# Properties

 Label 1805.2.a.o.1.4 Level $1805$ Weight $2$ Character 1805.1 Self dual yes Analytic conductor $14.413$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,2,Mod(1,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, 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("1805.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.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: $$14.4129975648$$ Analytic rank: $$0$$ 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 1.4 Root $$-0.491918$$ of defining polynomial Character $$\chi$$ $$=$$ 1805.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.75802 q^{2} +1.49192 q^{3} +5.60665 q^{4} +1.00000 q^{5} +4.11474 q^{6} -2.84864 q^{7} +9.94721 q^{8} -0.774179 q^{9} +O(q^{10})$$ $$q+2.75802 q^{2} +1.49192 q^{3} +5.60665 q^{4} +1.00000 q^{5} +4.11474 q^{6} -2.84864 q^{7} +9.94721 q^{8} -0.774179 q^{9} +2.75802 q^{10} -0.864801 q^{11} +8.36467 q^{12} +0.643281 q^{13} -7.85659 q^{14} +1.49192 q^{15} +16.2213 q^{16} +3.74185 q^{17} -2.13520 q^{18} +5.60665 q^{20} -4.24993 q^{21} -2.38513 q^{22} +0.417460 q^{23} +14.8404 q^{24} +1.00000 q^{25} +1.77418 q^{26} -5.63077 q^{27} -15.9713 q^{28} -9.70523 q^{29} +4.11474 q^{30} +4.93349 q^{31} +24.8441 q^{32} -1.29021 q^{33} +10.3201 q^{34} -2.84864 q^{35} -4.34056 q^{36} +6.36467 q^{37} +0.959723 q^{39} +9.94721 q^{40} -4.01372 q^{41} -11.7214 q^{42} -2.05829 q^{43} -4.84864 q^{44} -0.774179 q^{45} +1.15136 q^{46} -3.95396 q^{47} +24.2008 q^{48} +1.11474 q^{49} +2.75802 q^{50} +5.58254 q^{51} +3.60665 q^{52} -10.9875 q^{53} -15.5297 q^{54} -0.864801 q^{55} -28.3360 q^{56} -26.7672 q^{58} +2.45959 q^{59} +8.36467 q^{60} +6.33479 q^{61} +13.6067 q^{62} +2.20536 q^{63} +36.0778 q^{64} +0.643281 q^{65} -3.55843 q^{66} -2.53220 q^{67} +20.9793 q^{68} +0.622817 q^{69} -7.85659 q^{70} -1.78213 q^{71} -7.70092 q^{72} -7.13090 q^{73} +17.5539 q^{74} +1.49192 q^{75} +2.46350 q^{77} +2.64693 q^{78} +1.82452 q^{79} +16.2213 q^{80} -6.07811 q^{81} -11.0699 q^{82} -7.43913 q^{83} -23.8279 q^{84} +3.74185 q^{85} -5.67681 q^{86} -14.4794 q^{87} -8.60235 q^{88} +4.44588 q^{89} -2.13520 q^{90} -1.83247 q^{91} +2.34056 q^{92} +7.36037 q^{93} -10.9051 q^{94} +37.0653 q^{96} -10.8541 q^{97} +3.07446 q^{98} +0.669511 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + 3 q^{3} + 5 q^{4} + 4 q^{5} + 2 q^{6} - 4 q^{7} + 12 q^{8} + q^{9}+O(q^{10})$$ 4 * q + q^2 + 3 * q^3 + 5 * q^4 + 4 * q^5 + 2 * q^6 - 4 * q^7 + 12 * q^8 + q^9 $$4 q + q^{2} + 3 q^{3} + 5 q^{4} + 4 q^{5} + 2 q^{6} - 4 q^{7} + 12 q^{8} + q^{9} + q^{10} - 2 q^{11} + 6 q^{12} + 7 q^{13} - q^{14} + 3 q^{15} + 7 q^{16} - q^{17} - 10 q^{18} + 5 q^{20} - 4 q^{21} + 2 q^{22} + 2 q^{23} + 23 q^{24} + 4 q^{25} + 3 q^{26} + 12 q^{27} - 19 q^{28} - q^{29} + 2 q^{30} + 30 q^{32} + 19 q^{33} + 15 q^{34} - 4 q^{35} - 7 q^{36} - 2 q^{37} + 15 q^{39} + 12 q^{40} - 8 q^{41} - 15 q^{42} + q^{43} - 12 q^{44} + q^{45} + 12 q^{46} - 12 q^{47} + 23 q^{48} - 10 q^{49} + q^{50} + 22 q^{51} - 3 q^{52} - 5 q^{53} - 34 q^{54} - 2 q^{55} - 41 q^{56} - 27 q^{58} - 5 q^{59} + 6 q^{60} + 37 q^{62} - 3 q^{63} + 56 q^{64} + 7 q^{65} - 31 q^{66} + 4 q^{67} + 16 q^{68} - 9 q^{69} - q^{70} + 20 q^{71} + 17 q^{72} - 20 q^{73} + 25 q^{74} + 3 q^{75} + 14 q^{77} - 18 q^{78} + 17 q^{79} + 7 q^{80} + 12 q^{81} + 21 q^{82} + q^{83} - 20 q^{84} - q^{85} + 8 q^{86} - 16 q^{87} - 7 q^{88} + 11 q^{89} - 10 q^{90} + 6 q^{91} - q^{92} - 8 q^{93} - 31 q^{94} + 21 q^{96} + q^{97} + 9 q^{98} + 38 q^{99}+O(q^{100})$$ 4 * q + q^2 + 3 * q^3 + 5 * q^4 + 4 * q^5 + 2 * q^6 - 4 * q^7 + 12 * q^8 + q^9 + q^10 - 2 * q^11 + 6 * q^12 + 7 * q^13 - q^14 + 3 * q^15 + 7 * q^16 - q^17 - 10 * q^18 + 5 * q^20 - 4 * q^21 + 2 * q^22 + 2 * q^23 + 23 * q^24 + 4 * q^25 + 3 * q^26 + 12 * q^27 - 19 * q^28 - q^29 + 2 * q^30 + 30 * q^32 + 19 * q^33 + 15 * q^34 - 4 * q^35 - 7 * q^36 - 2 * q^37 + 15 * q^39 + 12 * q^40 - 8 * q^41 - 15 * q^42 + q^43 - 12 * q^44 + q^45 + 12 * q^46 - 12 * q^47 + 23 * q^48 - 10 * q^49 + q^50 + 22 * q^51 - 3 * q^52 - 5 * q^53 - 34 * q^54 - 2 * q^55 - 41 * q^56 - 27 * q^58 - 5 * q^59 + 6 * q^60 + 37 * q^62 - 3 * q^63 + 56 * q^64 + 7 * q^65 - 31 * q^66 + 4 * q^67 + 16 * q^68 - 9 * q^69 - q^70 + 20 * q^71 + 17 * q^72 - 20 * q^73 + 25 * q^74 + 3 * q^75 + 14 * q^77 - 18 * q^78 + 17 * q^79 + 7 * q^80 + 12 * q^81 + 21 * q^82 + q^83 - 20 * q^84 - q^85 + 8 * q^86 - 16 * q^87 - 7 * q^88 + 11 * q^89 - 10 * q^90 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.75802 1.95021 0.975106 0.221739i $$-0.0711734\pi$$
0.975106 + 0.221739i $$0.0711734\pi$$
$$3$$ 1.49192 0.861360 0.430680 0.902505i $$-0.358274\pi$$
0.430680 + 0.902505i $$0.358274\pi$$
$$4$$ 5.60665 2.80333
$$5$$ 1.00000 0.447214
$$6$$ 4.11474 1.67983
$$7$$ −2.84864 −1.07668 −0.538342 0.842727i $$-0.680949\pi$$
−0.538342 + 0.842727i $$0.680949\pi$$
$$8$$ 9.94721 3.51687
$$9$$ −0.774179 −0.258060
$$10$$ 2.75802 0.872161
$$11$$ −0.864801 −0.260747 −0.130374 0.991465i $$-0.541618\pi$$
−0.130374 + 0.991465i $$0.541618\pi$$
$$12$$ 8.36467 2.41467
$$13$$ 0.643281 0.178414 0.0892070 0.996013i $$-0.471567\pi$$
0.0892070 + 0.996013i $$0.471567\pi$$
$$14$$ −7.85659 −2.09976
$$15$$ 1.49192 0.385212
$$16$$ 16.2213 4.05531
$$17$$ 3.74185 0.907533 0.453766 0.891121i $$-0.350080\pi$$
0.453766 + 0.891121i $$0.350080\pi$$
$$18$$ −2.13520 −0.503271
$$19$$ 0 0
$$20$$ 5.60665 1.25369
$$21$$ −4.24993 −0.927412
