LMFDB - Embedded newform 85.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [85,2,Mod(1,85)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(85, 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("85.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$85 = 5 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	85.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:	$0.678728417181$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 2$ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	85.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} -0.585786 q^{3} +3.82843 q^{4} -1.00000 q^{5} +1.41421 q^{6} -3.41421 q^{7} -4.41421 q^{8} -2.65685 q^{9} +2.41421 q^{10} -5.41421 q^{11} -2.24264 q^{12} +2.82843 q^{13} +8.24264 q^{14} +0.585786 q^{15} +3.00000 q^{16} -1.00000 q^{17} +6.41421 q^{18} +2.82843 q^{19} -3.82843 q^{20} +2.00000 q^{21} +13.0711 q^{22} -0.585786 q^{23} +2.58579 q^{24} +1.00000 q^{25} -6.82843 q^{26} +3.31371 q^{27} -13.0711 q^{28} +0.828427 q^{29} -1.41421 q^{30} -4.24264 q^{31} +1.58579 q^{32} +3.17157 q^{33} +2.41421 q^{34} +3.41421 q^{35} -10.1716 q^{36} -10.4853 q^{37} -6.82843 q^{38} -1.65685 q^{39} +4.41421 q^{40} +10.4853 q^{41} -4.82843 q^{42} -3.65685 q^{43} -20.7279 q^{44} +2.65685 q^{45} +1.41421 q^{46} +0.828427 q^{47} -1.75736 q^{48} +4.65685 q^{49} -2.41421 q^{50} +0.585786 q^{51} +10.8284 q^{52} +11.6569 q^{53} -8.00000 q^{54} +5.41421 q^{55} +15.0711 q^{56} -1.65685 q^{57} -2.00000 q^{58} -14.8284 q^{59} +2.24264 q^{60} -3.65685 q^{61} +10.2426 q^{62} +9.07107 q^{63} -9.82843 q^{64} -2.82843 q^{65} -7.65685 q^{66} -8.82843 q^{67} -3.82843 q^{68} +0.343146 q^{69} -8.24264 q^{70} +4.24264 q^{71} +11.7279 q^{72} +0.828427 q^{73} +25.3137 q^{74} -0.585786 q^{75} +10.8284 q^{76} +18.4853 q^{77} +4.00000 q^{78} +2.58579 q^{79} -3.00000 q^{80} +6.02944 q^{81} -25.3137 q^{82} -13.3137 q^{83} +7.65685 q^{84} +1.00000 q^{85} +8.82843 q^{86} -0.485281 q^{87} +23.8995 q^{88} -13.6569 q^{89} -6.41421 q^{90} -9.65685 q^{91} -2.24264 q^{92} +2.48528 q^{93} -2.00000 q^{94} -2.82843 q^{95} -0.928932 q^{96} -7.65685 q^{97} -11.2426 q^{98} +14.3848 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 6 q^{9} + 2 q^{10} - 8 q^{11} + 4 q^{12} + 8 q^{14} + 4 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 4 q^{23}+ \cdots - 8 q^{99}+O(q^{100})$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^7 - 6 * q^8 + 6 * q^9 + 2 * q^10 - 8 * q^11 + 4 * q^12 + 8 * q^14 + 4 * q^15 + 6 * q^16 - 2 * q^17 + 10 * q^18 - 2 * q^20 + 4 * q^21 + 12 * q^22 - 4 * q^23 + 8 * q^24 + 2 * q^25 - 8 * q^26 - 16 * q^27 - 12 * q^28 - 4 * q^29 + 6 * q^32 + 12 * q^33 + 2 * q^34 + 4 * q^35 - 26 * q^36 - 4 * q^37 - 8 * q^38 + 8 * q^39 + 6 * q^40 + 4 * q^41 - 4 * q^42 + 4 * q^43 - 16 * q^44 - 6 * q^45 - 4 * q^47 - 12 * q^48 - 2 * q^49 - 2 * q^50 + 4 * q^51 + 16 * q^52 + 12 * q^53 - 16 * q^54 + 8 * q^55 + 16 * q^56 + 8 * q^57 - 4 * q^58 - 24 * q^59 - 4 * q^60 + 4 * q^61 + 12 * q^62 + 4 * q^63 - 14 * q^64 - 4 * q^66 - 12 * q^67 - 2 * q^68 + 12 * q^69 - 8 * q^70 - 2 * q^72 - 4 * q^73 + 28 * q^74 - 4 * q^75 + 16 * q^76 + 20 * q^77 + 8 * q^78 + 8 * q^79 - 6 * q^80 + 46 * q^81 - 28 * q^82 - 4 * q^83 + 4 * q^84 + 2 * q^85 + 12 * q^86 + 16 * q^87 + 28 * q^88 - 16 * q^89 - 10 * q^90 - 8 * q^91 + 4 * q^92 - 12 * q^93 - 4 * q^94 - 16 * q^96 - 4 * q^97 - 14 * q^98 - 8 * 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 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	−0.585786	−0.338204	−0.169102	−	0.985599i	$-0.554087\pi$
$3$	−0.585786	−0.338204	−0.169102	+	0.985599i	$0.554087\pi$
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.41421	0.577350
$7$	−3.41421	−1.29045	−0.645226	−	0.763992i	$-0.723237\pi$
$7$	−3.41421	−1.29045	−0.645226	+	0.763992i	$0.723237\pi$
$8$	−4.41421	−1.56066
$9$	−2.65685	−0.885618
$10$	2.41421	0.763441
$11$	−5.41421	−1.63245	−0.816223	−	0.577736i	$-0.803936\pi$
$11$	−5.41421	−1.63245	−0.816223	+	0.577736i	$0.803936\pi$
$12$	−2.24264	−0.647395
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	8.24264	2.20294
$15$	0.585786	0.151249
$16$	3.00000	0.750000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	6.41421	1.51184
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	−3.82843	−0.856062
$21$	2.00000	0.436436
$22$	13.0711	2.78676
$23$	−0.585786	−0.122145	−0.0610725	−	0.998133i	$-0.519452\pi$
$23$	−0.585786	−0.122145	−0.0610725	+	0.998133i	$0.519452\pi$
$24$	2.58579	0.527821
$25$	1.00000	0.200000
$26$	−6.82843	−1.33916
$27$	3.31371	0.637723
$28$	−13.0711	−2.47020
$29$	0.828427	0.153835	0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427	0.153835	0.0769175	+	0.997037i	$0.475492\pi$
$30$	−1.41421	−0.258199
$31$	−4.24264	−0.762001	−0.381000	−	0.924575i	$-0.624420\pi$
$31$	−4.24264	−0.762001	−0.381000	+	0.924575i	$0.624420\pi$
$32$	1.58579	0.280330
$33$	3.17157	0.552100
$34$	2.41421	0.414034
$35$	3.41421	0.577107
$36$	−10.1716	−1.69526
$37$	−10.4853	−1.72377	−0.861885	−	0.507104i	$-0.830716\pi$
$37$	−10.4853	−1.72377	−0.861885	+	0.507104i	$0.830716\pi$
$38$	−6.82843	−1.10772
$39$	−1.65685	−0.265309
$40$	4.41421	0.697948
$41$	10.4853	1.63753	0.818763	−	0.574132i	$-0.194660\pi$
$41$	10.4853	1.63753	0.818763	+	0.574132i	$0.194660\pi$
$42$	−4.82843	−0.745042
$43$	−3.65685	−0.557665	−0.278833	−	0.960340i	$-0.589947\pi$
$43$	−3.65685	−0.557665	−0.278833	+	0.960340i	$0.589947\pi$
$44$	−20.7279	−3.12485
$45$	2.65685	0.396060
$46$	1.41421	0.208514
