# Properties

 Label 9360.2.a.cw.1.1 Level $9360$ Weight $2$ Character 9360.1 Self dual yes Analytic conductor $74.740$ 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] = [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 1.1 Root $$2.56155$$ 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} +0.438447 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} +0.438447 q^{7} +1.56155 q^{11} -1.00000 q^{13} +1.56155 q^{17} +5.12311 q^{19} -2.43845 q^{23} +1.00000 q^{25} -7.12311 q^{29} -6.00000 q^{31} +0.438447 q^{35} -10.6847 q^{37} +3.56155 q^{41} -3.12311 q^{43} -11.1231 q^{47} -6.80776 q^{49} -4.68466 q^{53} +1.56155 q^{55} -12.0000 q^{59} -6.68466 q^{61} -1.00000 q^{65} +11.3693 q^{67} -10.4384 q^{71} -6.00000 q^{73} +0.684658 q^{77} -4.68466 q^{79} +16.4924 q^{83} +1.56155 q^{85} +10.6847 q^{89} -0.438447 q^{91} +5.12311 q^{95} +16.9309 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 5 q^{7}+O(q^{10})$$ 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
$$7$$ 0.438447 0.165717 0.0828587 0.996561i $$-0.473595\pi$$
0.0828587 + 0.996561i $$0.473595\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.56155 0.470826 0.235413 0.971895i $$-0.424356\pi$$
0.235413 + 0.971895i $$0.424356\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.56155 0.378732 0.189366 0.981907i $$-0.439357\pi$$
0.189366 + 0.981907i $$0.439357\pi$$
$$18$$ 0 0
$$19$$ 5.12311 1.17532 0.587661 0.809108i $$-0.300049\pi$$
0.587661 + 0.809108i $$0.300049\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2.43845 −0.508451 −0.254226 0.967145i $$-0.581821\pi$$
−0.254226 + 0.967145i $$0.581821\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −7.12311 −1.32273 −0.661364 0.750065i $$-0.730022\pi$$
−0.661364 + 0.750065i $$0.730022\pi$$
$$30$$ 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
$$35$$ 0.438447 0.0741111
$$36$$ 0 0
$$37$$ −10.6847 −1.75655 −0.878274 0.478159i $$-0.841304\pi$$
−0.878274 + 0.478159i $$0.841304\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.56155 0.556221 0.278111 0.960549i $$-0.410292\pi$$
0.278111 + 0.960549i $$0.410292\pi$$
$$42$$ 0 0
$$43$$ −3.12311 −0.476269 −0.238135 0.971232i $$-0.576536\pi$$
−0.238135 + 0.971232i $$0.576536\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −11.1231 −1.62247 −0.811236 0.584719i $$-0.801205\pi$$
−0.811236 + 0.584719i $$0.801205\pi$$
$$48$$ 0 0
$$49$$ −6.80776 −0.972538
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −4.68466 −0.643487 −0.321744 0.946827i $$-0.604269\pi$$
−0.321744 + 0.946827i $$0.604269\pi$$
$$54$$ 0 0
$$55$$ 1.56155 0.210560
$$56$$ 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
$$61$$ −6.68466 −0.855883 −0.427941 0.903806i $$-0.640761\pi$$
−0.427941 + 0.903806i $$0.640761\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ 11.3693 1.38898 0.694492 0.719501i $$-0.255629\pi$$
0.694492 + 0.719501i $$0.255629\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −10.4384 −1.23882 −0.619408 0.785069i $$-0.712627\pi$$
−0.619408 + 0.785069i $$0.712627\pi$$
$$72$$ 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
$$77$$ 0.684658 0.0780241
$$78$$ 0 0
$$79$$ −4.68466 −0.527065 −0.263533 0.964650i $$-0.584888\pi$$
−0.263533 + 0.964650i $$0.584888\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 16.4924 1.81028 0.905139 0.425115i $$-0.139766\pi$$
0.905139 + 0.425115i $$0.139766\pi$$
$$84$$ 0 0
$$85$$ 1.56155 0.169374
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 10.6847 1.13257 0.566286 0.824209i $$-0.308380\pi$$
0.566286 + 0.824209i $$0.308380\pi$$
$$90$$ 0 0
$$91$$ −0.438447 −0.0459618
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 5.12311 0.525620
$$96$$ 0 0
$$97$$ 16.9309 1.71907 0.859535 0.511077i $$-0.170753\pi$$
