# Properties

 Label 7616.2.a.v.1.1 Level $7616$ Weight $2$ Character 7616.1 Self dual yes Analytic conductor $60.814$ 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] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7616 = 2^{6} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7616.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: $$60.8140661794$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 476) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 7616.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.30278 q^{3} +2.30278 q^{5} -1.00000 q^{7} -1.30278 q^{9} +O(q^{10})$$ $$q-1.30278 q^{3} +2.30278 q^{5} -1.00000 q^{7} -1.30278 q^{9} -6.60555 q^{13} -3.00000 q^{15} +1.00000 q^{17} +6.60555 q^{19} +1.30278 q^{21} +0.302776 q^{25} +5.60555 q^{27} -4.30278 q^{31} -2.30278 q^{35} +2.60555 q^{37} +8.60555 q^{39} +3.90833 q^{41} +7.30278 q^{43} -3.00000 q^{45} -4.60555 q^{47} +1.00000 q^{49} -1.30278 q^{51} +3.69722 q^{53} -8.60555 q^{57} +9.21110 q^{59} +7.90833 q^{61} +1.30278 q^{63} -15.2111 q^{65} -1.69722 q^{67} +7.81665 q^{71} -7.90833 q^{73} -0.394449 q^{75} -12.6056 q^{79} -3.39445 q^{81} +6.00000 q^{83} +2.30278 q^{85} -16.6056 q^{89} +6.60555 q^{91} +5.60555 q^{93} +15.2111 q^{95} -6.30278 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{5} - 2 q^{7} + q^{9}+O(q^{10})$$ 2 * q + q^3 + q^5 - 2 * q^7 + q^9 $$2 q + q^{3} + q^{5} - 2 q^{7} + q^{9} - 6 q^{13} - 6 q^{15} + 2 q^{17} + 6 q^{19} - q^{21} - 3 q^{25} + 4 q^{27} - 5 q^{31} - q^{35} - 2 q^{37} + 10 q^{39} - 3 q^{41} + 11 q^{43} - 6 q^{45} - 2 q^{47} + 2 q^{49} + q^{51} + 11 q^{53} - 10 q^{57} + 4 q^{59} + 5 q^{61} - q^{63} - 16 q^{65} - 7 q^{67} - 6 q^{71} - 5 q^{73} - 8 q^{75} - 18 q^{79} - 14 q^{81} + 12 q^{83} + q^{85} - 26 q^{89} + 6 q^{91} + 4 q^{93} + 16 q^{95} - 9 q^{97}+O(q^{100})$$ 2 * q + q^3 + q^5 - 2 * q^7 + q^9 - 6 * q^13 - 6 * q^15 + 2 * q^17 + 6 * q^19 - q^21 - 3 * q^25 + 4 * q^27 - 5 * q^31 - q^35 - 2 * q^37 + 10 * q^39 - 3 * q^41 + 11 * q^43 - 6 * q^45 - 2 * q^47 + 2 * q^49 + q^51 + 11 * q^53 - 10 * q^57 + 4 * q^59 + 5 * q^61 - q^63 - 16 * q^65 - 7 * q^67 - 6 * q^71 - 5 * q^73 - 8 * q^75 - 18 * q^79 - 14 * q^81 + 12 * q^83 + q^85 - 26 * q^89 + 6 * q^91 + 4 * q^93 + 16 * q^95 - 9 * 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
$$3$$ −1.30278 −0.752158 −0.376079 0.926588i $$-0.622728\pi$$
−0.376079 + 0.926588i $$0.622728\pi$$
$$4$$ 0 0
$$5$$ 2.30278 1.02983 0.514916 0.857240i $$-0.327823\pi$$
0.514916 + 0.857240i $$0.327823\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ −1.30278 −0.434259
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −6.60555 −1.83205 −0.916025 0.401121i $$-0.868621\pi$$
−0.916025 + 0.401121i $$0.868621\pi$$
$$14$$ 0 0
$$15$$ −3.00000 −0.774597
$$16$$ 0 0
$$17$$ 1.00000 0.242536
$$18$$ 0 0
$$19$$ 6.60555 1.51542 0.757709 0.652593i $$-0.226319\pi$$
0.757709 + 0.652593i $$0.226319\pi$$
$$20$$ 0 0
$$21$$ 1.30278 0.284289
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 0.302776 0.0605551
$$26$$ 0 0
$$27$$ 5.60555 1.07879
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −4.30278 −0.772801 −0.386401 0.922331i $$-0.626282\pi$$
−0.386401 + 0.922331i $$0.626282\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.30278 −0.389240
$$36$$ 0 0
$$37$$ 2.60555 0.428350 0.214175 0.976795i $$-0.431294\pi$$
0.214175 + 0.976795i $$0.431294\pi$$
$$38$$ 0 0
$$39$$ 8.60555 1.37799
$$40$$ 0 0
$$41$$ 3.90833 0.610378 0.305189 0.952292i $$-0.401280\pi$$
0.305189 + 0.952292i $$0.401280\pi$$
$$42$$ 0 0
$$43$$ 7.30278 1.11366 0.556831 0.830626i $$-0.312017\pi$$
0.556831 + 0.830626i $$0.312017\pi$$
$$44$$ 0 0
$$45$$ −3.00000 −0.447214
$$46$$ 0 0
$$47$$ −4.60555 −0.671789 −0.335894 0.941900i $$-0.609039\pi$$
−0.335894 + 0.941900i $$0.609039\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −1.30278 −0.182425
$$52$$ 0 0
$$53$$ 3.69722 0.507853 0.253926 0.967224i $$-0.418278\pi$$
0.253926 + 0.967224i $$0.418278\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −8.60555 −1.13983
$$58$$ 0 0
$$59$$ 9.21110 1.19918 0.599592 0.800306i $$-0.295330\pi$$
