# Properties

 Label 4680.2.a.bb.1.1 Level $4680$ Weight $2$ Character 4680.1 Self dual yes Analytic conductor $37.370$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(1,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, 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("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4680.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: $$37.3699881460$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1560) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 4680.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} -3.70156 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} -3.70156 q^{7} +5.70156 q^{11} -1.00000 q^{13} -5.70156 q^{17} -2.00000 q^{19} -0.298438 q^{23} +1.00000 q^{25} +3.40312 q^{29} +3.40312 q^{31} -3.70156 q^{35} -0.298438 q^{37} -4.29844 q^{41} +4.00000 q^{43} -11.4031 q^{47} +6.70156 q^{49} -4.29844 q^{53} +5.70156 q^{55} -10.8062 q^{59} -3.70156 q^{61} -1.00000 q^{65} -4.00000 q^{67} +16.5078 q^{71} +0.596876 q^{73} -21.1047 q^{77} -1.70156 q^{79} -4.00000 q^{83} -5.70156 q^{85} +11.1047 q^{89} +3.70156 q^{91} -2.00000 q^{95} -1.10469 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 - q^7 $$2 q + 2 q^{5} - q^{7} + 5 q^{11} - 2 q^{13} - 5 q^{17} - 4 q^{19} - 7 q^{23} + 2 q^{25} - 6 q^{29} - 6 q^{31} - q^{35} - 7 q^{37} - 15 q^{41} + 8 q^{43} - 10 q^{47} + 7 q^{49} - 15 q^{53} + 5 q^{55} + 4 q^{59} - q^{61} - 2 q^{65} - 8 q^{67} + q^{71} + 14 q^{73} - 23 q^{77} + 3 q^{79} - 8 q^{83} - 5 q^{85} + 3 q^{89} + q^{91} - 4 q^{95} + 17 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - q^7 + 5 * q^11 - 2 * q^13 - 5 * q^17 - 4 * q^19 - 7 * q^23 + 2 * q^25 - 6 * q^29 - 6 * q^31 - q^35 - 7 * q^37 - 15 * q^41 + 8 * q^43 - 10 * q^47 + 7 * q^49 - 15 * q^53 + 5 * q^55 + 4 * q^59 - q^61 - 2 * q^65 - 8 * q^67 + q^71 + 14 * q^73 - 23 * q^77 + 3 * q^79 - 8 * q^83 - 5 * q^85 + 3 * q^89 + q^91 - 4 * q^95 + 17 * 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
$$7$$ −3.70156 −1.39906 −0.699529 0.714604i $$-0.746607\pi$$
−0.699529 + 0.714604i $$0.746607\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.70156 1.71909 0.859543 0.511064i $$-0.170748\pi$$
0.859543 + 0.511064i $$0.170748\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −5.70156 −1.38283 −0.691416 0.722457i $$-0.743013\pi$$
−0.691416 + 0.722457i $$0.743013\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −0.298438 −0.0622286 −0.0311143 0.999516i $$-0.509906\pi$$
−0.0311143 + 0.999516i $$0.509906\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 3.40312 0.631944 0.315972 0.948768i $$-0.397669\pi$$
0.315972 + 0.948768i $$0.397669\pi$$
$$30$$ 0 0
$$31$$ 3.40312 0.611219 0.305610 0.952157i $$-0.401140\pi$$
0.305610 + 0.952157i $$0.401140\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.70156 −0.625678
$$36$$ 0 0
$$37$$ −0.298438 −0.0490629 −0.0245314 0.999699i $$-0.507809\pi$$
−0.0245314 + 0.999699i $$0.507809\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.29844 −0.671303 −0.335652 0.941986i $$-0.608956\pi$$
−0.335652 + 0.941986i $$0.608956\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −11.4031 −1.66332 −0.831658 0.555288i $$-0.812608\pi$$
−0.831658 + 0.555288i $$0.812608\pi$$
$$48$$ 0 0
$$49$$ 6.70156 0.957366
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −4.29844 −0.590436 −0.295218 0.955430i $$-0.595392\pi$$
−0.295218 + 0.955430i $$0.595392\pi$$
$$54$$ 0 0
$$55$$ 5.70156 0.768798
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −10.8062 −1.40685 −0.703427 0.710768i $$-0.748348\pi$$
−0.703427 + 0.710768i $$0.748348\pi$$
$$60$$ 0 0
$$61$$ −3.70156 −0.473936 −0.236968 0.971517i $$-0.576154\pi$$