$47$	0.828427	0.120839	0.0604193	−	0.998173i	$-0.480756\pi$
$47$	0.828427	0.120839	0.0604193	+	0.998173i	$0.480756\pi$
$48$	−1.75736	−0.253653
$49$	4.65685	0.665265
$50$	−2.41421	−0.341421
$51$	0.585786	0.0820265
$52$	10.8284	1.50163
$53$	11.6569	1.60119	0.800596	−	0.599204i	$-0.204516\pi$
$53$	11.6569	1.60119	0.800596	+	0.599204i	$0.204516\pi$
$54$	−8.00000	−1.08866
$55$	5.41421	0.730052
$56$	15.0711	2.01396
$57$	−1.65685	−0.219456
$58$	−2.00000	−0.262613
$59$	−14.8284	−1.93050	−0.965248	−	0.261334i	$-0.915838\pi$
$59$	−14.8284	−1.93050	−0.965248	+	0.261334i	$0.915838\pi$
$60$	2.24264	0.289524
$61$	−3.65685	−0.468212	−0.234106	−	0.972211i	$-0.575216\pi$
$61$	−3.65685	−0.468212	−0.234106	+	0.972211i	$0.575216\pi$
$62$	10.2426	1.30082
$63$	9.07107	1.14285
$64$	−9.82843	−1.22855
$65$	−2.82843	−0.350823
$66$	−7.65685	−0.942494
$67$	−8.82843	−1.07856	−0.539282	−	0.842125i	$-0.681304\pi$
$67$	−8.82843	−1.07856	−0.539282	+	0.842125i	$0.681304\pi$
$68$	−3.82843	−0.464265
$69$	0.343146	0.0413099
$70$	−8.24264	−0.985184
$71$	4.24264	0.503509	0.251754	−	0.967791i	$-0.418992\pi$
$71$	4.24264	0.503509	0.251754	+	0.967791i	$0.418992\pi$
$72$	11.7279	1.38215
$73$	0.828427	0.0969601	0.0484800	−	0.998824i	$-0.484562\pi$
$73$	0.828427	0.0969601	0.0484800	+	0.998824i	$0.484562\pi$
$74$	25.3137	2.94266
$75$	−0.585786	−0.0676408
$76$	10.8284	1.24211
$77$	18.4853	2.10659
$78$	4.00000	0.452911
$79$	2.58579	0.290924	0.145462	−	0.989364i	$-0.453533\pi$
$79$	2.58579	0.290924	0.145462	+	0.989364i	$0.453533\pi$
$80$	−3.00000	−0.335410
$81$	6.02944	0.669937
$82$	−25.3137	−2.79543
$83$	−13.3137	−1.46137	−0.730685	−	0.682715i	$-0.760799\pi$
$83$	−13.3137	−1.46137	−0.730685	+	0.682715i	$0.760799\pi$
$84$	7.65685	0.835431
$85$	1.00000	0.108465
$86$	8.82843	0.951994
$87$	−0.485281	−0.0520276
$88$	23.8995	2.54769
$89$	−13.6569	−1.44762	−0.723812	−	0.689997i	$-0.757612\pi$
$89$	−13.6569	−1.44762	−0.723812	+	0.689997i	$0.757612\pi$
$90$	−6.41421	−0.676117
$91$	−9.65685	−1.01231
$92$	−2.24264	−0.233811
$93$	2.48528	0.257712
$94$	−2.00000	−0.206284
$95$	−2.82843	−0.290191
$96$	−0.928932	−0.0948087
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	−11.2426	−1.13568
$99$	14.3848	1.44572
$100$	3.82843	0.382843
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	−1.41421	−0.140028
$103$	4.82843	0.475759	0.237880	−	0.971295i	$-0.423548\pi$
$103$	4.82843	0.475759	0.237880	+	0.971295i	$0.423548\pi$
$104$	−12.4853	−1.22428
$105$	−2.00000	−0.195180
$106$	−28.1421	−2.73341
$107$	−7.89949	−0.763673	−0.381837	−	0.924230i	$-0.624708\pi$
$107$	−7.89949	−0.763673	−0.381837	+	0.924230i	$0.624708\pi$
$108$	12.6863	1.22074
$109$	5.31371	0.508961	0.254480	−	0.967078i	$-0.418096\pi$
$109$	5.31371	0.508961	0.254480	+	0.967078i	$0.418096\pi$
$110$	−13.0711	−1.24628
$111$	6.14214	0.582986
$112$	−10.2426	−0.967839
$113$	−8.82843	−0.830509	−0.415254	−	0.909705i	$-0.636307\pi$
$113$	−8.82843	−0.830509	−0.415254	+	0.909705i	$0.636307\pi$
$114$	4.00000	0.374634
$115$	0.585786	0.0546249
$116$	3.17157	0.294473
$117$	−7.51472	−0.694736
$118$	35.7990	3.29556
$119$	3.41421	0.312980
$120$	−2.58579	−0.236049
$121$	18.3137	1.66488
$122$	8.82843	0.799288
$123$	−6.14214	−0.553818
$124$	−16.2426	−1.45863
$125$	−1.00000	−0.0894427
$126$	−21.8995	−1.95096
$127$	5.31371	0.471515	0.235758	−	0.971812i	$-0.424243\pi$
$127$	5.31371	0.471515	0.235758	+	0.971812i	$0.424243\pi$
$128$	20.5563	1.81694
$129$	2.14214	0.188605
$130$	6.82843	0.598893
$131$	5.89949	0.515441	0.257721	−	0.966219i	$-0.417029\pi$
$131$	5.89949	0.515441	0.257721	+	0.966219i	$0.417029\pi$
$132$	12.1421	1.05684
$133$	−9.65685	−0.837355
$134$	21.3137	1.84122
$135$	−3.31371	−0.285199
$136$	4.41421	0.378516
$137$	6.82843	0.583392	0.291696	−	0.956511i	$-0.405780\pi$
$137$	6.82843	0.583392	0.291696	+	0.956511i	$0.405780\pi$
$138$	−0.828427	−0.0705204
$139$	−1.89949	−0.161113	−0.0805565	−	0.996750i	$-0.525670\pi$
$139$	−1.89949	−0.161113	−0.0805565	+	0.996750i	$0.525670\pi$
$140$	13.0711	1.10471
$141$	−0.485281	−0.0408681
$142$	−10.2426	−0.859543
$143$	−15.3137	−1.28060
$144$	−7.97056	−0.664214
$145$	−0.828427	−0.0687971
$146$	−2.00000	−0.165521
$147$	−2.72792	−0.224995
$148$	−40.1421	−3.29966
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	1.41421	0.115470
$151$	24.4853	1.99258	0.996292	−	0.0860367i	$-0.0274202\pi$
$151$	24.4853	1.99258	0.996292	+	0.0860367i	$0.0274202\pi$
$152$	−12.4853	−1.01269
$153$	2.65685	0.214794
$154$	−44.6274	−3.59618
$155$	4.24264	0.340777
$156$	−6.34315	−0.507858
$157$	1.31371	0.104845	0.0524227	−	0.998625i	$-0.483306\pi$
$157$	1.31371	0.104845	0.0524227	+	0.998625i	$0.483306\pi$
$158$	−6.24264	−0.496638
$159$	−6.82843	−0.541529
$160$	−1.58579	−0.125367
$161$	2.00000	0.157622
$162$	−14.5563	−1.14365
$163$	3.41421	0.267422	0.133711	−	0.991020i	$-0.457311\pi$
$163$	3.41421	0.267422	0.133711	+	0.991020i	$0.457311\pi$
$164$	40.1421	3.13457
$165$	−3.17157	−0.246907
$166$	32.1421	2.49471
$167$	−6.24264	−0.483070	−0.241535	−	0.970392i	$-0.577651\pi$
$167$	−6.24264	−0.483070	−0.241535	+	0.970392i	$0.577651\pi$
$168$	−8.82843	−0.681128
$169$	−5.00000	−0.384615
$170$	−2.41421	−0.185162
$171$	−7.51472	−0.574665
$172$	−14.0000	−1.06749
$173$	7.17157	0.545245	0.272622	−	0.962121i	$-0.412109\pi$
$173$	7.17157	0.545245	0.272622	+	0.962121i	$0.412109\pi$
$174$	1.17157	0.0888167
$175$	−3.41421	−0.258090
$176$	−16.2426	−1.22434
$177$	8.68629	0.652902
$178$	32.9706	2.47125
