# Properties

 Label 5520.2.a.cc.1.2 Level $5520$ Weight $2$ Character 5520.1 Self dual yes Analytic conductor $44.077$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, 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("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5520.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: $$44.0774219157$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.20087896.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 21x^{3} + 5x^{2} + 84x - 4$$ x^5 - x^4 - 21*x^3 + 5*x^2 + 84*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 2760) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-3.43588$$ of defining polynomial Character $$\chi$$ $$=$$ 5520.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} -2.62057 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} -2.62057 q^{7} +1.00000 q^{9} +6.55627 q^{11} +7.06828 q^{13} +1.00000 q^{15} +6.42405 q^{17} +1.80348 q^{19} -2.62057 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.17684 q^{29} -4.10856 q^{31} +6.55627 q^{33} -2.62057 q^{35} +4.62057 q^{37} +7.06828 q^{39} +4.30508 q^{41} -6.87176 q^{43} +1.00000 q^{45} -1.80348 q^{47} -0.132589 q^{49} +6.42405 q^{51} -4.93606 q^{53} +6.55627 q^{55} +1.80348 q^{57} +7.54623 q^{59} +2.70854 q^{61} -2.62057 q^{63} +7.06828 q^{65} -7.37337 q^{67} +1.00000 q^{69} -3.93569 q^{71} -5.04462 q^{73} +1.00000 q^{75} -17.1812 q^{77} -6.11897 q^{79} +1.00000 q^{81} +8.42405 q^{83} +6.42405 q^{85} -7.17684 q^{87} +11.3838 q^{89} -18.5230 q^{91} -4.10856 q^{93} +1.80348 q^{95} -17.9843 q^{97} +6.55627 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{3} + 5 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10})$$ 5 * q + 5 * q^3 + 5 * q^5 + 4 * q^7 + 5 * q^9 $$5 q + 5 q^{3} + 5 q^{5} + 4 q^{7} + 5 q^{9} + 4 q^{11} + 4 q^{13} + 5 q^{15} + 10 q^{17} + 4 q^{19} + 4 q^{21} + 5 q^{23} + 5 q^{25} + 5 q^{27} + 10 q^{29} - 6 q^{31} + 4 q^{33} + 4 q^{35} + 6 q^{37} + 4 q^{39} + 12 q^{41} + 2 q^{43} + 5 q^{45} - 4 q^{47} + 19 q^{49} + 10 q^{51} + 4 q^{55} + 4 q^{57} - 6 q^{59} + 16 q^{61} + 4 q^{63} + 4 q^{65} + 4 q^{67} + 5 q^{69} - 8 q^{71} + 14 q^{73} + 5 q^{75} + 16 q^{77} - 18 q^{79} + 5 q^{81} + 20 q^{83} + 10 q^{85} + 10 q^{87} + 18 q^{89} - 20 q^{91} - 6 q^{93} + 4 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100})$$ 5 * q + 5 * q^3 + 5 * q^5 + 4 * q^7 + 5 * q^9 + 4 * q^11 + 4 * q^13 + 5 * q^15 + 10 * q^17 + 4 * q^19 + 4 * q^21 + 5 * q^23 + 5 * q^25 + 5 * q^27 + 10 * q^29 - 6 * q^31 + 4 * q^33 + 4 * q^35 + 6 * q^37 + 4 * q^39 + 12 * q^41 + 2 * q^43 + 5 * q^45 - 4 * q^47 + 19 * q^49 + 10 * q^51 + 4 * q^55 + 4 * q^57 - 6 * q^59 + 16 * q^61 + 4 * q^63 + 4 * q^65 + 4 * q^67 + 5 * q^69 - 8 * q^71 + 14 * q^73 + 5 * q^75 + 16 * q^77 - 18 * q^79 + 5 * q^81 + 20 * q^83 + 10 * q^85 + 10 * q^87 + 18 * q^89 - 20 * q^91 - 6 * q^93 + 4 * q^95 + 4 * q^97 + 4 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.62057 −0.990484 −0.495242 0.868755i $$-0.664921\pi$$
−0.495242 + 0.868755i $$0.664921\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 6.55627 1.97679 0.988395 0.151907i $$-0.0485414\pi$$
0.988395 + 0.151907i $$0.0485414\pi$$
$$12$$ 0 0
$$13$$ 7.06828 1.96039 0.980195 0.198037i $$-0.0634565\pi$$
0.980195 + 0.198037i $$0.0634565\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 6.42405 1.55806 0.779030 0.626986i $$-0.215712\pi$$
0.779030 + 0.626986i $$0.215712\pi$$
$$18$$ 0 0
$$19$$ 1.80348 0.413746 0.206873 0.978368i $$-0.433671\pi$$
0.206873 + 0.978368i $$0.433671\pi$$
$$20$$ 0 0
$$21$$ −2.62057 −0.571856
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −7.17684 −1.33271 −0.666353 0.745636i $$-0.732146\pi$$
−0.666353 + 0.745636i $$0.732146\pi$$
$$30$$ 0 0
$$31$$ −4.10856 −0.737919 −0.368960 0.929445i $$-0.620286\pi$$
−0.368960 + 0.929445i $$0.620286\pi$$
$$32$$ 0 0
$$33$$ 6.55627 1.14130
$$34$$ 0 0
$$35$$ −2.62057 −0.442958
$$36$$ 0 0
$$37$$ 4.62057 0.759618 0.379809 0.925065i $$-0.375990\pi$$
0.379809 + 0.925065i $$0.375990\pi$$
$$38$$ 0 0
$$39$$ 7.06828 1.13183
$$40$$ 0 0
$$41$$ 4.30508 0.672341 0.336171 0.941801i $$-0.390868\pi$$
0.336171 + 0.941801i $$0.390868\pi$$
$$42$$ 0 0
$$43$$ −6.87176 −1.04793 −0.523967 0.851739i $$-0.675548\pi$$
