# Properties

 Label 9025.2.a.bp.1.4 Level $9025$ Weight $2$ Character 9025.1 Self dual yes Analytic conductor $72.065$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$72.0649878242$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 9025.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.75802 q^{2} +1.49192 q^{3} +5.60665 q^{4} +4.11474 q^{6} +2.84864 q^{7} +9.94721 q^{8} -0.774179 q^{9} +O(q^{10})$$ $$q+2.75802 q^{2} +1.49192 q^{3} +5.60665 q^{4} +4.11474 q^{6} +2.84864 q^{7} +9.94721 q^{8} -0.774179 q^{9} -0.864801 q^{11} +8.36467 q^{12} +0.643281 q^{13} +7.85659 q^{14} +16.2213 q^{16} -3.74185 q^{17} -2.13520 q^{18} +4.24993 q^{21} -2.38513 q^{22} -0.417460 q^{23} +14.8404 q^{24} +1.77418 q^{26} -5.63077 q^{27} +15.9713 q^{28} +9.70523 q^{29} -4.93349 q^{31} +24.8441 q^{32} -1.29021 q^{33} -10.3201 q^{34} -4.34056 q^{36} +6.36467 q^{37} +0.959723 q^{39} +4.01372 q^{41} +11.7214 q^{42} +2.05829 q^{43} -4.84864 q^{44} -1.15136 q^{46} +3.95396 q^{47} +24.2008 q^{48} +1.11474 q^{49} -5.58254 q^{51} +3.60665 q^{52} -10.9875 q^{53} -15.5297 q^{54} +28.3360 q^{56} +26.7672 q^{58} -2.45959 q^{59} +6.33479 q^{61} -13.6067 q^{62} -2.20536 q^{63} +36.0778 q^{64} -3.55843 q^{66} -2.53220 q^{67} -20.9793 q^{68} -0.622817 q^{69} +1.78213 q^{71} -7.70092 q^{72} +7.13090 q^{73} +17.5539 q^{74} -2.46350 q^{77} +2.64693 q^{78} -1.82452 q^{79} -6.07811 q^{81} +11.0699 q^{82} +7.43913 q^{83} +23.8279 q^{84} +5.67681 q^{86} +14.4794 q^{87} -8.60235 q^{88} -4.44588 q^{89} +1.83247 q^{91} -2.34056 q^{92} -7.36037 q^{93} +10.9051 q^{94} +37.0653 q^{96} -10.8541 q^{97} +3.07446 q^{98} +0.669511 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} + 12 q^{8} + q^{9}+O(q^{10})$$ 4 * q + q^2 + 3 * q^3 + 5 * q^4 + 2 * q^6 + 4 * q^7 + 12 * q^8 + q^9 $$4 q + q^{2} + 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} + 12 q^{8} + q^{9} - 2 q^{11} + 6 q^{12} + 7 q^{13} + q^{14} + 7 q^{16} + q^{17} - 10 q^{18} + 4 q^{21} + 2 q^{22} - 2 q^{23} + 23 q^{24} + 3 q^{26} + 12 q^{27} + 19 q^{28} + q^{29} + 30 q^{32} + 19 q^{33} - 15 q^{34} - 7 q^{36} - 2 q^{37} + 15 q^{39} + 8 q^{41} + 15 q^{42} - q^{43} - 12 q^{44} - 12 q^{46} + 12 q^{47} + 23 q^{48} - 10 q^{49} - 22 q^{51} - 3 q^{52} - 5 q^{53} - 34 q^{54} + 41 q^{56} + 27 q^{58} + 5 q^{59} - 37 q^{62} + 3 q^{63} + 56 q^{64} - 31 q^{66} + 4 q^{67} - 16 q^{68} + 9 q^{69} - 20 q^{71} + 17 q^{72} + 20 q^{73} + 25 q^{74} - 14 q^{77} - 18 q^{78} - 17 q^{79} + 12 q^{81} - 21 q^{82} - q^{83} + 20 q^{84} - 8 q^{86} + 16 q^{87} - 7 q^{88} - 11 q^{89} - 6 q^{91} + q^{92} + 8 q^{93} + 31 q^{94} + 21 q^{96} + q^{97} + 9 q^{98} + 38 q^{99}+O(q^{100})$$ 4 * q + q^2 + 3 * q^3 + 5 * q^4 + 2 * q^6 + 4 * q^7 + 12 * q^8 + q^9 - 2 * q^11 + 6 * q^12 + 7 * q^13 + q^14 + 7 * q^16 + q^17 - 10 * q^18 + 4 * q^21 + 2 * q^22 - 2 * q^23 + 23 * q^24 + 3 * q^26 + 12 * q^27 + 19 * q^28 + q^29 + 30 * q^32 + 19 * q^33 - 15 * q^34 - 7 * q^36 - 2 * q^37 + 15 * q^39 + 8 * q^41 + 15 * q^42 - q^43 - 12 * q^44 - 12 * q^46 + 12 * q^47 + 23 * q^48 - 10 * q^49 - 22 * q^51 - 3 * q^52 - 5 * q^53 - 34 * q^54 + 41 * q^56 + 27 * q^58 + 5 * q^59 - 37 * q^62 + 3 * q^63 + 56 * q^64 - 31 * q^66 + 4 * q^67 - 16 * q^68 + 9 * q^69 - 20 * q^71 + 17 * q^72 + 20 * q^73 + 25 * q^74 - 14 * q^77 - 18 * q^78 - 17 * q^79 + 12 * q^81 - 21 * q^82 - q^83 + 20 * q^84 - 8 * q^86 + 16 * q^87 - 7 * q^88 - 11 * q^89 - 6 * q^91 + q^92 + 8 * q^93 + 31 * q^94 + 21 * q^96 + q^97 + 9 * q^98 + 38 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 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
$$5$$ 0 0
$$6$$ 4.11474 1.67983
$$7$$ 2.84864 1.07668 0.538342 0.842727i $$-0.319051\pi$$
0.538342 + 0.842727i $$0.319051\pi$$
$$8$$ 9.94721 3.51687
$$9$$ −0.774179 −0.258060
$$10$$ 0 0
$$11$$ −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
$$15$$ 0 0
