# Properties

 Label 9360.2.a.cw.1.2 Level $9360$ Weight $2$ Character 9360.1 Self dual yes Analytic conductor $74.740$ 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] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9360.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: $$74.7399762919$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 585) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 9360.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} +4.56155 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} +4.56155 q^{7} -2.56155 q^{11} -1.00000 q^{13} -2.56155 q^{17} -3.12311 q^{19} -6.56155 q^{23} +1.00000 q^{25} +1.12311 q^{29} -6.00000 q^{31} +4.56155 q^{35} +1.68466 q^{37} -0.561553 q^{41} +5.12311 q^{43} -2.87689 q^{47} +13.8078 q^{49} +7.68466 q^{53} -2.56155 q^{55} -12.0000 q^{59} +5.68466 q^{61} -1.00000 q^{65} -13.3693 q^{67} -14.5616 q^{71} -6.00000 q^{73} -11.6847 q^{77} +7.68466 q^{79} -16.4924 q^{83} -2.56155 q^{85} -1.68466 q^{89} -4.56155 q^{91} -3.12311 q^{95} -11.9309 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 5 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + 5 * q^7 $$2 q + 2 q^{5} + 5 q^{7} - q^{11} - 2 q^{13} - q^{17} + 2 q^{19} - 9 q^{23} + 2 q^{25} - 6 q^{29} - 12 q^{31} + 5 q^{35} - 9 q^{37} + 3 q^{41} + 2 q^{43} - 14 q^{47} + 7 q^{49} + 3 q^{53} - q^{55} - 24 q^{59} - q^{61} - 2 q^{65} - 2 q^{67} - 25 q^{71} - 12 q^{73} - 11 q^{77} + 3 q^{79} - q^{85} + 9 q^{89} - 5 q^{91} + 2 q^{95} + 5 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + 5 * q^7 - q^11 - 2 * q^13 - q^17 + 2 * q^19 - 9 * q^23 + 2 * q^25 - 6 * q^29 - 12 * q^31 + 5 * q^35 - 9 * q^37 + 3 * q^41 + 2 * q^43 - 14 * q^47 + 7 * q^49 + 3 * q^53 - q^55 - 24 * q^59 - q^61 - 2 * q^65 - 2 * q^67 - 25 * q^71 - 12 * q^73 - 11 * q^77 + 3 * q^79 - q^85 + 9 * q^89 - 5 * q^91 + 2 * q^95 + 5 * q^97

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 4.56155 1.72410 0.862052 0.506819i $$-0.169179\pi$$
0.862052 + 0.506819i $$0.169179\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.56155 −0.772337 −0.386169 0.922428i $$-0.626202\pi$$
−0.386169 + 0.922428i $$0.626202\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.56155 −0.621268 −0.310634 0.950530i $$-0.600541\pi$$
−0.310634 + 0.950530i $$0.600541\pi$$
$$18$$ 0 0
$$19$$ −3.12311 −0.716490 −0.358245 0.933628i $$-0.616625\pi$$
−0.358245 + 0.933628i $$0.616625\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −6.56155 −1.36818 −0.684089 0.729398i $$-0.739800\pi$$
−0.684089 + 0.729398i $$0.739800\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1.12311 0.208555 0.104278 0.994548i $$-0.466747\pi$$
0.104278 + 0.994548i $$0.466747\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 4.56155 0.771043
$$36$$ 0 0
$$37$$ 1.68466 0.276956 0.138478 0.990366i $$-0.455779\pi$$
0.138478 + 0.990366i $$0.455779\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −0.561553 −0.0876998 −0.0438499 0.999038i $$-0.513962\pi$$
−0.0438499 + 0.999038i $$0.513962\pi$$
$$42$$ 0 0
$$43$$ 5.12311 0.781266 0.390633 0.920546i $$-0.372256\pi$$
0.390633 + 0.920546i $$0.372256\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −2.87689 −0.419638 −0.209819 0.977740i $$-0.567288\pi$$
−0.209819 + 0.977740i $$0.567288\pi$$
$$48$$ 0 0
$$49$$ 13.8078 1.97254
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 7.68466 1.05557 0.527785 0.849378i $$-0.323023\pi$$
0.527785 + 0.849378i $$0.323023\pi$$
$$54$$ 0 0
$$55$$ −2.56155 −0.345400
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ 5.68466 0.727846 0.363923 0.931429i $$-0.381437\pi$$
0.363923 + 0.931429i $$0.381437\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −13.3693 −1.63332 −0.816661 0.577118i $$-0.804177\pi$$
−0.816661 + 0.577118i $$0.804177\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −14.5616 −1.72814 −0.864069 0.503373i $$-0.832092\pi$$
