LMFDB - Embedded newform 80.20.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,20,Mod(1,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("80.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	80.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:	$183.053357245$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 1351720x + 139588750 $ x^3 - x^2 - 1351720*x + 139588750
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$104.094$ of defining polynomial
Character	$\chi$	$=$	80.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16342.5 q^{3} -1.95312e6 q^{5} -6.88942e7 q^{7} -8.95184e8 q^{9} +O(q^{10})$	$q-16342.5 q^{3} -1.95312e6 q^{5} -6.88942e7 q^{7} -8.95184e8 q^{9} -2.10125e9 q^{11} +2.82599e10 q^{13} +3.19190e10 q^{15} -4.97336e11 q^{17} +9.50809e11 q^{19} +1.12590e12 q^{21} +4.43914e11 q^{23} +3.81470e12 q^{25} +3.36238e13 q^{27} +8.85323e13 q^{29} +4.77762e13 q^{31} +3.43396e13 q^{33} +1.34559e14 q^{35} -3.94706e14 q^{37} -4.61837e14 q^{39} -1.34775e15 q^{41} +4.89845e15 q^{43} +1.74841e15 q^{45} +4.56510e15 q^{47} -6.65249e15 q^{49} +8.12772e15 q^{51} -1.34862e16 q^{53} +4.10400e15 q^{55} -1.55386e16 q^{57} +9.95497e16 q^{59} -1.00537e17 q^{61} +6.16730e16 q^{63} -5.51951e16 q^{65} +3.34175e17 q^{67} -7.25466e15 q^{69} -3.02457e17 q^{71} +1.56726e16 q^{73} -6.23417e16 q^{75} +1.44764e17 q^{77} +5.08507e17 q^{79} +4.90940e17 q^{81} -3.81205e17 q^{83} +9.71359e17 q^{85} -1.44684e18 q^{87} +4.13177e18 q^{89} -1.94694e18 q^{91} -7.80783e17 q^{93} -1.85705e18 q^{95} -1.49904e18 q^{97} +1.88100e18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 34086 q^{3} - 5859375 q^{5} + 115130574 q^{7} + 3129228027 q^{9}+O(q^{10}) $ 3 * q - 34086 * q^3 - 5859375 * q^5 + 115130574 * q^7 + 3129228027 * q^9	$ 3 q - 34086 q^{3} - 5859375 q^{5} + 115130574 q^{7} + 3129228027 q^{9} - 6359213280 q^{11} - 33996539826 q^{13} + 66574218750 q^{15} - 60515110578 q^{17} + 1436218405908 q^{19} - 7119772762332 q^{21} - 1682066292342 q^{23} + 11444091796875 q^{25} - 91316335875732 q^{27} - 181580995192278 q^{29} + 3566464773252 q^{31} - 90839870844480 q^{33} - 224864402343750 q^{35} + 653714901466206 q^{37} + 43\!\cdots\!08 q^{39}+ \cdots - 37\!\cdots\!20 q^{99}+O(q^{100}) $ 3 * q - 34086 * q^3 - 5859375 * q^5 + 115130574 * q^7 + 3129228027 * q^9 - 6359213280 * q^11 - 33996539826 * q^13 + 66574218750 * q^15 - 60515110578 * q^17 + 1436218405908 * q^19 - 7119772762332 * q^21 - 1682066292342 * q^23 + 11444091796875 * q^25 - 91316335875732 * q^27 - 181580995192278 * q^29 + 3566464773252 * q^31 - 90839870844480 * q^33 - 224864402343750 * q^35 + 653714901466206 * q^37 + 4339017394217508 * q^39 - 1257752912210622 * q^41 + 4361040178994610 * q^43 - 6111773490234375 * q^45 + 23174987171661798 * q^47 - 5454428919168921 * q^49 + 33371959424606244 * q^51 - 12180381712114362 * q^53 + 12420338437500000 * q^55 - 177419957801365416 * q^57 + 180023573560918116 * q^59 - 213491197168562514 * q^61 + 549307385783925246 * q^63 + 66399491847656250 * q^65 + 32494317103156134 * q^67 - 498099089220386004 * q^69 + 245957452394007276 * q^71 - 94437681636898506 * q^73 - 130027770996093750 * q^75 - 72972512123263680 * q^77 - 819623047726342536 * q^79 + 3159048884904329967 * q^81 - 2812828487653823982 * q^83 + 118193575347656250 * q^85 + 2510285506713629676 * q^87 + 4879439966436006078 * q^89 - 12213088562709029652 * q^91 + 16020238495381453176 * q^93 - 2805114074039062500 * q^95 + 8189756396617107846 * q^97 - 3795512975671396320 * 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
$3$	−16342.5	−0.479366	−0.239683	−	0.970851i	$-0.577043\pi$
$3$	−16342.5	−0.479366	−0.239683	+	0.970851i	$0.577043\pi$
$4$	0	0
$5$	−1.95312e6	−0.447214
$5$	−1.95312e6	−0.447214
$6$	0	0
$7$	−6.88942e7	−0.645284	−0.322642	−	0.946521i	$-0.604571\pi$
$7$	−6.88942e7	−0.645284	−0.322642	+	0.946521i	$0.604571\pi$
$8$	0	0
$9$	−8.95184e8	−0.770209
$10$	0	0
$11$	−2.10125e9	−0.268687	−0.134343	−	0.990935i	$-0.542893\pi$
$11$	−2.10125e9	−0.268687	−0.134343	+	0.990935i	$0.542893\pi$
$12$	0	0
$13$	2.82599e10	0.739109	0.369555	−	0.929209i	$-0.379510\pi$
$13$	2.82599e10	0.739109	0.369555	+	0.929209i	$0.379510\pi$
$14$	0	0
$15$	3.19190e10	0.214379
$16$	0	0
$17$	−4.97336e11	−1.01715	−0.508575	−	0.861018i	$-0.669828\pi$
$17$	−4.97336e11	−1.01715	−0.508575	+	0.861018i	$0.669828\pi$
$18$	0	0
$19$	9.50809e11	0.675981	0.337990	−	0.941150i	$-0.390253\pi$
$19$	9.50809e11	0.675981	0.337990	+	0.941150i	$0.390253\pi$
$20$	0	0
$21$	1.12590e12	0.309327
$22$	0	0
$23$	4.43914e11	0.0513906	0.0256953	−	0.999670i	$-0.491820\pi$
$23$	4.43914e11	0.0513906	0.0256953	+	0.999670i	$0.491820\pi$
$24$	0	0
$25$	3.81470e12	0.200000
$26$	0	0
$27$	3.36238e13	0.848577
$28$	0	0
$29$	8.85323e13	1.13324	0.566619	−	0.823980i	$-0.308251\pi$
$29$	8.85323e13	1.13324	0.566619	+	0.823980i	$0.308251\pi$
$30$	0	0
$31$	4.77762e13	0.324545	0.162273	−	0.986746i	$-0.448118\pi$
$31$	4.77762e13	0.324545	0.162273	+	0.986746i	$0.448118\pi$
$32$	0	0
$33$	3.43396e13	0.128799
$34$	0	0
$35$	1.34559e14	0.288580
$36$	0	0
$37$	−3.94706e14	−0.499295	−0.249647	−	0.968337i	$-0.580315\pi$
$37$	−3.94706e14	−0.499295	−0.249647	+	0.968337i	$0.580315\pi$
$38$	0	0
$39$	−4.61837e14	−0.354303
$40$	0	0
$41$	−1.34775e15	−0.642927	−0.321463	−	0.946922i	$-0.604175\pi$
