LMFDB - Embedded newform 80.18.a.l.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [80,18,Mod(1,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.1"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	80.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,15384] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$146.577669876$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 2573495x^{3} + 1741012708x^{2} - 129847160472x - 10015787544672 $ x^5 - 2x^4 - 2573495x^3 + 1741012708x^2 - 129847160472x - 10015787544672
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{4}\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$848.005$ 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)

$f(q)$	$=$	$q+15641.7 q^{3} +390625. q^{5} -2.95100e7 q^{7} +1.15523e8 q^{9} -5.55492e8 q^{11} +5.05164e9 q^{13} +6.11004e9 q^{15} +2.65821e10 q^{17} -6.99046e10 q^{19} -4.61586e11 q^{21} +2.17498e11 q^{23} +1.52588e11 q^{25} -2.13002e11 q^{27} -2.64418e12 q^{29} +5.06233e12 q^{31} -8.68884e12 q^{33} -1.15273e13 q^{35} -1.35581e13 q^{37} +7.90162e13 q^{39} +6.27018e13 q^{41} +1.30526e13 q^{43} +4.51260e13 q^{45} -1.38441e14 q^{47} +6.38209e14 q^{49} +4.15789e14 q^{51} +4.87513e14 q^{53} -2.16989e14 q^{55} -1.09343e15 q^{57} -1.22412e14 q^{59} -8.94475e14 q^{61} -3.40907e15 q^{63} +1.97330e15 q^{65} +4.14519e15 q^{67} +3.40203e15 q^{69} +6.78083e15 q^{71} +1.00658e16 q^{73} +2.38673e15 q^{75} +1.63926e16 q^{77} +1.92777e16 q^{79} -1.82503e16 q^{81} -1.95626e16 q^{83} +1.03836e16 q^{85} -4.13594e16 q^{87} +5.91887e16 q^{89} -1.49074e17 q^{91} +7.91834e16 q^{93} -2.73065e16 q^{95} -1.03756e17 q^{97} -6.41719e16 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 15384 q^{3} + 1953125 q^{5} - 312868 q^{7} + 410286417 q^{9} - 757119716 q^{11} + 2827963478 q^{13} + 6009375000 q^{15} + 24776563114 q^{17} - 811272116 q^{19} + 332870780352 q^{21} + 73100431588 q^{23}+ \cdots - 35\!\cdots\!44 q^{99}+O(q^{100}) $ 5 * q + 15384 * q^3 + 1953125 * q^5 - 312868 * q^7 + 410286417 * q^9 - 757119716 * q^11 + 2827963478 * q^13 + 6009375000 * q^15 + 24776563114 * q^17 - 811272116 * q^19 + 332870780352 * q^21 + 73100431588 * q^23 + 762939453125 * q^25 + 3166101346032 * q^27 - 3954378361738 * q^29 + 12340046636056 * q^31 - 11191253546784 * q^33 - 122214062500 * q^35 + 19853810999454 * q^37 - 26906648601264 * q^39 + 70876127296194 * q^41 + 133955994850144 * q^43 + 160268131640625 * q^45 + 113374769936500 * q^47 + 734065948219149 * q^49 - 91587467914320 * q^51 - 232473412424130 * q^53 - 295749889062500 * q^55 + 192839487986976 * q^57 - 2891505753119596 * q^59 - 120759403445194 * q^61 + 75256025229420 * q^63 + 1104673233593750 * q^65 - 7139319995171008 * q^67 + 12678898772781888 * q^69 - 5858971441767504 * q^71 + 22237456819273602 * q^73 + 2347412109375000 * q^75 + 19042533632763664 * q^77 + 11793240059085792 * q^79 + 52885709091819405 * q^81 - 49375024849026904 * q^83 + 9678344966406250 * q^85 - 129320807184056112 * q^87 + 102588609132622626 * q^89 - 260090514257055160 * q^91 + 342096413075911680 * q^93 - 316903170312500 * q^95 + 157786395292410650 * q^97 - 355232381581465044 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0	0
$2$	0	0
$3$	15641.7	1.37643	0.688214	−	0.725508i	$-0.258395\pi$
$3$	15641.7	1.37643	0.688214	+	0.725508i	$0.258395\pi$
$4$	0	0
$5$	390625.	0.447214
$5$	390625.	0.447214
$6$	0	0
$7$	−2.95100e7	−1.93480	−0.967399	−	0.253257i	$-0.918498\pi$
$7$	−2.95100e7	−1.93480	−0.967399	+	0.253257i	$0.918498\pi$
$8$	0	0
$9$	1.15523e8	0.894552
$10$	0	0
$11$	−5.55492e8	−0.781340	−0.390670	−	0.920531i	$-0.627757\pi$
$11$	−5.55492e8	−0.781340	−0.390670	+	0.920531i	$0.627757\pi$
$12$	0	0
$13$	5.05164e9	1.71757	0.858784	−	0.512339i	$-0.171221\pi$
$13$	5.05164e9	1.71757	0.858784	+	0.512339i	$0.171221\pi$
$14$	0	0
$15$	6.11004e9	0.615557
$16$	0	0
$17$	2.65821e10	0.924216	0.462108	−	0.886824i	$-0.347093\pi$
$17$	2.65821e10	0.924216	0.462108	+	0.886824i	$0.347093\pi$
$18$	0	0
$19$	−6.99046e10	−0.944278	−0.472139	−	0.881524i	$-0.656518\pi$
$19$	−6.99046e10	−0.944278	−0.472139	+	0.881524i	$0.656518\pi$
$20$	0	0
$21$	−4.61586e11	−2.66311
$22$	0	0
$23$	2.17498e11	0.579119	0.289560	−	0.957160i	$-0.406491\pi$
$23$	2.17498e11	0.579119	0.289560	+	0.957160i	$0.406491\pi$
$24$	0	0
$25$	1.52588e11	0.200000
$26$	0	0
$27$	−2.13002e11	−0.145141
$28$	0	0
$29$	−2.64418e12	−0.981538	−0.490769	−	0.871290i	$-0.663284\pi$
$29$	−2.64418e12	−0.981538	−0.490769	+	0.871290i	$0.663284\pi$
$30$	0	0
$31$	5.06233e12	1.06605	0.533023	−	0.846101i	$-0.321056\pi$
$31$	5.06233e12	1.06605	0.533023	+	0.846101i	$0.321056\pi$
$32$	0	0
$33$	−8.68884e12	−1.07546
$34$	0	0
$35$	−1.15273e13	−0.865268
$36$	0	0
$37$	−1.35581e13	−0.634575	−0.317288	−	0.948329i	$-0.602772\pi$
$37$	−1.35581e13	−0.634575	−0.317288	+	0.948329i	$0.602772\pi$
$38$	0	0
$39$	7.90162e13	2.36411
$40$	0	0
$41$	6.27018e13	1.22636	0.613179	−	0.789944i	$-0.289891\pi$
$41$	6.27018e13	1.22636	0.613179	+	0.789944i	$0.289891\pi$
$42$	0	0
$43$	1.30526e13	0.170300	0.0851499	−	0.996368i	$-0.472863\pi$
$43$	1.30526e13	0.170300	0.0851499	+	0.996368i	$0.472863\pi$
$44$	0	0