$$22$$ −2.38513 −0.508512
$$23$$ 0.417460 0.0870465 0.0435233 0.999052i $$-0.486142\pi$$
0.0435233 + 0.999052i $$0.486142\pi$$
$$24$$ 14.8404 3.02929
$$25$$ 1.00000 0.200000
$$26$$ 1.77418 0.347945
$$27$$ −5.63077 −1.08364
$$28$$ −15.9713 −3.01830
$$29$$ −9.70523 −1.80222 −0.901108 0.433596i $$-0.857245\pi$$
−0.901108 + 0.433596i $$0.857245\pi$$
$$30$$ 4.11474 0.751244
$$31$$ 4.93349 0.886081 0.443041 0.896501i $$-0.353900\pi$$
0.443041 + 0.896501i $$0.353900\pi$$
$$32$$ 24.8441 4.39185
$$33$$ −1.29021 −0.224597
$$34$$ 10.3201 1.76988
$$35$$ −2.84864 −0.481508
$$36$$ −4.34056 −0.723426
$$37$$ 6.36467 1.04635 0.523173 0.852227i $$-0.324748\pi$$
0.523173 + 0.852227i $$0.324748\pi$$
$$38$$ 0 0
$$39$$ 0.959723 0.153679
$$40$$ 9.94721 1.57279
$$41$$ −4.01372 −0.626837 −0.313419 0.949615i $$-0.601474\pi$$
−0.313419 + 0.949615i $$0.601474\pi$$
$$42$$ −11.7214 −1.80865
$$43$$ −2.05829 −0.313887 −0.156944 0.987608i $$-0.550164\pi$$
−0.156944 + 0.987608i $$0.550164\pi$$
$$44$$ −4.84864 −0.730960
$$45$$ −0.774179 −0.115408
$$46$$ 1.15136 0.169759
$$47$$ −3.95396 −0.576744 −0.288372 0.957518i $$-0.593114\pi$$
−0.288372 + 0.957518i $$0.593114\pi$$
$$48$$ 24.2008 3.49308
$$49$$ 1.11474 0.159248
$$50$$ 2.75802 0.390042
$$51$$ 5.58254 0.781712
$$52$$ 3.60665 0.500153
$$53$$ −10.9875 −1.50925 −0.754624 0.656158i $$-0.772181\pi$$
−0.754624 + 0.656158i $$0.772181\pi$$
$$54$$ −15.5297 −2.11333
$$55$$ −0.864801 −0.116610
$$56$$ −28.3360 −3.78656
$$57$$ 0 0
$$58$$ −26.7672 −3.51470
$$59$$ 2.45959 0.320212 0.160106 0.987100i $$-0.448816\pi$$
0.160106 + 0.987100i $$0.448816\pi$$
$$60$$ 8.36467 1.07987
$$61$$ 6.33479 0.811087 0.405543 0.914076i $$-0.367082\pi$$
0.405543 + 0.914076i $$0.367082\pi$$
$$62$$ 13.6067 1.72805
$$63$$ 2.20536 0.277849
$$64$$ 36.0778 4.50973
$$65$$ 0.643281 0.0797892
$$66$$ −3.55843 −0.438012
$$67$$ −2.53220 −0.309357 −0.154678 0.987965i $$-0.549434\pi$$
−0.154678 + 0.987965i $$0.549434\pi$$
$$68$$ 20.9793 2.54411
$$69$$ 0.622817 0.0749783
$$70$$ −7.85659 −0.939042
$$71$$ −1.78213 −0.211500 −0.105750 0.994393i $$-0.533724\pi$$
−0.105750 + 0.994393i $$0.533724\pi$$
$$72$$ −7.70092 −0.907563
$$73$$ −7.13090 −0.834609 −0.417304 0.908767i $$-0.637025\pi$$
−0.417304 + 0.908767i $$0.637025\pi$$
$$74$$ 17.5539 2.04060
$$75$$ 1.49192 0.172272
$$76$$ 0 0
$$77$$ 2.46350 0.280742
$$78$$ 2.64693 0.299706
$$79$$ 1.82452 0.205275 0.102637 0.994719i $$-0.467272\pi$$
0.102637 + 0.994719i $$0.467272\pi$$
$$80$$ 16.2213 1.81359
$$81$$ −6.07811 −0.675345
$$82$$ −11.0699 −1.22247
$$83$$ −7.43913 −0.816550 −0.408275 0.912859i $$-0.633870\pi$$
−0.408275 + 0.912859i $$0.633870\pi$$
$$84$$ −23.8279 −2.59984
$$85$$ 3.74185 0.405861
$$86$$ −5.67681 −0.612146
$$87$$ −14.4794 −1.55236
$$88$$ −8.60235 −0.917014
$$89$$ 4.44588 0.471262 0.235631 0.971843i $$-0.424284\pi$$
0.235631 + 0.971843i $$0.424284\pi$$
$$90$$ −2.13520 −0.225070
$$91$$ −1.83247 −0.192096
$$92$$ 2.34056 0.244020
$$93$$ 7.36037 0.763235
$$94$$ −10.9051 −1.12477
$$95$$ 0 0
$$96$$ 37.0653 3.78296
$$97$$ −10.8541 −1.10207 −0.551036 0.834482i $$-0.685767\pi$$
−0.551036 + 0.834482i $$0.685767\pi$$
$$98$$ 3.07446 0.310567
$$99$$ 0.669511 0.0672884
$$100$$ 5.60665 0.560665
$$101$$ −5.29598 −0.526969 −0.263485 0.964664i $$-0.584872\pi$$
−0.263485 + 0.964664i $$0.584872\pi$$
$$102$$ 15.3967 1.52450
$$103$$ −0.385134 −0.0379484 −0.0189742 0.999820i $$-0.506040\pi$$
−0.0189742 + 0.999820i $$0.506040\pi$$
$$104$$ 6.39885 0.627459
$$105$$ −4.24993 −0.414751
$$106$$ −30.3037 −2.94335
$$107$$ −6.43336 −0.621937 −0.310968 0.950420i $$-0.600653\pi$$
−0.310968 + 0.950420i $$0.600653\pi$$
$$108$$ −31.5698 −3.03780
$$109$$ 6.56882 0.629179 0.314590 0.949228i $$-0.398133\pi$$
0.314590 + 0.949228i $$0.398133\pi$$
$$110$$ −2.38513 −0.227414
$$111$$ 9.49557 0.901279
$$112$$ −46.2085 −4.36629
$$113$$ 0.294513 0.0277054 0.0138527 0.999904i $$-0.495590\pi$$
0.0138527 + 0.999904i $$0.495590\pi$$
$$114$$ 0 0
$$115$$ 0.417460 0.0389284
$$116$$ −54.4138 −5.05220
$$117$$ −0.498015 −0.0460415
$$118$$ 6.78360 0.624481
$$119$$ −10.6592 −0.977126
$$120$$ 14.8404 1.35474
$$121$$ −10.2521 −0.932011
$$122$$ 17.4715 1.58179
$$123$$ −5.98814 −0.539932
$$124$$ 27.6604 2.48398
$$125$$ 1.00000 0.0894427
$$126$$ 6.08241 0.541864
$$127$$ 8.83492 0.783972 0.391986 0.919971i $$-0.371788\pi$$
0.391986 + 0.919971i $$0.371788\pi$$
$$128$$ 49.8151 4.40308
$$129$$ −3.07081 −0.270370
$$130$$ 1.77418 0.155606
$$131$$ 20.9128 1.82716 0.913578 0.406662i $$-0.133307\pi$$
0.913578 + 0.406662i $$0.133307\pi$$
$$132$$ −7.23377 −0.629619
$$133$$ 0 0
$$134$$ −6.98384 −0.603312
$$135$$ −5.63077 −0.484619
$$136$$ 37.2210 3.19167
$$137$$ 5.21477 0.445528 0.222764 0.974872i $$-0.428492\pi$$
0.222764 + 0.974872i $$0.428492\pi$$
$$138$$ 1.71774 0.146224
$$139$$ −10.7238 −0.909584 −0.454792 0.890598i $$-0.650286\pi$$
−0.454792 + 0.890598i $$0.650286\pi$$
$$140$$ −15.9713 −1.34982
$$141$$ −5.89898 −0.496784
$$142$$ −4.91514 −0.412470
$$143$$ −0.556310 −0.0465210
$$144$$ −12.5582 −1.04651
$$145$$ −9.70523 −0.805975
$$146$$ −19.6671 −1.62766
$$147$$ 1.66309 0.137170
$$148$$ 35.6845 2.93325
$$149$$ −14.9116 −1.22160 −0.610801 0.791784i $$-0.709153\pi$$
−0.610801 + 0.791784i $$0.709153\pi$$
$$150$$ 4.11474 0.335967
$$151$$ 21.4589 1.74630 0.873152 0.487448i $$-0.162072\pi$$
0.873152 + 0.487448i $$0.162072\pi$$
$$152$$ 0 0
$$153$$ −2.89687 −0.234198
$$154$$ 6.79438 0.547507
