# Properties

 Label 175.2.a.d.1.2 Level $175$ Weight $2$ Character 175.1 Self dual yes Analytic conductor $1.397$ 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] = [175,2,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.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: $$1.39738203537$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.618034 q^{2} +3.23607 q^{3} -1.61803 q^{4} +2.00000 q^{6} -1.00000 q^{7} -2.23607 q^{8} +7.47214 q^{9} +O(q^{10})$$ $$q+0.618034 q^{2} +3.23607 q^{3} -1.61803 q^{4} +2.00000 q^{6} -1.00000 q^{7} -2.23607 q^{8} +7.47214 q^{9} -0.236068 q^{11} -5.23607 q^{12} -1.23607 q^{13} -0.618034 q^{14} +1.85410 q^{16} -2.47214 q^{17} +4.61803 q^{18} -4.47214 q^{19} -3.23607 q^{21} -0.145898 q^{22} -6.23607 q^{23} -7.23607 q^{24} -0.763932 q^{26} +14.4721 q^{27} +1.61803 q^{28} +5.00000 q^{29} +3.70820 q^{31} +5.61803 q^{32} -0.763932 q^{33} -1.52786 q^{34} -12.0902 q^{36} -3.00000 q^{37} -2.76393 q^{38} -4.00000 q^{39} +4.76393 q^{41} -2.00000 q^{42} -1.76393 q^{43} +0.381966 q^{44} -3.85410 q^{46} +2.00000 q^{47} +6.00000 q^{48} +1.00000 q^{49} -8.00000 q^{51} +2.00000 q^{52} -8.47214 q^{53} +8.94427 q^{54} +2.23607 q^{56} -14.4721 q^{57} +3.09017 q^{58} +11.7082 q^{59} -9.70820 q^{61} +2.29180 q^{62} -7.47214 q^{63} -0.236068 q^{64} -0.472136 q^{66} +4.23607 q^{67} +4.00000 q^{68} -20.1803 q^{69} +8.70820 q^{71} -16.7082 q^{72} +8.76393 q^{73} -1.85410 q^{74} +7.23607 q^{76} +0.236068 q^{77} -2.47214 q^{78} -11.1803 q^{79} +24.4164 q^{81} +2.94427 q^{82} +7.70820 q^{83} +5.23607 q^{84} -1.09017 q^{86} +16.1803 q^{87} +0.527864 q^{88} +17.2361 q^{89} +1.23607 q^{91} +10.0902 q^{92} +12.0000 q^{93} +1.23607 q^{94} +18.1803 q^{96} -5.23607 q^{97} +0.618034 q^{98} -1.76393 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 2 q^{3} - q^{4} + 4 q^{6} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q - q^2 + 2 * q^3 - q^4 + 4 * q^6 - 2 * q^7 + 6 * q^9 $$2 q - q^{2} + 2 q^{3} - q^{4} + 4 q^{6} - 2 q^{7} + 6 q^{9} + 4 q^{11} - 6 q^{12} + 2 q^{13} + q^{14} - 3 q^{16} + 4 q^{17} + 7 q^{18} - 2 q^{21} - 7 q^{22} - 8 q^{23} - 10 q^{24} - 6 q^{26} + 20 q^{27} + q^{28} + 10 q^{29} - 6 q^{31} + 9 q^{32} - 6 q^{33} - 12 q^{34} - 13 q^{36} - 6 q^{37} - 10 q^{38} - 8 q^{39} + 14 q^{41} - 4 q^{42} - 8 q^{43} + 3 q^{44} - q^{46} + 4 q^{47} + 12 q^{48} + 2 q^{49} - 16 q^{51} + 4 q^{52} - 8 q^{53} - 20 q^{57} - 5 q^{58} + 10 q^{59} - 6 q^{61} + 18 q^{62} - 6 q^{63} + 4 q^{64} + 8 q^{66} + 4 q^{67} + 8 q^{68} - 18 q^{69} + 4 q^{71} - 20 q^{72} + 22 q^{73} + 3 q^{74} + 10 q^{76} - 4 q^{77} + 4 q^{78} + 22 q^{81} - 12 q^{82} + 2 q^{83} + 6 q^{84} + 9 q^{86} + 10 q^{87} + 10 q^{88} + 30 q^{89} - 2 q^{91} + 9 q^{92} + 24 q^{93} - 2 q^{94} + 14 q^{96} - 6 q^{97} - q^{98} - 8 q^{99}+O(q^{100})$$ 2 * q - q^2 + 2 * q^3 - q^4 + 4 * q^6 - 2 * q^7 + 6 * q^9 + 4 * q^11 - 6 * q^12 + 2 * q^13 + q^14 - 3 * q^16 + 4 * q^17 + 7 * q^18 - 2 * q^21 - 7 * q^22 - 8 * q^23 - 10 * q^24 - 6 * q^26 + 20 * q^27 + q^28 + 10 * q^29 - 6 * q^31 + 9 * q^32 - 6 * q^33 - 12 * q^34 - 13 * q^36 - 6 * q^37 - 10 * q^38 - 8 * q^39 + 14 * q^41 - 4 * q^42 - 8 * q^43 + 3 * q^44 - q^46 + 4 * q^47 + 12 * q^48 + 2 * q^49 - 16 * q^51 + 4 * q^52 - 8 * q^53 - 20 * q^57 - 5 * q^58 + 10 * q^59 - 6 * q^61 + 18 * q^62 - 6 * q^63 + 4 * q^64 + 8 * q^66 + 4 * q^67 + 8 * q^68 - 18 * q^69 + 4 * q^71 - 20 * q^72 + 22 * q^73 + 3 * q^74 + 10 * q^76 - 4 * q^77 + 4 * q^78 + 22 * q^81 - 12 * q^82 + 2 * q^83 + 6 * q^84 + 9 * q^86 + 10 * q^87 + 10 * q^88 + 30 * q^89 - 2 * q^91 + 9 * q^92 + 24 * q^93 - 2 * q^94 + 14 * q^96 - 6 * q^97 - q^98 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.618034 0.437016 0.218508 0.975835i $$-0.429881\pi$$
0.218508 + 0.975835i $$0.429881\pi$$
$$3$$ 3.23607 1.86834 0.934172 0.356822i $$-0.116140\pi$$
0.934172 + 0.356822i $$0.116140\pi$$
$$4$$ −1.61803 −0.809017
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ −1.00000 −0.377964
$$8$$ −2.23607 −0.790569
$$9$$ 7.47214 2.49071
$$10$$ 0 0
$$11$$ −0.236068 −0.0711772 −0.0355886 0.999367i $$-0.511331\pi$$
−0.0355886 + 0.999367i $$0.511331\pi$$
$$12$$ −5.23607 −1.51152
$$13$$ −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 $$-0.596917\pi$$
−0.299791 + 0.954005i $$0.596917\pi$$
