# Properties

 Label 1560.2.a.q.1.1 Level $1560$ Weight $2$ Character 1560.1 Self dual yes Analytic conductor $12.457$ Analytic rank $0$ 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] = [1560,2,Mod(1,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, 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("1560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1560.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: $$12.4566627153$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.940.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 7x - 4$$ x^3 - 7*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.29240$$ of defining polynomial Character $$\chi$$ $$=$$ 1560.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} -4.83991 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} -4.83991 q^{7} +1.00000 q^{9} +2.25511 q^{11} -1.00000 q^{13} +1.00000 q^{15} +6.32970 q^{17} -2.58480 q^{19} -4.83991 q^{21} +4.83991 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.09501 q^{29} -0.510210 q^{31} +2.25511 q^{33} -4.83991 q^{35} +4.25511 q^{37} -1.00000 q^{39} +0.255105 q^{41} +9.16961 q^{43} +1.00000 q^{45} -4.51021 q^{47} +16.4247 q^{49} +6.32970 q^{51} -9.93492 q^{53} +2.25511 q^{55} -2.58480 q^{57} +14.1900 q^{59} +9.93492 q^{61} -4.83991 q^{63} -1.00000 q^{65} -13.1696 q^{67} +4.83991 q^{69} -5.74489 q^{71} +2.90499 q^{73} +1.00000 q^{75} -10.9145 q^{77} +5.74489 q^{79} +1.00000 q^{81} +10.1900 q^{83} +6.32970 q^{85} +9.09501 q^{87} +3.74489 q^{89} +4.83991 q^{91} -0.510210 q^{93} -2.58480 q^{95} -12.5197 q^{97} +2.25511 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{5} + q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 3 * q^5 + q^7 + 3 * q^9 $$3 q + 3 q^{3} + 3 q^{5} + q^{7} + 3 q^{9} + 5 q^{11} - 3 q^{13} + 3 q^{15} + 7 q^{17} + 6 q^{19} + q^{21} - q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{29} + 2 q^{31} + 5 q^{33} + q^{35} + 11 q^{37} - 3 q^{39} - q^{41} + 3 q^{45} - 10 q^{47} + 20 q^{49} + 7 q^{51} + 3 q^{53} + 5 q^{55} + 6 q^{57} + 8 q^{59} - 3 q^{61} + q^{63} - 3 q^{65} - 12 q^{67} - q^{69} - 19 q^{71} + 26 q^{73} + 3 q^{75} - 7 q^{77} + 19 q^{79} + 3 q^{81} - 4 q^{83} + 7 q^{85} + 10 q^{87} + 13 q^{89} - q^{91} + 2 q^{93} + 6 q^{95} + 9 q^{97} + 5 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 3 * q^5 + q^7 + 3 * q^9 + 5 * q^11 - 3 * q^13 + 3 * q^15 + 7 * q^17 + 6 * q^19 + q^21 - q^23 + 3 * q^25 + 3 * q^27 + 10 * q^29 + 2 * q^31 + 5 * q^33 + q^35 + 11 * q^37 - 3 * q^39 - q^41 + 3 * q^45 - 10 * q^47 + 20 * q^49 + 7 * q^51 + 3 * q^53 + 5 * q^55 + 6 * q^57 + 8 * q^59 - 3 * q^61 + q^63 - 3 * q^65 - 12 * q^67 - q^69 - 19 * q^71 + 26 * q^73 + 3 * q^75 - 7 * q^77 + 19 * q^79 + 3 * q^81 - 4 * q^83 + 7 * q^85 + 10 * q^87 + 13 * q^89 - q^91 + 2 * q^93 + 6 * q^95 + 9 * q^97 + 5 * 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −4.83991 −1.82931 −0.914657 0.404232i $$-0.867539\pi$$
−0.914657 + 0.404232i $$0.867539\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.25511 0.679940 0.339970 0.940436i $$-0.389583\pi$$
0.339970 + 0.940436i $$0.389583\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 6.32970 1.53518 0.767589 0.640943i $$-0.221456\pi$$
0.767589 + 0.640943i $$0.221456\pi$$
$$18$$ 0 0
$$19$$ −2.58480 −0.592995 −0.296497 0.955034i $$-0.595819\pi$$
−0.296497 + 0.955034i $$0.595819\pi$$
$$20$$ 0 0
$$21$$ −4.83991 −1.05615
$$22$$ 0 0
$$23$$ 4.83991 1.00919 0.504595 0.863356i $$-0.331642\pi$$
0.504595 + 0.863356i $$0.331642\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 9.09501 1.68890 0.844451 0.535633i $$-0.179927\pi$$
0.844451 + 0.535633i $$0.179927\pi$$
$$30$$ 0 0
$$31$$ −0.510210 −0.0916364 −0.0458182 0.998950i $$-0.514589\pi$$
−0.0458182 + 0.998950i $$0.514589\pi$$
$$32$$ 0 0
$$33$$ 2.25511 0.392563
$$34$$ 0 0
$$35$$ −4.83991 −0.818094
$$36$$ 0 0
$$37$$ 4.25511 0.699535 0.349767 0.936837i $$-0.386261\pi$$
0.349767 + 0.936837i $$0.386261\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 0.255105 0.0398407 0.0199204 0.999802i $$-0.493659\pi$$
0.0199204 + 0.999802i $$0.493659\pi$$
$$42$$ 0 0
$$43$$ 9.16961 1.39835 0.699176 0.714950i $$-0.253550\pi$$
0.699176 + 0.714950i $$0.253550\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ −4.51021 −0.657882 −0.328941 0.944351i $$-0.606692\pi$$
