LMFDB - Embedded newform 84.18.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,18,Mod(1,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("84.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	84.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:	$153.906553369$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9535884613x^{2} - 112984728346643x + 14354337121591770480 $ x^4 - x^3 - 9535884613x^2 - 112984728346643x + 14354337121591770480
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{5}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$95646.0$ of defining polynomial
Character	$\chi$	$=$	84.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6561.00 q^{3} +1.27982e6 q^{5} +5.76480e6 q^{7} +4.30467e7 q^{9} +O(q^{10})$	$q-6561.00 q^{3} +1.27982e6 q^{5} +5.76480e6 q^{7} +4.30467e7 q^{9} +1.49988e8 q^{11} +4.18855e9 q^{13} -8.39689e9 q^{15} -3.27433e10 q^{17} -1.27446e11 q^{19} -3.78229e10 q^{21} +2.51844e11 q^{23} +8.74996e11 q^{25} -2.82430e11 q^{27} -4.48896e12 q^{29} -4.90401e12 q^{31} -9.84073e11 q^{33} +7.37790e12 q^{35} -2.51079e13 q^{37} -2.74811e13 q^{39} -5.76265e13 q^{41} +1.03582e14 q^{43} +5.50920e13 q^{45} -2.03941e14 q^{47} +3.32329e13 q^{49} +2.14829e14 q^{51} -6.32409e14 q^{53} +1.91958e14 q^{55} +8.36175e14 q^{57} +1.55150e15 q^{59} -1.58692e15 q^{61} +2.48156e14 q^{63} +5.36059e15 q^{65} +4.74535e15 q^{67} -1.65235e15 q^{69} +2.24346e15 q^{71} +9.16486e14 q^{73} -5.74085e15 q^{75} +8.64652e14 q^{77} +5.05681e15 q^{79} +1.85302e15 q^{81} -1.67439e16 q^{83} -4.19055e16 q^{85} +2.94521e16 q^{87} +2.52351e16 q^{89} +2.41462e16 q^{91} +3.21752e16 q^{93} -1.63108e17 q^{95} -8.57721e16 q^{97} +6.45650e15 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 26244 q^{3} + 528276 q^{5} + 23059204 q^{7} + 172186884 q^{9}+O(q^{10}) $ 4 * q - 26244 * q^3 + 528276 * q^5 + 23059204 * q^7 + 172186884 * q^9	$ 4 q - 26244 q^{3} + 528276 q^{5} + 23059204 q^{7} + 172186884 q^{9} - 269632020 q^{11} + 1984752056 q^{13} - 3466018836 q^{15} - 29765040036 q^{17} - 91677835096 q^{19} - 151291437444 q^{21} + 38384818716 q^{23} - 235654160804 q^{25} - 1129718145924 q^{27} - 3815624693904 q^{29} - 4907739231784 q^{31} + 1769055683220 q^{33} + 3045406013076 q^{35} + 13670824759208 q^{37} - 13021958239416 q^{39} + 10307194403604 q^{41} + 125363311182368 q^{43} + 22740549582996 q^{45} - 25278931593000 q^{47} + 132931722278404 q^{49} + 195288427676196 q^{51} - 239725859759304 q^{53} + 694573116031368 q^{55} + 601498276064856 q^{57} + 13\!\cdots\!00 q^{59}+ \cdots - 11\!\cdots\!20 q^{99}+O(q^{100}) $ 4 * q - 26244 * q^3 + 528276 * q^5 + 23059204 * q^7 + 172186884 * q^9 - 269632020 * q^11 + 1984752056 * q^13 - 3466018836 * q^15 - 29765040036 * q^17 - 91677835096 * q^19 - 151291437444 * q^21 + 38384818716 * q^23 - 235654160804 * q^25 - 1129718145924 * q^27 - 3815624693904 * q^29 - 4907739231784 * q^31 + 1769055683220 * q^33 + 3045406013076 * q^35 + 13670824759208 * q^37 - 13021958239416 * q^39 + 10307194403604 * q^41 + 125363311182368 * q^43 + 22740549582996 * q^45 - 25278931593000 * q^47 + 132931722278404 * q^49 + 195288427676196 * q^51 - 239725859759304 * q^53 + 694573116031368 * q^55 + 601498276064856 * q^57 + 1374756208497000 * q^59 + 652646281638080 * q^61 + 992623121070084 * q^63 + 4153293849278904 * q^65 + 6617359123302392 * q^67 - 251842795595676 * q^69 + 2405715947056548 * q^71 + 4250185178265824 * q^73 + 1546126949035044 * q^75 - 1554374938528020 * q^77 + 20052572228473688 * q^79 + 7412080755407364 * q^81 - 5593593799375584 * q^83 - 13012862591692368 * q^85 + 25034313616704144 * q^87 + 1518004216828212 * q^89 + 11441700637180856 * q^91 + 32199677099734824 * q^93 - 149304801904677408 * q^95 - 102152247026326000 * q^97 - 11606774337606420 * 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
$3$	−6561.00	−0.577350
$3$	−6561.00	−0.577350
$4$	0	0
$5$	1.27982e6	1.46522	0.732611	−	0.680648i	$-0.238302\pi$
$5$	1.27982e6	1.46522	0.732611	+	0.680648i	$0.238302\pi$
$6$	0	0
$7$	5.76480e6	0.377964
$7$	5.76480e6	0.377964
$8$	0	0
$9$	4.30467e7	0.333333
$10$	0	0
$11$	1.49988e8	0.210969	0.105485	−	0.994421i	$-0.466361\pi$
$11$	1.49988e8	0.210969	0.105485	+	0.994421i	$0.466361\pi$
$12$	0	0
$13$	4.18855e9	1.42412	0.712058	−	0.702120i	$-0.247763\pi$
$13$	4.18855e9	1.42412	0.712058	+	0.702120i	$0.247763\pi$
$14$	0	0
$15$	−8.39689e9	−0.845946
$16$	0	0
$17$	−3.27433e10	−1.13843	−0.569216	−	0.822188i	$-0.692753\pi$