−0.523967 + 0.851739i $$0.675548\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ −1.80348 −0.263064 −0.131532 0.991312i $$-0.541990\pi$$
−0.131532 + 0.991312i $$0.541990\pi$$
$$48$$ 0 0
$$49$$ −0.132589 −0.0189413
$$50$$ 0 0
$$51$$ 6.42405 0.899547
$$52$$ 0 0
$$53$$ −4.93606 −0.678021 −0.339010 0.940783i $$-0.610092\pi$$
−0.339010 + 0.940783i $$0.610092\pi$$
$$54$$ 0 0
$$55$$ 6.55627 0.884047
$$56$$ 0 0
$$57$$ 1.80348 0.238876
$$58$$ 0 0
$$59$$ 7.54623 0.982436 0.491218 0.871037i $$-0.336552\pi$$
0.491218 + 0.871037i $$0.336552\pi$$
$$60$$ 0 0
$$61$$ 2.70854 0.346793 0.173396 0.984852i $$-0.444526\pi$$
0.173396 + 0.984852i $$0.444526\pi$$
$$62$$ 0 0
$$63$$ −2.62057 −0.330161
$$64$$ 0 0
$$65$$ 7.06828 0.876713
$$66$$ 0 0
$$67$$ −7.37337 −0.900800 −0.450400 0.892827i $$-0.648719\pi$$
−0.450400 + 0.892827i $$0.648719\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −3.93569 −0.467081 −0.233541 0.972347i $$-0.575031\pi$$
−0.233541 + 0.972347i $$0.575031\pi$$
$$72$$ 0 0
$$73$$ −5.04462 −0.590429 −0.295214 0.955431i $$-0.595391\pi$$
−0.295214 + 0.955431i $$0.595391\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −17.1812 −1.95798
$$78$$ 0 0
$$79$$ −6.11897 −0.688437 −0.344219 0.938889i $$-0.611856\pi$$
−0.344219 + 0.938889i $$0.611856\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 8.42405 0.924660 0.462330 0.886708i $$-0.347014\pi$$
0.462330 + 0.886708i $$0.347014\pi$$
$$84$$ 0 0
$$85$$ 6.42405 0.696786
$$86$$ 0 0
$$87$$ −7.17684 −0.769438
$$88$$ 0 0
$$89$$ 11.3838 1.20668 0.603339 0.797485i $$-0.293837\pi$$
0.603339 + 0.797485i $$0.293837\pi$$
$$90$$ 0 0
$$91$$ −18.5230 −1.94173
$$92$$ 0 0
$$93$$ −4.10856 −0.426038
$$94$$ 0 0
$$95$$ 1.80348 0.185033
$$96$$ 0 0
$$97$$ −17.9843 −1.82603 −0.913014 0.407927i $$-0.866252\pi$$
−0.913014 + 0.407927i $$0.866252\pi$$
$$98$$ 0 0
$$99$$ 6.55627 0.658930
$$100$$ 0 0
$$101$$ 1.67126 0.166296 0.0831481 0.996537i $$-0.473503\pi$$
0.0831481 + 0.996537i $$0.473503\pi$$
$$102$$ 0 0
$$103$$ −6.72913 −0.663041 −0.331521 0.943448i $$-0.607562\pi$$
−0.331521 + 0.943448i $$0.607562\pi$$
$$104$$ 0 0
$$105$$ −2.62057 −0.255742
$$106$$ 0 0
$$107$$ −17.9296 −1.73332 −0.866662 0.498896i $$-0.833739\pi$$
−0.866662 + 0.498896i $$0.833739\pi$$
$$108$$ 0 0
$$109$$ −12.6692 −1.21349 −0.606744 0.794898i $$-0.707525\pi$$
−0.606744 + 0.794898i $$0.707525\pi$$
$$110$$ 0 0
$$111$$ 4.62057 0.438566
$$112$$ 0 0
$$113$$ 21.1708 1.99158 0.995790 0.0916634i $$-0.0292184\pi$$
0.995790 + 0.0916634i $$0.0292184\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 7.06828 0.653463
$$118$$ 0 0
$$119$$ −16.8347 −1.54323
$$120$$ 0 0
$$121$$ 31.9847 2.90770
$$122$$ 0 0
$$123$$ 4.30508 0.388176
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 4.04425 0.358870 0.179435 0.983770i $$-0.442573\pi$$
0.179435 + 0.983770i $$0.442573\pi$$
$$128$$ 0 0
$$129$$ −6.87176 −0.605025
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −4.72614 −0.409808
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 5.75316 0.491526 0.245763 0.969330i $$-0.420962\pi$$
0.245763 + 0.969330i $$0.420962\pi$$
$$138$$ 0 0
$$139$$ −4.50161 −0.381821 −0.190911 0.981607i $$-0.561144\pi$$
−0.190911 + 0.981607i $$0.561144\pi$$
$$140$$ 0 0
$$141$$ −1.80348 −0.151880
$$142$$ 0 0
$$143$$ 46.3416 3.87528
$$144$$ 0 0
$$145$$ −7.17684 −0.596004
$$146$$ 0 0
$$147$$ −0.132589 −0.0109358
$$148$$ 0 0
$$149$$ −14.1572 −1.15980 −0.579900 0.814688i $$-0.696908\pi$$
−0.579900 + 0.814688i $$0.696908\pi$$
$$150$$ 0 0
$$151$$ 18.9847 1.54495 0.772475 0.635045i $$-0.219018\pi$$
0.772475 + 0.635045i $$0.219018\pi$$
$$152$$ 0 0
$$153$$ 6.42405 0.519354
$$154$$ 0 0
$$155$$ −4.10856 −0.330007
$$156$$ 0 0
$$157$$ 8.22753 0.656628 0.328314 0.944569i $$-0.393520\pi$$
0.328314 + 0.944569i $$0.393520\pi$$
$$158$$ 0 0
$$159$$ −4.93606 −0.391455
$$160$$ 0 0
$$161$$ −2.62057 −0.206530
$$162$$ 0 0
$$163$$ −0.264439 −0.0207124 −0.0103562 0.999946i $$-0.503297\pi$$
−0.0103562 + 0.999946i $$0.503297\pi$$
$$164$$ 0 0
$$165$$ 6.55627 0.510405
$$166$$ 0 0
$$167$$ −9.45775 −0.731862 −0.365931 0.930642i $$-0.619249\pi$$
−0.365931 + 0.930642i $$0.619249\pi$$
$$168$$ 0 0
$$169$$ 36.9606 2.84313
$$170$$ 0 0
$$171$$ 1.80348 0.137915
$$172$$ 0 0
$$173$$ −0.758851 −0.0576944 −0.0288472 0.999584i $$-0.509184\pi$$
−0.0288472 + 0.999584i $$0.509184\pi$$
