LMFDB - Embedded newform 80.18.a.l.1.3

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			1.3
Root			$-46.3535$ 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)

$f(q)$	$=$	$q+1318.46 q^{3} +390625. q^{5} +1.28833e7 q^{7} -1.27402e8 q^{9} +1.29449e9 q^{11} +2.46098e9 q^{13} +5.15023e8 q^{15} +2.06308e10 q^{17} +2.35658e10 q^{19} +1.69860e10 q^{21} -1.62325e11 q^{23} +1.52588e11 q^{25} -3.38240e11 q^{27} -1.55032e12 q^{29} +6.44205e12 q^{31} +1.70673e12 q^{33} +5.03252e12 q^{35} +3.71799e13 q^{37} +3.24470e12 q^{39} -3.11582e13 q^{41} -1.13535e12 q^{43} -4.97663e13 q^{45} -2.18435e14 q^{47} -6.66523e13 q^{49} +2.72009e13 q^{51} +1.24200e14 q^{53} +5.05659e14 q^{55} +3.10705e13 q^{57} +3.27101e13 q^{59} +2.09504e15 q^{61} -1.64135e15 q^{63} +9.61318e14 q^{65} +2.90186e15 q^{67} -2.14018e14 q^{69} -8.60335e15 q^{71} -1.13767e16 q^{73} +2.01181e14 q^{75} +1.66772e16 q^{77} -1.09792e16 q^{79} +1.60067e16 q^{81} -1.22943e16 q^{83} +8.05890e15 q^{85} -2.04404e15 q^{87} +3.12723e16 q^{89} +3.17054e16 q^{91} +8.49359e15 q^{93} +9.20537e15 q^{95} +1.38194e17 q^{97} -1.64920e17 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1318.46	0.116021	0.0580104	−	0.998316i	$-0.481524\pi$
$3$	1318.46	0.116021	0.0580104	+	0.998316i	$0.481524\pi$
$4$	0	0
$5$	390625.	0.447214
$5$	390625.	0.447214
$6$	0	0
$7$	1.28833e7	0.844680	0.422340	−	0.906438i	$-0.361209\pi$
$7$	1.28833e7	0.844680	0.422340	+	0.906438i	$0.361209\pi$
$8$	0	0
$9$	−1.27402e8	−0.986539
$10$	0	0
$11$	1.29449e9	1.82079	0.910396	−	0.413739i	$-0.135777\pi$
$11$	1.29449e9	1.82079	0.910396	+	0.413739i	$0.135777\pi$
$12$	0	0
$13$	2.46098e9	0.836737	0.418368	−	0.908277i	$-0.362602\pi$
$13$	2.46098e9	0.836737	0.418368	+	0.908277i	$0.362602\pi$
$14$	0	0
$15$	5.15023e8	0.0518861
$16$	0	0
$17$	2.06308e10	0.717298	0.358649	−	0.933472i	$-0.383237\pi$
$17$	2.06308e10	0.717298	0.358649	+	0.933472i	$0.383237\pi$
$18$	0	0
$19$	2.35658e10	0.318329	0.159164	−	0.987252i	$-0.449120\pi$
$19$	2.35658e10	0.318329	0.159164	+	0.987252i	$0.449120\pi$
$20$	0	0
$21$	1.69860e10	0.0980005
$22$	0	0
$23$	−1.62325e11	−0.432213	−0.216106	−	0.976370i	$-0.569336\pi$
$23$	−1.62325e11	−0.432213	−0.216106	+	0.976370i	$0.569336\pi$
$24$	0	0
$25$	1.52588e11	0.200000
$26$	0	0
$27$	−3.38240e11	−0.230480
$28$	0	0
$29$	−1.55032e12	−0.575491	−0.287746	−	0.957707i	$-0.592906\pi$
$29$	−1.55032e12	−0.575491	−0.287746	+	0.957707i	$0.592906\pi$
$30$	0	0
$31$	6.44205e12	1.35659	0.678297	−	0.734788i	$-0.262718\pi$
$31$	6.44205e12	1.35659	0.678297	+	0.734788i	$0.262718\pi$
$32$	0	0
$33$	1.70673e12	0.211250
$34$	0	0
$35$	5.03252e12	0.377752
$36$	0	0
$37$	3.71799e13	1.74018	0.870089	−	0.492894i	$-0.164061\pi$
$37$	3.71799e13	1.74018	0.870089	+	0.492894i	$0.164061\pi$
$38$	0	0
$39$	3.24470e12	0.0970789
$40$	0	0
$41$	−3.11582e13	−0.609410	−0.304705	−	0.952447i	$-0.598558\pi$
$41$	−3.11582e13	−0.609410	−0.304705	+	0.952447i	$0.598558\pi$
$42$	0	0
$43$	−1.13535e12	−0.0148131	−0.00740656	−	0.999973i	$-0.502358\pi$
$43$	−1.13535e12	−0.0148131	−0.00740656	+	0.999973i	$0.502358\pi$
$44$	0	0
$45$	−4.97663e13	−0.441194
$46$	0	0
$47$	−2.18435e14	−1.33811	−0.669053	−	0.743214i	$-0.733300\pi$
$47$	−2.18435e14	−1.33811	−0.669053	+	0.743214i	$0.733300\pi$
$48$	0	0
$49$	−6.66523e13	−0.286516
$50$	0	0
$51$	2.72009e13	0.0832216
$52$	0	0
$53$	1.24200e14	0.274016	0.137008	−	0.990570i	$-0.456251\pi$
$53$	1.24200e14	0.274016	0.137008	+	0.990570i	$0.456251\pi$
$54$	0	0
$55$	5.05659e14	0.814283
$56$	0	0
$57$	3.10705e13	0.0369328
$58$	0	0
$59$	3.27101e13	0.0290028	0.0145014	−	0.999895i	$-0.495384\pi$
$59$	3.27101e13	0.0290028	0.0145014	+	0.999895i	$0.495384\pi$
$60$	0	0
$61$	2.09504e15	1.39923	0.699614	−	0.714521i	$-0.253355\pi$
$61$	2.09504e15	1.39923	0.699614	+	0.714521i	$0.253355\pi$
$62$	0	0
$63$	−1.64135e15	−0.833310
$64$	0	0
$65$	9.61318e14	0.374200
$66$	0	0
$67$	2.90186e15	0.873052	0.436526	−	0.899692i	$-0.356209\pi$
$67$	2.90186e15	0.873052	0.436526	+	0.899692i	$0.356209\pi$
$68$	0	0
$69$	−2.14018e14	−0.0501457
$70$	0	0
$71$	−8.60335e15	−1.58114	−0.790572	−	0.612369i	$-0.790217\pi$
$71$	−8.60335e15	−1.58114	−0.790572	+	0.612369i	$0.790217\pi$
$72$	0	0
$73$	−1.13767e16	−1.65109	−0.825546	−	0.564335i	$-0.809133\pi$
$73$	−1.13767e16	−1.65109	−0.825546	+	0.564335i	$0.809133\pi$
$74$	0	0
$75$	2.01181e14	0.0232042
$76$	0	0
$77$	1.66772e16	1.53799
$78$	0	0
$79$	−1.09792e16	−0.814217	−0.407109	−	0.913380i	$-0.633463\pi$
$79$	−1.09792e16	−0.814217	−0.407109	+	0.913380i	$0.633463\pi$
$80$	0	0
$81$	1.60067e16	0.959799
$82$	0	0
$83$	−1.22943e16	−0.599154	−0.299577	−	0.954072i	$-0.596846\pi$
$83$	−1.22943e16	−0.599154	−0.299577	+	0.954072i	$0.596846\pi$
$84$	0	0
$85$	8.05890e15	0.320786
$86$	0	0
$87$	−2.04404e15	−0.0667690
