# Properties

 Label 1875.2.a.p.1.4 Level $1875$ Weight $2$ Character 1875.1 Self dual yes Analytic conductor $14.972$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, 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("1875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1875.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: $$14.9719503790$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.8.5444000000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1$$ x^8 - 4*x^7 - 2*x^6 + 20*x^5 - 4*x^4 - 30*x^3 + 7*x^2 + 12*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-0.0898194$$ of defining polynomial Character $$\chi$$ $$=$$ 1875.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.0898194 q^{2} -1.00000 q^{3} -1.99193 q^{4} +0.0898194 q^{6} +4.36070 q^{7} +0.358553 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.0898194 q^{2} -1.00000 q^{3} -1.99193 q^{4} +0.0898194 q^{6} +4.36070 q^{7} +0.358553 q^{8} +1.00000 q^{9} +4.39094 q^{11} +1.99193 q^{12} -1.98166 q^{13} -0.391676 q^{14} +3.95166 q^{16} +0.997022 q^{17} -0.0898194 q^{18} +1.35096 q^{19} -4.36070 q^{21} -0.394392 q^{22} +2.35651 q^{23} -0.358553 q^{24} +0.177991 q^{26} -1.00000 q^{27} -8.68622 q^{28} -7.97856 q^{29} -3.67761 q^{31} -1.07204 q^{32} -4.39094 q^{33} -0.0895519 q^{34} -1.99193 q^{36} -1.43706 q^{37} -0.121342 q^{38} +1.98166 q^{39} -5.98248 q^{41} +0.391676 q^{42} +2.68554 q^{43} -8.74646 q^{44} -0.211660 q^{46} +10.9393 q^{47} -3.95166 q^{48} +12.0157 q^{49} -0.997022 q^{51} +3.94732 q^{52} +11.0510 q^{53} +0.0898194 q^{54} +1.56354 q^{56} -1.35096 q^{57} +0.716629 q^{58} +6.68895 q^{59} -9.45570 q^{61} +0.330321 q^{62} +4.36070 q^{63} -7.80703 q^{64} +0.394392 q^{66} +12.9219 q^{67} -1.98600 q^{68} -2.35651 q^{69} +7.32257 q^{71} +0.358553 q^{72} -0.424804 q^{73} +0.129076 q^{74} -2.69101 q^{76} +19.1476 q^{77} -0.177991 q^{78} -6.35531 q^{79} +1.00000 q^{81} +0.537343 q^{82} +0.737011 q^{83} +8.68622 q^{84} -0.241213 q^{86} +7.97856 q^{87} +1.57439 q^{88} -9.78736 q^{89} -8.64141 q^{91} -4.69401 q^{92} +3.67761 q^{93} -0.982560 q^{94} +1.07204 q^{96} +0.0337081 q^{97} -1.07925 q^{98} +4.39094 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10})$$ 8 * q + 4 * q^2 - 8 * q^3 + 4 * q^4 - 4 * q^6 + 8 * q^7 + 12 * q^8 + 8 * q^9 $$8 q + 4 q^{2} - 8 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{7} + 12 q^{8} + 8 q^{9} + 2 q^{11} - 4 q^{12} + 16 q^{13} + 6 q^{14} + 16 q^{17} + 4 q^{18} - 14 q^{19} - 8 q^{21} + 12 q^{22} + 14 q^{23} - 12 q^{24} + 6 q^{26} - 8 q^{27} + 16 q^{28} + 2 q^{29} - 22 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{34} + 4 q^{36} + 28 q^{37} - 16 q^{38} - 16 q^{39} + 8 q^{41} - 6 q^{42} + 20 q^{43} + 22 q^{44} - 2 q^{46} + 10 q^{47} - 16 q^{51} + 16 q^{52} + 44 q^{53} - 4 q^{54} + 30 q^{56} + 14 q^{57} + 8 q^{58} + 14 q^{59} - 20 q^{61} + 16 q^{62} + 8 q^{63} + 6 q^{64} - 12 q^{66} + 16 q^{67} - 2 q^{68} - 14 q^{69} + 16 q^{71} + 12 q^{72} + 24 q^{73} + 26 q^{74} - 16 q^{76} + 46 q^{77} - 6 q^{78} - 30 q^{79} + 8 q^{81} + 16 q^{82} + 12 q^{83} - 16 q^{84} + 32 q^{86} - 2 q^{87} + 32 q^{88} + 16 q^{89} - 12 q^{91} - 2 q^{92} + 22 q^{93} + 14 q^{94} + 2 q^{96} + 16 q^{97} + 4 q^{98} + 2 q^{99}+O(q^{100})$$ 8 * q + 4 * q^2 - 8 * q^3 + 4 * q^4 - 4 * q^6 + 8 * q^7 + 12 * q^8 + 8 * q^9 + 2 * q^11 - 4 * q^12 + 16 * q^13 + 6 * q^14 + 16 * q^17 + 4 * q^18 - 14 * q^19 - 8 * q^21 + 12 * q^22 + 14 * q^23 - 12 * q^24 + 6 * q^26 - 8 * q^27 + 16 * q^28 + 2 * q^29 - 22 * q^31 - 2 * q^32 - 2 * q^33 - 12 * q^34 + 4 * q^36 + 28 * q^37 - 16 * q^38 - 16 * q^39 + 8 * q^41 - 6 * q^42 + 20 * q^43 + 22 * q^44 - 2 * q^46 + 10 * q^47 - 16 * q^51 + 16 * q^52 + 44 * q^53 - 4 * q^54 + 30 * q^56 + 14 * q^57 + 8 * q^58 + 14 * q^59 - 20 * q^61 + 16 * q^62 + 8 * q^63 + 6 * q^64 - 12 * q^66 + 16 * q^67 - 2 * q^68 - 14 * q^69 + 16 * q^71 + 12 * q^72 + 24 * q^73 + 26 * q^74 - 16 * q^76 + 46 * q^77 - 6 * q^78 - 30 * q^79 + 8 * q^81 + 16 * q^82 + 12 * q^83 - 16 * q^84 + 32 * q^86 - 2 * q^87 + 32 * q^88 + 16 * q^89 - 12 * q^91 - 2 * q^92 + 22 * q^93 + 14 * q^94 + 2 * q^96 + 16 * q^97 + 4 * q^98 + 2 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.0898194 −0.0635119 −0.0317560 0.999496i $$-0.510110\pi$$
−0.0317560 + 0.999496i $$0.510110\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.99193 −0.995966
$$5$$ 0 0
$$6$$ 0.0898194 0.0366686
$$7$$ 4.36070 1.64819 0.824095 0.566451i $$-0.191684\pi$$
0.824095 + 0.566451i $$0.191684\pi$$
$$8$$ 0.358553 0.126768
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.39094 1.32392 0.661959 0.749540i $$-0.269725\pi$$
0.661959 + 0.749540i $$0.269725\pi$$
$$12$$ 1.99193 0.575021
$$13$$ −1.98166 −0.549612 −0.274806 0.961500i $$-0.588614\pi$$
−0.274806 + 0.961500i $$0.588614\pi$$
$$14$$ −0.391676 −0.104680
$$15$$ 0 0
$$16$$ 3.95166 0.987915
$$17$$ 0.997022 0.241813 0.120907 0.992664i $$-0.461420\pi$$
0.120907 + 0.992664i $$0.461420\pi$$
$$18$$ −0.0898194 −0.0211706
$$19$$ 1.35096 0.309931 0.154965 0.987920i $$-0.450473\pi$$
0.154965 + 0.987920i $$0.450473\pi$$
$$20$$ 0 0
$$21$$ −4.36070 −0.951583
$$22$$ −0.394392 −0.0840846
$$23$$ 2.35651 0.491366 0.245683 0.969350i $$-0.420988\pi$$