−0.236968 + 0.971517i $$0.576154\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 16.5078 1.95912 0.979558 0.201160i $$-0.0644712\pi$$
0.979558 + 0.201160i $$0.0644712\pi$$
$$72$$ 0 0
$$73$$ 0.596876 0.0698590 0.0349295 0.999390i $$-0.488879\pi$$
0.0349295 + 0.999390i $$0.488879\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −21.1047 −2.40510
$$78$$ 0 0
$$79$$ −1.70156 −0.191441 −0.0957203 0.995408i $$-0.530515\pi$$
−0.0957203 + 0.995408i $$0.530515\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ −5.70156 −0.618421
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 11.1047 1.17709 0.588547 0.808463i $$-0.299700\pi$$
0.588547 + 0.808463i $$0.299700\pi$$
$$90$$ 0 0
$$91$$ 3.70156 0.388029
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −2.00000 −0.205196
$$96$$ 0 0
$$97$$ −1.10469 −0.112164 −0.0560820 0.998426i $$-0.517861\pi$$
−0.0560820 + 0.998426i $$0.517861\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −18.8062 −1.87129 −0.935646 0.352940i $$-0.885182\pi$$
−0.935646 + 0.352940i $$0.885182\pi$$
$$102$$ 0 0
$$103$$ 12.0000 1.18240 0.591198 0.806527i $$-0.298655\pi$$
0.591198 + 0.806527i $$0.298655\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1.10469 0.106794 0.0533970 0.998573i $$-0.482995\pi$$
0.0533970 + 0.998573i $$0.482995\pi$$
$$108$$ 0 0
$$109$$ −14.8062 −1.41818 −0.709091 0.705117i $$-0.750894\pi$$
−0.709091 + 0.705117i $$0.750894\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −19.4031 −1.82529 −0.912646 0.408750i $$-0.865965\pi$$
−0.912646 + 0.408750i $$0.865965\pi$$
$$114$$ 0 0
$$115$$ −0.298438 −0.0278295
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 21.1047 1.93466
$$120$$ 0 0
$$121$$ 21.5078 1.95526
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −0.596876 −0.0529642 −0.0264821 0.999649i $$-0.508430\pi$$
−0.0264821 + 0.999649i $$0.508430\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.8062 −1.11889 −0.559444 0.828868i $$-0.688985\pi$$
−0.559444 + 0.828868i $$0.688985\pi$$
$$132$$ 0 0
$$133$$ 7.40312 0.641932
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ 1.70156 0.144325 0.0721623 0.997393i $$-0.477010\pi$$
0.0721623 + 0.997393i $$0.477010\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −5.70156 −0.476789
$$144$$ 0 0
$$145$$ 3.40312 0.282614
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −7.70156 −0.630937 −0.315468 0.948936i $$-0.602162\pi$$
−0.315468 + 0.948936i $$0.602162\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 3.40312 0.273346
$$156$$ 0 0
$$157$$ −16.8062 −1.34128 −0.670642 0.741781i $$-0.733981\pi$$
−0.670642 + 0.741781i $$0.733981\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1.10469 0.0870615
$$162$$ 0 0
$$163$$ 17.7016 1.38649 0.693247 0.720700i $$-0.256180\pi$$
0.693247 + 0.720700i $$0.256180\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2.80625 −0.217154 −0.108577 0.994088i $$-0.534629\pi$$
−0.108577 + 0.994088i $$0.534629\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ −3.70156 −0.279812
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 22.0000 1.64436 0.822179 0.569230i $$-0.192758\pi$$
0.822179 + 0.569230i $$0.192758\pi$$
$$180$$ 0 0
$$181$$ −7.70156 −0.572453 −0.286226 0.958162i $$-0.592401\pi$$
−0.286226 + 0.958162i $$0.592401\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −0.298438 −0.0219416
$$186$$ 0 0
$$187$$ −32.5078 −2.37721
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −2.89531 −0.208409 −0.104205 0.994556i $$-0.533230\pi$$
−0.104205 + 0.994556i $$0.533230\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −24.2094 −1.72485 −0.862423 0.506188i $$-0.831054\pi$$
−0.862423 + 0.506188i $$0.831054\pi$$
$$198$$ 0 0
$$199$$ −22.8062 −1.61669 −0.808346 0.588708i $$-0.799637\pi$$