$17$	−3.27433e10	−1.13843	−0.569216	+	0.822188i	$0.692753\pi$
$18$	0	0
$19$	−1.27446e11	−1.72156	−0.860779	−	0.508979i	$-0.830023\pi$
$19$	−1.27446e11	−1.72156	−0.860779	+	0.508979i	$0.830023\pi$
$20$	0	0
$21$	−3.78229e10	−0.218218
$22$	0	0
$23$	2.51844e11	0.670571	0.335286	−	0.942117i	$-0.391167\pi$
$23$	2.51844e11	0.670571	0.335286	+	0.942117i	$0.391167\pi$
$24$	0	0
$25$	8.74996e11	1.14687
$26$	0	0
$27$	−2.82430e11	−0.192450
$28$	0	0
$29$	−4.48896e12	−1.66634	−0.833168	−	0.553020i	$-0.813475\pi$
$29$	−4.48896e12	−1.66634	−0.833168	+	0.553020i	$0.813475\pi$
$30$	0	0
$31$	−4.90401e12	−1.03271	−0.516353	−	0.856376i	$-0.672711\pi$
$31$	−4.90401e12	−1.03271	−0.516353	+	0.856376i	$0.672711\pi$
$32$	0	0
$33$	−9.84073e11	−0.121803
$34$	0	0
$35$	7.37790e12	0.553802
$36$	0	0
$37$	−2.51079e13	−1.17516	−0.587579	−	0.809167i	$-0.699919\pi$
$37$	−2.51079e13	−1.17516	−0.587579	+	0.809167i	$0.699919\pi$
$38$	0	0
$39$	−2.74811e13	−0.822214
$40$	0	0
$41$	−5.76265e13	−1.12709	−0.563546	−	0.826085i	$-0.690563\pi$
$41$	−5.76265e13	−1.12709	−0.563546	+	0.826085i	$0.690563\pi$
$42$	0	0
$43$	1.03582e14	1.35145	0.675727	−	0.737152i	$-0.263830\pi$
$43$	1.03582e14	1.35145	0.675727	+	0.737152i	$0.263830\pi$
$44$	0	0
$45$	5.50920e13	0.488407
$46$	0	0
$47$	−2.03941e14	−1.24932	−0.624659	−	0.780897i	$-0.714762\pi$
$47$	−2.03941e14	−1.24932	−0.624659	+	0.780897i	$0.714762\pi$
$48$	0	0
$49$	3.32329e13	0.142857
$50$	0	0
$51$	2.14829e14	0.657274
$52$	0	0
$53$	−6.32409e14	−1.39525	−0.697627	−	0.716461i	$-0.745761\pi$
$53$	−6.32409e14	−1.39525	−0.697627	+	0.716461i	$0.745761\pi$
$54$	0	0
$55$	1.91958e14	0.309117
$56$	0	0
$57$	8.36175e14	0.993942
$58$	0	0
$59$	1.55150e15	1.37566	0.687828	−	0.725873i	$-0.258564\pi$
$59$	1.55150e15	1.37566	0.687828	+	0.725873i	$0.258564\pi$
$60$	0	0
$61$	−1.58692e15	−1.05987	−0.529933	−	0.848040i	$-0.677783\pi$
$61$	−1.58692e15	−1.05987	−0.529933	+	0.848040i	$0.677783\pi$
$62$	0	0
$63$	2.48156e14	0.125988
$64$	0	0
$65$	5.36059e15	2.08665
$66$	0	0
$67$	4.74535e15	1.42768	0.713842	−	0.700306i	$-0.246953\pi$
$67$	4.74535e15	1.42768	0.713842	+	0.700306i	$0.246953\pi$
$68$	0	0
$69$	−1.65235e15	−0.387154
$70$	0	0
$71$	2.24346e15	0.412309	0.206155	−	0.978519i	$-0.433905\pi$
$71$	2.24346e15	0.412309	0.206155	+	0.978519i	$0.433905\pi$
$72$	0	0
$73$	9.16486e14	0.133009	0.0665046	−	0.997786i	$-0.478815\pi$
$73$	9.16486e14	0.133009	0.0665046	+	0.997786i	$0.478815\pi$
$74$	0	0
$75$	−5.74085e15	−0.662148
$76$	0	0
$77$	8.64652e14	0.0797389
$78$	0	0
$79$	5.05681e15	0.375013	0.187507	−	0.982263i	$-0.439959\pi$
$79$	5.05681e15	0.375013	0.187507	+	0.982263i	$0.439959\pi$
$80$	0	0
$81$	1.85302e15	0.111111
$82$	0	0
$83$	−1.67439e16	−0.816005	−0.408003	−	0.912981i	$-0.633775\pi$
$83$	−1.67439e16	−0.816005	−0.408003	+	0.912981i	$0.633775\pi$
$84$	0	0
$85$	−4.19055e16	−1.66805
$86$	0	0
$87$	2.94521e16	0.962059
$88$	0	0
$89$	2.52351e16	0.679501	0.339751	−	0.940516i	$-0.389657\pi$
$89$	2.52351e16	0.679501	0.339751	+	0.940516i	$0.389657\pi$
$90$	0	0
$91$	2.41462e16	0.538265
$92$	0	0
$93$	3.21752e16	0.596233
$94$	0	0
$95$	−1.63108e17	−2.52246
$96$	0	0
$97$	−8.57721e16	−1.11119	−0.555593	−	0.831455i	$-0.687509\pi$
$97$	−8.57721e16	−1.11119	−0.555593	+	0.831455i	$0.687509\pi$
$98$	0	0
$99$	6.45650e15	0.0703231
$100$	0	0
$101$	9.82992e16	0.903271	0.451636	−	0.892202i	$-0.350841\pi$
$101$	9.82992e16	0.903271	0.451636	+	0.892202i	$0.350841\pi$
$102$	0	0
$103$	−1.00283e17	−0.780028	−0.390014	−	0.920809i	$-0.627530\pi$
$103$	−1.00283e17	−0.780028	−0.390014	+	0.920809i	$0.627530\pi$
$104$	0	0
$105$	−4.84064e16	−0.319738
$106$	0	0
$107$	4.44951e16	0.250351	0.125176	−	0.992135i	$-0.460051\pi$
$107$	4.44951e16	0.250351	0.125176	+	0.992135i	$0.460051\pi$
$108$	0	0
$109$	−2.59675e17	−1.24826	−0.624131	−	0.781320i	$-0.714547\pi$
$109$	−2.59675e17	−1.24826	−0.624131	+	0.781320i	$0.714547\pi$
$110$	0	0
$111$	1.64733e17	0.678478
$112$	0	0
$113$	3.18535e17	1.12717	0.563587	−	0.826057i	$-0.309421\pi$
$113$	3.18535e17	1.12717	0.563587	+	0.826057i	$0.309421\pi$
$114$	0	0
$115$	3.22315e17	0.982536
$116$	0	0
$117$	1.80303e17	0.474705
$118$	0	0
$119$	−1.88759e17	−0.430287
$120$	0	0
$121$	−4.82951e17	−0.955492
$122$	0	0
$123$	3.78087e17	0.650727
$124$	0	0
$125$	1.43412e17	0.215204
$126$	0	0
$127$	−1.15314e18	−1.51200	−0.755999	−	0.654572i	$-0.772849\pi$
$127$	−1.15314e18	−1.51200	−0.755999	+	0.654572i	$0.772849\pi$
$128$	0	0
$129$	−6.79599e17	−0.780262
$130$	0	0
$131$	1.23296e18	1.24206	0.621031	−	0.783786i	$-0.286714\pi$