−0.328941 + 0.944351i $$0.606692\pi$$
$$48$$ 0 0
$$49$$ 16.4247 2.34639
$$50$$ 0 0
$$51$$ 6.32970 0.886335
$$52$$ 0 0
$$53$$ −9.93492 −1.36467 −0.682333 0.731041i $$-0.739035\pi$$
−0.682333 + 0.731041i $$0.739035\pi$$
$$54$$ 0 0
$$55$$ 2.25511 0.304078
$$56$$ 0 0
$$57$$ −2.58480 −0.342366
$$58$$ 0 0
$$59$$ 14.1900 1.84738 0.923692 0.383136i $$-0.125156\pi$$
0.923692 + 0.383136i $$0.125156\pi$$
$$60$$ 0 0
$$61$$ 9.93492 1.27204 0.636018 0.771674i $$-0.280580\pi$$
0.636018 + 0.771674i $$0.280580\pi$$
$$62$$ 0 0
$$63$$ −4.83991 −0.609771
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −13.1696 −1.60892 −0.804462 0.594004i $$-0.797546\pi$$
−0.804462 + 0.594004i $$0.797546\pi$$
$$68$$ 0 0
$$69$$ 4.83991 0.582656
$$70$$ 0 0
$$71$$ −5.74489 −0.681794 −0.340897 0.940101i $$-0.610731\pi$$
−0.340897 + 0.940101i $$0.610731\pi$$
$$72$$ 0 0
$$73$$ 2.90499 0.340003 0.170001 0.985444i $$-0.445623\pi$$
0.170001 + 0.985444i $$0.445623\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −10.9145 −1.24382
$$78$$ 0 0
$$79$$ 5.74489 0.646351 0.323176 0.946339i $$-0.395250\pi$$
0.323176 + 0.946339i $$0.395250\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 10.1900 1.11850 0.559250 0.828999i $$-0.311089\pi$$
0.559250 + 0.828999i $$0.311089\pi$$
$$84$$ 0 0
$$85$$ 6.32970 0.686552
$$86$$ 0 0
$$87$$ 9.09501 0.975088
$$88$$ 0 0
$$89$$ 3.74489 0.396958 0.198479 0.980105i $$-0.436400\pi$$
0.198479 + 0.980105i $$0.436400\pi$$
$$90$$ 0 0
$$91$$ 4.83991 0.507360
$$92$$ 0 0
$$93$$ −0.510210 −0.0529063
$$94$$ 0 0
$$95$$ −2.58480 −0.265195
$$96$$ 0 0
$$97$$ −12.5197 −1.27119 −0.635593 0.772025i $$-0.719244\pi$$
−0.635593 + 0.772025i $$0.719244\pi$$
$$98$$ 0 0
$$99$$ 2.25511 0.226647
$$100$$ 0 0
$$101$$ −17.6052 −1.75179 −0.875893 0.482506i $$-0.839727\pi$$
−0.875893 + 0.482506i $$0.839727\pi$$
$$102$$ 0 0
$$103$$ −17.1696 −1.69177 −0.845886 0.533364i $$-0.820928\pi$$
−0.845886 + 0.533364i $$0.820928\pi$$
$$104$$ 0 0
$$105$$ −4.83991 −0.472327
$$106$$ 0 0
$$107$$ 6.91450 0.668450 0.334225 0.942493i $$-0.391525\pi$$
0.334225 + 0.942493i $$0.391525\pi$$
$$108$$ 0 0
$$109$$ 3.41520 0.327117 0.163558 0.986534i $$-0.447703\pi$$
0.163558 + 0.986534i $$0.447703\pi$$
$$110$$ 0 0
$$111$$ 4.25511 0.403877
$$112$$ 0 0
$$113$$ 5.09501 0.479299 0.239649 0.970860i $$-0.422968\pi$$
0.239649 + 0.970860i $$0.422968\pi$$
$$114$$ 0 0
$$115$$ 4.83991 0.451324
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ −30.6352 −2.80832
$$120$$ 0 0
$$121$$ −5.91450 −0.537682
$$122$$ 0 0
$$123$$ 0.255105 0.0230020
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −12.6594 −1.12334 −0.561670 0.827361i $$-0.689841\pi$$
−0.561670 + 0.827361i $$0.689841\pi$$
$$128$$ 0 0
$$129$$ 9.16961 0.807339
$$130$$ 0 0
$$131$$ −16.7748 −1.46562 −0.732812 0.680431i $$-0.761792\pi$$
−0.732812 + 0.680431i $$0.761792\pi$$
$$132$$ 0 0
$$133$$ 12.5102 1.08477
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ 5.23468 0.444000 0.222000 0.975047i $$-0.428741\pi$$
0.222000 + 0.975047i $$0.428741\pi$$
$$140$$ 0 0
$$141$$ −4.51021 −0.379828
$$142$$ 0 0
$$143$$ −2.25511 −0.188581
$$144$$ 0 0
$$145$$ 9.09501 0.755300
$$146$$ 0 0
$$147$$ 16.4247 1.35469
$$148$$ 0 0
$$149$$ 3.74489 0.306794 0.153397 0.988165i $$-0.450979\pi$$
0.153397 + 0.988165i $$0.450979\pi$$
$$150$$ 0 0
$$151$$ 15.3596 1.24995 0.624975 0.780645i $$-0.285109\pi$$
0.624975 + 0.780645i $$0.285109\pi$$
$$152$$ 0 0
$$153$$ 6.32970 0.511726
$$154$$ 0 0
$$155$$ −0.510210 −0.0409811
$$156$$ 0 0
$$157$$ 0.979580 0.0781790 0.0390895 0.999236i $$-0.487554\pi$$
0.0390895 + 0.999236i $$0.487554\pi$$
$$158$$ 0 0
$$159$$ −9.93492 −0.787891
$$160$$ 0 0
$$161$$ −23.4247 −1.84613
$$162$$ 0 0
$$163$$ 7.42471 0.581548 0.290774 0.956792i $$-0.406087\pi$$
0.290774 + 0.956792i $$0.406087\pi$$
$$164$$ 0 0
$$165$$ 2.25511 0.175560
$$166$$ 0 0
$$167$$ −21.1696 −1.63815 −0.819077 0.573684i $$-0.805514\pi$$
−0.819077 + 0.573684i $$0.805514\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −2.58480 −0.197665
$$172$$ 0 0
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ −4.83991 −0.365863
$$176$$ 0 0
$$177$$ 14.1900 1.06659
$$178$$ 0 0
$$179$$ 8.77483 0.655862 0.327931 0.944702i $$-0.393649\pi$$