$88$	0	0
$89$	3.12723e16	0.842063	0.421031	−	0.907046i	$-0.361668\pi$
$89$	3.12723e16	0.842063	0.421031	+	0.907046i	$0.361668\pi$
$90$	0	0
$91$	3.17054e16	0.706775
$92$	0	0
$93$	8.49359e15	0.157393
$94$	0	0
$95$	9.20537e15	0.142361
$96$	0	0
$97$	1.38194e17	1.79032	0.895160	−	0.445745i	$-0.147061\pi$
$97$	1.38194e17	1.79032	0.895160	+	0.445745i	$0.147061\pi$
$98$	0	0
$99$	−1.64920e17	−1.79628
$100$	0	0
$101$	−9.59703e13	−0.000881872 0	−0.000440936	−	1.00000i	$-0.500140\pi$
$101$	−9.59703e13	−0.000881872 0	−0.000440936		1.00000i	$0.500140\pi$
$102$	0	0
$103$	−2.49608e17	−1.94152	−0.970761	−	0.240049i	$-0.922837\pi$
$103$	−2.49608e17	−1.94152	−0.970761	+	0.240049i	$0.922837\pi$
$104$	0	0
$105$	6.63518e15	0.0438272
$106$	0	0
$107$	7.23274e16	0.406950	0.203475	−	0.979080i	$-0.434777\pi$
$107$	7.23274e16	0.406950	0.203475	+	0.979080i	$0.434777\pi$
$108$	0	0
$109$	−1.11700e17	−0.536942	−0.268471	−	0.963288i	$-0.586518\pi$
$109$	−1.11700e17	−0.536942	−0.268471	+	0.963288i	$0.586518\pi$
$110$	0	0
$111$	4.90202e16	0.201897
$112$	0	0
$113$	1.49665e17	0.529607	0.264804	−	0.964302i	$-0.414693\pi$
$113$	1.49665e17	0.529607	0.264804	+	0.964302i	$0.414693\pi$
$114$	0	0
$115$	−6.34080e16	−0.193291
$116$	0	0
$117$	−3.13533e17	−0.825473
$118$	0	0
$119$	2.65792e17	0.605887
$120$	0	0
$121$	1.17025e18	2.31528
$122$	0	0
$123$	−4.10808e16	−0.0707043
$124$	0	0
$125$	5.96046e16	0.0894427
$126$	0	0
$127$	−5.87093e17	−0.769796	−0.384898	−	0.922959i	$-0.625763\pi$
$127$	−5.87093e17	−0.769796	−0.384898	+	0.922959i	$0.625763\pi$
$128$	0	0
$129$	−1.49691e15	−0.00171863
$130$	0	0
$131$	1.16170e18	1.17027	0.585137	−	0.810934i	$-0.301041\pi$
$131$	1.16170e18	1.17027	0.585137	+	0.810934i	$0.301041\pi$
$132$	0	0
$133$	3.03604e17	0.268886
$134$	0	0
$135$	−1.32125e17	−0.103074
$136$	0	0
$137$	−1.22337e18	−0.842234	−0.421117	−	0.907006i	$-0.638362\pi$
$137$	−1.22337e18	−0.842234	−0.421117	+	0.907006i	$0.638362\pi$
$138$	0	0
$139$	1.13922e18	0.693398	0.346699	−	0.937976i	$-0.387303\pi$
$139$	1.13922e18	0.693398	0.346699	+	0.937976i	$0.387303\pi$
$140$	0	0
$141$	−2.87998e17	−0.155248
$142$	0	0
$143$	3.18570e18	1.52352
$144$	0	0
$145$	−6.05594e17	−0.257367
$146$	0	0
$147$	−8.78784e16	−0.0332418
$148$	0	0
$149$	3.78128e18	1.27513	0.637567	−	0.770395i	$-0.279941\pi$
$149$	3.78128e18	1.27513	0.637567	+	0.770395i	$0.279941\pi$
$150$	0	0
$151$	3.56935e18	1.07469	0.537347	−	0.843361i	$-0.319426\pi$
$151$	3.56935e18	1.07469	0.537347	+	0.843361i	$0.319426\pi$
$152$	0	0
$153$	−2.62840e18	−0.707643
$154$	0	0
$155$	2.51643e18	0.606687
$156$	0	0
$157$	−1.24660e18	−0.269513	−0.134756	−	0.990879i	$-0.543025\pi$
$157$	−1.24660e18	−0.269513	−0.134756	+	0.990879i	$0.543025\pi$
$158$	0	0
$159$	1.63752e17	0.0317916
$160$	0	0
$161$	−2.09127e18	−0.365081
$162$	0	0
$163$	4.77033e18	0.749814	0.374907	−	0.927062i	$-0.377675\pi$
$163$	4.77033e18	0.749814	0.374907	+	0.927062i	$0.377675\pi$
$164$	0	0
$165$	6.66691e17	0.0944738
$166$	0	0
$167$	3.38636e18	0.433155	0.216577	−	0.976265i	$-0.430511\pi$
$167$	3.38636e18	0.433155	0.216577	+	0.976265i	$0.430511\pi$
$168$	0	0
$169$	−2.59402e18	−0.299872
$170$	0	0
$171$	−3.00232e18	−0.314044
$172$	0	0
$173$	6.87772e18	0.651707	0.325854	−	0.945420i	$-0.394348\pi$
$173$	6.87772e18	0.651707	0.325854	+	0.945420i	$0.394348\pi$
$174$	0	0
$175$	1.96583e18	0.168936
$176$	0	0
$177$	4.31269e16	0.00336493
$178$	0	0
$179$	2.09387e19	1.48491	0.742455	−	0.669896i	$-0.233661\pi$
$179$	2.09387e19	1.48491	0.742455	+	0.669896i	$0.233661\pi$
$180$	0	0
$181$	1.20204e19	0.775622	0.387811	−	0.921739i	$-0.373231\pi$
$181$	1.20204e19	0.775622	0.387811	+	0.921739i	$0.373231\pi$
$182$	0	0
$183$	2.76223e18	0.162340
$184$	0	0
$185$	1.45234e19	0.778232
$186$	0	0
$187$	2.67063e19	1.30605
$188$	0	0
$189$	−4.35763e18	−0.194682
$190$	0	0
$191$	1.47162e18	0.0601190	0.0300595	−	0.999548i	$-0.490430\pi$
$191$	1.47162e18	0.0601190	0.0300595	+	0.999548i	$0.490430\pi$
$192$	0	0
$193$	4.39550e19	1.64351	0.821753	−	0.569844i	$-0.192996\pi$
$193$	4.39550e19	1.64351	0.821753	+	0.569844i	$0.192996\pi$
$194$	0	0
$195$	1.26746e18	0.0434150
$196$	0	0
$197$	4.62419e19	1.45235	0.726177	−	0.687508i	$-0.241295\pi$
$197$	4.62419e19	1.45235	0.726177	+	0.687508i	$0.241295\pi$
$198$	0	0
$199$	−4.24592e19	−1.22383	−0.611914	−	0.790924i	$-0.709600\pi$
$199$	−4.24592e19	−1.22383	−0.611914	+	0.790924i	$0.709600\pi$
$200$	0	0
$201$	3.82598e18	0.101292
$202$	0	0
$203$	−1.99732e19	−0.486106
$204$	0	0
$205$	−1.21712e19	−0.272536
$206$	0	0
$207$	2.06804e19	0.426395
$208$	0	0
$209$	3.05056e19	0.579610
$210$	0	0
$211$	2.11934e19	0.371364	0.185682	−	0.982610i	$-0.440551\pi$
$211$	2.11934e19	0.371364	0.185682	+	0.982610i	$0.440551\pi$
$212$	0	0
$213$	−1.13432e19	−0.183446
