LMFDB - Embedded newform 160.8.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,8,Mod(1,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, 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("160.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	160.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:	$49.9816040775$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.77052.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 63x + 183 $ x^3 - x^2 - 63*x + 183
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.68291$ of defining polynomial
Character	$\chi$	$=$	160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-42.7859 q^{3} -125.000 q^{5} -911.015 q^{7} -356.364 q^{9} +O(q^{10})$	$q-42.7859 q^{3} -125.000 q^{5} -911.015 q^{7} -356.364 q^{9} +1460.11 q^{11} -8084.77 q^{13} +5348.24 q^{15} -29926.5 q^{17} -50707.3 q^{19} +38978.6 q^{21} +41984.8 q^{23} +15625.0 q^{25} +108820. q^{27} -226949. q^{29} -127965. q^{31} -62472.0 q^{33} +113877. q^{35} +230539. q^{37} +345914. q^{39} -617391. q^{41} +430783. q^{43} +44545.5 q^{45} -630747. q^{47} +6405.88 q^{49} +1.28043e6 q^{51} +634301. q^{53} -182513. q^{55} +2.16956e6 q^{57} +1.08074e6 q^{59} +1.48562e6 q^{61} +324653. q^{63} +1.01060e6 q^{65} +3.54147e6 q^{67} -1.79636e6 q^{69} +2.18324e6 q^{71} -927343. q^{73} -668530. q^{75} -1.33018e6 q^{77} +2.59115e6 q^{79} -3.87661e6 q^{81} +4.46651e6 q^{83} +3.74082e6 q^{85} +9.71022e6 q^{87} -4.65769e6 q^{89} +7.36535e6 q^{91} +5.47511e6 q^{93} +6.33841e6 q^{95} -9.82058e6 q^{97} -520330. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 70 q^{3} - 375 q^{5} + 62 q^{7} + 3403 q^{9}+O(q^{10}) $ 3 * q + 70 * q^3 - 375 * q^5 + 62 * q^7 + 3403 * q^9	$ 3 q + 70 q^{3} - 375 q^{5} + 62 q^{7} + 3403 q^{9} + 1964 q^{11} + 4742 q^{13} - 8750 q^{15} - 31066 q^{17} - 88792 q^{19} + 145500 q^{21} + 40282 q^{23} + 46875 q^{25} + 274132 q^{27} - 31966 q^{29} - 6044 q^{31} + 184344 q^{33} - 7750 q^{35} + 573614 q^{37} + 1381772 q^{39} - 530030 q^{41} + 44982 q^{43} - 425375 q^{45} + 1191086 q^{47} + 337319 q^{49} - 18996 q^{51} - 632754 q^{53} - 245500 q^{55} - 2265392 q^{57} + 2294080 q^{59} + 512222 q^{61} + 7979022 q^{63} - 592750 q^{65} + 5861970 q^{67} - 2044844 q^{69} + 8594932 q^{71} - 6778770 q^{73} + 1093750 q^{75} + 5277176 q^{77} + 5291192 q^{79} + 7956271 q^{81} + 11482686 q^{83} + 3883250 q^{85} + 15732676 q^{87} + 2747614 q^{89} + 22709628 q^{91} + 3088136 q^{93} + 11099000 q^{95} - 6939170 q^{97} + 25059276 q^{99}+O(q^{100}) $ 3 * q + 70 * q^3 - 375 * q^5 + 62 * q^7 + 3403 * q^9 + 1964 * q^11 + 4742 * q^13 - 8750 * q^15 - 31066 * q^17 - 88792 * q^19 + 145500 * q^21 + 40282 * q^23 + 46875 * q^25 + 274132 * q^27 - 31966 * q^29 - 6044 * q^31 + 184344 * q^33 - 7750 * q^35 + 573614 * q^37 + 1381772 * q^39 - 530030 * q^41 + 44982 * q^43 - 425375 * q^45 + 1191086 * q^47 + 337319 * q^49 - 18996 * q^51 - 632754 * q^53 - 245500 * q^55 - 2265392 * q^57 + 2294080 * q^59 + 512222 * q^61 + 7979022 * q^63 - 592750 * q^65 + 5861970 * q^67 - 2044844 * q^69 + 8594932 * q^71 - 6778770 * q^73 + 1093750 * q^75 + 5277176 * q^77 + 5291192 * q^79 + 7956271 * q^81 + 11482686 * q^83 + 3883250 * q^85 + 15732676 * q^87 + 2747614 * q^89 + 22709628 * q^91 + 3088136 * q^93 + 11099000 * q^95 - 6939170 * q^97 + 25059276 * 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
$3$	−42.7859	−0.914906	−0.457453	−	0.889234i	$-0.651238\pi$
$3$	−42.7859	−0.914906	−0.457453	+	0.889234i	$0.651238\pi$
$4$	0	0
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	−911.015	−1.00388	−0.501941	−	0.864902i	$-0.667380\pi$
$7$	−911.015	−1.00388	−0.501941	+	0.864902i	$0.667380\pi$
$8$	0	0
$9$	−356.364	−0.162947
$10$	0	0
$11$	1460.11	0.330758	0.165379	−	0.986230i	$-0.447115\pi$
$11$	1460.11	0.330758	0.165379	+	0.986230i	$0.447115\pi$
$12$	0	0
$13$	−8084.77	−1.02062	−0.510312	−	0.859989i	$-0.670470\pi$
$13$	−8084.77	−1.02062	−0.510312	+	0.859989i	$0.670470\pi$
$14$	0	0
$15$	5348.24	0.409159
$16$	0	0
$17$	−29926.5	−1.47736	−0.738678	−	0.674058i	$-0.764550\pi$
$17$	−29926.5	−1.47736	−0.738678	+	0.674058i	$0.764550\pi$
$18$	0	0
$19$	−50707.3	−1.69603	−0.848013	−	0.529975i	$-0.822201\pi$
$19$	−50707.3	−1.69603	−0.848013	+	0.529975i	$0.822201\pi$
$20$	0	0
$21$	38978.6	0.918458
$22$	0	0
$23$	41984.8	0.719523	0.359762	−	0.933044i	$-0.382858\pi$
$23$	41984.8	0.719523	0.359762	+	0.933044i	$0.382858\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	108820.	1.06399
$28$	0	0
$29$	−226949.	−1.72797	−0.863983	−	0.503521i	$-0.832038\pi$
$29$	−226949.	−1.72797	−0.863983	+	0.503521i	$0.832038\pi$
$30$	0	0
$31$	−127965.	−0.771482	−0.385741	−	0.922607i	$-0.626054\pi$
$31$	−127965.	−0.771482	−0.385741	+	0.922607i	$0.626054\pi$
$32$	0	0
$33$	−62472.0	−0.302612
$34$	0	0
$35$	113877.	0.448950
$36$	0	0
$37$	230539.	0.748236	0.374118	−	0.927381i	$-0.377946\pi$
$37$	230539.	0.748236	0.374118	+	0.927381i	$0.377946\pi$
$38$	0	0
$39$	345914.	0.933776
$40$	0	0
$41$	−617391.	−1.39900	−0.699499	−	0.714634i	$-0.746593\pi$
$41$	−617391.	−1.39900	−0.699499	+	0.714634i	$0.746593\pi$
$42$	0	0
$43$	430783.	0.826264	0.413132	−	0.910671i	$-0.364435\pi$