$131$	1.23296e18	1.24206	0.621031	+	0.783786i	$0.286714\pi$
$132$	0	0
$133$	−7.34703e17	−0.650688
$134$	0	0
$135$	−3.61459e17	−0.281982
$136$	0	0
$137$	−1.81912e18	−1.25238	−0.626189	−	0.779671i	$-0.715386\pi$
$137$	−1.81912e18	−1.25238	−0.626189	+	0.779671i	$0.715386\pi$
$138$	0	0
$139$	−4.61969e17	−0.281182	−0.140591	−	0.990068i	$-0.544900\pi$
$139$	−4.61969e17	−0.281182	−0.140591	+	0.990068i	$0.544900\pi$
$140$	0	0
$141$	1.33806e18	0.721295
$142$	0	0
$143$	6.28233e17	0.300445
$144$	0	0
$145$	−5.74505e18	−2.44155
$146$	0	0
$147$	−2.18041e17	−0.0824786
$148$	0	0
$149$	2.37490e18	0.800870	0.400435	−	0.916325i	$-0.368859\pi$
$149$	2.37490e18	0.800870	0.400435	+	0.916325i	$0.368859\pi$
$150$	0	0
$151$	5.49616e18	1.65484	0.827419	−	0.561585i	$-0.189808\pi$
$151$	5.49616e18	1.65484	0.827419	+	0.561585i	$0.189808\pi$
$152$	0	0
$153$	−1.40949e18	−0.379477
$154$	0	0
$155$	−6.27624e18	−1.51314
$156$	0	0
$157$	−6.45092e18	−1.39468	−0.697340	−	0.716741i	$-0.745633\pi$
$157$	−6.45092e18	−1.39468	−0.697340	+	0.716741i	$0.745633\pi$
$158$	0	0
$159$	4.14923e18	0.805550
$160$	0	0
$161$	1.45183e18	0.253452
$162$	0	0
$163$	1.19105e17	0.0187213	0.00936066	−	0.999956i	$-0.497020\pi$
$163$	1.19105e17	0.0187213	0.00936066	+	0.999956i	$0.497020\pi$
$164$	0	0
$165$	−1.25943e18	−0.178469
$166$	0	0
$167$	−5.49454e17	−0.0702816	−0.0351408	−	0.999382i	$-0.511188\pi$
$167$	−5.49454e17	−0.0702816	−0.0351408	+	0.999382i	$0.511188\pi$
$168$	0	0
$169$	8.89356e18	1.02811
$170$	0	0
$171$	−5.48614e18	−0.573853
$172$	0	0
$173$	−5.18280e18	−0.491103	−0.245552	−	0.969384i	$-0.578969\pi$
$173$	−5.18280e18	−0.491103	−0.245552	+	0.969384i	$0.578969\pi$
$174$	0	0
$175$	5.04418e18	0.433478
$176$	0	0
$177$	−1.01794e19	−0.794236
$178$	0	0
$179$	−1.21981e19	−0.865053	−0.432526	−	0.901621i	$-0.642378\pi$
$179$	−1.21981e19	−0.865053	−0.432526	+	0.901621i	$0.642378\pi$
$180$	0	0
$181$	1.57029e18	0.101324	0.0506620	−	0.998716i	$-0.483867\pi$
$181$	1.57029e18	0.101324	0.0506620	+	0.998716i	$0.483867\pi$
$182$	0	0
$183$	1.04118e19	0.611913
$184$	0	0
$185$	−3.21336e19	−1.72187
$186$	0	0
$187$	−4.91111e18	−0.240174
$188$	0	0
$189$	−1.62815e18	−0.0727393
$190$	0	0
$191$	−2.13543e19	−0.872370	−0.436185	−	0.899857i	$-0.643671\pi$
$191$	−2.13543e19	−0.872370	−0.436185	+	0.899857i	$0.643671\pi$
$192$	0	0
$193$	−4.54571e19	−1.69967	−0.849835	−	0.527049i	$-0.823298\pi$
$193$	−4.54571e19	−1.69967	−0.849835	+	0.527049i	$0.823298\pi$
$194$	0	0
$195$	−3.51708e19	−1.20473
$196$	0	0
$197$	−3.83172e19	−1.20346	−0.601730	−	0.798700i	$-0.705522\pi$
$197$	−3.83172e19	−1.20346	−0.601730	+	0.798700i	$0.705522\pi$
$198$	0	0
$199$	−7.02414e18	−0.202461	−0.101231	−	0.994863i	$-0.532278\pi$
$199$	−7.02414e18	−0.202461	−0.101231	+	0.994863i	$0.532278\pi$
$200$	0	0
$201$	−3.11342e19	−0.824274
$202$	0	0
$203$	−2.58779e19	−0.629816
$204$	0	0
$205$	−7.37514e19	−1.65144
$206$	0	0
$207$	1.08411e19	0.223524
$208$	0	0
$209$	−1.91154e19	−0.363196
$210$	0	0
$211$	4.17585e19	0.731719	0.365860	−	0.930670i	$-0.380775\pi$
$211$	4.17585e19	0.731719	0.365860	+	0.930670i	$0.380775\pi$
$212$	0	0
$213$	−1.47194e19	−0.238047
$214$	0	0
$215$	1.32566e20	1.98018
$216$	0	0
$217$	−2.82706e19	−0.390326
$218$	0	0
$219$	−6.01306e18	−0.0767929
$220$	0	0
$221$	−1.37147e20	−1.62126
$222$	0	0
$223$	1.12828e20	1.23545	0.617725	−	0.786394i	$-0.288054\pi$
$223$	1.12828e20	1.23545	0.617725	+	0.786394i	$0.288054\pi$
$224$	0	0
$225$	3.76657e19	0.382292
$226$	0	0
$227$	−1.62147e20	−1.52647	−0.763235	−	0.646121i	$-0.776390\pi$
$227$	−1.62147e20	−1.52647	−0.763235	+	0.646121i	$0.776390\pi$
$228$	0	0
$229$	1.01743e20	0.889001	0.444501	−	0.895779i	$-0.353381\pi$
$229$	1.01743e20	0.889001	0.444501	+	0.895779i	$0.353381\pi$
$230$	0	0
$231$	−5.67298e18	−0.0460373
$232$	0	0
$233$	1.60636e20	1.21148	0.605741	−	0.795662i	$-0.292877\pi$
$233$	1.60636e20	1.21148	0.605741	+	0.795662i	$0.292877\pi$
$234$	0	0
$235$	−2.61008e20	−1.83053
$236$	0	0
$237$	−3.31778e19	−0.216514
$238$	0	0
$239$	−2.57134e20	−1.56234	−0.781172	−	0.624316i	$-0.785378\pi$
$239$	−2.57134e20	−1.56234	−0.781172	+	0.624316i	$0.785378\pi$
$240$	0	0
$241$	3.21955e20	1.82243	0.911215	−	0.411932i	$-0.135146\pi$
$241$	3.21955e20	1.82243	0.911215	+	0.411932i	$0.135146\pi$
$242$	0	0
$243$	−1.21577e19	−0.0641500
$244$	0	0
$245$	4.25321e19	0.209317
$246$	0	0
$247$	−5.33815e20	−2.45170
$248$	0	0
$249$	1.09857e20	0.471121
$250$	0	0
$251$	2.01415e20	0.806983	0.403492	−	0.914983i	$-0.367796\pi$