$214$	0	0
$215$	−4.43495e17	−0.00662463
$216$	0	0
$217$	8.29946e19	1.14589
$218$	0	0
$219$	−1.49997e19	−0.191561
$220$	0	0
$221$	5.07718e19	0.600190
$222$	0	0
$223$	4.41630e19	0.483578	0.241789	−	0.970329i	$-0.422266\pi$
$223$	4.41630e19	0.483578	0.241789	+	0.970329i	$0.422266\pi$
$224$	0	0
$225$	−1.94400e19	−0.197308
$226$	0	0
$227$	−2.83332e19	−0.266732	−0.133366	−	0.991067i	$-0.542579\pi$
$227$	−2.83332e19	−0.266732	−0.133366	+	0.991067i	$0.542579\pi$
$228$	0	0
$229$	1.80362e20	1.57596	0.787978	−	0.615704i	$-0.211128\pi$
$229$	1.80362e20	1.57596	0.787978	+	0.615704i	$0.211128\pi$
$230$	0	0
$231$	2.19882e19	0.178438
$232$	0	0
$233$	5.28102e18	0.0398284	0.0199142	−	0.999802i	$-0.493661\pi$
$233$	5.28102e18	0.0398284	0.0199142	+	0.999802i	$0.493661\pi$
$234$	0	0
$235$	−8.53263e19	−0.598420
$236$	0	0
$237$	−1.44756e19	−0.0944662
$238$	0	0
$239$	2.33305e20	1.41756	0.708780	−	0.705429i	$-0.249246\pi$
$239$	2.33305e20	1.41756	0.708780	+	0.705429i	$0.249246\pi$
$240$	0	0
$241$	−2.44816e18	−0.0138578	−0.00692890	−	0.999976i	$-0.502206\pi$
$241$	−2.44816e18	−0.0138578	−0.00692890	+	0.999976i	$0.502206\pi$
$242$	0	0
$243$	6.47846e19	0.341837
$244$	0	0
$245$	−2.60361e19	−0.128134
$246$	0	0
$247$	5.79947e19	0.266357
$248$	0	0
$249$	−1.62095e19	−0.0695144
$250$	0	0
$251$	−3.81063e20	−1.52676	−0.763380	−	0.645950i	$-0.776461\pi$
$251$	−3.81063e20	−1.52676	−0.763380	+	0.645950i	$0.776461\pi$
$252$	0	0
$253$	−2.10127e20	−0.786969
$254$	0	0
$255$	1.06253e19	0.0372178
$256$	0	0
$257$	−2.95275e20	−0.967820	−0.483910	−	0.875118i	$-0.660784\pi$
$257$	−2.95275e20	−0.967820	−0.483910	+	0.875118i	$0.660784\pi$
$258$	0	0
$259$	4.78998e20	1.46989
$260$	0	0
$261$	1.97514e20	0.567744
$262$	0	0
$263$	−3.37915e19	−0.0910299	−0.0455150	−	0.998964i	$-0.514493\pi$
$263$	−3.37915e19	−0.0910299	−0.0455150	+	0.998964i	$0.514493\pi$
$264$	0	0
$265$	4.85155e19	0.122544
$266$	0	0
$267$	4.12312e19	0.0976969
$268$	0	0
$269$	−5.37595e20	−1.19553	−0.597766	−	0.801671i	$-0.703945\pi$
$269$	−5.37595e20	−1.19553	−0.597766	+	0.801671i	$0.703945\pi$
$270$	0	0
$271$	6.17960e20	1.29039	0.645196	−	0.764017i	$-0.276776\pi$
$271$	6.17960e20	1.29039	0.645196	+	0.764017i	$0.276776\pi$
$272$	0	0
$273$	4.18022e19	0.0820006
$274$	0	0
$275$	1.97523e20	0.364158
$276$	0	0
$277$	−4.52933e20	−0.785157	−0.392578	−	0.919719i	$-0.628417\pi$
$277$	−4.52933e20	−0.785157	−0.392578	+	0.919719i	$0.628417\pi$
$278$	0	0
$279$	−8.20729e20	−1.33833
$280$	0	0
$281$	−7.92146e20	−1.21563	−0.607815	−	0.794079i	$-0.707954\pi$
$281$	−7.92146e20	−1.21563	−0.607815	+	0.794079i	$0.707954\pi$
$282$	0	0
$283$	−4.00061e20	−0.578019	−0.289009	−	0.957326i	$-0.593326\pi$
$283$	−4.00061e20	−0.578019	−0.289009	+	0.957326i	$0.593326\pi$
$284$	0	0
$285$	1.21369e19	0.0165168
$286$	0	0
$287$	−4.01419e20	−0.514756
$288$	0	0
$289$	−4.01611e20	−0.485483
$290$	0	0
$291$	1.82204e20	0.207715
$292$	0	0
$293$	−5.43391e20	−0.584437	−0.292218	−	0.956352i	$-0.594393\pi$
$293$	−5.43391e20	−0.584437	−0.292218	+	0.956352i	$0.594393\pi$
$294$	0	0
$295$	1.27774e19	0.0129704
$296$	0	0
$297$	−4.37848e20	−0.419656
$298$	0	0
$299$	−3.99477e20	−0.361648
$300$	0	0
$301$	−1.46270e19	−0.0125123
$302$	0	0
$303$	−1.26533e17	−0.000102316 0
$304$	0	0
$305$	8.18375e20	0.625754
$306$	0	0
$307$	4.88990e20	0.353691	0.176845	−	0.984239i	$-0.443411\pi$
$307$	4.88990e20	0.353691	0.176845	+	0.984239i	$0.443411\pi$
$308$	0	0
$309$	−3.29098e20	−0.225257
$310$	0	0
$311$	−1.87387e21	−1.21416	−0.607079	−	0.794642i	$-0.707659\pi$
$311$	−1.87387e21	−1.21416	−0.607079	+	0.794642i	$0.707659\pi$
$312$	0	0
$313$	1.09750e21	0.673408	0.336704	−	0.941611i	$-0.390688\pi$
$313$	1.09750e21	0.673408	0.336704	+	0.941611i	$0.390688\pi$
$314$	0	0
$315$	−6.41152e20	−0.372667
$316$	0	0
$317$	1.14233e21	0.629201	0.314600	−	0.949224i	$-0.398129\pi$
$317$	1.14233e21	0.629201	0.314600	+	0.949224i	$0.398129\pi$
$318$	0	0
$319$	−2.00687e21	−1.04785
$320$	0	0
$321$	9.53607e19	0.0472147
$322$	0	0
$323$	4.86180e20	0.228337
$324$	0	0
$325$	3.75515e20	0.167347
$326$	0	0
$327$	−1.47272e20	−0.0622965
$328$	0	0
$329$	−2.81416e21	−1.13027
$330$	0	0
$331$	−4.78454e21	−1.82517	−0.912583	−	0.408892i	$-0.865915\pi$
$331$	−4.78454e21	−1.82517	−0.912583	+	0.408892i	$0.865915\pi$
$332$	0	0
$333$	−4.73679e21	−1.71675
$334$	0	0
$335$	1.13354e21	0.390441
$336$	0	0
$337$	−4.59118e21	−1.50338	−0.751692	−	0.659515i	$-0.770762\pi$
$337$	−4.59118e21	−1.50338	−0.751692	+	0.659515i	$0.770762\pi$
$338$	0	0
$339$	1.97327e20	0.0614455
$340$	0	0
$341$	8.33916e21	2.47007
$342$	0	0
$343$	−3.85574e21	−1.08669
$344$	0	0
$345$	−8.36009e19	−0.0224258
$346$	0	0
$347$	−1.40187e21	−0.358020	−0.179010	−	0.983847i	$-0.557289\pi$
$347$	−1.40187e21	−0.358020	−0.179010	+	0.983847i	$0.557289\pi$
