# Properties

 Label 1225.2.a.u.1.1 Level $1225$ Weight $2$ Character 1225.1 Self dual yes Analytic conductor $9.782$ 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] = [1225,2,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, 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("1225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1225.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: $$9.78167424761$$ 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: 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 $$\chi$$ $$=$$ 1225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.618034 q^{2} +3.23607 q^{3} -1.61803 q^{4} -2.00000 q^{6} +2.23607 q^{8} +7.47214 q^{9} +O(q^{10})$$ $$q-0.618034 q^{2} +3.23607 q^{3} -1.61803 q^{4} -2.00000 q^{6} +2.23607 q^{8} +7.47214 q^{9} -0.236068 q^{11} -5.23607 q^{12} -1.23607 q^{13} +1.85410 q^{16} -2.47214 q^{17} -4.61803 q^{18} +4.47214 q^{19} +0.145898 q^{22} +6.23607 q^{23} +7.23607 q^{24} +0.763932 q^{26} +14.4721 q^{27} +5.00000 q^{29} -3.70820 q^{31} -5.61803 q^{32} -0.763932 q^{33} +1.52786 q^{34} -12.0902 q^{36} +3.00000 q^{37} -2.76393 q^{38} -4.00000 q^{39} -4.76393 q^{41} +1.76393 q^{43} +0.381966 q^{44} -3.85410 q^{46} +2.00000 q^{47} +6.00000 q^{48} -8.00000 q^{51} +2.00000 q^{52} +8.47214 q^{53} -8.94427 q^{54} +14.4721 q^{57} -3.09017 q^{58} -11.7082 q^{59} +9.70820 q^{61} +2.29180 q^{62} -0.236068 q^{64} +0.472136 q^{66} -4.23607 q^{67} +4.00000 q^{68} +20.1803 q^{69} +8.70820 q^{71} +16.7082 q^{72} +8.76393 q^{73} -1.85410 q^{74} -7.23607 q^{76} +2.47214 q^{78} -11.1803 q^{79} +24.4164 q^{81} +2.94427 q^{82} +7.70820 q^{83} -1.09017 q^{86} +16.1803 q^{87} -0.527864 q^{88} -17.2361 q^{89} -10.0902 q^{92} -12.0000 q^{93} -1.23607 q^{94} -18.1803 q^{96} -5.23607 q^{97} -1.76393 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 2 q^{3} - q^{4} - 4 q^{6} + 6 q^{9}+O(q^{10})$$ 2 * q + q^2 + 2 * q^3 - q^4 - 4 * q^6 + 6 * q^9 $$2 q + q^{2} + 2 q^{3} - q^{4} - 4 q^{6} + 6 q^{9} + 4 q^{11} - 6 q^{12} + 2 q^{13} - 3 q^{16} + 4 q^{17} - 7 q^{18} + 7 q^{22} + 8 q^{23} + 10 q^{24} + 6 q^{26} + 20 q^{27} + 10 q^{29} + 6 q^{31} - 9 q^{32} - 6 q^{33} + 12 q^{34} - 13 q^{36} + 6 q^{37} - 10 q^{38} - 8 q^{39} - 14 q^{41} + 8 q^{43} + 3 q^{44} - q^{46} + 4 q^{47} + 12 q^{48} - 16 q^{51} + 4 q^{52} + 8 q^{53} + 20 q^{57} + 5 q^{58} - 10 q^{59} + 6 q^{61} + 18 q^{62} + 4 q^{64} - 8 q^{66} - 4 q^{67} + 8 q^{68} + 18 q^{69} + 4 q^{71} + 20 q^{72} + 22 q^{73} + 3 q^{74} - 10 q^{76} - 4 q^{78} + 22 q^{81} - 12 q^{82} + 2 q^{83} + 9 q^{86} + 10 q^{87} - 10 q^{88} - 30 q^{89} - 9 q^{92} - 24 q^{93} + 2 q^{94} - 14 q^{96} - 6 q^{97} - 8 q^{99}+O(q^{100})$$ 2 * q + q^2 + 2 * q^3 - q^4 - 4 * q^6 + 6 * q^9 + 4 * q^11 - 6 * q^12 + 2 * q^13 - 3 * q^16 + 4 * q^17 - 7 * q^18 + 7 * q^22 + 8 * q^23 + 10 * q^24 + 6 * q^26 + 20 * q^27 + 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 + 8 * q^43 + 3 * q^44 - q^46 + 4 * q^47 + 12 * q^48 - 16 * q^51 + 4 * q^52 + 8 * q^53 + 20 * q^57 + 5 * q^58 - 10 * q^59 + 6 * q^61 + 18 * q^62 + 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^78 + 22 * q^81 - 12 * q^82 + 2 * q^83 + 9 * q^86 + 10 * q^87 - 10 * q^88 - 30 * q^89 - 9 * q^92 - 24 * q^93 + 2 * q^94 - 14 * q^96 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.618034 −0.437016 −0.218508 0.975835i $$-0.570119\pi$$
−0.218508 + 0.975835i $$0.570119\pi$$
$$3$$ 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
$$7$$ 0 0
$$8$$ 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 0
$$15$$ 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 $$-0.328538\pi$$
