# Properties

 Label 2800.2.a.bp.1.2 Level $2800$ Weight $2$ Character 2800.1 Self dual yes Analytic conductor $22.358$ Analytic rank $0$ 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] = [2800,2,Mod(1,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, 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("2800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.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: $$22.3581125660$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 175) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 2800.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.23607 q^{3} -1.00000 q^{7} +7.47214 q^{9} +O(q^{10})$$ $$q+3.23607 q^{3} -1.00000 q^{7} +7.47214 q^{9} +0.236068 q^{11} +1.23607 q^{13} +2.47214 q^{17} +4.47214 q^{19} -3.23607 q^{21} -6.23607 q^{23} +14.4721 q^{27} +5.00000 q^{29} -3.70820 q^{31} +0.763932 q^{33} +3.00000 q^{37} +4.00000 q^{39} +4.76393 q^{41} -1.76393 q^{43} +2.00000 q^{47} +1.00000 q^{49} +8.00000 q^{51} +8.47214 q^{53} +14.4721 q^{57} -11.7082 q^{59} -9.70820 q^{61} -7.47214 q^{63} +4.23607 q^{67} -20.1803 q^{69} -8.70820 q^{71} -8.76393 q^{73} -0.236068 q^{77} +11.1803 q^{79} +24.4164 q^{81} +7.70820 q^{83} +16.1803 q^{87} +17.2361 q^{89} -1.23607 q^{91} -12.0000 q^{93} +5.23607 q^{97} +1.76393 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^7 + 6 * q^9 $$2 q + 2 q^{3} - 2 q^{7} + 6 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{17} - 2 q^{21} - 8 q^{23} + 20 q^{27} + 10 q^{29} + 6 q^{31} + 6 q^{33} + 6 q^{37} + 8 q^{39} + 14 q^{41} - 8 q^{43} + 4 q^{47} + 2 q^{49} + 16 q^{51} + 8 q^{53} + 20 q^{57} - 10 q^{59} - 6 q^{61} - 6 q^{63} + 4 q^{67} - 18 q^{69} - 4 q^{71} - 22 q^{73} + 4 q^{77} + 22 q^{81} + 2 q^{83} + 10 q^{87} + 30 q^{89} + 2 q^{91} - 24 q^{93} + 6 q^{97} + 8 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^7 + 6 * q^9 - 4 * q^11 - 2 * q^13 - 4 * q^17 - 2 * q^21 - 8 * q^23 + 20 * q^27 + 10 * q^29 + 6 * q^31 + 6 * q^33 + 6 * q^37 + 8 * q^39 + 14 * q^41 - 8 * q^43 + 4 * q^47 + 2 * q^49 + 16 * q^51 + 8 * q^53 + 20 * q^57 - 10 * q^59 - 6 * q^61 - 6 * q^63 + 4 * q^67 - 18 * q^69 - 4 * q^71 - 22 * q^73 + 4 * q^77 + 22 * q^81 + 2 * q^83 + 10 * q^87 + 30 * q^89 + 2 * q^91 - 24 * q^93 + 6 * q^97 + 8 * 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
$$3$$ 3.23607 1.86834 0.934172 0.356822i $$-0.116140\pi$$
0.934172 + 0.356822i $$0.116140\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 7.47214 2.49071
$$10$$ 0 0
$$11$$ 0.236068 0.0711772 0.0355886 0.999367i $$-0.488669\pi$$
0.0355886 + 0.999367i $$0.488669\pi$$
$$12$$ 0 0
$$13$$ 1.23607 0.342824 0.171412 0.985199i $$-0.445167\pi$$
0.171412 + 0.985199i $$0.445167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.47214 0.599581 0.299791 0.954005i $$-0.403083\pi$$
0.299791 + 0.954005i $$0.403083\pi$$
$$18$$ 0 0
$$19$$ 4.47214 1.02598 0.512989 0.858395i $$-0.328538\pi$$
0.512989 + 0.858395i $$0.328538\pi$$
$$20$$ 0 0
$$21$$ −3.23607 −0.706168
$$22$$ 0 0
$$23$$ −6.23607 −1.30031 −0.650155 0.759802i $$-0.725296\pi$$
−0.650155 + 0.759802i $$0.725296\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 14.4721 2.78516
$$28$$ 0 0
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ −3.70820 −0.666013 −0.333007 0.942925i $$-0.608063\pi$$
−0.333007 + 0.942925i $$0.608063\pi$$
$$32$$ 0 0
$$33$$ 0.763932 0.132983
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ 0 0
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ 4.76393 0.744001 0.372001 0.928232i $$-0.378672\pi$$
0.372001 + 0.928232i $$0.378672\pi$$
$$42$$ 0 0
$$43$$ −1.76393 −0.268997 −0.134499 0.990914i $$-0.542942\pi$$
−0.134499 + 0.990914i $$0.542942\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 8.00000 1.12022
$$52$$ 0 0
$$53$$ 8.47214 1.16374 0.581869 0.813283i $$-0.302322\pi$$
0.581869 + 0.813283i $$0.302322\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 14.4721 1.91688
$$58$$ 0 0
$$59$$ −11.7082 −1.52428 −0.762139 0.647413i $$-0.775851\pi$$
−0.762139 + 0.647413i $$0.775851\pi$$
$$60$$ 0 0
$$61$$ −9.70820 −1.24301 −0.621504 0.783411i $$-0.713478\pi$$
−0.621504 + 0.783411i $$0.713478\pi$$
$$62$$ 0 0