$348$	0	0
$349$	5.24405e21	1.27541	0.637707	−	0.770279i	$-0.279883\pi$
$349$	5.24405e21	1.27541	0.637707	+	0.770279i	$0.279883\pi$
$350$	0	0
$351$	−8.32401e20	−0.192851
$352$	0	0
$353$	8.36954e21	1.84764	0.923818	−	0.382832i	$-0.125051\pi$
$353$	8.36954e21	1.84764	0.923818	+	0.382832i	$0.125051\pi$
$354$	0	0
$355$	−3.36068e21	−0.707109
$356$	0	0
$357$	3.50435e20	0.0702956
$358$	0	0
$359$	6.38455e21	1.22131	0.610657	−	0.791895i	$-0.290905\pi$
$359$	6.38455e21	1.22131	0.610657	+	0.791895i	$0.290905\pi$
$360$	0	0
$361$	−4.92504e21	−0.898667
$362$	0	0
$363$	1.54293e21	0.268621
$364$	0	0
$365$	−4.44402e21	−0.738391
$366$	0	0
$367$	3.11866e21	0.494659	0.247330	−	0.968931i	$-0.420447\pi$
$367$	3.11866e21	0.494659	0.247330	+	0.968931i	$0.420447\pi$
$368$	0	0
$369$	3.96961e21	0.601207
$370$	0	0
$371$	1.60010e21	0.231456
$372$	0	0
$373$	−1.12730e22	−1.55781	−0.778904	−	0.627143i	$-0.784224\pi$
$373$	−1.12730e22	−1.55781	−0.778904	+	0.627143i	$0.784224\pi$
$374$	0	0
$375$	7.85863e19	0.0103772
$376$	0	0
$377$	−3.81530e21	−0.481534
$378$	0	0
$379$	6.82454e21	0.823456	0.411728	−	0.911307i	$-0.364925\pi$
$379$	6.82454e21	0.823456	0.411728	+	0.911307i	$0.364925\pi$
$380$	0	0
$381$	−7.74059e20	−0.0893124
$382$	0	0
$383$	−5.10368e21	−0.563240	−0.281620	−	0.959526i	$-0.590872\pi$
$383$	−5.10368e21	−0.563240	−0.281620	+	0.959526i	$0.590872\pi$
$384$	0	0
$385$	6.51454e21	0.687808
$386$	0	0
$387$	1.44645e20	0.0146137
$388$	0	0
$389$	−8.13898e21	−0.787043	−0.393522	−	0.919315i	$-0.628743\pi$
$389$	−8.13898e21	−0.787043	−0.393522	+	0.919315i	$0.628743\pi$
$390$	0	0
$391$	−3.34888e21	−0.310025
$392$	0	0
$393$	1.53165e21	0.135776
$394$	0	0
$395$	−4.28875e21	−0.364129
$396$	0	0
$397$	−7.03201e21	−0.571953	−0.285977	−	0.958237i	$-0.592318\pi$
$397$	−7.03201e21	−0.571953	−0.285977	+	0.958237i	$0.592318\pi$
$398$	0	0
$399$	4.00289e20	0.0311964
$400$	0	0
$401$	1.67041e22	1.24766	0.623830	−	0.781560i	$-0.285576\pi$
$401$	1.67041e22	1.24766	0.623830	+	0.781560i	$0.285576\pi$
$402$	0	0
$403$	1.58537e22	1.13511
$404$	0	0
$405$	6.25263e21	0.429235
$406$	0	0
$407$	4.81290e22	3.16850
$408$	0	0
$409$	−5.17984e21	−0.327091	−0.163545	−	0.986536i	$-0.552293\pi$
$409$	−5.17984e21	−0.327091	−0.163545	+	0.986536i	$0.552293\pi$
$410$	0	0
$411$	−1.61296e21	−0.0977168
$412$	0	0
$413$	4.21412e20	0.0244981
$414$	0	0
$415$	−4.80244e21	−0.267950
$416$	0	0
$417$	1.50202e21	0.0804486
$418$	0	0
$419$	2.11912e22	1.08977	0.544887	−	0.838509i	$-0.316572\pi$
$419$	2.11912e22	1.08977	0.544887	+	0.838509i	$0.316572\pi$
$420$	0	0
$421$	1.47972e22	0.730774	0.365387	−	0.930856i	$-0.380937\pi$
$421$	1.47972e22	0.730774	0.365387	+	0.930856i	$0.380937\pi$
$422$	0	0
$423$	2.78290e22	1.32009
$424$	0	0
$425$	3.14801e21	0.143460
$426$	0	0
$427$	2.69909e22	1.18190
$428$	0	0
$429$	4.20022e21	0.176760
$430$	0	0
$431$	−3.11948e22	−1.26190	−0.630950	−	0.775823i	$-0.717335\pi$
$431$	−3.11948e22	−1.26190	−0.630950	+	0.775823i	$0.717335\pi$
$432$	0	0
$433$	1.92999e22	0.750597	0.375299	−	0.926904i	$-0.377540\pi$
$433$	1.92999e22	0.750597	0.375299	+	0.926904i	$0.377540\pi$
$434$	0	0
$435$	−7.98451e20	−0.0298600
$436$	0	0
$437$	−3.82530e21	−0.137586
$438$	0	0
$439$	−1.29380e22	−0.447630	−0.223815	−	0.974632i	$-0.571851\pi$
$439$	−1.29380e22	−0.447630	−0.223815	+	0.974632i	$0.571851\pi$
$440$	0	0
$441$	8.49163e21	0.282659
$442$	0	0
$443$	−1.95846e22	−0.627312	−0.313656	−	0.949537i	$-0.601554\pi$
$443$	−1.95846e22	−0.627312	−0.313656	+	0.949537i	$0.601554\pi$
$444$	0	0
$445$	1.22157e22	0.376582
$446$	0	0
$447$	4.98547e21	0.147942
$448$	0	0
$449$	−6.08659e22	−1.73892	−0.869460	−	0.494003i	$-0.835533\pi$
$449$	−6.08659e22	−1.73892	−0.869460	+	0.494003i	$0.835533\pi$
$450$	0	0
$451$	−4.03339e22	−1.10961
$452$	0	0
$453$	4.70604e21	0.124687
$454$	0	0
$455$	1.23849e22	0.316079
$456$	0	0
$457$	−1.58093e21	−0.0388710	−0.0194355	−	0.999811i	$-0.506187\pi$
$457$	−1.58093e21	−0.0388710	−0.0194355	+	0.999811i	$0.506187\pi$
$458$	0	0
$459$	−6.97816e21	−0.165323
$460$	0	0
$461$	7.43310e22	1.69712	0.848560	−	0.529100i	$-0.177470\pi$
$461$	7.43310e22	1.69712	0.848560	+	0.529100i	$0.177470\pi$
$462$	0	0
$463$	1.75196e22	0.385556	0.192778	−	0.981242i	$-0.438250\pi$
$463$	1.75196e22	0.385556	0.192778	+	0.981242i	$0.438250\pi$
$464$	0	0
$465$	3.31781e21	0.0703884
$466$	0	0
$467$	−7.70965e22	−1.57703	−0.788517	−	0.615013i	$-0.789151\pi$
$467$	−7.70965e22	−1.57703	−0.788517	+	0.615013i	$0.789151\pi$
$468$	0	0
$469$	3.73853e22	0.737449
$470$	0	0
$471$	−1.64359e21	−0.0312691
$472$	0	0
$473$	−1.46969e21	−0.0269716
$474$	0	0
$475$	3.59585e21	0.0636657
$476$	0	0
$477$	−1.58233e22	−0.270328
$478$	0	0
$479$	2.06396e22	0.340290	0.170145	−	0.985419i	$-0.445576\pi$