$43$	430783.	0.826264	0.413132	+	0.910671i	$0.364435\pi$
$44$	0	0
$45$	44545.5	0.0728719
$46$	0	0
$47$	−630747.	−0.886161	−0.443080	−	0.896482i	$-0.646114\pi$
$47$	−630747.	−0.886161	−0.443080	+	0.896482i	$0.646114\pi$
$48$	0	0
$49$	6405.88	0.00777844
$50$	0	0
$51$	1.28043e6	1.35164
$52$	0	0
$53$	634301.	0.585234	0.292617	−	0.956230i	$-0.405474\pi$
$53$	634301.	0.585234	0.292617	+	0.956230i	$0.405474\pi$
$54$	0	0
$55$	−182513.	−0.147919
$56$	0	0
$57$	2.16956e6	1.55170
$58$	0	0
$59$	1.08074e6	0.685077	0.342539	−	0.939504i	$-0.388713\pi$
$59$	1.08074e6	0.685077	0.342539	+	0.939504i	$0.388713\pi$
$60$	0	0
$61$	1.48562e6	0.838020	0.419010	−	0.907982i	$-0.362377\pi$
$61$	1.48562e6	0.838020	0.419010	+	0.907982i	$0.362377\pi$
$62$	0	0
$63$	324653.	0.163579
$64$	0	0
$65$	1.01060e6	0.456437
$66$	0	0
$67$	3.54147e6	1.43854	0.719269	−	0.694731i	$-0.244477\pi$
$67$	3.54147e6	1.43854	0.719269	+	0.694731i	$0.244477\pi$
$68$	0	0
$69$	−1.79636e6	−0.658296
$70$	0	0
$71$	2.18324e6	0.723932	0.361966	−	0.932191i	$-0.382106\pi$
$71$	2.18324e6	0.723932	0.361966	+	0.932191i	$0.382106\pi$
$72$	0	0
$73$	−927343.	−0.279004	−0.139502	−	0.990222i	$-0.544550\pi$
$73$	−927343.	−0.279004	−0.139502	+	0.990222i	$0.544550\pi$
$74$	0	0
$75$	−668530.	−0.182981
$76$	0	0
$77$	−1.33018e6	−0.332042
$78$	0	0
$79$	2.59115e6	0.591287	0.295643	−	0.955298i	$-0.404466\pi$
$79$	2.59115e6	0.591287	0.295643	+	0.955298i	$0.404466\pi$
$80$	0	0
$81$	−3.87661e6	−0.810502
$82$	0	0
$83$	4.46651e6	0.857423	0.428712	−	0.903441i	$-0.358968\pi$
$83$	4.46651e6	0.857423	0.428712	+	0.903441i	$0.358968\pi$
$84$	0	0
$85$	3.74082e6	0.660694
$86$	0	0
$87$	9.71022e6	1.58093
$88$	0	0
$89$	−4.65769e6	−0.700335	−0.350167	−	0.936687i	$-0.613875\pi$
$89$	−4.65769e6	−0.700335	−0.350167	+	0.936687i	$0.613875\pi$
$90$	0	0
$91$	7.36535e6	1.02459
$92$	0	0
$93$	5.47511e6	0.705834
$94$	0	0
$95$	6.33841e6	0.758486
$96$	0	0
$97$	−9.82058e6	−1.09254	−0.546268	−	0.837610i	$-0.683952\pi$
$97$	−9.82058e6	−1.09254	−0.546268	+	0.837610i	$0.683952\pi$
$98$	0	0
$99$	−520330.	−0.0538959
$100$	0	0
$101$	−1.27069e7	−1.22720	−0.613600	−	0.789617i	$-0.710279\pi$
$101$	−1.27069e7	−1.22720	−0.613600	+	0.789617i	$0.710279\pi$
$102$	0	0
$103$	3.53275e6	0.318554	0.159277	−	0.987234i	$-0.449084\pi$
$103$	3.53275e6	0.318554	0.159277	+	0.987234i	$0.449084\pi$
$104$	0	0
$105$	−4.87233e6	−0.410747
$106$	0	0
$107$	−8.32890e6	−0.657271	−0.328635	−	0.944457i	$-0.606589\pi$
$107$	−8.32890e6	−0.657271	−0.328635	+	0.944457i	$0.606589\pi$
$108$	0	0
$109$	1.94826e7	1.44097	0.720483	−	0.693473i	$-0.243920\pi$
$109$	1.94826e7	1.44097	0.720483	+	0.693473i	$0.243920\pi$
$110$	0	0
$111$	−9.86383e6	−0.684566
$112$	0	0
$113$	1.61364e7	1.05204	0.526020	−	0.850472i	$-0.323684\pi$
$113$	1.61364e7	1.05204	0.526020	+	0.850472i	$0.323684\pi$
$114$	0	0
$115$	−5.24810e6	−0.321781
$116$	0	0
$117$	2.88112e6	0.166307
$118$	0	0
$119$	2.72635e7	1.48309
$120$	0	0
$121$	−1.73553e7	−0.890599
$122$	0	0
$123$	2.64156e7	1.27995
$124$	0	0
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−3.43724e6	−0.148901	−0.0744504	−	0.997225i	$-0.523720\pi$
$127$	−3.43724e6	−0.148901	−0.0744504	+	0.997225i	$0.523720\pi$
$128$	0	0
$129$	−1.84314e7	−0.755954
$130$	0	0
$131$	2.56923e6	0.0998514	0.0499257	−	0.998753i	$-0.484102\pi$
$131$	2.56923e6	0.0998514	0.0499257	+	0.998753i	$0.484102\pi$
$132$	0	0
$133$	4.61951e7	1.70261
$134$	0	0
$135$	−1.36025e7	−0.475829
$136$	0	0
$137$	−4.43854e7	−1.47475	−0.737375	−	0.675484i	$-0.763935\pi$
$137$	−4.43854e7	−1.47475	−0.737375	+	0.675484i	$0.763935\pi$
$138$	0	0
$139$	−5.61199e7	−1.77241	−0.886207	−	0.463290i	$-0.846669\pi$
$139$	−5.61199e7	−1.77241	−0.886207	+	0.463290i	$0.846669\pi$
$140$	0	0
$141$	2.69871e7	0.810754
$142$	0	0
$143$	−1.18046e7	−0.337580
$144$	0	0
$145$	2.83686e7	0.772770
$146$	0	0
$147$	−274082.	−0.00711655
$148$	0	0
$149$	2.83549e7	0.702224	0.351112	−	0.936333i	$-0.385804\pi$
$149$	2.83549e7	0.702224	0.351112	+	0.936333i	$0.385804\pi$
$150$	0	0
$151$	1.33677e7	0.315964	0.157982	−	0.987442i	$-0.449501\pi$
$151$	1.33677e7	0.315964	0.157982	+	0.987442i	$0.449501\pi$
$152$	0	0
$153$	1.06647e7	0.240730
$154$	0	0
$155$	1.59956e7	0.345017
$156$	0	0
$157$	−4.12475e6	−0.0850646	−0.0425323	−	0.999095i	$-0.513543\pi$
$157$	−4.12475e6	−0.0850646	−0.0425323	+	0.999095i	$0.513543\pi$
$158$	0	0
$159$	−2.71392e7	−0.535435
$160$	0	0
$161$	−3.82488e7	−0.722316
$162$	0	0
$163$	−8.68634e7	−1.57102	−0.785508	−	0.618852i	$-0.787598\pi$
$163$	−8.68634e7	−1.57102	−0.785508	+	0.618852i	$0.787598\pi$
$164$	0	0
$165$	7.80900e6	0.135332
$166$	0	0
$167$	−1.15786e7	−0.192374	−0.0961872	−	0.995363i	$-0.530665\pi$
$167$	−1.15786e7	−0.192374	−0.0961872	+	0.995363i	$0.530665\pi$
$168$	0	0
$169$	2.61500e6	0.0416742
$170$	0	0
$171$	1.80702e7	0.276362
$172$	0	0
$173$	1.02454e8	1.50442	0.752209	−	0.658924i	$-0.228988\pi$
$173$	1.02454e8	1.50442	0.752209	+	0.658924i	$0.228988\pi$
$174$	0	0
$175$	−1.42346e7	−0.200776
$176$	0	0
$177$	−4.62405e7	−0.626782
$178$	0	0
$179$	−6.01787e7	−0.784255	−0.392127	−	0.919911i	$-0.628261\pi$
$179$	−6.01787e7	−0.784255	−0.392127	+	0.919911i	$0.628261\pi$