−0.864069 + 0.503373i $$0.832092\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −11.6847 −1.33159
$$78$$ 0 0
$$79$$ 7.68466 0.864592 0.432296 0.901732i $$-0.357704\pi$$
0.432296 + 0.901732i $$0.357704\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −16.4924 −1.81028 −0.905139 0.425115i $$-0.860234\pi$$
−0.905139 + 0.425115i $$0.860234\pi$$
$$84$$ 0 0
$$85$$ −2.56155 −0.277839
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −1.68466 −0.178573 −0.0892867 0.996006i $$-0.528459\pi$$
−0.0892867 + 0.996006i $$0.528459\pi$$
$$90$$ 0 0
$$91$$ −4.56155 −0.478181
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −3.12311 −0.320424
$$96$$ 0 0
$$97$$ −11.9309 −1.21140 −0.605698 0.795695i $$-0.707106\pi$$
−0.605698 + 0.795695i $$0.707106\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.24621 −0.621521 −0.310761 0.950488i $$-0.600584\pi$$
−0.310761 + 0.950488i $$0.600584\pi$$
$$102$$ 0 0
$$103$$ 6.87689 0.677601 0.338800 0.940858i $$-0.389979\pi$$
0.338800 + 0.940858i $$0.389979\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 17.9309 1.73344 0.866721 0.498793i $$-0.166223\pi$$
0.866721 + 0.498793i $$0.166223\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −13.1231 −1.23452 −0.617259 0.786760i $$-0.711757\pi$$
−0.617259 + 0.786760i $$0.711757\pi$$
$$114$$ 0 0
$$115$$ −6.56155 −0.611868
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −11.6847 −1.07113
$$120$$ 0 0
$$121$$ −4.43845 −0.403495
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −18.2462 −1.61909 −0.809545 0.587058i $$-0.800286\pi$$
−0.809545 + 0.587058i $$0.800286\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −14.2462 −1.23530
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −7.12311 −0.608568 −0.304284 0.952581i $$-0.598417\pi$$
−0.304284 + 0.952581i $$0.598417\pi$$
$$138$$ 0 0
$$139$$ −15.6847 −1.33036 −0.665178 0.746685i $$-0.731644\pi$$
−0.665178 + 0.746685i $$0.731644\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 2.56155 0.214208
$$144$$ 0 0
$$145$$ 1.12311 0.0932688
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2.80776 0.230021 0.115010 0.993364i $$-0.463310\pi$$
0.115010 + 0.993364i $$0.463310\pi$$
$$150$$ 0 0
$$151$$ 13.3693 1.08798 0.543990 0.839092i $$-0.316913\pi$$
0.543990 + 0.839092i $$0.316913\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −6.00000 −0.481932
$$156$$ 0 0
$$157$$ −21.3693 −1.70546 −0.852729 0.522354i $$-0.825054\pi$$
−0.852729 + 0.522354i $$0.825054\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −29.9309 −2.35888
$$162$$ 0 0
$$163$$ 21.0540 1.64907 0.824537 0.565808i $$-0.191436\pi$$
0.824537 + 0.565808i $$0.191436\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −13.1231 −1.01550 −0.507748 0.861506i $$-0.669522\pi$$
−0.507748 + 0.861506i $$0.669522\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 21.1231 1.60596 0.802980 0.596006i $$-0.203247\pi$$
0.802980 + 0.596006i $$0.203247\pi$$
$$174$$ 0 0
$$175$$ 4.56155 0.344821
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 13.1231 0.980867 0.490433 0.871479i $$-0.336838\pi$$
0.490433 + 0.871479i $$0.336838\pi$$
$$180$$ 0 0
$$181$$ 25.6847 1.90913 0.954563 0.298010i $$-0.0963228\pi$$
0.954563 + 0.298010i $$0.0963228\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1.68466 0.123859
$$186$$ 0 0
$$187$$ 6.56155 0.479828
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 21.6155 1.56404 0.782022 0.623250i $$-0.214188\pi$$
0.782022 + 0.623250i $$0.214188\pi$$
$$192$$ 0 0
$$193$$ 15.4384 1.11128 0.555642 0.831422i $$-0.312473\pi$$
0.555642 + 0.831422i $$0.312473\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 21.3693 1.52250 0.761250 0.648458i $$-0.224586\pi$$
0.761250 + 0.648458i $$0.224586\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 5.12311 0.359572
$$204$$ 0 0
$$205$$ −0.561553 −0.0392205
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ 10.2462 0.705378 0.352689 0.935741i $$-0.385267\pi$$
