LMFDB - Embedded newform 592.8.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,8,Mod(1,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("592.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	592.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:	$184.931935087$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 405x^{2} - 2998x - 4396 $ x^4 - 2x^3 - 405x^2 - 2998*x - 4396
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.83090$ of defining polynomial
Character	$\chi$	$=$	592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.4940 q^{3} +375.302 q^{5} +980.897 q^{7} -1766.99 q^{9} +O(q^{10})$	$q-20.4940 q^{3} +375.302 q^{5} +980.897 q^{7} -1766.99 q^{9} +3376.51 q^{11} +3436.44 q^{13} -7691.45 q^{15} -4744.28 q^{17} +11268.9 q^{19} -20102.6 q^{21} +17737.1 q^{23} +62726.4 q^{25} +81033.3 q^{27} -106898. q^{29} +31372.3 q^{31} -69198.3 q^{33} +368132. q^{35} +50653.0 q^{37} -70426.5 q^{39} +278656. q^{41} +606906. q^{43} -663156. q^{45} +1.13131e6 q^{47} +138617. q^{49} +97229.5 q^{51} -894201. q^{53} +1.26721e6 q^{55} -230946. q^{57} +897569. q^{59} +1.69191e6 q^{61} -1.73324e6 q^{63} +1.28970e6 q^{65} -2.02846e6 q^{67} -363504. q^{69} +585796. q^{71} +428683. q^{73} -1.28552e6 q^{75} +3.31201e6 q^{77} +1.57559e6 q^{79} +2.20372e6 q^{81} +537839. q^{83} -1.78054e6 q^{85} +2.19078e6 q^{87} -1.84127e6 q^{89} +3.37079e6 q^{91} -642946. q^{93} +4.22924e6 q^{95} -6.63828e6 q^{97} -5.96627e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 53 q^{3} + 111 q^{5} + 1666 q^{7} + 4609 q^{9}+O(q^{10}) $ 4 * q + 53 * q^3 + 111 * q^5 + 1666 * q^7 + 4609 * q^9	$ 4 q + 53 q^{3} + 111 q^{5} + 1666 q^{7} + 4609 q^{9} + 4593 q^{11} + 7847 q^{13} - 18900 q^{15} + 23172 q^{17} - 23696 q^{19} + 69416 q^{21} - 24105 q^{23} - 138149 q^{25} + 433646 q^{27} - 140949 q^{29} + 664609 q^{31} - 240450 q^{33} + 248544 q^{35} + 202612 q^{37} + 2288827 q^{39} - 709737 q^{41} + 128962 q^{43} - 1755342 q^{45} + 445842 q^{47} - 1602774 q^{49} + 2883630 q^{51} - 975870 q^{53} + 644145 q^{55} + 3494630 q^{57} + 1812858 q^{59} - 2955031 q^{61} + 3362482 q^{63} + 666 q^{65} - 2737235 q^{67} - 1781673 q^{69} - 4958184 q^{71} - 931591 q^{73} - 4945810 q^{75} + 4352514 q^{77} - 5813561 q^{79} + 16394896 q^{81} - 2120460 q^{83} - 4845402 q^{85} - 7965333 q^{87} + 8833716 q^{89} + 18886274 q^{91} + 3024182 q^{93} + 3151794 q^{95} - 22666876 q^{97} + 17931894 q^{99}+O(q^{100}) $ 4 * q + 53 * q^3 + 111 * q^5 + 1666 * q^7 + 4609 * q^9 + 4593 * q^11 + 7847 * q^13 - 18900 * q^15 + 23172 * q^17 - 23696 * q^19 + 69416 * q^21 - 24105 * q^23 - 138149 * q^25 + 433646 * q^27 - 140949 * q^29 + 664609 * q^31 - 240450 * q^33 + 248544 * q^35 + 202612 * q^37 + 2288827 * q^39 - 709737 * q^41 + 128962 * q^43 - 1755342 * q^45 + 445842 * q^47 - 1602774 * q^49 + 2883630 * q^51 - 975870 * q^53 + 644145 * q^55 + 3494630 * q^57 + 1812858 * q^59 - 2955031 * q^61 + 3362482 * q^63 + 666 * q^65 - 2737235 * q^67 - 1781673 * q^69 - 4958184 * q^71 - 931591 * q^73 - 4945810 * q^75 + 4352514 * q^77 - 5813561 * q^79 + 16394896 * q^81 - 2120460 * q^83 - 4845402 * q^85 - 7965333 * q^87 + 8833716 * q^89 + 18886274 * q^91 + 3024182 * q^93 + 3151794 * q^95 - 22666876 * q^97 + 17931894 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−20.4940	−0.438231	−0.219116	−	0.975699i	$-0.570317\pi$
$3$	−20.4940	−0.438231	−0.219116	+	0.975699i	$0.570317\pi$
$4$	0	0
$5$	375.302	1.34272	0.671360	−	0.741131i	$-0.265710\pi$
$5$	375.302	1.34272	0.671360	+	0.741131i	$0.265710\pi$
$6$	0	0
$7$	980.897	1.08089	0.540444	−	0.841380i	$-0.318256\pi$
$7$	980.897	1.08089	0.540444	+	0.841380i	$0.318256\pi$
$8$	0	0
$9$	−1766.99	−0.807953
$10$	0	0
$11$	3376.51	0.764881	0.382440	−	0.923980i	$-0.375084\pi$
$11$	3376.51	0.764881	0.382440	+	0.923980i	$0.375084\pi$
$12$	0	0
$13$	3436.44	0.433817	0.216909	−	0.976192i	$-0.430403\pi$
$13$	3436.44	0.433817	0.216909	+	0.976192i	$0.430403\pi$
$14$	0	0
$15$	−7691.45	−0.588422
$16$	0	0
$17$	−4744.28	−0.234207	−0.117103	−	0.993120i	$-0.537361\pi$
$17$	−4744.28	−0.234207	−0.117103	+	0.993120i	$0.537361\pi$
$18$	0	0
$19$	11268.9	0.376916	0.188458	−	0.982081i	$-0.439651\pi$
$19$	11268.9	0.376916	0.188458	+	0.982081i	$0.439651\pi$
$20$	0	0
$21$	−20102.6	−0.473678
$22$	0	0
$23$	17737.1	0.303973	0.151986	−	0.988383i	$-0.451433\pi$
$23$	17737.1	0.303973	0.151986	+	0.988383i	$0.451433\pi$
$24$	0	0
$25$	62726.4	0.802898
$26$	0	0
$27$	81033.3	0.792302
$28$	0	0
$29$	−106898.	−0.813914	−0.406957	−	0.913447i	$-0.633410\pi$
$29$	−106898.	−0.813914	−0.406957	+	0.913447i	$0.633410\pi$
$30$	0	0
$31$	31372.3	0.189139	0.0945695	−	0.995518i	$-0.469853\pi$
$31$	31372.3	0.189139	0.0945695	+	0.995518i	$0.469853\pi$
$32$	0	0
$33$	−69198.3	−0.335194
$34$	0	0
$35$	368132.	1.45133
$36$	0	0
$37$	50653.0	0.164399
$37$	50653.0	0.164399
$38$	0	0
$39$	−70426.5	−0.190112
$40$	0	0
$41$	278656.	0.631430	0.315715	−	0.948854i	$-0.397756\pi$
$41$	278656.	0.631430	0.315715	+	0.948854i	$0.397756\pi$
$42$	0	0
$43$	606906.	1.16408	0.582039	−	0.813161i	$-0.302255\pi$
$43$	606906.	1.16408	0.582039	+	0.813161i	$0.302255\pi$