0.599592 + 0.800306i $$0.295330\pi$$
$$60$$ 0 0
$$61$$ 7.90833 1.01256 0.506279 0.862370i $$-0.331021\pi$$
0.506279 + 0.862370i $$0.331021\pi$$
$$62$$ 0 0
$$63$$ 1.30278 0.164134
$$64$$ 0 0
$$65$$ −15.2111 −1.88671
$$66$$ 0 0
$$67$$ −1.69722 −0.207349 −0.103674 0.994611i $$-0.533060\pi$$
−0.103674 + 0.994611i $$0.533060\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 7.81665 0.927666 0.463833 0.885923i $$-0.346474\pi$$
0.463833 + 0.885923i $$0.346474\pi$$
$$72$$ 0 0
$$73$$ −7.90833 −0.925600 −0.462800 0.886463i $$-0.653155\pi$$
−0.462800 + 0.886463i $$0.653155\pi$$
$$74$$ 0 0
$$75$$ −0.394449 −0.0455470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −12.6056 −1.41824 −0.709118 0.705090i $$-0.750907\pi$$
−0.709118 + 0.705090i $$0.750907\pi$$
$$80$$ 0 0
$$81$$ −3.39445 −0.377161
$$82$$ 0 0
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 2.30278 0.249771
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −16.6056 −1.76018 −0.880092 0.474802i $$-0.842520\pi$$
−0.880092 + 0.474802i $$0.842520\pi$$
$$90$$ 0 0
$$91$$ 6.60555 0.692450
$$92$$ 0 0
$$93$$ 5.60555 0.581269
$$94$$ 0 0
$$95$$ 15.2111 1.56063
$$96$$ 0 0
$$97$$ −6.30278 −0.639950 −0.319975 0.947426i $$-0.603675\pi$$
−0.319975 + 0.947426i $$0.603675\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −19.8167 −1.97183 −0.985915 0.167245i $$-0.946513\pi$$
−0.985915 + 0.167245i $$0.946513\pi$$
$$102$$ 0 0
$$103$$ 7.21110 0.710531 0.355266 0.934765i $$-0.384390\pi$$
0.355266 + 0.934765i $$0.384390\pi$$
$$104$$ 0 0
$$105$$ 3.00000 0.292770
$$106$$ 0 0
$$107$$ −1.39445 −0.134806 −0.0674032 0.997726i $$-0.521471\pi$$
−0.0674032 + 0.997726i $$0.521471\pi$$
$$108$$ 0 0
$$109$$ −8.42221 −0.806701 −0.403350 0.915046i $$-0.632154\pi$$
−0.403350 + 0.915046i $$0.632154\pi$$
$$110$$ 0 0
$$111$$ −3.39445 −0.322187
$$112$$ 0 0
$$113$$ 4.60555 0.433254 0.216627 0.976254i $$-0.430494\pi$$
0.216627 + 0.976254i $$0.430494\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 8.60555 0.795583
$$118$$ 0 0
$$119$$ −1.00000 −0.0916698
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ −5.09167 −0.459101
$$124$$ 0 0
$$125$$ −10.8167 −0.967471
$$126$$ 0 0
$$127$$ −14.9083 −1.32290 −0.661450 0.749989i $$-0.730059\pi$$
−0.661450 + 0.749989i $$0.730059\pi$$
$$128$$ 0 0
$$129$$ −9.51388 −0.837650
$$130$$ 0 0
$$131$$ −21.2111 −1.85322 −0.926611 0.376021i $$-0.877292\pi$$
−0.926611 + 0.376021i $$0.877292\pi$$
$$132$$ 0 0
$$133$$ −6.60555 −0.572774
$$134$$ 0 0
$$135$$ 12.9083 1.11097
$$136$$ 0 0
$$137$$ −12.6972 −1.08480 −0.542399 0.840121i $$-0.682484\pi$$
−0.542399 + 0.840121i $$0.682484\pi$$
$$138$$ 0 0
$$139$$ 7.09167 0.601508 0.300754 0.953702i $$-0.402762\pi$$
0.300754 + 0.953702i $$0.402762\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −1.30278 −0.107451
$$148$$ 0 0
$$149$$ −17.7250 −1.45209 −0.726044 0.687649i $$-0.758643\pi$$
−0.726044 + 0.687649i $$0.758643\pi$$
$$150$$ 0 0
$$151$$ 11.1194 0.904886 0.452443 0.891793i $$-0.350553\pi$$
0.452443 + 0.891793i $$0.350553\pi$$
$$152$$ 0 0
$$153$$ −1.30278 −0.105323
$$154$$ 0 0
$$155$$ −9.90833 −0.795856
$$156$$ 0 0
$$157$$ 7.21110 0.575509 0.287754 0.957704i $$-0.407091\pi$$
0.287754 + 0.957704i $$0.407091\pi$$
$$158$$ 0 0
$$159$$ −4.81665 −0.381985
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 12.6056 0.987343 0.493671 0.869648i $$-0.335655\pi$$
0.493671 + 0.869648i $$0.335655\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 23.7250 1.83589 0.917947 0.396703i $$-0.129846\pi$$
0.917947 + 0.396703i $$0.129846\pi$$
$$168$$ 0 0
$$169$$ 30.6333 2.35641
$$170$$ 0 0
$$171$$ −8.60555 −0.658083
$$172$$ 0 0
$$173$$ −17.3028 −1.31551 −0.657753 0.753234i $$-0.728493\pi$$
−0.657753 + 0.753234i $$0.728493\pi$$
$$174$$ 0 0
$$175$$ −0.302776 −0.0228877
$$176$$ 0 0
$$177$$ −12.0000 −0.901975
$$178$$ 0 0
$$179$$ 9.90833 0.740583 0.370292 0.928916i $$-0.379258\pi$$
0.370292 + 0.928916i $$0.379258\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ −10.3028 −0.761603
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −5.60555 −0.407744
$$190$$ 0 0