0.512989 + 0.858395i $$0.328538\pi$$
$$20$$ 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 $$0.274704\pi$$
$$24$$ 7.23607 1.47706
$$25$$ 0 0
$$26$$ 0.763932 0.149819
$$27$$ 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 $$0.608063\pi$$
$$32$$ −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 $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ −2.76393 −0.448369
$$39$$ −4.00000 −0.640513
$$40$$ 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 $$-0.457058\pi$$
0.134499 + 0.990914i $$0.457058\pi$$
$$44$$ 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
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −8.00000 −1.12022
$$52$$ 2.00000 0.277350
$$53$$ 8.47214 1.16374 0.581869 0.813283i $$-0.302322\pi$$
0.581869 + 0.813283i $$0.302322\pi$$
$$54$$ −8.94427 −1.21716
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 14.4721 1.91688
$$58$$ −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 $$-0.286522\pi$$
0.621504 + 0.783411i $$0.286522\pi$$
$$62$$ 2.29180 0.291058
$$63$$ 0 0
$$64$$ −0.236068 −0.0295085
$$65$$ 0 0
$$66$$ 0.472136 0.0581159
$$67$$ −4.23607 −0.517518 −0.258759 0.965942i $$-0.583314\pi$$
−0.258759 + 0.965942i $$0.583314\pi$$
$$68$$ 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 0
$$78$$ 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 $$0.360962\pi$$
$$84$$ 0 0
$$85$$ 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 $$-0.866639\pi$$
−0.913510 + 0.406817i $$0.866639\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −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 0
$$99$$ −1.76393 −0.177282
$$100$$ 0 0
$$101$$ −4.76393 −0.474029 −0.237014 0.971506i $$-0.576169\pi$$
−0.237014 + 0.971506i $$0.576169\pi$$
$$102$$ 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 $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ −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
$$112$$ 0 0
$$113$$ −14.4164 −1.35618 −0.678091 0.734978i $$-0.737192\pi$$
−0.678091 + 0.734978i $$0.737192\pi$$
$$114$$ −8.94427 −0.837708
$$115$$ 0 0
$$116$$ −8.09017 −0.751153
$$117$$ −9.23607 −0.853875
$$118$$ 7.23607 0.666134
$$119$$ 0 0
$$120$$ 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
$$126$$ 0 0
$$127$$ 13.6525 1.21146 0.605731 0.795670i $$-0.292881\pi$$
0.605731 + 0.795670i $$0.292881\pi$$
$$128$$ 11.3820 1.00603
$$129$$ 5.70820 0.502579
$$130$$ 0 0
$$131$$ 16.9443 1.48043 0.740214 0.672371i $$-0.234724\pi$$
0.740214 + 0.672371i $$0.234724\pi$$
$$132$$ 1.23607 0.107586
$$133$$ 0 0
$$134$$ 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 $$0.654851\pi$$
$$138$$ −12.4721 −1.06170
$$139$$ −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
$$142$$ −5.38197 −0.451645
$$143$$ 0.291796 0.0244012
$$144$$ 13.8541 1.15451
$$145$$ 0 0
$$146$$ −5.41641 −0.448265
$$147$$ 0 0
$$148$$ −4.85410 −0.399005
$$149$$ 3.94427 0.323127 0.161564 0.986862i $$-0.448346\pi$$
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 0
$$155$$ 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
$$161$$ 0 0
$$162$$ −15.0902 −1.18560
$$163$$ −1.52786 −0.119672 −0.0598358 0.998208i $$-0.519058\pi$$
−0.0598358 + 0.998208i $$0.519058\pi$$
$$164$$ 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 $$0.564936\pi$$
$$168$$ 0 0
$$169$$ −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 $$-0.598329\pi$$
−0.304020 + 0.952666i $$0.598329\pi$$
$$182$$ 0 0
$$183$$ 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
$$189$$ 0 0
$$190$$ 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 $$-0.352536\pi$$
0.446876 + 0.894596i $$0.352536\pi$$
$$194$$ 3.23607 0.232336
