LMFDB - Embedded newform 171.10.a.f.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [171,10,Mod(1,171)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("171.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	171.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$88.0711279840$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 3356 x^{6} - 1330 x^{5} + 3186388 x^{4} - 1801192 x^{3} - 758043152 x^{2} + \cdots - 16080668672 $ x^8 - x^7 - 3356x^6 - 1330x^5 + 3186388x^4 - 1801192x^3 - 758043152x^2 + 6645763712x - 16080668672
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$45.0760$ of defining polynomial
Character	$\chi$	$=$	171.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+43.0760 q^{2} +1343.54 q^{4} -1106.01 q^{5} +1237.92 q^{7} +35819.6 q^{8} -47642.6 q^{10} -77454.8 q^{11} -166982. q^{13} +53324.8 q^{14} +855070. q^{16} -202288. q^{17} +130321. q^{19} -1.48598e6 q^{20} -3.33644e6 q^{22} -1.07984e6 q^{23} -729860. q^{25} -7.19293e6 q^{26} +1.66320e6 q^{28} -4.07412e6 q^{29} +5.94389e6 q^{31} +1.84934e7 q^{32} -8.71375e6 q^{34} -1.36916e6 q^{35} +6.26051e6 q^{37} +5.61371e6 q^{38} -3.96169e7 q^{40} +2.45995e7 q^{41} +2.97996e6 q^{43} -1.04064e8 q^{44} -4.65150e7 q^{46} -1.29594e7 q^{47} -3.88212e7 q^{49} -3.14395e7 q^{50} -2.24348e8 q^{52} +2.55151e7 q^{53} +8.56660e7 q^{55} +4.43419e7 q^{56} -1.75497e8 q^{58} -7.81216e7 q^{59} -4.14570e7 q^{61} +2.56039e8 q^{62} +3.58826e8 q^{64} +1.84685e8 q^{65} -6.16078e7 q^{67} -2.71782e8 q^{68} -5.89779e7 q^{70} -1.97167e8 q^{71} -9.45381e7 q^{73} +2.69678e8 q^{74} +1.75092e8 q^{76} -9.58832e7 q^{77} -3.53552e7 q^{79} -9.45718e8 q^{80} +1.05965e9 q^{82} -4.46185e7 q^{83} +2.23733e8 q^{85} +1.28365e8 q^{86} -2.77440e9 q^{88} +3.95259e8 q^{89} -2.06711e8 q^{91} -1.45081e9 q^{92} -5.58238e8 q^{94} -1.44137e8 q^{95} -1.73986e9 q^{97} -1.67226e9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 15 q^{2} + 2645 q^{4} - 3894 q^{5} - 7133 q^{7} - 10911 q^{8} + 113172 q^{10} - 172818 q^{11} + 109291 q^{13} - 250959 q^{14} + 1590377 q^{16} - 583575 q^{17} + 1042568 q^{19} - 3191676 q^{20} + 57234 q^{22}+ \cdots - 4178510532 q^{98}+O(q^{100}) $ 8 * q - 15 * q^2 + 2645 * q^4 - 3894 * q^5 - 7133 * q^7 - 10911 * q^8 + 113172 * q^10 - 172818 * q^11 + 109291 * q^13 - 250959 * q^14 + 1590377 * q^16 - 583575 * q^17 + 1042568 * q^19 - 3191676 * q^20 + 57234 * q^22 + 2292405 * q^23 + 205328 * q^25 + 821247 * q^26 - 10217945 * q^28 - 4546335 * q^29 - 8277416 * q^31 + 15773553 * q^32 - 22393029 * q^34 - 15633768 * q^35 + 24485536 * q^37 - 1954815 * q^38 + 2543472 * q^40 - 35051946 * q^41 + 36624268 * q^43 - 215210070 * q^44 - 66065277 * q^46 - 9865524 * q^47 + 111765087 * q^49 - 50599293 * q^50 - 258827351 * q^52 - 13963683 * q^53 + 261540660 * q^55 + 517172871 * q^56 - 399459963 * q^58 - 21215463 * q^59 + 180552112 * q^61 + 451241340 * q^62 + 180711569 * q^64 + 21864624 * q^65 + 1767355 * q^67 + 41475117 * q^68 - 516193752 * q^70 - 273631578 * q^71 - 263711543 * q^73 - 677785566 * q^74 + 344699045 * q^76 - 441243702 * q^77 + 744472174 * q^79 - 1904949528 * q^80 + 1284295572 * q^82 - 832139910 * q^83 + 1013793492 * q^85 - 956757432 * q^86 - 585082794 * q^88 + 291457734 * q^89 - 1351077415 * q^91 - 2042025591 * q^92 - 2759974128 * q^94 - 507469974 * q^95 - 3483837140 * q^97 - 4178510532 * q^98

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 $

$2$	43.0760	1.90371	0.951854	−	0.306550i	$-0.0991748\pi$
$2$	43.0760	1.90371	0.951854	+	0.306550i	$0.0991748\pi$
$3$	0	0
$3$	0	0
$4$	1343.54	2.62411
$5$	−1106.01	−0.791398	−0.395699	−	0.918380i	$-0.629498\pi$
$5$	−1106.01	−0.791398	−0.395699	+	0.918380i	$0.629498\pi$
$6$	0	0
$7$	1237.92	0.194873	0.0974367	−	0.995242i	$-0.468936\pi$
$7$	1237.92	0.194873	0.0974367	+	0.995242i	$0.468936\pi$
$8$	35819.6	3.09183
$9$	0	0
$10$	−47642.6	−1.50659
$11$	−77454.8	−1.59508	−0.797538	−	0.603269i	$-0.793865\pi$
$11$	−77454.8	−1.59508	−0.797538	+	0.603269i	$0.793865\pi$
$12$	0	0
$13$	−166982.	−1.62153	−0.810766	−	0.585370i	$-0.800949\pi$
$13$	−166982.	−1.62153	−0.810766	+	0.585370i	$0.800949\pi$
$14$	53324.8	0.370982
$15$	0	0
$16$	855070.	3.26183
$17$	−202288.	−0.587421	−0.293710	−	0.955894i	$-0.594890\pi$
$17$	−202288.	−0.587421	−0.293710	+	0.955894i	$0.594890\pi$
$18$	0	0
$19$	130321.	0.229416
$19$	130321.	0.229416
$20$	−1.48598e6	−2.07671
$21$	0	0
$22$	−3.33644e6	−3.03656
$23$	−1.07984e6	−0.804604	−0.402302	−	0.915507i	$-0.631790\pi$
$23$	−1.07984e6	−0.804604	−0.402302	+	0.915507i	$0.631790\pi$
$24$	0	0
$25$	−729860.	−0.373689
$26$	−7.19293e6	−3.08692
$27$	0	0
$28$	1.66320e6	0.511369
$29$	−4.07412e6	−1.06965	−0.534826	−	0.844962i	$-0.679623\pi$
$29$	−4.07412e6	−1.06965	−0.534826	+	0.844962i	$0.679623\pi$
$30$	0	0
$31$	5.94389e6	1.15596	0.577981	−	0.816050i	$-0.303841\pi$
$31$	5.94389e6	1.15596	0.577981	+	0.816050i	$0.303841\pi$
$32$	1.84934e7	3.11775
$33$	0	0
$34$	−8.71375e6	−1.11828
$35$	−1.36916e6	−0.154223
$36$	0	0
$37$	6.26051e6	0.549163	0.274582	−	0.961564i	$-0.411461\pi$
$37$	6.26051e6	0.549163	0.274582	+	0.961564i	$0.411461\pi$
$38$	5.61371e6	0.436741
$39$	0	0
$40$	−3.96169e7	−2.44687
$41$	2.45995e7	1.35956	0.679780	−	0.733416i	$-0.262075\pi$
$41$	2.45995e7	1.35956	0.679780	+	0.733416i	$0.262075\pi$
$42$	0	0
$43$	2.97996e6	0.132924	0.0664619	−	0.997789i	$-0.478829\pi$
$43$	2.97996e6	0.132924	0.0664619	+	0.997789i	$0.478829\pi$
$44$	−1.04064e8	−4.18565
$45$	0	0
$46$	−4.65150e7	−1.53173
$47$	−1.29594e7	−0.387386	−0.193693	−	0.981062i	$-0.562047\pi$
