Properties

 Label 1575.2.a.s.1.1 Level $1575$ Weight $2$ Character 1575.1 Self dual yes Analytic conductor $12.576$ 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] = [1575,2,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.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: $$12.5764383184$$ Analytic rank: $$1$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 175) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})$$ $$q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{7} +2.23607 q^{8} +0.236068 q^{11} -1.23607 q^{13} +0.618034 q^{14} +1.85410 q^{16} +2.47214 q^{17} -4.47214 q^{19} -0.145898 q^{22} +6.23607 q^{23} +0.763932 q^{26} +1.61803 q^{28} -5.00000 q^{29} +3.70820 q^{31} -5.61803 q^{32} -1.52786 q^{34} -3.00000 q^{37} +2.76393 q^{38} -4.76393 q^{41} -1.76393 q^{43} -0.381966 q^{44} -3.85410 q^{46} -2.00000 q^{47} +1.00000 q^{49} +2.00000 q^{52} +8.47214 q^{53} -2.23607 q^{56} +3.09017 q^{58} -11.7082 q^{59} -9.70820 q^{61} -2.29180 q^{62} -0.236068 q^{64} +4.23607 q^{67} -4.00000 q^{68} -8.70820 q^{71} +8.76393 q^{73} +1.85410 q^{74} +7.23607 q^{76} -0.236068 q^{77} -11.1803 q^{79} +2.94427 q^{82} -7.70820 q^{83} +1.09017 q^{86} +0.527864 q^{88} -17.2361 q^{89} +1.23607 q^{91} -10.0902 q^{92} +1.23607 q^{94} -5.23607 q^{97} -0.618034 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - 2 q^{7}+O(q^{10})$$ 2 * q + q^2 - q^4 - 2 * q^7 $$2 q + q^{2} - q^{4} - 2 q^{7} - 4 q^{11} + 2 q^{13} - q^{14} - 3 q^{16} - 4 q^{17} - 7 q^{22} + 8 q^{23} + 6 q^{26} + q^{28} - 10 q^{29} - 6 q^{31} - 9 q^{32} - 12 q^{34} - 6 q^{37} + 10 q^{38} - 14 q^{41} - 8 q^{43} - 3 q^{44} - q^{46} - 4 q^{47} + 2 q^{49} + 4 q^{52} + 8 q^{53} - 5 q^{58} - 10 q^{59} - 6 q^{61} - 18 q^{62} + 4 q^{64} + 4 q^{67} - 8 q^{68} - 4 q^{71} + 22 q^{73} - 3 q^{74} + 10 q^{76} + 4 q^{77} - 12 q^{82} - 2 q^{83} - 9 q^{86} + 10 q^{88} - 30 q^{89} - 2 q^{91} - 9 q^{92} - 2 q^{94} - 6 q^{97} + q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 - 2 * q^7 - 4 * q^11 + 2 * q^13 - q^14 - 3 * q^16 - 4 * q^17 - 7 * q^22 + 8 * q^23 + 6 * q^26 + q^28 - 10 * q^29 - 6 * q^31 - 9 * q^32 - 12 * q^34 - 6 * q^37 + 10 * q^38 - 14 * q^41 - 8 * q^43 - 3 * q^44 - q^46 - 4 * q^47 + 2 * q^49 + 4 * q^52 + 8 * q^53 - 5 * q^58 - 10 * q^59 - 6 * q^61 - 18 * q^62 + 4 * q^64 + 4 * q^67 - 8 * q^68 - 4 * q^71 + 22 * q^73 - 3 * q^74 + 10 * q^76 + 4 * q^77 - 12 * q^82 - 2 * q^83 - 9 * q^86 + 10 * q^88 - 30 * q^89 - 2 * q^91 - 9 * q^92 - 2 * q^94 - 6 * q^97 + q^98

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.618034 −0.437016 −0.218508 0.975835i $$-0.570119\pi$$
−0.218508 + 0.975835i $$0.570119\pi$$
$$3$$ 0 0
$$4$$ −1.61803 −0.809017
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 2.23607 0.790569
$$9$$ 0 0
$$10$$ 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 $$-0.554833\pi$$
−0.171412 + 0.985199i $$0.554833\pi$$
$$14$$ 0.618034 0.165177
$$15$$ 0 0
$$16$$ 1.85410 0.463525
$$17$$ 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 $$-0.671462\pi$$
−0.512989 + 0.858395i $$0.671462\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −0.145898 −0.0311056
$$23$$ 6.23607 1.30031 0.650155 0.759802i $$-0.274704\pi$$
0.650155 + 0.759802i $$0.274704\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0.763932 0.149819
$$27$$ 0 0
$$28$$ 1.61803 0.305780
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ 3.70820 0.666013 0.333007 0.942925i $$-0.391937\pi$$
0.333007 + 0.942925i $$0.391937\pi$$
$$32$$ −5.61803 −0.993137
$$33$$ 0 0
$$34$$ −1.52786 −0.262027
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ 2.76393 0.448369
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.76393 −0.744001 −0.372001 0.928232i $$-0.621328\pi$$
−0.372001 + 0.928232i $$0.621328\pi$$
$$42$$ 0 0
$$43$$ −1.76393 −0.268997 −0.134499 0.990914i $$-0.542942\pi$$
−0.134499 + 0.990914i $$0.542942\pi$$
$$44$$ −0.381966 −0.0575835
$$45$$ 0 0
$$46$$ −3.85410 −0.568256
$$47$$ −2.00000 −0.291730 −0.145865 0.989305i $$-0.546597\pi$$
−0.145865 + 0.989305i $$0.546597\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ 8.47214 1.16374 0.581869 0.813283i $$-0.302322\pi$$
0.581869 + 0.813283i $$0.302322\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −2.23607 −0.298807