0.352689 + 0.935741i $$0.385267\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 5.12311 0.349393
$$216$$ 0 0
$$217$$ −27.3693 −1.85795
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 2.56155 0.172309
$$222$$ 0 0
$$223$$ −9.36932 −0.627416 −0.313708 0.949520i $$-0.601571\pi$$
−0.313708 + 0.949520i $$0.601571\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −22.2462 −1.47653 −0.738266 0.674509i $$-0.764355\pi$$
−0.738266 + 0.674509i $$0.764355\pi$$
$$228$$ 0 0
$$229$$ 8.87689 0.586602 0.293301 0.956020i $$-0.405246\pi$$
0.293301 + 0.956020i $$0.405246\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −7.19224 −0.471179 −0.235590 0.971853i $$-0.575702\pi$$
−0.235590 + 0.971853i $$0.575702\pi$$
$$234$$ 0 0
$$235$$ −2.87689 −0.187668
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 5.93087 0.383636 0.191818 0.981431i $$-0.438562\pi$$
0.191818 + 0.981431i $$0.438562\pi$$
$$240$$ 0 0
$$241$$ −24.7386 −1.59356 −0.796778 0.604272i $$-0.793464\pi$$
−0.796778 + 0.604272i $$0.793464\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 13.8078 0.882146
$$246$$ 0 0
$$247$$ 3.12311 0.198718
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −9.75379 −0.615654 −0.307827 0.951442i $$-0.599602\pi$$
−0.307827 + 0.951442i $$0.599602\pi$$
$$252$$ 0 0
$$253$$ 16.8078 1.05670
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −21.1231 −1.31762 −0.658812 0.752308i $$-0.728941\pi$$
−0.658812 + 0.752308i $$0.728941\pi$$
$$258$$ 0 0
$$259$$ 7.68466 0.477501
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −14.2462 −0.878459 −0.439230 0.898375i $$-0.644749\pi$$
−0.439230 + 0.898375i $$0.644749\pi$$
$$264$$ 0 0
$$265$$ 7.68466 0.472065
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −9.12311 −0.556246 −0.278123 0.960546i $$-0.589712\pi$$
−0.278123 + 0.960546i $$0.589712\pi$$
$$270$$ 0 0
$$271$$ −5.36932 −0.326163 −0.163081 0.986613i $$-0.552143\pi$$
−0.163081 + 0.986613i $$0.552143\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −2.56155 −0.154467
$$276$$ 0 0
$$277$$ 4.24621 0.255130 0.127565 0.991830i $$-0.459284\pi$$
0.127565 + 0.991830i $$0.459284\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −12.2462 −0.730548 −0.365274 0.930900i $$-0.619025\pi$$
−0.365274 + 0.930900i $$0.619025\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −2.56155 −0.151204
$$288$$ 0 0
$$289$$ −10.4384 −0.614026
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −3.75379 −0.219299 −0.109649 0.993970i $$-0.534973\pi$$
−0.109649 + 0.993970i $$0.534973\pi$$
$$294$$ 0 0
$$295$$ −12.0000 −0.698667
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 6.56155 0.379464
$$300$$ 0 0
$$301$$ 23.3693 1.34699
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 5.68466 0.325503
$$306$$ 0 0
$$307$$ −3.43845 −0.196243 −0.0981213 0.995174i $$-0.531283\pi$$
−0.0981213 + 0.995174i $$0.531283\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −27.3693 −1.55197 −0.775986 0.630750i $$-0.782747\pi$$
−0.775986 + 0.630750i $$0.782747\pi$$
$$312$$ 0 0
$$313$$ 20.8769 1.18003 0.590016 0.807392i $$-0.299121\pi$$
0.590016 + 0.807392i $$0.299121\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 34.4924 1.93729 0.968644 0.248454i $$-0.0799224\pi$$
0.968644 + 0.248454i $$0.0799224\pi$$
$$318$$ 0 0
$$319$$ −2.87689 −0.161075
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 8.00000 0.445132
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −13.1231 −0.723500
$$330$$ 0 0
$$331$$ −20.8769 −1.14750 −0.573749 0.819031i $$-0.694511\pi$$
−0.573749 + 0.819031i $$0.694511\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −13.3693 −0.730444
$$336$$ 0 0
$$337$$ 2.49242 0.135771 0.0678855 0.997693i $$-0.478375\pi$$
0.0678855 + 0.997693i $$0.478375\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 15.3693 0.832295
$$342$$ 0 0
$$343$$ 31.0540 1.67676