$47$	−1.29594e7	−0.387386	−0.193693	+	0.981062i	$0.562047\pi$
$48$	0	0
$49$	−3.88212e7	−0.962024
$50$	−3.14395e7	−0.711394
$51$	0	0
$52$	−2.24348e8	−4.25507
$53$	2.55151e7	0.444177	0.222089	−	0.975026i	$-0.428713\pi$
$53$	2.55151e7	0.444177	0.222089	+	0.975026i	$0.428713\pi$
$54$	0	0
$55$	8.56660e7	1.26234
$56$	4.43419e7	0.602515
$57$	0	0
$58$	−1.75497e8	−2.03631
$59$	−7.81216e7	−0.839338	−0.419669	−	0.907677i	$-0.637854\pi$
$59$	−7.81216e7	−0.839338	−0.419669	+	0.907677i	$0.637854\pi$
$60$	0	0
$61$	−4.14570e7	−0.383366	−0.191683	−	0.981457i	$-0.561395\pi$
$61$	−4.14570e7	−0.383366	−0.191683	+	0.981457i	$0.561395\pi$
$62$	2.56039e8	2.20061
$63$	0	0
$64$	3.58826e8	2.67346
$65$	1.84685e8	1.28328
$66$	0	0
$67$	−6.16078e7	−0.373507	−0.186754	−	0.982407i	$-0.559797\pi$
$67$	−6.16078e7	−0.373507	−0.186754	+	0.982407i	$0.559797\pi$
$68$	−2.71782e8	−1.54146
$69$	0	0
$70$	−5.89779e7	−0.293595
$71$	−1.97167e8	−0.920815	−0.460407	−	0.887708i	$-0.652297\pi$
$71$	−1.97167e8	−0.920815	−0.460407	+	0.887708i	$0.652297\pi$
$72$	0	0
$73$	−9.45381e7	−0.389632	−0.194816	−	0.980840i	$-0.562411\pi$
$73$	−9.45381e7	−0.389632	−0.194816	+	0.980840i	$0.562411\pi$
$74$	2.69678e8	1.04545
$75$	0	0
$76$	1.75092e8	0.602012
$77$	−9.58832e7	−0.310838
$78$	0	0
$79$	−3.53552e7	−0.102125	−0.0510625	−	0.998695i	$-0.516261\pi$
$79$	−3.53552e7	−0.102125	−0.0510625	+	0.998695i	$0.516261\pi$
$80$	−9.45718e8	−2.58141
$81$	0	0
$82$	1.05965e9	2.58820
$83$	−4.46185e7	−0.103196	−0.0515981	−	0.998668i	$-0.516431\pi$
$83$	−4.46185e7	−0.103196	−0.0515981	+	0.998668i	$0.516431\pi$
$84$	0	0
$85$	2.23733e8	0.464884
$86$	1.28365e8	0.253048
$87$	0	0
$88$	−2.77440e9	−4.93170
$89$	3.95259e8	0.667770	0.333885	−	0.942614i	$-0.391640\pi$
$89$	3.95259e8	0.667770	0.333885	+	0.942614i	$0.391640\pi$
$90$	0	0
$91$	−2.06711e8	−0.315994
$92$	−1.45081e9	−2.11137
$93$	0	0
$94$	−5.58238e8	−0.737470
$95$	−1.44137e8	−0.181559
$96$	0	0
$97$	−1.73986e9	−1.99546	−0.997728	−	0.0673747i	$-0.978538\pi$
$97$	−1.73986e9	−1.99546	−0.997728	+	0.0673747i	$0.978538\pi$
$98$	−1.67226e9	−1.83141
$99$	0	0
$100$	−9.80599e8	−0.980599
$101$	1.58117e9	1.51194	0.755968	−	0.654609i	$-0.227167\pi$
$101$	1.58117e9	1.51194	0.755968	+	0.654609i	$0.227167\pi$
$102$	0	0
$103$	−2.48998e8	−0.217986	−0.108993	−	0.994043i	$-0.534763\pi$
$103$	−2.48998e8	−0.217986	−0.108993	+	0.994043i	$0.534763\pi$
$104$	−5.98123e9	−5.01350
$105$	0	0
$106$	1.09909e9	0.845584
$107$	1.49426e9	1.10204	0.551022	−	0.834491i	$-0.314238\pi$
$107$	1.49426e9	1.10204	0.551022	+	0.834491i	$0.314238\pi$
$108$	0	0
$109$	1.24254e9	0.843123	0.421562	−	0.906800i	$-0.361482\pi$
$109$	1.24254e9	0.843123	0.421562	+	0.906800i	$0.361482\pi$
$110$	3.69015e9	2.40313
$111$	0	0
$112$	1.05851e9	0.635645
$113$	1.65242e8	0.0953383	0.0476692	−	0.998863i	$-0.484821\pi$
$113$	1.65242e8	0.0953383	0.0476692	+	0.998863i	$0.484821\pi$
$114$	0	0
$115$	1.19431e9	0.636763
$116$	−5.47375e9	−2.80688
$117$	0	0
$118$	−3.36517e9	−1.59786
$119$	−2.50417e8	−0.114473
$120$	0	0
$121$	3.64130e9	1.54427
$122$	−1.78580e9	−0.729818
$123$	0	0
$124$	7.98588e9	3.03337
$125$	2.96742e9	1.08713
$126$	0	0
$127$	−2.09297e9	−0.713916	−0.356958	−	0.934121i	$-0.616186\pi$
$127$	−2.09297e9	−0.713916	−0.356958	+	0.934121i	$0.616186\pi$
$128$	5.98816e9	1.97174
$129$	0	0
$130$	7.95548e9	2.44299
$131$	1.95319e9	0.579459	0.289730	−	0.957109i	$-0.406435\pi$
$131$	1.95319e9	0.579459	0.289730	+	0.957109i	$0.406435\pi$
$132$	0	0
$133$	1.61327e8	0.0447070
$134$	−2.65382e9	−0.711049
$135$	0	0
$136$	−7.24586e9	−1.81620
$137$	9.41931e8	0.228442	0.114221	−	0.993455i	$-0.463563\pi$
$137$	9.41931e8	0.228442	0.114221	+	0.993455i	$0.463563\pi$
$138$	0	0
$139$	−5.29062e9	−1.20210	−0.601049	−	0.799212i	$-0.705250\pi$
$139$	−5.29062e9	−1.20210	−0.601049	+	0.799212i	$0.705250\pi$
$140$	−1.83953e9	−0.404697
$141$	0	0
$142$	−8.49318e9	−1.75296
$143$	1.29336e10	2.58647
$144$	0	0
$145$	4.50602e9	0.846521
$146$	−4.07233e9	−0.741745
$147$	0	0
$148$	8.41126e9	1.44106
$149$	−1.09321e10	−1.81705	−0.908524	−	0.417832i	$-0.862790\pi$
$149$	−1.09321e10	−1.81705	−0.908524	+	0.417832i	$0.862790\pi$
$150$	0	0
$151$	7.06959e9	1.10662	0.553310	−	0.832976i	$-0.313365\pi$
$151$	7.06959e9	1.10662	0.553310	+	0.832976i	$0.313365\pi$
$152$	4.66804e9	0.709314
$153$	0	0
$154$	−4.13026e9	−0.591745
$155$	−6.57402e9	−0.914826
$156$	0	0
$157$	−1.05476e9	−0.138550	−0.0692748	−	0.997598i	$-0.522069\pi$
$157$	−1.05476e9	−0.138550	−0.0692748	+	0.997598i	$0.522069\pi$
$158$	−1.52296e9	−0.194416
$159$	0	0
$160$	−2.04539e10	−2.46738
$161$	−1.33675e9	−0.156796
$162$	0	0
$163$	3.82509e9	0.424422	0.212211	−	0.977224i	$-0.431934\pi$
$163$	3.82509e9	0.424422	0.212211	+	0.977224i	$0.431934\pi$
$164$	3.30504e10	3.56763
$165$	0	0
$166$	−1.92199e9	−0.196456
$167$	5.94366e9	0.591330	0.295665	−	0.955292i	$-0.404459\pi$
$167$	5.94366e9	0.591330	0.295665	+	0.955292i	$0.404459\pi$
$168$	0	0
$169$	1.72786e10	1.62937
$170$	9.63752e9	0.885004
$171$	0	0
$172$	4.00371e9	0.348806
$173$	−5.16427e8	−0.0438330	−0.0219165	−	0.999760i	$-0.506977\pi$
$173$	−5.16427e8	−0.0438330	−0.0219165	+	0.999760i	$0.506977\pi$
$174$	0	0
$175$	−9.03512e8	−0.0728220
$176$	−6.62293e10	−5.20287
$177$	0	0
$178$	1.70262e10	1.27124
$179$	−1.54846e10	−1.12736	−0.563678	−	0.825995i	$-0.690614\pi$
$179$	−1.54846e10	−1.12736	−0.563678	+	0.825995i	$0.690614\pi$
$180$	0	0
$181$	−1.72451e10	−1.19430	−0.597150	−	0.802130i	$-0.703700\pi$
$181$	−1.72451e10	−1.19430	−0.597150	+	0.802130i	$0.703700\pi$
$182$	−8.90430e9	−0.601560
$183$	0	0
$184$	−3.86792e10	−2.48770