$$57$$ 0 0
$$58$$ 3.09017 0.405759
$$59$$ −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 $$0.713478\pi$$
$$62$$ −2.29180 −0.291058
$$63$$ 0 0
$$64$$ −0.236068 −0.0295085
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.23607 0.517518 0.258759 0.965942i $$-0.416686\pi$$
0.258759 + 0.965942i $$0.416686\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 0 0
$$70$$ 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 $$-0.328582\pi$$
0.512870 + 0.858466i $$0.328582\pi$$
$$74$$ 1.85410 0.215535
$$75$$ 0 0
$$76$$ 7.23607 0.830034
$$77$$ −0.236068 −0.0269024
$$78$$ 0 0
$$79$$ −11.1803 −1.25789 −0.628943 0.777451i $$-0.716512\pi$$
−0.628943 + 0.777451i $$0.716512\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 2.94427 0.325140
$$83$$ −7.70820 −0.846085 −0.423043 0.906110i $$-0.639038\pi$$
−0.423043 + 0.906110i $$0.639038\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1.09017 0.117556
$$87$$ 0 0
$$88$$ 0.527864 0.0562705
$$89$$ −17.2361 −1.82702 −0.913510 0.406817i $$-0.866639\pi$$
−0.913510 + 0.406817i $$0.866639\pi$$
$$90$$ 0 0
$$91$$ 1.23607 0.129575
$$92$$ −10.0902 −1.05197
$$93$$ 0 0
$$94$$ 1.23607 0.127491
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −5.23607 −0.531642 −0.265821 0.964022i $$-0.585643\pi$$
−0.265821 + 0.964022i $$0.585643\pi$$
$$98$$ −0.618034 −0.0624309
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.76393 −0.474029 −0.237014 0.971506i $$-0.576169\pi$$
−0.237014 + 0.971506i $$0.576169\pi$$
$$102$$ 0 0
$$103$$ −8.47214 −0.834784 −0.417392 0.908726i $$-0.637056\pi$$
−0.417392 + 0.908726i $$0.637056\pi$$
$$104$$ −2.76393 −0.271026
$$105$$ 0 0
$$106$$ −5.23607 −0.508572
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ 8.41641 0.806146 0.403073 0.915168i $$-0.367942\pi$$
0.403073 + 0.915168i $$0.367942\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −1.85410 −0.175196
$$113$$ −14.4164 −1.35618 −0.678091 0.734978i $$-0.737192\pi$$
−0.678091 + 0.734978i $$0.737192\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 8.09017 0.751153
$$117$$ 0 0
$$118$$ 7.23607 0.666134
$$119$$ −2.47214 −0.226620
$$120$$ 0 0
$$121$$ −10.9443 −0.994934
$$122$$ 6.00000 0.543214
$$123$$ 0 0
$$124$$ −6.00000 −0.538816
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −13.6525 −1.21146 −0.605731 0.795670i $$-0.707119\pi$$
−0.605731 + 0.795670i $$0.707119\pi$$
$$128$$ 11.3820 1.00603
$$129$$ 0 0
$$130$$ 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
$$134$$ −2.61803 −0.226164
$$135$$ 0 0
$$136$$ 5.52786 0.474010
$$137$$ −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 $$-0.350794\pi$$
0.451766 + 0.892137i $$0.350794\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 5.38197 0.451645
$$143$$ −0.291796 −0.0244012
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −5.41641 −0.448265
$$147$$ 0 0
$$148$$ 4.85410 0.399005
$$149$$ −3.94427 −0.323127 −0.161564 0.986862i $$-0.551654\pi$$
−0.161564 + 0.986862i $$0.551654\pi$$
$$150$$ 0 0
$$151$$ −20.2361 −1.64679 −0.823394 0.567470i $$-0.807922\pi$$
−0.823394 + 0.567470i $$0.807922\pi$$
$$152$$ −10.0000 −0.811107
$$153$$ 0 0
$$154$$ 0.145898 0.0117568
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −0.763932 −0.0609684 −0.0304842 0.999535i $$-0.509705\pi$$
−0.0304842 + 0.999535i $$0.509705\pi$$
$$158$$ 6.90983 0.549717
$$159$$ 0 0
$$160$$ 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 $$0.480942\pi$$
$$164$$ 7.70820 0.601910
$$165$$ 0 0
$$166$$ 4.76393 0.369753
$$167$$ 5.23607 0.405179 0.202590 0.979264i $$-0.435064\pi$$
0.202590 + 0.979264i $$0.435064\pi$$
$$168$$ 0 0
$$169$$ −11.4721 −0.882472
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 2.85410 0.217623
$$173$$ −11.5279 −0.876447 −0.438224 0.898866i $$-0.644392\pi$$
−0.438224 + 0.898866i $$0.644392\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0.437694 0.0329924
$$177$$ 0 0
$$178$$ 10.6525 0.798437
$$179$$ 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 $$0.401671\pi$$
$$182$$ −0.763932 −0.0566264
$$183$$ 0 0
$$184$$ 13.9443 1.02799
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.583592 0.0426765
$$188$$ 3.23607 0.236015
$$189$$ 0 0