$$191$$ −20.7250 −1.49961 −0.749803 0.661661i $$-0.769852\pi$$
−0.749803 + 0.661661i $$0.769852\pi$$
$$192$$ 0 0
$$193$$ −11.8167 −0.850581 −0.425291 0.905057i $$-0.639828\pi$$
−0.425291 + 0.905057i $$0.639828\pi$$
$$194$$ 0 0
$$195$$ 19.8167 1.41910
$$196$$ 0 0
$$197$$ 4.60555 0.328132 0.164066 0.986449i $$-0.447539\pi$$
0.164066 + 0.986449i $$0.447539\pi$$
$$198$$ 0 0
$$199$$ −21.1194 −1.49712 −0.748558 0.663069i $$-0.769254\pi$$
−0.748558 + 0.663069i $$0.769254\pi$$
$$200$$ 0 0
$$201$$ 2.21110 0.155959
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 9.00000 0.628587
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −21.0278 −1.44761 −0.723805 0.690004i $$-0.757609\pi$$
−0.723805 + 0.690004i $$0.757609\pi$$
$$212$$ 0 0
$$213$$ −10.1833 −0.697751
$$214$$ 0 0
$$215$$ 16.8167 1.14689
$$216$$ 0 0
$$217$$ 4.30278 0.292091
$$218$$ 0 0
$$219$$ 10.3028 0.696197
$$220$$ 0 0
$$221$$ −6.60555 −0.444337
$$222$$ 0 0
$$223$$ 9.02776 0.604543 0.302272 0.953222i $$-0.402255\pi$$
0.302272 + 0.953222i $$0.402255\pi$$
$$224$$ 0 0
$$225$$ −0.394449 −0.0262966
$$226$$ 0 0
$$227$$ 5.51388 0.365969 0.182984 0.983116i $$-0.441424\pi$$
0.182984 + 0.983116i $$0.441424\pi$$
$$228$$ 0 0
$$229$$ −10.7889 −0.712950 −0.356475 0.934305i $$-0.616022\pi$$
−0.356475 + 0.934305i $$0.616022\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −13.8167 −0.905159 −0.452580 0.891724i $$-0.649496\pi$$
−0.452580 + 0.891724i $$0.649496\pi$$
$$234$$ 0 0
$$235$$ −10.6056 −0.691830
$$236$$ 0 0
$$237$$ 16.4222 1.06674
$$238$$ 0 0
$$239$$ −11.3028 −0.731116 −0.365558 0.930789i $$-0.619122\pi$$
−0.365558 + 0.930789i $$0.619122\pi$$
$$240$$ 0 0
$$241$$ 8.48612 0.546639 0.273320 0.961923i $$-0.411878\pi$$
0.273320 + 0.961923i $$0.411878\pi$$
$$242$$ 0 0
$$243$$ −12.3944 −0.795104
$$244$$ 0 0
$$245$$ 2.30278 0.147119
$$246$$ 0 0
$$247$$ −43.6333 −2.77632
$$248$$ 0 0
$$249$$ −7.81665 −0.495360
$$250$$ 0 0
$$251$$ 4.18335 0.264050 0.132025 0.991246i $$-0.457852\pi$$
0.132025 + 0.991246i $$0.457852\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −3.00000 −0.187867
$$256$$ 0 0
$$257$$ 3.21110 0.200303 0.100152 0.994972i $$-0.468067\pi$$
0.100152 + 0.994972i $$0.468067\pi$$
$$258$$ 0 0
$$259$$ −2.60555 −0.161901
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 18.4222 1.13596 0.567981 0.823042i $$-0.307725\pi$$
0.567981 + 0.823042i $$0.307725\pi$$
$$264$$ 0 0
$$265$$ 8.51388 0.523003
$$266$$ 0 0
$$267$$ 21.6333 1.32394
$$268$$ 0 0
$$269$$ −15.2111 −0.927437 −0.463719 0.885983i $$-0.653485\pi$$
−0.463719 + 0.885983i $$0.653485\pi$$
$$270$$ 0 0
$$271$$ 1.21110 0.0735692 0.0367846 0.999323i $$-0.488288\pi$$
0.0367846 + 0.999323i $$0.488288\pi$$
$$272$$ 0 0
$$273$$ −8.60555 −0.520832
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.4222 1.34722 0.673610 0.739087i $$-0.264743\pi$$
0.673610 + 0.739087i $$0.264743\pi$$
$$278$$ 0 0
$$279$$ 5.60555 0.335596
$$280$$ 0 0
$$281$$ 3.69722 0.220558 0.110279 0.993901i $$-0.464826\pi$$
0.110279 + 0.993901i $$0.464826\pi$$
$$282$$ 0 0
$$283$$ −9.51388 −0.565541 −0.282771 0.959188i $$-0.591254\pi$$
−0.282771 + 0.959188i $$0.591254\pi$$
$$284$$ 0 0
$$285$$ −19.8167 −1.17384
$$286$$ 0 0
$$287$$ −3.90833 −0.230701
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 8.21110 0.481343
$$292$$ 0 0
$$293$$ 21.6333 1.26383 0.631916 0.775037i $$-0.282269\pi$$
0.631916 + 0.775037i $$0.282269\pi$$
$$294$$ 0 0
$$295$$ 21.2111 1.23496
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −7.30278 −0.420925
$$302$$ 0 0
$$303$$ 25.8167 1.48313
$$304$$ 0 0
$$305$$ 18.2111 1.04276
$$306$$ 0 0
$$307$$ −1.21110 −0.0691213 −0.0345606 0.999403i $$-0.511003\pi$$
−0.0345606 + 0.999403i $$0.511003\pi$$
$$308$$ 0 0
$$309$$ −9.39445 −0.534432
$$310$$ 0 0
$$311$$ 1.88057 0.106637 0.0533187 0.998578i $$-0.483020\pi$$
0.0533187 + 0.998578i $$0.483020\pi$$
$$312$$ 0 0
$$313$$ 33.3305 1.88395 0.941977 0.335679i $$-0.108966\pi$$
0.941977 + 0.335679i $$0.108966\pi$$
$$314$$ 0 0
$$315$$ 3.00000 0.169031
$$316$$ 0 0
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 1.81665 0.101396
$$322$$ 0 0