$179$	−1.17157	−0.0875675	−0.0437837	−	0.999041i	$-0.513941\pi$
$179$	−1.17157	−0.0875675	−0.0437837	+	0.999041i	$0.513941\pi$
$180$	10.1716	0.758144
$181$	−14.4853	−1.07668	−0.538341	−	0.842727i	$-0.680949\pi$
$181$	−14.4853	−1.07668	−0.538341	+	0.842727i	$0.680949\pi$
$182$	23.3137	1.72813
$183$	2.14214	0.158351
$184$	2.58579	0.190627
$185$	10.4853	0.770893
$186$	−6.00000	−0.439941
$187$	5.41421	0.395927
$188$	3.17157	0.231311
$189$	−11.3137	−0.822951
$190$	6.82843	0.495386
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	5.75736	0.415502
$193$	−15.1716	−1.09207	−0.546037	−	0.837761i	$-0.683864\pi$
$193$	−15.1716	−1.09207	−0.546037	+	0.837761i	$0.683864\pi$
$194$	18.4853	1.32717
$195$	1.65685	0.118650
$196$	17.8284	1.27346
$197$	7.17157	0.510953	0.255477	−	0.966815i	$-0.417768\pi$
$197$	7.17157	0.510953	0.255477	+	0.966815i	$0.417768\pi$
$198$	−34.7279	−2.46801
$199$	15.7574	1.11701	0.558505	−	0.829501i	$-0.311375\pi$
$199$	15.7574	1.11701	0.558505	+	0.829501i	$0.311375\pi$
$200$	−4.41421	−0.312132
$201$	5.17157	0.364775
$202$	19.3137	1.35891
$203$	−2.82843	−0.198517
$204$	2.24264	0.157016
$205$	−10.4853	−0.732324
$206$	−11.6569	−0.812172
$207$	1.55635	0.108174
$208$	8.48528	0.588348
$209$	−15.3137	−1.05927
$210$	4.82843	0.333193
$211$	−21.8995	−1.50762	−0.753812	−	0.657090i	$-0.771787\pi$
$211$	−21.8995	−1.50762	−0.753812	+	0.657090i	$0.771787\pi$
$212$	44.6274	3.06502
$213$	−2.48528	−0.170289
$214$	19.0711	1.30367
$215$	3.65685	0.249395
$216$	−14.6274	−0.995270
$217$	14.4853	0.983325
$218$	−12.8284	−0.868851
$219$	−0.485281	−0.0327923
$220$	20.7279	1.39748
$221$	−2.82843	−0.190261
$222$	−14.8284	−0.995219
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	−5.41421	−0.361752
$225$	−2.65685	−0.177124
$226$	21.3137	1.41777
$227$	2.72792	0.181059	0.0905293	−	0.995894i	$-0.471144\pi$
$227$	2.72792	0.181059	0.0905293	+	0.995894i	$0.471144\pi$
$228$	−6.34315	−0.420085
$229$	−23.3137	−1.54061	−0.770307	−	0.637674i	$-0.779897\pi$
$229$	−23.3137	−1.54061	−0.770307	+	0.637674i	$0.779897\pi$
$230$	−1.41421	−0.0932505
$231$	−10.8284	−0.712458
$232$	−3.65685	−0.240084
$233$	−13.3137	−0.872210	−0.436105	−	0.899896i	$-0.643642\pi$
$233$	−13.3137	−0.872210	−0.436105	+	0.899896i	$0.643642\pi$
$234$	18.1421	1.18599
$235$	−0.828427	−0.0540406
$236$	−56.7696	−3.69538
$237$	−1.51472	−0.0983915
$238$	−8.24264	−0.534291
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	1.75736	0.113437
$241$	7.17157	0.461962	0.230981	−	0.972958i	$-0.425807\pi$
$241$	7.17157	0.461962	0.230981	+	0.972958i	$0.425807\pi$
$242$	−44.2132	−2.84213
$243$	−13.4731	−0.864299
$244$	−14.0000	−0.896258
$245$	−4.65685	−0.297516
$246$	14.8284	0.945426
$247$	8.00000	0.509028
$248$	18.7279	1.18922
$249$	7.79899	0.494241
$250$	2.41421	0.152688
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	34.7279	2.18765
$253$	3.17157	0.199395
$254$	−12.8284	−0.804927
$255$	−0.585786	−0.0366834
$256$	−29.9706	−1.87316
$257$	2.82843	0.176432	0.0882162	−	0.996101i	$-0.471883\pi$
$257$	2.82843	0.176432	0.0882162	+	0.996101i	$0.471883\pi$
$258$	−5.17157	−0.321968
$259$	35.7990	2.22444
$260$	−10.8284	−0.671551
$261$	−2.20101	−0.136239
$262$	−14.2426	−0.879913
$263$	−13.3137	−0.820958	−0.410479	−	0.911870i	$-0.634639\pi$
$263$	−13.3137	−0.820958	−0.410479	+	0.911870i	$0.634639\pi$
$264$	−14.0000	−0.861640
$265$	−11.6569	−0.716075
$266$	23.3137	1.42946
$267$	8.00000	0.489592
$268$	−33.7990	−2.06460
$269$	−14.9706	−0.912771	−0.456386	−	0.889782i	$-0.650856\pi$
$269$	−14.9706	−0.912771	−0.456386	+	0.889782i	$0.650856\pi$
$270$	8.00000	0.486864
$271$	−2.34315	−0.142336	−0.0711680	−	0.997464i	$-0.522673\pi$
$271$	−2.34315	−0.142336	−0.0711680	+	0.997464i	$0.522673\pi$
$272$	−3.00000	−0.181902
$273$	5.65685	0.342368
$274$	−16.4853	−0.995912
$275$	−5.41421	−0.326489
$276$	1.31371	0.0790760
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	4.58579	0.275037
$279$	11.2721	0.674842
$280$	−15.0711	−0.900669
$281$	−15.6569	−0.934010	−0.467005	−	0.884255i	$-0.654667\pi$
$281$	−15.6569	−0.934010	−0.467005	+	0.884255i	$0.654667\pi$
$282$	1.17157	0.0697661
$283$	5.75736	0.342239	0.171120	−	0.985250i	$-0.445262\pi$
$283$	5.75736	0.342239	0.171120	+	0.985250i	$0.445262\pi$
$284$	16.2426	0.963823
$285$	1.65685	0.0981436
$286$	36.9706	2.18612
$287$	−35.7990	−2.11315
$288$	−4.21320	−0.248265
$289$	1.00000	0.0588235
$290$	2.00000	0.117444
$291$	4.48528	0.262932
$292$	3.17157	0.185602
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	6.58579	0.384091
$295$	14.8284	0.863344
$296$	46.2843	2.69022
$297$	−17.9411	−1.04105
$298$	−4.82843	−0.279703
$299$	−1.65685	−0.0958184
$300$	−2.24264	−0.129479
$301$	12.4853	0.719640
$302$	−59.1127	−3.40155
$303$	4.68629	0.269220
$304$	8.48528	0.486664
$305$	3.65685	0.209391
$306$	−6.41421	−0.366676
$307$	−11.1716	−0.637595	−0.318798	−	0.947823i	$-0.603279\pi$
$307$	−11.1716	−0.637595	−0.318798	+	0.947823i	$0.603279\pi$
$308$	70.7696	4.03247
$309$	−2.82843	−0.160904
$310$	−10.2426	−0.581743
$311$	0.928932	0.0526749	0.0263375	−	0.999653i	$-0.491616\pi$
$311$	0.928932	0.0526749	0.0263375	+	0.999653i	$0.491616\pi$
$312$	7.31371	0.414057
$313$	33.7990	1.91043	0.955216	−	0.295910i	$-0.0956227\pi$
$313$	33.7990	1.91043	0.955216	+	0.295910i	$0.0956227\pi$
$314$	−3.17157	−0.178982
$315$	−9.07107	−0.511097
$316$	9.89949	0.556890
$317$	1.31371	0.0737852	0.0368926	−	0.999319i	$-0.488254\pi$
$317$	1.31371	0.0737852	0.0368926	+	0.999319i	$0.488254\pi$
$318$	16.4853	0.924449
