# Properties

 Label 6080.2.a.bp.1.3 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6080 = 2^{6} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6080.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: $$48.5490444289$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3040) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.51414$$ of defining polynomial Character $$\chi$$ $$=$$ 6080.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.51414 q^{3} +1.00000 q^{5} +1.32088 q^{7} -0.707389 q^{9} +O(q^{10})$$ $$q+1.51414 q^{3} +1.00000 q^{5} +1.32088 q^{7} -0.707389 q^{9} -3.32088 q^{11} -1.80675 q^{13} +1.51414 q^{15} +2.00000 q^{17} +1.00000 q^{19} +2.00000 q^{21} -5.32088 q^{23} +1.00000 q^{25} -5.61350 q^{27} -5.02827 q^{29} -5.02827 q^{31} -5.02827 q^{33} +1.32088 q^{35} -0.193252 q^{37} -2.73566 q^{39} +0.971726 q^{41} +4.34916 q^{43} -0.707389 q^{45} +1.32088 q^{47} -5.25526 q^{49} +3.02827 q^{51} -3.80675 q^{53} -3.32088 q^{55} +1.51414 q^{57} +8.05655 q^{59} -12.9909 q^{61} -0.934380 q^{63} -1.80675 q^{65} -6.92892 q^{67} -8.05655 q^{69} -13.0283 q^{71} +5.02827 q^{73} +1.51414 q^{75} -4.38650 q^{77} +2.00000 q^{79} -6.37743 q^{81} -2.29261 q^{83} +2.00000 q^{85} -7.61350 q^{87} +13.0283 q^{89} -2.38650 q^{91} -7.61350 q^{93} +1.00000 q^{95} +1.16498 q^{97} +2.34916 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 $$3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 2 q^{11} - 4 q^{13} - 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} - 8 q^{23} + 3 q^{25} - 14 q^{27} - 2 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{35} - 2 q^{37} + 10 q^{39} + 16 q^{41} - 8 q^{43} + 3 q^{45} - 4 q^{47} + 3 q^{49} - 4 q^{51} - 10 q^{53} - 2 q^{55} - 2 q^{57} - 2 q^{59} - 2 q^{61} + 8 q^{63} - 4 q^{65} - 4 q^{67} + 2 q^{69} - 26 q^{71} + 2 q^{73} - 2 q^{75} - 16 q^{77} + 6 q^{79} + 15 q^{81} - 12 q^{83} + 6 q^{85} - 20 q^{87} + 26 q^{89} - 10 q^{91} - 20 q^{93} + 3 q^{95} + 18 q^{97} - 14 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 2 * q^11 - 4 * q^13 - 2 * q^15 + 6 * q^17 + 3 * q^19 + 6 * q^21 - 8 * q^23 + 3 * q^25 - 14 * q^27 - 2 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^35 - 2 * q^37 + 10 * q^39 + 16 * q^41 - 8 * q^43 + 3 * q^45 - 4 * q^47 + 3 * q^49 - 4 * q^51 - 10 * q^53 - 2 * q^55 - 2 * q^57 - 2 * q^59 - 2 * q^61 + 8 * q^63 - 4 * q^65 - 4 * q^67 + 2 * q^69 - 26 * q^71 + 2 * q^73 - 2 * q^75 - 16 * q^77 + 6 * q^79 + 15 * q^81 - 12 * q^83 + 6 * q^85 - 20 * q^87 + 26 * q^89 - 10 * q^91 - 20 * q^93 + 3 * q^95 + 18 * q^97 - 14 * 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
$$3$$ 1.51414 0.874187 0.437094 0.899416i $$-0.356008\pi$$
0.437094 + 0.899416i $$0.356008\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 1.32088 0.499247 0.249624 0.968343i $$-0.419693\pi$$
0.249624 + 0.968343i $$0.419693\pi$$
$$8$$ 0 0
$$9$$ −0.707389 −0.235796
$$10$$ 0 0
$$11$$ −3.32088 −1.00128 −0.500642 0.865654i $$-0.666903\pi$$
−0.500642 + 0.865654i $$0.666903\pi$$
$$12$$ 0 0
$$13$$ −1.80675 −0.501102 −0.250551 0.968103i $$-0.580612\pi$$
−0.250551 + 0.968103i $$0.580612\pi$$
$$14$$ 0 0
$$15$$ 1.51414 0.390948
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ −5.32088 −1.10948 −0.554741 0.832023i $$-0.687182\pi$$
−0.554741 + 0.832023i $$0.687182\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.61350 −1.08032
$$28$$ 0 0
$$29$$ −5.02827 −0.933727 −0.466864 0.884329i $$-0.654616\pi$$
−0.466864 + 0.884329i $$0.654616\pi$$
$$30$$ 0 0
$$31$$ −5.02827 −0.903105 −0.451552 0.892245i $$-0.649130\pi$$
−0.451552 + 0.892245i $$0.649130\pi$$
$$32$$ 0 0
$$33$$ −5.02827 −0.875310
$$34$$ 0 0
$$35$$ 1.32088 0.223270
$$36$$ 0 0
$$37$$ −0.193252 −0.0317705 −0.0158853 0.999874i $$-0.505057\pi$$
−0.0158853 + 0.999874i $$0.505057\pi$$
$$38$$ 0 0
$$39$$ −2.73566 −0.438057
$$40$$ 0 0
$$41$$ 0.971726 0.151758 0.0758791 0.997117i $$-0.475824\pi$$
0.0758791 + 0.997117i $$0.475824\pi$$
$$42$$ 0 0
$$43$$ 4.34916 0.663240 0.331620 0.943413i $$-0.392405\pi$$
0.331620 + 0.943413i $$0.392405\pi$$
$$44$$ 0 0
$$45$$ −0.707389 −0.105451
$$46$$ 0 0
$$47$$ 1.32088 0.192671 0.0963354 0.995349i $$-0.469288\pi$$
0.0963354 + 0.995349i $$0.469288\pi$$
$$48$$ 0 0
$$49$$ −5.25526 −0.750752
$$50$$ 0 0
$$51$$ 3.02827 0.424043