$41$	−1.34775e15	−0.642927	−0.321463	+	0.946922i	$0.604175\pi$
$42$	0	0
$43$	4.89845e15	1.48631	0.743153	−	0.669121i	$-0.233329\pi$
$43$	4.89845e15	1.48631	0.743153	+	0.669121i	$0.233329\pi$
$44$	0	0
$45$	1.74841e15	0.344448
$46$	0	0
$47$	4.56510e15	0.595005	0.297503	−	0.954721i	$-0.403846\pi$
$47$	4.56510e15	0.595005	0.297503	+	0.954721i	$0.403846\pi$
$48$	0	0
$49$	−6.65249e15	−0.583608
$50$	0	0
$51$	8.12772e15	0.487587
$52$	0	0
$53$	−1.34862e16	−0.561394	−0.280697	−	0.959796i	$-0.590566\pi$
$53$	−1.34862e16	−0.561394	−0.280697	+	0.959796i	$0.590566\pi$
$54$	0	0
$55$	4.10400e15	0.120160
$56$	0	0
$57$	−1.55386e16	−0.324042
$58$	0	0
$59$	9.95497e16	1.49605	0.748025	−	0.663671i	$-0.231002\pi$
$59$	9.95497e16	1.49605	0.748025	+	0.663671i	$0.231002\pi$
$60$	0	0
$61$	−1.00537e17	−1.10076	−0.550378	−	0.834916i	$-0.685516\pi$
$61$	−1.00537e17	−1.10076	−0.550378	+	0.834916i	$0.685516\pi$
$62$	0	0
$63$	6.16730e16	0.497004
$64$	0	0
$65$	−5.51951e16	−0.330540
$66$	0	0
$67$	3.34175e17	1.50059	0.750297	−	0.661101i	$-0.229911\pi$
$67$	3.34175e17	1.50059	0.750297	+	0.661101i	$0.229911\pi$
$68$	0	0
$69$	−7.25466e15	−0.0246349
$70$	0	0
$71$	−3.02457e17	−0.782907	−0.391453	−	0.920198i	$-0.628028\pi$
$71$	−3.02457e17	−0.782907	−0.391453	+	0.920198i	$0.628028\pi$
$72$	0	0
$73$	1.56726e16	0.0311583	0.0155792	−	0.999879i	$-0.495041\pi$
$73$	1.56726e16	0.0311583	0.0155792	+	0.999879i	$0.495041\pi$
$74$	0	0
$75$	−6.23417e16	−0.0958731
$76$	0	0
$77$	1.44764e17	0.173379
$78$	0	0
$79$	5.08507e17	0.477353	0.238676	−	0.971099i	$-0.423286\pi$
$79$	5.08507e17	0.477353	0.238676	+	0.971099i	$0.423286\pi$
$80$	0	0
$81$	4.90940e17	0.363430
$82$	0	0
$83$	−3.81205e17	−0.223829	−0.111915	−	0.993718i	$-0.535698\pi$
$83$	−3.81205e17	−0.223829	−0.111915	+	0.993718i	$0.535698\pi$
$84$	0	0
$85$	9.71359e17	0.454883
$86$	0	0
$87$	−1.44684e18	−0.543235
$88$	0	0
$89$	4.13177e18	1.25006	0.625031	−	0.780600i	$-0.285086\pi$
$89$	4.13177e18	1.25006	0.625031	+	0.780600i	$0.285086\pi$
$90$	0	0
$91$	−1.94694e18	−0.476936
$92$	0	0
$93$	−7.80783e17	−0.155576
$94$	0	0
$95$	−1.85705e18	−0.302308
$96$	0	0
$97$	−1.49904e18	−0.200208	−0.100104	−	0.994977i	$-0.531918\pi$
$97$	−1.49904e18	−0.200208	−0.100104	+	0.994977i	$0.531918\pi$
$98$	0	0
$99$	1.88100e18	0.206945
$100$	0	0
$101$	1.80857e19	1.64544	0.822720	−	0.568448i	$-0.192456\pi$
$101$	1.80857e19	1.64544	0.822720	+	0.568448i	$0.192456\pi$
$102$	0	0
$103$	−1.38480e19	−1.04576	−0.522882	−	0.852405i	$-0.675143\pi$
$103$	−1.38480e19	−1.04576	−0.522882	+	0.852405i	$0.675143\pi$
$104$	0	0
$105$	−2.19903e18	−0.138335
$106$	0	0
$107$	1.19291e19	0.627282	0.313641	−	0.949542i	$-0.398451\pi$
$107$	1.19291e19	0.627282	0.313641	+	0.949542i	$0.398451\pi$
$108$	0	0
$109$	−7.79360e18	−0.343705	−0.171853	−	0.985123i	$-0.554975\pi$
$109$	−7.79360e18	−0.343705	−0.171853	+	0.985123i	$0.554975\pi$
$110$	0	0
$111$	6.45048e18	0.239345
$112$	0	0
$113$	−2.35670e19	−0.738006	−0.369003	−	0.929428i	$-0.620301\pi$
$113$	−2.35670e19	−0.738006	−0.369003	+	0.929428i	$0.620301\pi$
$114$	0	0
$115$	−8.67019e17	−0.0229826
$116$	0	0
$117$	−2.52978e19	−0.569268
$118$	0	0
$119$	3.42636e19	0.656351
$120$	0	0
$121$	−5.67439e19	−0.927807
$122$	0	0
$123$	2.20256e19	0.308197
$124$	0	0
$125$	−7.45058e18	−0.0894427
$126$	0	0
$127$	2.68145e18	0.0276844	0.0138422	−	0.999904i	$-0.495594\pi$
$127$	2.68145e18	0.0276844	0.0138422	+	0.999904i	$0.495594\pi$
$128$	0	0
$129$	−8.00529e19	−0.712484
$130$	0	0
$131$	−1.05018e19	−0.0807581	−0.0403790	−	0.999184i	$-0.512857\pi$
$131$	−1.05018e19	−0.0807581	−0.0403790	+	0.999184i	$0.512857\pi$
$132$	0	0
$133$	−6.55052e19	−0.436200
$134$	0	0
$135$	−6.56715e19	−0.379495
$136$	0	0
$137$	4.44799e19	0.223521	0.111761	−	0.993735i	$-0.464351\pi$
$137$	4.44799e19	0.223521	0.111761	+	0.993735i	$0.464351\pi$
$138$	0	0
$139$	−2.06346e20	−0.903557	−0.451778	−	0.892130i	$-0.649210\pi$
$139$	−2.06346e20	−0.903557	−0.451778	+	0.892130i	$0.649210\pi$
$140$	0	0
$141$	−7.46052e19	−0.285225
$142$	0	0
$143$	−5.93810e19	−0.198589
$144$	0	0
$145$	−1.72915e20	−0.506799
$146$	0	0
$147$	1.08718e20	0.279762
$148$	0	0
$149$	4.33074e20	0.980150	0.490075	−	0.871680i	$-0.336969\pi$
$149$	4.33074e20	0.980150	0.490075	+	0.871680i	$0.336969\pi$
$150$	0	0
$151$	9.68999e19	0.193216	0.0966079	−	0.995323i	$-0.469201\pi$
$151$	9.68999e19	0.193216	0.0966079	+	0.995323i	$0.469201\pi$
$152$	0	0
$153$	4.45207e20	0.783418
$154$	0	0
$155$	−9.33129e19	−0.145141
$156$	0	0
$157$	1.12982e21	1.55583	0.777914	−	0.628370i	$-0.216278\pi$
$157$	1.12982e21	1.55583	0.777914	+	0.628370i	$0.216278\pi$
$158$	0	0
$159$	2.20398e20	0.269113
$160$	0	0
$161$	−3.05831e19	−0.0331616
$162$	0	0
$163$	−1.36836e21	−1.31953	−0.659764	−	0.751473i	$-0.729344\pi$
$163$	−1.36836e21	−1.31953	−0.659764	+	0.751473i	$0.729344\pi$
$164$	0	0
$165$	−6.70696e19	−0.0576008
$166$	0	0
$167$	−2.02198e20	−0.154871	−0.0774354	−	0.996997i	$-0.524673\pi$
$167$	−2.02198e20	−0.154871	−0.0774354	+	0.996997i	$0.524673\pi$
$168$	0	0
$169$	−6.63299e20	−0.453718
$170$	0	0
$171$	−8.51149e20	−0.520646
$172$	0	0
$173$	−1.18559e21	−0.649379	−0.324689	−	0.945821i	$-0.605260\pi$
$173$	−1.18559e21	−0.649379	−0.324689	+	0.945821i	$0.605260\pi$
$174$	0	0
$175$	−2.62810e20	−0.129057
$176$	0	0
$177$	−1.62689e21	−0.717155
$178$	0	0