$$155$$ 4.93349 0.396268
$$156$$ 5.38083 0.430811
$$157$$ −2.43118 −0.194029 −0.0970145 0.995283i $$-0.530929\pi$$
−0.0970145 + 0.995283i $$0.530929\pi$$
$$158$$ 5.03207 0.400330
$$159$$ −16.3924 −1.30000
$$160$$ 24.8441 1.96410
$$161$$ −1.18919 −0.0937216
$$162$$ −16.7635 −1.31707
$$163$$ 17.8175 1.39558 0.697788 0.716305i $$-0.254168\pi$$
0.697788 + 0.716305i $$0.254168\pi$$
$$164$$ −22.5035 −1.75723
$$165$$ −1.29021 −0.100443
$$166$$ −20.5172 −1.59245
$$167$$ −0.405598 −0.0313861 −0.0156931 0.999877i $$-0.504995\pi$$
−0.0156931 + 0.999877i $$0.504995\pi$$
$$168$$ −42.2750 −3.26159
$$169$$ −12.5862 −0.968168
$$170$$ 10.3201 0.791515
$$171$$ 0 0
$$172$$ −11.5401 −0.879928
$$173$$ 18.0210 1.37011 0.685056 0.728490i $$-0.259778\pi$$
0.685056 + 0.728490i $$0.259778\pi$$
$$174$$ −39.9344 −3.02742
$$175$$ −2.84864 −0.215337
$$176$$ −14.0282 −1.05741
$$177$$ 3.66951 0.275817
$$178$$ 12.2618 0.919061
$$179$$ 20.1523 1.50625 0.753127 0.657875i $$-0.228545\pi$$
0.753127 + 0.657875i $$0.228545\pi$$
$$180$$ −4.34056 −0.323526
$$181$$ −17.1108 −1.27184 −0.635919 0.771756i $$-0.719379\pi$$
−0.635919 + 0.771756i $$0.719379\pi$$
$$182$$ −5.05399 −0.374627
$$183$$ 9.45099 0.698637
$$184$$ 4.15257 0.306131
$$185$$ 6.36467 0.467940
$$186$$ 20.3000 1.48847
$$187$$ −3.23596 −0.236637
$$188$$ −22.1685 −1.61680
$$189$$ 16.0400 1.16674
$$190$$ 0 0
$$191$$ 5.28080 0.382105 0.191053 0.981580i $$-0.438810\pi$$
0.191053 + 0.981580i $$0.438810\pi$$
$$192$$ 53.8252 3.88450
$$193$$ 18.0036 1.29593 0.647966 0.761670i $$-0.275620\pi$$
0.647966 + 0.761670i $$0.275620\pi$$
$$194$$ −29.9359 −2.14927
$$195$$ 0.959723 0.0687272
$$196$$ 6.24993 0.446424
$$197$$ 8.07785 0.575523 0.287761 0.957702i $$-0.407089\pi$$
0.287761 + 0.957702i $$0.407089\pi$$
$$198$$ 1.84652 0.131227
$$199$$ 1.40374 0.0995088 0.0497544 0.998761i $$-0.484156\pi$$
0.0497544 + 0.998761i $$0.484156\pi$$
$$200$$ 9.94721 0.703374
$$201$$ −3.77783 −0.266468
$$202$$ −14.6064 −1.02770
$$203$$ 27.6467 1.94042
$$204$$ 31.2994 2.19139
$$205$$ −4.01372 −0.280330
$$206$$ −1.06221 −0.0740074
$$207$$ −0.323189 −0.0224632
$$208$$ 10.4348 0.723525
$$209$$ 0 0
$$210$$ −11.7214 −0.808853
$$211$$ 18.9163 1.30226 0.651128 0.758968i $$-0.274296\pi$$
0.651128 + 0.758968i $$0.274296\pi$$
$$212$$ −61.6030 −4.23091
$$213$$ −2.65879 −0.182178
$$214$$ −17.7433 −1.21291
$$215$$ −2.05829 −0.140375
$$216$$ −56.0104 −3.81103
$$217$$ −14.0537 −0.954030
$$218$$ 18.1169 1.22703
$$219$$ −10.6387 −0.718898
$$220$$ −4.84864 −0.326895
$$221$$ 2.40706 0.161917
$$222$$ 26.1889 1.75769
$$223$$ −16.1480 −1.08135 −0.540675 0.841231i $$-0.681831\pi$$
−0.540675 + 0.841231i $$0.681831\pi$$
$$224$$ −70.7718 −4.72864
$$225$$ −0.774179 −0.0516120
$$226$$ 0.812271 0.0540315
$$227$$ 26.3186 1.74683 0.873414 0.486978i $$-0.161901\pi$$
0.873414 + 0.486978i $$0.161901\pi$$
$$228$$ 0 0
$$229$$ −13.3323 −0.881026 −0.440513 0.897746i $$-0.645203\pi$$
−0.440513 + 0.897746i $$0.645203\pi$$
$$230$$ 1.15136 0.0759186
$$231$$ 3.67535 0.241820
$$232$$ −96.5399 −6.33816
$$233$$ 25.3094 1.65808 0.829038 0.559192i $$-0.188889\pi$$
0.829038 + 0.559192i $$0.188889\pi$$
$$234$$ −1.37353 −0.0897907
$$235$$ −3.95396 −0.257928
$$236$$ 13.7901 0.897658
$$237$$ 2.72204 0.176815
$$238$$ −29.3982 −1.90560
$$239$$ −23.5500 −1.52332 −0.761660 0.647977i $$-0.775615\pi$$
−0.761660 + 0.647977i $$0.775615\pi$$
$$240$$ 24.2008 1.56215
$$241$$ 8.38415 0.540071 0.270035 0.962850i $$-0.412965\pi$$
0.270035 + 0.962850i $$0.412965\pi$$
$$242$$ −28.2755 −1.81762
$$243$$ 7.82426 0.501927
$$244$$ 35.5170 2.27374
$$245$$ 1.11474 0.0712178
$$246$$ −16.5154 −1.05298
$$247$$ 0 0
$$248$$ 49.0745 3.11623
$$249$$ −11.0986 −0.703343
$$250$$ 2.75802 0.174432
$$251$$ 18.2478 1.15179 0.575896 0.817523i $$-0.304653\pi$$
0.575896 + 0.817523i $$0.304653\pi$$
$$252$$ 12.3647 0.778901
$$253$$ −0.361020 −0.0226971
$$254$$ 24.3669 1.52891
$$255$$ 5.58254 0.349592
$$256$$ 65.2353 4.07720
$$257$$ 14.0998 0.879520 0.439760 0.898115i $$-0.355064\pi$$
0.439760 + 0.898115i $$0.355064\pi$$
$$258$$ −8.46934 −0.527278
$$259$$ −18.1306 −1.12658
$$260$$ 3.60665 0.223675
$$261$$ 7.51359 0.465079
$$262$$ 57.6778 3.56334
$$263$$ −6.41071 −0.395301 −0.197651 0.980273i $$-0.563331\pi$$
−0.197651 + 0.980273i $$0.563331\pi$$
$$264$$ −12.8340 −0.789879
$$265$$ −10.9875 −0.674956
$$266$$ 0 0
$$267$$ 6.63288 0.405926
$$268$$ −14.1971 −0.867229
$$269$$ 17.9911 1.09694 0.548469 0.836171i $$-0.315211\pi$$
0.548469 + 0.836171i $$0.315211\pi$$
$$270$$ −15.5297 −0.945110
$$271$$ 11.8819 0.721774 0.360887 0.932609i $$-0.382474\pi$$
0.360887 + 0.932609i $$0.382474\pi$$
$$272$$ 60.6976 3.68033
$$273$$ −2.73390 −0.165463
$$274$$ 14.3824 0.868874
$$275$$ −0.864801 −0.0521494
$$276$$ 3.49192 0.210189
$$277$$ 23.6240 1.41943 0.709715 0.704489i $$-0.248824\pi$$
0.709715 + 0.704489i $$0.248824\pi$$
$$278$$ −29.5765 −1.77388
$$279$$ −3.81941 −0.228662
$$280$$ −28.3360 −1.69340
$$281$$ 13.8093 0.823794 0.411897 0.911230i $$-0.364866\pi$$
0.411897 + 0.911230i $$0.364866\pi$$
$$282$$ −16.2695 −0.968834
$$283$$ −11.7574 −0.698903 −0.349451 0.936954i $$-0.613632\pi$$
−0.349451 + 0.936954i $$0.613632\pi$$
$$284$$ −9.99179 −0.592904
$$285$$ 0 0
$$286$$ −1.53431 −0.0907257
$$287$$ 11.4336 0.674905
$$288$$ −19.2338 −1.13336
$$289$$ −2.99854 −0.176384
$$290$$ −26.7672 −1.57182
$$291$$ −16.1935 −0.949279
$$292$$ −39.9805 −2.33968
$$293$$ −27.0576 −1.58072 −0.790362 0.612640i $$-0.790108\pi$$