$$52$$ 0 0
$$53$$ −3.80675 −0.522897 −0.261448 0.965217i $$-0.584200\pi$$
−0.261448 + 0.965217i $$0.584200\pi$$
$$54$$ 0 0
$$55$$ −3.32088 −0.447788
$$56$$ 0 0
$$57$$ 1.51414 0.200552
$$58$$ 0 0
$$59$$ 8.05655 1.04887 0.524437 0.851450i $$-0.324276\pi$$
0.524437 + 0.851450i $$0.324276\pi$$
$$60$$ 0 0
$$61$$ −12.9909 −1.66332 −0.831659 0.555287i $$-0.812608\pi$$
−0.831659 + 0.555287i $$0.812608\pi$$
$$62$$ 0 0
$$63$$ −0.934380 −0.117721
$$64$$ 0 0
$$65$$ −1.80675 −0.224099
$$66$$ 0 0
$$67$$ −6.92892 −0.846502 −0.423251 0.906013i $$-0.639111\pi$$
−0.423251 + 0.906013i $$0.639111\pi$$
$$68$$ 0 0
$$69$$ −8.05655 −0.969894
$$70$$ 0 0
$$71$$ −13.0283 −1.54617 −0.773086 0.634301i $$-0.781288\pi$$
−0.773086 + 0.634301i $$0.781288\pi$$
$$72$$ 0 0
$$73$$ 5.02827 0.588515 0.294257 0.955726i $$-0.404928\pi$$
0.294257 + 0.955726i $$0.404928\pi$$
$$74$$ 0 0
$$75$$ 1.51414 0.174837
$$76$$ 0 0
$$77$$ −4.38650 −0.499889
$$78$$ 0 0
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ −6.37743 −0.708604
$$82$$ 0 0
$$83$$ −2.29261 −0.251647 −0.125823 0.992053i $$-0.540157\pi$$
−0.125823 + 0.992053i $$0.540157\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ 0 0
$$87$$ −7.61350 −0.816252
$$88$$ 0 0
$$89$$ 13.0283 1.38099 0.690497 0.723335i $$-0.257392\pi$$
0.690497 + 0.723335i $$0.257392\pi$$
$$90$$ 0 0
$$91$$ −2.38650 −0.250174
$$92$$ 0 0
$$93$$ −7.61350 −0.789483
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 1.16498 0.118286 0.0591428 0.998250i $$-0.481163\pi$$
0.0591428 + 0.998250i $$0.481163\pi$$
$$98$$ 0 0
$$99$$ 2.34916 0.236099
$$100$$ 0 0
$$101$$ 3.32088 0.330440 0.165220 0.986257i $$-0.447167\pi$$
0.165220 + 0.986257i $$0.447167\pi$$
$$102$$ 0 0
$$103$$ −3.12763 −0.308175 −0.154087 0.988057i $$-0.549244\pi$$
−0.154087 + 0.988057i $$0.549244\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 0 0
$$107$$ −9.18418 −0.887868 −0.443934 0.896059i $$-0.646418\pi$$
−0.443934 + 0.896059i $$0.646418\pi$$
$$108$$ 0 0
$$109$$ −7.41478 −0.710207 −0.355103 0.934827i $$-0.615554\pi$$
−0.355103 + 0.934827i $$0.615554\pi$$
$$110$$ 0 0
$$111$$ −0.292611 −0.0277734
$$112$$ 0 0
$$113$$ 6.77847 0.637665 0.318833 0.947811i $$-0.396709\pi$$
0.318833 + 0.947811i $$0.396709\pi$$
$$114$$ 0 0
$$115$$ −5.32088 −0.496175
$$116$$ 0 0
$$117$$ 1.27807 0.118158
$$118$$ 0 0
$$119$$ 2.64177 0.242171
$$120$$ 0 0
$$121$$ 0.0282739 0.00257035
$$122$$ 0 0
$$123$$ 1.47133 0.132665
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 1.12763 0.100061 0.0500306 0.998748i $$-0.484068\pi$$
0.0500306 + 0.998748i $$0.484068\pi$$
$$128$$ 0 0
$$129$$ 6.58522 0.579796
$$130$$ 0 0
$$131$$ −6.44305 −0.562932 −0.281466 0.959571i $$-0.590821\pi$$
−0.281466 + 0.959571i $$0.590821\pi$$
$$132$$ 0 0
$$133$$ 1.32088 0.114535
$$134$$ 0 0
$$135$$ −5.61350 −0.483133
$$136$$ 0 0
$$137$$ 21.9253 1.87321 0.936603 0.350393i $$-0.113952\pi$$
0.936603 + 0.350393i $$0.113952\pi$$
$$138$$ 0 0
$$139$$ −6.73566 −0.571311 −0.285656 0.958332i $$-0.592211\pi$$
−0.285656 + 0.958332i $$0.592211\pi$$
$$140$$ 0 0
$$141$$ 2.00000 0.168430
$$142$$ 0 0
$$143$$ 6.00000 0.501745
$$144$$ 0 0
$$145$$ −5.02827 −0.417575
$$146$$ 0 0
$$147$$ −7.95719 −0.656298
$$148$$ 0 0
$$149$$ −20.9909 −1.71964 −0.859822 0.510594i $$-0.829426\pi$$
−0.859822 + 0.510594i $$0.829426\pi$$
$$150$$ 0 0
$$151$$ 2.84049 0.231155 0.115578 0.993298i $$-0.463128\pi$$
0.115578 + 0.993298i $$0.463128\pi$$
$$152$$ 0 0
$$153$$ −1.41478 −0.114378
$$154$$ 0 0
$$155$$ −5.02827 −0.403881
$$156$$ 0 0
$$157$$ −15.4713 −1.23475 −0.617373 0.786670i $$-0.711803\pi$$
−0.617373 + 0.786670i $$0.711803\pi$$
$$158$$ 0 0
$$159$$ −5.76394 −0.457110
$$160$$ 0 0
$$161$$ −7.02827 −0.553906
$$162$$ 0 0
$$163$$ −12.3492 −0.967261 −0.483630 0.875272i $$-0.660682\pi$$
−0.483630 + 0.875272i $$0.660682\pi$$
$$164$$ 0 0
$$165$$ −5.02827 −0.391451
$$166$$ 0 0
$$167$$ 2.21245 0.171205 0.0856024 0.996329i $$-0.472719\pi$$
0.0856024 + 0.996329i $$0.472719\pi$$
$$168$$ 0 0
$$169$$ −9.73566 −0.748897
$$170$$ 0 0
$$171$$ −0.707389 −0.0540954
$$172$$ 0 0
$$173$$ 2.50506 0.190457 0.0952283 0.995455i $$-0.469642\pi$$
0.0952283 + 0.995455i $$0.469642\pi$$
$$174$$ 0 0
$$175$$ 1.32088 0.0998495
$$176$$ 0 0
$$177$$ 12.1987 0.916912
$$178$$ 0 0