$479$	2.06396e22	0.340290	0.170145	+	0.985419i	$0.445576\pi$
$480$	0	0
$481$	9.14989e22	1.45607
$482$	0	0
$483$	−2.75725e21	−0.0423571
$484$	0	0
$485$	5.39822e22	0.800656
$486$	0	0
$487$	9.29138e22	1.33071	0.665356	−	0.746527i	$-0.268280\pi$
$487$	9.29138e22	1.33071	0.665356	+	0.746527i	$0.268280\pi$
$488$	0	0
$489$	6.28949e21	0.0869941
$490$	0	0
$491$	−4.31577e22	−0.576587	−0.288294	−	0.957542i	$-0.593088\pi$
$491$	−4.31577e22	−0.576587	−0.288294	+	0.957542i	$0.593088\pi$
$492$	0	0
$493$	−3.19843e22	−0.412799
$494$	0	0
$495$	−6.44219e22	−0.803322
$496$	0	0
$497$	−1.10839e23	−1.33556
$498$	0	0
$499$	4.81122e22	0.560274	0.280137	−	0.959960i	$-0.409620\pi$
$499$	4.81122e22	0.560274	0.280137	+	0.959960i	$0.409620\pi$
$500$	0	0
$501$	4.46478e21	0.0502550
$502$	0	0
$503$	−6.43576e22	−0.700280	−0.350140	−	0.936697i	$-0.613866\pi$
$503$	−6.43576e22	−0.700280	−0.350140	+	0.936697i	$0.613866\pi$
$504$	0	0
$505$	−3.74884e19	−0.000394385 0
$506$	0	0
$507$	−3.42011e21	−0.0347914
$508$	0	0
$509$	6.98569e21	0.0687240	0.0343620	−	0.999409i	$-0.489060\pi$
$509$	6.98569e21	0.0687240	0.0343620	+	0.999409i	$0.489060\pi$
$510$	0	0
$511$	−1.46569e23	−1.39464
$512$	0	0
$513$	−7.97089e21	−0.0733684
$514$	0	0
$515$	−9.75032e22	−0.868275
$516$	0	0
$517$	−2.82762e23	−2.43641
$518$	0	0
$519$	9.06799e21	0.0756116
$520$	0	0
$521$	1.59270e23	1.28533	0.642664	−	0.766148i	$-0.277829\pi$
$521$	1.59270e23	1.28533	0.642664	+	0.766148i	$0.277829\pi$
$522$	0	0
$523$	−2.37823e22	−0.185776	−0.0928880	−	0.995677i	$-0.529610\pi$
$523$	−2.37823e22	−0.185776	−0.0928880	+	0.995677i	$0.529610\pi$
$524$	0	0
$525$	2.59187e21	0.0196001
$526$	0	0
$527$	1.32905e23	0.973083
$528$	0	0
$529$	−1.14701e23	−0.813192
$530$	0	0
$531$	−4.16732e21	−0.0286124
$532$	0	0
$533$	−7.66795e22	−0.509915
$534$	0	0
$535$	2.82529e22	0.181993
$536$	0	0
$537$	2.76069e22	0.172280
$538$	0	0
$539$	−8.62806e22	−0.521686
$540$	0	0
$541$	−1.78303e23	−1.04468	−0.522339	−	0.852738i	$-0.674941\pi$
$541$	−1.78303e23	−1.04468	−0.522339	+	0.852738i	$0.674941\pi$
$542$	0	0
$543$	1.58484e22	0.0899884
$544$	0	0
$545$	−4.36328e22	−0.240128
$546$	0	0
$547$	1.48213e23	0.790668	0.395334	−	0.918538i	$-0.370629\pi$
$547$	1.48213e23	0.790668	0.395334	+	0.918538i	$0.370629\pi$
$548$	0	0
$549$	−2.66912e23	−1.38039
$550$	0	0
$551$	−3.65345e22	−0.183195
$552$	0	0
$553$	−1.41448e23	−0.687753
$554$	0	0
$555$	1.91485e22	0.0902911
$556$	0	0
$557$	1.77111e23	0.809983	0.404992	−	0.914320i	$-0.367274\pi$
$557$	1.77111e23	0.809983	0.404992	+	0.914320i	$0.367274\pi$
$558$	0	0
$559$	−2.79406e21	−0.0123947
$560$	0	0
$561$	3.52112e22	0.151529
$562$	0	0
$563$	4.04590e21	0.0168925	0.00844625	−	0.999964i	$-0.497311\pi$
$563$	4.04590e21	0.0168925	0.00844625	+	0.999964i	$0.497311\pi$
$564$	0	0
$565$	5.84629e22	0.236847
$566$	0	0
$567$	2.06219e23	0.810723
$568$	0	0
$569$	−3.66551e23	−1.39856	−0.699278	−	0.714850i	$-0.746495\pi$
$569$	−3.66551e23	−1.39856	−0.699278	+	0.714850i	$0.746495\pi$
$570$	0	0
$571$	−4.00635e23	−1.48369	−0.741843	−	0.670573i	$-0.766048\pi$
$571$	−4.00635e23	−1.48369	−0.741843	+	0.670573i	$0.766048\pi$
$572$	0	0
$573$	1.94027e21	0.00697506
$574$	0	0
$575$	−2.47688e22	−0.0864425
$576$	0	0
$577$	8.04220e22	0.272509	0.136254	−	0.990674i	$-0.456494\pi$
$577$	8.04220e22	0.272509	0.136254	+	0.990674i	$0.456494\pi$
$578$	0	0
$579$	5.79529e22	0.190681
$580$	0	0
$581$	−1.58390e23	−0.506093
$582$	0	0
$583$	1.60775e23	0.498926
$584$	0	0
$585$	−1.22474e23	−0.369163
$586$	0	0
$587$	1.16687e23	0.341663	0.170832	−	0.985300i	$-0.445355\pi$
$587$	1.16687e23	0.341663	0.170832	+	0.985300i	$0.445355\pi$
$588$	0	0
$589$	1.51812e23	0.431843
$590$	0	0
$591$	6.09680e22	0.168503
$592$	0	0
$593$	−3.44705e23	−0.925726	−0.462863	−	0.886430i	$-0.653178\pi$
$593$	−3.44705e23	−0.925726	−0.462863	+	0.886430i	$0.653178\pi$
$594$	0	0
$595$	1.03825e23	0.270961
$596$	0	0
$597$	−5.59807e22	−0.141990
$598$	0	0
$599$	−1.64013e23	−0.404344	−0.202172	−	0.979350i	$-0.564800\pi$
$599$	−1.64013e23	−0.404344	−0.202172	+	0.979350i	$0.564800\pi$
$600$	0	0
$601$	5.27655e23	1.26449	0.632247	−	0.774767i	$-0.282133\pi$
$601$	5.27655e23	1.26449	0.632247	+	0.774767i	$0.282133\pi$
$602$	0	0
$603$	−3.69702e23	−0.861300
$604$	0	0
$605$	4.57129e23	1.03542
$606$	0	0
$607$	3.31440e23	0.729963	0.364981	−	0.931015i	$-0.381075\pi$
$607$	3.31440e23	0.729963	0.364981	+	0.931015i	$0.381075\pi$
$608$	0	0
$609$	−2.63338e22	−0.0563984
$610$	0	0
$611$	−5.37564e23	−1.11964
$612$	0	0
$613$	5.48999e23	1.11214	0.556068	−	0.831137i	$-0.312309\pi$
$613$	5.48999e23	1.11214	0.556068	+	0.831137i	$0.312309\pi$
$614$	0	0
$615$	−1.60472e22	−0.0316199
$616$	0	0
$617$	−1.71333e23	−0.328410	−0.164205	−	0.986426i	$-0.552506\pi$
$617$	−1.71333e23	−0.328410	−0.164205	+	0.986426i	$0.552506\pi$