$251$	2.01415e20	0.806983	0.403492	+	0.914983i	$0.367796\pi$
$252$	0	0
$253$	3.77736e19	0.141470
$254$	0	0
$255$	2.74942e20	0.963052
$256$	0	0
$257$	1.62752e20	0.533453	0.266726	−	0.963772i	$-0.414058\pi$
$257$	1.62752e20	0.533453	0.266726	+	0.963772i	$0.414058\pi$
$258$	0	0
$259$	−1.44742e20	−0.444168
$260$	0	0
$261$	−1.93235e20	−0.555445
$262$	0	0
$263$	3.08165e20	0.830157	0.415078	−	0.909786i	$-0.363754\pi$
$263$	3.08165e20	0.830157	0.415078	+	0.909786i	$0.363754\pi$
$264$	0	0
$265$	−8.09369e20	−2.04436
$266$	0	0
$267$	−1.65568e20	−0.392310
$268$	0	0
$269$	−6.22709e20	−1.38481	−0.692406	−	0.721508i	$-0.743449\pi$
$269$	−6.22709e20	−1.38481	−0.692406	+	0.721508i	$0.743449\pi$
$270$	0	0
$271$	3.30814e20	0.690788	0.345394	−	0.938458i	$-0.387745\pi$
$271$	3.30814e20	0.690788	0.345394	+	0.938458i	$0.387745\pi$
$272$	0	0
$273$	−1.58423e20	−0.310768
$274$	0	0
$275$	1.31239e20	0.241955
$276$	0	0
$277$	−6.43079e20	−1.11477	−0.557386	−	0.830253i	$-0.688196\pi$
$277$	−6.43079e20	−1.11477	−0.557386	+	0.830253i	$0.688196\pi$
$278$	0	0
$279$	−2.11101e20	−0.344235
$280$	0	0
$281$	1.08826e21	1.67004	0.835021	−	0.550218i	$-0.185456\pi$
$281$	1.08826e21	1.67004	0.835021	+	0.550218i	$0.185456\pi$
$282$	0	0
$283$	−1.05552e21	−1.52504	−0.762519	−	0.646965i	$-0.776038\pi$
$283$	−1.05552e21	−1.52504	−0.762519	+	0.646965i	$0.776038\pi$
$284$	0	0
$285$	1.07015e21	1.45635
$286$	0	0
$287$	−3.32205e20	−0.426001
$288$	0	0
$289$	2.44884e20	0.296026
$290$	0	0
$291$	5.62751e20	0.641543
$292$	0	0
$293$	2.99726e20	0.322366	0.161183	−	0.986925i	$-0.448469\pi$
$293$	2.99726e20	0.322366	0.161183	+	0.986925i	$0.448469\pi$
$294$	0	0
$295$	1.98564e21	2.01564
$296$	0	0
$297$	−4.23611e19	−0.0406011
$298$	0	0
$299$	1.05486e21	0.954971
$300$	0	0
$301$	5.97128e20	0.510801
$302$	0	0
$303$	−6.44941e20	−0.521504
$304$	0	0
$305$	−2.03097e21	−1.55294
$306$	0	0
$307$	2.49241e21	1.80278	0.901392	−	0.433003i	$-0.142546\pi$
$307$	2.49241e21	1.80278	0.901392	+	0.433003i	$0.142546\pi$
$308$	0	0
$309$	6.57956e20	0.450349
$310$	0	0
$311$	−1.63732e21	−1.06089	−0.530446	−	0.847719i	$-0.677976\pi$
$311$	−1.63732e21	−1.06089	−0.530446	+	0.847719i	$0.677976\pi$
$312$	0	0
$313$	3.14779e21	1.93143	0.965716	−	0.259600i	$-0.0835907\pi$
$313$	3.14779e21	1.93143	0.965716	+	0.259600i	$0.0835907\pi$
$314$	0	0
$315$	3.17594e20	0.184601
$316$	0	0
$317$	−5.66441e20	−0.311998	−0.155999	−	0.987757i	$-0.549860\pi$
$317$	−5.66441e20	−0.311998	−0.155999	+	0.987757i	$0.549860\pi$
$318$	0	0
$319$	−6.73291e20	−0.351546
$320$	0	0
$321$	−2.91932e20	−0.144540
$322$	0	0
$323$	4.17301e21	1.95988
$324$	0	0
$325$	3.66497e21	1.63328
$326$	0	0
$327$	1.70373e21	0.720684
$328$	0	0
$329$	−1.17568e21	−0.472198
$330$	0	0
$331$	4.08262e20	0.155740	0.0778701	−	0.996964i	$-0.475188\pi$
$331$	4.08262e20	0.155740	0.0778701	+	0.996964i	$0.475188\pi$
$332$	0	0
$333$	−1.08081e21	−0.391719
$334$	0	0
$335$	6.07319e21	2.09187
$336$	0	0
$337$	−3.02124e21	−0.989305	−0.494653	−	0.869091i	$-0.664705\pi$
$337$	−3.02124e21	−0.989305	−0.494653	+	0.869091i	$0.664705\pi$
$338$	0	0
$339$	−2.08991e21	−0.650774
$340$	0	0
$341$	−7.35543e20	−0.217869
$342$	0	0
$343$	1.91581e20	0.0539949
$344$	0	0
$345$	−2.11471e21	−0.567267
$346$	0	0
$347$	−1.90379e21	−0.486204	−0.243102	−	0.970001i	$-0.578165\pi$
$347$	−1.90379e21	−0.486204	−0.243102	+	0.970001i	$0.578165\pi$
$348$	0	0
$349$	−3.31547e21	−0.806360	−0.403180	−	0.915121i	$-0.632095\pi$
$349$	−3.31547e21	−0.806360	−0.403180	+	0.915121i	$0.632095\pi$
$350$	0	0
$351$	−1.18297e21	−0.274071
$352$	0	0
$353$	4.96524e20	0.109611	0.0548056	−	0.998497i	$-0.482546\pi$
$353$	4.96524e20	0.109611	0.0548056	+	0.998497i	$0.482546\pi$
$354$	0	0
$355$	2.87123e21	0.604124
$356$	0	0
$357$	1.23845e21	0.248426
$358$	0	0
$359$	6.11499e21	1.16975	0.584875	−	0.811123i	$-0.301143\pi$
$359$	6.11499e21	1.16975	0.584875	+	0.811123i	$0.301143\pi$
$360$	0	0
$361$	1.07622e22	1.96376
$362$	0	0
$363$	3.16864e21	0.551654
$364$	0	0
$365$	1.17294e21	0.194888
$366$	0	0
$367$	9.17571e20	0.145539	0.0727693	−	0.997349i	$-0.476816\pi$
$367$	9.17571e20	0.145539	0.0727693	+	0.997349i	$0.476816\pi$
$368$	0	0
$369$	−2.48063e21	−0.375697
$370$	0	0
$371$	−3.64571e21	−0.527356
$372$	0	0
$373$	4.59571e21	0.635079	0.317540	−	0.948245i	$-0.397143\pi$
$373$	4.59571e21	0.635079	0.317540	+	0.948245i	$0.397143\pi$
$374$	0	0
$375$	−9.40926e20	−0.124248
$376$	0	0
$377$	−1.88022e22	−2.37306
$378$	0	0
$379$	3.07939e21	0.371562	0.185781	−	0.982591i	$-0.440519\pi$