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1.47214 −0.104885 −0.0524427 0.998624i $$-0.516701\pi$$
−0.0524427 + 0.998624i $$0.516701\pi$$
$$198$$ 1.09017 0.0774750
$$199$$ −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
$$202$$ 2.94427 0.207158
$$203$$ 0 0
$$204$$ 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
$$217$$ 0 0
$$218$$ −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 $$0.736153\pi$$
$$224$$ 0 0
$$225$$ 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 $$-0.452793\pi$$
0.147764 + 0.989023i $$0.452793\pi$$
$$230$$ 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 $$0.416204\pi$$
$$234$$ 5.70820 0.373157
$$235$$ 0 0
$$236$$ 18.9443 1.23317
$$237$$ −36.1803 −2.35017
$$238$$ 0 0
$$239$$ −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 $$-0.463754\pi$$
0.113625 + 0.993524i $$0.463754\pi$$
$$242$$ 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 $$-0.565478\pi$$
−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 $$-0.370876\pi$$
0.394620 + 0.918844i $$0.370876\pi$$
$$258$$ −3.52786 −0.219635
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 37.3607 2.31257
$$262$$ −10.4721 −0.646971
$$263$$ 16.2361 1.00116 0.500579 0.865691i $$-0.333120\pi$$
0.500579 + 0.865691i $$0.333120\pi$$
$$264$$ −1.70820 −0.105133
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −55.7771 −3.41350
$$268$$ 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 $$-0.755897\pi$$
−0.720085 + 0.693885i $$0.755897\pi$$
$$272$$ −4.58359 −0.277921
$$273$$ 0 0
$$274$$ 6.76393 0.408624
$$275$$ 0 0
$$276$$ −32.6525 −1.96545
$$277$$ −19.8885 −1.19499 −0.597493 0.801874i $$-0.703837\pi$$
−0.597493 + 0.801874i $$0.703837\pi$$
$$278$$ 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
$$287$$ 0 0
$$288$$ −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 $$0.136725\pi$$
$$294$$ 0 0
$$295$$ 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
$$301$$ 0 0
$$302$$ 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 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 $$0.742690\pi$$
$$312$$ −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 $$-0.247699\pi$$
0.712199 + 0.701978i $$0.247699\pi$$
$$318$$ −16.9443 −0.950188
$$319$$ −1.18034 −0.0660863
$$320$$ 0 0
$$321$$ 25.8885 1.44496
$$322$$ 0 0
$$323$$ −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
$$329$$ 0 0
$$330$$ 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
$$336$$ 0 0
$$337$$ −16.4721 −0.897294 −0.448647 0.893709i $$-0.648094\pi$$
−0.448647 + 0.893709i $$0.648094\pi$$
$$338$$ 7.09017 0.385654
$$339$$ −46.6525 −2.53381
$$340$$ 0 0
$$341$$ 0.875388 0.0474049
$$342$$ −20.6525 −1.11676
$$343$$ 0 0
$$344$$ 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 $$0.317226\pi$$
$$348$$ −26.1803 −1.40341
$$349$$ −4.47214 −0.239388 −0.119694 0.992811i $$-0.538191\pi$$
−0.119694 + 0.992811i $$0.538191\pi$$
$$350$$ 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
$$357$$ 0 0
$$358$$ 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
$$364$$ 0 0
$$365$$ 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
$$371$$ 0 0
$$372$$ 19.4164 1.00669
$$373$$ −37.8328 −1.95891 −0.979454 0.201665i $$-0.935365\pi$$
−0.979454 + 0.201665i $$0.935365\pi$$
$$374$$ −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
$$381$$ 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
$$392$$ 0 0
$$393$$ 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
$$399$$ 0 0
$$400$$ 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
$$406$$ 0 0
$$407$$ −0.708204 −0.0351044
$$408$$ −17.8885 −0.885615