−0.808346 + 0.588708i $$0.799637\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −12.5969 −0.884127
$$204$$ 0 0
$$205$$ −4.29844 −0.300216
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −11.4031 −0.788771
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ −12.5969 −0.855132
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 5.70156 0.383529
$$222$$ 0 0
$$223$$ −9.40312 −0.629680 −0.314840 0.949145i $$-0.601951\pi$$
−0.314840 + 0.949145i $$0.601951\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 23.4031 1.55332 0.776660 0.629920i $$-0.216912\pi$$
0.776660 + 0.629920i $$0.216912\pi$$
$$228$$ 0 0
$$229$$ −12.5969 −0.832425 −0.416212 0.909267i $$-0.636643\pi$$
−0.416212 + 0.909267i $$0.636643\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −9.10469 −0.596468 −0.298234 0.954493i $$-0.596398\pi$$
−0.298234 + 0.954493i $$0.596398\pi$$
$$234$$ 0 0
$$235$$ −11.4031 −0.743858
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −13.1047 −0.847672 −0.423836 0.905739i $$-0.639317\pi$$
−0.423836 + 0.905739i $$0.639317\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 6.70156 0.428147
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −20.2094 −1.27560 −0.637802 0.770200i $$-0.720156\pi$$
−0.637802 + 0.770200i $$0.720156\pi$$
$$252$$ 0 0
$$253$$ −1.70156 −0.106976
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 18.2094 1.13587 0.567935 0.823074i $$-0.307743\pi$$
0.567935 + 0.823074i $$0.307743\pi$$
$$258$$ 0 0
$$259$$ 1.10469 0.0686419
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −17.4031 −1.07312 −0.536561 0.843861i $$-0.680277\pi$$
−0.536561 + 0.843861i $$0.680277\pi$$
$$264$$ 0 0
$$265$$ −4.29844 −0.264051
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −10.8062 −0.658869 −0.329434 0.944179i $$-0.606858\pi$$
−0.329434 + 0.944179i $$0.606858\pi$$
$$270$$ 0 0
$$271$$ −19.4031 −1.17866 −0.589328 0.807894i $$-0.700607\pi$$
−0.589328 + 0.807894i $$0.700607\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 5.70156 0.343817
$$276$$ 0 0
$$277$$ −16.8062 −1.00979 −0.504895 0.863181i $$-0.668469\pi$$
−0.504895 + 0.863181i $$0.668469\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ 20.0000 1.18888 0.594438 0.804141i $$-0.297374\pi$$
0.594438 + 0.804141i $$0.297374\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 15.9109 0.939193
$$288$$ 0 0
$$289$$ 15.5078 0.912224
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −2.59688 −0.151711 −0.0758556 0.997119i $$-0.524169\pi$$
−0.0758556 + 0.997119i $$0.524169\pi$$
$$294$$ 0 0
$$295$$ −10.8062 −0.629164
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0.298438 0.0172591
$$300$$ 0 0
$$301$$ −14.8062 −0.853418
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −3.70156 −0.211951
$$306$$ 0 0
$$307$$ −33.1047 −1.88938 −0.944692 0.327959i $$-0.893639\pi$$
−0.944692 + 0.327959i $$0.893639\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ −8.80625 −0.497759 −0.248879 0.968535i $$-0.580062\pi$$
−0.248879 + 0.968535i $$0.580062\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 24.2094 1.35973 0.679867 0.733336i $$-0.262038\pi$$
0.679867 + 0.733336i $$0.262038\pi$$
$$318$$ 0 0
$$319$$ 19.4031 1.08637
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 11.4031 0.634487
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 42.2094 2.32708
$$330$$ 0 0
$$331$$ 24.2094 1.33067 0.665334 0.746546i $$-0.268289\pi$$
0.665334 + 0.746546i $$0.268289\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ 32.2094 1.75456 0.877278 0.479982i $$-0.159357\pi$$
0.877278 + 0.479982i $$0.159357\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 19.4031 1.05074
$$342$$ 0 0
$$343$$ 1.10469 0.0596475