$319$	−4.48528	−0.251128
$320$	9.82843	0.549426
$321$	4.62742	0.258277
$322$	−4.82843	−0.269078
$323$	−2.82843	−0.157378
$324$	23.0833	1.28240
$325$	2.82843	0.156893
$326$	−8.24264	−0.456518
$327$	−3.11270	−0.172133
$328$	−46.2843	−2.55562
$329$	−2.82843	−0.155936
$330$	7.65685	0.421496
$331$	21.1716	1.16369	0.581847	−	0.813298i	$-0.302330\pi$
$331$	21.1716	1.16369	0.581847	+	0.813298i	$0.302330\pi$
$332$	−50.9706	−2.79737
$333$	27.8579	1.52660
$334$	15.0711	0.824652
$335$	8.82843	0.482349
$336$	6.00000	0.327327
$337$	24.6274	1.34154	0.670770	−	0.741665i	$-0.265964\pi$
$337$	24.6274	1.34154	0.670770	+	0.741665i	$0.265964\pi$
$338$	12.0711	0.656580
$339$	5.17157	0.280881
$340$	3.82843	0.207626
$341$	22.9706	1.24393
$342$	18.1421	0.981014
$343$	8.00000	0.431959
$344$	16.1421	0.870326
$345$	−0.343146	−0.0184743
$346$	−17.3137	−0.930791
$347$	35.6985	1.91640	0.958198	−	0.286107i	$-0.0923614\pi$
$347$	35.6985	1.91640	0.958198	+	0.286107i	$0.0923614\pi$
$348$	−1.85786	−0.0995920
$349$	20.3431	1.08894	0.544472	−	0.838779i	$-0.316730\pi$
$349$	20.3431	1.08894	0.544472	+	0.838779i	$0.316730\pi$
$350$	8.24264	0.440588
$351$	9.37258	0.500271
$352$	−8.58579	−0.457624
$353$	−16.3431	−0.869858	−0.434929	−	0.900465i	$-0.643227\pi$
$353$	−16.3431	−0.869858	−0.434929	+	0.900465i	$0.643227\pi$
$354$	−20.9706	−1.11457
$355$	−4.24264	−0.225176
$356$	−52.2843	−2.77106
$357$	−2.00000	−0.105851
$358$	2.82843	0.149487
$359$	7.79899	0.411615	0.205807	−	0.978593i	$-0.434018\pi$
$359$	7.79899	0.411615	0.205807	+	0.978593i	$0.434018\pi$
$360$	−11.7279	−0.618116
$361$	−11.0000	−0.578947
$362$	34.9706	1.83801
$363$	−10.7279	−0.563070
$364$	−36.9706	−1.93778
$365$	−0.828427	−0.0433619
$366$	−5.17157	−0.270322
$367$	−5.75736	−0.300532	−0.150266	−	0.988646i	$-0.548013\pi$
$367$	−5.75736	−0.300532	−0.150266	+	0.988646i	$0.548013\pi$
$368$	−1.75736	−0.0916087
$369$	−27.8579	−1.45022
$370$	−25.3137	−1.31600
$371$	−39.7990	−2.06626
$372$	9.51472	0.493315
$373$	−27.7990	−1.43938	−0.719689	−	0.694297i	$-0.755715\pi$
$373$	−27.7990	−1.43938	−0.719689	+	0.694297i	$0.755715\pi$
$374$	−13.0711	−0.675889
$375$	0.585786	0.0302499
$376$	−3.65685	−0.188588
$377$	2.34315	0.120678
$378$	27.3137	1.40487
$379$	26.8701	1.38022	0.690111	−	0.723703i	$-0.257562\pi$
$379$	26.8701	1.38022	0.690111	+	0.723703i	$0.257562\pi$
$380$	−10.8284	−0.555487
$381$	−3.11270	−0.159468
$382$	−28.9706	−1.48226
$383$	22.2843	1.13867	0.569337	−	0.822105i	$-0.307200\pi$
$383$	22.2843	1.13867	0.569337	+	0.822105i	$0.307200\pi$
$384$	−12.0416	−0.614497
$385$	−18.4853	−0.942097
$386$	36.6274	1.86429
$387$	9.71573	0.493878
$388$	−29.3137	−1.48818
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	−4.00000	−0.202548
$391$	0.585786	0.0296245
$392$	−20.5563	−1.03825
$393$	−3.45584	−0.174324
$394$	−17.3137	−0.872252
$395$	−2.58579	−0.130105
$396$	55.0711	2.76743
$397$	−9.31371	−0.467442	−0.233721	−	0.972304i	$-0.575090\pi$
$397$	−9.31371	−0.467442	−0.233721	+	0.972304i	$0.575090\pi$
$398$	−38.0416	−1.90685
$399$	5.65685	0.283197
$400$	3.00000	0.150000
$401$	−27.6569	−1.38112	−0.690559	−	0.723276i	$-0.742635\pi$
$401$	−27.6569	−1.38112	−0.690559	+	0.723276i	$0.742635\pi$
$402$	−12.4853	−0.622709
$403$	−12.0000	−0.597763
$404$	−30.6274	−1.52377
$405$	−6.02944	−0.299605
$406$	6.82843	0.338889
$407$	56.7696	2.81396
$408$	−2.58579	−0.128016
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	25.3137	1.25015
$411$	−4.00000	−0.197305
$412$	18.4853	0.910704
$413$	50.6274	2.49121
$414$	−3.75736	−0.184664
$415$	13.3137	0.653544
$416$	4.48528	0.219909
$417$	1.11270	0.0544891
$418$	36.9706	1.80829
$419$	3.75736	0.183559	0.0917795	−	0.995779i	$-0.470745\pi$
$419$	3.75736	0.183559	0.0917795	+	0.995779i	$0.470745\pi$
$420$	−7.65685	−0.373616
$421$	4.97056	0.242250	0.121125	−	0.992637i	$-0.461350\pi$
$421$	4.97056	0.242250	0.121125	+	0.992637i	$0.461350\pi$
$422$	52.8701	2.57367
$423$	−2.20101	−0.107017
$424$	−51.4558	−2.49892
$425$	−1.00000	−0.0485071
$426$	6.00000	0.290701
$427$	12.4853	0.604205
$428$	−30.2426	−1.46183
$429$	8.97056	0.433103
$430$	−8.82843	−0.425745
$431$	−1.41421	−0.0681203	−0.0340601	−	0.999420i	$-0.510844\pi$
$431$	−1.41421	−0.0681203	−0.0340601	+	0.999420i	$0.510844\pi$
$432$	9.94113	0.478293
$433$	−2.82843	−0.135926	−0.0679628	−	0.997688i	$-0.521650\pi$
$433$	−2.82843	−0.135926	−0.0679628	+	0.997688i	$0.521650\pi$
$434$	−34.9706	−1.67864
$435$	0.485281	0.0232675
$436$	20.3431	0.974260
$437$	−1.65685	−0.0792581
$438$	1.17157	0.0559799
$439$	2.58579	0.123413	0.0617064	−	0.998094i	$-0.480346\pi$
$439$	2.58579	0.123413	0.0617064	+	0.998094i	$0.480346\pi$
$440$	−23.8995	−1.13936
$441$	−12.3726	−0.589171
$442$	6.82843	0.324795
$443$	2.48528	0.118079	0.0590396	−	0.998256i	$-0.481196\pi$
$443$	2.48528	0.118079	0.0590396	+	0.998256i	$0.481196\pi$
$444$	23.5147	1.11596
$445$	13.6569	0.647397
$446$	−14.4853	−0.685898
$447$	−1.17157	−0.0554135
$448$	33.5563	1.58539
$449$	−9.51472	−0.449027	−0.224514	−	0.974471i	$-0.572079\pi$
$449$	−9.51472	−0.449027	−0.224514	+	0.974471i	$0.572079\pi$
$450$	6.41421	0.302369
$451$	−56.7696	−2.67317
$452$	−33.7990	−1.58977
$453$	−14.3431	−0.673900
$454$	−6.58579	−0.309086
$455$	9.65685	0.452720
$456$	7.31371	0.342496
$457$	−23.1127	−1.08117	−0.540583	−	0.841291i	$-0.681796\pi$
$457$	−23.1127	−1.08117	−0.540583	+	0.841291i	$0.681796\pi$
$458$	56.2843	2.62999
$459$	−3.31371	−0.154671
$460$	2.24264	0.104564
$461$	−37.5980	−1.75111	−0.875556	−	0.483116i	$-0.839505\pi$