$179$	−7.21571e20	−0.285874	−0.142937	−	0.989732i	$-0.545655\pi$
$179$	−7.21571e20	−0.285874	−0.142937	+	0.989732i	$0.545655\pi$
$180$	0	0
$181$	−4.18545e21	−1.49209	−0.746045	−	0.665895i	$-0.768050\pi$
$181$	−4.18545e21	−1.49209	−0.746045	+	0.665895i	$0.768050\pi$
$182$	0	0
$183$	1.64302e21	0.527664
$184$	0	0
$185$	7.70909e20	0.223291
$186$	0	0
$187$	1.04503e21	0.273295
$188$	0	0
$189$	−2.31649e21	−0.547574
$190$	0	0
$191$	6.11584e19	0.0130809	0.00654047	−	0.999979i	$-0.497918\pi$
$191$	6.11584e19	0.0130809	0.00654047	+	0.999979i	$0.497918\pi$
$192$	0	0
$193$	8.55367e20	0.165714	0.0828568	−	0.996561i	$-0.473596\pi$
$193$	8.55367e20	0.165714	0.0828568	+	0.996561i	$0.473596\pi$
$194$	0	0
$195$	9.02026e20	0.158449
$196$	0	0
$197$	−8.95557e21	−1.42779	−0.713895	−	0.700253i	$-0.753071\pi$
$197$	−8.95557e21	−1.42779	−0.713895	+	0.700253i	$0.753071\pi$
$198$	0	0
$199$	1.38919e20	0.0201214	0.0100607	−	0.999949i	$-0.496798\pi$
$199$	1.38919e20	0.0201214	0.0100607	+	0.999949i	$0.496798\pi$
$200$	0	0
$201$	−5.46126e21	−0.719333
$202$	0	0
$203$	−6.09936e21	−0.731261
$204$	0	0
$205$	2.63232e21	0.287526
$206$	0	0
$207$	−3.97384e20	−0.0395815
$208$	0	0
$209$	−1.99788e21	−0.181627
$210$	0	0
$211$	−1.33210e22	−1.10625	−0.553124	−	0.833099i	$-0.686564\pi$
$211$	−1.33210e22	−1.10625	−0.553124	+	0.833099i	$0.686564\pi$
$212$	0	0
$213$	4.94291e21	0.375298
$214$	0	0
$215$	−9.56728e21	−0.664696
$216$	0	0
$217$	−3.29150e21	−0.209424
$218$	0	0
$219$	−2.56130e20	−0.0149362
$220$	0	0
$221$	−1.40547e22	−0.751785
$222$	0	0
$223$	−2.51483e22	−1.23485	−0.617424	−	0.786631i	$-0.711823\pi$
$223$	−2.51483e22	−1.23485	−0.617424	+	0.786631i	$0.711823\pi$
$224$	0	0
$225$	−3.41486e21	−0.154042
$226$	0	0
$227$	−2.20188e22	−0.913164	−0.456582	−	0.889681i	$-0.650927\pi$
$227$	−2.20188e22	−0.913164	−0.456582	+	0.889681i	$0.650927\pi$
$228$	0	0
$229$	−2.41034e22	−0.919688	−0.459844	−	0.888000i	$-0.652095\pi$
$229$	−2.41034e22	−0.919688	−0.459844	+	0.888000i	$0.652095\pi$
$230$	0	0
$231$	−2.36580e21	−0.0831121
$232$	0	0
$233$	3.34467e22	1.08261	0.541306	−	0.840826i	$-0.317930\pi$
$233$	3.34467e22	1.08261	0.541306	+	0.840826i	$0.317930\pi$
$234$	0	0
$235$	−8.91622e21	−0.266094
$236$	0	0
$237$	−8.31028e21	−0.228827
$238$	0	0
$239$	−6.20556e22	−1.57762	−0.788808	−	0.614640i	$-0.789301\pi$
$239$	−6.20556e22	−1.57762	−0.788808	+	0.614640i	$0.789301\pi$
$240$	0	0
$241$	−3.52166e22	−0.827152	−0.413576	−	0.910470i	$-0.635720\pi$
$241$	−3.52166e22	−0.827152	−0.413576	+	0.910470i	$0.635720\pi$
$242$	0	0
$243$	−4.71029e22	−1.02279
$244$	0	0
$245$	1.29931e22	0.260997
$246$	0	0
$247$	2.68698e22	0.499624
$248$	0	0
$249$	6.22985e21	0.107296
$250$	0	0
$251$	1.06473e23	1.69957	0.849784	−	0.527131i	$-0.176732\pi$
$251$	1.06473e23	1.69957	0.849784	+	0.527131i	$0.176732\pi$
$252$	0	0
$253$	−9.32772e20	−0.0138080
$254$	0	0
$255$	−1.58744e22	−0.218055
$256$	0	0
$257$	−1.19941e23	−1.52969	−0.764847	−	0.644212i	$-0.777186\pi$
$257$	−1.19941e23	−1.52969	−0.764847	+	0.644212i	$0.777186\pi$
$258$	0	0
$259$	2.71929e22	0.322187
$260$	0	0
$261$	−7.92527e22	−0.872829
$262$	0	0
$263$	−8.81019e22	−0.902414	−0.451207	−	0.892419i	$-0.649006\pi$
$263$	−8.81019e22	−0.902414	−0.451207	+	0.892419i	$0.649006\pi$
$264$	0	0
$265$	2.63402e22	0.251063
$266$	0	0
$267$	−6.75235e22	−0.599236
$268$	0	0
$269$	−1.91691e23	−1.58473	−0.792366	−	0.610046i	$-0.791151\pi$
$269$	−1.91691e23	−1.58473	−0.792366	+	0.610046i	$0.791151\pi$
$270$	0	0
$271$	2.36163e23	1.81971	0.909857	−	0.414923i	$-0.136191\pi$
$271$	2.36163e23	1.81971	0.909857	+	0.414923i	$0.136191\pi$
$272$	0	0
$273$	3.18179e22	0.228627
$274$	0	0
$275$	−8.01562e21	−0.0537374
$276$	0	0
$277$	−4.06870e22	−0.254623	−0.127312	−	0.991863i	$-0.540635\pi$
$277$	−4.06870e22	−0.254623	−0.127312	+	0.991863i	$0.540635\pi$
$278$	0	0
$279$	−4.27685e22	−0.249968
$280$	0	0
$281$	−9.55968e22	−0.522076	−0.261038	−	0.965329i	$-0.584065\pi$
$281$	−9.55968e22	−0.522076	−0.261038	+	0.965329i	$0.584065\pi$
$282$	0	0
$283$	−3.71379e23	−1.89604	−0.948018	−	0.318218i	$-0.896916\pi$
$283$	−3.71379e23	−1.89604	−0.948018	+	0.318218i	$0.896916\pi$
$284$	0	0
$285$	3.03489e22	0.144916
$286$	0	0
$287$	9.28519e22	0.414871
$288$	0	0
$289$	8.27069e21	0.0345949
$290$	0	0
$291$	2.44981e22	0.0959728
$292$	0	0
$293$	3.39231e23	1.24524	0.622619	−	0.782525i	$-0.286069\pi$
$293$	3.39231e23	1.24524	0.622619	+	0.782525i	$0.286069\pi$
$294$	0	0
$295$	−1.94433e23	−0.669054
$296$	0	0
$297$	−7.06519e22	−0.228001
$298$	0	0
$299$	1.25450e22	0.0379833
$300$	0	0
$301$	−3.37475e23	−0.959090
$302$	0	0
$303$	−2.95565e23	−0.788767
$304$	0	0
$305$	1.96361e23	0.492273
$306$	0	0
$307$	3.31992e23	0.782191	0.391096	−	0.920350i	$-0.372096\pi$
$307$	3.31992e23	0.782191	0.391096	+	0.920350i	$0.372096\pi$
$308$	0	0
$309$	2.26311e23	0.501303
$310$	0	0
$311$	−2.89037e23	−0.602185	−0.301093	−	0.953595i	$-0.597351\pi$
$311$	−2.89037e23	−0.602185	−0.301093	+	0.953595i	$0.597351\pi$
$312$	0	0
$313$	7.68038e23	1.50561	0.752803	−	0.658246i	$-0.228701\pi$
$313$	7.68038e23	1.50561	0.752803	+	0.658246i	$0.228701\pi$
$314$	0	0
$315$	−1.20455e23	−0.222267
$316$	0	0
$317$	−2.86169e23	−0.497233	−0.248617	−	0.968602i	$-0.579976\pi$
$317$	−2.86169e23	−0.497233	−0.248617	+	0.968602i	$0.579976\pi$
$318$	0	0
$319$	−1.86028e23	−0.304486
$320$	0	0
$321$	−1.94952e23	−0.300698