$185$	−6.92420e9	−0.434607
$186$	0	0
$187$	1.56682e10	0.936981
$188$	−1.74115e10	−1.01654
$189$	0	0
$190$	−6.20883e9	−0.345636
$191$	−3.06828e9	−0.166819	−0.0834093	−	0.996515i	$-0.526581\pi$
$191$	−3.06828e9	−0.166819	−0.0834093	+	0.996515i	$0.526581\pi$
$192$	0	0
$193$	5.94109e9	0.308218	0.154109	−	0.988054i	$-0.450749\pi$
$193$	5.94109e9	0.308218	0.154109	+	0.988054i	$0.450749\pi$
$194$	−7.49463e10	−3.79877
$195$	0	0
$196$	−5.21579e10	−2.52446
$197$	2.88485e10	1.36466	0.682332	−	0.731042i	$-0.260966\pi$
$197$	2.88485e10	1.36466	0.682332	+	0.731042i	$0.260966\pi$
$198$	0	0
$199$	−1.01293e10	−0.457868	−0.228934	−	0.973442i	$-0.573524\pi$
$199$	−1.01293e10	−0.457868	−0.228934	+	0.973442i	$0.573524\pi$
$200$	−2.61433e10	−1.15538
$201$	0	0
$202$	6.81106e10	2.87828
$203$	−5.04344e9	−0.208447
$204$	0	0
$205$	−2.72073e10	−1.07595
$206$	−1.07258e10	−0.414981
$207$	0	0
$208$	−1.42782e11	−5.28917
$209$	−1.00940e10	−0.365935
$210$	0	0
$211$	−3.41201e10	−1.18506	−0.592528	−	0.805550i	$-0.701870\pi$
$211$	−3.41201e10	−1.18506	−0.592528	+	0.805550i	$0.701870\pi$
$212$	3.42807e10	1.16557
$213$	0	0
$214$	6.43667e10	2.09797
$215$	−3.29588e9	−0.105196
$216$	0	0
$217$	7.35809e9	0.225266
$218$	5.35236e10	1.60506
$219$	0	0
$220$	1.15096e11	3.31252
$221$	3.37785e10	0.952522
$222$	0	0
$223$	9.38041e9	0.254010	0.127005	−	0.991902i	$-0.459464\pi$
$223$	9.38041e9	0.254010	0.127005	+	0.991902i	$0.459464\pi$
$224$	2.28934e10	0.607567
$225$	0	0
$226$	7.11797e9	0.181496
$227$	3.27963e10	0.819800	0.409900	−	0.912130i	$-0.365564\pi$
$227$	3.27963e10	0.819800	0.409900	+	0.912130i	$0.365564\pi$
$228$	0	0
$229$	3.30142e10	0.793307	0.396653	−	0.917968i	$-0.370172\pi$
$229$	3.30142e10	0.793307	0.396653	+	0.917968i	$0.370172\pi$
$230$	5.14462e10	1.21221
$231$	0	0
$232$	−1.45933e11	−3.30718
$233$	−2.01073e10	−0.446944	−0.223472	−	0.974710i	$-0.571739\pi$
$233$	−2.01073e10	−0.446944	−0.223472	+	0.974710i	$0.571739\pi$
$234$	0	0
$235$	1.43332e10	0.306577
$236$	−1.04960e11	−2.20251
$237$	0	0
$238$	−1.07870e10	−0.217923
$239$	4.41998e10	0.876253	0.438126	−	0.898913i	$-0.355642\pi$
$239$	4.41998e10	0.876253	0.438126	+	0.898913i	$0.355642\pi$
$240$	0	0
$241$	5.60361e10	1.07002	0.535009	−	0.844846i	$-0.320308\pi$
$241$	5.60361e10	1.07002	0.535009	+	0.844846i	$0.320308\pi$
$242$	1.56853e11	2.93983
$243$	0	0
$244$	−5.56993e10	−1.00599
$245$	4.29367e10	0.761345
$246$	0	0
$247$	−2.17613e10	−0.372005
$248$	2.12908e11	3.57403
$249$	0	0
$250$	1.27824e11	2.06959
$251$	5.21561e10	0.829417	0.414709	−	0.909954i	$-0.363883\pi$
$251$	5.21561e10	0.829417	0.414709	+	0.909954i	$0.363883\pi$
$252$	0	0
$253$	8.36385e10	1.28341
$254$	−9.01569e10	−1.35909
$255$	0	0
$256$	7.42275e10	1.08015
$257$	5.58927e10	0.799201	0.399600	−	0.916689i	$-0.369149\pi$
$257$	5.58927e10	0.799201	0.399600	+	0.916689i	$0.369149\pi$
$258$	0	0
$259$	7.75003e9	0.107017
$260$	2.48132e11	3.36746
$261$	0	0
$262$	8.41355e10	1.10312
$263$	−1.16761e11	−1.50487	−0.752433	−	0.658669i	$-0.771120\pi$
$263$	−1.16761e11	−1.50487	−0.752433	+	0.658669i	$0.771120\pi$
$264$	0	0
$265$	−2.82201e10	−0.351521
$266$	6.94934e9	0.0851092
$267$	0	0
$268$	−8.27728e10	−0.980124
$269$	−2.13880e10	−0.249049	−0.124525	−	0.992217i	$-0.539741\pi$
$269$	−2.13880e10	−0.249049	−0.124525	+	0.992217i	$0.539741\pi$
$270$	0	0
$271$	8.91662e10	1.00424	0.502121	−	0.864797i	$-0.332553\pi$
$271$	8.91662e10	1.00424	0.502121	+	0.864797i	$0.332553\pi$
$272$	−1.72970e11	−1.91607
$273$	0	0
$274$	4.05746e10	0.434888
$275$	5.65312e10	0.596062
$276$	0	0
$277$	−1.73135e11	−1.76695	−0.883477	−	0.468474i	$-0.844804\pi$
$277$	−1.73135e11	−1.76695	−0.883477	+	0.468474i	$0.844804\pi$
$278$	−2.27899e11	−2.28844
$279$	0	0
$280$	−4.90427e10	−0.476830
$281$	−1.99315e11	−1.90705	−0.953525	−	0.301314i	$-0.902575\pi$
$281$	−1.99315e11	−1.90705	−0.953525	+	0.301314i	$0.902575\pi$
$282$	0	0
$283$	1.53833e11	1.42565	0.712823	−	0.701344i	$-0.247416\pi$
$283$	1.53833e11	1.42565	0.712823	+	0.701344i	$0.247416\pi$
$284$	−2.64903e11	−2.41632
$285$	0	0
$286$	5.57127e11	4.92388
$287$	3.04523e10	0.264942
$288$	0	0
$289$	−7.76676e10	−0.654937
$290$	1.94102e11	1.61153
$291$	0	0
$292$	−1.27016e11	−1.02244
$293$	−3.10497e9	−0.0246123	−0.0123062	−	0.999924i	$-0.503917\pi$
$293$	−3.10497e9	−0.0246123	−0.0123062	+	0.999924i	$0.503917\pi$
$294$	0	0
$295$	8.64035e10	0.664251
$296$	2.24249e11	1.69792
$297$	0	0
$298$	−4.70913e11	−3.45913
$299$	1.80313e11	1.30469
$300$	0	0
$301$	3.68897e9	0.0259033
$302$	3.04530e11	2.10668
$303$	0	0
$304$	1.11434e11	0.748316
$305$	4.58520e10	0.303396
$306$	0	0
$307$	−2.42673e11	−1.55919	−0.779593	−	0.626286i	$-0.784574\pi$
$307$	−2.42673e11	−1.55919	−0.779593	+	0.626286i	$0.784574\pi$
$308$	−1.28823e11	−0.815672
$309$	0	0
$310$	−2.83183e11	−1.74156
$311$	5.16832e9	0.0313276	0.0156638	−	0.999877i	$-0.495014\pi$
$311$	5.16832e9	0.0313276	0.0156638	+	0.999877i	$0.495014\pi$
$312$	0	0
$313$	2.71239e11	1.59736	0.798680	−	0.601757i	$-0.205532\pi$
$313$	2.71239e11	1.59736	0.798680	+	0.601757i	$0.205532\pi$
$314$	−4.54349e10	−0.263758
$315$	0	0
$316$	−4.75013e10	−0.267987
$317$	−1.20224e11	−0.668690	−0.334345	−	0.942451i	$-0.608515\pi$
$317$	−1.20224e11	−0.668690	−0.334345	+	0.942451i	$0.608515\pi$
$318$	0	0
$319$	3.15560e11	1.70618
$320$	−3.96866e11	−2.11577
$321$	0	0
$322$	−5.75820e10	−0.298494
$323$	−2.63623e10	−0.134764
$324$	0	0
$325$	1.21874e11	0.605948
$326$	1.64770e11	0.807976
$327$	0	0
$328$	8.81142e11	4.20352
$329$	−1.60427e10	−0.0754913
$330$	0	0