$180$	0	0
$181$	−8.91956e7	−1.11807	−0.559034	−	0.829145i	$-0.688828\pi$
$181$	−8.91956e7	−1.11807	−0.559034	+	0.829145i	$0.688828\pi$
$182$	0	0
$183$	−6.35638e7	−0.766710
$184$	0	0
$185$	−2.88174e7	−0.334621
$186$	0	0
$187$	−4.36959e7	−0.488647
$188$	0	0
$189$	−9.91369e7	−1.06812
$190$	0	0
$191$	−1.79707e8	−1.86616	−0.933079	−	0.359671i	$-0.882889\pi$
$191$	−1.79707e8	−1.86616	−0.933079	+	0.359671i	$0.882889\pi$
$192$	0	0
$193$	−1.50864e8	−1.51055	−0.755277	−	0.655406i	$-0.772498\pi$
$193$	−1.50864e8	−1.51055	−0.755277	+	0.655406i	$0.772498\pi$
$194$	0	0
$195$	−4.32393e7	−0.417597
$196$	0	0
$197$	1.51354e8	1.41046	0.705231	−	0.708977i	$-0.250843\pi$
$197$	1.51354e8	1.41046	0.705231	+	0.708977i	$0.250843\pi$
$198$	0	0
$199$	−1.33193e8	−1.19811	−0.599053	−	0.800710i	$-0.704456\pi$
$199$	−1.33193e8	−1.19811	−0.599053	+	0.800710i	$0.704456\pi$
$200$	0	0
$201$	−1.51525e8	−1.31613
$202$	0	0
$203$	2.06754e8	1.73467
$204$	0	0
$205$	7.71739e7	0.625650
$206$	0	0
$207$	−1.49619e7	−0.117244
$208$	0	0
$209$	−7.40380e7	−0.560974
$210$	0	0
$211$	6.82471e7	0.500144	0.250072	−	0.968227i	$-0.419546\pi$
$211$	6.82471e7	0.500144	0.250072	+	0.968227i	$0.419546\pi$
$212$	0	0
$213$	−9.34121e7	−0.662330
$214$	0	0
$215$	−5.38478e7	−0.369516
$216$	0	0
$217$	1.16578e8	0.774477
$218$	0	0
$219$	3.96772e7	0.255262
$220$	0	0
$221$	2.41949e8	1.50783
$222$	0	0
$223$	1.74199e8	1.05191	0.525954	−	0.850513i	$-0.323708\pi$
$223$	1.74199e8	1.05191	0.525954	+	0.850513i	$0.323708\pi$
$224$	0	0
$225$	−5.56819e6	−0.0325893
$226$	0	0
$227$	−2.66554e8	−1.51250	−0.756250	−	0.654283i	$-0.772970\pi$
$227$	−2.66554e8	−1.51250	−0.756250	+	0.654283i	$0.772970\pi$
$228$	0	0
$229$	1.75772e8	0.967220	0.483610	−	0.875283i	$-0.339325\pi$
$229$	1.75772e8	0.967220	0.483610	+	0.875283i	$0.339325\pi$
$230$	0	0
$231$	5.69130e7	0.303787
$232$	0	0
$233$	1.55358e8	0.804613	0.402307	−	0.915505i	$-0.368209\pi$
$233$	1.55358e8	0.804613	0.402307	+	0.915505i	$0.368209\pi$
$234$	0	0
$235$	7.88433e7	0.396303
$236$	0	0
$237$	−1.10865e8	−0.540972
$238$	0	0
$239$	−1.54339e8	−0.731280	−0.365640	−	0.930756i	$-0.619150\pi$
$239$	−1.54339e8	−0.731280	−0.365640	+	0.930756i	$0.619150\pi$
$240$	0	0
$241$	1.84278e8	0.848034	0.424017	−	0.905654i	$-0.360620\pi$
$241$	1.84278e8	0.848034	0.424017	+	0.905654i	$0.360620\pi$
$242$	0	0
$243$	−7.21256e7	−0.322454
$244$	0	0
$245$	−800735.	−0.00347863
$246$	0	0
$247$	4.09957e8	1.73101
$248$	0	0
$249$	−1.91104e8	−0.784462
$250$	0	0
$251$	2.68145e8	1.07032	0.535158	−	0.844752i	$-0.320252\pi$
$251$	2.68145e8	1.07032	0.535158	+	0.844752i	$0.320252\pi$
$252$	0	0
$253$	6.13023e7	0.237988
$254$	0	0
$255$	−1.60054e8	−0.604473
$256$	0	0
$257$	5.83031e6	0.0214253	0.0107126	−	0.999943i	$-0.496590\pi$
$257$	5.83031e6	0.0214253	0.0107126	+	0.999943i	$0.496590\pi$
$258$	0	0
$259$	−2.10025e8	−0.751140
$260$	0	0
$261$	8.08764e7	0.281566
$262$	0	0
$263$	5.67940e8	1.92512	0.962559	−	0.271073i	$-0.0873784\pi$
$263$	5.67940e8	1.92512	0.962559	+	0.271073i	$0.0873784\pi$
$264$	0	0
$265$	−7.92876e7	−0.261725
$266$	0	0
$267$	1.99284e8	0.640741
$268$	0	0
$269$	7.05897e7	0.221110	0.110555	−	0.993870i	$-0.464737\pi$
$269$	7.05897e7	0.221110	0.110555	+	0.993870i	$0.464737\pi$
$270$	0	0
$271$	−2.19853e7	−0.0671027	−0.0335514	−	0.999437i	$-0.510682\pi$
$271$	−2.19853e7	−0.0671027	−0.0335514	+	0.999437i	$0.510682\pi$
$272$	0	0
$273$	−3.15133e8	−0.937400
$274$	0	0
$275$	2.28142e7	0.0661516
$276$	0	0
$277$	−1.06671e8	−0.301556	−0.150778	−	0.988568i	$-0.548178\pi$
$277$	−1.06671e8	−0.301556	−0.150778	+	0.988568i	$0.548178\pi$
$278$	0	0
$279$	4.56022e7	0.125710
$280$	0	0
$281$	3.15821e8	0.849119	0.424559	−	0.905400i	$-0.360429\pi$
$281$	3.15821e8	0.849119	0.424559	+	0.905400i	$0.360429\pi$
$282$	0	0
$283$	−3.28884e8	−0.862562	−0.431281	−	0.902218i	$-0.641938\pi$
$283$	−3.28884e8	−0.862562	−0.431281	+	0.902218i	$0.641938\pi$
$284$	0	0
$285$	−2.71195e8	−0.693944
$286$	0	0
$287$	5.62453e8	1.40443
$288$	0	0
$289$	4.85259e8	1.18258
$290$	0	0
$291$	4.20183e8	0.999569
$292$	0	0
$293$	−5.93422e8	−1.37825	−0.689123	−	0.724645i	$-0.742004\pi$
$293$	−5.93422e8	−1.37825	−0.689123	+	0.724645i	$0.742004\pi$
$294$	0	0
$295$	−1.35093e8	−0.306376
$296$	0	0
$297$	1.58889e8	0.351922
$298$	0	0
$299$	−3.39438e8	−0.734363
$300$	0	0
$301$	−3.92450e8	−0.829471
$302$	0	0
$303$	5.43677e8	1.12277
$304$	0	0
$305$	−1.85703e8	−0.374774
$306$	0	0
$307$	2.09760e8	0.413750	0.206875	−	0.978367i	$-0.433671\pi$
$307$	2.09760e8	0.413750	0.206875	+	0.978367i	$0.433671\pi$
$308$	0	0
$309$	−1.51152e8	−0.291447
$310$	0	0
$311$	6.95917e8	1.31189	0.655943	−	0.754810i	$-0.272271\pi$
$311$	6.95917e8	1.31189	0.655943	+	0.754810i	$0.272271\pi$
$312$	0	0
$313$	−4.63876e7	−0.0855060	−0.0427530	−	0.999086i	$-0.513613\pi$
$313$	−4.63876e7	−0.0855060	−0.0427530	+	0.999086i	$0.513613\pi$
$314$	0	0
$315$	−4.05816e7	−0.0731548
$316$	0	0
$317$	2.05392e8	0.362139	0.181070	−	0.983470i	$-0.442044\pi$
$317$	2.05392e8	0.362139	0.181070	+	0.983470i	$0.442044\pi$
$318$	0	0
$319$	−3.31370e8	−0.571538
$320$	0	0
$321$	3.56360e8	0.601341
$322$	0	0
$323$	1.51749e9	2.50564
$324$	0	0
$325$	−1.26325e8	−0.204125
