# Properties

 Label 9025.2.a.bg.1.2 Level $9025$ Weight $2$ Character 9025.1 Self dual yes Analytic conductor $72.065$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.7537.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 4x + 3$$ x^4 - x^3 - 5*x^2 + 4*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.37933$$ 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)

 $$f(q)$$ $$=$$ $$q-1.09744 q^{2} +0.379334 q^{3} -0.795629 q^{4} -0.416295 q^{6} -1.89307 q^{7} +3.06803 q^{8} -2.85611 q^{9} +O(q^{10})$$ $$q-1.09744 q^{2} +0.379334 q^{3} -0.795629 q^{4} -0.416295 q^{6} -1.89307 q^{7} +3.06803 q^{8} -2.85611 q^{9} +0.134400 q^{11} -0.301809 q^{12} -3.51373 q^{13} +2.07752 q^{14} -1.77572 q^{16} +1.66123 q^{17} +3.13440 q^{18} -0.718104 q^{21} -0.147496 q^{22} -5.36984 q^{23} +1.16381 q^{24} +3.85611 q^{26} -2.22142 q^{27} +1.50618 q^{28} +4.97059 q^{29} +6.56472 q^{31} -4.18732 q^{32} +0.0509824 q^{33} -1.82310 q^{34} +2.27240 q^{36} +1.69819 q^{37} -1.33288 q^{39} +10.6327 q^{41} +0.788075 q^{42} -8.50784 q^{43} -0.106932 q^{44} +5.89307 q^{46} +11.1154 q^{47} -0.673589 q^{48} -3.41630 q^{49} +0.630160 q^{51} +2.79563 q^{52} +0.264847 q^{53} +2.43787 q^{54} -5.80799 q^{56} -5.45492 q^{58} -6.89667 q^{59} +9.17589 q^{61} -7.20437 q^{62} +5.40680 q^{63} +8.14676 q^{64} -0.0559501 q^{66} +2.95354 q^{67} -1.32172 q^{68} -2.03696 q^{69} +1.32835 q^{71} -8.76262 q^{72} +6.34237 q^{73} -1.86366 q^{74} -0.254428 q^{77} +1.46275 q^{78} -1.46728 q^{79} +7.72566 q^{81} -11.6688 q^{82} -7.44736 q^{83} +0.571345 q^{84} +9.33683 q^{86} +1.88551 q^{87} +0.412343 q^{88} +9.73608 q^{89} +6.65174 q^{91} +4.27240 q^{92} +2.49022 q^{93} -12.1985 q^{94} -1.58839 q^{96} -17.4689 q^{97} +3.74917 q^{98} -0.383860 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} - 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} - 12 q^{8} + q^{9}+O(q^{10})$$ 4 * q - q^2 - 3 * q^3 + 5 * q^4 + 2 * q^6 + 4 * q^7 - 12 * q^8 + q^9 $$4 q - q^{2} - 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} - 12 q^{8} + q^{9} - 2 q^{11} - 6 q^{12} - 7 q^{13} - q^{14} + 7 q^{16} + q^{17} + 10 q^{18} - 4 q^{21} - 2 q^{22} - 2 q^{23} + 23 q^{24} + 3 q^{26} - 12 q^{27} + 19 q^{28} - q^{29} - 30 q^{32} - 19 q^{33} + 15 q^{34} - 7 q^{36} + 2 q^{37} + 15 q^{39} - 8 q^{41} + 15 q^{42} - q^{43} - 12 q^{44} + 12 q^{46} + 12 q^{47} - 23 q^{48} - 10 q^{49} + 22 q^{51} + 3 q^{52} + 5 q^{53} - 34 q^{54} - 41 q^{56} + 27 q^{58} - 5 q^{59} - 37 q^{62} + 3 q^{63} + 56 q^{64} - 31 q^{66} - 4 q^{67} - 16 q^{68} - 9 q^{69} + 20 q^{71} - 17 q^{72} + 20 q^{73} + 25 q^{74} - 14 q^{77} + 18 q^{78} + 17 q^{79} + 12 q^{81} - 21 q^{82} - q^{83} - 20 q^{84} + 8 q^{86} + 16 q^{87} + 7 q^{88} + 11 q^{89} + 6 q^{91} + q^{92} + 8 q^{93} - 31 q^{94} + 21 q^{96} - q^{97} - 9 q^{98} + 38 q^{99}+O(q^{100})$$ 4 * q - q^2 - 3 * q^3 + 5 * q^4 + 2 * q^6 + 4 * q^7 - 12 * q^8 + q^9 - 2 * q^11 - 6 * q^12 - 7 * q^13 - q^14 + 7 * q^16 + q^17 + 10 * q^18 - 4 * q^21 - 2 * q^22 - 2 * q^23 + 23 * q^24 + 3 * q^26 - 12 * q^27 + 19 * q^28 - q^29 - 30 * q^32 - 19 * q^33 + 15 * q^34 - 7 * q^36 + 2 * q^37 + 15 * q^39 - 8 * q^41 + 15 * q^42 - q^43 - 12 * q^44 + 12 * q^46 + 12 * q^47 - 23 * q^48 - 10 * q^49 + 22 * q^51 + 3 * q^52 + 5 * q^53 - 34 * q^54 - 41 * q^56 + 27 * q^58 - 5 * q^59 - 37 * q^62 + 3 * q^63 + 56 * q^64 - 31 * q^66 - 4 * q^67 - 16 * q^68 - 9 * q^69 + 20 * q^71 - 17 * q^72 + 20 * q^73 + 25 * q^74 - 14 * q^77 + 18 * q^78 + 17 * q^79 + 12 * q^81 - 21 * q^82 - q^83 - 20 * q^84 + 8 * q^86 + 16 * q^87 + 7 * q^88 + 11 * q^89 + 6 * q^91 + q^92 + 8 * q^93 - 31 * q^94 + 21 * q^96 - q^97 - 9 * q^98 + 38 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.09744 −0.776006 −0.388003 0.921658i $$-0.626835\pi$$
−0.388003 + 0.921658i $$0.626835\pi$$
$$3$$ 0.379334 0.219008 0.109504 0.993986i $$-0.465074\pi$$
0.109504 + 0.993986i $$0.465074\pi$$
$$4$$ −0.795629 −0.397815
$$5$$ 0 0
$$6$$ −0.416295 −0.169952
$$7$$ −1.89307 −0.715512 −0.357756 0.933815i $$-0.616458\pi$$
−0.357756 + 0.933815i $$0.616458\pi$$
$$8$$ 3.06803 1.08471
$$9$$ −2.85611 −0.952035
$$10$$ 0 0
$$11$$ 0.134400 0.0405231 0.0202615 0.999795i $$-0.493550\pi$$
0.0202615 + 0.999795i $$0.493550\pi$$
$$12$$ −0.301809 −0.0871248
$$13$$ −3.51373 −0.974534 −0.487267 0.873253i $$-0.662006\pi$$
−0.487267 + 0.873253i $$0.662006\pi$$
$$14$$ 2.07752 0.555242
$$15$$ 0 0
$$16$$ −1.77572 −0.443929
$$17$$ 1.66123 0.402907 0.201454 0.979498i $$-0.435433\pi$$
0.201454 + 0.979498i $$0.435433\pi$$
$$18$$ 3.13440 0.738785
$$19$$ 0 0
$$20$$ 0 0
$$21$$ −0.718104 −0.156703
$$22$$ −0.147496 −0.0314462
$$23$$ −5.36984 −1.11969 −0.559844 0.828598i $$-0.689139\pi$$
−0.559844 + 0.828598i $$0.689139\pi$$
$$24$$ 1.16381 0.237561
$$25$$ 0 0
$$26$$ 3.85611 0.756245
$$27$$ −2.22142 −0.427512
$$28$$ 1.50618 0.284641
$$29$$ 4.97059 0.923016 0.461508 0.887136i $$-0.347309\pi$$
0.461508 + 0.887136i $$0.347309\pi$$
$$30$$ 0 0
$$31$$ 6.56472 1.17906 0.589529 0.807747i $$-0.299313\pi$$
0.589529 + 0.807747i $$0.299313\pi$$
$$32$$ −4.18732 −0.740221
$$33$$ 0.0509824 0.00887490