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −11.0540 −0.593408 −0.296704 0.954969i $$-0.595888\pi$$
−0.296704 + 0.954969i $$0.595888\pi$$
$$348$$ 0 0
$$349$$ 1.36932 0.0732979 0.0366489 0.999328i $$-0.488332\pi$$
0.0366489 + 0.999328i $$0.488332\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 10.4924 0.558455 0.279228 0.960225i $$-0.409922\pi$$
0.279228 + 0.960225i $$0.409922\pi$$
$$354$$ 0 0
$$355$$ −14.5616 −0.772847
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −2.24621 −0.118550 −0.0592752 0.998242i $$-0.518879\pi$$
−0.0592752 + 0.998242i $$0.518879\pi$$
$$360$$ 0 0
$$361$$ −9.24621 −0.486643
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 18.2462 0.952444 0.476222 0.879325i $$-0.342006\pi$$
0.476222 + 0.879325i $$0.342006\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 35.0540 1.81991
$$372$$ 0 0
$$373$$ −4.24621 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1.12311 −0.0578429
$$378$$ 0 0
$$379$$ −2.49242 −0.128027 −0.0640136 0.997949i $$-0.520390\pi$$
−0.0640136 + 0.997949i $$0.520390\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −36.4924 −1.86468 −0.932338 0.361588i $$-0.882235\pi$$
−0.932338 + 0.361588i $$0.882235\pi$$
$$384$$ 0 0
$$385$$ −11.6847 −0.595505
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −19.8617 −1.00703 −0.503515 0.863986i $$-0.667960\pi$$
−0.503515 + 0.863986i $$0.667960\pi$$
$$390$$ 0 0
$$391$$ 16.8078 0.850005
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 7.68466 0.386657
$$396$$ 0 0
$$397$$ −8.56155 −0.429692 −0.214846 0.976648i $$-0.568925\pi$$
−0.214846 + 0.976648i $$0.568925\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 20.2462 1.01105 0.505524 0.862813i $$-0.331299\pi$$
0.505524 + 0.862813i $$0.331299\pi$$
$$402$$ 0 0
$$403$$ 6.00000 0.298881
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4.31534 −0.213904
$$408$$ 0 0
$$409$$ −15.1231 −0.747789 −0.373895 0.927471i $$-0.621978\pi$$
−0.373895 + 0.927471i $$0.621978\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −54.7386 −2.69351
$$414$$ 0 0
$$415$$ −16.4924 −0.809581
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 33.6155 1.64223 0.821113 0.570766i $$-0.193354\pi$$
0.821113 + 0.570766i $$0.193354\pi$$
$$420$$ 0 0
$$421$$ 14.6307 0.713056 0.356528 0.934285i $$-0.383960\pi$$
0.356528 + 0.934285i $$0.383960\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −2.56155 −0.124254
$$426$$ 0 0
$$427$$ 25.9309 1.25488
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 36.4924 1.75778 0.878889 0.477026i $$-0.158285\pi$$
0.878889 + 0.477026i $$0.158285\pi$$
$$432$$ 0 0
$$433$$ 31.6155 1.51935 0.759673 0.650306i $$-0.225359\pi$$
0.759673 + 0.650306i $$0.225359\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 20.4924 0.980286
$$438$$ 0 0
$$439$$ 15.0540 0.718487 0.359244 0.933244i $$-0.383035\pi$$
0.359244 + 0.933244i $$0.383035\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −12.8078 −0.608515 −0.304258 0.952590i $$-0.598408\pi$$
−0.304258 + 0.952590i $$0.598408\pi$$
$$444$$ 0 0
$$445$$ −1.68466 −0.0798605
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 10.1771 0.480286 0.240143 0.970738i $$-0.422806\pi$$
0.240143 + 0.970738i $$0.422806\pi$$
$$450$$ 0 0
$$451$$ 1.43845 0.0677338
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −4.56155 −0.213849
$$456$$ 0 0
$$457$$ −25.6847 −1.20148 −0.600739 0.799445i $$-0.705127\pi$$
−0.600739 + 0.799445i $$0.705127\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −29.0540 −1.35318 −0.676589 0.736361i $$-0.736543\pi$$
−0.676589 + 0.736361i $$0.736543\pi$$
$$462$$ 0 0
$$463$$ 0.0691303 0.00321276 0.00160638 0.999999i $$-0.499489\pi$$
0.00160638 + 0.999999i $$0.499489\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 21.9309 1.01484 0.507420 0.861699i $$-0.330599\pi$$
0.507420 + 0.861699i $$0.330599\pi$$
$$468$$ 0 0
$$469$$ −60.9848 −2.81602
