# Properties

 Label 1064.2.a.b.1.2 Level $1064$ Weight $2$ Character 1064.1 Self dual yes Analytic conductor $8.496$ 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] = [1064,2,Mod(1,1064)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1064, 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("1064.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1064 = 2^{3} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1064.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: $$8.49608277506$$ 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: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 1064.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.30278 q^{3} -1.00000 q^{5} -1.00000 q^{7} -1.30278 q^{9} +O(q^{10})$$ $$q+1.30278 q^{3} -1.00000 q^{5} -1.00000 q^{7} -1.30278 q^{9} -2.30278 q^{11} -3.60555 q^{13} -1.30278 q^{15} +1.69722 q^{17} +1.00000 q^{19} -1.30278 q^{21} -0.394449 q^{23} -4.00000 q^{25} -5.60555 q^{27} -4.30278 q^{29} -8.30278 q^{31} -3.00000 q^{33} +1.00000 q^{35} +3.60555 q^{37} -4.69722 q^{39} +0.302776 q^{41} +7.21110 q^{43} +1.30278 q^{45} -7.60555 q^{47} +1.00000 q^{49} +2.21110 q^{51} -3.90833 q^{53} +2.30278 q^{55} +1.30278 q^{57} +5.60555 q^{59} +8.21110 q^{61} +1.30278 q^{63} +3.60555 q^{65} -10.9083 q^{67} -0.513878 q^{69} +8.81665 q^{71} -5.90833 q^{73} -5.21110 q^{75} +2.30278 q^{77} -14.0000 q^{79} -3.39445 q^{81} -10.5139 q^{83} -1.69722 q^{85} -5.60555 q^{87} +13.8167 q^{89} +3.60555 q^{91} -10.8167 q^{93} -1.00000 q^{95} +16.8167 q^{97} +3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{5} - 2 q^{7} + q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^5 - 2 * q^7 + q^9 $$2 q - q^{3} - 2 q^{5} - 2 q^{7} + q^{9} - q^{11} + q^{15} + 7 q^{17} + 2 q^{19} + q^{21} - 8 q^{23} - 8 q^{25} - 4 q^{27} - 5 q^{29} - 13 q^{31} - 6 q^{33} + 2 q^{35} - 13 q^{39} - 3 q^{41} - q^{45} - 8 q^{47} + 2 q^{49} - 10 q^{51} + 3 q^{53} + q^{55} - q^{57} + 4 q^{59} + 2 q^{61} - q^{63} - 11 q^{67} + 17 q^{69} - 4 q^{71} - q^{73} + 4 q^{75} + q^{77} - 28 q^{79} - 14 q^{81} - 3 q^{83} - 7 q^{85} - 4 q^{87} + 6 q^{89} - 2 q^{95} + 12 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^5 - 2 * q^7 + q^9 - q^11 + q^15 + 7 * q^17 + 2 * q^19 + q^21 - 8 * q^23 - 8 * q^25 - 4 * q^27 - 5 * q^29 - 13 * q^31 - 6 * q^33 + 2 * q^35 - 13 * q^39 - 3 * q^41 - q^45 - 8 * q^47 + 2 * q^49 - 10 * q^51 + 3 * q^53 + q^55 - q^57 + 4 * q^59 + 2 * q^61 - q^63 - 11 * q^67 + 17 * q^69 - 4 * q^71 - q^73 + 4 * q^75 + q^77 - 28 * q^79 - 14 * q^81 - 3 * q^83 - 7 * q^85 - 4 * q^87 + 6 * q^89 - 2 * q^95 + 12 * q^97 + 6 * 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
$$3$$ 1.30278 0.752158 0.376079 0.926588i $$-0.377272\pi$$
0.376079 + 0.926588i $$0.377272\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ −1.30278 −0.434259
$$10$$ 0 0
$$11$$ −2.30278 −0.694313 −0.347156 0.937807i $$-0.612853\pi$$
−0.347156 + 0.937807i $$0.612853\pi$$
$$12$$ 0 0
$$13$$ −3.60555 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$14$$ 0 0
$$15$$ −1.30278 −0.336375
$$16$$ 0 0
$$17$$ 1.69722 0.411637 0.205819 0.978590i $$-0.434014\pi$$
0.205819 + 0.978590i $$0.434014\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −1.30278 −0.284289
$$22$$ 0 0
$$23$$ −0.394449 −0.0822482 −0.0411241 0.999154i $$-0.513094\pi$$
−0.0411241 + 0.999154i $$0.513094\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ −5.60555 −1.07879
$$28$$ 0 0
$$29$$ −4.30278 −0.799005 −0.399503 0.916732i $$-0.630817\pi$$
−0.399503 + 0.916732i $$0.630817\pi$$
$$30$$ 0 0
$$31$$ −8.30278 −1.49122 −0.745611 0.666381i $$-0.767842\pi$$
−0.745611 + 0.666381i $$0.767842\pi$$
$$32$$ 0 0
$$33$$ −3.00000 −0.522233
$$34$$ 0 0
$$35$$ 1.00000 0.169031
$$36$$ 0 0
$$37$$ 3.60555 0.592749 0.296374 0.955072i $$-0.404222\pi$$
0.296374 + 0.955072i $$0.404222\pi$$
$$38$$ 0 0
$$39$$ −4.69722 −0.752158
$$40$$ 0 0
$$41$$ 0.302776 0.0472856 0.0236428 0.999720i $$-0.492474\pi$$
0.0236428 + 0.999720i $$0.492474\pi$$
$$42$$ 0 0
$$43$$ 7.21110 1.09968 0.549841 0.835269i $$-0.314688\pi$$
0.549841 + 0.835269i $$0.314688\pi$$
$$44$$ 0 0
$$45$$ 1.30278 0.194206
$$46$$ 0 0
$$47$$ −7.60555 −1.10938 −0.554692 0.832056i $$-0.687164\pi$$
−0.554692 + 0.832056i $$0.687164\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 2.21110 0.309616