0.859535 + 0.511077i $$0.170753\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 10.2462 1.01954 0.509768 0.860312i $$-0.329731\pi$$
0.509768 + 0.860312i $$0.329731\pi$$
$$102$$ 0 0
$$103$$ 15.1231 1.49012 0.745062 0.666995i $$-0.232420\pi$$
0.745062 + 0.666995i $$0.232420\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −10.9309 −1.05673 −0.528364 0.849018i $$-0.677194\pi$$
−0.528364 + 0.849018i $$0.677194\pi$$
$$108$$ 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
$$113$$ −4.87689 −0.458780 −0.229390 0.973335i $$-0.573673\pi$$
−0.229390 + 0.973335i $$0.573673\pi$$
$$114$$ 0 0
$$115$$ −2.43845 −0.227386
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0.684658 0.0627625
$$120$$ 0 0
$$121$$ −8.56155 −0.778323
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −1.75379 −0.155624 −0.0778118 0.996968i $$-0.524793\pi$$
−0.0778118 + 0.996968i $$0.524793\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 2.24621 0.194771
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1.12311 0.0959534 0.0479767 0.998848i $$-0.484723\pi$$
0.0479767 + 0.998848i $$0.484723\pi$$
$$138$$ 0 0
$$139$$ −3.31534 −0.281204 −0.140602 0.990066i $$-0.544904\pi$$
−0.140602 + 0.990066i $$0.544904\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −1.56155 −0.130584
$$144$$ 0 0
$$145$$ −7.12311 −0.591542
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −17.8078 −1.45887 −0.729434 0.684051i $$-0.760217\pi$$
−0.729434 + 0.684051i $$0.760217\pi$$
$$150$$ 0 0
$$151$$ −11.3693 −0.925222 −0.462611 0.886561i $$-0.653087\pi$$
−0.462611 + 0.886561i $$0.653087\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −6.00000 −0.481932
$$156$$ 0 0
$$157$$ 3.36932 0.268901 0.134450 0.990920i $$-0.457073\pi$$
0.134450 + 0.990920i $$0.457073\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −1.06913 −0.0842593
$$162$$ 0 0
$$163$$ −16.0540 −1.25744 −0.628722 0.777630i $$-0.716422\pi$$
−0.628722 + 0.777630i $$0.716422\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −4.87689 −0.377385 −0.188693 0.982036i $$-0.560425\pi$$
−0.188693 + 0.982036i $$0.560425\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 12.8769 0.979012 0.489506 0.872000i $$-0.337177\pi$$
0.489506 + 0.872000i $$0.337177\pi$$
$$174$$ 0 0
$$175$$ 0.438447 0.0331435
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 4.87689 0.364516 0.182258 0.983251i $$-0.441659\pi$$
0.182258 + 0.983251i $$0.441659\pi$$
$$180$$ 0 0
$$181$$ 13.3153 0.989722 0.494861 0.868972i $$-0.335219\pi$$
0.494861 + 0.868972i $$0.335219\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −10.6847 −0.785552
$$186$$ 0 0
$$187$$ 2.43845 0.178317
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −19.6155 −1.41933 −0.709665 0.704539i $$-0.751154\pi$$
−0.709665 + 0.704539i $$0.751154\pi$$
$$192$$ 0 0
$$193$$ 19.5616 1.40807 0.704036 0.710165i $$-0.251379\pi$$
0.704036 + 0.710165i $$0.251379\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3.36932 −0.240054 −0.120027 0.992771i $$-0.538298\pi$$
−0.120027 + 0.992771i $$0.538298\pi$$
$$198$$ 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
$$203$$ −3.12311 −0.219199
$$204$$ 0 0
$$205$$ 3.56155 0.248750
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ −6.24621 −0.430007 −0.215003 0.976613i $$-0.568976\pi$$
−0.215003 + 0.976613i $$0.568976\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −3.12311 −0.212994
$$216$$ 0 0
$$217$$ −2.63068 −0.178582
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1.56155 −0.105041
$$222$$ 0 0
$$223$$ 15.3693 1.02921 0.514603 0.857429i $$-0.327939\pi$$
0.514603 + 0.857429i $$0.327939\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −5.75379 −0.381892 −0.190946 0.981601i $$-0.561156\pi$$
−0.190946 + 0.981601i $$0.561156\pi$$
$$228$$ 0 0
$$229$$ 17.1231 1.13153 0.565763 0.824568i $$-0.308582\pi$$
