# Properties

 Label 5225.2.a.l.1.1 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.7281497.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1$$ x^6 - 2*x^5 - 5*x^4 + 7*x^3 + 6*x^2 - 2*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.326248$$ of defining polynomial Character $$\chi$$ $$=$$ 5225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73890 q^{2} +0.901323 q^{3} +1.02379 q^{4} -1.56731 q^{6} -2.22757 q^{7} +1.69754 q^{8} -2.18762 q^{9} +O(q^{10})$$ $$q-1.73890 q^{2} +0.901323 q^{3} +1.02379 q^{4} -1.56731 q^{6} -2.22757 q^{7} +1.69754 q^{8} -2.18762 q^{9} -1.00000 q^{11} +0.922763 q^{12} +3.64023 q^{13} +3.87353 q^{14} -4.99943 q^{16} -0.644551 q^{17} +3.80406 q^{18} -1.00000 q^{19} -2.00776 q^{21} +1.73890 q^{22} +6.15983 q^{23} +1.53003 q^{24} -6.33001 q^{26} -4.67572 q^{27} -2.28056 q^{28} -1.88204 q^{29} -0.183675 q^{31} +5.29846 q^{32} -0.901323 q^{33} +1.12081 q^{34} -2.23966 q^{36} +4.11428 q^{37} +1.73890 q^{38} +3.28102 q^{39} -1.53371 q^{41} +3.49130 q^{42} +1.53577 q^{43} -1.02379 q^{44} -10.7113 q^{46} -1.75837 q^{47} -4.50610 q^{48} -2.03793 q^{49} -0.580948 q^{51} +3.72682 q^{52} +2.81491 q^{53} +8.13063 q^{54} -3.78139 q^{56} -0.901323 q^{57} +3.27269 q^{58} -3.90068 q^{59} +0.734057 q^{61} +0.319392 q^{62} +4.87307 q^{63} +0.785358 q^{64} +1.56731 q^{66} +1.30264 q^{67} -0.659883 q^{68} +5.55199 q^{69} +10.5493 q^{71} -3.71357 q^{72} +5.17599 q^{73} -7.15433 q^{74} -1.02379 q^{76} +2.22757 q^{77} -5.70538 q^{78} +2.89974 q^{79} +2.34852 q^{81} +2.66698 q^{82} -13.8209 q^{83} -2.05552 q^{84} -2.67055 q^{86} -1.69633 q^{87} -1.69754 q^{88} -6.39884 q^{89} -8.10886 q^{91} +6.30635 q^{92} -0.165550 q^{93} +3.05763 q^{94} +4.77562 q^{96} -6.23352 q^{97} +3.54376 q^{98} +2.18762 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{7} + 12 q^{8} + q^{9}+O(q^{10})$$ 6 * q + 2 * q^2 + q^3 + 4 * q^4 - 5 * q^7 + 12 * q^8 + q^9 $$6 q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{7} + 12 q^{8} + q^{9} - 6 q^{11} - q^{12} + 5 q^{13} - 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{18} - 6 q^{19} - 21 q^{21} - 2 q^{22} - 4 q^{23} - q^{24} - 14 q^{26} + 16 q^{27} - 10 q^{28} - 9 q^{29} - 21 q^{31} + q^{32} - q^{33} - 28 q^{36} + 3 q^{37} - 2 q^{38} + 20 q^{39} - 23 q^{41} - q^{42} - 7 q^{43} - 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} - 16 q^{51} - 13 q^{52} + 17 q^{53} + q^{54} - 2 q^{56} - q^{57} - 23 q^{58} - 29 q^{59} + 17 q^{61} - 2 q^{62} - 6 q^{63} - 18 q^{64} - 8 q^{67} + q^{68} - 38 q^{69} - 12 q^{71} - 13 q^{72} - 2 q^{73} - 37 q^{74} - 4 q^{76} + 5 q^{77} - q^{78} + 3 q^{79} - 2 q^{81} - 24 q^{82} + 11 q^{83} - 3 q^{84} - 12 q^{86} + 12 q^{87} - 12 q^{88} - 22 q^{89} - 18 q^{91} + 15 q^{92} - 18 q^{93} + 22 q^{94} - 17 q^{96} + 2 q^{97} + q^{98} - q^{99}+O(q^{100})$$ 6 * q + 2 * q^2 + q^3 + 4 * q^4 - 5 * q^7 + 12 * q^8 + q^9 - 6 * q^11 - q^12 + 5 * q^13 - 8 * q^14 + 4 * q^16 - q^17 - 6 * q^18 - 6 * q^19 - 21 * q^21 - 2 * q^22 - 4 * q^23 - q^24 - 14 * q^26 + 16 * q^27 - 10 * q^28 - 9 * q^29 - 21 * q^31 + q^32 - q^33 - 28 * q^36 + 3 * q^37 - 2 * q^38 + 20 * q^39 - 23 * q^41 - q^42 - 7 * q^43 - 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 - 16 * q^51 - 13 * q^52 + 17 * q^53 + q^54 - 2 * q^56 - q^57 - 23 * q^58 - 29 * q^59 + 17 * q^61 - 2 * q^62 - 6 * q^63 - 18 * q^64 - 8 * q^67 + q^68 - 38 * q^69 - 12 * q^71 - 13 * q^72 - 2 * q^73 - 37 * q^74 - 4 * q^76 + 5 * q^77 - q^78 + 3 * q^79 - 2 * q^81 - 24 * q^82 + 11 * q^83 - 3 * q^84 - 12 * q^86 + 12 * q^87 - 12 * q^88 - 22 * q^89 - 18 * q^91 + 15 * q^92 - 18 * q^93 + 22 * q^94 - 17 * q^96 + 2 * q^97 + q^98 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.73890 −1.22959 −0.614795 0.788687i $$-0.710761\pi$$
−0.614795 + 0.788687i $$0.710761\pi$$
$$3$$ 0.901323 0.520379 0.260190 0.965558i $$-0.416215\pi$$
0.260190 + 0.965558i $$0.416215\pi$$
$$4$$ 1.02379 0.511894
$$5$$ 0 0
$$6$$ −1.56731 −0.639853
$$7$$ −2.22757 −0.841943 −0.420971 0.907074i $$-0.638311\pi$$
−0.420971 + 0.907074i $$0.638311\pi$$
$$8$$ 1.69754 0.600171
$$9$$ −2.18762 −0.729206
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0.922763 0.266379
$$13$$ 3.64023 1.00962 0.504809 0.863231i $$-0.331563\pi$$
0.504809 + 0.863231i $$0.331563\pi$$
$$14$$ 3.87353 1.03525
$$15$$ 0 0
$$16$$ −4.99943 −1.24986
$$17$$ −0.644551 −0.156327 −0.0781633 0.996941i $$-0.524906\pi$$
−0.0781633 + 0.996941i $$0.524906\pi$$
$$18$$ 3.80406 0.896625
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −2.00776 −0.438129
$$22$$ 1.73890 0.370736
$$23$$ 6.15983 1.28441 0.642206 0.766532i $$-0.278019\pi$$
0.642206 + 0.766532i $$0.278019\pi$$
$$24$$ 1.53003 0.312316
$$25$$ 0 0
$$26$$ −6.33001 −1.24142
$$27$$ −4.67572 −0.899842
$$28$$ −2.28056 −0.430985
$$29$$ −1.88204 −0.349487 −0.174743 0.984614i $$-0.555910\pi$$
−0.174743 + 0.984614i $$0.555910\pi$$
$$30$$ 0 0
$$31$$ −0.183675 −0.0329889 −0.0164945 0.999864i $$-0.505251\pi$$
−0.0164945 + 0.999864i $$0.505251\pi$$
$$32$$ 5.29846 0.936644