$$63$$ −7.47214 −0.941401
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.23607 0.517518 0.258759 0.965942i $$-0.416686\pi$$
0.258759 + 0.965942i $$0.416686\pi$$
$$68$$ 0 0
$$69$$ −20.1803 −2.42943
$$70$$ 0 0
$$71$$ −8.70820 −1.03347 −0.516737 0.856144i $$-0.672853\pi$$
−0.516737 + 0.856144i $$0.672853\pi$$
$$72$$ 0 0
$$73$$ −8.76393 −1.02574 −0.512870 0.858466i $$-0.671418\pi$$
−0.512870 + 0.858466i $$0.671418\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −0.236068 −0.0269024
$$78$$ 0 0
$$79$$ 11.1803 1.25789 0.628943 0.777451i $$-0.283488\pi$$
0.628943 + 0.777451i $$0.283488\pi$$
$$80$$ 0 0
$$81$$ 24.4164 2.71293
$$82$$ 0 0
$$83$$ 7.70820 0.846085 0.423043 0.906110i $$-0.360962\pi$$
0.423043 + 0.906110i $$0.360962\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 16.1803 1.73471
$$88$$ 0 0
$$89$$ 17.2361 1.82702 0.913510 0.406817i $$-0.133361\pi$$
0.913510 + 0.406817i $$0.133361\pi$$
$$90$$ 0 0
$$91$$ −1.23607 −0.129575
$$92$$ 0 0
$$93$$ −12.0000 −1.24434
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 5.23607 0.531642 0.265821 0.964022i $$-0.414357\pi$$
0.265821 + 0.964022i $$0.414357\pi$$
$$98$$ 0 0
$$99$$ 1.76393 0.177282
$$100$$ 0 0
$$101$$ 4.76393 0.474029 0.237014 0.971506i $$-0.423831\pi$$
0.237014 + 0.971506i $$0.423831\pi$$
$$102$$ 0 0
$$103$$ −8.47214 −0.834784 −0.417392 0.908726i $$-0.637056\pi$$
−0.417392 + 0.908726i $$0.637056\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ 0 0
$$109$$ 8.41641 0.806146 0.403073 0.915168i $$-0.367942\pi$$
0.403073 + 0.915168i $$0.367942\pi$$
$$110$$ 0 0
$$111$$ 9.70820 0.921462
$$112$$ 0 0
$$113$$ −14.4164 −1.35618 −0.678091 0.734978i $$-0.737192\pi$$
−0.678091 + 0.734978i $$0.737192\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 9.23607 0.853875
$$118$$ 0 0
$$119$$ −2.47214 −0.226620
$$120$$ 0 0
$$121$$ −10.9443 −0.994934
$$122$$ 0 0
$$123$$ 15.4164 1.39005
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −13.6525 −1.21146 −0.605731 0.795670i $$-0.707119\pi$$
−0.605731 + 0.795670i $$0.707119\pi$$
$$128$$ 0 0
$$129$$ −5.70820 −0.502579
$$130$$ 0 0
$$131$$ 16.9443 1.48043 0.740214 0.672371i $$-0.234724\pi$$
0.740214 + 0.672371i $$0.234724\pi$$
$$132$$ 0 0
$$133$$ −4.47214 −0.387783
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −10.9443 −0.935032 −0.467516 0.883985i $$-0.654851\pi$$
−0.467516 + 0.883985i $$0.654851\pi$$
$$138$$ 0 0
$$139$$ −10.6525 −0.903531 −0.451766 0.892137i $$-0.649206\pi$$
−0.451766 + 0.892137i $$0.649206\pi$$
$$140$$ 0 0
$$141$$ 6.47214 0.545052
$$142$$ 0 0
$$143$$ 0.291796 0.0244012
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 3.23607 0.266906
$$148$$ 0 0
$$149$$ 3.94427 0.323127 0.161564 0.986862i $$-0.448346\pi$$
0.161564 + 0.986862i $$0.448346\pi$$
$$150$$ 0 0
$$151$$ 20.2361 1.64679 0.823394 0.567470i $$-0.192078\pi$$
0.823394 + 0.567470i $$0.192078\pi$$
$$152$$ 0 0
$$153$$ 18.4721 1.49338
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0.763932 0.0609684 0.0304842 0.999535i $$-0.490295\pi$$
0.0304842 + 0.999535i $$0.490295\pi$$
$$158$$ 0 0
$$159$$ 27.4164 2.17426
$$160$$ 0 0
$$161$$ 6.23607 0.491471
$$162$$ 0 0
$$163$$ 1.52786 0.119672 0.0598358 0.998208i $$-0.480942\pi$$
0.0598358 + 0.998208i $$0.480942\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −5.23607 −0.405179 −0.202590 0.979264i $$-0.564936\pi$$
−0.202590 + 0.979264i $$0.564936\pi$$
$$168$$ 0 0
$$169$$ −11.4721 −0.882472
$$170$$ 0 0
$$171$$ 33.4164 2.55542
$$172$$ 0 0
$$173$$ −11.5279 −0.876447 −0.438224 0.898866i $$-0.644392\pi$$
−0.438224 + 0.898866i $$0.644392\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −37.8885 −2.84788
$$178$$ 0 0
$$179$$ 23.4164 1.75022 0.875112 0.483920i $$-0.160787\pi$$
0.875112 + 0.483920i $$0.160787\pi$$
$$180$$ 0 0
$$181$$ 8.18034 0.608040 0.304020 0.952666i $$-0.401671\pi$$
0.304020 + 0.952666i $$0.401671\pi$$
$$182$$ 0 0
$$183$$ −31.4164 −2.32237
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.583592 0.0426765
$$188$$ 0 0
$$189$$ −14.4721 −1.05269
$$190$$ 0 0
$$191$$ −6.47214 −0.468307 −0.234154 0.972200i $$-0.575232\pi$$
−0.234154 + 0.972200i $$0.575232\pi$$
$$192$$ 0 0
$$193$$ 12.4164 0.893753 0.446876 0.894596i $$-0.352536\pi$$