$379$	3.07939e21	0.371562	0.185781	+	0.982591i	$0.440519\pi$
$380$	0	0
$381$	7.56577e21	0.872953
$382$	0	0
$383$	−8.43121e21	−0.930466	−0.465233	−	0.885188i	$-0.654029\pi$
$383$	−8.43121e21	−0.930466	−0.465233	+	0.885188i	$0.654029\pi$
$384$	0	0
$385$	1.10660e21	0.116835
$386$	0	0
$387$	4.45885e21	0.450485
$388$	0	0
$389$	−3.71219e21	−0.358971	−0.179485	−	0.983761i	$-0.557443\pi$
$389$	−3.71219e21	−0.358971	−0.179485	+	0.983761i	$0.557443\pi$
$390$	0	0
$391$	−8.24621e21	−0.763399
$392$	0	0
$393$	−8.08946e21	−0.717105
$394$	0	0
$395$	6.47180e21	0.549478
$396$	0	0
$397$	6.78633e21	0.551970	0.275985	−	0.961162i	$-0.410996\pi$
$397$	6.78633e21	0.551970	0.275985	+	0.961162i	$0.410996\pi$
$398$	0	0
$399$	4.82038e21	0.375675
$400$	0	0
$401$	2.29375e22	1.71324	0.856622	−	0.515944i	$-0.172559\pi$
$401$	2.29375e22	1.71324	0.856622	+	0.515944i	$0.172559\pi$
$402$	0	0
$403$	−2.05407e22	−1.47069
$404$	0	0
$405$	2.37153e21	0.162802
$406$	0	0
$407$	−3.76589e21	−0.247922
$408$	0	0
$409$	−1.45314e22	−0.917615	−0.458807	−	0.888536i	$-0.651723\pi$
$409$	−1.45314e22	−0.917615	−0.458807	+	0.888536i	$0.651723\pi$
$410$	0	0
$411$	1.19352e22	0.723061
$412$	0	0
$413$	8.94410e21	0.519949
$414$	0	0
$415$	−2.14292e22	−1.19563
$416$	0	0
$417$	3.03098e21	0.162340
$418$	0	0
$419$	−2.94806e22	−1.51606	−0.758031	−	0.652219i	$-0.773838\pi$
$419$	−2.94806e22	−1.51606	−0.758031	+	0.652219i	$0.773838\pi$
$420$	0	0
$421$	−1.71458e22	−0.846758	−0.423379	−	0.905953i	$-0.639156\pi$
$421$	−1.71458e22	−0.846758	−0.423379	+	0.905953i	$0.639156\pi$
$422$	0	0
$423$	−8.77900e21	−0.416440
$424$	0	0
$425$	−2.86503e22	−1.30564
$426$	0	0
$427$	−9.14826e21	−0.400591
$428$	0	0
$429$	−4.12184e21	−0.173462
$430$	0	0
$431$	−9.13600e21	−0.369572	−0.184786	−	0.982779i	$-0.559159\pi$
$431$	−9.13600e21	−0.369572	−0.184786	+	0.982779i	$0.559159\pi$
$432$	0	0
$433$	−4.48218e21	−0.174318	−0.0871590	−	0.996194i	$-0.527779\pi$
$433$	−4.48218e21	−0.174318	−0.0871590	+	0.996194i	$0.527779\pi$
$434$	0	0
$435$	3.76933e22	1.40963
$436$	0	0
$437$	−3.20966e22	−1.15443
$438$	0	0
$439$	−2.01973e22	−0.698787	−0.349394	−	0.936976i	$-0.613612\pi$
$439$	−2.01973e22	−0.698787	−0.349394	+	0.936976i	$0.613612\pi$
$440$	0	0
$441$	1.43057e21	0.0476190
$442$	0	0
$443$	−3.53945e22	−1.13372	−0.566858	−	0.823815i	$-0.691841\pi$
$443$	−3.53945e22	−1.13372	−0.566858	+	0.823815i	$0.691841\pi$
$444$	0	0
$445$	3.22964e22	0.995620
$446$	0	0
$447$	−1.55817e22	−0.462383
$448$	0	0
$449$	−1.81146e22	−0.517529	−0.258764	−	0.965940i	$-0.583315\pi$
$449$	−1.81146e22	−0.517529	−0.258764	+	0.965940i	$0.583315\pi$
$450$	0	0
$451$	−8.64329e21	−0.237782
$452$	0	0
$453$	−3.60603e22	−0.955422
$454$	0	0
$455$	3.09027e22	0.788678
$456$	0	0
$457$	6.42910e22	1.58075	0.790374	−	0.612625i	$-0.209886\pi$
$457$	6.42910e22	1.58075	0.790374	+	0.612625i	$0.209886\pi$
$458$	0	0
$459$	9.24768e21	0.219091
$460$	0	0
$461$	2.70614e22	0.617863	0.308932	−	0.951084i	$-0.400029\pi$
$461$	2.70614e22	0.617863	0.308932	+	0.951084i	$0.400029\pi$
$462$	0	0
$463$	1.32252e22	0.291049	0.145524	−	0.989355i	$-0.453513\pi$
$463$	1.32252e22	0.291049	0.145524	+	0.989355i	$0.453513\pi$
$464$	0	0
$465$	4.11784e22	0.873613
$466$	0	0
$467$	−1.21272e22	−0.248066	−0.124033	−	0.992278i	$-0.539583\pi$
$467$	−1.21272e22	−0.248066	−0.124033	+	0.992278i	$0.539583\pi$
$468$	0	0
$469$	2.73560e22	0.539614
$470$	0	0
$471$	4.23245e22	0.805218
$472$	0	0
$473$	1.55360e22	0.285115
$474$	0	0
$475$	−1.11515e23	−1.97441
$476$	0	0
$477$	−2.72231e22	−0.465084
$478$	0	0
$479$	−2.77719e22	−0.457882	−0.228941	−	0.973440i	$-0.573526\pi$
$479$	−2.77719e22	−0.457882	−0.228941	+	0.973440i	$0.573526\pi$
$480$	0	0
$481$	−1.05166e23	−1.67356
$482$	0	0
$483$	−9.52546e21	−0.146331
$484$	0	0
$485$	−1.09773e23	−1.62813
$486$	0	0
$487$	5.28188e22	0.756471	0.378236	−	0.925709i	$-0.376531\pi$
$487$	5.28188e22	0.756471	0.378236	+	0.925709i	$0.376531\pi$
$488$	0	0
$489$	−7.81450e20	−0.0108088
$490$	0	0
$491$	−1.06674e23	−1.42517	−0.712583	−	0.701588i	$-0.752475\pi$
$491$	−1.06674e23	−1.42517	−0.712583	+	0.701588i	$0.752475\pi$
$492$	0	0
$493$	1.46983e23	1.89701
$494$	0	0
$495$	8.26315e21	0.103039
$496$	0	0
$497$	1.29331e22	0.155838
$498$	0	0
$499$	1.12483e23	1.30988	0.654940	−	0.755681i	$-0.272694\pi$
$499$	1.12483e23	1.30988	0.654940	+	0.755681i	$0.272694\pi$
$500$	0	0
$501$	3.60497e21	0.0405771
$502$	0	0
$503$	−4.56597e22	−0.496827	−0.248414	−	0.968654i	$-0.579909\pi$