$$16$$ 16.2213 4.05531
$$17$$ −3.74185 −0.907533 −0.453766 0.891121i $$-0.649920\pi$$
−0.453766 + 0.891121i $$0.649920\pi$$
$$18$$ −2.13520 −0.503271
$$19$$ 0 0
$$20$$ 0 0
$$21$$ 4.24993 0.927412
$$22$$ −2.38513 −0.508512
$$23$$ −0.417460 −0.0870465 −0.0435233 0.999052i $$-0.513858\pi$$
−0.0435233 + 0.999052i $$0.513858\pi$$
$$24$$ 14.8404 3.02929
$$25$$ 0 0
$$26$$ 1.77418 0.347945
$$27$$ −5.63077 −1.08364
$$28$$ 15.9713 3.01830
$$29$$ 9.70523 1.80222 0.901108 0.433596i $$-0.142755\pi$$
0.901108 + 0.433596i $$0.142755\pi$$
$$30$$ 0 0
$$31$$ −4.93349 −0.886081 −0.443041 0.896501i $$-0.646100\pi$$
−0.443041 + 0.896501i $$0.646100\pi$$
$$32$$ 24.8441 4.39185
$$33$$ −1.29021 −0.224597
$$34$$ −10.3201 −1.76988
$$35$$ 0 0
$$36$$ −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
$$40$$ 0 0
$$41$$ 4.01372 0.626837 0.313419 0.949615i $$-0.398526\pi$$
0.313419 + 0.949615i $$0.398526\pi$$
$$42$$ 11.7214 1.80865
$$43$$ 2.05829 0.313887 0.156944 0.987608i $$-0.449836\pi$$
0.156944 + 0.987608i $$0.449836\pi$$
$$44$$ −4.84864 −0.730960
$$45$$ 0 0
$$46$$ −1.15136 −0.169759
$$47$$ 3.95396 0.576744 0.288372 0.957518i $$-0.406886\pi$$
0.288372 + 0.957518i $$0.406886\pi$$
$$48$$ 24.2008 3.49308
$$49$$ 1.11474 0.159248
$$50$$ 0 0
$$51$$ −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 0
$$56$$ 28.3360 3.78656
$$57$$ 0 0
$$58$$ 26.7672 3.51470
$$59$$ −2.45959 −0.320212 −0.160106 0.987100i $$-0.551184\pi$$
−0.160106 + 0.987100i $$0.551184\pi$$
$$60$$ 0 0
$$61$$ 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 0
$$66$$ −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
$$70$$ 0 0
$$71$$ 1.78213 0.211500 0.105750 0.994393i $$-0.466276\pi$$
0.105750 + 0.994393i $$0.466276\pi$$
$$72$$ −7.70092 −0.907563
$$73$$ 7.13090 0.834609 0.417304 0.908767i $$-0.362975\pi$$
0.417304 + 0.908767i $$0.362975\pi$$
$$74$$ 17.5539 2.04060
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −2.46350 −0.280742
$$78$$ 2.64693 0.299706
$$79$$ −1.82452 −0.205275 −0.102637 0.994719i $$-0.532728\pi$$
−0.102637 + 0.994719i $$0.532728\pi$$
$$80$$ 0 0
$$81$$ −6.07811 −0.675345
$$82$$ 11.0699 1.22247
$$83$$ 7.43913 0.816550 0.408275 0.912859i $$-0.366130\pi$$
0.408275 + 0.912859i $$0.366130\pi$$
$$84$$ 23.8279 2.59984
$$85$$ 0 0
$$86$$ 5.67681 0.612146
$$87$$ 14.4794 1.55236
$$88$$ −8.60235 −0.917014
$$89$$ −4.44588 −0.471262 −0.235631 0.971843i $$-0.575716\pi$$
−0.235631 + 0.971843i $$0.575716\pi$$
$$90$$ 0 0
$$91$$ 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
$$100$$ 0 0
$$101$$ −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
$$105$$ 0 0
$$106$$ −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 $$-0.601867\pi$$
−0.314590 + 0.949228i $$0.601867\pi$$
$$110$$ 0 0
$$111$$ 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 0
$$116$$ 54.4138 5.05220
$$117$$ −0.498015 −0.0460415
$$118$$ −6.78360 −0.624481
$$119$$ −10.6592 −0.977126
$$120$$ 0 0
$$121$$ −10.2521 −0.932011
$$122$$ 17.4715 1.58179
$$123$$ 5.98814 0.539932
$$124$$ −27.6604 −2.48398
$$125$$ 0 0
$$126$$ −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
$$130$$ 0 0
$$131$$ 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
$$135$$ 0 0
$$136$$ −37.2210 −3.19167
$$137$$ −5.21477 −0.445528 −0.222764 0.974872i $$-0.571508\pi$$
−0.222764 + 0.974872i $$0.571508\pi$$
$$138$$ −1.71774 −0.146224
$$139$$ −10.7238 −0.909584 −0.454792 0.890598i $$-0.650286\pi$$
−0.454792 + 0.890598i $$0.650286\pi$$
$$140$$ 0 0
$$141$$ 5.89898 0.496784
$$142$$ 4.91514 0.412470
$$143$$ −0.556310 −0.0465210
$$144$$ −12.5582 −1.04651
$$145$$ 0 0
$$146$$ 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 $$0.709153\pi$$
$$150$$ 0 0
$$151$$ −21.4589 −1.74630 −0.873152 0.487448i $$-0.837928\pi$$
−0.873152 + 0.487448i $$0.837928\pi$$
$$152$$ 0 0
$$153$$ 2.89687 0.234198
$$154$$ −6.79438 −0.547507
$$155$$ 0 0