$$33$$ −0.901323 −0.156900
$$34$$ 1.12081 0.192218
$$35$$ 0 0
$$36$$ −2.23966 −0.373276
$$37$$ 4.11428 0.676383 0.338192 0.941077i $$-0.390185\pi$$
0.338192 + 0.941077i $$0.390185\pi$$
$$38$$ 1.73890 0.282088
$$39$$ 3.28102 0.525384
$$40$$ 0 0
$$41$$ −1.53371 −0.239525 −0.119763 0.992803i $$-0.538213\pi$$
−0.119763 + 0.992803i $$0.538213\pi$$
$$42$$ 3.49130 0.538720
$$43$$ 1.53577 0.234202 0.117101 0.993120i $$-0.462640\pi$$
0.117101 + 0.993120i $$0.462640\pi$$
$$44$$ −1.02379 −0.154342
$$45$$ 0 0
$$46$$ −10.7113 −1.57930
$$47$$ −1.75837 −0.256484 −0.128242 0.991743i $$-0.540933\pi$$
−0.128242 + 0.991743i $$0.540933\pi$$
$$48$$ −4.50610 −0.650400
$$49$$ −2.03793 −0.291133
$$50$$ 0 0
$$51$$ −0.580948 −0.0813491
$$52$$ 3.72682 0.516817
$$53$$ 2.81491 0.386658 0.193329 0.981134i $$-0.438071\pi$$
0.193329 + 0.981134i $$0.438071\pi$$
$$54$$ 8.13063 1.10644
$$55$$ 0 0
$$56$$ −3.78139 −0.505309
$$57$$ −0.901323 −0.119383
$$58$$ 3.27269 0.429726
$$59$$ −3.90068 −0.507825 −0.253913 0.967227i $$-0.581718\pi$$
−0.253913 + 0.967227i $$0.581718\pi$$
$$60$$ 0 0
$$61$$ 0.734057 0.0939863 0.0469932 0.998895i $$-0.485036\pi$$
0.0469932 + 0.998895i $$0.485036\pi$$
$$62$$ 0.319392 0.0405629
$$63$$ 4.87307 0.613949
$$64$$ 0.785358 0.0981698
$$65$$ 0 0
$$66$$ 1.56731 0.192923
$$67$$ 1.30264 0.159143 0.0795717 0.996829i $$-0.474645\pi$$
0.0795717 + 0.996829i $$0.474645\pi$$
$$68$$ −0.659883 −0.0800226
$$69$$ 5.55199 0.668381
$$70$$ 0 0
$$71$$ 10.5493 1.25197 0.625983 0.779837i $$-0.284698\pi$$
0.625983 + 0.779837i $$0.284698\pi$$
$$72$$ −3.71357 −0.437648
$$73$$ 5.17599 0.605804 0.302902 0.953022i $$-0.402044\pi$$
0.302902 + 0.953022i $$0.402044\pi$$
$$74$$ −7.15433 −0.831674
$$75$$ 0 0
$$76$$ −1.02379 −0.117437
$$77$$ 2.22757 0.253855
$$78$$ −5.70538 −0.646007
$$79$$ 2.89974 0.326246 0.163123 0.986606i $$-0.447843\pi$$
0.163123 + 0.986606i $$0.447843\pi$$
$$80$$ 0 0
$$81$$ 2.34852 0.260947
$$82$$ 2.66698 0.294518
$$83$$ −13.8209 −1.51704 −0.758522 0.651647i $$-0.774078\pi$$
−0.758522 + 0.651647i $$0.774078\pi$$
$$84$$ −2.05552 −0.224276
$$85$$ 0 0
$$86$$ −2.67055 −0.287973
$$87$$ −1.69633 −0.181866
$$88$$ −1.69754 −0.180958
$$89$$ −6.39884 −0.678275 −0.339138 0.940737i $$-0.610135\pi$$
−0.339138 + 0.940737i $$0.610135\pi$$
$$90$$ 0 0
$$91$$ −8.10886 −0.850040
$$92$$ 6.30635 0.657483
$$93$$ −0.165550 −0.0171667
$$94$$ 3.05763 0.315371
$$95$$ 0 0
$$96$$ 4.77562 0.487410
$$97$$ −6.23352 −0.632918 −0.316459 0.948606i $$-0.602494\pi$$
−0.316459 + 0.948606i $$0.602494\pi$$
$$98$$ 3.54376 0.357974
$$99$$ 2.18762 0.219864
$$100$$ 0 0
$$101$$ −16.3563 −1.62751 −0.813756 0.581206i $$-0.802581\pi$$
−0.813756 + 0.581206i $$0.802581\pi$$
$$102$$ 1.01021 0.100026
$$103$$ −0.645519 −0.0636049 −0.0318025 0.999494i $$-0.510125\pi$$
−0.0318025 + 0.999494i $$0.510125\pi$$
$$104$$ 6.17943 0.605943
$$105$$ 0 0
$$106$$ −4.89487 −0.475432
$$107$$ 17.5900 1.70049 0.850243 0.526390i $$-0.176455\pi$$
0.850243 + 0.526390i $$0.176455\pi$$
$$108$$ −4.78694 −0.460624
$$109$$ 19.4359 1.86162 0.930808 0.365507i $$-0.119104\pi$$
0.930808 + 0.365507i $$0.119104\pi$$
$$110$$ 0 0
$$111$$ 3.70829 0.351976
$$112$$ 11.1366 1.05231
$$113$$ 5.88383 0.553504 0.276752 0.960941i $$-0.410742\pi$$
0.276752 + 0.960941i $$0.410742\pi$$
$$114$$ 1.56731 0.146792
$$115$$ 0 0
$$116$$ −1.92681 −0.178900
$$117$$ −7.96342 −0.736219
$$118$$ 6.78291 0.624417
$$119$$ 1.43578 0.131618
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −1.27645 −0.115565
$$123$$ −1.38237 −0.124644
$$124$$ −0.188044 −0.0168868
$$125$$ 0 0
$$126$$ −8.47381 −0.754907
$$127$$ −15.9676 −1.41690 −0.708450 0.705762i $$-0.750605\pi$$
−0.708450 + 0.705762i $$0.750605\pi$$
$$128$$ −11.9626 −1.05735
$$129$$ 1.38422 0.121874
$$130$$ 0 0
$$131$$ −3.93173 −0.343517 −0.171759 0.985139i $$-0.554945\pi$$
−0.171759 + 0.985139i $$0.554945\pi$$
$$132$$ −0.922763 −0.0803162
$$133$$ 2.22757 0.193155
$$134$$ −2.26517 −0.195681
$$135$$ 0 0
$$136$$ −1.09415 −0.0938226
$$137$$ −15.5422 −1.32786 −0.663929 0.747796i $$-0.731112\pi$$
−0.663929 + 0.747796i $$0.731112\pi$$
$$138$$ −9.65438 −0.821836
$$139$$ −6.69900 −0.568202 −0.284101 0.958794i $$-0.591695\pi$$
−0.284101 + 0.958794i $$0.591695\pi$$
$$140$$ 0 0
$$141$$ −1.58486 −0.133469
$$142$$ −18.3441 −1.53941
$$143$$ −3.64023 −0.304411
$$144$$ 10.9368 0.911404
$$145$$ 0 0
$$146$$ −9.00056 −0.744891
$$147$$ −1.83683 −0.151499
$$148$$ 4.21215 0.346236
$$149$$ 0.783045 0.0641496 0.0320748 0.999485i $$-0.489789\pi$$
0.0320748 + 0.999485i $$0.489789\pi$$
$$150$$ 0 0
$$151$$ 8.46749 0.689075 0.344537 0.938773i $$-0.388036\pi$$
0.344537 + 0.938773i $$0.388036\pi$$
$$152$$ −1.69754 −0.137689
$$153$$ 1.41003 0.113994
$$154$$ −3.87353 −0.312138
$$155$$ 0 0
$$156$$ 3.35907 0.268941
$$157$$ −8.27024 −0.660037 −0.330019 0.943974i $$-0.607055\pi$$
−0.330019 + 0.943974i $$0.607055\pi$$
$$158$$ −5.04237 −0.401150
$$159$$ 2.53715 0.201209
$$160$$ 0 0
$$161$$ −13.7215 −1.08140
$$162$$ −4.08385 −0.320858