$461$	−37.5980	−1.75111	−0.875556	+	0.483116i	$0.839505\pi$
$462$	26.1421	1.21624
$463$	−8.82843	−0.410292	−0.205146	−	0.978731i	$-0.565767\pi$
$463$	−8.82843	−0.410292	−0.205146	+	0.978731i	$0.565767\pi$
$464$	2.48528	0.115376
$465$	−2.48528	−0.115252
$466$	32.1421	1.48896
$467$	−11.6569	−0.539415	−0.269707	−	0.962942i	$-0.586927\pi$
$467$	−11.6569	−0.539415	−0.269707	+	0.962942i	$0.586927\pi$
$468$	−28.7696	−1.32987
$469$	30.1421	1.39183
$470$	2.00000	0.0922531
$471$	−0.769553	−0.0354591
$472$	65.4558	3.01285
$473$	19.7990	0.910359
$474$	3.65685	0.167965
$475$	2.82843	0.129777
$476$	13.0711	0.599111
$477$	−30.9706	−1.41804
$478$	−48.2843	−2.20847
$479$	−24.2426	−1.10767	−0.553837	−	0.832625i	$-0.686837\pi$
$479$	−24.2426	−1.10767	−0.553837	+	0.832625i	$0.686837\pi$
$480$	0.928932	0.0423998
$481$	−29.6569	−1.35224
$482$	−17.3137	−0.788618
$483$	−1.17157	−0.0533084
$484$	70.1127	3.18694
$485$	7.65685	0.347680
$486$	32.5269	1.47545
$487$	15.8995	0.720475	0.360237	−	0.932861i	$-0.382696\pi$
$487$	15.8995	0.720475	0.360237	+	0.932861i	$0.382696\pi$
$488$	16.1421	0.730720
$489$	−2.00000	−0.0904431
$490$	11.2426	0.507891
$491$	0.485281	0.0219004	0.0109502	−	0.999940i	$-0.496514\pi$
$491$	0.485281	0.0219004	0.0109502	+	0.999940i	$0.496514\pi$
$492$	−23.5147	−1.06013
$493$	−0.828427	−0.0373105
$494$	−19.3137	−0.868965
$495$	−14.3848	−0.646548
$496$	−12.7279	−0.571501
$497$	−14.4853	−0.649754
$498$	−18.8284	−0.843722
$499$	3.75736	0.168203	0.0841013	−	0.996457i	$-0.473198\pi$
$499$	3.75736	0.168203	0.0841013	+	0.996457i	$0.473198\pi$
$500$	−3.82843	−0.171212
$501$	3.65685	0.163376
$502$	28.9706	1.29302
$503$	−27.6985	−1.23501	−0.617507	−	0.786565i	$-0.711857\pi$
$503$	−27.6985	−1.23501	−0.617507	+	0.786565i	$0.711857\pi$
$504$	−40.0416	−1.78360
$505$	8.00000	0.355995
$506$	−7.65685	−0.340389
$507$	2.92893	0.130078
$508$	20.3431	0.902581
$509$	−24.6274	−1.09159	−0.545796	−	0.837918i	$-0.683772\pi$
$509$	−24.6274	−1.09159	−0.545796	+	0.837918i	$0.683772\pi$
$510$	1.41421	0.0626224
$511$	−2.82843	−0.125122
$512$	31.2426	1.38074
$513$	9.37258	0.413810
$514$	−6.82843	−0.301189
$515$	−4.82843	−0.212766
$516$	8.20101	0.361029
$517$	−4.48528	−0.197262
$518$	−86.4264	−3.79736
$519$	−4.20101	−0.184404
$520$	12.4853	0.547516
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	5.31371	0.232575
$523$	28.1421	1.23057	0.615285	−	0.788305i	$-0.289041\pi$
$523$	28.1421	1.23057	0.615285	+	0.788305i	$0.289041\pi$
$524$	22.5858	0.986665
$525$	2.00000	0.0872872
$526$	32.1421	1.40146
$527$	4.24264	0.184812
$528$	9.51472	0.414075
$529$	−22.6569	−0.985081
$530$	28.1421	1.22242
$531$	39.3970	1.70968
$532$	−36.9706	−1.60288
$533$	29.6569	1.28458
$534$	−19.3137	−0.835786
$535$	7.89949	0.341525
$536$	38.9706	1.68327
$537$	0.686292	0.0296157
$538$	36.1421	1.55820
$539$	−25.2132	−1.08601
$540$	−12.6863	−0.545931
$541$	30.7696	1.32289	0.661443	−	0.749995i	$-0.269944\pi$
$541$	30.7696	1.32289	0.661443	+	0.749995i	$0.269944\pi$
$542$	5.65685	0.242983
$543$	8.48528	0.364138
$544$	−1.58579	−0.0679900
$545$	−5.31371	−0.227614
$546$	−13.6569	−0.584459
$547$	−35.2132	−1.50561	−0.752804	−	0.658245i	$-0.771299\pi$
$547$	−35.2132	−1.50561	−0.752804	+	0.658245i	$0.771299\pi$
$548$	26.1421	1.11674
$549$	9.71573	0.414657
$550$	13.0711	0.557352
$551$	2.34315	0.0998214
$552$	−1.51472	−0.0644707
$553$	−8.82843	−0.375423
$554$	24.1421	1.02570
$555$	−6.14214	−0.260719
$556$	−7.27208	−0.308405
$557$	21.1716	0.897068	0.448534	−	0.893766i	$-0.351946\pi$
$557$	21.1716	0.897068	0.448534	+	0.893766i	$0.351946\pi$
$558$	−27.2132	−1.15203
$559$	−10.3431	−0.437468
$560$	10.2426	0.432831
$561$	−3.17157	−0.133904
$562$	37.7990	1.59445
$563$	−31.6569	−1.33418	−0.667089	−	0.744978i	$-0.732460\pi$
$563$	−31.6569	−1.33418	−0.667089	+	0.744978i	$0.732460\pi$
$564$	−1.85786	−0.0782302
$565$	8.82843	0.371415
$566$	−13.8995	−0.584239
$567$	−20.5858	−0.864522
$568$	−18.7279	−0.785806
$569$	30.2843	1.26958	0.634791	−	0.772684i	$-0.281086\pi$
$569$	30.2843	1.26958	0.634791	+	0.772684i	$0.281086\pi$
$570$	−4.00000	−0.167542
$571$	−35.5563	−1.48799	−0.743993	−	0.668187i	$-0.767071\pi$
$571$	−35.5563	−1.48799	−0.743993	+	0.668187i	$0.767071\pi$
$572$	−58.6274	−2.45134
$573$	−7.02944	−0.293659
$574$	86.4264	3.60737
$575$	−0.585786	−0.0244290
$576$	26.1127	1.08803
$577$	17.1716	0.714862	0.357431	−	0.933940i	$-0.383653\pi$
$577$	17.1716	0.714862	0.357431	+	0.933940i	$0.383653\pi$
$578$	−2.41421	−0.100418
$579$	8.88730	0.369344
$580$	−3.17157	−0.131692
$581$	45.4558	1.88583
$582$	−10.8284	−0.448853
$583$	−63.1127	−2.61386
$584$	−3.65685	−0.151322
$585$	7.51472	0.310695
$586$	−43.4558	−1.79514
$587$	−16.6274	−0.686287	−0.343143	−	0.939283i	$-0.611492\pi$
$587$	−16.6274	−0.686287	−0.343143	+	0.939283i	$0.611492\pi$
$588$	−10.4437	−0.430689
$589$	−12.0000	−0.494451
$590$	−35.7990	−1.47382
$591$	−4.20101	−0.172806
$592$	−31.4558	−1.29283
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	43.3137	1.77718
$595$	−3.41421	−0.139969
$596$	7.65685	0.313637
$597$	−9.23045	−0.377777
$598$	4.00000	0.163572
$599$	−21.9411	−0.896490	−0.448245	−	0.893911i	$-0.647951\pi$
$599$	−21.9411	−0.896490	−0.448245	+	0.893911i	$0.647951\pi$
$600$	2.58579	0.105564
$601$	13.7990	0.562873	0.281436	−	0.959580i	$-0.409189\pi$
$601$	13.7990	0.562873	0.281436	+	0.959580i	$0.409189\pi$
$602$	−30.1421	−1.22850
$603$	23.4558	0.955196
$604$	93.7401	3.81423
$605$	−18.3137	−0.744558