$$156$$ 5.38083 0.430811
$$157$$ 2.43118 0.194029 0.0970145 0.995283i $$-0.469071\pi$$
0.0970145 + 0.995283i $$0.469071\pi$$
$$158$$ −5.03207 −0.400330
$$159$$ −16.3924 −1.30000
$$160$$ 0 0
$$161$$ −1.18919 −0.0937216
$$162$$ −16.7635 −1.31707
$$163$$ −17.8175 −1.39558 −0.697788 0.716305i $$-0.745832\pi$$
−0.697788 + 0.716305i $$0.745832\pi$$
$$164$$ 22.5035 1.75723
$$165$$ 0 0
$$166$$ 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
$$170$$ 0 0
$$171$$ 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
$$175$$ 0 0
$$176$$ −14.0282 −1.05741
$$177$$ −3.66951 −0.275817
$$178$$ −12.2618 −0.919061
$$179$$ −20.1523 −1.50625 −0.753127 0.657875i $$-0.771455\pi$$
−0.753127 + 0.657875i $$0.771455\pi$$
$$180$$ 0 0
$$181$$ 17.1108 1.27184 0.635919 0.771756i $$-0.280621\pi$$
0.635919 + 0.771756i $$0.280621\pi$$
$$182$$ 5.05399 0.374627
$$183$$ 9.45099 0.698637
$$184$$ −4.15257 −0.306131
$$185$$ 0 0
$$186$$ −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 0
$$196$$ 6.24993 0.446424
$$197$$ −8.07785 −0.575523 −0.287761 0.957702i $$-0.592911\pi$$
−0.287761 + 0.957702i $$0.592911\pi$$
$$198$$ 1.84652 0.131227
$$199$$ 1.40374 0.0995088 0.0497544 0.998761i $$-0.484156\pi$$
0.0497544 + 0.998761i $$0.484156\pi$$
$$200$$ 0 0
$$201$$ −3.77783 −0.266468
$$202$$ −14.6064 −1.02770
$$203$$ 27.6467 1.94042
$$204$$ −31.2994 −2.19139
$$205$$ 0 0
$$206$$ −1.06221 −0.0740074
$$207$$ 0.323189 0.0224632
$$208$$ 10.4348 0.723525
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −18.9163 −1.30226 −0.651128 0.758968i $$-0.725704\pi$$
−0.651128 + 0.758968i $$0.725704\pi$$
$$212$$ −61.6030 −4.23091
$$213$$ 2.65879 0.182178
$$214$$ −17.7433 −1.21291
$$215$$ 0 0
$$216$$ −56.0104 −3.81103
$$217$$ −14.0537 −0.954030
$$218$$ −18.1169 −1.22703
$$219$$ 10.6387 0.718898
$$220$$ 0 0
$$221$$ −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 0
$$226$$ 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 $$0.645203\pi$$
$$230$$ 0 0
$$231$$ −3.67535 −0.241820
$$232$$ 96.5399 6.33816
$$233$$ −25.3094 −1.65808 −0.829038 0.559192i $$-0.811111\pi$$
−0.829038 + 0.559192i $$0.811111\pi$$
$$234$$ −1.37353 −0.0897907
$$235$$ 0 0
$$236$$ −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 $$0.775615\pi$$
$$240$$ 0 0
$$241$$ −8.38415 −0.540071 −0.270035 0.962850i $$-0.587035\pi$$
−0.270035 + 0.962850i $$0.587035\pi$$
$$242$$ −28.2755 −1.81762
$$243$$ 7.82426 0.501927
$$244$$ 35.5170 2.27374
$$245$$ 0 0
$$246$$ 16.5154 1.05298
$$247$$ 0 0
$$248$$ −49.0745 −3.11623
$$249$$ 11.0986 0.703343
$$250$$ 0 0
$$251$$ 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
$$255$$ 0 0
$$256$$ 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
$$260$$ 0 0
$$261$$ −7.51359 −0.465079
$$262$$ 57.6778 3.56334
$$263$$ 6.41071 0.395301 0.197651 0.980273i $$-0.436669\pi$$
0.197651 + 0.980273i $$0.436669\pi$$
$$264$$ −12.8340 −0.789879
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −6.63288 −0.405926
$$268$$ −14.1971 −0.867229
$$269$$ −17.9911 −1.09694 −0.548469 0.836171i $$-0.684789\pi$$
−0.548469 + 0.836171i $$0.684789\pi$$
$$270$$ 0 0
$$271$$ 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 0
$$276$$ −3.49192 −0.210189
$$277$$ −23.6240 −1.41943 −0.709715 0.704489i $$-0.751176\pi$$
−0.709715 + 0.704489i $$0.751176\pi$$
$$278$$ −29.5765 −1.77388
$$279$$ 3.81941 0.228662
$$280$$ 0 0
$$281$$ −13.8093 −0.823794 −0.411897 0.911230i $$-0.635134\pi$$
−0.411897 + 0.911230i $$0.635134\pi$$
$$282$$ 16.2695 0.968834
$$283$$ 11.7574 0.698903 0.349451 0.936954i $$-0.386368\pi$$
0.349451 + 0.936954i $$0.386368\pi$$
$$284$$ 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
$$290$$ 0 0
$$291$$ −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
$$295$$ 0 0
$$296$$ 63.3107 3.67986
$$297$$ 4.86949 0.282557
$$298$$ −41.1263 −2.38238
$$299$$ −0.268544 −0.0155303
$$300$$ 0 0