$$163$$ −23.9030 −1.87223 −0.936113 0.351699i $$-0.885604\pi$$
−0.936113 + 0.351699i $$0.885604\pi$$
$$164$$ −1.57019 −0.122612
$$165$$ 0 0
$$166$$ 24.0333 1.86534
$$167$$ 19.4405 1.50435 0.752176 0.658962i $$-0.229004\pi$$
0.752176 + 0.658962i $$0.229004\pi$$
$$168$$ −3.40825 −0.262952
$$169$$ 0.251253 0.0193272
$$170$$ 0 0
$$171$$ 2.18762 0.167291
$$172$$ 1.57230 0.119887
$$173$$ 11.6888 0.888682 0.444341 0.895858i $$-0.353438\pi$$
0.444341 + 0.895858i $$0.353438\pi$$
$$174$$ 2.94975 0.223620
$$175$$ 0 0
$$176$$ 4.99943 0.376847
$$177$$ −3.51577 −0.264261
$$178$$ 11.1270 0.834001
$$179$$ −6.77452 −0.506351 −0.253176 0.967420i $$-0.581475\pi$$
−0.253176 + 0.967420i $$0.581475\pi$$
$$180$$ 0 0
$$181$$ −4.34551 −0.322999 −0.161500 0.986873i $$-0.551633\pi$$
−0.161500 + 0.986873i $$0.551633\pi$$
$$182$$ 14.1005 1.04520
$$183$$ 0.661622 0.0489085
$$184$$ 10.4565 0.770867
$$185$$ 0 0
$$186$$ 0.287876 0.0211081
$$187$$ 0.644551 0.0471342
$$188$$ −1.80020 −0.131293
$$189$$ 10.4155 0.757616
$$190$$ 0 0
$$191$$ −1.37612 −0.0995728 −0.0497864 0.998760i $$-0.515854\pi$$
−0.0497864 + 0.998760i $$0.515854\pi$$
$$192$$ 0.707861 0.0510855
$$193$$ −5.15711 −0.371217 −0.185609 0.982624i $$-0.559426\pi$$
−0.185609 + 0.982624i $$0.559426\pi$$
$$194$$ 10.8395 0.778231
$$195$$ 0 0
$$196$$ −2.08641 −0.149029
$$197$$ −12.0998 −0.862072 −0.431036 0.902335i $$-0.641852\pi$$
−0.431036 + 0.902335i $$0.641852\pi$$
$$198$$ −3.80406 −0.270343
$$199$$ −13.2948 −0.942445 −0.471223 0.882014i $$-0.656187\pi$$
−0.471223 + 0.882014i $$0.656187\pi$$
$$200$$ 0 0
$$201$$ 1.17410 0.0828149
$$202$$ 28.4420 2.00117
$$203$$ 4.19239 0.294248
$$204$$ −0.594768 −0.0416421
$$205$$ 0 0
$$206$$ 1.12250 0.0782080
$$207$$ −13.4753 −0.936601
$$208$$ −18.1991 −1.26188
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ −26.0866 −1.79588 −0.897939 0.440120i $$-0.854936\pi$$
−0.897939 + 0.440120i $$0.854936\pi$$
$$212$$ 2.88188 0.197928
$$213$$ 9.50828 0.651497
$$214$$ −30.5873 −2.09090
$$215$$ 0 0
$$216$$ −7.93722 −0.540059
$$217$$ 0.409148 0.0277748
$$218$$ −33.7971 −2.28903
$$219$$ 4.66524 0.315248
$$220$$ 0 0
$$221$$ −2.34631 −0.157830
$$222$$ −6.44837 −0.432786
$$223$$ −13.8606 −0.928175 −0.464087 0.885789i $$-0.653618\pi$$
−0.464087 + 0.885789i $$0.653618\pi$$
$$224$$ −11.8027 −0.788600
$$225$$ 0 0
$$226$$ −10.2314 −0.680583
$$227$$ 12.9104 0.856893 0.428446 0.903567i $$-0.359061\pi$$
0.428446 + 0.903567i $$0.359061\pi$$
$$228$$ −0.922763 −0.0611115
$$229$$ 7.93710 0.524499 0.262249 0.965000i $$-0.415536\pi$$
0.262249 + 0.965000i $$0.415536\pi$$
$$230$$ 0 0
$$231$$ 2.00776 0.132101
$$232$$ −3.19484 −0.209752
$$233$$ −17.6265 −1.15475 −0.577374 0.816480i $$-0.695922\pi$$
−0.577374 + 0.816480i $$0.695922\pi$$
$$234$$ 13.8476 0.905248
$$235$$ 0 0
$$236$$ −3.99347 −0.259953
$$237$$ 2.61360 0.169772
$$238$$ −2.49669 −0.161836
$$239$$ −18.6909 −1.20901 −0.604507 0.796600i $$-0.706630\pi$$
−0.604507 + 0.796600i $$0.706630\pi$$
$$240$$ 0 0
$$241$$ −0.877285 −0.0565109 −0.0282555 0.999601i $$-0.508995\pi$$
−0.0282555 + 0.999601i $$0.508995\pi$$
$$242$$ −1.73890 −0.111781
$$243$$ 16.1439 1.03563
$$244$$ 0.751518 0.0481110
$$245$$ 0 0
$$246$$ 2.40381 0.153261
$$247$$ −3.64023 −0.231622
$$248$$ −0.311795 −0.0197990
$$249$$ −12.4571 −0.789438
$$250$$ 0 0
$$251$$ −0.448818 −0.0283291 −0.0141646 0.999900i $$-0.504509\pi$$
−0.0141646 + 0.999900i $$0.504509\pi$$
$$252$$ 4.98899 0.314277
$$253$$ −6.15983 −0.387265
$$254$$ 27.7662 1.74221
$$255$$ 0 0
$$256$$ 19.2311 1.20194
$$257$$ 7.85809 0.490174 0.245087 0.969501i $$-0.421183\pi$$
0.245087 + 0.969501i $$0.421183\pi$$
$$258$$ −2.40703 −0.149855
$$259$$ −9.16484 −0.569476
$$260$$ 0 0
$$261$$ 4.11719 0.254848
$$262$$ 6.83691 0.422386
$$263$$ −0.585215 −0.0360859 −0.0180429 0.999837i $$-0.505744\pi$$
−0.0180429 + 0.999837i $$0.505744\pi$$
$$264$$ −1.53003 −0.0941669
$$265$$ 0 0
$$266$$ −3.87353 −0.237502
$$267$$ −5.76742 −0.352960
$$268$$ 1.33363 0.0814645
$$269$$ −3.59078 −0.218934 −0.109467 0.993990i $$-0.534914\pi$$
−0.109467 + 0.993990i $$0.534914\pi$$
$$270$$ 0 0
$$271$$ −23.4466 −1.42428 −0.712139 0.702038i $$-0.752274\pi$$
−0.712139 + 0.702038i $$0.752274\pi$$
$$272$$ 3.22239 0.195386
$$273$$ −7.30870 −0.442343
$$274$$ 27.0264 1.63272
$$275$$ 0 0
$$276$$ 5.68406 0.342140
$$277$$ 19.2912 1.15909 0.579547 0.814939i $$-0.303229\pi$$
0.579547 + 0.814939i $$0.303229\pi$$
$$278$$ 11.6489 0.698656
$$279$$ 0.401809 0.0240557
$$280$$ 0 0
$$281$$ −5.82550 −0.347520 −0.173760 0.984788i $$-0.555592\pi$$
−0.173760 + 0.984788i $$0.555592\pi$$
$$282$$ 2.75592 0.164112
$$283$$ 15.5166 0.922368 0.461184 0.887305i $$-0.347425\pi$$
0.461184 + 0.887305i $$0.347425\pi$$
$$284$$ 10.8002 0.640874
$$285$$ 0 0
$$286$$ 6.33001 0.374301
$$287$$ 3.41645 0.201667
$$288$$ −11.5910 −0.683006
$$289$$ −16.5846 −0.975562
$$290$$ 0 0
$$291$$ −5.61842 −0.329357
$$292$$ 5.29912 0.310107
$$293$$ 12.8089 0.748305 0.374152 0.927367i $$-0.377934\pi$$
0.374152 + 0.927367i $$0.377934\pi$$