$618$	0	0
$619$	−5.04276e23	−0.940368	−0.470184	−	0.882568i	$-0.655812\pi$
$619$	−5.04276e23	−0.940368	−0.470184	+	0.882568i	$0.655812\pi$
$620$	0	0
$621$	5.49047e22	0.0996164
$622$	0	0
$623$	4.02889e23	0.711273
$624$	0	0
$625$	2.32831e22	0.0400000
$626$	0	0
$627$	4.02204e22	0.0672469
$628$	0	0
$629$	7.67051e23	1.24823
$630$	0	0
$631$	−7.36135e22	−0.116602	−0.0583012	−	0.998299i	$-0.518568\pi$
$631$	−7.36135e22	−0.116602	−0.0583012	+	0.998299i	$0.518568\pi$
$632$	0	0
$633$	2.79427e22	0.0430860
$634$	0	0
$635$	−2.29333e23	−0.344263
$636$	0	0
$637$	−1.64030e23	−0.239738
$638$	0	0
$639$	1.09608e24	1.55986
$640$	0	0
$641$	1.37693e23	0.190818	0.0954090	−	0.995438i	$-0.469584\pi$
$641$	1.37693e23	0.190818	0.0954090	+	0.995438i	$0.469584\pi$
$642$	0	0
$643$	−1.06980e23	−0.144380	−0.0721901	−	0.997391i	$-0.522999\pi$
$643$	−1.06980e23	−0.144380	−0.0721901	+	0.997391i	$0.522999\pi$
$644$	0	0
$645$	−5.84730e20	−0.000768595 0
$646$	0	0
$647$	−1.05305e24	−1.34822	−0.674112	−	0.738629i	$-0.735474\pi$
$647$	−1.05305e24	−1.34822	−0.674112	+	0.738629i	$0.735474\pi$
$648$	0	0
$649$	4.23428e22	0.0528080
$650$	0	0
$651$	1.09425e23	0.132947
$652$	0	0
$653$	2.49950e23	0.295863	0.147932	−	0.988998i	$-0.452738\pi$
$653$	2.49950e23	0.295863	0.147932	+	0.988998i	$0.452738\pi$
$654$	0	0
$655$	4.53789e23	0.523363
$656$	0	0
$657$	1.44941e24	1.62887
$658$	0	0
$659$	−1.48770e24	−1.62926	−0.814631	−	0.579980i	$-0.803060\pi$
$659$	−1.48770e24	−1.62926	−0.814631	+	0.579980i	$0.803060\pi$
$660$	0	0
$661$	−1.01867e24	−1.08723	−0.543615	−	0.839335i	$-0.682945\pi$
$661$	−1.01867e24	−1.08723	−0.543615	+	0.839335i	$0.682945\pi$
$662$	0	0
$663$	6.69406e22	0.0696345
$664$	0	0
$665$	1.18595e23	0.120249
$666$	0	0
$667$	2.51655e23	0.248735
$668$	0	0
$669$	5.82271e22	0.0561052
$670$	0	0
$671$	2.71200e24	2.54770
$672$	0	0
$673$	1.93408e24	1.77152	0.885762	−	0.464140i	$-0.153637\pi$
$673$	1.93408e24	1.77152	0.885762	+	0.464140i	$0.153637\pi$
$674$	0	0
$675$	−5.16114e22	−0.0460960
$676$	0	0
$677$	1.40989e24	1.22795	0.613976	−	0.789325i	$-0.289569\pi$
$677$	1.40989e24	1.22795	0.613976	+	0.789325i	$0.289569\pi$
$678$	0	0
$679$	1.78039e24	1.51225
$680$	0	0
$681$	−3.73561e22	−0.0309465
$682$	0	0
$683$	−1.08698e24	−0.878305	−0.439153	−	0.898413i	$-0.644721\pi$
$683$	−1.08698e24	−0.878305	−0.439153	+	0.898413i	$0.644721\pi$
$684$	0	0
$685$	−4.77879e23	−0.376659
$686$	0	0
$687$	2.37800e23	0.182844
$688$	0	0
$689$	3.05653e23	0.229279
$690$	0	0
$691$	4.28347e23	0.313497	0.156748	−	0.987639i	$-0.449899\pi$
$691$	4.28347e23	0.313497	0.156748	+	0.987639i	$0.449899\pi$
$692$	0	0
$693$	−2.12471e24	−1.51728
$694$	0	0
$695$	4.45008e23	0.310097
$696$	0	0
$697$	−6.42818e23	−0.437129
$698$	0	0
$699$	6.96281e21	0.00462092
$700$	0	0
$701$	2.71394e24	1.75791	0.878955	−	0.476905i	$-0.158241\pi$
$701$	2.71394e24	1.75791	0.878955	+	0.476905i	$0.158241\pi$
$702$	0	0
$703$	8.76173e23	0.553949
$704$	0	0
$705$	−1.12499e23	−0.0694292
$706$	0	0
$707$	−1.23641e21	−0.000744899 0
$708$	0	0
$709$	−1.69687e24	−0.998056	−0.499028	−	0.866586i	$-0.666310\pi$
$709$	−1.69687e24	−0.998056	−0.499028	+	0.866586i	$0.666310\pi$
$710$	0	0
$711$	1.39877e24	0.803257
$712$	0	0
$713$	−1.04570e24	−0.586337
$714$	0	0
$715$	1.24441e24	0.681340
$716$	0	0
$717$	3.07603e23	0.164467
$718$	0	0
$719$	−1.87362e24	−0.978333	−0.489167	−	0.872190i	$-0.662699\pi$
$719$	−1.87362e24	−0.978333	−0.489167	+	0.872190i	$0.662699\pi$
$720$	0	0
$721$	−3.21576e24	−1.63996
$722$	0	0
$723$	−3.22779e21	−0.00160779
$724$	0	0
$725$	−2.36560e23	−0.115098
$726$	0	0
$727$	2.29336e24	1.09001	0.545003	−	0.838434i	$-0.316528\pi$
$727$	2.29336e24	1.09001	0.545003	+	0.838434i	$0.316528\pi$
$728$	0	0
$729$	−1.98170e24	−0.920138
$730$	0	0
$731$	−2.34231e22	−0.0106254
$732$	0	0
$733$	−2.61406e24	−1.15859	−0.579297	−	0.815117i	$-0.696673\pi$
$733$	−2.61406e24	−1.15859	−0.579297	+	0.815117i	$0.696673\pi$
$734$	0	0
$735$	−3.43275e22	−0.0148662
$736$	0	0
$737$	3.75642e24	1.58964
$738$	0	0
$739$	−4.47027e24	−1.84866	−0.924329	−	0.381598i	$-0.875374\pi$
$739$	−4.47027e24	−1.84866	−0.924329	+	0.381598i	$0.875374\pi$
$740$	0	0
$741$	7.64637e22	0.0309030
$742$	0	0
$743$	−2.49359e23	−0.0984963	−0.0492482	−	0.998787i	$-0.515683\pi$
$743$	−2.49359e23	−0.0984963	−0.0492482	+	0.998787i	$0.515683\pi$
$744$	0	0
$745$	1.47706e24	0.570258
$746$	0	0
$747$	1.56631e24	0.591089
$748$	0	0
$749$	9.31812e23	0.343742
$750$	0	0
$751$	8.20625e23	0.295941	0.147971	−	0.988992i	$-0.452726\pi$
$751$	8.20625e23	0.295941	0.147971	+	0.988992i	$0.452726\pi$
$752$	0	0
$753$	−5.02417e23	−0.177136
$754$	0	0
$755$	1.39428e24	0.480618
$756$	0	0
$757$	4.01821e24	1.35431	0.677154	−	0.735841i	$-0.263213\pi$
$757$	4.01821e24	1.35431	0.677154	+	0.735841i	$0.263213\pi$