$606$	−11.3137	−0.459588
$607$	−7.41421	−0.300934	−0.150467	−	0.988615i	$-0.548078\pi$
$607$	−7.41421	−0.300934	−0.150467	+	0.988615i	$0.548078\pi$
$608$	4.48528	0.181902
$609$	1.65685	0.0671391
$610$	−8.82843	−0.357453
$611$	2.34315	0.0947935
$612$	10.1716	0.411161
$613$	−13.0294	−0.526254	−0.263127	−	0.964761i	$-0.584754\pi$
$613$	−13.0294	−0.526254	−0.263127	+	0.964761i	$0.584754\pi$
$614$	26.9706	1.08844
$615$	6.14214	0.247675
$616$	−81.5980	−3.28768
$617$	42.4853	1.71039	0.855197	−	0.518304i	$-0.173436\pi$
$617$	42.4853	1.71039	0.855197	+	0.518304i	$0.173436\pi$
$618$	6.82843	0.274680
$619$	−31.0711	−1.24885	−0.624426	−	0.781084i	$-0.714667\pi$
$619$	−31.0711	−1.24885	−0.624426	+	0.781084i	$0.714667\pi$
$620$	16.2426	0.652320
$621$	−1.94113	−0.0778947
$622$	−2.24264	−0.0899217
$623$	46.6274	1.86809
$624$	−4.97056	−0.198982
$625$	1.00000	0.0400000
$626$	−81.5980	−3.26131
$627$	8.97056	0.358250
$628$	5.02944	0.200696
$629$	10.4853	0.418076
$630$	21.8995	0.872497
$631$	23.7990	0.947423	0.473711	−	0.880680i	$-0.342914\pi$
$631$	23.7990	0.947423	0.473711	+	0.880680i	$0.342914\pi$
$632$	−11.4142	−0.454033
$633$	12.8284	0.509884
$634$	−3.17157	−0.125959
$635$	−5.31371	−0.210868
$636$	−26.1421	−1.03660
$637$	13.1716	0.521877
$638$	10.8284	0.428702
$639$	−11.2721	−0.445917
$640$	−20.5563	−0.812561
$641$	−0.142136	−0.00561402	−0.00280701	−	0.999996i	$-0.500894\pi$
$641$	−0.142136	−0.00561402	−0.00280701	+	0.999996i	$0.500894\pi$
$642$	−11.1716	−0.440907
$643$	11.6985	0.461343	0.230672	−	0.973032i	$-0.425908\pi$
$643$	11.6985	0.461343	0.230672	+	0.973032i	$0.425908\pi$
$644$	7.65685	0.301722
$645$	−2.14214	−0.0843465
$646$	6.82843	0.268661
$647$	−15.1716	−0.596456	−0.298228	−	0.954495i	$-0.596396\pi$
$647$	−15.1716	−0.596456	−0.298228	+	0.954495i	$0.596396\pi$
$648$	−26.6152	−1.04554
$649$	80.2843	3.15143
$650$	−6.82843	−0.267833
$651$	−8.48528	−0.332564
$652$	13.0711	0.511903
$653$	1.51472	0.0592755	0.0296378	−	0.999561i	$-0.490565\pi$
$653$	1.51472	0.0592755	0.0296378	+	0.999561i	$0.490565\pi$
$654$	7.51472	0.293849
$655$	−5.89949	−0.230512
$656$	31.4558	1.22814
$657$	−2.20101	−0.0858696
$658$	6.82843	0.266200
$659$	−27.3137	−1.06399	−0.531996	−	0.846747i	$-0.678558\pi$
$659$	−27.3137	−1.06399	−0.531996	+	0.846747i	$0.678558\pi$
$660$	−12.1421	−0.472632
$661$	−9.31371	−0.362261	−0.181131	−	0.983459i	$-0.557976\pi$
$661$	−9.31371	−0.362261	−0.181131	+	0.983459i	$0.557976\pi$
$662$	−51.1127	−1.98655
$663$	1.65685	0.0643469
$664$	58.7696	2.28070
$665$	9.65685	0.374477
$666$	−67.2548	−2.60607
$667$	−0.485281	−0.0187902
$668$	−23.8995	−0.924699
$669$	−3.51472	−0.135887
$670$	−21.3137	−0.823420
$671$	19.7990	0.764332
$672$	3.17157	0.122346
$673$	0.142136	0.00547893	0.00273946	−	0.999996i	$-0.499128\pi$
$673$	0.142136	0.00547893	0.00273946	+	0.999996i	$0.499128\pi$
$674$	−59.4558	−2.29015
$675$	3.31371	0.127545
$676$	−19.1421	−0.736236
$677$	−40.6274	−1.56144	−0.780719	−	0.624882i	$-0.785147\pi$
$677$	−40.6274	−1.56144	−0.780719	+	0.624882i	$0.785147\pi$
$678$	−12.4853	−0.479494
$679$	26.1421	1.00324
$680$	−4.41421	−0.169277
$681$	−1.59798	−0.0612347
$682$	−55.4558	−2.12351
$683$	10.7279	0.410493	0.205246	−	0.978710i	$-0.434200\pi$
$683$	10.7279	0.410493	0.205246	+	0.978710i	$0.434200\pi$
$684$	−28.7696	−1.10003
$685$	−6.82843	−0.260901
$686$	−19.3137	−0.737401
$687$	13.6569	0.521041
$688$	−10.9706	−0.418249
$689$	32.9706	1.25608
$690$	0.828427	0.0315377
$691$	25.2132	0.959155	0.479578	−	0.877499i	$-0.340790\pi$
$691$	25.2132	0.959155	0.479578	+	0.877499i	$0.340790\pi$
$692$	27.4558	1.04371
$693$	−49.1127	−1.86564
$694$	−86.1838	−3.27149
$695$	1.89949	0.0720519
$696$	2.14214	0.0811974
$697$	−10.4853	−0.397158
$698$	−49.1127	−1.85894
$699$	7.79899	0.294985
$700$	−13.0711	−0.494040
$701$	41.6569	1.57336	0.786679	−	0.617362i	$-0.211799\pi$
$701$	41.6569	1.57336	0.786679	+	0.617362i	$0.211799\pi$
$702$	−22.6274	−0.854017
$703$	−29.6569	−1.11853
$704$	53.2132	2.00555
$705$	0.485281	0.0182768
$706$	39.4558	1.48494
$707$	27.3137	1.02724
$708$	33.2548	1.24979
$709$	−13.7990	−0.518232	−0.259116	−	0.965846i	$-0.583431\pi$
$709$	−13.7990	−0.518232	−0.259116	+	0.965846i	$0.583431\pi$
$710$	10.2426	0.384399
$711$	−6.87006	−0.257647
$712$	60.2843	2.25925
$713$	2.48528	0.0930745
$714$	4.82843	0.180699
$715$	15.3137	0.572700
$716$	−4.48528	−0.167623
$717$	−11.7157	−0.437532
$718$	−18.8284	−0.702671
$719$	45.4975	1.69677	0.848385	−	0.529380i	$-0.177575\pi$
$719$	45.4975	1.69677	0.848385	+	0.529380i	$0.177575\pi$
$720$	7.97056	0.297045
$721$	−16.4853	−0.613944
$722$	26.5563	0.988325
$723$	−4.20101	−0.156237
$724$	−55.4558	−2.06100
$725$	0.828427	0.0307670
$726$	25.8995	0.961220
$727$	−39.4558	−1.46334	−0.731668	−	0.681661i	$-0.761258\pi$
$727$	−39.4558	−1.46334	−0.731668	+	0.681661i	$0.761258\pi$
$728$	42.6274	1.57988
$729$	−10.1960	−0.377628
$730$	2.00000	0.0740233
$731$	3.65685	0.135254
$732$	8.20101	0.303118
$733$	−10.2843	−0.379858	−0.189929	−	0.981798i	$-0.560826\pi$
$733$	−10.2843	−0.379858	−0.189929	+	0.981798i	$0.560826\pi$
$734$	13.8995	0.513040
$735$	2.72792	0.100621
$736$	−0.928932	−0.0342409
$737$	47.7990	1.76070
$738$	67.2548	2.47568
$739$	−11.7990	−0.434033	−0.217016	−	0.976168i	$-0.569633\pi$
$739$	−11.7990	−0.434033	−0.217016	+	0.976168i	$0.569633\pi$
$740$	40.1421	1.47565
$741$	−4.68629	−0.172155
$742$	96.0833	3.52733
$743$	−34.0416	−1.24887	−0.624433	−	0.781078i	$-0.714670\pi$
