# Properties

 Label 9025.2.a.l.1.1 Level $9025$ Weight $2$ Character 9025.1 Self dual yes Analytic conductor $72.065$ 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] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.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: $$72.0649878242$$ 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: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 9025.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.61803 q^{2} +2.23607 q^{3} +4.85410 q^{4} -5.85410 q^{6} -4.23607 q^{7} -7.47214 q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q-2.61803 q^{2} +2.23607 q^{3} +4.85410 q^{4} -5.85410 q^{6} -4.23607 q^{7} -7.47214 q^{8} +2.00000 q^{9} +5.47214 q^{11} +10.8541 q^{12} +2.00000 q^{13} +11.0902 q^{14} +9.85410 q^{16} -4.85410 q^{17} -5.23607 q^{18} -9.47214 q^{21} -14.3262 q^{22} -1.23607 q^{23} -16.7082 q^{24} -5.23607 q^{26} -2.23607 q^{27} -20.5623 q^{28} -6.00000 q^{29} +1.85410 q^{31} -10.8541 q^{32} +12.2361 q^{33} +12.7082 q^{34} +9.70820 q^{36} -4.14590 q^{37} +4.47214 q^{39} +6.38197 q^{41} +24.7984 q^{42} +10.5623 q^{43} +26.5623 q^{44} +3.23607 q^{46} -7.61803 q^{47} +22.0344 q^{48} +10.9443 q^{49} -10.8541 q^{51} +9.70820 q^{52} +6.38197 q^{53} +5.85410 q^{54} +31.6525 q^{56} +15.7082 q^{58} +2.76393 q^{59} -5.56231 q^{61} -4.85410 q^{62} -8.47214 q^{63} +8.70820 q^{64} -32.0344 q^{66} +8.70820 q^{67} -23.5623 q^{68} -2.76393 q^{69} +4.52786 q^{71} -14.9443 q^{72} -10.7082 q^{73} +10.8541 q^{74} -23.1803 q^{77} -11.7082 q^{78} -1.00000 q^{79} -11.0000 q^{81} -16.7082 q^{82} +1.09017 q^{83} -45.9787 q^{84} -27.6525 q^{86} -13.4164 q^{87} -40.8885 q^{88} -3.70820 q^{89} -8.47214 q^{91} -6.00000 q^{92} +4.14590 q^{93} +19.9443 q^{94} -24.2705 q^{96} -7.38197 q^{97} -28.6525 q^{98} +10.9443 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} - 4 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 3 * q^4 - 5 * q^6 - 4 * q^7 - 6 * q^8 + 4 * q^9 $$2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} - 4 q^{7} - 6 q^{8} + 4 q^{9} + 2 q^{11} + 15 q^{12} + 4 q^{13} + 11 q^{14} + 13 q^{16} - 3 q^{17} - 6 q^{18} - 10 q^{21} - 13 q^{22} + 2 q^{23} - 20 q^{24} - 6 q^{26} - 21 q^{28} - 12 q^{29} - 3 q^{31} - 15 q^{32} + 20 q^{33} + 12 q^{34} + 6 q^{36} - 15 q^{37} + 15 q^{41} + 25 q^{42} + q^{43} + 33 q^{44} + 2 q^{46} - 13 q^{47} + 15 q^{48} + 4 q^{49} - 15 q^{51} + 6 q^{52} + 15 q^{53} + 5 q^{54} + 32 q^{56} + 18 q^{58} + 10 q^{59} + 9 q^{61} - 3 q^{62} - 8 q^{63} + 4 q^{64} - 35 q^{66} + 4 q^{67} - 27 q^{68} - 10 q^{69} + 18 q^{71} - 12 q^{72} - 8 q^{73} + 15 q^{74} - 24 q^{77} - 10 q^{78} - 2 q^{79} - 22 q^{81} - 20 q^{82} - 9 q^{83} - 45 q^{84} - 24 q^{86} - 46 q^{88} + 6 q^{89} - 8 q^{91} - 12 q^{92} + 15 q^{93} + 22 q^{94} - 15 q^{96} - 17 q^{97} - 26 q^{98} + 4 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 + 3 * q^4 - 5 * q^6 - 4 * q^7 - 6 * q^8 + 4 * q^9 + 2 * q^11 + 15 * q^12 + 4 * q^13 + 11 * q^14 + 13 * q^16 - 3 * q^17 - 6 * q^18 - 10 * q^21 - 13 * q^22 + 2 * q^23 - 20 * q^24 - 6 * q^26 - 21 * q^28 - 12 * q^29 - 3 * q^31 - 15 * q^32 + 20 * q^33 + 12 * q^34 + 6 * q^36 - 15 * q^37 + 15 * q^41 + 25 * q^42 + q^43 + 33 * q^44 + 2 * q^46 - 13 * q^47 + 15 * q^48 + 4 * q^49 - 15 * q^51 + 6 * q^52 + 15 * q^53 + 5 * q^54 + 32 * q^56 + 18 * q^58 + 10 * q^59 + 9 * q^61 - 3 * q^62 - 8 * q^63 + 4 * q^64 - 35 * q^66 + 4 * q^67 - 27 * q^68 - 10 * q^69 + 18 * q^71 - 12 * q^72 - 8 * q^73 + 15 * q^74 - 24 * q^77 - 10 * q^78 - 2 * q^79 - 22 * q^81 - 20 * q^82 - 9 * q^83 - 45 * q^84 - 24 * q^86 - 46 * q^88 + 6 * q^89 - 8 * q^91 - 12 * q^92 + 15 * q^93 + 22 * q^94 - 15 * q^96 - 17 * q^97 - 26 * q^98 + 4 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.61803 −1.85123 −0.925615 0.378467i $$-0.876451\pi$$
−0.925615 + 0.378467i $$0.876451\pi$$
$$3$$ 2.23607 1.29099 0.645497 0.763763i $$-0.276650\pi$$
0.645497 + 0.763763i $$0.276650\pi$$
$$4$$ 4.85410 2.42705
$$5$$ 0 0
$$6$$ −5.85410 −2.38993
$$7$$ −4.23607 −1.60108 −0.800542 0.599277i $$-0.795455\pi$$
−0.800542 + 0.599277i $$0.795455\pi$$
$$8$$ −7.47214 −2.64180
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ 5.47214 1.64991 0.824956 0.565198i $$-0.191200\pi$$
0.824956 + 0.565198i $$0.191200\pi$$
$$12$$ 10.8541 3.13331
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 11.0902 2.96397
$$15$$ 0 0
$$16$$ 9.85410 2.46353
$$17$$ −4.85410 −1.17729 −0.588646 0.808391i $$-0.700339\pi$$
−0.588646 + 0.808391i $$0.700339\pi$$
$$18$$ −5.23607 −1.23415
$$19$$ 0 0
$$20$$ 0 0
$$21$$ −9.47214 −2.06699
$$22$$ −14.3262 −3.05436
$$23$$ −1.23607 −0.257738 −0.128869 0.991662i $$-0.541135\pi$$
−0.128869 + 0.991662i $$0.541135\pi$$
$$24$$ −16.7082 −3.41055
$$25$$ 0 0
$$26$$ −5.23607 −1.02688
$$27$$ −2.23607 −0.430331
$$28$$ −20.5623 −3.88591
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 1.85410 0.333007 0.166503 0.986041i $$-0.446752\pi$$