$$190$$ 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 $$-0.647464\pi$$
−0.446876 + 0.894596i $$0.647464\pi$$
$$194$$ 3.23607 0.232336
$$195$$ 0 0
$$196$$ −1.61803 −0.115574
$$197$$ −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 $$-0.417439\pi$$
0.256476 + 0.966551i $$0.417439\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 2.94427 0.207158
$$203$$ 5.00000 0.350931
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 5.23607 0.364814
$$207$$ 0 0
$$208$$ −2.29180 −0.158907
$$209$$ −1.05573 −0.0730262
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ −13.7082 −0.941483
$$213$$ 0 0
$$214$$ −4.94427 −0.337983
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −3.70820 −0.251729
$$218$$ −5.20163 −0.352299
$$219$$ 0 0
$$220$$ 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 $$0.736153\pi$$
$$224$$ 5.61803 0.375371
$$225$$ 0 0
$$226$$ 8.90983 0.592673
$$227$$ 21.4164 1.42146 0.710728 0.703466i $$-0.248365\pi$$
0.710728 + 0.703466i $$0.248365\pi$$
$$228$$ 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 0
$$232$$ −11.1803 −0.734025
$$233$$ 7.94427 0.520447 0.260223 0.965548i $$-0.416204\pi$$
0.260223 + 0.965548i $$0.416204\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 18.9443 1.23317
$$237$$ 0 0
$$238$$ 1.52786 0.0990367
$$239$$ 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 $$0.536246\pi$$
$$242$$ 6.76393 0.434802
$$243$$ 0 0
$$244$$ 15.7082 1.00561
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.52786 0.351730
$$248$$ 8.29180 0.526530
$$249$$ 0 0
$$250$$ 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
$$254$$ 8.43769 0.529428
$$255$$ 0 0
$$256$$ −6.56231 −0.410144
$$257$$ −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
$$261$$ 0 0
$$262$$ −10.4721 −0.646971
$$263$$ 16.2361 1.00116 0.500579 0.865691i $$-0.333120\pi$$
0.500579 + 0.865691i $$0.333120\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.76393 −0.169468
$$267$$ 0 0
$$268$$ −6.85410 −0.418681
$$269$$ 11.7082 0.713862 0.356931 0.934131i $$-0.383823\pi$$
0.356931 + 0.934131i $$0.383823\pi$$
$$270$$ 0 0
$$271$$ 23.7082 1.44017 0.720085 0.693885i $$-0.244103\pi$$
0.720085 + 0.693885i $$0.244103\pi$$
$$272$$ 4.58359 0.277921
$$273$$ 0 0
$$274$$ 6.76393 0.408624
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 19.8885 1.19499 0.597493 0.801874i $$-0.296163\pi$$
0.597493 + 0.801874i $$0.296163\pi$$
$$278$$ −6.58359 −0.394858
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 15.3607 0.916341 0.458171 0.888864i $$-0.348505\pi$$
0.458171 + 0.888864i $$0.348505\pi$$
$$282$$ 0 0
$$283$$ −17.4164 −1.03530 −0.517649 0.855593i $$-0.673193\pi$$
−0.517649 + 0.855593i $$0.673193\pi$$
$$284$$ 14.0902 0.836098
$$285$$ 0 0
$$286$$ 0.180340 0.0106637
$$287$$ 4.76393 0.281206
$$288$$ 0 0
$$289$$ −10.8885 −0.640503
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −14.1803 −0.829842
$$293$$ −31.1246 −1.81832 −0.909160 0.416448i $$-0.863275\pi$$
−0.909160 + 0.416448i $$0.863275\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −6.70820 −0.389906
$$297$$ 0 0
$$298$$ 2.43769 0.141212
$$299$$ −7.70820 −0.445777
$$300$$ 0 0
$$301$$ 1.76393 0.101671
$$302$$ 12.5066 0.719673
$$303$$ 0 0
$$304$$ −8.29180 −0.475567
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −4.58359 −0.261599 −0.130800 0.991409i $$-0.541754\pi$$
−0.130800 + 0.991409i $$0.541754\pi$$
$$308$$ 0.381966 0.0217645
$$309$$ 0 0
$$310$$ 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 $$-0.686093\pi$$
−0.551890 + 0.833917i $$0.686093\pi$$
$$314$$ 0.472136 0.0266442
$$315$$ 0 0
$$316$$ 18.0902 1.01765
$$317$$ 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
$$321$$ 0 0
$$322$$ 3.85410 0.214781
$$323$$ −11.0557 −0.615157
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −0.944272 −0.0522984
$$327$$ 0 0
$$328$$ −10.6525 −0.588185
$$329$$ 2.00000 0.110264
$$330$$ 0 0
$$331$$ −24.7082 −1.35809 −0.679043 0.734099i $$-0.737605\pi$$
−0.679043 + 0.734099i $$0.737605\pi$$
$$332$$ 12.4721 0.684497
$$333$$ 0 0
$$334$$ −3.23607 −0.177070
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 16.4721 0.897294 0.448647 0.893709i $$-0.351906\pi$$
0.448647 + 0.893709i $$0.351906\pi$$
$$338$$ 7.09017 0.385654