0.245683 + 0.969350i $$0.420988\pi$$
$$24$$ −0.358553 −0.0731893
$$25$$ 0 0
$$26$$ 0.177991 0.0349069
$$27$$ −1.00000 −0.192450
$$28$$ −8.68622 −1.64154
$$29$$ −7.97856 −1.48158 −0.740790 0.671736i $$-0.765549\pi$$
−0.740790 + 0.671736i $$0.765549\pi$$
$$30$$ 0 0
$$31$$ −3.67761 −0.660519 −0.330259 0.943890i $$-0.607136\pi$$
−0.330259 + 0.943890i $$0.607136\pi$$
$$32$$ −1.07204 −0.189512
$$33$$ −4.39094 −0.764365
$$34$$ −0.0895519 −0.0153580
$$35$$ 0 0
$$36$$ −1.99193 −0.331989
$$37$$ −1.43706 −0.236251 −0.118125 0.992999i $$-0.537688\pi$$
−0.118125 + 0.992999i $$0.537688\pi$$
$$38$$ −0.121342 −0.0196843
$$39$$ 1.98166 0.317319
$$40$$ 0 0
$$41$$ −5.98248 −0.934306 −0.467153 0.884177i $$-0.654720\pi$$
−0.467153 + 0.884177i $$0.654720\pi$$
$$42$$ 0.391676 0.0604369
$$43$$ 2.68554 0.409541 0.204770 0.978810i $$-0.434355\pi$$
0.204770 + 0.978810i $$0.434355\pi$$
$$44$$ −8.74646 −1.31858
$$45$$ 0 0
$$46$$ −0.211660 −0.0312076
$$47$$ 10.9393 1.59566 0.797829 0.602883i $$-0.205982\pi$$
0.797829 + 0.602883i $$0.205982\pi$$
$$48$$ −3.95166 −0.570373
$$49$$ 12.0157 1.71653
$$50$$ 0 0
$$51$$ −0.997022 −0.139611
$$52$$ 3.94732 0.547395
$$53$$ 11.0510 1.51798 0.758989 0.651104i $$-0.225694\pi$$
0.758989 + 0.651104i $$0.225694\pi$$
$$54$$ 0.0898194 0.0122229
$$55$$ 0 0
$$56$$ 1.56354 0.208937
$$57$$ −1.35096 −0.178938
$$58$$ 0.716629 0.0940980
$$59$$ 6.68895 0.870827 0.435414 0.900231i $$-0.356602\pi$$
0.435414 + 0.900231i $$0.356602\pi$$
$$60$$ 0 0
$$61$$ −9.45570 −1.21068 −0.605339 0.795967i $$-0.706963\pi$$
−0.605339 + 0.795967i $$0.706963\pi$$
$$62$$ 0.330321 0.0419508
$$63$$ 4.36070 0.549397
$$64$$ −7.80703 −0.975879
$$65$$ 0 0
$$66$$ 0.394392 0.0485463
$$67$$ 12.9219 1.57866 0.789328 0.613972i $$-0.210429\pi$$
0.789328 + 0.613972i $$0.210429\pi$$
$$68$$ −1.98600 −0.240838
$$69$$ −2.35651 −0.283690
$$70$$ 0 0
$$71$$ 7.32257 0.869029 0.434515 0.900665i $$-0.356920\pi$$
0.434515 + 0.900665i $$0.356920\pi$$
$$72$$ 0.358553 0.0422559
$$73$$ −0.424804 −0.0497195 −0.0248598 0.999691i $$-0.507914\pi$$
−0.0248598 + 0.999691i $$0.507914\pi$$
$$74$$ 0.129076 0.0150047
$$75$$ 0 0
$$76$$ −2.69101 −0.308680
$$77$$ 19.1476 2.18207
$$78$$ −0.177991 −0.0201535
$$79$$ −6.35531 −0.715028 −0.357514 0.933908i $$-0.616376\pi$$
−0.357514 + 0.933908i $$0.616376\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0.537343 0.0593396
$$83$$ 0.737011 0.0808975 0.0404487 0.999182i $$-0.487121\pi$$
0.0404487 + 0.999182i $$0.487121\pi$$
$$84$$ 8.68622 0.947745
$$85$$ 0 0
$$86$$ −0.241213 −0.0260107
$$87$$ 7.97856 0.855391
$$88$$ 1.57439 0.167830
$$89$$ −9.78736 −1.03746 −0.518729 0.854939i $$-0.673595\pi$$
−0.518729 + 0.854939i $$0.673595\pi$$
$$90$$ 0 0
$$91$$ −8.64141 −0.905866
$$92$$ −4.69401 −0.489384
$$93$$ 3.67761 0.381351
$$94$$ −0.982560 −0.101343
$$95$$ 0 0
$$96$$ 1.07204 0.109415
$$97$$ 0.0337081 0.00342254 0.00171127 0.999999i $$-0.499455\pi$$
0.00171127 + 0.999999i $$0.499455\pi$$
$$98$$ −1.07925 −0.109020
$$99$$ 4.39094 0.441306
$$100$$ 0 0
$$101$$ 3.19390 0.317805 0.158902 0.987294i $$-0.449204\pi$$
0.158902 + 0.987294i $$0.449204\pi$$
$$102$$ 0.0895519 0.00886696
$$103$$ 8.55342 0.842794 0.421397 0.906876i $$-0.361540\pi$$
0.421397 + 0.906876i $$0.361540\pi$$
$$104$$ −0.710529 −0.0696731
$$105$$ 0 0
$$106$$ −0.992598 −0.0964097
$$107$$ 2.22136 0.214747 0.107373 0.994219i $$-0.465756\pi$$
0.107373 + 0.994219i $$0.465756\pi$$
$$108$$ 1.99193 0.191674
$$109$$ 11.0023 1.05383 0.526916 0.849917i $$-0.323348\pi$$
0.526916 + 0.849917i $$0.323348\pi$$
$$110$$ 0 0
$$111$$ 1.43706 0.136399
$$112$$ 17.2320 1.62827
$$113$$ 1.71021 0.160883 0.0804415 0.996759i $$-0.474367\pi$$
0.0804415 + 0.996759i $$0.474367\pi$$
$$114$$ 0.121342 0.0113647
$$115$$ 0 0
$$116$$ 15.8927 1.47560
$$117$$ −1.98166 −0.183204
$$118$$ −0.600798 −0.0553079
$$119$$ 4.34771 0.398554
$$120$$ 0 0
$$121$$ 8.28037 0.752761
$$122$$ 0.849306 0.0768925
$$123$$ 5.98248 0.539422
$$124$$ 7.32556 0.657855
$$125$$ 0 0
$$126$$ −0.391676 −0.0348933
$$127$$ −12.5570 −1.11425 −0.557125 0.830429i $$-0.688095\pi$$
−0.557125 + 0.830429i $$0.688095\pi$$
$$128$$ 2.84531 0.251492
$$129$$ −2.68554 −0.236448
$$130$$ 0 0
$$131$$ 16.4718 1.43915 0.719574 0.694416i $$-0.244337\pi$$
0.719574 + 0.694416i $$0.244337\pi$$
$$132$$ 8.74646 0.761282
$$133$$ 5.89112 0.510825
$$134$$ −1.16063 −0.100263
$$135$$ 0 0
$$136$$ 0.357485 0.0306541
$$137$$ −9.66732 −0.825935 −0.412967 0.910746i $$-0.635508\pi$$
−0.412967 + 0.910746i $$0.635508\pi$$
$$138$$ 0.211660 0.0180177
$$139$$ 13.5327 1.14783 0.573913 0.818916i $$-0.305425\pi$$
0.573913 + 0.818916i $$0.305425\pi$$
$$140$$ 0 0
$$141$$ −10.9393 −0.921254
$$142$$ −0.657709 −0.0551937
$$143$$ −8.70133 −0.727642
$$144$$ 3.95166 0.329305
$$145$$ 0 0
$$146$$ 0.0381557 0.00315778
$$147$$ −12.0157 −0.991040
$$148$$ 2.86252 0.235298
$$149$$ 13.6843 1.12106 0.560529 0.828134i $$-0.310598\pi$$
0.560529 + 0.828134i $$0.310598\pi$$
$$150$$ 0 0
$$151$$ −11.3204 −0.921237 −0.460619 0.887598i $$-0.652372\pi$$
−0.460619 + 0.887598i $$0.652372\pi$$
$$152$$ 0.484389 0.0392892
$$153$$ 0.997022 0.0806044
$$154$$ −1.71983 −0.138587
$$155$$ 0 0
$$156$$ −3.94732 −0.316039
$$157$$ 8.56070 0.683219 0.341609 0.939842i $$-0.389028\pi$$