0.166503 + 0.986041i $$0.446752\pi$$
$$32$$ −10.8541 −1.91875
$$33$$ 12.2361 2.13003
$$34$$ 12.7082 2.17944
$$35$$ 0 0
$$36$$ 9.70820 1.61803
$$37$$ −4.14590 −0.681581 −0.340791 0.940139i $$-0.610695\pi$$
−0.340791 + 0.940139i $$0.610695\pi$$
$$38$$ 0 0
$$39$$ 4.47214 0.716115
$$40$$ 0 0
$$41$$ 6.38197 0.996696 0.498348 0.866977i $$-0.333940\pi$$
0.498348 + 0.866977i $$0.333940\pi$$
$$42$$ 24.7984 3.82647
$$43$$ 10.5623 1.61074 0.805368 0.592775i $$-0.201968\pi$$
0.805368 + 0.592775i $$0.201968\pi$$
$$44$$ 26.5623 4.00442
$$45$$ 0 0
$$46$$ 3.23607 0.477132
$$47$$ −7.61803 −1.11120 −0.555602 0.831448i $$-0.687512\pi$$
−0.555602 + 0.831448i $$0.687512\pi$$
$$48$$ 22.0344 3.18040
$$49$$ 10.9443 1.56347
$$50$$ 0 0
$$51$$ −10.8541 −1.51988
$$52$$ 9.70820 1.34629
$$53$$ 6.38197 0.876630 0.438315 0.898821i $$-0.355575\pi$$
0.438315 + 0.898821i $$0.355575\pi$$
$$54$$ 5.85410 0.796642
$$55$$ 0 0
$$56$$ 31.6525 4.22974
$$57$$ 0 0
$$58$$ 15.7082 2.06259
$$59$$ 2.76393 0.359833 0.179917 0.983682i $$-0.442417\pi$$
0.179917 + 0.983682i $$0.442417\pi$$
$$60$$ 0 0
$$61$$ −5.56231 −0.712180 −0.356090 0.934452i $$-0.615890\pi$$
−0.356090 + 0.934452i $$0.615890\pi$$
$$62$$ −4.85410 −0.616472
$$63$$ −8.47214 −1.06739
$$64$$ 8.70820 1.08853
$$65$$ 0 0
$$66$$ −32.0344 −3.94317
$$67$$ 8.70820 1.06388 0.531938 0.846783i $$-0.321464\pi$$
0.531938 + 0.846783i $$0.321464\pi$$
$$68$$ −23.5623 −2.85735
$$69$$ −2.76393 −0.332738
$$70$$ 0 0
$$71$$ 4.52786 0.537359 0.268679 0.963230i $$-0.413413\pi$$
0.268679 + 0.963230i $$0.413413\pi$$
$$72$$ −14.9443 −1.76120
$$73$$ −10.7082 −1.25330 −0.626650 0.779301i $$-0.715575\pi$$
−0.626650 + 0.779301i $$0.715575\pi$$
$$74$$ 10.8541 1.26176
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −23.1803 −2.64164
$$78$$ −11.7082 −1.32569
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ −16.7082 −1.84511
$$83$$ 1.09017 0.119662 0.0598308 0.998209i $$-0.480944\pi$$
0.0598308 + 0.998209i $$0.480944\pi$$
$$84$$ −45.9787 −5.01669
$$85$$ 0 0
$$86$$ −27.6525 −2.98184
$$87$$ −13.4164 −1.43839
$$88$$ −40.8885 −4.35873
$$89$$ −3.70820 −0.393069 −0.196534 0.980497i $$-0.562969\pi$$
−0.196534 + 0.980497i $$0.562969\pi$$
$$90$$ 0 0
$$91$$ −8.47214 −0.888121
$$92$$ −6.00000 −0.625543
$$93$$ 4.14590 0.429910
$$94$$ 19.9443 2.05709
$$95$$ 0 0
$$96$$ −24.2705 −2.47710
$$97$$ −7.38197 −0.749525 −0.374763 0.927121i $$-0.622276\pi$$
−0.374763 + 0.927121i $$0.622276\pi$$
$$98$$ −28.6525 −2.89434
$$99$$ 10.9443 1.09994
$$100$$ 0 0
$$101$$ −13.0344 −1.29698 −0.648488 0.761225i $$-0.724598\pi$$
−0.648488 + 0.761225i $$0.724598\pi$$
$$102$$ 28.4164 2.81364
$$103$$ 1.23607 0.121793 0.0608967 0.998144i $$-0.480604\pi$$
0.0608967 + 0.998144i $$0.480604\pi$$
$$104$$ −14.9443 −1.46541
$$105$$ 0 0
$$106$$ −16.7082 −1.62284
$$107$$ 19.8541 1.91937 0.959684 0.281080i $$-0.0906927\pi$$
0.959684 + 0.281080i $$0.0906927\pi$$
$$108$$ −10.8541 −1.04444
$$109$$ −7.94427 −0.760923 −0.380462 0.924797i $$-0.624235\pi$$
−0.380462 + 0.924797i $$0.624235\pi$$
$$110$$ 0 0
$$111$$ −9.27051 −0.879918
$$112$$ −41.7426 −3.94431
$$113$$ 9.32624 0.877339 0.438669 0.898649i $$-0.355450\pi$$
0.438669 + 0.898649i $$0.355450\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −29.1246 −2.70415
$$117$$ 4.00000 0.369800
$$118$$ −7.23607 −0.666134
$$119$$ 20.5623 1.88494
$$120$$ 0 0
$$121$$ 18.9443 1.72221
$$122$$ 14.5623 1.31841
$$123$$ 14.2705 1.28673
$$124$$ 9.00000 0.808224
$$125$$ 0 0
$$126$$ 22.1803 1.97598
$$127$$ −0.854102 −0.0757893 −0.0378946 0.999282i $$-0.512065\pi$$
−0.0378946 + 0.999282i $$0.512065\pi$$
$$128$$ −1.09017 −0.0963583
$$129$$ 23.6180 2.07945
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 59.3951 5.16968
$$133$$ 0 0
$$134$$ −22.7984 −1.96948
$$135$$ 0 0
$$136$$ 36.2705 3.11017
$$137$$ −9.23607 −0.789091 −0.394545 0.918877i $$-0.629098\pi$$
−0.394545 + 0.918877i $$0.629098\pi$$
$$138$$ 7.23607 0.615975
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ −17.0344 −1.43456
$$142$$ −11.8541 −0.994774
$$143$$ 10.9443 0.915206
$$144$$ 19.7082 1.64235
$$145$$ 0 0
$$146$$ 28.0344 2.32015
$$147$$ 24.4721 2.01843
$$148$$ −20.1246 −1.65423
$$149$$ −9.47214 −0.775988 −0.387994 0.921662i $$-0.626832\pi$$
−0.387994 + 0.921662i $$0.626832\pi$$
$$150$$ 0 0
$$151$$ −5.79837 −0.471865 −0.235932 0.971769i $$-0.575814\pi$$
−0.235932 + 0.971769i $$0.575814\pi$$
$$152$$ 0 0
$$153$$ −9.70820 −0.784862
$$154$$ 60.6869 4.89029
$$155$$ 0 0
$$156$$ 21.7082 1.73805
$$157$$ 1.81966 0.145225 0.0726123 0.997360i $$-0.476866\pi$$
0.0726123 + 0.997360i $$0.476866\pi$$
$$158$$ 2.61803 0.208280
$$159$$ 14.2705 1.13173
$$160$$ 0 0
$$161$$ 5.23607 0.412660
$$162$$ 28.7984 2.26261
$$163$$ −2.76393 −0.216488 −0.108244 0.994124i $$-0.534523\pi$$
−0.108244 + 0.994124i $$0.534523\pi$$
$$164$$ 30.9787 2.41903
$$165$$ 0 0
$$166$$ −2.85410 −0.221521