$45$	4.51260e13	0.400056
$46$	0	0
$47$	−1.38441e14	−0.848074	−0.424037	−	0.905645i	$-0.639387\pi$
$47$	−1.38441e14	−0.848074	−0.424037	+	0.905645i	$0.639387\pi$
$48$	0	0
$49$	6.38209e14	2.74344
$50$	0	0
$51$	4.15789e14	1.27212
$52$	0	0
$53$	4.87513e14	1.07558	0.537788	−	0.843080i	$-0.319260\pi$
$53$	4.87513e14	1.07558	0.537788	+	0.843080i	$0.319260\pi$
$54$	0	0
$55$	−2.16989e14	−0.349426
$56$	0	0
$57$	−1.09343e15	−1.29973
$58$	0	0
$59$	−1.22412e14	−0.108538	−0.0542689	−	0.998526i	$-0.517283\pi$
$59$	−1.22412e14	−0.108538	−0.0542689	+	0.998526i	$0.517283\pi$
$60$	0	0
$61$	−8.94475e14	−0.597399	−0.298700	−	0.954347i	$-0.596553\pi$
$61$	−8.94475e14	−0.597399	−0.298700	+	0.954347i	$0.596553\pi$
$62$	0	0
$63$	−3.40907e15	−1.73078
$64$	0	0
$65$	1.97330e15	0.768119
$66$	0	0
$67$	4.14519e15	1.24712	0.623561	−	0.781775i	$-0.285685\pi$
$67$	4.14519e15	1.24712	0.623561	+	0.781775i	$0.285685\pi$
$68$	0	0
$69$	3.40203e15	0.797116
$70$	0	0
$71$	6.78083e15	1.24620	0.623099	−	0.782143i	$-0.285873\pi$
$71$	6.78083e15	1.24620	0.623099	+	0.782143i	$0.285873\pi$
$72$	0	0
$73$	1.00658e16	1.46085	0.730423	−	0.682995i	$-0.239323\pi$
$73$	1.00658e16	1.46085	0.730423	+	0.682995i	$0.239323\pi$
$74$	0	0
$75$	2.38673e15	0.275285
$76$	0	0
$77$	1.63926e16	1.51174
$78$	0	0
$79$	1.92777e16	1.42963	0.714815	−	0.699313i	$-0.246511\pi$
$79$	1.92777e16	1.42963	0.714815	+	0.699313i	$0.246511\pi$
$80$	0	0
$81$	−1.82503e16	−1.09433
$82$	0	0
$83$	−1.95626e16	−0.953374	−0.476687	−	0.879073i	$-0.658163\pi$
$83$	−1.95626e16	−0.953374	−0.476687	+	0.879073i	$0.658163\pi$
$84$	0	0
$85$	1.03836e16	0.413322
$86$	0	0
$87$	−4.13594e16	−1.35102
$88$	0	0
$89$	5.91887e16	1.59376	0.796881	−	0.604136i	$-0.206482\pi$
$89$	5.91887e16	1.59376	0.796881	+	0.604136i	$0.206482\pi$
$90$	0	0
$91$	−1.49074e17	−3.32315
$92$	0	0
$93$	7.91834e16	1.46733
$94$	0	0
$95$	−2.73065e16	−0.422294
$96$	0	0
$97$	−1.03756e17	−1.34417	−0.672085	−	0.740474i	$-0.734601\pi$
$97$	−1.03756e17	−1.34417	−0.672085	+	0.740474i	$0.734601\pi$
$98$	0	0
$99$	−6.41719e16	−0.698949
$100$	0	0
$101$	4.82709e16	0.443562	0.221781	−	0.975097i	$-0.428813\pi$
$101$	4.82709e16	0.443562	0.221781	+	0.975097i	$0.428813\pi$
$102$	0	0
$103$	6.82307e16	0.530717	0.265359	−	0.964150i	$-0.414510\pi$
$103$	6.82307e16	0.530717	0.265359	+	0.964150i	$0.414510\pi$
$104$	0	0
$105$	−1.80307e17	−1.19098
$106$	0	0
$107$	2.80815e16	0.158000	0.0790002	−	0.996875i	$-0.474827\pi$
$107$	2.80815e16	0.158000	0.0790002	+	0.996875i	$0.474827\pi$
$108$	0	0
$109$	1.79201e17	0.861421	0.430710	−	0.902490i	$-0.358263\pi$
$109$	1.79201e17	0.861421	0.430710	+	0.902490i	$0.358263\pi$
$110$	0	0
$111$	−2.12071e17	−0.873447
$112$	0	0
$113$	−1.65683e17	−0.586289	−0.293145	−	0.956068i	$-0.594702\pi$
$113$	−1.65683e17	−0.586289	−0.293145	+	0.956068i	$0.594702\pi$
$114$	0	0
$115$	8.49600e16	0.258990
$116$	0	0
$117$	5.83578e17	1.53645
$118$	0	0
$119$	−7.84437e17	−1.78817
$120$	0	0
$121$	−1.96876e17	−0.389508
$122$	0	0
$123$	9.80762e17	1.68799
$124$	0	0
$125$	5.96046e16	0.0894427
$126$	0	0
$127$	1.29174e18	1.69373	0.846866	−	0.531806i	$-0.178487\pi$
$127$	1.29174e18	1.69373	0.846866	+	0.531806i	$0.178487\pi$
$128$	0	0
$129$	2.04164e17	0.234405
$130$	0	0
$131$	−1.02415e18	−1.03171	−0.515856	−	0.856675i	$-0.672526\pi$
$131$	−1.02415e18	−1.03171	−0.515856	+	0.856675i	$0.672526\pi$
$132$	0	0
$133$	2.06288e18	1.82699
$134$	0	0
$135$	−8.32039e16	−0.0649092
$136$	0	0
$137$	5.34170e17	0.367751	0.183876	−	0.982950i	$-0.441136\pi$
$137$	5.34170e17	0.367751	0.183876	+	0.982950i	$0.441136\pi$
$138$	0	0
$139$	1.18809e18	0.723140	0.361570	−	0.932345i	$-0.382241\pi$
$139$	1.18809e18	0.723140	0.361570	+	0.932345i	$0.382241\pi$
$140$	0	0
$141$	−2.16546e18	−1.16731
$142$	0	0
$143$	−2.80615e18	−1.34200
$144$	0	0
$145$	−1.03288e18	−0.438957
$146$	0	0
$147$	9.98267e18	3.77615
$148$	0	0
$149$	4.52594e18	1.52625	0.763125	−	0.646251i	$-0.223664\pi$
$149$	4.52594e18	1.52625	0.763125	+	0.646251i	$0.223664\pi$
$150$	0	0
$151$	3.16614e18	0.953293	0.476647	−	0.879095i	$-0.341852\pi$
$151$	3.16614e18	0.953293	0.476647	+	0.879095i	$0.341852\pi$
$152$	0	0
$153$	3.07083e18	0.826759
$154$	0	0
$155$	1.97747e18	0.476750
$156$	0	0
$157$	−7.10229e18	−1.53551	−0.767753	−	0.640746i	$-0.778625\pi$
$157$	−7.10229e18	−1.53551	−0.767753	+	0.640746i	$0.778625\pi$
$158$	0	0
$159$	7.62553e18	1.48045
$160$	0	0
$161$	−6.41835e18	−1.12048
$162$	0	0
$163$	3.33159e17	0.0523670	0.0261835	−	0.999657i	$-0.491665\pi$
$163$	3.33159e17	0.0523670	0.0261835	+	0.999657i	$0.491665\pi$
$164$	0	0
$165$	−3.39408e18	−0.480959
$166$	0	0
$167$	1.26039e19	1.61218	0.806092	−	0.591791i	$-0.201579\pi$
$167$	1.26039e19	1.61218	0.806092	+	0.591791i	$0.201579\pi$
$168$	0	0
$169$	1.68686e19	1.95004
$170$	0	0
$171$	−8.07556e18	−0.844706
$172$	0	0
$173$	−1.16500e19	−1.10391	−0.551957	−	0.833872i	$-0.686119\pi$
$173$	−1.16500e19	−1.10391	−0.551957	+	0.833872i	$0.686119\pi$
$174$	0	0
$175$	−4.50287e18	−0.386960
$176$	0	0
$177$	−1.91473e18	−0.149394
$178$	0	0
$179$	−7.59523e18	−0.538629	−0.269315	−	0.963052i	$-0.586797\pi$
$179$	−7.59523e18	−0.538629	−0.269315	+	0.963052i	$0.586797\pi$