$$323$$ 6.60555 0.367543
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 0 0
$$327$$ 10.9722 0.606766
$$328$$ 0 0
$$329$$ 4.60555 0.253912
$$330$$ 0 0
$$331$$ 5.69722 0.313148 0.156574 0.987666i $$-0.449955\pi$$
0.156574 + 0.987666i $$0.449955\pi$$
$$332$$ 0 0
$$333$$ −3.39445 −0.186015
$$334$$ 0 0
$$335$$ −3.90833 −0.213535
$$336$$ 0 0
$$337$$ −16.0000 −0.871576 −0.435788 0.900049i $$-0.643530\pi$$
−0.435788 + 0.900049i $$0.643530\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −30.0000 −1.61048 −0.805242 0.592946i $$-0.797965\pi$$
−0.805242 + 0.592946i $$0.797965\pi$$
$$348$$ 0 0
$$349$$ −29.2111 −1.56363 −0.781817 0.623508i $$-0.785707\pi$$
−0.781817 + 0.623508i $$0.785707\pi$$
$$350$$ 0 0
$$351$$ −37.0278 −1.97640
$$352$$ 0 0
$$353$$ −15.6333 −0.832077 −0.416039 0.909347i $$-0.636582\pi$$
−0.416039 + 0.909347i $$0.636582\pi$$
$$354$$ 0 0
$$355$$ 18.0000 0.955341
$$356$$ 0 0
$$357$$ 1.30278 0.0689502
$$358$$ 0 0
$$359$$ −6.90833 −0.364608 −0.182304 0.983242i $$-0.558355\pi$$
−0.182304 + 0.983242i $$0.558355\pi$$
$$360$$ 0 0
$$361$$ 24.6333 1.29649
$$362$$ 0 0
$$363$$ 14.3305 0.752158
$$364$$ 0 0
$$365$$ −18.2111 −0.953213
$$366$$ 0 0
$$367$$ −6.33053 −0.330451 −0.165226 0.986256i $$-0.552835\pi$$
−0.165226 + 0.986256i $$0.552835\pi$$
$$368$$ 0 0
$$369$$ −5.09167 −0.265062
$$370$$ 0 0
$$371$$ −3.69722 −0.191950
$$372$$ 0 0
$$373$$ 30.9361 1.60181 0.800905 0.598792i $$-0.204352\pi$$
0.800905 + 0.598792i $$0.204352\pi$$
$$374$$ 0 0
$$375$$ 14.0917 0.727691
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −6.78890 −0.348722 −0.174361 0.984682i $$-0.555786\pi$$
−0.174361 + 0.984682i $$0.555786\pi$$
$$380$$ 0 0
$$381$$ 19.4222 0.995030
$$382$$ 0 0
$$383$$ −18.4222 −0.941331 −0.470665 0.882312i $$-0.655986\pi$$
−0.470665 + 0.882312i $$0.655986\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −9.51388 −0.483618
$$388$$ 0 0
$$389$$ 10.1194 0.513075 0.256538 0.966534i $$-0.417418\pi$$
0.256538 + 0.966534i $$0.417418\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 27.6333 1.39392
$$394$$ 0 0
$$395$$ −29.0278 −1.46054
$$396$$ 0 0
$$397$$ 21.9361 1.10094 0.550470 0.834855i $$-0.314448\pi$$
0.550470 + 0.834855i $$0.314448\pi$$
$$398$$ 0 0
$$399$$ 8.60555 0.430816
$$400$$ 0 0
$$401$$ −16.6056 −0.829242 −0.414621 0.909994i $$-0.636086\pi$$
−0.414621 + 0.909994i $$0.636086\pi$$
$$402$$ 0 0
$$403$$ 28.4222 1.41581
$$404$$ 0 0
$$405$$ −7.81665 −0.388413
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −26.6056 −1.31556 −0.657780 0.753210i $$-0.728504\pi$$
−0.657780 + 0.753210i $$0.728504\pi$$
$$410$$ 0 0
$$411$$ 16.5416 0.815939
$$412$$ 0 0
$$413$$ −9.21110 −0.453249
$$414$$ 0 0
$$415$$ 13.8167 0.678233
$$416$$ 0 0
$$417$$ −9.23886 −0.452429
$$418$$ 0 0
$$419$$ −17.5139 −0.855609 −0.427804 0.903871i $$-0.640713\pi$$
−0.427804 + 0.903871i $$0.640713\pi$$
$$420$$ 0 0
$$421$$ 30.9361 1.50773 0.753866 0.657028i $$-0.228187\pi$$
0.753866 + 0.657028i $$0.228187\pi$$
$$422$$ 0 0
$$423$$ 6.00000 0.291730
$$424$$ 0 0
$$425$$ 0.302776 0.0146868
$$426$$ 0 0
$$427$$ −7.90833 −0.382711
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 15.6333 0.753030 0.376515 0.926411i $$-0.377122\pi$$
0.376515 + 0.926411i $$0.377122\pi$$
$$432$$ 0 0
$$433$$ 3.81665 0.183417 0.0917083 0.995786i $$-0.470767\pi$$
0.0917083 + 0.995786i $$0.470767\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 7.90833 0.377444 0.188722 0.982031i $$-0.439566\pi$$
0.188722 + 0.982031i $$0.439566\pi$$
$$440$$ 0 0
$$441$$ −1.30278 −0.0620369
$$442$$ 0 0
$$443$$ −21.2111 −1.00777 −0.503885 0.863771i $$-0.668096\pi$$
−0.503885 + 0.863771i $$0.668096\pi$$
$$444$$ 0 0
$$445$$ −38.2389 −1.81270
$$446$$ 0 0
$$447$$ 23.0917 1.09220
$$448$$ 0 0
$$449$$ −32.2389 −1.52145 −0.760723 0.649077i $$-0.775155\pi$$
−0.760723 + 0.649077i $$0.775155\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −14.4861 −0.680617
$$454$$ 0 0
$$455$$ 15.2111 0.713107
$$456$$ 0 0
$$457$$ −3.09167 −0.144622 −0.0723112 0.997382i $$-0.523037\pi$$
−0.0723112 + 0.997382i $$0.523037\pi$$
$$458$$ 0 0
$$459$$ 5.60555 0.261645
$$460$$ 0 0