−0.790362 + 0.612640i $$0.790108\pi$$
$$294$$ 4.58684 0.267510
$$295$$ 2.45959 0.143203
$$296$$ 63.3107 3.67986
$$297$$ 4.86949 0.282557
$$298$$ −41.1263 −2.38238
$$299$$ 0.268544 0.0155303
$$300$$ 8.36467 0.482934
$$301$$ 5.86334 0.337957
$$302$$ 59.1841 3.40566
$$303$$ −7.90117 −0.453910
$$304$$ 0 0
$$305$$ 6.33479 0.362729
$$306$$ −7.98960 −0.456735
$$307$$ 8.83824 0.504425 0.252212 0.967672i $$-0.418842\pi$$
0.252212 + 0.967672i $$0.418842\pi$$
$$308$$ 13.8120 0.787012
$$309$$ −0.574589 −0.0326872
$$310$$ 13.6067 0.772806
$$311$$ −0.651493 −0.0369428 −0.0184714 0.999829i $$-0.505880\pi$$
−0.0184714 + 0.999829i $$0.505880\pi$$
$$312$$ 9.54656 0.540468
$$313$$ −2.96556 −0.167623 −0.0838116 0.996482i $$-0.526709\pi$$
−0.0838116 + 0.996482i $$0.526709\pi$$
$$314$$ −6.70523 −0.378398
$$315$$ 2.20536 0.124258
$$316$$ 10.2295 0.575453
$$317$$ −10.3799 −0.582991 −0.291495 0.956572i $$-0.594153\pi$$
−0.291495 + 0.956572i $$0.594153\pi$$
$$318$$ −45.2106 −2.53528
$$319$$ 8.39309 0.469923
$$320$$ 36.0778 2.01681
$$321$$ −9.59805 −0.535711
$$322$$ −3.27981 −0.182777
$$323$$ 0 0
$$324$$ −34.0778 −1.89321
$$325$$ 0.643281 0.0356828
$$326$$ 49.1410 2.72167
$$327$$ 9.80015 0.541949
$$328$$ −39.9253 −2.20450
$$329$$ 11.2634 0.620971
$$330$$ −3.55843 −0.195885
$$331$$ 15.0922 0.829543 0.414772 0.909926i $$-0.363861\pi$$
0.414772 + 0.909926i $$0.363861\pi$$
$$332$$ −41.7086 −2.28906
$$333$$ −4.92740 −0.270020
$$334$$ −1.11865 −0.0612096
$$335$$ −2.53220 −0.138349
$$336$$ −68.9393 −3.76095
$$337$$ 15.7974 0.860541 0.430271 0.902700i $$-0.358418\pi$$
0.430271 + 0.902700i $$0.358418\pi$$
$$338$$ −34.7129 −1.88813
$$339$$ 0.439389 0.0238643
$$340$$ 20.9793 1.13776
$$341$$ −4.26649 −0.231043
$$342$$ 0 0
$$343$$ 16.7650 0.905224
$$344$$ −20.4743 −1.10390
$$345$$ 0.622817 0.0335313
$$346$$ 49.7023 2.67201
$$347$$ −21.3522 −1.14624 −0.573122 0.819470i $$-0.694268\pi$$
−0.573122 + 0.819470i $$0.694268\pi$$
$$348$$ −81.1810 −4.35176
$$349$$ −32.3897 −1.73378 −0.866891 0.498497i $$-0.833885\pi$$
−0.866891 + 0.498497i $$0.833885\pi$$
$$350$$ −7.85659 −0.419952
$$351$$ −3.62217 −0.193337
$$352$$ −21.4852 −1.14516
$$353$$ 0.730583 0.0388850 0.0194425 0.999811i $$-0.493811\pi$$
0.0194425 + 0.999811i $$0.493811\pi$$
$$354$$ 10.1206 0.537902
$$355$$ −1.78213 −0.0945857
$$356$$ 24.9265 1.32110
$$357$$ −15.9026 −0.841657
$$358$$ 55.5804 2.93751
$$359$$ −26.8496 −1.41707 −0.708533 0.705677i $$-0.750643\pi$$
−0.708533 + 0.705677i $$0.750643\pi$$
$$360$$ −7.70092 −0.405874
$$361$$ 0 0
$$362$$ −47.1919 −2.48035
$$363$$ −15.2953 −0.802796
$$364$$ −10.2740 −0.538506
$$365$$ −7.13090 −0.373248
$$366$$ 26.0660 1.36249
$$367$$ −22.9643 −1.19873 −0.599364 0.800477i $$-0.704580\pi$$
−0.599364 + 0.800477i $$0.704580\pi$$
$$368$$ 6.77173 0.353001
$$369$$ 3.10734 0.161761
$$370$$ 17.5539 0.912582
$$371$$ 31.2994 1.62498
$$372$$ 41.2670 2.13960
$$373$$ 29.5305 1.52903 0.764515 0.644606i $$-0.222979\pi$$
0.764515 + 0.644606i $$0.222979\pi$$
$$374$$ −8.92482 −0.461492
$$375$$ 1.49192 0.0770423
$$376$$ −39.3308 −2.02833
$$377$$ −6.24319 −0.321541
$$378$$ 44.2386 2.27539
$$379$$ 17.5117 0.899517 0.449759 0.893150i $$-0.351510\pi$$
0.449759 + 0.893150i $$0.351510\pi$$
$$380$$ 0 0
$$381$$ 13.1810 0.675282
$$382$$ 14.5645 0.745186
$$383$$ 8.10652 0.414224 0.207112 0.978317i $$-0.433594\pi$$
0.207112 + 0.978317i $$0.433594\pi$$
$$384$$ 74.3201 3.79263
$$385$$ 2.46350 0.125552
$$386$$ 49.6544 2.52734
$$387$$ 1.59349 0.0810016
$$388$$ −60.8554 −3.08947
$$389$$ 17.3078 0.877542 0.438771 0.898599i $$-0.355414\pi$$
0.438771 + 0.898599i $$0.355414\pi$$
$$390$$ 2.64693 0.134033
$$391$$ 1.56208 0.0789976
$$392$$ 11.0885 0.560054
$$393$$ 31.2001 1.57384
$$394$$ 22.2788 1.12239
$$395$$ 1.82452 0.0918017
$$396$$ 3.75372 0.188631
$$397$$ −11.3894 −0.571619 −0.285810 0.958286i $$-0.592263\pi$$
−0.285810 + 0.958286i $$0.592263\pi$$
$$398$$ 3.87155 0.194063
$$399$$ 0 0
$$400$$ 16.2213 0.811063
$$401$$ −8.93861 −0.446373 −0.223186 0.974776i $$-0.571646\pi$$
−0.223186 + 0.974776i $$0.571646\pi$$
$$402$$ −10.4193 −0.519668
$$403$$ 3.17362 0.158089
$$404$$ −29.6927 −1.47727
$$405$$ −6.07811 −0.302024
$$406$$ 76.2500 3.78422
$$407$$ −5.50417 −0.272832
$$408$$ 55.5307 2.74918
$$409$$ −6.54471 −0.323615 −0.161808 0.986822i $$-0.551732\pi$$
−0.161808 + 0.986822i $$0.551732\pi$$
$$410$$ −11.0699 −0.546703
$$411$$ 7.78001 0.383760
$$412$$ −2.15931 −0.106382
$$413$$ −7.00649 −0.344767
$$414$$ −0.891361 −0.0438080
$$415$$ −7.43913 −0.365172
$$416$$ 15.9817 0.783568
$$417$$ −15.9991 −0.783479
$$418$$ 0 0
$$419$$ 21.8441 1.06715 0.533576 0.845752i $$-0.320848\pi$$
0.533576 + 0.845752i $$0.320848\pi$$
$$420$$ −23.8279 −1.16268
$$421$$ −29.3434 −1.43011 −0.715054 0.699069i $$-0.753598\pi$$
−0.715054 + 0.699069i $$0.753598\pi$$
$$422$$ 52.1716 2.53967
$$423$$ 3.06107 0.148834
$$424$$ −109.295 −5.30783
$$425$$ 3.74185 0.181507
$$426$$ −7.33299 −0.355285
$$427$$ −18.0455 −0.873284
$$428$$ −36.0696 −1.74349
$$429$$ −0.829969 −0.0400713
$$430$$ −5.67681 −0.273760
$$431$$ −12.8867 −0.620732 −0.310366 0.950617i $$-0.600452\pi$$
−0.310366 + 0.950617i $$0.600452\pi$$
$$432$$ −91.3381 −4.39451
$$433$$ −13.8429 −0.665246 −0.332623 0.943060i $$-0.607934\pi$$
−0.332623 + 0.943060i $$0.607934\pi$$
$$434$$ −38.7604 −1.86056
$$435$$ −14.4794 −0.694234
$$436$$ 36.8291 1.76379
$$437$$ 0 0
$$438$$ −29.3418 −1.40200