$$34$$ −1.82310 −0.312658
$$35$$ 0 0
$$36$$ 2.27240 0.378734
$$37$$ 1.69819 0.279181 0.139590 0.990209i $$-0.455421\pi$$
0.139590 + 0.990209i $$0.455421\pi$$
$$38$$ 0 0
$$39$$ −1.33288 −0.213431
$$40$$ 0 0
$$41$$ 10.6327 1.66056 0.830278 0.557349i $$-0.188182\pi$$
0.830278 + 0.557349i $$0.188182\pi$$
$$42$$ 0.788075 0.121603
$$43$$ −8.50784 −1.29743 −0.648717 0.761030i $$-0.724694\pi$$
−0.648717 + 0.761030i $$0.724694\pi$$
$$44$$ −0.106932 −0.0161207
$$45$$ 0 0
$$46$$ 5.89307 0.868885
$$47$$ 11.1154 1.62135 0.810675 0.585497i $$-0.199101\pi$$
0.810675 + 0.585497i $$0.199101\pi$$
$$48$$ −0.673589 −0.0972242
$$49$$ −3.41630 −0.488042
$$50$$ 0 0
$$51$$ 0.630160 0.0882401
$$52$$ 2.79563 0.387684
$$53$$ 0.264847 0.0363796 0.0181898 0.999835i $$-0.494210\pi$$
0.0181898 + 0.999835i $$0.494210\pi$$
$$54$$ 2.43787 0.331752
$$55$$ 0 0
$$56$$ −5.80799 −0.776125
$$57$$ 0 0
$$58$$ −5.45492 −0.716266
$$59$$ −6.89667 −0.897870 −0.448935 0.893564i $$-0.648196\pi$$
−0.448935 + 0.893564i $$0.648196\pi$$
$$60$$ 0 0
$$61$$ 9.17589 1.17485 0.587426 0.809278i $$-0.300141\pi$$
0.587426 + 0.809278i $$0.300141\pi$$
$$62$$ −7.20437 −0.914956
$$63$$ 5.40680 0.681193
$$64$$ 8.14676 1.01834
$$65$$ 0 0
$$66$$ −0.0559501 −0.00688698
$$67$$ 2.95354 0.360833 0.180416 0.983590i $$-0.442255\pi$$
0.180416 + 0.983590i $$0.442255\pi$$
$$68$$ −1.32172 −0.160282
$$69$$ −2.03696 −0.245221
$$70$$ 0 0
$$71$$ 1.32835 0.157646 0.0788232 0.996889i $$-0.474884\pi$$
0.0788232 + 0.996889i $$0.474884\pi$$
$$72$$ −8.76262 −1.03268
$$73$$ 6.34237 0.742319 0.371159 0.928569i $$-0.378960\pi$$
0.371159 + 0.928569i $$0.378960\pi$$
$$74$$ −1.86366 −0.216646
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −0.254428 −0.0289948
$$78$$ 1.46275 0.165624
$$79$$ −1.46728 −0.165082 −0.0825408 0.996588i $$-0.526303\pi$$
−0.0825408 + 0.996588i $$0.526303\pi$$
$$80$$ 0 0
$$81$$ 7.72566 0.858406
$$82$$ −11.6688 −1.28860
$$83$$ −7.44736 −0.817454 −0.408727 0.912657i $$-0.634027\pi$$
−0.408727 + 0.912657i $$0.634027\pi$$
$$84$$ 0.571345 0.0623388
$$85$$ 0 0
$$86$$ 9.33683 1.00682
$$87$$ 1.88551 0.202148
$$88$$ 0.412343 0.0439559
$$89$$ 9.73608 1.03202 0.516011 0.856582i $$-0.327416\pi$$
0.516011 + 0.856582i $$0.327416\pi$$
$$90$$ 0 0
$$91$$ 6.65174 0.697291
$$92$$ 4.27240 0.445429
$$93$$ 2.49022 0.258224
$$94$$ −12.1985 −1.25818
$$95$$ 0 0
$$96$$ −1.58839 −0.162115
$$97$$ −17.4689 −1.77370 −0.886851 0.462055i $$-0.847112\pi$$
−0.886851 + 0.462055i $$0.847112\pi$$
$$98$$ 3.74917 0.378724
$$99$$ −0.383860 −0.0385794
$$100$$ 0 0
$$101$$ 5.39731 0.537052 0.268526 0.963272i $$-0.413463\pi$$
0.268526 + 0.963272i $$0.413463\pi$$
$$102$$ −0.691562 −0.0684749
$$103$$ −2.14750 −0.211599 −0.105800 0.994387i $$-0.533740\pi$$
−0.105800 + 0.994387i $$0.533740\pi$$
$$104$$ −10.7802 −1.05709
$$105$$ 0 0
$$106$$ −0.290654 −0.0282308
$$107$$ 1.00093 0.0967631 0.0483815 0.998829i $$-0.484594\pi$$
0.0483815 + 0.998829i $$0.484594\pi$$
$$108$$ 1.76743 0.170071
$$109$$ 16.2629 1.55770 0.778852 0.627208i $$-0.215802\pi$$
0.778852 + 0.627208i $$0.215802\pi$$
$$110$$ 0 0
$$111$$ 0.644181 0.0611430
$$112$$ 3.36155 0.317637
$$113$$ −0.843010 −0.0793037 −0.0396519 0.999214i $$-0.512625\pi$$
−0.0396519 + 0.999214i $$0.512625\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −3.95475 −0.367189
$$117$$ 10.0356 0.927791
$$118$$ 7.56867 0.696752
$$119$$ −3.14482 −0.288285
$$120$$ 0 0
$$121$$ −10.9819 −0.998358
$$122$$ −10.0700 −0.911692
$$123$$ 4.03336 0.363676
$$124$$ −5.22308 −0.469046
$$125$$ 0 0
$$126$$ −5.93363 −0.528610
$$127$$ −18.7397 −1.66288 −0.831439 0.555616i $$-0.812482\pi$$
−0.831439 + 0.555616i $$0.812482\pi$$
$$128$$ −0.565920 −0.0500208
$$129$$ −3.22731 −0.284149
$$130$$ 0 0
$$131$$ 2.88644 0.252189 0.126095 0.992018i $$-0.459756\pi$$
0.126095 + 0.992018i $$0.459756\pi$$
$$132$$ −0.0405631 −0.00353057
$$133$$ 0 0
$$134$$ −3.24133 −0.280008
$$135$$ 0 0
$$136$$ 5.09670 0.437039
$$137$$ 18.8316 1.60889 0.804445 0.594027i $$-0.202463\pi$$
0.804445 + 0.594027i $$0.202463\pi$$
$$138$$ 2.23544 0.190293
$$139$$ −18.1795 −1.54196 −0.770982 0.636857i $$-0.780234\pi$$
−0.770982 + 0.636857i $$0.780234\pi$$
$$140$$ 0 0
$$141$$ 4.21645 0.355089
$$142$$ −1.45778 −0.122334
$$143$$ −0.472245 −0.0394912
$$144$$ 5.07163 0.422636
$$145$$ 0 0
$$146$$ −6.96036 −0.576044
$$147$$ −1.29592 −0.106885
$$148$$ −1.35113 −0.111062
$$149$$ −22.2543 −1.82315 −0.911573 0.411138i $$-0.865131\pi$$
−0.911573 + 0.411138i $$0.865131\pi$$
$$150$$ 0 0
$$151$$ 3.33482 0.271384 0.135692 0.990751i $$-0.456674\pi$$
0.135692 + 0.990751i $$0.456674\pi$$
$$152$$ 0 0
$$153$$ −4.74465 −0.383582
$$154$$ 0.279219 0.0225001
$$155$$ 0 0
$$156$$ 1.06048 0.0849061
$$157$$ −7.26291 −0.579643 −0.289822 0.957081i $$-0.593596\pi$$
−0.289822 + 0.957081i $$0.593596\pi$$
$$158$$ 1.61025 0.128104
$$159$$ 0.100466 0.00796744
$$160$$ 0 0
$$161$$ 10.1655 0.801151
$$162$$ −8.47843 −0.666129
$$163$$ 19.7783 1.54916 0.774578 0.632478i $$-0.217962\pi$$
0.774578 + 0.632478i $$0.217962\pi$$
$$164$$ −8.45972 −0.660593
$$165$$ 0 0
$$166$$ 8.17302 0.634350