$326$	0	0
$327$	−8.33580e8	−1.31835
$328$	0	0
$329$	5.74620e8	0.889601
$330$	0	0
$331$	9.71962e8	1.47316	0.736582	−	0.676348i	$-0.236438\pi$
$331$	9.71962e8	1.47316	0.736582	+	0.676348i	$0.236438\pi$
$332$	0	0
$333$	−8.21559e7	−0.121922
$334$	0	0
$335$	−4.42684e8	−0.643334
$336$	0	0
$337$	−5.14775e8	−0.732677	−0.366339	−	0.930482i	$-0.619389\pi$
$337$	−5.14775e8	−0.732677	−0.366339	+	0.930482i	$0.619389\pi$
$338$	0	0
$339$	−6.90412e8	−0.962519
$340$	0	0
$341$	−1.86843e8	−0.255174
$342$	0	0
$343$	7.44424e8	0.996073
$344$	0	0
$345$	2.24545e8	0.294399
$346$	0	0
$347$	−6.55551e8	−0.842274	−0.421137	−	0.906997i	$-0.638369\pi$
$347$	−6.55551e8	−0.842274	−0.421137	+	0.906997i	$0.638369\pi$
$348$	0	0
$349$	−6.34432e8	−0.798906	−0.399453	−	0.916754i	$-0.630800\pi$
$349$	−6.34432e8	−0.798906	−0.399453	+	0.916754i	$0.630800\pi$
$350$	0	0
$351$	−8.79786e8	−1.08593
$352$	0	0
$353$	−7.21331e8	−0.872816	−0.436408	−	0.899749i	$-0.643750\pi$
$353$	−7.21331e8	−0.872816	−0.436408	+	0.899749i	$0.643750\pi$
$354$	0	0
$355$	−2.72905e8	−0.323752
$356$	0	0
$357$	−1.16650e9	−1.35689
$358$	0	0
$359$	−5.74090e8	−0.654862	−0.327431	−	0.944875i	$-0.606183\pi$
$359$	−5.74090e8	−0.654862	−0.327431	+	0.944875i	$0.606183\pi$
$360$	0	0
$361$	1.67735e9	1.87650
$362$	0	0
$363$	7.42561e8	0.814815
$364$	0	0
$365$	1.15918e8	0.124774
$366$	0	0
$367$	1.31197e9	1.38545	0.692727	−	0.721200i	$-0.256409\pi$
$367$	1.31197e9	1.38545	0.692727	+	0.721200i	$0.256409\pi$
$368$	0	0
$369$	2.20016e8	0.227962
$370$	0	0
$371$	−5.77858e8	−0.587506
$372$	0	0
$373$	1.03384e9	1.03150	0.515752	−	0.856738i	$-0.327513\pi$
$373$	1.03384e9	1.03150	0.515752	+	0.856738i	$0.327513\pi$
$374$	0	0
$375$	8.35663e7	0.0818317
$376$	0	0
$377$	1.83483e9	1.76360
$378$	0	0
$379$	−1.88676e9	−1.78025	−0.890123	−	0.455719i	$-0.849382\pi$
$379$	−1.88676e9	−1.78025	−0.890123	+	0.455719i	$0.849382\pi$
$380$	0	0
$381$	1.47065e8	0.136230
$382$	0	0
$383$	−1.12630e9	−1.02438	−0.512188	−	0.858873i	$-0.671165\pi$
$383$	−1.12630e9	−1.02438	−0.512188	+	0.858873i	$0.671165\pi$
$384$	0	0
$385$	1.66272e8	0.148494
$386$	0	0
$387$	−1.53516e8	−0.134637
$388$	0	0
$389$	−3.29029e8	−0.283407	−0.141704	−	0.989909i	$-0.545258\pi$
$389$	−3.29029e8	−0.283407	−0.141704	+	0.989909i	$0.545258\pi$
$390$	0	0
$391$	−1.25646e9	−1.06299
$392$	0	0
$393$	−1.09927e8	−0.0913547
$394$	0	0
$395$	−3.23894e8	−0.264431
$396$	0	0
$397$	1.22197e9	0.980149	0.490075	−	0.871680i	$-0.336970\pi$
$397$	1.22197e9	0.980149	0.490075	+	0.871680i	$0.336970\pi$
$398$	0	0
$399$	−1.97650e9	−1.55773
$400$	0	0
$401$	−1.14033e9	−0.883135	−0.441567	−	0.897228i	$-0.645577\pi$
$401$	−1.14033e9	−0.883135	−0.441567	+	0.897228i	$0.645577\pi$
$402$	0	0
$403$	1.03457e9	0.787394
$404$	0	0
$405$	4.84576e8	0.362467
$406$	0	0
$407$	3.36612e8	0.247485
$408$	0	0
$409$	−9.14536e8	−0.660951	−0.330475	−	0.943815i	$-0.607209\pi$
$409$	−9.14536e8	−0.660951	−0.330475	+	0.943815i	$0.607209\pi$
$410$	0	0
$411$	1.89907e9	1.34926
$412$	0	0
$413$	−9.84572e8	−0.687737
$414$	0	0
$415$	−5.58314e8	−0.383451
$416$	0	0
$417$	2.40114e9	1.62159
$418$	0	0
$419$	−1.58643e9	−1.05359	−0.526796	−	0.849992i	$-0.676607\pi$
$419$	−1.58643e9	−1.05359	−0.526796	+	0.849992i	$0.676607\pi$
$420$	0	0
$421$	−2.43252e9	−1.58880	−0.794401	−	0.607393i	$-0.792215\pi$
$421$	−2.43252e9	−1.58880	−0.794401	+	0.607393i	$0.792215\pi$
$422$	0	0
$423$	2.24775e8	0.144397
$424$	0	0
$425$	−4.67602e8	−0.295471
$426$	0	0
$427$	−1.35343e9	−0.841273
$428$	0	0
$429$	5.05072e8	0.308854
$430$	0	0
$431$	2.19588e9	1.32111	0.660553	−	0.750779i	$-0.270322\pi$
$431$	2.19588e9	1.32111	0.660553	+	0.750779i	$0.270322\pi$
$432$	0	0
$433$	2.47499e9	1.46510	0.732549	−	0.680714i	$-0.238331\pi$
$433$	2.47499e9	1.46510	0.732549	+	0.680714i	$0.238331\pi$
$434$	0	0
$435$	−1.21378e9	−0.707012
$436$	0	0
$437$	−2.12893e9	−1.22033
$438$	0	0
$439$	−3.24699e9	−1.83170	−0.915851	−	0.401519i	$-0.868482\pi$
$439$	−3.24699e9	−1.83170	−0.915851	+	0.401519i	$0.868482\pi$
$440$	0	0
$441$	−2.28283e6	−0.00126747
$442$	0	0
$443$	9.66174e7	0.0528010	0.0264005	−	0.999651i	$-0.491595\pi$
$443$	9.66174e7	0.0528010	0.0264005	+	0.999651i	$0.491595\pi$
$444$	0	0
$445$	5.82211e8	0.313199
$446$	0	0
$447$	−1.21319e9	−0.642470
$448$	0	0
$449$	−1.20956e9	−0.630614	−0.315307	−	0.948990i	$-0.602107\pi$
$449$	−1.20956e9	−0.630614	−0.315307	+	0.948990i	$0.602107\pi$
$450$	0	0
$451$	−9.01457e8	−0.462729
$452$	0	0
$453$	−5.71949e8	−0.289077
$454$	0	0
$455$	−9.20669e8	−0.458209
$456$	0	0
$457$	−3.55263e9	−1.74118	−0.870590	−	0.492010i	$-0.836262\pi$
$457$	−3.55263e9	−1.74118	−0.870590	+	0.492010i	$0.836262\pi$
$458$	0	0
$459$	−3.25661e9	−1.57189
$460$	0	0
$461$	−6.27051e8	−0.298091	−0.149046	−	0.988830i	$-0.547620\pi$
$461$	−6.27051e8	−0.298091	−0.149046	+	0.988830i	$0.547620\pi$
$462$	0	0
$463$	2.34842e9	1.09962	0.549809	−	0.835290i	$-0.314700\pi$
$463$	2.34842e9	1.09962	0.549809	+	0.835290i	$0.314700\pi$
$464$	0	0
$465$	−6.84389e8	−0.315658
$466$	0	0
$467$	3.37582e9	1.53381	0.766903	−	0.641763i	$-0.221797\pi$
$467$	3.37582e9	1.53381	0.766903	+	0.641763i	$0.221797\pi$
$468$	0	0
$469$	−3.22633e9	−1.44412
$470$	0	0