$$439$$ −0.0708081 −0.00337948 −0.00168974 0.999999i $$-0.500538\pi$$
−0.00168974 + 0.999999i $$0.500538\pi$$
$$440$$ −8.60235 −0.410101
$$441$$ −0.863005 −0.0410955
$$442$$ 6.63872 0.315772
$$443$$ −3.78914 −0.180027 −0.0900137 0.995941i $$-0.528691\pi$$
−0.0900137 + 0.995941i $$0.528691\pi$$
$$444$$ 53.2384 2.52658
$$445$$ 4.44588 0.210755
$$446$$ −44.5365 −2.10886
$$447$$ −22.2468 −1.05224
$$448$$ −102.773 −4.85555
$$449$$ −26.5765 −1.25422 −0.627112 0.778929i $$-0.715763\pi$$
−0.627112 + 0.778929i $$0.715763\pi$$
$$450$$ −2.13520 −0.100654
$$451$$ 3.47106 0.163446
$$452$$ 1.65123 0.0776674
$$453$$ 32.0150 1.50420
$$454$$ 72.5872 3.40669
$$455$$ −1.83247 −0.0859077
$$456$$ 0 0
$$457$$ −33.1523 −1.55080 −0.775400 0.631471i $$-0.782452\pi$$
−0.775400 + 0.631471i $$0.782452\pi$$
$$458$$ −36.7708 −1.71819
$$459$$ −21.0695 −0.983440
$$460$$ 2.34056 0.109129
$$461$$ −19.2536 −0.896729 −0.448364 0.893851i $$-0.647993\pi$$
−0.448364 + 0.893851i $$0.647993\pi$$
$$462$$ 10.1367 0.471600
$$463$$ 39.1713 1.82044 0.910222 0.414120i $$-0.135911\pi$$
0.910222 + 0.414120i $$0.135911\pi$$
$$464$$ −157.431 −7.30855
$$465$$ 7.36037 0.341329
$$466$$ 69.8038 3.23360
$$467$$ −39.0650 −1.80771 −0.903856 0.427836i $$-0.859276\pi$$
−0.903856 + 0.427836i $$0.859276\pi$$
$$468$$ −2.79220 −0.129069
$$469$$ 7.21331 0.333080
$$470$$ −10.9051 −0.503014
$$471$$ −3.62712 −0.167129
$$472$$ 24.4661 1.12614
$$473$$ 1.78001 0.0818452
$$474$$ 7.50743 0.344828
$$475$$ 0 0
$$476$$ −59.7623 −2.73920
$$477$$ 8.50629 0.389476
$$478$$ −64.9512 −2.97080
$$479$$ −24.7550 −1.13109 −0.565543 0.824718i $$-0.691334\pi$$
−0.565543 + 0.824718i $$0.691334\pi$$
$$480$$ 37.0653 1.69179
$$481$$ 4.09427 0.186683
$$482$$ 23.1236 1.05325
$$483$$ −1.77418 −0.0807280
$$484$$ −57.4801 −2.61273
$$485$$ −10.8541 −0.492861
$$486$$ 21.5794 0.978863
$$487$$ 21.8871 0.991797 0.495899 0.868380i $$-0.334839\pi$$
0.495899 + 0.868380i $$0.334839\pi$$
$$488$$ 63.0135 2.85249
$$489$$ 26.5823 1.20209
$$490$$ 3.07446 0.138890
$$491$$ 9.39553 0.424014 0.212007 0.977268i $$-0.432000\pi$$
0.212007 + 0.977268i $$0.432000\pi$$
$$492$$ −33.5734 −1.51361
$$493$$ −36.3155 −1.63557
$$494$$ 0 0
$$495$$ 0.669511 0.0300923
$$496$$ 80.0275 3.59334
$$497$$ 5.07664 0.227719
$$498$$ −30.6100 −1.37167
$$499$$ 24.9115 1.11519 0.557596 0.830112i $$-0.311724\pi$$
0.557596 + 0.830112i $$0.311724\pi$$
$$500$$ 5.60665 0.250737
$$501$$ −0.605119 −0.0270347
$$502$$ 50.3278 2.24624
$$503$$ −31.3180 −1.39640 −0.698200 0.715903i $$-0.746015\pi$$
−0.698200 + 0.715903i $$0.746015\pi$$
$$504$$ 21.9371 0.977158
$$505$$ −5.29598 −0.235668
$$506$$ −0.995699 −0.0442642
$$507$$ −18.7776 −0.833941
$$508$$ 49.5343 2.19773
$$509$$ −9.66619 −0.428446 −0.214223 0.976785i $$-0.568722\pi$$
−0.214223 + 0.976785i $$0.568722\pi$$
$$510$$ 15.3967 0.681779
$$511$$ 20.3133 0.898609
$$512$$ 80.2896 3.54833
$$513$$ 0 0
$$514$$ 38.8874 1.71525
$$515$$ −0.385134 −0.0169710
$$516$$ −17.2170 −0.757934
$$517$$ 3.41938 0.150384
$$518$$ −50.0046 −2.19708
$$519$$ 26.8859 1.18016
$$520$$ 6.39885 0.280608
$$521$$ −0.982633 −0.0430499 −0.0215250 0.999768i $$-0.506852\pi$$
−0.0215250 + 0.999768i $$0.506852\pi$$
$$522$$ 20.7226 0.907003
$$523$$ 39.7209 1.73687 0.868436 0.495801i $$-0.165125\pi$$
0.868436 + 0.495801i $$0.165125\pi$$
$$524$$ 117.251 5.12212
$$525$$ −4.24993 −0.185482
$$526$$ −17.6809 −0.770922
$$527$$ 18.4604 0.804148
$$528$$ −20.9289 −0.910812
$$529$$ −22.8257 −0.992423
$$530$$ −30.3037 −1.31631
$$531$$ −1.90417 −0.0826337
$$532$$ 0 0
$$533$$ −2.58195 −0.111837
$$534$$ 18.2936 0.791642
$$535$$ −6.43336 −0.278139
$$536$$ −25.1883 −1.08797
$$537$$ 30.0656 1.29743
$$538$$ 49.6198 2.13926
$$539$$ −0.964024 −0.0415234
$$540$$ −31.5698 −1.35855
$$541$$ 30.7775 1.32323 0.661614 0.749845i $$-0.269872\pi$$
0.661614 + 0.749845i $$0.269872\pi$$
$$542$$ 32.7705 1.40761
$$543$$ −25.5280 −1.09551
$$544$$ 92.9629 3.98575
$$545$$ 6.56882 0.281377
$$546$$ −7.54015 −0.322688
$$547$$ −17.8657 −0.763884 −0.381942 0.924186i $$-0.624745\pi$$
−0.381942 + 0.924186i $$0.624745\pi$$
$$548$$ 29.2374 1.24896
$$549$$ −4.90426 −0.209309
$$550$$ −2.38513 −0.101702
$$551$$ 0 0
$$552$$ 6.19529 0.263689
$$553$$ −5.19741 −0.221016
$$554$$ 65.1554 2.76819
$$555$$ 9.49557 0.403064
$$556$$ −60.1248 −2.54986
$$557$$ −10.6576 −0.451575 −0.225787 0.974177i $$-0.572495\pi$$
−0.225787 + 0.974177i $$0.572495\pi$$
$$558$$ −10.5340 −0.445939
$$559$$ −1.32406 −0.0560019
$$560$$ −46.2085 −1.95266
$$561$$ −4.82778 −0.203829
$$562$$ 38.0863 1.60657
$$563$$ 7.75961 0.327029 0.163514 0.986541i $$-0.447717\pi$$
0.163514 + 0.986541i $$0.447717\pi$$
$$564$$ −33.0735 −1.39265
$$565$$ 0.294513 0.0123903
$$566$$ −32.4270 −1.36301
$$567$$ 17.3143 0.727133
$$568$$ −17.7272 −0.743818
$$569$$ 5.72754 0.240111 0.120056 0.992767i $$-0.461693\pi$$
0.120056 + 0.992767i $$0.461693\pi$$
$$570$$ 0 0
$$571$$ −20.8347 −0.871903 −0.435952 0.899970i $$-0.643588\pi$$
−0.435952 + 0.899970i $$0.643588\pi$$
$$572$$ −3.11904 −0.130413
$$573$$ 7.87852 0.329130
$$574$$ 31.5341 1.31621
$$575$$ 0.417460 0.0174093
$$576$$ −27.9307 −1.16378
$$577$$ −5.11190 −0.212811 −0.106406 0.994323i $$-0.533934\pi$$
−0.106406 + 0.994323i $$0.533934\pi$$
$$578$$ −8.27001 −0.343987
$$579$$ 26.8600 1.11626
$$580$$ −54.4138 −2.25941
$$581$$ 21.1914 0.879167
$$582$$ −44.6619 −1.85130
$$583$$ 9.50199 0.393532
$$584$$ −70.9325 −2.93521
$$585$$ −0.498015 −0.0205904
$$586$$ −74.6254 −3.08275