$743$	−34.0416	−1.24887	−0.624433	+	0.781078i	$0.714670\pi$
$744$	−10.9706	−0.402200
$745$	−2.00000	−0.0732743
$746$	67.1127	2.45717
$747$	35.3726	1.29422
$748$	20.7279	0.757888
$749$	26.9706	0.985483
$750$	−1.41421	−0.0516398
$751$	30.1838	1.10142	0.550711	−	0.834696i	$-0.314357\pi$
$751$	30.1838	1.10142	0.550711	+	0.834696i	$0.314357\pi$
$752$	2.48528	0.0906289
$753$	7.02944	0.256167
$754$	−5.65685	−0.206010
$755$	−24.4853	−0.891111
$756$	−43.3137	−1.57530
$757$	46.8284	1.70201	0.851004	−	0.525159i	$-0.175994\pi$
$757$	46.8284	1.70201	0.851004	+	0.525159i	$0.175994\pi$
$758$	−64.8701	−2.35619
$759$	−1.85786	−0.0674362
$760$	12.4853	0.452889
$761$	35.3137	1.28012	0.640060	−	0.768325i	$-0.278909\pi$
$761$	35.3137	1.28012	0.640060	+	0.768325i	$0.278909\pi$
$762$	7.51472	0.272230
$763$	−18.1421	−0.656789
$764$	45.9411	1.66209
$765$	−2.65685	−0.0960588
$766$	−53.7990	−1.94384
$767$	−41.9411	−1.51441
$768$	17.5563	0.633510
$769$	−45.6569	−1.64643	−0.823214	−	0.567731i	$-0.807821\pi$
$769$	−45.6569	−1.64643	−0.823214	+	0.567731i	$0.807821\pi$
$770$	44.6274	1.60826
$771$	−1.65685	−0.0596701
$772$	−58.0833	−2.09046
$773$	−35.1127	−1.26292	−0.631458	−	0.775410i	$-0.717543\pi$
$773$	−35.1127	−1.26292	−0.631458	+	0.775410i	$0.717543\pi$
$774$	−23.4558	−0.843103
$775$	−4.24264	−0.152400
$776$	33.7990	1.21331
$777$	−20.9706	−0.752315
$778$	38.6274	1.38486
$779$	29.6569	1.06257
$780$	6.34315	0.227121
$781$	−22.9706	−0.821951
$782$	−1.41421	−0.0505722
$783$	2.74517	0.0981042
$784$	13.9706	0.498949
$785$	−1.31371	−0.0468883
$786$	8.34315	0.297590
$787$	11.2132	0.399708	0.199854	−	0.979826i	$-0.435953\pi$
$787$	11.2132	0.399708	0.199854	+	0.979826i	$0.435953\pi$
$788$	27.4558	0.978074
$789$	7.79899	0.277651
$790$	6.24264	0.222103
$791$	30.1421	1.07173
$792$	−63.4975	−2.25628
$793$	−10.3431	−0.367296
$794$	22.4853	0.797973
$795$	6.82843	0.242179
$796$	60.3259	2.13819
$797$	8.62742	0.305599	0.152799	−	0.988257i	$-0.451171\pi$
$797$	8.62742	0.305599	0.152799	+	0.988257i	$0.451171\pi$
$798$	−13.6569	−0.483447
$799$	−0.828427	−0.0293076
$800$	1.58579	0.0560660
$801$	36.2843	1.28204
$802$	66.7696	2.35771
$803$	−4.48528	−0.158282
$804$	19.7990	0.698257
$805$	−2.00000	−0.0704907
$806$	28.9706	1.02044
$807$	8.76955	0.308703
$808$	35.3137	1.24233
$809$	−20.3431	−0.715227	−0.357613	−	0.933870i	$-0.616409\pi$
$809$	−20.3431	−0.715227	−0.357613	+	0.933870i	$0.616409\pi$
$810$	14.5563	0.511458
$811$	21.4142	0.751955	0.375977	−	0.926629i	$-0.377307\pi$
$811$	21.4142	0.751955	0.375977	+	0.926629i	$0.377307\pi$
$812$	−10.8284	−0.380003
$813$	1.37258	0.0481386
$814$	−137.054	−4.80373
$815$	−3.41421	−0.119595
$816$	1.75736	0.0615199
$817$	−10.3431	−0.361861
$818$	14.4853	0.506466
$819$	25.6569	0.896523
$820$	−40.1421	−1.40182
$821$	−8.62742	−0.301099	−0.150549	−	0.988602i	$-0.548104\pi$
$821$	−8.62742	−0.301099	−0.150549	+	0.988602i	$0.548104\pi$
$822$	9.65685	0.336821
$823$	−10.9289	−0.380959	−0.190479	−	0.981691i	$-0.561004\pi$
$823$	−10.9289	−0.380959	−0.190479	+	0.981691i	$0.561004\pi$
$824$	−21.3137	−0.742498
$825$	3.17157	0.110420
$826$	−122.225	−4.25277
$827$	1.55635	0.0541196	0.0270598	−	0.999634i	$-0.491386\pi$
$827$	1.55635	0.0541196	0.0270598	+	0.999634i	$0.491386\pi$
$828$	5.95837	0.207068
$829$	−27.9411	−0.970435	−0.485218	−	0.874393i	$-0.661260\pi$
$829$	−27.9411	−0.970435	−0.485218	+	0.874393i	$0.661260\pi$
$830$	−32.1421	−1.11567
$831$	5.85786	0.203207
$832$	−27.7990	−0.963757
$833$	−4.65685	−0.161350
$834$	−2.68629	−0.0930187
$835$	6.24264	0.216035
$836$	−58.6274	−2.02767
$837$	−14.0589	−0.485946
$838$	−9.07107	−0.313355
$839$	−34.1838	−1.18015	−0.590077	−	0.807347i	$-0.700903\pi$
$839$	−34.1838	−1.18015	−0.590077	+	0.807347i	$0.700903\pi$
$840$	8.82843	0.304610
$841$	−28.3137	−0.976335
$842$	−12.0000	−0.413547
$843$	9.17157	0.315886
$844$	−83.8406	−2.88591
$845$	5.00000	0.172005
$846$	5.31371	0.182689
$847$	−62.5269	−2.14845
$848$	34.9706	1.20089
$849$	−3.37258	−0.115747
$850$	2.41421	0.0828068
$851$	6.14214	0.210550
$852$	−9.51472	−0.325969
$853$	−36.8284	−1.26098	−0.630491	−	0.776197i	$-0.717146\pi$
$853$	−36.8284	−1.26098	−0.630491	+	0.776197i	$0.717146\pi$
$854$	−30.1421	−1.03144
$855$	7.51472	0.256998
$856$	34.8701	1.19183
$857$	30.4853	1.04136	0.520679	−	0.853753i	$-0.325679\pi$
$857$	30.4853	1.04136	0.520679	+	0.853753i	$0.325679\pi$
$858$	−21.6569	−0.739353
$859$	−36.7696	−1.25456	−0.627280	−	0.778793i	$-0.715832\pi$
$859$	−36.7696	−1.25456	−0.627280	+	0.778793i	$0.715832\pi$
$860$	14.0000	0.477396
$861$	20.9706	0.714675
$862$	3.41421	0.116289
$863$	−10.4853	−0.356923	−0.178462	−	0.983947i	$-0.557112\pi$
$863$	−10.4853	−0.356923	−0.178462	+	0.983947i	$0.557112\pi$
$864$	5.25483	0.178773
$865$	−7.17157	−0.243841
$866$	6.82843	0.232039
$867$	−0.585786	−0.0198944
$868$	55.4558	1.88229
$869$	−14.0000	−0.474917
$870$	−1.17157	−0.0397200
$871$	−24.9706	−0.846095
$872$	−23.4558	−0.794315
$873$	20.3431	0.688511
$874$	4.00000	0.135302
$875$	3.41421	0.115421
$876$	−1.85786	−0.0627714
$877$	54.2843	1.83305	0.916525	−	0.399978i	$-0.130982\pi$
$877$	54.2843	1.83305	0.916525	+	0.399978i	$0.130982\pi$
$878$	−6.24264	−0.210679
$879$	−10.5442	−0.355646
$880$	16.2426	0.547539
$881$	19.8579	0.669028	0.334514	−	0.942391i	$-0.391428\pi$
$881$	19.8579	0.669028	0.334514	+	0.942391i	$0.391428\pi$
$882$	29.8701	1.00578
$883$	20.8284	0.700932	0.350466	−	0.936575i	$-0.386023\pi$
$883$	20.8284	0.700932	0.350466	+	0.936575i	$0.386023\pi$