$$52$$ 0 0
$$53$$ −3.90833 −0.536850 −0.268425 0.963301i $$-0.586503\pi$$
−0.268425 + 0.963301i $$0.586503\pi$$
$$54$$ 0 0
$$55$$ 2.30278 0.310506
$$56$$ 0 0
$$57$$ 1.30278 0.172557
$$58$$ 0 0
$$59$$ 5.60555 0.729781 0.364890 0.931051i $$-0.381106\pi$$
0.364890 + 0.931051i $$0.381106\pi$$
$$60$$ 0 0
$$61$$ 8.21110 1.05132 0.525662 0.850694i $$-0.323818\pi$$
0.525662 + 0.850694i $$0.323818\pi$$
$$62$$ 0 0
$$63$$ 1.30278 0.164134
$$64$$ 0 0
$$65$$ 3.60555 0.447214
$$66$$ 0 0
$$67$$ −10.9083 −1.33266 −0.666332 0.745655i $$-0.732137\pi$$
−0.666332 + 0.745655i $$0.732137\pi$$
$$68$$ 0 0
$$69$$ −0.513878 −0.0618637
$$70$$ 0 0
$$71$$ 8.81665 1.04634 0.523172 0.852227i $$-0.324748\pi$$
0.523172 + 0.852227i $$0.324748\pi$$
$$72$$ 0 0
$$73$$ −5.90833 −0.691517 −0.345759 0.938323i $$-0.612378\pi$$
−0.345759 + 0.938323i $$0.612378\pi$$
$$74$$ 0 0
$$75$$ −5.21110 −0.601726
$$76$$ 0 0
$$77$$ 2.30278 0.262426
$$78$$ 0 0
$$79$$ −14.0000 −1.57512 −0.787562 0.616236i $$-0.788657\pi$$
−0.787562 + 0.616236i $$0.788657\pi$$
$$80$$ 0 0
$$81$$ −3.39445 −0.377161
$$82$$ 0 0
$$83$$ −10.5139 −1.15405 −0.577024 0.816727i $$-0.695786\pi$$
−0.577024 + 0.816727i $$0.695786\pi$$
$$84$$ 0 0
$$85$$ −1.69722 −0.184090
$$86$$ 0 0
$$87$$ −5.60555 −0.600978
$$88$$ 0 0
$$89$$ 13.8167 1.46456 0.732281 0.681002i $$-0.238456\pi$$
0.732281 + 0.681002i $$0.238456\pi$$
$$90$$ 0 0
$$91$$ 3.60555 0.377964
$$92$$ 0 0
$$93$$ −10.8167 −1.12163
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 16.8167 1.70747 0.853736 0.520706i $$-0.174331\pi$$
0.853736 + 0.520706i $$0.174331\pi$$
$$98$$ 0 0
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ 2.39445 0.238257 0.119128 0.992879i $$-0.461990\pi$$
0.119128 + 0.992879i $$0.461990\pi$$
$$102$$ 0 0
$$103$$ −4.39445 −0.432998 −0.216499 0.976283i $$-0.569464\pi$$
−0.216499 + 0.976283i $$0.569464\pi$$
$$104$$ 0 0
$$105$$ 1.30278 0.127138
$$106$$ 0 0
$$107$$ 10.8167 1.04569 0.522843 0.852429i $$-0.324872\pi$$
0.522843 + 0.852429i $$0.324872\pi$$
$$108$$ 0 0
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ 0 0
$$111$$ 4.69722 0.445841
$$112$$ 0 0
$$113$$ 17.1194 1.61046 0.805230 0.592962i $$-0.202042\pi$$
0.805230 + 0.592962i $$0.202042\pi$$
$$114$$ 0 0
$$115$$ 0.394449 0.0367825
$$116$$ 0 0
$$117$$ 4.69722 0.434259
$$118$$ 0 0
$$119$$ −1.69722 −0.155584
$$120$$ 0 0
$$121$$ −5.69722 −0.517929
$$122$$ 0 0
$$123$$ 0.394449 0.0355662
$$124$$ 0 0
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ −15.0278 −1.33350 −0.666749 0.745282i $$-0.732315\pi$$
−0.666749 + 0.745282i $$0.732315\pi$$
$$128$$ 0 0
$$129$$ 9.39445 0.827135
$$130$$ 0 0
$$131$$ −14.3028 −1.24964 −0.624820 0.780769i $$-0.714828\pi$$
−0.624820 + 0.780769i $$0.714828\pi$$
$$132$$ 0 0
$$133$$ −1.00000 −0.0867110
$$134$$ 0 0
$$135$$ 5.60555 0.482449
$$136$$ 0 0
$$137$$ 0.788897 0.0674001 0.0337000 0.999432i $$-0.489271\pi$$
0.0337000 + 0.999432i $$0.489271\pi$$
$$138$$ 0 0
$$139$$ 4.39445 0.372732 0.186366 0.982480i $$-0.440329\pi$$
0.186366 + 0.982480i $$0.440329\pi$$
$$140$$ 0 0
$$141$$ −9.90833 −0.834432
$$142$$ 0 0
$$143$$ 8.30278 0.694313
$$144$$ 0 0
$$145$$ 4.30278 0.357326
$$146$$ 0 0
$$147$$ 1.30278 0.107451
$$148$$ 0 0
$$149$$ 3.00000 0.245770 0.122885 0.992421i $$-0.460785\pi$$
0.122885 + 0.992421i $$0.460785\pi$$
$$150$$ 0 0
$$151$$ −5.09167 −0.414354 −0.207177 0.978303i $$-0.566428\pi$$
−0.207177 + 0.978303i $$0.566428\pi$$
$$152$$ 0 0
$$153$$ −2.21110 −0.178757
$$154$$ 0 0
$$155$$ 8.30278 0.666895
$$156$$ 0 0
$$157$$ −2.48612 −0.198414 −0.0992071 0.995067i $$-0.531631\pi$$
−0.0992071 + 0.995067i $$0.531631\pi$$
$$158$$ 0 0
$$159$$ −5.09167 −0.403796
$$160$$ 0 0
$$161$$ 0.394449 0.0310869
$$162$$ 0 0
$$163$$ −0.513878 −0.0402500 −0.0201250 0.999797i $$-0.506406\pi$$
−0.0201250 + 0.999797i $$0.506406\pi$$
$$164$$ 0 0
$$165$$ 3.00000 0.233550
$$166$$ 0 0
$$167$$ −21.4222 −1.65770 −0.828850 0.559471i $$-0.811004\pi$$
−0.828850 + 0.559471i $$0.811004\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ −1.30278 −0.0996257
$$172$$ 0 0
$$173$$ 13.8167 1.05046 0.525230 0.850960i $$-0.323979\pi$$
0.525230 + 0.850960i $$0.323979\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ 7.30278 0.548910
$$178$$ 0 0
$$179$$ −8.30278 −0.620579 −0.310289 0.950642i $$-0.600426\pi$$