$331$	−3.62897e11	−1.66172	−0.830860	−	0.556482i	$-0.812151\pi$
$331$	−3.62897e11	−1.66172	−0.830860	+	0.556482i	$0.812151\pi$
$332$	−5.99469e10	−0.270798
$333$	0	0
$334$	2.56029e11	1.12572
$335$	6.81390e10	0.295593
$336$	0	0
$337$	2.40253e11	1.01469	0.507347	−	0.861742i	$-0.330626\pi$
$337$	2.40253e11	1.01469	0.507347	+	0.861742i	$0.330626\pi$
$338$	7.44293e11	3.10184
$339$	0	0
$340$	3.00595e11	1.21991
$341$	−4.60383e11	−1.84385
$342$	0	0
$343$	−9.80123e10	−0.382347
$344$	1.06741e11	0.410977
$345$	0	0
$346$	−2.22456e10	−0.0834453
$347$	−1.87422e11	−0.693967	−0.346984	−	0.937871i	$-0.612794\pi$
$347$	−1.87422e11	−0.693967	−0.346984	+	0.937871i	$0.612794\pi$
$348$	0	0
$349$	1.99385e11	0.719412	0.359706	−	0.933066i	$-0.382877\pi$
$349$	1.99385e11	0.719412	0.359706	+	0.933066i	$0.382877\pi$
$350$	−3.89197e10	−0.138632
$351$	0	0
$352$	−1.43240e12	−4.97305
$353$	3.13590e11	1.07492	0.537459	−	0.843290i	$-0.319384\pi$
$353$	3.13590e11	1.07492	0.537459	+	0.843290i	$0.319384\pi$
$354$	0	0
$355$	2.18070e11	0.728731
$356$	5.31048e11	1.75230
$357$	0	0
$358$	−6.67014e11	−2.14616
$359$	5.91463e9	0.0187933	0.00939664	−	0.999956i	$-0.497009\pi$
$359$	5.91463e9	0.0187933	0.00939664	+	0.999956i	$0.497009\pi$
$360$	0	0
$361$	1.69836e10	0.0526316
$362$	−7.42852e11	−2.27360
$363$	0	0
$364$	−2.77726e11	−0.829201
$365$	1.04560e11	0.308354
$366$	0	0
$367$	−3.59433e11	−1.03424	−0.517119	−	0.855914i	$-0.672995\pi$
$367$	−3.59433e11	−1.03424	−0.517119	+	0.855914i	$0.672995\pi$
$368$	−9.23335e11	−2.62448
$369$	0	0
$370$	−2.98267e11	−0.827365
$371$	3.15858e10	0.0865584
$372$	0	0
$373$	5.44784e11	1.45725	0.728626	−	0.684912i	$-0.240159\pi$
$373$	5.44784e11	1.45725	0.728626	+	0.684912i	$0.240159\pi$
$374$	6.74922e11	1.78374
$375$	0	0
$376$	−4.64199e11	−1.19773
$377$	6.80305e11	1.73447
$378$	0	0
$379$	2.17993e11	0.542708	0.271354	−	0.962480i	$-0.412529\pi$
$379$	2.17993e11	0.542708	0.271354	+	0.962480i	$0.412529\pi$
$380$	−1.93654e11	−0.476431
$381$	0	0
$382$	−1.32169e11	−0.317574
$383$	−7.78646e11	−1.84904	−0.924518	−	0.381138i	$-0.875532\pi$
$383$	−7.78646e11	−1.84904	−0.924518	+	0.381138i	$0.875532\pi$
$384$	0	0
$385$	1.06048e11	0.245997
$386$	2.55919e11	0.586758
$387$	0	0
$388$	−2.33758e12	−5.23629
$389$	3.58920e10	0.0794739	0.0397370	−	0.999210i	$-0.487348\pi$
$389$	3.58920e10	0.0794739	0.0397370	+	0.999210i	$0.487348\pi$
$390$	0	0
$391$	2.18437e11	0.472641
$392$	−1.39056e12	−2.97441
$393$	0	0
$394$	1.24268e12	2.59792
$395$	3.91034e10	0.0808216
$396$	0	0
$397$	2.77227e11	0.560116	0.280058	−	0.959983i	$-0.409646\pi$
$397$	2.77227e11	0.560116	0.280058	+	0.959983i	$0.409646\pi$
$398$	−4.36330e11	−0.871647
$399$	0	0
$400$	−6.24082e11	−1.21891
$401$	−7.76515e11	−1.49969	−0.749843	−	0.661616i	$-0.769871\pi$
$401$	−7.76515e11	−1.49969	−0.749843	+	0.661616i	$0.769871\pi$
$402$	0	0
$403$	−9.92525e11	−1.87443
$404$	2.12437e12	3.96748
$405$	0	0
$406$	−2.17251e11	−0.396822
$407$	−4.84906e11	−0.875957
$408$	0	0
$409$	7.79697e11	1.37775	0.688876	−	0.724880i	$-0.258105\pi$
$409$	7.79697e11	1.37775	0.688876	+	0.724880i	$0.258105\pi$
$410$	−1.17198e12	−2.04830
$411$	0	0
$412$	−3.34539e11	−0.572018
$413$	−9.67085e10	−0.163565
$414$	0	0
$415$	4.93487e10	0.0816693
$416$	−3.08807e12	−5.05553
$417$	0	0
$418$	−4.34809e11	−0.696635
$419$	3.67894e11	0.583122	0.291561	−	0.956552i	$-0.405825\pi$
$419$	3.67894e11	0.583122	0.291561	+	0.956552i	$0.405825\pi$
$420$	0	0
$421$	−1.66297e11	−0.257997	−0.128998	−	0.991645i	$-0.541176\pi$
$421$	−1.66297e11	−0.257997	−0.128998	+	0.991645i	$0.541176\pi$
$422$	−1.46976e12	−2.25600
$423$	0	0
$424$	9.13941e11	1.37332
$425$	1.47642e11	0.219512
$426$	0	0
$427$	−5.13207e10	−0.0747079
$428$	2.00760e12	2.89188
$429$	0	0
$430$	−1.41973e11	−0.200262
$431$	−1.10815e12	−1.54686	−0.773431	−	0.633880i	$-0.781461\pi$
$431$	−1.10815e12	−1.54686	−0.773431	+	0.633880i	$0.781461\pi$
$432$	0	0
$433$	2.07579e10	0.0283784	0.0141892	−	0.999899i	$-0.495483\pi$
$433$	2.07579e10	0.0283784	0.0141892	+	0.999899i	$0.495483\pi$
$434$	3.16957e11	0.428841
$435$	0	0
$436$	1.66941e12	2.21245
$437$	−1.40725e11	−0.184589
$438$	0	0
$439$	−8.98118e11	−1.15410	−0.577050	−	0.816709i	$-0.695796\pi$
$439$	−8.98118e11	−1.15410	−0.577050	+	0.816709i	$0.695796\pi$
$440$	3.06852e12	3.90294
$441$	0	0
$442$	1.45504e12	1.81332
$443$	6.20774e11	0.765803	0.382901	−	0.923789i	$-0.374925\pi$
$443$	6.20774e11	0.765803	0.382901	+	0.923789i	$0.374925\pi$
$444$	0	0
$445$	−4.37162e11	−0.528472
$446$	4.04071e11	0.483560
$447$	0	0
$448$	4.44199e11	0.520986
$449$	5.22475e11	0.606677	0.303338	−	0.952883i	$-0.401899\pi$
$449$	5.22475e11	0.606677	0.303338	+	0.952883i	$0.401899\pi$
$450$	0	0
$451$	−1.90535e12	−2.16860
$452$	2.22010e11	0.250178
$453$	0	0
$454$	1.41273e12	1.56066
$455$	2.28626e11	0.250077
$456$	0	0
$457$	−1.61179e12	−1.72856	−0.864281	−	0.503010i	$-0.832226\pi$
$457$	−1.61179e12	−1.72856	−0.864281	+	0.503010i	$0.832226\pi$
$458$	1.42212e12	1.51022
$459$	0	0
$460$	1.60461e12	1.67093
$461$	−1.17117e12	−1.20772	−0.603861	−	0.797090i	$-0.706372\pi$
$461$	−1.17117e12	−1.20772	−0.603861	+	0.797090i	$0.706372\pi$
$462$	0	0
$463$	−5.19973e11	−0.525856	−0.262928	−	0.964815i	$-0.584688\pi$
$463$	−5.19973e11	−0.525856	−0.262928	+	0.964815i	$0.584688\pi$
$464$	−3.48365e12	−3.48902
$465$	0	0
$466$	−8.66144e11	−0.850851
$467$	−1.08655e12	−1.05712	−0.528559	−	0.848897i	$-0.677267\pi$
$467$	−1.08655e12	−1.05712	−0.528559	+	0.848897i	$0.677267\pi$
$468$	0	0
$469$	−7.62658e10	−0.0727867
$470$	6.17419e11	0.583633
$471$	0	0
$472$	−2.79828e12	−2.59509