$$174$$ 0 0
$$175$$ −2.62057 −0.198097
$$176$$ 0 0
$$177$$ 7.54623 0.567210
$$178$$ 0 0
$$179$$ −21.9606 −1.64142 −0.820708 0.571348i $$-0.806420\pi$$
−0.820708 + 0.571348i $$0.806420\pi$$
$$180$$ 0 0
$$181$$ −14.4727 −1.07574 −0.537872 0.843027i $$-0.680772\pi$$
−0.537872 + 0.843027i $$0.680772\pi$$
$$182$$ 0 0
$$183$$ 2.70854 0.200221
$$184$$ 0 0
$$185$$ 4.62057 0.339711
$$186$$ 0 0
$$187$$ 42.1178 3.07996
$$188$$ 0 0
$$189$$ −2.62057 −0.190619
$$190$$ 0 0
$$191$$ 11.9400 0.863951 0.431976 0.901885i $$-0.357817\pi$$
0.431976 + 0.901885i $$0.357817\pi$$
$$192$$ 0 0
$$193$$ 11.6342 0.837448 0.418724 0.908114i $$-0.362477\pi$$
0.418724 + 0.908114i $$0.362477\pi$$
$$194$$ 0 0
$$195$$ 7.06828 0.506170
$$196$$ 0 0
$$197$$ 24.6181 1.75397 0.876984 0.480519i $$-0.159552\pi$$
0.876984 + 0.480519i $$0.159552\pi$$
$$198$$ 0 0
$$199$$ −18.7291 −1.32767 −0.663837 0.747878i $$-0.731073\pi$$
−0.663837 + 0.747878i $$0.731073\pi$$
$$200$$ 0 0
$$201$$ −7.37337 −0.520077
$$202$$ 0 0
$$203$$ 18.8075 1.32002
$$204$$ 0 0
$$205$$ 4.30508 0.300680
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ 11.8241 0.817888
$$210$$ 0 0
$$211$$ −3.49839 −0.240839 −0.120420 0.992723i $$-0.538424\pi$$
−0.120420 + 0.992723i $$0.538424\pi$$
$$212$$ 0 0
$$213$$ −3.93569 −0.269669
$$214$$ 0 0
$$215$$ −6.87176 −0.468650
$$216$$ 0 0
$$217$$ 10.7668 0.730897
$$218$$ 0 0
$$219$$ −5.04462 −0.340884
$$220$$ 0 0
$$221$$ 45.4070 3.05441
$$222$$ 0 0
$$223$$ 13.8477 0.927313 0.463656 0.886015i $$-0.346537\pi$$
0.463656 + 0.886015i $$0.346537\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −19.5772 −1.29939 −0.649693 0.760196i $$-0.725103\pi$$
−0.649693 + 0.760196i $$0.725103\pi$$
$$228$$ 0 0
$$229$$ 25.4959 1.68482 0.842410 0.538838i $$-0.181136\pi$$
0.842410 + 0.538838i $$0.181136\pi$$
$$230$$ 0 0
$$231$$ −17.1812 −1.13044
$$232$$ 0 0
$$233$$ 16.9847 1.11270 0.556351 0.830947i $$-0.312201\pi$$
0.556351 + 0.830947i $$0.312201\pi$$
$$234$$ 0 0
$$235$$ −1.80348 −0.117646
$$236$$ 0 0
$$237$$ −6.11897 −0.397470
$$238$$ 0 0
$$239$$ −16.5459 −1.07026 −0.535131 0.844769i $$-0.679738\pi$$
−0.535131 + 0.844769i $$0.679738\pi$$
$$240$$ 0 0
$$241$$ −10.1572 −0.654280 −0.327140 0.944976i $$-0.606085\pi$$
−0.327140 + 0.944976i $$0.606085\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −0.132589 −0.00847081
$$246$$ 0 0
$$247$$ 12.7475 0.811103
$$248$$ 0 0
$$249$$ 8.42405 0.533852
$$250$$ 0 0
$$251$$ −5.72950 −0.361643 −0.180822 0.983516i $$-0.557876\pi$$
−0.180822 + 0.983516i $$0.557876\pi$$
$$252$$ 0 0
$$253$$ 6.55627 0.412189
$$254$$ 0 0
$$255$$ 6.42405 0.402290
$$256$$ 0 0
$$257$$ −19.5269 −1.21806 −0.609028 0.793149i $$-0.708440\pi$$
−0.609028 + 0.793149i $$0.708440\pi$$
$$258$$ 0 0
$$259$$ −12.1086 −0.752389
$$260$$ 0 0
$$261$$ −7.17684 −0.444235
$$262$$ 0 0
$$263$$ 29.0726 1.79270 0.896348 0.443352i $$-0.146211\pi$$
0.896348 + 0.443352i $$0.146211\pi$$
$$264$$ 0 0
$$265$$ −4.93606 −0.303220
$$266$$ 0 0
$$267$$ 11.3838 0.696676
$$268$$ 0 0
$$269$$ 11.8078 0.719936 0.359968 0.932965i $$-0.382788\pi$$
0.359968 + 0.932965i $$0.382788\pi$$
$$270$$ 0 0
$$271$$ −9.13259 −0.554765 −0.277383 0.960760i $$-0.589467\pi$$
−0.277383 + 0.960760i $$0.589467\pi$$
$$272$$ 0 0
$$273$$ −18.5230 −1.12106
$$274$$ 0 0
$$275$$ 6.55627 0.395358
$$276$$ 0 0
$$277$$ 27.8564 1.67373 0.836865 0.547409i $$-0.184386\pi$$
0.836865 + 0.547409i $$0.184386\pi$$
$$278$$ 0 0
$$279$$ −4.10856 −0.245973
$$280$$ 0 0
$$281$$ −11.9461 −0.712645 −0.356322 0.934363i $$-0.615969\pi$$
−0.356322 + 0.934363i $$0.615969\pi$$
$$282$$ 0 0
$$283$$ 17.0076 1.01099 0.505497 0.862828i $$-0.331309\pi$$
0.505497 + 0.862828i $$0.331309\pi$$
$$284$$ 0 0
$$285$$ 1.80348 0.106829
$$286$$ 0 0
$$287$$ −11.2818 −0.665943
$$288$$ 0 0
$$289$$ 24.2684 1.42755
$$290$$ 0 0
$$291$$ −17.9843 −1.05426
$$292$$ 0 0
$$293$$ −23.9472 −1.39901 −0.699506 0.714626i $$-0.746597\pi$$
−0.699506 + 0.714626i $$0.746597\pi$$
$$294$$ 0 0
$$295$$ 7.54623 0.439359
$$296$$ 0 0
$$297$$ 6.55627 0.380433
$$298$$ 0 0
$$299$$ 7.06828 0.408769
$$300$$ 0 0
$$301$$ 18.0080 1.03796
$$302$$ 0 0
$$303$$ 1.67126 0.0960111
$$304$$ 0 0
$$305$$ 2.70854 0.155091
$$306$$ 0 0
$$307$$ −15.9400 −0.909746 −0.454873 0.890556i $$-0.650315\pi$$
−0.454873 + 0.890556i $$0.650315\pi$$
$$308$$ 0 0