−0.310289 + 0.950642i $$0.600426\pi$$
$$180$$ 0 0
$$181$$ 5.69722 0.423471 0.211736 0.977327i $$-0.432088\pi$$
0.211736 + 0.977327i $$0.432088\pi$$
$$182$$ 0 0
$$183$$ 10.6972 0.790762
$$184$$ 0 0
$$185$$ −3.60555 −0.265085
$$186$$ 0 0
$$187$$ −3.90833 −0.285805
$$188$$ 0 0
$$189$$ 5.60555 0.407744
$$190$$ 0 0
$$191$$ −9.72498 −0.703675 −0.351837 0.936061i $$-0.614443\pi$$
−0.351837 + 0.936061i $$0.614443\pi$$
$$192$$ 0 0
$$193$$ −15.6972 −1.12991 −0.564955 0.825121i $$-0.691107\pi$$
−0.564955 + 0.825121i $$0.691107\pi$$
$$194$$ 0 0
$$195$$ 4.69722 0.336375
$$196$$ 0 0
$$197$$ 3.90833 0.278457 0.139228 0.990260i $$-0.455538\pi$$
0.139228 + 0.990260i $$0.455538\pi$$
$$198$$ 0 0
$$199$$ 5.60555 0.397367 0.198683 0.980064i $$-0.436333\pi$$
0.198683 + 0.980064i $$0.436333\pi$$
$$200$$ 0 0
$$201$$ −14.2111 −1.00237
$$202$$ 0 0
$$203$$ 4.30278 0.301996
$$204$$ 0 0
$$205$$ −0.302776 −0.0211468
$$206$$ 0 0
$$207$$ 0.513878 0.0357170
$$208$$ 0 0
$$209$$ −2.30278 −0.159286
$$210$$ 0 0
$$211$$ 10.6972 0.736427 0.368214 0.929741i $$-0.379969\pi$$
0.368214 + 0.929741i $$0.379969\pi$$
$$212$$ 0 0
$$213$$ 11.4861 0.787016
$$214$$ 0 0
$$215$$ −7.21110 −0.491793
$$216$$ 0 0
$$217$$ 8.30278 0.563629
$$218$$ 0 0
$$219$$ −7.69722 −0.520130
$$220$$ 0 0
$$221$$ −6.11943 −0.411637
$$222$$ 0 0
$$223$$ 8.21110 0.549856 0.274928 0.961465i $$-0.411346\pi$$
0.274928 + 0.961465i $$0.411346\pi$$
$$224$$ 0 0
$$225$$ 5.21110 0.347407
$$226$$ 0 0
$$227$$ 11.6972 0.776372 0.388186 0.921581i $$-0.373102\pi$$
0.388186 + 0.921581i $$0.373102\pi$$
$$228$$ 0 0
$$229$$ 3.21110 0.212196 0.106098 0.994356i $$-0.466164\pi$$
0.106098 + 0.994356i $$0.466164\pi$$
$$230$$ 0 0
$$231$$ 3.00000 0.197386
$$232$$ 0 0
$$233$$ 29.5139 1.93352 0.966759 0.255688i $$-0.0823021\pi$$
0.966759 + 0.255688i $$0.0823021\pi$$
$$234$$ 0 0
$$235$$ 7.60555 0.496131
$$236$$ 0 0
$$237$$ −18.2389 −1.18474
$$238$$ 0 0
$$239$$ −14.0278 −0.907380 −0.453690 0.891160i $$-0.649893\pi$$
−0.453690 + 0.891160i $$0.649893\pi$$
$$240$$ 0 0
$$241$$ −0.183346 −0.0118104 −0.00590518 0.999983i $$-0.501880\pi$$
−0.00590518 + 0.999983i $$0.501880\pi$$
$$242$$ 0 0
$$243$$ 12.3944 0.795104
$$244$$ 0 0
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ −3.60555 −0.229416
$$248$$ 0 0
$$249$$ −13.6972 −0.868026
$$250$$ 0 0
$$251$$ 26.7250 1.68687 0.843433 0.537235i $$-0.180531\pi$$
0.843433 + 0.537235i $$0.180531\pi$$
$$252$$ 0 0
$$253$$ 0.908327 0.0571060
$$254$$ 0 0
$$255$$ −2.21110 −0.138465
$$256$$ 0 0
$$257$$ 5.90833 0.368551 0.184276 0.982875i $$-0.441006\pi$$
0.184276 + 0.982875i $$0.441006\pi$$
$$258$$ 0 0
$$259$$ −3.60555 −0.224038
$$260$$ 0 0
$$261$$ 5.60555 0.346975
$$262$$ 0 0
$$263$$ −4.69722 −0.289643 −0.144822 0.989458i $$-0.546261\pi$$
−0.144822 + 0.989458i $$0.546261\pi$$
$$264$$ 0 0
$$265$$ 3.90833 0.240087
$$266$$ 0 0
$$267$$ 18.0000 1.10158
$$268$$ 0 0
$$269$$ −2.51388 −0.153274 −0.0766369 0.997059i $$-0.524418\pi$$
−0.0766369 + 0.997059i $$0.524418\pi$$
$$270$$ 0 0
$$271$$ −5.09167 −0.309297 −0.154649 0.987970i $$-0.549425\pi$$
−0.154649 + 0.987970i $$0.549425\pi$$
$$272$$ 0 0
$$273$$ 4.69722 0.284289
$$274$$ 0 0
$$275$$ 9.21110 0.555450
$$276$$ 0 0
$$277$$ −27.8167 −1.67134 −0.835670 0.549231i $$-0.814921\pi$$
−0.835670 + 0.549231i $$0.814921\pi$$
$$278$$ 0 0
$$279$$ 10.8167 0.647576
$$280$$ 0 0
$$281$$ 19.4222 1.15863 0.579316 0.815103i $$-0.303320\pi$$
0.579316 + 0.815103i $$0.303320\pi$$
$$282$$ 0 0
$$283$$ 22.5139 1.33831 0.669156 0.743122i $$-0.266656\pi$$
0.669156 + 0.743122i $$0.266656\pi$$
$$284$$ 0 0
$$285$$ −1.30278 −0.0771698
$$286$$ 0 0
$$287$$ −0.302776 −0.0178723
$$288$$ 0 0
$$289$$ −14.1194 −0.830555
$$290$$ 0 0
$$291$$ 21.9083 1.28429
$$292$$ 0 0
$$293$$ 18.6333 1.08857 0.544285 0.838901i $$-0.316801\pi$$
0.544285 + 0.838901i $$0.316801\pi$$
$$294$$ 0 0
$$295$$ −5.60555 −0.326368
$$296$$ 0 0
$$297$$ 12.9083 0.749017
$$298$$ 0 0
$$299$$ 1.42221 0.0822482
$$300$$ 0 0
$$301$$ −7.21110 −0.415641
$$302$$ 0 0
$$303$$ 3.11943 0.179207
$$304$$ 0 0
$$305$$ −8.21110 −0.470166
$$306$$ 0 0
$$307$$ −14.1194 −0.805838 −0.402919 0.915236i $$-0.632004\pi$$
−0.402919 + 0.915236i $$0.632004\pi$$
$$308$$ 0 0
$$309$$ −5.72498 −0.325683
$$310$$ 0 0
$$311$$ 7.33053 0.415676 0.207838 0.978163i $$-0.433357\pi$$
