pi$$
0.780717 + 0.624884i $$0.214854\pi$$
$$998$$ 0 0
$$999$$ 66.2389 2.09570
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.1.1 2
17.16 even 2 476.2.a.c.1.2 2
51.50 odd 2 4284.2.a.l.1.1 2
68.67 odd 2 1904.2.a.k.1.1 2
119.118 odd 2 3332.2.a.k.1.1 2
136.67 odd 2 7616.2.a.o.1.2 2
136.101 even 2 7616.2.a.t.1.1 2

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