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ 12.8296 0.953613 0.476807 0.879008i $$-0.341794\pi$$
0.476807 + 0.879008i $$0.341794\pi$$
$$182$$ 0 0
$$183$$ −19.6700 −1.45405
$$184$$ 0 0
$$185$$ −0.193252 −0.0142082
$$186$$ 0 0
$$187$$ −6.64177 −0.485694
$$188$$ 0 0
$$189$$ −7.41478 −0.539346
$$190$$ 0 0
$$191$$ 11.9253 0.862885 0.431442 0.902140i $$-0.358005\pi$$
0.431442 + 0.902140i $$0.358005\pi$$
$$192$$ 0 0
$$193$$ −2.06201 −0.148427 −0.0742134 0.997242i $$-0.523645\pi$$
−0.0742134 + 0.997242i $$0.523645\pi$$
$$194$$ 0 0
$$195$$ −2.73566 −0.195905
$$196$$ 0 0
$$197$$ 2.31181 0.164710 0.0823549 0.996603i $$-0.473756\pi$$
0.0823549 + 0.996603i $$0.473756\pi$$
$$198$$ 0 0
$$199$$ −0.773010 −0.0547972 −0.0273986 0.999625i $$-0.508722\pi$$
−0.0273986 + 0.999625i $$0.508722\pi$$
$$200$$ 0 0
$$201$$ −10.4913 −0.740001
$$202$$ 0 0
$$203$$ −6.64177 −0.466161
$$204$$ 0 0
$$205$$ 0.971726 0.0678683
$$206$$ 0 0
$$207$$ 3.76394 0.261612
$$208$$ 0 0
$$209$$ −3.32088 −0.229710
$$210$$ 0 0
$$211$$ 0.386505 0.0266081 0.0133040 0.999911i $$-0.495765\pi$$
0.0133040 + 0.999911i $$0.495765\pi$$
$$212$$ 0 0
$$213$$ −19.7266 −1.35164
$$214$$ 0 0
$$215$$ 4.34916 0.296610
$$216$$ 0 0
$$217$$ −6.64177 −0.450873
$$218$$ 0 0
$$219$$ 7.61350 0.514472
$$220$$ 0 0
$$221$$ −3.61350 −0.243070
$$222$$ 0 0
$$223$$ −5.71285 −0.382561 −0.191280 0.981535i $$-0.561264\pi$$
−0.191280 + 0.981535i $$0.561264\pi$$
$$224$$ 0 0
$$225$$ −0.707389 −0.0471593
$$226$$ 0 0
$$227$$ −17.3720 −1.15302 −0.576509 0.817091i $$-0.695585\pi$$
−0.576509 + 0.817091i $$0.695585\pi$$
$$228$$ 0 0
$$229$$ 8.79221 0.581006 0.290503 0.956874i $$-0.406177\pi$$
0.290503 + 0.956874i $$0.406177\pi$$
$$230$$ 0 0
$$231$$ −6.64177 −0.436996
$$232$$ 0 0
$$233$$ 20.3684 1.33438 0.667188 0.744890i $$-0.267498\pi$$
0.667188 + 0.744890i $$0.267498\pi$$
$$234$$ 0 0
$$235$$ 1.32088 0.0861650
$$236$$ 0 0
$$237$$ 3.02827 0.196708
$$238$$ 0 0
$$239$$ −27.5388 −1.78134 −0.890669 0.454653i $$-0.849763\pi$$
−0.890669 + 0.454653i $$0.849763\pi$$
$$240$$ 0 0
$$241$$ −3.61350 −0.232766 −0.116383 0.993204i $$-0.537130\pi$$
−0.116383 + 0.993204i $$0.537130\pi$$
$$242$$ 0 0
$$243$$ 7.18418 0.460865
$$244$$ 0 0
$$245$$ −5.25526 −0.335747
$$246$$ 0 0
$$247$$ −1.80675 −0.114961
$$248$$ 0 0
$$249$$ −3.47133 −0.219986
$$250$$ 0 0
$$251$$ −15.9253 −1.00520 −0.502598 0.864520i $$-0.667622\pi$$
−0.502598 + 0.864520i $$0.667622\pi$$
$$252$$ 0 0
$$253$$ 17.6700 1.11091
$$254$$ 0 0
$$255$$ 3.02827 0.189638
$$256$$ 0 0
$$257$$ 23.5333 1.46797 0.733985 0.679166i $$-0.237658\pi$$
0.733985 + 0.679166i $$0.237658\pi$$
$$258$$ 0 0
$$259$$ −0.255264 −0.0158613
$$260$$ 0 0
$$261$$ 3.55695 0.220170
$$262$$ 0 0
$$263$$ 20.6610 1.27401 0.637005 0.770860i $$-0.280173\pi$$
0.637005 + 0.770860i $$0.280173\pi$$
$$264$$ 0 0
$$265$$ −3.80675 −0.233847
$$266$$ 0 0
$$267$$ 19.7266 1.20725
$$268$$ 0 0
$$269$$ 18.5671 1.13205 0.566027 0.824386i $$-0.308480\pi$$
0.566027 + 0.824386i $$0.308480\pi$$
$$270$$ 0 0
$$271$$ 14.1504 0.859578 0.429789 0.902929i $$-0.358588\pi$$
0.429789 + 0.902929i $$0.358588\pi$$
$$272$$ 0 0
$$273$$ −3.61350 −0.218699
$$274$$ 0 0
$$275$$ −3.32088 −0.200257
$$276$$ 0 0
$$277$$ −10.5852 −0.636004 −0.318002 0.948090i $$-0.603012\pi$$
−0.318002 + 0.948090i $$0.603012\pi$$
$$278$$ 0 0
$$279$$ 3.55695 0.212949
$$280$$ 0 0
$$281$$ 19.1414 1.14188 0.570939 0.820992i $$-0.306579\pi$$
0.570939 + 0.820992i $$0.306579\pi$$
$$282$$ 0 0
$$283$$ 14.2926 0.849608 0.424804 0.905285i $$-0.360343\pi$$
0.424804 + 0.905285i $$0.360343\pi$$
$$284$$ 0 0
$$285$$ 1.51414 0.0896897
$$286$$ 0 0
$$287$$ 1.28354 0.0757649
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 1.76394 0.103404
$$292$$ 0 0
$$293$$ 14.1751 0.828119 0.414059 0.910250i $$-0.364111\pi$$
0.414059 + 0.910250i $$0.364111\pi$$
$$294$$ 0 0
$$295$$ 8.05655 0.469070
$$296$$ 0 0
$$297$$ 18.6418 1.08171
$$298$$ 0 0
$$299$$ 9.61350 0.555963
$$300$$ 0 0
$$301$$ 5.74474 0.331121
$$302$$ 0 0
$$303$$ 5.02827 0.288867
$$304$$ 0 0
$$305$$ −12.9909 −0.743858
$$306$$ 0 0
$$307$$ −17.9572 −1.02487 −0.512435 0.858726i $$-0.671257\pi$$
−0.512435 + 0.858726i $$0.671257\pi$$
$$308$$ 0 0
$$309$$ −4.73566 −0.269402
$$310$$ 0 0
$$311$$ 0.0938942 0.00532425 0.00266213 0.999996i $$-0.499153\pi$$
0.00266213 + 0.999996i $$0.499153\pi$$