$$309$$ −6.72913 −0.382807
$$310$$ 0 0
$$311$$ −8.16023 −0.462724 −0.231362 0.972868i $$-0.574318\pi$$
−0.231362 + 0.972868i $$0.574318\pi$$
$$312$$ 0 0
$$313$$ 2.84375 0.160738 0.0803692 0.996765i $$-0.474390\pi$$
0.0803692 + 0.996765i $$0.474390\pi$$
$$314$$ 0 0
$$315$$ −2.62057 −0.147653
$$316$$ 0 0
$$317$$ −23.5269 −1.32140 −0.660702 0.750648i $$-0.729741\pi$$
−0.660702 + 0.750648i $$0.729741\pi$$
$$318$$ 0 0
$$319$$ −47.0533 −2.63448
$$320$$ 0 0
$$321$$ −17.9296 −1.00073
$$322$$ 0 0
$$323$$ 11.5856 0.644641
$$324$$ 0 0
$$325$$ 7.06828 0.392078
$$326$$ 0 0
$$327$$ −12.6692 −0.700607
$$328$$ 0 0
$$329$$ 4.72614 0.260561
$$330$$ 0 0
$$331$$ 11.9326 0.655877 0.327938 0.944699i $$-0.393646\pi$$
0.327938 + 0.944699i $$0.393646\pi$$
$$332$$ 0 0
$$333$$ 4.62057 0.253206
$$334$$ 0 0
$$335$$ −7.37337 −0.402850
$$336$$ 0 0
$$337$$ 12.6073 0.686765 0.343382 0.939196i $$-0.388427\pi$$
0.343382 + 0.939196i $$0.388427\pi$$
$$338$$ 0 0
$$339$$ 21.1708 1.14984
$$340$$ 0 0
$$341$$ −26.9368 −1.45871
$$342$$ 0 0
$$343$$ 18.6915 1.00925
$$344$$ 0 0
$$345$$ 1.00000 0.0538382
$$346$$ 0 0
$$347$$ 5.85132 0.314115 0.157058 0.987589i $$-0.449799\pi$$
0.157058 + 0.987589i $$0.449799\pi$$
$$348$$ 0 0
$$349$$ −29.7034 −1.58999 −0.794993 0.606618i $$-0.792526\pi$$
−0.794993 + 0.606618i $$0.792526\pi$$
$$350$$ 0 0
$$351$$ 7.06828 0.377277
$$352$$ 0 0
$$353$$ −10.1160 −0.538419 −0.269209 0.963082i $$-0.586762\pi$$
−0.269209 + 0.963082i $$0.586762\pi$$
$$354$$ 0 0
$$355$$ −3.93569 −0.208885
$$356$$ 0 0
$$357$$ −16.8347 −0.890987
$$358$$ 0 0
$$359$$ −1.45775 −0.0769369 −0.0384684 0.999260i $$-0.512248\pi$$
−0.0384684 + 0.999260i $$0.512248\pi$$
$$360$$ 0 0
$$361$$ −15.7475 −0.828815
$$362$$ 0 0
$$363$$ 31.9847 1.67876
$$364$$ 0 0
$$365$$ −5.04462 −0.264048
$$366$$ 0 0
$$367$$ −1.98959 −0.103856 −0.0519280 0.998651i $$-0.516537\pi$$
−0.0519280 + 0.998651i $$0.516537\pi$$
$$368$$ 0 0
$$369$$ 4.30508 0.224114
$$370$$ 0 0
$$371$$ 12.9353 0.671569
$$372$$ 0 0
$$373$$ 5.12218 0.265217 0.132608 0.991169i $$-0.457665\pi$$
0.132608 + 0.991169i $$0.457665\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ −50.7280 −2.61262
$$378$$ 0 0
$$379$$ 9.79084 0.502922 0.251461 0.967867i $$-0.419089\pi$$
0.251461 + 0.967867i $$0.419089\pi$$
$$380$$ 0 0
$$381$$ 4.04425 0.207193
$$382$$ 0 0
$$383$$ −15.4176 −0.787804 −0.393902 0.919152i $$-0.628875\pi$$
−0.393902 + 0.919152i $$0.628875\pi$$
$$384$$ 0 0
$$385$$ −17.1812 −0.875635
$$386$$ 0 0
$$387$$ −6.87176 −0.349311
$$388$$ 0 0
$$389$$ −38.3616 −1.94501 −0.972506 0.232877i $$-0.925186\pi$$
−0.972506 + 0.232877i $$0.925186\pi$$
$$390$$ 0 0
$$391$$ 6.42405 0.324878
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −6.11897 −0.307879
$$396$$ 0 0
$$397$$ 2.87102 0.144092 0.0720462 0.997401i $$-0.477047\pi$$
0.0720462 + 0.997401i $$0.477047\pi$$
$$398$$ 0 0
$$399$$ −4.72614 −0.236603
$$400$$ 0 0
$$401$$ −29.8453 −1.49040 −0.745201 0.666840i $$-0.767646\pi$$
−0.745201 + 0.666840i $$0.767646\pi$$
$$402$$ 0 0
$$403$$ −29.0405 −1.44661
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 30.2937 1.50160
$$408$$ 0 0
$$409$$ −6.21788 −0.307454 −0.153727 0.988113i $$-0.549128\pi$$
−0.153727 + 0.988113i $$0.549128\pi$$
$$410$$ 0 0
$$411$$ 5.75316 0.283783
$$412$$ 0 0
$$413$$ −19.7755 −0.973087
$$414$$ 0 0
$$415$$ 8.42405 0.413520
$$416$$ 0 0
$$417$$ −4.50161 −0.220445
$$418$$ 0 0
$$419$$ 33.6695 1.64487 0.822433 0.568863i $$-0.192616\pi$$
0.822433 + 0.568863i $$0.192616\pi$$
$$420$$ 0 0
$$421$$ 17.0149 0.829256 0.414628 0.909991i $$-0.363912\pi$$
0.414628 + 0.909991i $$0.363912\pi$$
$$422$$ 0 0
$$423$$ −1.80348 −0.0876880
$$424$$ 0 0
$$425$$ 6.42405 0.311612
$$426$$ 0 0
$$427$$ −7.09793 −0.343493
$$428$$ 0 0
$$429$$ 46.3416 2.23739
$$430$$ 0 0
$$431$$ −19.6422 −0.946129 −0.473065 0.881028i $$-0.656852\pi$$
−0.473065 + 0.881028i $$0.656852\pi$$
$$432$$ 0 0
$$433$$ −5.24476 −0.252047 −0.126023 0.992027i $$-0.540221\pi$$
−0.126023 + 0.992027i $$0.540221\pi$$
$$434$$ 0 0
$$435$$ −7.17684 −0.344103
$$436$$ 0 0
$$437$$ 1.80348 0.0862719
$$438$$ 0 0
$$439$$ −17.9194 −0.855249 −0.427624 0.903957i $$-0.640649\pi$$
−0.427624 + 0.903957i $$0.640649\pi$$
$$440$$ 0 0
$$441$$ −0.132589 −0.00631377
$$442$$ 0 0
$$443$$ 9.75616 0.463529 0.231764 0.972772i $$-0.425550\pi$$