0.341609 + 0.939842i $$0.389028\pi$$
$$158$$ 0.570830 0.0454128
$$159$$ −11.0510 −0.876405
$$160$$ 0 0
$$161$$ 10.2760 0.809865
$$162$$ −0.0898194 −0.00705688
$$163$$ 4.58509 0.359132 0.179566 0.983746i $$-0.442531\pi$$
0.179566 + 0.983746i $$0.442531\pi$$
$$164$$ 11.9167 0.930537
$$165$$ 0 0
$$166$$ −0.0661979 −0.00513795
$$167$$ 7.21792 0.558540 0.279270 0.960213i $$-0.409908\pi$$
0.279270 + 0.960213i $$0.409908\pi$$
$$168$$ −1.56354 −0.120630
$$169$$ −9.07304 −0.697926
$$170$$ 0 0
$$171$$ 1.35096 0.103310
$$172$$ −5.34941 −0.407889
$$173$$ −4.65009 −0.353540 −0.176770 0.984252i $$-0.556565\pi$$
−0.176770 + 0.984252i $$0.556565\pi$$
$$174$$ −0.716629 −0.0543275
$$175$$ 0 0
$$176$$ 17.3515 1.30792
$$177$$ −6.68895 −0.502772
$$178$$ 0.879095 0.0658909
$$179$$ −7.20338 −0.538406 −0.269203 0.963083i $$-0.586760\pi$$
−0.269203 + 0.963083i $$0.586760\pi$$
$$180$$ 0 0
$$181$$ −3.80424 −0.282767 −0.141383 0.989955i $$-0.545155\pi$$
−0.141383 + 0.989955i $$0.545155\pi$$
$$182$$ 0.776167 0.0575333
$$183$$ 9.45570 0.698986
$$184$$ 0.844934 0.0622893
$$185$$ 0 0
$$186$$ −0.330321 −0.0242203
$$187$$ 4.37786 0.320141
$$188$$ −21.7903 −1.58922
$$189$$ −4.36070 −0.317194
$$190$$ 0 0
$$191$$ 21.5541 1.55960 0.779801 0.626027i $$-0.215320\pi$$
0.779801 + 0.626027i $$0.215320\pi$$
$$192$$ 7.80703 0.563424
$$193$$ 3.15029 0.226763 0.113381 0.993552i $$-0.463832\pi$$
0.113381 + 0.993552i $$0.463832\pi$$
$$194$$ −0.00302764 −0.000217372 0
$$195$$ 0 0
$$196$$ −23.9345 −1.70961
$$197$$ 26.0837 1.85839 0.929195 0.369590i $$-0.120502\pi$$
0.929195 + 0.369590i $$0.120502\pi$$
$$198$$ −0.394392 −0.0280282
$$199$$ −24.2662 −1.72018 −0.860092 0.510139i $$-0.829594\pi$$
−0.860092 + 0.510139i $$0.829594\pi$$
$$200$$ 0 0
$$201$$ −12.9219 −0.911437
$$202$$ −0.286874 −0.0201844
$$203$$ −34.7921 −2.44193
$$204$$ 1.98600 0.139048
$$205$$ 0 0
$$206$$ −0.768264 −0.0535275
$$207$$ 2.35651 0.163789
$$208$$ −7.83083 −0.542970
$$209$$ 5.93197 0.410323
$$210$$ 0 0
$$211$$ 16.3783 1.12753 0.563765 0.825935i $$-0.309352\pi$$
0.563765 + 0.825935i $$0.309352\pi$$
$$212$$ −22.0129 −1.51185
$$213$$ −7.32257 −0.501734
$$214$$ −0.199521 −0.0136390
$$215$$ 0 0
$$216$$ −0.358553 −0.0243964
$$217$$ −16.0370 −1.08866
$$218$$ −0.988224 −0.0669310
$$219$$ 0.424804 0.0287056
$$220$$ 0 0
$$221$$ −1.97575 −0.132904
$$222$$ −0.129076 −0.00866299
$$223$$ −5.68295 −0.380558 −0.190279 0.981730i $$-0.560939\pi$$
−0.190279 + 0.981730i $$0.560939\pi$$
$$224$$ −4.67486 −0.312352
$$225$$ 0 0
$$226$$ −0.153610 −0.0102180
$$227$$ 3.64374 0.241844 0.120922 0.992662i $$-0.461415\pi$$
0.120922 + 0.992662i $$0.461415\pi$$
$$228$$ 2.69101 0.178217
$$229$$ 1.66125 0.109779 0.0548893 0.998492i $$-0.482519\pi$$
0.0548893 + 0.998492i $$0.482519\pi$$
$$230$$ 0 0
$$231$$ −19.1476 −1.25982
$$232$$ −2.86074 −0.187817
$$233$$ 7.85899 0.514860 0.257430 0.966297i $$-0.417124\pi$$
0.257430 + 0.966297i $$0.417124\pi$$
$$234$$ 0.177991 0.0116356
$$235$$ 0 0
$$236$$ −13.3239 −0.867314
$$237$$ 6.35531 0.412821
$$238$$ −0.390509 −0.0253130
$$239$$ −0.567301 −0.0366956 −0.0183478 0.999832i $$-0.505841\pi$$
−0.0183478 + 0.999832i $$0.505841\pi$$
$$240$$ 0 0
$$241$$ −19.0081 −1.22442 −0.612211 0.790695i $$-0.709720\pi$$
−0.612211 + 0.790695i $$0.709720\pi$$
$$242$$ −0.743738 −0.0478093
$$243$$ −1.00000 −0.0641500
$$244$$ 18.8351 1.20580
$$245$$ 0 0
$$246$$ −0.537343 −0.0342597
$$247$$ −2.67713 −0.170342
$$248$$ −1.31862 −0.0837324
$$249$$ −0.737011 −0.0467062
$$250$$ 0 0
$$251$$ 3.02533 0.190957 0.0954787 0.995431i $$-0.469562\pi$$
0.0954787 + 0.995431i $$0.469562\pi$$
$$252$$ −8.68622 −0.547181
$$253$$ 10.3473 0.650529
$$254$$ 1.12786 0.0707681
$$255$$ 0 0
$$256$$ 15.3585 0.959906
$$257$$ −19.8613 −1.23891 −0.619456 0.785032i $$-0.712647\pi$$
−0.619456 + 0.785032i $$0.712647\pi$$
$$258$$ 0.241213 0.0150173
$$259$$ −6.26658 −0.389386
$$260$$ 0 0
$$261$$ −7.97856 −0.493860
$$262$$ −1.47949 −0.0914030
$$263$$ 22.8299 1.40775 0.703876 0.710323i $$-0.251451\pi$$
0.703876 + 0.710323i $$0.251451\pi$$
$$264$$ −1.57439 −0.0968968
$$265$$ 0 0
$$266$$ −0.529137 −0.0324435
$$267$$ 9.78736 0.598977
$$268$$ −25.7395 −1.57229
$$269$$ 14.8324 0.904347 0.452174 0.891930i $$-0.350649\pi$$
0.452174 + 0.891930i $$0.350649\pi$$
$$270$$ 0 0
$$271$$ −6.43720 −0.391032 −0.195516 0.980701i $$-0.562638\pi$$
−0.195516 + 0.980701i $$0.562638\pi$$
$$272$$ 3.93989 0.238891
$$273$$ 8.64141 0.523002
$$274$$ 0.868313 0.0524567
$$275$$ 0 0
$$276$$ 4.69401 0.282546
$$277$$ 6.70976 0.403150 0.201575 0.979473i $$-0.435394\pi$$
0.201575 + 0.979473i $$0.435394\pi$$
$$278$$ −1.21550 −0.0729006
$$279$$ −3.67761 −0.220173
$$280$$ 0 0
$$281$$ −20.4867 −1.22214 −0.611068 0.791578i $$-0.709260\pi$$
−0.611068 + 0.791578i $$0.709260\pi$$
$$282$$ 0.982560 0.0585106
$$283$$ −11.4177 −0.678715 −0.339357 0.940658i $$-0.610210\pi$$
−0.339357 + 0.940658i $$0.610210\pi$$
$$284$$ −14.5861 −0.865524
$$285$$ 0 0
$$286$$ 0.781549 0.0462140
$$287$$ −26.0878 −1.53991
$$288$$ −1.07204 −0.0631707
$$289$$ −16.0059 −0.941526
$$290$$ 0 0
$$291$$ −0.0337081 −0.00197600
$$292$$ 0.846181 0.0495190
$$293$$ 28.5505 1.66794 0.833968 0.551812i $$-0.186063\pi$$