$$167$$ 3.40320 0.263348 0.131674 0.991293i $$-0.457965\pi$$
0.131674 + 0.991293i $$0.457965\pi$$
$$168$$ −2.20317 −0.169978
$$169$$ −0.653675 −0.0502827
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 6.76909 0.516138
$$173$$ 10.5857 0.804817 0.402409 0.915460i $$-0.368173\pi$$
0.402409 + 0.915460i $$0.368173\pi$$
$$174$$ −2.06923 −0.156868
$$175$$ 0 0
$$176$$ −0.238656 −0.0179894
$$177$$ −2.61614 −0.196641
$$178$$ −10.6847 −0.800855
$$179$$ −14.6024 −1.09144 −0.545718 0.837969i $$-0.683743\pi$$
−0.545718 + 0.837969i $$0.683743\pi$$
$$180$$ 0 0
$$181$$ −5.43261 −0.403803 −0.201901 0.979406i $$-0.564712\pi$$
−0.201901 + 0.979406i $$0.564712\pi$$
$$182$$ −7.29987 −0.541102
$$183$$ 3.48072 0.257303
$$184$$ −16.4748 −1.21454
$$185$$ 0 0
$$186$$ −2.73286 −0.200383
$$187$$ 0.223269 0.0163271
$$188$$ −8.84375 −0.644996
$$189$$ 4.20530 0.305890
$$190$$ 0 0
$$191$$ 20.4758 1.48157 0.740787 0.671740i $$-0.234453\pi$$
0.740787 + 0.671740i $$0.234453\pi$$
$$192$$ 3.09034 0.223026
$$193$$ −11.0235 −0.793490 −0.396745 0.917929i $$-0.629860\pi$$
−0.396745 + 0.917929i $$0.629860\pi$$
$$194$$ 19.1711 1.37640
$$195$$ 0 0
$$196$$ 2.71810 0.194150
$$197$$ 19.8532 1.41448 0.707242 0.706971i $$-0.249939\pi$$
0.707242 + 0.706971i $$0.249939\pi$$
$$198$$ 0.421263 0.0299379
$$199$$ 21.0026 1.48883 0.744417 0.667715i $$-0.232728\pi$$
0.744417 + 0.667715i $$0.232728\pi$$
$$200$$ 0 0
$$201$$ 1.12038 0.0790254
$$202$$ −5.92321 −0.416756
$$203$$ −9.40967 −0.660429
$$204$$ −0.501374 −0.0351032
$$205$$ 0 0
$$206$$ 2.35674 0.164202
$$207$$ 15.3368 1.06598
$$208$$ 6.23939 0.432624
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.8257 −0.882956 −0.441478 0.897272i $$-0.645546\pi$$
−0.441478 + 0.897272i $$0.645546\pi$$
$$212$$ −0.210720 −0.0144723
$$213$$ 0.503889 0.0345259
$$214$$ −1.09845 −0.0750887
$$215$$ 0 0
$$216$$ −6.81538 −0.463728
$$217$$ −12.4275 −0.843630
$$218$$ −17.8475 −1.20879
$$219$$ 2.40588 0.162574
$$220$$ 0 0
$$221$$ −5.83712 −0.392647
$$222$$ −0.706949 −0.0474473
$$223$$ −20.3944 −1.36571 −0.682856 0.730553i $$-0.739263\pi$$
−0.682856 + 0.730553i $$0.739263\pi$$
$$224$$ 7.92688 0.529637
$$225$$ 0 0
$$226$$ 0.925152 0.0615402
$$227$$ −25.4172 −1.68700 −0.843500 0.537129i $$-0.819509\pi$$
−0.843500 + 0.537129i $$0.819509\pi$$
$$228$$ 0 0
$$229$$ 2.21553 0.146406 0.0732030 0.997317i $$-0.476678\pi$$
0.0732030 + 0.997317i $$0.476678\pi$$
$$230$$ 0 0
$$231$$ −0.0965132 −0.00635010
$$232$$ 15.2499 1.00121
$$233$$ 14.1576 0.927498 0.463749 0.885967i $$-0.346504\pi$$
0.463749 + 0.885967i $$0.346504\pi$$
$$234$$ −11.0134 −0.719972
$$235$$ 0 0
$$236$$ 5.48719 0.357186
$$237$$ −0.556588 −0.0361543
$$238$$ 3.45125 0.223711
$$239$$ 3.01476 0.195008 0.0975042 0.995235i $$-0.468914\pi$$
0.0975042 + 0.995235i $$0.468914\pi$$
$$240$$ 0 0
$$241$$ −23.7792 −1.53175 −0.765877 0.642987i $$-0.777695\pi$$
−0.765877 + 0.642987i $$0.777695\pi$$
$$242$$ 12.0520 0.774732
$$243$$ 9.59486 0.615511
$$244$$ −7.30060 −0.467373
$$245$$ 0 0
$$246$$ −4.42636 −0.282215
$$247$$ 0 0
$$248$$ 20.1407 1.27894
$$249$$ −2.82504 −0.179029
$$250$$ 0 0
$$251$$ 17.1899 1.08502 0.542509 0.840050i $$-0.317475\pi$$
0.542509 + 0.840050i $$0.317475\pi$$
$$252$$ −4.30181 −0.270988
$$253$$ −0.721706 −0.0453733
$$254$$ 20.5656 1.29040
$$255$$ 0 0
$$256$$ −15.6725 −0.979529
$$257$$ 19.5429 1.21905 0.609525 0.792767i $$-0.291360\pi$$
0.609525 + 0.792767i $$0.291360\pi$$
$$258$$ 3.54178 0.220501
$$259$$ −3.21479 −0.199757
$$260$$ 0 0
$$261$$ −14.1965 −0.878744
$$262$$ −3.16769 −0.195700
$$263$$ −8.81360 −0.543470 −0.271735 0.962372i $$-0.587597\pi$$
−0.271735 + 0.962372i $$0.587597\pi$$
$$264$$ 0.156416 0.00962672
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 3.69322 0.226022
$$268$$ −2.34993 −0.143545
$$269$$ 0.288362 0.0175818 0.00879088 0.999961i $$-0.497202\pi$$
0.00879088 + 0.999961i $$0.497202\pi$$
$$270$$ 0 0
$$271$$ −24.8712 −1.51082 −0.755409 0.655253i $$-0.772562\pi$$
−0.755409 + 0.655253i $$0.772562\pi$$
$$272$$ −2.94987 −0.178862
$$273$$ 2.52323 0.152713
$$274$$ −20.6665 −1.24851
$$275$$ 0 0
$$276$$ 1.62067 0.0975526
$$277$$ 4.40486 0.264662 0.132331 0.991206i $$-0.457754\pi$$
0.132331 + 0.991206i $$0.457754\pi$$
$$278$$ 19.9509 1.19657
$$279$$ −18.7495 −1.12250
$$280$$ 0 0
$$281$$ −32.7214 −1.95200 −0.975998 0.217778i $$-0.930119\pi$$
−0.975998 + 0.217778i $$0.930119\pi$$
$$282$$ −4.62729 −0.275551
$$283$$ 1.32893 0.0789964 0.0394982 0.999220i $$-0.487424\pi$$
0.0394982 + 0.999220i $$0.487424\pi$$
$$284$$ −1.05688 −0.0627140
$$285$$ 0 0
$$286$$ 0.518260 0.0306454
$$287$$ −20.1285 −1.18815
$$288$$ 11.9594 0.704717
$$289$$ −14.2403 −0.837666
$$290$$ 0 0
$$291$$ −6.62656 −0.388456
$$292$$ −5.04618 −0.295305
$$293$$ 7.72365 0.451220 0.225610 0.974218i $$-0.427562\pi$$
0.225610 + 0.974218i $$0.427562\pi$$
$$294$$ 1.42219 0.0829437
$$295$$ 0 0
$$296$$ 5.21010 0.302831
$$297$$ −0.298558 −0.0173241
$$298$$ 24.4228 1.41477
$$299$$ 18.8682 1.09118
$$300$$ 0 0
$$301$$ 16.1059 0.928330
$$302$$ −3.65976 −0.210595
$$303$$ 2.04738 0.117619
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 5.20696 0.297662