0.565763 + 0.824568i $$0.308582\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −27.8078 −1.82175 −0.910874 0.412685i $$-0.864591\pi$$
−0.910874 + 0.412685i $$0.864591\pi$$
$$234$$ 0 0
$$235$$ −11.1231 −0.725591
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −22.9309 −1.48327 −0.741637 0.670801i $$-0.765950\pi$$
−0.741637 + 0.670801i $$0.765950\pi$$
$$240$$ 0 0
$$241$$ 24.7386 1.59356 0.796778 0.604272i $$-0.206536\pi$$
0.796778 + 0.604272i $$0.206536\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −6.80776 −0.434932
$$246$$ 0 0
$$247$$ −5.12311 −0.325975
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −26.2462 −1.65665 −0.828323 0.560251i $$-0.810705\pi$$
−0.828323 + 0.560251i $$0.810705\pi$$
$$252$$ 0 0
$$253$$ −3.80776 −0.239392
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −12.8769 −0.803239 −0.401619 0.915807i $$-0.631552\pi$$
−0.401619 + 0.915807i $$0.631552\pi$$
$$258$$ 0 0
$$259$$ −4.68466 −0.291091
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 2.24621 0.138507 0.0692537 0.997599i $$-0.477938\pi$$
0.0692537 + 0.997599i $$0.477938\pi$$
$$264$$ 0 0
$$265$$ −4.68466 −0.287776
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −0.876894 −0.0534652 −0.0267326 0.999643i $$-0.508510\pi$$
−0.0267326 + 0.999643i $$0.508510\pi$$
$$270$$ 0 0
$$271$$ 19.3693 1.17660 0.588301 0.808642i $$-0.299797\pi$$
0.588301 + 0.808642i $$0.299797\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.56155 0.0941652
$$276$$ 0 0
$$277$$ −12.2462 −0.735804 −0.367902 0.929865i $$-0.619924\pi$$
−0.367902 + 0.929865i $$0.619924\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4.24621 0.253308 0.126654 0.991947i $$-0.459576\pi$$
0.126654 + 0.991947i $$0.459576\pi$$
$$282$$ 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
$$287$$ 1.56155 0.0921755
$$288$$ 0 0
$$289$$ −14.5616 −0.856562
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −20.2462 −1.18280 −0.591398 0.806380i $$-0.701424\pi$$
−0.591398 + 0.806380i $$0.701424\pi$$
$$294$$ 0 0
$$295$$ −12.0000 −0.698667
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 2.43845 0.141019
$$300$$ 0 0
$$301$$ −1.36932 −0.0789261
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −6.68466 −0.382762
$$306$$ 0 0
$$307$$ −7.56155 −0.431561 −0.215780 0.976442i $$-0.569230\pi$$
−0.215780 + 0.976442i $$0.569230\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2.63068 −0.149172 −0.0745862 0.997215i $$-0.523764\pi$$
−0.0745862 + 0.997215i $$0.523764\pi$$
$$312$$ 0 0
$$313$$ 29.1231 1.64614 0.823068 0.567943i $$-0.192261\pi$$
0.823068 + 0.567943i $$0.192261\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.50758 0.0846740 0.0423370 0.999103i $$-0.486520\pi$$
0.0423370 + 0.999103i $$0.486520\pi$$
$$318$$ 0 0
$$319$$ −11.1231 −0.622774
$$320$$ 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
$$329$$ −4.87689 −0.268872
$$330$$ 0 0
$$331$$ −29.1231 −1.60075 −0.800375 0.599499i $$-0.795366\pi$$
−0.800375 + 0.599499i $$0.795366\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 11.3693 0.621172
$$336$$ 0 0
$$337$$ −30.4924 −1.66103 −0.830514 0.556998i $$-0.811953\pi$$
−0.830514 + 0.556998i $$0.811953\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −9.36932 −0.507377
$$342$$ 0 0
$$343$$ −6.05398 −0.326884
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 26.0540 1.39865 0.699325 0.714804i $$-0.253484\pi$$
0.699325 + 0.714804i $$0.253484\pi$$
$$348$$ 0 0
$$349$$ −23.3693 −1.25093 −0.625465 0.780252i $$-0.715091\pi$$
−0.625465 + 0.780252i $$0.715091\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −22.4924 −1.19715 −0.598575 0.801066i $$-0.704266\pi$$
−0.598575 + 0.801066i $$0.704266\pi$$
$$354$$ 0 0
$$355$$ −10.4384 −0.554015
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 14.2462 0.751886 0.375943 0.926643i $$-0.377319\pi$$