$44$	0	0
$45$	−663156.	−1.08486
$46$	0	0
$47$	1.13131e6	1.58943	0.794713	−	0.606986i	$-0.207621\pi$
$47$	1.13131e6	1.58943	0.794713	+	0.606986i	$0.207621\pi$
$48$	0	0
$49$	138617.	0.168317
$50$	0	0
$51$	97229.5	0.102637
$52$	0	0
$53$	−894201.	−0.825030	−0.412515	−	0.910951i	$-0.635350\pi$
$53$	−894201.	−0.825030	−0.412515	+	0.910951i	$0.635350\pi$
$54$	0	0
$55$	1.26721e6	1.02702
$56$	0	0
$57$	−230946.	−0.165176
$58$	0	0
$59$	897569.	0.568966	0.284483	−	0.958681i	$-0.408178\pi$
$59$	897569.	0.568966	0.284483	+	0.958681i	$0.408178\pi$
$60$	0	0
$61$	1.69191e6	0.954381	0.477191	−	0.878800i	$-0.341655\pi$
$61$	1.69191e6	0.954381	0.477191	+	0.878800i	$0.341655\pi$
$62$	0	0
$63$	−1.73324e6	−0.873307
$64$	0	0
$65$	1.28970e6	0.582495
$66$	0	0
$67$	−2.02846e6	−0.823957	−0.411979	−	0.911194i	$-0.635162\pi$
$67$	−2.02846e6	−0.823957	−0.411979	+	0.911194i	$0.635162\pi$
$68$	0	0
$69$	−363504.	−0.133210
$70$	0	0
$71$	585796.	0.194242	0.0971208	−	0.995273i	$-0.469037\pi$
$71$	585796.	0.194242	0.0971208	+	0.995273i	$0.469037\pi$
$72$	0	0
$73$	428683.	0.128975	0.0644875	−	0.997919i	$-0.479459\pi$
$73$	428683.	0.128975	0.0644875	+	0.997919i	$0.479459\pi$
$74$	0	0
$75$	−1.28552e6	−0.351855
$76$	0	0
$77$	3.31201e6	0.826750
$78$	0	0
$79$	1.57559e6	0.359540	0.179770	−	0.983709i	$-0.442465\pi$
$79$	1.57559e6	0.359540	0.179770	+	0.983709i	$0.442465\pi$
$80$	0	0
$81$	2.20372e6	0.460742
$82$	0	0
$83$	537839.	0.103247	0.0516236	−	0.998667i	$-0.483560\pi$
$83$	537839.	0.103247	0.0516236	+	0.998667i	$0.483560\pi$
$84$	0	0
$85$	−1.78054e6	−0.314474
$86$	0	0
$87$	2.19078e6	0.356682
$88$	0	0
$89$	−1.84127e6	−0.276856	−0.138428	−	0.990373i	$-0.544205\pi$
$89$	−1.84127e6	−0.276856	−0.138428	+	0.990373i	$0.544205\pi$
$90$	0	0
$91$	3.37079e6	0.468907
$92$	0	0
$93$	−642946.	−0.0828866
$94$	0	0
$95$	4.22924e6	0.506093
$96$	0	0
$97$	−6.63828e6	−0.738506	−0.369253	−	0.929329i	$-0.620387\pi$
$97$	−6.63828e6	−0.738506	−0.369253	+	0.929329i	$0.620387\pi$
$98$	0	0
$99$	−5.96627e6	−0.617988
$100$	0	0
$101$	2.01181e6	0.194295	0.0971475	−	0.995270i	$-0.469028\pi$
$101$	2.01181e6	0.194295	0.0971475	+	0.995270i	$0.469028\pi$
$102$	0	0
$103$	8.44464e6	0.761466	0.380733	−	0.924685i	$-0.375672\pi$
$103$	8.44464e6	0.761466	0.380733	+	0.924685i	$0.375672\pi$
$104$	0	0
$105$	−7.54452e6	−0.636018
$106$	0	0
$107$	−2.02730e7	−1.59984	−0.799919	−	0.600108i	$-0.795124\pi$
$107$	−2.02730e7	−1.59984	−0.799919	+	0.600108i	$0.795124\pi$
$108$	0	0
$109$	−2.49898e7	−1.84829	−0.924145	−	0.382042i	$-0.875221\pi$
$109$	−2.49898e7	−1.84829	−0.924145	+	0.382042i	$0.875221\pi$
$110$	0	0
$111$	−1.03808e6	−0.0720448
$112$	0	0
$113$	1.39234e7	0.907760	0.453880	−	0.891063i	$-0.350039\pi$
$113$	1.39234e7	0.907760	0.453880	+	0.891063i	$0.350039\pi$
$114$	0	0
$115$	6.65675e6	0.408150
$116$	0	0
$117$	−6.07217e6	−0.350504
$118$	0	0
$119$	−4.65365e6	−0.253151
$120$	0	0
$121$	−8.08635e6	−0.414958
$122$	0	0
$123$	−5.71079e6	−0.276712
$124$	0	0
$125$	−5.77912e6	−0.264653
$126$	0	0
$127$	−3.66831e6	−0.158911	−0.0794553	−	0.996838i	$-0.525318\pi$
$127$	−3.66831e6	−0.158911	−0.0794553	+	0.996838i	$0.525318\pi$
$128$	0	0
$129$	−1.24380e7	−0.510135
$130$	0	0
$131$	−4.68710e7	−1.82161	−0.910803	−	0.412842i	$-0.864536\pi$
$131$	−4.68710e7	−1.82161	−0.910803	+	0.412842i	$0.864536\pi$
$132$	0	0
$133$	1.10536e7	0.407404
$134$	0	0
$135$	3.04119e7	1.06384
$136$	0	0
$137$	1.68178e7	0.558787	0.279393	−	0.960177i	$-0.409867\pi$
$137$	1.68178e7	0.558787	0.279393	+	0.960177i	$0.409867\pi$
$138$	0	0
$139$	−1.17238e7	−0.370270	−0.185135	−	0.982713i	$-0.559272\pi$
$139$	−1.17238e7	−0.370270	−0.185135	+	0.982713i	$0.559272\pi$
$140$	0	0
$141$	−2.31852e7	−0.696536
$142$	0	0
$143$	1.16032e7	0.331818
$144$	0	0
$145$	−4.01192e7	−1.09286
$146$	0	0
$147$	−2.84081e6	−0.0737619
$148$	0	0
$149$	7.64948e7	1.89444	0.947219	−	0.320588i	$-0.103881\pi$
$149$	7.64948e7	1.89444	0.947219	+	0.320588i	$0.103881\pi$
$150$	0	0
$151$	2.19120e7	0.517921	0.258960	−	0.965888i	$-0.416620\pi$
$151$	2.19120e7	0.517921	0.258960	+	0.965888i	$0.416620\pi$
$152$	0	0
$153$	8.38311e6	0.189228
$154$	0	0
$155$	1.17741e7	0.253961
$156$	0	0
$157$	1.91872e6	0.0395696	0.0197848	−	0.999804i	$-0.493702\pi$
$157$	1.91872e6	0.0395696	0.0197848	+	0.999804i	$0.493702\pi$
$158$	0	0
$159$	1.83258e7	0.361554
$160$	0	0
$161$	1.73982e7	0.328560
$162$	0	0
$163$	2.21304e7	0.400251	0.200126	−	0.979770i	$-0.435865\pi$
$163$	2.21304e7	0.400251	0.200126	+	0.979770i	$0.435865\pi$
$164$	0	0
$165$	−2.59703e7	−0.450072
$166$	0	0
$167$	−4.43183e7	−0.736334	−0.368167	−	0.929760i	$-0.620015\pi$
$167$	−4.43183e7	−0.736334	−0.368167	+	0.929760i	$0.620015\pi$
$168$	0	0
$169$	−5.09394e7	−0.811803
$170$	0	0
$171$	−1.99121e7	−0.304531
$172$	0	0
$173$	7.47705e7	1.09792	0.548958	−	0.835850i	$-0.315025\pi$
$173$	7.47705e7	1.09792	0.548958	+	0.835850i	$0.315025\pi$
$174$	0	0
$175$	6.15282e7	0.867842
$176$	0	0
$177$	−1.83948e7	−0.249338
$178$	0	0
$179$	1.20143e8	1.56571	0.782855	−	0.622204i	$-0.213763\pi$
$179$	1.20143e8	1.56571	0.782855	+	0.622204i	$0.213763\pi$
$180$	0	0
$181$	1.49404e8	1.87278	0.936390	−	0.350962i	$-0.114146\pi$