$503$	−4.56597e22	−0.496827	−0.248414	+	0.968654i	$0.579909\pi$
$504$	0	0
$505$	1.25805e23	1.32349
$506$	0	0
$507$	−5.83506e22	−0.593578
$508$	0	0
$509$	−6.98967e22	−0.687632	−0.343816	−	0.939037i	$-0.611720\pi$
$509$	−6.98967e22	−0.687632	−0.343816	+	0.939037i	$0.611720\pi$
$510$	0	0
$511$	5.28336e21	0.0502727
$512$	0	0
$513$	3.59946e22	0.331314
$514$	0	0
$515$	−1.28344e23	−1.14291
$516$	0	0
$517$	−3.05888e22	−0.263568
$518$	0	0
$519$	3.40044e22	0.283538
$520$	0	0
$521$	−1.14485e23	−0.923905	−0.461952	−	0.886905i	$-0.652851\pi$
$521$	−1.14485e23	−0.923905	−0.461952	+	0.886905i	$0.652851\pi$
$522$	0	0
$523$	−4.91362e22	−0.383828	−0.191914	−	0.981412i	$-0.561470\pi$
$523$	−4.91362e22	−0.383828	−0.191914	+	0.981412i	$0.561470\pi$
$524$	0	0
$525$	−3.30948e22	−0.250269
$526$	0	0
$527$	1.60573e23	1.17566
$528$	0	0
$529$	−7.76247e22	−0.550334
$530$	0	0
$531$	6.67870e22	0.458552
$532$	0	0
$533$	−2.41371e23	−1.60511
$534$	0	0
$535$	5.69456e22	0.366820
$536$	0	0
$537$	8.00319e22	0.499438
$538$	0	0
$539$	4.98455e21	0.0301385
$540$	0	0
$541$	1.22632e23	0.718499	0.359249	−	0.933242i	$-0.383033\pi$
$541$	1.22632e23	0.718499	0.359249	+	0.933242i	$0.383033\pi$
$542$	0	0
$543$	−1.03027e22	−0.0584994
$544$	0	0
$545$	−3.32338e23	−1.82898
$546$	0	0
$547$	2.32932e23	1.24261	0.621307	−	0.783567i	$-0.286602\pi$
$547$	2.32932e23	1.24261	0.621307	+	0.783567i	$0.286602\pi$
$548$	0	0
$549$	−6.83116e22	−0.353288
$550$	0	0
$551$	5.72101e23	2.86869
$552$	0	0
$553$	2.91515e22	0.141742
$554$	0	0
$555$	2.10829e23	0.994120
$556$	0	0
$557$	7.63103e22	0.348991	0.174495	−	0.984658i	$-0.444171\pi$
$557$	7.63103e22	0.348991	0.174495	+	0.984658i	$0.444171\pi$
$558$	0	0
$559$	4.33857e23	1.92463
$560$	0	0
$561$	3.22218e22	0.138665
$562$	0	0
$563$	2.86912e23	1.19792	0.598960	−	0.800779i	$-0.295581\pi$
$563$	2.86912e23	1.19792	0.598960	+	0.800779i	$0.295581\pi$
$564$	0	0
$565$	4.07667e23	1.65156
$566$	0	0
$567$	1.06823e22	0.0419961
$568$	0	0
$569$	−9.27285e22	−0.353801	−0.176900	−	0.984229i	$-0.556607\pi$
$569$	−9.27285e22	−0.353801	−0.176900	+	0.984229i	$0.556607\pi$
$570$	0	0
$571$	−6.92683e22	−0.256524	−0.128262	−	0.991740i	$-0.540940\pi$
$571$	−6.92683e22	−0.256524	−0.128262	+	0.991740i	$0.540940\pi$
$572$	0	0
$573$	1.40105e23	0.503663
$574$	0	0
$575$	2.20362e23	0.769061
$576$	0	0
$577$	−1.77034e23	−0.599876	−0.299938	−	0.953959i	$-0.596966\pi$
$577$	−1.77034e23	−0.599876	−0.299938	+	0.953959i	$0.596966\pi$
$578$	0	0
$579$	2.98244e23	0.981305
$580$	0	0
$581$	−9.65253e22	−0.308421
$582$	0	0
$583$	−9.48539e22	−0.294356
$584$	0	0
$585$	2.30756e23	0.695549
$586$	0	0
$587$	−4.31921e23	−1.26468	−0.632340	−	0.774691i	$-0.717906\pi$
$587$	−4.31921e23	−1.26468	−0.632340	+	0.774691i	$0.717906\pi$
$588$	0	0
$589$	6.24997e23	1.77786
$590$	0	0
$591$	2.51399e23	0.694818
$592$	0	0
$593$	−6.27114e22	−0.168415	−0.0842076	−	0.996448i	$-0.526836\pi$
$593$	−6.27114e22	−0.168415	−0.0842076	+	0.996448i	$0.526836\pi$
$594$	0	0
$595$	−2.41577e23	−0.630465
$596$	0	0
$597$	4.60854e22	0.116891
$598$	0	0
$599$	4.36148e22	0.107524	0.0537621	−	0.998554i	$-0.482879\pi$
$599$	4.36148e22	0.107524	0.0537621	+	0.998554i	$0.482879\pi$
$600$	0	0
$601$	−2.50920e23	−0.601316	−0.300658	−	0.953732i	$-0.597206\pi$
$601$	−2.50920e23	−0.601316	−0.300658	+	0.953732i	$0.597206\pi$
$602$	0	0
$603$	2.04272e23	0.475895
$604$	0	0
$605$	−6.18089e23	−1.40001
$606$	0	0
$607$	7.64198e22	0.168307	0.0841535	−	0.996453i	$-0.473181\pi$
$607$	7.64198e22	0.168307	0.0841535	+	0.996453i	$0.473181\pi$
$608$	0	0
$609$	1.69785e23	0.363624
$610$	0	0
$611$	−8.54219e23	−1.77918
$612$	0	0
$613$	−5.32557e23	−1.07883	−0.539414	−	0.842041i	$-0.681354\pi$
$613$	−5.32557e23	−1.07883	−0.539414	+	0.842041i	$0.681354\pi$
$614$	0	0
$615$	4.83883e23	0.953459
$616$	0	0
$617$	7.18425e22	0.137708	0.0688538	−	0.997627i	$-0.478066\pi$
$617$	7.18425e22	0.137708	0.0688538	+	0.997627i	$0.478066\pi$
$618$	0	0
$619$	8.69183e22	0.162084	0.0810421	−	0.996711i	$-0.474175\pi$
$619$	8.69183e22	0.162084	0.0810421	+	0.996711i	$0.474175\pi$
$620$	0	0
$621$	−7.11282e22	−0.129051
$622$	0	0
$623$	1.45475e23	0.256827
$624$	0	0
$625$	−4.84028e23	−0.831553
$626$	0	0
$627$	1.25416e23	0.209691
$628$	0	0
$629$	8.22117e23	1.33784
$630$	0	0
$631$	−3.86630e22	−0.0612415	−0.0306208	−	0.999531i	$-0.509748\pi$
$631$	−3.86630e22	−0.0612415	−0.0306208	+	0.999531i	$0.509748\pi$
$632$	0	0
$633$	−2.73978e23	−0.422458
$634$	0	0
$635$	−1.47581e24	−2.21541
$636$	0	0
$637$	1.39198e23	0.203445
$638$	0	0