0.375943 + 0.926643i $$0.377319\pi$$
$$360$$ 0 0
$$361$$ 7.24621 0.381380
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 1.75379 0.0915470 0.0457735 0.998952i $$-0.485425\pi$$
0.0457735 + 0.998952i $$0.485425\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2.05398 −0.106637
$$372$$ 0 0
$$373$$ 12.2462 0.634085 0.317042 0.948411i $$-0.397310\pi$$
0.317042 + 0.948411i $$0.397310\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7.12311 0.366859
$$378$$ 0 0
$$379$$ 30.4924 1.56629 0.783145 0.621839i $$-0.213614\pi$$
0.783145 + 0.621839i $$0.213614\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −3.50758 −0.179229 −0.0896144 0.995977i $$-0.528563\pi$$
−0.0896144 + 0.995977i $$0.528563\pi$$
$$384$$ 0 0
$$385$$ 0.684658 0.0348934
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 37.8617 1.91967 0.959833 0.280571i $$-0.0905239\pi$$
0.959833 + 0.280571i $$0.0905239\pi$$
$$390$$ 0 0
$$391$$ −3.80776 −0.192567
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −4.68466 −0.235711
$$396$$ 0 0
$$397$$ −4.43845 −0.222759 −0.111380 0.993778i $$-0.535527\pi$$
−0.111380 + 0.993778i $$0.535527\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3.75379 0.187455 0.0937276 0.995598i $$-0.470122\pi$$
0.0937276 + 0.995598i $$0.470122\pi$$
$$402$$ 0 0
$$403$$ 6.00000 0.298881
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −16.6847 −0.827028
$$408$$ 0 0
$$409$$ −6.87689 −0.340041 −0.170020 0.985441i $$-0.554383\pi$$
−0.170020 + 0.985441i $$0.554383\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −5.26137 −0.258895
$$414$$ 0 0
$$415$$ 16.4924 0.809581
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −7.61553 −0.372043 −0.186021 0.982546i $$-0.559559\pi$$
−0.186021 + 0.982546i $$0.559559\pi$$
$$420$$ 0 0
$$421$$ 39.3693 1.91874 0.959372 0.282146i $$-0.0910462\pi$$
0.959372 + 0.282146i $$0.0910462\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1.56155 0.0757464
$$426$$ 0 0
$$427$$ −2.93087 −0.141835
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 3.50758 0.168954 0.0844770 0.996425i $$-0.473078\pi$$
0.0844770 + 0.996425i $$0.473078\pi$$
$$432$$ 0 0
$$433$$ −9.61553 −0.462093 −0.231046 0.972943i $$-0.574215\pi$$
−0.231046 + 0.972943i $$0.574215\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −12.4924 −0.597594
$$438$$ 0 0
$$439$$ −22.0540 −1.05258 −0.526289 0.850306i $$-0.676417\pi$$
−0.526289 + 0.850306i $$0.676417\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 7.80776 0.370958 0.185479 0.982648i $$-0.440616\pi$$
0.185479 + 0.982648i $$0.440616\pi$$
$$444$$ 0 0
$$445$$ 10.6847 0.506501
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −35.1771 −1.66011 −0.830055 0.557682i $$-0.811691\pi$$
−0.830055 + 0.557682i $$0.811691\pi$$
$$450$$ 0 0
$$451$$ 5.56155 0.261883
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −0.438447 −0.0205547
$$456$$ 0 0
$$457$$ −13.3153 −0.622865 −0.311433 0.950268i $$-0.600809\pi$$
−0.311433 + 0.950268i $$0.600809\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 8.05398 0.375111 0.187556 0.982254i $$-0.439944\pi$$
0.187556 + 0.982254i $$0.439944\pi$$
$$462$$ 0 0
$$463$$ 28.9309 1.34453 0.672266 0.740310i $$-0.265321\pi$$
0.672266 + 0.740310i $$0.265321\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −6.93087 −0.320722 −0.160361 0.987058i $$-0.551266\pi$$
−0.160361 + 0.987058i $$0.551266\pi$$
$$468$$ 0 0
$$469$$ 4.98485 0.230179
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −4.87689 −0.224240
$$474$$ 0 0
$$475$$ 5.12311 0.235064
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −26.0540 −1.19044 −0.595218 0.803564i $$-0.702934\pi$$
−0.595218 + 0.803564i $$0.702934\pi$$
$$480$$ 0 0
$$481$$ 10.6847 0.487178
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 16.9309 0.768791