$$167$$ 5.23607 0.405179 0.202590 0.979264i $$-0.435064\pi$$
0.202590 + 0.979264i $$0.435064\pi$$
$$168$$ 70.7771 5.46057
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 51.2705 3.90934
$$173$$ −19.6525 −1.49415 −0.747075 0.664740i $$-0.768542\pi$$
−0.747075 + 0.664740i $$0.768542\pi$$
$$174$$ 35.1246 2.66279
$$175$$ 0 0
$$176$$ 53.9230 4.06460
$$177$$ 6.18034 0.464543
$$178$$ 9.70820 0.727661
$$179$$ −11.1803 −0.835658 −0.417829 0.908526i $$-0.637209\pi$$
−0.417829 + 0.908526i $$0.637209\pi$$
$$180$$ 0 0
$$181$$ 18.7082 1.39057 0.695285 0.718734i $$-0.255278\pi$$
0.695285 + 0.718734i $$0.255278\pi$$
$$182$$ 22.1803 1.64412
$$183$$ −12.4377 −0.919421
$$184$$ 9.23607 0.680892
$$185$$ 0 0
$$186$$ −10.8541 −0.795861
$$187$$ −26.5623 −1.94243
$$188$$ −36.9787 −2.69695
$$189$$ 9.47214 0.688997
$$190$$ 0 0
$$191$$ −17.9443 −1.29840 −0.649201 0.760617i $$-0.724897\pi$$
−0.649201 + 0.760617i $$0.724897\pi$$
$$192$$ 19.4721 1.40528
$$193$$ −23.2705 −1.67505 −0.837524 0.546401i $$-0.815998\pi$$
−0.837524 + 0.546401i $$0.815998\pi$$
$$194$$ 19.3262 1.38754
$$195$$ 0 0
$$196$$ 53.1246 3.79462
$$197$$ −7.47214 −0.532368 −0.266184 0.963922i $$-0.585763\pi$$
−0.266184 + 0.963922i $$0.585763\pi$$
$$198$$ −28.6525 −2.03624
$$199$$ 19.4164 1.37639 0.688196 0.725525i $$-0.258403\pi$$
0.688196 + 0.725525i $$0.258403\pi$$
$$200$$ 0 0
$$201$$ 19.4721 1.37346
$$202$$ 34.1246 2.40100
$$203$$ 25.4164 1.78388
$$204$$ −52.6869 −3.68882
$$205$$ 0 0
$$206$$ −3.23607 −0.225468
$$207$$ −2.47214 −0.171825
$$208$$ 19.7082 1.36652
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 12.2361 0.842366 0.421183 0.906976i $$-0.361615\pi$$
0.421183 + 0.906976i $$0.361615\pi$$
$$212$$ 30.9787 2.12763
$$213$$ 10.1246 0.693727
$$214$$ −51.9787 −3.55319
$$215$$ 0 0
$$216$$ 16.7082 1.13685
$$217$$ −7.85410 −0.533171
$$218$$ 20.7984 1.40864
$$219$$ −23.9443 −1.61800
$$220$$ 0 0
$$221$$ −9.70820 −0.653044
$$222$$ 24.2705 1.62893
$$223$$ −18.8885 −1.26487 −0.632435 0.774613i $$-0.717945\pi$$
−0.632435 + 0.774613i $$0.717945\pi$$
$$224$$ 45.9787 3.07208
$$225$$ 0 0
$$226$$ −24.4164 −1.62416
$$227$$ −14.4721 −0.960549 −0.480275 0.877118i $$-0.659463\pi$$
−0.480275 + 0.877118i $$0.659463\pi$$
$$228$$ 0 0
$$229$$ 20.6525 1.36475 0.682377 0.731000i $$-0.260946\pi$$
0.682377 + 0.731000i $$0.260946\pi$$
$$230$$ 0 0
$$231$$ −51.8328 −3.41035
$$232$$ 44.8328 2.94342
$$233$$ −16.5279 −1.08278 −0.541388 0.840773i $$-0.682101\pi$$
−0.541388 + 0.840773i $$0.682101\pi$$
$$234$$ −10.4721 −0.684585
$$235$$ 0 0
$$236$$ 13.4164 0.873334
$$237$$ −2.23607 −0.145248
$$238$$ −53.8328 −3.48946
$$239$$ 0.708204 0.0458099 0.0229050 0.999738i $$-0.492708\pi$$
0.0229050 + 0.999738i $$0.492708\pi$$
$$240$$ 0 0
$$241$$ 13.0000 0.837404 0.418702 0.908124i $$-0.362485\pi$$
0.418702 + 0.908124i $$0.362485\pi$$
$$242$$ −49.5967 −3.18820
$$243$$ −17.8885 −1.14755
$$244$$ −27.0000 −1.72850
$$245$$ 0 0
$$246$$ −37.3607 −2.38203
$$247$$ 0 0
$$248$$ −13.8541 −0.879736
$$249$$ 2.43769 0.154483
$$250$$ 0 0
$$251$$ 5.90983 0.373025 0.186513 0.982453i $$-0.440281\pi$$
0.186513 + 0.982453i $$0.440281\pi$$
$$252$$ −41.1246 −2.59061
$$253$$ −6.76393 −0.425245
$$254$$ 2.23607 0.140303
$$255$$ 0 0
$$256$$ −14.5623 −0.910144
$$257$$ −4.20163 −0.262090 −0.131045 0.991376i $$-0.541833\pi$$
−0.131045 + 0.991376i $$0.541833\pi$$
$$258$$ −61.8328 −3.84954
$$259$$ 17.5623 1.09127
$$260$$ 0 0
$$261$$ −12.0000 −0.742781
$$262$$ 31.4164 1.94091
$$263$$ −22.6180 −1.39469 −0.697344 0.716737i $$-0.745635\pi$$
−0.697344 + 0.716737i $$0.745635\pi$$
$$264$$ −91.4296 −5.62710
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −8.29180 −0.507450
$$268$$ 42.2705 2.58208
$$269$$ −9.18034 −0.559735 −0.279868 0.960039i $$-0.590291\pi$$
−0.279868 + 0.960039i $$0.590291\pi$$
$$270$$ 0 0
$$271$$ −11.3262 −0.688020 −0.344010 0.938966i $$-0.611785\pi$$
−0.344010 + 0.938966i $$0.611785\pi$$
$$272$$ −47.8328 −2.90029
$$273$$ −18.9443 −1.14656
$$274$$ 24.1803 1.46079
$$275$$ 0 0
$$276$$ −13.4164 −0.807573
$$277$$ 12.0000 0.721010 0.360505 0.932757i $$-0.382604\pi$$
0.360505 + 0.932757i $$0.382604\pi$$
$$278$$ 36.6525 2.19827
$$279$$ 3.70820 0.222004
$$280$$ 0 0
$$281$$ −4.61803 −0.275489 −0.137744 0.990468i $$-0.543985\pi$$
−0.137744 + 0.990468i $$0.543985\pi$$
$$282$$ 44.5967 2.65570
$$283$$ 23.7639 1.41262 0.706310 0.707903i $$-0.250359\pi$$
0.706310 + 0.707903i $$0.250359\pi$$
$$284$$ 21.9787 1.30420
$$285$$ 0 0
$$286$$ −28.6525 −1.69426
$$287$$ −27.0344 −1.59579
$$288$$ −21.7082 −1.27917
$$289$$ 6.56231 0.386018
$$290$$ 0 0
$$291$$ −16.5066 −0.967633
$$292$$ −51.9787 −3.04182
$$293$$ 13.4721 0.787051 0.393525 0.919314i $$-0.371255\pi$$
0.393525 + 0.919314i $$0.371255\pi$$
$$294$$ −64.0689 −3.73657
$$295$$ 0 0
$$296$$ 30.9787 1.80060
$$297$$ −12.2361 −0.710009
$$298$$ 24.7984 1.43653
$$299$$ −2.47214 −0.142967
$$300$$ 0 0