$473$	−2.30812e11	−0.212023
$474$	0	0
$475$	−9.51161e10	−0.0857300
$476$	−3.36446e11	−0.300389
$477$	0	0
$478$	1.90395e12	1.66813
$479$	2.08556e9	0.00181015	0.000905073	−	1.00000i	$-0.499712\pi$
$479$	2.08556e9	0.00181015	0.000905073		1.00000i	$0.499712\pi$
$480$	0	0
$481$	−1.04539e12	−0.890486
$482$	2.41381e12	2.03700
$483$	0	0
$484$	4.89224e12	4.05232
$485$	1.92431e12	1.57920
$486$	0	0
$487$	1.93534e12	1.55911	0.779554	−	0.626335i	$-0.215446\pi$
$487$	1.93534e12	1.55911	0.779554	+	0.626335i	$0.215446\pi$
$488$	−1.48497e12	−1.18530
$489$	0	0
$490$	1.84954e12	1.44938
$491$	−2.34837e12	−1.82347	−0.911736	−	0.410777i	$-0.865257\pi$
$491$	−2.34837e12	−1.82347	−0.911736	+	0.410777i	$0.865257\pi$
$492$	0	0
$493$	8.24143e11	0.628336
$494$	−9.37390e11	−0.708189
$495$	0	0
$496$	5.08244e12	3.77055
$497$	−2.44078e11	−0.179442
$498$	0	0
$499$	−8.80765e11	−0.635927	−0.317964	−	0.948103i	$-0.602999\pi$
$499$	−8.80765e11	−0.635927	−0.317964	+	0.948103i	$0.602999\pi$
$500$	3.98685e12	2.85276
$501$	0	0
$502$	2.24668e12	1.57897
$503$	9.56263e11	0.666072	0.333036	−	0.942914i	$-0.391927\pi$
$503$	9.56263e11	0.666072	0.333036	+	0.942914i	$0.391927\pi$
$504$	0	0
$505$	−1.74880e12	−1.19654
$506$	3.60281e12	2.44323
$507$	0	0
$508$	−2.81200e12	−1.87339
$509$	8.02783e11	0.530112	0.265056	−	0.964233i	$-0.414610\pi$
$509$	8.02783e11	0.530112	0.265056	+	0.964233i	$0.414610\pi$
$510$	0	0
$511$	−1.17031e11	−0.0759289
$512$	1.31485e11	0.0845591
$513$	0	0
$514$	2.40763e12	1.52145
$515$	2.75395e11	0.172514
$516$	0	0
$517$	1.00377e12	0.617910
$518$	3.33840e11	0.203730
$519$	0	0
$520$	6.61532e12	3.96767
$521$	1.35878e12	0.807939	0.403969	−	0.914773i	$-0.367630\pi$
$521$	1.35878e12	0.807939	0.403969	+	0.914773i	$0.367630\pi$
$522$	0	0
$523$	1.01846e12	0.595234	0.297617	−	0.954685i	$-0.403808\pi$
$523$	1.01846e12	0.595234	0.297617	+	0.954685i	$0.403808\pi$
$524$	2.62419e12	1.52056
$525$	0	0
$526$	−5.02961e12	−2.86483
$527$	−1.20238e12	−0.679036
$528$	0	0
$529$	−6.35108e11	−0.352612
$530$	−1.21561e12	−0.669194
$531$	0	0
$532$	2.16750e11	0.117316
$533$	−4.10767e12	−2.20457
$534$	0	0
$535$	−1.65267e12	−0.872155
$536$	−2.20676e12	−1.15482
$537$	0	0
$538$	−9.21311e11	−0.474118
$539$	3.00689e12	1.53450
$540$	0	0
$541$	−3.14397e12	−1.57794	−0.788971	−	0.614431i	$-0.789386\pi$
$541$	−3.14397e12	−1.57794	−0.788971	+	0.614431i	$0.789386\pi$
$542$	3.84092e12	1.91178
$543$	0	0
$544$	−3.74098e12	−1.83143
$545$	−1.37426e12	−0.667246
$546$	0	0
$547$	−2.40457e12	−1.14840	−0.574201	−	0.818714i	$-0.694687\pi$
$547$	−2.40457e12	−1.14840	−0.574201	+	0.818714i	$0.694687\pi$
$548$	1.26552e12	0.599457
$549$	0	0
$550$	2.43514e12	1.13473
$551$	−5.30943e11	−0.245395
$552$	0	0
$553$	−4.37671e10	−0.0199015
$554$	−7.45796e12	−3.36377
$555$	0	0
$556$	−7.10817e12	−3.15443
$557$	−1.99076e12	−0.876337	−0.438168	−	0.898893i	$-0.644373\pi$
$557$	−1.99076e12	−0.876337	−0.438168	+	0.898893i	$0.644373\pi$
$558$	0	0
$559$	−4.97601e11	−0.215540
$560$	−1.17073e12	−0.503048
$561$	0	0
$562$	−8.58571e12	−3.63047
$563$	−2.14736e12	−0.900778	−0.450389	−	0.892832i	$-0.648715\pi$
$563$	−2.14736e12	−0.900778	−0.450389	+	0.892832i	$0.648715\pi$
$564$	0	0
$565$	−1.82760e11	−0.0754506
$566$	6.62653e12	2.71402
$567$	0	0
$568$	−7.06245e12	−2.84700
$569$	−3.97619e12	−1.59024	−0.795120	−	0.606453i	$-0.792592\pi$
$569$	−3.97619e12	−1.59024	−0.795120	+	0.606453i	$0.792592\pi$
$570$	0	0
$571$	−1.06959e12	−0.421072	−0.210536	−	0.977586i	$-0.567521\pi$
$571$	−1.06959e12	−0.421072	−0.210536	+	0.977586i	$0.567521\pi$
$572$	1.73768e13	6.78717
$573$	0	0
$574$	1.31176e12	0.504372
$575$	7.88129e11	0.300671
$576$	0	0
$577$	1.68962e12	0.634596	0.317298	−	0.948326i	$-0.397224\pi$
$577$	1.68962e12	0.634596	0.317298	+	0.948326i	$0.397224\pi$
$578$	−3.34561e12	−1.24681
$579$	0	0
$580$	6.05404e12	2.22136
$581$	−5.52344e10	−0.0201102
$582$	0	0
$583$	−1.97627e12	−0.708497
$584$	−3.38631e12	−1.20467
$585$	0	0
$586$	−1.33750e11	−0.0468547
$587$	−2.74627e12	−0.954709	−0.477355	−	0.878711i	$-0.658404\pi$
$587$	−2.74627e12	−0.954709	−0.477355	+	0.878711i	$0.658404\pi$
$588$	0	0
$589$	7.74614e11	0.265196
$590$	3.72192e12	1.26454
$591$	0	0
$592$	5.35317e12	1.79128
$593$	1.76038e12	0.584603	0.292302	−	0.956326i	$-0.405579\pi$
$593$	1.76038e12	0.584603	0.292302	+	0.956326i	$0.405579\pi$
$594$	0	0
$595$	2.76964e11	0.0905935
$596$	−1.46878e13	−4.76813
$597$	0	0
$598$	7.76719e12	2.48375
$599$	3.01029e12	0.955406	0.477703	−	0.878521i	$-0.341469\pi$
$599$	3.01029e12	0.955406	0.477703	+	0.878521i	$0.341469\pi$
$600$	0	0
$601$	−4.62803e12	−1.44698	−0.723488	−	0.690337i	$-0.757462\pi$
$601$	−4.62803e12	−1.44698	−0.723488	+	0.690337i	$0.757462\pi$
$602$	1.58906e11	0.0493124
$603$	0	0
$604$	9.49830e12	2.90389
$605$	−4.02733e12	−1.22213
$606$	0	0
$607$	−4.99925e12	−1.49471	−0.747353	−	0.664427i	$-0.768675\pi$
$607$	−4.99925e12	−1.49471	−0.747353	+	0.664427i	$0.768675\pi$
$608$	2.41008e12	0.715261
$609$	0	0
$610$	1.97512e12	0.577577
$611$	2.16399e12	0.628159
$612$	0	0
$613$	−2.54862e11	−0.0729010	−0.0364505	−	0.999335i	$-0.511605\pi$
$613$	−2.54862e11	−0.0729010	−0.0364505	+	0.999335i	$0.511605\pi$
$614$	−1.04534e13	−2.96824
$615$	0	0
$616$	−3.43449e12	−0.961057
$617$	6.48851e12	1.80244	0.901222	−	0.433357i	$-0.142671\pi$
$617$	6.48851e12	1.80244	0.901222	+	0.433357i	$0.142671\pi$
$618$	0	0
$619$	−6.56476e11	−0.179726	−0.0898630	−	0.995954i	$-0.528643\pi$
$619$	−6.56476e11	−0.179726	−0.0898630	+	0.995954i	$0.528643\pi$
$620$	−8.83248e12	−2.40060
$621$	0	0
$622$	2.22630e11	0.0596387