$$587$$ −10.6692 −0.440367 −0.220184 0.975458i $$-0.570666\pi$$
−0.220184 + 0.975458i $$0.570666\pi$$
$$588$$ 9.32439 0.384531
$$589$$ 0 0
$$590$$ 6.78360 0.279276
$$591$$ 12.0515 0.495732
$$592$$ 103.243 4.24326
$$593$$ −17.0027 −0.698216 −0.349108 0.937083i $$-0.613515\pi$$
−0.349108 + 0.937083i $$0.613515\pi$$
$$594$$ 13.4301 0.551045
$$595$$ −10.6592 −0.436984
$$596$$ −83.6040 −3.42455
$$597$$ 2.09427 0.0857128
$$598$$ 0.740650 0.0302874
$$599$$ 28.6751 1.17163 0.585816 0.810444i $$-0.300774\pi$$
0.585816 + 0.810444i $$0.300774\pi$$
$$600$$ 14.8404 0.605858
$$601$$ 27.4370 1.11918 0.559590 0.828770i $$-0.310959\pi$$
0.559590 + 0.828770i $$0.310959\pi$$
$$602$$ 16.1712 0.659088
$$603$$ 1.96037 0.0798326
$$604$$ 120.313 4.89546
$$605$$ −10.2521 −0.416808
$$606$$ −21.7915 −0.885221
$$607$$ −17.7547 −0.720639 −0.360320 0.932829i $$-0.617332\pi$$
−0.360320 + 0.932829i $$0.617332\pi$$
$$608$$ 0 0
$$609$$ 41.2466 1.67140
$$610$$ 17.4715 0.707399
$$611$$ −2.54351 −0.102899
$$612$$ −16.2417 −0.656533
$$613$$ 34.6391 1.39906 0.699530 0.714603i $$-0.253393\pi$$
0.699530 + 0.714603i $$0.253393\pi$$
$$614$$ 24.3760 0.983736
$$615$$ −5.98814 −0.241465
$$616$$ 24.5050 0.987334
$$617$$ 4.46569 0.179782 0.0898909 0.995952i $$-0.471348\pi$$
0.0898909 + 0.995952i $$0.471348\pi$$
$$618$$ −1.58472 −0.0637470
$$619$$ −17.9112 −0.719913 −0.359957 0.932969i $$-0.617209\pi$$
−0.359957 + 0.932969i $$0.617209\pi$$
$$620$$ 27.6604 1.11087
$$621$$ −2.35062 −0.0943272
$$622$$ −1.79683 −0.0720463
$$623$$ −12.6647 −0.507400
$$624$$ 15.5679 0.623215
$$625$$ 1.00000 0.0400000
$$626$$ −8.17906 −0.326901
$$627$$ 0 0
$$628$$ −13.6308 −0.543927
$$629$$ 23.8157 0.949593
$$630$$ 6.08241 0.242329
$$631$$ 4.96881 0.197805 0.0989026 0.995097i $$-0.468467\pi$$
0.0989026 + 0.995097i $$0.468467\pi$$
$$632$$ 18.1489 0.721925
$$633$$ 28.2216 1.12171
$$634$$ −28.6278 −1.13696
$$635$$ 8.83492 0.350603
$$636$$ −91.9067 −3.64434
$$637$$ 0.717088 0.0284121
$$638$$ 23.1483 0.916449
$$639$$ 1.37969 0.0545796
$$640$$ 49.8151 1.96912
$$641$$ −37.9521 −1.49902 −0.749508 0.661995i $$-0.769710\pi$$
−0.749508 + 0.661995i $$0.769710\pi$$
$$642$$ −26.4716 −1.04475
$$643$$ 35.2502 1.39013 0.695067 0.718945i $$-0.255375\pi$$
0.695067 + 0.718945i $$0.255375\pi$$
$$644$$ −6.66740 −0.262732
$$645$$ −3.07081 −0.120913
$$646$$ 0 0
$$647$$ 35.5219 1.39651 0.698254 0.715850i $$-0.253960\pi$$
0.698254 + 0.715850i $$0.253960\pi$$
$$648$$ −60.4602 −2.37510
$$649$$ −2.12706 −0.0834943
$$650$$ 1.77418 0.0695890
$$651$$ −20.9670 −0.821762
$$652$$ 99.8966 3.91225
$$653$$ 8.02411 0.314008 0.157004 0.987598i $$-0.449816\pi$$
0.157004 + 0.987598i $$0.449816\pi$$
$$654$$ 27.0290 1.05692
$$655$$ 20.9128 0.817129
$$656$$ −65.1075 −2.54202
$$657$$ 5.52059 0.215379
$$658$$ 31.0646 1.21102
$$659$$ −47.2195 −1.83941 −0.919706 0.392608i $$-0.871573\pi$$
−0.919706 + 0.392608i $$0.871573\pi$$
$$660$$ −7.23377 −0.281574
$$661$$ −26.1159 −1.01579 −0.507896 0.861418i $$-0.669577\pi$$
−0.507896 + 0.861418i $$0.669577\pi$$
$$662$$ 41.6246 1.61778
$$663$$ 3.59114 0.139468
$$664$$ −73.9986 −2.87170
$$665$$ 0 0
$$666$$ −13.5898 −0.526596
$$667$$ −4.05155 −0.156877
$$668$$ −2.27405 −0.0879856
$$669$$ −24.0915 −0.931431
$$670$$ −6.98384 −0.269809
$$671$$ −5.47833 −0.211489
$$672$$ −105.586 −4.07306
$$673$$ 15.3820 0.592931 0.296466 0.955044i $$-0.404192\pi$$
0.296466 + 0.955044i $$0.404192\pi$$
$$674$$ 43.5696 1.67824
$$675$$ −5.63077 −0.216728
$$676$$ −70.5664 −2.71409
$$677$$ 24.4763 0.940701 0.470350 0.882480i $$-0.344128\pi$$
0.470350 + 0.882480i $$0.344128\pi$$
$$678$$ 1.21184 0.0465405
$$679$$ 30.9195 1.18658
$$680$$ 37.2210 1.42736
$$681$$ 39.2652 1.50465
$$682$$ −11.7670 −0.450583
$$683$$ −17.8502 −0.683018 −0.341509 0.939879i $$-0.610938\pi$$
−0.341509 + 0.939879i $$0.610938\pi$$
$$684$$ 0 0
$$685$$ 5.21477 0.199246
$$686$$ 46.2381 1.76538
$$687$$ −19.8908 −0.758880
$$688$$ −33.3881 −1.27291
$$689$$ −7.06804 −0.269271
$$690$$ 1.71774 0.0653932
$$691$$ −9.27242 −0.352739 −0.176370 0.984324i $$-0.556435\pi$$
−0.176370 + 0.984324i $$0.556435\pi$$
$$692$$ 101.038 3.84087
$$693$$ −1.90719 −0.0724483
$$694$$ −58.8896 −2.23542
$$695$$ −10.7238 −0.406778
$$696$$ −144.030 −5.45943
$$697$$ −15.0187 −0.568875
$$698$$ −89.3314 −3.38124
$$699$$ 37.7596 1.42820
$$700$$ −15.9713 −0.603659
$$701$$ −7.68906 −0.290412 −0.145206 0.989401i $$-0.546384\pi$$
−0.145206 + 0.989401i $$0.546384\pi$$
$$702$$ −9.98999 −0.377048
$$703$$ 0 0
$$704$$ −31.2001 −1.17590
$$705$$ −5.89898 −0.222168
$$706$$ 2.01496 0.0758340
$$707$$ 15.0863 0.567379
$$708$$ 20.5737 0.773206
$$709$$ 24.4375 0.917769 0.458885 0.888496i $$-0.348249\pi$$
0.458885 + 0.888496i $$0.348249\pi$$
$$710$$ −4.91514 −0.184462
$$711$$ −1.41251 −0.0529732
$$712$$ 44.2241 1.65737
$$713$$ 2.05954 0.0771303
$$714$$ −43.8597 −1.64141
$$715$$ −0.556310 −0.0208048
$$716$$ 112.987 4.22252
$$717$$ −35.1346 −1.31213
$$718$$ −74.0516 −2.76358
$$719$$ −22.1126 −0.824662 −0.412331 0.911034i $$-0.635285\pi$$
−0.412331 + 0.911034i $$0.635285\pi$$
$$720$$ −12.5582 −0.468015
$$721$$ 1.09711 0.0408584
$$722$$ 0 0
$$723$$ 12.5085 0.465195
$$724$$ −95.9345 −3.56538
$$725$$ −9.70523 −0.360443
$$726$$ −42.1848 −1.56562
$$727$$ −29.0494 −1.07738 −0.538692 0.842503i $$-0.681081\pi$$
−0.538692 + 0.842503i $$0.681081\pi$$
$$728$$ −18.2280 −0.675575
$$729$$ 29.9075 1.10768
$$730$$ −19.6671 −0.727913
$$731$$ −7.70184 −0.284863