$$312$$ 0 0
$$313$$ 26.4249 1.49362 0.746812 0.665035i $$-0.231583\pi$$
0.746812 + 0.665035i $$0.231583\pi$$
$$314$$ 0 0
$$315$$ −0.934380 −0.0526463
$$316$$ 0 0
$$317$$ 1.23245 0.0692215 0.0346108 0.999401i $$-0.488981\pi$$
0.0346108 + 0.999401i $$0.488981\pi$$
$$318$$ 0 0
$$319$$ 16.6983 0.934926
$$320$$ 0 0
$$321$$ −13.9061 −0.776163
$$322$$ 0 0
$$323$$ 2.00000 0.111283
$$324$$ 0 0
$$325$$ −1.80675 −0.100220
$$326$$ 0 0
$$327$$ −11.2270 −0.620854
$$328$$ 0 0
$$329$$ 1.74474 0.0961904
$$330$$ 0 0
$$331$$ −21.8578 −1.20142 −0.600708 0.799469i $$-0.705114\pi$$
−0.600708 + 0.799469i $$0.705114\pi$$
$$332$$ 0 0
$$333$$ 0.136705 0.00749137
$$334$$ 0 0
$$335$$ −6.92892 −0.378567
$$336$$ 0 0
$$337$$ 4.64723 0.253151 0.126575 0.991957i $$-0.459601\pi$$
0.126575 + 0.991957i $$0.459601\pi$$
$$338$$ 0 0
$$339$$ 10.2635 0.557439
$$340$$ 0 0
$$341$$ 16.6983 0.904265
$$342$$ 0 0
$$343$$ −16.1878 −0.874058
$$344$$ 0 0
$$345$$ −8.05655 −0.433750
$$346$$ 0 0
$$347$$ −13.9061 −0.746519 −0.373259 0.927727i $$-0.621760\pi$$
−0.373259 + 0.927727i $$0.621760\pi$$
$$348$$ 0 0
$$349$$ −18.4249 −0.986263 −0.493131 0.869955i $$-0.664148\pi$$
−0.493131 + 0.869955i $$0.664148\pi$$
$$350$$ 0 0
$$351$$ 10.1422 0.541349
$$352$$ 0 0
$$353$$ 36.1696 1.92512 0.962558 0.271076i $$-0.0873795\pi$$
0.962558 + 0.271076i $$0.0873795\pi$$
$$354$$ 0 0
$$355$$ −13.0283 −0.691469
$$356$$ 0 0
$$357$$ 4.00000 0.211702
$$358$$ 0 0
$$359$$ −21.8880 −1.15520 −0.577601 0.816319i $$-0.696011\pi$$
−0.577601 + 0.816319i $$0.696011\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0.0428105 0.00224697
$$364$$ 0 0
$$365$$ 5.02827 0.263192
$$366$$ 0 0
$$367$$ −2.10482 −0.109871 −0.0549354 0.998490i $$-0.517495\pi$$
−0.0549354 + 0.998490i $$0.517495\pi$$
$$368$$ 0 0
$$369$$ −0.687389 −0.0357840
$$370$$ 0 0
$$371$$ −5.02827 −0.261055
$$372$$ 0 0
$$373$$ 34.1751 1.76952 0.884760 0.466047i $$-0.154322\pi$$
0.884760 + 0.466047i $$0.154322\pi$$
$$374$$ 0 0
$$375$$ 1.51414 0.0781897
$$376$$ 0 0
$$377$$ 9.08482 0.467892
$$378$$ 0 0
$$379$$ −2.71646 −0.139535 −0.0697676 0.997563i $$-0.522226\pi$$
−0.0697676 + 0.997563i $$0.522226\pi$$
$$380$$ 0 0
$$381$$ 1.70739 0.0874722
$$382$$ 0 0
$$383$$ −6.15591 −0.314552 −0.157276 0.987555i $$-0.550271\pi$$
−0.157276 + 0.987555i $$0.550271\pi$$
$$384$$ 0 0
$$385$$ −4.38650 −0.223557
$$386$$ 0 0
$$387$$ −3.07655 −0.156390
$$388$$ 0 0
$$389$$ −11.4823 −0.582173 −0.291087 0.956697i $$-0.594017\pi$$
−0.291087 + 0.956697i $$0.594017\pi$$
$$390$$ 0 0
$$391$$ −10.6418 −0.538177
$$392$$ 0 0
$$393$$ −9.75566 −0.492108
$$394$$ 0 0
$$395$$ 2.00000 0.100631
$$396$$ 0 0
$$397$$ −8.06748 −0.404895 −0.202447 0.979293i $$-0.564890\pi$$
−0.202447 + 0.979293i $$0.564890\pi$$
$$398$$ 0 0
$$399$$ 2.00000 0.100125
$$400$$ 0 0
$$401$$ −34.8114 −1.73840 −0.869199 0.494462i $$-0.835365\pi$$
−0.869199 + 0.494462i $$0.835365\pi$$
$$402$$ 0 0
$$403$$ 9.08482 0.452547
$$404$$ 0 0
$$405$$ −6.37743 −0.316897
$$406$$ 0 0
$$407$$ 0.641769 0.0318113
$$408$$ 0 0
$$409$$ 0.773010 0.0382229 0.0191114 0.999817i $$-0.493916\pi$$
0.0191114 + 0.999817i $$0.493916\pi$$
$$410$$ 0 0
$$411$$ 33.1979 1.63753
$$412$$ 0 0
$$413$$ 10.6418 0.523647
$$414$$ 0 0
$$415$$ −2.29261 −0.112540
$$416$$ 0 0
$$417$$ −10.1987 −0.499433
$$418$$ 0 0
$$419$$ 21.0848 1.03006 0.515030 0.857172i $$-0.327781\pi$$
0.515030 + 0.857172i $$0.327781\pi$$
$$420$$ 0 0
$$421$$ 37.0101 1.80376 0.901882 0.431983i $$-0.142186\pi$$
0.901882 + 0.431983i $$0.142186\pi$$
$$422$$ 0 0
$$423$$ −0.934380 −0.0454311
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ −17.1595 −0.830407
$$428$$ 0 0
$$429$$ 9.08482 0.438619
$$430$$ 0 0
$$431$$ −22.3684 −1.07745 −0.538723 0.842483i $$-0.681093\pi$$
−0.538723 + 0.842483i $$0.681093\pi$$
$$432$$ 0 0
$$433$$ −11.6755 −0.561089 −0.280545 0.959841i $$-0.590515\pi$$
−0.280545 + 0.959841i $$0.590515\pi$$
$$434$$ 0 0
$$435$$ −7.61350 −0.365039
$$436$$ 0 0
$$437$$ −5.32088 −0.254532
$$438$$ 0 0
$$439$$ −8.52867 −0.407051 −0.203526 0.979070i $$-0.565240\pi$$
−0.203526 + 0.979070i $$0.565240\pi$$
$$440$$ 0 0
$$441$$ 3.71752 0.177025
$$442$$ 0 0
$$443$$ −15.9627 −0.758409 −0.379204 0.925313i $$-0.623802\pi$$
−0.379204 + 0.925313i $$0.623802\pi$$
$$444$$ 0 0
$$445$$ 13.0283 0.617599