$623$	4.89301e11	0.130131
$624$	0	0
$625$	−1.85649e12	−0.486668
$626$	1.16839e13	3.04091
$627$	0	0
$628$	−1.41712e12	−0.363569
$629$	−1.26642e12	−0.322590
$630$	0	0
$631$	5.77708e12	1.45069	0.725347	−	0.688383i	$-0.241679\pi$
$631$	5.77708e12	1.45069	0.725347	+	0.688383i	$0.241679\pi$
$632$	−1.26641e12	−0.315753
$633$	0	0
$634$	−5.17877e12	−1.27299
$635$	2.31486e12	0.564992
$636$	0	0
$637$	6.48245e12	1.55995
$638$	1.35931e13	3.24806
$639$	0	0
$640$	−6.62298e12	−1.56043
$641$	7.53058e11	0.176184	0.0880922	−	0.996112i	$-0.471923\pi$
$641$	7.53058e11	0.176184	0.0880922	+	0.996112i	$0.471923\pi$
$642$	0	0
$643$	5.17768e12	1.19450	0.597249	−	0.802056i	$-0.296260\pi$
$643$	5.17768e12	1.19450	0.597249	+	0.802056i	$0.296260\pi$
$644$	−1.79599e12	−0.411450
$645$	0	0
$646$	−1.13558e12	−0.256551
$647$	−8.30839e12	−1.86401	−0.932004	−	0.362449i	$-0.881941\pi$
$647$	−8.30839e12	−1.86401	−0.932004	+	0.362449i	$0.881941\pi$
$648$	0	0
$649$	6.05089e12	1.33881
$650$	5.24984e12	1.15355
$651$	0	0
$652$	5.13918e12	1.11373
$653$	−7.02989e12	−1.51300	−0.756500	−	0.653993i	$-0.773092\pi$
$653$	−7.02989e12	−1.51300	−0.756500	+	0.653993i	$0.773092\pi$
$654$	0	0
$655$	−2.16025e12	−0.458583
$656$	2.10343e13	4.43465
$657$	0	0
$658$	−6.91057e11	−0.143713
$659$	−2.62898e12	−0.543003	−0.271501	−	0.962438i	$-0.587520\pi$
$659$	−2.62898e12	−0.543003	−0.271501	+	0.962438i	$0.587520\pi$
$660$	0	0
$661$	−5.20751e12	−1.06102	−0.530510	−	0.847679i	$-0.678000\pi$
$661$	−5.20751e12	−1.06102	−0.530510	+	0.847679i	$0.678000\pi$
$662$	−1.56322e13	−3.16343
$663$	0	0
$664$	−1.59822e12	−0.319065
$665$	−1.78430e11	−0.0353811
$666$	0	0
$667$	4.39938e12	0.860646
$668$	7.98556e12	1.55171
$669$	0	0
$670$	2.93516e12	0.562723
$671$	3.21105e12	0.611498
$672$	0	0
$673$	4.84982e12	0.911292	0.455646	−	0.890161i	$-0.349408\pi$
$673$	4.84982e12	0.911292	0.455646	+	0.890161i	$0.349408\pi$
$674$	1.03492e13	1.93168
$675$	0	0
$676$	2.32146e13	4.27563
$677$	−2.55117e12	−0.466756	−0.233378	−	0.972386i	$-0.574978\pi$
$677$	−2.55117e12	−0.466756	−0.233378	+	0.972386i	$0.574978\pi$
$678$	0	0
$679$	−2.15382e12	−0.388861
$680$	8.01401e12	1.43734
$681$	0	0
$682$	−1.98315e13	−3.51015
$683$	−2.83001e12	−0.497617	−0.248808	−	0.968553i	$-0.580039\pi$
$683$	−2.83001e12	−0.497617	−0.248808	+	0.968553i	$0.580039\pi$
$684$	0	0
$685$	−1.04179e12	−0.180789
$686$	−4.22198e12	−0.727876
$687$	0	0
$688$	2.54808e12	0.433575
$689$	−4.26058e12	−0.720248
$690$	0	0
$691$	−3.09239e12	−0.515993	−0.257996	−	0.966146i	$-0.583062\pi$
$691$	−3.09239e12	−0.515993	−0.257996	+	0.966146i	$0.583062\pi$
$692$	−6.93842e11	−0.115023
$693$	0	0
$694$	−8.07341e12	−1.32111
$695$	5.85149e12	0.951338
$696$	0	0
$697$	−4.97617e12	−0.798633
$698$	8.58870e12	1.36955
$699$	0	0
$700$	−1.21391e12	−0.191093
$701$	−1.12689e13	−1.76259	−0.881293	−	0.472570i	$-0.843327\pi$
$701$	−1.12689e13	−1.76259	−0.881293	+	0.472570i	$0.843327\pi$
$702$	0	0
$703$	8.15875e11	0.125987
$704$	−2.77928e13	−4.26437
$705$	0	0
$706$	1.35082e13	2.04633
$707$	1.95737e12	0.294636
$708$	0	0
$709$	−4.29500e12	−0.638344	−0.319172	−	0.947697i	$-0.603405\pi$
$709$	−4.29500e12	−0.638344	−0.319172	+	0.947697i	$0.603405\pi$
$710$	9.39357e12	1.38729
$711$	0	0
$712$	1.41580e13	2.06463
$713$	−6.41843e12	−0.930092
$714$	0	0
$715$	−1.43047e13	−2.04693
$716$	−2.08042e13	−2.95830
$717$	0	0
$718$	2.54779e11	0.0357769
$719$	1.11078e13	1.55006	0.775028	−	0.631927i	$-0.217736\pi$
$719$	1.11078e13	1.55006	0.775028	+	0.631927i	$0.217736\pi$
$720$	0	0
$721$	−3.08240e11	−0.0424796
$722$	7.31584e11	0.100195
$723$	0	0
$724$	−2.31696e13	−3.13397
$725$	2.97354e12	0.399717
$726$	0	0
$727$	9.05905e12	1.20276	0.601379	−	0.798964i	$-0.294618\pi$
$727$	9.05905e12	1.20276	0.601379	+	0.798964i	$0.294618\pi$
$728$	−7.40431e12	−0.976998
$729$	0	0
$730$	4.50405e12	0.587016
$731$	−6.02810e11	−0.0780822
$732$	0	0
$733$	6.02219e12	0.770525	0.385262	−	0.922807i	$-0.374111\pi$
$733$	6.02219e12	0.770525	0.385262	+	0.922807i	$0.374111\pi$
$734$	−1.54829e13	−1.96889
$735$	0	0
$736$	−1.99698e13	−2.50856
$737$	4.77182e12	0.595773
$738$	0	0
$739$	−7.03467e11	−0.0867649	−0.0433824	−	0.999059i	$-0.513813\pi$
$739$	−7.03467e11	−0.0867649	−0.0433824	+	0.999059i	$0.513813\pi$
$740$	−9.30296e12	−1.14046
$741$	0	0
$742$	1.36059e12	0.164782
$743$	7.85573e12	0.945664	0.472832	−	0.881153i	$-0.343232\pi$
$743$	7.85573e12	0.945664	0.472832	+	0.881153i	$0.343232\pi$
$744$	0	0
$745$	1.20911e13	1.43801
$746$	2.34671e13	2.77418
$747$	0	0
$748$	2.10508e13	2.45874
$749$	1.84978e12	0.214759
$750$	0	0
$751$	1.08319e13	1.24259	0.621293	−	0.783578i	$-0.286607\pi$
$751$	1.08319e13	1.24259	0.621293	+	0.783578i	$0.286607\pi$
$752$	−1.10812e13	−1.26359
$753$	0	0
$754$	2.93048e13	3.30193
$755$	−7.81906e12	−0.875777
$756$	0	0
$757$	−1.32080e13	−1.46186	−0.730930	−	0.682453i	$-0.760913\pi$
$757$	−1.32080e13	−1.46186	−0.730930	+	0.682453i	$0.760913\pi$
$758$	9.39027e12	1.03316
$759$	0	0
$760$	−5.16291e12	−0.561350
$761$	2.17084e12	0.234637	0.117318	−	0.993094i	$-0.462570\pi$
$761$	2.17084e12	0.234637	0.117318	+	0.993094i	$0.462570\pi$
$762$	0	0
$763$	1.53817e12	0.164302
$764$	−4.12236e12	−0.437750
$765$	0	0
$766$	−3.35410e13	−3.52003
$767$	1.30449e13	1.36101
$768$	0	0
$769$	1.19754e13	1.23487	0.617435	−	0.786622i	$-0.288172\pi$
$769$	1.19754e13	1.23487	0.617435	+	0.786622i	$0.288172\pi$
$770$	4.56813e12	0.468306
$771$	0	0
$772$	7.98211e12	0.808798
$773$	−2.52680e12	−0.254544	−0.127272	−	0.991868i	$-0.540622\pi$
$773$	−2.52680e12	−0.254544	−0.127272	+	0.991868i	$0.540622\pi$