$$486$$ 0 0
$$487$$ 32.0540 1.45250 0.726252 0.687428i $$-0.241260\pi$$
0.726252 + 0.687428i $$0.241260\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 27.6155 1.24627 0.623136 0.782114i $$-0.285858\pi$$
0.623136 + 0.782114i $$0.285858\pi$$
$$492$$ 0 0
$$493$$ −11.1231 −0.500959
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −4.57671 −0.205293
$$498$$ 0 0
$$499$$ −34.9848 −1.56614 −0.783068 0.621936i $$-0.786347\pi$$
−0.783068 + 0.621936i $$0.786347\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −13.7538 −0.613251 −0.306626 0.951830i $$-0.599200\pi$$
−0.306626 + 0.951830i $$0.599200\pi$$
$$504$$ 0 0
$$505$$ 10.2462 0.455950
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −23.5616 −1.04435 −0.522174 0.852839i $$-0.674879\pi$$
−0.522174 + 0.852839i $$0.674879\pi$$
$$510$$ 0 0
$$511$$ −2.63068 −0.116375
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 15.1231 0.666404
$$516$$ 0 0
$$517$$ −17.3693 −0.763902
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −25.8617 −1.13302 −0.566512 0.824054i $$-0.691707\pi$$
−0.566512 + 0.824054i $$0.691707\pi$$
$$522$$ 0 0
$$523$$ 30.7386 1.34411 0.672053 0.740503i $$-0.265413\pi$$
0.672053 + 0.740503i $$0.265413\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −9.36932 −0.408134
$$528$$ 0 0
$$529$$ −17.0540 −0.741477
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −3.56155 −0.154268
$$534$$ 0 0
$$535$$ −10.9309 −0.472583
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −10.6307 −0.457896
$$540$$ 0 0
$$541$$ 18.8769 0.811581 0.405791 0.913966i $$-0.366996\pi$$
0.405791 + 0.913966i $$0.366996\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −2.00000 −0.0856706
$$546$$ 0 0
$$547$$ 5.36932 0.229575 0.114788 0.993390i $$-0.463381\pi$$
0.114788 + 0.993390i $$0.463381\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −36.4924 −1.55463
$$552$$ 0 0
$$553$$ −2.05398 −0.0873439
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 6.49242 0.275093 0.137546 0.990495i $$-0.456078\pi$$
0.137546 + 0.990495i $$0.456078\pi$$
$$558$$ 0 0
$$559$$ 3.12311 0.132093
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 19.3153 0.814045 0.407022 0.913418i $$-0.366567\pi$$
0.407022 + 0.913418i $$0.366567\pi$$
$$564$$ 0 0
$$565$$ −4.87689 −0.205172
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −32.8769 −1.37827 −0.689136 0.724632i $$-0.742010\pi$$
−0.689136 + 0.724632i $$0.742010\pi$$
$$570$$ 0 0
$$571$$ 22.0540 0.922930 0.461465 0.887158i $$-0.347324\pi$$
0.461465 + 0.887158i $$0.347324\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2.43845 −0.101690
$$576$$ 0 0
$$577$$ 24.4384 1.01739 0.508693 0.860948i $$-0.330129\pi$$
0.508693 + 0.860948i $$0.330129\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 7.23106 0.299995
$$582$$ 0 0
$$583$$ −7.31534 −0.302970
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 32.4924 1.34111 0.670553 0.741862i $$-0.266057\pi$$
0.670553 + 0.741862i $$0.266057\pi$$
$$588$$ 0 0
$$589$$ −30.7386 −1.26656
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 24.2462 0.995673 0.497836 0.867271i $$-0.334128\pi$$
0.497836 + 0.867271i $$0.334128\pi$$
$$594$$ 0 0
$$595$$ 0.684658 0.0280683
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 9.36932 0.382820 0.191410 0.981510i $$-0.438694\pi$$
0.191410 + 0.981510i $$0.438694\pi$$
$$600$$ 0 0
$$601$$ −28.5464 −1.16443 −0.582216 0.813034i $$-0.697814\pi$$
−0.582216 + 0.813034i $$0.697814\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −8.56155 −0.348077
$$606$$ 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
$$611$$ 11.1231 0.449993
$$612$$ 0 0
$$613$$ −32.5464 −1.31454 −0.657268 0.753657i $$-0.728288\pi$$
−0.657268 + 0.753657i $$0.728288\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0.738634 0.0297363 0.0148681 0.999889i $$-0.495267\pi$$