$$461$$ 31.8167 1.48185 0.740925 0.671588i $$-0.234388\pi$$
0.740925 + 0.671588i $$0.234388\pi$$
$$462$$ 0 0
$$463$$ 16.6972 0.775986 0.387993 0.921662i $$-0.373168\pi$$
0.387993 + 0.921662i $$0.373168\pi$$
$$464$$ 0 0
$$465$$ 12.9083 0.598609
$$466$$ 0 0
$$467$$ −35.4500 −1.64043 −0.820214 0.572056i $$-0.806146\pi$$
−0.820214 + 0.572056i $$0.806146\pi$$
$$468$$ 0 0
$$469$$ 1.69722 0.0783705
$$470$$ 0 0
$$471$$ −9.39445 −0.432873
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 2.00000 0.0917663
$$476$$ 0 0
$$477$$ −4.81665 −0.220539
$$478$$ 0 0
$$479$$ −14.5139 −0.663156 −0.331578 0.943428i $$-0.607581\pi$$
−0.331578 + 0.943428i $$0.607581\pi$$
$$480$$ 0 0
$$481$$ −17.2111 −0.784759
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −14.5139 −0.659041
$$486$$ 0 0
$$487$$ −5.21110 −0.236138 −0.118069 0.993005i $$-0.537670\pi$$
−0.118069 + 0.993005i $$0.537670\pi$$
$$488$$ 0 0
$$489$$ −16.4222 −0.742638
$$490$$ 0 0
$$491$$ −25.3305 −1.14315 −0.571575 0.820550i $$-0.693668\pi$$
−0.571575 + 0.820550i $$0.693668\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −7.81665 −0.350625
$$498$$ 0 0
$$499$$ 1.57779 0.0706318 0.0353159 0.999376i $$-0.488756\pi$$
0.0353159 + 0.999376i $$0.488756\pi$$
$$500$$ 0 0
$$501$$ −30.9083 −1.38088
$$502$$ 0 0
$$503$$ −21.1472 −0.942906 −0.471453 0.881891i $$-0.656270\pi$$
−0.471453 + 0.881891i $$0.656270\pi$$
$$504$$ 0 0
$$505$$ −45.6333 −2.03066
$$506$$ 0 0
$$507$$ −39.9083 −1.77239
$$508$$ 0 0
$$509$$ 11.0278 0.488797 0.244398 0.969675i $$-0.421410\pi$$
0.244398 + 0.969675i $$0.421410\pi$$
$$510$$ 0 0
$$511$$ 7.90833 0.349844
$$512$$ 0 0
$$513$$ 37.0278 1.63482
$$514$$ 0 0
$$515$$ 16.6056 0.731728
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 22.5416 0.989468
$$520$$ 0 0
$$521$$ 39.3583 1.72432 0.862159 0.506638i $$-0.169112\pi$$
0.862159 + 0.506638i $$0.169112\pi$$
$$522$$ 0 0
$$523$$ 10.7889 0.471766 0.235883 0.971782i $$-0.424202\pi$$
0.235883 + 0.971782i $$0.424202\pi$$
$$524$$ 0 0
$$525$$ 0.394449 0.0172152
$$526$$ 0 0
$$527$$ −4.30278 −0.187432
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ −25.8167 −1.11824
$$534$$ 0 0
$$535$$ −3.21110 −0.138828
$$536$$ 0 0
$$537$$ −12.9083 −0.557035
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −42.0555 −1.80811 −0.904054 0.427419i $$-0.859423\pi$$
−0.904054 + 0.427419i $$0.859423\pi$$
$$542$$ 0 0
$$543$$ 18.2389 0.782704
$$544$$ 0 0
$$545$$ −19.3944 −0.830767
$$546$$ 0 0
$$547$$ −31.2111 −1.33449 −0.667245 0.744838i $$-0.732527\pi$$
−0.667245 + 0.744838i $$0.732527\pi$$
$$548$$ 0 0
$$549$$ −10.3028 −0.439712
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 12.6056 0.536043
$$554$$ 0 0
$$555$$ −7.81665 −0.331798
$$556$$ 0 0
$$557$$ −33.6333 −1.42509 −0.712544 0.701627i $$-0.752457\pi$$
−0.712544 + 0.701627i $$0.752457\pi$$
$$558$$ 0 0
$$559$$ −48.2389 −2.04029
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −27.6333 −1.16461 −0.582303 0.812972i $$-0.697848\pi$$
−0.582303 + 0.812972i $$0.697848\pi$$
$$564$$ 0 0
$$565$$ 10.6056 0.446179
$$566$$ 0 0
$$567$$ 3.39445 0.142553
$$568$$ 0 0
$$569$$ 4.88057 0.204604 0.102302 0.994753i $$-0.467379\pi$$
0.102302 + 0.994753i $$0.467379\pi$$
$$570$$ 0 0
$$571$$ −9.02776 −0.377800 −0.188900 0.981996i $$-0.560492\pi$$
−0.188900 + 0.981996i $$0.560492\pi$$
$$572$$ 0 0
$$573$$ 27.0000 1.12794
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 28.2389 1.17560 0.587800 0.809007i $$-0.299994\pi$$
0.587800 + 0.809007i $$0.299994\pi$$
$$578$$ 0 0
$$579$$ 15.3944 0.639771
$$580$$ 0 0
$$581$$ −6.00000 −0.248922
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 19.8167 0.819318
$$586$$ 0 0
$$587$$ −10.6056 −0.437738 −0.218869 0.975754i $$-0.570237\pi$$
−0.218869 + 0.975754i $$0.570237\pi$$
$$588$$ 0 0
$$589$$ −28.4222 −1.17112
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ 43.8167 1.79933 0.899667 0.436576i $$-0.143809\pi$$
0.899667 + 0.436576i $$0.143809\pi$$
$$594$$ 0 0
$$595$$ −2.30278 −0.0944046
$$596$$ 0 0
$$597$$ 27.5139 1.12607
$$598$$ 0 0
$$599$$ −43.9638 −1.79631 −0.898157 0.439675i $$-0.855094\pi$$