$$18$$ 4.61803 1.08848
$$19$$ −4.47214 −1.02598 −0.512989 0.858395i $$-0.671462\pi$$
−0.512989 + 0.858395i $$0.671462\pi$$
$$20$$ 0 0
$$21$$ −3.23607 −0.706168
$$22$$ −0.145898 −0.0311056
$$23$$ −6.23607 −1.30031 −0.650155 0.759802i $$-0.725296\pi$$
−0.650155 + 0.759802i $$0.725296\pi$$
$$24$$ −7.23607 −1.47706
$$25$$ 0 0
$$26$$ −0.763932 −0.149819
$$27$$ 14.4721 2.78516
$$28$$ 1.61803 0.305780
$$29$$ 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 $$-0.391937\pi$$
0.333007 + 0.942925i $$0.391937\pi$$
$$32$$ 5.61803 0.993137
$$33$$ −0.763932 −0.132983
$$34$$ −1.52786 −0.262027
$$35$$ 0 0
$$36$$ −12.0902 −2.01503
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ −2.76393 −0.448369
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ 4.76393 0.744001 0.372001 0.928232i $$-0.378672\pi$$
0.372001 + 0.928232i $$0.378672\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −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 $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 6.00000 0.866025
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −8.00000 −1.12022
$$52$$ 2.00000 0.277350
$$53$$ −8.47214 −1.16374 −0.581869 0.813283i $$-0.697678\pi$$
−0.581869 + 0.813283i $$0.697678\pi$$
$$54$$ 8.94427 1.21716
$$55$$ 0 0
$$56$$ 2.23607 0.298807
$$57$$ −14.4721 −1.91688
$$58$$ 3.09017 0.405759
$$59$$ 11.7082 1.52428 0.762139 0.647413i $$-0.224149\pi$$
0.762139 + 0.647413i $$0.224149\pi$$
$$60$$ 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
$$63$$ −7.47214 −0.941401
$$64$$ −0.236068 −0.0295085
$$65$$ 0 0
$$66$$ −0.472136 −0.0581159
$$67$$ 4.23607 0.517518 0.258759 0.965942i $$-0.416686\pi$$
0.258759 + 0.965942i $$0.416686\pi$$
$$68$$ 4.00000 0.485071
$$69$$ −20.1803 −2.42943
$$70$$ 0 0
$$71$$ 8.70820 1.03347 0.516737 0.856144i $$-0.327147\pi$$
0.516737 + 0.856144i $$0.327147\pi$$
$$72$$ −16.7082 −1.96908
$$73$$ 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
$$78$$ −2.47214 −0.279914
$$79$$ −11.1803 −1.25789 −0.628943 0.777451i $$-0.716512\pi$$
−0.628943 + 0.777451i $$0.716512\pi$$
$$80$$ 0 0
$$81$$ 24.4164 2.71293
$$82$$ 2.94427 0.325140
$$83$$ 7.70820 0.846085 0.423043 0.906110i $$-0.360962\pi$$
0.423043 + 0.906110i $$0.360962\pi$$
$$84$$ 5.23607 0.571302
$$85$$ 0 0
$$86$$ −1.09017 −0.117556
$$87$$ 16.1803 1.73471
$$88$$ 0.527864 0.0562705
$$89$$ 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
$$92$$ 10.0902 1.05197
$$93$$ 12.0000 1.24434
$$94$$ 1.23607 0.127491
$$95$$ 0 0
$$96$$ 18.1803 1.85552
$$97$$ −5.23607 −0.531642 −0.265821 0.964022i $$-0.585643\pi$$
−0.265821 + 0.964022i $$0.585643\pi$$
$$98$$ 0.618034 0.0624309
$$99$$ −1.76393 −0.177282
$$100$$ 0 0
$$101$$ 4.76393 0.474029 0.237014 0.971506i $$-0.423831\pi$$
0.237014 + 0.971506i $$0.423831\pi$$
$$102$$ −4.94427 −0.489556
$$103$$ −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 $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ −23.4164 −2.25324
$$109$$ 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
$$112$$ −1.85410 −0.175196
$$113$$ 14.4164 1.35618 0.678091 0.734978i $$-0.262808\pi$$
0.678091 + 0.734978i $$0.262808\pi$$
$$114$$ −8.94427 −0.837708
$$115$$ 0 0
$$116$$ −8.09017 −0.751153
$$117$$ −9.23607 −0.853875
$$118$$ 7.23607 0.666134
$$119$$ 2.47214 0.226620
$$120$$ 0 0
$$121$$ −10.9443 −0.994934
$$122$$ −6.00000 −0.543214
$$123$$ 15.4164 1.39005
$$124$$ −6.00000 −0.538816
$$125$$ 0 0
$$126$$ −4.61803 −0.411407
$$127$$ −13.6525 −1.21146 −0.605731 0.795670i $$-0.707119\pi$$
−0.605731 + 0.795670i $$0.707119\pi$$
$$128$$ −11.3820 −1.00603
$$129$$ −5.70820 −0.502579
$$130$$ 0 0
$$131$$ −16.9443 −1.48043 −0.740214 0.672371i $$-0.765276\pi$$
−0.740214 + 0.672371i $$0.765276\pi$$
$$132$$ 1.23607 0.107586
$$133$$ 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 $$-0.345149\pi$$
0.467516 + 0.883985i $$0.345149\pi$$
$$138$$ −12.4721 −1.06170
$$139$$ 10.6525 0.903531 0.451766 0.892137i $$-0.350794\pi$$
0.451766 + 0.892137i $$0.350794\pi$$
$$140$$ 0 0
$$141$$ 6.47214 0.545052
$$142$$ 5.38197 0.451645
$$143$$ 0.291796 0.0244012
$$144$$ 13.8541 1.15451
$$145$$ 0 0
$$146$$ 5.41641 0.448265
$$147$$ 3.23607 0.266906
$$148$$ 4.85410 0.399005
$$149$$ 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 $$-0.807922\pi$$
−0.823394 + 0.567470i $$0.807922\pi$$