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 13.1047 0.703496 0.351748 0.936095i $$-0.385587\pi$$
0.351748 + 0.936095i $$0.385587\pi$$
$$348$$ 0 0
$$349$$ −9.19375 −0.492130 −0.246065 0.969253i $$-0.579138\pi$$
−0.246065 + 0.969253i $$0.579138\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 12.8062 0.681608 0.340804 0.940134i $$-0.389301\pi$$
0.340804 + 0.940134i $$0.389301\pi$$
$$354$$ 0 0
$$355$$ 16.5078 0.876144
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −29.6125 −1.56289 −0.781444 0.623975i $$-0.785516\pi$$
−0.781444 + 0.623975i $$0.785516\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0.596876 0.0312419
$$366$$ 0 0
$$367$$ 26.8062 1.39927 0.699637 0.714498i $$-0.253345\pi$$
0.699637 + 0.714498i $$0.253345\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 15.9109 0.826055
$$372$$ 0 0
$$373$$ −21.4031 −1.10821 −0.554106 0.832446i $$-0.686940\pi$$
−0.554106 + 0.832446i $$0.686940\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −3.40312 −0.175270
$$378$$ 0 0
$$379$$ −35.6125 −1.82929 −0.914646 0.404257i $$-0.867530\pi$$
−0.914646 + 0.404257i $$0.867530\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −7.40312 −0.378282 −0.189141 0.981950i $$-0.560570\pi$$
−0.189141 + 0.981950i $$0.560570\pi$$
$$384$$ 0 0
$$385$$ −21.1047 −1.07559
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 25.6125 1.29861 0.649303 0.760530i $$-0.275061\pi$$
0.649303 + 0.760530i $$0.275061\pi$$
$$390$$ 0 0
$$391$$ 1.70156 0.0860517
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −1.70156 −0.0856149
$$396$$ 0 0
$$397$$ −16.2984 −0.817995 −0.408998 0.912535i $$-0.634122\pi$$
−0.408998 + 0.912535i $$0.634122\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 24.8062 1.23876 0.619382 0.785089i $$-0.287383\pi$$
0.619382 + 0.785089i $$0.287383\pi$$
$$402$$ 0 0
$$403$$ −3.40312 −0.169522
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1.70156 −0.0843433
$$408$$ 0 0
$$409$$ 20.8062 1.02880 0.514401 0.857550i $$-0.328014\pi$$
0.514401 + 0.857550i $$0.328014\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 40.0000 1.96827
$$414$$ 0 0
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 23.6125 1.15355 0.576773 0.816904i $$-0.304312\pi$$
0.576773 + 0.816904i $$0.304312\pi$$
$$420$$ 0 0
$$421$$ 25.6125 1.24828 0.624138 0.781314i $$-0.285450\pi$$
0.624138 + 0.781314i $$0.285450\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −5.70156 −0.276566
$$426$$ 0 0
$$427$$ 13.7016 0.663065
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 0 0
$$433$$ 25.4031 1.22080 0.610398 0.792095i $$-0.291009\pi$$
0.610398 + 0.792095i $$0.291009\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0.596876 0.0285524
$$438$$ 0 0
$$439$$ −21.1047 −1.00727 −0.503636 0.863916i $$-0.668005\pi$$
−0.503636 + 0.863916i $$0.668005\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −31.3141 −1.48778 −0.743888 0.668304i $$-0.767020\pi$$
−0.743888 + 0.668304i $$0.767020\pi$$
$$444$$ 0 0
$$445$$ 11.1047 0.526413
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −23.1047 −1.09038 −0.545189 0.838313i $$-0.683542\pi$$
−0.545189 + 0.838313i $$0.683542\pi$$
$$450$$ 0 0
$$451$$ −24.5078 −1.15403
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 3.70156 0.173532
$$456$$ 0 0
$$457$$ 13.7016 0.640932 0.320466 0.947260i $$-0.396160\pi$$
0.320466 + 0.947260i $$0.396160\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −39.1047 −1.82129 −0.910643 0.413193i $$-0.864413\pi$$
−0.910643 + 0.413193i $$0.864413\pi$$
$$462$$ 0 0
$$463$$ 15.1047 0.701974 0.350987 0.936380i $$-0.385846\pi$$
0.350987 + 0.936380i $$0.385846\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 1.10469 0.0511188 0.0255594 0.999673i $$-0.491863\pi$$