$774$	0	0
$775$	−4.33821e12	−0.431970
$776$	−6.23211e13	−6.16960
$777$	0	0
$778$	1.54608e12	0.151295
$779$	3.20583e12	0.311904
$780$	0	0
$781$	1.52716e13	1.46877
$782$	9.40942e12	0.899772
$783$	0	0
$784$	−3.31948e13	−3.13796
$785$	1.16658e12	0.109648
$786$	0	0
$787$	−8.75997e12	−0.813985	−0.406992	−	0.913432i	$-0.633422\pi$
$787$	−8.75997e12	−0.813985	−0.406992	+	0.913432i	$0.633422\pi$
$788$	3.87593e13	3.58103
$789$	0	0
$790$	1.68442e12	0.153861
$791$	2.04557e11	0.0185789
$792$	0	0
$793$	6.92259e12	0.621641
$794$	1.19418e13	1.06630
$795$	0	0
$796$	−1.36091e13	−1.20149
$797$	−3.27725e12	−0.287705	−0.143852	−	0.989599i	$-0.545949\pi$
$797$	−3.27725e12	−0.287705	−0.143852	+	0.989599i	$0.545949\pi$
$798$	0	0
$799$	2.62152e12	0.227559
$800$	−1.34976e13	−1.16507
$801$	0	0
$802$	−3.34492e13	−2.85496
$803$	7.32243e12	0.621492
$804$	0	0
$805$	1.47847e12	0.124088
$806$	−4.27540e13	−3.56837
$807$	0	0
$808$	5.66369e13	4.67464
$809$	−1.21000e13	−0.993159	−0.496579	−	0.867991i	$-0.665411\pi$
$809$	−1.21000e13	−0.993159	−0.496579	+	0.867991i	$0.665411\pi$
$810$	0	0
$811$	−1.32448e13	−1.07511	−0.537555	−	0.843229i	$-0.680652\pi$
$811$	−1.32448e13	−1.07511	−0.537555	+	0.843229i	$0.680652\pi$
$812$	−6.77608e12	−0.546987
$813$	0	0
$814$	−2.08878e13	−1.66757
$815$	−4.23060e12	−0.335887
$816$	0	0
$817$	3.88352e11	0.0304948
$818$	3.35862e13	2.62284
$819$	0	0
$820$	−3.65542e13	−2.82342
$821$	−1.15927e12	−0.0890511	−0.0445256	−	0.999008i	$-0.514178\pi$
$821$	−1.15927e12	−0.0890511	−0.0445256	+	0.999008i	$0.514178\pi$
$822$	0	0
$823$	1.70322e13	1.29411	0.647056	−	0.762443i	$-0.276000\pi$
$823$	1.70322e13	1.29411	0.647056	+	0.762443i	$0.276000\pi$
$824$	−8.91899e12	−0.673974
$825$	0	0
$826$	−4.16582e12	−0.311380
$827$	3.98089e12	0.295941	0.147970	−	0.988992i	$-0.452726\pi$
$827$	3.98089e12	0.295941	0.147970	+	0.988992i	$0.452726\pi$
$828$	0	0
$829$	8.79574e12	0.646810	0.323405	−	0.946261i	$-0.395172\pi$
$829$	8.79574e12	0.646810	0.323405	+	0.946261i	$0.395172\pi$
$830$	2.12574e12	0.155475
$831$	0	0
$832$	−5.99175e13	−4.33510
$833$	7.85304e12	0.565113
$834$	0	0
$835$	−6.57377e12	−0.467978
$836$	−1.35617e13	−0.960254
$837$	0	0
$838$	1.58474e13	1.11010
$839$	1.26751e13	0.883127	0.441564	−	0.897230i	$-0.354424\pi$
$839$	1.26751e13	0.883127	0.441564	+	0.897230i	$0.354424\pi$
$840$	0	0
$841$	2.09127e12	0.144154
$842$	−7.16340e12	−0.491150
$843$	0	0
$844$	−4.58418e13	−3.10971
$845$	−1.91104e13	−1.28948
$846$	0	0
$847$	4.50765e12	0.300937
$848$	2.18172e13	1.44883
$849$	0	0
$850$	6.35982e12	0.417888
$851$	−6.76032e12	−0.441859
$852$	0	0
$853$	−8.10425e12	−0.524134	−0.262067	−	0.965050i	$-0.584404\pi$
$853$	−8.10425e12	−0.524134	−0.262067	+	0.965050i	$0.584404\pi$
$854$	−2.21069e12	−0.142222
$855$	0	0
$856$	5.35237e13	3.40733
$857$	9.24804e12	0.585647	0.292824	−	0.956166i	$-0.405405\pi$
$857$	9.24804e12	0.585647	0.292824	+	0.956166i	$0.405405\pi$
$858$	0	0
$859$	−3.30863e12	−0.207338	−0.103669	−	0.994612i	$-0.533058\pi$
$859$	−3.30863e12	−0.207338	−0.103669	+	0.994612i	$0.533058\pi$
$860$	−4.42815e12	−0.276045
$861$	0	0
$862$	−4.77348e13	−2.94478
$863$	2.69935e13	1.65657	0.828287	−	0.560305i	$-0.189316\pi$
$863$	2.69935e13	1.65657	0.828287	+	0.560305i	$0.189316\pi$
$864$	0	0
$865$	5.71175e11	0.0346894
$866$	8.94169e11	0.0540243
$867$	0	0
$868$	9.88591e12	0.591123
$869$	2.73843e12	0.162897
$870$	0	0
$871$	1.02874e13	0.605654
$872$	4.45072e13	2.60679
$873$	0	0
$874$	−6.06188e12	−0.351404
$875$	3.67344e12	0.211854
$876$	0	0
$877$	2.37336e13	1.35477	0.677386	−	0.735628i	$-0.263113\pi$
$877$	2.37336e13	1.35477	0.677386	+	0.735628i	$0.263113\pi$
$878$	−3.86874e13	−2.19707
$879$	0	0
$880$	7.32504e13	4.11754
$881$	2.27869e13	1.27437	0.637183	−	0.770712i	$-0.280099\pi$
$881$	2.27869e13	1.27437	0.637183	+	0.770712i	$0.280099\pi$
$882$	0	0
$883$	−2.77944e13	−1.53863	−0.769315	−	0.638870i	$-0.779402\pi$
$883$	−2.77944e13	−1.53863	−0.769315	+	0.638870i	$0.779402\pi$
$884$	4.53828e13	2.49952
$885$	0	0
$886$	2.67405e13	1.45787
$887$	−1.16529e13	−0.632091	−0.316045	−	0.948744i	$-0.602355\pi$
$887$	−1.16529e13	−0.632091	−0.316045	+	0.948744i	$0.602355\pi$
$888$	0	0
$889$	−2.59094e12	−0.139123
$890$	−1.88312e13	−1.00606
$891$	0	0
$892$	1.26030e13	0.666549
$893$	−1.68888e12	−0.0888724
$894$	0	0
$895$	1.71262e13	0.892188
$896$	7.41289e12	0.384239
$897$	0	0
$898$	2.25062e13	1.15494
$899$	−2.42161e13	−1.23648
$900$	0	0
$901$	−5.16140e12	−0.260919
$902$	−8.20747e13	−4.12838
$903$	0	0
$904$	5.91890e12	0.294770
$905$	1.90733e13	0.945166
$906$	0	0
$907$	1.39840e13	0.686120	0.343060	−	0.939314i	$-0.388537\pi$
$907$	1.39840e13	0.686120	0.343060	+	0.939314i	$0.388537\pi$
$908$	4.40632e13	2.15124
$909$	0	0
$910$	9.84828e12	0.476073
$911$	−8.94857e11	−0.0430448	−0.0215224	−	0.999768i	$-0.506851\pi$
$911$	−8.94857e11	−0.0430448	−0.0215224	+	0.999768i	$0.506851\pi$
$912$	0	0
$913$	3.45592e12	0.164606
$914$	−6.94294e13	−3.29068
$915$	0	0
$916$	4.43560e13	2.08172
$917$	2.41790e12	0.112921
$918$	0	0
$919$	1.60525e13	0.742374	0.371187	−	0.928558i	$-0.378951\pi$
$919$	1.60525e13	0.742374	0.371187	+	0.928558i	$0.378951\pi$
$920$	4.27797e13	1.96876
$921$	0	0
$922$	−5.04495e13	−2.29915
$923$	3.29235e13	1.49313
$924$	0	0
$925$	−4.56930e12	−0.205216
$926$	−2.23984e13	−1.00108
$927$	0	0
$928$	−7.53442e13	−3.33491
$929$	5.95071e12	0.262119	0.131059	−	0.991375i	$-0.458162\pi$
$929$	5.95071e12	0.262119	0.131059	+	0.991375i	$0.458162\pi$
$930$	0	0
$931$	−5.05921e12	−0.220704
$932$	−2.70151e13	−1.17283
$933$	0	0
$934$	−4.68042e13	−2.01244