$758$	0	0
$759$	−2.77044e23	−0.0913049
$760$	0	0
$761$	−4.19948e24	−1.35340	−0.676700	−	0.736259i	$-0.736591\pi$
$761$	−4.19948e24	−1.35340	−0.676700	+	0.736259i	$0.736591\pi$
$762$	0	0
$763$	−1.43906e24	−0.453544
$764$	0	0
$765$	−1.02672e24	−0.316468
$766$	0	0
$767$	8.04987e22	0.0242677
$768$	0	0
$769$	−8.42992e23	−0.248571	−0.124285	−	0.992247i	$-0.539664\pi$
$769$	−8.42992e23	−0.248571	−0.124285	+	0.992247i	$0.539664\pi$
$770$	0	0
$771$	−3.89308e23	−0.112287
$772$	0	0
$773$	−7.90088e23	−0.222920	−0.111460	−	0.993769i	$-0.535553\pi$
$773$	−7.90088e23	−0.222920	−0.111460	+	0.993769i	$0.535553\pi$
$774$	0	0
$775$	9.82979e23	0.271319
$776$	0	0
$777$	6.31540e23	0.170538
$778$	0	0
$779$	−7.34266e23	−0.193993
$780$	0	0
$781$	−1.11369e25	−2.87893
$782$	0	0
$783$	5.24381e23	0.132639
$784$	0	0
$785$	−4.86952e23	−0.120530
$786$	0	0
$787$	−6.59484e24	−1.59742	−0.798710	−	0.601717i	$-0.794484\pi$
$787$	−6.59484e24	−1.59742	−0.798710	+	0.601717i	$0.794484\pi$
$788$	0	0
$789$	−4.45528e22	−0.0105614
$790$	0	0
$791$	1.92817e24	0.447348
$792$	0	0
$793$	5.15584e24	1.17079
$794$	0	0
$795$	6.39658e22	0.0142176
$796$	0	0
$797$	7.26238e24	1.58010	0.790048	−	0.613045i	$-0.210056\pi$
$797$	7.26238e24	1.58010	0.790048	+	0.613045i	$0.210056\pi$
$798$	0	0
$799$	−4.50649e24	−0.959822
$800$	0	0
$801$	−3.98415e24	−0.830728
$802$	0	0
$803$	−1.47270e25	−3.00629
$804$	0	0
$805$	−8.16901e23	−0.163269
$806$	0	0
$807$	−7.08798e23	−0.138707
$808$	0	0
$809$	4.34267e24	0.832137	0.416068	−	0.909333i	$-0.363408\pi$
$809$	4.34267e24	0.832137	0.416068	+	0.909333i	$0.363408\pi$
$810$	0	0
$811$	−2.42001e24	−0.454088	−0.227044	−	0.973885i	$-0.572906\pi$
$811$	−2.42001e24	−0.454088	−0.227044	+	0.973885i	$0.572906\pi$
$812$	0	0
$813$	8.14756e23	0.149712
$814$	0	0
$815$	1.86341e24	0.335327
$816$	0	0
$817$	−2.67553e22	−0.00471544
$818$	0	0
$819$	−4.03932e24	−0.697261
$820$	0	0
$821$	7.14459e24	1.20798	0.603991	−	0.796991i	$-0.293576\pi$
$821$	7.14459e24	1.20798	0.603991	+	0.796991i	$0.293576\pi$
$822$	0	0
$823$	8.98461e24	1.48799	0.743996	−	0.668184i	$-0.232928\pi$
$823$	8.98461e24	1.48799	0.743996	+	0.668184i	$0.232928\pi$
$824$	0	0
$825$	2.60426e23	0.0422500
$826$	0	0
$827$	−6.67657e24	−1.06110	−0.530550	−	0.847654i	$-0.678015\pi$
$827$	−6.67657e24	−1.06110	−0.530550	+	0.847654i	$0.678015\pi$
$828$	0	0
$829$	−4.56314e24	−0.710477	−0.355238	−	0.934776i	$-0.615600\pi$
$829$	−4.56314e24	−0.710477	−0.355238	+	0.934776i	$0.615600\pi$
$830$	0	0
$831$	−5.97175e23	−0.0910946
$832$	0	0
$833$	−1.37509e24	−0.205517
$834$	0	0
$835$	1.32280e24	0.193713
$836$	0	0
$837$	−2.17896e24	−0.312668
$838$	0	0
$839$	1.99989e24	0.281209	0.140604	−	0.990066i	$-0.455095\pi$
$839$	1.99989e24	0.281209	0.140604	+	0.990066i	$0.455095\pi$
$840$	0	0
$841$	−4.85365e24	−0.668810
$842$	0	0
$843$	−1.04441e24	−0.141038
$844$	0	0
$845$	−1.01329e24	−0.134107
$846$	0	0
$847$	1.50766e25	1.95567
$848$	0	0
$849$	−5.27465e23	−0.0670623
$850$	0	0
$851$	−6.03521e24	−0.752127
$852$	0	0
$853$	6.35284e24	0.776071	0.388035	−	0.921645i	$-0.373154\pi$
$853$	6.35284e24	0.776071	0.388035	+	0.921645i	$0.373154\pi$
$854$	0	0
$855$	−1.17278e24	−0.140445
$856$	0	0
$857$	6.73960e24	0.791220	0.395610	−	0.918419i	$-0.370533\pi$
$857$	6.73960e24	0.791220	0.395610	+	0.918419i	$0.370533\pi$
$858$	0	0
$859$	−1.54449e25	−1.77763	−0.888817	−	0.458262i	$-0.848472\pi$
$859$	−1.54449e25	−1.77763	−0.888817	+	0.458262i	$0.848472\pi$
$860$	0	0
$861$	−5.29255e23	−0.0597225
$862$	0	0
$863$	−1.22551e25	−1.35590	−0.677948	−	0.735110i	$-0.737131\pi$
$863$	−1.22551e25	−1.35590	−0.677948	+	0.735110i	$0.737131\pi$
$864$	0	0
$865$	2.68661e24	0.291452
$866$	0	0
$867$	−5.29508e23	−0.0563262
$868$	0	0
$869$	−1.42124e25	−1.48252
$870$	0	0
$871$	7.14139e24	0.730514
$872$	0	0
$873$	−1.76062e25	−1.76622
$874$	0	0
$875$	7.67902e23	0.0755505
$876$	0	0
$877$	1.24989e25	1.20608	0.603041	−	0.797710i	$-0.293955\pi$
$877$	1.24989e25	1.20608	0.603041	+	0.797710i	$0.293955\pi$
$878$	0	0
$879$	−7.16439e23	−0.0678069
$880$	0	0
$881$	4.70446e24	0.436732	0.218366	−	0.975867i	$-0.429927\pi$
$881$	4.70446e24	0.436732	0.218366	+	0.975867i	$0.429927\pi$
$882$	0	0
$883$	2.05718e25	1.87330	0.936650	−	0.350267i	$-0.113909\pi$
$883$	2.05718e25	1.87330	0.936650	+	0.350267i	$0.113909\pi$
$884$	0	0
$885$	1.68465e22	0.00150484
$886$	0	0
$887$	1.45774e23	0.0127741	0.00638705	−	0.999980i	$-0.497967\pi$
$887$	1.45774e23	0.0127741	0.00638705	+	0.999980i	$0.497967\pi$
$888$	0	0
$889$	−7.56367e24	−0.650231
$890$	0	0
$891$	2.07205e25	1.74759
$892$	0	0
$893$	−5.14759e24	−0.425958
$894$	0	0
$895$	8.17920e24	0.664072
$896$	0	0
$897$	−5.26694e23	−0.0419587
$898$	0	0
$899$	−9.98725e24	−0.780708
$900$	0	0
$901$	2.56234e24	0.196551
$902$	0	0