0.231764 + 0.972772i $$0.425550\pi$$
$$444$$ 0 0
$$445$$ 11.3838 0.539643
$$446$$ 0 0
$$447$$ −14.1572 −0.669611
$$448$$ 0 0
$$449$$ −34.5703 −1.63147 −0.815736 0.578425i $$-0.803668\pi$$
−0.815736 + 0.578425i $$0.803668\pi$$
$$450$$ 0 0
$$451$$ 28.2253 1.32908
$$452$$ 0 0
$$453$$ 18.9847 0.891978
$$454$$ 0 0
$$455$$ −18.5230 −0.868370
$$456$$ 0 0
$$457$$ 30.0455 1.40547 0.702736 0.711451i $$-0.251962\pi$$
0.702736 + 0.711451i $$0.251962\pi$$
$$458$$ 0 0
$$459$$ 6.42405 0.299849
$$460$$ 0 0
$$461$$ −15.2849 −0.711888 −0.355944 0.934507i $$-0.615841\pi$$
−0.355944 + 0.934507i $$0.615841\pi$$
$$462$$ 0 0
$$463$$ −4.89235 −0.227367 −0.113684 0.993517i $$-0.536265\pi$$
−0.113684 + 0.993517i $$0.536265\pi$$
$$464$$ 0 0
$$465$$ −4.10856 −0.190530
$$466$$ 0 0
$$467$$ 35.4972 1.64262 0.821308 0.570485i $$-0.193245\pi$$
0.821308 + 0.570485i $$0.193245\pi$$
$$468$$ 0 0
$$469$$ 19.3225 0.892228
$$470$$ 0 0
$$471$$ 8.22753 0.379104
$$472$$ 0 0
$$473$$ −45.0531 −2.07154
$$474$$ 0 0
$$475$$ 1.80348 0.0827491
$$476$$ 0 0
$$477$$ −4.93606 −0.226007
$$478$$ 0 0
$$479$$ −7.80421 −0.356584 −0.178292 0.983978i $$-0.557057\pi$$
−0.178292 + 0.983978i $$0.557057\pi$$
$$480$$ 0 0
$$481$$ 32.6595 1.48915
$$482$$ 0 0
$$483$$ −2.62057 −0.119240
$$484$$ 0 0
$$485$$ −17.9843 −0.816625
$$486$$ 0 0
$$487$$ 21.5498 0.976517 0.488258 0.872699i $$-0.337632\pi$$
0.488258 + 0.872699i $$0.337632\pi$$
$$488$$ 0 0
$$489$$ −0.264439 −0.0119583
$$490$$ 0 0
$$491$$ 15.8315 0.714465 0.357232 0.934016i $$-0.383720\pi$$
0.357232 + 0.934016i $$0.383720\pi$$
$$492$$ 0 0
$$493$$ −46.1044 −2.07644
$$494$$ 0 0
$$495$$ 6.55627 0.294682
$$496$$ 0 0
$$497$$ 10.3138 0.462636
$$498$$ 0 0
$$499$$ 5.76357 0.258013 0.129006 0.991644i $$-0.458821\pi$$
0.129006 + 0.991644i $$0.458821\pi$$
$$500$$ 0 0
$$501$$ −9.45775 −0.422541
$$502$$ 0 0
$$503$$ 18.8075 0.838583 0.419291 0.907852i $$-0.362279\pi$$
0.419291 + 0.907852i $$0.362279\pi$$
$$504$$ 0 0
$$505$$ 1.67126 0.0743699
$$506$$ 0 0
$$507$$ 36.9606 1.64148
$$508$$ 0 0
$$509$$ 3.21675 0.142580 0.0712900 0.997456i $$-0.477288\pi$$
0.0712900 + 0.997456i $$0.477288\pi$$
$$510$$ 0 0
$$511$$ 13.2198 0.584810
$$512$$ 0 0
$$513$$ 1.80348 0.0796254
$$514$$ 0 0
$$515$$ −6.72913 −0.296521
$$516$$ 0 0
$$517$$ −11.8241 −0.520022
$$518$$ 0 0
$$519$$ −0.758851 −0.0333099
$$520$$ 0 0
$$521$$ 14.9100 0.653217 0.326609 0.945160i $$-0.394094\pi$$
0.326609 + 0.945160i $$0.394094\pi$$
$$522$$ 0 0
$$523$$ −12.3982 −0.542134 −0.271067 0.962561i $$-0.587376\pi$$
−0.271067 + 0.962561i $$0.587376\pi$$
$$524$$ 0 0
$$525$$ −2.62057 −0.114371
$$526$$ 0 0
$$527$$ −26.3936 −1.14972
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 7.54623 0.327479
$$532$$ 0 0
$$533$$ 30.4296 1.31805
$$534$$ 0 0
$$535$$ −17.9296 −0.775166
$$536$$ 0 0
$$537$$ −21.9606 −0.947672
$$538$$ 0 0
$$539$$ −0.869290 −0.0374430
$$540$$ 0 0
$$541$$ −30.8954 −1.32830 −0.664149 0.747600i $$-0.731206\pi$$
−0.664149 + 0.747600i $$0.731206\pi$$
$$542$$ 0 0
$$543$$ −14.4727 −0.621081
$$544$$ 0 0
$$545$$ −12.6692 −0.542688
$$546$$ 0 0
$$547$$ −14.2251 −0.608220 −0.304110 0.952637i $$-0.598359\pi$$
−0.304110 + 0.952637i $$0.598359\pi$$
$$548$$ 0 0
$$549$$ 2.70854 0.115598
$$550$$ 0 0
$$551$$ −12.9433 −0.551401
$$552$$ 0 0
$$553$$ 16.0352 0.681886
$$554$$ 0 0
$$555$$ 4.62057 0.196132
$$556$$ 0 0
$$557$$ −14.6989 −0.622811 −0.311406 0.950277i $$-0.600800\pi$$
−0.311406 + 0.950277i $$0.600800\pi$$
$$558$$ 0 0
$$559$$ −48.5715 −2.05436
$$560$$ 0 0
$$561$$ 42.1178 1.77821
$$562$$ 0 0
$$563$$ −19.4972 −0.821710 −0.410855 0.911701i $$-0.634770\pi$$
−0.410855 + 0.911701i $$0.634770\pi$$
$$564$$ 0 0
$$565$$ 21.1708 0.890662
$$566$$ 0 0
$$567$$ −2.62057 −0.110054
$$568$$ 0 0
$$569$$ −17.8854 −0.749795 −0.374898 0.927066i $$-0.622322\pi$$
−0.374898 + 0.927066i $$0.622322\pi$$
$$570$$ 0 0
$$571$$ 23.3904 0.978856 0.489428 0.872044i $$-0.337206\pi$$
0.489428 + 0.872044i $$0.337206\pi$$
$$572$$ 0 0
$$573$$ 11.9400 0.498802
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ −45.7515 −1.90466 −0.952329 0.305072i $$-0.901320\pi$$
−0.952329 + 0.305072i $$0.901320\pi$$
$$578$$ 0 0
$$579$$ 11.6342 0.483501
$$580$$ 0 0
$$581$$ −22.0759 −0.915861
$$582$$ 0 0
$$583$$ −32.3622 −1.34030
$$584$$ 0 0
$$585$$ 7.06828 0.292238