$$446$$ 0 0
$$447$$ −31.7831 −1.50329
$$448$$ 0 0
$$449$$ 23.9253 1.12911 0.564553 0.825397i $$-0.309049\pi$$
0.564553 + 0.825397i $$0.309049\pi$$
$$450$$ 0 0
$$451$$ −3.22699 −0.151953
$$452$$ 0 0
$$453$$ 4.30088 0.202073
$$454$$ 0 0
$$455$$ −2.38650 −0.111881
$$456$$ 0 0
$$457$$ −17.7375 −0.829726 −0.414863 0.909884i $$-0.636171\pi$$
−0.414863 + 0.909884i $$0.636171\pi$$
$$458$$ 0 0
$$459$$ −11.2270 −0.524031
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ −18.2070 −0.846151 −0.423075 0.906095i $$-0.639049\pi$$
−0.423075 + 0.906095i $$0.639049\pi$$
$$464$$ 0 0
$$465$$ −7.61350 −0.353067
$$466$$ 0 0
$$467$$ −22.7922 −1.05470 −0.527349 0.849649i $$-0.676814\pi$$
−0.527349 + 0.849649i $$0.676814\pi$$
$$468$$ 0 0
$$469$$ −9.15230 −0.422614
$$470$$ 0 0
$$471$$ −23.4257 −1.07940
$$472$$ 0 0
$$473$$ −14.4431 −0.664092
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 2.69285 0.123297
$$478$$ 0 0
$$479$$ 26.7357 1.22158 0.610792 0.791791i $$-0.290851\pi$$
0.610792 + 0.791791i $$0.290851\pi$$
$$480$$ 0 0
$$481$$ 0.349158 0.0159203
$$482$$ 0 0
$$483$$ −10.6418 −0.484217
$$484$$ 0 0
$$485$$ 1.16498 0.0528990
$$486$$ 0 0
$$487$$ −25.7694 −1.16772 −0.583862 0.811853i $$-0.698459\pi$$
−0.583862 + 0.811853i $$0.698459\pi$$
$$488$$ 0 0
$$489$$ −18.6983 −0.845567
$$490$$ 0 0
$$491$$ 26.3793 1.19048 0.595240 0.803548i $$-0.297057\pi$$
0.595240 + 0.803548i $$0.297057\pi$$
$$492$$ 0 0
$$493$$ −10.0565 −0.452924
$$494$$ 0 0
$$495$$ 2.34916 0.105587
$$496$$ 0 0
$$497$$ −17.2088 −0.771922
$$498$$ 0 0
$$499$$ 5.96265 0.266925 0.133463 0.991054i $$-0.457390\pi$$
0.133463 + 0.991054i $$0.457390\pi$$
$$500$$ 0 0
$$501$$ 3.34996 0.149665
$$502$$ 0 0
$$503$$ 19.1896 0.855624 0.427812 0.903868i $$-0.359285\pi$$
0.427812 + 0.903868i $$0.359285\pi$$
$$504$$ 0 0
$$505$$ 3.32088 0.147777
$$506$$ 0 0
$$507$$ −14.7411 −0.654676
$$508$$ 0 0
$$509$$ −23.6700 −1.04916 −0.524578 0.851362i $$-0.675777\pi$$
−0.524578 + 0.851362i $$0.675777\pi$$
$$510$$ 0 0
$$511$$ 6.64177 0.293815
$$512$$ 0 0
$$513$$ −5.61350 −0.247842
$$514$$ 0 0
$$515$$ −3.12763 −0.137820
$$516$$ 0 0
$$517$$ −4.38650 −0.192918
$$518$$ 0 0
$$519$$ 3.79301 0.166495
$$520$$ 0 0
$$521$$ −7.35823 −0.322370 −0.161185 0.986924i $$-0.551532\pi$$
−0.161185 + 0.986924i $$0.551532\pi$$
$$522$$ 0 0
$$523$$ 6.48586 0.283607 0.141803 0.989895i $$-0.454710\pi$$
0.141803 + 0.989895i $$0.454710\pi$$
$$524$$ 0 0
$$525$$ 2.00000 0.0872872
$$526$$ 0 0
$$527$$ −10.0565 −0.438070
$$528$$ 0 0
$$529$$ 5.31181 0.230948
$$530$$ 0 0
$$531$$ −5.69912 −0.247321
$$532$$ 0 0
$$533$$ −1.75566 −0.0760462
$$534$$ 0 0
$$535$$ −9.18418 −0.397067
$$536$$ 0 0
$$537$$ −9.08482 −0.392039
$$538$$ 0 0
$$539$$ 17.4521 0.751716
$$540$$ 0 0
$$541$$ 2.22513 0.0956660 0.0478330 0.998855i $$-0.484768\pi$$
0.0478330 + 0.998855i $$0.484768\pi$$
$$542$$ 0 0
$$543$$ 19.4257 0.833637
$$544$$ 0 0
$$545$$ −7.41478 −0.317614
$$546$$ 0 0
$$547$$ 22.1378 0.946542 0.473271 0.880917i $$-0.343073\pi$$
0.473271 + 0.880917i $$0.343073\pi$$
$$548$$ 0 0
$$549$$ 9.18964 0.392204
$$550$$ 0 0
$$551$$ −5.02827 −0.214212
$$552$$ 0 0
$$553$$ 2.64177 0.112339
$$554$$ 0 0
$$555$$ −0.292611 −0.0124206
$$556$$ 0 0
$$557$$ −31.8688 −1.35032 −0.675161 0.737670i $$-0.735926\pi$$
−0.675161 + 0.737670i $$0.735926\pi$$
$$558$$ 0 0
$$559$$ −7.85783 −0.332351
$$560$$ 0 0
$$561$$ −10.0565 −0.424588
$$562$$ 0 0
$$563$$ −23.9681 −1.01014 −0.505068 0.863080i $$-0.668533\pi$$
−0.505068 + 0.863080i $$0.668533\pi$$
$$564$$ 0 0
$$565$$ 6.77847 0.285173
$$566$$ 0 0
$$567$$ −8.42385 −0.353769
$$568$$ 0 0
$$569$$ 0.659914 0.0276650 0.0138325 0.999904i $$-0.495597\pi$$
0.0138325 + 0.999904i $$0.495597\pi$$
$$570$$ 0 0
$$571$$ 35.2462 1.47501 0.737504 0.675343i $$-0.236004\pi$$
0.737504 + 0.675343i $$0.236004\pi$$
$$572$$ 0 0
$$573$$ 18.0565 0.754323
$$574$$ 0 0
$$575$$ −5.32088 −0.221896
$$576$$ 0 0
$$577$$ 33.1414 1.37969 0.689847 0.723956i $$-0.257678\pi$$
0.689847 + 0.723956i $$0.257678\pi$$
$$578$$ 0 0
$$579$$ −3.12217 −0.129753
$$580$$ 0 0
$$581$$ −3.02827 −0.125634
$$582$$ 0 0
$$583$$ 12.6418 0.523569
$$584$$ 0 0
$$585$$ 1.27807 0.0528419
$$586$$ 0 0
$$587$$ −11.7639 −0.485550 −0.242775 0.970083i $$-0.578058\pi$$
−0.242775 + 0.970083i $$0.578058\pi$$
$$588$$ 0 0