0.833968 + 0.551812i $$0.186063\pi$$
$$294$$ 1.07925 0.0629429
$$295$$ 0 0
$$296$$ −0.515261 −0.0299489
$$297$$ −4.39094 −0.254788
$$298$$ −1.22911 −0.0712006
$$299$$ −4.66979 −0.270061
$$300$$ 0 0
$$301$$ 11.7108 0.675001
$$302$$ 1.01679 0.0585095
$$303$$ −3.19390 −0.183485
$$304$$ 5.33852 0.306185
$$305$$ 0 0
$$306$$ −0.0895519 −0.00511934
$$307$$ −20.5417 −1.17238 −0.586188 0.810175i $$-0.699372\pi$$
−0.586188 + 0.810175i $$0.699372\pi$$
$$308$$ −38.1407 −2.17327
$$309$$ −8.55342 −0.486587
$$310$$ 0 0
$$311$$ 17.5496 0.995146 0.497573 0.867422i $$-0.334225\pi$$
0.497573 + 0.867422i $$0.334225\pi$$
$$312$$ 0.710529 0.0402258
$$313$$ −2.98564 −0.168758 −0.0843790 0.996434i $$-0.526891\pi$$
−0.0843790 + 0.996434i $$0.526891\pi$$
$$314$$ −0.768918 −0.0433925
$$315$$ 0 0
$$316$$ 12.6593 0.712143
$$317$$ −16.1708 −0.908244 −0.454122 0.890940i $$-0.650047\pi$$
−0.454122 + 0.890940i $$0.650047\pi$$
$$318$$ 0.992598 0.0556621
$$319$$ −35.0334 −1.96149
$$320$$ 0 0
$$321$$ −2.22136 −0.123984
$$322$$ −0.922988 −0.0514361
$$323$$ 1.34693 0.0749453
$$324$$ −1.99193 −0.110663
$$325$$ 0 0
$$326$$ −0.411830 −0.0228092
$$327$$ −11.0023 −0.608431
$$328$$ −2.14504 −0.118440
$$329$$ 47.7030 2.62995
$$330$$ 0 0
$$331$$ −13.0705 −0.718418 −0.359209 0.933257i $$-0.616953\pi$$
−0.359209 + 0.933257i $$0.616953\pi$$
$$332$$ −1.46808 −0.0805711
$$333$$ −1.43706 −0.0787502
$$334$$ −0.648310 −0.0354739
$$335$$ 0 0
$$336$$ −17.2320 −0.940083
$$337$$ 26.8049 1.46015 0.730077 0.683365i $$-0.239484\pi$$
0.730077 + 0.683365i $$0.239484\pi$$
$$338$$ 0.814935 0.0443266
$$339$$ −1.71021 −0.0928858
$$340$$ 0 0
$$341$$ −16.1482 −0.874473
$$342$$ −0.121342 −0.00656143
$$343$$ 21.8721 1.18098
$$344$$ 0.962908 0.0519165
$$345$$ 0 0
$$346$$ 0.417669 0.0224540
$$347$$ −25.4859 −1.36815 −0.684077 0.729410i $$-0.739795\pi$$
−0.684077 + 0.729410i $$0.739795\pi$$
$$348$$ −15.8927 −0.851941
$$349$$ −28.0435 −1.50113 −0.750566 0.660795i $$-0.770219\pi$$
−0.750566 + 0.660795i $$0.770219\pi$$
$$350$$ 0 0
$$351$$ 1.98166 0.105773
$$352$$ −4.70727 −0.250899
$$353$$ 14.6667 0.780630 0.390315 0.920681i $$-0.372366\pi$$
0.390315 + 0.920681i $$0.372366\pi$$
$$354$$ 0.600798 0.0319320
$$355$$ 0 0
$$356$$ 19.4958 1.03327
$$357$$ −4.34771 −0.230105
$$358$$ 0.647004 0.0341952
$$359$$ −14.6205 −0.771642 −0.385821 0.922574i $$-0.626082\pi$$
−0.385821 + 0.922574i $$0.626082\pi$$
$$360$$ 0 0
$$361$$ −17.1749 −0.903943
$$362$$ 0.341694 0.0179591
$$363$$ −8.28037 −0.434607
$$364$$ 17.2131 0.902212
$$365$$ 0 0
$$366$$ −0.849306 −0.0443939
$$367$$ −17.6940 −0.923617 −0.461808 0.886980i $$-0.652799\pi$$
−0.461808 + 0.886980i $$0.652799\pi$$
$$368$$ 9.31212 0.485428
$$369$$ −5.98248 −0.311435
$$370$$ 0 0
$$371$$ 48.1903 2.50192
$$372$$ −7.32556 −0.379813
$$373$$ 12.2293 0.633210 0.316605 0.948557i $$-0.397457\pi$$
0.316605 + 0.948557i $$0.397457\pi$$
$$374$$ −0.393217 −0.0203328
$$375$$ 0 0
$$376$$ 3.92231 0.202278
$$377$$ 15.8107 0.814295
$$378$$ 0.391676 0.0201456
$$379$$ 28.0951 1.44315 0.721574 0.692338i $$-0.243419\pi$$
0.721574 + 0.692338i $$0.243419\pi$$
$$380$$ 0 0
$$381$$ 12.5570 0.643312
$$382$$ −1.93598 −0.0990533
$$383$$ 34.4334 1.75947 0.879733 0.475468i $$-0.157721\pi$$
0.879733 + 0.475468i $$0.157721\pi$$
$$384$$ −2.84531 −0.145199
$$385$$ 0 0
$$386$$ −0.282957 −0.0144021
$$387$$ 2.68554 0.136514
$$388$$ −0.0671442 −0.00340873
$$389$$ 13.5225 0.685619 0.342809 0.939405i $$-0.388622\pi$$
0.342809 + 0.939405i $$0.388622\pi$$
$$390$$ 0 0
$$391$$ 2.34949 0.118819
$$392$$ 4.30827 0.217601
$$393$$ −16.4718 −0.830892
$$394$$ −2.34283 −0.118030
$$395$$ 0 0
$$396$$ −8.74646 −0.439526
$$397$$ −35.9744 −1.80550 −0.902751 0.430164i $$-0.858456\pi$$
−0.902751 + 0.430164i $$0.858456\pi$$
$$398$$ 2.17958 0.109252
$$399$$ −5.89112 −0.294925
$$400$$ 0 0
$$401$$ 4.35977 0.217717 0.108858 0.994057i $$-0.465281\pi$$
0.108858 + 0.994057i $$0.465281\pi$$
$$402$$ 1.16063 0.0578871
$$403$$ 7.28776 0.363029
$$404$$ −6.36203 −0.316523
$$405$$ 0 0
$$406$$ 3.12501 0.155092
$$407$$ −6.31003 −0.312777
$$408$$ −0.357485 −0.0176982
$$409$$ −18.1138 −0.895668 −0.447834 0.894117i $$-0.647804\pi$$
−0.447834 + 0.894117i $$0.647804\pi$$
$$410$$ 0 0
$$411$$ 9.66732 0.476854
$$412$$ −17.0378 −0.839394
$$413$$ 29.1685 1.43529
$$414$$ −0.211660 −0.0104025
$$415$$ 0 0
$$416$$ 2.12442 0.104158
$$417$$ −13.5327 −0.662698
$$418$$ −0.532806 −0.0260604
$$419$$ 0.527867 0.0257880 0.0128940 0.999917i $$-0.495896\pi$$
0.0128940 + 0.999917i $$0.495896\pi$$
$$420$$ 0 0
$$421$$ 18.6586 0.909365 0.454682 0.890654i $$-0.349753\pi$$
0.454682 + 0.890654i $$0.349753\pi$$
$$422$$ −1.47109 −0.0716117
$$423$$ 10.9393 0.531886
$$424$$ 3.96239 0.192430
$$425$$ 0 0
$$426$$ 0.657709 0.0318661
$$427$$ −41.2335 −1.99543
$$428$$ −4.42480 −0.213881
$$429$$ 8.70133 0.420104
$$430$$ 0 0
$$431$$ 20.9913 1.01112 0.505559 0.862792i $$-0.331286\pi$$
0.505559 + 0.862792i $$0.331286\pi$$
$$432$$ −3.95166 −0.190124
$$433$$ −13.6639 −0.656647 −0.328324 0.944565i $$-0.606484\pi$$
−0.328324 + 0.944565i $$0.606484\pi$$
$$434$$ 1.44043 0.0691430
$$435$$ 0 0
$$436$$ −21.9159 −1.04958
$$437$$ 3.18354 0.152289
$$438$$ −0.0381557 −0.00182315
$$439$$ −4.33339 −0.206821 −0.103411 0.994639i $$-0.532976\pi$$