$471$	1.76481e8	0.0778261
$472$	0	0
$473$	6.28989e8	0.273293
$474$	0	0
$475$	−7.92301e8	−0.339205
$476$	0	0
$477$	−2.26042e8	−0.0953619
$478$	0	0
$479$	4.10635e9	1.70719	0.853594	−	0.520939i	$-0.174418\pi$
$479$	4.10635e9	1.70719	0.853594	+	0.520939i	$0.174418\pi$
$480$	0	0
$481$	−1.86386e9	−0.763668
$482$	0	0
$483$	1.63651e9	0.660852
$484$	0	0
$485$	1.22757e9	0.488597
$486$	0	0
$487$	−4.00248e9	−1.57028	−0.785141	−	0.619318i	$-0.787409\pi$
$487$	−4.00248e9	−1.57028	−0.785141	+	0.619318i	$0.787409\pi$
$488$	0	0
$489$	3.71653e9	1.43733
$490$	0	0
$491$	−6.04238e8	−0.230368	−0.115184	−	0.993344i	$-0.536746\pi$
$491$	−6.04238e8	−0.230368	−0.115184	+	0.993344i	$0.536746\pi$
$492$	0	0
$493$	6.79180e9	2.55282
$494$	0	0
$495$	6.50412e7	0.0241030
$496$	0	0
$497$	−1.98897e9	−0.726742
$498$	0	0
$499$	−2.06678e9	−0.744632	−0.372316	−	0.928106i	$-0.621436\pi$
$499$	−2.06678e9	−0.744632	−0.372316	+	0.928106i	$0.621436\pi$
$500$	0	0
$501$	4.95400e8	0.176005
$502$	0	0
$503$	2.70359e9	0.947223	0.473612	−	0.880734i	$-0.342950\pi$
$503$	2.70359e9	0.947223	0.473612	+	0.880734i	$0.342950\pi$
$504$	0	0
$505$	1.58836e9	0.548820
$506$	0	0
$507$	−1.11885e8	−0.0381280
$508$	0	0
$509$	4.64705e9	1.56194	0.780971	−	0.624567i	$-0.214725\pi$
$509$	4.64705e9	1.56194	0.780971	+	0.624567i	$0.214725\pi$
$510$	0	0
$511$	8.44824e8	0.280087
$512$	0	0
$513$	−5.51797e9	−1.80455
$514$	0	0
$515$	−4.41594e8	−0.142462
$516$	0	0
$517$	−9.20958e8	−0.293105
$518$	0	0
$519$	−4.38360e9	−1.37640
$520$	0	0
$521$	2.87744e9	0.891403	0.445702	−	0.895182i	$-0.352954\pi$
$521$	2.87744e9	0.891403	0.445702	+	0.895182i	$0.352954\pi$
$522$	0	0
$523$	2.03697e9	0.622629	0.311315	−	0.950307i	$-0.399231\pi$
$523$	2.03697e9	0.622629	0.311315	+	0.950307i	$0.399231\pi$
$524$	0	0
$525$	6.09041e8	0.183692
$526$	0	0
$527$	3.82956e9	1.13975
$528$	0	0
$529$	−1.64210e9	−0.482286
$530$	0	0
$531$	−3.85137e8	−0.111631
$532$	0	0
$533$	4.99146e9	1.42785
$534$	0	0
$535$	1.04111e9	0.293940
$536$	0	0
$537$	2.57480e9	0.717520
$538$	0	0
$539$	9.35327e6	0.00257278
$540$	0	0
$541$	−3.19398e9	−0.867246	−0.433623	−	0.901094i	$-0.642765\pi$
$541$	−3.19398e9	−0.867246	−0.433623	+	0.901094i	$0.642765\pi$
$542$	0	0
$543$	3.81632e9	1.02293
$544$	0	0
$545$	−2.43532e9	−0.644420
$546$	0	0
$547$	1.50396e9	0.392900	0.196450	−	0.980514i	$-0.437059\pi$
$547$	1.50396e9	0.392900	0.196450	+	0.980514i	$0.437059\pi$
$548$	0	0
$549$	−5.29423e8	−0.136552
$550$	0	0
$551$	1.15080e10	2.93067
$552$	0	0
$553$	−2.36058e9	−0.593582
$554$	0	0
$555$	1.23298e9	0.306147
$556$	0	0
$557$	−1.01295e9	−0.248367	−0.124184	−	0.992259i	$-0.539631\pi$
$557$	−1.01295e9	−0.248367	−0.124184	+	0.992259i	$0.539631\pi$
$558$	0	0
$559$	−3.48278e9	−0.843305
$560$	0	0
$561$	1.86957e9	0.447066
$562$	0	0
$563$	3.38007e9	0.798263	0.399132	−	0.916894i	$-0.369312\pi$
$563$	3.38007e9	0.798263	0.399132	+	0.916894i	$0.369312\pi$
$564$	0	0
$565$	−2.01705e9	−0.470487
$566$	0	0
$567$	3.53165e9	0.813648
$568$	0	0
$569$	1.54684e9	0.352009	0.176004	−	0.984389i	$-0.443683\pi$
$569$	1.54684e9	0.352009	0.176004	+	0.984389i	$0.443683\pi$
$570$	0	0
$571$	−9.51174e8	−0.213813	−0.106906	−	0.994269i	$-0.534095\pi$
$571$	−9.51174e8	−0.213813	−0.106906	+	0.994269i	$0.534095\pi$
$572$	0	0
$573$	7.68894e9	1.70736
$574$	0	0
$575$	6.56013e8	0.143905
$576$	0	0
$577$	−4.69613e9	−1.01771	−0.508856	−	0.860852i	$-0.669931\pi$
$577$	−4.69613e9	−1.01771	−0.508856	+	0.860852i	$0.669931\pi$
$578$	0	0
$579$	6.45487e9	1.38201
$580$	0	0
$581$	−4.06906e9	−0.860751
$582$	0	0
$583$	9.26147e8	0.193571
$584$	0	0
$585$	−3.60140e8	−0.0743748
$586$	0	0
$587$	5.05479e9	1.03150	0.515750	−	0.856739i	$-0.327513\pi$
$587$	5.05479e9	1.03150	0.515750	+	0.856739i	$0.327513\pi$
$588$	0	0
$589$	6.48876e9	1.30845
$590$	0	0
$591$	−6.47582e9	−1.29044
$592$	0	0
$593$	−5.92524e9	−1.16685	−0.583424	−	0.812168i	$-0.698287\pi$
$593$	−5.92524e9	−1.16685	−0.583424	+	0.812168i	$0.698287\pi$
$594$	0	0
$595$	−3.40794e9	−0.663259
$596$	0	0
$597$	5.69878e9	1.09615
$598$	0	0
$599$	−7.45956e9	−1.41814	−0.709071	−	0.705137i	$-0.750885\pi$
$599$	−7.45956e9	−1.41814	−0.709071	+	0.705137i	$0.750885\pi$
$600$	0	0
$601$	−4.22885e9	−0.794624	−0.397312	−	0.917684i	$-0.630057\pi$
$601$	−4.22885e9	−0.794624	−0.397312	+	0.917684i	$0.630057\pi$
$602$	0	0
$603$	−1.26205e9	−0.234405
$604$	0	0
$605$	2.16941e9	0.398288
$606$	0	0
$607$	−2.56303e9	−0.465151	−0.232576	−	0.972578i	$-0.574715\pi$
$607$	−2.56303e9	−0.465151	−0.232576	+	0.972578i	$0.574715\pi$
$608$	0	0
$609$	−8.84616e9	−1.58706
$610$	0	0
$611$	5.09944e9	0.904437
$612$	0	0
$613$	−1.92679e9	−0.337848	−0.168924	−	0.985629i	$-0.554029\pi$
$613$	−1.92679e9	−0.337848	−0.168924	+	0.985629i	$0.554029\pi$
$614$	0	0
$615$	−3.30196e9	−0.572412
$616$	0	0
$617$	2.87197e9	0.492246	0.246123	−	0.969239i	$-0.420843\pi$
$617$	2.87197e9	0.492246	0.246123	+	0.969239i	$0.420843\pi$
$618$	0	0
$619$	9.45544e9	1.60238	0.801188	−	0.598413i	$-0.204202\pi$
$619$	9.45544e9	1.60238	0.801188	+	0.598413i	$0.204202\pi$
$620$	0	0
$621$	4.56880e9	0.765563
$622$	0	0
$623$	4.24323e9	0.703053
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	3.16779e9	0.513239