$903$	−1.92851e22	−0.00145169
$904$	0	0
$905$	4.69546e24	0.346869
$906$	0	0
$907$	3.03117e24	0.219760	0.109880	−	0.993945i	$-0.464953\pi$
$907$	3.03117e24	0.219760	0.109880	+	0.993945i	$0.464953\pi$
$908$	0	0
$909$	1.22268e22	0.000870001 0
$910$	0	0
$911$	7.30782e24	0.510366	0.255183	−	0.966893i	$-0.417864\pi$
$911$	7.30782e24	0.510366	0.255183	+	0.966893i	$0.417864\pi$
$912$	0	0
$913$	−1.59148e25	−1.09093
$914$	0	0
$915$	1.07899e24	0.0726005
$916$	0	0
$917$	1.49665e25	0.988507
$918$	0	0
$919$	1.55368e24	0.100735	0.0503673	−	0.998731i	$-0.483961\pi$
$919$	1.55368e24	0.100735	0.0503673	+	0.998731i	$0.483961\pi$
$920$	0	0
$921$	6.44713e23	0.0410355
$922$	0	0
$923$	−2.11726e25	−1.32300
$924$	0	0
$925$	5.67321e24	0.348036
$926$	0	0
$927$	3.18005e25	1.91539
$928$	0	0
$929$	−4.57365e24	−0.270476	−0.135238	−	0.990813i	$-0.543180\pi$
$929$	−4.57365e24	−0.270476	−0.135238	+	0.990813i	$0.543180\pi$
$930$	0	0
$931$	−1.57071e24	−0.0912062
$932$	0	0
$933$	−2.47062e24	−0.140868
$934$	0	0
$935$	1.04321e25	0.584084
$936$	0	0
$937$	2.61843e25	1.43964	0.719821	−	0.694159i	$-0.244224\pi$
$937$	2.61843e25	1.43964	0.719821	+	0.694159i	$0.244224\pi$
$938$	0	0
$939$	1.44701e24	0.0781294
$940$	0	0
$941$	−1.02278e25	−0.542341	−0.271170	−	0.962531i	$-0.587411\pi$
$941$	−1.02278e25	−0.542341	−0.271170	+	0.962531i	$0.587411\pi$
$942$	0	0
$943$	5.05774e24	0.263395
$944$	0	0
$945$	−1.70220e24	−0.0870644
$946$	0	0
$947$	3.57120e25	1.79407	0.897034	−	0.441962i	$-0.145717\pi$
$947$	3.57120e25	1.79407	0.897034	+	0.441962i	$0.145717\pi$
$948$	0	0
$949$	−2.79977e25	−1.38153
$950$	0	0
$951$	1.50612e24	0.0730005
$952$	0	0
$953$	1.63928e25	0.780483	0.390241	−	0.920713i	$-0.372392\pi$
$953$	1.63928e25	0.780483	0.390241	+	0.920713i	$0.372392\pi$
$954$	0	0
$955$	5.74851e23	0.0268860
$956$	0	0
$957$	−2.64598e24	−0.121572
$958$	0	0
$959$	−1.57610e25	−0.711418
$960$	0	0
$961$	1.89499e25	0.840347
$962$	0	0
$963$	−9.21464e24	−0.401472
$964$	0	0
$965$	1.71699e25	0.734998
$966$	0	0
$967$	4.18323e25	1.75949	0.879746	−	0.475444i	$-0.157713\pi$
$967$	4.18323e25	1.75949	0.879746	+	0.475444i	$0.157713\pi$
$968$	0	0
$969$	6.41009e23	0.0264918
$970$	0	0
$971$	−4.48245e25	−1.82034	−0.910169	−	0.414238i	$-0.864048\pi$
$971$	−4.48245e25	−1.82034	−0.910169	+	0.414238i	$0.864048\pi$
$972$	0	0
$973$	1.46769e25	0.585699
$974$	0	0
$975$	4.95101e23	0.0194158
$976$	0	0
$977$	2.15059e25	0.828808	0.414404	−	0.910093i	$-0.363990\pi$
$977$	2.15059e25	0.828808	0.414404	+	0.910093i	$0.363990\pi$
$978$	0	0
$979$	4.04816e25	1.53322
$980$	0	0
$981$	1.42308e25	0.529714
$982$	0	0
$983$	−6.63523e24	−0.242745	−0.121373	−	0.992607i	$-0.538730\pi$
$983$	−6.63523e24	−0.242745	−0.121373	+	0.992607i	$0.538730\pi$
$984$	0	0
$985$	1.80632e25	0.649512
$986$	0	0
$987$	−3.71035e24	−0.131135
$988$	0	0
$989$	1.84295e23	0.00640242
$990$	0	0
$991$	−1.56507e25	−0.534450	−0.267225	−	0.963634i	$-0.586107\pi$
$991$	−1.56507e25	−0.534450	−0.267225	+	0.963634i	$0.586107\pi$
$992$	0	0
$993$	−6.30822e24	−0.211757
$994$	0	0
$995$	−1.65856e25	−0.547312
$996$	0	0
$997$	−3.41918e25	−1.10921	−0.554605	−	0.832114i	$-0.687130\pi$
$997$	−3.41918e25	−1.10921	−0.554605	+	0.832114i	$0.687130\pi$
$998$	0	0
$999$	−1.25757e25	−0.401076

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.18.a.l.1.3		5
4.3	odd	2		40.18.a.d.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.18.a.d.1.3	✓	5	4.3	odd	2
80.18.a.l.1.3		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)

\(f(q)\)	\(=\)	\(q+1318.46 q^{3} +390625. q^{5} +1.28833e7 q^{7} -1.27402e8 q^{9} +1.29449e9 q^{11} +2.46098e9 q^{13} +5.15023e8 q^{15} +2.06308e10 q^{17} +2.35658e10 q^{19} +1.69860e10 q^{21} -1.62325e11 q^{23} +1.52588e11 q^{25} -3.38240e11 q^{27} -1.55032e12 q^{29} +6.44205e12 q^{31} +1.70673e12 q^{33} +5.03252e12 q^{35} +3.71799e13 q^{37} +3.24470e12 q^{39} -3.11582e13 q^{41} -1.13535e12 q^{43} -4.97663e13 q^{45} -2.18435e14 q^{47} -6.66523e13 q^{49} +2.72009e13 q^{51} +1.24200e14 q^{53} +5.05659e14 q^{55} +3.10705e13 q^{57} +3.27101e13 q^{59} +2.09504e15 q^{61} -1.64135e15 q^{63} +9.61318e14 q^{65} +2.90186e15 q^{67} -2.14018e14 q^{69} -8.60335e15 q^{71} -1.13767e16 q^{73} +2.01181e14 q^{75} +1.66772e16 q^{77} -1.09792e16 q^{79} +1.60067e16 q^{81} -1.22943e16 q^{83} +8.05890e15 q^{85} -2.04404e15 q^{87} +3.12723e16 q^{89} +3.17054e16 q^{91} +8.49359e15 q^{93} +9.20537e15 q^{95} +1.38194e17 q^{97} -1.64920e17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 15384 q^{3} + 1953125 q^{5} - 312868 q^{7} + 410286417 q^{9} - 757119716 q^{11} + 2827963478 q^{13} + 6009375000 q^{15} + 24776563114 q^{17} - 811272116 q^{19} + 332870780352 q^{21} + 73100431588 q^{23}+ \cdots - 35\!\cdots\!44 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more