$181$	1.49404e8	1.87278	0.936390	+	0.350962i	$0.114146\pi$
$182$	0	0
$183$	−3.46740e7	−0.418240
$184$	0	0
$185$	1.90102e7	0.220742
$186$	0	0
$187$	−1.60191e7	−0.179140
$188$	0	0
$189$	7.94854e7	0.856389
$190$	0	0
$191$	2.34648e7	0.243669	0.121835	−	0.992550i	$-0.461122\pi$
$191$	2.34648e7	0.243669	0.121835	+	0.992550i	$0.461122\pi$
$192$	0	0
$193$	1.01584e8	1.01713	0.508565	−	0.861024i	$-0.330176\pi$
$193$	1.01584e8	1.01713	0.508565	+	0.861024i	$0.330176\pi$
$194$	0	0
$195$	−2.64312e7	−0.255268
$196$	0	0
$197$	1.85514e8	1.72880	0.864402	−	0.502801i	$-0.167697\pi$
$197$	1.85514e8	1.72880	0.864402	+	0.502801i	$0.167697\pi$
$198$	0	0
$199$	2.86267e7	0.257505	0.128752	−	0.991677i	$-0.458903\pi$
$199$	2.86267e7	0.257505	0.128752	+	0.991677i	$0.458903\pi$
$200$	0	0
$201$	4.15713e7	0.361084
$202$	0	0
$203$	−1.04856e8	−0.879749
$204$	0	0
$205$	1.04580e8	0.847834
$206$	0	0
$207$	−3.13413e7	−0.245596
$208$	0	0
$209$	3.80496e7	0.288296
$210$	0	0
$211$	−1.57443e7	−0.115381	−0.0576907	−	0.998335i	$-0.518374\pi$
$211$	−1.57443e7	−0.115381	−0.0576907	+	0.998335i	$0.518374\pi$
$212$	0	0
$213$	−1.20053e7	−0.0851227
$214$	0	0
$215$	2.27773e8	1.56303
$216$	0	0
$217$	3.07730e7	0.204438
$218$	0	0
$219$	−8.78544e6	−0.0565209
$220$	0	0
$221$	−1.63034e7	−0.101603
$222$	0	0
$223$	3.23566e7	0.195387	0.0976937	−	0.995217i	$-0.468853\pi$
$223$	3.23566e7	0.195387	0.0976937	+	0.995217i	$0.468853\pi$
$224$	0	0
$225$	−1.10837e8	−0.648704
$226$	0	0
$227$	−2.94511e8	−1.67113	−0.835566	−	0.549390i	$-0.814860\pi$
$227$	−2.94511e8	−1.67113	−0.835566	+	0.549390i	$0.814860\pi$
$228$	0	0
$229$	−2.66064e8	−1.46407	−0.732034	−	0.681268i	$-0.761429\pi$
$229$	−2.66064e8	−1.46407	−0.732034	+	0.681268i	$0.761429\pi$
$230$	0	0
$231$	−6.78765e7	−0.362307
$232$	0	0
$233$	−2.14468e8	−1.11075	−0.555377	−	0.831599i	$-0.687426\pi$
$233$	−2.14468e8	−1.11075	−0.555377	+	0.831599i	$0.687426\pi$
$234$	0	0
$235$	4.24584e8	2.13415
$236$	0	0
$237$	−3.22901e7	−0.157562
$238$	0	0
$239$	3.30136e8	1.56423	0.782115	−	0.623134i	$-0.214141\pi$
$239$	3.30136e8	1.56423	0.782115	+	0.623134i	$0.214141\pi$
$240$	0	0
$241$	3.56063e8	1.63858	0.819290	−	0.573379i	$-0.194368\pi$
$241$	3.56063e8	1.63858	0.819290	+	0.573379i	$0.194368\pi$
$242$	0	0
$243$	−2.22383e8	−0.994213
$244$	0	0
$245$	5.20230e7	0.226003
$246$	0	0
$247$	3.87249e7	0.163513
$248$	0	0
$249$	−1.10225e7	−0.0452462
$250$	0	0
$251$	6.90749e7	0.275716	0.137858	−	0.990452i	$-0.455978\pi$
$251$	6.90749e7	0.275716	0.137858	+	0.990452i	$0.455978\pi$
$252$	0	0
$253$	5.98894e7	0.232503
$254$	0	0
$255$	3.64904e7	0.137812
$256$	0	0
$257$	−1.82080e8	−0.669110	−0.334555	−	0.942376i	$-0.608586\pi$
$257$	−1.82080e8	−0.669110	−0.334555	+	0.942376i	$0.608586\pi$
$258$	0	0
$259$	4.96854e7	0.177697
$260$	0	0
$261$	1.88889e8	0.657604
$262$	0	0
$263$	3.32466e8	1.12694	0.563471	−	0.826136i	$-0.309466\pi$
$263$	3.32466e8	1.12694	0.563471	+	0.826136i	$0.309466\pi$
$264$	0	0
$265$	−3.35595e8	−1.10778
$266$	0	0
$267$	3.77352e7	0.121327
$268$	0	0
$269$	3.56196e8	1.11572	0.557862	−	0.829934i	$-0.311622\pi$
$269$	3.56196e8	1.11572	0.557862	+	0.829934i	$0.311622\pi$
$270$	0	0
$271$	6.50894e8	1.98663	0.993316	−	0.115423i	$-0.0368223\pi$
$271$	6.50894e8	1.98663	0.993316	+	0.115423i	$0.0368223\pi$
$272$	0	0
$273$	−6.90812e7	−0.205490
$274$	0	0
$275$	2.11796e8	0.614121
$276$	0	0
$277$	1.11215e8	0.314402	0.157201	−	0.987567i	$-0.449753\pi$
$277$	1.11215e8	0.314402	0.157201	+	0.987567i	$0.449753\pi$
$278$	0	0
$279$	−5.54347e7	−0.152815
$280$	0	0
$281$	−5.93017e7	−0.159439	−0.0797196	−	0.996817i	$-0.525403\pi$
$281$	−5.93017e7	−0.159439	−0.0797196	+	0.996817i	$0.525403\pi$
$282$	0	0
$283$	6.90811e8	1.81179	0.905893	−	0.423508i	$-0.139201\pi$
$283$	6.90811e8	1.81179	0.905893	+	0.423508i	$0.139201\pi$
$284$	0	0
$285$	−8.66743e7	−0.221786
$286$	0	0
$287$	2.73333e8	0.682504
$288$	0	0
$289$	−3.87830e8	−0.945147
$290$	0	0
$291$	1.36045e8	0.323637
$292$	0	0
$293$	−6.33150e8	−1.47052	−0.735258	−	0.677787i	$-0.762939\pi$
$293$	−6.33150e8	−1.47052	−0.735258	+	0.677787i	$0.762939\pi$
$294$	0	0
$295$	3.36859e8	0.763962
$296$	0	0
$297$	2.73610e8	0.606016
$298$	0	0
$299$	6.09523e7	0.131869
$300$	0	0
$301$	5.95313e8	1.25824
$302$	0	0
$303$	−4.12301e7	−0.0851461
$304$	0	0
$305$	6.34975e8	1.28147
$306$	0	0
$307$	−3.19674e8	−0.630554	−0.315277	−	0.949000i	$-0.602097\pi$
$307$	−3.19674e8	−0.630554	−0.315277	+	0.949000i	$0.602097\pi$
$308$	0	0
$309$	−1.73065e8	−0.333698
$310$	0	0
$311$	−9.05480e8	−1.70694	−0.853469	−	0.521144i	$-0.825505\pi$
$311$	−9.05480e8	−1.70694	−0.853469	+	0.521144i	$0.825505\pi$
$312$	0	0
$313$	4.16880e8	0.768433	0.384216	−	0.923243i	$-0.374472\pi$
$313$	4.16880e8	0.768433	0.384216	+	0.923243i	$0.374472\pi$
$314$	0	0
$315$	−6.50488e8	−1.17261
$316$	0	0
$317$	2.87485e8	0.506883	0.253442	−	0.967351i	$-0.418437\pi$
$317$	2.87485e8	0.506883	0.253442	+	0.967351i	$0.418437\pi$
$318$	0	0
$319$	−3.60944e8	−0.622547
$320$	0	0
$321$	4.15477e8	0.701099
$322$	0	0
$323$	−5.34629e7	−0.0882762
$324$	0	0
$325$	2.15555e8	0.348311
$326$	0	0
$327$	5.12142e8	0.809978
$328$	0	0
$329$	1.10970e9	1.71799