0.327931 + 0.944702i $$0.393649\pi$$
$$180$$ 0 0
$$181$$ 23.1045 1.71735 0.858673 0.512524i $$-0.171289\pi$$
0.858673 + 0.512524i $$0.171289\pi$$
$$182$$ 0 0
$$183$$ 9.93492 0.734411
$$184$$ 0 0
$$185$$ 4.25511 0.312842
$$186$$ 0 0
$$187$$ 14.2741 1.04383
$$188$$ 0 0
$$189$$ −4.83991 −0.352052
$$190$$ 0 0
$$191$$ 19.3596 1.40081 0.700407 0.713744i $$-0.253002\pi$$
0.700407 + 0.713744i $$0.253002\pi$$
$$192$$ 0 0
$$193$$ 6.18051 0.444883 0.222441 0.974946i $$-0.428597\pi$$
0.222441 + 0.974946i $$0.428597\pi$$
$$194$$ 0 0
$$195$$ −1.00000 −0.0716115
$$196$$ 0 0
$$197$$ 21.8698 1.55816 0.779081 0.626923i $$-0.215686\pi$$
0.779081 + 0.626923i $$0.215686\pi$$
$$198$$ 0 0
$$199$$ 2.83039 0.200641 0.100321 0.994955i $$-0.468013\pi$$
0.100321 + 0.994955i $$0.468013\pi$$
$$200$$ 0 0
$$201$$ −13.1696 −0.928912
$$202$$ 0 0
$$203$$ −44.0190 −3.08953
$$204$$ 0 0
$$205$$ 0.255105 0.0178173
$$206$$ 0 0
$$207$$ 4.83991 0.336397
$$208$$ 0 0
$$209$$ −5.82900 −0.403201
$$210$$ 0 0
$$211$$ −22.3392 −1.53789 −0.768947 0.639312i $$-0.779219\pi$$
−0.768947 + 0.639312i $$0.779219\pi$$
$$212$$ 0 0
$$213$$ −5.74489 −0.393634
$$214$$ 0 0
$$215$$ 9.16961 0.625362
$$216$$ 0 0
$$217$$ 2.46937 0.167632
$$218$$ 0 0
$$219$$ 2.90499 0.196301
$$220$$ 0 0
$$221$$ −6.32970 −0.425782
$$222$$ 0 0
$$223$$ −23.0950 −1.54656 −0.773278 0.634067i $$-0.781384\pi$$
−0.773278 + 0.634067i $$0.781384\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −0.510210 −0.0338638 −0.0169319 0.999857i $$-0.505390\pi$$
−0.0169319 + 0.999857i $$0.505390\pi$$
$$228$$ 0 0
$$229$$ −1.09501 −0.0723605 −0.0361803 0.999345i $$-0.511519\pi$$
−0.0361803 + 0.999345i $$0.511519\pi$$
$$230$$ 0 0
$$231$$ −10.9145 −0.718121
$$232$$ 0 0
$$233$$ 1.81949 0.119199 0.0595993 0.998222i $$-0.481018\pi$$
0.0595993 + 0.998222i $$0.481018\pi$$
$$234$$ 0 0
$$235$$ −4.51021 −0.294214
$$236$$ 0 0
$$237$$ 5.74489 0.373171
$$238$$ 0 0
$$239$$ 8.44513 0.546270 0.273135 0.961976i $$-0.411939\pi$$
0.273135 + 0.961976i $$0.411939\pi$$
$$240$$ 0 0
$$241$$ 20.3392 1.31016 0.655082 0.755558i $$-0.272634\pi$$
0.655082 + 0.755558i $$0.272634\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 16.4247 1.04934
$$246$$ 0 0
$$247$$ 2.58480 0.164467
$$248$$ 0 0
$$249$$ 10.1900 0.645767
$$250$$ 0 0
$$251$$ −17.4342 −1.10044 −0.550219 0.835020i $$-0.685456\pi$$
−0.550219 + 0.835020i $$0.685456\pi$$
$$252$$ 0 0
$$253$$ 10.9145 0.686189
$$254$$ 0 0
$$255$$ 6.32970 0.396381
$$256$$ 0 0
$$257$$ 15.4342 0.962761 0.481380 0.876512i $$-0.340136\pi$$
0.481380 + 0.876512i $$0.340136\pi$$
$$258$$ 0 0
$$259$$ −20.5943 −1.27967
$$260$$ 0 0
$$261$$ 9.09501 0.562967
$$262$$ 0 0
$$263$$ −8.90499 −0.549105 −0.274553 0.961572i $$-0.588530\pi$$
−0.274553 + 0.961572i $$0.588530\pi$$
$$264$$ 0 0
$$265$$ −9.93492 −0.610297
$$266$$ 0 0
$$267$$ 3.74489 0.229184
$$268$$ 0 0
$$269$$ −16.5848 −1.01119 −0.505597 0.862770i $$-0.668728\pi$$
−0.505597 + 0.862770i $$0.668728\pi$$
$$270$$ 0 0
$$271$$ 17.8290 1.08303 0.541517 0.840690i $$-0.317850\pi$$
0.541517 + 0.840690i $$0.317850\pi$$
$$272$$ 0 0
$$273$$ 4.83991 0.292925
$$274$$ 0 0
$$275$$ 2.25511 0.135988
$$276$$ 0 0
$$277$$ 22.3801 1.34469 0.672344 0.740239i $$-0.265288\pi$$
0.672344 + 0.740239i $$0.265288\pi$$
$$278$$ 0 0
$$279$$ −0.510210 −0.0305455
$$280$$ 0 0
$$281$$ −24.1900 −1.44306 −0.721528 0.692385i $$-0.756560\pi$$
−0.721528 + 0.692385i $$0.756560\pi$$
$$282$$ 0 0
$$283$$ −26.1900 −1.55684 −0.778418 0.627747i $$-0.783977\pi$$
−0.778418 + 0.627747i $$0.783977\pi$$
$$284$$ 0 0
$$285$$ −2.58480 −0.153111
$$286$$ 0 0
$$287$$ −1.23468 −0.0728811
$$288$$ 0 0
$$289$$ 23.0651 1.35677
$$290$$ 0 0
$$291$$ −12.5197 −0.733919
$$292$$ 0 0
$$293$$ −5.48979 −0.320717 −0.160358 0.987059i $$-0.551265\pi$$
−0.160358 + 0.987059i $$0.551265\pi$$
$$294$$ 0 0
$$295$$ 14.1900 0.826175
$$296$$ 0 0
$$297$$ 2.25511 0.130854
$$298$$ 0 0
$$299$$ −4.83991 −0.279899
$$300$$ 0 0
$$301$$ −44.3801 −2.55802
$$302$$ 0 0
$$303$$ −17.6052 −1.01139
$$304$$ 0 0
$$305$$ 9.93492 0.568872
$$306$$ 0 0
$$307$$ −6.76532 −0.386117 −0.193058 0.981187i $$-0.561841\pi$$
−0.193058 + 0.981187i $$0.561841\pi$$
$$308$$ 0 0
$$309$$ −17.1696 −0.976745
$$310$$ 0 0
$$311$$ −17.0204 −0.965139 −0.482570 0.875858i $$-0.660297\pi$$