$$409$$ −24.4721 −1.21007 −0.605035 0.796199i $$-0.706841\pi$$
−0.605035 + 0.796199i $$0.706841\pi$$
$$410$$ 0 0
$$411$$ −35.4164 −1.74696
$$412$$ 13.7082 0.675355
$$413$$ 0 0
$$414$$ −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 $$-0.279143\pi$$
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
$$423$$ 14.9443 0.726615
$$424$$ 18.9443 0.920015
$$425$$ 0 0
$$426$$ −17.4164 −0.843828
$$427$$ 0 0
$$428$$ −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 $$0.261344\pi$$
$$434$$ 0 0
$$435$$ 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 $$-0.436597\pi$$
0.197873 + 0.980228i $$0.436597\pi$$
$$440$$ 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 $$0.652599\pi$$
$$444$$ −15.7082 −0.745478
$$445$$ 0 0
$$446$$ 12.4721 0.590573
$$447$$ 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
$$452$$ 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 $$-0.594658\pi$$
−0.293014 + 0.956108i $$0.594658\pi$$
$$458$$ −2.76393 −0.129150
$$459$$ −35.7771 −1.66993
$$460$$ 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 $$-0.604600\pi$$
−0.322728 + 0.946492i $$0.604600\pi$$
$$464$$ 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
$$469$$ 0 0
$$470$$ 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
$$476$$ 0 0
$$477$$ 63.3050 2.89853
$$478$$ 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
$$486$$ −22.0000 −0.997940
$$487$$ 5.76393 0.261189 0.130594 0.991436i $$-0.458311\pi$$
0.130594 + 0.991436i $$0.458311\pi$$
$$488$$ 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
$$497$$ 0 0
$$498$$ −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 $$0.442120\pi$$
$$504$$ 0 0
$$505$$ 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 $$-0.857125\pi$$
−0.900945 + 0.433934i $$0.857125\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −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
$$518$$ 0 0
$$519$$ 37.3050 1.63751
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ −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
$$532$$ 0 0
$$533$$ 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 0
$$540$$ 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
$$546$$ 0 0
$$547$$ −9.76393 −0.417476 −0.208738 0.977972i $$-0.566936\pi$$
−0.208738 + 0.977972i $$0.566936\pi$$
$$548$$ 17.7082 0.756457
$$549$$ 72.5410 3.09598
$$550$$ 0 0
$$551$$ 22.3607 0.952597
$$552$$ 45.1246 1.92063
$$553$$ 0 0
$$554$$ 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 $$0.561832\pi$$
$$558$$ 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
$$567$$ 0 0
$$568$$ 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
$$574$$ 0 0
$$575$$ 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
$$581$$ 0 0
$$582$$ 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 $$0.329254\pi$$
$$588$$ 0 0
$$589$$ −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 $$-0.228367\pi$$
0.753494 + 0.657455i $$0.228367\pi$$
$$602$$ 0 0
$$603$$ −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
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2.47214 −0.100012
$$612$$ 29.8885 1.20817
$$613$$ −44.4164 −1.79396 −0.896981 0.442069i $$-0.854245\pi$$
−0.896981 + 0.442069i $$0.854245\pi$$
$$614$$ 2.83282 0.114323
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −5.94427 −0.239307 −0.119654 0.992816i $$-0.538178\pi$$
−0.119654 + 0.992816i $$0.538178\pi$$
$$618$$ 16.9443 0.681599
$$619$$ 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
$$622$$ 15.0557 0.603680
$$623$$ 0 0
$$624$$ −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
$$637$$ 0 0
$$638$$ 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 $$0.618669\pi$$