$322$	0	0
$323$	−4.72872e23	−0.687574
$324$	0	0
$325$	1.07803e23	0.147822
$326$	0	0
$327$	1.27367e23	0.164761
$328$	0	0
$329$	−3.14509e23	−0.383948
$330$	0	0
$331$	1.48377e24	1.71002	0.855008	−	0.518615i	$-0.173552\pi$
$331$	1.48377e24	1.71002	0.855008	+	0.518615i	$0.173552\pi$
$332$	0	0
$333$	3.53334e23	0.384561
$334$	0	0
$335$	−6.52685e23	−0.671086
$336$	0	0
$337$	4.45818e23	0.433185	0.216592	−	0.976262i	$-0.430506\pi$
$337$	4.45818e23	0.433185	0.216592	+	0.976262i	$0.430506\pi$
$338$	0	0
$339$	3.85144e23	0.353774
$340$	0	0
$341$	−1.00390e23	−0.0872011
$342$	0	0
$343$	1.24364e24	1.02188
$344$	0	0
$345$	1.41693e22	0.0110171
$346$	0	0
$347$	1.51475e24	1.11483	0.557417	−	0.830233i	$-0.311792\pi$
$347$	1.51475e24	1.11483	0.557417	+	0.830233i	$0.311792\pi$
$348$	0	0
$349$	−1.16880e24	−0.814515	−0.407257	−	0.913313i	$-0.633515\pi$
$349$	−1.16880e24	−0.814515	−0.407257	+	0.913313i	$0.633515\pi$
$350$	0	0
$351$	9.50205e23	0.627191
$352$	0	0
$353$	3.10436e24	1.94139	0.970695	−	0.240314i	$-0.0772503\pi$
$353$	3.10436e24	1.94139	0.970695	+	0.240314i	$0.0772503\pi$
$354$	0	0
$355$	5.90737e23	0.350126
$356$	0	0
$357$	−5.59953e23	−0.314632
$358$	0	0
$359$	1.76958e24	0.942918	0.471459	−	0.881888i	$-0.343728\pi$
$359$	1.76958e24	0.942918	0.471459	+	0.881888i	$0.343728\pi$
$360$	0	0
$361$	−1.07438e24	−0.543050
$362$	0	0
$363$	9.27337e23	0.444759
$364$	0	0
$365$	−3.06106e22	−0.0139344
$366$	0	0
$367$	1.70808e24	0.738210	0.369105	−	0.929388i	$-0.379664\pi$
$367$	1.70808e24	0.738210	0.369105	+	0.929388i	$0.379664\pi$
$368$	0	0
$369$	1.20648e24	0.495188
$370$	0	0
$371$	9.29119e23	0.362259
$372$	0	0
$373$	1.58722e24	0.588036	0.294018	−	0.955800i	$-0.405007\pi$
$373$	1.58722e24	0.588036	0.294018	+	0.955800i	$0.405007\pi$
$374$	0	0
$375$	1.21761e23	0.0428758
$376$	0	0
$377$	2.50191e24	0.837586
$378$	0	0
$379$	−3.42565e24	−1.09061	−0.545306	−	0.838237i	$-0.683586\pi$
$379$	−3.42565e24	−1.09061	−0.545306	+	0.838237i	$0.683586\pi$
$380$	0	0
$381$	−4.38217e22	−0.0132709
$382$	0	0
$383$	1.83552e24	0.528897	0.264449	−	0.964400i	$-0.414810\pi$
$383$	1.83552e24	0.528897	0.264449	+	0.964400i	$0.414810\pi$
$384$	0	0
$385$	−2.82741e23	−0.0775376
$386$	0	0
$387$	−4.38501e24	−1.14477
$388$	0	0
$389$	4.27661e24	1.06311	0.531555	−	0.847024i	$-0.321608\pi$
$389$	4.27661e24	1.06311	0.531555	+	0.847024i	$0.321608\pi$
$390$	0	0
$391$	−2.20774e23	−0.0522720
$392$	0	0
$393$	1.71626e23	0.0387126
$394$	0	0
$395$	−9.93178e23	−0.213479
$396$	0	0
$397$	2.88290e24	0.590635	0.295318	−	0.955399i	$-0.404575\pi$
$397$	2.88290e24	0.590635	0.295318	+	0.955399i	$0.404575\pi$
$398$	0	0
$399$	1.07052e24	0.209099
$400$	0	0
$401$	6.95475e24	1.29542	0.647709	−	0.761888i	$-0.275727\pi$
$401$	6.95475e24	1.29542	0.647709	+	0.761888i	$0.275727\pi$
$402$	0	0
$403$	1.35015e24	0.239875
$404$	0	0
$405$	−9.58868e23	−0.162531
$406$	0	0
$407$	8.29374e23	0.134154
$408$	0	0
$409$	−6.18673e24	−0.955191	−0.477595	−	0.878580i	$-0.658491\pi$
$409$	−6.18673e24	−0.955191	−0.477595	+	0.878580i	$0.658491\pi$
$410$	0	0
$411$	−7.26913e23	−0.107148
$412$	0	0
$413$	−6.85840e24	−0.965378
$414$	0	0
$415$	7.44542e23	0.100100
$416$	0	0
$417$	3.37221e24	0.433134
$418$	0	0
$419$	−5.63401e24	−0.691488	−0.345744	−	0.938329i	$-0.612373\pi$
$419$	−5.63401e24	−0.691488	−0.345744	+	0.938329i	$0.612373\pi$
$420$	0	0
$421$	6.26639e24	0.735085	0.367542	−	0.930007i	$-0.380199\pi$
$421$	6.26639e24	0.735085	0.367542	+	0.930007i	$0.380199\pi$
$422$	0	0
$423$	−4.08661e24	−0.458278
$424$	0	0
$425$	−1.89719e24	−0.203430
$426$	0	0
$427$	6.92639e24	0.710301
$428$	0	0
$429$	9.70434e23	0.0951967
$430$	0	0
$431$	−8.49090e24	−0.796929	−0.398465	−	0.917184i	$-0.630457\pi$
$431$	−8.49090e24	−0.796929	−0.398465	+	0.917184i	$0.630457\pi$
$432$	0	0
$433$	−1.59933e25	−1.43649	−0.718246	−	0.695790i	$-0.755055\pi$
$433$	−1.59933e25	−1.43649	−0.718246	+	0.695790i	$0.755055\pi$
$434$	0	0
$435$	2.82586e24	0.242942
$436$	0	0
$437$	4.22077e23	0.0347391
$438$	0	0
$439$	8.92543e24	0.703422	0.351711	−	0.936109i	$-0.385600\pi$
$439$	8.92543e24	0.703422	0.351711	+	0.936109i	$0.385600\pi$
$440$	0	0
$441$	5.95520e24	0.449500
$442$	0	0
$443$	1.93421e25	1.39852	0.699258	−	0.714869i	$-0.253514\pi$
$443$	1.93421e25	1.39852	0.699258	+	0.714869i	$0.253514\pi$
$444$	0	0
$445$	−8.06987e24	−0.559044
$446$	0	0
$447$	−7.07752e24	−0.469850
$448$	0	0
$449$	−9.70625e24	−0.617606	−0.308803	−	0.951126i	$-0.599928\pi$
$449$	−9.70625e24	−0.617606	−0.308803	+	0.951126i	$0.599928\pi$
$450$	0	0
$451$	2.83195e24	0.172746
$452$	0	0
$453$	−1.58359e24	−0.0926209
$454$	0	0
$455$	3.80262e24	0.213292
$456$	0	0
$457$	−2.30896e25	−1.24226	−0.621130	−	0.783708i	$-0.713326\pi$
$457$	−2.30896e25	−1.24226	−0.621130	+	0.783708i	$0.713326\pi$
$458$	0	0
$459$	−1.67223e25	−0.863130
$460$	0	0
$461$	2.85447e25	1.41373	0.706866	−	0.707347i	$-0.250108\pi$
$461$	2.85447e25	1.41373	0.706866	+	0.707347i	$0.250108\pi$
$462$	0	0
$463$	3.94099e25	1.87321	0.936603	−	0.350391i	$-0.113951\pi$
$463$	3.94099e25	1.87321	0.936603	+	0.350391i	$0.113951\pi$
$464$	0	0
$465$	1.52497e24	0.0695757
$466$	0	0
$467$	−1.57297e25	−0.688984	−0.344492	−	0.938789i	$-0.611949\pi$
$467$	−1.57297e25	−0.688984	−0.344492	+	0.938789i	$0.611949\pi$
$468$	0	0
$469$	−2.30227e25	−0.968310
$470$	0	0
$471$	−1.84641e25	−0.745811
$472$	0	0