$639$	9.65737e22	0.137436
$640$	0	0
$641$	−1.22656e24	−1.69979	−0.849897	−	0.526949i	$-0.823336\pi$
$641$	−1.22656e24	−1.69979	−0.849897	+	0.526949i	$0.823336\pi$
$642$	0	0
$643$	1.01913e24	1.37542	0.687708	−	0.725987i	$-0.258617\pi$
$643$	1.01913e24	1.37542	0.687708	+	0.725987i	$0.258617\pi$
$644$	0	0
$645$	−8.69764e23	−1.14326
$646$	0	0
$647$	3.34123e23	0.427779	0.213890	−	0.976858i	$-0.431387\pi$
$647$	3.34123e23	0.427779	0.213890	+	0.976858i	$0.431387\pi$
$648$	0	0
$649$	2.32707e23	0.290221
$650$	0	0
$651$	1.85483e23	0.225355
$652$	0	0
$653$	−1.00653e20	−0.000119143 0	−5.95713e−5	−	1.00000i	$-0.500019\pi$
$653$	−1.00653e20	−0.000119143 0	−5.95713e−5		1.00000i	$0.500019\pi$
$654$	0	0
$655$	1.57797e24	1.81990
$656$	0	0
$657$	3.94517e22	0.0443364
$658$	0	0
$659$	−2.47090e23	−0.270600	−0.135300	−	0.990805i	$-0.543200\pi$
$659$	−2.47090e23	−0.270600	−0.135300	+	0.990805i	$0.543200\pi$
$660$	0	0
$661$	−7.05069e23	−0.752522	−0.376261	−	0.926514i	$-0.622790\pi$
$661$	−7.05069e23	−0.752522	−0.376261	+	0.926514i	$0.622790\pi$
$662$	0	0
$663$	8.99822e23	0.936034
$664$	0	0
$665$	−9.40286e23	−0.953402
$666$	0	0
$667$	−1.13052e24	−1.11740
$668$	0	0
$669$	−7.40264e23	−0.713288
$670$	0	0
$671$	−2.38019e23	−0.223599
$672$	0	0
$673$	1.02923e24	0.942720	0.471360	−	0.881941i	$-0.343763\pi$
$673$	1.02923e24	0.942720	0.471360	+	0.881941i	$0.343763\pi$
$674$	0	0
$675$	−2.47125e23	−0.220716
$676$	0	0
$677$	2.01633e24	1.75614	0.878069	−	0.478534i	$-0.158832\pi$
$677$	2.01633e24	1.75614	0.878069	+	0.478534i	$0.158832\pi$
$678$	0	0
$679$	−4.94459e23	−0.419988
$680$	0	0
$681$	1.06384e24	0.881308
$682$	0	0
$683$	−8.13667e23	−0.657462	−0.328731	−	0.944424i	$-0.606621\pi$
$683$	−8.13667e23	−0.657462	−0.328731	+	0.944424i	$0.606621\pi$
$684$	0	0
$685$	−2.32814e24	−1.83501
$686$	0	0
$687$	−6.67535e23	−0.513265
$688$	0	0
$689$	−2.64888e24	−1.98700
$690$	0	0
$691$	8.41969e23	0.616216	0.308108	−	0.951351i	$-0.400304\pi$
$691$	8.41969e23	0.616216	0.308108	+	0.951351i	$0.400304\pi$
$692$	0	0
$693$	3.72204e22	0.0265796
$694$	0	0
$695$	−5.91236e23	−0.411994
$696$	0	0
$697$	1.88688e24	1.28312
$698$	0	0
$699$	−1.05393e24	−0.699450
$700$	0	0
$701$	−1.71845e24	−1.11310	−0.556551	−	0.830813i	$-0.687876\pi$
$701$	−1.71845e24	−1.11310	−0.556551	+	0.830813i	$0.687876\pi$
$702$	0	0
$703$	3.19991e24	2.02310
$704$	0	0
$705$	1.71247e24	1.05686
$706$	0	0
$707$	5.66675e23	0.341405
$708$	0	0
$709$	−5.99743e23	−0.352754	−0.176377	−	0.984323i	$-0.556438\pi$
$709$	−5.99743e23	−0.352754	−0.176377	+	0.984323i	$0.556438\pi$
$710$	0	0
$711$	2.17679e23	0.125004
$712$	0	0
$713$	−1.23504e24	−0.692503
$714$	0	0
$715$	8.04025e23	0.440218
$716$	0	0
$717$	1.68705e24	0.902020
$718$	0	0
$719$	−3.27303e24	−1.70905	−0.854524	−	0.519412i	$-0.826151\pi$
$719$	−3.27303e24	−1.70905	−0.854524	+	0.519412i	$0.826151\pi$
$720$	0	0
$721$	−5.78111e23	−0.294823
$722$	0	0
$723$	−2.11235e24	−1.05218
$724$	0	0
$725$	−3.92782e24	−1.91108
$726$	0	0
$727$	−1.85970e24	−0.883894	−0.441947	−	0.897041i	$-0.645712\pi$
$727$	−1.85970e24	−0.883894	−0.441947	+	0.897041i	$0.645712\pi$
$728$	0	0
$729$	7.97664e22	0.0370370
$730$	0	0
$731$	−3.39161e24	−1.53854
$732$	0	0
$733$	−3.65519e24	−1.62004	−0.810021	−	0.586401i	$-0.800544\pi$
$733$	−3.65519e24	−1.62004	−0.810021	+	0.586401i	$0.800544\pi$
$734$	0	0
$735$	−2.79053e23	−0.120849
$736$	0	0
$737$	7.11746e23	0.301198
$738$	0	0
$739$	1.66864e24	0.690057	0.345029	−	0.938592i	$-0.387869\pi$
$739$	1.66864e24	0.690057	0.345029	+	0.938592i	$0.387869\pi$
$740$	0	0
$741$	3.50236e24	1.41549
$742$	0	0
$743$	−3.53080e24	−1.39466	−0.697330	−	0.716750i	$-0.745629\pi$
$743$	−3.53080e24	−1.39466	−0.697330	+	0.716750i	$0.745629\pi$
$744$	0	0
$745$	3.03944e24	1.17345
$746$	0	0
$747$	−7.20770e23	−0.272002
$748$	0	0
$749$	2.56505e23	0.0946239
$750$	0	0
$751$	3.46978e24	1.25130	0.625651	−	0.780103i	$-0.284834\pi$
$751$	3.46978e24	1.25130	0.625651	+	0.780103i	$0.284834\pi$
$752$	0	0
$753$	−1.32148e24	−0.465912
$754$	0	0
$755$	7.03409e24	2.42471
$756$	0	0
$757$	−1.30475e24	−0.439757	−0.219878	−	0.975527i	$-0.570566\pi$
$757$	−1.30475e24	−0.439757	−0.219878	+	0.975527i	$0.570566\pi$
$758$	0	0
$759$	−2.47833e23	−0.0816777
$760$	0	0
$761$	−1.45265e24	−0.468156	−0.234078	−	0.972218i	$-0.575207\pi$
$761$	−1.45265e24	−0.468156	−0.234078	+	0.972218i	$0.575207\pi$
$762$	0	0
$763$	−1.49698e24	−0.471799
$764$	0	0
$765$	−1.80389e24	−0.556018
$766$	0	0
$767$	6.49854e24	1.95910
$768$	0	0