$330$	0	0
$331$	1.43202e7	0.0217046	0.0108523	−	0.999941i	$-0.496546\pi$
$331$	1.43202e7	0.0217046	0.0108523	+	0.999941i	$0.496546\pi$
$332$	0	0
$333$	−8.95036e7	−0.132827
$334$	0	0
$335$	−7.61285e8	−1.10634
$336$	0	0
$337$	−7.28640e8	−1.03707	−0.518535	−	0.855056i	$-0.673522\pi$
$337$	−7.28640e8	−1.03707	−0.518535	+	0.855056i	$0.673522\pi$
$338$	0	0
$339$	−2.85347e8	−0.397809
$340$	0	0
$341$	1.05929e8	0.144669
$342$	0	0
$343$	−6.71843e8	−0.898955
$344$	0	0
$345$	−1.36424e8	−0.178864
$346$	0	0
$347$	5.87298e8	0.754581	0.377290	−	0.926095i	$-0.376856\pi$
$347$	5.87298e8	0.754581	0.377290	+	0.926095i	$0.376856\pi$
$348$	0	0
$349$	−1.89025e8	−0.238029	−0.119015	−	0.992892i	$-0.537974\pi$
$349$	−1.89025e8	−0.238029	−0.119015	+	0.992892i	$0.537974\pi$
$350$	0	0
$351$	2.78466e8	0.343714
$352$	0	0
$353$	1.09082e9	1.31990	0.659951	−	0.751309i	$-0.270577\pi$
$353$	1.09082e9	1.31990	0.659951	+	0.751309i	$0.270577\pi$
$354$	0	0
$355$	2.19850e8	0.260812
$356$	0	0
$357$	9.53721e7	0.110939
$358$	0	0
$359$	1.29066e9	1.47225	0.736123	−	0.676848i	$-0.236655\pi$
$359$	1.29066e9	1.47225	0.736123	+	0.676848i	$0.236655\pi$
$360$	0	0
$361$	−7.66883e8	−0.857934
$362$	0	0
$363$	1.65722e8	0.181847
$364$	0	0
$365$	1.60885e8	0.173177
$366$	0	0
$367$	6.31310e8	0.666671	0.333335	−	0.942808i	$-0.391826\pi$
$367$	6.31310e8	0.666671	0.333335	+	0.942808i	$0.391826\pi$
$368$	0	0
$369$	−4.92384e8	−0.510166
$370$	0	0
$371$	−8.77120e8	−0.891765
$372$	0	0
$373$	−1.48854e8	−0.148518	−0.0742589	−	0.997239i	$-0.523659\pi$
$373$	−1.48854e8	−0.148518	−0.0742589	+	0.997239i	$0.523659\pi$
$374$	0	0
$375$	1.18438e8	0.115979
$376$	0	0
$377$	−3.67350e8	−0.353090
$378$	0	0
$379$	−1.20416e8	−0.113618	−0.0568092	−	0.998385i	$-0.518093\pi$
$379$	−1.20416e8	−0.113618	−0.0568092	+	0.998385i	$0.518093\pi$
$380$	0	0
$381$	7.51785e7	0.0696396
$382$	0	0
$383$	−7.76564e8	−0.706287	−0.353144	−	0.935569i	$-0.614887\pi$
$383$	−7.76564e8	−0.706287	−0.353144	+	0.935569i	$0.614887\pi$
$384$	0	0
$385$	1.24300e9	1.11009
$386$	0	0
$387$	−1.07240e9	−0.940521
$388$	0	0
$389$	−3.59623e8	−0.309759	−0.154880	−	0.987933i	$-0.549499\pi$
$389$	−3.59623e8	−0.309759	−0.154880	+	0.987933i	$0.549499\pi$
$390$	0	0
$391$	−8.41496e7	−0.0711924
$392$	0	0
$393$	9.60575e8	0.798284
$394$	0	0
$395$	5.91320e8	0.482762
$396$	0	0
$397$	−6.04729e8	−0.485058	−0.242529	−	0.970144i	$-0.577977\pi$
$397$	−6.04729e8	−0.485058	−0.242529	+	0.970144i	$0.577977\pi$
$398$	0	0
$399$	−2.26534e8	−0.178537
$400$	0	0
$401$	−1.22031e9	−0.945068	−0.472534	−	0.881312i	$-0.656661\pi$
$401$	−1.22031e9	−0.945068	−0.472534	+	0.881312i	$0.656661\pi$
$402$	0	0
$403$	1.07809e8	0.0820517
$404$	0	0
$405$	8.27058e8	0.618648
$406$	0	0
$407$	1.71030e8	0.125746
$408$	0	0
$409$	1.32264e9	0.955895	0.477948	−	0.878388i	$-0.341381\pi$
$409$	1.32264e9	0.955895	0.477948	+	0.878388i	$0.341381\pi$
$410$	0	0
$411$	−3.44664e8	−0.244878
$412$	0	0
$413$	8.80423e8	0.614988
$414$	0	0
$415$	2.01852e8	0.138632
$416$	0	0
$417$	2.40269e8	0.162264
$418$	0	0
$419$	2.21575e9	1.47154	0.735769	−	0.677232i	$-0.236821\pi$
$419$	2.21575e9	1.47154	0.735769	+	0.677232i	$0.236821\pi$
$420$	0	0
$421$	1.14567e9	0.748291	0.374146	−	0.927370i	$-0.377936\pi$
$421$	1.14567e9	0.748291	0.374146	+	0.927370i	$0.377936\pi$
$422$	0	0
$423$	−1.99902e9	−1.28418
$424$	0	0
$425$	−2.97592e8	−0.188044
$426$	0	0
$427$	1.65959e9	1.03158
$428$	0	0
$429$	−2.37796e8	−0.145413
$430$	0	0
$431$	9.03397e8	0.543511	0.271755	−	0.962366i	$-0.412396\pi$
$431$	9.03397e8	0.543511	0.271755	+	0.962366i	$0.412396\pi$
$432$	0	0
$433$	2.69931e8	0.159788	0.0798940	−	0.996803i	$-0.474542\pi$
$433$	2.69931e8	0.159788	0.0798940	+	0.996803i	$0.474542\pi$
$434$	0	0
$435$	8.22204e8	0.478925
$436$	0	0
$437$	1.99878e8	0.114572
$438$	0	0
$439$	1.79656e9	1.01348	0.506741	−	0.862098i	$-0.330850\pi$
$439$	1.79656e9	1.01348	0.506741	+	0.862098i	$0.330850\pi$
$440$	0	0
$441$	−2.44935e8	−0.135993
$442$	0	0
$443$	1.78755e7	0.00976891	0.00488445	−	0.999988i	$-0.498445\pi$
$443$	1.78755e7	0.00976891	0.00488445	+	0.999988i	$0.498445\pi$
$444$	0	0
$445$	−6.91034e8	−0.371740
$446$	0	0
$447$	−1.56769e9	−0.830201
$448$	0	0
$449$	−2.18310e9	−1.13818	−0.569090	−	0.822275i	$-0.692704\pi$
$449$	−2.18310e9	−1.13818	−0.569090	+	0.822275i	$0.692704\pi$
$450$	0	0
$451$	9.40885e8	0.482968
$452$	0	0
$453$	−4.49066e8	−0.226969
$454$	0	0
$455$	1.26506e9	0.629612
$456$	0	0
$457$	1.51733e9	0.743658	0.371829	−	0.928301i	$-0.378731\pi$
$457$	1.51733e9	0.743658	0.371829	+	0.928301i	$0.378731\pi$
$458$	0	0
$459$	−3.84445e8	−0.185562
$460$	0	0
$461$	2.03360e9	0.966748	0.483374	−	0.875414i	$-0.339411\pi$
$461$	2.03360e9	0.966748	0.483374	+	0.875414i	$0.339411\pi$
$462$	0	0
$463$	2.70950e9	1.26869	0.634345	−	0.773050i	$-0.281270\pi$
$463$	2.70950e9	1.26869	0.634345	+	0.773050i	$0.281270\pi$
$464$	0	0
$465$	−2.41299e8	−0.111293
$466$	0	0
$467$	8.79465e8	0.399585	0.199793	−	0.979838i	$-0.435973\pi$
$467$	8.79465e8	0.399585	0.199793	+	0.979838i	$0.435973\pi$
$468$	0	0
$469$	−1.98971e9	−0.890605
$470$	0	0
$471$	−3.93223e7	−0.0173406
$472$	0	0
$473$	2.04922e9	0.890380