$$589$$ −5.02827 −0.207186
$$590$$ 0 0
$$591$$ 3.50040 0.143987
$$592$$ 0 0
$$593$$ 5.34009 0.219291 0.109646 0.993971i $$-0.465028\pi$$
0.109646 + 0.993971i $$0.465028\pi$$
$$594$$ 0 0
$$595$$ 2.64177 0.108302
$$596$$ 0 0
$$597$$ −1.17044 −0.0479030
$$598$$ 0 0
$$599$$ −4.64177 −0.189658 −0.0948288 0.995494i $$-0.530230\pi$$
−0.0948288 + 0.995494i $$0.530230\pi$$
$$600$$ 0 0
$$601$$ −18.4431 −0.752308 −0.376154 0.926557i $$-0.622754\pi$$
−0.376154 + 0.926557i $$0.622754\pi$$
$$602$$ 0 0
$$603$$ 4.90144 0.199602
$$604$$ 0 0
$$605$$ 0.0282739 0.00114950
$$606$$ 0 0
$$607$$ 2.92892 0.118881 0.0594405 0.998232i $$-0.481068\pi$$
0.0594405 + 0.998232i $$0.481068\pi$$
$$608$$ 0 0
$$609$$ −10.0565 −0.407512
$$610$$ 0 0
$$611$$ −2.38650 −0.0965477
$$612$$ 0 0
$$613$$ 7.48225 0.302205 0.151103 0.988518i $$-0.451718\pi$$
0.151103 + 0.988518i $$0.451718\pi$$
$$614$$ 0 0
$$615$$ 1.47133 0.0593296
$$616$$ 0 0
$$617$$ −5.53880 −0.222984 −0.111492 0.993765i $$-0.535563\pi$$
−0.111492 + 0.993765i $$0.535563\pi$$
$$618$$ 0 0
$$619$$ −13.3027 −0.534682 −0.267341 0.963602i $$-0.586145\pi$$
−0.267341 + 0.963602i $$0.586145\pi$$
$$620$$ 0 0
$$621$$ 29.8688 1.19859
$$622$$ 0 0
$$623$$ 17.2088 0.689458
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −5.02827 −0.200810
$$628$$ 0 0
$$629$$ −0.386505 −0.0154110
$$630$$ 0 0
$$631$$ −31.2462 −1.24389 −0.621946 0.783060i $$-0.713658\pi$$
−0.621946 + 0.783060i $$0.713658\pi$$
$$632$$ 0 0
$$633$$ 0.585221 0.0232605
$$634$$ 0 0
$$635$$ 1.12763 0.0447487
$$636$$ 0 0
$$637$$ 9.49494 0.376203
$$638$$ 0 0
$$639$$ 9.21606 0.364582
$$640$$ 0 0
$$641$$ 17.0101 0.671860 0.335930 0.941887i $$-0.390949\pi$$
0.335930 + 0.941887i $$0.390949\pi$$
$$642$$ 0 0
$$643$$ 20.0757 0.791710 0.395855 0.918313i $$-0.370448\pi$$
0.395855 + 0.918313i $$0.370448\pi$$
$$644$$ 0 0
$$645$$ 6.58522 0.259293
$$646$$ 0 0
$$647$$ −23.6508 −0.929811 −0.464905 0.885360i $$-0.653912\pi$$
−0.464905 + 0.885360i $$0.653912\pi$$
$$648$$ 0 0
$$649$$ −26.7549 −1.05022
$$650$$ 0 0
$$651$$ −10.0565 −0.394147
$$652$$ 0 0
$$653$$ −12.3684 −0.484011 −0.242006 0.970275i $$-0.577805\pi$$
−0.242006 + 0.970275i $$0.577805\pi$$
$$654$$ 0 0
$$655$$ −6.44305 −0.251751
$$656$$ 0 0
$$657$$ −3.55695 −0.138770
$$658$$ 0 0
$$659$$ −14.7730 −0.575475 −0.287737 0.957709i $$-0.592903\pi$$
−0.287737 + 0.957709i $$0.592903\pi$$
$$660$$ 0 0
$$661$$ 41.7831 1.62518 0.812588 0.582839i $$-0.198058\pi$$
0.812588 + 0.582839i $$0.198058\pi$$
$$662$$ 0 0
$$663$$ −5.47133 −0.212489
$$664$$ 0 0
$$665$$ 1.32088 0.0512217
$$666$$ 0 0
$$667$$ 26.7549 1.03595
$$668$$ 0 0
$$669$$ −8.65004 −0.334430
$$670$$ 0 0
$$671$$ 43.1414 1.66545
$$672$$ 0 0
$$673$$ −29.1359 −1.12311 −0.561553 0.827441i $$-0.689796\pi$$
−0.561553 + 0.827441i $$0.689796\pi$$
$$674$$ 0 0
$$675$$ −5.61350 −0.216064
$$676$$ 0 0
$$677$$ 10.9481 0.420770 0.210385 0.977619i $$-0.432528\pi$$
0.210385 + 0.977619i $$0.432528\pi$$
$$678$$ 0 0
$$679$$ 1.53880 0.0590538
$$680$$ 0 0
$$681$$ −26.3035 −1.00795
$$682$$ 0 0
$$683$$ 36.2799 1.38821 0.694106 0.719872i $$-0.255800\pi$$
0.694106 + 0.719872i $$0.255800\pi$$
$$684$$ 0 0
$$685$$ 21.9253 0.837723
$$686$$ 0 0
$$687$$ 13.3126 0.507908
$$688$$ 0 0
$$689$$ 6.87783 0.262025
$$690$$ 0 0
$$691$$ −31.5471 −1.20011 −0.600054 0.799960i $$-0.704854\pi$$
−0.600054 + 0.799960i $$0.704854\pi$$
$$692$$ 0 0
$$693$$ 3.10297 0.117872
$$694$$ 0 0
$$695$$ −6.73566 −0.255498
$$696$$ 0 0
$$697$$ 1.94345 0.0736135
$$698$$ 0 0
$$699$$ 30.8405 1.16649
$$700$$ 0 0
$$701$$ 51.9336 1.96150 0.980752 0.195257i $$-0.0625541\pi$$
0.980752 + 0.195257i $$0.0625541\pi$$
$$702$$ 0 0
$$703$$ −0.193252 −0.00728865
$$704$$ 0 0
$$705$$ 2.00000 0.0753244
$$706$$ 0 0
$$707$$ 4.38650 0.164971
$$708$$ 0 0
$$709$$ 35.5953 1.33681 0.668406 0.743797i $$-0.266977\pi$$
0.668406 + 0.743797i $$0.266977\pi$$
$$710$$ 0 0
$$711$$ −1.41478 −0.0530583
$$712$$ 0 0
$$713$$ 26.7549 1.00198
$$714$$ 0 0
$$715$$ 6.00000 0.224387
$$716$$ 0 0
$$717$$ −41.6975 −1.55722
$$718$$ 0 0
$$719$$ 26.2635 0.979465 0.489732 0.871873i $$-0.337094\pi$$
0.489732 + 0.871873i $$0.337094\pi$$
$$720$$ 0 0
$$721$$ −4.13124 −0.153855
$$722$$ 0 0
$$723$$ −5.47133 −0.203481
$$724$$ 0 0
$$725$$ −5.02827 −0.186745
$$726$$ 0 0
$$727$$ −22.4804 −0.833752 −0.416876 0.908963i $$-0.636875\pi$$