$$307$$ 9.10002 0.519366 0.259683 0.965694i $$-0.416382\pi$$
0.259683 + 0.965694i $$0.416382\pi$$
$$308$$ 0.202430 0.0115345
$$309$$ −0.814618 −0.0463420
$$310$$ 0 0
$$311$$ −12.4569 −0.706364 −0.353182 0.935555i $$-0.614900\pi$$
−0.353182 + 0.935555i $$0.614900\pi$$
$$312$$ −4.08931 −0.231512
$$313$$ −2.04553 −0.115620 −0.0578101 0.998328i $$-0.518412\pi$$
−0.0578101 + 0.998328i $$0.518412\pi$$
$$314$$ 7.97059 0.449807
$$315$$ 0 0
$$316$$ 1.16741 0.0656719
$$317$$ −23.5713 −1.32389 −0.661947 0.749551i $$-0.730270\pi$$
−0.661947 + 0.749551i $$0.730270\pi$$
$$318$$ −0.110255 −0.00618278
$$319$$ 0.668047 0.0374035
$$320$$ 0 0
$$321$$ 0.379685 0.0211919
$$322$$ −11.1560 −0.621698
$$323$$ 0 0
$$324$$ −6.14676 −0.341487
$$325$$ 0 0
$$326$$ −21.7055 −1.20215
$$327$$ 6.16907 0.341150
$$328$$ 32.6216 1.80123
$$329$$ −21.0422 −1.16010
$$330$$ 0 0
$$331$$ −18.7175 −1.02881 −0.514403 0.857549i $$-0.671986\pi$$
−0.514403 + 0.857549i $$0.671986\pi$$
$$332$$ 5.92534 0.325195
$$333$$ −4.85021 −0.265790
$$334$$ −3.73480 −0.204359
$$335$$ 0 0
$$336$$ 1.27515 0.0695651
$$337$$ 32.6881 1.78063 0.890316 0.455343i $$-0.150483\pi$$
0.890316 + 0.455343i $$0.150483\pi$$
$$338$$ 0.717368 0.0390197
$$339$$ −0.319782 −0.0173682
$$340$$ 0 0
$$341$$ 0.882297 0.0477791
$$342$$ 0 0
$$343$$ 19.7188 1.06471
$$344$$ −26.1023 −1.40734
$$345$$ 0 0
$$346$$ −11.6172 −0.624543
$$347$$ 2.56666 0.137785 0.0688927 0.997624i $$-0.478053\pi$$
0.0688927 + 0.997624i $$0.478053\pi$$
$$348$$ −1.50017 −0.0804175
$$349$$ 16.6195 0.889619 0.444810 0.895625i $$-0.353271\pi$$
0.444810 + 0.895625i $$0.353271\pi$$
$$350$$ 0 0
$$351$$ 7.80547 0.416625
$$352$$ −0.562776 −0.0299961
$$353$$ −28.3629 −1.50961 −0.754803 0.655951i $$-0.772268\pi$$
−0.754803 + 0.655951i $$0.772268\pi$$
$$354$$ 2.87105 0.152595
$$355$$ 0 0
$$356$$ −7.74631 −0.410553
$$357$$ −1.19294 −0.0631369
$$358$$ 16.0252 0.846961
$$359$$ 17.3885 0.917733 0.458866 0.888505i $$-0.348256\pi$$
0.458866 + 0.888505i $$0.348256\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ 5.96195 0.313353
$$363$$ −4.16582 −0.218649
$$364$$ −5.29231 −0.277393
$$365$$ 0 0
$$366$$ −3.81988 −0.199668
$$367$$ −25.8048 −1.34700 −0.673501 0.739186i $$-0.735210\pi$$
−0.673501 + 0.739186i $$0.735210\pi$$
$$368$$ 9.53531 0.497062
$$369$$ −30.3683 −1.58091
$$370$$ 0 0
$$371$$ −0.501374 −0.0260300
$$372$$ −1.98129 −0.102725
$$373$$ −27.0663 −1.40144 −0.700719 0.713437i $$-0.747138\pi$$
−0.700719 + 0.713437i $$0.747138\pi$$
$$374$$ −0.245024 −0.0126699
$$375$$ 0 0
$$376$$ 34.1024 1.75870
$$377$$ −17.4653 −0.899511
$$378$$ −4.61505 −0.237373
$$379$$ 12.4028 0.637092 0.318546 0.947907i $$-0.396806\pi$$
0.318546 + 0.947907i $$0.396806\pi$$
$$380$$ 0 0
$$381$$ −7.10859 −0.364184
$$382$$ −22.4709 −1.14971
$$383$$ 5.35942 0.273854 0.136927 0.990581i $$-0.456277\pi$$
0.136927 + 0.990581i $$0.456277\pi$$
$$384$$ −0.214673 −0.0109550
$$385$$ 0 0
$$386$$ 12.0976 0.615753
$$387$$ 24.2993 1.23520
$$388$$ 13.8988 0.705605
$$389$$ 8.56933 0.434482 0.217241 0.976118i $$-0.430294\pi$$
0.217241 + 0.976118i $$0.430294\pi$$
$$390$$ 0 0
$$391$$ −8.92053 −0.451131
$$392$$ −10.4813 −0.529386
$$393$$ 1.09492 0.0552316
$$394$$ −21.7877 −1.09765
$$395$$ 0 0
$$396$$ 0.305411 0.0153475
$$397$$ 10.6445 0.534234 0.267117 0.963664i $$-0.413929\pi$$
0.267117 + 0.963664i $$0.413929\pi$$
$$398$$ −23.0490 −1.15534
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 7.65209 0.382127 0.191063 0.981578i $$-0.438806\pi$$
0.191063 + 0.981578i $$0.438806\pi$$
$$402$$ −1.22955 −0.0613242
$$403$$ −23.0667 −1.14903
$$404$$ −4.29426 −0.213647
$$405$$ 0 0
$$406$$ 10.3265 0.512497
$$407$$ 0.228237 0.0113133
$$408$$ 1.93335 0.0957152
$$409$$ −17.6887 −0.874650 −0.437325 0.899304i $$-0.644074\pi$$
−0.437325 + 0.899304i $$0.644074\pi$$
$$410$$ 0 0
$$411$$ 7.14345 0.352361
$$412$$ 1.70861 0.0841772
$$413$$ 13.0559 0.642437
$$414$$ −16.8312 −0.827210
$$415$$ 0 0
$$416$$ 14.7131 0.721371
$$417$$ −6.89609 −0.337703
$$418$$ 0 0
$$419$$ 1.18732 0.0580045 0.0290023 0.999579i $$-0.490767\pi$$
0.0290023 + 0.999579i $$0.490767\pi$$
$$420$$ 0 0
$$421$$ 33.3673 1.62622 0.813111 0.582109i $$-0.197772\pi$$
0.813111 + 0.582109i $$0.197772\pi$$
$$422$$ 14.0754 0.685180
$$423$$ −31.7468 −1.54358
$$424$$ 0.812560 0.0394614
$$425$$ 0 0
$$426$$ −0.552987 −0.0267923
$$427$$ −17.3706 −0.840621
$$428$$ −0.796365 −0.0384938
$$429$$ −0.179139 −0.00864890
$$430$$ 0 0
$$431$$ −6.17598 −0.297486 −0.148743 0.988876i $$-0.547523\pi$$
−0.148743 + 0.988876i $$0.547523\pi$$
$$432$$ 3.94461 0.189785
$$433$$ 18.5552 0.891707 0.445854 0.895106i $$-0.352900\pi$$
0.445854 + 0.895106i $$0.352900\pi$$
$$434$$ 13.6384 0.654662
$$435$$ 0 0
$$436$$ −12.9392 −0.619677
$$437$$ 0 0
$$438$$ −2.64030 −0.126158
$$439$$ −0.227312 −0.0108490 −0.00542450 0.999985i $$-0.501727\pi$$
−0.00542450 + 0.999985i $$0.501727\pi$$
$$440$$ 0 0
$$441$$ 9.75730 0.464633
$$442$$ 6.40588 0.304696
$$443$$ 34.9827 1.66208 0.831038 0.556215i $$-0.187747\pi$$
0.831038 + 0.556215i $$0.187747\pi$$
$$444$$ −0.512529 −0.0243236
$$445$$ 0 0