−0.482570 + 0.875858i $$0.660297\pi$$
$$312$$ 0 0
$$313$$ 12.8304 0.725217 0.362608 0.931942i $$-0.381886\pi$$
0.362608 + 0.931942i $$0.381886\pi$$
$$314$$ 0 0
$$315$$ −4.83991 −0.272698
$$316$$ 0 0
$$317$$ −13.4898 −0.757662 −0.378831 0.925466i $$-0.623674\pi$$
−0.378831 + 0.925466i $$0.623674\pi$$
$$318$$ 0 0
$$319$$ 20.5102 1.14835
$$320$$ 0 0
$$321$$ 6.91450 0.385930
$$322$$ 0 0
$$323$$ −16.3610 −0.910352
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ 3.41520 0.188861
$$328$$ 0 0
$$329$$ 21.8290 1.20347
$$330$$ 0 0
$$331$$ −14.0746 −0.773610 −0.386805 0.922162i $$-0.626421\pi$$
−0.386805 + 0.922162i $$0.626421\pi$$
$$332$$ 0 0
$$333$$ 4.25511 0.233178
$$334$$ 0 0
$$335$$ −13.1696 −0.719532
$$336$$ 0 0
$$337$$ −22.6594 −1.23434 −0.617168 0.786831i $$-0.711720\pi$$
−0.617168 + 0.786831i $$0.711720\pi$$
$$338$$ 0 0
$$339$$ 5.09501 0.276723
$$340$$ 0 0
$$341$$ −1.15058 −0.0623073
$$342$$ 0 0
$$343$$ −45.6147 −2.46296
$$344$$ 0 0
$$345$$ 4.83991 0.260572
$$346$$ 0 0
$$347$$ −8.59432 −0.461367 −0.230684 0.973029i $$-0.574096\pi$$
−0.230684 + 0.973029i $$0.574096\pi$$
$$348$$ 0 0
$$349$$ −23.9444 −1.28172 −0.640858 0.767659i $$-0.721421\pi$$
−0.640858 + 0.767659i $$0.721421\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ −5.74489 −0.304907
$$356$$ 0 0
$$357$$ −30.6352 −1.62138
$$358$$ 0 0
$$359$$ −1.80997 −0.0955268 −0.0477634 0.998859i $$-0.515209\pi$$
−0.0477634 + 0.998859i $$0.515209\pi$$
$$360$$ 0 0
$$361$$ −12.3188 −0.648358
$$362$$ 0 0
$$363$$ −5.91450 −0.310431
$$364$$ 0 0
$$365$$ 2.90499 0.152054
$$366$$ 0 0
$$367$$ 9.16961 0.478650 0.239325 0.970940i $$-0.423074\pi$$
0.239325 + 0.970940i $$0.423074\pi$$
$$368$$ 0 0
$$369$$ 0.255105 0.0132802
$$370$$ 0 0
$$371$$ 48.0841 2.49640
$$372$$ 0 0
$$373$$ −28.8494 −1.49377 −0.746883 0.664955i $$-0.768451\pi$$
−0.746883 + 0.664955i $$0.768451\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ −9.09501 −0.468417
$$378$$ 0 0
$$379$$ 21.9444 1.12721 0.563605 0.826044i $$-0.309414\pi$$
0.563605 + 0.826044i $$0.309414\pi$$
$$380$$ 0 0
$$381$$ −12.6594 −0.648561
$$382$$ 0 0
$$383$$ −6.32018 −0.322946 −0.161473 0.986877i $$-0.551625\pi$$
−0.161473 + 0.986877i $$0.551625\pi$$
$$384$$ 0 0
$$385$$ −10.9145 −0.556254
$$386$$ 0 0
$$387$$ 9.16961 0.466117
$$388$$ 0 0
$$389$$ −17.6052 −0.892620 −0.446310 0.894878i $$-0.647262\pi$$
−0.446310 + 0.894878i $$0.647262\pi$$
$$390$$ 0 0
$$391$$ 30.6352 1.54929
$$392$$ 0 0
$$393$$ −16.7748 −0.846178
$$394$$ 0 0
$$395$$ 5.74489 0.289057
$$396$$ 0 0
$$397$$ −30.0841 −1.50988 −0.754939 0.655795i $$-0.772334\pi$$
−0.754939 + 0.655795i $$0.772334\pi$$
$$398$$ 0 0
$$399$$ 12.5102 0.626294
$$400$$ 0 0
$$401$$ 22.5292 1.12506 0.562528 0.826778i $$-0.309829\pi$$
0.562528 + 0.826778i $$0.309829\pi$$
$$402$$ 0 0
$$403$$ 0.510210 0.0254154
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 9.59571 0.475642
$$408$$ 0 0
$$409$$ 23.1696 1.14566 0.572832 0.819673i $$-0.305845\pi$$
0.572832 + 0.819673i $$0.305845\pi$$
$$410$$ 0 0
$$411$$ 2.00000 0.0986527
$$412$$ 0 0
$$413$$ −68.6784 −3.37944
$$414$$ 0 0
$$415$$ 10.1900 0.500209
$$416$$ 0 0
$$417$$ 5.23468 0.256344
$$418$$ 0 0
$$419$$ 9.56438 0.467251 0.233625 0.972327i $$-0.424941\pi$$
0.233625 + 0.972327i $$0.424941\pi$$
$$420$$ 0 0
$$421$$ −35.7952 −1.74455 −0.872277 0.489012i $$-0.837357\pi$$
−0.872277 + 0.489012i $$0.837357\pi$$
$$422$$ 0 0
$$423$$ −4.51021 −0.219294
$$424$$ 0 0
$$425$$ 6.32970 0.307035
$$426$$ 0 0
$$427$$ −48.0841 −2.32695
$$428$$ 0 0
$$429$$ −2.25511 −0.108877
$$430$$ 0 0
$$431$$ 28.3801 1.36702 0.683510 0.729942i $$-0.260453\pi$$
0.683510 + 0.729942i $$0.260453\pi$$
$$432$$ 0 0
$$433$$ 19.6798 0.945752 0.472876 0.881129i $$-0.343216\pi$$
0.472876 + 0.881129i $$0.343216\pi$$
$$434$$ 0 0
$$435$$ 9.09501 0.436073
$$436$$ 0 0
$$437$$ −12.5102 −0.598445
$$438$$ 0 0
$$439$$ 33.7639 1.61146 0.805732 0.592280i $$-0.201772\pi$$
0.805732 + 0.592280i $$0.201772\pi$$
$$440$$ 0 0
$$441$$ 16.4247 0.782129
$$442$$ 0 0
$$443$$ 29.1045 1.38280 0.691399 0.722473i $$-0.256995\pi$$
0.691399 + 0.722473i $$0.256995\pi$$
$$444$$ 0 0
$$445$$ 3.74489 0.177525
$$446$$ 0 0
$$447$$ 3.74489 0.177127
$$448$$ 0 0
$$449$$ −22.4642 −1.06015 −0.530075 0.847951i $$-0.677836\pi$$