$$301$$ 5.86334 0.337957
$$302$$ −59.1841 −3.40566
$$303$$ −7.90117 −0.453910
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 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
$$310$$ 0 0
$$311$$ −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 $$-0.473291\pi$$
0.0838116 + 0.996482i $$0.473291\pi$$
$$314$$ 6.70523 0.378398
$$315$$ 0 0
$$316$$ −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
$$320$$ 0 0
$$321$$ −9.59805 −0.535711
$$322$$ −3.27981 −0.182777
$$323$$ 0 0
$$324$$ −34.0778 −1.89321
$$325$$ 0 0
$$326$$ −49.1410 −2.72167
$$327$$ −9.80015 −0.541949
$$328$$ 39.9253 2.20450
$$329$$ 11.2634 0.620971
$$330$$ 0 0
$$331$$ −15.0922 −0.829543 −0.414772 0.909926i $$-0.636139\pi$$
−0.414772 + 0.909926i $$0.636139\pi$$
$$332$$ 41.7086 2.28906
$$333$$ −4.92740 −0.270020
$$334$$ −1.11865 −0.0612096
$$335$$ 0 0
$$336$$ 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
$$340$$ 0 0
$$341$$ 4.26649 0.231043
$$342$$ 0 0
$$343$$ −16.7650 −0.905224
$$344$$ 20.4743 1.10390
$$345$$ 0 0
$$346$$ 49.7023 2.67201
$$347$$ 21.3522 1.14624 0.573122 0.819470i $$-0.305732\pi$$
0.573122 + 0.819470i $$0.305732\pi$$
$$348$$ 81.1810 4.35176
$$349$$ −32.3897 −1.73378 −0.866891 0.498497i $$-0.833885\pi$$
−0.866891 + 0.498497i $$0.833885\pi$$
$$350$$ 0 0
$$351$$ −3.62217 −0.193337
$$352$$ −21.4852 −1.14516
$$353$$ −0.730583 −0.0388850 −0.0194425 0.999811i $$-0.506189\pi$$
−0.0194425 + 0.999811i $$0.506189\pi$$
$$354$$ −10.1206 −0.537902
$$355$$ 0 0
$$356$$ −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 $$0.750643\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ 47.1919 2.48035
$$363$$ −15.2953 −0.802796
$$364$$ 10.2740 0.538506
$$365$$ 0 0
$$366$$ 26.0660 1.36249
$$367$$ 22.9643 1.19873 0.599364 0.800477i $$-0.295420\pi$$
0.599364 + 0.800477i $$0.295420\pi$$
$$368$$ −6.77173 −0.353001
$$369$$ −3.10734 −0.161761
$$370$$ 0 0
$$371$$ −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
$$375$$ 0 0
$$376$$ 39.3308 2.02833
$$377$$ 6.24319 0.321541
$$378$$ −44.2386 −2.27539
$$379$$ −17.5117 −0.899517 −0.449759 0.893150i $$-0.648490\pi$$
−0.449759 + 0.893150i $$0.648490\pi$$
$$380$$ 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
$$385$$ 0 0
$$386$$ 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 $$0.355414\pi$$
$$390$$ 0 0
$$391$$ 1.56208 0.0789976
$$392$$ 11.0885 0.560054
$$393$$ 31.2001 1.57384
$$394$$ −22.2788 −1.12239
$$395$$ 0 0
$$396$$ 3.75372 0.188631
$$397$$ 11.3894 0.571619 0.285810 0.958286i $$-0.407737\pi$$
0.285810 + 0.958286i $$0.407737\pi$$
$$398$$ 3.87155 0.194063
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 8.93861 0.446373 0.223186 0.974776i $$-0.428354\pi$$
0.223186 + 0.974776i $$0.428354\pi$$
$$402$$ −10.4193 −0.519668
$$403$$ −3.17362 −0.158089
$$404$$ −29.6927 −1.47727
$$405$$ 0 0
$$406$$ 76.2500 3.78422
$$407$$ −5.50417 −0.272832
$$408$$ −55.5307 −2.74918
$$409$$ 6.54471 0.323615 0.161808 0.986822i $$-0.448268\pi$$
0.161808 + 0.986822i $$0.448268\pi$$
$$410$$ 0 0
$$411$$ −7.78001 −0.383760
$$412$$ −2.15931 −0.106382
$$413$$ −7.00649 −0.344767
$$414$$ 0.891361 0.0438080
$$415$$ 0 0
$$416$$ 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 $$0.320848\pi$$
$$420$$ 0 0
$$421$$ 29.3434 1.43011 0.715054 0.699069i $$-0.246402\pi$$
0.715054 + 0.699069i $$0.246402\pi$$
$$422$$ −52.1716 −2.53967
$$423$$ −3.06107 −0.148834
$$424$$ −109.295 −5.30783
$$425$$ 0 0
$$426$$ 7.33299 0.355285
$$427$$ 18.0455 0.873284
$$428$$ −36.0696 −1.74349
$$429$$ −0.829969 −0.0400713
$$430$$ 0 0
$$431$$ 12.8867 0.620732 0.310366 0.950617i $$-0.399548\pi$$
0.310366 + 0.950617i $$0.399548\pi$$
$$432$$ −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
$$435$$ 0 0
$$436$$ −36.8291 −1.76379
$$437$$ 0 0
$$438$$ 29.3418 1.40200
$$439$$ 0.0708081 0.00337948 0.00168974 0.999999i $$-0.499462\pi$$