$$586$$ 0 0
$$587$$ −24.9839 −1.03120 −0.515599 0.856830i $$-0.672430\pi$$
−0.515599 + 0.856830i $$0.672430\pi$$
$$588$$ 0 0
$$589$$ −7.40969 −0.305311
$$590$$ 0 0
$$591$$ 24.6181 1.01265
$$592$$ 0 0
$$593$$ 25.3178 1.03968 0.519838 0.854265i $$-0.325992\pi$$
0.519838 + 0.854265i $$0.325992\pi$$
$$594$$ 0 0
$$595$$ −16.8347 −0.690155
$$596$$ 0 0
$$597$$ −18.7291 −0.766532
$$598$$ 0 0
$$599$$ −5.23756 −0.214001 −0.107000 0.994259i $$-0.534125\pi$$
−0.107000 + 0.994259i $$0.534125\pi$$
$$600$$ 0 0
$$601$$ −26.1173 −1.06535 −0.532673 0.846321i $$-0.678812\pi$$
−0.532673 + 0.846321i $$0.678812\pi$$
$$602$$ 0 0
$$603$$ −7.37337 −0.300267
$$604$$ 0 0
$$605$$ 31.9847 1.30036
$$606$$ 0 0
$$607$$ 24.6631 1.00105 0.500523 0.865723i $$-0.333141\pi$$
0.500523 + 0.865723i $$0.333141\pi$$
$$608$$ 0 0
$$609$$ 18.8075 0.762117
$$610$$ 0 0
$$611$$ −12.7475 −0.515708
$$612$$ 0 0
$$613$$ −4.60052 −0.185813 −0.0929067 0.995675i $$-0.529616\pi$$
−0.0929067 + 0.995675i $$0.529616\pi$$
$$614$$ 0 0
$$615$$ 4.30508 0.173598
$$616$$ 0 0
$$617$$ 18.9143 0.761461 0.380731 0.924686i $$-0.375673\pi$$
0.380731 + 0.924686i $$0.375673\pi$$
$$618$$ 0 0
$$619$$ 47.9686 1.92802 0.964010 0.265865i $$-0.0856575\pi$$
0.964010 + 0.265865i $$0.0856575\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ −29.8320 −1.19519
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 11.8241 0.472208
$$628$$ 0 0
$$629$$ 29.6828 1.18353
$$630$$ 0 0
$$631$$ 29.5445 1.17615 0.588074 0.808807i $$-0.299886\pi$$
0.588074 + 0.808807i $$0.299886\pi$$
$$632$$ 0 0
$$633$$ −3.49839 −0.139049
$$634$$ 0 0
$$635$$ 4.04425 0.160491
$$636$$ 0 0
$$637$$ −0.937178 −0.0371323
$$638$$ 0 0
$$639$$ −3.93569 −0.155694
$$640$$ 0 0
$$641$$ −43.2557 −1.70850 −0.854248 0.519865i $$-0.825982\pi$$
−0.854248 + 0.519865i $$0.825982\pi$$
$$642$$ 0 0
$$643$$ 37.2455 1.46882 0.734410 0.678707i $$-0.237459\pi$$
0.734410 + 0.678707i $$0.237459\pi$$
$$644$$ 0 0
$$645$$ −6.87176 −0.270575
$$646$$ 0 0
$$647$$ −41.9007 −1.64729 −0.823643 0.567109i $$-0.808062\pi$$
−0.823643 + 0.567109i $$0.808062\pi$$
$$648$$ 0 0
$$649$$ 49.4751 1.94207
$$650$$ 0 0
$$651$$ 10.7668 0.421984
$$652$$ 0 0
$$653$$ 36.4697 1.42717 0.713584 0.700570i $$-0.247071\pi$$
0.713584 + 0.700570i $$0.247071\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −5.04462 −0.196810
$$658$$ 0 0
$$659$$ 17.8247 0.694350 0.347175 0.937800i $$-0.387141\pi$$
0.347175 + 0.937800i $$0.387141\pi$$
$$660$$ 0 0
$$661$$ −2.70189 −0.105091 −0.0525456 0.998619i $$-0.516733\pi$$
−0.0525456 + 0.998619i $$0.516733\pi$$
$$662$$ 0 0
$$663$$ 45.4070 1.76346
$$664$$ 0 0
$$665$$ −4.72614 −0.183272
$$666$$ 0 0
$$667$$ −7.17684 −0.277889
$$668$$ 0 0
$$669$$ 13.8477 0.535384
$$670$$ 0 0
$$671$$ 17.7579 0.685537
$$672$$ 0 0
$$673$$ 8.19652 0.315953 0.157976 0.987443i $$-0.449503\pi$$
0.157976 + 0.987443i $$0.449503\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 26.8759 1.03292 0.516462 0.856310i $$-0.327249\pi$$
0.516462 + 0.856310i $$0.327249\pi$$
$$678$$ 0 0
$$679$$ 47.1292 1.80865
$$680$$ 0 0
$$681$$ −19.5772 −0.750201
$$682$$ 0 0
$$683$$ 17.1252 0.655277 0.327638 0.944803i $$-0.393747\pi$$
0.327638 + 0.944803i $$0.393747\pi$$
$$684$$ 0 0
$$685$$ 5.75316 0.219817
$$686$$ 0 0
$$687$$ 25.4959 0.972731
$$688$$ 0 0
$$689$$ −34.8895 −1.32918
$$690$$ 0 0
$$691$$ −15.7042 −0.597414 −0.298707 0.954345i $$-0.596555\pi$$
−0.298707 + 0.954345i $$0.596555\pi$$
$$692$$ 0 0
$$693$$ −17.1812 −0.652660
$$694$$ 0 0
$$695$$ −4.50161 −0.170756
$$696$$ 0 0
$$697$$ 27.6561 1.04755
$$698$$ 0 0
$$699$$ 16.9847 0.642419
$$700$$ 0 0
$$701$$ −6.81955 −0.257571 −0.128785 0.991672i $$-0.541108\pi$$
−0.128785 + 0.991672i $$0.541108\pi$$
$$702$$ 0 0
$$703$$ 8.33309 0.314289
$$704$$ 0 0
$$705$$ −1.80348 −0.0679228
$$706$$ 0 0
$$707$$ −4.37965 −0.164714
$$708$$ 0 0
$$709$$ 23.0955 0.867368 0.433684 0.901065i $$-0.357213\pi$$
0.433684 + 0.901065i $$0.357213\pi$$
$$710$$ 0 0
$$711$$ −6.11897 −0.229479
$$712$$ 0 0
$$713$$ −4.10856 −0.153867
$$714$$ 0 0
$$715$$ 46.3416 1.73308
$$716$$ 0 0
$$717$$ −16.5459 −0.617917
$$718$$ 0 0
$$719$$ 22.5152 0.839675 0.419838 0.907599i $$-0.362087\pi$$
0.419838 + 0.907599i $$0.362087\pi$$
$$720$$ 0 0
$$721$$ 17.6342 0.656732
$$722$$ 0 0
$$723$$ −10.1572 −0.377749
$$724$$ 0 0
$$725$$ −7.17684 −0.266541