$$152$$ 10.0000 0.811107
$$153$$ −18.4721 −1.49338
$$154$$ 0.145898 0.0117568
$$155$$ 0 0
$$156$$ 6.47214 0.518186
$$157$$ −0.763932 −0.0609684 −0.0304842 0.999535i $$-0.509705\pi$$
−0.0304842 + 0.999535i $$0.509705\pi$$
$$158$$ −6.90983 −0.549717
$$159$$ −27.4164 −2.17426
$$160$$ 0 0
$$161$$ 6.23607 0.491471
$$162$$ 15.0902 1.18560
$$163$$ 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 $$-0.564936\pi$$
−0.202590 + 0.979264i $$0.564936\pi$$
$$168$$ 7.23607 0.558275
$$169$$ −11.4721 −0.882472
$$170$$ 0 0
$$171$$ −33.4164 −2.55542
$$172$$ 2.85410 0.217623
$$173$$ 11.5279 0.876447 0.438224 0.898866i $$-0.355608\pi$$
0.438224 + 0.898866i $$0.355608\pi$$
$$174$$ 10.0000 0.758098
$$175$$ 0 0
$$176$$ −0.437694 −0.0329924
$$177$$ 37.8885 2.84788
$$178$$ 10.6525 0.798437
$$179$$ −23.4164 −1.75022 −0.875112 0.483920i $$-0.839213\pi$$
−0.875112 + 0.483920i $$0.839213\pi$$
$$180$$ 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
$$183$$ −31.4164 −2.32237
$$184$$ 13.9443 1.02799
$$185$$ 0 0
$$186$$ 7.41641 0.543797
$$187$$ 0.583592 0.0426765
$$188$$ −3.23607 −0.236015
$$189$$ −14.4721 −1.05269
$$190$$ 0 0
$$191$$ 6.47214 0.468307 0.234154 0.972200i $$-0.424768\pi$$
0.234154 + 0.972200i $$0.424768\pi$$
$$192$$ −0.763932 −0.0551320
$$193$$ −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 $$-0.483299\pi$$
0.0524427 + 0.998624i $$0.483299\pi$$
$$198$$ −1.09017 −0.0774750
$$199$$ 7.23607 0.512951 0.256476 0.966551i $$-0.417439\pi$$
0.256476 + 0.966551i $$0.417439\pi$$
$$200$$ 0 0
$$201$$ 13.7082 0.966902
$$202$$ 2.94427 0.207158
$$203$$ −5.00000 −0.350931
$$204$$ 12.9443 0.906280
$$205$$ 0 0
$$206$$ −5.23607 −0.364814
$$207$$ −46.5967 −3.23870
$$208$$ −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
$$213$$ 28.1803 1.93089
$$214$$ −4.94427 −0.337983
$$215$$ 0 0
$$216$$ −32.3607 −2.20187
$$217$$ −3.70820 −0.251729
$$218$$ 5.20163 0.352299
$$219$$ 28.3607 1.91644
$$220$$ 0 0
$$221$$ 3.05573 0.205551
$$222$$ −6.00000 −0.402694
$$223$$ −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 $$-0.751635\pi$$
−0.710728 + 0.703466i $$0.751635\pi$$
$$228$$ 23.4164 1.55079
$$229$$ −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
$$232$$ −11.1803 −0.734025
$$233$$ −7.94427 −0.520447 −0.260223 0.965548i $$-0.583796\pi$$
−0.260223 + 0.965548i $$0.583796\pi$$
$$234$$ −5.70820 −0.373157
$$235$$ 0 0
$$236$$ −18.9443 −1.23317
$$237$$ −36.1803 −2.35017
$$238$$ 1.52786 0.0990367
$$239$$ −5.52786 −0.357568 −0.178784 0.983888i $$-0.557216\pi$$
−0.178784 + 0.983888i $$0.557216\pi$$
$$240$$ 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
$$243$$ 35.5967 2.28353
$$244$$ 15.7082 1.00561
$$245$$ 0 0
$$246$$ 9.52786 0.607474
$$247$$ 5.52786 0.351730
$$248$$ −8.29180 −0.526530
$$249$$ 24.9443 1.58078
$$250$$ 0 0
$$251$$ 6.47214 0.408518 0.204259 0.978917i $$-0.434522\pi$$
0.204259 + 0.978917i $$0.434522\pi$$
$$252$$ 12.0902 0.761609
$$253$$ 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 $$-0.370876\pi$$
0.394620 + 0.918844i $$0.370876\pi$$
$$258$$ −3.52786 −0.219635
$$259$$ 3.00000 0.186411
$$260$$ 0 0
$$261$$ 37.3607 2.31257
$$262$$ −10.4721 −0.646971
$$263$$ −16.2361 −1.00116 −0.500579 0.865691i $$-0.666880\pi$$
−0.500579 + 0.865691i $$0.666880\pi$$
$$264$$ 1.70820 0.105133
$$265$$ 0 0
$$266$$ 2.76393 0.169468
$$267$$ 55.7771 3.41350
$$268$$ −6.85410 −0.418681
$$269$$ −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 $$-0.244103\pi$$
0.720085 + 0.693885i $$0.244103\pi$$
$$272$$ −4.58359 −0.277921
$$273$$ 4.00000 0.242091
$$274$$ 6.76393 0.408624
$$275$$ 0 0
$$276$$ 32.6525 1.96545
$$277$$ 19.8885 1.19499 0.597493 0.801874i $$-0.296163\pi$$
0.597493 + 0.801874i $$0.296163\pi$$
$$278$$ 6.58359 0.394858
$$279$$ 27.7082 1.65885
$$280$$ 0 0
$$281$$ −15.3607 −0.916341 −0.458171 0.888864i $$-0.651495\pi$$
−0.458171 + 0.888864i $$0.651495\pi$$
$$282$$ 4.00000 0.238197
$$283$$ −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
$$288$$ 41.9787 2.47362
$$289$$ −10.8885 −0.640503
$$290$$ 0 0
$$291$$ −16.9443 −0.993291
$$292$$ −14.1803 −0.829842
$$293$$ 31.1246 1.81832 0.909160 0.416448i $$-0.136725\pi$$
0.909160 + 0.416448i $$0.136725\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 0 0
$$296$$ 6.70820 0.389906