0.207838 + 0.978163i $$0.433357\pi$$
$$312$$ 0 0
$$313$$ −27.8167 −1.57229 −0.786145 0.618042i $$-0.787926\pi$$
−0.786145 + 0.618042i $$0.787926\pi$$
$$314$$ 0 0
$$315$$ −1.30278 −0.0734031
$$316$$ 0 0
$$317$$ −11.2111 −0.629678 −0.314839 0.949145i $$-0.601951\pi$$
−0.314839 + 0.949145i $$0.601951\pi$$
$$318$$ 0 0
$$319$$ 9.90833 0.554760
$$320$$ 0 0
$$321$$ 14.0917 0.786520
$$322$$ 0 0
$$323$$ 1.69722 0.0944361
$$324$$ 0 0
$$325$$ 14.4222 0.800000
$$326$$ 0 0
$$327$$ −9.11943 −0.504306
$$328$$ 0 0
$$329$$ 7.60555 0.419308
$$330$$ 0 0
$$331$$ −9.48612 −0.521404 −0.260702 0.965419i $$-0.583954\pi$$
−0.260702 + 0.965419i $$0.583954\pi$$
$$332$$ 0 0
$$333$$ −4.69722 −0.257406
$$334$$ 0 0
$$335$$ 10.9083 0.595986
$$336$$ 0 0
$$337$$ −29.6972 −1.61771 −0.808855 0.588008i $$-0.799913\pi$$
−0.808855 + 0.588008i $$0.799913\pi$$
$$338$$ 0 0
$$339$$ 22.3028 1.21132
$$340$$ 0 0
$$341$$ 19.1194 1.03538
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0.513878 0.0276663
$$346$$ 0 0
$$347$$ −10.6972 −0.574257 −0.287129 0.957892i $$-0.592701\pi$$
−0.287129 + 0.957892i $$0.592701\pi$$
$$348$$ 0 0
$$349$$ 6.51388 0.348680 0.174340 0.984686i $$-0.444221\pi$$
0.174340 + 0.984686i $$0.444221\pi$$
$$350$$ 0 0
$$351$$ 20.2111 1.07879
$$352$$ 0 0
$$353$$ 1.27502 0.0678624 0.0339312 0.999424i $$-0.489197\pi$$
0.0339312 + 0.999424i $$0.489197\pi$$
$$354$$ 0 0
$$355$$ −8.81665 −0.467939
$$356$$ 0 0
$$357$$ −2.21110 −0.117024
$$358$$ 0 0
$$359$$ −10.3028 −0.543760 −0.271880 0.962331i $$-0.587645\pi$$
−0.271880 + 0.962331i $$0.587645\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −7.42221 −0.389565
$$364$$ 0 0
$$365$$ 5.90833 0.309256
$$366$$ 0 0
$$367$$ 18.2111 0.950612 0.475306 0.879821i $$-0.342337\pi$$
0.475306 + 0.879821i $$0.342337\pi$$
$$368$$ 0 0
$$369$$ −0.394449 −0.0205342
$$370$$ 0 0
$$371$$ 3.90833 0.202910
$$372$$ 0 0
$$373$$ 19.9083 1.03081 0.515407 0.856945i $$-0.327641\pi$$
0.515407 + 0.856945i $$0.327641\pi$$
$$374$$ 0 0
$$375$$ 11.7250 0.605475
$$376$$ 0 0
$$377$$ 15.5139 0.799005
$$378$$ 0 0
$$379$$ 21.4222 1.10038 0.550192 0.835038i $$-0.314554\pi$$
0.550192 + 0.835038i $$0.314554\pi$$
$$380$$ 0 0
$$381$$ −19.5778 −1.00300
$$382$$ 0 0
$$383$$ −32.4500 −1.65812 −0.829058 0.559163i $$-0.811123\pi$$
−0.829058 + 0.559163i $$0.811123\pi$$
$$384$$ 0 0
$$385$$ −2.30278 −0.117360
$$386$$ 0 0
$$387$$ −9.39445 −0.477547
$$388$$ 0 0
$$389$$ −3.69722 −0.187457 −0.0937284 0.995598i $$-0.529879\pi$$
−0.0937284 + 0.995598i $$0.529879\pi$$
$$390$$ 0 0
$$391$$ −0.669468 −0.0338565
$$392$$ 0 0
$$393$$ −18.6333 −0.939926
$$394$$ 0 0
$$395$$ 14.0000 0.704416
$$396$$ 0 0
$$397$$ −4.60555 −0.231146 −0.115573 0.993299i $$-0.536870\pi$$
−0.115573 + 0.993299i $$0.536870\pi$$
$$398$$ 0 0
$$399$$ −1.30278 −0.0652204
$$400$$ 0 0
$$401$$ −12.9083 −0.644611 −0.322306 0.946636i $$-0.604458\pi$$
−0.322306 + 0.946636i $$0.604458\pi$$
$$402$$ 0 0
$$403$$ 29.9361 1.49122
$$404$$ 0 0
$$405$$ 3.39445 0.168672
$$406$$ 0 0
$$407$$ −8.30278 −0.411553
$$408$$ 0 0
$$409$$ −38.7527 −1.91620 −0.958100 0.286435i $$-0.907530\pi$$
−0.958100 + 0.286435i $$0.907530\pi$$
$$410$$ 0 0
$$411$$ 1.02776 0.0506955
$$412$$ 0 0
$$413$$ −5.60555 −0.275831
$$414$$ 0 0
$$415$$ 10.5139 0.516106
$$416$$ 0 0
$$417$$ 5.72498 0.280354
$$418$$ 0 0
$$419$$ −2.81665 −0.137603 −0.0688013 0.997630i $$-0.521917\pi$$
−0.0688013 + 0.997630i $$0.521917\pi$$
$$420$$ 0 0
$$421$$ −15.1833 −0.739991 −0.369996 0.929034i $$-0.620641\pi$$
−0.369996 + 0.929034i $$0.620641\pi$$
$$422$$ 0 0
$$423$$ 9.90833 0.481759
$$424$$ 0 0
$$425$$ −6.78890 −0.329310
$$426$$ 0 0
$$427$$ −8.21110 −0.397363
$$428$$ 0 0
$$429$$ 10.8167 0.522233
$$430$$ 0 0
$$431$$ −36.4222 −1.75440 −0.877198 0.480129i $$-0.840590\pi$$
−0.877198 + 0.480129i $$0.840590\pi$$
$$432$$ 0 0
$$433$$ 7.78890 0.374311 0.187155 0.982330i $$-0.440073\pi$$
0.187155 + 0.982330i $$0.440073\pi$$
$$434$$ 0 0
$$435$$ 5.60555 0.268766
$$436$$ 0 0
$$437$$ −0.394449 −0.0188690
$$438$$ 0 0
$$439$$ 5.21110 0.248712 0.124356 0.992238i $$-0.460313\pi$$
0.124356 + 0.992238i $$0.460313\pi$$
$$440$$ 0 0
$$441$$ −1.30278 −0.0620369
$$442$$ 0 0
$$443$$ 14.0917 0.669516 0.334758 0.942304i $$-0.391345\pi$$
0.334758 + 0.942304i $$0.391345\pi$$
$$444$$ 0 0
$$445$$ −13.8167 −0.654972
$$446$$ 0 0