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −13.1231 −0.603401
$$474$$ 0 0
$$475$$ −3.12311 −0.143298
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 11.0540 0.505069 0.252535 0.967588i $$-0.418736\pi$$
0.252535 + 0.967588i $$0.418736\pi$$
$$480$$ 0 0
$$481$$ −1.68466 −0.0768138
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −11.9309 −0.541753
$$486$$ 0 0
$$487$$ −5.05398 −0.229017 −0.114509 0.993422i $$-0.536529\pi$$
−0.114509 + 0.993422i $$0.536529\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −13.6155 −0.614460 −0.307230 0.951635i $$-0.599402\pi$$
−0.307230 + 0.951635i $$0.599402\pi$$
$$492$$ 0 0
$$493$$ −2.87689 −0.129569
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −66.4233 −2.97949
$$498$$ 0 0
$$499$$ 30.9848 1.38707 0.693536 0.720422i $$-0.256052\pi$$
0.693536 + 0.720422i $$0.256052\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −30.2462 −1.34861 −0.674306 0.738452i $$-0.735557\pi$$
−0.674306 + 0.738452i $$0.735557\pi$$
$$504$$ 0 0
$$505$$ −6.24621 −0.277953
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −19.4384 −0.861594 −0.430797 0.902449i $$-0.641768\pi$$
−0.430797 + 0.902449i $$0.641768\pi$$
$$510$$ 0 0
$$511$$ −27.3693 −1.21075
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 6.87689 0.303032
$$516$$ 0 0
$$517$$ 7.36932 0.324102
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 31.8617 1.39589 0.697944 0.716152i $$-0.254098\pi$$
0.697944 + 0.716152i $$0.254098\pi$$
$$522$$ 0 0
$$523$$ −18.7386 −0.819383 −0.409692 0.912224i $$-0.634364\pi$$
−0.409692 + 0.912224i $$0.634364\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 15.3693 0.669498
$$528$$ 0 0
$$529$$ 20.0540 0.871912
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0.561553 0.0243236
$$534$$ 0 0
$$535$$ 17.9309 0.775219
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −35.3693 −1.52346
$$540$$ 0 0
$$541$$ 27.1231 1.16611 0.583057 0.812431i $$-0.301857\pi$$
0.583057 + 0.812431i $$0.301857\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −2.00000 −0.0856706
$$546$$ 0 0
$$547$$ −19.3693 −0.828172 −0.414086 0.910238i $$-0.635899\pi$$
−0.414086 + 0.910238i $$0.635899\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −3.50758 −0.149428
$$552$$ 0 0
$$553$$ 35.0540 1.49065
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −26.4924 −1.12252 −0.561260 0.827640i $$-0.689683\pi$$
−0.561260 + 0.827640i $$0.689683\pi$$
$$558$$ 0 0
$$559$$ −5.12311 −0.216684
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 31.6847 1.33535 0.667675 0.744453i $$-0.267290\pi$$
0.667675 + 0.744453i $$0.267290\pi$$
$$564$$ 0 0
$$565$$ −13.1231 −0.552093
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −41.1231 −1.72397 −0.861985 0.506934i $$-0.830779\pi$$
−0.861985 + 0.506934i $$0.830779\pi$$
$$570$$ 0 0
$$571$$ −15.0540 −0.629989 −0.314995 0.949093i $$-0.602003\pi$$
−0.314995 + 0.949093i $$0.602003\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −6.56155 −0.273636
$$576$$ 0 0
$$577$$ 28.5616 1.18903 0.594517 0.804083i $$-0.297343\pi$$
0.594517 + 0.804083i $$0.297343\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −75.2311 −3.12111
$$582$$ 0 0
$$583$$ −19.6847 −0.815255
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −0.492423 −0.0203245 −0.0101622 0.999948i $$-0.503235\pi$$
−0.0101622 + 0.999948i $$0.503235\pi$$
$$588$$ 0 0
$$589$$ 18.7386 0.772112
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 7.75379 0.318410 0.159205 0.987246i $$-0.449107\pi$$
0.159205 + 0.987246i $$0.449107\pi$$
$$594$$ 0 0
$$595$$ −11.6847 −0.479024
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −15.3693 −0.627973 −0.313987 0.949427i $$-0.601665\pi$$
−0.313987 + 0.949427i $$0.601665\pi$$
$$600$$ 0 0
$$601$$ 41.5464 1.69471 0.847356 0.531025i $$-0.178193\pi$$
0.847356 + 0.531025i $$0.178193\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −4.43845 −0.180449
$$606$$ 0 0