$180$	0	0
$181$	2.11303e19	1.36344	0.681722	−	0.731612i	$-0.261231\pi$
$181$	2.11303e19	1.36344	0.681722	+	0.731612i	$0.261231\pi$
$182$	0	0
$183$	−1.39911e19	−0.822276
$184$	0	0
$185$	−5.29612e18	−0.283791
$186$	0	0
$187$	−1.47661e19	−0.722127
$188$	0	0
$189$	6.28568e18	0.280819
$190$	0	0
$191$	3.67685e19	1.50208	0.751039	−	0.660258i	$-0.229553\pi$
$191$	3.67685e19	1.50208	0.751039	+	0.660258i	$0.229553\pi$
$192$	0	0
$193$	−6.98109e17	−0.0261027	−0.0130514	−	0.999915i	$-0.504154\pi$
$193$	−6.98109e17	−0.0261027	−0.0130514	+	0.999915i	$0.504154\pi$
$194$	0	0
$195$	3.08657e19	1.05726
$196$	0	0
$197$	3.61885e19	1.13660	0.568300	−	0.822821i	$-0.307601\pi$
$197$	3.61885e19	1.13660	0.568300	+	0.822821i	$0.307601\pi$
$198$	0	0
$199$	−5.37339e19	−1.54881	−0.774404	−	0.632692i	$-0.781950\pi$
$199$	−5.37339e19	−1.54881	−0.774404	+	0.632692i	$0.781950\pi$
$200$	0	0
$201$	6.48379e19	1.71657
$202$	0	0
$203$	7.80296e19	1.89908
$204$	0	0
$205$	2.44929e19	0.548444
$206$	0	0
$207$	2.51259e19	0.518052
$208$	0	0
$209$	3.88314e19	0.737802
$210$	0	0
$211$	−5.33853e19	−0.935451	−0.467726	−	0.883874i	$-0.654926\pi$
$211$	−5.33853e19	−0.935451	−0.467726	+	0.883874i	$0.654926\pi$
$212$	0	0
$213$	1.06064e20	1.71530
$214$	0	0
$215$	5.09866e18	0.0761604
$216$	0	0
$217$	−1.49389e20	−2.06258
$218$	0	0
$219$	1.57446e20	2.01075
$220$	0	0
$221$	1.34283e20	1.58740
$222$	0	0
$223$	−5.66288e19	−0.620078	−0.310039	−	0.950724i	$-0.600342\pi$
$223$	−5.66288e19	−0.620078	−0.310039	+	0.950724i	$0.600342\pi$
$224$	0	0
$225$	1.76273e19	0.178910
$226$	0	0
$227$	−1.08803e20	−1.02429	−0.512144	−	0.858900i	$-0.671148\pi$
$227$	−1.08803e20	−1.02429	−0.512144	+	0.858900i	$0.671148\pi$
$228$	0	0
$229$	5.69405e19	0.497530	0.248765	−	0.968564i	$-0.419975\pi$
$229$	5.69405e19	0.497530	0.248765	+	0.968564i	$0.419975\pi$
$230$	0	0
$231$	2.56408e20	2.08079
$232$	0	0
$233$	2.80449e19	0.211509	0.105754	−	0.994392i	$-0.466274\pi$
$233$	2.80449e19	0.211509	0.105754	+	0.994392i	$0.466274\pi$
$234$	0	0
$235$	−5.40786e19	−0.379270
$236$	0	0
$237$	3.01535e20	1.96778
$238$	0	0
$239$	1.04425e20	0.634485	0.317243	−	0.948344i	$-0.397243\pi$
$239$	1.04425e20	0.634485	0.317243	+	0.948344i	$0.397243\pi$
$240$	0	0
$241$	−2.71252e20	−1.53542	−0.767710	−	0.640797i	$-0.778604\pi$
$241$	−2.71252e20	−1.53542	−0.767710	+	0.640797i	$0.778604\pi$
$242$	0	0
$243$	−2.57959e20	−1.36112
$244$	0	0
$245$	2.49300e20	1.22690
$246$	0	0
$247$	−3.53133e20	−1.62186
$248$	0	0
$249$	−3.05993e20	−1.31225
$250$	0	0
$251$	−1.57255e20	−0.630053	−0.315026	−	0.949083i	$-0.602013\pi$
$251$	−1.57255e20	−0.630053	−0.315026	+	0.949083i	$0.602013\pi$
$252$	0	0
$253$	−1.20818e20	−0.452489
$254$	0	0
$255$	1.62418e20	0.568908
$256$	0	0
$257$	7.56388e19	0.247921	0.123960	−	0.992287i	$-0.460440\pi$
$257$	7.56388e19	0.247921	0.123960	+	0.992287i	$0.460440\pi$
$258$	0	0
$259$	4.00098e20	1.22777
$260$	0	0
$261$	−3.05462e20	−0.878037
$262$	0	0
$263$	4.38635e20	1.18162	0.590812	−	0.806809i	$-0.298808\pi$
$263$	4.38635e20	1.18162	0.590812	+	0.806809i	$0.298808\pi$
$264$	0	0
$265$	1.90435e20	0.481013
$266$	0	0
$267$	9.25811e20	2.19370
$268$	0	0
$269$	2.87224e20	0.638743	0.319371	−	0.947630i	$-0.396528\pi$
$269$	2.87224e20	0.638743	0.319371	+	0.947630i	$0.396528\pi$
$270$	0	0
$271$	−2.83260e20	−0.591488	−0.295744	−	0.955267i	$-0.595567\pi$
$271$	−2.83260e20	−0.591488	−0.295744	+	0.955267i	$0.595567\pi$
$272$	0	0
$273$	−2.33177e21	−4.57407
$274$	0	0
$275$	−8.47614e19	−0.156268
$276$	0	0
$277$	−4.79051e20	−0.830431	−0.415215	−	0.909723i	$-0.636294\pi$
$277$	−4.79051e20	−0.830431	−0.415215	+	0.909723i	$0.636294\pi$
$278$	0	0
$279$	5.84813e20	0.953633
$280$	0	0
$281$	−1.51762e20	−0.232894	−0.116447	−	0.993197i	$-0.537150\pi$
$281$	−1.51762e20	−0.232894	−0.116447	+	0.993197i	$0.537150\pi$
$282$	0	0
$283$	−1.08098e21	−1.56183	−0.780916	−	0.624637i	$-0.785247\pi$
$283$	−1.08098e21	−1.56183	−0.780916	+	0.624637i	$0.785247\pi$
$284$	0	0
$285$	−4.27120e20	−0.581257
$286$	0	0
$287$	−1.85033e21	−2.37275
$288$	0	0
$289$	−1.20632e20	−0.145825
$290$	0	0
$291$	−1.62292e21	−1.85015
$292$	0	0
$293$	5.64181e20	0.606797	0.303398	−	0.952864i	$-0.401879\pi$
$293$	5.64181e20	0.606797	0.303398	+	0.952864i	$0.401879\pi$
$294$	0	0
$295$	−4.78171e19	−0.0485396
$296$	0	0
$297$	1.18321e20	0.113405
$298$	0	0
$299$	1.09872e21	0.994676
$300$	0	0
$301$	−3.85181e20	−0.329496
$302$	0	0
$303$	7.55039e20	0.610530
$304$	0	0
$305$	−3.49404e20	−0.267165
$306$	0	0
$307$	7.86538e20	0.568910	0.284455	−	0.958689i	$-0.408187\pi$
$307$	7.86538e20	0.568910	0.284455	+	0.958689i	$0.408187\pi$
$308$	0	0
$309$	1.06724e21	0.730494
$310$	0	0
$311$	−1.85897e21	−1.20451	−0.602254	−	0.798304i	$-0.705731\pi$
$311$	−1.85897e21	−1.20451	−0.602254	+	0.798304i	$0.705731\pi$
$312$	0	0
$313$	1.01811e21	0.624694	0.312347	−	0.949968i	$-0.398885\pi$
$313$	1.01811e21	0.624694	0.312347	+	0.949968i	$0.398885\pi$
$314$	0	0
$315$	−1.33167e21	−0.774027
$316$	0	0
$317$	3.16819e20	0.174505	0.0872523	−	0.996186i	$-0.472191\pi$
$317$	3.16819e20	0.174505	0.0872523	+	0.996186i	$0.472191\pi$
$318$	0	0
$319$	1.46882e21	0.766915
$320$	0	0
$321$	4.39243e20	0.217476