0.00168974 + 0.999999i $$0.499462\pi$$
$$440$$ 0 0
$$441$$ −0.863005 −0.0410955
$$442$$ −6.63872 −0.315772
$$443$$ 3.78914 0.180027 0.0900137 0.995941i $$-0.471309\pi$$
0.0900137 + 0.995941i $$0.471309\pi$$
$$444$$ 53.2384 2.52658
$$445$$ 0 0
$$446$$ −44.5365 −2.10886
$$447$$ −22.2468 −1.05224
$$448$$ 102.773 4.85555
$$449$$ 26.5765 1.25422 0.627112 0.778929i $$-0.284237\pi$$
0.627112 + 0.778929i $$0.284237\pi$$
$$450$$ 0 0
$$451$$ −3.47106 −0.163446
$$452$$ 1.65123 0.0776674
$$453$$ −32.0150 −1.50420
$$454$$ 72.5872 3.40669
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 33.1523 1.55080 0.775400 0.631471i $$-0.217548\pi$$
0.775400 + 0.631471i $$0.217548\pi$$
$$458$$ −36.7708 −1.71819
$$459$$ 21.0695 0.983440
$$460$$ 0 0
$$461$$ −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 $$-0.864089\pi$$
−0.910222 + 0.414120i $$0.864089\pi$$
$$464$$ 157.431 7.30855
$$465$$ 0 0
$$466$$ −69.8038 −3.23360
$$467$$ 39.0650 1.80771 0.903856 0.427836i $$-0.140724\pi$$
0.903856 + 0.427836i $$0.140724\pi$$
$$468$$ −2.79220 −0.129069
$$469$$ −7.21331 −0.333080
$$470$$ 0 0
$$471$$ 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 $$0.691334\pi$$
$$480$$ 0 0
$$481$$ 4.09427 0.186683
$$482$$ −23.1236 −1.05325
$$483$$ −1.77418 −0.0807280
$$484$$ −57.4801 −2.61273
$$485$$ 0 0
$$486$$ 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
$$490$$ 0 0
$$491$$ 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 0
$$496$$ −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 $$0.311724\pi$$
$$500$$ 0 0
$$501$$ −0.605119 −0.0270347
$$502$$ 50.3278 2.24624
$$503$$ 31.3180 1.39640 0.698200 0.715903i $$-0.253985\pi$$
0.698200 + 0.715903i $$0.253985\pi$$
$$504$$ −21.9371 −0.977158
$$505$$ 0 0
$$506$$ 0.995699 0.0442642
$$507$$ −18.7776 −0.833941
$$508$$ 49.5343 2.19773
$$509$$ 9.66619 0.428446 0.214223 0.976785i $$-0.431278\pi$$
0.214223 + 0.976785i $$0.431278\pi$$
$$510$$ 0 0
$$511$$ 20.3133 0.898609
$$512$$ 80.2896 3.54833
$$513$$ 0 0
$$514$$ 38.8874 1.71525
$$515$$ 0 0
$$516$$ 17.2170 0.757934
$$517$$ −3.41938 −0.150384
$$518$$ 50.0046 2.19708
$$519$$ 26.8859 1.18016
$$520$$ 0 0
$$521$$ 0.982633 0.0430499 0.0215250 0.999768i $$-0.493148\pi$$
0.0215250 + 0.999768i $$0.493148\pi$$
$$522$$ −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
$$525$$ 0 0
$$526$$ 17.6809 0.770922
$$527$$ 18.4604 0.804148
$$528$$ −20.9289 −0.910812
$$529$$ −22.8257 −0.992423
$$530$$ 0 0
$$531$$ 1.90417 0.0826337
$$532$$ 0 0
$$533$$ 2.58195 0.111837
$$534$$ −18.2936 −0.791642
$$535$$ 0 0
$$536$$ −25.1883 −1.08797
$$537$$ −30.0656 −1.29743
$$538$$ −49.6198 −2.13926
$$539$$ −0.964024 −0.0415234
$$540$$ 0 0
$$541$$ 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
$$545$$ 0 0
$$546$$ 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
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −6.19529 −0.263689
$$553$$ −5.19741 −0.221016
$$554$$ −65.1554 −2.76819
$$555$$ 0 0
$$556$$ −60.1248 −2.54986
$$557$$ 10.6576 0.451575 0.225787 0.974177i $$-0.427505\pi$$
0.225787 + 0.974177i $$0.427505\pi$$
$$558$$ 10.5340 0.445939
$$559$$ 1.32406 0.0560019
$$560$$ 0 0
$$561$$ 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 0
$$566$$ 32.4270 1.36301
$$567$$ −17.3143 −0.727133
$$568$$ 17.7272 0.743818
$$569$$ −5.72754 −0.240111 −0.120056 0.992767i $$-0.538307\pi$$
−0.120056 + 0.992767i $$0.538307\pi$$
$$570$$ 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 0
$$576$$ −27.9307 −1.16378
$$577$$ 5.11190 0.212811 0.106406 0.994323i $$-0.466066\pi$$
0.106406 + 0.994323i $$0.466066\pi$$
$$578$$ −8.27001 −0.343987
$$579$$ 26.8600 1.11626
$$580$$ 0 0
$$581$$ 21.1914 0.879167
$$582$$ −44.6619 −1.85130
$$583$$ 9.50199 0.393532
$$584$$ 70.9325 2.93521
$$585$$ 0 0
$$586$$ −74.6254 −3.08275
$$587$$ 10.6692 0.440367 0.220184 0.975458i $$-0.429334\pi$$
0.220184 + 0.975458i $$0.429334\pi$$