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0.875388 0.0474049
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ −3.94427 −0.212661
$$345$$ 0 0
$$346$$ 7.12461 0.383022
$$347$$ 20.2361 1.08633 0.543165 0.839626i $$-0.317226\pi$$
0.543165 + 0.839626i $$0.317226\pi$$
$$348$$ 0 0
$$349$$ 4.47214 0.239388 0.119694 0.992811i $$-0.461809\pi$$
0.119694 + 0.992811i $$0.461809\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.32624 −0.0706887
$$353$$ −2.18034 −0.116048 −0.0580239 0.998315i $$-0.518480\pi$$
−0.0580239 + 0.998315i $$0.518480\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 27.8885 1.47809
$$357$$ 0 0
$$358$$ −14.4721 −0.764876
$$359$$ 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
$$362$$ −5.05573 −0.265723
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 37.1246 1.93789 0.968944 0.247278i $$-0.0795362\pi$$
0.968944 + 0.247278i $$0.0795362\pi$$
$$368$$ 11.5623 0.602727
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −8.47214 −0.439851
$$372$$ 0 0
$$373$$ 37.8328 1.95891 0.979454 0.201665i $$-0.0646354\pi$$
0.979454 + 0.201665i $$0.0646354\pi$$
$$374$$ −0.360680 −0.0186503
$$375$$ 0 0
$$376$$ −4.47214 −0.230633
$$377$$ 6.18034 0.318304
$$378$$ 0 0
$$379$$ −11.1803 −0.574295 −0.287148 0.957886i $$-0.592707\pi$$
−0.287148 + 0.957886i $$0.592707\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 4.00000 0.204658
$$383$$ −33.2361 −1.69828 −0.849142 0.528165i $$-0.822880\pi$$
−0.849142 + 0.528165i $$0.822880\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 7.67376 0.390584
$$387$$ 0 0
$$388$$ 8.47214 0.430108
$$389$$ −2.88854 −0.146455 −0.0732275 0.997315i $$-0.523330\pi$$
−0.0732275 + 0.997315i $$0.523330\pi$$
$$390$$ 0 0
$$391$$ 15.4164 0.779641
$$392$$ 2.23607 0.112938
$$393$$ 0 0
$$394$$ 0.909830 0.0458366
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −9.05573 −0.454494 −0.227247 0.973837i $$-0.572972\pi$$
−0.227247 + 0.973837i $$0.572972\pi$$
$$398$$ −4.47214 −0.224168
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.52786 −0.126236 −0.0631178 0.998006i $$-0.520104\pi$$
−0.0631178 + 0.998006i $$0.520104\pi$$
$$402$$ 0 0
$$403$$ −4.58359 −0.228325
$$404$$ 7.70820 0.383497
$$405$$ 0 0
$$406$$ −3.09017 −0.153363
$$407$$ −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
$$411$$ 0 0
$$412$$ 13.7082 0.675355
$$413$$ 11.7082 0.576123
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 6.94427 0.340471
$$417$$ 0 0
$$418$$ 0.652476 0.0319136
$$419$$ 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 $$0.602605\pi$$
$$422$$ −7.41641 −0.361025
$$423$$ 0 0
$$424$$ 18.9443 0.920015
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 9.70820 0.469813
$$428$$ −12.9443 −0.625685
$$429$$ 0 0
$$430$$ 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 $$-0.261344\pi$$
0.681464 + 0.731852i $$0.261344\pi$$
$$434$$ 2.29180 0.110010
$$435$$ 0 0
$$436$$ −13.6180 −0.652186
$$437$$ −27.8885 −1.33409
$$438$$ 0 0
$$439$$ −8.29180 −0.395746 −0.197873 0.980228i $$-0.563403\pi$$
−0.197873 + 0.980228i $$0.563403\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 1.88854 0.0898289
$$443$$ −19.4164 −0.922501 −0.461251 0.887270i $$-0.652599\pi$$
−0.461251 + 0.887270i $$0.652599\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 12.4721 0.590573
$$447$$ 0 0
$$448$$ 0.236068 0.0111532
$$449$$ −20.5279 −0.968770 −0.484385 0.874855i $$-0.660957\pi$$
−0.484385 + 0.874855i $$0.660957\pi$$
$$450$$ 0 0
$$451$$ −1.12461 −0.0529559
$$452$$ 23.3262 1.09717
$$453$$ 0 0
$$454$$ −13.2361 −0.621199
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 12.5279 0.586029 0.293014 0.956108i $$-0.405342\pi$$
0.293014 + 0.956108i $$0.405342\pi$$
$$458$$ 2.76393 0.129150
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 14.1803 0.660444 0.330222 0.943903i $$-0.392876\pi$$
0.330222 + 0.943903i $$0.392876\pi$$
$$462$$ 0 0
$$463$$ 13.8885 0.645455 0.322728 0.946492i $$-0.395400\pi$$
0.322728 + 0.946492i $$0.395400\pi$$
$$464$$ −9.27051 −0.430373
$$465$$ 0 0
$$466$$ −4.90983 −0.227443
$$467$$ 6.94427 0.321343 0.160671 0.987008i $$-0.448634\pi$$
0.160671 + 0.987008i $$0.448634\pi$$
$$468$$ 0 0
$$469$$ −4.23607 −0.195603
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −26.1803 −1.20505