$$294$$ 3.19407 0.186282
$$295$$ 0 0
$$296$$ 6.98415 0.405945
$$297$$ 4.67572 0.271313
$$298$$ −1.36164 −0.0788777
$$299$$ 22.4232 1.29677
$$300$$ 0 0
$$301$$ −3.42103 −0.197185
$$302$$ −14.7242 −0.847280
$$303$$ −14.7423 −0.846923
$$304$$ 4.99943 0.286737
$$305$$ 0 0
$$306$$ −2.45191 −0.140166
$$307$$ 14.0052 0.799321 0.399661 0.916663i $$-0.369128\pi$$
0.399661 + 0.916663i $$0.369128\pi$$
$$308$$ 2.28056 0.129947
$$309$$ −0.581821 −0.0330987
$$310$$ 0 0
$$311$$ 4.10325 0.232674 0.116337 0.993210i $$-0.462885\pi$$
0.116337 + 0.993210i $$0.462885\pi$$
$$312$$ 5.56966 0.315320
$$313$$ −21.5394 −1.21748 −0.608740 0.793370i $$-0.708325\pi$$
−0.608740 + 0.793370i $$0.708325\pi$$
$$314$$ 14.3812 0.811576
$$315$$ 0 0
$$316$$ 2.96872 0.167004
$$317$$ 8.36917 0.470059 0.235030 0.971988i $$-0.424481\pi$$
0.235030 + 0.971988i $$0.424481\pi$$
$$318$$ −4.41186 −0.247405
$$319$$ 1.88204 0.105374
$$320$$ 0 0
$$321$$ 15.8542 0.884898
$$322$$ 23.8603 1.32968
$$323$$ 0.644551 0.0358638
$$324$$ 2.40439 0.133577
$$325$$ 0 0
$$326$$ 41.5650 2.30207
$$327$$ 17.5180 0.968746
$$328$$ −2.60353 −0.143756
$$329$$ 3.91689 0.215945
$$330$$ 0 0
$$331$$ −29.1194 −1.60055 −0.800274 0.599635i $$-0.795312\pi$$
−0.800274 + 0.599635i $$0.795312\pi$$
$$332$$ −14.1497 −0.776565
$$333$$ −9.00046 −0.493222
$$334$$ −33.8052 −1.84974
$$335$$ 0 0
$$336$$ 10.0377 0.547600
$$337$$ −3.36844 −0.183491 −0.0917453 0.995783i $$-0.529245\pi$$
−0.0917453 + 0.995783i $$0.529245\pi$$
$$338$$ −0.436905 −0.0237645
$$339$$ 5.30323 0.288032
$$340$$ 0 0
$$341$$ 0.183675 0.00994653
$$342$$ −3.80406 −0.205700
$$343$$ 20.1326 1.08706
$$344$$ 2.60702 0.140561
$$345$$ 0 0
$$346$$ −20.3257 −1.09272
$$347$$ −3.97272 −0.213267 −0.106633 0.994298i $$-0.534007\pi$$
−0.106633 + 0.994298i $$0.534007\pi$$
$$348$$ −1.73668 −0.0930959
$$349$$ 1.51616 0.0811581 0.0405790 0.999176i $$-0.487080\pi$$
0.0405790 + 0.999176i $$0.487080\pi$$
$$350$$ 0 0
$$351$$ −17.0207 −0.908496
$$352$$ −5.29846 −0.282409
$$353$$ −4.76876 −0.253816 −0.126908 0.991915i $$-0.540505\pi$$
−0.126908 + 0.991915i $$0.540505\pi$$
$$354$$ 6.11359 0.324934
$$355$$ 0 0
$$356$$ −6.55105 −0.347205
$$357$$ 1.29410 0.0684912
$$358$$ 11.7802 0.622605
$$359$$ 0.542750 0.0286453 0.0143226 0.999897i $$-0.495441\pi$$
0.0143226 + 0.999897i $$0.495441\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 7.55643 0.397157
$$363$$ 0.901323 0.0473072
$$364$$ −8.30176 −0.435130
$$365$$ 0 0
$$366$$ −1.15050 −0.0601375
$$367$$ 26.1792 1.36654 0.683271 0.730165i $$-0.260557\pi$$
0.683271 + 0.730165i $$0.260557\pi$$
$$368$$ −30.7956 −1.60533
$$369$$ 3.35517 0.174663
$$370$$ 0 0
$$371$$ −6.27042 −0.325544
$$372$$ −0.169488 −0.00878755
$$373$$ 12.7807 0.661760 0.330880 0.943673i $$-0.392655\pi$$
0.330880 + 0.943673i $$0.392655\pi$$
$$374$$ −1.12081 −0.0579558
$$375$$ 0 0
$$376$$ −2.98490 −0.153934
$$377$$ −6.85107 −0.352848
$$378$$ −18.1115 −0.931557
$$379$$ −26.0931 −1.34031 −0.670157 0.742219i $$-0.733773\pi$$
−0.670157 + 0.742219i $$0.733773\pi$$
$$380$$ 0 0
$$381$$ −14.3920 −0.737325
$$382$$ 2.39295 0.122434
$$383$$ 12.7812 0.653091 0.326545 0.945182i $$-0.394115\pi$$
0.326545 + 0.945182i $$0.394115\pi$$
$$384$$ −10.7821 −0.550224
$$385$$ 0 0
$$386$$ 8.96773 0.456445
$$387$$ −3.35967 −0.170781
$$388$$ −6.38180 −0.323987
$$389$$ −8.97979 −0.455293 −0.227647 0.973744i $$-0.573103\pi$$
−0.227647 + 0.973744i $$0.573103\pi$$
$$390$$ 0 0
$$391$$ −3.97032 −0.200788
$$392$$ −3.45946 −0.174729
$$393$$ −3.54376 −0.178759
$$394$$ 21.0403 1.06000
$$395$$ 0 0
$$396$$ 2.23966 0.112547
$$397$$ −26.8417 −1.34715 −0.673573 0.739120i $$-0.735242\pi$$
−0.673573 + 0.739120i $$0.735242\pi$$
$$398$$ 23.1184 1.15882
$$399$$ 2.00776 0.100514
$$400$$ 0 0
$$401$$ −9.25518 −0.462182 −0.231091 0.972932i $$-0.574229\pi$$
−0.231091 + 0.972932i $$0.574229\pi$$
$$402$$ −2.04165 −0.101828
$$403$$ −0.668617 −0.0333062
$$404$$ −16.7454 −0.833114
$$405$$ 0 0
$$406$$ −7.29016 −0.361804
$$407$$ −4.11428 −0.203937
$$408$$ −0.986183 −0.0488233
$$409$$ 26.0835 1.28975 0.644874 0.764289i $$-0.276910\pi$$
0.644874 + 0.764289i $$0.276910\pi$$
$$410$$ 0 0
$$411$$ −14.0085 −0.690990
$$412$$ −0.660875 −0.0325590
$$413$$ 8.68904 0.427560
$$414$$ 23.4323 1.15164
$$415$$ 0 0
$$416$$ 19.2876 0.945652
$$417$$ −6.03796 −0.295680
$$418$$ −1.73890 −0.0850526
$$419$$ −32.5181 −1.58861 −0.794307 0.607517i $$-0.792166\pi$$
−0.794307 + 0.607517i $$0.792166\pi$$
$$420$$ 0 0
$$421$$ 3.16934 0.154464 0.0772320 0.997013i $$-0.475392\pi$$
0.0772320 + 0.997013i $$0.475392\pi$$
$$422$$ 45.3621 2.20819
$$423$$ 3.84664 0.187030
$$424$$ 4.77843 0.232061
$$425$$ 0 0
$$426$$ −16.5340 −0.801074
$$427$$ −1.63516 −0.0791311
$$428$$ 18.0084 0.870469
$$429$$ −3.28102 −0.158409
$$430$$ 0 0
$$431$$ −1.19700 −0.0576575 −0.0288288 0.999584i $$-0.509178\pi$$
−0.0288288 + 0.999584i $$0.509178\pi$$
$$432$$ 23.3759 1.12468
$$433$$ −23.4218 −1.12558 −0.562789 0.826601i $$-0.690272\pi$$
−0.562789 + 0.826601i $$0.690272\pi$$