$322$	0	0
$323$	−1.85821e21	−0.872717
$324$	0	0
$325$	7.70819e20	0.343513
$326$	0	0
$327$	2.80301e21	1.18568
$328$	0	0
$329$	4.08540e21	1.64085
$330$	0	0
$331$	−9.33600e20	−0.356142	−0.178071	−	0.984018i	$-0.556986\pi$
$331$	−9.33600e20	−0.356142	−0.178071	+	0.984018i	$0.556986\pi$
$332$	0	0
$333$	−1.56626e21	−0.567661
$334$	0	0
$335$	1.61922e21	0.557730
$336$	0	0
$337$	−1.02437e21	−0.335429	−0.167715	−	0.985836i	$-0.553639\pi$
$337$	−1.02437e21	−0.335429	−0.167715	+	0.985836i	$0.553639\pi$
$338$	0	0
$339$	−2.59157e21	−0.806984
$340$	0	0
$341$	−2.81208e21	−0.832944
$342$	0	0
$343$	−1.19686e22	−3.37321
$344$	0	0
$345$	1.32892e21	0.356481
$346$	0	0
$347$	6.75262e21	1.72453	0.862267	−	0.506453i	$-0.169044\pi$
$347$	6.75262e21	1.72453	0.862267	+	0.506453i	$0.169044\pi$
$348$	0	0
$349$	1.40155e21	0.340872	0.170436	−	0.985369i	$-0.445482\pi$
$349$	1.40155e21	0.340872	0.170436	+	0.985369i	$0.445482\pi$
$350$	0	0
$351$	−1.07601e21	−0.249290
$352$	0	0
$353$	−5.00648e21	−1.10522	−0.552609	−	0.833441i	$-0.686368\pi$
$353$	−5.00648e21	−1.10522	−0.552609	+	0.833441i	$0.686368\pi$
$354$	0	0
$355$	2.64876e21	0.557316
$356$	0	0
$357$	−1.22699e22	−2.46129
$358$	0	0
$359$	−3.40671e21	−0.651677	−0.325839	−	0.945425i	$-0.605647\pi$
$359$	−3.40671e21	−0.651677	−0.325839	+	0.945425i	$0.605647\pi$
$360$	0	0
$361$	−5.93738e20	−0.108339
$362$	0	0
$363$	−3.07947e21	−0.536129
$364$	0	0
$365$	3.93196e21	0.653310
$366$	0	0
$367$	−6.57465e21	−1.04282	−0.521412	−	0.853305i	$-0.674594\pi$
$367$	−6.57465e21	−1.04282	−0.521412	+	0.853305i	$0.674594\pi$
$368$	0	0
$369$	7.24347e21	1.09704
$370$	0	0
$371$	−1.43865e22	−2.08102
$372$	0	0
$373$	4.36334e21	0.602968	0.301484	−	0.953471i	$-0.402518\pi$
$373$	4.36334e21	0.602968	0.301484	+	0.953471i	$0.402518\pi$
$374$	0	0
$375$	9.32318e20	0.123111
$376$	0	0
$377$	−1.33574e22	−1.68586
$378$	0	0
$379$	6.34370e21	0.765436	0.382718	−	0.923865i	$-0.374988\pi$
$379$	6.34370e21	0.765436	0.382718	+	0.923865i	$0.374988\pi$
$380$	0	0
$381$	2.02051e22	2.33130
$382$	0	0
$383$	1.39508e22	1.53960	0.769802	−	0.638283i	$-0.220355\pi$
$383$	1.39508e22	1.53960	0.769802	+	0.638283i	$0.220355\pi$
$384$	0	0
$385$	6.40334e21	0.676068
$386$	0	0
$387$	1.50787e21	0.152342
$388$	0	0
$389$	−8.36900e21	−0.809286	−0.404643	−	0.914475i	$-0.632604\pi$
$389$	−8.36900e21	−0.809286	−0.404643	+	0.914475i	$0.632604\pi$
$390$	0	0
$391$	5.78155e21	0.535231
$392$	0	0
$393$	−1.60195e22	−1.42008
$394$	0	0
$395$	7.53033e21	0.639350
$396$	0	0
$397$	−2.83041e21	−0.230213	−0.115107	−	0.993353i	$-0.536721\pi$
$397$	−2.83041e21	−0.230213	−0.115107	+	0.993353i	$0.536721\pi$
$398$	0	0
$399$	3.22670e22	2.51472
$400$	0	0
$401$	1.80370e22	1.34722	0.673608	−	0.739089i	$-0.264744\pi$
$401$	1.80370e22	1.34722	0.673608	+	0.739089i	$0.264744\pi$
$402$	0	0
$403$	2.55730e22	1.83100
$404$	0	0
$405$	−7.12903e21	−0.489399
$406$	0	0
$407$	7.53140e21	0.495819
$408$	0	0
$409$	−1.90532e21	−0.120315	−0.0601575	−	0.998189i	$-0.519160\pi$
$409$	−1.90532e21	−0.120315	−0.0601575	+	0.998189i	$0.519160\pi$
$410$	0	0
$411$	8.35532e21	0.506183
$412$	0	0
$413$	3.61237e21	0.209999
$414$	0	0
$415$	−7.64165e21	−0.426362
$416$	0	0
$417$	1.85837e22	0.995349
$418$	0	0
$419$	−1.60121e22	−0.823436	−0.411718	−	0.911311i	$-0.635071\pi$
$419$	−1.60121e22	−0.823436	−0.411718	+	0.911311i	$0.635071\pi$
$420$	0	0
$421$	−7.60456e21	−0.375557	−0.187779	−	0.982211i	$-0.560129\pi$
$421$	−7.60456e21	−0.375557	−0.187779	+	0.982211i	$0.560129\pi$
$422$	0	0
$423$	−1.59931e22	−0.758646
$424$	0	0
$425$	4.05611e21	0.184843
$426$	0	0
$427$	2.63959e22	1.15585
$428$	0	0
$429$	−4.38929e22	−1.84717
$430$	0	0
$431$	1.12410e22	0.454725	0.227363	−	0.973810i	$-0.426990\pi$
$431$	1.12410e22	0.454725	0.227363	+	0.973810i	$0.426990\pi$
$432$	0	0
$433$	5.47417e21	0.212898	0.106449	−	0.994318i	$-0.466052\pi$
$433$	5.47417e21	0.212898	0.106449	+	0.994318i	$0.466052\pi$
$434$	0	0
$435$	−1.61560e22	−0.604193
$436$	0	0
$437$	−1.52041e22	−0.546850
$438$	0	0
$439$	3.59418e22	1.24352	0.621759	−	0.783209i	$-0.286418\pi$
$439$	3.59418e22	1.24352	0.621759	+	0.783209i	$0.286418\pi$
$440$	0	0
$441$	7.37275e22	2.45415
$442$	0	0
$443$	−1.59250e20	−0.00510093	−0.00255046	−	0.999997i	$-0.500812\pi$
$443$	−1.59250e20	−0.00510093	−0.00255046	+	0.999997i	$0.500812\pi$
$444$	0	0
$445$	2.31206e22	0.712752
$446$	0	0
$447$	7.07934e22	2.10077
$448$	0	0
$449$	6.10942e22	1.74544	0.872722	−	0.488217i	$-0.162353\pi$
$449$	6.10942e22	1.74544	0.872722	+	0.488217i	$0.162353\pi$
$450$	0	0
$451$	−3.48303e22	−0.958202
$452$	0	0
$453$	4.95239e22	1.31214
$454$	0	0
$455$	−5.82319e22	−1.48616
$456$	0	0
$457$	3.96641e22	0.975237	0.487618	−	0.873057i	$-0.337866\pi$
$457$	3.96641e22	0.975237	0.487618	+	0.873057i	$0.337866\pi$
$458$	0	0
$459$	−5.66204e21	−0.134142
$460$	0	0
$461$	−5.01853e21	−0.114583	−0.0572913	−	0.998358i	$-0.518246\pi$
$461$	−5.01853e21	−0.114583	−0.0572913	+	0.998358i	$0.518246\pi$
$462$	0	0
$463$	6.37654e22	1.40329	0.701644	−	0.712528i	$-0.252450\pi$
$463$	6.37654e22	1.40329	0.701644	+	0.712528i	$0.252450\pi$
$464$	0	0
$465$	3.09310e22	0.656212
$466$	0	0