$$732$$ 52.9884 1.95851
$$733$$ 14.5428 0.537151 0.268576 0.963259i $$-0.413447\pi$$
0.268576 + 0.963259i $$0.413447\pi$$
$$734$$ −63.3360 −2.33777
$$735$$ 1.66309 0.0613442
$$736$$ 10.3714 0.382296
$$737$$ 2.18984 0.0806640
$$738$$ 8.57009 0.315469
$$739$$ −4.75596 −0.174951 −0.0874754 0.996167i $$-0.527880\pi$$
−0.0874754 + 0.996167i $$0.527880\pi$$
$$740$$ 35.6845 1.31179
$$741$$ 0 0
$$742$$ 86.3242 3.16906
$$743$$ 5.87705 0.215608 0.107804 0.994172i $$-0.465618\pi$$
0.107804 + 0.994172i $$0.465618\pi$$
$$744$$ 73.2151 2.68420
$$745$$ −14.9116 −0.546317
$$746$$ 81.4455 2.98193
$$747$$ 5.75922 0.210719
$$748$$ −18.1429 −0.663370
$$749$$ 18.3263 0.669629
$$750$$ 4.11474 0.150249
$$751$$ 1.62096 0.0591498 0.0295749 0.999563i $$-0.490585\pi$$
0.0295749 + 0.999563i $$0.490585\pi$$
$$752$$ −64.1382 −2.33888
$$753$$ 27.2243 0.992107
$$754$$ −17.2188 −0.627072
$$755$$ 21.4589 0.780971
$$756$$ 89.9308 3.27075
$$757$$ −28.1135 −1.02180 −0.510901 0.859639i $$-0.670688\pi$$
−0.510901 + 0.859639i $$0.670688\pi$$
$$758$$ 48.2976 1.75425
$$759$$ −0.538612 −0.0195504
$$760$$ 0 0
$$761$$ 20.1663 0.731027 0.365514 0.930806i $$-0.380893\pi$$
0.365514 + 0.930806i $$0.380893\pi$$
$$762$$ 36.3534 1.31694
$$763$$ −18.7122 −0.677427
$$764$$ 29.6076 1.07117
$$765$$ −2.89687 −0.104736
$$766$$ 22.3579 0.807825
$$767$$ 1.58221 0.0571303
$$768$$ 97.3257 3.51194
$$769$$ 45.3047 1.63373 0.816865 0.576829i $$-0.195710\pi$$
0.816865 + 0.576829i $$0.195710\pi$$
$$770$$ 6.79438 0.244853
$$771$$ 21.0357 0.757583
$$772$$ 100.940 3.63292
$$773$$ −20.1762 −0.725686 −0.362843 0.931850i $$-0.618194\pi$$
−0.362843 + 0.931850i $$0.618194\pi$$
$$774$$ 4.39487 0.157970
$$775$$ 4.93349 0.177216
$$776$$ −107.968 −3.87584
$$777$$ −27.0494 −0.970393
$$778$$ 47.7353 1.71139
$$779$$ 0 0
$$780$$ 5.38083 0.192665
$$781$$ 1.54119 0.0551480
$$782$$ 4.30823 0.154062
$$783$$ 54.6479 1.95296
$$784$$ 18.0824 0.645800
$$785$$ −2.43118 −0.0867724
$$786$$ 86.0505 3.06932
$$787$$ −46.1385 −1.64466 −0.822331 0.569010i $$-0.807327\pi$$
−0.822331 + 0.569010i $$0.807327\pi$$
$$788$$ 45.2897 1.61338
$$789$$ −9.56426 −0.340497
$$790$$ 5.03207 0.179033
$$791$$ −0.838961 −0.0298300
$$792$$ 6.65976 0.236644
$$793$$ 4.07505 0.144709
$$794$$ −31.4122 −1.11478
$$795$$ −16.3924 −0.581380
$$796$$ 7.87031 0.278956
$$797$$ −1.86497 −0.0660606 −0.0330303 0.999454i $$-0.510516\pi$$
−0.0330303 + 0.999454i $$0.510516\pi$$
$$798$$ 0 0
$$799$$ −14.7951 −0.523414
$$800$$ 24.8441 0.878371
$$801$$ −3.44191 −0.121614
$$802$$ −24.6528 −0.870522
$$803$$ 6.16681 0.217622
$$804$$ −21.1810 −0.746996
$$805$$ −1.18919 −0.0419136
$$806$$ 8.75290 0.308308
$$807$$ 26.8413 0.944859
$$808$$ −52.6802 −1.85328
$$809$$ 18.2267 0.640816 0.320408 0.947280i $$-0.396180\pi$$
0.320408 + 0.947280i $$0.396180\pi$$
$$810$$ −16.7635 −0.589010
$$811$$ −20.9779 −0.736634 −0.368317 0.929700i $$-0.620066\pi$$
−0.368317 + 0.929700i $$0.620066\pi$$
$$812$$ 155.005 5.43962
$$813$$ 17.7268 0.621707
$$814$$ −15.1806 −0.532079
$$815$$ 17.8175 0.624120
$$816$$ 90.5558 3.17009
$$817$$ 0 0
$$818$$ −18.0504 −0.631118
$$819$$ 1.41866 0.0495721
$$820$$ −22.5035 −0.785857
$$821$$ −22.0867 −0.770829 −0.385415 0.922743i $$-0.625942\pi$$
−0.385415 + 0.922743i $$0.625942\pi$$
$$822$$ 21.4574 0.748413
$$823$$ 15.9769 0.556921 0.278461 0.960448i $$-0.410176\pi$$
0.278461 + 0.960448i $$0.410176\pi$$
$$824$$ −3.83101 −0.133460
$$825$$ −1.29021 −0.0449194
$$826$$ −19.3240 −0.672368
$$827$$ 49.2009 1.71088 0.855441 0.517901i $$-0.173286\pi$$
0.855441 + 0.517901i $$0.173286\pi$$
$$828$$ −1.81201 −0.0629717
$$829$$ −35.8564 −1.24534 −0.622672 0.782483i $$-0.713953\pi$$
−0.622672 + 0.782483i $$0.713953\pi$$
$$830$$ −20.5172 −0.712164
$$831$$ 35.2451 1.22264
$$832$$ 23.2082 0.804599
$$833$$ 4.17118 0.144523
$$834$$ −44.1257 −1.52795
$$835$$ −0.405598 −0.0140363
$$836$$ 0 0
$$837$$ −27.7794 −0.960195
$$838$$ 60.2463 2.08117
$$839$$ −25.4830 −0.879770 −0.439885 0.898054i $$-0.644981\pi$$
−0.439885 + 0.898054i $$0.644981\pi$$
$$840$$ −42.2750 −1.45863
$$841$$ 65.1914 2.24798
$$842$$ −80.9294 −2.78901
$$843$$ 20.6024 0.709583
$$844$$ 106.057 3.65065
$$845$$ −12.5862 −0.432978
$$846$$ 8.44249 0.290259
$$847$$ 29.2046 1.00348
$$848$$ −178.231 −6.12047
$$849$$ −17.5410 −0.602007
$$850$$ 10.3201 0.353976
$$851$$ 2.65700 0.0910807
$$852$$ −14.9069 −0.510703
$$853$$ −57.1622 −1.95720 −0.978598 0.205782i $$-0.934026\pi$$
−0.978598 + 0.205782i $$0.934026\pi$$
$$854$$ −49.7698 −1.70309
$$855$$ 0 0
$$856$$ −63.9940 −2.18727
$$857$$ 26.1974 0.894885 0.447442 0.894313i $$-0.352335\pi$$
0.447442 + 0.894313i $$0.352335\pi$$
$$858$$ −2.28907 −0.0781475
$$859$$ −17.7449 −0.605449 −0.302724 0.953078i $$-0.597896\pi$$
−0.302724 + 0.953078i $$0.597896\pi$$
$$860$$ −11.5401 −0.393516
$$861$$ 17.0580 0.581336
$$862$$ −35.5418 −1.21056
$$863$$ 25.0867 0.853960 0.426980 0.904261i $$-0.359578\pi$$
0.426980 + 0.904261i $$0.359578\pi$$
$$864$$ −139.891 −4.75920
$$865$$ 18.0210 0.612733
$$866$$ −38.1789 −1.29737
$$867$$ −4.47357 −0.151930
$$868$$ −78.7944 −2.67446
$$869$$ −1.57785 −0.0535249
$$870$$ −39.9344 −1.35390
$$871$$ −1.62891 −0.0551936
$$872$$ 65.3415 2.21274
$$873$$ 8.40305 0.284400
$$874$$ 0 0
$$875$$ −2.84864 −0.0963015
$$876$$ −59.6476 −2.01531
$$877$$ 10.0157 0.338205 0.169103 0.985598i $$-0.445913\pi$$
0.169103 + 0.985598i $$0.445913\pi$$
$$878$$ −0.195290 −0.00659071
$$879$$ −40.3678 −1.36157