$474$	0	0
$475$	7.06858e8	0.302625
$476$	0	0
$477$	1.58005e9	0.666586
$478$	0	0
$479$	6.53811e8	0.271818	0.135909	−	0.990721i	$-0.456605\pi$
$479$	6.53811e8	0.271818	0.135909	+	0.990721i	$0.456605\pi$
$480$	0	0
$481$	1.74066e8	0.0713191
$482$	0	0
$483$	−3.56560e8	−0.143985
$484$	0	0
$485$	−2.49136e9	−0.991608
$486$	0	0
$487$	1.56097e9	0.612413	0.306207	−	0.951965i	$-0.400940\pi$
$487$	1.56097e9	0.612413	0.306207	+	0.951965i	$0.400940\pi$
$488$	0	0
$489$	−4.53542e8	−0.175403
$490$	0	0
$491$	1.64069e9	0.625520	0.312760	−	0.949832i	$-0.398746\pi$
$491$	1.64069e9	0.625520	0.312760	+	0.949832i	$0.398746\pi$
$492$	0	0
$493$	5.07156e8	0.190624
$494$	0	0
$495$	−2.23915e9	−0.829785
$496$	0	0
$497$	5.74606e8	0.209953
$498$	0	0
$499$	−2.04350e9	−0.736245	−0.368123	−	0.929777i	$-0.619999\pi$
$499$	−2.04350e9	−0.736245	−0.368123	+	0.929777i	$0.619999\pi$
$500$	0	0
$501$	9.08260e8	0.322685
$502$	0	0
$503$	−5.03945e9	−1.76561	−0.882805	−	0.469739i	$-0.844348\pi$
$503$	−5.03945e9	−1.76561	−0.882805	+	0.469739i	$0.844348\pi$
$504$	0	0
$505$	7.55035e8	0.260884
$506$	0	0
$507$	1.04395e9	0.355757
$508$	0	0
$509$	1.43927e9	0.483761	0.241881	−	0.970306i	$-0.422236\pi$
$509$	1.43927e9	0.483761	0.241881	+	0.970306i	$0.422236\pi$
$510$	0	0
$511$	4.20494e8	0.139408
$512$	0	0
$513$	9.13158e8	0.298631
$514$	0	0
$515$	3.16929e9	1.02244
$516$	0	0
$517$	3.81989e9	1.21572
$518$	0	0
$519$	−1.53235e9	−0.481141
$520$	0	0
$521$	3.31327e8	0.102642	0.0513210	−	0.998682i	$-0.483657\pi$
$521$	3.31327e8	0.102642	0.0513210	+	0.998682i	$0.483657\pi$
$522$	0	0
$523$	3.41666e9	1.04435	0.522175	−	0.852838i	$-0.325121\pi$
$523$	3.41666e9	1.04435	0.522175	+	0.852838i	$0.325121\pi$
$524$	0	0
$525$	−1.26096e9	−0.380315
$526$	0	0
$527$	−1.48839e8	−0.0442976
$528$	0	0
$529$	−3.09022e9	−0.907601
$530$	0	0
$531$	−1.58600e9	−0.459698
$532$	0	0
$533$	9.57584e8	0.273925
$534$	0	0
$535$	−7.60851e9	−2.14813
$536$	0	0
$537$	−2.46221e9	−0.686143
$538$	0	0
$539$	4.68040e8	0.128743
$540$	0	0
$541$	−3.31862e9	−0.901087	−0.450544	−	0.892754i	$-0.648770\pi$
$541$	−3.31862e9	−0.901087	−0.450544	+	0.892754i	$0.648770\pi$
$542$	0	0
$543$	−3.06189e9	−0.820710
$544$	0	0
$545$	−9.37872e9	−2.48174
$546$	0	0
$547$	−3.77978e9	−0.987440	−0.493720	−	0.869621i	$-0.664363\pi$
$547$	−3.77978e9	−0.987440	−0.493720	+	0.869621i	$0.664363\pi$
$548$	0	0
$549$	−2.98959e9	−0.771096
$550$	0	0
$551$	−1.20463e9	−0.306777
$552$	0	0
$553$	1.54549e9	0.388622
$554$	0	0
$555$	−3.89595e8	−0.0967360
$556$	0	0
$557$	−2.29803e9	−0.563460	−0.281730	−	0.959494i	$-0.590908\pi$
$557$	−2.29803e9	−0.563460	−0.281730	+	0.959494i	$0.590908\pi$
$558$	0	0
$559$	2.08560e9	0.504997
$560$	0	0
$561$	3.28296e8	0.0785047
$562$	0	0
$563$	−4.61923e9	−1.09091	−0.545456	−	0.838139i	$-0.683644\pi$
$563$	−4.61923e9	−1.09091	−0.545456	+	0.838139i	$0.683644\pi$
$564$	0	0
$565$	5.22548e9	1.21887
$566$	0	0
$567$	2.16162e9	0.498010
$568$	0	0
$569$	−1.50413e8	−0.0342289	−0.0171145	−	0.999854i	$-0.505448\pi$
$569$	−1.50413e8	−0.0342289	−0.0171145	+	0.999854i	$0.505448\pi$
$570$	0	0
$571$	1.41430e9	0.317918	0.158959	−	0.987285i	$-0.449186\pi$
$571$	1.41430e9	0.317918	0.158959	+	0.987285i	$0.449186\pi$
$572$	0	0
$573$	−4.80890e8	−0.106784
$574$	0	0
$575$	1.11258e9	0.244059
$576$	0	0
$577$	−5.23894e9	−1.13535	−0.567673	−	0.823254i	$-0.692156\pi$
$577$	−5.23894e9	−1.13535	−0.567673	+	0.823254i	$0.692156\pi$
$578$	0	0
$579$	−2.08187e9	−0.445738
$580$	0	0
$581$	5.27564e8	0.111599
$582$	0	0
$583$	−3.01928e9	−0.631049
$584$	0	0
$585$	−2.27889e9	−0.470629
$586$	0	0
$587$	−9.01754e8	−0.184016	−0.0920078	−	0.995758i	$-0.529328\pi$
$587$	−9.01754e8	−0.184016	−0.0920078	+	0.995758i	$0.529328\pi$
$588$	0	0
$589$	3.53532e8	0.0712895
$590$	0	0
$591$	−3.80194e9	−0.757616
$592$	0	0
$593$	1.90592e9	0.375329	0.187665	−	0.982233i	$-0.439908\pi$
$593$	1.90592e9	0.375329	0.187665	+	0.982233i	$0.439908\pi$
$594$	0	0
$595$	−1.74652e9	−0.339911
$596$	0	0
$597$	−5.86676e8	−0.112847
$598$	0	0
$599$	−1.30391e9	−0.247887	−0.123944	−	0.992289i	$-0.539554\pi$
$599$	−1.30391e9	−0.247887	−0.123944	+	0.992289i	$0.539554\pi$
$600$	0	0
$601$	−2.06025e9	−0.387131	−0.193566	−	0.981087i	$-0.562005\pi$
$601$	−2.06025e9	−0.387131	−0.193566	+	0.981087i	$0.562005\pi$
$602$	0	0
$603$	3.58428e9	0.665719
$604$	0	0
$605$	−3.03482e9	−0.557172
$606$	0	0
$607$	−3.47284e9	−0.630267	−0.315134	−	0.949047i	$-0.602049\pi$
$607$	−3.47284e9	−0.630267	−0.315134	+	0.949047i	$0.602049\pi$
$608$	0	0
$609$	2.14893e9	0.385533
$610$	0	0
$611$	3.88769e9	0.689520
$612$	0	0
$613$	−6.62090e9	−1.16093	−0.580464	−	0.814286i	$-0.697129\pi$
$613$	−6.62090e9	−1.16093	−0.580464	+	0.814286i	$0.697129\pi$
$614$	0	0
$615$	−2.14327e9	−0.371547
$616$	0	0
$617$	−7.31262e9	−1.25336	−0.626679	−	0.779278i	$-0.715586\pi$
$617$	−7.31262e9	−1.25336	−0.626679	+	0.779278i	$0.715586\pi$
$618$	0	0
$619$	1.16269e10	1.97037	0.985185	−	0.171494i	$-0.0548596\pi$
$619$	1.16269e10	1.97037	0.985185	+	0.171494i	$0.0548596\pi$
$620$	0	0
$621$	1.43729e9	0.240838
$622$	0	0
$623$	−1.80610e9	−0.299250
$624$	0	0
$625$	−7.06941e9	−1.15825
$626$	0	0
$627$	−7.79790e8	−0.126340