−0.103411 + 0.994639i $$0.532976\pi$$
$$440$$ 0 0
$$441$$ 12.0157 0.572177
$$442$$ 0.177461 0.00844096
$$443$$ −1.60742 −0.0763707 −0.0381854 0.999271i $$-0.512158\pi$$
−0.0381854 + 0.999271i $$0.512158\pi$$
$$444$$ −2.86252 −0.135849
$$445$$ 0 0
$$446$$ 0.510439 0.0241700
$$447$$ −13.6843 −0.647244
$$448$$ −34.0441 −1.60843
$$449$$ −13.8291 −0.652634 −0.326317 0.945260i $$-0.605808\pi$$
−0.326317 + 0.945260i $$0.605808\pi$$
$$450$$ 0 0
$$451$$ −26.2687 −1.23694
$$452$$ −3.40662 −0.160234
$$453$$ 11.3204 0.531876
$$454$$ −0.327279 −0.0153600
$$455$$ 0 0
$$456$$ −0.484389 −0.0226836
$$457$$ 20.1345 0.941850 0.470925 0.882173i $$-0.343920\pi$$
0.470925 + 0.882173i $$0.343920\pi$$
$$458$$ −0.149213 −0.00697225
$$459$$ −0.997022 −0.0465370
$$460$$ 0 0
$$461$$ 31.5264 1.46833 0.734166 0.678970i $$-0.237573\pi$$
0.734166 + 0.678970i $$0.237573\pi$$
$$462$$ 1.71983 0.0800135
$$463$$ −0.0451697 −0.00209921 −0.00104961 0.999999i $$-0.500334\pi$$
−0.00104961 + 0.999999i $$0.500334\pi$$
$$464$$ −31.5285 −1.46368
$$465$$ 0 0
$$466$$ −0.705890 −0.0326997
$$467$$ −32.8349 −1.51942 −0.759708 0.650265i $$-0.774658\pi$$
−0.759708 + 0.650265i $$0.774658\pi$$
$$468$$ 3.94732 0.182465
$$469$$ 56.3484 2.60193
$$470$$ 0 0
$$471$$ −8.56070 −0.394456
$$472$$ 2.39834 0.110393
$$473$$ 11.7920 0.542198
$$474$$ −0.570830 −0.0262191
$$475$$ 0 0
$$476$$ −8.66035 −0.396947
$$477$$ 11.0510 0.505992
$$478$$ 0.0509546 0.00233061
$$479$$ 7.48576 0.342033 0.171017 0.985268i $$-0.445295\pi$$
0.171017 + 0.985268i $$0.445295\pi$$
$$480$$ 0 0
$$481$$ 2.84775 0.129846
$$482$$ 1.70730 0.0777654
$$483$$ −10.2760 −0.467576
$$484$$ −16.4939 −0.749724
$$485$$ 0 0
$$486$$ 0.0898194 0.00407429
$$487$$ −25.4295 −1.15232 −0.576160 0.817337i $$-0.695449\pi$$
−0.576160 + 0.817337i $$0.695449\pi$$
$$488$$ −3.39037 −0.153475
$$489$$ −4.58509 −0.207345
$$490$$ 0 0
$$491$$ −11.0668 −0.499438 −0.249719 0.968318i $$-0.580338\pi$$
−0.249719 + 0.968318i $$0.580338\pi$$
$$492$$ −11.9167 −0.537246
$$493$$ −7.95479 −0.358266
$$494$$ 0.240458 0.0108187
$$495$$ 0 0
$$496$$ −14.5327 −0.652537
$$497$$ 31.9315 1.43233
$$498$$ 0.0661979 0.00296640
$$499$$ 4.68157 0.209576 0.104788 0.994495i $$-0.466584\pi$$
0.104788 + 0.994495i $$0.466584\pi$$
$$500$$ 0 0
$$501$$ −7.21792 −0.322473
$$502$$ −0.271734 −0.0121281
$$503$$ 10.8285 0.482818 0.241409 0.970423i $$-0.422391\pi$$
0.241409 + 0.970423i $$0.422391\pi$$
$$504$$ 1.56354 0.0696458
$$505$$ 0 0
$$506$$ −0.929388 −0.0413163
$$507$$ 9.07304 0.402948
$$508$$ 25.0126 1.10975
$$509$$ 11.5007 0.509757 0.254879 0.966973i $$-0.417964\pi$$
0.254879 + 0.966973i $$0.417964\pi$$
$$510$$ 0 0
$$511$$ −1.85244 −0.0819473
$$512$$ −7.07011 −0.312457
$$513$$ −1.35096 −0.0596462
$$514$$ 1.78393 0.0786856
$$515$$ 0 0
$$516$$ 5.34941 0.235495
$$517$$ 48.0338 2.11252
$$518$$ 0.562860 0.0247307
$$519$$ 4.65009 0.204116
$$520$$ 0 0
$$521$$ 0.797807 0.0349525 0.0174763 0.999847i $$-0.494437\pi$$
0.0174763 + 0.999847i $$0.494437\pi$$
$$522$$ 0.716629 0.0313660
$$523$$ 40.1429 1.75533 0.877663 0.479279i $$-0.159102\pi$$
0.877663 + 0.479279i $$0.159102\pi$$
$$524$$ −32.8107 −1.43334
$$525$$ 0 0
$$526$$ −2.05057 −0.0894090
$$527$$ −3.66666 −0.159722
$$528$$ −17.3515 −0.755127
$$529$$ −17.4469 −0.758559
$$530$$ 0 0
$$531$$ 6.68895 0.290276
$$532$$ −11.7347 −0.508764
$$533$$ 11.8552 0.513506
$$534$$ −0.879095 −0.0380422
$$535$$ 0 0
$$536$$ 4.63317 0.200123
$$537$$ 7.20338 0.310849
$$538$$ −1.33224 −0.0574369
$$539$$ 52.7603 2.27255
$$540$$ 0 0
$$541$$ 1.41016 0.0606277 0.0303138 0.999540i $$-0.490349\pi$$
0.0303138 + 0.999540i $$0.490349\pi$$
$$542$$ 0.578185 0.0248352
$$543$$ 3.80424 0.163255
$$544$$ −1.06885 −0.0458265
$$545$$ 0 0
$$546$$ −0.776167 −0.0332169
$$547$$ 15.4621 0.661110 0.330555 0.943787i $$-0.392764\pi$$
0.330555 + 0.943787i $$0.392764\pi$$
$$548$$ 19.2566 0.822603
$$549$$ −9.45570 −0.403560
$$550$$ 0 0
$$551$$ −10.7787 −0.459187
$$552$$ −0.844934 −0.0359628
$$553$$ −27.7136 −1.17850
$$554$$ −0.602667 −0.0256049
$$555$$ 0 0
$$556$$ −26.9562 −1.14320
$$557$$ −18.0445 −0.764568 −0.382284 0.924045i $$-0.624862\pi$$
−0.382284 + 0.924045i $$0.624862\pi$$
$$558$$ 0.330321 0.0139836
$$559$$ −5.32181 −0.225089
$$560$$ 0 0
$$561$$ −4.37786 −0.184834
$$562$$ 1.84011 0.0776202
$$563$$ −28.5327 −1.20251 −0.601254 0.799058i $$-0.705332\pi$$
−0.601254 + 0.799058i $$0.705332\pi$$
$$564$$ 21.7903 0.917538
$$565$$ 0 0
$$566$$ 1.02554 0.0431065
$$567$$ 4.36070 0.183132
$$568$$ 2.62553 0.110165
$$569$$ 16.4072 0.687826 0.343913 0.939001i $$-0.388247\pi$$
0.343913 + 0.939001i $$0.388247\pi$$
$$570$$ 0 0
$$571$$ −8.34705 −0.349313 −0.174657 0.984629i $$-0.555882\pi$$
−0.174657 + 0.984629i $$0.555882\pi$$
$$572$$ 17.3325 0.724707
$$573$$ −21.5541 −0.900437
$$574$$ 2.34319 0.0978029
$$575$$ 0 0
$$576$$ −7.80703 −0.325293
$$577$$ 8.43531 0.351167 0.175583 0.984465i $$-0.443819\pi$$
0.175583 + 0.984465i $$0.443819\pi$$
$$578$$ 1.43765 0.0597982
$$579$$ −3.15029 −0.130921
$$580$$ 0 0
$$581$$ 3.21389 0.133334
$$582$$ 0.00302764 0.000125500 0
$$583$$ 48.5245 2.00968
$$584$$ −0.152315 −0.00630283
$$585$$ 0 0
$$586$$ −2.56439 −0.105934