$$434$$ −0.711469 −0.0341516
$$435$$ 0 0
$$436$$ 19.8982 0.952950
$$437$$ −6.15983 −0.294664
$$438$$ −8.11241 −0.387626
$$439$$ −21.4961 −1.02595 −0.512975 0.858403i $$-0.671457\pi$$
−0.512975 + 0.858403i $$0.671457\pi$$
$$440$$ 0 0
$$441$$ 4.45821 0.212295
$$442$$ 4.08001 0.194066
$$443$$ −23.6133 −1.12190 −0.560950 0.827850i $$-0.689564\pi$$
−0.560950 + 0.827850i $$0.689564\pi$$
$$444$$ 3.79650 0.180174
$$445$$ 0 0
$$446$$ 24.1023 1.14128
$$447$$ 0.705777 0.0333821
$$448$$ −1.74944 −0.0826533
$$449$$ −5.35004 −0.252484 −0.126242 0.991999i $$-0.540292\pi$$
−0.126242 + 0.991999i $$0.540292\pi$$
$$450$$ 0 0
$$451$$ 1.53371 0.0722196
$$452$$ 6.02379 0.283335
$$453$$ 7.63194 0.358580
$$454$$ −22.4499 −1.05363
$$455$$ 0 0
$$456$$ −1.53003 −0.0716503
$$457$$ −39.7929 −1.86143 −0.930716 0.365742i $$-0.880815\pi$$
−0.930716 + 0.365742i $$0.880815\pi$$
$$458$$ −13.8019 −0.644919
$$459$$ 3.01374 0.140669
$$460$$ 0 0
$$461$$ −3.21566 −0.149768 −0.0748840 0.997192i $$-0.523859\pi$$
−0.0748840 + 0.997192i $$0.523859\pi$$
$$462$$ −3.49130 −0.162430
$$463$$ −16.7065 −0.776416 −0.388208 0.921572i $$-0.626906\pi$$
−0.388208 + 0.921572i $$0.626906\pi$$
$$464$$ 9.40915 0.436809
$$465$$ 0 0
$$466$$ 30.6507 1.41987
$$467$$ 37.8565 1.75179 0.875894 0.482503i $$-0.160272\pi$$
0.875894 + 0.482503i $$0.160272\pi$$
$$468$$ −8.15285 −0.376866
$$469$$ −2.90173 −0.133990
$$470$$ 0 0
$$471$$ −7.45416 −0.343469
$$472$$ −6.62156 −0.304782
$$473$$ −1.53577 −0.0706146
$$474$$ −4.54481 −0.208750
$$475$$ 0 0
$$476$$ 1.46994 0.0673744
$$477$$ −6.15796 −0.281953
$$478$$ 32.5017 1.48659
$$479$$ −16.6101 −0.758936 −0.379468 0.925205i $$-0.623893\pi$$
−0.379468 + 0.925205i $$0.623893\pi$$
$$480$$ 0 0
$$481$$ 14.9769 0.682888
$$482$$ 1.52552 0.0694853
$$483$$ −12.3675 −0.562739
$$484$$ 1.02379 0.0465358
$$485$$ 0 0
$$486$$ −28.0727 −1.27341
$$487$$ −29.2975 −1.32760 −0.663798 0.747912i $$-0.731056\pi$$
−0.663798 + 0.747912i $$0.731056\pi$$
$$488$$ 1.24609 0.0564079
$$489$$ −21.5443 −0.974267
$$490$$ 0 0
$$491$$ −0.806717 −0.0364066 −0.0182033 0.999834i $$-0.505795\pi$$
−0.0182033 + 0.999834i $$0.505795\pi$$
$$492$$ −1.41525 −0.0638045
$$493$$ 1.21307 0.0546340
$$494$$ 6.33001 0.284800
$$495$$ 0 0
$$496$$ 0.918269 0.0412315
$$497$$ −23.4992 −1.05408
$$498$$ 21.6617 0.970686
$$499$$ −30.7212 −1.37527 −0.687634 0.726058i $$-0.741351\pi$$
−0.687634 + 0.726058i $$0.741351\pi$$
$$500$$ 0 0
$$501$$ 17.5222 0.782833
$$502$$ 0.780451 0.0348332
$$503$$ 13.0527 0.581991 0.290995 0.956724i $$-0.406014\pi$$
0.290995 + 0.956724i $$0.406014\pi$$
$$504$$ 8.27223 0.368475
$$505$$ 0 0
$$506$$ 10.7113 0.476178
$$507$$ 0.226460 0.0100574
$$508$$ −16.3475 −0.725302
$$509$$ −22.6615 −1.00445 −0.502226 0.864736i $$-0.667485\pi$$
−0.502226 + 0.864736i $$0.667485\pi$$
$$510$$ 0 0
$$511$$ −11.5299 −0.510052
$$512$$ −9.51581 −0.420544
$$513$$ 4.67572 0.206438
$$514$$ −13.6645 −0.602713
$$515$$ 0 0
$$516$$ 1.41715 0.0623865
$$517$$ 1.75837 0.0773329
$$518$$ 15.9368 0.700222
$$519$$ 10.5354 0.462451
$$520$$ 0 0
$$521$$ 19.0608 0.835070 0.417535 0.908661i $$-0.362894\pi$$
0.417535 + 0.908661i $$0.362894\pi$$
$$522$$ −7.15940 −0.313358
$$523$$ −15.1864 −0.664054 −0.332027 0.943270i $$-0.607732\pi$$
−0.332027 + 0.943270i $$0.607732\pi$$
$$524$$ −4.02526 −0.175844
$$525$$ 0 0
$$526$$ 1.01763 0.0443709
$$527$$ 0.118388 0.00515704
$$528$$ 4.50610 0.196103
$$529$$ 14.9435 0.649716
$$530$$ 0 0
$$531$$ 8.53319 0.370309
$$532$$ 2.28056 0.0988748
$$533$$ −5.58305 −0.241829
$$534$$ 10.0290 0.433997
$$535$$ 0 0
$$536$$ 2.21129 0.0955132
$$537$$ −6.10603 −0.263495
$$538$$ 6.24402 0.269199
$$539$$ 2.03793 0.0877798
$$540$$ 0 0
$$541$$ 41.1432 1.76888 0.884442 0.466650i $$-0.154539\pi$$
0.884442 + 0.466650i $$0.154539\pi$$
$$542$$ 40.7714 1.75128
$$543$$ −3.91671 −0.168082
$$544$$ −3.41513 −0.146422
$$545$$ 0 0
$$546$$ 12.7091 0.543901
$$547$$ −40.7714 −1.74326 −0.871630 0.490164i $$-0.836937\pi$$
−0.871630 + 0.490164i $$0.836937\pi$$
$$548$$ −15.9119 −0.679722
$$549$$ −1.60583 −0.0685354
$$550$$ 0 0
$$551$$ 1.88204 0.0801778
$$552$$ 9.42473 0.401143
$$553$$ −6.45938 −0.274681
$$554$$ −33.5455 −1.42521
$$555$$ 0 0
$$556$$ −6.85835 −0.290859
$$557$$ 14.2315 0.603009 0.301504 0.953465i $$-0.402511\pi$$
0.301504 + 0.953465i $$0.402511\pi$$
$$558$$ −0.698708 −0.0295787
$$559$$ 5.59054 0.236454
$$560$$ 0 0
$$561$$ 0.580948 0.0245277
$$562$$ 10.1300 0.427308
$$563$$ 11.1511 0.469963 0.234982 0.972000i $$-0.424497\pi$$
0.234982 + 0.972000i $$0.424497\pi$$
$$564$$ −1.62256 −0.0683220
$$565$$ 0 0
$$566$$ −26.9819 −1.13414
$$567$$ −5.23149 −0.219702
$$568$$ 17.9078 0.751393
$$569$$ 11.4022 0.478005 0.239003 0.971019i $$-0.423180\pi$$
0.239003 + 0.971019i $$0.423180\pi$$
$$570$$ 0 0
$$571$$ 12.5111 0.523573 0.261787 0.965126i $$-0.415688\pi$$
0.261787 + 0.965126i $$0.415688\pi$$
$$572$$ −3.72682 −0.155826
$$573$$ −1.24033 −0.0518156
$$574$$ −5.94088 −0.247967
$$575$$ 0 0
$$576$$ −1.71806 −0.0715860
$$577$$ −7.95520 −0.331179 −0.165590 0.986195i $$-0.552953\pi$$