0.0148681 + 0.999889i $$0.495267\pi$$
$$618$$ 0 0
$$619$$ 8.63068 0.346896 0.173448 0.984843i $$-0.444509\pi$$
0.173448 + 0.984843i $$0.444509\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 4.68466 0.187687
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −16.6847 −0.665261
$$630$$ 0 0
$$631$$ 32.7386 1.30330 0.651652 0.758518i $$-0.274076\pi$$
0.651652 + 0.758518i $$0.274076\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −1.75379 −0.0695970
$$636$$ 0 0
$$637$$ 6.80776 0.269733
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −32.9848 −1.30282 −0.651412 0.758725i $$-0.725823\pi$$
−0.651412 + 0.758725i $$0.725823\pi$$
$$642$$ 0 0
$$643$$ 10.6847 0.421362 0.210681 0.977555i $$-0.432432\pi$$
0.210681 + 0.977555i $$0.432432\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −31.8078 −1.25049 −0.625246 0.780428i $$-0.715001\pi$$
−0.625246 + 0.780428i $$0.715001\pi$$
$$648$$ 0 0
$$649$$ −18.7386 −0.735556
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −33.3693 −1.30584 −0.652921 0.757426i $$-0.726457\pi$$
−0.652921 + 0.757426i $$0.726457\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −0.876894 −0.0341590 −0.0170795 0.999854i $$-0.505437\pi$$
−0.0170795 + 0.999854i $$0.505437\pi$$
$$660$$ 0 0
$$661$$ 6.49242 0.252526 0.126263 0.991997i $$-0.459702\pi$$
0.126263 + 0.991997i $$0.459702\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2.24621 0.0871043
$$666$$ 0 0
$$667$$ 17.3693 0.672543
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −10.4384 −0.402972
$$672$$ 0 0
$$673$$ 16.7386 0.645227 0.322613 0.946531i $$-0.395439\pi$$
0.322613 + 0.946531i $$0.395439\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −14.4384 −0.554915 −0.277457 0.960738i $$-0.589492\pi$$
−0.277457 + 0.960738i $$0.589492\pi$$
$$678$$ 0 0
$$679$$ 7.42329 0.284880
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −32.4924 −1.24329 −0.621644 0.783300i $$-0.713535\pi$$
−0.621644 + 0.783300i $$0.713535\pi$$
$$684$$ 0 0
$$685$$ 1.12311 0.0429117
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 4.68466 0.178471
$$690$$ 0 0
$$691$$ 21.6155 0.822293 0.411147 0.911569i $$-0.365128\pi$$
0.411147 + 0.911569i $$0.365128\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −3.31534 −0.125758
$$696$$ 0 0
$$697$$ 5.56155 0.210659
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −48.9848 −1.85013 −0.925066 0.379806i $$-0.875991\pi$$
−0.925066 + 0.379806i $$0.875991\pi$$
$$702$$ 0 0
$$703$$ −54.7386 −2.06451
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 4.49242 0.168955
$$708$$ 0 0
$$709$$ 9.12311 0.342625 0.171313 0.985217i $$-0.445199\pi$$
0.171313 + 0.985217i $$0.445199\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 14.6307 0.547923
$$714$$ 0 0
$$715$$ −1.56155 −0.0583988
$$716$$ 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
$$721$$ 6.63068 0.246940
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −7.12311 −0.264546
$$726$$ 0 0
$$727$$ −12.8769 −0.477578 −0.238789 0.971072i $$-0.576750\pi$$
−0.238789 + 0.971072i $$0.576750\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −4.87689 −0.180378
$$732$$ 0 0
$$733$$ −16.4384 −0.607168 −0.303584 0.952805i $$-0.598183\pi$$
−0.303584 + 0.952805i $$0.598183\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 17.7538 0.653969
$$738$$ 0 0
$$739$$ −48.7386 −1.79288 −0.896440 0.443166i $$-0.853855\pi$$
−0.896440 + 0.443166i $$0.853855\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −18.7386 −0.687454 −0.343727 0.939070i $$-0.611689\pi$$
−0.343727 + 0.939070i $$0.611689\pi$$
$$744$$ 0 0
$$745$$ −17.8078 −0.652426
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −4.79261 −0.175118
$$750$$ 0 0
$$751$$ −14.0540 −0.512837 −0.256418 0.966566i $$-0.582542\pi$$