$$587$$ −24.2852 −1.00236 −0.501179 0.865344i $$-0.667100\pi$$
−0.501179 + 0.865344i $$0.667100\pi$$
$$588$$ 23.9345 0.987042
$$589$$ −4.96829 −0.204715
$$590$$ 0 0
$$591$$ −26.0837 −1.07294
$$592$$ −5.67876 −0.233396
$$593$$ −28.4653 −1.16893 −0.584466 0.811418i $$-0.698696\pi$$
−0.584466 + 0.811418i $$0.698696\pi$$
$$594$$ 0.394392 0.0161821
$$595$$ 0 0
$$596$$ −27.2581 −1.11654
$$597$$ 24.2662 0.993149
$$598$$ 0.419438 0.0171521
$$599$$ 16.0387 0.655323 0.327662 0.944795i $$-0.393739\pi$$
0.327662 + 0.944795i $$0.393739\pi$$
$$600$$ 0 0
$$601$$ −8.09005 −0.330000 −0.165000 0.986294i $$-0.552762\pi$$
−0.165000 + 0.986294i $$0.552762\pi$$
$$602$$ −1.05186 −0.0428706
$$603$$ 12.9219 0.526219
$$604$$ 22.5494 0.917521
$$605$$ 0 0
$$606$$ 0.286874 0.0116535
$$607$$ −0.434608 −0.0176402 −0.00882010 0.999961i $$-0.502808\pi$$
−0.00882010 + 0.999961i $$0.502808\pi$$
$$608$$ −1.44828 −0.0587356
$$609$$ 34.7921 1.40985
$$610$$ 0 0
$$611$$ −21.6779 −0.876994
$$612$$ −1.98600 −0.0802793
$$613$$ −39.3962 −1.59120 −0.795599 0.605824i $$-0.792844\pi$$
−0.795599 + 0.605824i $$0.792844\pi$$
$$614$$ 1.84504 0.0744599
$$615$$ 0 0
$$616$$ 6.86543 0.276616
$$617$$ 15.4146 0.620568 0.310284 0.950644i $$-0.399576\pi$$
0.310284 + 0.950644i $$0.399576\pi$$
$$618$$ 0.768264 0.0309041
$$619$$ −10.6718 −0.428935 −0.214468 0.976731i $$-0.568802\pi$$
−0.214468 + 0.976731i $$0.568802\pi$$
$$620$$ 0 0
$$621$$ −2.35651 −0.0945635
$$622$$ −1.57629 −0.0632036
$$623$$ −42.6797 −1.70993
$$624$$ 7.83083 0.313484
$$625$$ 0 0
$$626$$ 0.268168 0.0107182
$$627$$ −5.93197 −0.236900
$$628$$ −17.0523 −0.680463
$$629$$ −1.43278 −0.0571285
$$630$$ 0 0
$$631$$ −18.4347 −0.733874 −0.366937 0.930246i $$-0.619594\pi$$
−0.366937 + 0.930246i $$0.619594\pi$$
$$632$$ −2.27871 −0.0906424
$$633$$ −16.3783 −0.650980
$$634$$ 1.45245 0.0576843
$$635$$ 0 0
$$636$$ 22.0129 0.872869
$$637$$ −23.8110 −0.943427
$$638$$ 3.14668 0.124578
$$639$$ 7.32257 0.289676
$$640$$ 0 0
$$641$$ 12.4281 0.490882 0.245441 0.969412i $$-0.421067\pi$$
0.245441 + 0.969412i $$0.421067\pi$$
$$642$$ 0.199521 0.00787447
$$643$$ −1.84657 −0.0728218 −0.0364109 0.999337i $$-0.511593\pi$$
−0.0364109 + 0.999337i $$0.511593\pi$$
$$644$$ −20.4692 −0.806598
$$645$$ 0 0
$$646$$ −0.120981 −0.00475992
$$647$$ −38.9760 −1.53230 −0.766152 0.642660i $$-0.777831\pi$$
−0.766152 + 0.642660i $$0.777831\pi$$
$$648$$ 0.358553 0.0140853
$$649$$ 29.3708 1.15290
$$650$$ 0 0
$$651$$ 16.0370 0.628539
$$652$$ −9.13319 −0.357683
$$653$$ 28.3389 1.10899 0.554493 0.832189i $$-0.312912\pi$$
0.554493 + 0.832189i $$0.312912\pi$$
$$654$$ 0.988224 0.0386426
$$655$$ 0 0
$$656$$ −23.6407 −0.923015
$$657$$ −0.424804 −0.0165732
$$658$$ −4.28465 −0.167033
$$659$$ −22.3561 −0.870868 −0.435434 0.900221i $$-0.643405\pi$$
−0.435434 + 0.900221i $$0.643405\pi$$
$$660$$ 0 0
$$661$$ 29.2143 1.13631 0.568153 0.822923i $$-0.307658\pi$$
0.568153 + 0.822923i $$0.307658\pi$$
$$662$$ 1.17398 0.0456281
$$663$$ 1.97575 0.0767319
$$664$$ 0.264258 0.0102552
$$665$$ 0 0
$$666$$ 0.129076 0.00500158
$$667$$ −18.8015 −0.727998
$$668$$ −14.3776 −0.556287
$$669$$ 5.68295 0.219715
$$670$$ 0 0
$$671$$ −41.5194 −1.60284
$$672$$ 4.67486 0.180336
$$673$$ 3.21546 0.123947 0.0619735 0.998078i $$-0.480261\pi$$
0.0619735 + 0.998078i $$0.480261\pi$$
$$674$$ −2.40760 −0.0927372
$$675$$ 0 0
$$676$$ 18.0729 0.695111
$$677$$ −47.2470 −1.81585 −0.907925 0.419133i $$-0.862334\pi$$
−0.907925 + 0.419133i $$0.862334\pi$$
$$678$$ 0.153610 0.00589936
$$679$$ 0.146991 0.00564099
$$680$$ 0 0
$$681$$ −3.64374 −0.139629
$$682$$ 1.45042 0.0555395
$$683$$ −37.2527 −1.42543 −0.712717 0.701452i $$-0.752536\pi$$
−0.712717 + 0.701452i $$0.752536\pi$$
$$684$$ −2.69101 −0.102893
$$685$$ 0 0
$$686$$ −1.96454 −0.0750064
$$687$$ −1.66125 −0.0633807
$$688$$ 10.6123 0.404591
$$689$$ −21.8994 −0.834299
$$690$$ 0 0
$$691$$ −10.9827 −0.417803 −0.208901 0.977937i $$-0.566989\pi$$
−0.208901 + 0.977937i $$0.566989\pi$$
$$692$$ 9.26267 0.352114
$$693$$ 19.1476 0.727357
$$694$$ 2.28913 0.0868941
$$695$$ 0 0
$$696$$ 2.86074 0.108436
$$697$$ −5.96466 −0.225928
$$698$$ 2.51885 0.0953399
$$699$$ −7.85899 −0.297254
$$700$$ 0 0
$$701$$ 22.4086 0.846361 0.423180 0.906046i $$-0.360914\pi$$
0.423180 + 0.906046i $$0.360914\pi$$
$$702$$ −0.177991 −0.00671784
$$703$$ −1.94140 −0.0732213
$$704$$ −34.2802 −1.29198
$$705$$ 0 0
$$706$$ −1.31736 −0.0495793
$$707$$ 13.9276 0.523803
$$708$$ 13.3239 0.500744
$$709$$ −14.6297 −0.549431 −0.274716 0.961526i $$-0.588584\pi$$
−0.274716 + 0.961526i $$0.588584\pi$$
$$710$$ 0 0
$$711$$ −6.35531 −0.238343
$$712$$ −3.50929 −0.131516
$$713$$ −8.66633 −0.324557
$$714$$ 0.390509 0.0146144
$$715$$ 0 0
$$716$$ 14.3486 0.536234
$$717$$ 0.567301 0.0211862
$$718$$ 1.31321 0.0490085
$$719$$ −28.4977 −1.06278 −0.531392 0.847126i $$-0.678331\pi$$
−0.531392 + 0.847126i $$0.678331\pi$$
$$720$$ 0 0
$$721$$ 37.2989 1.38908
$$722$$ 1.54264 0.0574112
$$723$$ 19.0081 0.706920
$$724$$ 7.57778 0.281626
$$725$$ 0 0
$$726$$ 0.743738 0.0276027
$$727$$ 44.0064 1.63211 0.816054 0.577976i $$-0.196157\pi$$
0.816054 + 0.577976i $$0.196157\pi$$
$$728$$ −3.09840 −0.114835
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 2.67754 0.0990324
$$732$$ −18.8351 −0.696166