−0.530075 + 0.847951i $$0.677836\pi$$
$$450$$ 0 0
$$451$$ 0.575289 0.0270893
$$452$$ 0 0
$$453$$ 15.3596 0.721659
$$454$$ 0 0
$$455$$ 4.83991 0.226898
$$456$$ 0 0
$$457$$ −11.4993 −0.537915 −0.268957 0.963152i $$-0.586679\pi$$
−0.268957 + 0.963152i $$0.586679\pi$$
$$458$$ 0 0
$$459$$ 6.32970 0.295445
$$460$$ 0 0
$$461$$ 16.7843 0.781725 0.390862 0.920449i $$-0.372177\pi$$
0.390862 + 0.920449i $$0.372177\pi$$
$$462$$ 0 0
$$463$$ 19.6893 0.915041 0.457520 0.889199i $$-0.348738\pi$$
0.457520 + 0.889199i $$0.348738\pi$$
$$464$$ 0 0
$$465$$ −0.510210 −0.0236604
$$466$$ 0 0
$$467$$ −1.38387 −0.0640379 −0.0320190 0.999487i $$-0.510194\pi$$
−0.0320190 + 0.999487i $$0.510194\pi$$
$$468$$ 0 0
$$469$$ 63.7397 2.94323
$$470$$ 0 0
$$471$$ 0.979580 0.0451367
$$472$$ 0 0
$$473$$ 20.6784 0.950795
$$474$$ 0 0
$$475$$ −2.58480 −0.118599
$$476$$ 0 0
$$477$$ −9.93492 −0.454889
$$478$$ 0 0
$$479$$ 28.9553 1.32300 0.661502 0.749944i $$-0.269919\pi$$
0.661502 + 0.749944i $$0.269919\pi$$
$$480$$ 0 0
$$481$$ −4.25511 −0.194016
$$482$$ 0 0
$$483$$ −23.4247 −1.06586
$$484$$ 0 0
$$485$$ −12.5197 −0.568491
$$486$$ 0 0
$$487$$ −26.3705 −1.19496 −0.597482 0.801883i $$-0.703832\pi$$
−0.597482 + 0.801883i $$0.703832\pi$$
$$488$$ 0 0
$$489$$ 7.42471 0.335757
$$490$$ 0 0
$$491$$ 5.05417 0.228092 0.114046 0.993475i $$-0.463619\pi$$
0.114046 + 0.993475i $$0.463619\pi$$
$$492$$ 0 0
$$493$$ 57.5687 2.59276
$$494$$ 0 0
$$495$$ 2.25511 0.101359
$$496$$ 0 0
$$497$$ 27.8048 1.24721
$$498$$ 0 0
$$499$$ −15.7544 −0.705264 −0.352632 0.935762i $$-0.614713\pi$$
−0.352632 + 0.935762i $$0.614713\pi$$
$$500$$ 0 0
$$501$$ −21.1696 −0.945788
$$502$$ 0 0
$$503$$ −9.92541 −0.442552 −0.221276 0.975211i $$-0.571022\pi$$
−0.221276 + 0.975211i $$0.571022\pi$$
$$504$$ 0 0
$$505$$ −17.6052 −0.783422
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ −25.4247 −1.12693 −0.563465 0.826140i $$-0.690532\pi$$
−0.563465 + 0.826140i $$0.690532\pi$$
$$510$$ 0 0
$$511$$ −14.0599 −0.621972
$$512$$ 0 0
$$513$$ −2.58480 −0.114122
$$514$$ 0 0
$$515$$ −17.1696 −0.756583
$$516$$ 0 0
$$517$$ −10.1710 −0.447320
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −14.5292 −0.636538 −0.318269 0.948001i $$-0.603101\pi$$
−0.318269 + 0.948001i $$0.603101\pi$$
$$522$$ 0 0
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 0 0
$$525$$ −4.83991 −0.211231
$$526$$ 0 0
$$527$$ −3.22948 −0.140678
$$528$$ 0 0
$$529$$ 0.424711 0.0184657
$$530$$ 0 0
$$531$$ 14.1900 0.615795
$$532$$ 0 0
$$533$$ −0.255105 −0.0110498
$$534$$ 0 0
$$535$$ 6.91450 0.298940
$$536$$ 0 0
$$537$$ 8.77483 0.378662
$$538$$ 0 0
$$539$$ 37.0394 1.59540
$$540$$ 0 0
$$541$$ 6.11543 0.262923 0.131462 0.991321i $$-0.458033\pi$$
0.131462 + 0.991321i $$0.458033\pi$$
$$542$$ 0 0
$$543$$ 23.1045 0.991510
$$544$$ 0 0
$$545$$ 3.41520 0.146291
$$546$$ 0 0
$$547$$ 15.3596 0.656730 0.328365 0.944551i $$-0.393502\pi$$
0.328365 + 0.944551i $$0.393502\pi$$
$$548$$ 0 0
$$549$$ 9.93492 0.424012
$$550$$ 0 0
$$551$$ −23.5088 −1.00151
$$552$$ 0 0
$$553$$ −27.8048 −1.18238
$$554$$ 0 0
$$555$$ 4.25511 0.180619
$$556$$ 0 0
$$557$$ −22.3801 −0.948273 −0.474137 0.880451i $$-0.657240\pi$$
−0.474137 + 0.880451i $$0.657240\pi$$
$$558$$ 0 0
$$559$$ −9.16961 −0.387833
$$560$$ 0 0
$$561$$ 14.2741 0.602654
$$562$$ 0 0
$$563$$ 17.7449 0.747858 0.373929 0.927457i $$-0.378010\pi$$
0.373929 + 0.927457i $$0.378010\pi$$
$$564$$ 0 0
$$565$$ 5.09501 0.214349
$$566$$ 0 0
$$567$$ −4.83991 −0.203257
$$568$$ 0 0
$$569$$ −10.0190 −0.420020 −0.210010 0.977699i $$-0.567350\pi$$
−0.210010 + 0.977699i $$0.567350\pi$$
$$570$$ 0 0
$$571$$ −17.6147 −0.737154 −0.368577 0.929597i $$-0.620155\pi$$
−0.368577 + 0.929597i $$0.620155\pi$$
$$572$$ 0 0
$$573$$ 19.3596 0.808760
$$574$$ 0 0
$$575$$ 4.83991 0.201838
$$576$$ 0 0
$$577$$ −15.3501 −0.639034 −0.319517 0.947581i $$-0.603521\pi$$
−0.319517 + 0.947581i $$0.603521\pi$$
$$578$$ 0 0
$$579$$ 6.18051 0.256853
$$580$$ 0 0
$$581$$ −49.3188 −2.04609
$$582$$ 0 0
$$583$$ −22.4043 −0.927891
$$584$$ 0 0
$$585$$ −1.00000 −0.0413449
$$586$$ 0 0
$$587$$ 23.4898 0.969527 0.484764 0.874645i $$-0.338906\pi$$
0.484764 + 0.874645i $$0.338906\pi$$
$$588$$ 0 0
$$589$$ 1.31879 0.0543399
$$590$$ 0 0
$$591$$ 21.8698 0.899605
$$592$$ 0 0