−0.165590 + 0.986195i $$0.552953\pi$$
$$578$$ 28.8390 1.19954
$$579$$ −4.64822 −0.193174
$$580$$ 0 0
$$581$$ 30.7871 1.27726
$$582$$ 9.76989 0.404975
$$583$$ −2.81491 −0.116582
$$584$$ 8.78645 0.363586
$$585$$ 0 0
$$586$$ −22.2735 −0.920109
$$587$$ −45.7680 −1.88905 −0.944524 0.328443i $$-0.893476\pi$$
−0.944524 + 0.328443i $$0.893476\pi$$
$$588$$ −1.88052 −0.0775515
$$589$$ 0.183675 0.00756818
$$590$$ 0 0
$$591$$ −10.9058 −0.448604
$$592$$ −20.5691 −0.845383
$$593$$ 26.3613 1.08253 0.541264 0.840853i $$-0.317946\pi$$
0.541264 + 0.840853i $$0.317946\pi$$
$$594$$ −8.13063 −0.333604
$$595$$ 0 0
$$596$$ 0.801672 0.0328378
$$597$$ −11.9829 −0.490429
$$598$$ −38.9917 −1.59449
$$599$$ 17.3368 0.708361 0.354180 0.935177i $$-0.384760\pi$$
0.354180 + 0.935177i $$0.384760\pi$$
$$600$$ 0 0
$$601$$ −15.7946 −0.644274 −0.322137 0.946693i $$-0.604401\pi$$
−0.322137 + 0.946693i $$0.604401\pi$$
$$602$$ 5.94884 0.242457
$$603$$ −2.84969 −0.116048
$$604$$ 8.66891 0.352733
$$605$$ 0 0
$$606$$ 25.6355 1.04137
$$607$$ 1.80754 0.0733659 0.0366829 0.999327i $$-0.488321\pi$$
0.0366829 + 0.999327i $$0.488321\pi$$
$$608$$ −5.29846 −0.214881
$$609$$ 3.77869 0.153120
$$610$$ 0 0
$$611$$ −6.40086 −0.258951
$$612$$ 1.44357 0.0583529
$$613$$ −10.9246 −0.441239 −0.220619 0.975360i $$-0.570808\pi$$
−0.220619 + 0.975360i $$0.570808\pi$$
$$614$$ −24.3538 −0.982838
$$615$$ 0 0
$$616$$ 3.78139 0.152357
$$617$$ 2.68742 0.108191 0.0540957 0.998536i $$-0.482772\pi$$
0.0540957 + 0.998536i $$0.482772\pi$$
$$618$$ 1.01173 0.0406978
$$619$$ 18.2439 0.733284 0.366642 0.930362i $$-0.380507\pi$$
0.366642 + 0.930362i $$0.380507\pi$$
$$620$$ 0 0
$$621$$ −28.8016 −1.15577
$$622$$ −7.13515 −0.286094
$$623$$ 14.2539 0.571069
$$624$$ −16.4032 −0.656655
$$625$$ 0 0
$$626$$ 37.4550 1.49700
$$627$$ 0.901323 0.0359954
$$628$$ −8.46697 −0.337869
$$629$$ −2.65186 −0.105737
$$630$$ 0 0
$$631$$ 9.12012 0.363066 0.181533 0.983385i $$-0.441894\pi$$
0.181533 + 0.983385i $$0.441894\pi$$
$$632$$ 4.92243 0.195804
$$633$$ −23.5125 −0.934537
$$634$$ −14.5532 −0.577981
$$635$$ 0 0
$$636$$ 2.59750 0.102998
$$637$$ −7.41852 −0.293932
$$638$$ −3.27269 −0.129567
$$639$$ −23.0777 −0.912940
$$640$$ 0 0
$$641$$ −2.90773 −0.114848 −0.0574241 0.998350i $$-0.518289\pi$$
−0.0574241 + 0.998350i $$0.518289\pi$$
$$642$$ −27.5690 −1.08806
$$643$$ 3.35603 0.132349 0.0661745 0.997808i $$-0.478921\pi$$
0.0661745 + 0.997808i $$0.478921\pi$$
$$644$$ −14.0479 −0.553563
$$645$$ 0 0
$$646$$ −1.12081 −0.0440978
$$647$$ −26.5428 −1.04351 −0.521753 0.853096i $$-0.674722\pi$$
−0.521753 + 0.853096i $$0.674722\pi$$
$$648$$ 3.98670 0.156613
$$649$$ 3.90068 0.153115
$$650$$ 0 0
$$651$$ 0.368774 0.0144534
$$652$$ −24.4716 −0.958381
$$653$$ −28.6844 −1.12251 −0.561253 0.827644i $$-0.689681\pi$$
−0.561253 + 0.827644i $$0.689681\pi$$
$$654$$ −30.4621 −1.19116
$$655$$ 0 0
$$656$$ 7.66768 0.299373
$$657$$ −11.3231 −0.441756
$$658$$ −6.81110 −0.265524
$$659$$ −39.0719 −1.52203 −0.761013 0.648737i $$-0.775297\pi$$
−0.761013 + 0.648737i $$0.775297\pi$$
$$660$$ 0 0
$$661$$ −21.1901 −0.824199 −0.412099 0.911139i $$-0.635204\pi$$
−0.412099 + 0.911139i $$0.635204\pi$$
$$662$$ 50.6358 1.96802
$$663$$ −2.11478 −0.0821314
$$664$$ −23.4616 −0.910486
$$665$$ 0 0
$$666$$ 15.6509 0.606462
$$667$$ −11.5931 −0.448885
$$668$$ 19.9030 0.770069
$$669$$ −12.4929 −0.483003
$$670$$ 0 0
$$671$$ −0.734057 −0.0283379
$$672$$ −10.6380 −0.410371
$$673$$ 24.6931 0.951849 0.475924 0.879486i $$-0.342114\pi$$
0.475924 + 0.879486i $$0.342114\pi$$
$$674$$ 5.85739 0.225618
$$675$$ 0 0
$$676$$ 0.257230 0.00989345
$$677$$ −5.41516 −0.208122 −0.104061 0.994571i $$-0.533184\pi$$
−0.104061 + 0.994571i $$0.533184\pi$$
$$678$$ −9.22180 −0.354161
$$679$$ 13.8856 0.532881
$$680$$ 0 0
$$681$$ 11.6364 0.445909
$$682$$ −0.319392 −0.0122302
$$683$$ 13.7885 0.527604 0.263802 0.964577i $$-0.415023\pi$$
0.263802 + 0.964577i $$0.415023\pi$$
$$684$$ 2.23966 0.0856354
$$685$$ 0 0
$$686$$ −35.0087 −1.33664
$$687$$ 7.15389 0.272938
$$688$$ −7.67796 −0.292719
$$689$$ 10.2469 0.390377
$$690$$ 0 0
$$691$$ −31.5750 −1.20117 −0.600586 0.799560i $$-0.705066\pi$$
−0.600586 + 0.799560i $$0.705066\pi$$
$$692$$ 11.9668 0.454911
$$693$$ −4.87307 −0.185113
$$694$$ 6.90818 0.262231
$$695$$ 0 0
$$696$$ −2.87959 −0.109150
$$697$$ 0.988554 0.0374442
$$698$$ −2.63645 −0.0997912
$$699$$ −15.8871 −0.600906
$$700$$ 0 0
$$701$$ −17.4459 −0.658924 −0.329462 0.944169i $$-0.606867\pi$$
−0.329462 + 0.944169i $$0.606867\pi$$
$$702$$ 29.5973 1.11708
$$703$$ −4.11428 −0.155173
$$704$$ −0.785358 −0.0295993
$$705$$ 0 0
$$706$$ 8.29242 0.312089
$$707$$ 36.4348 1.37027
$$708$$ −3.59940 −0.135274
$$709$$ −38.7266 −1.45441 −0.727205 0.686421i $$-0.759181\pi$$
−0.727205 + 0.686421i $$0.759181\pi$$
$$710$$ 0 0
$$711$$ −6.34352 −0.237901
$$712$$ −10.8623 −0.407081
$$713$$ −1.13140 −0.0423714
$$714$$ −2.25032 −0.0842162
$$715$$ 0 0
$$716$$ −6.93567 −0.259198
$$717$$ −16.8465 −0.629145
$$718$$ −0.943791 −0.0352220