$$726$$ 0 0
$$727$$ 3.45998 0.128323 0.0641617 0.997940i $$-0.479563\pi$$
0.0641617 + 0.997940i $$0.479563\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −44.1445 −1.63274
$$732$$ 0 0
$$733$$ −16.2835 −0.601446 −0.300723 0.953711i $$-0.597228\pi$$
−0.300723 + 0.953711i $$0.597228\pi$$
$$734$$ 0 0
$$735$$ −0.132589 −0.00489062
$$736$$ 0 0
$$737$$ −48.3418 −1.78069
$$738$$ 0 0
$$739$$ −24.2459 −0.891899 −0.445949 0.895058i $$-0.647134\pi$$
−0.445949 + 0.895058i $$0.647134\pi$$
$$740$$ 0 0
$$741$$ 12.7475 0.468290
$$742$$ 0 0
$$743$$ −30.7202 −1.12702 −0.563508 0.826111i $$-0.690549\pi$$
−0.563508 + 0.826111i $$0.690549\pi$$
$$744$$ 0 0
$$745$$ −14.1572 −0.518678
$$746$$ 0 0
$$747$$ 8.42405 0.308220
$$748$$ 0 0
$$749$$ 46.9860 1.71683
$$750$$ 0 0
$$751$$ 37.3493 1.36290 0.681448 0.731867i $$-0.261351\pi$$
0.681448 + 0.731867i $$0.261351\pi$$
$$752$$ 0 0
$$753$$ −5.72950 −0.208795
$$754$$ 0 0
$$755$$ 18.9847 0.690923
$$756$$ 0 0
$$757$$ 18.8930 0.686677 0.343338 0.939212i $$-0.388442\pi$$
0.343338 + 0.939212i $$0.388442\pi$$
$$758$$ 0 0
$$759$$ 6.55627 0.237978
$$760$$ 0 0
$$761$$ −37.5549 −1.36137 −0.680683 0.732578i $$-0.738317\pi$$
−0.680683 + 0.732578i $$0.738317\pi$$
$$762$$ 0 0
$$763$$ 33.2005 1.20194
$$764$$ 0 0
$$765$$ 6.42405 0.232262
$$766$$ 0 0
$$767$$ 53.3389 1.92596
$$768$$ 0 0
$$769$$ −35.8799 −1.29386 −0.646931 0.762549i $$-0.723948\pi$$
−0.646931 + 0.762549i $$0.723948\pi$$
$$770$$ 0 0
$$771$$ −19.5269 −0.703245
$$772$$ 0 0
$$773$$ −32.3745 −1.16443 −0.582215 0.813035i $$-0.697814\pi$$
−0.582215 + 0.813035i $$0.697814\pi$$
$$774$$ 0 0
$$775$$ −4.10856 −0.147584
$$776$$ 0 0
$$777$$ −12.1086 −0.434392
$$778$$ 0 0
$$779$$ 7.76411 0.278178
$$780$$ 0 0
$$781$$ −25.8035 −0.923321
$$782$$ 0 0
$$783$$ −7.17684 −0.256479
$$784$$ 0 0
$$785$$ 8.22753 0.293653
$$786$$ 0 0
$$787$$ 47.7278 1.70131 0.850656 0.525723i $$-0.176205\pi$$
0.850656 + 0.525723i $$0.176205\pi$$
$$788$$ 0 0
$$789$$ 29.0726 1.03501
$$790$$ 0 0
$$791$$ −55.4796 −1.97263
$$792$$ 0 0
$$793$$ 19.1447 0.679849
$$794$$ 0 0
$$795$$ −4.93606 −0.175064
$$796$$ 0 0
$$797$$ −12.8427 −0.454910 −0.227455 0.973789i $$-0.573041\pi$$
−0.227455 + 0.973789i $$0.573041\pi$$
$$798$$ 0 0
$$799$$ −11.5856 −0.409870
$$800$$ 0 0
$$801$$ 11.3838 0.402226
$$802$$ 0 0
$$803$$ −33.0739 −1.16715
$$804$$ 0 0
$$805$$ −2.62057 −0.0923631
$$806$$ 0 0
$$807$$ 11.8078 0.415655
$$808$$ 0 0
$$809$$ 8.84864 0.311102 0.155551 0.987828i $$-0.450285\pi$$
0.155551 + 0.987828i $$0.450285\pi$$
$$810$$ 0 0
$$811$$ −30.4149 −1.06801 −0.534006 0.845480i $$-0.679314\pi$$
−0.534006 + 0.845480i $$0.679314\pi$$
$$812$$ 0 0
$$813$$ −9.13259 −0.320294
$$814$$ 0 0
$$815$$ −0.264439 −0.00926289
$$816$$ 0 0
$$817$$ −12.3930 −0.433578
$$818$$ 0 0
$$819$$ −18.5230 −0.647245
$$820$$ 0 0
$$821$$ 31.2463 1.09050 0.545251 0.838273i $$-0.316434\pi$$
0.545251 + 0.838273i $$0.316434\pi$$
$$822$$ 0 0
$$823$$ 50.6232 1.76462 0.882308 0.470673i $$-0.155989\pi$$
0.882308 + 0.470673i $$0.155989\pi$$
$$824$$ 0 0
$$825$$ 6.55627 0.228260
$$826$$ 0 0
$$827$$ 32.5478 1.13180 0.565898 0.824475i $$-0.308529\pi$$
0.565898 + 0.824475i $$0.308529\pi$$
$$828$$ 0 0
$$829$$ −29.0119 −1.00763 −0.503813 0.863813i $$-0.668070\pi$$
−0.503813 + 0.863813i $$0.668070\pi$$
$$830$$ 0 0
$$831$$ 27.8564 0.966329
$$832$$ 0 0
$$833$$ −0.851759 −0.0295117
$$834$$ 0 0
$$835$$ −9.45775 −0.327299
$$836$$ 0 0
$$837$$ −4.10856 −0.142013
$$838$$ 0 0
$$839$$ 1.97993 0.0683547 0.0341774 0.999416i $$-0.489119\pi$$
0.0341774 + 0.999416i $$0.489119\pi$$
$$840$$ 0 0
$$841$$ 22.5071 0.776106
$$842$$ 0 0
$$843$$ −11.9461 −0.411446
$$844$$ 0 0
$$845$$ 36.9606 1.27148
$$846$$ 0 0
$$847$$ −83.8182 −2.88003
$$848$$ 0 0
$$849$$ 17.0076 0.583698
$$850$$ 0 0
$$851$$ 4.62057 0.158391
$$852$$ 0 0
$$853$$ −11.1781 −0.382732 −0.191366 0.981519i $$-0.561292\pi$$
−0.191366 + 0.981519i $$0.561292\pi$$
$$854$$ 0 0
$$855$$ 1.80348 0.0616776
$$856$$ 0 0
$$857$$ 29.2491 0.999130 0.499565 0.866276i $$-0.333493\pi$$
0.499565 + 0.866276i $$0.333493\pi$$
$$858$$ 0 0
$$859$$ −47.5143 −1.62117 −0.810584 0.585623i $$-0.800850\pi$$
−0.810584 + 0.585623i $$0.800850\pi$$
$$860$$ 0 0
$$861$$ −11.2818 −0.384483
$$862$$ 0 0
$$863$$ 16.5628 0.563806 0.281903 0.959443i $$-0.409034\pi$$