−0.256418 + 0.966566i $$0.582542\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −11.3693 −0.413772
$$756$$ 0 0
$$757$$ −14.4924 −0.526736 −0.263368 0.964695i $$-0.584833\pi$$
−0.263368 + 0.964695i $$0.584833\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 45.2311 1.63962 0.819812 0.572632i $$-0.194078\pi$$
0.819812 + 0.572632i $$0.194078\pi$$
$$762$$ 0 0
$$763$$ −0.876894 −0.0317457
$$764$$ 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
$$773$$ −9.12311 −0.328135 −0.164068 0.986449i $$-0.552462\pi$$
−0.164068 + 0.986449i $$0.552462\pi$$
$$774$$ 0 0
$$775$$ −6.00000 −0.215526
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 18.2462 0.653738
$$780$$ 0 0
$$781$$ −16.3002 −0.583267
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 3.36932 0.120256
$$786$$ 0 0
$$787$$ −11.3693 −0.405272 −0.202636 0.979254i $$-0.564951\pi$$
−0.202636 + 0.979254i $$0.564951\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −2.13826 −0.0760278
$$792$$ 0 0
$$793$$ 6.68466 0.237379
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 4.68466 0.165939 0.0829696 0.996552i $$-0.473560\pi$$
0.0829696 + 0.996552i $$0.473560\pi$$
$$798$$ 0 0
$$799$$ −17.3693 −0.614482
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −9.36932 −0.330636
$$804$$ 0 0
$$805$$ −1.06913 −0.0376819
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 7.50758 0.263952 0.131976 0.991253i $$-0.457868\pi$$
0.131976 + 0.991253i $$0.457868\pi$$
$$810$$ 0 0
$$811$$ −4.24621 −0.149105 −0.0745523 0.997217i $$-0.523753\pi$$
−0.0745523 + 0.997217i $$0.523753\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −16.0540 −0.562346
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 48.5464 1.69428 0.847140 0.531369i $$-0.178322\pi$$
0.847140 + 0.531369i $$0.178322\pi$$
$$822$$ 0 0
$$823$$ 29.7538 1.03715 0.518576 0.855032i $$-0.326462\pi$$
0.518576 + 0.855032i $$0.326462\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −7.12311 −0.247695 −0.123847 0.992301i $$-0.539523\pi$$
−0.123847 + 0.992301i $$0.539523\pi$$
$$828$$ 0 0
$$829$$ 0.738634 0.0256538 0.0128269 0.999918i $$-0.495917\pi$$
0.0128269 + 0.999918i $$0.495917\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −10.6307 −0.368331
$$834$$ 0 0
$$835$$ −4.87689 −0.168772
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 44.7926 1.54641 0.773206 0.634155i $$-0.218652\pi$$
0.773206 + 0.634155i $$0.218652\pi$$
$$840$$ 0 0
$$841$$ 21.7386 0.749608
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ −3.75379 −0.128982
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 26.0540 0.893119
$$852$$ 0 0
$$853$$ 41.4233 1.41831 0.709153 0.705054i $$-0.249077\pi$$
0.709153 + 0.705054i $$0.249077\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 2.43845 0.0832958 0.0416479 0.999132i $$-0.486739\pi$$
0.0416479 + 0.999132i $$0.486739\pi$$
$$858$$ 0 0
$$859$$ −3.80776 −0.129919 −0.0649596 0.997888i $$-0.520692\pi$$
−0.0649596 + 0.997888i $$0.520692\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 9.36932 0.318935 0.159468 0.987203i $$-0.449022\pi$$
0.159468 + 0.987203i $$0.449022\pi$$
$$864$$ 0 0
$$865$$ 12.8769 0.437828
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −7.31534 −0.248156
$$870$$ 0 0
$$871$$ −11.3693 −0.385235
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0.438447 0.0148222
$$876$$ 0 0
$$877$$ 46.9848 1.58657 0.793283 0.608853i $$-0.208370\pi$$
0.793283 + 0.608853i $$0.208370\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −21.3693 −0.719951 −0.359975 0.932962i $$-0.617215\pi$$
−0.359975 + 0.932962i $$0.617215\pi$$
$$882$$ 0 0
$$883$$ 56.1080 1.88818 0.944091 0.329684i $$-0.106942\pi$$
0.944091 + 0.329684i $$0.106942\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −5.56155 −0.186739 −0.0933693 0.995632i $$-0.529764\pi$$