$628$	0	0
$629$	−2.40312e8	−0.0385033
$630$	0	0
$631$	1.21058e10	1.91818	0.959091	−	0.283097i	$-0.0913618\pi$
$631$	1.21058e10	1.91818	0.959091	+	0.283097i	$0.0913618\pi$
$632$	0	0
$633$	3.22665e8	0.0505637
$634$	0	0
$635$	−1.37672e9	−0.213373
$636$	0	0
$637$	4.76347e8	0.0730189
$638$	0	0
$639$	−1.03510e9	−0.156938
$640$	0	0
$641$	−1.02354e10	−1.53498	−0.767489	−	0.641062i	$-0.778494\pi$
$641$	−1.02354e10	−1.53498	−0.767489	+	0.641062i	$0.778494\pi$
$642$	0	0
$643$	−5.79264e9	−0.859287	−0.429643	−	0.902999i	$-0.641361\pi$
$643$	−5.79264e9	−0.859287	−0.429643	+	0.902999i	$0.641361\pi$
$644$	0	0
$645$	−4.66799e9	−0.684969
$646$	0	0
$647$	−5.49161e9	−0.797140	−0.398570	−	0.917138i	$-0.630493\pi$
$647$	−5.49161e9	−0.797140	−0.398570	+	0.917138i	$0.630493\pi$
$648$	0	0
$649$	3.03065e9	0.435191
$650$	0	0
$651$	−6.30664e8	−0.0895910
$652$	0	0
$653$	1.23545e10	1.73632	0.868159	−	0.496286i	$-0.165303\pi$
$653$	1.23545e10	1.73632	0.868159	+	0.496286i	$0.165303\pi$
$654$	0	0
$655$	−1.75908e10	−2.44591
$656$	0	0
$657$	−7.57480e8	−0.104206
$658$	0	0
$659$	1.28561e9	0.174989	0.0874943	−	0.996165i	$-0.472114\pi$
$659$	1.28561e9	0.174989	0.0874943	+	0.996165i	$0.472114\pi$
$660$	0	0
$661$	3.05232e9	0.411078	0.205539	−	0.978649i	$-0.434105\pi$
$661$	3.05232e9	0.411078	0.205539	+	0.978649i	$0.434105\pi$
$662$	0	0
$663$	3.34123e8	0.0445255
$664$	0	0
$665$	4.14845e9	0.547029
$666$	0	0
$667$	−1.89607e9	−0.247407
$668$	0	0
$669$	−6.63118e8	−0.0856248
$670$	0	0
$671$	5.71274e9	0.729988
$672$	0	0
$673$	1.07819e10	1.36346	0.681730	−	0.731604i	$-0.261228\pi$
$673$	1.07819e10	1.36346	0.681730	+	0.731604i	$0.261228\pi$
$674$	0	0
$675$	5.08293e9	0.636137
$676$	0	0
$677$	1.22831e10	1.52142	0.760709	−	0.649093i	$-0.224851\pi$
$677$	1.22831e10	1.52142	0.760709	+	0.649093i	$0.224851\pi$
$678$	0	0
$679$	−6.51147e9	−0.798242
$680$	0	0
$681$	6.03572e9	0.732342
$682$	0	0
$683$	−7.76468e8	−0.0932505	−0.0466252	−	0.998912i	$-0.514847\pi$
$683$	−7.76468e8	−0.0932505	−0.0466252	+	0.998912i	$0.514847\pi$
$684$	0	0
$685$	6.31173e9	0.750294
$686$	0	0
$687$	5.45272e9	0.641601
$688$	0	0
$689$	−3.07287e9	−0.357912
$690$	0	0
$691$	1.01602e10	1.17147	0.585733	−	0.810504i	$-0.300807\pi$
$691$	1.01602e10	1.17147	0.585733	+	0.810504i	$0.300807\pi$
$692$	0	0
$693$	−5.85230e9	−0.667975
$694$	0	0
$695$	−4.39998e9	−0.497169
$696$	0	0
$697$	−1.32202e9	−0.147885
$698$	0	0
$699$	4.39532e9	0.486767
$700$	0	0
$701$	−1.47467e10	−1.61689	−0.808446	−	0.588571i	$-0.799691\pi$
$701$	−1.47467e10	−1.61689	−0.808446	+	0.588571i	$0.799691\pi$
$702$	0	0
$703$	5.70804e8	0.0619646
$704$	0	0
$705$	−8.70144e9	−0.935253
$706$	0	0
$707$	1.97338e9	0.210011
$708$	0	0
$709$	1.12511e10	1.18559	0.592795	−	0.805353i	$-0.298024\pi$
$709$	1.12511e10	1.18559	0.592795	+	0.805353i	$0.298024\pi$
$710$	0	0
$711$	−2.78405e9	−0.290492
$712$	0	0
$713$	5.56453e8	0.0574930
$714$	0	0
$715$	4.35469e9	0.445539
$716$	0	0
$717$	−6.76583e9	−0.685495
$718$	0	0
$719$	−1.05695e10	−1.06048	−0.530242	−	0.847846i	$-0.677899\pi$
$719$	−1.05695e10	−1.06048	−0.530242	+	0.847846i	$0.677899\pi$
$720$	0	0
$721$	8.28332e9	0.823059
$722$	0	0
$723$	−7.29718e9	−0.718077
$724$	0	0
$725$	−6.70535e9	−0.653490
$726$	0	0
$727$	−1.28794e10	−1.24316	−0.621578	−	0.783352i	$-0.713508\pi$
$727$	−1.28794e10	−1.24316	−0.621578	+	0.783352i	$0.713508\pi$
$728$	0	0
$729$	−2.62002e8	−0.0250471
$730$	0	0
$731$	−2.87933e9	−0.272635
$732$	0	0
$733$	4.04814e9	0.379657	0.189828	−	0.981817i	$-0.439207\pi$
$733$	4.04814e9	0.379657	0.189828	+	0.981817i	$0.439207\pi$
$734$	0	0
$735$	−1.06616e9	−0.0990416
$736$	0	0
$737$	−6.84912e9	−0.630229
$738$	0	0
$739$	−2.07897e10	−1.89493	−0.947465	−	0.319860i	$-0.896364\pi$
$739$	−2.07897e10	−1.89493	−0.947465	+	0.319860i	$0.896364\pi$
$740$	0	0
$741$	−7.93630e8	−0.0716563
$742$	0	0
$743$	1.77336e9	0.158612	0.0793059	−	0.996850i	$-0.474730\pi$
$743$	1.77336e9	0.158612	0.0793059	+	0.996850i	$0.474730\pi$
$744$	0	0
$745$	2.87086e10	2.54370
$746$	0	0
$747$	−9.50358e8	−0.0834190
$748$	0	0
$749$	−1.98858e10	−1.72924
$750$	0	0
$751$	−1.57353e9	−0.135561	−0.0677805	−	0.997700i	$-0.521592\pi$
$751$	−1.57353e9	−0.135561	−0.0677805	+	0.997700i	$0.521592\pi$
$752$	0	0
$753$	−1.41562e9	−0.120827
$754$	0	0
$755$	8.22363e9	0.695423
$756$	0	0
$757$	3.09511e9	0.259322	0.129661	−	0.991558i	$-0.458611\pi$
$757$	3.09511e9	0.259322	0.129661	+	0.991558i	$0.458611\pi$
$758$	0	0
$759$	−1.22738e9	−0.101890
$760$	0	0
$761$	8.84425e9	0.727470	0.363735	−	0.931503i	$-0.381501\pi$
$761$	8.84425e9	0.727470	0.363735	+	0.931503i	$0.381501\pi$
$762$	0	0
$763$	−2.45124e10	−1.99779
$764$	0	0
$765$	3.14620e9	0.254080
$766$	0	0
$767$	3.08444e9	0.246827
$768$	0	0
$769$	−1.74163e10	−1.38106	−0.690532	−	0.723302i	$-0.742624\pi$
$769$	−1.74163e10	−1.38106	−0.690532	+	0.723302i	$0.742624\pi$
$770$	0	0
$771$	3.73156e9	0.293225
$772$	0	0
$773$	9.23354e9	0.719018	0.359509	−	0.933142i	$-0.382944\pi$
$773$	9.23354e9	0.719018	0.359509	+	0.933142i	$0.382944\pi$
$774$	0	0
$775$	1.96787e9	0.151859
$776$	0	0
$777$	−1.01825e9	−0.0778723
$778$	0	0
$779$	3.14015e9	0.237996
$780$	0	0