$$588$$ 9.32439 0.384531
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −12.0515 −0.495732
$$592$$ 103.243 4.24326
$$593$$ 17.0027 0.698216 0.349108 0.937083i $$-0.386485\pi$$
0.349108 + 0.937083i $$0.386485\pi$$
$$594$$ 13.4301 0.551045
$$595$$ 0 0
$$596$$ −83.6040 −3.42455
$$597$$ 2.09427 0.0857128
$$598$$ −0.740650 −0.0302874
$$599$$ −28.6751 −1.17163 −0.585816 0.810444i $$-0.699226\pi$$
−0.585816 + 0.810444i $$0.699226\pi$$
$$600$$ 0 0
$$601$$ −27.4370 −1.11918 −0.559590 0.828770i $$-0.689041\pi$$
−0.559590 + 0.828770i $$0.689041\pi$$
$$602$$ 16.1712 0.659088
$$603$$ 1.96037 0.0798326
$$604$$ −120.313 −4.89546
$$605$$ 0 0
$$606$$ −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
$$610$$ 0 0
$$611$$ 2.54351 0.102899
$$612$$ 16.2417 0.656533
$$613$$ −34.6391 −1.39906 −0.699530 0.714603i $$-0.746607\pi$$
−0.699530 + 0.714603i $$0.746607\pi$$
$$614$$ 24.3760 0.983736
$$615$$ 0 0
$$616$$ −24.5050 −0.987334
$$617$$ −4.46569 −0.179782 −0.0898909 0.995952i $$-0.528652\pi$$
−0.0898909 + 0.995952i $$0.528652\pi$$
$$618$$ −1.58472 −0.0637470
$$619$$ −17.9112 −0.719913 −0.359957 0.932969i $$-0.617209\pi$$
−0.359957 + 0.932969i $$0.617209\pi$$
$$620$$ 0 0
$$621$$ 2.35062 0.0943272
$$622$$ −1.79683 −0.0720463
$$623$$ −12.6647 −0.507400
$$624$$ 15.5679 0.623215
$$625$$ 0 0
$$626$$ 8.17906 0.326901
$$627$$ 0 0
$$628$$ 13.6308 0.543927
$$629$$ −23.8157 −0.949593
$$630$$ 0 0
$$631$$ 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
$$635$$ 0 0
$$636$$ −91.9067 −3.64434
$$637$$ 0.717088 0.0284121
$$638$$ −23.1483 −0.916449
$$639$$ −1.37969 −0.0545796
$$640$$ 0 0
$$641$$ 37.9521 1.49902 0.749508 0.661995i $$-0.230290\pi$$
0.749508 + 0.661995i $$0.230290\pi$$
$$642$$ −26.4716 −1.04475
$$643$$ −35.2502 −1.39013 −0.695067 0.718945i $$-0.744625\pi$$
−0.695067 + 0.718945i $$0.744625\pi$$
$$644$$ −6.66740 −0.262732
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −35.5219 −1.39651 −0.698254 0.715850i $$-0.746040\pi$$
−0.698254 + 0.715850i $$0.746040\pi$$
$$648$$ −60.4602 −2.37510
$$649$$ 2.12706 0.0834943
$$650$$ 0 0
$$651$$ −20.9670 −0.821762
$$652$$ −99.8966 −3.91225
$$653$$ −8.02411 −0.314008 −0.157004 0.987598i $$-0.550184\pi$$
−0.157004 + 0.987598i $$0.550184\pi$$
$$654$$ −27.0290 −1.05692
$$655$$ 0 0
$$656$$ 65.1075 2.54202
$$657$$ −5.52059 −0.215379
$$658$$ 31.0646 1.21102
$$659$$ 47.2195 1.83941 0.919706 0.392608i $$-0.128427\pi$$
0.919706 + 0.392608i $$0.128427\pi$$
$$660$$ 0 0
$$661$$ 26.1159 1.01579 0.507896 0.861418i $$-0.330423\pi$$
0.507896 + 0.861418i $$0.330423\pi$$
$$662$$ −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
$$670$$ 0 0
$$671$$ −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
$$675$$ 0 0
$$676$$ −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
$$680$$ 0 0
$$681$$ 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
$$685$$ 0 0
$$686$$ −46.2381 −1.76538
$$687$$ −19.8908 −0.758880
$$688$$ 33.3881 1.27291
$$689$$ −7.06804 −0.269271
$$690$$ 0 0
$$691$$ −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
$$695$$ 0 0
$$696$$ 144.030 5.45943
$$697$$ −15.0187 −0.568875
$$698$$ −89.3314 −3.38124
$$699$$ −37.7596 −1.42820
$$700$$ 0 0
$$701$$ −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
$$705$$ 0 0
$$706$$ −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 $$0.348249\pi$$
$$710$$ 0 0
$$711$$ 1.41251 0.0529732
$$712$$ −44.2241 −1.65737
$$713$$ 2.05954 0.0771303
$$714$$ −43.8597 −1.64141
$$715$$ 0 0
$$716$$ −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 $$0.635285\pi$$
$$720$$ 0 0
$$721$$ −1.09711 −0.0408584
$$722$$ 0 0
$$723$$ −12.5085 −0.465195
$$724$$ 95.9345 3.56538
$$725$$ 0 0
$$726$$ −42.1848 −1.56562
$$727$$ 29.0494 1.07738 0.538692 0.842503i $$-0.318919\pi$$
0.538692 + 0.842503i $$0.318919\pi$$
$$728$$ 18.2280 0.675575
$$729$$ 29.9075 1.10768