$467$	−6.69888e22	−1.37028	−0.685139	−	0.728412i	$-0.740258\pi$
$467$	−6.69888e22	−1.37028	−0.685139	+	0.728412i	$0.740258\pi$
$468$	0	0
$469$	−1.22325e23	−2.41293
$470$	0	0
$471$	−1.11092e23	−2.11351
$472$	0	0
$473$	−7.25060e21	−0.133062
$474$	0	0
$475$	−1.06666e22	−0.188856
$476$	0	0
$477$	5.63188e22	0.962159
$478$	0	0
$479$	2.66400e22	0.439220	0.219610	−	0.975588i	$-0.429522\pi$
$479$	2.66400e22	0.439220	0.219610	+	0.975588i	$0.429522\pi$
$480$	0	0
$481$	−6.84905e22	−1.08993
$482$	0	0
$483$	−1.00394e23	−1.54226
$484$	0	0
$485$	−4.05298e22	−0.601131
$486$	0	0
$487$	−5.25859e22	−0.753136	−0.376568	−	0.926389i	$-0.622896\pi$
$487$	−5.25859e22	−0.753136	−0.376568	+	0.926389i	$0.622896\pi$
$488$	0	0
$489$	5.21118e21	0.0720793
$490$	0	0
$491$	−3.83652e21	−0.0512561	−0.0256280	−	0.999672i	$-0.508159\pi$
$491$	−3.83652e21	−0.0512561	−0.0256280	+	0.999672i	$0.508159\pi$
$492$	0	0
$493$	−7.02878e22	−0.907154
$494$	0	0
$495$	−2.50671e22	−0.312580
$496$	0	0
$497$	−2.00102e23	−2.41114
$498$	0	0
$499$	1.29674e22	0.151007	0.0755037	−	0.997146i	$-0.475944\pi$
$499$	1.29674e22	0.151007	0.0755037	+	0.997146i	$0.475944\pi$
$500$	0	0
$501$	1.97146e23	2.21905
$502$	0	0
$503$	1.28633e23	1.39967	0.699835	−	0.714304i	$-0.253257\pi$
$503$	1.28633e23	1.39967	0.699835	+	0.714304i	$0.253257\pi$
$504$	0	0
$505$	1.88558e22	0.198367
$506$	0	0
$507$	2.63854e23	2.68408
$508$	0	0
$509$	−1.25797e23	−1.23756	−0.618782	−	0.785562i	$-0.712374\pi$
$509$	−1.25797e23	−1.23756	−0.618782	+	0.785562i	$0.712374\pi$
$510$	0	0
$511$	−2.97042e23	−2.82644
$512$	0	0
$513$	1.48898e22	0.137054
$514$	0	0
$515$	2.66526e22	0.237344
$516$	0	0
$517$	7.69030e22	0.662634
$518$	0	0
$519$	−1.82226e23	−1.51946
$520$	0	0
$521$	1.17618e22	0.0949189	0.0474595	−	0.998873i	$-0.484888\pi$
$521$	1.17618e22	0.0949189	0.0474595	+	0.998873i	$0.484888\pi$
$522$	0	0
$523$	−2.86621e21	−0.0223895	−0.0111947	−	0.999937i	$-0.503563\pi$
$523$	−2.86621e21	−0.0223895	−0.0111947	+	0.999937i	$0.503563\pi$
$524$	0	0
$525$	−7.04325e22	−0.532622
$526$	0	0
$527$	1.34567e23	0.985256
$528$	0	0
$529$	−9.37448e22	−0.664621
$530$	0	0
$531$	−1.41413e22	−0.0970927
$532$	0	0
$533$	3.16747e23	2.10635
$534$	0	0
$535$	1.09693e22	0.0706600
$536$	0	0
$537$	−1.18802e23	−0.741384
$538$	0	0
$539$	−3.54520e23	−2.14356
$540$	0	0
$541$	2.71778e23	1.59235	0.796174	−	0.605068i	$-0.206854\pi$
$541$	2.71778e23	1.59235	0.796174	+	0.605068i	$0.206854\pi$
$542$	0	0
$543$	3.30513e23	1.87668
$544$	0	0
$545$	7.00004e22	0.385239
$546$	0	0
$547$	7.28132e21	0.0388434	0.0194217	−	0.999811i	$-0.493817\pi$
$547$	7.28132e21	0.0388434	0.0194217	+	0.999811i	$0.493817\pi$
$548$	0	0
$549$	−1.03332e23	−0.534405
$550$	0	0
$551$	1.84840e23	0.926845
$552$	0	0
$553$	−5.68883e23	−2.76605
$554$	0	0
$555$	−8.28403e22	−0.390617
$556$	0	0
$557$	−4.03298e23	−1.84441	−0.922203	−	0.386706i	$-0.873613\pi$
$557$	−4.03298e23	−1.84441	−0.922203	+	0.386706i	$0.873613\pi$
$558$	0	0
$559$	6.59369e22	0.292501
$560$	0	0
$561$	−2.30968e23	−0.993955
$562$	0	0
$563$	−1.97223e23	−0.823449	−0.411724	−	0.911308i	$-0.635073\pi$
$563$	−1.97223e23	−0.823449	−0.411724	+	0.911308i	$0.635073\pi$
$564$	0	0
$565$	−6.47200e22	−0.262196
$566$	0	0
$567$	5.38567e23	2.11730
$568$	0	0
$569$	−3.84081e23	−1.46544	−0.732721	−	0.680529i	$-0.761750\pi$
$569$	−3.84081e23	−1.46544	−0.732721	+	0.680529i	$0.761750\pi$
$570$	0	0
$571$	3.82690e20	0.00141723	0.000708615	−	1.00000i	$-0.499774\pi$
$571$	3.82690e20	0.00141723	0.000708615		1.00000i	$0.499774\pi$
$572$	0	0
$573$	5.75122e23	2.06750
$574$	0	0
$575$	3.31875e22	0.115824
$576$	0	0
$577$	−4.83185e23	−1.63727	−0.818633	−	0.574317i	$-0.805268\pi$
$577$	−4.83185e23	−1.63727	−0.818633	+	0.574317i	$0.805268\pi$
$578$	0	0
$579$	−1.09196e22	−0.0359285
$580$	0	0
$581$	5.77293e23	1.84459
$582$	0	0
$583$	−2.70810e23	−0.840391
$584$	0	0
$585$	2.27960e23	0.687123
$586$	0	0
$587$	−4.44210e23	−1.30066	−0.650330	−	0.759651i	$-0.725370\pi$
$587$	−4.44210e23	−1.30066	−0.650330	+	0.759651i	$0.725370\pi$
$588$	0	0
$589$	−3.53880e23	−1.00664
$590$	0	0
$591$	5.66050e23	1.56445
$592$	0	0
$593$	2.55235e23	0.685449	0.342724	−	0.939436i	$-0.388650\pi$
$593$	2.55235e23	0.685449	0.342724	+	0.939436i	$0.388650\pi$
$594$	0	0
$595$	−3.06421e23	−0.799695
$596$	0	0
$597$	−8.40490e23	−2.13182
$598$	0	0
$599$	−3.68729e23	−0.909032	−0.454516	−	0.890739i	$-0.650188\pi$
$599$	−3.68729e23	−0.909032	−0.454516	+	0.890739i	$0.650188\pi$
$600$	0	0
$601$	−1.30500e23	−0.312735	−0.156368	−	0.987699i	$-0.549978\pi$
$601$	−1.30500e23	−0.312735	−0.156368	+	0.987699i	$0.549978\pi$
$602$	0	0
$603$	4.78864e23	1.11562
$604$	0	0
$605$	−7.69045e22	−0.174193
$606$	0	0
$607$	−4.88721e23	−1.07636	−0.538180	−	0.842830i	$-0.680888\pi$
$607$	−4.88721e23	−1.07636	−0.538180	+	0.842830i	$0.680888\pi$
$608$	0	0
$609$	1.22052e24	2.61394
$610$	0	0
$611$	−6.99355e23	−1.45662
$612$	0	0
$613$	−2.91453e23	−0.590411	−0.295205	−	0.955434i	$-0.595388\pi$
$613$	−2.91453e23	−0.590411	−0.295205	+	0.955434i	$0.595388\pi$
$614$	0	0
$615$	3.83110e23	0.754893
$616$	0	0
$617$	4.96535e23	0.951756	0.475878	−	0.879511i	$-0.342130\pi$
$617$	4.96535e23	0.951756	0.475878	+	0.879511i	$0.342130\pi$