−0.898157 + 0.439675i $$0.855094\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 0 0
$$603$$ 2.21110 0.0900431
$$604$$ 0 0
$$605$$ −25.3305 −1.02983
$$606$$ 0 0
$$607$$ 21.5139 0.873221 0.436611 0.899651i $$-0.356179\pi$$
0.436611 + 0.899651i $$0.356179\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 30.4222 1.23075
$$612$$ 0 0
$$613$$ −29.1472 −1.17724 −0.588622 0.808408i $$-0.700329\pi$$
−0.588622 + 0.808408i $$0.700329\pi$$
$$614$$ 0 0
$$615$$ −11.7250 −0.472797
$$616$$ 0 0
$$617$$ −24.0000 −0.966204 −0.483102 0.875564i $$-0.660490\pi$$
−0.483102 + 0.875564i $$0.660490\pi$$
$$618$$ 0 0
$$619$$ −43.6333 −1.75377 −0.876885 0.480700i $$-0.840383\pi$$
−0.876885 + 0.480700i $$0.840383\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 16.6056 0.665287
$$624$$ 0 0
$$625$$ −26.4222 −1.05689
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 2.60555 0.103890
$$630$$ 0 0
$$631$$ 8.11943 0.323229 0.161615 0.986854i $$-0.448330\pi$$
0.161615 + 0.986854i $$0.448330\pi$$
$$632$$ 0 0
$$633$$ 27.3944 1.08883
$$634$$ 0 0
$$635$$ −34.3305 −1.36237
$$636$$ 0 0
$$637$$ −6.60555 −0.261721
$$638$$ 0 0
$$639$$ −10.1833 −0.402847
$$640$$ 0 0
$$641$$ −12.8444 −0.507324 −0.253662 0.967293i $$-0.581635\pi$$
−0.253662 + 0.967293i $$0.581635\pi$$
$$642$$ 0 0
$$643$$ 38.1472 1.50438 0.752189 0.658947i $$-0.228998\pi$$
0.752189 + 0.658947i $$0.228998\pi$$
$$644$$ 0 0
$$645$$ −21.9083 −0.862640
$$646$$ 0 0
$$647$$ −18.4222 −0.724252 −0.362126 0.932129i $$-0.617949\pi$$
−0.362126 + 0.932129i $$0.617949\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −5.60555 −0.219699
$$652$$ 0 0
$$653$$ −28.0555 −1.09790 −0.548949 0.835856i $$-0.684972\pi$$
−0.548949 + 0.835856i $$0.684972\pi$$
$$654$$ 0 0
$$655$$ −48.8444 −1.90851
$$656$$ 0 0
$$657$$ 10.3028 0.401950
$$658$$ 0 0
$$659$$ 38.3028 1.49206 0.746032 0.665910i $$-0.231957\pi$$
0.746032 + 0.665910i $$0.231957\pi$$
$$660$$ 0 0
$$661$$ 14.1833 0.551668 0.275834 0.961205i $$-0.411046\pi$$
0.275834 + 0.961205i $$0.411046\pi$$
$$662$$ 0 0
$$663$$ 8.60555 0.334212
$$664$$ 0 0
$$665$$ −15.2111 −0.589861
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −11.7611 −0.454712
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −20.1833 −0.778011 −0.389005 0.921235i $$-0.627181\pi$$
−0.389005 + 0.921235i $$0.627181\pi$$
$$674$$ 0 0
$$675$$ 1.69722 0.0653262
$$676$$ 0 0
$$677$$ −48.4222 −1.86102 −0.930508 0.366271i $$-0.880634\pi$$
−0.930508 + 0.366271i $$0.880634\pi$$
$$678$$ 0 0
$$679$$ 6.30278 0.241878
$$680$$ 0 0
$$681$$ −7.18335 −0.275266
$$682$$ 0 0
$$683$$ 41.4500 1.58604 0.793019 0.609196i $$-0.208508\pi$$
0.793019 + 0.609196i $$0.208508\pi$$
$$684$$ 0 0
$$685$$ −29.2389 −1.11716
$$686$$ 0 0
$$687$$ 14.0555 0.536251
$$688$$ 0 0
$$689$$ −24.4222 −0.930412
$$690$$ 0 0
$$691$$ 29.4861 1.12170 0.560852 0.827916i $$-0.310473\pi$$
0.560852 + 0.827916i $$0.310473\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 16.3305 0.619452
$$696$$ 0 0
$$697$$ 3.90833 0.148038
$$698$$ 0 0
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ 18.8444 0.711744 0.355872 0.934535i $$-0.384184\pi$$
0.355872 + 0.934535i $$0.384184\pi$$
$$702$$ 0 0
$$703$$ 17.2111 0.649129
$$704$$ 0 0
$$705$$ 13.8167 0.520365
$$706$$ 0 0
$$707$$ 19.8167 0.745282
$$708$$ 0 0
$$709$$ −12.1833 −0.457555 −0.228778 0.973479i $$-0.573473\pi$$
−0.228778 + 0.973479i $$0.573473\pi$$
$$710$$ 0 0
$$711$$ 16.4222 0.615881
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 14.7250 0.549914
$$718$$ 0 0
$$719$$ 29.3028 1.09281 0.546405 0.837521i $$-0.315996\pi$$
0.546405 + 0.837521i $$0.315996\pi$$
$$720$$ 0 0
$$721$$ −7.21110 −0.268555
$$722$$ 0 0
$$723$$ −11.0555 −0.411159
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 22.4222 0.831594 0.415797 0.909458i $$-0.363503\pi$$
0.415797 + 0.909458i $$0.363503\pi$$
$$728$$ 0 0
$$729$$ 26.3305 0.975205
$$730$$ 0 0
$$731$$ 7.30278 0.270103
$$732$$ 0 0
$$733$$ 30.2389 1.11690 0.558449 0.829539i $$-0.311397\pi$$
0.558449 + 0.829539i $$0.311397\pi$$
$$734$$ 0 0
$$735$$ −3.00000 −0.110657
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −25.3583 −0.932820 −0.466410 0.884569i $$-0.654453\pi$$