$$730$$ 0 0
$$731$$ −7.70184 −0.284863
$$732$$ 52.9884 1.95851
$$733$$ −14.5428 −0.537151 −0.268576 0.963259i $$-0.586553\pi$$
−0.268576 + 0.963259i $$0.586553\pi$$
$$734$$ 63.3360 2.33777
$$735$$ 0 0
$$736$$ −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 $$0.527880\pi$$
$$740$$ 0 0
$$741$$ 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
$$745$$ 0 0
$$746$$ 81.4455 2.98193
$$747$$ −5.75922 −0.210719
$$748$$ 18.1429 0.663370
$$749$$ −18.3263 −0.669629
$$750$$ 0 0
$$751$$ −1.62096 −0.0591498 −0.0295749 0.999563i $$-0.509415\pi$$
−0.0295749 + 0.999563i $$0.509415\pi$$
$$752$$ 64.1382 2.33888
$$753$$ 27.2243 0.992107
$$754$$ 17.2188 0.627072
$$755$$ 0 0
$$756$$ −89.9308 −3.27075
$$757$$ 28.1135 1.02180 0.510901 0.859639i $$-0.329312\pi$$
0.510901 + 0.859639i $$0.329312\pi$$
$$758$$ −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
$$765$$ 0 0
$$766$$ 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 $$0.195710\pi$$
$$770$$ 0 0
$$771$$ 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
$$775$$ 0 0
$$776$$ −107.968 −3.87584
$$777$$ 27.0494 0.970393
$$778$$ 47.7353 1.71139
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −1.54119 −0.0551480
$$782$$ 4.30823 0.154062
$$783$$ −54.6479 −1.95296
$$784$$ 18.0824 0.645800
$$785$$ 0 0
$$786$$ 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
$$790$$ 0 0
$$791$$ 0.838961 0.0298300
$$792$$ 6.65976 0.236644
$$793$$ 4.07505 0.144709
$$794$$ 31.4122 1.11478
$$795$$ 0 0
$$796$$ 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
$$800$$ 0 0
$$801$$ 3.44191 0.121614
$$802$$ 24.6528 0.870522
$$803$$ −6.16681 −0.217622
$$804$$ −21.1810 −0.746996
$$805$$ 0 0
$$806$$ −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 $$0.396180\pi$$
$$810$$ 0 0
$$811$$ 20.9779 0.736634 0.368317 0.929700i $$-0.379934\pi$$
0.368317 + 0.929700i $$0.379934\pi$$
$$812$$ 155.005 5.43962
$$813$$ 17.7268 0.621707
$$814$$ −15.1806 −0.532079
$$815$$ 0 0
$$816$$ −90.5558 −3.17009
$$817$$ 0 0
$$818$$ 18.0504 0.631118
$$819$$ −1.41866 −0.0495721
$$820$$ 0 0
$$821$$ −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 $$-0.589824\pi$$
−0.278461 + 0.960448i $$0.589824\pi$$
$$824$$ −3.83101 −0.133460
$$825$$ 0 0
$$826$$ −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 $$-0.286047\pi$$
0.622672 + 0.782483i $$0.286047\pi$$
$$830$$ 0 0
$$831$$ −35.2451 −1.22264
$$832$$ 23.2082 0.804599
$$833$$ −4.17118 −0.144523
$$834$$ −44.1257 −1.52795
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 27.7794 0.960195
$$838$$ 60.2463 2.08117
$$839$$ 25.4830 0.879770 0.439885 0.898054i $$-0.355019\pi$$
0.439885 + 0.898054i $$0.355019\pi$$
$$840$$ 0 0
$$841$$ 65.1914 2.24798
$$842$$ 80.9294 2.78901
$$843$$ −20.6024 −0.709583
$$844$$ −106.057 −3.65065
$$845$$ 0 0
$$846$$ −8.44249 −0.290259
$$847$$ −29.2046 −1.00348
$$848$$ −178.231 −6.12047
$$849$$ 17.5410 0.602007
$$850$$ 0 0
$$851$$ −2.65700 −0.0910807
$$852$$ 14.9069 0.510703
$$853$$ 57.1622 1.95720 0.978598 0.205782i $$-0.0659737\pi$$
0.978598 + 0.205782i $$0.0659737\pi$$
$$854$$ 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 $$0.597896\pi$$
$$860$$ 0 0
$$861$$ 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
$$865$$ 0 0
$$866$$ −38.1789 −1.29737
$$867$$ −4.47357 −0.151930
$$868$$ −78.7944 −2.67446
$$869$$ 1.57785 0.0535249
$$870$$ 0 0
$$871$$ −1.62891 −0.0551936
$$872$$ −65.3415 −2.21274
$$873$$ 8.40305 0.284400
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 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
$$880$$ 0 0
$$881$$ 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 $$-0.347735\pi$$
0.460319 + 0.887754i $$0.347735\pi$$
$$884$$ −13.4956 −0.453905
$$885$$ 0 0
$$886$$ 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
$$890$$ 0 0
$$891$$ 5.25635 0.176094
$$892$$ −90.5363 −3.03138
$$893$$ 0 0