$618$	0	0
$619$	1.38017e23	0.257372	0.128686	−	0.991685i	$-0.458924\pi$
$619$	1.38017e23	0.257372	0.128686	+	0.991685i	$0.458924\pi$
$620$	0	0
$621$	−4.63274e22	−0.0840542
$622$	0	0
$623$	−1.74666e24	−3.08361
$624$	0	0
$625$	2.32831e22	0.0400000
$626$	0	0
$627$	6.07390e23	1.01553
$628$	0	0
$629$	−3.60402e23	−0.586485
$630$	0	0
$631$	4.54842e23	0.720463	0.360231	−	0.932863i	$-0.382698\pi$
$631$	4.54842e23	0.720463	0.360231	+	0.932863i	$0.382698\pi$
$632$	0	0
$633$	−8.35038e23	−1.28758
$634$	0	0
$635$	5.04587e23	0.757460
$636$	0	0
$637$	3.22400e24	4.71205
$638$	0	0
$639$	7.83339e23	1.11479
$640$	0	0
$641$	1.01467e24	1.40614	0.703072	−	0.711118i	$-0.251811\pi$
$641$	1.01467e24	1.40614	0.703072	+	0.711118i	$0.251811\pi$
$642$	0	0
$643$	−2.72123e23	−0.367258	−0.183629	−	0.982996i	$-0.558785\pi$
$643$	−2.72123e23	−0.367258	−0.183629	+	0.982996i	$0.558785\pi$
$644$	0	0
$645$	7.97517e22	0.104829
$646$	0	0
$647$	6.92887e23	0.887108	0.443554	−	0.896248i	$-0.353718\pi$
$647$	6.92887e23	0.887108	0.443554	+	0.896248i	$0.353718\pi$
$648$	0	0
$649$	6.79988e22	0.0848049
$650$	0	0
$651$	−2.33670e24	−2.83899
$652$	0	0
$653$	−1.09165e24	−1.29218	−0.646088	−	0.763263i	$-0.723596\pi$
$653$	−1.09165e24	−1.29218	−0.646088	+	0.763263i	$0.723596\pi$
$654$	0	0
$655$	−4.00059e23	−0.461395
$656$	0	0
$657$	1.16283e24	1.30680
$658$	0	0
$659$	−3.23117e23	−0.353861	−0.176931	−	0.984223i	$-0.556617\pi$
$659$	−3.23117e23	−0.353861	−0.176931	+	0.984223i	$0.556617\pi$
$660$	0	0
$661$	−1.07030e24	−1.14233	−0.571166	−	0.820835i	$-0.693509\pi$
$661$	−1.07030e24	−1.14233	−0.571166	+	0.820835i	$0.693509\pi$
$662$	0	0
$663$	2.10042e24	2.18495
$664$	0	0
$665$	8.05814e23	0.817054
$666$	0	0
$667$	−5.75102e23	−0.568428
$668$	0	0
$669$	−8.85771e23	−0.853492
$670$	0	0
$671$	4.96874e23	0.466772
$672$	0	0
$673$	−3.00782e23	−0.275502	−0.137751	−	0.990467i	$-0.543987\pi$
$673$	−3.00782e23	−0.275502	−0.137751	+	0.990467i	$0.543987\pi$
$674$	0	0
$675$	−3.25015e22	−0.0290283
$676$	0	0
$677$	1.63077e23	0.142033	0.0710164	−	0.997475i	$-0.477376\pi$
$677$	1.63077e23	0.142033	0.0710164	+	0.997475i	$0.477376\pi$
$678$	0	0
$679$	3.06184e24	2.60070
$680$	0	0
$681$	−1.70187e24	−1.40986
$682$	0	0
$683$	9.16610e23	0.740643	0.370321	−	0.928904i	$-0.379248\pi$
$683$	9.16610e23	0.740643	0.370321	+	0.928904i	$0.379248\pi$
$684$	0	0
$685$	2.08660e23	0.164463
$686$	0	0
$687$	8.90646e23	0.684814
$688$	0	0
$689$	2.46274e24	1.84738
$690$	0	0
$691$	1.83129e24	1.34028	0.670139	−	0.742236i	$-0.266235\pi$
$691$	1.83129e24	1.34028	0.670139	+	0.742236i	$0.266235\pi$
$692$	0	0
$693$	1.89371e24	1.35233
$694$	0	0
$695$	4.64096e23	0.323398
$696$	0	0
$697$	1.66675e24	1.13342
$698$	0	0
$699$	4.38670e23	0.291126
$700$	0	0
$701$	−1.01906e24	−0.660080	−0.330040	−	0.943967i	$-0.607062\pi$
$701$	−1.01906e24	−0.660080	−0.330040	+	0.943967i	$0.607062\pi$
$702$	0	0
$703$	9.47771e23	0.599216
$704$	0	0
$705$	−8.45881e23	−0.522038
$706$	0	0
$707$	−1.42447e24	−0.858202
$708$	0	0
$709$	9.36499e23	0.550826	0.275413	−	0.961326i	$-0.411185\pi$
$709$	9.36499e23	0.550826	0.275413	+	0.961326i	$0.411185\pi$
$710$	0	0
$711$	2.22700e24	1.27888
$712$	0	0
$713$	1.10104e24	0.617367
$714$	0	0
$715$	−1.09615e24	−0.600162
$716$	0	0
$717$	1.63338e24	0.873323
$718$	0	0
$719$	−3.46287e24	−1.80817	−0.904087	−	0.427348i	$-0.859448\pi$
$719$	−3.46287e24	−1.80817	−0.904087	+	0.427348i	$0.859448\pi$
$720$	0	0
$721$	−2.01349e24	−1.02683
$722$	0	0
$723$	−4.24283e24	−2.11339
$724$	0	0
$725$	−4.03469e23	−0.196308
$726$	0	0
$727$	1.95222e24	0.927866	0.463933	−	0.885870i	$-0.346438\pi$
$727$	1.95222e24	0.927866	0.463933	+	0.885870i	$0.346438\pi$
$728$	0	0
$729$	−1.67807e24	−0.779157
$730$	0	0
$731$	3.46965e23	0.157394
$732$	0	0
$733$	−1.57883e24	−0.699765	−0.349883	−	0.936794i	$-0.613779\pi$
$733$	−1.57883e24	−0.699765	−0.349883	+	0.936794i	$0.613779\pi$
$734$	0	0
$735$	3.89948e24	1.68875
$736$	0	0
$737$	−2.30262e24	−0.974427
$738$	0	0
$739$	1.68319e24	0.696074	0.348037	−	0.937481i	$-0.386848\pi$
$739$	1.68319e24	0.696074	0.348037	+	0.937481i	$0.386848\pi$
$740$	0	0
$741$	−5.52359e24	−2.23237
$742$	0	0
$743$	−1.94960e24	−0.770088	−0.385044	−	0.922898i	$-0.625814\pi$
$743$	−1.94960e24	−0.770088	−0.385044	+	0.922898i	$0.625814\pi$
$744$	0	0
$745$	1.76795e24	0.682560
$746$	0	0
$747$	−2.25993e24	−0.852843
$748$	0	0
$749$	−8.28685e23	−0.305699
$750$	0	0
$751$	−1.05627e24	−0.380922	−0.190461	−	0.981695i	$-0.560998\pi$
$751$	−1.05627e24	−0.380922	−0.190461	+	0.981695i	$0.560998\pi$
$752$	0	0
$753$	−2.45973e24	−0.867222
$754$	0	0
$755$	1.23677e24	0.426326
$756$	0	0
$757$	−4.04949e24	−1.36485	−0.682425	−	0.730955i	$-0.739075\pi$
$757$	−4.04949e24	−1.36485	−0.682425	+	0.730955i	$0.739075\pi$
$758$	0	0
$759$	−1.88980e24	−0.622818
$760$	0	0
$761$	1.99728e24	0.643679	0.321840	−	0.946794i	$-0.395699\pi$
$761$	1.99728e24	0.643679	0.321840	+	0.946794i	$0.395699\pi$
$762$	0	0
$763$	−5.28822e24	−1.66668
$764$	0	0
$765$	1.19954e24	0.369738
$766$	0	0
$767$	−6.18380e23	−0.186421
$768$	0	0
$769$	−3.39610e24	−1.00140	−0.500699	−	0.865621i	$-0.666924\pi$
$769$	−3.39610e24	−1.00140	−0.500699	+	0.865621i	$0.666924\pi$