$$473$$ −0.416408 −0.0191465
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ 0 0
$$478$$ −3.41641 −0.156263
$$479$$ 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
$$482$$ 2.18034 0.0993118
$$483$$ 0 0
$$484$$ 17.7082 0.804918
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −5.76393 −0.261189 −0.130594 0.991436i $$-0.541689\pi$$
−0.130594 + 0.991436i $$0.541689\pi$$
$$488$$ −21.7082 −0.982684
$$489$$ 0 0
$$490$$ 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
$$494$$ −3.41641 −0.153711
$$495$$ 0 0
$$496$$ 6.87539 0.308714
$$497$$ 8.70820 0.390616
$$498$$ 0 0
$$499$$ 11.0557 0.494922 0.247461 0.968898i $$-0.420404\pi$$
0.247461 + 0.968898i $$0.420404\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 4.00000 0.178529
$$503$$ −8.11146 −0.361672 −0.180836 0.983513i $$-0.557880\pi$$
−0.180836 + 0.983513i $$0.557880\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −0.909830 −0.0404469
$$507$$ 0 0
$$508$$ 22.0902 0.980093
$$509$$ −40.6525 −1.80189 −0.900945 0.433934i $$-0.857125\pi$$
−0.900945 + 0.433934i $$0.857125\pi$$
$$510$$ 0 0
$$511$$ −8.76393 −0.387694
$$512$$ −18.7082 −0.826794
$$513$$ 0 0
$$514$$ 7.81966 0.344910
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −0.472136 −0.0207645
$$518$$ −1.85410 −0.0814646
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 0 0
$$523$$ −16.3607 −0.715403 −0.357701 0.933836i $$-0.616439\pi$$
−0.357701 + 0.933836i $$0.616439\pi$$
$$524$$ −27.4164 −1.19769
$$525$$ 0 0
$$526$$ −10.0344 −0.437522
$$527$$ 9.16718 0.399329
$$528$$ 0 0
$$529$$ 15.8885 0.690806
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −7.23607 −0.313723
$$533$$ 5.88854 0.255061
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 9.47214 0.409134
$$537$$ 0 0
$$538$$ −7.23607 −0.311969
$$539$$ 0.236068 0.0101682
$$540$$ 0 0
$$541$$ 15.9443 0.685498 0.342749 0.939427i $$-0.388642\pi$$
0.342749 + 0.939427i $$0.388642\pi$$
$$542$$ −14.6525 −0.629378
$$543$$ 0 0
$$544$$ −13.8885 −0.595466
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 9.76393 0.417476 0.208738 0.977972i $$-0.433064\pi$$
0.208738 + 0.977972i $$0.433064\pi$$
$$548$$ 17.7082 0.756457
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 22.3607 0.952597
$$552$$ 0 0
$$553$$ 11.1803 0.475436
$$554$$ −12.2918 −0.522228
$$555$$ 0 0
$$556$$ −17.2361 −0.730972
$$557$$ −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
$$561$$ 0 0
$$562$$ −9.49342 −0.400456
$$563$$ 17.4164 0.734014 0.367007 0.930218i $$-0.380382\pi$$
0.367007 + 0.930218i $$0.380382\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 10.7639 0.452442
$$567$$ 0 0
$$568$$ −19.4721 −0.817033
$$569$$ 3.94427 0.165352 0.0826762 0.996576i $$-0.473653\pi$$
0.0826762 + 0.996576i $$0.473653\pi$$
$$570$$ 0 0
$$571$$ 36.5967 1.53153 0.765763 0.643123i $$-0.222362\pi$$
0.765763 + 0.643123i $$0.222362\pi$$
$$572$$ 0.472136 0.0197410
$$573$$ 0 0
$$574$$ −2.94427 −0.122892
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 6.72949 0.279910
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 7.70820 0.319790
$$582$$ 0 0
$$583$$ 2.00000 0.0828315
$$584$$ 19.5967 0.810919
$$585$$ 0 0
$$586$$ 19.2361 0.794635
$$587$$ −24.7639 −1.02212 −0.511058 0.859546i $$-0.670746\pi$$
−0.511058 + 0.859546i $$0.670746\pi$$
$$588$$ 0 0
$$589$$ −16.5836 −0.683315
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −5.56231 −0.228609
$$593$$ −37.3050 −1.53193 −0.765965 0.642882i $$-0.777739\pi$$
−0.765965 + 0.642882i $$0.777739\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.38197 0.261416
$$597$$ 0 0
$$598$$ 4.76393 0.194812
$$599$$ 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 $$0.771633\pi$$
$$602$$ −1.09017 −0.0444320
$$603$$ 0 0
$$604$$ 32.7426 1.33228
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 7.12461 0.289179 0.144590 0.989492i $$-0.453814\pi$$
0.144590 + 0.989492i $$0.453814\pi$$
$$608$$ 25.1246 1.01894
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 2.47214 0.100012
$$612$$ 0 0
$$613$$ 44.4164 1.79396 0.896981 0.442069i $$-0.145755\pi$$
0.896981 + 0.442069i $$0.145755\pi$$
$$614$$ 2.83282 0.114323
$$615$$ 0 0
$$616$$ −0.527864 −0.0212682