0.446876 + 0.894596i $$0.352536\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1.47214 −0.104885 −0.0524427 0.998624i $$-0.516701\pi$$
−0.0524427 + 0.998624i $$0.516701\pi$$
$$198$$ 0 0
$$199$$ −7.23607 −0.512951 −0.256476 0.966551i $$-0.582561\pi$$
−0.256476 + 0.966551i $$0.582561\pi$$
$$200$$ 0 0
$$201$$ 13.7082 0.966902
$$202$$ 0 0
$$203$$ −5.00000 −0.350931
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −46.5967 −3.23870
$$208$$ 0 0
$$209$$ 1.05573 0.0730262
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 0 0
$$213$$ −28.1803 −1.93089
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3.70820 0.251729
$$218$$ 0 0
$$219$$ −28.3607 −1.91644
$$220$$ 0 0
$$221$$ 3.05573 0.205551
$$222$$ 0 0
$$223$$ −20.1803 −1.35138 −0.675688 0.737188i $$-0.736153\pi$$
−0.675688 + 0.737188i $$0.736153\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −21.4164 −1.42146 −0.710728 0.703466i $$-0.751635\pi$$
−0.710728 + 0.703466i $$0.751635\pi$$
$$228$$ 0 0
$$229$$ −4.47214 −0.295527 −0.147764 0.989023i $$-0.547207\pi$$
−0.147764 + 0.989023i $$0.547207\pi$$
$$230$$ 0 0
$$231$$ −0.763932 −0.0502630
$$232$$ 0 0
$$233$$ 7.94427 0.520447 0.260223 0.965548i $$-0.416204\pi$$
0.260223 + 0.965548i $$0.416204\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 36.1803 2.35017
$$238$$ 0 0
$$239$$ 5.52786 0.357568 0.178784 0.983888i $$-0.442784\pi$$
0.178784 + 0.983888i $$0.442784\pi$$
$$240$$ 0 0
$$241$$ −3.52786 −0.227250 −0.113625 0.993524i $$-0.536246\pi$$
−0.113625 + 0.993524i $$0.536246\pi$$
$$242$$ 0 0
$$243$$ 35.5967 2.28353
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.52786 0.351730
$$248$$ 0 0
$$249$$ 24.9443 1.58078
$$250$$ 0 0
$$251$$ −6.47214 −0.408518 −0.204259 0.978917i $$-0.565478\pi$$
−0.204259 + 0.978917i $$0.565478\pi$$
$$252$$ 0 0
$$253$$ −1.47214 −0.0925524
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −12.6525 −0.789240 −0.394620 0.918844i $$-0.629124\pi$$
−0.394620 + 0.918844i $$0.629124\pi$$
$$258$$ 0 0
$$259$$ −3.00000 −0.186411
$$260$$ 0 0
$$261$$ 37.3607 2.31257
$$262$$ 0 0
$$263$$ −16.2361 −1.00116 −0.500579 0.865691i $$-0.666880\pi$$
−0.500579 + 0.865691i $$0.666880\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 55.7771 3.41350
$$268$$ 0 0
$$269$$ −11.7082 −0.713862 −0.356931 0.934131i $$-0.616177\pi$$
−0.356931 + 0.934131i $$0.616177\pi$$
$$270$$ 0 0
$$271$$ −23.7082 −1.44017 −0.720085 0.693885i $$-0.755897\pi$$
−0.720085 + 0.693885i $$0.755897\pi$$
$$272$$ 0 0
$$273$$ −4.00000 −0.242091
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −19.8885 −1.19499 −0.597493 0.801874i $$-0.703837\pi$$
−0.597493 + 0.801874i $$0.703837\pi$$
$$278$$ 0 0
$$279$$ −27.7082 −1.65885
$$280$$ 0 0
$$281$$ −15.3607 −0.916341 −0.458171 0.888864i $$-0.651495\pi$$
−0.458171 + 0.888864i $$0.651495\pi$$
$$282$$ 0 0
$$283$$ −17.4164 −1.03530 −0.517649 0.855593i $$-0.673193\pi$$
−0.517649 + 0.855593i $$0.673193\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −4.76393 −0.281206
$$288$$ 0 0
$$289$$ −10.8885 −0.640503
$$290$$ 0 0
$$291$$ 16.9443 0.993291
$$292$$ 0 0
$$293$$ −31.1246 −1.81832 −0.909160 0.416448i $$-0.863275\pi$$
−0.909160 + 0.416448i $$0.863275\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3.41641 0.198240
$$298$$ 0 0
$$299$$ −7.70820 −0.445777
$$300$$ 0 0
$$301$$ 1.76393 0.101671
$$302$$ 0 0
$$303$$ 15.4164 0.885649
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −4.58359 −0.261599 −0.130800 0.991409i $$-0.541754\pi$$
−0.130800 + 0.991409i $$0.541754\pi$$
$$308$$ 0 0
$$309$$ −27.4164 −1.55966
$$310$$ 0 0
$$311$$ −24.3607 −1.38137 −0.690684 0.723157i $$-0.742690\pi$$
−0.690684 + 0.723157i $$0.742690\pi$$
$$312$$ 0 0
$$313$$ 19.5279 1.10378 0.551890 0.833917i $$-0.313907\pi$$
0.551890 + 0.833917i $$0.313907\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 25.3607 1.42440 0.712199 0.701978i $$-0.247699\pi$$
0.712199 + 0.701978i $$0.247699\pi$$
$$318$$ 0 0
$$319$$ 1.18034 0.0660863
$$320$$ 0 0
$$321$$ −25.8885 −1.44496
$$322$$ 0 0
$$323$$ 11.0557 0.615157
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 27.2361 1.50616
$$328$$ 0 0
$$329$$ −2.00000 −0.110264
$$330$$ 0 0