−0.416876 + 0.908963i $$0.636875\pi$$
$$728$$ 0 0
$$729$$ 30.0101 1.11149
$$730$$ 0 0
$$731$$ 8.69832 0.321719
$$732$$ 0 0
$$733$$ −28.1806 −1.04087 −0.520437 0.853900i $$-0.674231\pi$$
−0.520437 + 0.853900i $$0.674231\pi$$
$$734$$ 0 0
$$735$$ −7.95719 −0.293505
$$736$$ 0 0
$$737$$ 23.0101 0.847589
$$738$$ 0 0
$$739$$ −35.8506 −1.31879 −0.659393 0.751798i $$-0.729187\pi$$
−0.659393 + 0.751798i $$0.729187\pi$$
$$740$$ 0 0
$$741$$ −2.73566 −0.100497
$$742$$ 0 0
$$743$$ −33.5525 −1.23092 −0.615462 0.788167i $$-0.711030\pi$$
−0.615462 + 0.788167i $$0.711030\pi$$
$$744$$ 0 0
$$745$$ −20.9909 −0.769048
$$746$$ 0 0
$$747$$ 1.62177 0.0593374
$$748$$ 0 0
$$749$$ −12.1312 −0.443266
$$750$$ 0 0
$$751$$ −11.4257 −0.416930 −0.208465 0.978030i $$-0.566847\pi$$
−0.208465 + 0.978030i $$0.566847\pi$$
$$752$$ 0 0
$$753$$ −24.1131 −0.878730
$$754$$ 0 0
$$755$$ 2.84049 0.103376
$$756$$ 0 0
$$757$$ −13.9144 −0.505727 −0.252863 0.967502i $$-0.581372\pi$$
−0.252863 + 0.967502i $$0.581372\pi$$
$$758$$ 0 0
$$759$$ 26.7549 0.971140
$$760$$ 0 0
$$761$$ 1.30274 0.0472243 0.0236121 0.999721i $$-0.492483\pi$$
0.0236121 + 0.999721i $$0.492483\pi$$
$$762$$ 0 0
$$763$$ −9.79407 −0.354569
$$764$$ 0 0
$$765$$ −1.41478 −0.0511514
$$766$$ 0 0
$$767$$ −14.5561 −0.525592
$$768$$ 0 0
$$769$$ 16.5297 0.596077 0.298039 0.954554i $$-0.403668\pi$$
0.298039 + 0.954554i $$0.403668\pi$$
$$770$$ 0 0
$$771$$ 35.6327 1.28328
$$772$$ 0 0
$$773$$ −17.1084 −0.615347 −0.307674 0.951492i $$-0.599551\pi$$
−0.307674 + 0.951492i $$0.599551\pi$$
$$774$$ 0 0
$$775$$ −5.02827 −0.180621
$$776$$ 0 0
$$777$$ −0.386505 −0.0138658
$$778$$ 0 0
$$779$$ 0.971726 0.0348157
$$780$$ 0 0
$$781$$ 43.2654 1.54816
$$782$$ 0 0
$$783$$ 28.2262 1.00872
$$784$$ 0 0
$$785$$ −15.4713 −0.552195
$$786$$ 0 0
$$787$$ −20.4293 −0.728226 −0.364113 0.931355i $$-0.618628\pi$$
−0.364113 + 0.931355i $$0.618628\pi$$
$$788$$ 0 0
$$789$$ 31.2835 1.11372
$$790$$ 0 0
$$791$$ 8.95358 0.318353
$$792$$ 0 0
$$793$$ 23.4713 0.833491
$$794$$ 0 0
$$795$$ −5.76394 −0.204426
$$796$$ 0 0
$$797$$ −4.69285 −0.166229 −0.0831147 0.996540i $$-0.526487\pi$$
−0.0831147 + 0.996540i $$0.526487\pi$$
$$798$$ 0 0
$$799$$ 2.64177 0.0934591
$$800$$ 0 0
$$801$$ −9.21606 −0.325634
$$802$$ 0 0
$$803$$ −16.6983 −0.589271
$$804$$ 0 0
$$805$$ −7.02827 −0.247714
$$806$$ 0 0
$$807$$ 28.1131 0.989628
$$808$$ 0 0
$$809$$ −24.7658 −0.870719 −0.435359 0.900257i $$-0.643379\pi$$
−0.435359 + 0.900257i $$0.643379\pi$$
$$810$$ 0 0
$$811$$ −48.8789 −1.71637 −0.858185 0.513341i $$-0.828408\pi$$
−0.858185 + 0.513341i $$0.828408\pi$$
$$812$$ 0 0
$$813$$ 21.4257 0.751432
$$814$$ 0 0
$$815$$ −12.3492 −0.432572
$$816$$ 0 0
$$817$$ 4.34916 0.152158
$$818$$ 0 0
$$819$$ 1.68819 0.0589901
$$820$$ 0 0
$$821$$ −17.0283 −0.594291 −0.297145 0.954832i $$-0.596035\pi$$
−0.297145 + 0.954832i $$0.596035\pi$$
$$822$$ 0 0
$$823$$ 53.8205 1.87606 0.938032 0.346548i $$-0.112646\pi$$
0.938032 + 0.346548i $$0.112646\pi$$
$$824$$ 0 0
$$825$$ −5.02827 −0.175062
$$826$$ 0 0
$$827$$ −23.1951 −0.806573 −0.403286 0.915074i $$-0.632132\pi$$
−0.403286 + 0.915074i $$0.632132\pi$$
$$828$$ 0 0
$$829$$ 17.3582 0.602876 0.301438 0.953486i $$-0.402533\pi$$
0.301438 + 0.953486i $$0.402533\pi$$
$$830$$ 0 0
$$831$$ −16.0275 −0.555987
$$832$$ 0 0
$$833$$ −10.5105 −0.364168
$$834$$ 0 0
$$835$$ 2.21245 0.0765651
$$836$$ 0 0
$$837$$ 28.2262 0.975640
$$838$$ 0 0
$$839$$ 45.4148 1.56789 0.783946 0.620829i $$-0.213204\pi$$
0.783946 + 0.620829i $$0.213204\pi$$
$$840$$ 0 0
$$841$$ −3.71646 −0.128154
$$842$$ 0 0
$$843$$ 28.9827 0.998216
$$844$$ 0 0
$$845$$ −9.73566 −0.334917
$$846$$ 0 0
$$847$$ 0.0373465 0.00128324
$$848$$ 0 0
$$849$$ 21.6410 0.742716
$$850$$ 0 0
$$851$$ 1.02827 0.0352488
$$852$$ 0 0
$$853$$ 28.0950 0.961953 0.480976 0.876734i $$-0.340282\pi$$
0.480976 + 0.876734i $$0.340282\pi$$
$$854$$ 0 0
$$855$$ −0.707389 −0.0241922
$$856$$ 0 0
$$857$$ −34.6000 −1.18191 −0.590957 0.806703i $$-0.701250\pi$$
−0.590957 + 0.806703i $$0.701250\pi$$
$$858$$ 0 0
$$859$$ 5.74474 0.196008 0.0980039 0.995186i $$-0.468754\pi$$
0.0980039 + 0.995186i $$0.468754\pi$$
$$860$$ 0 0
$$861$$ 1.94345 0.0662327
$$862$$ 0 0
$$863$$ 4.03188 0.137247 0.0686234 0.997643i $$-0.478139\pi$$