$628$	0	0
$629$	−6.89924e9	−1.10541
$630$	0	0
$631$	6.42065e9	1.01736	0.508682	−	0.860955i	$-0.330133\pi$
$631$	6.42065e9	1.01736	0.508682	+	0.860955i	$0.330133\pi$
$632$	0	0
$633$	−2.92001e9	−0.457585
$634$	0	0
$635$	4.29655e8	0.0665904
$636$	0	0
$637$	−5.17901e7	−0.00793887
$638$	0	0
$639$	−7.78029e8	−0.117962
$640$	0	0
$641$	9.74593e9	1.46157	0.730786	−	0.682606i	$-0.239154\pi$
$641$	9.74593e9	1.46157	0.730786	+	0.682606i	$0.239154\pi$
$642$	0	0
$643$	−3.08535e9	−0.457684	−0.228842	−	0.973464i	$-0.573494\pi$
$643$	−3.08535e9	−0.457684	−0.228842	+	0.973464i	$0.573494\pi$
$644$	0	0
$645$	2.30393e9	0.338073
$646$	0	0
$647$	−5.58156e9	−0.810197	−0.405099	−	0.914273i	$-0.632763\pi$
$647$	−5.58156e9	−0.810197	−0.405099	+	0.914273i	$0.632763\pi$
$648$	0	0
$649$	1.57800e9	0.226595
$650$	0	0
$651$	−4.98791e9	−0.708574
$652$	0	0
$653$	−6.71884e9	−0.944274	−0.472137	−	0.881525i	$-0.656517\pi$
$653$	−6.71884e9	−0.944274	−0.472137	+	0.881525i	$0.656517\pi$
$654$	0	0
$655$	−3.21154e8	−0.0446549
$656$	0	0
$657$	3.30472e8	0.0454627
$658$	0	0
$659$	−7.67533e8	−0.104472	−0.0522358	−	0.998635i	$-0.516635\pi$
$659$	−7.67533e8	−0.104472	−0.0522358	+	0.998635i	$0.516635\pi$
$660$	0	0
$661$	−2.28905e9	−0.308284	−0.154142	−	0.988049i	$-0.549261\pi$
$661$	−2.28905e9	−0.308284	−0.154142	+	0.988049i	$0.549261\pi$
$662$	0	0
$663$	−1.03520e10	−1.37952
$664$	0	0
$665$	−5.77439e9	−0.761430
$666$	0	0
$667$	−9.52841e9	−1.24331
$668$	0	0
$669$	−7.45325e9	−0.962398
$670$	0	0
$671$	2.16917e9	0.277182
$672$	0	0
$673$	−1.29457e10	−1.63710	−0.818548	−	0.574438i	$-0.805221\pi$
$673$	−1.29457e10	−1.63710	−0.818548	+	0.574438i	$0.805221\pi$
$674$	0	0
$675$	1.70032e9	0.212797
$676$	0	0
$677$	−4.32406e9	−0.535589	−0.267795	−	0.963476i	$-0.586295\pi$
$677$	−4.32406e9	−0.535589	−0.267795	+	0.963476i	$0.586295\pi$
$678$	0	0
$679$	8.94670e9	1.09678
$680$	0	0
$681$	1.14048e10	1.38380
$682$	0	0
$683$	4.32619e9	0.519557	0.259778	−	0.965668i	$-0.416351\pi$
$683$	4.32619e9	0.519557	0.259778	+	0.965668i	$0.416351\pi$
$684$	0	0
$685$	5.54818e9	0.659528
$686$	0	0
$687$	−7.52057e9	−0.884916
$688$	0	0
$689$	−5.12818e9	−0.597305
$690$	0	0
$691$	−1.32198e10	−1.52423	−0.762117	−	0.647439i	$-0.775840\pi$
$691$	−1.32198e10	−1.52423	−0.762117	+	0.647439i	$0.775840\pi$
$692$	0	0
$693$	4.74028e8	0.0541051
$694$	0	0
$695$	7.01499e9	0.792647
$696$	0	0
$697$	1.84764e10	2.06682
$698$	0	0
$699$	−6.64713e9	−0.736146
$700$	0	0
$701$	4.90925e9	0.538272	0.269136	−	0.963102i	$-0.413262\pi$
$701$	4.90925e9	0.538272	0.269136	+	0.963102i	$0.413262\pi$
$702$	0	0
$703$	−1.16900e10	−1.26903
$704$	0	0
$705$	−3.37339e9	−0.362580
$706$	0	0
$707$	1.15762e10	1.23196
$708$	0	0
$709$	1.28054e10	1.34937	0.674684	−	0.738107i	$-0.264280\pi$
$709$	1.28054e10	1.34937	0.674684	+	0.738107i	$0.264280\pi$
$710$	0	0
$711$	−9.23393e8	−0.0963481
$712$	0	0
$713$	−5.37259e9	−0.555099
$714$	0	0
$715$	1.47558e9	0.150970
$716$	0	0
$717$	6.60355e9	0.669053
$718$	0	0
$719$	3.29759e9	0.330861	0.165430	−	0.986221i	$-0.447099\pi$
$719$	3.29759e9	0.330861	0.165430	+	0.986221i	$0.447099\pi$
$720$	0	0
$721$	−3.21839e9	−0.319790
$722$	0	0
$723$	−7.88450e9	−0.775872
$724$	0	0
$725$	−3.54608e9	−0.345593
$726$	0	0
$727$	3.57720e9	0.345281	0.172641	−	0.984985i	$-0.444770\pi$
$727$	3.57720e9	0.345281	0.172641	+	0.984985i	$0.444770\pi$
$728$	0	0
$729$	1.15641e10	1.10552
$730$	0	0
$731$	−1.28918e10	−1.22069
$732$	0	0
$733$	1.76656e10	1.65678	0.828391	−	0.560151i	$-0.189257\pi$
$733$	1.76656e10	1.65678	0.828391	+	0.560151i	$0.189257\pi$
$734$	0	0
$735$	3.42602e7	0.00318262
$736$	0	0
$737$	5.17092e9	0.475808
$738$	0	0
$739$	−2.18361e10	−1.99030	−0.995151	−	0.0983618i	$-0.968640\pi$
$739$	−2.18361e10	−1.99030	−0.995151	+	0.0983618i	$0.968640\pi$
$740$	0	0
$741$	−1.75404e10	−1.58371
$742$	0	0
$743$	1.68977e10	1.51135	0.755676	−	0.654945i	$-0.227308\pi$
$743$	1.68977e10	1.51135	0.755676	+	0.654945i	$0.227308\pi$
$744$	0	0
$745$	−3.54436e9	−0.314044
$746$	0	0
$747$	−1.59170e9	−0.139714
$748$	0	0
$749$	7.58775e9	0.659822
$750$	0	0
$751$	−9.46641e9	−0.815541	−0.407770	−	0.913084i	$-0.633694\pi$
$751$	−9.46641e9	−0.815541	−0.407770	+	0.913084i	$0.633694\pi$
$752$	0	0
$753$	−1.14729e10	−0.979240
$754$	0	0
$755$	−1.67096e9	−0.141303
$756$	0	0
$757$	−1.76198e10	−1.47627	−0.738135	−	0.674654i	$-0.764293\pi$
$757$	−1.76198e10	−1.47627	−0.738135	+	0.674654i	$0.764293\pi$
$758$	0	0
$759$	−2.62288e9	−0.217737
$760$	0	0
$761$	1.82289e10	1.49939	0.749694	−	0.661785i	$-0.230201\pi$
$761$	1.82289e10	1.49939	0.749694	+	0.661785i	$0.230201\pi$
$762$	0	0
$763$	−1.77489e10	−1.44656
$764$	0	0
$765$	−1.33309e9	−0.107658
$766$	0	0
$767$	−8.73754e9	−0.699207
$768$	0	0
$769$	1.23985e10	0.983164	0.491582	−	0.870831i	$-0.336419\pi$
$769$	1.23985e10	0.983164	0.491582	+	0.870831i	$0.336419\pi$
$770$	0	0
$771$	−2.49455e8	−0.0196021
$772$	0	0
$773$	3.02879e9	0.235853	0.117926	−	0.993022i	$-0.462375\pi$
$773$	3.02879e9	0.235853	0.117926	+	0.993022i	$0.462375\pi$
$774$	0	0
$775$	−1.99946e9	−0.154296
$776$	0	0
$777$	8.98610e9	0.687223
$778$	0	0
$779$	3.13062e10	2.37274
$780$	0	0
$781$	3.18777e9	0.239446
$782$	0	0