$$446$$ 22.3816 1.05980
$$447$$ −8.44182 −0.399285
$$448$$ −15.4224 −0.728638
$$449$$ −16.9509 −0.799961 −0.399980 0.916524i $$-0.630983\pi$$
−0.399980 + 0.916524i $$0.630983\pi$$
$$450$$ 0 0
$$451$$ 1.42904 0.0672909
$$452$$ 0.670724 0.0315482
$$453$$ 1.26501 0.0594353
$$454$$ 27.8938 1.30912
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.60241 −0.0749578 −0.0374789 0.999297i $$-0.511933\pi$$
−0.0374789 + 0.999297i $$0.511933\pi$$
$$458$$ −2.43140 −0.113612
$$459$$ −3.69029 −0.172248
$$460$$ 0 0
$$461$$ −8.74162 −0.407138 −0.203569 0.979061i $$-0.565254\pi$$
−0.203569 + 0.979061i $$0.565254\pi$$
$$462$$ 0.105917 0.00492772
$$463$$ −21.1886 −0.984718 −0.492359 0.870392i $$-0.663865\pi$$
−0.492359 + 0.870392i $$0.663865\pi$$
$$464$$ −8.82636 −0.409753
$$465$$ 0 0
$$466$$ −15.5371 −0.719744
$$467$$ −20.4516 −0.946388 −0.473194 0.880958i $$-0.656899\pi$$
−0.473194 + 0.880958i $$0.656899\pi$$
$$468$$ −7.98461 −0.369089
$$469$$ −5.59126 −0.258180
$$470$$ 0 0
$$471$$ −2.75507 −0.126947
$$472$$ −21.1592 −0.973931
$$473$$ −1.14345 −0.0525760
$$474$$ 0.610821 0.0280559
$$475$$ 0 0
$$476$$ 2.50211 0.114684
$$477$$ −0.756432 −0.0346347
$$478$$ −3.30851 −0.151328
$$479$$ 23.5491 1.07599 0.537994 0.842949i $$-0.319182\pi$$
0.537994 + 0.842949i $$0.319182\pi$$
$$480$$ 0 0
$$481$$ −5.96699 −0.272071
$$482$$ 26.0962 1.18865
$$483$$ 3.85611 0.175459
$$484$$ 8.73755 0.397161
$$485$$ 0 0
$$486$$ −10.5298 −0.477640
$$487$$ −36.0392 −1.63309 −0.816546 0.577280i $$-0.804114\pi$$
−0.816546 + 0.577280i $$0.804114\pi$$
$$488$$ 28.1519 1.27438
$$489$$ 7.50258 0.339278
$$490$$ 0 0
$$491$$ 20.0595 0.905271 0.452635 0.891696i $$-0.350484\pi$$
0.452635 + 0.891696i $$0.350484\pi$$
$$492$$ −3.20906 −0.144676
$$493$$ 8.25729 0.371890
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −11.6571 −0.523418
$$497$$ −2.51466 −0.112798
$$498$$ 3.10030 0.138928
$$499$$ −36.8729 −1.65066 −0.825328 0.564653i $$-0.809010\pi$$
−0.825328 + 0.564653i $$0.809010\pi$$
$$500$$ 0 0
$$501$$ 1.29095 0.0576753
$$502$$ −18.8649 −0.841980
$$503$$ 21.6487 0.965268 0.482634 0.875822i $$-0.339680\pi$$
0.482634 + 0.875822i $$0.339680\pi$$
$$504$$ 16.5882 0.738899
$$505$$ 0 0
$$506$$ 0.792028 0.0352099
$$507$$ −0.247961 −0.0110123
$$508$$ 14.9098 0.661517
$$509$$ −36.4558 −1.61588 −0.807938 0.589267i $$-0.799417\pi$$
−0.807938 + 0.589267i $$0.799417\pi$$
$$510$$ 0 0
$$511$$ −12.0065 −0.531138
$$512$$ 18.3314 0.810141
$$513$$ 0 0
$$514$$ −21.4471 −0.945990
$$515$$ 0 0
$$516$$ 2.56774 0.113039
$$517$$ 1.49391 0.0657021
$$518$$ 3.52803 0.155013
$$519$$ 4.01552 0.176262
$$520$$ 0 0
$$521$$ −22.6092 −0.990528 −0.495264 0.868742i $$-0.664929\pi$$
−0.495264 + 0.868742i $$0.664929\pi$$
$$522$$ 15.5798 0.681910
$$523$$ −0.532911 −0.0233026 −0.0116513 0.999932i $$-0.503709\pi$$
−0.0116513 + 0.999932i $$0.503709\pi$$
$$524$$ −2.29654 −0.100325
$$525$$ 0 0
$$526$$ 9.67238 0.421736
$$527$$ 10.9055 0.475051
$$528$$ −0.0905303 −0.00393983
$$529$$ 5.83518 0.253703
$$530$$ 0 0
$$531$$ 19.6976 0.854804
$$532$$ 0 0
$$533$$ −37.3606 −1.61827
$$534$$ −4.05308 −0.175394
$$535$$ 0 0
$$536$$ 9.06156 0.391400
$$537$$ −5.53919 −0.239034
$$538$$ −0.316460 −0.0136436
$$539$$ −0.459150 −0.0197770
$$540$$ 0 0
$$541$$ 5.01640 0.215672 0.107836 0.994169i $$-0.465608\pi$$
0.107836 + 0.994169i $$0.465608\pi$$
$$542$$ 27.2946 1.17240
$$543$$ −2.06077 −0.0884362
$$544$$ −6.95610 −0.298240
$$545$$ 0 0
$$546$$ −2.76909 −0.118506
$$547$$ −22.6299 −0.967584 −0.483792 0.875183i $$-0.660741\pi$$
−0.483792 + 0.875183i $$0.660741\pi$$
$$548$$ −14.9830 −0.640040
$$549$$ −26.2073 −1.11850
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −6.24946 −0.265995
$$553$$ 2.77766 0.118118
$$554$$ −4.83406 −0.205380
$$555$$ 0 0
$$556$$ 14.4641 0.613416
$$557$$ −35.2554 −1.49382 −0.746910 0.664925i $$-0.768463\pi$$
−0.746910 + 0.664925i $$0.768463\pi$$
$$558$$ 20.5764 0.871070
$$559$$ 29.8943 1.26439
$$560$$ 0 0
$$561$$ 0.0846935 0.00357576
$$562$$ 35.9097 1.51476
$$563$$ 24.6295 1.03801 0.519005 0.854771i $$-0.326302\pi$$
0.519005 + 0.854771i $$0.326302\pi$$
$$564$$ −3.35473 −0.141260
$$565$$ 0 0
$$566$$ −1.45841 −0.0613017
$$567$$ −14.6252 −0.614200
$$568$$ 4.07542 0.171001
$$569$$ −20.0193 −0.839252 −0.419626 0.907697i $$-0.637839\pi$$
−0.419626 + 0.907697i $$0.637839\pi$$
$$570$$ 0 0
$$571$$ −16.6121 −0.695195 −0.347597 0.937644i $$-0.613002\pi$$
−0.347597 + 0.937644i $$0.613002\pi$$
$$572$$ 0.375732 0.0157102
$$573$$ 7.76715 0.324477
$$574$$ 22.0898 0.922010
$$575$$ 0 0
$$576$$ −23.2680 −0.969500
$$577$$ −12.4486 −0.518244 −0.259122 0.965845i $$-0.583433\pi$$
−0.259122 + 0.965845i $$0.583433\pi$$
$$578$$ 15.6279 0.650034
$$579$$ −4.18159 −0.173781
$$580$$ 0 0
$$581$$ 14.0984 0.584899
$$582$$ 7.27224 0.301444
$$583$$ 0.0355955 0.00147421
$$584$$ 19.4586 0.805202
$$585$$ 0 0
$$586$$ −8.47623 −0.350150
$$587$$ −4.51144 −0.186207 −0.0931036 0.995656i $$-0.529679\pi$$
−0.0931036 + 0.995656i $$0.529679\pi$$
$$588$$ 1.03107 0.0425206
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 7.53101 0.309784
$$592$$ −3.01550 −0.123936