$473$	−1.02928e25	−0.399351
$474$	0	0
$475$	3.62705e24	0.135196
$476$	0	0
$477$	1.20726e25	0.432391
$478$	0	0
$479$	−2.47573e25	−0.852149	−0.426074	−	0.904688i	$-0.640104\pi$
$479$	−2.47573e25	−0.852149	−0.426074	+	0.904688i	$0.640104\pi$
$480$	0	0
$481$	−1.11543e25	−0.369033
$482$	0	0
$483$	4.99804e23	0.0158965
$484$	0	0
$485$	2.92781e24	0.0895358
$486$	0	0
$487$	−5.53488e25	−1.62773	−0.813866	−	0.581053i	$-0.802641\pi$
$487$	−5.53488e25	−1.62773	−0.813866	+	0.581053i	$0.802641\pi$
$488$	0	0
$489$	2.23625e25	0.632536
$490$	0	0
$491$	−4.53655e25	−1.23439	−0.617193	−	0.786812i	$-0.711730\pi$
$491$	−4.53655e25	−1.23439	−0.617193	+	0.786812i	$0.711730\pi$
$492$	0	0
$493$	−4.40303e25	−1.15267
$494$	0	0
$495$	−3.67383e24	−0.0925486
$496$	0	0
$497$	2.08376e25	0.505197
$498$	0	0
$499$	−2.96310e25	−0.691497	−0.345749	−	0.938327i	$-0.612375\pi$
$499$	−2.96310e25	−0.691497	−0.345749	+	0.938327i	$0.612375\pi$
$500$	0	0
$501$	3.30442e24	0.0742398
$502$	0	0
$503$	−4.35477e25	−0.942040	−0.471020	−	0.882122i	$-0.656114\pi$
$503$	−4.35477e25	−0.942040	−0.471020	+	0.882122i	$0.656114\pi$
$504$	0	0
$505$	−3.53236e25	−0.735863
$506$	0	0
$507$	1.08400e25	0.217497
$508$	0	0
$509$	1.77095e25	0.342285	0.171143	−	0.985246i	$-0.445254\pi$
$509$	1.77095e25	0.342285	0.171143	+	0.985246i	$0.445254\pi$
$510$	0	0
$511$	−1.07975e24	−0.0201060
$512$	0	0
$513$	3.19698e25	0.573622
$514$	0	0
$515$	2.70469e25	0.467680
$516$	0	0
$517$	−9.59240e24	−0.159870
$518$	0	0
$519$	1.93756e25	0.311290
$520$	0	0
$521$	−2.24177e25	−0.347242	−0.173621	−	0.984813i	$-0.555547\pi$
$521$	−2.24177e25	−0.347242	−0.173621	+	0.984813i	$0.555547\pi$
$522$	0	0
$523$	1.32426e25	0.197792	0.0988959	−	0.995098i	$-0.468469\pi$
$523$	1.32426e25	0.197792	0.0988959	+	0.995098i	$0.468469\pi$
$524$	0	0
$525$	4.29498e24	0.0618654
$526$	0	0
$527$	−2.37608e25	−0.330112
$528$	0	0
$529$	−7.44184e25	−0.997359
$530$	0	0
$531$	−8.91153e25	−1.15227
$532$	0	0
$533$	−3.80872e25	−0.475193
$534$	0	0
$535$	−2.32991e25	−0.280529
$536$	0	0
$537$	1.17923e25	0.137038
$538$	0	0
$539$	1.39785e25	0.156808
$540$	0	0
$541$	−1.46186e26	−1.58319	−0.791594	−	0.611047i	$-0.790749\pi$
$541$	−1.46186e26	−1.58319	−0.791594	+	0.611047i	$0.790749\pi$
$542$	0	0
$543$	6.84007e25	0.715257
$544$	0	0
$545$	1.52219e25	0.153710
$546$	0	0
$547$	1.50096e26	1.46383	0.731913	−	0.681398i	$-0.238628\pi$
$547$	1.50096e26	1.46383	0.731913	+	0.681398i	$0.238628\pi$
$548$	0	0
$549$	8.99988e25	0.847812
$550$	0	0
$551$	8.41774e25	0.766047
$552$	0	0
$553$	−3.50332e25	−0.308028
$554$	0	0
$555$	−1.25986e25	−0.107038
$556$	0	0
$557$	4.24648e25	0.348662	0.174331	−	0.984687i	$-0.444224\pi$
$557$	4.24648e25	0.348662	0.174331	+	0.984687i	$0.444224\pi$
$558$	0	0
$559$	1.38430e26	1.09854
$560$	0	0
$561$	−1.70783e25	−0.131008
$562$	0	0
$563$	−2.34701e26	−1.74055	−0.870274	−	0.492569i	$-0.836058\pi$
$563$	−2.34701e26	−1.74055	−0.870274	+	0.492569i	$0.836058\pi$
$564$	0	0
$565$	4.60294e25	0.330046
$566$	0	0
$567$	−3.38229e25	−0.234516
$568$	0	0
$569$	−2.35991e25	−0.158245	−0.0791223	−	0.996865i	$-0.525212\pi$
$569$	−2.35991e25	−0.158245	−0.0791223	+	0.996865i	$0.525212\pi$
$570$	0	0
$571$	8.51927e25	0.552534	0.276267	−	0.961081i	$-0.410903\pi$
$571$	8.51927e25	0.552534	0.276267	+	0.961081i	$0.410903\pi$
$572$	0	0
$573$	−9.99482e23	−0.00627056
$574$	0	0
$575$	1.69340e24	0.0102781
$576$	0	0
$577$	−2.01229e26	−1.18173	−0.590867	−	0.806769i	$-0.701214\pi$
$577$	−2.01229e26	−1.18173	−0.590867	+	0.806769i	$0.701214\pi$
$578$	0	0
$579$	−1.39788e25	−0.0794374
$580$	0	0
$581$	2.62628e25	0.144434
$582$	0	0
$583$	2.83378e25	0.150839
$584$	0	0
$585$	4.94098e25	0.254585
$586$	0	0
$587$	7.07603e25	0.352962	0.176481	−	0.984304i	$-0.443529\pi$
$587$	7.07603e25	0.352962	0.176481	+	0.984304i	$0.443529\pi$
$588$	0	0
$589$	4.54261e25	0.219386
$590$	0	0
$591$	1.46357e26	0.684433
$592$	0	0
$593$	2.85195e26	1.29158	0.645792	−	0.763514i	$-0.276527\pi$
$593$	2.85195e26	1.29158	0.645792	+	0.763514i	$0.276527\pi$
$594$	0	0
$595$	−6.69210e25	−0.293529
$596$	0	0
$597$	−2.27029e24	−0.00964550
$598$	0	0
$599$	−1.03040e26	−0.424084	−0.212042	−	0.977261i	$-0.568011\pi$
$599$	−1.03040e26	−0.424084	−0.212042	+	0.977261i	$0.568011\pi$
$600$	0	0
$601$	2.73991e26	1.09252	0.546259	−	0.837616i	$-0.316051\pi$
$601$	2.73991e26	1.09252	0.546259	+	0.837616i	$0.316051\pi$
$602$	0	0
$603$	−2.99148e26	−1.15577
$604$	0	0
$605$	1.10828e26	0.414928
$606$	0	0
$607$	2.62590e26	0.952764	0.476382	−	0.879238i	$-0.341948\pi$
$607$	2.62590e26	0.952764	0.476382	+	0.879238i	$0.341948\pi$
$608$	0	0
$609$	9.96789e25	0.350541
$610$	0	0
$611$	1.29009e26	0.439774
$612$	0	0
$613$	4.36178e26	1.44142	0.720709	−	0.693238i	$-0.243817\pi$
$613$	4.36178e26	1.44142	0.720709	+	0.693238i	$0.243817\pi$
$614$	0	0
$615$	−4.30187e25	−0.137830
$616$	0	0
$617$	4.29936e26	1.33566	0.667829	−	0.744315i	$-0.267224\pi$
$617$	4.29936e26	1.33566	0.667829	+	0.744315i	$0.267224\pi$
$618$	0	0
$619$	3.31071e26	0.997378	0.498689	−	0.866781i	$-0.333815\pi$
$619$	3.31071e26	0.997378	0.498689	+	0.866781i	$0.333815\pi$
$620$	0	0
$621$	1.49261e25	0.0436089
$622$	0	0
$623$	−2.84655e26	−0.806645
$624$	0	0
$625$	1.45519e25	0.0400000
$626$	0	0
$627$	3.26504e25	0.0870658
$628$	0	0
$629$	1.96301e26	0.507858
$630$	0	0
$631$	5.41260e26	1.35871	0.679355	−	0.733810i	$-0.262260\pi$