$$301$$ −44.7426 −2.57892
$$302$$ 15.1803 0.873530
$$303$$ −29.1459 −1.67439
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 25.4164 1.45296
$$307$$ 32.5623 1.85843 0.929214 0.369541i $$-0.120485\pi$$
0.929214 + 0.369541i $$0.120485\pi$$
$$308$$ −112.520 −6.41141
$$309$$ 2.76393 0.157235
$$310$$ 0 0
$$311$$ 16.2361 0.920663 0.460331 0.887747i $$-0.347731\pi$$
0.460331 + 0.887747i $$0.347731\pi$$
$$312$$ −33.4164 −1.89183
$$313$$ −19.9443 −1.12732 −0.563658 0.826008i $$-0.690607\pi$$
−0.563658 + 0.826008i $$0.690607\pi$$
$$314$$ −4.76393 −0.268844
$$315$$ 0 0
$$316$$ −4.85410 −0.273065
$$317$$ −10.9443 −0.614692 −0.307346 0.951598i $$-0.599441\pi$$
−0.307346 + 0.951598i $$0.599441\pi$$
$$318$$ −37.3607 −2.09508
$$319$$ −32.8328 −1.83828
$$320$$ 0 0
$$321$$ 44.3951 2.47789
$$322$$ −13.7082 −0.763928
$$323$$ 0 0
$$324$$ −53.3951 −2.96640
$$325$$ 0 0
$$326$$ 7.23607 0.400769
$$327$$ −17.7639 −0.982348
$$328$$ −47.6869 −2.63307
$$329$$ 32.2705 1.77913
$$330$$ 0 0
$$331$$ 13.5623 0.745452 0.372726 0.927941i $$-0.378423\pi$$
0.372726 + 0.927941i $$0.378423\pi$$
$$332$$ 5.29180 0.290425
$$333$$ −8.29180 −0.454388
$$334$$ −13.7082 −0.750080
$$335$$ 0 0
$$336$$ −93.3394 −5.09208
$$337$$ 4.70820 0.256472 0.128236 0.991744i $$-0.459068\pi$$
0.128236 + 0.991744i $$0.459068\pi$$
$$338$$ 23.5623 1.28162
$$339$$ 20.8541 1.13264
$$340$$ 0 0
$$341$$ 10.1459 0.549431
$$342$$ 0 0
$$343$$ −16.7082 −0.902158
$$344$$ −78.9230 −4.25524
$$345$$ 0 0
$$346$$ 51.4508 2.76601
$$347$$ −33.4721 −1.79688 −0.898439 0.439098i $$-0.855298\pi$$
−0.898439 + 0.439098i $$0.855298\pi$$
$$348$$ −65.1246 −3.49105
$$349$$ −27.0689 −1.44896 −0.724482 0.689294i $$-0.757921\pi$$
−0.724482 + 0.689294i $$0.757921\pi$$
$$350$$ 0 0
$$351$$ −4.47214 −0.238705
$$352$$ −59.3951 −3.16577
$$353$$ 16.5279 0.879689 0.439845 0.898074i $$-0.355033\pi$$
0.439845 + 0.898074i $$0.355033\pi$$
$$354$$ −16.1803 −0.859975
$$355$$ 0 0
$$356$$ −18.0000 −0.953998
$$357$$ 45.9787 2.43345
$$358$$ 29.2705 1.54699
$$359$$ −22.5967 −1.19261 −0.596305 0.802758i $$-0.703365\pi$$
−0.596305 + 0.802758i $$0.703365\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ −48.9787 −2.57426
$$363$$ 42.3607 2.22336
$$364$$ −41.1246 −2.15552
$$365$$ 0 0
$$366$$ 32.5623 1.70206
$$367$$ −12.4721 −0.651040 −0.325520 0.945535i $$-0.605539\pi$$
−0.325520 + 0.945535i $$0.605539\pi$$
$$368$$ −12.1803 −0.634944
$$369$$ 12.7639 0.664464
$$370$$ 0 0
$$371$$ −27.0344 −1.40356
$$372$$ 20.1246 1.04341
$$373$$ −13.1459 −0.680669 −0.340334 0.940304i $$-0.610540\pi$$
−0.340334 + 0.940304i $$0.610540\pi$$
$$374$$ 69.5410 3.59588
$$375$$ 0 0
$$376$$ 56.9230 2.93558
$$377$$ −12.0000 −0.618031
$$378$$ −24.7984 −1.27549
$$379$$ 36.5410 1.87699 0.938493 0.345298i $$-0.112222\pi$$
0.938493 + 0.345298i $$0.112222\pi$$
$$380$$ 0 0
$$381$$ −1.90983 −0.0978436
$$382$$ 46.9787 2.40364
$$383$$ 22.0689 1.12767 0.563834 0.825888i $$-0.309326\pi$$
0.563834 + 0.825888i $$0.309326\pi$$
$$384$$ −2.43769 −0.124398
$$385$$ 0 0
$$386$$ 60.9230 3.10090
$$387$$ 21.1246 1.07382
$$388$$ −35.8328 −1.81914
$$389$$ −4.47214 −0.226746 −0.113373 0.993552i $$-0.536166\pi$$
−0.113373 + 0.993552i $$0.536166\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ −81.7771 −4.13037
$$393$$ −26.8328 −1.35354
$$394$$ 19.5623 0.985535
$$395$$ 0 0
$$396$$ 53.1246 2.66961
$$397$$ −9.03444 −0.453426 −0.226713 0.973962i $$-0.572798\pi$$
−0.226713 + 0.973962i $$0.572798\pi$$
$$398$$ −50.8328 −2.54802
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 5.88854 0.294060 0.147030 0.989132i $$-0.453029\pi$$
0.147030 + 0.989132i $$0.453029\pi$$
$$402$$ −50.9787 −2.54259
$$403$$ 3.70820 0.184719
$$404$$ −63.2705 −3.14783
$$405$$ 0 0
$$406$$ −66.5410 −3.30238
$$407$$ −22.6869 −1.12455
$$408$$ 81.1033 4.01521
$$409$$ −16.4377 −0.812792 −0.406396 0.913697i $$-0.633215\pi$$
−0.406396 + 0.913697i $$0.633215\pi$$
$$410$$ 0 0
$$411$$ −20.6525 −1.01871
$$412$$ 6.00000 0.295599
$$413$$ −11.7082 −0.576123
$$414$$ 6.47214 0.318088
$$415$$ 0 0
$$416$$ −21.7082 −1.06433
$$417$$ −31.3050 −1.53301
$$418$$ 0 0
$$419$$ −9.52786 −0.465467 −0.232733 0.972541i $$-0.574767\pi$$
−0.232733 + 0.972541i $$0.574767\pi$$
$$420$$ 0 0
$$421$$ 15.0000 0.731055 0.365528 0.930800i $$-0.380889\pi$$
0.365528 + 0.930800i $$0.380889\pi$$
$$422$$ −32.0344 −1.55941
$$423$$ −15.2361 −0.740803
$$424$$ −47.6869 −2.31588
$$425$$ 0 0
$$426$$ −26.5066 −1.28425
$$427$$ 23.5623 1.14026
$$428$$ 96.3738 4.65841
$$429$$ 24.4721 1.18153
$$430$$ 0 0
$$431$$ 10.6525 0.513112 0.256556 0.966529i $$-0.417412\pi$$
0.256556 + 0.966529i $$0.417412\pi$$
$$432$$ −22.0344 −1.06013
$$433$$ −33.3607 −1.60321 −0.801606 0.597853i $$-0.796021\pi$$
−0.801606 + 0.597853i $$0.796021\pi$$
$$434$$ 20.5623 0.987022
$$435$$ 0 0
$$436$$ −38.5623 −1.84680
$$437$$ 0 0
$$438$$ 62.6869 2.99530
$$439$$ −36.5066 −1.74236 −0.871182 0.490960i $$-0.836646\pi$$