$781$	1.97795e9	0.148572
$782$	0	0
$783$	−8.66234e9	−0.644865
$784$	0	0
$785$	7.20098e8	0.0531309
$786$	0	0
$787$	−1.14265e10	−0.835609	−0.417805	−	0.908537i	$-0.637200\pi$
$787$	−1.14265e10	−0.835609	−0.417805	+	0.908537i	$0.637200\pi$
$788$	0	0
$789$	−6.81356e9	−0.493861
$790$	0	0
$791$	1.36574e10	0.981186
$792$	0	0
$793$	5.81413e9	0.414027
$794$	0	0
$795$	6.87770e9	0.485466
$796$	0	0
$797$	−2.63234e10	−1.84178	−0.920891	−	0.389820i	$-0.872537\pi$
$797$	−2.63234e10	−1.84178	−0.920891	+	0.389820i	$0.872537\pi$
$798$	0	0
$799$	−5.36726e9	−0.372254
$800$	0	0
$801$	3.25352e9	0.223687
$802$	0	0
$803$	1.44745e9	0.0986505
$804$	0	0
$805$	6.52959e9	0.441164
$806$	0	0
$807$	−7.29990e9	−0.488945
$808$	0	0
$809$	−7.47328e9	−0.496240	−0.248120	−	0.968729i	$-0.579813\pi$
$809$	−7.47328e9	−0.496240	−0.248120	+	0.968729i	$0.579813\pi$
$810$	0	0
$811$	1.32084e10	0.869513	0.434756	−	0.900548i	$-0.356834\pi$
$811$	1.32084e10	0.869513	0.434756	+	0.900548i	$0.356834\pi$
$812$	0	0
$813$	−1.33394e10	−0.870604
$814$	0	0
$815$	8.30558e9	0.537426
$816$	0	0
$817$	6.83917e9	0.438759
$818$	0	0
$819$	−5.95617e9	−0.378855
$820$	0	0
$821$	−2.22151e10	−1.40103	−0.700516	−	0.713637i	$-0.747047\pi$
$821$	−2.22151e10	−1.40103	−0.700516	+	0.713637i	$0.747047\pi$
$822$	0	0
$823$	5.39596e9	0.337419	0.168710	−	0.985666i	$-0.446040\pi$
$823$	5.39596e9	0.337419	0.168710	+	0.985666i	$0.446040\pi$
$824$	0	0
$825$	−4.34056e9	−0.269127
$826$	0	0
$827$	1.31682e10	0.809575	0.404787	−	0.914411i	$-0.367346\pi$
$827$	1.31682e10	0.809575	0.404787	+	0.914411i	$0.367346\pi$
$828$	0	0
$829$	−1.33819e10	−0.815790	−0.407895	−	0.913029i	$-0.633737\pi$
$829$	−1.33819e10	−0.815790	−0.407895	+	0.913029i	$0.633737\pi$
$830$	0	0
$831$	−2.27925e9	−0.137781
$832$	0	0
$833$	−6.57636e8	−0.0394210
$834$	0	0
$835$	−1.66327e10	−0.988691
$836$	0	0
$837$	2.54220e9	0.149855
$838$	0	0
$839$	2.87088e9	0.167822	0.0839108	−	0.996473i	$-0.473259\pi$
$839$	2.87088e9	0.167822	0.0839108	+	0.996473i	$0.473259\pi$
$840$	0	0
$841$	−5.82260e9	−0.337544
$842$	0	0
$843$	1.21533e9	0.0698713
$844$	0	0
$845$	−1.91177e10	−1.09002
$846$	0	0
$847$	−7.93188e9	−0.448523
$848$	0	0
$849$	−1.41575e10	−0.793981
$850$	0	0
$851$	8.98436e8	0.0499728
$852$	0	0
$853$	2.37600e9	0.131077	0.0655383	−	0.997850i	$-0.479124\pi$
$853$	2.37600e9	0.131077	0.0655383	+	0.997850i	$0.479124\pi$
$854$	0	0
$855$	−7.47305e9	−0.408899
$856$	0	0
$857$	−3.42331e10	−1.85786	−0.928931	−	0.370253i	$-0.879271\pi$
$857$	−3.42331e10	−1.85786	−0.928931	+	0.370253i	$0.879271\pi$
$858$	0	0
$859$	3.38131e9	0.182016	0.0910078	−	0.995850i	$-0.470991\pi$
$859$	3.38131e9	0.182016	0.0910078	+	0.995850i	$0.470991\pi$
$860$	0	0
$861$	−5.60170e9	−0.299095
$862$	0	0
$863$	1.21792e10	0.645030	0.322515	−	0.946564i	$-0.395472\pi$
$863$	1.21792e10	0.645030	0.322515	+	0.946564i	$0.395472\pi$
$864$	0	0
$865$	2.80615e10	1.47419
$866$	0	0
$867$	7.94821e9	0.414193
$868$	0	0
$869$	5.31998e9	0.275005
$870$	0	0
$871$	−6.97068e9	−0.357447
$872$	0	0
$873$	1.17298e10	0.596679
$874$	0	0
$875$	−5.66872e9	−0.286060
$876$	0	0
$877$	1.71298e10	0.857539	0.428770	−	0.903414i	$-0.358947\pi$
$877$	1.71298e10	0.857539	0.428770	+	0.903414i	$0.358947\pi$
$878$	0	0
$879$	1.29758e10	0.644426
$880$	0	0
$881$	−1.99520e10	−0.983039	−0.491520	−	0.870867i	$-0.663558\pi$
$881$	−1.99520e10	−0.983039	−0.491520	+	0.870867i	$0.663558\pi$
$882$	0	0
$883$	3.12819e10	1.52908	0.764542	−	0.644574i	$-0.222965\pi$
$883$	3.12819e10	1.52908	0.764542	+	0.644574i	$0.222965\pi$
$884$	0	0
$885$	−6.90361e9	−0.334792
$886$	0	0
$887$	−1.92928e10	−0.928244	−0.464122	−	0.885771i	$-0.653630\pi$
$887$	−1.92928e10	−0.928244	−0.464122	+	0.885771i	$0.653630\pi$
$888$	0	0
$889$	−3.59824e9	−0.171765
$890$	0	0
$891$	7.44087e9	0.352413
$892$	0	0
$893$	1.27487e10	0.599080
$894$	0	0
$895$	4.50897e10	2.10231
$896$	0	0
$897$	−1.24916e9	−0.0577889
$898$	0	0
$899$	−3.35365e9	−0.153943
$900$	0	0
$901$	4.24234e9	0.193227
$902$	0	0
$903$	−1.22004e10	−0.551399
$904$	0	0
$905$	5.60715e10	2.51462
$906$	0	0
$907$	2.33637e10	1.03972	0.519860	−	0.854251i	$-0.325984\pi$
$907$	2.33637e10	1.03972	0.519860	+	0.854251i	$0.325984\pi$
$908$	0	0
$909$	−3.55485e9	−0.156981
$910$	0	0
$911$	1.56760e10	0.686943	0.343471	−	0.939163i	$-0.388397\pi$
$911$	1.56760e10	0.686943	0.343471	+	0.939163i	$0.388397\pi$
$912$	0	0
$913$	1.81602e9	0.0789718
$914$	0	0
$915$	−1.30132e10	−0.561579
$916$	0	0
$917$	−4.59756e10	−1.96895
$918$	0	0
$919$	2.15624e10	0.916418	0.458209	−	0.888845i	$-0.348491\pi$
$919$	2.15624e10	0.916418	0.458209	+	0.888845i	$0.348491\pi$
$920$	0	0
$921$	6.55141e9	0.276329
$922$	0	0
$923$	2.01305e9	0.0842653
$924$	0	0
$925$	3.17728e9	0.131996
$926$	0	0
$927$	−1.49216e10	−0.615229
$928$	0	0
$929$	8.87728e9	0.363266	0.181633	−	0.983366i	$-0.441862\pi$
$929$	8.87728e9	0.363266	0.181633	+	0.983366i	$0.441862\pi$
$930$	0	0
$931$	1.56206e9	0.0634415
$932$	0	0
$933$	1.85570e10	0.748033
$934$	0	0
$935$	−6.01200e9	−0.240535
$936$	0	0
$937$	−5.24977e9	−0.208474	−0.104237	−	0.994552i	$-0.533240\pi$
$937$	−5.24977e9	−0.208474	−0.104237	+	0.994552i	$0.533240\pi$
$938$	0	0