$$447$$ 3.90833 0.184858
$$448$$ 0 0
$$449$$ −17.1194 −0.807916 −0.403958 0.914778i $$-0.632366\pi$$
−0.403958 + 0.914778i $$0.632366\pi$$
$$450$$ 0 0
$$451$$ −0.697224 −0.0328310
$$452$$ 0 0
$$453$$ −6.63331 −0.311660
$$454$$ 0 0
$$455$$ −3.60555 −0.169031
$$456$$ 0 0
$$457$$ −24.9361 −1.16646 −0.583230 0.812307i $$-0.698212\pi$$
−0.583230 + 0.812307i $$0.698212\pi$$
$$458$$ 0 0
$$459$$ −9.51388 −0.444070
$$460$$ 0 0
$$461$$ 15.1194 0.704182 0.352091 0.935966i $$-0.385471\pi$$
0.352091 + 0.935966i $$0.385471\pi$$
$$462$$ 0 0
$$463$$ −12.2111 −0.567498 −0.283749 0.958899i $$-0.591578\pi$$
−0.283749 + 0.958899i $$0.591578\pi$$
$$464$$ 0 0
$$465$$ 10.8167 0.501610
$$466$$ 0 0
$$467$$ −29.1194 −1.34749 −0.673743 0.738966i $$-0.735315\pi$$
−0.673743 + 0.738966i $$0.735315\pi$$
$$468$$ 0 0
$$469$$ 10.9083 0.503700
$$470$$ 0 0
$$471$$ −3.23886 −0.149239
$$472$$ 0 0
$$473$$ −16.6056 −0.763524
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 5.09167 0.233132
$$478$$ 0 0
$$479$$ 10.3028 0.470746 0.235373 0.971905i $$-0.424369\pi$$
0.235373 + 0.971905i $$0.424369\pi$$
$$480$$ 0 0
$$481$$ −13.0000 −0.592749
$$482$$ 0 0
$$483$$ 0.513878 0.0233823
$$484$$ 0 0
$$485$$ −16.8167 −0.763605
$$486$$ 0 0
$$487$$ −37.2389 −1.68745 −0.843727 0.536773i $$-0.819643\pi$$
−0.843727 + 0.536773i $$0.819643\pi$$
$$488$$ 0 0
$$489$$ −0.669468 −0.0302744
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 0 0
$$493$$ −7.30278 −0.328900
$$494$$ 0 0
$$495$$ −3.00000 −0.134840
$$496$$ 0 0
$$497$$ −8.81665 −0.395481
$$498$$ 0 0
$$499$$ −18.3305 −0.820587 −0.410294 0.911953i $$-0.634574\pi$$
−0.410294 + 0.911953i $$0.634574\pi$$
$$500$$ 0 0
$$501$$ −27.9083 −1.24685
$$502$$ 0 0
$$503$$ 24.8444 1.10776 0.553879 0.832597i $$-0.313147\pi$$
0.553879 + 0.832597i $$0.313147\pi$$
$$504$$ 0 0
$$505$$ −2.39445 −0.106552
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −8.78890 −0.389561 −0.194781 0.980847i $$-0.562399\pi$$
−0.194781 + 0.980847i $$0.562399\pi$$
$$510$$ 0 0
$$511$$ 5.90833 0.261369
$$512$$ 0 0
$$513$$ −5.60555 −0.247491
$$514$$ 0 0
$$515$$ 4.39445 0.193643
$$516$$ 0 0
$$517$$ 17.5139 0.770259
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −13.4222 −0.588037 −0.294019 0.955800i $$-0.594993\pi$$
−0.294019 + 0.955800i $$0.594993\pi$$
$$522$$ 0 0
$$523$$ 5.42221 0.237096 0.118548 0.992948i $$-0.462176\pi$$
0.118548 + 0.992948i $$0.462176\pi$$
$$524$$ 0 0
$$525$$ 5.21110 0.227431
$$526$$ 0 0
$$527$$ −14.0917 −0.613843
$$528$$ 0 0
$$529$$ −22.8444 −0.993235
$$530$$ 0 0
$$531$$ −7.30278 −0.316913
$$532$$ 0 0
$$533$$ −1.09167 −0.0472856
$$534$$ 0 0
$$535$$ −10.8167 −0.467645
$$536$$ 0 0
$$537$$ −10.8167 −0.466773
$$538$$ 0 0
$$539$$ −2.30278 −0.0991876
$$540$$ 0 0
$$541$$ −39.2111 −1.68582 −0.842908 0.538057i $$-0.819159\pi$$
−0.842908 + 0.538057i $$0.819159\pi$$
$$542$$ 0 0
$$543$$ 7.42221 0.318517
$$544$$ 0 0
$$545$$ 7.00000 0.299847
$$546$$ 0 0
$$547$$ −6.51388 −0.278513 −0.139257 0.990256i $$-0.544471\pi$$
−0.139257 + 0.990256i $$0.544471\pi$$
$$548$$ 0 0
$$549$$ −10.6972 −0.456546
$$550$$ 0 0
$$551$$ −4.30278 −0.183304
$$552$$ 0 0
$$553$$ 14.0000 0.595341
$$554$$ 0 0
$$555$$ −4.69722 −0.199386
$$556$$ 0 0
$$557$$ 35.7527 1.51489 0.757446 0.652898i $$-0.226447\pi$$
0.757446 + 0.652898i $$0.226447\pi$$
$$558$$ 0 0
$$559$$ −26.0000 −1.09968
$$560$$ 0 0
$$561$$ −5.09167 −0.214971
$$562$$ 0 0
$$563$$ −20.0278 −0.844069 −0.422035 0.906580i $$-0.638684\pi$$
−0.422035 + 0.906580i $$0.638684\pi$$
$$564$$ 0 0
$$565$$ −17.1194 −0.720220
$$566$$ 0 0
$$567$$ 3.39445 0.142553
$$568$$ 0 0
$$569$$ −14.0278 −0.588074 −0.294037 0.955794i $$-0.594999\pi$$
−0.294037 + 0.955794i $$0.594999\pi$$
$$570$$ 0 0
$$571$$ −17.7889 −0.744442 −0.372221 0.928144i $$-0.621404\pi$$
−0.372221 + 0.928144i $$0.621404\pi$$
$$572$$ 0 0
$$573$$ −12.6695 −0.529275
$$574$$ 0 0
$$575$$ 1.57779 0.0657986
$$576$$ 0 0
$$577$$ 3.72498 0.155073 0.0775365 0.996990i $$-0.475295\pi$$
0.0775365 + 0.996990i $$0.475295\pi$$
$$578$$ 0 0
$$579$$ −20.4500 −0.849871
$$580$$ 0 0
$$581$$ 10.5139 0.436189
$$582$$ 0 0
$$583$$ 9.00000 0.372742
$$584$$ 0 0
$$585$$ −4.69722 −0.194206
$$586$$ 0 0
$$587$$ −9.63331 −0.397609 −0.198805 0.980039i $$-0.563706\pi$$
−0.198805 + 0.980039i $$0.563706\pi$$
$$588$$ 0 0
$$589$$ −8.30278 −0.342110