$$297$$ −3.41641 −0.198240
$$298$$ 2.43769 0.141212
$$299$$ 7.70820 0.445777
$$300$$ 0 0
$$301$$ 1.76393 0.101671
$$302$$ −12.5066 −0.719673
$$303$$ 15.4164 0.885649
$$304$$ −8.29180 −0.475567
$$305$$ 0 0
$$306$$ −11.4164 −0.652633
$$307$$ −4.58359 −0.261599 −0.130800 0.991409i $$-0.541754\pi$$
−0.130800 + 0.991409i $$0.541754\pi$$
$$308$$ −0.381966 −0.0217645
$$309$$ −27.4164 −1.55966
$$310$$ 0 0
$$311$$ 24.3607 1.38137 0.690684 0.723157i $$-0.257310\pi$$
0.690684 + 0.723157i $$0.257310\pi$$
$$312$$ 8.94427 0.506370
$$313$$ −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 $$-0.752301\pi$$
−0.712199 + 0.701978i $$0.752301\pi$$
$$318$$ −16.9443 −0.950188
$$319$$ −1.18034 −0.0660863
$$320$$ 0 0
$$321$$ −25.8885 −1.44496
$$322$$ 3.85410 0.214781
$$323$$ 11.0557 0.615157
$$324$$ −39.5066 −2.19481
$$325$$ 0 0
$$326$$ 0.944272 0.0522984
$$327$$ 27.2361 1.50616
$$328$$ −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
$$333$$ −22.4164 −1.22841
$$334$$ −3.23607 −0.177070
$$335$$ 0 0
$$336$$ −6.00000 −0.327327
$$337$$ 16.4721 0.897294 0.448647 0.893709i $$-0.351906\pi$$
0.448647 + 0.893709i $$0.351906\pi$$
$$338$$ −7.09017 −0.385654
$$339$$ 46.6525 2.53381
$$340$$ 0 0
$$341$$ −0.875388 −0.0474049
$$342$$ −20.6525 −1.11676
$$343$$ −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 $$-0.682774\pi$$
−0.543165 + 0.839626i $$0.682774\pi$$
$$348$$ −26.1803 −1.40341
$$349$$ 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
$$352$$ −1.32624 −0.0706887
$$353$$ 2.18034 0.116048 0.0580239 0.998315i $$-0.481520\pi$$
0.0580239 + 0.998315i $$0.481520\pi$$
$$354$$ 23.4164 1.24457
$$355$$ 0 0
$$356$$ −27.8885 −1.47809
$$357$$ 8.00000 0.423405
$$358$$ −14.4721 −0.764876
$$359$$ −30.1246 −1.58992 −0.794958 0.606664i $$-0.792507\pi$$
−0.794958 + 0.606664i $$0.792507\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 5.05573 0.265723
$$363$$ −35.4164 −1.85888
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ −19.4164 −1.01491
$$367$$ 37.1246 1.93789 0.968944 0.247278i $$-0.0795362\pi$$
0.968944 + 0.247278i $$0.0795362\pi$$
$$368$$ −11.5623 −0.602727
$$369$$ 35.5967 1.85309
$$370$$ 0 0
$$371$$ 8.47214 0.439851
$$372$$ −19.4164 −1.00669
$$373$$ 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
$$378$$ −8.94427 −0.460044
$$379$$ −11.1803 −0.574295 −0.287148 0.957886i $$-0.592707\pi$$
−0.287148 + 0.957886i $$0.592707\pi$$
$$380$$ 0 0
$$381$$ −44.1803 −2.26343
$$382$$ 4.00000 0.204658
$$383$$ 33.2361 1.69828 0.849142 0.528165i $$-0.177120\pi$$
0.849142 + 0.528165i $$0.177120\pi$$
$$384$$ −36.8328 −1.87962
$$385$$ 0 0
$$386$$ −7.67376 −0.390584
$$387$$ −13.1803 −0.669994
$$388$$ 8.47214 0.430108
$$389$$ 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
$$392$$ −2.23607 −0.112938
$$393$$ −54.8328 −2.76595
$$394$$ 0.909830 0.0458366
$$395$$ 0 0
$$396$$ 2.85410 0.143424
$$397$$ −9.05573 −0.454494 −0.227247 0.973837i $$-0.572972\pi$$
−0.227247 + 0.973837i $$0.572972\pi$$
$$398$$ 4.47214 0.224168
$$399$$ 14.4721 0.724513
$$400$$ 0 0
$$401$$ 2.52786 0.126236 0.0631178 0.998006i $$-0.479896\pi$$
0.0631178 + 0.998006i $$0.479896\pi$$
$$402$$ 8.47214 0.422552
$$403$$ −4.58359 −0.228325
$$404$$ −7.70820 −0.383497
$$405$$ 0 0
$$406$$ −3.09017 −0.153363
$$407$$ 0.708204 0.0351044
$$408$$ 17.8885 0.885615
$$409$$ 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
$$412$$ 13.7082 0.675355
$$413$$ −11.7082 −0.576123
$$414$$ −28.7984 −1.41536
$$415$$ 0 0
$$416$$ −6.94427 −0.340471
$$417$$ 34.4721 1.68811
$$418$$ 0.652476 0.0319136
$$419$$ −26.1803 −1.27899 −0.639497 0.768794i $$-0.720857\pi$$
−0.639497 + 0.768794i $$0.720857\pi$$
$$420$$ 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
$$423$$ 14.9443 0.726615
$$424$$ 18.9443 0.920015
$$425$$ 0 0
$$426$$ 17.4164 0.843828
$$427$$ 9.70820 0.469813
$$428$$ 12.9443 0.625685
$$429$$ 0.944272 0.0455899
$$430$$ 0 0
$$431$$ 17.5279 0.844288 0.422144 0.906529i $$-0.361278\pi$$
0.422144 + 0.906529i $$0.361278\pi$$
$$432$$ 26.8328 1.29099
$$433$$ 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
$$438$$ 17.5279 0.837514
$$439$$ −8.29180 −0.395746 −0.197873 0.980228i $$-0.563403\pi$$
−0.197873 + 0.980228i $$0.563403\pi$$
$$440$$ 0 0
$$441$$ 7.47214 0.355816
$$442$$ 1.88854 0.0898289