$$593$$ −19.6082 −0.805213 −0.402606 0.915373i $$-0.631896\pi$$
−0.402606 + 0.915373i $$0.631896\pi$$
$$594$$ 0.327650 0.0134436
$$595$$ 0 0
$$596$$ 17.7062 0.725274
$$597$$ 7.96699 0.326067
$$598$$ −20.7067 −0.846759
$$599$$ 10.7759 0.440292 0.220146 0.975467i $$-0.429347\pi$$
0.220146 + 0.975467i $$0.429347\pi$$
$$600$$ 0 0
$$601$$ 15.0244 0.612860 0.306430 0.951893i $$-0.400866\pi$$
0.306430 + 0.951893i $$0.400866\pi$$
$$602$$ −17.6753 −0.720389
$$603$$ −8.43563 −0.343526
$$604$$ −2.65328 −0.107960
$$605$$ 0 0
$$606$$ −2.24687 −0.0912730
$$607$$ −25.1901 −1.02243 −0.511217 0.859452i $$-0.670805\pi$$
−0.511217 + 0.859452i $$0.670805\pi$$
$$608$$ 0 0
$$609$$ −3.56940 −0.144640
$$610$$ 0 0
$$611$$ −39.0566 −1.58006
$$612$$ 3.77498 0.152595
$$613$$ −16.2351 −0.655728 −0.327864 0.944725i $$-0.606329\pi$$
−0.327864 + 0.944725i $$0.606329\pi$$
$$614$$ −9.98672 −0.403031
$$615$$ 0 0
$$616$$ −0.780593 −0.0314510
$$617$$ −6.51826 −0.262415 −0.131208 0.991355i $$-0.541885\pi$$
−0.131208 + 0.991355i $$0.541885\pi$$
$$618$$ 0.893993 0.0359617
$$619$$ −4.39112 −0.176494 −0.0882470 0.996099i $$-0.528126\pi$$
−0.0882470 + 0.996099i $$0.528126\pi$$
$$620$$ 0 0
$$621$$ 11.9287 0.478681
$$622$$ 13.6706 0.548142
$$623$$ −18.4311 −0.738425
$$624$$ 2.36681 0.0947483
$$625$$ 0 0
$$626$$ 2.24484 0.0897220
$$627$$ 0 0
$$628$$ 5.77858 0.230590
$$629$$ 2.82108 0.112484
$$630$$ 0 0
$$631$$ −34.6209 −1.37824 −0.689118 0.724650i $$-0.742002\pi$$
−0.689118 + 0.724650i $$0.742002\pi$$
$$632$$ −4.50165 −0.179066
$$633$$ −4.86521 −0.193375
$$634$$ 25.8680 1.02735
$$635$$ 0 0
$$636$$ −0.0799333 −0.00316956
$$637$$ 12.0040 0.475614
$$638$$ −0.733141 −0.0290253
$$639$$ −3.79391 −0.150085
$$640$$ 0 0
$$641$$ 7.41241 0.292773 0.146386 0.989227i $$-0.453236\pi$$
0.146386 + 0.989227i $$0.453236\pi$$
$$642$$ −0.416681 −0.0164451
$$643$$ 16.5459 0.652506 0.326253 0.945282i $$-0.394214\pi$$
0.326253 + 0.945282i $$0.394214\pi$$
$$644$$ −8.08794 −0.318710
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −29.4822 −1.15907 −0.579533 0.814949i $$-0.696765\pi$$
−0.579533 + 0.814949i $$0.696765\pi$$
$$648$$ 23.7026 0.931124
$$649$$ −0.926912 −0.0363845
$$650$$ 0 0
$$651$$ −4.71415 −0.184762
$$652$$ −15.7362 −0.616277
$$653$$ −6.57421 −0.257269 −0.128634 0.991692i $$-0.541059\pi$$
−0.128634 + 0.991692i $$0.541059\pi$$
$$654$$ −6.77017 −0.264735
$$655$$ 0 0
$$656$$ −18.8807 −0.737169
$$657$$ −18.1145 −0.706713
$$658$$ 23.0925 0.900241
$$659$$ −26.3370 −1.02594 −0.512972 0.858405i $$-0.671456\pi$$
−0.512972 + 0.858405i $$0.671456\pi$$
$$660$$ 0 0
$$661$$ 3.78419 0.147188 0.0735941 0.997288i $$-0.476553\pi$$
0.0735941 + 0.997288i $$0.476553\pi$$
$$662$$ 20.5413 0.798359
$$663$$ −2.21422 −0.0859930
$$664$$ −22.8487 −0.886703
$$665$$ 0 0
$$666$$ 5.32281 0.206255
$$667$$ −26.6913 −1.03349
$$668$$ −2.70769 −0.104763
$$669$$ −7.73630 −0.299103
$$670$$ 0 0
$$671$$ 1.23324 0.0476086
$$672$$ 3.00694 0.115995
$$673$$ 21.0431 0.811150 0.405575 0.914062i $$-0.367071\pi$$
0.405575 + 0.914062i $$0.367071\pi$$
$$674$$ −35.8731 −1.38178
$$675$$ 0 0
$$676$$ 0.520083 0.0200032
$$677$$ 15.2744 0.587043 0.293522 0.955952i $$-0.405173\pi$$
0.293522 + 0.955952i $$0.405173\pi$$
$$678$$ 0.350941 0.0134778
$$679$$ 33.0699 1.26911
$$680$$ 0 0
$$681$$ −9.64161 −0.369467
$$682$$ −0.968267 −0.0370769
$$683$$ 8.60225 0.329156 0.164578 0.986364i $$-0.447374\pi$$
0.164578 + 0.986364i $$0.447374\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −21.6401 −0.826223
$$687$$ 0.840424 0.0320642
$$688$$ 15.1075 0.575968
$$689$$ −0.930603 −0.0354532
$$690$$ 0 0
$$691$$ 34.1079 1.29753 0.648763 0.760990i $$-0.275286\pi$$
0.648763 + 0.760990i $$0.275286\pi$$
$$692$$ −8.42231 −0.320168
$$693$$ 0.726674 0.0276040
$$694$$ −2.81675 −0.106922
$$695$$ 0 0
$$696$$ 5.78481 0.219273
$$697$$ 17.6634 0.669050
$$698$$ −18.2388 −0.690350
$$699$$ 5.37047 0.203130
$$700$$ 0 0
$$701$$ 10.7293 0.405239 0.202619 0.979258i $$-0.435055\pi$$
0.202619 + 0.979258i $$0.435055\pi$$
$$702$$ −8.56603 −0.323304
$$703$$ 0 0
$$704$$ 1.09492 0.0412665
$$705$$ 0 0
$$706$$ 31.1266 1.17146
$$707$$ −10.2175 −0.384267
$$708$$ 2.08148 0.0782267
$$709$$ −28.8476 −1.08339 −0.541697 0.840574i $$-0.682218\pi$$
−0.541697 + 0.840574i $$0.682218\pi$$
$$710$$ 0 0
$$711$$ 4.19070 0.157164
$$712$$ 29.8706 1.11945
$$713$$ −35.2515 −1.32018
$$714$$ 1.30917 0.0489946
$$715$$ 0 0
$$716$$ 11.6181 0.434189
$$717$$ 1.14360 0.0427085
$$718$$ −19.0829 −0.712166
$$719$$ −20.0555 −0.747944 −0.373972 0.927440i $$-0.622004\pi$$
−0.373972 + 0.927440i $$0.622004\pi$$
$$720$$ 0 0
$$721$$ 4.06535 0.151402
$$722$$ 0 0
$$723$$ −9.02026 −0.335467
$$724$$ 4.32234 0.160639
$$725$$ 0 0
$$726$$ 4.57173 0.169673
$$727$$ 0.780521 0.0289479 0.0144740 0.999895i $$-0.495393\pi$$
0.0144740 + 0.999895i $$0.495393\pi$$
$$728$$ 20.4077 0.756361
$$729$$ −19.5373 −0.723604
$$730$$ 0 0
$$731$$ −14.1335 −0.522745
$$732$$ −2.76937 −0.102359
$$733$$ 26.8391 0.991326 0.495663 0.868515i $$-0.334925\pi$$
0.495663 + 0.868515i $$0.334925\pi$$
$$734$$ 28.3192 1.04528
$$735$$ 0 0
$$736$$ 22.4853 0.828817