$$593$$ −5.63898 −0.231565 −0.115782 0.993275i $$-0.536938\pi$$
−0.115782 + 0.993275i $$0.536938\pi$$
$$594$$ 0 0
$$595$$ −30.6352 −1.25592
$$596$$ 0 0
$$597$$ 2.83039 0.115840
$$598$$ 0 0
$$599$$ −10.0408 −0.410258 −0.205129 0.978735i $$-0.565761\pi$$
−0.205129 + 0.978735i $$0.565761\pi$$
$$600$$ 0 0
$$601$$ 5.04466 0.205776 0.102888 0.994693i $$-0.467192\pi$$
0.102888 + 0.994693i $$0.467192\pi$$
$$602$$ 0 0
$$603$$ −13.1696 −0.536308
$$604$$ 0 0
$$605$$ −5.91450 −0.240459
$$606$$ 0 0
$$607$$ −2.19003 −0.0888904 −0.0444452 0.999012i $$-0.514152\pi$$
−0.0444452 + 0.999012i $$0.514152\pi$$
$$608$$ 0 0
$$609$$ −44.0190 −1.78374
$$610$$ 0 0
$$611$$ 4.51021 0.182464
$$612$$ 0 0
$$613$$ 45.4437 1.83546 0.917728 0.397210i $$-0.130022\pi$$
0.917728 + 0.397210i $$0.130022\pi$$
$$614$$ 0 0
$$615$$ 0.255105 0.0102868
$$616$$ 0 0
$$617$$ −7.31879 −0.294643 −0.147322 0.989089i $$-0.547065\pi$$
−0.147322 + 0.989089i $$0.547065\pi$$
$$618$$ 0 0
$$619$$ 20.6256 0.829015 0.414507 0.910046i $$-0.363954\pi$$
0.414507 + 0.910046i $$0.363954\pi$$
$$620$$ 0 0
$$621$$ 4.83991 0.194219
$$622$$ 0 0
$$623$$ −18.1249 −0.726161
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −5.82900 −0.232788
$$628$$ 0 0
$$629$$ 26.9335 1.07391
$$630$$ 0 0
$$631$$ 14.8304 0.590389 0.295194 0.955437i $$-0.404616\pi$$
0.295194 + 0.955437i $$0.404616\pi$$
$$632$$ 0 0
$$633$$ −22.3392 −0.887904
$$634$$ 0 0
$$635$$ −12.6594 −0.502373
$$636$$ 0 0
$$637$$ −16.4247 −0.650771
$$638$$ 0 0
$$639$$ −5.74489 −0.227265
$$640$$ 0 0
$$641$$ −28.8494 −1.13948 −0.569742 0.821824i $$-0.692957\pi$$
−0.569742 + 0.821824i $$0.692957\pi$$
$$642$$ 0 0
$$643$$ −29.7449 −1.17302 −0.586512 0.809940i $$-0.699499\pi$$
−0.586512 + 0.809940i $$0.699499\pi$$
$$644$$ 0 0
$$645$$ 9.16961 0.361053
$$646$$ 0 0
$$647$$ 2.13967 0.0841192 0.0420596 0.999115i $$-0.486608\pi$$
0.0420596 + 0.999115i $$0.486608\pi$$
$$648$$ 0 0
$$649$$ 32.0000 1.25611
$$650$$ 0 0
$$651$$ 2.46937 0.0967822
$$652$$ 0 0
$$653$$ −13.2104 −0.516965 −0.258482 0.966016i $$-0.583222\pi$$
−0.258482 + 0.966016i $$0.583222\pi$$
$$654$$ 0 0
$$655$$ −16.7748 −0.655447
$$656$$ 0 0
$$657$$ 2.90499 0.113334
$$658$$ 0 0
$$659$$ −48.6447 −1.89493 −0.947464 0.319863i $$-0.896363\pi$$
−0.947464 + 0.319863i $$0.896363\pi$$
$$660$$ 0 0
$$661$$ −24.7340 −0.962041 −0.481020 0.876709i $$-0.659734\pi$$
−0.481020 + 0.876709i $$0.659734\pi$$
$$662$$ 0 0
$$663$$ −6.32970 −0.245825
$$664$$ 0 0
$$665$$ 12.5102 0.485125
$$666$$ 0 0
$$667$$ 44.0190 1.70442
$$668$$ 0 0
$$669$$ −23.0950 −0.892905
$$670$$ 0 0
$$671$$ 22.4043 0.864908
$$672$$ 0 0
$$673$$ 39.6988 1.53028 0.765139 0.643865i $$-0.222670\pi$$
0.765139 + 0.643865i $$0.222670\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 28.5535 1.09740 0.548700 0.836020i $$-0.315123\pi$$
0.548700 + 0.836020i $$0.315123\pi$$
$$678$$ 0 0
$$679$$ 60.5943 2.32540
$$680$$ 0 0
$$681$$ −0.510210 −0.0195513
$$682$$ 0 0
$$683$$ 15.8508 0.606515 0.303257 0.952909i $$-0.401926\pi$$
0.303257 + 0.952909i $$0.401926\pi$$
$$684$$ 0 0
$$685$$ 2.00000 0.0764161
$$686$$ 0 0
$$687$$ −1.09501 −0.0417774
$$688$$ 0 0
$$689$$ 9.93492 0.378490
$$690$$ 0 0
$$691$$ −28.6256 −1.08897 −0.544485 0.838770i $$-0.683275\pi$$
−0.544485 + 0.838770i $$0.683275\pi$$
$$692$$ 0 0
$$693$$ −10.9145 −0.414608
$$694$$ 0 0
$$695$$ 5.23468 0.198563
$$696$$ 0 0
$$697$$ 1.61474 0.0611626
$$698$$ 0 0
$$699$$ 1.81949 0.0688194
$$700$$ 0 0
$$701$$ −8.58480 −0.324244 −0.162122 0.986771i $$-0.551834\pi$$
−0.162122 + 0.986771i $$0.551834\pi$$
$$702$$ 0 0
$$703$$ −10.9986 −0.414820
$$704$$ 0 0
$$705$$ −4.51021 −0.169864
$$706$$ 0 0
$$707$$ 85.2077 3.20456
$$708$$ 0 0
$$709$$ −3.56438 −0.133863 −0.0669316 0.997758i $$-0.521321\pi$$
−0.0669316 + 0.997758i $$0.521321\pi$$
$$710$$ 0 0
$$711$$ 5.74489 0.215450
$$712$$ 0 0
$$713$$ −2.46937 −0.0924786
$$714$$ 0 0
$$715$$ −2.25511 −0.0843361
$$716$$ 0 0
$$717$$ 8.44513 0.315389
$$718$$ 0 0
$$719$$ 22.1900 0.827548 0.413774 0.910380i $$-0.364210\pi$$
0.413774 + 0.910380i $$0.364210\pi$$
$$720$$ 0 0
$$721$$ 83.0993 3.09478
$$722$$ 0 0
$$723$$ 20.3392 0.756423
$$724$$ 0 0
$$725$$ 9.09501 0.337780
$$726$$ 0 0
$$727$$ 9.82900 0.364538 0.182269 0.983249i $$-0.441656\pi$$
0.182269 + 0.983249i $$0.441656\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 58.0408 2.14672