0.0686234 + 0.997643i $$0.478139\pi$$
$$864$$ 0 0
$$865$$ 2.50506 0.0851747
$$866$$ 0 0
$$867$$ −19.6838 −0.668496
$$868$$ 0 0
$$869$$ −6.64177 −0.225307
$$870$$ 0 0
$$871$$ 12.5188 0.424183
$$872$$ 0 0
$$873$$ −0.824093 −0.0278913
$$874$$ 0 0
$$875$$ 1.32088 0.0446540
$$876$$ 0 0
$$877$$ −38.9590 −1.31555 −0.657777 0.753213i $$-0.728503\pi$$
−0.657777 + 0.753213i $$0.728503\pi$$
$$878$$ 0 0
$$879$$ 21.4631 0.723931
$$880$$ 0 0
$$881$$ −11.7074 −0.394432 −0.197216 0.980360i $$-0.563190\pi$$
−0.197216 + 0.980360i $$0.563190\pi$$
$$882$$ 0 0
$$883$$ −28.3382 −0.953657 −0.476829 0.878996i $$-0.658214\pi$$
−0.476829 + 0.878996i $$0.658214\pi$$
$$884$$ 0 0
$$885$$ 12.1987 0.410055
$$886$$ 0 0
$$887$$ −3.13856 −0.105383 −0.0526913 0.998611i $$-0.516780\pi$$
−0.0526913 + 0.998611i $$0.516780\pi$$
$$888$$ 0 0
$$889$$ 1.48947 0.0499553
$$890$$ 0 0
$$891$$ 21.1787 0.709514
$$892$$ 0 0
$$893$$ 1.32088 0.0442017
$$894$$ 0 0
$$895$$ −6.00000 −0.200558
$$896$$ 0 0
$$897$$ 14.5561 0.486016
$$898$$ 0 0
$$899$$ 25.2835 0.843253
$$900$$ 0 0
$$901$$ −7.61350 −0.253642
$$902$$ 0 0
$$903$$ 8.69832 0.289462
$$904$$ 0 0
$$905$$ 12.8296 0.426469
$$906$$ 0 0
$$907$$ −13.1842 −0.437774 −0.218887 0.975750i $$-0.570243\pi$$
−0.218887 + 0.975750i $$0.570243\pi$$
$$908$$ 0 0
$$909$$ −2.34916 −0.0779167
$$910$$ 0 0
$$911$$ 43.6519 1.44625 0.723126 0.690716i $$-0.242705\pi$$
0.723126 + 0.690716i $$0.242705\pi$$
$$912$$ 0 0
$$913$$ 7.61350 0.251970
$$914$$ 0 0
$$915$$ −19.6700 −0.650272
$$916$$ 0 0
$$917$$ −8.51053 −0.281042
$$918$$ 0 0
$$919$$ −19.8013 −0.653184 −0.326592 0.945165i $$-0.605900\pi$$
−0.326592 + 0.945165i $$0.605900\pi$$
$$920$$ 0 0
$$921$$ −27.1896 −0.895929
$$922$$ 0 0
$$923$$ 23.5388 0.774789
$$924$$ 0 0
$$925$$ −0.193252 −0.00635410
$$926$$ 0 0
$$927$$ 2.21245 0.0726665
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ −5.25526 −0.172234
$$932$$ 0 0
$$933$$ 0.142169 0.00465439
$$934$$ 0 0
$$935$$ −6.64177 −0.217209
$$936$$ 0 0
$$937$$ −31.9709 −1.04444 −0.522222 0.852809i $$-0.674897\pi$$
−0.522222 + 0.852809i $$0.674897\pi$$
$$938$$ 0 0
$$939$$ 40.0109 1.30571
$$940$$ 0 0
$$941$$ 38.8114 1.26522 0.632608 0.774472i $$-0.281984\pi$$
0.632608 + 0.774472i $$0.281984\pi$$
$$942$$ 0 0
$$943$$ −5.17044 −0.168373
$$944$$ 0 0
$$945$$ −7.41478 −0.241203
$$946$$ 0 0
$$947$$ 32.9619 1.07112 0.535558 0.844498i $$-0.320101\pi$$
0.535558 + 0.844498i $$0.320101\pi$$
$$948$$ 0 0
$$949$$ −9.08482 −0.294906
$$950$$ 0 0
$$951$$ 1.86610 0.0605126
$$952$$ 0 0
$$953$$ −6.82409 −0.221054 −0.110527 0.993873i $$-0.535254\pi$$
−0.110527 + 0.993873i $$0.535254\pi$$
$$954$$ 0 0
$$955$$ 11.9253 0.385894
$$956$$ 0 0
$$957$$ 25.2835 0.817301
$$958$$ 0 0
$$959$$ 28.9608 0.935193
$$960$$ 0 0
$$961$$ −5.71646 −0.184402
$$962$$ 0 0
$$963$$ 6.49679 0.209356
$$964$$ 0 0
$$965$$ −2.06201 −0.0663785
$$966$$ 0 0
$$967$$ −17.7458 −0.570666 −0.285333 0.958428i $$-0.592104\pi$$
−0.285333 + 0.958428i $$0.592104\pi$$
$$968$$ 0 0
$$969$$ 3.02827 0.0972822
$$970$$ 0 0
$$971$$ 31.1595 0.999956 0.499978 0.866038i $$-0.333341\pi$$
0.499978 + 0.866038i $$0.333341\pi$$
$$972$$ 0 0
$$973$$ −8.89703 −0.285226
$$974$$ 0 0
$$975$$ −2.73566 −0.0876113
$$976$$ 0 0
$$977$$ −14.3063 −0.457701 −0.228850 0.973462i $$-0.573497\pi$$
−0.228850 + 0.973462i $$0.573497\pi$$
$$978$$ 0 0
$$979$$ −43.2654 −1.38277
$$980$$ 0 0
$$981$$ 5.24514 0.167464
$$982$$ 0 0
$$983$$ −48.2509 −1.53896 −0.769482 0.638669i $$-0.779485\pi$$
−0.769482 + 0.638669i $$0.779485\pi$$
$$984$$ 0 0
$$985$$ 2.31181 0.0736605
$$986$$ 0 0
$$987$$ 2.64177 0.0840884
$$988$$ 0 0
$$989$$ −23.1414 −0.735853
$$990$$ 0 0
$$991$$ 6.04562 0.192045 0.0960227 0.995379i $$-0.469388\pi$$
0.0960227 + 0.995379i $$0.469388\pi$$
$$992$$ 0 0
$$993$$ −33.0957 −1.05026
$$994$$ 0 0
$$995$$ −0.773010 −0.0245061
$$996$$ 0 0
$$997$$ 23.9072 0.757147 0.378574 0.925571i $$-0.376415\pi$$
0.378574 + 0.925571i $$0.376415\pi$$
$$998$$ 0 0
$$999$$ 1.08482 0.0343222
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 6080.2.a.bp.1.3 3
4.3 odd 2 6080.2.a.bz.1.1 3
8.3 odd 2 3040.2.a.k.1.3 3
8.5 even 2 3040.2.a.n.1.1 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.k.1.3 3 8.3 odd 2
3040.2.a.n.1.1 yes 3 8.5 even 2
6080.2.a.bp.1.3 3 1.1 even 1 trivial
6080.2.a.bz.1.1 3 4.3 odd 2