$$590$$ 0 0
$$591$$ 5.09167 0.209443
$$592$$ 0 0
$$593$$ 23.1833 0.952026 0.476013 0.879438i $$-0.342082\pi$$
0.476013 + 0.879438i $$0.342082\pi$$
$$594$$ 0 0
$$595$$ 1.69722 0.0695794
$$596$$ 0 0
$$597$$ 7.30278 0.298883
$$598$$ 0 0
$$599$$ 14.3305 0.585530 0.292765 0.956184i $$-0.405425\pi$$
0.292765 + 0.956184i $$0.405425\pi$$
$$600$$ 0 0
$$601$$ 30.1472 1.22973 0.614865 0.788633i $$-0.289211\pi$$
0.614865 + 0.788633i $$0.289211\pi$$
$$602$$ 0 0
$$603$$ 14.2111 0.578721
$$604$$ 0 0
$$605$$ 5.69722 0.231625
$$606$$ 0 0
$$607$$ −8.81665 −0.357857 −0.178928 0.983862i $$-0.557263\pi$$
−0.178928 + 0.983862i $$0.557263\pi$$
$$608$$ 0 0
$$609$$ 5.60555 0.227148
$$610$$ 0 0
$$611$$ 27.4222 1.10938
$$612$$ 0 0
$$613$$ 39.3583 1.58967 0.794833 0.606828i $$-0.207558\pi$$
0.794833 + 0.606828i $$0.207558\pi$$
$$614$$ 0 0
$$615$$ −0.394449 −0.0159057
$$616$$ 0 0
$$617$$ −42.5416 −1.71266 −0.856331 0.516428i $$-0.827262\pi$$
−0.856331 + 0.516428i $$0.827262\pi$$
$$618$$ 0 0
$$619$$ −46.5694 −1.87178 −0.935891 0.352290i $$-0.885403\pi$$
−0.935891 + 0.352290i $$0.885403\pi$$
$$620$$ 0 0
$$621$$ 2.21110 0.0887285
$$622$$ 0 0
$$623$$ −13.8167 −0.553553
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ −3.00000 −0.119808
$$628$$ 0 0
$$629$$ 6.11943 0.243998
$$630$$ 0 0
$$631$$ −8.21110 −0.326879 −0.163439 0.986553i $$-0.552259\pi$$
−0.163439 + 0.986553i $$0.552259\pi$$
$$632$$ 0 0
$$633$$ 13.9361 0.553910
$$634$$ 0 0
$$635$$ 15.0278 0.596358
$$636$$ 0 0
$$637$$ −3.60555 −0.142857
$$638$$ 0 0
$$639$$ −11.4861 −0.454384
$$640$$ 0 0
$$641$$ −40.3028 −1.59186 −0.795932 0.605386i $$-0.793019\pi$$
−0.795932 + 0.605386i $$0.793019\pi$$
$$642$$ 0 0
$$643$$ 14.5778 0.574892 0.287446 0.957797i $$-0.407194\pi$$
0.287446 + 0.957797i $$0.407194\pi$$
$$644$$ 0 0
$$645$$ −9.39445 −0.369906
$$646$$ 0 0
$$647$$ −0.816654 −0.0321060 −0.0160530 0.999871i $$-0.505110\pi$$
−0.0160530 + 0.999871i $$0.505110\pi$$
$$648$$ 0 0
$$649$$ −12.9083 −0.506696
$$650$$ 0 0
$$651$$ 10.8167 0.423938
$$652$$ 0 0
$$653$$ −24.0278 −0.940279 −0.470139 0.882592i $$-0.655796\pi$$
−0.470139 + 0.882592i $$0.655796\pi$$
$$654$$ 0 0
$$655$$ 14.3028 0.558856
$$656$$ 0 0
$$657$$ 7.69722 0.300297
$$658$$ 0 0
$$659$$ 6.90833 0.269110 0.134555 0.990906i $$-0.457039\pi$$
0.134555 + 0.990906i $$0.457039\pi$$
$$660$$ 0 0
$$661$$ −16.7889 −0.653012 −0.326506 0.945195i $$-0.605871\pi$$
−0.326506 + 0.945195i $$0.605871\pi$$
$$662$$ 0 0
$$663$$ −7.97224 −0.309616
$$664$$ 0 0
$$665$$ 1.00000 0.0387783
$$666$$ 0 0
$$667$$ 1.69722 0.0657168
$$668$$ 0 0
$$669$$ 10.6972 0.413579
$$670$$ 0 0
$$671$$ −18.9083 −0.729948
$$672$$ 0 0
$$673$$ −31.7250 −1.22291 −0.611454 0.791280i $$-0.709415\pi$$
−0.611454 + 0.791280i $$0.709415\pi$$
$$674$$ 0 0
$$675$$ 22.4222 0.863031
$$676$$ 0 0
$$677$$ 17.4861 0.672046 0.336023 0.941854i $$-0.390918\pi$$
0.336023 + 0.941854i $$0.390918\pi$$
$$678$$ 0 0
$$679$$ −16.8167 −0.645364
$$680$$ 0 0
$$681$$ 15.2389 0.583954
$$682$$ 0 0
$$683$$ −21.2389 −0.812682 −0.406341 0.913721i $$-0.633196\pi$$
−0.406341 + 0.913721i $$0.633196\pi$$
$$684$$ 0 0
$$685$$ −0.788897 −0.0301422
$$686$$ 0 0
$$687$$ 4.18335 0.159605
$$688$$ 0 0
$$689$$ 14.0917 0.536850
$$690$$ 0 0
$$691$$ −5.39445 −0.205215 −0.102607 0.994722i $$-0.532718\pi$$
−0.102607 + 0.994722i $$0.532718\pi$$
$$692$$ 0 0
$$693$$ −3.00000 −0.113961
$$694$$ 0 0
$$695$$ −4.39445 −0.166691
$$696$$ 0 0
$$697$$ 0.513878 0.0194645
$$698$$ 0 0
$$699$$ 38.4500 1.45431
$$700$$ 0 0
$$701$$ −27.6333 −1.04370 −0.521848 0.853039i $$-0.674757\pi$$
−0.521848 + 0.853039i $$0.674757\pi$$
$$702$$ 0 0
$$703$$ 3.60555 0.135986
$$704$$ 0 0
$$705$$ 9.90833 0.373169
$$706$$ 0 0
$$707$$ −2.39445 −0.0900525
$$708$$ 0 0
$$709$$ −21.2389 −0.797642 −0.398821 0.917029i $$-0.630581\pi$$
−0.398821 + 0.917029i $$0.630581\pi$$
$$710$$ 0 0
$$711$$ 18.2389 0.684011
$$712$$ 0 0
$$713$$ 3.27502 0.122650
$$714$$ 0 0
$$715$$ −8.30278 −0.310506
$$716$$ 0 0
$$717$$ −18.2750 −0.682493
$$718$$ 0 0
$$719$$ 23.6333 0.881374 0.440687 0.897661i $$-0.354735\pi$$
0.440687 + 0.897661i $$0.354735\pi$$
$$720$$ 0 0
$$721$$ 4.39445 0.163658
$$722$$ 0 0
$$723$$ −0.238859 −0.00888326
$$724$$ 0 0
$$725$$ 17.2111 0.639204
$$726$$ 0 0
$$727$$ −0.577795 −0.0214292 −0.0107146 0.999943i $$-0.503411\pi$$