−0.466410 + 0.884569i $$0.654453\pi$$
$$740$$ 0 0
$$741$$ 56.8444 2.08823
$$742$$ 0 0
$$743$$ −1.81665 −0.0666466 −0.0333233 0.999445i $$-0.510609\pi$$
−0.0333233 + 0.999445i $$0.510609\pi$$
$$744$$ 0 0
$$745$$ −40.8167 −1.49541
$$746$$ 0 0
$$747$$ −7.81665 −0.285996
$$748$$ 0 0
$$749$$ 1.39445 0.0509520
$$750$$ 0 0
$$751$$ −18.6056 −0.678926 −0.339463 0.940619i $$-0.610245\pi$$
−0.339463 + 0.940619i $$0.610245\pi$$
$$752$$ 0 0
$$753$$ −5.44996 −0.198608
$$754$$ 0 0
$$755$$ 25.6056 0.931881
$$756$$ 0 0
$$757$$ 45.7250 1.66190 0.830951 0.556345i $$-0.187797\pi$$
0.830951 + 0.556345i $$0.187797\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 37.8167 1.37085 0.685426 0.728142i $$-0.259616\pi$$
0.685426 + 0.728142i $$0.259616\pi$$
$$762$$ 0 0
$$763$$ 8.42221 0.304904
$$764$$ 0 0
$$765$$ −3.00000 −0.108465
$$766$$ 0 0
$$767$$ −60.8444 −2.19696
$$768$$ 0 0
$$769$$ −53.2666 −1.92084 −0.960422 0.278550i $$-0.910146\pi$$
−0.960422 + 0.278550i $$0.910146\pi$$
$$770$$ 0 0
$$771$$ −4.18335 −0.150660
$$772$$ 0 0
$$773$$ −25.8167 −0.928560 −0.464280 0.885688i $$-0.653687\pi$$
−0.464280 + 0.885688i $$0.653687\pi$$
$$774$$ 0 0
$$775$$ −1.30278 −0.0467971
$$776$$ 0 0
$$777$$ 3.39445 0.121775
$$778$$ 0 0
$$779$$ 25.8167 0.924978
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 16.6056 0.592678
$$786$$ 0 0
$$787$$ 2.42221 0.0863423 0.0431711 0.999068i $$-0.486254\pi$$
0.0431711 + 0.999068i $$0.486254\pi$$
$$788$$ 0 0
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ −4.60555 −0.163755
$$792$$ 0 0
$$793$$ −52.2389 −1.85506
$$794$$ 0 0
$$795$$ −11.0917 −0.393381
$$796$$ 0 0
$$797$$ −0.422205 −0.0149553 −0.00747764 0.999972i $$-0.502380\pi$$
−0.00747764 + 0.999972i $$0.502380\pi$$
$$798$$ 0 0
$$799$$ −4.60555 −0.162933
$$800$$ 0 0
$$801$$ 21.6333 0.764375
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 19.8167 0.697579
$$808$$ 0 0
$$809$$ 40.6056 1.42762 0.713808 0.700342i $$-0.246969\pi$$
0.713808 + 0.700342i $$0.246969\pi$$
$$810$$ 0 0
$$811$$ 4.09167 0.143678 0.0718390 0.997416i $$-0.477113\pi$$
0.0718390 + 0.997416i $$0.477113\pi$$
$$812$$ 0 0
$$813$$ −1.57779 −0.0553357
$$814$$ 0 0
$$815$$ 29.0278 1.01680
$$816$$ 0 0
$$817$$ 48.2389 1.68766
$$818$$ 0 0
$$819$$ −8.60555 −0.300702
$$820$$ 0 0
$$821$$ 10.6056 0.370136 0.185068 0.982726i $$-0.440749\pi$$
0.185068 + 0.982726i $$0.440749\pi$$
$$822$$ 0 0
$$823$$ −43.8722 −1.52929 −0.764644 0.644453i $$-0.777085\pi$$
−0.764644 + 0.644453i $$0.777085\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 27.6333 0.960904 0.480452 0.877021i $$-0.340473\pi$$
0.480452 + 0.877021i $$0.340473\pi$$
$$828$$ 0 0
$$829$$ 26.6056 0.924049 0.462024 0.886867i $$-0.347123\pi$$
0.462024 + 0.886867i $$0.347123\pi$$
$$830$$ 0 0
$$831$$ −29.2111 −1.01332
$$832$$ 0 0
$$833$$ 1.00000 0.0346479
$$834$$ 0 0
$$835$$ 54.6333 1.89066
$$836$$ 0 0
$$837$$ −24.1194 −0.833689
$$838$$ 0 0
$$839$$ −18.4222 −0.636005 −0.318003 0.948090i $$-0.603012\pi$$
−0.318003 + 0.948090i $$0.603012\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 0 0
$$843$$ −4.81665 −0.165894
$$844$$ 0 0
$$845$$ 70.5416 2.42671
$$846$$ 0 0
$$847$$ 11.0000 0.377964
$$848$$ 0 0
$$849$$ 12.3944 0.425376
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 16.4222 0.562286 0.281143 0.959666i $$-0.409287\pi$$
0.281143 + 0.959666i $$0.409287\pi$$
$$854$$ 0 0
$$855$$ −19.8167 −0.677715
$$856$$ 0 0
$$857$$ −34.5416 −1.17992 −0.589960 0.807433i $$-0.700856\pi$$
−0.589960 + 0.807433i $$0.700856\pi$$
$$858$$ 0 0
$$859$$ 37.4500 1.27778 0.638888 0.769300i $$-0.279395\pi$$
0.638888 + 0.769300i $$0.279395\pi$$
$$860$$ 0 0
$$861$$ 5.09167 0.173524
$$862$$ 0 0
$$863$$ −18.9083 −0.643647 −0.321823 0.946800i $$-0.604296\pi$$
−0.321823 + 0.946800i $$0.604296\pi$$
$$864$$ 0 0
$$865$$ −39.8444 −1.35475
$$866$$ 0 0
$$867$$ −1.30278 −0.0442446
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 11.2111 0.379874
$$872$$ 0 0
$$873$$ 8.21110 0.277904
$$874$$ 0 0
$$875$$ 10.8167 0.365670
$$876$$ 0 0
$$877$$ 32.6056 1.10101 0.550506 0.834831i $$-0.314435\pi$$