$$443$$ 19.4164 0.922501 0.461251 0.887270i $$-0.347401\pi$$
0.461251 + 0.887270i $$0.347401\pi$$
$$444$$ 15.7082 0.745478
$$445$$ 0 0
$$446$$ −12.4721 −0.590573
$$447$$ 12.7639 0.603713
$$448$$ 0.236068 0.0111532
$$449$$ 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
$$452$$ −23.3262 −1.09717
$$453$$ −65.4853 −3.07677
$$454$$ −13.2361 −0.621199
$$455$$ 0 0
$$456$$ 32.3607 1.51543
$$457$$ 12.5279 0.586029 0.293014 0.956108i $$-0.405342\pi$$
0.293014 + 0.956108i $$0.405342\pi$$
$$458$$ −2.76393 −0.129150
$$459$$ −35.7771 −1.66993
$$460$$ 0 0
$$461$$ −14.1803 −0.660444 −0.330222 0.943903i $$-0.607124\pi$$
−0.330222 + 0.943903i $$0.607124\pi$$
$$462$$ 0.472136 0.0219658
$$463$$ 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 $$-0.551366\pi$$
−0.160671 + 0.987008i $$0.551366\pi$$
$$468$$ 14.9443 0.690799
$$469$$ −4.23607 −0.195603
$$470$$ 0 0
$$471$$ −2.47214 −0.113910
$$472$$ −26.1803 −1.20505
$$473$$ 0.416408 0.0191465
$$474$$ −22.3607 −1.02706
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ −63.3050 −2.89853
$$478$$ −3.41641 −0.156263
$$479$$ −26.1803 −1.19621 −0.598105 0.801418i $$-0.704079\pi$$
−0.598105 + 0.801418i $$0.704079\pi$$
$$480$$ 0 0
$$481$$ 3.70820 0.169080
$$482$$ −2.18034 −0.0993118
$$483$$ 20.1803 0.918237
$$484$$ 17.7082 0.804918
$$485$$ 0 0
$$486$$ 22.0000 0.997940
$$487$$ −5.76393 −0.261189 −0.130594 0.991436i $$-0.541689\pi$$
−0.130594 + 0.991436i $$0.541689\pi$$
$$488$$ 21.7082 0.982684
$$489$$ 4.94427 0.223588
$$490$$ 0 0
$$491$$ −5.76393 −0.260123 −0.130061 0.991506i $$-0.541517\pi$$
−0.130061 + 0.991506i $$0.541517\pi$$
$$492$$ −24.9443 −1.12457
$$493$$ −12.3607 −0.556697
$$494$$ 3.41641 0.153711
$$495$$ 0 0
$$496$$ 6.87539 0.308714
$$497$$ −8.70820 −0.390616
$$498$$ 15.4164 0.690826
$$499$$ 11.0557 0.494922 0.247461 0.968898i $$-0.420404\pi$$
0.247461 + 0.968898i $$0.420404\pi$$
$$500$$ 0 0
$$501$$ −16.9443 −0.757014
$$502$$ 4.00000 0.178529
$$503$$ 8.11146 0.361672 0.180836 0.983513i $$-0.442120\pi$$
0.180836 + 0.983513i $$0.442120\pi$$
$$504$$ 16.7082 0.744243
$$505$$ 0 0
$$506$$ 0.909830 0.0404469
$$507$$ −37.1246 −1.64876
$$508$$ 22.0902 0.980093
$$509$$ 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
$$512$$ 18.7082 0.826794
$$513$$ −64.7214 −2.85752
$$514$$ 7.81966 0.344910
$$515$$ 0 0
$$516$$ 9.23607 0.406595
$$517$$ −0.472136 −0.0207645
$$518$$ 1.85410 0.0814646
$$519$$ 37.3050 1.63751
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 23.0902 1.01063
$$523$$ −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
$$528$$ −1.41641 −0.0616412
$$529$$ 15.8885 0.690806
$$530$$ 0 0
$$531$$ 87.4853 3.79654
$$532$$ −7.23607 −0.313723
$$533$$ −5.88854 −0.255061
$$534$$ 34.4721 1.49176
$$535$$ 0 0
$$536$$ −9.47214 −0.409134
$$537$$ −75.7771 −3.27002
$$538$$ −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
$$543$$ 26.4721 1.13603
$$544$$ −13.8885 −0.595466
$$545$$ 0 0
$$546$$ 2.47214 0.105798
$$547$$ 9.76393 0.417476 0.208738 0.977972i $$-0.433064\pi$$
0.208738 + 0.977972i $$0.433064\pi$$
$$548$$ −17.7082 −0.756457
$$549$$ −72.5410 −3.09598
$$550$$ 0 0
$$551$$ −22.3607 −0.952597
$$552$$ 45.1246 1.92063
$$553$$ 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 $$-0.438168\pi$$
0.193032 + 0.981192i $$0.438168\pi$$
$$558$$ 17.1246 0.724943
$$559$$ 2.18034 0.0922186
$$560$$ 0 0
$$561$$ 1.88854 0.0797344
$$562$$ −9.49342 −0.400456
$$563$$ −17.4164 −0.734014 −0.367007 0.930218i $$-0.619618\pi$$
−0.367007 + 0.930218i $$0.619618\pi$$
$$564$$ −10.4721 −0.440956
$$565$$ 0 0
$$566$$ −10.7639 −0.452442
$$567$$ −24.4164 −1.02539
$$568$$ −19.4721 −0.817033
$$569$$ −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 $$-0.222362\pi$$
0.765763 + 0.643123i $$0.222362\pi$$
$$572$$ −0.472136 −0.0197410
$$573$$ 20.9443 0.874960
$$574$$ −2.94427 −0.122892
$$575$$ 0 0
$$576$$ −1.76393 −0.0734972
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −6.72949 −0.279910
$$579$$ −40.1803 −1.66984
$$580$$ 0 0
$$581$$ −7.70820 −0.319790
$$582$$ −10.4721 −0.434084
$$583$$ 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 $$-0.329254\pi$$
0.511058 + 0.859546i $$0.329254\pi$$
$$588$$ −5.23607 −0.215932
$$589$$ −16.5836 −0.683315
$$590$$ 0 0
$$591$$ 4.76393 0.195962
$$592$$ −5.56231 −0.228609
$$593$$ 37.3050 1.53193 0.765965 0.642882i $$-0.222261\pi$$