−0.0107146 + 0.999943i $$0.503411\pi$$
$$728$$ 0 0
$$729$$ 26.3305 0.975205
$$730$$ 0 0
$$731$$ 12.2389 0.452671
$$732$$ 0 0
$$733$$ 52.8444 1.95185 0.975926 0.218100i $$-0.0699859\pi$$
0.975926 + 0.218100i $$0.0699859\pi$$
$$734$$ 0 0
$$735$$ −1.30278 −0.0480536
$$736$$ 0 0
$$737$$ 25.1194 0.925286
$$738$$ 0 0
$$739$$ −24.8167 −0.912895 −0.456448 0.889750i $$-0.650878\pi$$
−0.456448 + 0.889750i $$0.650878\pi$$
$$740$$ 0 0
$$741$$ −4.69722 −0.172557
$$742$$ 0 0
$$743$$ 23.2389 0.852551 0.426276 0.904593i $$-0.359825\pi$$
0.426276 + 0.904593i $$0.359825\pi$$
$$744$$ 0 0
$$745$$ −3.00000 −0.109911
$$746$$ 0 0
$$747$$ 13.6972 0.501155
$$748$$ 0 0
$$749$$ −10.8167 −0.395232
$$750$$ 0 0
$$751$$ −42.1472 −1.53797 −0.768986 0.639265i $$-0.779239\pi$$
−0.768986 + 0.639265i $$0.779239\pi$$
$$752$$ 0 0
$$753$$ 34.8167 1.26879
$$754$$ 0 0
$$755$$ 5.09167 0.185305
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 0 0
$$759$$ 1.18335 0.0429527
$$760$$ 0 0
$$761$$ 3.00000 0.108750 0.0543750 0.998521i $$-0.482683\pi$$
0.0543750 + 0.998521i $$0.482683\pi$$
$$762$$ 0 0
$$763$$ 7.00000 0.253417
$$764$$ 0 0
$$765$$ 2.21110 0.0799426
$$766$$ 0 0
$$767$$ −20.2111 −0.729781
$$768$$ 0 0
$$769$$ 26.7889 0.966032 0.483016 0.875611i $$-0.339541\pi$$
0.483016 + 0.875611i $$0.339541\pi$$
$$770$$ 0 0
$$771$$ 7.69722 0.277209
$$772$$ 0 0
$$773$$ 27.8167 1.00050 0.500248 0.865882i $$-0.333242\pi$$
0.500248 + 0.865882i $$0.333242\pi$$
$$774$$ 0 0
$$775$$ 33.2111 1.19298
$$776$$ 0 0
$$777$$ −4.69722 −0.168512
$$778$$ 0 0
$$779$$ 0.302776 0.0108481
$$780$$ 0 0
$$781$$ −20.3028 −0.726490
$$782$$ 0 0
$$783$$ 24.1194 0.861958
$$784$$ 0 0
$$785$$ 2.48612 0.0887335
$$786$$ 0 0
$$787$$ 11.6333 0.414683 0.207341 0.978269i $$-0.433519\pi$$
0.207341 + 0.978269i $$0.433519\pi$$
$$788$$ 0 0
$$789$$ −6.11943 −0.217857
$$790$$ 0 0
$$791$$ −17.1194 −0.608697
$$792$$ 0 0
$$793$$ −29.6056 −1.05132
$$794$$ 0 0
$$795$$ 5.09167 0.180583
$$796$$ 0 0
$$797$$ −38.7250 −1.37171 −0.685855 0.727739i $$-0.740571\pi$$
−0.685855 + 0.727739i $$0.740571\pi$$
$$798$$ 0 0
$$799$$ −12.9083 −0.456664
$$800$$ 0 0
$$801$$ −18.0000 −0.635999
$$802$$ 0 0
$$803$$ 13.6056 0.480129
$$804$$ 0 0
$$805$$ −0.394449 −0.0139025
$$806$$ 0 0
$$807$$ −3.27502 −0.115286
$$808$$ 0 0
$$809$$ −11.1833 −0.393186 −0.196593 0.980485i $$-0.562988\pi$$
−0.196593 + 0.980485i $$0.562988\pi$$
$$810$$ 0 0
$$811$$ −24.4222 −0.857580 −0.428790 0.903404i $$-0.641060\pi$$
−0.428790 + 0.903404i $$0.641060\pi$$
$$812$$ 0 0
$$813$$ −6.63331 −0.232640
$$814$$ 0 0
$$815$$ 0.513878 0.0180004
$$816$$ 0 0
$$817$$ 7.21110 0.252285
$$818$$ 0 0
$$819$$ −4.69722 −0.164134
$$820$$ 0 0
$$821$$ 47.6333 1.66241 0.831207 0.555963i $$-0.187650\pi$$
0.831207 + 0.555963i $$0.187650\pi$$
$$822$$ 0 0
$$823$$ −6.23886 −0.217473 −0.108736 0.994071i $$-0.534680\pi$$
−0.108736 + 0.994071i $$0.534680\pi$$
$$824$$ 0 0
$$825$$ 12.0000 0.417786
$$826$$ 0 0
$$827$$ −9.60555 −0.334018 −0.167009 0.985955i $$-0.553411\pi$$
−0.167009 + 0.985955i $$0.553411\pi$$
$$828$$ 0 0
$$829$$ −42.0555 −1.46065 −0.730324 0.683101i $$-0.760631\pi$$
−0.730324 + 0.683101i $$0.760631\pi$$
$$830$$ 0 0
$$831$$ −36.2389 −1.25711
$$832$$ 0 0
$$833$$ 1.69722 0.0588053
$$834$$ 0 0
$$835$$ 21.4222 0.741346
$$836$$ 0 0
$$837$$ 46.5416 1.60871
$$838$$ 0 0
$$839$$ −8.36669 −0.288850 −0.144425 0.989516i $$-0.546133\pi$$
−0.144425 + 0.989516i $$0.546133\pi$$
$$840$$ 0 0
$$841$$ −10.4861 −0.361590
$$842$$ 0 0
$$843$$ 25.3028 0.871474
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.69722 0.195759
$$848$$ 0 0
$$849$$ 29.3305 1.00662
$$850$$ 0 0
$$851$$ −1.42221 −0.0487526
$$852$$ 0 0
$$853$$ 7.48612 0.256320 0.128160 0.991754i $$-0.459093\pi$$
0.128160 + 0.991754i $$0.459093\pi$$
$$854$$ 0 0
$$855$$ 1.30278 0.0445540
$$856$$ 0 0
$$857$$ 39.7527 1.35793 0.678964 0.734172i $$-0.262429\pi$$
0.678964 + 0.734172i $$0.262429\pi$$
$$858$$ 0 0
$$859$$ 35.7527 1.21987 0.609934 0.792452i $$-0.291196\pi$$
0.609934 + 0.792452i $$0.291196\pi$$
$$860$$ 0 0
$$861$$ −0.394449 −0.0134428
$$862$$ 0 0
$$863$$ −50.5139 −1.71951 −0.859756 0.510705i $$-0.829385\pi$$
−0.859756 + 0.510705i $$0.829385\pi$$
$$864$$ 0 0
$$865$$ −13.8167 −0.469780
$$866$$ 0 0
$$867$$ −18.3944 −0.624708
$$868$$ 0 0
$$869$$ 32.2389 1.09363
$$870$$ 0 0