$783$	−2.46966e10	−1.83853
$784$	0	0
$785$	5.15594e8	0.0380420
$786$	0	0
$787$	−2.25511e10	−1.64913	−0.824567	−	0.565765i	$-0.808581\pi$
$787$	−2.25511e10	−1.64913	−0.824567	+	0.565765i	$0.808581\pi$
$788$	0	0
$789$	−2.42998e10	−1.76130
$790$	0	0
$791$	−1.47005e10	−1.05612
$792$	0	0
$793$	−1.20109e10	−0.855304
$794$	0	0
$795$	3.39240e9	0.239454
$796$	0	0
$797$	−1.19794e10	−0.838169	−0.419084	−	0.907947i	$-0.637649\pi$
$797$	−1.19794e10	−0.838169	−0.419084	+	0.907947i	$0.637649\pi$
$798$	0	0
$799$	1.88761e10	1.30918
$800$	0	0
$801$	1.65983e9	0.114117
$802$	0	0
$803$	−1.35402e9	−0.0922828
$804$	0	0
$805$	4.78110e9	0.323030
$806$	0	0
$807$	−3.02025e9	−0.202295
$808$	0	0
$809$	2.73709e9	0.181748	0.0908739	−	0.995862i	$-0.471034\pi$
$809$	2.73709e9	0.181748	0.0908739	+	0.995862i	$0.471034\pi$
$810$	0	0
$811$	2.25024e10	1.48134	0.740672	−	0.671867i	$-0.234507\pi$
$811$	2.25024e10	1.48134	0.740672	+	0.671867i	$0.234507\pi$
$812$	0	0
$813$	9.40662e8	0.0613927
$814$	0	0
$815$	1.08579e10	0.702579
$816$	0	0
$817$	−2.18438e10	−1.40137
$818$	0	0
$819$	−2.62475e9	−0.166953
$820$	0	0
$821$	−2.27324e10	−1.43365	−0.716826	−	0.697252i	$-0.754406\pi$
$821$	−2.27324e10	−1.43365	−0.716826	+	0.697252i	$0.754406\pi$
$822$	0	0
$823$	−1.63240e10	−1.02077	−0.510383	−	0.859947i	$-0.670496\pi$
$823$	−1.63240e10	−1.02077	−0.510383	+	0.859947i	$0.670496\pi$
$824$	0	0
$825$	−9.76125e8	−0.0605225
$826$	0	0
$827$	4.42747e9	0.272199	0.136099	−	0.990695i	$-0.456543\pi$
$827$	4.42747e9	0.272199	0.136099	+	0.990695i	$0.456543\pi$
$828$	0	0
$829$	−1.83486e10	−1.11857	−0.559283	−	0.828977i	$-0.688923\pi$
$829$	−1.83486e10	−1.11857	−0.559283	+	0.828977i	$0.688923\pi$
$830$	0	0
$831$	4.56402e9	0.275895
$832$	0	0
$833$	−1.91706e8	−0.0114915
$834$	0	0
$835$	1.44732e9	0.0860325
$836$	0	0
$837$	−1.39252e10	−0.820847
$838$	0	0
$839$	2.76487e10	1.61625	0.808123	−	0.589013i	$-0.200483\pi$
$839$	2.76487e10	1.61625	0.808123	+	0.589013i	$0.200483\pi$
$840$	0	0
$841$	3.42559e10	1.98587
$842$	0	0
$843$	−1.35127e10	−0.776864
$844$	0	0
$845$	−3.26875e8	−0.0186373
$846$	0	0
$847$	1.58109e10	0.894056
$848$	0	0
$849$	1.40716e10	0.789164
$850$	0	0
$851$	9.67914e9	0.538373
$852$	0	0
$853$	1.91680e10	1.05744	0.528718	−	0.848797i	$-0.322673\pi$
$853$	1.91680e10	1.05744	0.528718	+	0.848797i	$0.322673\pi$
$854$	0	0
$855$	−2.25878e9	−0.123593
$856$	0	0
$857$	−3.66143e8	−0.0198709	−0.00993547	−	0.999951i	$-0.503163\pi$
$857$	−3.66143e8	−0.0198709	−0.00993547	+	0.999951i	$0.503163\pi$
$858$	0	0
$859$	2.15503e10	1.16005	0.580027	−	0.814597i	$-0.303042\pi$
$859$	2.15503e10	1.16005	0.580027	+	0.814597i	$0.303042\pi$
$860$	0	0
$861$	−2.40651e10	−1.28492
$862$	0	0
$863$	−1.78064e10	−0.943059	−0.471529	−	0.881850i	$-0.656298\pi$
$863$	−1.78064e10	−0.943059	−0.471529	+	0.881850i	$0.656298\pi$
$864$	0	0
$865$	−1.28068e10	−0.672796
$866$	0	0
$867$	−2.07623e10	−1.08195
$868$	0	0
$869$	3.78336e9	0.195573
$870$	0	0
$871$	−2.86320e10	−1.46821
$872$	0	0
$873$	3.49970e9	0.178025
$874$	0	0
$875$	1.77933e9	0.0897899
$876$	0	0
$877$	1.70859e9	0.0855339	0.0427670	−	0.999085i	$-0.486383\pi$
$877$	1.70859e9	0.0855339	0.0427670	+	0.999085i	$0.486383\pi$
$878$	0	0
$879$	2.53901e10	1.26097
$880$	0	0
$881$	1.15342e10	0.568294	0.284147	−	0.958781i	$-0.408290\pi$
$881$	1.15342e10	0.568294	0.284147	+	0.958781i	$0.408290\pi$
$882$	0	0
$883$	2.28319e10	1.11604	0.558020	−	0.829828i	$-0.311561\pi$
$883$	2.28319e10	1.11604	0.558020	+	0.829828i	$0.311561\pi$
$884$	0	0
$885$	5.78006e9	0.280305
$886$	0	0
$887$	2.76649e10	1.33106	0.665528	−	0.746373i	$-0.268206\pi$
$887$	2.76649e10	1.33106	0.665528	+	0.746373i	$0.268206\pi$
$888$	0	0
$889$	3.13138e9	0.149479
$890$	0	0
$891$	−5.66026e9	−0.268080
$892$	0	0
$893$	3.19834e10	1.50295
$894$	0	0
$895$	7.52234e9	0.350729
$896$	0	0
$897$	1.45232e10	0.671873
$898$	0	0
$899$	2.90416e10	1.33309
$900$	0	0
$901$	−1.89824e10	−0.864600
$902$	0	0
$903$	1.67913e10	0.758888
$904$	0	0
$905$	1.11494e10	0.500015
$906$	0	0
$907$	2.91517e10	1.29729	0.648647	−	0.761089i	$-0.275335\pi$
$907$	2.91517e10	1.29729	0.648647	+	0.761089i	$0.275335\pi$
$908$	0	0
$909$	4.52829e9	0.199968
$910$	0	0
$911$	5.04406e9	0.221037	0.110519	−	0.993874i	$-0.464749\pi$
$911$	5.04406e9	0.221037	0.110519	+	0.993874i	$0.464749\pi$
$912$	0	0
$913$	6.52158e9	0.283599
$914$	0	0
$915$	7.94547e9	0.342883
$916$	0	0
$917$	−2.34061e9	−0.100239
$918$	0	0
$919$	−1.48636e10	−0.631714	−0.315857	−	0.948807i	$-0.602292\pi$
$919$	−1.48636e10	−0.631714	−0.315857	+	0.948807i	$0.602292\pi$
$920$	0	0
$921$	−8.97476e9	−0.378542
$922$	0	0
$923$	−1.76510e10	−0.738863
$924$	0	0
$925$	3.60217e9	0.149647
$926$	0	0
$927$	−1.25895e9	−0.0519073
$928$	0	0
$929$	4.53935e9	0.185754	0.0928771	−	0.995678i	$-0.470394\pi$
$929$	4.53935e9	0.185754	0.0928771	+	0.995678i	$0.470394\pi$
$930$	0	0
$931$	−3.24825e8	−0.0131924
$932$	0	0
$933$	−2.97755e10	−1.20025
$934$	0	0
$935$	5.46199e9	0.218530
$936$	0	0
$937$	2.50210e10	0.993612	0.496806	−	0.867862i	$-0.334506\pi$
$937$	2.50210e10	0.993612	0.496806	+	0.867862i	$0.334506\pi$
$938$	0	0
$939$	1.98474e9	0.0782299
$940$	0	0
$941$	−4.14156e10	−1.62032	−0.810159	−	0.586210i	$-0.800619\pi$