$$644$$ 0 0
$$645$$ 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
$$651$$ 0 0
$$652$$ 2.47214 0.0968163
$$653$$ 25.0557 0.980506 0.490253 0.871580i $$-0.336904\pi$$
0.490253 + 0.871580i $$0.336904\pi$$
$$654$$ −16.8328 −0.658215
$$655$$ 0 0
$$656$$ −8.83282 −0.344864
$$657$$ 65.4853 2.55482
$$658$$ 0 0
$$659$$ 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 $$-0.187864\pi$$
0.830834 + 0.556520i $$0.187864\pi$$
$$662$$ 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
$$672$$ 0 0
$$673$$ 19.5279 0.752744 0.376372 0.926469i $$-0.377171\pi$$
0.376372 + 0.926469i $$0.377171\pi$$
$$674$$ 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
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −69.3050 −2.65577
$$682$$ −0.541020 −0.0207167
$$683$$ 14.1246 0.540463 0.270232 0.962795i $$-0.412900\pi$$
0.270232 + 0.962795i $$0.412900\pi$$
$$684$$ −54.0689 −2.06738
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 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 $$-0.474663\pi$$
0.0795138 + 0.996834i $$0.474663\pi$$
$$692$$ −18.6525 −0.709061
$$693$$ 0 0
$$694$$ −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
$$707$$ 0 0
$$708$$ 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
$$714$$ 0 0
$$715$$ 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 $$-0.597558\pi$$
−0.301712 + 0.953399i $$0.597558\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −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 $$0.481953\pi$$
$$728$$ 0 0
$$729$$ 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
$$742$$ 0 0
$$743$$ −10.4721 −0.384185 −0.192093 0.981377i $$-0.561527\pi$$
−0.192093 + 0.981377i $$0.561527\pi$$
$$744$$ −26.8328 −0.983739
$$745$$ 0 0
$$746$$ 23.3820 0.856075
$$747$$ 57.5967 2.10735
$$748$$ −0.944272 −0.0345260
$$749$$ 0 0
$$750$$ 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
$$756$$ 0 0
$$757$$ 19.5836 0.711778 0.355889 0.934528i $$-0.384178\pi$$
0.355889 + 0.934528i $$0.384178\pi$$
$$758$$ 6.90983 0.250976
$$759$$ −4.76393 −0.172920
$$760$$ 0 0
$$761$$ −27.7771 −1.00692 −0.503459 0.864019i $$-0.667940\pi$$
−0.503459 + 0.864019i $$0.667940\pi$$
$$762$$ −27.3050 −0.989154
$$763$$ 0 0
$$764$$ −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 $$-0.217474\pi$$
0.775547 + 0.631290i $$0.217474\pi$$
$$770$$ 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
$$777$$ 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
$$783$$ 72.3607 2.58596
$$784$$ 0 0
$$785$$ 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
$$791$$ 0 0
$$792$$ −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 $$0.284178\pi$$
$$798$$ 0 0
$$799$$ −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 $$-0.230017\pi$$
0.750075 + 0.661353i $$0.230017\pi$$
$$812$$ 0 0
$$813$$ −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
$$819$$ 0 0
$$820$$ 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 $$-0.686008\pi$$
−0.551668 + 0.834064i $$0.686008\pi$$
$$824$$ −18.9443 −0.659955
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 41.5410 1.44452 0.722261 0.691620i $$-0.243103\pi$$
0.722261 + 0.691620i $$0.243103\pi$$
$$828$$ −75.3951 −2.62016
$$829$$ 7.63932 0.265325 0.132662 0.991161i $$-0.457647\pi$$
0.132662 + 0.991161i $$0.457647\pi$$
$$830$$ 0 0
$$831$$ −64.3607 −2.23265
$$832$$ 0.291796 0.0101162
$$833$$ 0 0
$$834$$ 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 $$-0.322522\pi$$
0.529120 + 0.848547i $$0.322522\pi$$
$$840$$ 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
$$847$$ 0 0
$$848$$ 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 $$0.655515\pi$$