$$894$$ −61.3571 −2.05209
$$895$$ 0 0
$$896$$ 141.905 4.74072
$$897$$ −0.400646 −0.0133772
$$898$$ 73.2985 2.44600
$$899$$ −47.8807 −1.59691
$$900$$ 0 0
$$901$$ 41.1136 1.36969
$$902$$ −9.57325 −0.318754
$$903$$ 8.74762 0.291103
$$904$$ 2.92958 0.0974364
$$905$$ 0 0
$$906$$ −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
$$910$$ 0 0
$$911$$ 19.5682 0.648324 0.324162 0.946002i $$-0.394918\pi$$
0.324162 + 0.946002i $$0.394918\pi$$
$$912$$ 0 0
$$913$$ −6.43336 −0.212913
$$914$$ 91.4346 3.02439
$$915$$ 0 0
$$916$$ −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 $$0.509526\pi$$
$$920$$ 0 0
$$921$$ 13.1859 0.434491
$$922$$ −53.1017 −1.74881
$$923$$ 1.14641 0.0377346
$$924$$ −20.6064 −0.677901
$$925$$ 0 0
$$926$$ −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 $$0.620196\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −141.901 −4.64813
$$933$$ −0.971975 −0.0318210
$$934$$ 107.742 3.52542
$$935$$ 0 0
$$936$$ −4.95386 −0.161922
$$937$$ −55.0385 −1.79803 −0.899015 0.437918i $$-0.855716\pi$$
−0.899015 + 0.437918i $$0.855716\pi$$
$$938$$ −19.8944 −0.649576
$$939$$ 4.42437 0.144384
$$940$$ 0 0
$$941$$ 12.1956 0.397566 0.198783 0.980044i $$-0.436301\pi$$
0.198783 + 0.980044i $$0.436301\pi$$
$$942$$ 10.0036 0.325937
$$943$$ −1.67557 −0.0545640
$$944$$ −39.8977 −1.29856
$$945$$ 0 0
$$946$$ −4.90931 −0.159615
$$947$$ −30.4307 −0.988863 −0.494432 0.869216i $$-0.664624\pi$$
−0.494432 + 0.869216i $$0.664624\pi$$
$$948$$ −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
$$955$$ 0 0
$$956$$ −132.036 −4.27036
$$957$$ −12.5218 −0.404772
$$958$$ −68.2748 −2.20586
$$959$$ −14.8550 −0.479693
$$960$$ 0 0
$$961$$ −6.66065 −0.214860
$$962$$ 11.2921 0.364071
$$963$$ 4.98058 0.160497
$$964$$ −47.0070 −1.51399
$$965$$ 0 0
$$966$$ −4.89322 −0.157437
$$967$$ −30.6373 −0.985228 −0.492614 0.870248i $$-0.663958\pi$$
−0.492614 + 0.870248i $$0.663958\pi$$
$$968$$ −101.980 −3.27776
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −16.4765 −0.528755 −0.264378 0.964419i $$-0.585167\pi$$
−0.264378 + 0.964419i $$0.585167\pi$$
$$972$$ 43.8679 1.40706
$$973$$ −30.5483 −0.979334
$$974$$ 60.3649 1.93422
$$975$$ 0 0
$$976$$ 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
$$980$$ 0 0
$$981$$ 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
$$985$$ 0 0
$$986$$ −100.159 −3.18971
$$987$$ 16.8041 0.534879
$$988$$ 0 0
$$989$$ −0.859257 −0.0273228
$$990$$ 0 0
$$991$$ 9.67421 0.307311 0.153656 0.988124i $$-0.450895\pi$$
0.153656 + 0.988124i $$0.450895\pi$$
$$992$$ −122.568 −3.89154
$$993$$ −22.5164 −0.714535
$$994$$ 14.0015 0.444099
$$995$$ 0 0
$$996$$ 62.2259 1.97170
$$997$$ −22.7311 −0.719902 −0.359951 0.932971i $$-0.617207\pi$$
−0.359951 + 0.932971i $$0.617207\pi$$
$$998$$ 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 9025.2.a.bp.1.4 4
5.4 even 2 1805.2.a.i.1.1 4
19.8 odd 6 475.2.e.e.26.4 8
19.12 odd 6 475.2.e.e.201.4 8
19.18 odd 2 9025.2.a.bg.1.1 4
95.8 even 12 475.2.j.c.349.8 16
95.12 even 12 475.2.j.c.49.8 16
95.27 even 12 475.2.j.c.349.1 16
95.69 odd 6 95.2.e.c.11.1 8
95.84 odd 6 95.2.e.c.26.1 yes 8
95.88 even 12 475.2.j.c.49.1 16
95.94 odd 2 1805.2.a.o.1.4 4
285.164 even 6 855.2.k.h.676.4 8
285.179 even 6 855.2.k.h.406.4 8
380.179 even 6 1520.2.q.o.881.3 8
380.259 even 6 1520.2.q.o.961.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.c.11.1 8 95.69 odd 6
95.2.e.c.26.1 yes 8 95.84 odd 6
475.2.e.e.26.4 8 19.8 odd 6
475.2.e.e.201.4 8 19.12 odd 6
475.2.j.c.49.1 16 95.88 even 12
475.2.j.c.49.8 16 95.12 even 12
475.2.j.c.349.1 16 95.27 even 12
475.2.j.c.349.8 16 95.8 even 12
855.2.k.h.406.4 8 285.179 even 6
855.2.k.h.676.4 8 285.164 even 6
1520.2.q.o.881.3 8 380.179 even 6
1520.2.q.o.961.3 8 380.259 even 6
1805.2.a.i.1.1 4 5.4 even 2
1805.2.a.o.1.4 4 95.94 odd 2
9025.2.a.bg.1.1 4 19.18 odd 2
9025.2.a.bp.1.4 4 1.1 even 1 trivial