$$737$$ 0.396956 0.0146221
$$738$$ 33.3273 1.22679
$$739$$ 20.6530 0.759735 0.379867 0.925041i $$-0.375970\pi$$
0.379867 + 0.925041i $$0.375970\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0.550227 0.0201995
$$743$$ −1.47317 −0.0540454 −0.0270227 0.999635i $$-0.508603\pi$$
−0.0270227 + 0.999635i $$0.508603\pi$$
$$744$$ 7.64007 0.280098
$$745$$ 0 0
$$746$$ 29.7036 1.08752
$$747$$ 21.2705 0.778245
$$748$$ −0.177639 −0.00649514
$$749$$ −1.89482 −0.0692352
$$750$$ 0 0
$$751$$ 15.5624 0.567881 0.283940 0.958842i $$-0.408358\pi$$
0.283940 + 0.958842i $$0.408358\pi$$
$$752$$ −19.7378 −0.719764
$$753$$ 6.52071 0.237628
$$754$$ 19.1671 0.698026
$$755$$ 0 0
$$756$$ −3.34586 −0.121688
$$757$$ −20.1756 −0.733295 −0.366647 0.930360i $$-0.619494\pi$$
−0.366647 + 0.930360i $$0.619494\pi$$
$$758$$ −13.6114 −0.494387
$$759$$ −0.273767 −0.00993713
$$760$$ 0 0
$$761$$ −15.1076 −0.547649 −0.273825 0.961780i $$-0.588289\pi$$
−0.273825 + 0.961780i $$0.588289\pi$$
$$762$$ 7.80124 0.282609
$$763$$ −30.7868 −1.11456
$$764$$ −16.2911 −0.589392
$$765$$ 0 0
$$766$$ −5.88163 −0.212512
$$767$$ 24.2331 0.875005
$$768$$ −5.94509 −0.214525
$$769$$ 51.6580 1.86284 0.931418 0.363951i $$-0.118573\pi$$
0.931418 + 0.363951i $$0.118573\pi$$
$$770$$ 0 0
$$771$$ 7.41327 0.266982
$$772$$ 8.77063 0.315662
$$773$$ −30.1558 −1.08463 −0.542314 0.840176i $$-0.682452\pi$$
−0.542314 + 0.840176i $$0.682452\pi$$
$$774$$ −26.6670 −0.958525
$$775$$ 0 0
$$776$$ −53.5952 −1.92396
$$777$$ −1.21948 −0.0437485
$$778$$ −9.40431 −0.337161
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0.178530 0.00638832
$$782$$ 9.78974 0.350080
$$783$$ −11.0418 −0.394601
$$784$$ 6.06637 0.216656
$$785$$ 0 0
$$786$$ −1.20161 −0.0428601
$$787$$ −43.0969 −1.53624 −0.768119 0.640307i $$-0.778807\pi$$
−0.768119 + 0.640307i $$0.778807\pi$$
$$788$$ −15.7958 −0.562703
$$789$$ −3.34330 −0.119025
$$790$$ 0 0
$$791$$ 1.59588 0.0567428
$$792$$ −1.17770 −0.0418476
$$793$$ −32.2416 −1.14493
$$794$$ −11.6817 −0.414569
$$795$$ 0 0
$$796$$ −16.7103 −0.592280
$$797$$ 50.5062 1.78902 0.894510 0.447048i $$-0.147525\pi$$
0.894510 + 0.447048i $$0.147525\pi$$
$$798$$ 0 0
$$799$$ 18.4652 0.653253
$$800$$ 0 0
$$801$$ −27.8073 −0.982522
$$802$$ −8.39769 −0.296533
$$803$$ 0.852414 0.0300810
$$804$$ −0.891406 −0.0314375
$$805$$ 0 0
$$806$$ 25.3142 0.891656
$$807$$ 0.109386 0.00385056
$$808$$ 16.5591 0.582547
$$809$$ −30.0872 −1.05781 −0.528905 0.848681i $$-0.677397\pi$$
−0.528905 + 0.848681i $$0.677397\pi$$
$$810$$ 0 0
$$811$$ 22.9509 0.805916 0.402958 0.915218i $$-0.367982\pi$$
0.402958 + 0.915218i $$0.367982\pi$$
$$812$$ 7.48661 0.262728
$$813$$ −9.43449 −0.330882
$$814$$ −0.250476 −0.00877917
$$815$$ 0 0
$$816$$ −1.11899 −0.0391723
$$817$$ 0 0
$$818$$ 19.4123 0.678734
$$819$$ −18.9981 −0.663846
$$820$$ 0 0
$$821$$ −38.0807 −1.32903 −0.664513 0.747277i $$-0.731361\pi$$
−0.664513 + 0.747277i $$0.731361\pi$$
$$822$$ −7.83950 −0.273434
$$823$$ −53.7932 −1.87511 −0.937556 0.347835i $$-0.886917\pi$$
−0.937556 + 0.347835i $$0.886917\pi$$
$$824$$ −6.58858 −0.229524
$$825$$ 0 0
$$826$$ −14.3280 −0.498535
$$827$$ −32.4092 −1.12698 −0.563490 0.826123i $$-0.690542\pi$$
−0.563490 + 0.826123i $$0.690542\pi$$
$$828$$ −12.2024 −0.424064
$$829$$ −18.5305 −0.643592 −0.321796 0.946809i $$-0.604287\pi$$
−0.321796 + 0.946809i $$0.604287\pi$$
$$830$$ 0 0
$$831$$ 1.67091 0.0579633
$$832$$ −28.6255 −0.992412
$$833$$ −5.67525 −0.196636
$$834$$ 7.56804 0.262060
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −14.5830 −0.504062
$$838$$ −1.30301 −0.0450118
$$839$$ 0.826608 0.0285377 0.0142688 0.999898i $$-0.495458\pi$$
0.0142688 + 0.999898i $$0.495458\pi$$
$$840$$ 0 0
$$841$$ −4.29321 −0.148042
$$842$$ −36.6185 −1.26196
$$843$$ −12.4123 −0.427504
$$844$$ 10.2045 0.351253
$$845$$ 0 0
$$846$$ 34.8401 1.19783
$$847$$ 20.7895 0.714337
$$848$$ −0.470294 −0.0161500
$$849$$ 0.504106 0.0173009
$$850$$ 0 0
$$851$$ −9.11901 −0.312596
$$852$$ −0.400908 −0.0137349
$$853$$ 7.34938 0.251638 0.125819 0.992053i $$-0.459844\pi$$
0.125819 + 0.992053i $$0.459844\pi$$
$$854$$ 19.0631 0.652327
$$855$$ 0 0
$$856$$ 3.07087 0.104960
$$857$$ 45.3496 1.54911 0.774556 0.632506i $$-0.217974\pi$$
0.774556 + 0.632506i $$0.217974\pi$$
$$858$$ 0.196594 0.00671160
$$859$$ 29.6285 1.01091 0.505456 0.862852i $$-0.331324\pi$$
0.505456 + 0.862852i $$0.331324\pi$$
$$860$$ 0 0
$$861$$ −7.63542 −0.260215
$$862$$ 6.77775 0.230851
$$863$$ −41.0807 −1.39840 −0.699201 0.714925i $$-0.746461\pi$$
−0.699201 + 0.714925i $$0.746461\pi$$
$$864$$ 9.30180 0.316454
$$865$$ 0 0
$$866$$ −20.3632 −0.691970
$$867$$ −5.40183 −0.183456
$$868$$ 9.88764 0.335608
$$869$$ −0.197202 −0.00668962
$$870$$ 0 0
$$871$$ −10.3780 −0.351644
$$872$$ 49.8951 1.68966
$$873$$ 49.8931 1.68863
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −1.91419 −0.0646743
$$877$$ −54.6307 −1.84475 −0.922374 0.386298i $$-0.873754\pi$$
−0.922374 + 0.386298i $$0.873754\pi$$
$$878$$ 0.249460 0.00841888
$$879$$ 2.92984 0.0988211
$$880$$ 0 0
$$881$$ −15.4805 −0.521552 −0.260776 0.965399i $$-0.583978\pi$$