−0.0933693 + 0.995632i $$0.529764\pi$$
$$888$$ 0 0
$$889$$ −0.768944 −0.0257895
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −56.9848 −1.90693
$$894$$ 0 0
$$895$$ 4.87689 0.163017
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 42.7386 1.42541
$$900$$ 0 0
$$901$$ −7.31534 −0.243709
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 13.3153 0.442617
$$906$$ 0 0
$$907$$ 2.63068 0.0873504 0.0436752 0.999046i $$-0.486093\pi$$
0.0436752 + 0.999046i $$0.486093\pi$$
$$908$$ 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
$$913$$ 25.7538 0.852326
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −1.94602 −0.0641934 −0.0320967 0.999485i $$-0.510218\pi$$
−0.0320967 + 0.999485i $$0.510218\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 10.4384 0.343586
$$924$$ 0 0
$$925$$ −10.6847 −0.351309
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 39.6695 1.30151 0.650757 0.759286i $$-0.274452\pi$$
0.650757 + 0.759286i $$0.274452\pi$$
$$930$$ 0 0
$$931$$ −34.8769 −1.14304
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 2.43845 0.0797458
$$936$$ 0 0
$$937$$ −27.3693 −0.894117 −0.447058 0.894505i $$-0.647528\pi$$
−0.447058 + 0.894505i $$0.647528\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −41.8078 −1.36289 −0.681447 0.731867i $$-0.738649\pi$$
−0.681447 + 0.731867i $$0.738649\pi$$
$$942$$ 0 0
$$943$$ −8.68466 −0.282811
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −60.6004 −1.96925 −0.984624 0.174688i $$-0.944108\pi$$
−0.984624 + 0.174688i $$0.944108\pi$$
$$948$$ 0 0
$$949$$ 6.00000 0.194768
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 29.5616 0.957593 0.478796 0.877926i $$-0.341073\pi$$
0.478796 + 0.877926i $$0.341073\pi$$
$$954$$ 0 0
$$955$$ −19.6155 −0.634744
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0.492423 0.0159012
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 19.5616 0.629709
$$966$$ 0 0
$$967$$ 7.36932 0.236981 0.118491 0.992955i $$-0.462194\pi$$
0.118491 + 0.992955i $$0.462194\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 24.4924 0.785999 0.393000 0.919539i $$-0.371437\pi$$
0.393000 + 0.919539i $$0.371437\pi$$
$$972$$ 0 0
$$973$$ −1.45360 −0.0466003
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −43.4773 −1.39096 −0.695481 0.718545i $$-0.744808\pi$$
−0.695481 + 0.718545i $$0.744808\pi$$
$$978$$ 0 0
$$979$$ 16.6847 0.533244
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −42.7386 −1.36315 −0.681575 0.731748i $$-0.738705\pi$$
−0.681575 + 0.731748i $$0.738705\pi$$
$$984$$ 0 0
$$985$$ −3.36932 −0.107355
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 7.61553 0.242160
$$990$$ 0 0
$$991$$ −10.0540 −0.319375 −0.159688 0.987168i $$-0.551049\pi$$
−0.159688 + 0.987168i $$0.551049\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 8.00000 0.253617
$$996$$ 0 0
$$997$$ 28.2462 0.894566 0.447283 0.894392i $$-0.352392\pi$$
0.447283 + 0.894392i $$0.352392\pi$$
$$998$$ 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.1.1 2
3.2 odd 2 9360.2.a.cl.1.1 2
4.3 odd 2 585.2.a.l.1.2 yes 2
12.11 even 2 585.2.a.j.1.1 2
20.3 even 4 2925.2.c.p.2224.1 4
20.7 even 4 2925.2.c.p.2224.4 4
20.19 odd 2 2925.2.a.x.1.1 2
52.51 odd 2 7605.2.a.bd.1.1 2
60.23 odd 4 2925.2.c.o.2224.4 4
60.47 odd 4 2925.2.c.o.2224.1 4
60.59 even 2 2925.2.a.bc.1.2 2
156.155 even 2 7605.2.a.bi.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.a.j.1.1 2 12.11 even 2
585.2.a.l.1.2 yes 2 4.3 odd 2
2925.2.a.x.1.1 2 20.19 odd 2
2925.2.a.bc.1.2 2 60.59 even 2
2925.2.c.o.2224.1 4 60.47 odd 4
2925.2.c.o.2224.4 4 60.23 odd 4
2925.2.c.p.2224.1 4 20.3 even 4
2925.2.c.p.2224.4 4 20.7 even 4
7605.2.a.bd.1.1 2 52.51 odd 2
7605.2.a.bi.1.2 2 156.155 even 2
9360.2.a.cl.1.1 2 3.2 odd 2
9360.2.a.cw.1.1 2 1.1 even 1 trivial