$$617$$ −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 $$-0.575606\pi$$
−0.235296 + 0.971924i $$0.575606\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 15.0557 0.603680
$$623$$ 17.2361 0.690548
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 12.0689 0.482370
$$627$$ 0 0
$$628$$ 1.23607 0.0493245
$$629$$ −7.41641 −0.295712
$$630$$ 0 0
$$631$$ 27.6525 1.10083 0.550414 0.834892i $$-0.314470\pi$$
0.550414 + 0.834892i $$0.314470\pi$$
$$632$$ −25.0000 −0.994447
$$633$$ 0 0
$$634$$ −15.6738 −0.622485
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −1.23607 −0.0489748
$$638$$ 0.729490 0.0288808
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −43.8328 −1.73129 −0.865646 0.500656i $$-0.833092\pi$$
−0.865646 + 0.500656i $$0.833092\pi$$
$$642$$ 0 0
$$643$$ −18.4721 −0.728470 −0.364235 0.931307i $$-0.618669\pi$$
−0.364235 + 0.931307i $$0.618669\pi$$
$$644$$ 10.0902 0.397608
$$645$$ 0 0
$$646$$ 6.83282 0.268834
$$647$$ −19.8885 −0.781899 −0.390950 0.920412i $$-0.627853\pi$$
−0.390950 + 0.920412i $$0.627853\pi$$
$$648$$ 0 0
$$649$$ −2.76393 −0.108494
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −2.47214 −0.0968163
$$653$$ 25.0557 0.980506 0.490253 0.871580i $$-0.336904\pi$$
0.490253 + 0.871580i $$0.336904\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −8.83282 −0.344864
$$657$$ 0 0
$$658$$ −1.23607 −0.0481869
$$659$$ −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 $$0.812136\pi$$
$$662$$ 15.2705 0.593505
$$663$$ 0 0
$$664$$ −17.2361 −0.668889
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −31.1803 −1.20731
$$668$$ −8.47214 −0.327797
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −2.29180 −0.0884738
$$672$$ 0 0
$$673$$ −19.5279 −0.752744 −0.376372 0.926469i $$-0.622829\pi$$
−0.376372 + 0.926469i $$0.622829\pi$$
$$674$$ −10.1803 −0.392132
$$675$$ 0 0
$$676$$ 18.5623 0.713935
$$677$$ −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
$$681$$ 0 0
$$682$$ −0.541020 −0.0207167
$$683$$ 14.1246 0.540463 0.270232 0.962795i $$-0.412900\pi$$
0.270232 + 0.962795i $$0.412900\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0.618034 0.0235966
$$687$$ 0 0
$$688$$ −3.27051 −0.124687
$$689$$ −10.4721 −0.398957
$$690$$ 0 0
$$691$$ −4.18034 −0.159028 −0.0795138 0.996834i $$-0.525337\pi$$
−0.0795138 + 0.996834i $$0.525337\pi$$
$$692$$ 18.6525 0.709061
$$693$$ 0 0
$$694$$ −12.5066 −0.474743
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −11.7771 −0.446089
$$698$$ −2.76393 −0.104616
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 29.0557 1.09742 0.548710 0.836013i $$-0.315119\pi$$
0.548710 + 0.836013i $$0.315119\pi$$
$$702$$ 0 0
$$703$$ 13.4164 0.506009
$$704$$ −0.0557281 −0.00210033
$$705$$ 0 0
$$706$$ 1.34752 0.0507147
$$707$$ 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
$$711$$ 0 0
$$712$$ −38.5410 −1.44439
$$713$$ 23.1246 0.866024
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −37.8885 −1.41596
$$717$$ 0 0
$$718$$ −18.6180 −0.694819
$$719$$ −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
$$722$$ −0.618034 −0.0230008
$$723$$ 0 0
$$724$$ −13.2361 −0.491915
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 3.05573 0.113331 0.0566653 0.998393i $$-0.481953\pi$$
0.0566653 + 0.998393i $$0.481953\pi$$
$$728$$ 2.76393 0.102438
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −4.36068 −0.161286
$$732$$ 0 0
$$733$$ −4.00000 −0.147743 −0.0738717 0.997268i $$-0.523536\pi$$
−0.0738717 + 0.997268i $$0.523536\pi$$
$$734$$ −22.9443 −0.846889
$$735$$ 0 0
$$736$$ −35.0344 −1.29139
$$737$$ 1.00000 0.0368355
$$738$$ 0 0
$$739$$ 25.6525 0.943642 0.471821 0.881694i $$-0.343597\pi$$
0.471821 + 0.881694i $$0.343597\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 5.23607 0.192222
$$743$$ −10.4721 −0.384185 −0.192093 0.981377i $$-0.561527\pi$$
−0.192093 + 0.981377i $$0.561527\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −23.3820 −0.856075
$$747$$ 0 0
$$748$$ −0.944272 −0.0345260
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ 3.05573 0.111505 0.0557526 0.998445i $$-0.482244\pi$$
0.0557526 + 0.998445i $$0.482244\pi$$
$$752$$ −3.70820 −0.135224
$$753$$ 0 0
$$754$$ −3.81966 −0.139104
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −19.5836 −0.711778 −0.355889 0.934528i $$-0.615822\pi$$