0.765965 + 0.642882i $$0.222261\pi$$
$$594$$ −2.11146 −0.0866341
$$595$$ 0 0
$$596$$ −6.38197 −0.261416
$$597$$ 23.4164 0.958370
$$598$$ 4.76393 0.194812
$$599$$ −11.1803 −0.456816 −0.228408 0.973565i $$-0.573352\pi$$
−0.228408 + 0.973565i $$0.573352\pi$$
$$600$$ 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
$$603$$ 31.6525 1.28899
$$604$$ 32.7426 1.33228
$$605$$ 0 0
$$606$$ 9.52786 0.387043
$$607$$ 7.12461 0.289179 0.144590 0.989492i $$-0.453814\pi$$
0.144590 + 0.989492i $$0.453814\pi$$
$$608$$ −25.1246 −1.01894
$$609$$ −16.1803 −0.655660
$$610$$ 0 0
$$611$$ −2.47214 −0.100012
$$612$$ 29.8885 1.20817
$$613$$ 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 $$-0.461822\pi$$
0.119654 + 0.992816i $$0.461822\pi$$
$$618$$ −16.9443 −0.681599
$$619$$ −11.7082 −0.470592 −0.235296 0.971924i $$-0.575606\pi$$
−0.235296 + 0.971924i $$0.575606\pi$$
$$620$$ 0 0
$$621$$ −90.2492 −3.62158
$$622$$ 15.0557 0.603680
$$623$$ −17.2361 −0.690548
$$624$$ −7.41641 −0.296894
$$625$$ 0 0
$$626$$ −12.0689 −0.482370
$$627$$ 3.41641 0.136438
$$628$$ 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
$$633$$ 38.8328 1.54347
$$634$$ −15.6738 −0.622485
$$635$$ 0 0
$$636$$ 44.3607 1.75902
$$637$$ −1.23607 −0.0489748
$$638$$ −0.729490 −0.0288808
$$639$$ 65.0689 2.57409
$$640$$ 0 0
$$641$$ 43.8328 1.73129 0.865646 0.500656i $$-0.166908\pi$$
0.865646 + 0.500656i $$0.166908\pi$$
$$642$$ −16.0000 −0.631470
$$643$$ −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 $$-0.372147\pi$$
0.390950 + 0.920412i $$0.372147\pi$$
$$648$$ −54.5967 −2.14476
$$649$$ −2.76393 −0.108494
$$650$$ 0 0
$$651$$ −12.0000 −0.470317
$$652$$ −2.47214 −0.0968163
$$653$$ −25.0557 −0.980506 −0.490253 0.871580i $$-0.663096\pi$$
−0.490253 + 0.871580i $$0.663096\pi$$
$$654$$ 16.8328 0.658215
$$655$$ 0 0
$$656$$ 8.83282 0.344864
$$657$$ 65.4853 2.55482
$$658$$ −1.23607 −0.0481869
$$659$$ 17.8885 0.696839 0.348419 0.937339i $$-0.386719\pi$$
0.348419 + 0.937339i $$0.386719\pi$$
$$660$$ 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
$$663$$ 9.88854 0.384039
$$664$$ −17.2361 −0.668889
$$665$$ 0 0
$$666$$ −13.8541 −0.536836
$$667$$ −31.1803 −1.20731
$$668$$ 8.47214 0.327797
$$669$$ −65.3050 −2.52484
$$670$$ 0 0
$$671$$ 2.29180 0.0884738
$$672$$ −18.1803 −0.701322
$$673$$ −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 $$-0.411003\pi$$
0.275963 + 0.961168i $$0.411003\pi$$
$$678$$ 28.8328 1.10732
$$679$$ 5.23607 0.200942
$$680$$ 0 0
$$681$$ −69.3050 −2.65577
$$682$$ −0.541020 −0.0207167
$$683$$ −14.1246 −0.540463 −0.270232 0.962795i $$-0.587100\pi$$
−0.270232 + 0.962795i $$0.587100\pi$$
$$684$$ 54.0689 2.06738
$$685$$ 0 0
$$686$$ −0.618034 −0.0235966
$$687$$ −14.4721 −0.552146
$$688$$ −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
$$693$$ 1.76393 0.0670062
$$694$$ −12.5066 −0.474743
$$695$$ 0 0
$$696$$ −36.1803 −1.37141
$$697$$ −11.7771 −0.446089
$$698$$ 2.76393 0.104616
$$699$$ −25.7082 −0.972374
$$700$$ 0 0
$$701$$ −29.0557 −1.09742 −0.548710 0.836013i $$-0.684881\pi$$
−0.548710 + 0.836013i $$0.684881\pi$$
$$702$$ −11.0557 −0.417272
$$703$$ 13.4164 0.506009
$$704$$ 0.0557281 0.00210033
$$705$$ 0 0
$$706$$ 1.34752 0.0507147
$$707$$ −4.76393 −0.179166
$$708$$ −61.3050 −2.30398
$$709$$ −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
$$712$$ −38.5410 −1.44439
$$713$$ −23.1246 −0.866024
$$714$$ 4.94427 0.185035
$$715$$ 0 0
$$716$$ 37.8885 1.41596
$$717$$ −17.8885 −0.668060
$$718$$ −18.6180 −0.694819
$$719$$ 16.1803 0.603425 0.301712 0.953399i $$-0.402442\pi$$
0.301712 + 0.953399i $$0.402442\pi$$
$$720$$ 0 0
$$721$$ 8.47214 0.315519
$$722$$ 0.618034 0.0230008
$$723$$ −11.4164 −0.424581
$$724$$ −13.2361 −0.491915
$$725$$ 0 0
$$726$$ −21.8885 −0.812360
$$727$$ 3.05573 0.113331 0.0566653 0.998393i $$-0.481953\pi$$
0.0566653 + 0.998393i $$0.481953\pi$$
$$728$$ −2.76393 −0.102438
$$729$$ 41.9443 1.55349
$$730$$ 0 0
$$731$$ 4.36068 0.161286
$$732$$ 50.8328 1.87883
$$733$$ −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
$$738$$ 22.0000 0.809831
$$739$$ 25.6525 0.943642 0.471821 0.881694i $$-0.343597\pi$$
0.471821 + 0.881694i $$0.343597\pi$$
$$740$$ 0 0
$$741$$ 17.8885 0.657152
$$742$$ 5.23607 0.192222
$$743$$ 10.4721 0.384185 0.192093 0.981377i $$-0.438473\pi$$