−0.871182 + 0.490960i $$0.836646\pi$$
$$440$$ 0 0
$$441$$ 21.8885 1.04231
$$442$$ 25.4164 1.20894
$$443$$ −12.6738 −0.602149 −0.301074 0.953601i $$-0.597345\pi$$
−0.301074 + 0.953601i $$0.597345\pi$$
$$444$$ −45.0000 −2.13561
$$445$$ 0 0
$$446$$ 49.4508 2.34157
$$447$$ −21.1803 −1.00180
$$448$$ −36.8885 −1.74282
$$449$$ −24.7984 −1.17031 −0.585154 0.810922i $$-0.698966\pi$$
−0.585154 + 0.810922i $$0.698966\pi$$
$$450$$ 0 0
$$451$$ 34.9230 1.64446
$$452$$ 45.2705 2.12935
$$453$$ −12.9656 −0.609175
$$454$$ 37.8885 1.77820
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5.56231 0.260194 0.130097 0.991501i $$-0.458471\pi$$
0.130097 + 0.991501i $$0.458471\pi$$
$$458$$ −54.0689 −2.52647
$$459$$ 10.8541 0.506626
$$460$$ 0 0
$$461$$ −8.03444 −0.374201 −0.187101 0.982341i $$-0.559909\pi$$
−0.187101 + 0.982341i $$0.559909\pi$$
$$462$$ 135.700 6.31334
$$463$$ 41.8673 1.94574 0.972868 0.231360i $$-0.0743175\pi$$
0.972868 + 0.231360i $$0.0743175\pi$$
$$464$$ −59.1246 −2.74479
$$465$$ 0 0
$$466$$ 43.2705 2.00447
$$467$$ −2.72949 −0.126306 −0.0631529 0.998004i $$-0.520116\pi$$
−0.0631529 + 0.998004i $$0.520116\pi$$
$$468$$ 19.4164 0.897524
$$469$$ −36.8885 −1.70335
$$470$$ 0 0
$$471$$ 4.06888 0.187484
$$472$$ −20.6525 −0.950607
$$473$$ 57.7984 2.65757
$$474$$ 5.85410 0.268888
$$475$$ 0 0
$$476$$ 99.8115 4.57485
$$477$$ 12.7639 0.584420
$$478$$ −1.85410 −0.0848047
$$479$$ −27.7426 −1.26759 −0.633797 0.773499i $$-0.718505\pi$$
−0.633797 + 0.773499i $$0.718505\pi$$
$$480$$ 0 0
$$481$$ −8.29180 −0.378073
$$482$$ −34.0344 −1.55023
$$483$$ 11.7082 0.532742
$$484$$ 91.9574 4.17988
$$485$$ 0 0
$$486$$ 46.8328 2.12438
$$487$$ 10.1246 0.458790 0.229395 0.973333i $$-0.426325\pi$$
0.229395 + 0.973333i $$0.426325\pi$$
$$488$$ 41.5623 1.88144
$$489$$ −6.18034 −0.279485
$$490$$ 0 0
$$491$$ −27.7639 −1.25297 −0.626484 0.779434i $$-0.715507\pi$$
−0.626484 + 0.779434i $$0.715507\pi$$
$$492$$ 69.2705 3.12296
$$493$$ 29.1246 1.31171
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 18.2705 0.820370
$$497$$ −19.1803 −0.860356
$$498$$ −6.38197 −0.285983
$$499$$ −10.5967 −0.474376 −0.237188 0.971464i $$-0.576226\pi$$
−0.237188 + 0.971464i $$0.576226\pi$$
$$500$$ 0 0
$$501$$ 11.7082 0.523084
$$502$$ −15.4721 −0.690555
$$503$$ −18.7426 −0.835693 −0.417847 0.908518i $$-0.637215\pi$$
−0.417847 + 0.908518i $$0.637215\pi$$
$$504$$ 63.3050 2.81983
$$505$$ 0 0
$$506$$ 17.7082 0.787226
$$507$$ −20.1246 −0.893765
$$508$$ −4.14590 −0.183944
$$509$$ −13.0344 −0.577741 −0.288871 0.957368i $$-0.593280\pi$$
−0.288871 + 0.957368i $$0.593280\pi$$
$$510$$ 0 0
$$511$$ 45.3607 2.00664
$$512$$ 40.3050 1.78124
$$513$$ 0 0
$$514$$ 11.0000 0.485189
$$515$$ 0 0
$$516$$ 114.644 5.04694
$$517$$ −41.6869 −1.83339
$$518$$ −45.9787 −2.02019
$$519$$ −43.9443 −1.92894
$$520$$ 0 0
$$521$$ −20.2361 −0.886558 −0.443279 0.896384i $$-0.646185\pi$$
−0.443279 + 0.896384i $$0.646185\pi$$
$$522$$ 31.4164 1.37506
$$523$$ −2.38197 −0.104156 −0.0520781 0.998643i $$-0.516584\pi$$
−0.0520781 + 0.998643i $$0.516584\pi$$
$$524$$ −58.2492 −2.54463
$$525$$ 0 0
$$526$$ 59.2148 2.58189
$$527$$ −9.00000 −0.392046
$$528$$ 120.575 5.24737
$$529$$ −21.4721 −0.933571
$$530$$ 0 0
$$531$$ 5.52786 0.239889
$$532$$ 0 0
$$533$$ 12.7639 0.552867
$$534$$ 21.7082 0.939406
$$535$$ 0 0
$$536$$ −65.0689 −2.81055
$$537$$ −25.0000 −1.07883
$$538$$ 24.0344 1.03620
$$539$$ 59.8885 2.57958
$$540$$ 0 0
$$541$$ 7.34752 0.315895 0.157947 0.987448i $$-0.449512\pi$$
0.157947 + 0.987448i $$0.449512\pi$$
$$542$$ 29.6525 1.27368
$$543$$ 41.8328 1.79522
$$544$$ 52.6869 2.25893
$$545$$ 0 0
$$546$$ 49.5967 2.12254
$$547$$ −6.41641 −0.274346 −0.137173 0.990547i $$-0.543802\pi$$
−0.137173 + 0.990547i $$0.543802\pi$$
$$548$$ −44.8328 −1.91516
$$549$$ −11.1246 −0.474787
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 20.6525 0.879028
$$553$$ 4.23607 0.180136
$$554$$ −31.4164 −1.33476
$$555$$ 0 0
$$556$$ −67.9574 −2.88204
$$557$$ 42.5066 1.80106 0.900531 0.434792i $$-0.143178\pi$$
0.900531 + 0.434792i $$0.143178\pi$$
$$558$$ −9.70820 −0.410981
$$559$$ 21.1246 0.893476
$$560$$ 0 0
$$561$$ −59.3951 −2.50766
$$562$$ 12.0902 0.509993
$$563$$ −16.5967 −0.699470 −0.349735 0.936849i $$-0.613728\pi$$
−0.349735 + 0.936849i $$0.613728\pi$$
$$564$$ −82.6869 −3.48175
$$565$$ 0 0
$$566$$ −62.2148 −2.61508
$$567$$ 46.5967 1.95688
$$568$$ −33.8328 −1.41959
$$569$$ −24.7082 −1.03582 −0.517911 0.855435i $$-0.673290\pi$$
−0.517911 + 0.855435i $$0.673290\pi$$
$$570$$ 0 0
$$571$$ 44.2492 1.85177 0.925886 0.377803i $$-0.123320\pi$$
0.925886 + 0.377803i $$0.123320\pi$$
$$572$$ 53.1246 2.22125
$$573$$ −40.1246 −1.67623
$$574$$ 70.7771 2.95418
$$575$$ 0 0
$$576$$ 17.4164 0.725684
$$577$$ 47.3050 1.96933 0.984665 0.174453i $$-0.0558158\pi$$
0.984665 + 0.174453i $$0.0558158\pi$$
$$578$$ −17.1803 −0.714608
$$579$$ −52.0344 −2.16248
$$580$$ 0 0