−0.355889 + 0.934528i $$0.615822\pi$$
$$758$$ 6.90983 0.250976
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −27.7771 −1.00692 −0.503459 0.864019i $$-0.667940\pi$$
−0.503459 + 0.864019i $$0.667940\pi$$
$$762$$ 0 0
$$763$$ −8.41641 −0.304694
$$764$$ 10.4721 0.378869
$$765$$ 0 0
$$766$$ 20.5410 0.742177
$$767$$ 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
$$771$$ 0 0
$$772$$ 20.0902 0.723061
$$773$$ 50.1803 1.80486 0.902431 0.430835i $$-0.141781\pi$$
0.902431 + 0.430835i $$0.141781\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −11.7082 −0.420300
$$777$$ 0 0
$$778$$ 1.78522 0.0640032
$$779$$ 21.3050 0.763329
$$780$$ 0 0
$$781$$ −2.05573 −0.0735597
$$782$$ −9.52786 −0.340716
$$783$$ 0 0
$$784$$ 1.85410 0.0662179
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −40.7639 −1.45308 −0.726539 0.687126i $$-0.758872\pi$$
−0.726539 + 0.687126i $$0.758872\pi$$
$$788$$ 2.38197 0.0848540
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 14.4164 0.512588
$$792$$ 0 0
$$793$$ 12.0000 0.426132
$$794$$ 5.59675 0.198621
$$795$$ 0 0
$$796$$ −11.7082 −0.414986
$$797$$ −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
$$801$$ 0 0
$$802$$ 1.56231 0.0551669
$$803$$ 2.06888 0.0730093
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 2.83282 0.0997817
$$807$$ 0 0
$$808$$ −10.6525 −0.374753
$$809$$ −29.4721 −1.03619 −0.518093 0.855325i $$-0.673358\pi$$
−0.518093 + 0.855325i $$0.673358\pi$$
$$810$$ 0 0
$$811$$ −42.7214 −1.50015 −0.750075 0.661353i $$-0.769983\pi$$
−0.750075 + 0.661353i $$0.769983\pi$$
$$812$$ −8.09017 −0.283909
$$813$$ 0 0
$$814$$ 0.437694 0.0153412
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7.88854 0.275985
$$818$$ −15.1246 −0.528820
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −28.8328 −1.00627 −0.503136 0.864207i $$-0.667821\pi$$
−0.503136 + 0.864207i $$0.667821\pi$$
$$822$$ 0 0
$$823$$ 31.6525 1.10334 0.551668 0.834064i $$-0.313992\pi$$
0.551668 + 0.834064i $$0.313992\pi$$
$$824$$ −18.9443 −0.659955
$$825$$ 0 0
$$826$$ −7.23607 −0.251775
$$827$$ 41.5410 1.44452 0.722261 0.691620i $$-0.243103\pi$$
0.722261 + 0.691620i $$0.243103\pi$$
$$828$$ 0 0
$$829$$ −7.63932 −0.265325 −0.132662 0.991161i $$-0.542353\pi$$
−0.132662 + 0.991161i $$0.542353\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0.291796 0.0101162
$$833$$ 2.47214 0.0856544
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 1.70820 0.0590795
$$837$$ 0 0
$$838$$ −16.1803 −0.558941
$$839$$ 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
$$842$$ 8.03444 0.276885
$$843$$ 0 0
$$844$$ −19.4164 −0.668340
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.9443 0.376050
$$848$$ 15.7082 0.539422
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −18.7082 −0.641309
$$852$$ 0 0
$$853$$ −27.4164 −0.938720 −0.469360 0.883007i $$-0.655515\pi$$
−0.469360 + 0.883007i $$0.655515\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 0 0
$$856$$ 17.8885 0.611418
$$857$$ −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 $$-0.375419\pi$$
0.381468 + 0.924382i $$0.375419\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 10.8328 0.368967
$$863$$ 18.3475 0.624557 0.312278 0.949991i $$-0.398908\pi$$
0.312278 + 0.949991i $$0.398908\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −17.5279 −0.595621
$$867$$ 0 0
$$868$$ 6.00000 0.203653
$$869$$ −2.63932 −0.0895328
$$870$$ 0 0
$$871$$ −5.23607 −0.177417
$$872$$ 18.8197 0.637314
$$873$$ 0 0
$$874$$ 17.2361 0.583019
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −30.3607 −1.02521 −0.512604 0.858625i $$-0.671319\pi$$
−0.512604 + 0.858625i $$0.671319\pi$$
$$878$$ 5.12461 0.172947
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −5.81966 −0.196069 −0.0980347 0.995183i $$-0.531256\pi$$
−0.0980347 + 0.995183i $$0.531256\pi$$
$$882$$ 0 0
$$883$$ 1.40325 0.0472232 0.0236116 0.999721i $$-0.492483\pi$$
0.0236116 + 0.999721i $$0.492483\pi$$
$$884$$ 4.94427 0.166294
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ −21.3475 −0.716780 −0.358390 0.933572i $$-0.616674\pi$$
−0.358390 + 0.933572i $$0.616674\pi$$
$$888$$ 0 0
$$889$$ 13.6525 0.457889
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 32.6525 1.09329