$$607$$ 24.0000 0.974130 0.487065 0.873366i $$-0.338067\pi$$
0.487065 + 0.873366i $$0.338067\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 2.87689 0.116387
$$612$$ 0 0
$$613$$ 37.5464 1.51648 0.758242 0.651973i $$-0.226058\pi$$
0.758242 + 0.651973i $$0.226058\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −48.7386 −1.96214 −0.981072 0.193645i $$-0.937969\pi$$
−0.981072 + 0.193645i $$0.937969\pi$$
$$618$$ 0 0
$$619$$ 33.3693 1.34123 0.670613 0.741807i $$-0.266031\pi$$
0.670613 + 0.741807i $$0.266031\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −7.68466 −0.307879
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −4.31534 −0.172064
$$630$$ 0 0
$$631$$ −16.7386 −0.666354 −0.333177 0.942864i $$-0.608121\pi$$
−0.333177 + 0.942864i $$0.608121\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −18.2462 −0.724079
$$636$$ 0 0
$$637$$ −13.8078 −0.547084
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 32.9848 1.30282 0.651412 0.758725i $$-0.274177\pi$$
0.651412 + 0.758725i $$0.274177\pi$$
$$642$$ 0 0
$$643$$ −1.68466 −0.0664364 −0.0332182 0.999448i $$-0.510576\pi$$
−0.0332182 + 0.999448i $$0.510576\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −11.1922 −0.440012 −0.220006 0.975498i $$-0.570608\pi$$
−0.220006 + 0.975498i $$0.570608\pi$$
$$648$$ 0 0
$$649$$ 30.7386 1.20660
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −8.63068 −0.337745 −0.168872 0.985638i $$-0.554013\pi$$
−0.168872 + 0.985638i $$0.554013\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −9.12311 −0.355386 −0.177693 0.984086i $$-0.556863\pi$$
−0.177693 + 0.984086i $$0.556863\pi$$
$$660$$ 0 0
$$661$$ −26.4924 −1.03044 −0.515218 0.857059i $$-0.672289\pi$$
−0.515218 + 0.857059i $$0.672289\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −14.2462 −0.552444
$$666$$ 0 0
$$667$$ −7.36932 −0.285341
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −14.5616 −0.562143
$$672$$ 0 0
$$673$$ −32.7386 −1.26198 −0.630991 0.775790i $$-0.717351\pi$$
−0.630991 + 0.775790i $$0.717351\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −18.5616 −0.713378 −0.356689 0.934223i $$-0.616094\pi$$
−0.356689 + 0.934223i $$0.616094\pi$$
$$678$$ 0 0
$$679$$ −54.4233 −2.08857
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0.492423 0.0188420 0.00942101 0.999956i $$-0.497001\pi$$
0.00942101 + 0.999956i $$0.497001\pi$$
$$684$$ 0 0
$$685$$ −7.12311 −0.272160
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −7.68466 −0.292762
$$690$$ 0 0
$$691$$ −19.6155 −0.746210 −0.373105 0.927789i $$-0.621707\pi$$
−0.373105 + 0.927789i $$0.621707\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −15.6847 −0.594953
$$696$$ 0 0
$$697$$ 1.43845 0.0544851
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 16.9848 0.641509 0.320754 0.947162i $$-0.396064\pi$$
0.320754 + 0.947162i $$0.396064\pi$$
$$702$$ 0 0
$$703$$ −5.26137 −0.198436
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −28.4924 −1.07157
$$708$$ 0 0
$$709$$ 0.876894 0.0329325 0.0164662 0.999864i $$-0.494758\pi$$
0.0164662 + 0.999864i $$0.494758\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 39.3693 1.47439
$$714$$ 0 0
$$715$$ 2.56155 0.0957966
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ 31.3693 1.16825
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1.12311 0.0417111
$$726$$ 0 0
$$727$$ −21.1231 −0.783413 −0.391706 0.920090i $$-0.628115\pi$$
−0.391706 + 0.920090i $$0.628115\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −13.1231 −0.485376
$$732$$ 0 0
$$733$$ −20.5616 −0.759458 −0.379729 0.925098i $$-0.623983\pi$$
−0.379729 + 0.925098i $$0.623983\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 34.2462 1.26148
$$738$$ 0 0
$$739$$ 0.738634 0.0271711 0.0135855 0.999908i $$-0.495675\pi$$
0.0135855 + 0.999908i $$0.495675\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 30.7386 1.12769 0.563846 0.825880i $$-0.309321\pi$$