$$732$$ 0 0
$$733$$ −29.0637 −1.07349 −0.536746 0.843744i $$-0.680347\pi$$
−0.536746 + 0.843744i $$0.680347\pi$$
$$734$$ 0 0
$$735$$ 16.4247 0.605835
$$736$$ 0 0
$$737$$ −29.6988 −1.09397
$$738$$ 0 0
$$739$$ 28.9240 1.06399 0.531994 0.846748i $$-0.321443\pi$$
0.531994 + 0.846748i $$0.321443\pi$$
$$740$$ 0 0
$$741$$ 2.58480 0.0949551
$$742$$ 0 0
$$743$$ −20.0190 −0.734427 −0.367213 0.930137i $$-0.619688\pi$$
−0.367213 + 0.930137i $$0.619688\pi$$
$$744$$ 0 0
$$745$$ 3.74489 0.137202
$$746$$ 0 0
$$747$$ 10.1900 0.372834
$$748$$ 0 0
$$749$$ −33.4656 −1.22280
$$750$$ 0 0
$$751$$ −22.2741 −0.812795 −0.406397 0.913696i $$-0.633215\pi$$
−0.406397 + 0.913696i $$0.633215\pi$$
$$752$$ 0 0
$$753$$ −17.4342 −0.635339
$$754$$ 0 0
$$755$$ 15.3596 0.558994
$$756$$ 0 0
$$757$$ −23.1886 −0.842805 −0.421403 0.906874i $$-0.638462\pi$$
−0.421403 + 0.906874i $$0.638462\pi$$
$$758$$ 0 0
$$759$$ 10.9145 0.396171
$$760$$ 0 0
$$761$$ 47.5497 1.72367 0.861837 0.507186i $$-0.169314\pi$$
0.861837 + 0.507186i $$0.169314\pi$$
$$762$$ 0 0
$$763$$ −16.5292 −0.598399
$$764$$ 0 0
$$765$$ 6.32970 0.228851
$$766$$ 0 0
$$767$$ −14.1900 −0.512372
$$768$$ 0 0
$$769$$ −21.8698 −0.788647 −0.394323 0.918972i $$-0.629021\pi$$
−0.394323 + 0.918972i $$0.629021\pi$$
$$770$$ 0 0
$$771$$ 15.4342 0.555850
$$772$$ 0 0
$$773$$ −8.84942 −0.318292 −0.159146 0.987255i $$-0.550874\pi$$
−0.159146 + 0.987255i $$0.550874\pi$$
$$774$$ 0 0
$$775$$ −0.510210 −0.0183273
$$776$$ 0 0
$$777$$ −20.5943 −0.738817
$$778$$ 0 0
$$779$$ −0.659396 −0.0236253
$$780$$ 0 0
$$781$$ −12.9553 −0.463579
$$782$$ 0 0
$$783$$ 9.09501 0.325029
$$784$$ 0 0
$$785$$ 0.979580 0.0349627
$$786$$ 0 0
$$787$$ 9.02042 0.321543 0.160772 0.986992i $$-0.448602\pi$$
0.160772 + 0.986992i $$0.448602\pi$$
$$788$$ 0 0
$$789$$ −8.90499 −0.317026
$$790$$ 0 0
$$791$$ −24.6594 −0.876787
$$792$$ 0 0
$$793$$ −9.93492 −0.352799
$$794$$ 0 0
$$795$$ −9.93492 −0.352355
$$796$$ 0 0
$$797$$ 9.55487 0.338451 0.169225 0.985577i $$-0.445873\pi$$
0.169225 + 0.985577i $$0.445873\pi$$
$$798$$ 0 0
$$799$$ −28.5483 −1.00997
$$800$$ 0 0
$$801$$ 3.74489 0.132319
$$802$$ 0 0
$$803$$ 6.55105 0.231182
$$804$$ 0 0
$$805$$ −23.4247 −0.825613
$$806$$ 0 0
$$807$$ −16.5848 −0.583813
$$808$$ 0 0
$$809$$ 53.3596 1.87602 0.938012 0.346602i $$-0.112664\pi$$
0.938012 + 0.346602i $$0.112664\pi$$
$$810$$ 0 0
$$811$$ 10.9458 0.384360 0.192180 0.981360i $$-0.438444\pi$$
0.192180 + 0.981360i $$0.438444\pi$$
$$812$$ 0 0
$$813$$ 17.8290 0.625290
$$814$$ 0 0
$$815$$ 7.42471 0.260076
$$816$$ 0 0
$$817$$ −23.7016 −0.829215
$$818$$ 0 0
$$819$$ 4.83991 0.169120
$$820$$ 0 0
$$821$$ 35.6147 1.24296 0.621481 0.783429i $$-0.286531\pi$$
0.621481 + 0.783429i $$0.286531\pi$$
$$822$$ 0 0
$$823$$ 10.1900 0.355202 0.177601 0.984103i $$-0.443166\pi$$
0.177601 + 0.984103i $$0.443166\pi$$
$$824$$ 0 0
$$825$$ 2.25511 0.0785127
$$826$$ 0 0
$$827$$ −25.0394 −0.870707 −0.435353 0.900260i $$-0.643377\pi$$
−0.435353 + 0.900260i $$0.643377\pi$$
$$828$$ 0 0
$$829$$ −36.3392 −1.26211 −0.631057 0.775737i $$-0.717378\pi$$
−0.631057 + 0.775737i $$0.717378\pi$$
$$830$$ 0 0
$$831$$ 22.3801 0.776355
$$832$$ 0 0
$$833$$ 103.963 3.60212
$$834$$ 0 0
$$835$$ −21.1696 −0.732604
$$836$$ 0 0
$$837$$ −0.510210 −0.0176354
$$838$$ 0 0
$$839$$ 38.4043 1.32586 0.662932 0.748680i $$-0.269312\pi$$
0.662932 + 0.748680i $$0.269312\pi$$
$$840$$ 0 0
$$841$$ 53.7193 1.85239
$$842$$ 0 0
$$843$$ −24.1900 −0.833149
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ 28.6256 0.983589
$$848$$ 0 0
$$849$$ −26.1900 −0.898839
$$850$$ 0 0
$$851$$ 20.5943 0.705964
$$852$$ 0 0
$$853$$ 22.7245 0.778071 0.389036 0.921223i $$-0.372808\pi$$
0.389036 + 0.921223i $$0.372808\pi$$
$$854$$ 0 0
$$855$$ −2.58480 −0.0883984
$$856$$ 0 0
$$857$$ −10.3297 −0.352856 −0.176428 0.984314i $$-0.556454\pi$$
−0.176428 + 0.984314i $$0.556454\pi$$
$$858$$ 0 0
$$859$$ 2.40429 0.0820334 0.0410167 0.999158i $$-0.486940\pi$$
0.0410167 + 0.999158i $$0.486940\pi$$
$$860$$ 0 0
$$861$$ −1.23468 −0.0420779
$$862$$ 0 0
$$863$$ 17.6798 0.601828 0.300914 0.953651i $$-0.402708\pi$$
0.300914 + 0.953651i $$0.402708\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ 0 0
$$867$$ 23.0651 0.783331
$$868$$ 0 0