$$893$$ 8.94427 0.299309
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −11.3820 −0.380245
$$897$$ 0 0
$$898$$ 12.6869 0.423368
$$899$$ −18.5410 −0.618378
$$900$$ 0 0
$$901$$ 20.9443 0.697755
$$902$$ 0.695048 0.0231426
$$903$$ 0 0
$$904$$ −32.2361 −1.07216
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −34.8328 −1.15660 −0.578302 0.815823i $$-0.696285\pi$$
−0.578302 + 0.815823i $$0.696285\pi$$
$$908$$ −34.6525 −1.14998
$$909$$ 0 0
$$910$$ 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
$$914$$ −7.74265 −0.256104
$$915$$ 0 0
$$916$$ 7.23607 0.239086
$$917$$ −16.9443 −0.559549
$$918$$ 0 0
$$919$$ −27.7639 −0.915848 −0.457924 0.888991i $$-0.651407\pi$$
−0.457924 + 0.888991i $$0.651407\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −8.76393 −0.288625
$$923$$ 10.7639 0.354299
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −8.58359 −0.282074
$$927$$ 0 0
$$928$$ 28.0902 0.922105
$$929$$ −38.2918 −1.25631 −0.628157 0.778087i $$-0.716190\pi$$
−0.628157 + 0.778087i $$0.716190\pi$$
$$930$$ 0 0
$$931$$ −4.47214 −0.146568
$$932$$ −12.8541 −0.421050
$$933$$ 0 0
$$934$$ −4.29180 −0.140432
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −35.2361 −1.15111 −0.575556 0.817762i $$-0.695214\pi$$
−0.575556 + 0.817762i $$0.695214\pi$$
$$938$$ 2.61803 0.0854818
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 5.23607 0.170691 0.0853455 0.996351i $$-0.472801\pi$$
0.0853455 + 0.996351i $$0.472801\pi$$
$$942$$ 0 0
$$943$$ −29.7082 −0.967432
$$944$$ −21.7082 −0.706542
$$945$$ 0 0
$$946$$ 0.257354 0.00836731
$$947$$ 34.8328 1.13191 0.565957 0.824435i $$-0.308507\pi$$
0.565957 + 0.824435i $$0.308507\pi$$
$$948$$ 0 0
$$949$$ −10.8328 −0.351648
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −5.52786 −0.179159
$$953$$ 3.47214 0.112474 0.0562368 0.998417i $$-0.482090\pi$$
0.0562368 + 0.998417i $$0.482090\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −8.94427 −0.289278
$$957$$ 0 0
$$958$$ −16.1803 −0.522763
$$959$$ 10.9443 0.353409
$$960$$ 0 0
$$961$$ −17.2492 −0.556427
$$962$$ −2.29180 −0.0738905
$$963$$ 0 0
$$964$$ 5.70820 0.183849
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 14.1115 0.453794 0.226897 0.973919i $$-0.427142\pi$$
0.226897 + 0.973919i $$0.427142\pi$$
$$968$$ −24.4721 −0.786564
$$969$$ 0 0
$$970$$ 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
$$974$$ 3.56231 0.114144
$$975$$ 0 0
$$976$$ −18.0000 −0.576166
$$977$$ −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
$$981$$ 0 0
$$982$$ −3.56231 −0.113678
$$983$$ −34.5410 −1.10169 −0.550844 0.834608i $$-0.685694\pi$$
−0.550844 + 0.834608i $$0.685694\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 7.63932 0.243286
$$987$$ 0 0
$$988$$ −8.94427 −0.284555
$$989$$ −11.0000 −0.349780
$$990$$ 0 0
$$991$$ 13.1803 0.418687 0.209344 0.977842i $$-0.432867\pi$$
0.209344 + 0.977842i $$0.432867\pi$$
$$992$$ −20.8328 −0.661443
$$993$$ 0 0
$$994$$ −5.38197 −0.170706
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 45.4164 1.43835 0.719176 0.694828i $$-0.244519\pi$$
0.719176 + 0.694828i $$0.244519\pi$$
$$998$$ −6.83282 −0.216289
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1575.2.a.s.1.1 2
3.2 odd 2 175.2.a.d.1.2 2
5.2 odd 4 1575.2.d.k.1324.2 4
5.3 odd 4 1575.2.d.k.1324.3 4
5.4 even 2 1575.2.a.n.1.2 2
12.11 even 2 2800.2.a.bh.1.1 2
15.2 even 4 175.2.b.c.99.3 4
15.8 even 4 175.2.b.c.99.2 4
15.14 odd 2 175.2.a.e.1.1 yes 2
21.20 even 2 1225.2.a.n.1.2 2
60.23 odd 4 2800.2.g.s.449.1 4
60.47 odd 4 2800.2.g.s.449.4 4
60.59 even 2 2800.2.a.bp.1.2 2
105.62 odd 4 1225.2.b.k.99.3 4
105.83 odd 4 1225.2.b.k.99.2 4
105.104 even 2 1225.2.a.u.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.2 2 3.2 odd 2
175.2.a.e.1.1 yes 2 15.14 odd 2
175.2.b.c.99.2 4 15.8 even 4
175.2.b.c.99.3 4 15.2 even 4
1225.2.a.n.1.2 2 21.20 even 2
1225.2.a.u.1.1 2 105.104 even 2
1225.2.b.k.99.2 4 105.83 odd 4
1225.2.b.k.99.3 4 105.62 odd 4
1575.2.a.n.1.2 2 5.4 even 2
1575.2.a.s.1.1 2 1.1 even 1 trivial
1575.2.d.k.1324.2 4 5.2 odd 4
1575.2.d.k.1324.3 4 5.3 odd 4
2800.2.a.bh.1.1 2 12.11 even 2
2800.2.a.bp.1.2 2 60.59 even 2
2800.2.g.s.449.1 4 60.23 odd 4
2800.2.g.s.449.4 4 60.47 odd 4