$941$	−4.14156e10	−1.62032	−0.810159	+	0.586210i	$0.800619\pi$
$942$	0	0
$943$	−2.59210e10	−1.00661
$944$	0	0
$945$	1.23921e10	0.477677
$946$	0	0
$947$	1.61121e10	0.616493	0.308246	−	0.951307i	$-0.400258\pi$
$947$	1.61121e10	0.616493	0.308246	+	0.951307i	$0.400258\pi$
$948$	0	0
$949$	7.49735e9	0.284758
$950$	0	0
$951$	−8.78788e9	−0.331323
$952$	0	0
$953$	2.66126e10	0.996007	0.498003	−	0.867175i	$-0.334067\pi$
$953$	2.66126e10	0.996007	0.498003	+	0.867175i	$0.334067\pi$
$954$	0	0
$955$	2.24634e10	0.834571
$956$	0	0
$957$	1.41780e10	0.522904
$958$	0	0
$959$	4.04358e10	1.48047
$960$	0	0
$961$	−1.11375e10	−0.404815
$962$	0	0
$963$	2.96812e9	0.107100
$964$	0	0
$965$	1.88580e10	0.675540
$966$	0	0
$967$	−3.81235e9	−0.135581	−0.0677907	−	0.997700i	$-0.521595\pi$
$967$	−3.81235e9	−0.135581	−0.0677907	+	0.997700i	$0.521595\pi$
$968$	0	0
$969$	−6.49273e10	−2.29242
$970$	0	0
$971$	1.62881e10	0.570956	0.285478	−	0.958385i	$-0.407848\pi$
$971$	1.62881e10	0.570956	0.285478	+	0.958385i	$0.407848\pi$
$972$	0	0
$973$	5.11261e10	1.77929
$974$	0	0
$975$	5.40491e9	0.186755
$976$	0	0
$977$	2.50249e10	0.858501	0.429251	−	0.903185i	$-0.358778\pi$
$977$	2.50249e10	0.858501	0.429251	+	0.903185i	$0.358778\pi$
$978$	0	0
$979$	−6.80073e9	−0.231641
$980$	0	0
$981$	−6.94289e9	−0.234800
$982$	0	0
$983$	2.17599e10	0.730668	0.365334	−	0.930877i	$-0.380955\pi$
$983$	2.17599e10	0.730668	0.365334	+	0.930877i	$0.380955\pi$
$984$	0	0
$985$	−1.89192e10	−0.630778
$986$	0	0
$987$	−2.45857e10	−0.813901
$988$	0	0
$989$	1.80863e10	0.594516
$990$	0	0
$991$	−2.91057e10	−0.949991	−0.474996	−	0.879988i	$-0.657550\pi$
$991$	−2.91057e10	−0.949991	−0.474996	+	0.879988i	$0.657550\pi$
$992$	0	0
$993$	−4.15863e10	−1.34781
$994$	0	0
$995$	1.66491e10	0.535809
$996$	0	0
$997$	−2.66758e10	−0.852480	−0.426240	−	0.904610i	$-0.640162\pi$
$997$	−2.66758e10	−0.852480	−0.426240	+	0.904610i	$0.640162\pi$
$998$	0	0
$999$	2.50873e10	0.796113

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.8.a.g.1.1	yes	3
4.3	odd	2		160.8.a.f.1.3	✓	3
8.3	odd	2		320.8.a.y.1.1		3
8.5	even	2		320.8.a.x.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.8.a.f.1.3	✓	3	4.3	odd	2
160.8.a.g.1.1	yes	3	1.1	even	1	trivial
320.8.a.x.1.3		3	8.5	even	2
320.8.a.y.1.1		3	8.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	160.a (trivial)

\(f(q)\)	\(=\)	\(q-42.7859 q^{3} -125.000 q^{5} -911.015 q^{7} -356.364 q^{9} +O(q^{10})\)	\(q-42.7859 q^{3} -125.000 q^{5} -911.015 q^{7} -356.364 q^{9} +1460.11 q^{11} -8084.77 q^{13} +5348.24 q^{15} -29926.5 q^{17} -50707.3 q^{19} +38978.6 q^{21} +41984.8 q^{23} +15625.0 q^{25} +108820. q^{27} -226949. q^{29} -127965. q^{31} -62472.0 q^{33} +113877. q^{35} +230539. q^{37} +345914. q^{39} -617391. q^{41} +430783. q^{43} +44545.5 q^{45} -630747. q^{47} +6405.88 q^{49} +1.28043e6 q^{51} +634301. q^{53} -182513. q^{55} +2.16956e6 q^{57} +1.08074e6 q^{59} +1.48562e6 q^{61} +324653. q^{63} +1.01060e6 q^{65} +3.54147e6 q^{67} -1.79636e6 q^{69} +2.18324e6 q^{71} -927343. q^{73} -668530. q^{75} -1.33018e6 q^{77} +2.59115e6 q^{79} -3.87661e6 q^{81} +4.46651e6 q^{83} +3.74082e6 q^{85} +9.71022e6 q^{87} -4.65769e6 q^{89} +7.36535e6 q^{91} +5.47511e6 q^{93} +6.33841e6 q^{95} -9.82058e6 q^{97} -520330. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 70 q^{3} - 375 q^{5} + 62 q^{7} + 3403 q^{9}+O(q^{10}) \) 3 * q + 70 * q^3 - 375 * q^5 + 62 * q^7 + 3403 * q^9	\( 3 q + 70 q^{3} - 375 q^{5} + 62 q^{7} + 3403 q^{9} + 1964 q^{11} + 4742 q^{13} - 8750 q^{15} - 31066 q^{17} - 88792 q^{19} + 145500 q^{21} + 40282 q^{23} + 46875 q^{25} + 274132 q^{27} - 31966 q^{29} - 6044 q^{31} + 184344 q^{33} - 7750 q^{35} + 573614 q^{37} + 1381772 q^{39} - 530030 q^{41} + 44982 q^{43} - 425375 q^{45} + 1191086 q^{47} + 337319 q^{49} - 18996 q^{51} - 632754 q^{53} - 245500 q^{55} - 2265392 q^{57} + 2294080 q^{59} + 512222 q^{61} + 7979022 q^{63} - 592750 q^{65} + 5861970 q^{67} - 2044844 q^{69} + 8594932 q^{71} - 6778770 q^{73} + 1093750 q^{75} + 5277176 q^{77} + 5291192 q^{79} + 7956271 q^{81} + 11482686 q^{83} + 3883250 q^{85} + 15732676 q^{87} + 2747614 q^{89} + 22709628 q^{91} + 3088136 q^{93} + 11099000 q^{95} - 6939170 q^{97} + 25059276 q^{99}+O(q^{100}) \) 3 * q + 70 * q^3 - 375 * q^5 + 62 * q^7 + 3403 * q^9 + 1964 * q^11 + 4742 * q^13 - 8750 * q^15 - 31066 * q^17 - 88792 * q^19 + 145500 * q^21 + 40282 * q^23 + 46875 * q^25 + 274132 * q^27 - 31966 * q^29 - 6044 * q^31 + 184344 * q^33 - 7750 * q^35 + 573614 * q^37 + 1381772 * q^39 - 530030 * q^41 + 44982 * q^43 - 425375 * q^45 + 1191086 * q^47 + 337319 * q^49 - 18996 * q^51 - 632754 * q^53 - 245500 * q^55 - 2265392 * q^57 + 2294080 * q^59 + 512222 * q^61 + 7979022 * q^63 - 592750 * q^65 + 5861970 * q^67 - 2044844 * q^69 + 8594932 * q^71 - 6778770 * q^73 + 1093750 * q^75 + 5277176 * q^77 + 5291192 * q^79 + 7956271 * q^81 + 11482686 * q^83 + 3883250 * q^85 + 15732676 * q^87 + 2747614 * q^89 + 22709628 * q^91 + 3088136 * q^93 + 11099000 * q^95 - 6939170 * q^97 + 25059276 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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