$$719$$ −28.9402 −1.07929 −0.539644 0.841894i $$-0.681441\pi$$
−0.539644 + 0.841894i $$0.681441\pi$$
$$720$$ 0 0
$$721$$ 1.43794 0.0535517
$$722$$ −1.73890 −0.0647153
$$723$$ −0.790717 −0.0294071
$$724$$ −4.44888 −0.165341
$$725$$ 0 0
$$726$$ −1.56731 −0.0581685
$$727$$ 0.331738 0.0123035 0.00615174 0.999981i $$-0.498042\pi$$
0.00615174 + 0.999981i $$0.498042\pi$$
$$728$$ −13.7651 −0.510169
$$729$$ 7.50533 0.277975
$$730$$ 0 0
$$731$$ −0.989879 −0.0366120
$$732$$ 0.677361 0.0250360
$$733$$ 30.5241 1.12743 0.563716 0.825969i $$-0.309371\pi$$
0.563716 + 0.825969i $$0.309371\pi$$
$$734$$ −45.5231 −1.68029
$$735$$ 0 0
$$736$$ 32.6376 1.20304
$$737$$ −1.30264 −0.0479835
$$738$$ −5.83432 −0.214764
$$739$$ −8.65314 −0.318311 −0.159155 0.987254i $$-0.550877\pi$$
−0.159155 + 0.987254i $$0.550877\pi$$
$$740$$ 0 0
$$741$$ −3.28102 −0.120531
$$742$$ 10.9037 0.400286
$$743$$ −33.9919 −1.24704 −0.623521 0.781807i $$-0.714298\pi$$
−0.623521 + 0.781807i $$0.714298\pi$$
$$744$$ −0.281028 −0.0103030
$$745$$ 0 0
$$746$$ −22.2244 −0.813694
$$747$$ 30.2349 1.10624
$$748$$ 0.659883 0.0241277
$$749$$ −39.1829 −1.43171
$$750$$ 0 0
$$751$$ −50.2581 −1.83394 −0.916972 0.398951i $$-0.869374\pi$$
−0.916972 + 0.398951i $$0.869374\pi$$
$$752$$ 8.79085 0.320569
$$753$$ −0.404530 −0.0147419
$$754$$ 11.9133 0.433859
$$755$$ 0 0
$$756$$ 10.6633 0.387819
$$757$$ −10.7948 −0.392344 −0.196172 0.980570i $$-0.562851\pi$$
−0.196172 + 0.980570i $$0.562851\pi$$
$$758$$ 45.3734 1.64804
$$759$$ −5.55199 −0.201525
$$760$$ 0 0
$$761$$ 15.5661 0.564272 0.282136 0.959374i $$-0.408957\pi$$
0.282136 + 0.959374i $$0.408957\pi$$
$$762$$ 25.0263 0.906608
$$763$$ −43.2947 −1.56737
$$764$$ −1.40886 −0.0509707
$$765$$ 0 0
$$766$$ −22.2254 −0.803035
$$767$$ −14.1994 −0.512709
$$768$$ 17.3334 0.625465
$$769$$ 19.6520 0.708671 0.354335 0.935118i $$-0.384707\pi$$
0.354335 + 0.935118i $$0.384707\pi$$
$$770$$ 0 0
$$771$$ 7.08267 0.255076
$$772$$ −5.27979 −0.190024
$$773$$ 41.0208 1.47541 0.737707 0.675121i $$-0.235908\pi$$
0.737707 + 0.675121i $$0.235908\pi$$
$$774$$ 5.84214 0.209991
$$775$$ 0 0
$$776$$ −10.5817 −0.379859
$$777$$ −8.26048 −0.296343
$$778$$ 15.6150 0.559825
$$779$$ 1.53371 0.0549509
$$780$$ 0 0
$$781$$ −10.5493 −0.377482
$$782$$ 6.90401 0.246887
$$783$$ 8.79991 0.314483
$$784$$ 10.1885 0.363874
$$785$$ 0 0
$$786$$ 6.16226 0.219801
$$787$$ −25.2114 −0.898688 −0.449344 0.893359i $$-0.648342\pi$$
−0.449344 + 0.893359i $$0.648342\pi$$
$$788$$ −12.3876 −0.441289
$$789$$ −0.527467 −0.0187783
$$790$$ 0 0
$$791$$ −13.1066 −0.466018
$$792$$ 3.71357 0.131956
$$793$$ 2.67213 0.0948902
$$794$$ 46.6752 1.65644
$$795$$ 0 0
$$796$$ −13.6111 −0.482432
$$797$$ 23.4053 0.829057 0.414528 0.910036i $$-0.363947\pi$$
0.414528 + 0.910036i $$0.363947\pi$$
$$798$$ −3.49130 −0.123591
$$799$$ 1.13336 0.0400953
$$800$$ 0 0
$$801$$ 13.9982 0.494602
$$802$$ 16.0939 0.568294
$$803$$ −5.17599 −0.182657
$$804$$ 1.20203 0.0423924
$$805$$ 0 0
$$806$$ 1.16266 0.0409530
$$807$$ −3.23645 −0.113929
$$808$$ −27.7655 −0.976786
$$809$$ −25.3235 −0.890325 −0.445163 0.895450i $$-0.646854\pi$$
−0.445163 + 0.895450i $$0.646854\pi$$
$$810$$ 0 0
$$811$$ −1.24610 −0.0437566 −0.0218783 0.999761i $$-0.506965\pi$$
−0.0218783 + 0.999761i $$0.506965\pi$$
$$812$$ 4.29211 0.150624
$$813$$ −21.1329 −0.741165
$$814$$ 7.15433 0.250759
$$815$$ 0 0
$$816$$ 2.90441 0.101675
$$817$$ −1.53577 −0.0537296
$$818$$ −45.3567 −1.58586
$$819$$ 17.7391 0.619854
$$820$$ 0 0
$$821$$ 20.4113 0.712359 0.356180 0.934418i $$-0.384079\pi$$
0.356180 + 0.934418i $$0.384079\pi$$
$$822$$ 24.3595 0.849634
$$823$$ 15.4655 0.539093 0.269546 0.962987i $$-0.413126\pi$$
0.269546 + 0.962987i $$0.413126\pi$$
$$824$$ −1.09579 −0.0381738
$$825$$ 0 0
$$826$$ −15.1094 −0.525723
$$827$$ 46.2008 1.60656 0.803280 0.595602i $$-0.203087\pi$$
0.803280 + 0.595602i $$0.203087\pi$$
$$828$$ −13.7959 −0.479440
$$829$$ 2.91356 0.101192 0.0505961 0.998719i $$-0.483888\pi$$
0.0505961 + 0.998719i $$0.483888\pi$$
$$830$$ 0 0
$$831$$ 17.3876 0.603168
$$832$$ 2.85888 0.0991139
$$833$$ 1.31355 0.0455117
$$834$$ 10.4994 0.363566
$$835$$ 0 0
$$836$$ 1.02379 0.0354084
$$837$$ 0.858810 0.0296848
$$838$$ 56.5459 1.95334
$$839$$ 28.2887 0.976633 0.488317 0.872667i $$-0.337611\pi$$
0.488317 + 0.872667i $$0.337611\pi$$
$$840$$ 0 0
$$841$$ −25.4579 −0.877859
$$842$$ −5.51117 −0.189928
$$843$$ −5.25066 −0.180842
$$844$$ −26.7072 −0.919299
$$845$$ 0 0
$$846$$ −6.68893 −0.229970
$$847$$ −2.22757 −0.0765402
$$848$$ −14.0730 −0.483268
$$849$$ 13.9855 0.479981
$$850$$ 0 0
$$851$$ 25.3432 0.868755
$$852$$ 9.73446 0.333497
$$853$$ −41.9234 −1.43543 −0.717715 0.696337i $$-0.754812\pi$$
−0.717715 + 0.696337i $$0.754812\pi$$
$$854$$ 2.84339 0.0972989
$$855$$ 0 0
$$856$$ 29.8597 1.02058
$$857$$ −52.4352 −1.79115 −0.895576 0.444909i $$-0.853236\pi$$
−0.895576 + 0.444909i $$0.853236\pi$$
$$858$$ 5.70538 0.194778
$$859$$ −21.2305 −0.724374 −0.362187 0.932105i $$-0.617970\pi$$
−0.362187 + 0.932105i $$0.617970\pi$$