$770$	0	0
$771$	1.18312e24	0.341245
$772$	0	0
$773$	−3.72165e24	−1.05005	−0.525024	−	0.851087i	$-0.675944\pi$
$773$	−3.72165e24	−1.05005	−0.525024	+	0.851087i	$0.675944\pi$
$774$	0	0
$775$	7.72450e23	0.213209
$776$	0	0
$777$	6.25822e24	1.68994
$778$	0	0
$779$	−4.38314e24	−1.15802
$780$	0	0
$781$	−3.76670e24	−0.973704
$782$	0	0
$783$	5.63214e23	0.142462
$784$	0	0
$785$	−2.77433e24	−0.686699
$786$	0	0
$787$	−4.44239e23	−0.107605	−0.0538023	−	0.998552i	$-0.517134\pi$
$787$	−4.44239e23	−0.107605	−0.0538023	+	0.998552i	$0.517134\pi$
$788$	0	0
$789$	6.86099e24	1.62642
$790$	0	0
$791$	4.88931e24	1.13435
$792$	0	0
$793$	−4.51856e24	−1.02607
$794$	0	0
$795$	2.97872e24	0.662079
$796$	0	0
$797$	6.81742e24	1.48328	0.741642	−	0.670796i	$-0.234047\pi$
$797$	6.81742e24	1.48328	0.741642	+	0.670796i	$0.234047\pi$
$798$	0	0
$799$	−3.68006e24	−0.783804
$800$	0	0
$801$	6.83763e24	1.42570
$802$	0	0
$803$	−5.59148e24	−1.14142
$804$	0	0
$805$	−2.50717e24	−0.501093
$806$	0	0
$807$	4.49267e24	0.879183
$808$	0	0
$809$	4.25297e24	0.814949	0.407474	−	0.913217i	$-0.366410\pi$
$809$	4.25297e24	0.814949	0.407474	+	0.913217i	$0.366410\pi$
$810$	0	0
$811$	4.18911e24	0.786040	0.393020	−	0.919530i	$-0.371430\pi$
$811$	4.18911e24	0.786040	0.393020	+	0.919530i	$0.371430\pi$
$812$	0	0
$813$	−4.43066e24	−0.814140
$814$	0	0
$815$	1.30140e23	0.0234192
$816$	0	0
$817$	−9.12435e23	−0.160810
$818$	0	0
$819$	−1.72214e25	−2.97273
$820$	0	0
$821$	7.48932e24	1.26627	0.633134	−	0.774042i	$-0.281768\pi$
$821$	7.48932e24	1.26627	0.633134	+	0.774042i	$0.281768\pi$
$822$	0	0
$823$	4.16223e24	0.689331	0.344666	−	0.938726i	$-0.387992\pi$
$823$	4.16223e24	0.689331	0.344666	+	0.938726i	$0.387992\pi$
$824$	0	0
$825$	−1.32581e24	−0.215092
$826$	0	0
$827$	−6.20940e24	−0.986853	−0.493427	−	0.869787i	$-0.664256\pi$
$827$	−6.20940e24	−0.986853	−0.493427	+	0.869787i	$0.664256\pi$
$828$	0	0
$829$	1.14805e24	0.178751	0.0893755	−	0.995998i	$-0.471513\pi$
$829$	1.14805e24	0.178751	0.0893755	+	0.995998i	$0.471513\pi$
$830$	0	0
$831$	−7.49316e24	−1.14303
$832$	0	0
$833$	1.69649e25	2.53553
$834$	0	0
$835$	4.92339e24	0.720990
$836$	0	0
$837$	−1.07829e24	−0.154727
$838$	0	0
$839$	−3.96214e24	−0.557126	−0.278563	−	0.960418i	$-0.589858\pi$
$839$	−3.96214e24	−0.557126	−0.278563	+	0.960418i	$0.589858\pi$
$840$	0	0
$841$	−2.65484e23	−0.0365824
$842$	0	0
$843$	−2.37381e24	−0.320561
$844$	0	0
$845$	6.58931e24	0.872083
$846$	0	0
$847$	5.80979e24	0.753619
$848$	0	0
$849$	−1.69084e25	−2.14975
$850$	0	0
$851$	−2.94885e24	−0.367495
$852$	0	0
$853$	5.80663e24	0.709345	0.354672	−	0.934991i	$-0.384592\pi$
$853$	5.80663e24	0.709345	0.354672	+	0.934991i	$0.384592\pi$
$854$	0	0
$855$	−3.15451e24	−0.377764
$856$	0	0
$857$	7.13830e24	0.838027	0.419013	−	0.907980i	$-0.362376\pi$
$857$	7.13830e24	0.838027	0.419013	+	0.907980i	$0.362376\pi$
$858$	0	0
$859$	−1.31344e25	−1.51171	−0.755857	−	0.654737i	$-0.772779\pi$
$859$	−1.31344e25	−1.51171	−0.755857	+	0.654737i	$0.772779\pi$
$860$	0	0
$861$	−2.89423e25	−3.26592
$862$	0	0
$863$	6.23473e24	0.689804	0.344902	−	0.938639i	$-0.387912\pi$
$863$	6.23473e24	0.689804	0.344902	+	0.938639i	$0.387912\pi$
$864$	0	0
$865$	−4.55080e24	−0.493686
$866$	0	0
$867$	−1.88689e24	−0.200717
$868$	0	0
$869$	−1.07086e25	−1.11703
$870$	0	0
$871$	2.09400e25	2.14202
$872$	0	0
$873$	−1.19862e25	−1.20243
$874$	0	0
$875$	−1.75893e24	−0.173054
$876$	0	0
$877$	1.80103e25	1.73790	0.868948	−	0.494903i	$-0.164796\pi$
$877$	1.80103e25	1.73790	0.868948	+	0.494903i	$0.164796\pi$
$878$	0	0
$879$	8.82474e24	0.835212
$880$	0	0
$881$	−1.44968e25	−1.34578	−0.672892	−	0.739741i	$-0.734948\pi$
$881$	−1.44968e25	−1.34578	−0.672892	+	0.739741i	$0.734948\pi$
$882$	0	0
$883$	−4.79871e24	−0.436977	−0.218489	−	0.975840i	$-0.570113\pi$
$883$	−4.79871e24	−0.436977	−0.218489	+	0.975840i	$0.570113\pi$
$884$	0	0
$885$	−7.47940e23	−0.0668112
$886$	0	0
$887$	1.60797e25	1.40905	0.704526	−	0.709679i	$-0.251160\pi$
$887$	1.60797e25	1.40905	0.704526	+	0.709679i	$0.251160\pi$
$888$	0	0
$889$	−3.81193e25	−3.27703
$890$	0	0
$891$	1.01379e25	0.855043
$892$	0	0
$893$	9.67768e24	0.800818
$894$	0	0
$895$	−2.96689e24	−0.240882
$896$	0	0
$897$	1.71858e25	1.36910
$898$	0	0
$899$	−1.33857e25	−1.04636
$900$	0	0
$901$	1.29591e25	0.994065
$902$	0	0
$903$	−6.02489e24	−0.453527
$904$	0	0
$905$	8.25401e24	0.609750
$906$	0	0
$907$	2.07733e25	1.50606	0.753031	−	0.657985i	$-0.228591\pi$
$907$	2.07733e25	1.50606	0.753031	+	0.657985i	$0.228591\pi$
$908$	0	0
$909$	5.57638e24	0.396789
$910$	0	0
$911$	−9.41556e24	−0.657567	−0.328783	−	0.944405i	$-0.606639\pi$
$911$	−9.41556e24	−0.657567	−0.328783	+	0.944405i	$0.606639\pi$
$912$	0	0
$913$	1.08669e25	0.744910
$914$	0	0
$915$	−5.46528e24	−0.367733
$916$	0	0
$917$	3.02227e25	1.99615
$918$	0	0
$919$	1.89739e25	1.23020	0.615098	−	0.788450i	$-0.289116\pi$
$919$	1.89739e25	1.23020	0.615098	+	0.788450i	$0.289116\pi$
$920$	0	0
$921$	1.23028e25	0.783063
$922$	0	0
$923$	3.42543e25	2.14043
$924$	0	0
$925$	−2.06880e24	−0.126915
$926$	0	0
$927$	7.88218e24	0.474754
$928$	0	0
$929$	−3.14410e25	−1.85936	−0.929679	−	0.368371i	$-0.879916\pi$