$631$	5.41260e26	1.35871	0.679355	+	0.733810i	$0.262260\pi$
$632$	0	0
$633$	2.17698e26	0.530297
$634$	0	0
$635$	−5.23722e24	−0.0123808
$636$	0	0
$637$	−1.87999e26	−0.431350
$638$	0	0
$639$	2.70755e26	0.603001
$640$	0	0
$641$	−6.39311e26	−1.38217	−0.691084	−	0.722774i	$-0.742867\pi$
$641$	−6.39311e26	−1.38217	−0.691084	+	0.722774i	$0.742867\pi$
$642$	0	0
$643$	−2.29716e25	−0.0482155	−0.0241077	−	0.999709i	$-0.507674\pi$
$643$	−2.29716e25	−0.0482155	−0.0241077	+	0.999709i	$0.507674\pi$
$644$	0	0
$645$	1.56353e26	0.318632
$646$	0	0
$647$	5.90509e25	0.116852	0.0584259	−	0.998292i	$-0.481392\pi$
$647$	5.90509e25	0.116852	0.0584259	+	0.998292i	$0.481392\pi$
$648$	0	0
$649$	−2.09178e26	−0.401969
$650$	0	0
$651$	5.37914e25	0.100391
$652$	0	0
$653$	−3.76079e26	−0.681717	−0.340859	−	0.940115i	$-0.610718\pi$
$653$	−3.76079e26	−0.681717	−0.340859	+	0.940115i	$0.610718\pi$
$654$	0	0
$655$	2.05113e25	0.0361161
$656$	0	0
$657$	−1.40299e25	−0.0239984
$658$	0	0
$659$	−1.62154e26	−0.269474	−0.134737	−	0.990881i	$-0.543019\pi$
$659$	−1.62154e26	−0.269474	−0.134737	+	0.990881i	$0.543019\pi$
$660$	0	0
$661$	−7.54016e26	−1.21749	−0.608747	−	0.793365i	$-0.708327\pi$
$661$	−7.54016e26	−1.21749	−0.608747	+	0.793365i	$0.708327\pi$
$662$	0	0
$663$	2.29688e26	0.360380
$664$	0	0
$665$	1.27940e26	0.195074
$666$	0	0
$667$	3.93007e25	0.0582378
$668$	0	0
$669$	4.10987e26	0.591943
$670$	0	0
$671$	2.11252e26	0.295759
$672$	0	0
$673$	2.07669e26	0.282636	0.141318	−	0.989964i	$-0.454866\pi$
$673$	2.07669e26	0.282636	0.141318	+	0.989964i	$0.454866\pi$
$674$	0	0
$675$	1.28265e26	0.169715
$676$	0	0
$677$	−9.35329e26	−1.20329	−0.601647	−	0.798762i	$-0.705489\pi$
$677$	−9.35329e26	−1.20329	−0.601647	+	0.798762i	$0.705489\pi$
$678$	0	0
$679$	1.03275e26	0.129191
$680$	0	0
$681$	3.59843e26	0.437740
$682$	0	0
$683$	−2.56125e26	−0.303009	−0.151504	−	0.988457i	$-0.548412\pi$
$683$	−2.56125e26	−0.303009	−0.151504	+	0.988457i	$0.548412\pi$
$684$	0	0
$685$	−8.68748e25	−0.0999617
$686$	0	0
$687$	3.93909e26	0.440867
$688$	0	0
$689$	−3.81118e26	−0.414932
$690$	0	0
$691$	3.05164e25	0.0323215	0.0161608	−	0.999869i	$-0.494856\pi$
$691$	3.05164e25	0.0323215	0.0161608	+	0.999869i	$0.494856\pi$
$692$	0	0
$693$	−1.29590e26	−0.133538
$694$	0	0
$695$	4.03020e26	0.404083
$696$	0	0
$697$	6.70283e26	0.653953
$698$	0	0
$699$	−5.46604e26	−0.518967
$700$	0	0
$701$	2.30606e26	0.213083	0.106542	−	0.994308i	$-0.466022\pi$
$701$	2.30606e26	0.213083	0.106542	+	0.994308i	$0.466022\pi$
$702$	0	0
$703$	−3.75290e26	−0.337514
$704$	0	0
$705$	1.45713e26	0.127557
$706$	0	0
$707$	−1.24600e27	−1.06178
$708$	0	0
$709$	−4.21230e26	−0.349446	−0.174723	−	0.984618i	$-0.555903\pi$
$709$	−4.21230e26	−0.349446	−0.174723	+	0.984618i	$0.555903\pi$
$710$	0	0
$711$	−4.55207e26	−0.367661
$712$	0	0
$713$	2.12085e25	0.0166786
$714$	0	0
$715$	1.15978e26	0.0888117
$716$	0	0
$717$	1.01414e27	0.756254
$718$	0	0
$719$	9.93772e26	0.721709	0.360855	−	0.932622i	$-0.382485\pi$
$719$	9.93772e26	0.721709	0.360855	+	0.932622i	$0.382485\pi$
$720$	0	0
$721$	9.54047e26	0.674815
$722$	0	0
$723$	5.75527e26	0.396508
$724$	0	0
$725$	3.37724e26	0.226647
$726$	0	0
$727$	−1.91977e26	−0.125508	−0.0627541	−	0.998029i	$-0.519988\pi$
$727$	−1.91977e26	−0.125508	−0.0627541	+	0.998029i	$0.519988\pi$
$728$	0	0
$729$	1.99178e26	0.126861
$730$	0	0
$731$	−2.43618e27	−1.51180
$732$	0	0
$733$	−9.25884e26	−0.559846	−0.279923	−	0.960022i	$-0.590309\pi$
$733$	−9.25884e26	−0.559846	−0.279923	+	0.960022i	$0.590309\pi$
$734$	0	0
$735$	−2.12340e26	−0.125113
$736$	0	0
$737$	−7.02184e26	−0.403190
$738$	0	0
$739$	−3.30962e27	−1.85206	−0.926031	−	0.377447i	$-0.876802\pi$
$739$	−3.30962e27	−1.85206	−0.926031	+	0.377447i	$0.876802\pi$
$740$	0	0
$741$	−4.39119e26	−0.239502
$742$	0	0
$743$	8.92167e26	0.474299	0.237150	−	0.971473i	$-0.423787\pi$
$743$	8.92167e26	0.474299	0.237150	+	0.971473i	$0.423787\pi$
$744$	0	0
$745$	−8.45848e26	−0.438336
$746$	0	0
$747$	3.41249e26	0.172395
$748$	0	0
$749$	−8.21848e26	−0.404776
$750$	0	0
$751$	−3.75519e27	−1.80323	−0.901617	−	0.432535i	$-0.857619\pi$
$751$	−3.75519e27	−1.80323	−0.901617	+	0.432535i	$0.857619\pi$
$752$	0	0
$753$	−1.74003e27	−0.814714
$754$	0	0
$755$	−1.89258e26	−0.0864087
$756$	0	0
$757$	2.55308e27	1.13672	0.568360	−	0.822780i	$-0.307578\pi$
$757$	2.55308e27	1.13672	0.568360	+	0.822780i	$0.307578\pi$
$758$	0	0
$759$	1.52438e25	0.00661907
$760$	0	0
$761$	2.32427e27	0.984312	0.492156	−	0.870507i	$-0.336209\pi$
$761$	2.32427e27	0.984312	0.492156	+	0.870507i	$0.336209\pi$
$762$	0	0
$763$	5.36934e26	0.221788
$764$	0	0
$765$	−8.69545e26	−0.350355
$766$	0	0
$767$	2.81326e27	1.10574
$768$	0	0
$769$	−2.29996e27	−0.881902	−0.440951	−	0.897531i	$-0.645359\pi$
$769$	−2.29996e27	−0.881902	−0.440951	+	0.897531i	$0.645359\pi$
$770$	0	0
$771$	1.96014e27	0.733283
$772$	0	0
$773$	−3.73098e27	−1.36181	−0.680907	−	0.732369i	$-0.738414\pi$
$773$	−3.73098e27	−1.36181	−0.680907	+	0.732369i	$0.738414\pi$
$774$	0	0
$775$	1.82252e26	0.0649091
$776$	0	0
$777$	−4.44401e26	−0.154445
$778$	0	0
$779$	−1.28145e27	−0.434606
$780$	0	0
$781$	6.35537e26	0.210357
$782$	0	0
$783$	2.97679e27	0.961639
$784$	0	0
$785$	−2.20668e27	−0.695788
$786$	0	0
$787$	−2.23051e27	−0.686504	−0.343252	−	0.939243i	$-0.611529\pi$