$$854$$ 0 0
$$855$$ 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 $$-0.624581\pi$$
−0.381468 + 0.924382i $$0.624581\pi$$
$$860$$ 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 $$0.398908\pi$$
$$864$$ −81.3050 −2.76605
$$865$$ 0 0
$$866$$ −17.5279 −0.595621
$$867$$ −35.2361 −1.19668
$$868$$ 0 0
$$869$$ 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 $$-0.328681\pi$$
0.512604 + 0.858625i $$0.328681\pi$$
$$878$$ −5.12461 −0.172947
$$879$$ 100.721 3.39725
$$880$$ 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 $$-0.507517\pi$$
−0.0236116 + 0.999721i $$0.507517\pi$$
$$884$$ −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
$$889$$ 0 0
$$890$$ 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
$$896$$ 0 0
$$897$$ −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
$$903$$ 0 0
$$904$$ −32.2361 −1.07216
$$905$$ 0 0
$$906$$ 40.4721 1.34460
$$907$$ 34.8328 1.15660 0.578302 0.815823i $$-0.303715\pi$$
0.578302 + 0.815823i $$0.303715\pi$$
$$908$$ 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
$$917$$ 0 0
$$918$$ 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
$$924$$ 0 0
$$925$$ 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 $$-0.716190\pi$$
−0.628157 + 0.778087i $$0.716190\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −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 $$0.695214\pi$$
$$938$$ 0 0
$$939$$ −63.1935 −2.06224
$$940$$ 0 0
$$941$$ 5.23607 0.170691 0.0853455 0.996351i $$-0.472801\pi$$
0.0853455 + 0.996351i $$0.472801\pi$$
$$942$$ 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 $$-0.308507\pi$$
0.565957 + 0.824435i $$0.308507\pi$$
$$948$$ 58.5410 1.90132
$$949$$ −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 $$0.482090\pi$$
$$954$$ −39.1246 −1.26671
$$955$$ 0 0
$$956$$ 8.94427 0.289278
$$957$$ −3.81966 −0.123472
$$958$$ −16.1803 −0.522763
$$959$$ 0 0
$$960$$ 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
$$966$$ 0 0
$$967$$ −14.1115 −0.453794 −0.226897 0.973919i $$-0.572858\pi$$
−0.226897 + 0.973919i $$0.572858\pi$$
$$968$$ −24.4721 −0.786564
$$969$$ −35.7771 −1.14933
$$970$$ 0 0
$$971$$ 18.0000 0.577647 0.288824 0.957382i $$-0.406736\pi$$
0.288824 + 0.957382i $$0.406736\pi$$
$$972$$ −57.5967 −1.84742
$$973$$ 0 0
$$974$$ −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 $$0.558747\pi$$
$$978$$ 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
$$987$$ 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
$$993$$ −79.9574 −2.53737
$$994$$ 0 0
$$995$$ 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 1225.2.a.u.1.1 2
5.2 odd 4 1225.2.b.k.99.2 4
5.3 odd 4 1225.2.b.k.99.3 4
5.4 even 2 1225.2.a.n.1.2 2
7.6 odd 2 175.2.a.e.1.1 yes 2
21.20 even 2 1575.2.a.n.1.2 2
28.27 even 2 2800.2.a.bp.1.2 2
35.13 even 4 175.2.b.c.99.3 4
35.27 even 4 175.2.b.c.99.2 4
35.34 odd 2 175.2.a.d.1.2 2
105.62 odd 4 1575.2.d.k.1324.3 4
105.83 odd 4 1575.2.d.k.1324.2 4
105.104 even 2 1575.2.a.s.1.1 2
140.27 odd 4 2800.2.g.s.449.1 4
140.83 odd 4 2800.2.g.s.449.4 4
140.139 even 2 2800.2.a.bh.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.2 2 35.34 odd 2
175.2.a.e.1.1 yes 2 7.6 odd 2
175.2.b.c.99.2 4 35.27 even 4
175.2.b.c.99.3 4 35.13 even 4
1225.2.a.n.1.2 2 5.4 even 2
1225.2.a.u.1.1 2 1.1 even 1 trivial
1225.2.b.k.99.2 4 5.2 odd 4
1225.2.b.k.99.3 4 5.3 odd 4
1575.2.a.n.1.2 2 21.20 even 2
1575.2.a.s.1.1 2 105.104 even 2
1575.2.d.k.1324.2 4 105.83 odd 4
1575.2.d.k.1324.3 4 105.62 odd 4
2800.2.a.bh.1.1 2 140.139 even 2
2800.2.a.bp.1.2 2 28.27 even 2
2800.2.g.s.449.1 4 140.27 odd 4
2800.2.g.s.449.4 4 140.83 odd 4