$$331$$ 24.7082 1.35809 0.679043 0.734099i $$-0.262395\pi$$
0.679043 + 0.734099i $$0.262395\pi$$
$$332$$ 0 0
$$333$$ 22.4164 1.22841
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −16.4721 −0.897294 −0.448647 0.893709i $$-0.648094\pi$$
−0.448647 + 0.893709i $$0.648094\pi$$
$$338$$ 0 0
$$339$$ −46.6525 −2.53381
$$340$$ 0 0
$$341$$ −0.875388 −0.0474049
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −20.2361 −1.08633 −0.543165 0.839626i $$-0.682774\pi$$
−0.543165 + 0.839626i $$0.682774\pi$$
$$348$$ 0 0
$$349$$ 4.47214 0.239388 0.119694 0.992811i $$-0.461809\pi$$
0.119694 + 0.992811i $$0.461809\pi$$
$$350$$ 0 0
$$351$$ 17.8885 0.954820
$$352$$ 0 0
$$353$$ −2.18034 −0.116048 −0.0580239 0.998315i $$-0.518480\pi$$
−0.0580239 + 0.998315i $$0.518480\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −8.00000 −0.423405
$$358$$ 0 0
$$359$$ 30.1246 1.58992 0.794958 0.606664i $$-0.207493\pi$$
0.794958 + 0.606664i $$0.207493\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −35.4164 −1.85888
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 37.1246 1.93789 0.968944 0.247278i $$-0.0795362\pi$$
0.968944 + 0.247278i $$0.0795362\pi$$
$$368$$ 0 0
$$369$$ 35.5967 1.85309
$$370$$ 0 0
$$371$$ −8.47214 −0.439851
$$372$$ 0 0
$$373$$ −37.8328 −1.95891 −0.979454 0.201665i $$-0.935365\pi$$
−0.979454 + 0.201665i $$0.935365\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.18034 0.318304
$$378$$ 0 0
$$379$$ 11.1803 0.574295 0.287148 0.957886i $$-0.407293\pi$$
0.287148 + 0.957886i $$0.407293\pi$$
$$380$$ 0 0
$$381$$ −44.1803 −2.26343
$$382$$ 0 0
$$383$$ 33.2361 1.69828 0.849142 0.528165i $$-0.177120\pi$$
0.849142 + 0.528165i $$0.177120\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −13.1803 −0.669994
$$388$$ 0 0
$$389$$ 2.88854 0.146455 0.0732275 0.997315i $$-0.476670\pi$$
0.0732275 + 0.997315i $$0.476670\pi$$
$$390$$ 0 0
$$391$$ −15.4164 −0.779641
$$392$$ 0 0
$$393$$ 54.8328 2.76595
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 9.05573 0.454494 0.227247 0.973837i $$-0.427028\pi$$
0.227247 + 0.973837i $$0.427028\pi$$
$$398$$ 0 0
$$399$$ −14.4721 −0.724513
$$400$$ 0 0
$$401$$ 2.52786 0.126236 0.0631178 0.998006i $$-0.479896\pi$$
0.0631178 + 0.998006i $$0.479896\pi$$
$$402$$ 0 0
$$403$$ −4.58359 −0.228325
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0.708204 0.0351044
$$408$$ 0 0
$$409$$ 24.4721 1.21007 0.605035 0.796199i $$-0.293159\pi$$
0.605035 + 0.796199i $$0.293159\pi$$
$$410$$ 0 0
$$411$$ −35.4164 −1.74696
$$412$$ 0 0
$$413$$ 11.7082 0.576123
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −34.4721 −1.68811
$$418$$ 0 0
$$419$$ 26.1803 1.27899 0.639497 0.768794i $$-0.279143\pi$$
0.639497 + 0.768794i $$0.279143\pi$$
$$420$$ 0 0
$$421$$ −13.0000 −0.633581 −0.316791 0.948495i $$-0.602605\pi$$
−0.316791 + 0.948495i $$0.602605\pi$$
$$422$$ 0 0
$$423$$ 14.9443 0.726615
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 9.70820 0.469813
$$428$$ 0 0
$$429$$ 0.944272 0.0455899
$$430$$ 0 0
$$431$$ −17.5279 −0.844288 −0.422144 0.906529i $$-0.638722\pi$$
−0.422144 + 0.906529i $$0.638722\pi$$
$$432$$ 0 0
$$433$$ −28.3607 −1.36293 −0.681464 0.731852i $$-0.738656\pi$$
−0.681464 + 0.731852i $$0.738656\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −27.8885 −1.33409
$$438$$ 0 0
$$439$$ 8.29180 0.395746 0.197873 0.980228i $$-0.436597\pi$$
0.197873 + 0.980228i $$0.436597\pi$$
$$440$$ 0 0
$$441$$ 7.47214 0.355816
$$442$$ 0 0
$$443$$ 19.4164 0.922501 0.461251 0.887270i $$-0.347401\pi$$
0.461251 + 0.887270i $$0.347401\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 12.7639 0.603713
$$448$$ 0 0
$$449$$ 20.5279 0.968770 0.484385 0.874855i $$-0.339043\pi$$
0.484385 + 0.874855i $$0.339043\pi$$
$$450$$ 0 0
$$451$$ 1.12461 0.0529559
$$452$$ 0 0
$$453$$ 65.4853 3.07677
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −12.5279 −0.586029 −0.293014 0.956108i $$-0.594658\pi$$
−0.293014 + 0.956108i $$0.594658\pi$$
$$458$$ 0 0
$$459$$ 35.7771 1.66993
$$460$$ 0 0
$$461$$ −14.1803 −0.660444 −0.330222 0.943903i $$-0.607124\pi$$
−0.330222 + 0.943903i $$0.607124\pi$$
$$462$$ 0 0
$$463$$ 13.8885 0.645455 0.322728 0.946492i $$-0.395400\pi$$