$935$	−1.73292e13	−0.741525
$936$	0	0
$937$	3.70309e12	0.156941	0.0784705	−	0.996916i	$-0.474996\pi$
$937$	3.70309e12	0.156941	0.0784705	+	0.996916i	$0.474996\pi$
$938$	−3.28523e12	−0.138565
$939$	0	0
$940$	1.92573e13	0.804490
$941$	−1.14833e13	−0.477435	−0.238717	−	0.971089i	$-0.576727\pi$
$941$	−1.14833e13	−0.477435	−0.238717	+	0.971089i	$0.576727\pi$
$942$	0	0
$943$	−2.65634e13	−1.09391
$944$	−6.67994e13	−2.73778
$945$	0	0
$946$	−9.94248e12	−0.403631
$947$	−1.87912e13	−0.759239	−0.379620	−	0.925143i	$-0.623945\pi$
$947$	−1.87912e13	−0.759239	−0.379620	+	0.925143i	$0.623945\pi$
$948$	0	0
$949$	1.57862e13	0.631800
$950$	−4.09722e12	−0.163205
$951$	0	0
$952$	−8.96982e12	−0.353930
$953$	3.02709e13	1.18880	0.594398	−	0.804171i	$-0.297391\pi$
$953$	3.02709e13	1.18880	0.594398	+	0.804171i	$0.297391\pi$
$954$	0	0
$955$	3.39356e12	0.132020
$956$	5.93843e13	2.29938
$957$	0	0
$958$	8.98378e10	0.00344599
$959$	1.16604e12	0.0445173
$960$	0	0
$961$	8.89025e12	0.336247
$962$	−4.50314e13	−1.69523
$963$	0	0
$964$	7.52869e13	2.80784
$965$	−6.57092e12	−0.243923
$966$	0	0
$967$	4.58324e13	1.68560	0.842799	−	0.538229i	$-0.180906\pi$
$967$	4.58324e13	1.68560	0.842799	+	0.538229i	$0.180906\pi$
$968$	1.30430e14	4.77461
$969$	0	0
$970$	8.28916e13	3.00634
$971$	4.80491e11	0.0173460	0.00867300	−	0.999962i	$-0.497239\pi$
$971$	4.80491e11	0.0173460	0.00867300	+	0.999962i	$0.497239\pi$
$972$	0	0
$973$	−6.54938e12	−0.234257
$974$	8.33666e13	2.96809
$975$	0	0
$976$	−3.54487e13	−1.25048
$977$	−4.03929e13	−1.41834	−0.709169	−	0.705039i	$-0.750930\pi$
$977$	−4.03929e13	−1.41834	−0.709169	+	0.705039i	$0.750930\pi$
$978$	0	0
$979$	−3.06147e13	−1.06514
$980$	5.76873e13	1.99785
$981$	0	0
$982$	−1.01158e14	−3.47136
$983$	−4.73016e13	−1.61579	−0.807895	−	0.589326i	$-0.799393\pi$
$983$	−4.73016e13	−1.61579	−0.807895	+	0.589326i	$0.799393\pi$
$984$	0	0
$985$	−3.19069e13	−1.07999
$986$	3.55008e13	1.19617
$987$	0	0
$988$	−2.92373e13	−0.976181
$989$	−3.21787e12	−0.106951
$990$	0	0
$991$	−1.02031e13	−0.336048	−0.168024	−	0.985783i	$-0.553739\pi$
$991$	−1.02031e13	−0.336048	−0.168024	+	0.985783i	$0.553739\pi$
$992$	1.09923e14	3.60400
$993$	0	0
$994$	−1.05139e13	−0.341606
$995$	1.12031e13	0.362356
$996$	0	0
$997$	5.02498e12	0.161067	0.0805335	−	0.996752i	$-0.474338\pi$
$997$	5.02498e12	0.161067	0.0805335	+	0.996752i	$0.474338\pi$
$998$	−3.79398e13	−1.21062
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.10.a.f.1.8		8
3.2	odd	2		19.10.a.b.1.1	✓	8
12.11	even	2		304.10.a.i.1.1		8
57.56	even	2		361.10.a.c.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.10.a.b.1.1	✓	8	3.2	odd	2
171.10.a.f.1.8		8	1.1	even	1	trivial
304.10.a.i.1.1		8	12.11	even	2
361.10.a.c.1.8		8	57.56	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

\(f(q)\)	\(=\)	\(q+43.0760 q^{2} +1343.54 q^{4} -1106.01 q^{5} +1237.92 q^{7} +35819.6 q^{8} -47642.6 q^{10} -77454.8 q^{11} -166982. q^{13} +53324.8 q^{14} +855070. q^{16} -202288. q^{17} +130321. q^{19} -1.48598e6 q^{20} -3.33644e6 q^{22} -1.07984e6 q^{23} -729860. q^{25} -7.19293e6 q^{26} +1.66320e6 q^{28} -4.07412e6 q^{29} +5.94389e6 q^{31} +1.84934e7 q^{32} -8.71375e6 q^{34} -1.36916e6 q^{35} +6.26051e6 q^{37} +5.61371e6 q^{38} -3.96169e7 q^{40} +2.45995e7 q^{41} +2.97996e6 q^{43} -1.04064e8 q^{44} -4.65150e7 q^{46} -1.29594e7 q^{47} -3.88212e7 q^{49} -3.14395e7 q^{50} -2.24348e8 q^{52} +2.55151e7 q^{53} +8.56660e7 q^{55} +4.43419e7 q^{56} -1.75497e8 q^{58} -7.81216e7 q^{59} -4.14570e7 q^{61} +2.56039e8 q^{62} +3.58826e8 q^{64} +1.84685e8 q^{65} -6.16078e7 q^{67} -2.71782e8 q^{68} -5.89779e7 q^{70} -1.97167e8 q^{71} -9.45381e7 q^{73} +2.69678e8 q^{74} +1.75092e8 q^{76} -9.58832e7 q^{77} -3.53552e7 q^{79} -9.45718e8 q^{80} +1.05965e9 q^{82} -4.46185e7 q^{83} +2.23733e8 q^{85} +1.28365e8 q^{86} -2.77440e9 q^{88} +3.95259e8 q^{89} -2.06711e8 q^{91} -1.45081e9 q^{92} -5.58238e8 q^{94} -1.44137e8 q^{95} -1.73986e9 q^{97} -1.67226e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 15 q^{2} + 2645 q^{4} - 3894 q^{5} - 7133 q^{7} - 10911 q^{8} + 113172 q^{10} - 172818 q^{11} + 109291 q^{13} - 250959 q^{14} + 1590377 q^{16} - 583575 q^{17} + 1042568 q^{19} - 3191676 q^{20} + 57234 q^{22}+ \cdots - 4178510532 q^{98}+O(q^{100}) \) 8 * q - 15 * q^2 + 2645 * q^4 - 3894 * q^5 - 7133 * q^7 - 10911 * q^8 + 113172 * q^10 - 172818 * q^11 + 109291 * q^13 - 250959 * q^14 + 1590377 * q^16 - 583575 * q^17 + 1042568 * q^19 - 3191676 * q^20 + 57234 * q^22 + 2292405 * q^23 + 205328 * q^25 + 821247 * q^26 - 10217945 * q^28 - 4546335 * q^29 - 8277416 * q^31 + 15773553 * q^32 - 22393029 * q^34 - 15633768 * q^35 + 24485536 * q^37 - 1954815 * q^38 + 2543472 * q^40 - 35051946 * q^41 + 36624268 * q^43 - 215210070 * q^44 - 66065277 * q^46 - 9865524 * q^47 + 111765087 * q^49 - 50599293 * q^50 - 258827351 * q^52 - 13963683 * q^53 + 261540660 * q^55 + 517172871 * q^56 - 399459963 * q^58 - 21215463 * q^59 + 180552112 * q^61 + 451241340 * q^62 + 180711569 * q^64 + 21864624 * q^65 + 1767355 * q^67 + 41475117 * q^68 - 516193752 * q^70 - 273631578 * q^71 - 263711543 * q^73 - 677785566 * q^74 + 344699045 * q^76 - 441243702 * q^77 + 744472174 * q^79 - 1904949528 * q^80 + 1284295572 * q^82 - 832139910 * q^83 + 1013793492 * q^85 - 956757432 * q^86 - 585082794 * q^88 + 291457734 * q^89 - 1351077415 * q^91 - 2042025591 * q^92 - 2759974128 * q^94 - 507469974 * q^95 - 3483837140 * q^97 - 4178510532 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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