$$581$$ −4.61803 −0.191588
$$582$$ 43.2148 1.79131
$$583$$ 34.9230 1.44636
$$584$$ 80.0132 3.31097
$$585$$ 0 0
$$586$$ −35.2705 −1.45701
$$587$$ −27.6738 −1.14222 −0.571109 0.820874i $$-0.693487\pi$$
−0.571109 + 0.820874i $$0.693487\pi$$
$$588$$ 118.790 4.89883
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −16.7082 −0.687284
$$592$$ −40.8541 −1.67909
$$593$$ 12.5967 0.517286 0.258643 0.965973i $$-0.416725\pi$$
0.258643 + 0.965973i $$0.416725\pi$$
$$594$$ 32.0344 1.31439
$$595$$ 0 0
$$596$$ −45.9787 −1.88336
$$597$$ 43.4164 1.77692
$$598$$ 6.47214 0.264665
$$599$$ 41.0689 1.67803 0.839015 0.544109i $$-0.183132\pi$$
0.839015 + 0.544109i $$0.183132\pi$$
$$600$$ 0 0
$$601$$ −18.6869 −0.762255 −0.381128 0.924522i $$-0.624464\pi$$
−0.381128 + 0.924522i $$0.624464\pi$$
$$602$$ 117.138 4.77418
$$603$$ 17.4164 0.709251
$$604$$ −28.1459 −1.14524
$$605$$ 0 0
$$606$$ 76.3050 3.09968
$$607$$ −26.9787 −1.09503 −0.547516 0.836795i $$-0.684427\pi$$
−0.547516 + 0.836795i $$0.684427\pi$$
$$608$$ 0 0
$$609$$ 56.8328 2.30298
$$610$$ 0 0
$$611$$ −15.2361 −0.616385
$$612$$ −47.1246 −1.90490
$$613$$ −7.50658 −0.303188 −0.151594 0.988443i $$-0.548441\pi$$
−0.151594 + 0.988443i $$0.548441\pi$$
$$614$$ −85.2492 −3.44038
$$615$$ 0 0
$$616$$ 173.207 6.97869
$$617$$ 9.50658 0.382720 0.191360 0.981520i $$-0.438710\pi$$
0.191360 + 0.981520i $$0.438710\pi$$
$$618$$ −7.23607 −0.291077
$$619$$ 36.5066 1.46732 0.733662 0.679515i $$-0.237810\pi$$
0.733662 + 0.679515i $$0.237810\pi$$
$$620$$ 0 0
$$621$$ 2.76393 0.110913
$$622$$ −42.5066 −1.70436
$$623$$ 15.7082 0.629336
$$624$$ 44.0689 1.76417
$$625$$ 0 0
$$626$$ 52.2148 2.08692
$$627$$ 0 0
$$628$$ 8.83282 0.352468
$$629$$ 20.1246 0.802421
$$630$$ 0 0
$$631$$ −33.9787 −1.35267 −0.676336 0.736594i $$-0.736433\pi$$
−0.676336 + 0.736594i $$0.736433\pi$$
$$632$$ 7.47214 0.297226
$$633$$ 27.3607 1.08749
$$634$$ 28.6525 1.13794
$$635$$ 0 0
$$636$$ 69.2705 2.74675
$$637$$ 21.8885 0.867256
$$638$$ 85.9574 3.40309
$$639$$ 9.05573 0.358239
$$640$$ 0 0
$$641$$ −17.2918 −0.682985 −0.341492 0.939885i $$-0.610932\pi$$
−0.341492 + 0.939885i $$0.610932\pi$$
$$642$$ −116.228 −4.58715
$$643$$ 38.3050 1.51060 0.755300 0.655379i $$-0.227491\pi$$
0.755300 + 0.655379i $$0.227491\pi$$
$$644$$ 25.4164 1.00155
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −21.5066 −0.845511 −0.422755 0.906244i $$-0.638937\pi$$
−0.422755 + 0.906244i $$0.638937\pi$$
$$648$$ 82.1935 3.22887
$$649$$ 15.1246 0.593693
$$650$$ 0 0
$$651$$ −17.5623 −0.688321
$$652$$ −13.4164 −0.525427
$$653$$ 14.1246 0.552739 0.276369 0.961051i $$-0.410869\pi$$
0.276369 + 0.961051i $$0.410869\pi$$
$$654$$ 46.5066 1.81855
$$655$$ 0 0
$$656$$ 62.8885 2.45539
$$657$$ −21.4164 −0.835534
$$658$$ −84.4853 −3.29358
$$659$$ −15.0689 −0.587000 −0.293500 0.955959i $$-0.594820\pi$$
−0.293500 + 0.955959i $$0.594820\pi$$
$$660$$ 0 0
$$661$$ 28.4164 1.10527 0.552635 0.833423i $$-0.313622\pi$$
0.552635 + 0.833423i $$0.313622\pi$$
$$662$$ −35.5066 −1.38000
$$663$$ −21.7082 −0.843077
$$664$$ −8.14590 −0.316122
$$665$$ 0 0
$$666$$ 21.7082 0.841176
$$667$$ 7.41641 0.287164
$$668$$ 25.4164 0.983390
$$669$$ −42.2361 −1.63294
$$670$$ 0 0
$$671$$ −30.4377 −1.17503
$$672$$ 102.812 3.96604
$$673$$ −2.81966 −0.108690 −0.0543450 0.998522i $$-0.517307\pi$$
−0.0543450 + 0.998522i $$0.517307\pi$$
$$674$$ −12.3262 −0.474789
$$675$$ 0 0
$$676$$ −43.6869 −1.68027
$$677$$ −36.0902 −1.38706 −0.693529 0.720429i $$-0.743945\pi$$
−0.693529 + 0.720429i $$0.743945\pi$$
$$678$$ −54.5967 −2.09678
$$679$$ 31.2705 1.20005
$$680$$ 0 0
$$681$$ −32.3607 −1.24006
$$682$$ −26.5623 −1.01712
$$683$$ −36.7984 −1.40805 −0.704025 0.710175i $$-0.748616\pi$$
−0.704025 + 0.710175i $$0.748616\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 43.7426 1.67010
$$687$$ 46.1803 1.76189
$$688$$ 104.082 3.96809
$$689$$ 12.7639 0.486267
$$690$$ 0 0
$$691$$ −7.65248 −0.291114 −0.145557 0.989350i $$-0.546497\pi$$
−0.145557 + 0.989350i $$0.546497\pi$$
$$692$$ −95.3951 −3.62638
$$693$$ −46.3607 −1.76110
$$694$$ 87.6312 3.32643
$$695$$ 0 0
$$696$$ 100.249 3.79994
$$697$$ −30.9787 −1.17340
$$698$$ 70.8673 2.68237
$$699$$ −36.9574 −1.39786
$$700$$ 0 0
$$701$$ −48.9443 −1.84860 −0.924300 0.381667i $$-0.875350\pi$$
−0.924300 + 0.381667i $$0.875350\pi$$
$$702$$ 11.7082 0.441898
$$703$$ 0 0
$$704$$ 47.6525 1.79597
$$705$$ 0 0
$$706$$ −43.2705 −1.62851
$$707$$ 55.2148 2.07657
$$708$$ 30.0000 1.12747
$$709$$ 8.83282 0.331723 0.165862 0.986149i $$-0.446959\pi$$
0.165862 + 0.986149i $$0.446959\pi$$
$$710$$ 0 0
$$711$$ −2.00000 −0.0750059
$$712$$ 27.7082 1.03841
$$713$$ −2.29180 −0.0858284
$$714$$ −120.374 −4.50488
$$715$$ 0 0
$$716$$ −54.2705 −2.02818
$$717$$ 1.58359 0.0591403
$$718$$ 59.1591 2.20780
$$719$$ −12.2361 −0.456328 −0.228164 0.973623i $$-0.573272\pi$$
−0.228164 + 0.973623i $$0.573272\pi$$
$$720$$ 0 0
$$721$$ −5.23607 −0.195001