0.281903 + 0.959443i $$0.409034\pi$$
$$864$$ 0 0
$$865$$ −0.758851 −0.0258017
$$866$$ 0 0
$$867$$ 24.2684 0.824199
$$868$$ 0 0
$$869$$ −40.1176 −1.36090
$$870$$ 0 0
$$871$$ −52.1171 −1.76592
$$872$$ 0 0
$$873$$ −17.9843 −0.608676
$$874$$ 0 0
$$875$$ −2.62057 −0.0885916
$$876$$ 0 0
$$877$$ −4.80976 −0.162414 −0.0812070 0.996697i $$-0.525877\pi$$
−0.0812070 + 0.996697i $$0.525877\pi$$
$$878$$ 0 0
$$879$$ −23.9472 −0.807720
$$880$$ 0 0
$$881$$ 29.8046 1.00414 0.502072 0.864826i $$-0.332571\pi$$
0.502072 + 0.864826i $$0.332571\pi$$
$$882$$ 0 0
$$883$$ 22.1579 0.745673 0.372836 0.927897i $$-0.378385\pi$$
0.372836 + 0.927897i $$0.378385\pi$$
$$884$$ 0 0
$$885$$ 7.54623 0.253664
$$886$$ 0 0
$$887$$ −31.3425 −1.05238 −0.526189 0.850367i $$-0.676380\pi$$
−0.526189 + 0.850367i $$0.676380\pi$$
$$888$$ 0 0
$$889$$ −10.5983 −0.355455
$$890$$ 0 0
$$891$$ 6.55627 0.219643
$$892$$ 0 0
$$893$$ −3.25252 −0.108842
$$894$$ 0 0
$$895$$ −21.9606 −0.734063
$$896$$ 0 0
$$897$$ 7.06828 0.236003
$$898$$ 0 0
$$899$$ 29.4865 0.983430
$$900$$ 0 0
$$901$$ −31.7095 −1.05640
$$902$$ 0 0
$$903$$ 18.0080 0.599267
$$904$$ 0 0
$$905$$ −14.4727 −0.481087
$$906$$ 0 0
$$907$$ −11.3541 −0.377006 −0.188503 0.982073i $$-0.560364\pi$$
−0.188503 + 0.982073i $$0.560364\pi$$
$$908$$ 0 0
$$909$$ 1.67126 0.0554321
$$910$$ 0 0
$$911$$ −5.23245 −0.173359 −0.0866794 0.996236i $$-0.527626\pi$$
−0.0866794 + 0.996236i $$0.527626\pi$$
$$912$$ 0 0
$$913$$ 55.2303 1.82786
$$914$$ 0 0
$$915$$ 2.70854 0.0895415
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −58.1349 −1.91769 −0.958846 0.283925i $$-0.908363\pi$$
−0.958846 + 0.283925i $$0.908363\pi$$
$$920$$ 0 0
$$921$$ −15.9400 −0.525242
$$922$$ 0 0
$$923$$ −27.8186 −0.915661
$$924$$ 0 0
$$925$$ 4.62057 0.151924
$$926$$ 0 0
$$927$$ −6.72913 −0.221014
$$928$$ 0 0
$$929$$ −31.9681 −1.04884 −0.524419 0.851460i $$-0.675717\pi$$
−0.524419 + 0.851460i $$0.675717\pi$$
$$930$$ 0 0
$$931$$ −0.239121 −0.00783688
$$932$$ 0 0
$$933$$ −8.16023 −0.267154
$$934$$ 0 0
$$935$$ 42.1178 1.37740
$$936$$ 0 0
$$937$$ 44.2567 1.44580 0.722902 0.690951i $$-0.242808\pi$$
0.722902 + 0.690951i $$0.242808\pi$$
$$938$$ 0 0
$$939$$ 2.84375 0.0928023
$$940$$ 0 0
$$941$$ 18.4144 0.600292 0.300146 0.953893i $$-0.402965\pi$$
0.300146 + 0.953893i $$0.402965\pi$$
$$942$$ 0 0
$$943$$ 4.30508 0.140193
$$944$$ 0 0
$$945$$ −2.62057 −0.0852473
$$946$$ 0 0
$$947$$ −48.8625 −1.58782 −0.793909 0.608037i $$-0.791957\pi$$
−0.793909 + 0.608037i $$0.791957\pi$$
$$948$$ 0 0
$$949$$ −35.6568 −1.15747
$$950$$ 0 0
$$951$$ −23.5269 −0.762913
$$952$$ 0 0
$$953$$ −49.0643 −1.58935 −0.794674 0.607037i $$-0.792358\pi$$
−0.794674 + 0.607037i $$0.792358\pi$$
$$954$$ 0 0
$$955$$ 11.9400 0.386371
$$956$$ 0 0
$$957$$ −47.0533 −1.52102
$$958$$ 0 0
$$959$$ −15.0766 −0.486849
$$960$$ 0 0
$$961$$ −14.1197 −0.455475
$$962$$ 0 0
$$963$$ −17.9296 −0.577775
$$964$$ 0 0
$$965$$ 11.6342 0.374518
$$966$$ 0 0
$$967$$ 20.4921 0.658981 0.329491 0.944159i $$-0.393123\pi$$
0.329491 + 0.944159i $$0.393123\pi$$
$$968$$ 0 0
$$969$$ 11.5856 0.372184
$$970$$ 0 0
$$971$$ 13.8581 0.444728 0.222364 0.974964i $$-0.428623\pi$$
0.222364 + 0.974964i $$0.428623\pi$$
$$972$$ 0 0
$$973$$ 11.7968 0.378188
$$974$$ 0 0
$$975$$ 7.06828 0.226366
$$976$$ 0 0
$$977$$ −10.1668 −0.325266 −0.162633 0.986687i $$-0.551999\pi$$
−0.162633 + 0.986687i $$0.551999\pi$$
$$978$$ 0 0
$$979$$ 74.6351 2.38535
$$980$$ 0 0
$$981$$ −12.6692 −0.404496
$$982$$ 0 0
$$983$$ −50.6875 −1.61668 −0.808341 0.588715i $$-0.799634\pi$$
−0.808341 + 0.588715i $$0.799634\pi$$
$$984$$ 0 0
$$985$$ 24.6181 0.784399
$$986$$ 0 0
$$987$$ 4.72614 0.150435
$$988$$ 0 0
$$989$$ −6.87176 −0.218509
$$990$$ 0 0
$$991$$ −19.9799 −0.634684 −0.317342 0.948311i $$-0.602790\pi$$
−0.317342 + 0.948311i $$0.602790\pi$$
$$992$$ 0 0
$$993$$ 11.9326 0.378671
$$994$$ 0 0
$$995$$ −18.7291 −0.593753
$$996$$ 0 0
$$997$$ −7.41454 −0.234821 −0.117410 0.993083i $$-0.537459\pi$$
−0.117410 + 0.993083i $$0.537459\pi$$
$$998$$ 0 0
$$999$$ 4.62057 0.146189
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 5520.2.a.cc.1.2 5
4.3 odd 2 2760.2.a.w.1.4 5
12.11 even 2 8280.2.a.br.1.4 5

By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.w.1.4 5 4.3 odd 2
5520.2.a.cc.1.2 5 1.1 even 1 trivial
8280.2.a.br.1.4 5 12.11 even 2