0.563846 + 0.825880i $$0.309321\pi$$
$$744$$ 0 0
$$745$$ 2.80776 0.102869
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 81.7926 2.98864
$$750$$ 0 0
$$751$$ 23.0540 0.841252 0.420626 0.907234i $$-0.361811\pi$$
0.420626 + 0.907234i $$0.361811\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 13.3693 0.486559
$$756$$ 0 0
$$757$$ 18.4924 0.672119 0.336059 0.941841i $$-0.390906\pi$$
0.336059 + 0.941841i $$0.390906\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −37.2311 −1.34962 −0.674812 0.737989i $$-0.735775\pi$$
−0.674812 + 0.737989i $$0.735775\pi$$
$$762$$ 0 0
$$763$$ −9.12311 −0.330279
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.0000 0.433295
$$768$$ 0 0
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −0.876894 −0.0315397 −0.0157698 0.999876i $$-0.505020\pi$$
−0.0157698 + 0.999876i $$0.505020\pi$$
$$774$$ 0 0
$$775$$ −6.00000 −0.215526
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 1.75379 0.0628360
$$780$$ 0 0
$$781$$ 37.3002 1.33471
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −21.3693 −0.762704
$$786$$ 0 0
$$787$$ 13.3693 0.476565 0.238282 0.971196i $$-0.423416\pi$$
0.238282 + 0.971196i $$0.423416\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −59.8617 −2.12844
$$792$$ 0 0
$$793$$ −5.68466 −0.201868
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −7.68466 −0.272205 −0.136102 0.990695i $$-0.543458\pi$$
−0.136102 + 0.990695i $$0.543458\pi$$
$$798$$ 0 0
$$799$$ 7.36932 0.260708
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 15.3693 0.542371
$$804$$ 0 0
$$805$$ −29.9309 −1.05492
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 40.4924 1.42364 0.711819 0.702363i $$-0.247872\pi$$
0.711819 + 0.702363i $$0.247872\pi$$
$$810$$ 0 0
$$811$$ 12.2462 0.430023 0.215011 0.976612i $$-0.431021\pi$$
0.215011 + 0.976612i $$0.431021\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 21.0540 0.737489
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −21.5464 −0.751974 −0.375987 0.926625i $$-0.622696\pi$$
−0.375987 + 0.926625i $$0.622696\pi$$
$$822$$ 0 0
$$823$$ 46.2462 1.61204 0.806021 0.591887i $$-0.201617\pi$$
0.806021 + 0.591887i $$0.201617\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 1.12311 0.0390542 0.0195271 0.999809i $$-0.493784\pi$$
0.0195271 + 0.999809i $$0.493784\pi$$
$$828$$ 0 0
$$829$$ −48.7386 −1.69276 −0.846381 0.532577i $$-0.821224\pi$$
−0.846381 + 0.532577i $$0.821224\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −35.3693 −1.22547
$$834$$ 0 0
$$835$$ −13.1231 −0.454144
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −41.7926 −1.44284 −0.721421 0.692497i $$-0.756510\pi$$
−0.721421 + 0.692497i $$0.756510\pi$$
$$840$$ 0 0
$$841$$ −27.7386 −0.956505
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ −20.2462 −0.695668
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −11.0540 −0.378925
$$852$$ 0 0
$$853$$ −20.4233 −0.699280 −0.349640 0.936884i $$-0.613696\pi$$
−0.349640 + 0.936884i $$0.613696\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 6.56155 0.224138 0.112069 0.993700i $$-0.464252\pi$$
0.112069 + 0.993700i $$0.464252\pi$$
$$858$$ 0 0
$$859$$ 16.8078 0.573474 0.286737 0.958009i $$-0.407429\pi$$
0.286737 + 0.958009i $$0.407429\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −15.3693 −0.523178 −0.261589 0.965179i $$-0.584246\pi$$
−0.261589 + 0.965179i $$0.584246\pi$$
$$864$$ 0 0
$$865$$ 21.1231 0.718207
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −19.6847 −0.667756
$$870$$ 0 0
$$871$$ 13.3693 0.453002
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 4.56155 0.154209
$$876$$ 0 0
$$877$$ −18.9848 −0.641073 −0.320536 0.947236i $$-0.603863\pi$$
−0.320536 + 0.947236i $$0.603863\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 3.36932 0.113515 0.0567576 0.998388i $$-0.481924\pi$$
0.0567576 + 0.998388i $$0.481924\pi$$
$$882$$ 0 0
$$883$$ −18.1080 −0.609381 −0.304691 0.952451i $$-0.598553\pi$$