$$733$$ −27.3227 −1.00919 −0.504593 0.863357i $$-0.668357\pi$$
−0.504593 + 0.863357i $$0.668357\pi$$
$$734$$ 1.58926 0.0586607
$$735$$ 0 0
$$736$$ −2.52628 −0.0931198
$$737$$ 56.7391 2.09001
$$738$$ 0.537343 0.0197799
$$739$$ 16.4678 0.605776 0.302888 0.953026i $$-0.402049\pi$$
0.302888 + 0.953026i $$0.402049\pi$$
$$740$$ 0 0
$$741$$ 2.67713 0.0983468
$$742$$ −4.32843 −0.158901
$$743$$ −35.6012 −1.30608 −0.653041 0.757322i $$-0.726507\pi$$
−0.653041 + 0.757322i $$0.726507\pi$$
$$744$$ 1.31862 0.0483430
$$745$$ 0 0
$$746$$ −1.09843 −0.0402164
$$747$$ 0.737011 0.0269658
$$748$$ −8.72041 −0.318850
$$749$$ 9.68668 0.353944
$$750$$ 0 0
$$751$$ 46.0748 1.68129 0.840647 0.541583i $$-0.182175\pi$$
0.840647 + 0.541583i $$0.182175\pi$$
$$752$$ 43.2283 1.57638
$$753$$ −3.02533 −0.110249
$$754$$ −1.42011 −0.0517174
$$755$$ 0 0
$$756$$ 8.68622 0.315915
$$757$$ 36.6482 1.33200 0.666000 0.745951i $$-0.268005\pi$$
0.666000 + 0.745951i $$0.268005\pi$$
$$758$$ −2.52348 −0.0916571
$$759$$ −10.3473 −0.375583
$$760$$ 0 0
$$761$$ −27.2078 −0.986282 −0.493141 0.869949i $$-0.664151\pi$$
−0.493141 + 0.869949i $$0.664151\pi$$
$$762$$ −1.12786 −0.0408580
$$763$$ 47.9779 1.73692
$$764$$ −42.9344 −1.55331
$$765$$ 0 0
$$766$$ −3.09279 −0.111747
$$767$$ −13.2552 −0.478617
$$768$$ −15.3585 −0.554202
$$769$$ 26.5327 0.956794 0.478397 0.878144i $$-0.341218\pi$$
0.478397 + 0.878144i $$0.341218\pi$$
$$770$$ 0 0
$$771$$ 19.8613 0.715286
$$772$$ −6.27516 −0.225848
$$773$$ −28.7677 −1.03470 −0.517351 0.855773i $$-0.673082\pi$$
−0.517351 + 0.855773i $$0.673082\pi$$
$$774$$ −0.241213 −0.00867024
$$775$$ 0 0
$$776$$ 0.0120861 0.000433867 0
$$777$$ 6.26658 0.224812
$$778$$ −1.21458 −0.0435450
$$779$$ −8.08206 −0.289570
$$780$$ 0 0
$$781$$ 32.1530 1.15052
$$782$$ −0.211030 −0.00754641
$$783$$ 7.97856 0.285130
$$784$$ 47.4820 1.69579
$$785$$ 0 0
$$786$$ 1.47949 0.0527716
$$787$$ 24.8663 0.886387 0.443193 0.896426i $$-0.353845\pi$$
0.443193 + 0.896426i $$0.353845\pi$$
$$788$$ −51.9571 −1.85089
$$789$$ −22.8299 −0.812766
$$790$$ 0 0
$$791$$ 7.45771 0.265166
$$792$$ 1.57439 0.0559434
$$793$$ 18.7379 0.665404
$$794$$ 3.23120 0.114671
$$795$$ 0 0
$$796$$ 48.3366 1.71325
$$797$$ −31.4206 −1.11298 −0.556488 0.830856i $$-0.687851\pi$$
−0.556488 + 0.830856i $$0.687851\pi$$
$$798$$ 0.529137 0.0187312
$$799$$ 10.9067 0.385851
$$800$$ 0 0
$$801$$ −9.78736 −0.345819
$$802$$ −0.391592 −0.0138276
$$803$$ −1.86529 −0.0658246
$$804$$ 25.7395 0.907761
$$805$$ 0 0
$$806$$ −0.654583 −0.0230567
$$807$$ −14.8324 −0.522125
$$808$$ 1.14518 0.0402874
$$809$$ 50.5027 1.77558 0.887790 0.460249i $$-0.152240\pi$$
0.887790 + 0.460249i $$0.152240\pi$$
$$810$$ 0 0
$$811$$ 25.1612 0.883528 0.441764 0.897131i $$-0.354353\pi$$
0.441764 + 0.897131i $$0.354353\pi$$
$$812$$ 69.3035 2.43208
$$813$$ 6.43720 0.225762
$$814$$ 0.566763 0.0198650
$$815$$ 0 0
$$816$$ −3.93989 −0.137924
$$817$$ 3.62804 0.126929
$$818$$ 1.62697 0.0568856
$$819$$ −8.64141 −0.301955
$$820$$ 0 0
$$821$$ −20.6307 −0.720017 −0.360009 0.932949i $$-0.617226\pi$$
−0.360009 + 0.932949i $$0.617226\pi$$
$$822$$ −0.868313 −0.0302859
$$823$$ −35.8618 −1.25006 −0.625032 0.780599i $$-0.714914\pi$$
−0.625032 + 0.780599i $$0.714914\pi$$
$$824$$ 3.06686 0.106839
$$825$$ 0 0
$$826$$ −2.61990 −0.0911580
$$827$$ 4.73642 0.164701 0.0823507 0.996603i $$-0.473757\pi$$
0.0823507 + 0.996603i $$0.473757\pi$$
$$828$$ −4.69401 −0.163128
$$829$$ −28.4768 −0.989039 −0.494520 0.869166i $$-0.664656\pi$$
−0.494520 + 0.869166i $$0.664656\pi$$
$$830$$ 0 0
$$831$$ −6.70976 −0.232759
$$832$$ 15.4708 0.536355
$$833$$ 11.9799 0.415080
$$834$$ 1.21550 0.0420892
$$835$$ 0 0
$$836$$ −11.8161 −0.408668
$$837$$ 3.67761 0.127117
$$838$$ −0.0474127 −0.00163784
$$839$$ −0.765715 −0.0264354 −0.0132177 0.999913i $$-0.504207\pi$$
−0.0132177 + 0.999913i $$0.504207\pi$$
$$840$$ 0 0
$$841$$ 34.6574 1.19508
$$842$$ −1.67591 −0.0577555
$$843$$ 20.4867 0.705600
$$844$$ −32.6245 −1.12298
$$845$$ 0 0
$$846$$ −0.982560 −0.0337811
$$847$$ 36.1082 1.24069
$$848$$ 43.6700 1.49963
$$849$$ 11.4177 0.391856
$$850$$ 0 0
$$851$$ −3.38644 −0.116086
$$852$$ 14.5861 0.499710
$$853$$ 34.7289 1.18910 0.594548 0.804060i $$-0.297331\pi$$
0.594548 + 0.804060i $$0.297331\pi$$
$$854$$ 3.70357 0.126734
$$855$$ 0 0
$$856$$ 0.796475 0.0272230
$$857$$ 54.2561 1.85335 0.926676 0.375860i $$-0.122653\pi$$
0.926676 + 0.375860i $$0.122653\pi$$
$$858$$ −0.781549 −0.0266816
$$859$$ 15.9095 0.542824 0.271412 0.962463i $$-0.412509\pi$$
0.271412 + 0.962463i $$0.412509\pi$$
$$860$$ 0 0
$$861$$ 26.0878 0.889070
$$862$$ −1.88543 −0.0642180
$$863$$ 22.4714 0.764935 0.382467 0.923969i $$-0.375074\pi$$
0.382467 + 0.923969i $$0.375074\pi$$
$$864$$ 1.07204 0.0364716
$$865$$ 0 0
$$866$$ 1.22729 0.0417049
$$867$$ 16.0059 0.543590
$$868$$ 31.9446 1.08427
$$869$$ −27.9058 −0.946639
$$870$$ 0 0
$$871$$ −25.6067 −0.867649
$$872$$ 3.94492 0.133592
$$873$$ 0.0337081 0.00114085
$$874$$ −0.285944 −0.00967219
$$875$$ 0 0
$$876$$ −0.846181 −0.0285898
$$877$$ 27.0756 0.914278 0.457139 0.889395i $$-0.348874\pi$$
0.457139 + 0.889395i $$0.348874\pi$$
$$878$$ 0.389222 0.0131356
$$879$$ −28.5505 −0.962984
$$880$$ 0 0