0.0255594 + 0.999673i $$0.491863\pi$$
$$468$$ 0 0
$$469$$ 14.8062 0.683689
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 22.8062 1.04863
$$474$$ 0 0
$$475$$ −2.00000 −0.0917663
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 22.2984 1.01884 0.509421 0.860518i $$-0.329860\pi$$
0.509421 + 0.860518i $$0.329860\pi$$
$$480$$ 0 0
$$481$$ 0.298438 0.0136076
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −1.10469 −0.0501612
$$486$$ 0 0
$$487$$ 21.3141 0.965832 0.482916 0.875667i $$-0.339578\pi$$
0.482916 + 0.875667i $$0.339578\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −32.2094 −1.45359 −0.726794 0.686855i $$-0.758991\pi$$
−0.726794 + 0.686855i $$0.758991\pi$$
$$492$$ 0 0
$$493$$ −19.4031 −0.873873
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −61.1047 −2.74092
$$498$$ 0 0
$$499$$ 40.8062 1.82674 0.913369 0.407132i $$-0.133471\pi$$
0.913369 + 0.407132i $$0.133471\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −18.5969 −0.829194 −0.414597 0.910005i $$-0.636077\pi$$
−0.414597 + 0.910005i $$0.636077\pi$$
$$504$$ 0 0
$$505$$ −18.8062 −0.836867
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −25.3141 −1.12203 −0.561013 0.827807i $$-0.689588\pi$$
−0.561013 + 0.827807i $$0.689588\pi$$
$$510$$ 0 0
$$511$$ −2.20937 −0.0977369
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 12.0000 0.528783
$$516$$ 0 0
$$517$$ −65.0156 −2.85938
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −19.4031 −0.845213
$$528$$ 0 0
$$529$$ −22.9109 −0.996128
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 4.29844 0.186186
$$534$$ 0 0
$$535$$ 1.10469 0.0477598
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 38.2094 1.64579
$$540$$ 0 0
$$541$$ 1.79063 0.0769851 0.0384925 0.999259i $$-0.487744\pi$$
0.0384925 + 0.999259i $$0.487744\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −14.8062 −0.634230
$$546$$ 0 0
$$547$$ 25.6125 1.09511 0.547556 0.836769i $$-0.315558\pi$$
0.547556 + 0.836769i $$0.315558\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −6.80625 −0.289956
$$552$$ 0 0
$$553$$ 6.29844 0.267837
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −2.89531 −0.122023 −0.0610115 0.998137i $$-0.519433\pi$$
−0.0610115 + 0.998137i $$0.519433\pi$$
$$564$$ 0 0
$$565$$ −19.4031 −0.816296
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 14.5969 0.611933 0.305966 0.952042i $$-0.401020\pi$$
0.305966 + 0.952042i $$0.401020\pi$$
$$570$$ 0 0
$$571$$ −31.3141 −1.31045 −0.655226 0.755433i $$-0.727427\pi$$
−0.655226 + 0.755433i $$0.727427\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −0.298438 −0.0124457
$$576$$ 0 0
$$577$$ 0.507811 0.0211404 0.0105702 0.999944i $$-0.496635\pi$$
0.0105702 + 0.999944i $$0.496635\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 14.8062 0.614267
$$582$$ 0 0
$$583$$ −24.5078 −1.01501
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −14.2094 −0.586484 −0.293242 0.956038i $$-0.594734\pi$$
−0.293242 + 0.956038i $$0.594734\pi$$
$$588$$ 0 0
$$589$$ −6.80625 −0.280447
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −9.40312 −0.386140 −0.193070 0.981185i $$-0.561844\pi$$
−0.193070 + 0.981185i $$0.561844\pi$$
$$594$$ 0 0
$$595$$ 21.1047 0.865208
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 17.1938 0.702518 0.351259 0.936278i $$-0.385754\pi$$
0.351259 + 0.936278i $$0.385754\pi$$
$$600$$ 0 0
$$601$$ 15.7016 0.640480 0.320240 0.947336i $$-0.396236\pi$$
0.320240 + 0.947336i $$0.396236\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 21.5078 0.874417
$$606$$ 0 0
$$607$$ 9.61250 0.390159 0.195080 0.980787i $$-0.437503\pi$$
0.195080 + 0.980787i $$0.437503\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 11.4031 0.461321
$$612$$ 0 0