$$722$$ 0 0
$$723$$ 29.0689 1.08108
$$724$$ 90.8115 3.37498
$$725$$ 0 0
$$726$$ −110.902 −4.11595
$$727$$ 40.6869 1.50899 0.754497 0.656303i $$-0.227881\pi$$
0.754497 + 0.656303i $$0.227881\pi$$
$$728$$ 63.3050 2.34624
$$729$$ −7.00000 −0.259259
$$730$$ 0 0
$$731$$ −51.2705 −1.89631
$$732$$ −60.3738 −2.23148
$$733$$ −50.1591 −1.85267 −0.926333 0.376705i $$-0.877057\pi$$
−0.926333 + 0.376705i $$0.877057\pi$$
$$734$$ 32.6525 1.20522
$$735$$ 0 0
$$736$$ 13.4164 0.494535
$$737$$ 47.6525 1.75530
$$738$$ −33.4164 −1.23007
$$739$$ 1.56231 0.0574704 0.0287352 0.999587i $$-0.490852\pi$$
0.0287352 + 0.999587i $$0.490852\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 70.7771 2.59831
$$743$$ −8.76393 −0.321517 −0.160759 0.986994i $$-0.551394\pi$$
−0.160759 + 0.986994i $$0.551394\pi$$
$$744$$ −30.9787 −1.13573
$$745$$ 0 0
$$746$$ 34.4164 1.26007
$$747$$ 2.18034 0.0797745
$$748$$ −128.936 −4.71437
$$749$$ −84.1033 −3.07307
$$750$$ 0 0
$$751$$ −27.4377 −1.00122 −0.500608 0.865674i $$-0.666890\pi$$
−0.500608 + 0.865674i $$0.666890\pi$$
$$752$$ −75.0689 −2.73748
$$753$$ 13.2148 0.481573
$$754$$ 31.4164 1.14412
$$755$$ 0 0
$$756$$ 45.9787 1.67223
$$757$$ 6.88854 0.250368 0.125184 0.992134i $$-0.460048\pi$$
0.125184 + 0.992134i $$0.460048\pi$$
$$758$$ −95.6656 −3.47473
$$759$$ −15.1246 −0.548989
$$760$$ 0 0
$$761$$ 27.3820 0.992595 0.496298 0.868152i $$-0.334692\pi$$
0.496298 + 0.868152i $$0.334692\pi$$
$$762$$ 5.00000 0.181131
$$763$$ 33.6525 1.21830
$$764$$ −87.1033 −3.15129
$$765$$ 0 0
$$766$$ −57.7771 −2.08757
$$767$$ 5.52786 0.199600
$$768$$ −32.5623 −1.17499
$$769$$ 15.2705 0.550669 0.275334 0.961349i $$-0.411211\pi$$
0.275334 + 0.961349i $$0.411211\pi$$
$$770$$ 0 0
$$771$$ −9.39512 −0.338357
$$772$$ −112.957 −4.06543
$$773$$ −48.0344 −1.72768 −0.863839 0.503767i $$-0.831947\pi$$
−0.863839 + 0.503767i $$0.831947\pi$$
$$774$$ −55.3050 −1.98790
$$775$$ 0 0
$$776$$ 55.1591 1.98009
$$777$$ 39.2705 1.40882
$$778$$ 11.7082 0.419759
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 24.7771 0.886594
$$782$$ −15.7082 −0.561724
$$783$$ 13.4164 0.479463
$$784$$ 107.846 3.85164
$$785$$ 0 0
$$786$$ 70.2492 2.50571
$$787$$ 43.5967 1.55406 0.777028 0.629466i $$-0.216726\pi$$
0.777028 + 0.629466i $$0.216726\pi$$
$$788$$ −36.2705 −1.29208
$$789$$ −50.5755 −1.80053
$$790$$ 0 0
$$791$$ −39.5066 −1.40469
$$792$$ −81.7771 −2.90582
$$793$$ −11.1246 −0.395047
$$794$$ 23.6525 0.839395
$$795$$ 0 0
$$796$$ 94.2492 3.34058
$$797$$ −27.8673 −0.987109 −0.493554 0.869715i $$-0.664303\pi$$
−0.493554 + 0.869715i $$0.664303\pi$$
$$798$$ 0 0
$$799$$ 36.9787 1.30821
$$800$$ 0 0
$$801$$ −7.41641 −0.262046
$$802$$ −15.4164 −0.544372
$$803$$ −58.5967 −2.06783
$$804$$ 94.5197 3.33345
$$805$$ 0 0
$$806$$ −9.70820 −0.341957
$$807$$ −20.5279 −0.722615
$$808$$ 97.3951 3.42635
$$809$$ −2.23607 −0.0786160 −0.0393080 0.999227i $$-0.512515\pi$$
−0.0393080 + 0.999227i $$0.512515\pi$$
$$810$$ 0 0
$$811$$ 9.61803 0.337735 0.168867 0.985639i $$-0.445989\pi$$
0.168867 + 0.985639i $$0.445989\pi$$
$$812$$ 123.374 4.32957
$$813$$ −25.3262 −0.888230
$$814$$ 59.3951 2.08180
$$815$$ 0 0
$$816$$ −106.957 −3.74426
$$817$$ 0 0
$$818$$ 43.0344 1.50466
$$819$$ −16.9443 −0.592081
$$820$$ 0 0
$$821$$ 27.9230 0.974519 0.487259 0.873257i $$-0.337997\pi$$
0.487259 + 0.873257i $$0.337997\pi$$
$$822$$ 54.0689 1.88587
$$823$$ −36.7426 −1.28077 −0.640384 0.768055i $$-0.721225\pi$$
−0.640384 + 0.768055i $$0.721225\pi$$
$$824$$ −9.23607 −0.321754
$$825$$ 0 0
$$826$$ 30.6525 1.06654
$$827$$ 31.2492 1.08664 0.543321 0.839525i $$-0.317167\pi$$
0.543321 + 0.839525i $$0.317167\pi$$
$$828$$ −12.0000 −0.417029
$$829$$ −1.00000 −0.0347314 −0.0173657 0.999849i $$-0.505528\pi$$
−0.0173657 + 0.999849i $$0.505528\pi$$
$$830$$ 0 0
$$831$$ 26.8328 0.930820
$$832$$ 17.4164 0.603805
$$833$$ −53.1246 −1.84066
$$834$$ 81.9574 2.83795
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −4.14590 −0.143303
$$838$$ 24.9443 0.861686
$$839$$ −54.2361 −1.87244 −0.936219 0.351418i $$-0.885699\pi$$
−0.936219 + 0.351418i $$0.885699\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −39.2705 −1.35335
$$843$$ −10.3262 −0.355655
$$844$$ 59.3951 2.04446
$$845$$ 0 0
$$846$$ 39.8885 1.37140
$$847$$ −80.2492 −2.75740
$$848$$ 62.8885 2.15960
$$849$$ 53.1378 1.82368
$$850$$ 0 0
$$851$$ 5.12461 0.175669
$$852$$ 49.1459 1.68371
$$853$$ −46.8541 −1.60425 −0.802127 0.597154i $$-0.796298\pi$$
−0.802127 + 0.597154i $$0.796298\pi$$
$$854$$ −61.6869 −2.11088
$$855$$ 0 0
$$856$$ −148.353 −5.07059
$$857$$ 24.8197 0.847823 0.423912 0.905704i $$-0.360657\pi$$
0.423912 + 0.905704i $$0.360657\pi$$
$$858$$ −64.0689 −2.18728
$$859$$ 11.9098 0.406358 0.203179 0.979142i $$-0.434873\pi$$
0.203179 + 0.979142i $$0.434873\pi$$
$$860$$ 0 0
$$861$$ −60.4508 −2.06016
$$862$$ −27.8885 −0.949888
$$863$$ −31.9230 −1.08667 −0.543336 0.839516i $$-0.682839\pi$$