0.322728 + 0.946492i $$0.395400\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −6.94427 −0.321343 −0.160671 0.987008i $$-0.551366\pi$$
−0.160671 + 0.987008i $$0.551366\pi$$
$$468$$ 0 0
$$469$$ −4.23607 −0.195603
$$470$$ 0 0
$$471$$ 2.47214 0.113910
$$472$$ 0 0
$$473$$ −0.416408 −0.0191465
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 63.3050 2.89853
$$478$$ 0 0
$$479$$ 26.1803 1.19621 0.598105 0.801418i $$-0.295921\pi$$
0.598105 + 0.801418i $$0.295921\pi$$
$$480$$ 0 0
$$481$$ 3.70820 0.169080
$$482$$ 0 0
$$483$$ 20.1803 0.918237
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −5.76393 −0.261189 −0.130594 0.991436i $$-0.541689\pi$$
−0.130594 + 0.991436i $$0.541689\pi$$
$$488$$ 0 0
$$489$$ 4.94427 0.223588
$$490$$ 0 0
$$491$$ 5.76393 0.260123 0.130061 0.991506i $$-0.458483\pi$$
0.130061 + 0.991506i $$0.458483\pi$$
$$492$$ 0 0
$$493$$ 12.3607 0.556697
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.70820 0.390616
$$498$$ 0 0
$$499$$ −11.0557 −0.494922 −0.247461 0.968898i $$-0.579596\pi$$
−0.247461 + 0.968898i $$0.579596\pi$$
$$500$$ 0 0
$$501$$ −16.9443 −0.757014
$$502$$ 0 0
$$503$$ 8.11146 0.361672 0.180836 0.983513i $$-0.442120\pi$$
0.180836 + 0.983513i $$0.442120\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −37.1246 −1.64876
$$508$$ 0 0
$$509$$ 40.6525 1.80189 0.900945 0.433934i $$-0.142875\pi$$
0.900945 + 0.433934i $$0.142875\pi$$
$$510$$ 0 0
$$511$$ 8.76393 0.387694
$$512$$ 0 0
$$513$$ 64.7214 2.85752
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0.472136 0.0207645
$$518$$ 0 0
$$519$$ −37.3050 −1.63751
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ −16.3607 −0.715403 −0.357701 0.933836i $$-0.616439\pi$$
−0.357701 + 0.933836i $$0.616439\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −9.16718 −0.399329
$$528$$ 0 0
$$529$$ 15.8885 0.690806
$$530$$ 0 0
$$531$$ −87.4853 −3.79654
$$532$$ 0 0
$$533$$ 5.88854 0.255061
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 75.7771 3.27002
$$538$$ 0 0
$$539$$ 0.236068 0.0101682
$$540$$ 0 0
$$541$$ 15.9443 0.685498 0.342749 0.939427i $$-0.388642\pi$$
0.342749 + 0.939427i $$0.388642\pi$$
$$542$$ 0 0
$$543$$ 26.4721 1.13603
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 9.76393 0.417476 0.208738 0.977972i $$-0.433064\pi$$
0.208738 + 0.977972i $$0.433064\pi$$
$$548$$ 0 0
$$549$$ −72.5410 −3.09598
$$550$$ 0 0
$$551$$ 22.3607 0.952597
$$552$$ 0 0
$$553$$ −11.1803 −0.475436
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −9.11146 −0.386065 −0.193032 0.981192i $$-0.561832\pi$$
−0.193032 + 0.981192i $$0.561832\pi$$
$$558$$ 0 0
$$559$$ −2.18034 −0.0922186
$$560$$ 0 0
$$561$$ 1.88854 0.0797344
$$562$$ 0 0
$$563$$ −17.4164 −0.734014 −0.367007 0.930218i $$-0.619618\pi$$
−0.367007 + 0.930218i $$0.619618\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −24.4164 −1.02539
$$568$$ 0 0
$$569$$ −3.94427 −0.165352 −0.0826762 0.996576i $$-0.526347\pi$$
−0.0826762 + 0.996576i $$0.526347\pi$$
$$570$$ 0 0
$$571$$ −36.5967 −1.53153 −0.765763 0.643123i $$-0.777638\pi$$
−0.765763 + 0.643123i $$0.777638\pi$$
$$572$$ 0 0
$$573$$ −20.9443 −0.874960
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 0 0
$$579$$ 40.1803 1.66984
$$580$$ 0 0
$$581$$ −7.70820 −0.319790
$$582$$ 0 0
$$583$$ 2.00000 0.0828315
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 24.7639 1.02212 0.511058 0.859546i $$-0.329254\pi$$
0.511058 + 0.859546i $$0.329254\pi$$
$$588$$ 0 0
$$589$$ −16.5836 −0.683315
$$590$$ 0 0
$$591$$ −4.76393 −0.195962
$$592$$ 0 0
$$593$$ −37.3050 −1.53193 −0.765965 0.642882i $$-0.777739\pi$$
−0.765965 + 0.642882i $$0.777739\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −23.4164 −0.958370
$$598$$ 0 0
$$599$$ 11.1803 0.456816 0.228408 0.973565i $$-0.426648\pi$$
0.228408 + 0.973565i $$0.426648\pi$$
$$600$$ 0 0
$$601$$ −36.9443 −1.50699 −0.753494 0.657455i $$-0.771633\pi$$
−0.753494 + 0.657455i $$0.771633\pi$$
$$602$$ 0 0
$$603$$ 31.6525 1.28899
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 7.12461 0.289179 0.144590 0.989492i $$-0.453814\pi$$
0.144590 + 0.989492i $$0.453814\pi$$
$$608$$ 0 0
$$609$$ −16.1803 −0.655660
$$610$$ 0 0
$$611$$ 2.47214 0.100012