0.192093 + 0.981377i $$0.438473\pi$$
$$744$$ −26.8328 −0.983739
$$745$$ 0 0
$$746$$ 23.3820 0.856075
$$747$$ 57.5967 2.10735
$$748$$ −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
$$753$$ 20.9443 0.763252
$$754$$ −3.81966 −0.139104
$$755$$ 0 0
$$756$$ 23.4164 0.851647
$$757$$ −19.5836 −0.711778 −0.355889 0.934528i $$-0.615822\pi$$
−0.355889 + 0.934528i $$0.615822\pi$$
$$758$$ −6.90983 −0.250976
$$759$$ 4.76393 0.172920
$$760$$ 0 0
$$761$$ 27.7771 1.00692 0.503459 0.864019i $$-0.332060\pi$$
0.503459 + 0.864019i $$0.332060\pi$$
$$762$$ −27.3050 −0.989154
$$763$$ −8.41641 −0.304694
$$764$$ −10.4721 −0.378869
$$765$$ 0 0
$$766$$ 20.5410 0.742177
$$767$$ −14.4721 −0.522559
$$768$$ −21.2361 −0.766291
$$769$$ −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
$$772$$ 20.0902 0.723061
$$773$$ −50.1803 −1.80486 −0.902431 0.430835i $$-0.858219\pi$$
−0.902431 + 0.430835i $$0.858219\pi$$
$$774$$ −8.14590 −0.292798
$$775$$ 0 0
$$776$$ 11.7082 0.420300
$$777$$ 9.70820 0.348280
$$778$$ 1.78522 0.0640032
$$779$$ −21.3050 −0.763329
$$780$$ 0 0
$$781$$ −2.05573 −0.0735597
$$782$$ 9.52786 0.340716
$$783$$ 72.3607 2.58596
$$784$$ 1.85410 0.0662179
$$785$$ 0 0
$$786$$ −33.8885 −1.20876
$$787$$ −40.7639 −1.45308 −0.726539 0.687126i $$-0.758872\pi$$
−0.726539 + 0.687126i $$0.758872\pi$$
$$788$$ −2.38197 −0.0848540
$$789$$ −52.5410 −1.87051
$$790$$ 0 0
$$791$$ −14.4164 −0.512588
$$792$$ 3.94427 0.140154
$$793$$ 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 $$-0.284178\pi$$
0.627257 + 0.778813i $$0.284178\pi$$
$$798$$ 8.94427 0.316624
$$799$$ −4.94427 −0.174916
$$800$$ 0 0
$$801$$ 128.790 4.55058
$$802$$ 1.56231 0.0551669
$$803$$ −2.06888 −0.0730093
$$804$$ −22.1803 −0.782240
$$805$$ 0 0
$$806$$ −2.83282 −0.0997817
$$807$$ −37.8885 −1.33374
$$808$$ −10.6525 −0.374753
$$809$$ 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 $$-0.769983\pi$$
−0.750075 + 0.661353i $$0.769983\pi$$
$$812$$ 8.09017 0.283909
$$813$$ 76.7214 2.69074
$$814$$ 0.437694 0.0153412
$$815$$ 0 0
$$816$$ −14.8328 −0.519252
$$817$$ 7.88854 0.275985
$$818$$ 15.1246 0.528820
$$819$$ 9.23607 0.322734
$$820$$ 0 0
$$821$$ 28.8328 1.00627 0.503136 0.864207i $$-0.332179\pi$$
0.503136 + 0.864207i $$0.332179\pi$$
$$822$$ 21.8885 0.763451
$$823$$ 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 $$-0.756897\pi$$
−0.722261 + 0.691620i $$0.756897\pi$$
$$828$$ 75.3951 2.62016
$$829$$ −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
$$832$$ 0.291796 0.0101162
$$833$$ −2.47214 −0.0856544
$$834$$ 21.3050 0.737730
$$835$$ 0 0
$$836$$ −1.70820 −0.0590795
$$837$$ 53.6656 1.85496
$$838$$ −16.1803 −0.558941
$$839$$ −30.6525 −1.05824 −0.529120 0.848547i $$-0.677478\pi$$
−0.529120 + 0.848547i $$0.677478\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ −8.03444 −0.276885
$$843$$ −49.7082 −1.71204
$$844$$ −19.4164 −0.668340
$$845$$ 0 0
$$846$$ 9.23607 0.317543
$$847$$ 10.9443 0.376050
$$848$$ −15.7082 −0.539422
$$849$$ −56.3607 −1.93429
$$850$$ 0 0
$$851$$ 18.7082 0.641309
$$852$$ −45.5967 −1.56212
$$853$$ −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 $$-0.412912\pi$$
0.270195 + 0.962806i $$0.412912\pi$$
$$858$$ 0.583592 0.0199235
$$859$$ 22.3607 0.762937 0.381468 0.924382i $$-0.375419\pi$$
0.381468 + 0.924382i $$0.375419\pi$$
$$860$$ 0 0
$$861$$ −15.4164 −0.525390
$$862$$ 10.8328 0.368967
$$863$$ −18.3475 −0.624557 −0.312278 0.949991i $$-0.601092\pi$$
−0.312278 + 0.949991i $$0.601092\pi$$
$$864$$ 81.3050 2.76605
$$865$$ 0 0
$$866$$ 17.5279 0.595621
$$867$$ −35.2361 −1.19668
$$868$$ 6.00000 0.203653
$$869$$ 2.63932 0.0895328
$$870$$ 0 0
$$871$$ −5.23607 −0.177417
$$872$$ −18.8197 −0.637314
$$873$$ −39.1246 −1.32417
$$874$$ 17.2361 0.583019
$$875$$ 0 0
$$876$$ −45.8885 −1.55043
$$877$$ −30.3607 −1.02521 −0.512604 0.858625i $$-0.671319\pi$$
−0.512604 + 0.858625i $$0.671319\pi$$
$$878$$ −5.12461 −0.172947
$$879$$ 100.721 3.39725
$$880$$ 0 0
$$881$$ 5.81966 0.196069 0.0980347 0.995183i $$-0.468744\pi$$
0.0980347 + 0.995183i $$0.468744\pi$$
$$882$$ 4.61803 0.155497
$$883$$ 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 $$-0.383326\pi$$
0.358390 + 0.933572i $$0.383326\pi$$
$$888$$ 21.7082 0.728480
$$889$$ 13.6525 0.457889
$$890$$ 0 0
$$891$$ −5.76393 −0.193099