$$881$$ −7.81317 −0.263232 −0.131616 0.991301i $$-0.542017\pi$$
−0.131616 + 0.991301i $$0.542017\pi$$
$$882$$ −1.07925 −0.0363401
$$883$$ −58.3010 −1.96199 −0.980993 0.194042i $$-0.937840\pi$$
−0.980993 + 0.194042i $$0.937840\pi$$
$$884$$ 3.93557 0.132367
$$885$$ 0 0
$$886$$ 0.144377 0.00485045
$$887$$ −8.64610 −0.290307 −0.145154 0.989409i $$-0.546368\pi$$
−0.145154 + 0.989409i $$0.546368\pi$$
$$888$$ 0.515261 0.0172910
$$889$$ −54.7571 −1.83650
$$890$$ 0 0
$$891$$ 4.39094 0.147102
$$892$$ 11.3200 0.379023
$$893$$ 14.7785 0.494543
$$894$$ 1.22911 0.0411077
$$895$$ 0 0
$$896$$ 12.4075 0.414507
$$897$$ 4.66979 0.155920
$$898$$ 1.24212 0.0414500
$$899$$ 29.3420 0.978612
$$900$$ 0 0
$$901$$ 11.0181 0.367067
$$902$$ 2.35944 0.0785608
$$903$$ −11.7108 −0.389712
$$904$$ 0.613201 0.0203948
$$905$$ 0 0
$$906$$ −1.01679 −0.0337805
$$907$$ −40.4367 −1.34268 −0.671339 0.741151i $$-0.734280\pi$$
−0.671339 + 0.741151i $$0.734280\pi$$
$$908$$ −7.25809 −0.240868
$$909$$ 3.19390 0.105935
$$910$$ 0 0
$$911$$ −50.5643 −1.67527 −0.837635 0.546230i $$-0.816063\pi$$
−0.837635 + 0.546230i $$0.816063\pi$$
$$912$$ −5.33852 −0.176776
$$913$$ 3.23617 0.107102
$$914$$ −1.80847 −0.0598187
$$915$$ 0 0
$$916$$ −3.30910 −0.109336
$$917$$ 71.8286 2.37199
$$918$$ 0.0895519 0.00295565
$$919$$ −43.3566 −1.43020 −0.715101 0.699021i $$-0.753619\pi$$
−0.715101 + 0.699021i $$0.753619\pi$$
$$920$$ 0 0
$$921$$ 20.5417 0.676871
$$922$$ −2.83169 −0.0932566
$$923$$ −14.5108 −0.477629
$$924$$ 38.1407 1.25474
$$925$$ 0 0
$$926$$ 0.00405712 0.000133325 0
$$927$$ 8.55342 0.280931
$$928$$ 8.55335 0.280777
$$929$$ −34.1746 −1.12123 −0.560617 0.828075i $$-0.689436\pi$$
−0.560617 + 0.828075i $$0.689436\pi$$
$$930$$ 0 0
$$931$$ 16.2327 0.532006
$$932$$ −15.6546 −0.512783
$$933$$ −17.5496 −0.574548
$$934$$ 2.94921 0.0965010
$$935$$ 0 0
$$936$$ −0.710529 −0.0232244
$$937$$ 27.8996 0.911441 0.455720 0.890123i $$-0.349382\pi$$
0.455720 + 0.890123i $$0.349382\pi$$
$$938$$ −5.06118 −0.165253
$$939$$ 2.98564 0.0974325
$$940$$ 0 0
$$941$$ −54.3261 −1.77098 −0.885490 0.464659i $$-0.846177\pi$$
−0.885490 + 0.464659i $$0.846177\pi$$
$$942$$ 0.768918 0.0250527
$$943$$ −14.0978 −0.459086
$$944$$ 26.4325 0.860303
$$945$$ 0 0
$$946$$ −1.05915 −0.0344361
$$947$$ 4.23171 0.137512 0.0687561 0.997633i $$-0.478097\pi$$
0.0687561 + 0.997633i $$0.478097\pi$$
$$948$$ −12.6593 −0.411156
$$949$$ 0.841815 0.0273265
$$950$$ 0 0
$$951$$ 16.1708 0.524375
$$952$$ 1.55889 0.0505238
$$953$$ −47.9307 −1.55263 −0.776314 0.630346i $$-0.782913\pi$$
−0.776314 + 0.630346i $$0.782913\pi$$
$$954$$ −0.992598 −0.0321366
$$955$$ 0 0
$$956$$ 1.13002 0.0365476
$$957$$ 35.0334 1.13247
$$958$$ −0.672367 −0.0217232
$$959$$ −42.1563 −1.36130
$$960$$ 0 0
$$961$$ −17.4752 −0.563715
$$962$$ −0.255783 −0.00824679
$$963$$ 2.22136 0.0715823
$$964$$ 37.8629 1.21948
$$965$$ 0 0
$$966$$ 0.922988 0.0296966
$$967$$ −20.8827 −0.671544 −0.335772 0.941943i $$-0.608997\pi$$
−0.335772 + 0.941943i $$0.608997\pi$$
$$968$$ 2.96895 0.0954257
$$969$$ −1.34693 −0.0432697
$$970$$ 0 0
$$971$$ 16.0740 0.515838 0.257919 0.966167i $$-0.416963\pi$$
0.257919 + 0.966167i $$0.416963\pi$$
$$972$$ 1.99193 0.0638913
$$973$$ 59.0119 1.89184
$$974$$ 2.28406 0.0731860
$$975$$ 0 0
$$976$$ −37.3657 −1.19605
$$977$$ −35.5665 −1.13787 −0.568937 0.822381i $$-0.692645\pi$$
−0.568937 + 0.822381i $$0.692645\pi$$
$$978$$ 0.411830 0.0131689
$$979$$ −42.9757 −1.37351
$$980$$ 0 0
$$981$$ 11.0023 0.351278
$$982$$ 0.994014 0.0317202
$$983$$ 18.4711 0.589138 0.294569 0.955630i $$-0.404824\pi$$
0.294569 + 0.955630i $$0.404824\pi$$
$$984$$ 2.14504 0.0683812
$$985$$ 0 0
$$986$$ 0.714495 0.0227542
$$987$$ −47.7030 −1.51840
$$988$$ 5.33266 0.169655
$$989$$ 6.32849 0.201234
$$990$$ 0 0
$$991$$ −41.5907 −1.32117 −0.660586 0.750750i $$-0.729692\pi$$
−0.660586 + 0.750750i $$0.729692\pi$$
$$992$$ 3.94256 0.125176
$$993$$ 13.0705 0.414779
$$994$$ −2.86807 −0.0909698
$$995$$ 0 0
$$996$$ 1.46808 0.0465178
$$997$$ 31.8374 1.00830 0.504151 0.863616i $$-0.331806\pi$$
0.504151 + 0.863616i $$0.331806\pi$$
$$998$$ −0.420496 −0.0133106
$$999$$ 1.43706 0.0454665
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 1875.2.a.p.1.4 8
3.2 odd 2 5625.2.a.t.1.5 8
5.2 odd 4 1875.2.b.h.1249.8 16
5.3 odd 4 1875.2.b.h.1249.9 16
5.4 even 2 1875.2.a.m.1.5 8
15.14 odd 2 5625.2.a.bd.1.4 8
25.3 odd 20 75.2.i.a.34.3 16
25.4 even 10 375.2.g.e.76.2 16
25.6 even 5 375.2.g.d.301.3 16
25.8 odd 20 375.2.i.c.199.2 16
25.17 odd 20 75.2.i.a.64.3 yes 16
25.19 even 10 375.2.g.e.301.2 16
25.21 even 5 375.2.g.d.76.3 16
25.22 odd 20 375.2.i.c.49.2 16
75.17 even 20 225.2.m.b.64.2 16
75.53 even 20 225.2.m.b.109.2 16

By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.i.a.34.3 16 25.3 odd 20
75.2.i.a.64.3 yes 16 25.17 odd 20
225.2.m.b.64.2 16 75.17 even 20
225.2.m.b.109.2 16 75.53 even 20
375.2.g.d.76.3 16 25.21 even 5
375.2.g.d.301.3 16 25.6 even 5
375.2.g.e.76.2 16 25.4 even 10
375.2.g.e.301.2 16 25.19 even 10
375.2.i.c.49.2 16 25.22 odd 20
375.2.i.c.199.2 16 25.8 odd 20
1875.2.a.m.1.5 8 5.4 even 2
1875.2.a.p.1.4 8 1.1 even 1 trivial
1875.2.b.h.1249.8 16 5.2 odd 4
1875.2.b.h.1249.9 16 5.3 odd 4
5625.2.a.t.1.5 8 3.2 odd 2
5625.2.a.bd.1.4 8 15.14 odd 2