$$612$$ 0 0
$$613$$ −44.4164 −1.79396 −0.896981 0.442069i $$-0.854245\pi$$
−0.896981 + 0.442069i $$0.854245\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −5.94427 −0.239307 −0.119654 0.992816i $$-0.538178\pi$$
−0.119654 + 0.992816i $$0.538178\pi$$
$$618$$ 0 0
$$619$$ 11.7082 0.470592 0.235296 0.971924i $$-0.424394\pi$$
0.235296 + 0.971924i $$0.424394\pi$$
$$620$$ 0 0
$$621$$ −90.2492 −3.62158
$$622$$ 0 0
$$623$$ −17.2361 −0.690548
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 3.41641 0.136438
$$628$$ 0 0
$$629$$ 7.41641 0.295712
$$630$$ 0 0
$$631$$ −27.6525 −1.10083 −0.550414 0.834892i $$-0.685530\pi$$
−0.550414 + 0.834892i $$0.685530\pi$$
$$632$$ 0 0
$$633$$ −38.8328 −1.54347
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.23607 0.0489748
$$638$$ 0 0
$$639$$ −65.0689 −2.57409
$$640$$ 0 0
$$641$$ 43.8328 1.73129 0.865646 0.500656i $$-0.166908\pi$$
0.865646 + 0.500656i $$0.166908\pi$$
$$642$$ 0 0
$$643$$ −18.4721 −0.728470 −0.364235 0.931307i $$-0.618669\pi$$
−0.364235 + 0.931307i $$0.618669\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 19.8885 0.781899 0.390950 0.920412i $$-0.372147\pi$$
0.390950 + 0.920412i $$0.372147\pi$$
$$648$$ 0 0
$$649$$ −2.76393 −0.108494
$$650$$ 0 0
$$651$$ 12.0000 0.470317
$$652$$ 0 0
$$653$$ 25.0557 0.980506 0.490253 0.871580i $$-0.336904\pi$$
0.490253 + 0.871580i $$0.336904\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −65.4853 −2.55482
$$658$$ 0 0
$$659$$ −17.8885 −0.696839 −0.348419 0.937339i $$-0.613281\pi$$
−0.348419 + 0.937339i $$0.613281\pi$$
$$660$$ 0 0
$$661$$ −42.7214 −1.66167 −0.830834 0.556520i $$-0.812136\pi$$
−0.830834 + 0.556520i $$0.812136\pi$$
$$662$$ 0 0
$$663$$ 9.88854 0.384039
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −31.1803 −1.20731
$$668$$ 0 0
$$669$$ −65.3050 −2.52484
$$670$$ 0 0
$$671$$ −2.29180 −0.0884738
$$672$$ 0 0
$$673$$ 19.5279 0.752744 0.376372 0.926469i $$-0.377171\pi$$
0.376372 + 0.926469i $$0.377171\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −14.3607 −0.551926 −0.275963 0.961168i $$-0.588997\pi$$
−0.275963 + 0.961168i $$0.588997\pi$$
$$678$$ 0 0
$$679$$ −5.23607 −0.200942
$$680$$ 0 0
$$681$$ −69.3050 −2.65577
$$682$$ 0 0
$$683$$ −14.1246 −0.540463 −0.270232 0.962795i $$-0.587100\pi$$
−0.270232 + 0.962795i $$0.587100\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −14.4721 −0.552146
$$688$$ 0 0
$$689$$ 10.4721 0.398957
$$690$$ 0 0
$$691$$ 4.18034 0.159028 0.0795138 0.996834i $$-0.474663\pi$$
0.0795138 + 0.996834i $$0.474663\pi$$
$$692$$ 0 0
$$693$$ −1.76393 −0.0670062
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 11.7771 0.446089
$$698$$ 0 0
$$699$$ 25.7082 0.972374
$$700$$ 0 0
$$701$$ −29.0557 −1.09742 −0.548710 0.836013i $$-0.684881\pi$$
−0.548710 + 0.836013i $$0.684881\pi$$
$$702$$ 0 0
$$703$$ 13.4164 0.506009
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −4.76393 −0.179166
$$708$$ 0 0
$$709$$ −12.1115 −0.454855 −0.227428 0.973795i $$-0.573032\pi$$
−0.227428 + 0.973795i $$0.573032\pi$$
$$710$$ 0 0
$$711$$ 83.5410 3.13303
$$712$$ 0 0
$$713$$ 23.1246 0.866024
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 17.8885 0.668060
$$718$$ 0 0
$$719$$ −16.1803 −0.603425 −0.301712 0.953399i $$-0.597558\pi$$
−0.301712 + 0.953399i $$0.597558\pi$$
$$720$$ 0 0
$$721$$ 8.47214 0.315519
$$722$$ 0 0
$$723$$ −11.4164 −0.424581
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 3.05573 0.113331 0.0566653 0.998393i $$-0.481953\pi$$
0.0566653 + 0.998393i $$0.481953\pi$$
$$728$$ 0 0
$$729$$ 41.9443 1.55349
$$730$$ 0 0
$$731$$ −4.36068 −0.161286
$$732$$ 0 0
$$733$$ 4.00000 0.147743 0.0738717 0.997268i $$-0.476464\pi$$
0.0738717 + 0.997268i $$0.476464\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.00000 0.0368355
$$738$$ 0 0
$$739$$ −25.6525 −0.943642 −0.471821 0.881694i $$-0.656403\pi$$
−0.471821 + 0.881694i $$0.656403\pi$$
$$740$$ 0 0
$$741$$ 17.8885 0.657152
$$742$$ 0 0
$$743$$ 10.4721 0.384185 0.192093 0.981377i $$-0.438473\pi$$
0.192093 + 0.981377i $$0.438473\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 57.5967 2.10735
$$748$$ 0 0
$$749$$ 8.00000 0.292314
$$750$$ 0 0