$787$	−2.23051e27	−0.686504	−0.343252	+	0.939243i	$0.611529\pi$
$788$	0	0
$789$	1.43981e27	0.432586
$790$	0	0
$791$	1.62363e27	0.476224
$792$	0	0
$793$	−2.84116e27	−0.813579
$794$	0	0
$795$	−4.30465e26	−0.120351
$796$	0	0
$797$	−7.68734e26	−0.209856	−0.104928	−	0.994480i	$-0.533461\pi$
$797$	−7.68734e26	−0.209856	−0.104928	+	0.994480i	$0.533461\pi$
$798$	0	0
$799$	−2.27039e27	−0.605210
$800$	0	0
$801$	−3.69870e27	−0.962808
$802$	0	0
$803$	−3.29320e25	−0.00837183
$804$	0	0
$805$	5.97326e25	0.0148303
$806$	0	0
$807$	3.13272e27	0.759666
$808$	0	0
$809$	−6.91078e27	−1.63688	−0.818439	−	0.574594i	$-0.805160\pi$
$809$	−6.91078e27	−1.63688	−0.818439	+	0.574594i	$0.805160\pi$
$810$	0	0
$811$	−6.31140e27	−1.46025	−0.730125	−	0.683313i	$-0.760538\pi$
$811$	−6.31140e27	−1.46025	−0.730125	+	0.683313i	$0.760538\pi$
$812$	0	0
$813$	−3.85949e27	−0.872308
$814$	0	0
$815$	2.67258e27	0.590111
$816$	0	0
$817$	4.65749e27	1.00471
$818$	0	0
$819$	1.74287e27	0.367340
$820$	0	0
$821$	4.01177e27	0.826182	0.413091	−	0.910690i	$-0.364449\pi$
$821$	4.01177e27	0.826182	0.413091	+	0.910690i	$0.364449\pi$
$822$	0	0
$823$	−1.40648e27	−0.283032	−0.141516	−	0.989936i	$-0.545198\pi$
$823$	−1.40648e27	−0.283032	−0.141516	+	0.989936i	$0.545198\pi$
$824$	0	0
$825$	1.30995e26	0.0257598
$826$	0	0
$827$	−9.27282e27	−1.78201	−0.891003	−	0.453997i	$-0.849998\pi$
$827$	−9.27282e27	−1.78201	−0.891003	+	0.453997i	$0.849998\pi$
$828$	0	0
$829$	1.32863e26	0.0249538	0.0124769	−	0.999922i	$-0.496028\pi$
$829$	1.32863e26	0.0249538	0.0124769	+	0.999922i	$0.496028\pi$
$830$	0	0
$831$	6.64928e26	0.122058
$832$	0	0
$833$	3.30852e27	0.593617
$834$	0	0
$835$	3.94917e26	0.0692604
$836$	0	0
$837$	1.60642e27	0.275402
$838$	0	0
$839$	−2.42246e27	−0.405993	−0.202996	−	0.979179i	$-0.565068\pi$
$839$	−2.42246e27	−0.405993	−0.202996	+	0.979179i	$0.565068\pi$
$840$	0	0
$841$	1.73471e27	0.284227
$842$	0	0
$843$	1.56229e27	0.250265
$844$	0	0
$845$	1.29551e27	0.202909
$846$	0	0
$847$	3.90932e27	0.598700
$848$	0	0
$849$	6.06927e27	0.908894
$850$	0	0
$851$	−1.75215e26	−0.0256591
$852$	0	0
$853$	7.77262e27	1.11315	0.556573	−	0.830799i	$-0.312116\pi$
$853$	7.77262e27	1.11315	0.556573	+	0.830799i	$0.312116\pi$
$854$	0	0
$855$	1.66240e27	0.232840
$856$	0	0
$857$	−1.21292e27	−0.166155	−0.0830776	−	0.996543i	$-0.526475\pi$
$857$	−1.21292e27	−0.166155	−0.0830776	+	0.996543i	$0.526475\pi$
$858$	0	0
$859$	−6.87412e27	−0.921047	−0.460523	−	0.887648i	$-0.652338\pi$
$859$	−6.87412e27	−0.921047	−0.460523	+	0.887648i	$0.652338\pi$
$860$	0	0
$861$	−1.51743e27	−0.198875
$862$	0	0
$863$	1.05557e28	1.35327	0.676633	−	0.736320i	$-0.263438\pi$
$863$	1.05557e28	1.35327	0.676633	+	0.736320i	$0.263438\pi$
$864$	0	0
$865$	2.31561e27	0.290411
$866$	0	0
$867$	−1.35164e26	−0.0165836
$868$	0	0
$869$	−1.06850e27	−0.128258
$870$	0	0
$871$	9.44375e27	1.10910
$872$	0	0
$873$	1.34192e27	0.154202
$874$	0	0
$875$	5.13302e26	0.0577160
$876$	0	0
$877$	9.64195e27	1.06089	0.530443	−	0.847720i	$-0.322025\pi$
$877$	9.64195e27	1.06089	0.530443	+	0.847720i	$0.322025\pi$
$878$	0	0
$879$	−5.54388e27	−0.596924
$880$	0	0
$881$	3.92103e27	0.413170	0.206585	−	0.978429i	$-0.433765\pi$
$881$	3.92103e27	0.413170	0.206585	+	0.978429i	$0.433765\pi$
$882$	0	0
$883$	−1.77780e28	−1.83340	−0.916700	−	0.399576i	$-0.869157\pi$
$883$	−1.77780e28	−1.83340	−0.916700	+	0.399576i	$0.869157\pi$
$884$	0	0
$885$	3.17752e27	0.320721
$886$	0	0
$887$	1.83872e28	1.81652	0.908262	−	0.418401i	$-0.137409\pi$
$887$	1.83872e28	1.81652	0.908262	+	0.418401i	$0.137409\pi$
$888$	0	0
$889$	−1.84737e26	−0.0178643
$890$	0	0
$891$	−1.03159e27	−0.0976489
$892$	0	0
$893$	4.34054e27	0.402212
$894$	0	0
$895$	1.40932e27	0.127847
$896$	0	0
$897$	−2.05016e26	−0.0182079
$898$	0	0
$899$	4.22974e27	0.367787
$900$	0	0
$901$	6.70716e27	0.571022
$902$	0	0
$903$	5.51518e27	0.459755
$904$	0	0
$905$	8.17470e27	0.667283
$906$	0	0
$907$	6.73755e27	0.538559	0.269279	−	0.963062i	$-0.413215\pi$
$907$	6.73755e27	0.538559	0.269279	+	0.963062i	$0.413215\pi$
$908$	0	0
$909$	−1.61900e28	−1.26733
$910$	0	0
$911$	2.20708e28	1.69198	0.845988	−	0.533201i	$-0.179011\pi$
$911$	2.20708e28	1.69198	0.845988	+	0.533201i	$0.179011\pi$
$912$	0	0
$913$	8.01006e26	0.0601400
$914$	0	0
$915$	−3.20903e27	−0.235979
$916$	0	0
$917$	7.23512e26	0.0521119
$918$	0	0
$919$	1.67229e28	1.17982	0.589910	−	0.807469i	$-0.299163\pi$
$919$	1.67229e28	1.17982	0.589910	+	0.807469i	$0.299163\pi$
$920$	0	0
$921$	−5.42558e27	−0.374956
$922$	0	0
$923$	−8.54741e27	−0.578654
$924$	0	0
$925$	−1.50568e27	−0.0998589
$926$	0	0
$927$	1.23965e28	0.805456
$928$	0	0
$929$	−1.29778e28	−0.826136	−0.413068	−	0.910700i	$-0.635543\pi$
$929$	−1.29778e28	−0.826136	−0.413068	+	0.910700i	$0.635543\pi$
$930$	0	0
$931$	−6.32525e27	−0.394508
$932$	0	0
$933$	4.72359e27	0.288667
$934$	0	0
$935$	−2.04107e27	−0.122221
$936$	0	0
$937$	−5.01055e26	−0.0294008	−0.0147004	−	0.999892i	$-0.504679\pi$
$937$	−5.01055e26	−0.0294008	−0.0147004	+	0.999892i	$0.504679\pi$
$938$	0	0
$939$	−1.25517e28	−0.721736
$940$	0	0
$941$	−2.53378e28	−1.42780	−0.713899	−	0.700249i	$-0.753072\pi$
$941$	−2.53378e28	−1.42780	−0.713899	+	0.700249i	$0.753072\pi$
$942$	0	0
$943$	−5.98283e26	−0.0330404
$944$	0	0
$945$	4.52439e27	0.244882
$946$	0	0
$947$	2.06671e28	1.09636	0.548182	−	0.836359i	$-0.315320\pi$