$$613$$ 24.7172 0.998318 0.499159 0.866511i $$-0.333642\pi$$
0.499159 + 0.866511i $$0.333642\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2.00000 −0.0805170 −0.0402585 0.999189i $$-0.512818\pi$$
−0.0402585 + 0.999189i $$0.512818\pi$$
$$618$$ 0 0
$$619$$ −11.6125 −0.466746 −0.233373 0.972387i $$-0.574976\pi$$
−0.233373 + 0.972387i $$0.574976\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −41.1047 −1.64682
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 1.70156 0.0678457
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −0.596876 −0.0236863
$$636$$ 0 0
$$637$$ −6.70156 −0.265526
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −28.2094 −1.11420 −0.557102 0.830444i $$-0.688087\pi$$
−0.557102 + 0.830444i $$0.688087\pi$$
$$642$$ 0 0
$$643$$ 37.7016 1.48680 0.743402 0.668845i $$-0.233211\pi$$
0.743402 + 0.668845i $$0.233211\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −0.895314 −0.0351984 −0.0175992 0.999845i $$-0.505602\pi$$
−0.0175992 + 0.999845i $$0.505602\pi$$
$$648$$ 0 0
$$649$$ −61.6125 −2.41850
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 35.6125 1.39362 0.696812 0.717253i $$-0.254601\pi$$
0.696812 + 0.717253i $$0.254601\pi$$
$$654$$ 0 0
$$655$$ −12.8062 −0.500382
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 14.5969 0.568614 0.284307 0.958733i $$-0.408237\pi$$
0.284307 + 0.958733i $$0.408237\pi$$
$$660$$ 0 0
$$661$$ −12.0000 −0.466746 −0.233373 0.972387i $$-0.574976\pi$$
−0.233373 + 0.972387i $$0.574976\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 7.40312 0.287081
$$666$$ 0 0
$$667$$ −1.01562 −0.0393250
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −21.1047 −0.814737
$$672$$ 0 0
$$673$$ −28.8062 −1.11040 −0.555200 0.831717i $$-0.687358\pi$$
−0.555200 + 0.831717i $$0.687358\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −15.1047 −0.580520 −0.290260 0.956948i $$-0.593742\pi$$
−0.290260 + 0.956948i $$0.593742\pi$$
$$678$$ 0 0
$$679$$ 4.08907 0.156924
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −28.0000 −1.07139 −0.535695 0.844411i $$-0.679950\pi$$
−0.535695 + 0.844411i $$0.679950\pi$$
$$684$$ 0 0
$$685$$ −18.0000 −0.687745
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 4.29844 0.163757
$$690$$ 0 0
$$691$$ −10.0000 −0.380418 −0.190209 0.981744i $$-0.560917\pi$$
−0.190209 + 0.981744i $$0.560917\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 1.70156 0.0645439
$$696$$ 0 0
$$697$$ 24.5078 0.928300
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −25.6125 −0.967371 −0.483685 0.875242i $$-0.660702\pi$$
−0.483685 + 0.875242i $$0.660702\pi$$
$$702$$ 0 0
$$703$$ 0.596876 0.0225116
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 69.6125 2.61805
$$708$$ 0 0
$$709$$ 17.1938 0.645725 0.322862 0.946446i $$-0.395355\pi$$
0.322862 + 0.946446i $$0.395355\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −1.01562 −0.0380353
$$714$$ 0 0
$$715$$ −5.70156 −0.213226
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −34.8062 −1.29805 −0.649027 0.760765i $$-0.724824\pi$$
−0.649027 + 0.760765i $$0.724824\pi$$
$$720$$ 0 0
$$721$$ −44.4187 −1.65424
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 3.40312 0.126389
$$726$$ 0 0
$$727$$ 3.40312 0.126215 0.0631074 0.998007i $$-0.479899\pi$$
0.0631074 + 0.998007i $$0.479899\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −22.8062 −0.843520
$$732$$ 0 0
$$733$$ −28.7172 −1.06069 −0.530347 0.847781i $$-0.677938\pi$$
−0.530347 + 0.847781i $$0.677938\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −22.8062 −0.840079
$$738$$ 0 0
$$739$$ 16.8062 0.618228 0.309114 0.951025i $$-0.399968\pi$$
0.309114 + 0.951025i $$0.399968\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 23.4031 0.858577 0.429289 0.903167i $$-0.358764\pi$$