$$869$$ 12.9553 0.439480
$$870$$ 0 0
$$871$$ 13.1696 0.446235
$$872$$ 0 0
$$873$$ −12.5197 −0.423728
$$874$$ 0 0
$$875$$ −4.83991 −0.163619
$$876$$ 0 0
$$877$$ 50.7601 1.71405 0.857023 0.515277i $$-0.172311\pi$$
0.857023 + 0.515277i $$0.172311\pi$$
$$878$$ 0 0
$$879$$ −5.48979 −0.185166
$$880$$ 0 0
$$881$$ −49.5905 −1.67075 −0.835373 0.549683i $$-0.814748\pi$$
−0.835373 + 0.549683i $$0.814748\pi$$
$$882$$ 0 0
$$883$$ 21.3188 0.717434 0.358717 0.933446i $$-0.383214\pi$$
0.358717 + 0.933446i $$0.383214\pi$$
$$884$$ 0 0
$$885$$ 14.1900 0.476993
$$886$$ 0 0
$$887$$ −16.9891 −0.570438 −0.285219 0.958462i $$-0.592066\pi$$
−0.285219 + 0.958462i $$0.592066\pi$$
$$888$$ 0 0
$$889$$ 61.2703 2.05494
$$890$$ 0 0
$$891$$ 2.25511 0.0755489
$$892$$ 0 0
$$893$$ 11.6580 0.390120
$$894$$ 0 0
$$895$$ 8.77483 0.293310
$$896$$ 0 0
$$897$$ −4.83991 −0.161600
$$898$$ 0 0
$$899$$ −4.64037 −0.154765
$$900$$ 0 0
$$901$$ −62.8851 −2.09500
$$902$$ 0 0
$$903$$ −44.3801 −1.47688
$$904$$ 0 0
$$905$$ 23.1045 0.768020
$$906$$ 0 0
$$907$$ −36.6594 −1.21726 −0.608628 0.793456i $$-0.708280\pi$$
−0.608628 + 0.793456i $$0.708280\pi$$
$$908$$ 0 0
$$909$$ −17.6052 −0.583928
$$910$$ 0 0
$$911$$ 22.1900 0.735188 0.367594 0.929986i $$-0.380182\pi$$
0.367594 + 0.929986i $$0.380182\pi$$
$$912$$ 0 0
$$913$$ 22.9796 0.760513
$$914$$ 0 0
$$915$$ 9.93492 0.328438
$$916$$ 0 0
$$917$$ 81.1886 2.68108
$$918$$ 0 0
$$919$$ 25.5957 0.844325 0.422162 0.906520i $$-0.361271\pi$$
0.422162 + 0.906520i $$0.361271\pi$$
$$920$$ 0 0
$$921$$ −6.76532 −0.222925
$$922$$ 0 0
$$923$$ 5.74489 0.189096
$$924$$ 0 0
$$925$$ 4.25511 0.139907
$$926$$ 0 0
$$927$$ −17.1696 −0.563924
$$928$$ 0 0
$$929$$ 9.40568 0.308590 0.154295 0.988025i $$-0.450689\pi$$
0.154295 + 0.988025i $$0.450689\pi$$
$$930$$ 0 0
$$931$$ −42.4546 −1.39139
$$932$$ 0 0
$$933$$ −17.0204 −0.557224
$$934$$ 0 0
$$935$$ 14.2741 0.466814
$$936$$ 0 0
$$937$$ 50.7601 1.65826 0.829130 0.559056i $$-0.188836\pi$$
0.829130 + 0.559056i $$0.188836\pi$$
$$938$$ 0 0
$$939$$ 12.8304 0.418704
$$940$$ 0 0
$$941$$ −30.5943 −0.997346 −0.498673 0.866790i $$-0.666179\pi$$
−0.498673 + 0.866790i $$0.666179\pi$$
$$942$$ 0 0
$$943$$ 1.23468 0.0402069
$$944$$ 0 0
$$945$$ −4.83991 −0.157442
$$946$$ 0 0
$$947$$ 7.12877 0.231654 0.115827 0.993269i $$-0.463048\pi$$
0.115827 + 0.993269i $$0.463048\pi$$
$$948$$ 0 0
$$949$$ −2.90499 −0.0942999
$$950$$ 0 0
$$951$$ −13.4898 −0.437436
$$952$$ 0 0
$$953$$ 3.99049 0.129265 0.0646323 0.997909i $$-0.479413\pi$$
0.0646323 + 0.997909i $$0.479413\pi$$
$$954$$ 0 0
$$955$$ 19.3596 0.626463
$$956$$ 0 0
$$957$$ 20.5102 0.663001
$$958$$ 0 0
$$959$$ −9.67982 −0.312578
$$960$$ 0 0
$$961$$ −30.7397 −0.991603
$$962$$ 0 0
$$963$$ 6.91450 0.222817
$$964$$ 0 0
$$965$$ 6.18051 0.198958
$$966$$ 0 0
$$967$$ 27.4751 0.883539 0.441769 0.897129i $$-0.354351\pi$$
0.441769 + 0.897129i $$0.354351\pi$$
$$968$$ 0 0
$$969$$ −16.3610 −0.525592
$$970$$ 0 0
$$971$$ 29.5834 0.949377 0.474688 0.880154i $$-0.342561\pi$$
0.474688 + 0.880154i $$0.342561\pi$$
$$972$$ 0 0
$$973$$ −25.3354 −0.812215
$$974$$ 0 0
$$975$$ −1.00000 −0.0320256
$$976$$ 0 0
$$977$$ −1.72066 −0.0550487 −0.0275243 0.999621i $$-0.508762\pi$$
−0.0275243 + 0.999621i $$0.508762\pi$$
$$978$$ 0 0
$$979$$ 8.44513 0.269908
$$980$$ 0 0
$$981$$ 3.41520 0.109039
$$982$$ 0 0
$$983$$ 27.8889 0.889517 0.444758 0.895651i $$-0.353290\pi$$
0.444758 + 0.895651i $$0.353290\pi$$
$$984$$ 0 0
$$985$$ 21.8698 0.696831
$$986$$ 0 0
$$987$$ 21.8290 0.694825
$$988$$ 0 0
$$989$$ 44.3801 1.41120
$$990$$ 0 0
$$991$$ 45.6147 1.44900 0.724500 0.689275i $$-0.242071\pi$$
0.724500 + 0.689275i $$0.242071\pi$$
$$992$$ 0 0
$$993$$ −14.0746 −0.446644
$$994$$ 0 0
$$995$$ 2.83039 0.0897295
$$996$$ 0 0
$$997$$ −19.8290 −0.627991 −0.313995 0.949425i $$-0.601668\pi$$
−0.313995 + 0.949425i $$0.601668\pi$$
$$998$$ 0 0
$$999$$ 4.25511 0.134626
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 1560.2.a.q.1.1 3
3.2 odd 2 4680.2.a.bh.1.1 3
4.3 odd 2 3120.2.a.bi.1.3 3
5.4 even 2 7800.2.a.bi.1.3 3
12.11 even 2 9360.2.a.cy.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.q.1.1 3 1.1 even 1 trivial
3120.2.a.bi.1.3 3 4.3 odd 2
4680.2.a.bh.1.1 3 3.2 odd 2
7800.2.a.bi.1.3 3 5.4 even 2
9360.2.a.cy.1.3 3 12.11 even 2