−0.260776 + 0.965399i $$0.583978\pi$$
$$882$$ −10.7080 −0.360558
$$883$$ 45.3495 1.52613 0.763066 0.646321i $$-0.223693\pi$$
0.763066 + 0.646321i $$0.223693\pi$$
$$884$$ 4.64418 0.156201
$$885$$ 0 0
$$886$$ −38.3913 −1.28978
$$887$$ 31.1399 1.04558 0.522788 0.852463i $$-0.324892\pi$$
0.522788 + 0.852463i $$0.324892\pi$$
$$888$$ 1.97637 0.0663226
$$889$$ 35.4755 1.18981
$$890$$ 0 0
$$891$$ 1.03833 0.0347853
$$892$$ 16.2264 0.543301
$$893$$ 0 0
$$894$$ 9.26438 0.309847
$$895$$ 0 0
$$896$$ 1.07133 0.0357905
$$897$$ 7.15734 0.238977
$$898$$ 18.6025 0.620775
$$899$$ 32.6305 1.08829
$$900$$ 0 0
$$901$$ 0.439972 0.0146576
$$902$$ −1.56828 −0.0522181
$$903$$ 6.10952 0.203312
$$904$$ −2.58638 −0.0860218
$$905$$ 0 0
$$906$$ −1.38827 −0.0461222
$$907$$ −43.5415 −1.44577 −0.722886 0.690967i $$-0.757185\pi$$
−0.722886 + 0.690967i $$0.757185\pi$$
$$908$$ 20.2227 0.671113
$$909$$ −15.4153 −0.511293
$$910$$ 0 0
$$911$$ 5.72789 0.189774 0.0948868 0.995488i $$-0.469751\pi$$
0.0948868 + 0.995488i $$0.469751\pi$$
$$912$$ 0 0
$$913$$ −1.00093 −0.0331258
$$914$$ 1.75855 0.0581677
$$915$$ 0 0
$$916$$ −1.76274 −0.0582425
$$917$$ −5.46422 −0.180445
$$918$$ 4.04986 0.133665
$$919$$ 7.93860 0.261870 0.130935 0.991391i $$-0.458202\pi$$
0.130935 + 0.991391i $$0.458202\pi$$
$$920$$ 0 0
$$921$$ 3.45195 0.113746
$$922$$ 9.59339 0.315941
$$923$$ −4.66747 −0.153632
$$924$$ 0.0767887 0.00252616
$$925$$ 0 0
$$926$$ 23.2532 0.764147
$$927$$ 6.13347 0.201450
$$928$$ −20.8135 −0.683236
$$929$$ −28.9567 −0.950039 −0.475019 0.879975i $$-0.657559\pi$$
−0.475019 + 0.879975i $$0.657559\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −11.2642 −0.368972
$$933$$ −4.72531 −0.154700
$$934$$ 22.4444 0.734403
$$935$$ 0 0
$$936$$ 30.7895 1.00639
$$937$$ 14.6816 0.479627 0.239813 0.970819i $$-0.422914\pi$$
0.239813 + 0.970819i $$0.422914\pi$$
$$938$$ 6.13606 0.200349
$$939$$ −0.775939 −0.0253218
$$940$$ 0 0
$$941$$ −0.154755 −0.00504485 −0.00252243 0.999997i $$-0.500803\pi$$
−0.00252243 + 0.999997i $$0.500803\pi$$
$$942$$ 3.02352 0.0985114
$$943$$ −57.0961 −1.85931
$$944$$ 12.2465 0.398590
$$945$$ 0 0
$$946$$ 1.25487 0.0407993
$$947$$ 7.51807 0.244304 0.122152 0.992511i $$-0.461020\pi$$
0.122152 + 0.992511i $$0.461020\pi$$
$$948$$ 0.442838 0.0143827
$$949$$ −22.2854 −0.723415
$$950$$ 0 0
$$951$$ −8.94137 −0.289944
$$952$$ −9.64840 −0.312706
$$953$$ −22.6743 −0.734493 −0.367247 0.930124i $$-0.619700\pi$$
−0.367247 + 0.930124i $$0.619700\pi$$
$$954$$ 0.830138 0.0268767
$$955$$ 0 0
$$956$$ −2.39863 −0.0775772
$$957$$ 0.253413 0.00819167
$$958$$ −25.8437 −0.834973
$$959$$ −35.6494 −1.15118
$$960$$ 0 0
$$961$$ 12.0955 0.390177
$$962$$ 6.54840 0.211129
$$963$$ −2.85875 −0.0921219
$$964$$ 18.9194 0.609354
$$965$$ 0 0
$$966$$ −4.23184 −0.136157
$$967$$ −28.8344 −0.927253 −0.463627 0.886031i $$-0.653452\pi$$
−0.463627 + 0.886031i $$0.653452\pi$$
$$968$$ −33.6929 −1.08293
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 26.3661 0.846130 0.423065 0.906099i $$-0.360954\pi$$
0.423065 + 0.906099i $$0.360954\pi$$
$$972$$ −7.63395 −0.244859
$$973$$ 34.4150 1.10329
$$974$$ 39.5508 1.26729
$$975$$ 0 0
$$976$$ −16.2938 −0.521551
$$977$$ −4.59218 −0.146917 −0.0734585 0.997298i $$-0.523404\pi$$
−0.0734585 + 0.997298i $$0.523404\pi$$
$$978$$ −8.23362 −0.263282
$$979$$ 1.30853 0.0418207
$$980$$ 0 0
$$981$$ −46.4486 −1.48299
$$982$$ −22.0140 −0.702496
$$983$$ −6.91473 −0.220546 −0.110273 0.993901i $$-0.535172\pi$$
−0.110273 + 0.993901i $$0.535172\pi$$
$$984$$ 12.3745 0.394484
$$985$$ 0 0
$$986$$ −9.06187 −0.288589
$$987$$ −7.98203 −0.254071
$$988$$ 0 0
$$989$$ 45.6857 1.45272
$$990$$ 0 0
$$991$$ −38.0070 −1.20733 −0.603666 0.797237i $$-0.706294\pi$$
−0.603666 + 0.797237i $$0.706294\pi$$
$$992$$ −27.4886 −0.872763
$$993$$ −7.10017 −0.225317
$$994$$ 2.75968 0.0875318
$$995$$ 0 0
$$996$$ 2.24768 0.0712205
$$997$$ −16.2265 −0.513899 −0.256950 0.966425i $$-0.582717\pi$$
−0.256950 + 0.966425i $$0.582717\pi$$
$$998$$ 40.4657 1.28092
$$999$$ −3.77239 −0.119353
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.bg.1.2 4
5.4 even 2 1805.2.a.o.1.3 4
19.7 even 3 475.2.e.e.201.3 8
19.11 even 3 475.2.e.e.26.3 8
19.18 odd 2 9025.2.a.bp.1.3 4
95.7 odd 12 475.2.j.c.49.5 16
95.49 even 6 95.2.e.c.26.2 yes 8
95.64 even 6 95.2.e.c.11.2 8
95.68 odd 12 475.2.j.c.349.5 16
95.83 odd 12 475.2.j.c.49.4 16
95.87 odd 12 475.2.j.c.349.4 16
95.94 odd 2 1805.2.a.i.1.2 4
285.239 odd 6 855.2.k.h.406.3 8
285.254 odd 6 855.2.k.h.676.3 8
380.159 odd 6 1520.2.q.o.961.2 8
380.239 odd 6 1520.2.q.o.881.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.c.11.2 8 95.64 even 6
95.2.e.c.26.2 yes 8 95.49 even 6
475.2.e.e.26.3 8 19.11 even 3
475.2.e.e.201.3 8 19.7 even 3
475.2.j.c.49.4 16 95.83 odd 12
475.2.j.c.49.5 16 95.7 odd 12
475.2.j.c.349.4 16 95.87 odd 12
475.2.j.c.349.5 16 95.68 odd 12
855.2.k.h.406.3 8 285.239 odd 6
855.2.k.h.676.3 8 285.254 odd 6
1520.2.q.o.881.2 8 380.239 odd 6
1520.2.q.o.961.2 8 380.159 odd 6
1805.2.a.i.1.2 4 95.94 odd 2
1805.2.a.o.1.3 4 5.4 even 2
9025.2.a.bg.1.2 4 1.1 even 1 trivial
9025.2.a.bp.1.3 4 19.18 odd 2