0.550506 + 0.834831i $$0.314435\pi$$
$$878$$ 0 0
$$879$$ −28.1833 −0.950601
$$880$$ 0 0
$$881$$ −9.69722 −0.326708 −0.163354 0.986568i $$-0.552231\pi$$
−0.163354 + 0.986568i $$0.552231\pi$$
$$882$$ 0 0
$$883$$ −14.6695 −0.493667 −0.246833 0.969058i $$-0.579390\pi$$
−0.246833 + 0.969058i $$0.579390\pi$$
$$884$$ 0 0
$$885$$ −27.6333 −0.928883
$$886$$ 0 0
$$887$$ 48.9083 1.64218 0.821090 0.570798i $$-0.193366\pi$$
0.821090 + 0.570798i $$0.193366\pi$$
$$888$$ 0 0
$$889$$ 14.9083 0.500009
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −30.4222 −1.01804
$$894$$ 0 0
$$895$$ 22.8167 0.762677
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 3.69722 0.123172
$$902$$ 0 0
$$903$$ 9.51388 0.316602
$$904$$ 0 0
$$905$$ −32.2389 −1.07166
$$906$$ 0 0
$$907$$ −22.8444 −0.758536 −0.379268 0.925287i $$-0.623824\pi$$
−0.379268 + 0.925287i $$0.623824\pi$$
$$908$$ 0 0
$$909$$ 25.8167 0.856284
$$910$$ 0 0
$$911$$ −37.8167 −1.25292 −0.626461 0.779453i $$-0.715497\pi$$
−0.626461 + 0.779453i $$0.715497\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −23.7250 −0.784324
$$916$$ 0 0
$$917$$ 21.2111 0.700452
$$918$$ 0 0
$$919$$ −24.3305 −0.802590 −0.401295 0.915949i $$-0.631440\pi$$
−0.401295 + 0.915949i $$0.631440\pi$$
$$920$$ 0 0
$$921$$ 1.57779 0.0519901
$$922$$ 0 0
$$923$$ −51.6333 −1.69953
$$924$$ 0 0
$$925$$ 0.788897 0.0259388
$$926$$ 0 0
$$927$$ −9.39445 −0.308554
$$928$$ 0 0
$$929$$ 53.9361 1.76959 0.884793 0.465985i $$-0.154300\pi$$
0.884793 + 0.465985i $$0.154300\pi$$
$$930$$ 0 0
$$931$$ 6.60555 0.216488
$$932$$ 0 0
$$933$$ −2.44996 −0.0802081
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −6.78890 −0.221784 −0.110892 0.993832i $$-0.535371\pi$$
−0.110892 + 0.993832i $$0.535371\pi$$
$$938$$ 0 0
$$939$$ −43.4222 −1.41703
$$940$$ 0 0
$$941$$ 0.275019 0.00896537 0.00448269 0.999990i $$-0.498573\pi$$
0.00448269 + 0.999990i $$0.498573\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ −12.9083 −0.419908
$$946$$ 0 0
$$947$$ 48.8444 1.58723 0.793615 0.608420i $$-0.208196\pi$$
0.793615 + 0.608420i $$0.208196\pi$$
$$948$$ 0 0
$$949$$ 52.2389 1.69575
$$950$$ 0 0
$$951$$ −7.81665 −0.253472
$$952$$ 0 0
$$953$$ 17.5139 0.567330 0.283665 0.958923i $$-0.408450\pi$$
0.283665 + 0.958923i $$0.408450\pi$$
$$954$$ 0 0
$$955$$ −47.7250 −1.54434
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 12.6972 0.410015
$$960$$ 0 0
$$961$$ −12.4861 −0.402778
$$962$$ 0 0
$$963$$ 1.81665 0.0585409
$$964$$ 0 0
$$965$$ −27.2111 −0.875956
$$966$$ 0 0
$$967$$ −16.5139 −0.531051 −0.265525 0.964104i $$-0.585545\pi$$
−0.265525 + 0.964104i $$0.585545\pi$$
$$968$$ 0 0
$$969$$ −8.60555 −0.276450
$$970$$ 0 0
$$971$$ 36.8444 1.18239 0.591197 0.806527i $$-0.298656\pi$$
0.591197 + 0.806527i $$0.298656\pi$$
$$972$$ 0 0
$$973$$ −7.09167 −0.227349
$$974$$ 0 0
$$975$$ 2.60555 0.0834444
$$976$$ 0 0
$$977$$ −33.1472 −1.06047 −0.530236 0.847850i $$-0.677897\pi$$
−0.530236 + 0.847850i $$0.677897\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 10.9722 0.350317
$$982$$ 0 0
$$983$$ −23.0917 −0.736510 −0.368255 0.929725i $$-0.620045\pi$$
−0.368255 + 0.929725i $$0.620045\pi$$
$$984$$ 0 0
$$985$$ 10.6056 0.337921
$$986$$ 0 0
$$987$$ −6.00000 −0.190982
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −14.0000 −0.444725 −0.222362 0.974964i $$-0.571377\pi$$
−0.222362 + 0.974964i $$0.571377\pi$$
$$992$$ 0 0
$$993$$ −7.42221 −0.235537
$$994$$ 0 0
$$995$$ −48.6333 −1.54178
$$996$$ 0 0
$$997$$ −14.1472 −0.448046 −0.224023 0.974584i $$-0.571919\pi$$
−0.224023 + 0.974584i $$0.571919\pi$$
$$998$$ 0 0
$$999$$ 14.6056 0.462099
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 7616.2.a.v.1.1 2
4.3 odd 2 7616.2.a.q.1.2 2
8.3 odd 2 476.2.a.d.1.1 2
8.5 even 2 1904.2.a.h.1.2 2
24.11 even 2 4284.2.a.n.1.2 2
56.27 even 2 3332.2.a.j.1.2 2
136.67 odd 2 8092.2.a.k.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.d.1.1 2 8.3 odd 2
1904.2.a.h.1.2 2 8.5 even 2
3332.2.a.j.1.2 2 56.27 even 2
4284.2.a.n.1.2 2 24.11 even 2
7616.2.a.q.1.2 2 4.3 odd 2
7616.2.a.v.1.1 2 1.1 even 1 trivial
8092.2.a.k.1.2 2 136.67 odd 2