$929$	−3.14410e25	−1.85936	−0.929679	+	0.368371i	$0.879916\pi$
$930$	0	0
$931$	−4.46137e25	−2.59057
$932$	0	0
$933$	−2.90775e25	−1.65792
$934$	0	0
$935$	−5.76803e24	−0.322945
$936$	0	0
$937$	−2.64305e25	−1.45318	−0.726589	−	0.687073i	$-0.758895\pi$
$937$	−2.64305e25	−1.45318	−0.726589	+	0.687073i	$0.758895\pi$
$938$	0	0
$939$	1.59250e25	0.859846
$940$	0	0
$941$	−1.16589e25	−0.618226	−0.309113	−	0.951025i	$-0.600032\pi$
$941$	−1.16589e25	−0.618226	−0.309113	+	0.951025i	$0.600032\pi$
$942$	0	0
$943$	1.36375e25	0.710207
$944$	0	0
$945$	2.45534e24	0.125586
$946$	0	0
$947$	−1.23570e25	−0.620780	−0.310390	−	0.950609i	$-0.600460\pi$
$947$	−1.23570e25	−0.620780	−0.310390	+	0.950609i	$0.600460\pi$
$948$	0	0
$949$	5.08488e25	2.50910
$950$	0	0
$951$	4.95558e24	0.240193
$952$	0	0
$953$	2.25388e25	1.07310	0.536550	−	0.843869i	$-0.319727\pi$
$953$	2.25388e25	1.07310	0.536550	+	0.843869i	$0.319727\pi$
$954$	0	0
$955$	1.43627e25	0.671749
$956$	0	0
$957$	2.29748e25	1.05560
$958$	0	0
$959$	−1.57633e25	−0.711525
$960$	0	0
$961$	3.07703e24	0.136453
$962$	0	0
$963$	3.24405e24	0.141340
$964$	0	0
$965$	−2.72699e23	−0.0116735
$966$	0	0
$967$	1.82941e25	0.769459	0.384730	−	0.923029i	$-0.374295\pi$
$967$	1.82941e25	0.769459	0.384730	+	0.923029i	$0.374295\pi$
$968$	0	0
$969$	−2.90656e25	−1.20123
$970$	0	0
$971$	−7.45406e22	−0.00302712	−0.00151356	−	0.999999i	$-0.500482\pi$
$971$	−7.45406e22	−0.00302712	−0.00151356	+	0.999999i	$0.500482\pi$
$972$	0	0
$973$	−3.50604e25	−1.39913
$974$	0	0
$975$	1.20569e25	0.472821
$976$	0	0
$977$	2.23755e25	0.862320	0.431160	−	0.902276i	$-0.358105\pi$
$977$	2.23755e25	0.862320	0.431160	+	0.902276i	$0.358105\pi$
$978$	0	0
$979$	−3.28788e25	−1.24527
$980$	0	0
$981$	2.07018e25	0.770586
$982$	0	0
$983$	7.32013e24	0.267802	0.133901	−	0.990995i	$-0.457250\pi$
$983$	7.32013e24	0.267802	0.133901	+	0.990995i	$0.457250\pi$
$984$	0	0
$985$	1.41361e25	0.508303
$986$	0	0
$987$	6.39026e25	2.25851
$988$	0	0
$989$	2.83890e24	0.0986239
$990$	0	0
$991$	4.05373e25	1.38429	0.692147	−	0.721757i	$-0.256665\pi$
$991$	4.05373e25	1.38429	0.692147	+	0.721757i	$0.256665\pi$
$992$	0	0
$993$	−1.46031e25	−0.490203
$994$	0	0
$995$	−2.09898e25	−0.692648
$996$	0	0
$997$	3.59617e25	1.16662	0.583312	−	0.812248i	$-0.301756\pi$
$997$	3.59617e25	1.16662	0.583312	+	0.812248i	$0.301756\pi$
$998$	0	0
$999$	2.88789e24	0.0921032

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.18.a.l.1.4		5
4.3	odd	2		40.18.a.d.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.18.a.d.1.2	✓	5	4.3	odd	2
80.18.a.l.1.4		5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(f(q)\)	\(=\)	\(q+15641.7 q^{3} +390625. q^{5} -2.95100e7 q^{7} +1.15523e8 q^{9} -5.55492e8 q^{11} +5.05164e9 q^{13} +6.11004e9 q^{15} +2.65821e10 q^{17} -6.99046e10 q^{19} -4.61586e11 q^{21} +2.17498e11 q^{23} +1.52588e11 q^{25} -2.13002e11 q^{27} -2.64418e12 q^{29} +5.06233e12 q^{31} -8.68884e12 q^{33} -1.15273e13 q^{35} -1.35581e13 q^{37} +7.90162e13 q^{39} +6.27018e13 q^{41} +1.30526e13 q^{43} +4.51260e13 q^{45} -1.38441e14 q^{47} +6.38209e14 q^{49} +4.15789e14 q^{51} +4.87513e14 q^{53} -2.16989e14 q^{55} -1.09343e15 q^{57} -1.22412e14 q^{59} -8.94475e14 q^{61} -3.40907e15 q^{63} +1.97330e15 q^{65} +4.14519e15 q^{67} +3.40203e15 q^{69} +6.78083e15 q^{71} +1.00658e16 q^{73} +2.38673e15 q^{75} +1.63926e16 q^{77} +1.92777e16 q^{79} -1.82503e16 q^{81} -1.95626e16 q^{83} +1.03836e16 q^{85} -4.13594e16 q^{87} +5.91887e16 q^{89} -1.49074e17 q^{91} +7.91834e16 q^{93} -2.73065e16 q^{95} -1.03756e17 q^{97} -6.41719e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 15384 q^{3} + 1953125 q^{5} - 312868 q^{7} + 410286417 q^{9} - 757119716 q^{11} + 2827963478 q^{13} + 6009375000 q^{15} + 24776563114 q^{17} - 811272116 q^{19} + 332870780352 q^{21} + 73100431588 q^{23}+ \cdots - 35\!\cdots\!44 q^{99}+O(q^{100}) \) 5 * q + 15384 * q^3 + 1953125 * q^5 - 312868 * q^7 + 410286417 * q^9 - 757119716 * q^11 + 2827963478 * q^13 + 6009375000 * q^15 + 24776563114 * q^17 - 811272116 * q^19 + 332870780352 * q^21 + 73100431588 * q^23 + 762939453125 * q^25 + 3166101346032 * q^27 - 3954378361738 * q^29 + 12340046636056 * q^31 - 11191253546784 * q^33 - 122214062500 * q^35 + 19853810999454 * q^37 - 26906648601264 * q^39 + 70876127296194 * q^41 + 133955994850144 * q^43 + 160268131640625 * q^45 + 113374769936500 * q^47 + 734065948219149 * q^49 - 91587467914320 * q^51 - 232473412424130 * q^53 - 295749889062500 * q^55 + 192839487986976 * q^57 - 2891505753119596 * q^59 - 120759403445194 * q^61 + 75256025229420 * q^63 + 1104673233593750 * q^65 - 7139319995171008 * q^67 + 12678898772781888 * q^69 - 5858971441767504 * q^71 + 22237456819273602 * q^73 + 2347412109375000 * q^75 + 19042533632763664 * q^77 + 11793240059085792 * q^79 + 52885709091819405 * q^81 - 49375024849026904 * q^83 + 9678344966406250 * q^85 - 129320807184056112 * q^87 + 102588609132622626 * q^89 - 260090514257055160 * q^91 + 342096413075911680 * q^93 - 316903170312500 * q^95 + 157786395292410650 * q^97 - 355232381581465044 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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