$939$	−8.54356e9	−0.336751
$940$	0	0
$941$	1.88471e10	0.737363	0.368682	−	0.929556i	$-0.379809\pi$
$941$	1.88471e10	0.737363	0.368682	+	0.929556i	$0.379809\pi$
$942$	0	0
$943$	4.94254e9	0.191937
$944$	0	0
$945$	2.98310e10	1.14989
$946$	0	0
$947$	−3.54978e9	−0.135824	−0.0679119	−	0.997691i	$-0.521634\pi$
$947$	−3.54978e9	−0.135824	−0.0679119	+	0.997691i	$0.521634\pi$
$948$	0	0
$949$	1.47314e9	0.0559516
$950$	0	0
$951$	−5.89173e9	−0.222132
$952$	0	0
$953$	−5.00673e9	−0.187383	−0.0936913	−	0.995601i	$-0.529867\pi$
$953$	−5.00673e9	−0.187383	−0.0936913	+	0.995601i	$0.529867\pi$
$954$	0	0
$955$	8.80640e9	0.327180
$956$	0	0
$957$	7.39719e9	0.272819
$958$	0	0
$959$	1.64965e10	0.603985
$960$	0	0
$961$	−2.65284e10	−0.964226
$962$	0	0
$963$	3.58224e10	1.29259
$964$	0	0
$965$	3.81248e10	1.36572
$966$	0	0
$967$	−1.13815e10	−0.404770	−0.202385	−	0.979306i	$-0.564869\pi$
$967$	−1.13815e10	−0.404770	−0.202385	+	0.979306i	$0.564869\pi$
$968$	0	0
$969$	1.09567e9	0.0386854
$970$	0	0
$971$	−2.87865e10	−1.00907	−0.504536	−	0.863391i	$-0.668336\pi$
$971$	−2.87865e10	−1.00907	−0.504536	+	0.863391i	$0.668336\pi$
$972$	0	0
$973$	−1.14999e10	−0.400220
$974$	0	0
$975$	−4.41760e9	−0.152641
$976$	0	0
$977$	−7.39126e9	−0.253564	−0.126782	−	0.991931i	$-0.540465\pi$
$977$	−7.39126e9	−0.253564	−0.126782	+	0.991931i	$0.540465\pi$
$978$	0	0
$979$	−6.21708e9	−0.211762
$980$	0	0
$981$	4.41568e10	1.49333
$982$	0	0
$983$	−2.07272e10	−0.695991	−0.347995	−	0.937496i	$-0.613138\pi$
$983$	−2.07272e10	−0.695991	−0.347995	+	0.937496i	$0.613138\pi$
$984$	0	0
$985$	6.96239e10	2.32130
$986$	0	0
$987$	−2.27423e10	−0.752877
$988$	0	0
$989$	1.07647e10	0.353848
$990$	0	0
$991$	3.71476e10	1.21248	0.606238	−	0.795283i	$-0.292678\pi$
$991$	3.71476e10	1.21248	0.606238	+	0.795283i	$0.292678\pi$
$992$	0	0
$993$	−2.93479e8	−0.00951162
$994$	0	0
$995$	1.07436e10	0.345757
$996$	0	0
$997$	−3.68049e10	−1.17618	−0.588088	−	0.808797i	$-0.700119\pi$
$997$	−3.68049e10	−1.17618	−0.588088	+	0.808797i	$0.700119\pi$
$998$	0	0
$999$	4.10458e9	0.130254

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.8.a.b.1.2		4
4.3	odd	2		74.8.a.a.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.8.a.a.1.3	✓	4	4.3	odd	2
592.8.a.b.1.2		4	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	592.a (trivial)

\(f(q)\)	\(=\)	\(q-20.4940 q^{3} +375.302 q^{5} +980.897 q^{7} -1766.99 q^{9} +O(q^{10})\)	\(q-20.4940 q^{3} +375.302 q^{5} +980.897 q^{7} -1766.99 q^{9} +3376.51 q^{11} +3436.44 q^{13} -7691.45 q^{15} -4744.28 q^{17} +11268.9 q^{19} -20102.6 q^{21} +17737.1 q^{23} +62726.4 q^{25} +81033.3 q^{27} -106898. q^{29} +31372.3 q^{31} -69198.3 q^{33} +368132. q^{35} +50653.0 q^{37} -70426.5 q^{39} +278656. q^{41} +606906. q^{43} -663156. q^{45} +1.13131e6 q^{47} +138617. q^{49} +97229.5 q^{51} -894201. q^{53} +1.26721e6 q^{55} -230946. q^{57} +897569. q^{59} +1.69191e6 q^{61} -1.73324e6 q^{63} +1.28970e6 q^{65} -2.02846e6 q^{67} -363504. q^{69} +585796. q^{71} +428683. q^{73} -1.28552e6 q^{75} +3.31201e6 q^{77} +1.57559e6 q^{79} +2.20372e6 q^{81} +537839. q^{83} -1.78054e6 q^{85} +2.19078e6 q^{87} -1.84127e6 q^{89} +3.37079e6 q^{91} -642946. q^{93} +4.22924e6 q^{95} -6.63828e6 q^{97} -5.96627e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 53 q^{3} + 111 q^{5} + 1666 q^{7} + 4609 q^{9}+O(q^{10}) \) 4 * q + 53 * q^3 + 111 * q^5 + 1666 * q^7 + 4609 * q^9	\( 4 q + 53 q^{3} + 111 q^{5} + 1666 q^{7} + 4609 q^{9} + 4593 q^{11} + 7847 q^{13} - 18900 q^{15} + 23172 q^{17} - 23696 q^{19} + 69416 q^{21} - 24105 q^{23} - 138149 q^{25} + 433646 q^{27} - 140949 q^{29} + 664609 q^{31} - 240450 q^{33} + 248544 q^{35} + 202612 q^{37} + 2288827 q^{39} - 709737 q^{41} + 128962 q^{43} - 1755342 q^{45} + 445842 q^{47} - 1602774 q^{49} + 2883630 q^{51} - 975870 q^{53} + 644145 q^{55} + 3494630 q^{57} + 1812858 q^{59} - 2955031 q^{61} + 3362482 q^{63} + 666 q^{65} - 2737235 q^{67} - 1781673 q^{69} - 4958184 q^{71} - 931591 q^{73} - 4945810 q^{75} + 4352514 q^{77} - 5813561 q^{79} + 16394896 q^{81} - 2120460 q^{83} - 4845402 q^{85} - 7965333 q^{87} + 8833716 q^{89} + 18886274 q^{91} + 3024182 q^{93} + 3151794 q^{95} - 22666876 q^{97} + 17931894 q^{99}+O(q^{100}) \) 4 * q + 53 * q^3 + 111 * q^5 + 1666 * q^7 + 4609 * q^9 + 4593 * q^11 + 7847 * q^13 - 18900 * q^15 + 23172 * q^17 - 23696 * q^19 + 69416 * q^21 - 24105 * q^23 - 138149 * q^25 + 433646 * q^27 - 140949 * q^29 + 664609 * q^31 - 240450 * q^33 + 248544 * q^35 + 202612 * q^37 + 2288827 * q^39 - 709737 * q^41 + 128962 * q^43 - 1755342 * q^45 + 445842 * q^47 - 1602774 * q^49 + 2883630 * q^51 - 975870 * q^53 + 644145 * q^55 + 3494630 * q^57 + 1812858 * q^59 - 2955031 * q^61 + 3362482 * q^63 + 666 * q^65 - 2737235 * q^67 - 1781673 * q^69 - 4958184 * q^71 - 931591 * q^73 - 4945810 * q^75 + 4352514 * q^77 - 5813561 * q^79 + 16394896 * q^81 - 2120460 * q^83 - 4845402 * q^85 - 7965333 * q^87 + 8833716 * q^89 + 18886274 * q^91 + 3024182 * q^93 + 3151794 * q^95 - 22666876 * q^97 + 17931894 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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