$947$	2.06671e28	1.09636	0.548182	+	0.836359i	$0.315320\pi$
$948$	0	0
$949$	4.42906e26	0.0230294
$950$	0	0
$951$	4.67673e27	0.238357
$952$	0	0
$953$	2.47193e28	1.23496	0.617481	−	0.786586i	$-0.288153\pi$
$953$	2.47193e28	1.23496	0.617481	+	0.786586i	$0.288153\pi$
$954$	0	0
$955$	−1.19450e26	−0.00584998
$956$	0	0
$957$	3.04017e27	0.145960
$958$	0	0
$959$	−3.06441e27	−0.144235
$960$	0	0
$961$	−1.93881e28	−0.894670
$962$	0	0
$963$	−1.06788e28	−0.483138
$964$	0	0
$965$	−1.67064e27	−0.0741094
$966$	0	0
$967$	1.41019e28	0.613376	0.306688	−	0.951810i	$-0.400779\pi$
$967$	1.41019e28	0.613376	0.306688	+	0.951810i	$0.400779\pi$
$968$	0	0
$969$	7.72791e27	0.329599
$970$	0	0
$971$	−4.18848e27	−0.175176	−0.0875878	−	0.996157i	$-0.527916\pi$
$971$	−4.18848e27	−0.175176	−0.0875878	+	0.996157i	$0.527916\pi$
$972$	0	0
$973$	1.42160e28	0.583051
$974$	0	0
$975$	−1.76177e27	−0.0708607
$976$	0	0
$977$	3.08455e28	1.21673	0.608364	−	0.793658i	$-0.291826\pi$
$977$	3.08455e28	1.21673	0.608364	+	0.793658i	$0.291826\pi$
$978$	0	0
$979$	−8.68187e27	−0.335875
$980$	0	0
$981$	6.97671e27	0.264725
$982$	0	0
$983$	−4.04845e28	−1.50671	−0.753356	−	0.657613i	$-0.771566\pi$
$983$	−4.04845e28	−1.50671	−0.753356	+	0.657613i	$0.771566\pi$
$984$	0	0
$985$	1.74914e28	0.638527
$986$	0	0
$987$	5.13987e27	0.184051
$988$	0	0
$989$	2.17449e27	0.0763822
$990$	0	0
$991$	1.82579e28	0.629144	0.314572	−	0.949234i	$-0.398139\pi$
$991$	1.82579e28	0.629144	0.314572	+	0.949234i	$0.398139\pi$
$992$	0	0
$993$	−2.42485e28	−0.819723
$994$	0	0
$995$	−2.71327e26	−0.00899856
$996$	0	0
$997$	−4.82287e27	−0.156928	−0.0784642	−	0.996917i	$-0.525002\pi$
$997$	−4.82287e27	−0.156928	−0.0784642	+	0.996917i	$0.525002\pi$
$998$	0	0
$999$	−1.32715e28	−0.423690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.20.a.e.1.2		3
4.3	odd	2		20.20.a.a.1.2	✓	3
20.3	even	4		100.20.c.b.49.4		6
20.7	even	4		100.20.c.b.49.3		6
20.19	odd	2		100.20.a.b.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.20.a.a.1.2	✓	3	4.3	odd	2
80.20.a.e.1.2		3	1.1	even	1	trivial
100.20.a.b.1.2		3	20.19	odd	2
100.20.c.b.49.3		6	20.7	even	4
100.20.c.b.49.4		6	20.3	even	4

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:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(f(q)\)	\(=\)	\(q-16342.5 q^{3} -1.95312e6 q^{5} -6.88942e7 q^{7} -8.95184e8 q^{9} +O(q^{10})\)	\(q-16342.5 q^{3} -1.95312e6 q^{5} -6.88942e7 q^{7} -8.95184e8 q^{9} -2.10125e9 q^{11} +2.82599e10 q^{13} +3.19190e10 q^{15} -4.97336e11 q^{17} +9.50809e11 q^{19} +1.12590e12 q^{21} +4.43914e11 q^{23} +3.81470e12 q^{25} +3.36238e13 q^{27} +8.85323e13 q^{29} +4.77762e13 q^{31} +3.43396e13 q^{33} +1.34559e14 q^{35} -3.94706e14 q^{37} -4.61837e14 q^{39} -1.34775e15 q^{41} +4.89845e15 q^{43} +1.74841e15 q^{45} +4.56510e15 q^{47} -6.65249e15 q^{49} +8.12772e15 q^{51} -1.34862e16 q^{53} +4.10400e15 q^{55} -1.55386e16 q^{57} +9.95497e16 q^{59} -1.00537e17 q^{61} +6.16730e16 q^{63} -5.51951e16 q^{65} +3.34175e17 q^{67} -7.25466e15 q^{69} -3.02457e17 q^{71} +1.56726e16 q^{73} -6.23417e16 q^{75} +1.44764e17 q^{77} +5.08507e17 q^{79} +4.90940e17 q^{81} -3.81205e17 q^{83} +9.71359e17 q^{85} -1.44684e18 q^{87} +4.13177e18 q^{89} -1.94694e18 q^{91} -7.80783e17 q^{93} -1.85705e18 q^{95} -1.49904e18 q^{97} +1.88100e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 34086 q^{3} - 5859375 q^{5} + 115130574 q^{7} + 3129228027 q^{9}+O(q^{10}) \) 3 * q - 34086 * q^3 - 5859375 * q^5 + 115130574 * q^7 + 3129228027 * q^9	\( 3 q - 34086 q^{3} - 5859375 q^{5} + 115130574 q^{7} + 3129228027 q^{9} - 6359213280 q^{11} - 33996539826 q^{13} + 66574218750 q^{15} - 60515110578 q^{17} + 1436218405908 q^{19} - 7119772762332 q^{21} - 1682066292342 q^{23} + 11444091796875 q^{25} - 91316335875732 q^{27} - 181580995192278 q^{29} + 3566464773252 q^{31} - 90839870844480 q^{33} - 224864402343750 q^{35} + 653714901466206 q^{37} + 43\!\cdots\!08 q^{39}+ \cdots - 37\!\cdots\!20 q^{99}+O(q^{100}) \) 3 * q - 34086 * q^3 - 5859375 * q^5 + 115130574 * q^7 + 3129228027 * q^9 - 6359213280 * q^11 - 33996539826 * q^13 + 66574218750 * q^15 - 60515110578 * q^17 + 1436218405908 * q^19 - 7119772762332 * q^21 - 1682066292342 * q^23 + 11444091796875 * q^25 - 91316335875732 * q^27 - 181580995192278 * q^29 + 3566464773252 * q^31 - 90839870844480 * q^33 - 224864402343750 * q^35 + 653714901466206 * q^37 + 4339017394217508 * q^39 - 1257752912210622 * q^41 + 4361040178994610 * q^43 - 6111773490234375 * q^45 + 23174987171661798 * q^47 - 5454428919168921 * q^49 + 33371959424606244 * q^51 - 12180381712114362 * q^53 + 12420338437500000 * q^55 - 177419957801365416 * q^57 + 180023573560918116 * q^59 - 213491197168562514 * q^61 + 549307385783925246 * q^63 + 66399491847656250 * q^65 + 32494317103156134 * q^67 - 498099089220386004 * q^69 + 245957452394007276 * q^71 - 94437681636898506 * q^73 - 130027770996093750 * q^75 - 72972512123263680 * q^77 - 819623047726342536 * q^79 + 3159048884904329967 * q^81 - 2812828487653823982 * q^83 + 118193575347656250 * q^85 + 2510285506713629676 * q^87 + 4879439966436006078 * q^89 - 12213088562709029652 * q^91 + 16020238495381453176 * q^93 - 2805114074039062500 * q^95 + 8189756396617107846 * q^97 - 3795512975671396320 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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