$$860$$ 0 0
$$861$$ 3.07932 0.104943
$$862$$ 2.08147 0.0708952
$$863$$ −1.75582 −0.0597690 −0.0298845 0.999553i $$-0.509514\pi$$
−0.0298845 + 0.999553i $$0.509514\pi$$
$$864$$ −24.7741 −0.842832
$$865$$ 0 0
$$866$$ 40.7282 1.38400
$$867$$ −14.9480 −0.507662
$$868$$ 0.418881 0.0142177
$$869$$ −2.89974 −0.0983670
$$870$$ 0 0
$$871$$ 4.74192 0.160674
$$872$$ 32.9931 1.11729
$$873$$ 13.6366 0.461528
$$874$$ 10.7113 0.362317
$$875$$ 0 0
$$876$$ 4.77622 0.161373
$$877$$ −27.7873 −0.938309 −0.469155 0.883116i $$-0.655441\pi$$
−0.469155 + 0.883116i $$0.655441\pi$$
$$878$$ 37.3796 1.26150
$$879$$ 11.5450 0.389402
$$880$$ 0 0
$$881$$ 38.8712 1.30960 0.654802 0.755800i $$-0.272752\pi$$
0.654802 + 0.755800i $$0.272752\pi$$
$$882$$ −7.75239 −0.261037
$$883$$ 1.20467 0.0405404 0.0202702 0.999795i $$-0.493547\pi$$
0.0202702 + 0.999795i $$0.493547\pi$$
$$884$$ −2.40212 −0.0807922
$$885$$ 0 0
$$886$$ 41.0612 1.37948
$$887$$ 44.0947 1.48056 0.740278 0.672301i $$-0.234694\pi$$
0.740278 + 0.672301i $$0.234694\pi$$
$$888$$ 6.29497 0.211245
$$889$$ 35.5691 1.19295
$$890$$ 0 0
$$891$$ −2.34852 −0.0786784
$$892$$ −14.1903 −0.475127
$$893$$ 1.75837 0.0588415
$$894$$ −1.22728 −0.0410463
$$895$$ 0 0
$$896$$ 26.6475 0.890230
$$897$$ 20.2105 0.674809
$$898$$ 9.30320 0.310452
$$899$$ 0.345683 0.0115292
$$900$$ 0 0
$$901$$ −1.81436 −0.0604450
$$902$$ −2.66698 −0.0888006
$$903$$ −3.08345 −0.102611
$$904$$ 9.98803 0.332197
$$905$$ 0 0
$$906$$ −13.2712 −0.440907
$$907$$ −36.2297 −1.20299 −0.601493 0.798878i $$-0.705427\pi$$
−0.601493 + 0.798878i $$0.705427\pi$$
$$908$$ 13.2175 0.438638
$$909$$ 35.7813 1.18679
$$910$$ 0 0
$$911$$ −5.41417 −0.179380 −0.0896898 0.995970i $$-0.528588\pi$$
−0.0896898 + 0.995970i $$0.528588\pi$$
$$912$$ 4.50610 0.149212
$$913$$ 13.8209 0.457406
$$914$$ 69.1960 2.28880
$$915$$ 0 0
$$916$$ 8.12591 0.268488
$$917$$ 8.75822 0.289222
$$918$$ −5.24060 −0.172966
$$919$$ 54.0931 1.78437 0.892183 0.451675i $$-0.149173\pi$$
0.892183 + 0.451675i $$0.149173\pi$$
$$920$$ 0 0
$$921$$ 12.6232 0.415950
$$922$$ 5.59172 0.184153
$$923$$ 38.4017 1.26401
$$924$$ 2.05552 0.0676217
$$925$$ 0 0
$$926$$ 29.0510 0.954674
$$927$$ 1.41215 0.0463811
$$928$$ −9.97193 −0.327345
$$929$$ 52.0785 1.70864 0.854320 0.519747i $$-0.173974\pi$$
0.854320 + 0.519747i $$0.173974\pi$$
$$930$$ 0 0
$$931$$ 2.03793 0.0667904
$$932$$ −18.0457 −0.591108
$$933$$ 3.69835 0.121079
$$934$$ −65.8288 −2.15398
$$935$$ 0 0
$$936$$ −13.5182 −0.441857
$$937$$ −35.9861 −1.17561 −0.587807 0.809002i $$-0.700008\pi$$
−0.587807 + 0.809002i $$0.700008\pi$$
$$938$$ 5.04584 0.164752
$$939$$ −19.4140 −0.633551
$$940$$ 0 0
$$941$$ 11.9972 0.391098 0.195549 0.980694i $$-0.437351\pi$$
0.195549 + 0.980694i $$0.437351\pi$$
$$942$$ 12.9621 0.422327
$$943$$ −9.44739 −0.307649
$$944$$ 19.5012 0.634709
$$945$$ 0 0
$$946$$ 2.67055 0.0868271
$$947$$ −37.1663 −1.20774 −0.603871 0.797082i $$-0.706376\pi$$
−0.603871 + 0.797082i $$0.706376\pi$$
$$948$$ 2.67577 0.0869051
$$949$$ 18.8418 0.611630
$$950$$ 0 0
$$951$$ 7.54332 0.244609
$$952$$ 2.43730 0.0789933
$$953$$ −16.5500 −0.536106 −0.268053 0.963404i $$-0.586380\pi$$
−0.268053 + 0.963404i $$0.586380\pi$$
$$954$$ 10.7081 0.346687
$$955$$ 0 0
$$956$$ −19.1355 −0.618886
$$957$$ 1.69633 0.0548345
$$958$$ 28.8834 0.933180
$$959$$ 34.6213 1.11798
$$960$$ 0 0
$$961$$ −30.9663 −0.998912
$$962$$ −26.0434 −0.839673
$$963$$ −38.4801 −1.24000
$$964$$ −0.898154 −0.0289276
$$965$$ 0 0
$$966$$ 21.5058 0.691939
$$967$$ −37.7443 −1.21377 −0.606887 0.794788i $$-0.707582\pi$$
−0.606887 + 0.794788i $$0.707582\pi$$
$$968$$ 1.69754 0.0545610
$$969$$ 0.580948 0.0186628
$$970$$ 0 0
$$971$$ −40.2912 −1.29301 −0.646504 0.762911i $$-0.723769\pi$$
−0.646504 + 0.762911i $$0.723769\pi$$
$$972$$ 16.5280 0.530134
$$973$$ 14.9225 0.478393
$$974$$ 50.9455 1.63240
$$975$$ 0 0
$$976$$ −3.66987 −0.117470
$$977$$ −1.24031 −0.0396812 −0.0198406 0.999803i $$-0.506316\pi$$
−0.0198406 + 0.999803i $$0.506316\pi$$
$$978$$ 37.4635 1.19795
$$979$$ 6.39884 0.204508
$$980$$ 0 0
$$981$$ −42.5182 −1.35750
$$982$$ 1.40280 0.0447653
$$983$$ −8.75471 −0.279232 −0.139616 0.990206i $$-0.544587\pi$$
−0.139616 + 0.990206i $$0.544587\pi$$
$$984$$ −2.34662 −0.0748077
$$985$$ 0 0
$$986$$ −2.10942 −0.0671775
$$987$$ 3.53038 0.112373
$$988$$ −3.72682 −0.118566
$$989$$ 9.46005 0.300812
$$990$$ 0 0
$$991$$ 49.7066 1.57898 0.789491 0.613763i $$-0.210345\pi$$
0.789491 + 0.613763i $$0.210345\pi$$
$$992$$ −0.973192 −0.0308989
$$993$$ −26.2460 −0.832891
$$994$$ 40.8629 1.29609
$$995$$ 0 0
$$996$$ −12.7534 −0.404108
$$997$$ −46.3005 −1.46635 −0.733176 0.680039i $$-0.761963\pi$$
−0.733176 + 0.680039i $$0.761963\pi$$
$$998$$ 53.4211 1.69102
$$999$$ −19.2372 −0.608638
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 5225.2.a.l.1.1 6
5.4 even 2 1045.2.a.f.1.6 6
15.14 odd 2 9405.2.a.z.1.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.f.1.6 6 5.4 even 2
5225.2.a.l.1.1 6 1.1 even 1 trivial
9405.2.a.z.1.1 6 15.14 odd 2