$884$	−10.8284	−0.364199
$885$	−8.68629	−0.291986
$886$	−6.00000	−0.201574
$887$	−29.0711	−0.976111	−0.488055	−	0.872813i	$-0.662294\pi$
$887$	−29.0711	−0.976111	−0.488055	+	0.872813i	$0.662294\pi$
$888$	−27.1127	−0.909843
$889$	−18.1421	−0.608468
$890$	−32.9706	−1.10518
$891$	−32.6447	−1.09364
$892$	22.9706	0.769111
$893$	2.34315	0.0784104
$894$	2.82843	0.0945968
$895$	1.17157	0.0391614
$896$	−70.1838	−2.34468
$897$	0.970563	0.0324061
$898$	22.9706	0.766538
$899$	−3.51472	−0.117222
$900$	−10.1716	−0.339052
$901$	−11.6569	−0.388346
$902$	137.054	4.56339
$903$	−7.31371	−0.243385
$904$	38.9706	1.29614
$905$	14.4853	0.481507
$906$	34.6274	1.15042
$907$	18.7279	0.621850	0.310925	−	0.950434i	$-0.399361\pi$
$907$	18.7279	0.621850	0.310925	+	0.950434i	$0.399361\pi$
$908$	10.4437	0.346585
$909$	21.2548	0.704978
$910$	−23.3137	−0.772842
$911$	3.75736	0.124487	0.0622434	−	0.998061i	$-0.480174\pi$
$911$	3.75736	0.124487	0.0622434	+	0.998061i	$0.480174\pi$
$912$	−4.97056	−0.164592
$913$	72.0833	2.38561
$914$	55.7990	1.84567
$915$	−2.14214	−0.0708168
$916$	−89.2548	−2.94906
$917$	−20.1421	−0.665152
$918$	8.00000	0.264039
$919$	7.02944	0.231880	0.115940	−	0.993256i	$-0.463012\pi$
$919$	7.02944	0.231880	0.115940	+	0.993256i	$0.463012\pi$
$920$	−2.58579	−0.0852509
$921$	6.54416	0.215637
$922$	90.7696	2.98934
$923$	12.0000	0.394985
$924$	−41.4558	−1.36380
$925$	−10.4853	−0.344754
$926$	21.3137	0.700412
$927$	−12.8284	−0.421341
$928$	1.31371	0.0431246
$929$	−15.4558	−0.507090	−0.253545	−	0.967324i	$-0.581597\pi$
$929$	−15.4558	−0.507090	−0.253545	+	0.967324i	$0.581597\pi$
$930$	6.00000	0.196748
$931$	13.1716	0.431681
$932$	−50.9706	−1.66960
$933$	−0.544156	−0.0178149
$934$	28.1421	0.920839
$935$	−5.41421	−0.177064
$936$	33.1716	1.08425
$937$	−22.2843	−0.727995	−0.363998	−	0.931400i	$-0.618588\pi$
$937$	−22.2843	−0.727995	−0.363998	+	0.931400i	$0.618588\pi$
$938$	−72.7696	−2.37601
$939$	−19.7990	−0.646116
$940$	−3.17157	−0.103445
$941$	47.4558	1.54702	0.773508	−	0.633786i	$-0.218500\pi$
$941$	47.4558	1.54702	0.773508	+	0.633786i	$0.218500\pi$
$942$	1.85786	0.0605325
$943$	−6.14214	−0.200015
$944$	−44.4853	−1.44787
$945$	11.3137	0.368035
$946$	−47.7990	−1.55408
$947$	−16.1005	−0.523196	−0.261598	−	0.965177i	$-0.584249\pi$
$947$	−16.1005	−0.523196	−0.261598	+	0.965177i	$0.584249\pi$
$948$	−5.79899	−0.188342
$949$	2.34315	0.0760617
$950$	−6.82843	−0.221543
$951$	−0.769553	−0.0249545
$952$	−15.0711	−0.488456
$953$	−50.1421	−1.62426	−0.812132	−	0.583474i	$-0.801693\pi$
$953$	−50.1421	−1.62426	−0.812132	+	0.583474i	$0.801693\pi$
$954$	74.7696	2.42075
$955$	−12.0000	−0.388311
$956$	76.5685	2.47640
$957$	2.62742	0.0849323
$958$	58.5269	1.89092
$959$	−23.3137	−0.752839
$960$	−5.75736	−0.185818
$961$	−13.0000	−0.419355
$962$	71.5980	2.30841
$963$	20.9878	0.676323
$964$	27.4558	0.884293
$965$	15.1716	0.488390
$966$	2.82843	0.0910032
$967$	−34.9706	−1.12458	−0.562289	−	0.826941i	$-0.690079\pi$
$967$	−34.9706	−1.12458	−0.562289	+	0.826941i	$0.690079\pi$
$968$	−80.8406	−2.59832
$969$	1.65685	0.0532258
$970$	−18.4853	−0.593527
$971$	−47.7990	−1.53394	−0.766971	−	0.641681i	$-0.778237\pi$
$971$	−47.7990	−1.53394	−0.766971	+	0.641681i	$0.778237\pi$
$972$	−51.5807	−1.65445
$973$	6.48528	0.207909
$974$	−38.3848	−1.22993
$975$	−1.65685	−0.0530618
$976$	−10.9706	−0.351159
$977$	−31.2548	−0.999931	−0.499965	−	0.866045i	$-0.666654\pi$
$977$	−31.2548	−0.999931	−0.499965	+	0.866045i	$0.666654\pi$
$978$	4.82843	0.154396
$979$	73.9411	2.36317
$980$	−17.8284	−0.569508
$981$	−14.1177	−0.450745
$982$	−1.17157	−0.0373864
$983$	29.3553	0.936290	0.468145	−	0.883652i	$-0.344923\pi$
$983$	29.3553	0.936290	0.468145	+	0.883652i	$0.344923\pi$
$984$	27.1127	0.864321
$985$	−7.17157	−0.228505
$986$	2.00000	0.0636930
$987$	1.65685	0.0527383
$988$	30.6274	0.974388
$989$	2.14214	0.0681159
$990$	34.7279	1.10373
$991$	30.5858	0.971590	0.485795	−	0.874073i	$-0.338530\pi$
$991$	30.5858	0.971590	0.485795	+	0.874073i	$0.338530\pi$
$992$	−6.72792	−0.213612
$993$	−12.4020	−0.393566
$994$	34.9706	1.10920
$995$	−15.7574	−0.499542
$996$	29.8579	0.946083
$997$	11.8579	0.375542	0.187771	−	0.982213i	$-0.439874\pi$
$997$	11.8579	0.375542	0.187771	+	0.982213i	$0.439874\pi$
$998$	−9.07107	−0.287140
$999$	−34.7452	−1.09929

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.2.a.b.1.1	✓	2
3.2	odd	2		765.2.a.i.1.2		2
4.3	odd	2		1360.2.a.o.1.1		2
5.2	odd	4		425.2.b.e.324.1		4
5.3	odd	4		425.2.b.e.324.4		4
5.4	even	2		425.2.a.f.1.2		2
7.6	odd	2		4165.2.a.q.1.1		2
8.3	odd	2		5440.2.a.ba.1.2		2
8.5	even	2		5440.2.a.bm.1.1		2
15.14	odd	2		3825.2.a.p.1.1		2
17.4	even	4		1445.2.d.f.866.4		4
17.13	even	4		1445.2.d.f.866.3		4
17.16	even	2		1445.2.a.f.1.1		2
20.19	odd	2		6800.2.a.ba.1.2		2
85.84	even	2		7225.2.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.b.1.1	✓	2	1.1	even	1	trivial
425.2.a.f.1.2		2	5.4	even	2
425.2.b.e.324.1		4	5.2	odd	4
425.2.b.e.324.4		4	5.3	odd	4
765.2.a.i.1.2		2	3.2	odd	2
1360.2.a.o.1.1		2	4.3	odd	2
1445.2.a.f.1.1		2	17.16	even	2
1445.2.d.f.866.3		4	17.13	even	4
1445.2.d.f.866.4		4	17.4	even	4
3825.2.a.p.1.1		2	15.14	odd	2
4165.2.a.q.1.1		2	7.6	odd	2
5440.2.a.ba.1.2		2	8.3	odd	2
5440.2.a.bm.1.1		2	8.5	even	2
6800.2.a.ba.1.2		2	20.19	odd	2
7225.2.a.o.1.2		2	85.84	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	85.2.a.b.1.1

Level	$85$
Weight	$2$
Character	85.1
Self dual	yes
Analytic conductor	$0.679$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$