$$751$$ −3.05573 −0.111505 −0.0557526 0.998445i $$-0.517756\pi$$
−0.0557526 + 0.998445i $$0.517756\pi$$
$$752$$ 0 0
$$753$$ −20.9443 −0.763252
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 19.5836 0.711778 0.355889 0.934528i $$-0.384178\pi$$
0.355889 + 0.934528i $$0.384178\pi$$
$$758$$ 0 0
$$759$$ −4.76393 −0.172920
$$760$$ 0 0
$$761$$ 27.7771 1.00692 0.503459 0.864019i $$-0.332060\pi$$
0.503459 + 0.864019i $$0.332060\pi$$
$$762$$ 0 0
$$763$$ −8.41641 −0.304694
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −14.4721 −0.522559
$$768$$ 0 0
$$769$$ −43.0132 −1.55109 −0.775547 0.631290i $$-0.782526\pi$$
−0.775547 + 0.631290i $$0.782526\pi$$
$$770$$ 0 0
$$771$$ −40.9443 −1.47457
$$772$$ 0 0
$$773$$ 50.1803 1.80486 0.902431 0.430835i $$-0.141781\pi$$
0.902431 + 0.430835i $$0.141781\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −9.70820 −0.348280
$$778$$ 0 0
$$779$$ 21.3050 0.763329
$$780$$ 0 0
$$781$$ −2.05573 −0.0735597
$$782$$ 0 0
$$783$$ 72.3607 2.58596
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −40.7639 −1.45308 −0.726539 0.687126i $$-0.758872\pi$$
−0.726539 + 0.687126i $$0.758872\pi$$
$$788$$ 0 0
$$789$$ −52.5410 −1.87051
$$790$$ 0 0
$$791$$ 14.4164 0.512588
$$792$$ 0 0
$$793$$ −12.0000 −0.426132
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −35.4164 −1.25451 −0.627257 0.778813i $$-0.715822\pi$$
−0.627257 + 0.778813i $$0.715822\pi$$
$$798$$ 0 0
$$799$$ 4.94427 0.174916
$$800$$ 0 0
$$801$$ 128.790 4.55058
$$802$$ 0 0
$$803$$ −2.06888 −0.0730093
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −37.8885 −1.33374
$$808$$ 0 0
$$809$$ 29.4721 1.03619 0.518093 0.855325i $$-0.326642\pi$$
0.518093 + 0.855325i $$0.326642\pi$$
$$810$$ 0 0
$$811$$ 42.7214 1.50015 0.750075 0.661353i $$-0.230017\pi$$
0.750075 + 0.661353i $$0.230017\pi$$
$$812$$ 0 0
$$813$$ −76.7214 −2.69074
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −7.88854 −0.275985
$$818$$ 0 0
$$819$$ −9.23607 −0.322734
$$820$$ 0 0
$$821$$ 28.8328 1.00627 0.503136 0.864207i $$-0.332179\pi$$
0.503136 + 0.864207i $$0.332179\pi$$
$$822$$ 0 0
$$823$$ 31.6525 1.10334 0.551668 0.834064i $$-0.313992\pi$$
0.551668 + 0.834064i $$0.313992\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −41.5410 −1.44452 −0.722261 0.691620i $$-0.756897\pi$$
−0.722261 + 0.691620i $$0.756897\pi$$
$$828$$ 0 0
$$829$$ −7.63932 −0.265325 −0.132662 0.991161i $$-0.542353\pi$$
−0.132662 + 0.991161i $$0.542353\pi$$
$$830$$ 0 0
$$831$$ −64.3607 −2.23265
$$832$$ 0 0
$$833$$ 2.47214 0.0856544
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −53.6656 −1.85496
$$838$$ 0 0
$$839$$ 30.6525 1.05824 0.529120 0.848547i $$-0.322522\pi$$
0.529120 + 0.848547i $$0.322522\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 0 0
$$843$$ −49.7082 −1.71204
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.9443 0.376050
$$848$$ 0 0
$$849$$ −56.3607 −1.93429
$$850$$ 0 0
$$851$$ −18.7082 −0.641309
$$852$$ 0 0
$$853$$ 27.4164 0.938720 0.469360 0.883007i $$-0.344485\pi$$
0.469360 + 0.883007i $$0.344485\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −15.8197 −0.540389 −0.270195 0.962806i $$-0.587088\pi$$
−0.270195 + 0.962806i $$0.587088\pi$$
$$858$$ 0 0
$$859$$ −22.3607 −0.762937 −0.381468 0.924382i $$-0.624581\pi$$
−0.381468 + 0.924382i $$0.624581\pi$$
$$860$$ 0 0
$$861$$ −15.4164 −0.525390
$$862$$ 0 0
$$863$$ −18.3475 −0.624557 −0.312278 0.949991i $$-0.601092\pi$$
−0.312278 + 0.949991i $$0.601092\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −35.2361 −1.19668
$$868$$ 0 0
$$869$$ 2.63932 0.0895328
$$870$$ 0 0
$$871$$ 5.23607 0.177417
$$872$$ 0 0
$$873$$ 39.1246 1.32417
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 30.3607 1.02521 0.512604 0.858625i $$-0.328681\pi$$
0.512604 + 0.858625i $$0.328681\pi$$
$$878$$ 0 0
$$879$$ −100.721 −3.39725
$$880$$ 0 0
$$881$$ 5.81966 0.196069 0.0980347 0.995183i $$-0.468744\pi$$
0.0980347 + 0.995183i $$0.468744\pi$$
$$882$$ 0 0
$$883$$ 1.40325 0.0472232 0.0236116 0.999721i $$-0.492483\pi$$
0.0236116 + 0.999721i $$0.492483\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 21.3475 0.716780 0.358390 0.933572i $$-0.383326\pi$$
0.358390 + 0.933572i $$0.383326\pi$$
$$888$$ 0 0