0.429289 + 0.903167i $$0.358764\pi$$
$$744$$ 0 0
$$745$$ −7.70156 −0.282163
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −4.08907 −0.149411
$$750$$ 0 0
$$751$$ 15.3141 0.558818 0.279409 0.960172i $$-0.409861\pi$$
0.279409 + 0.960172i $$0.409861\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −34.5969 −1.25744 −0.628722 0.777630i $$-0.716422\pi$$
−0.628722 + 0.777630i $$0.716422\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ 54.8062 1.98412
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 10.8062 0.390191
$$768$$ 0 0
$$769$$ 38.5969 1.39184 0.695919 0.718120i $$-0.254997\pi$$
0.695919 + 0.718120i $$0.254997\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 45.4031 1.63304 0.816518 0.577319i $$-0.195901\pi$$
0.816518 + 0.577319i $$0.195901\pi$$
$$774$$ 0 0
$$775$$ 3.40312 0.122244
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 8.59688 0.308015
$$780$$ 0 0
$$781$$ 94.1203 3.36789
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −16.8062 −0.599841
$$786$$ 0 0
$$787$$ 30.8062 1.09812 0.549062 0.835782i $$-0.314985\pi$$
0.549062 + 0.835782i $$0.314985\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 71.8219 2.55369
$$792$$ 0 0
$$793$$ 3.70156 0.131446
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 8.29844 0.293946 0.146973 0.989141i $$-0.453047\pi$$
0.146973 + 0.989141i $$0.453047\pi$$
$$798$$ 0 0
$$799$$ 65.0156 2.30009
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 3.40312 0.120094
$$804$$ 0 0
$$805$$ 1.10469 0.0389351
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −24.8062 −0.872141 −0.436071 0.899912i $$-0.643630\pi$$
−0.436071 + 0.899912i $$0.643630\pi$$
$$810$$ 0 0
$$811$$ 24.2094 0.850106 0.425053 0.905168i $$-0.360255\pi$$
0.425053 + 0.905168i $$0.360255\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 17.7016 0.620059
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 34.5078 1.20433 0.602165 0.798371i $$-0.294305\pi$$
0.602165 + 0.798371i $$0.294305\pi$$
$$822$$ 0 0
$$823$$ 4.00000 0.139431 0.0697156 0.997567i $$-0.477791\pi$$
0.0697156 + 0.997567i $$0.477791\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −47.4031 −1.64837 −0.824184 0.566322i $$-0.808366\pi$$
−0.824184 + 0.566322i $$0.808366\pi$$
$$828$$ 0 0
$$829$$ 3.61250 0.125467 0.0627336 0.998030i $$-0.480018\pi$$
0.0627336 + 0.998030i $$0.480018\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −38.2094 −1.32388
$$834$$ 0 0
$$835$$ −2.80625 −0.0971142
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 38.1203 1.31606 0.658030 0.752992i $$-0.271390\pi$$
0.658030 + 0.752992i $$0.271390\pi$$
$$840$$ 0 0
$$841$$ −17.4187 −0.600646
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ −79.6125 −2.73552
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0.0890652 0.00305311
$$852$$ 0 0
$$853$$ 48.7172 1.66804 0.834022 0.551731i $$-0.186032\pi$$
0.834022 + 0.551731i $$0.186032\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −6.29844 −0.215151 −0.107575 0.994197i $$-0.534309\pi$$
−0.107575 + 0.994197i $$0.534309\pi$$
$$858$$ 0 0
$$859$$ 37.7016 1.28636 0.643180 0.765715i $$-0.277615\pi$$
0.643180 + 0.765715i $$0.277615\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −0.596876 −0.0203179 −0.0101589 0.999948i $$-0.503234\pi$$
−0.0101589 + 0.999948i $$0.503234\pi$$
$$864$$ 0 0
$$865$$ 2.00000 0.0680020
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −9.70156 −0.329103
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −3.70156 −0.125136
$$876$$ 0 0
$$877$$ 47.6125 1.60776 0.803880 0.594792i $$-0.202765\pi$$
0.803880 + 0.594792i $$0.202765\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −12.8062 −0.431453 −0.215727 0.976454i $$-0.569212\pi$$