−0.543336 + 0.839516i $$0.682839\pi$$
$$864$$ 24.2705 0.825700
$$865$$ 0 0
$$866$$ 87.3394 2.96791
$$867$$ 14.6738 0.498347
$$868$$ −38.1246 −1.29403
$$869$$ −5.47214 −0.185629
$$870$$ 0 0
$$871$$ 17.4164 0.590132
$$872$$ 59.3607 2.01021
$$873$$ −14.7639 −0.499683
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −116.228 −3.92698
$$877$$ 45.0344 1.52071 0.760353 0.649511i $$-0.225026\pi$$
0.760353 + 0.649511i $$0.225026\pi$$
$$878$$ 95.5755 3.22552
$$879$$ 30.1246 1.01608
$$880$$ 0 0
$$881$$ 7.47214 0.251743 0.125871 0.992047i $$-0.459827\pi$$
0.125871 + 0.992047i $$0.459827\pi$$
$$882$$ −57.3050 −1.92956
$$883$$ 21.7771 0.732857 0.366429 0.930446i $$-0.380580\pi$$
0.366429 + 0.930446i $$0.380580\pi$$
$$884$$ −47.1246 −1.58497
$$885$$ 0 0
$$886$$ 33.1803 1.11472
$$887$$ −36.9443 −1.24047 −0.620234 0.784417i $$-0.712962\pi$$
−0.620234 + 0.784417i $$0.712962\pi$$
$$888$$ 69.2705 2.32457
$$889$$ 3.61803 0.121345
$$890$$ 0 0
$$891$$ −60.1935 −2.01656
$$892$$ −91.6869 −3.06991
$$893$$ 0 0
$$894$$ 55.4508 1.85455
$$895$$ 0 0
$$896$$ 4.61803 0.154278
$$897$$ −5.52786 −0.184570
$$898$$ 64.9230 2.16651
$$899$$ −11.1246 −0.371027
$$900$$ 0 0
$$901$$ −30.9787 −1.03205
$$902$$ −91.4296 −3.04427
$$903$$ −100.048 −3.32938
$$904$$ −69.6869 −2.31775
$$905$$ 0 0
$$906$$ 33.9443 1.12772
$$907$$ −38.9574 −1.29356 −0.646780 0.762677i $$-0.723885\pi$$
−0.646780 + 0.762677i $$0.723885\pi$$
$$908$$ −70.2492 −2.33130
$$909$$ −26.0689 −0.864650
$$910$$ 0 0
$$911$$ −22.8541 −0.757190 −0.378595 0.925562i $$-0.623593\pi$$
−0.378595 + 0.925562i $$0.623593\pi$$
$$912$$ 0 0
$$913$$ 5.96556 0.197431
$$914$$ −14.5623 −0.481678
$$915$$ 0 0
$$916$$ 100.249 3.31233
$$917$$ 50.8328 1.67865
$$918$$ −28.4164 −0.937881
$$919$$ −30.5066 −1.00632 −0.503160 0.864194i $$-0.667829\pi$$
−0.503160 + 0.864194i $$0.667829\pi$$
$$920$$ 0 0
$$921$$ 72.8115 2.39922
$$922$$ 21.0344 0.692732
$$923$$ 9.05573 0.298073
$$924$$ −251.602 −8.27709
$$925$$ 0 0
$$926$$ −109.610 −3.60200
$$927$$ 2.47214 0.0811956
$$928$$ 65.1246 2.13782
$$929$$ 25.4721 0.835714 0.417857 0.908513i $$-0.362781\pi$$
0.417857 + 0.908513i $$0.362781\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −80.2279 −2.62795
$$933$$ 36.3050 1.18857
$$934$$ 7.14590 0.233821
$$935$$ 0 0
$$936$$ −29.8885 −0.976938
$$937$$ −26.8328 −0.876590 −0.438295 0.898831i $$-0.644417\pi$$
−0.438295 + 0.898831i $$0.644417\pi$$
$$938$$ 96.5755 3.15330
$$939$$ −44.5967 −1.45536
$$940$$ 0 0
$$941$$ −59.7426 −1.94755 −0.973777 0.227503i $$-0.926944\pi$$
−0.973777 + 0.227503i $$0.926944\pi$$
$$942$$ −10.6525 −0.347076
$$943$$ −7.88854 −0.256886
$$944$$ 27.2361 0.886459
$$945$$ 0 0
$$946$$ −151.318 −4.91978
$$947$$ 37.6869 1.22466 0.612330 0.790602i $$-0.290232\pi$$
0.612330 + 0.790602i $$0.290232\pi$$
$$948$$ −10.8541 −0.352525
$$949$$ −21.4164 −0.695206
$$950$$ 0 0
$$951$$ −24.4721 −0.793563
$$952$$ −153.644 −4.97964
$$953$$ −13.4164 −0.434600 −0.217300 0.976105i $$-0.569725\pi$$
−0.217300 + 0.976105i $$0.569725\pi$$
$$954$$ −33.4164 −1.08190
$$955$$ 0 0
$$956$$ 3.43769 0.111183
$$957$$ −73.4164 −2.37322
$$958$$ 72.6312 2.34661
$$959$$ 39.1246 1.26340
$$960$$ 0 0
$$961$$ −27.5623 −0.889107
$$962$$ 21.7082 0.699901
$$963$$ 39.7082 1.27958
$$964$$ 63.1033 2.03242
$$965$$ 0 0
$$966$$ −30.6525 −0.986227
$$967$$ −6.09017 −0.195847 −0.0979233 0.995194i $$-0.531220\pi$$
−0.0979233 + 0.995194i $$0.531220\pi$$
$$968$$ −141.554 −4.54972
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 19.2016 0.616210 0.308105 0.951352i $$-0.400305\pi$$
0.308105 + 0.951352i $$0.400305\pi$$
$$972$$ −86.8328 −2.78516
$$973$$ 59.3050 1.90123
$$974$$ −26.5066 −0.849326
$$975$$ 0 0
$$976$$ −54.8115 −1.75447
$$977$$ 32.2148 1.03064 0.515321 0.856997i $$-0.327673\pi$$
0.515321 + 0.856997i $$0.327673\pi$$
$$978$$ 16.1803 0.517390
$$979$$ −20.2918 −0.648529
$$980$$ 0 0
$$981$$ −15.8885 −0.507282
$$982$$ 72.6869 2.31953
$$983$$ 12.9787 0.413957 0.206978 0.978346i $$-0.433637\pi$$
0.206978 + 0.978346i $$0.433637\pi$$
$$984$$ −106.631 −3.39928
$$985$$ 0 0
$$986$$ −76.2492 −2.42827
$$987$$ 72.1591 2.29685
$$988$$ 0 0
$$989$$ −13.0557 −0.415148
$$990$$ 0 0
$$991$$ 15.4164 0.489718 0.244859 0.969559i $$-0.421258\pi$$
0.244859 + 0.969559i $$0.421258\pi$$
$$992$$ −20.1246 −0.638957
$$993$$ 30.3262 0.962374
$$994$$ 50.2148 1.59272
$$995$$ 0 0
$$996$$ 11.8328 0.374937
$$997$$ 6.74265 0.213542 0.106771 0.994284i $$-0.465949\pi$$
0.106771 + 0.994284i $$0.465949\pi$$
$$998$$ 27.7426 0.878178
$$999$$ 9.27051 0.293306
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 9025.2.a.l.1.1 2
5.4 even 2 9025.2.a.u.1.2 yes 2
19.18 odd 2 9025.2.a.t.1.2 yes 2
95.94 odd 2 9025.2.a.m.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
9025.2.a.l.1.1 2 1.1 even 1 trivial
9025.2.a.m.1.1 yes 2 95.94 odd 2
9025.2.a.t.1.2 yes 2 19.18 odd 2
9025.2.a.u.1.2 yes 2 5.4 even 2