$$889$$ 13.6525 0.457889
$$890$$ 0 0
$$891$$ 5.76393 0.193099
$$892$$ 0 0
$$893$$ 8.94427 0.299309
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −24.9443 −0.832865
$$898$$ 0 0
$$899$$ −18.5410 −0.618378
$$900$$ 0 0
$$901$$ 20.9443 0.697755
$$902$$ 0 0
$$903$$ 5.70820 0.189957
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −34.8328 −1.15660 −0.578302 0.815823i $$-0.696285\pi$$
−0.578302 + 0.815823i $$0.696285\pi$$
$$908$$ 0 0
$$909$$ 35.5967 1.18067
$$910$$ 0 0
$$911$$ −0.819660 −0.0271566 −0.0135783 0.999908i $$-0.504322\pi$$
−0.0135783 + 0.999908i $$0.504322\pi$$
$$912$$ 0 0
$$913$$ 1.81966 0.0602220
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −16.9443 −0.559549
$$918$$ 0 0
$$919$$ 27.7639 0.915848 0.457924 0.888991i $$-0.348593\pi$$
0.457924 + 0.888991i $$0.348593\pi$$
$$920$$ 0 0
$$921$$ −14.8328 −0.488758
$$922$$ 0 0
$$923$$ −10.7639 −0.354299
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −63.3050 −2.07921
$$928$$ 0 0
$$929$$ 38.2918 1.25631 0.628157 0.778087i $$-0.283810\pi$$
0.628157 + 0.778087i $$0.283810\pi$$
$$930$$ 0 0
$$931$$ 4.47214 0.146568
$$932$$ 0 0
$$933$$ −78.8328 −2.58087
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 35.2361 1.15111 0.575556 0.817762i $$-0.304786\pi$$
0.575556 + 0.817762i $$0.304786\pi$$
$$938$$ 0 0
$$939$$ 63.1935 2.06224
$$940$$ 0 0
$$941$$ −5.23607 −0.170691 −0.0853455 0.996351i $$-0.527199\pi$$
−0.0853455 + 0.996351i $$0.527199\pi$$
$$942$$ 0 0
$$943$$ −29.7082 −0.967432
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −34.8328 −1.13191 −0.565957 0.824435i $$-0.691493\pi$$
−0.565957 + 0.824435i $$0.691493\pi$$
$$948$$ 0 0
$$949$$ −10.8328 −0.351648
$$950$$ 0 0
$$951$$ 82.0689 2.66127
$$952$$ 0 0
$$953$$ 3.47214 0.112474 0.0562368 0.998417i $$-0.482090\pi$$
0.0562368 + 0.998417i $$0.482090\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 3.81966 0.123472
$$958$$ 0 0
$$959$$ 10.9443 0.353409
$$960$$ 0 0
$$961$$ −17.2492 −0.556427
$$962$$ 0 0
$$963$$ −59.7771 −1.92629
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 14.1115 0.453794 0.226897 0.973919i $$-0.427142\pi$$
0.226897 + 0.973919i $$0.427142\pi$$
$$968$$ 0 0
$$969$$ 35.7771 1.14933
$$970$$ 0 0
$$971$$ 18.0000 0.577647 0.288824 0.957382i $$-0.406736\pi$$
0.288824 + 0.957382i $$0.406736\pi$$
$$972$$ 0 0
$$973$$ 10.6525 0.341503
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −11.4721 −0.367026 −0.183513 0.983017i $$-0.558747\pi$$
−0.183513 + 0.983017i $$0.558747\pi$$
$$978$$ 0 0
$$979$$ 4.06888 0.130042
$$980$$ 0 0
$$981$$ 62.8885 2.00788
$$982$$ 0 0
$$983$$ 34.5410 1.10169 0.550844 0.834608i $$-0.314306\pi$$
0.550844 + 0.834608i $$0.314306\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −6.47214 −0.206010
$$988$$ 0 0
$$989$$ 11.0000 0.349780
$$990$$ 0 0
$$991$$ −13.1803 −0.418687 −0.209344 0.977842i $$-0.567133\pi$$
−0.209344 + 0.977842i $$0.567133\pi$$
$$992$$ 0 0
$$993$$ 79.9574 2.53737
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −45.4164 −1.43835 −0.719176 0.694828i $$-0.755481\pi$$
−0.719176 + 0.694828i $$0.755481\pi$$
$$998$$ 0 0
$$999$$ 43.4164 1.37363
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 2800.2.a.bp.1.2 2
4.3 odd 2 175.2.a.e.1.1 yes 2
5.2 odd 4 2800.2.g.s.449.1 4
5.3 odd 4 2800.2.g.s.449.4 4
5.4 even 2 2800.2.a.bh.1.1 2
12.11 even 2 1575.2.a.n.1.2 2
20.3 even 4 175.2.b.c.99.3 4
20.7 even 4 175.2.b.c.99.2 4
20.19 odd 2 175.2.a.d.1.2 2
28.27 even 2 1225.2.a.u.1.1 2
60.23 odd 4 1575.2.d.k.1324.2 4
60.47 odd 4 1575.2.d.k.1324.3 4
60.59 even 2 1575.2.a.s.1.1 2
140.27 odd 4 1225.2.b.k.99.2 4
140.83 odd 4 1225.2.b.k.99.3 4
140.139 even 2 1225.2.a.n.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.2 2 20.19 odd 2
175.2.a.e.1.1 yes 2 4.3 odd 2
175.2.b.c.99.2 4 20.7 even 4
175.2.b.c.99.3 4 20.3 even 4
1225.2.a.n.1.2 2 140.139 even 2
1225.2.a.u.1.1 2 28.27 even 2
1225.2.b.k.99.2 4 140.27 odd 4
1225.2.b.k.99.3 4 140.83 odd 4
1575.2.a.n.1.2 2 12.11 even 2
1575.2.a.s.1.1 2 60.59 even 2
1575.2.d.k.1324.2 4 60.23 odd 4
1575.2.d.k.1324.3 4 60.47 odd 4
2800.2.a.bh.1.1 2 5.4 even 2
2800.2.a.bp.1.2 2 1.1 even 1 trivial
2800.2.g.s.449.1 4 5.2 odd 4
2800.2.g.s.449.4 4 5.3 odd 4