−0.304691 + 0.952451i $$0.598553\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −1.43845 −0.0482983 −0.0241492 0.999708i $$-0.507688\pi$$
−0.0241492 + 0.999708i $$0.507688\pi$$
$$888$$ 0 0
$$889$$ −83.2311 −2.79148
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 8.98485 0.300666
$$894$$ 0 0
$$895$$ 13.1231 0.438657
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −6.73863 −0.224746
$$900$$ 0 0
$$901$$ −19.6847 −0.655791
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 25.6847 0.853787
$$906$$ 0 0
$$907$$ 27.3693 0.908783 0.454392 0.890802i $$-0.349857\pi$$
0.454392 + 0.890802i $$0.349857\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ 42.2462 1.39815
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −39.0540 −1.28827 −0.644136 0.764911i $$-0.722783\pi$$
−0.644136 + 0.764911i $$0.722783\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 14.5616 0.479299
$$924$$ 0 0
$$925$$ 1.68466 0.0553912
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −38.6695 −1.26871 −0.634353 0.773044i $$-0.718733\pi$$
−0.634353 + 0.773044i $$0.718733\pi$$
$$930$$ 0 0
$$931$$ −43.1231 −1.41330
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 6.56155 0.214586
$$936$$ 0 0
$$937$$ −2.63068 −0.0859407 −0.0429703 0.999076i $$-0.513682\pi$$
−0.0429703 + 0.999076i $$0.513682\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −21.1922 −0.690847 −0.345424 0.938447i $$-0.612265\pi$$
−0.345424 + 0.938447i $$0.612265\pi$$
$$942$$ 0 0
$$943$$ 3.68466 0.119989
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 46.6004 1.51431 0.757154 0.653236i $$-0.226589\pi$$
0.757154 + 0.653236i $$0.226589\pi$$
$$948$$ 0 0
$$949$$ 6.00000 0.194768
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 25.4384 0.824032 0.412016 0.911177i $$-0.364825\pi$$
0.412016 + 0.911177i $$0.364825\pi$$
$$954$$ 0 0
$$955$$ 21.6155 0.699462
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −32.4924 −1.04924
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 15.4384 0.496981
$$966$$ 0 0
$$967$$ −17.3693 −0.558560 −0.279280 0.960210i $$-0.590096\pi$$
−0.279280 + 0.960210i $$0.590096\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −8.49242 −0.272535 −0.136267 0.990672i $$-0.543511\pi$$
−0.136267 + 0.990672i $$0.543511\pi$$
$$972$$ 0 0
$$973$$ −71.5464 −2.29367
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 55.4773 1.77488 0.887438 0.460928i $$-0.152483\pi$$
0.887438 + 0.460928i $$0.152483\pi$$
$$978$$ 0 0
$$979$$ 4.31534 0.137919
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 6.73863 0.214929 0.107465 0.994209i $$-0.465727\pi$$
0.107465 + 0.994209i $$0.465727\pi$$
$$984$$ 0 0
$$985$$ 21.3693 0.680883
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −33.6155 −1.06891
$$990$$ 0 0
$$991$$ 27.0540 0.859398 0.429699 0.902972i $$-0.358620\pi$$
0.429699 + 0.902972i $$0.358620\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 8.00000 0.253617
$$996$$ 0 0
$$997$$ 11.7538 0.372246 0.186123 0.982526i $$-0.440408\pi$$
0.186123 + 0.982526i $$0.440408\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 9360.2.a.cw.1.2 2
3.2 odd 2 9360.2.a.cl.1.2 2
4.3 odd 2 585.2.a.l.1.1 yes 2
12.11 even 2 585.2.a.j.1.2 2
20.3 even 4 2925.2.c.p.2224.3 4
20.7 even 4 2925.2.c.p.2224.2 4
20.19 odd 2 2925.2.a.x.1.2 2
52.51 odd 2 7605.2.a.bd.1.2 2
60.23 odd 4 2925.2.c.o.2224.2 4
60.47 odd 4 2925.2.c.o.2224.3 4
60.59 even 2 2925.2.a.bc.1.1 2
156.155 even 2 7605.2.a.bi.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.a.j.1.2 2 12.11 even 2
585.2.a.l.1.1 yes 2 4.3 odd 2
2925.2.a.x.1.2 2 20.19 odd 2
2925.2.a.bc.1.1 2 60.59 even 2
2925.2.c.o.2224.2 4 60.23 odd 4
2925.2.c.o.2224.3 4 60.47 odd 4
2925.2.c.p.2224.2 4 20.7 even 4
2925.2.c.p.2224.3 4 20.3 even 4
7605.2.a.bd.1.2 2 52.51 odd 2
7605.2.a.bi.1.1 2 156.155 even 2
9360.2.a.cl.1.2 2 3.2 odd 2
9360.2.a.cw.1.2 2 1.1 even 1 trivial