$$892$$ 32.6525 1.09329
$$893$$ −8.94427 −0.299309
$$894$$ 7.88854 0.263832
$$895$$ 0 0
$$896$$ 11.3820 0.380245
$$897$$ 24.9443 0.832865
$$898$$ 12.6869 0.423368
$$899$$ 18.5410 0.618378
$$900$$ 0 0
$$901$$ 20.9443 0.697755
$$902$$ −0.695048 −0.0231426
$$903$$ 5.70820 0.189957
$$904$$ −32.2361 −1.07216
$$905$$ 0 0
$$906$$ −40.4721 −1.34460
$$907$$ −34.8328 −1.15660 −0.578302 0.815823i $$-0.696285\pi$$
−0.578302 + 0.815823i $$0.696285\pi$$
$$908$$ 34.6525 1.14998
$$909$$ 35.5967 1.18067
$$910$$ 0 0
$$911$$ 0.819660 0.0271566 0.0135783 0.999908i $$-0.495678\pi$$
0.0135783 + 0.999908i $$0.495678\pi$$
$$912$$ −26.8328 −0.888523
$$913$$ −1.81966 −0.0602220
$$914$$ 7.74265 0.256104
$$915$$ 0 0
$$916$$ 7.23607 0.239086
$$917$$ 16.9443 0.559549
$$918$$ −22.1115 −0.729787
$$919$$ −27.7639 −0.915848 −0.457924 0.888991i $$-0.651407\pi$$
−0.457924 + 0.888991i $$0.651407\pi$$
$$920$$ 0 0
$$921$$ −14.8328 −0.488758
$$922$$ −8.76393 −0.288625
$$923$$ −10.7639 −0.354299
$$924$$ −1.23607 −0.0406637
$$925$$ 0 0
$$926$$ 8.58359 0.282074
$$927$$ −63.3050 −2.07921
$$928$$ 28.0902 0.922105
$$929$$ 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
$$932$$ 12.8541 0.421050
$$933$$ 78.8328 2.58087
$$934$$ −4.29180 −0.140432
$$935$$ 0 0
$$936$$ 20.6525 0.675047
$$937$$ −35.2361 −1.15111 −0.575556 0.817762i $$-0.695214\pi$$
−0.575556 + 0.817762i $$0.695214\pi$$
$$938$$ −2.61803 −0.0854818
$$939$$ −63.1935 −2.06224
$$940$$ 0 0
$$941$$ −5.23607 −0.170691 −0.0853455 0.996351i $$-0.527199\pi$$
−0.0853455 + 0.996351i $$0.527199\pi$$
$$942$$ −1.52786 −0.0497805
$$943$$ −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 $$-0.691493\pi$$
−0.565957 + 0.824435i $$0.691493\pi$$
$$948$$ 58.5410 1.90132
$$949$$ −10.8328 −0.351648
$$950$$ 0 0
$$951$$ −82.0689 −2.66127
$$952$$ −5.52786 −0.179159
$$953$$ −3.47214 −0.112474 −0.0562368 0.998417i $$-0.517910\pi$$
−0.0562368 + 0.998417i $$0.517910\pi$$
$$954$$ −39.1246 −1.26671
$$955$$ 0 0
$$956$$ 8.94427 0.289278
$$957$$ −3.81966 −0.123472
$$958$$ −16.1803 −0.522763
$$959$$ −10.9443 −0.353409
$$960$$ 0 0
$$961$$ −17.2492 −0.556427
$$962$$ 2.29180 0.0738905
$$963$$ −59.7771 −1.92629
$$964$$ 5.70820 0.183849
$$965$$ 0 0
$$966$$ 12.4721 0.401284
$$967$$ 14.1115 0.453794 0.226897 0.973919i $$-0.427142\pi$$
0.226897 + 0.973919i $$0.427142\pi$$
$$968$$ 24.4721 0.786564
$$969$$ 35.7771 1.14933
$$970$$ 0 0
$$971$$ −18.0000 −0.577647 −0.288824 0.957382i $$-0.593264\pi$$
−0.288824 + 0.957382i $$0.593264\pi$$
$$972$$ −57.5967 −1.84742
$$973$$ −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 $$-0.441253\pi$$
0.183513 + 0.983017i $$0.441253\pi$$
$$978$$ 3.05573 0.0977114
$$979$$ −4.06888 −0.130042
$$980$$ 0 0
$$981$$ 62.8885 2.00788
$$982$$ −3.56231 −0.113678
$$983$$ 34.5410 1.10169 0.550844 0.834608i $$-0.314306\pi$$
0.550844 + 0.834608i $$0.314306\pi$$
$$984$$ −34.4721 −1.09893
$$985$$ 0 0
$$986$$ −7.63932 −0.243286
$$987$$ −6.47214 −0.206010
$$988$$ −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
$$993$$ −79.9574 −2.53737
$$994$$ −5.38197 −0.170706
$$995$$ 0 0
$$996$$ −40.3607 −1.27888
$$997$$ 45.4164 1.43835 0.719176 0.694828i $$-0.244519\pi$$
0.719176 + 0.694828i $$0.244519\pi$$
$$998$$ 6.83282 0.216289
$$999$$ −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 175.2.a.d.1.2 2
3.2 odd 2 1575.2.a.s.1.1 2
4.3 odd 2 2800.2.a.bh.1.1 2
5.2 odd 4 175.2.b.c.99.3 4
5.3 odd 4 175.2.b.c.99.2 4
5.4 even 2 175.2.a.e.1.1 yes 2
7.6 odd 2 1225.2.a.n.1.2 2
15.2 even 4 1575.2.d.k.1324.2 4
15.8 even 4 1575.2.d.k.1324.3 4
15.14 odd 2 1575.2.a.n.1.2 2
20.3 even 4 2800.2.g.s.449.1 4
20.7 even 4 2800.2.g.s.449.4 4
20.19 odd 2 2800.2.a.bp.1.2 2
35.13 even 4 1225.2.b.k.99.2 4
35.27 even 4 1225.2.b.k.99.3 4
35.34 odd 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 1.1 even 1 trivial
175.2.a.e.1.1 yes 2 5.4 even 2
175.2.b.c.99.2 4 5.3 odd 4
175.2.b.c.99.3 4 5.2 odd 4
1225.2.a.n.1.2 2 7.6 odd 2
1225.2.a.u.1.1 2 35.34 odd 2
1225.2.b.k.99.2 4 35.13 even 4
1225.2.b.k.99.3 4 35.27 even 4
1575.2.a.n.1.2 2 15.14 odd 2
1575.2.a.s.1.1 2 3.2 odd 2
1575.2.d.k.1324.2 4 15.2 even 4
1575.2.d.k.1324.3 4 15.8 even 4
2800.2.a.bh.1.1 2 4.3 odd 2
2800.2.a.bp.1.2 2 20.19 odd 2
2800.2.g.s.449.1 4 20.3 even 4
2800.2.g.s.449.4 4 20.7 even 4