$$871$$ 39.3305 1.33266
$$872$$ 0 0
$$873$$ −21.9083 −0.741485
$$874$$ 0 0
$$875$$ −9.00000 −0.304256
$$876$$ 0 0
$$877$$ 51.8722 1.75160 0.875799 0.482675i $$-0.160335\pi$$
0.875799 + 0.482675i $$0.160335\pi$$
$$878$$ 0 0
$$879$$ 24.2750 0.818776
$$880$$ 0 0
$$881$$ −29.9638 −1.00951 −0.504754 0.863263i $$-0.668417\pi$$
−0.504754 + 0.863263i $$0.668417\pi$$
$$882$$ 0 0
$$883$$ 35.6611 1.20009 0.600045 0.799966i $$-0.295149\pi$$
0.600045 + 0.799966i $$0.295149\pi$$
$$884$$ 0 0
$$885$$ −7.30278 −0.245480
$$886$$ 0 0
$$887$$ −37.7889 −1.26883 −0.634413 0.772994i $$-0.718758\pi$$
−0.634413 + 0.772994i $$0.718758\pi$$
$$888$$ 0 0
$$889$$ 15.0278 0.504015
$$890$$ 0 0
$$891$$ 7.81665 0.261868
$$892$$ 0 0
$$893$$ −7.60555 −0.254510
$$894$$ 0 0
$$895$$ 8.30278 0.277531
$$896$$ 0 0
$$897$$ 1.85281 0.0618637
$$898$$ 0 0
$$899$$ 35.7250 1.19149
$$900$$ 0 0
$$901$$ −6.63331 −0.220988
$$902$$ 0 0
$$903$$ −9.39445 −0.312628
$$904$$ 0 0
$$905$$ −5.69722 −0.189382
$$906$$ 0 0
$$907$$ 0.577795 0.0191854 0.00959268 0.999954i $$-0.496947\pi$$
0.00959268 + 0.999954i $$0.496947\pi$$
$$908$$ 0 0
$$909$$ −3.11943 −0.103465
$$910$$ 0 0
$$911$$ 12.7889 0.423715 0.211858 0.977301i $$-0.432049\pi$$
0.211858 + 0.977301i $$0.432049\pi$$
$$912$$ 0 0
$$913$$ 24.2111 0.801271
$$914$$ 0 0
$$915$$ −10.6972 −0.353639
$$916$$ 0 0
$$917$$ 14.3028 0.472319
$$918$$ 0 0
$$919$$ −3.84441 −0.126815 −0.0634077 0.997988i $$-0.520197\pi$$
−0.0634077 + 0.997988i $$0.520197\pi$$
$$920$$ 0 0
$$921$$ −18.3944 −0.606118
$$922$$ 0 0
$$923$$ −31.7889 −1.04634
$$924$$ 0 0
$$925$$ −14.4222 −0.474199
$$926$$ 0 0
$$927$$ 5.72498 0.188033
$$928$$ 0 0
$$929$$ 17.3028 0.567686 0.283843 0.958871i $$-0.408391\pi$$
0.283843 + 0.958871i $$0.408391\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ 9.55004 0.312654
$$934$$ 0 0
$$935$$ 3.90833 0.127816
$$936$$ 0 0
$$937$$ −9.93608 −0.324598 −0.162299 0.986742i $$-0.551891\pi$$
−0.162299 + 0.986742i $$0.551891\pi$$
$$938$$ 0 0
$$939$$ −36.2389 −1.18261
$$940$$ 0 0
$$941$$ 6.00000 0.195594 0.0977972 0.995206i $$-0.468820\pi$$
0.0977972 + 0.995206i $$0.468820\pi$$
$$942$$ 0 0
$$943$$ −0.119429 −0.00388916
$$944$$ 0 0
$$945$$ −5.60555 −0.182349
$$946$$ 0 0
$$947$$ −14.5139 −0.471638 −0.235819 0.971797i $$-0.575777\pi$$
−0.235819 + 0.971797i $$0.575777\pi$$
$$948$$ 0 0
$$949$$ 21.3028 0.691517
$$950$$ 0 0
$$951$$ −14.6056 −0.473617
$$952$$ 0 0
$$953$$ 42.1194 1.36438 0.682191 0.731174i $$-0.261027\pi$$
0.682191 + 0.731174i $$0.261027\pi$$
$$954$$ 0 0
$$955$$ 9.72498 0.314693
$$956$$ 0 0
$$957$$ 12.9083 0.417267
$$958$$ 0 0
$$959$$ −0.788897 −0.0254748
$$960$$ 0 0
$$961$$ 37.9361 1.22374
$$962$$ 0 0
$$963$$ −14.0917 −0.454098
$$964$$ 0 0
$$965$$ 15.6972 0.505312
$$966$$ 0 0
$$967$$ −6.48612 −0.208580 −0.104290 0.994547i $$-0.533257\pi$$
−0.104290 + 0.994547i $$0.533257\pi$$
$$968$$ 0 0
$$969$$ 2.21110 0.0710308
$$970$$ 0 0
$$971$$ 4.81665 0.154574 0.0772869 0.997009i $$-0.475374\pi$$
0.0772869 + 0.997009i $$0.475374\pi$$
$$972$$ 0 0
$$973$$ −4.39445 −0.140880
$$974$$ 0 0
$$975$$ 18.7889 0.601726
$$976$$ 0 0
$$977$$ 34.6056 1.10713 0.553565 0.832806i $$-0.313267\pi$$
0.553565 + 0.832806i $$0.313267\pi$$
$$978$$ 0 0
$$979$$ −31.8167 −1.01686
$$980$$ 0 0
$$981$$ 9.11943 0.291161
$$982$$ 0 0
$$983$$ −43.6611 −1.39257 −0.696286 0.717765i $$-0.745165\pi$$
−0.696286 + 0.717765i $$0.745165\pi$$
$$984$$ 0 0
$$985$$ −3.90833 −0.124530
$$986$$ 0 0
$$987$$ 9.90833 0.315386
$$988$$ 0 0
$$989$$ −2.84441 −0.0904470
$$990$$ 0 0
$$991$$ −22.2389 −0.706441 −0.353220 0.935540i $$-0.614913\pi$$
−0.353220 + 0.935540i $$0.614913\pi$$
$$992$$ 0 0
$$993$$ −12.3583 −0.392178
$$994$$ 0 0
$$995$$ −5.60555 −0.177708
$$996$$ 0 0
$$997$$ −13.7250 −0.434675 −0.217337 0.976097i $$-0.569737\pi$$
−0.217337 + 0.976097i $$0.569737\pi$$
$$998$$ 0 0
$$999$$ −20.2111 −0.639451
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 1064.2.a.b.1.2 2
3.2 odd 2 9576.2.a.bs.1.2 2
4.3 odd 2 2128.2.a.i.1.1 2
7.6 odd 2 7448.2.a.bb.1.1 2
8.3 odd 2 8512.2.a.r.1.2 2
8.5 even 2 8512.2.a.x.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1064.2.a.b.1.2 2 1.1 even 1 trivial
2128.2.a.i.1.1 2 4.3 odd 2
7448.2.a.bb.1.1 2 7.6 odd 2
8512.2.a.r.1.2 2 8.3 odd 2
8512.2.a.x.1.1 2 8.5 even 2
9576.2.a.bs.1.2 2 3.2 odd 2