$$880$$ −14.0282 −0.472889
$$881$$ 33.3473 1.12350 0.561750 0.827307i $$-0.310128\pi$$
0.561750 + 0.827307i $$0.310128\pi$$
$$882$$ −2.38018 −0.0801449
$$883$$ −27.3570 −0.920638 −0.460319 0.887754i $$-0.652265\pi$$
−0.460319 + 0.887754i $$0.652265\pi$$
$$884$$ 13.4956 0.453905
$$885$$ 3.66951 0.123349
$$886$$ −10.4505 −0.351092
$$887$$ 17.1634 0.576292 0.288146 0.957586i $$-0.406961\pi$$
0.288146 + 0.957586i $$0.406961\pi$$
$$888$$ 94.4544 3.16968
$$889$$ −25.1675 −0.844090
$$890$$ 12.2618 0.411016
$$891$$ 5.25635 0.176094
$$892$$ −90.5363 −3.03138
$$893$$ 0 0
$$894$$ −61.3571 −2.05209
$$895$$ 20.1523 0.673617
$$896$$ −141.905 −4.74072
$$897$$ 0.400646 0.0133772
$$898$$ −73.2985 −2.44600
$$899$$ −47.8807 −1.59691
$$900$$ −4.34056 −0.144685
$$901$$ −41.1136 −1.36969
$$902$$ 9.57325 0.318754
$$903$$ 8.74762 0.291103
$$904$$ 2.92958 0.0974364
$$905$$ −17.1108 −0.568783
$$906$$ 88.2979 2.93350
$$907$$ 3.02106 0.100313 0.0501563 0.998741i $$-0.484028\pi$$
0.0501563 + 0.998741i $$0.484028\pi$$
$$908$$ 147.559 4.89693
$$909$$ 4.10004 0.135990
$$910$$ −5.05399 −0.167538
$$911$$ −19.5682 −0.648324 −0.324162 0.946002i $$-0.605082\pi$$
−0.324162 + 0.946002i $$0.605082\pi$$
$$912$$ 0 0
$$913$$ 6.43336 0.212913
$$914$$ −91.4346 −3.02439
$$915$$ 9.45099 0.312440
$$916$$ −74.7498 −2.46980
$$917$$ −59.5729 −1.96727
$$918$$ −58.1100 −1.91792
$$919$$ −1.81420 −0.0598448 −0.0299224 0.999552i $$-0.509526\pi$$
−0.0299224 + 0.999552i $$0.509526\pi$$
$$920$$ 4.15257 0.136906
$$921$$ 13.1859 0.434491
$$922$$ −53.1017 −1.74881
$$923$$ −1.14641 −0.0377346
$$924$$ 20.6064 0.677901
$$925$$ 6.36467 0.209269
$$926$$ 108.035 3.55025
$$927$$ 0.298163 0.00979295
$$928$$ −241.117 −7.91507
$$929$$ −22.4754 −0.737395 −0.368698 0.929549i $$-0.620196\pi$$
−0.368698 + 0.929549i $$0.620196\pi$$
$$930$$ 20.3000 0.665664
$$931$$ 0 0
$$932$$ 141.901 4.64813
$$933$$ −0.971975 −0.0318210
$$934$$ −107.742 −3.52542
$$935$$ −3.23596 −0.105827
$$936$$ −4.95386 −0.161922
$$937$$ 55.0385 1.79803 0.899015 0.437918i $$-0.144284\pi$$
0.899015 + 0.437918i $$0.144284\pi$$
$$938$$ 19.8944 0.649576
$$939$$ −4.42437 −0.144384
$$940$$ −22.1685 −0.723056
$$941$$ −12.1956 −0.397566 −0.198783 0.980044i $$-0.563699\pi$$
−0.198783 + 0.980044i $$0.563699\pi$$
$$942$$ −10.0036 −0.325937
$$943$$ −1.67557 −0.0545640
$$944$$ 39.8977 1.29856
$$945$$ 16.0400 0.521782
$$946$$ 4.90931 0.159615
$$947$$ 30.4307 0.988863 0.494432 0.869216i $$-0.335376\pi$$
0.494432 + 0.869216i $$0.335376\pi$$
$$948$$ 15.2615 0.495672
$$949$$ −4.58717 −0.148906
$$950$$ 0 0
$$951$$ −15.4859 −0.502164
$$952$$ −106.029 −3.43642
$$953$$ 15.7378 0.509798 0.254899 0.966968i $$-0.417958\pi$$
0.254899 + 0.966968i $$0.417958\pi$$
$$954$$ 23.4605 0.759561
$$955$$ 5.28080 0.170883
$$956$$ −132.036 −4.27036
$$957$$ 12.5218 0.404772
$$958$$ −68.2748 −2.20586
$$959$$ −14.8550 −0.479693
$$960$$ 53.8252 1.73720
$$961$$ −6.66065 −0.214860
$$962$$ 11.2921 0.364071
$$963$$ 4.98058 0.160497
$$964$$ 47.0070 1.51399
$$965$$ 18.0036 0.579558
$$966$$ −4.89322 −0.157437
$$967$$ 30.6373 0.985228 0.492614 0.870248i $$-0.336042\pi$$
0.492614 + 0.870248i $$0.336042\pi$$
$$968$$ −101.980 −3.27776
$$969$$ 0 0
$$970$$ −29.9359 −0.961184
$$971$$ 16.4765 0.528755 0.264378 0.964419i $$-0.414833\pi$$
0.264378 + 0.964419i $$0.414833\pi$$
$$972$$ 43.8679 1.40706
$$973$$ 30.5483 0.979334
$$974$$ 60.3649 1.93422
$$975$$ 0.959723 0.0307357
$$976$$ 102.758 3.28921
$$977$$ −2.77995 −0.0889383 −0.0444692 0.999011i $$-0.514160\pi$$
−0.0444692 + 0.999011i $$0.514160\pi$$
$$978$$ 73.3144 2.34433
$$979$$ −3.84480 −0.122880
$$980$$ 6.24993 0.199647
$$981$$ −5.08545 −0.162366
$$982$$ 25.9130 0.826918
$$983$$ −1.71171 −0.0545951 −0.0272976 0.999627i $$-0.508690\pi$$
−0.0272976 + 0.999627i $$0.508690\pi$$
$$984$$ −59.5653 −1.89887
$$985$$ 8.07785 0.257382
$$986$$ −100.159 −3.18971
$$987$$ 16.8041 0.534879
$$988$$ 0 0
$$989$$ −0.859257 −0.0273228
$$990$$ 1.84652 0.0586863
$$991$$ −9.67421 −0.307311 −0.153656 0.988124i $$-0.549105\pi$$
−0.153656 + 0.988124i $$0.549105\pi$$
$$992$$ 122.568 3.89154
$$993$$ 22.5164 0.714535
$$994$$ 14.0015 0.444099
$$995$$ 1.40374 0.0445017
$$996$$ −62.2259 −1.97170
$$997$$ 22.7311 0.719902 0.359951 0.932971i $$-0.382793\pi$$
0.359951 + 0.932971i $$0.382793\pi$$
$$998$$ 68.7064 2.17486
$$999$$ −35.8380 −1.13386
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 1805.2.a.o.1.4 4
5.4 even 2 9025.2.a.bg.1.1 4
19.7 even 3 95.2.e.c.11.1 8
19.11 even 3 95.2.e.c.26.1 yes 8
19.18 odd 2 1805.2.a.i.1.1 4
57.11 odd 6 855.2.k.h.406.4 8
57.26 odd 6 855.2.k.h.676.4 8
76.7 odd 6 1520.2.q.o.961.3 8
76.11 odd 6 1520.2.q.o.881.3 8
95.7 odd 12 475.2.j.c.49.1 16
95.49 even 6 475.2.e.e.26.4 8
95.64 even 6 475.2.e.e.201.4 8
95.68 odd 12 475.2.j.c.349.1 16
95.83 odd 12 475.2.j.c.49.8 16
95.87 odd 12 475.2.j.c.349.8 16
95.94 odd 2 9025.2.a.bp.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.c.11.1 8 19.7 even 3
95.2.e.c.26.1 yes 8 19.11 even 3
475.2.e.e.26.4 8 95.49 even 6
475.2.e.e.201.4 8 95.64 even 6
475.2.j.c.49.1 16 95.7 odd 12
475.2.j.c.49.8 16 95.83 odd 12
475.2.j.c.349.1 16 95.68 odd 12
475.2.j.c.349.8 16 95.87 odd 12
855.2.k.h.406.4 8 57.11 odd 6
855.2.k.h.676.4 8 57.26 odd 6
1520.2.q.o.881.3 8 76.11 odd 6
1520.2.q.o.961.3 8 76.7 odd 6
1805.2.a.i.1.1 4 19.18 odd 2
1805.2.a.o.1.4 4 1.1 even 1 trivial
9025.2.a.bg.1.1 4 5.4 even 2
9025.2.a.bp.1.4 4 95.94 odd 2