−0.215727 + 0.976454i $$0.569212\pi$$
$$882$$ 0 0
$$883$$ 33.6125 1.13115 0.565575 0.824697i $$-0.308654\pi$$
0.565575 + 0.824697i $$0.308654\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −50.5078 −1.69589 −0.847943 0.530087i $$-0.822159\pi$$
−0.847943 + 0.530087i $$0.822159\pi$$
$$888$$ 0 0
$$889$$ 2.20937 0.0741000
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 22.8062 0.763182
$$894$$ 0 0
$$895$$ 22.0000 0.735379
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 11.5813 0.386256
$$900$$ 0 0
$$901$$ 24.5078 0.816474
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −7.70156 −0.256009
$$906$$ 0 0
$$907$$ 22.2094 0.737450 0.368725 0.929539i $$-0.379794\pi$$
0.368725 + 0.929539i $$0.379794\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 18.8062 0.623079 0.311539 0.950233i $$-0.399155\pi$$
0.311539 + 0.950233i $$0.399155\pi$$
$$912$$ 0 0
$$913$$ −22.8062 −0.754777
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 47.4031 1.56539
$$918$$ 0 0
$$919$$ 51.9109 1.71238 0.856192 0.516658i $$-0.172824\pi$$
0.856192 + 0.516658i $$0.172824\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −16.5078 −0.543361
$$924$$ 0 0
$$925$$ −0.298438 −0.00981258
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 11.1047 0.364333 0.182166 0.983268i $$-0.441689\pi$$
0.182166 + 0.983268i $$0.441689\pi$$
$$930$$ 0 0
$$931$$ −13.4031 −0.439270
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −32.5078 −1.06312
$$936$$ 0 0
$$937$$ 27.1938 0.888381 0.444191 0.895932i $$-0.353491\pi$$
0.444191 + 0.895932i $$0.353491\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 48.2984 1.57448 0.787242 0.616644i $$-0.211508\pi$$
0.787242 + 0.616644i $$0.211508\pi$$
$$942$$ 0 0
$$943$$ 1.28282 0.0417743
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −41.6125 −1.35222 −0.676112 0.736799i $$-0.736337\pi$$
−0.676112 + 0.736799i $$0.736337\pi$$
$$948$$ 0 0
$$949$$ −0.596876 −0.0193754
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 39.9109 1.29284 0.646421 0.762981i $$-0.276265\pi$$
0.646421 + 0.762981i $$0.276265\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 66.6281 2.15153
$$960$$ 0 0
$$961$$ −19.4187 −0.626411
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −2.89531 −0.0932034
$$966$$ 0 0
$$967$$ −49.8219 −1.60216 −0.801082 0.598555i $$-0.795742\pi$$
−0.801082 + 0.598555i $$0.795742\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 7.01562 0.225142 0.112571 0.993644i $$-0.464091\pi$$
0.112571 + 0.993644i $$0.464091\pi$$
$$972$$ 0 0
$$973$$ −6.29844 −0.201919
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 12.2094 0.390612 0.195306 0.980742i $$-0.437430\pi$$
0.195306 + 0.980742i $$0.437430\pi$$
$$978$$ 0 0
$$979$$ 63.3141 2.02353
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 20.0000 0.637901 0.318950 0.947771i $$-0.396670\pi$$
0.318950 + 0.947771i $$0.396670\pi$$
$$984$$ 0 0
$$985$$ −24.2094 −0.771375
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −1.19375 −0.0379591
$$990$$ 0 0
$$991$$ −42.7172 −1.35696 −0.678478 0.734621i $$-0.737360\pi$$
−0.678478 + 0.734621i $$0.737360\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −22.8062 −0.723007
$$996$$ 0 0
$$997$$ −38.5969 −1.22238 −0.611188 0.791486i $$-0.709308\pi$$
−0.611188 + 0.791486i $$0.709308\pi$$
$$998$$ 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 4680.2.a.bb.1.1 2
3.2 odd 2 1560.2.a.m.1.1 2
4.3 odd 2 9360.2.a.ct.1.2 2
12.11 even 2 3120.2.a.bf.1.2 2
15.14 odd 2 7800.2.a.be.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.m.1.1 2 3.2 odd 2
3120.2.a.bf.1.2 2 12.11 even 2
4680.2.a.bb.1.1 2 1.1 even 1 trivial
7800.2.a.be.1.2 2 15.14 odd 2
9360.2.a.ct.1.2 2 4.3 odd 2