$769$	−2.49416e24	−0.735444	−0.367722	−	0.929936i	$-0.619862\pi$
$769$	−2.49416e24	−0.735444	−0.367722	+	0.929936i	$0.619862\pi$
$770$	0	0
$771$	−1.06782e24	−0.307989
$772$	0	0
$773$	1.97174e24	0.556319	0.278159	−	0.960535i	$-0.410276\pi$
$773$	1.97174e24	0.556319	0.278159	+	0.960535i	$0.410276\pi$
$774$	0	0
$775$	−4.29098e24	−1.18438
$776$	0	0
$777$	9.49654e23	0.256440
$778$	0	0
$779$	7.34428e24	1.94035
$780$	0	0
$781$	3.36493e23	0.0869846
$782$	0	0
$783$	1.26781e24	0.320686
$784$	0	0
$785$	−8.25600e24	−2.04351
$786$	0	0
$787$	−5.42371e23	−0.131375	−0.0656873	−	0.997840i	$-0.520924\pi$
$787$	−5.42371e23	−0.131375	−0.0656873	+	0.997840i	$0.520924\pi$
$788$	0	0
$789$	−2.02187e24	−0.479291
$790$	0	0
$791$	1.83629e24	0.426032
$792$	0	0
$793$	−6.64689e24	−1.50937
$794$	0	0
$795$	5.31027e24	1.18031
$796$	0	0
$797$	−2.11014e24	−0.459108	−0.229554	−	0.973296i	$-0.573727\pi$
$797$	−2.11014e24	−0.459108	−0.229554	+	0.973296i	$0.573727\pi$
$798$	0	0
$799$	6.67771e24	1.42226
$800$	0	0
$801$	1.08629e24	0.226500
$802$	0	0
$803$	1.37462e23	0.0280608
$804$	0	0
$805$	1.85808e24	0.371364
$806$	0	0
$807$	4.08559e24	0.799521
$808$	0	0
$809$	7.89951e24	1.51369	0.756846	−	0.653593i	$-0.226739\pi$
$809$	7.89951e24	1.51369	0.756846	+	0.653593i	$0.226739\pi$
$810$	0	0
$811$	2.67257e24	0.501478	0.250739	−	0.968055i	$-0.419326\pi$
$811$	2.67257e24	0.501478	0.250739	+	0.968055i	$0.419326\pi$
$812$	0	0
$813$	−2.17047e24	−0.398827
$814$	0	0
$815$	1.52433e23	0.0274309
$816$	0	0
$817$	−1.32011e25	−2.32661
$818$	0	0
$819$	1.03941e24	0.179422
$820$	0	0
$821$	7.77812e23	0.131510	0.0657548	−	0.997836i	$-0.479054\pi$
$821$	7.77812e23	0.131510	0.0657548	+	0.997836i	$0.479054\pi$
$822$	0	0
$823$	8.91303e24	1.47614	0.738069	−	0.674725i	$-0.235738\pi$
$823$	8.91303e24	1.47614	0.738069	+	0.674725i	$0.235738\pi$
$824$	0	0
$825$	−8.61060e23	−0.139693
$826$	0	0
$827$	−7.71148e23	−0.122558	−0.0612789	−	0.998121i	$-0.519518\pi$
$827$	−7.71148e23	−0.122558	−0.0612789	+	0.998121i	$0.519518\pi$
$828$	0	0
$829$	−1.47139e24	−0.229094	−0.114547	−	0.993418i	$-0.536542\pi$
$829$	−1.47139e24	−0.229094	−0.114547	+	0.993418i	$0.536542\pi$
$830$	0	0
$831$	4.21924e24	0.643614
$832$	0	0
$833$	−1.08816e24	−0.162633
$834$	0	0
$835$	−7.03202e23	−0.102978
$836$	0	0
$837$	1.38504e24	0.198744
$838$	0	0
$839$	9.00089e24	1.26564	0.632818	−	0.774300i	$-0.281898\pi$
$839$	9.00089e24	1.26564	0.632818	+	0.774300i	$0.281898\pi$
$840$	0	0
$841$	1.28936e25	1.77667
$842$	0	0
$843$	−7.14005e24	−0.964199
$844$	0	0
$845$	1.13821e25	1.50641
$846$	0	0
$847$	−2.78411e24	−0.361142
$848$	0	0
$849$	6.92525e24	0.880482
$850$	0	0
$851$	−6.32328e24	−0.788027
$852$	0	0
$853$	−8.87129e23	−0.108373	−0.0541864	−	0.998531i	$-0.517257\pi$
$853$	−8.87129e23	−0.108373	−0.0541864	+	0.998531i	$0.517257\pi$
$854$	0	0
$855$	−7.02127e24	−0.840821
$856$	0	0
$857$	−1.75747e23	−0.0206325	−0.0103162	−	0.999947i	$-0.503284\pi$
$857$	−1.75747e23	−0.0206325	−0.0103162	+	0.999947i	$0.503284\pi$
$858$	0	0
$859$	−4.36368e23	−0.0502240	−0.0251120	−	0.999685i	$-0.507994\pi$
$859$	−4.36368e23	−0.0502240	−0.0251120	+	0.999685i	$0.507994\pi$
$860$	0	0
$861$	2.17960e24	0.245952
$862$	0	0
$863$	6.27063e23	0.0693776	0.0346888	−	0.999398i	$-0.488956\pi$
$863$	6.27063e23	0.0693776	0.0346888	+	0.999398i	$0.488956\pi$
$864$	0	0
$865$	−6.63304e24	−0.719575
$866$	0	0
$867$	−1.60669e24	−0.170911
$868$	0	0
$869$	7.58462e23	0.0791163
$870$	0	0
$871$	1.98761e25	2.03319
$872$	0	0
$873$	−3.69221e24	−0.370395
$874$	0	0
$875$	8.26741e23	0.0813395
$876$	0	0
$877$	−1.15775e25	−1.11717	−0.558584	−	0.829448i	$-0.688655\pi$
$877$	−1.15775e25	−1.11717	−0.558584	+	0.829448i	$0.688655\pi$
$878$	0	0
$879$	−1.96650e24	−0.186118
$880$	0	0
$881$	−9.84970e24	−0.914383	−0.457191	−	0.889368i	$-0.651145\pi$
$881$	−9.84970e24	−0.914383	−0.457191	+	0.889368i	$0.651145\pi$
$882$	0	0
$883$	2.13833e25	1.94719	0.973595	−	0.228284i	$-0.0733114\pi$
$883$	2.13833e25	1.94719	0.973595	+	0.228284i	$0.0733114\pi$
$884$	0	0
$885$	−1.30278e25	−1.16373
$886$	0	0
$887$	1.57904e25	1.38370	0.691850	−	0.722042i	$-0.256796\pi$
$887$	1.57904e25	1.38370	0.691850	+	0.722042i	$0.256796\pi$
$888$	0	0
$889$	−6.64764e24	−0.571482
$890$	0	0
$891$	2.77931e23	0.0234410
$892$	0	0
$893$	2.59916e25	2.15077
$894$	0	0
$895$	−1.56114e25	−1.26749
$896$	0	0
$897$	−6.92095e24	−0.551353
$898$	0	0
$899$	2.20139e25	1.72083
$900$	0	0
$901$	2.07072e25	1.58840
$902$	0	0
$903$	−3.91776e24	−0.294911
$904$	0	0
$905$	2.00969e24	0.148462
$906$	0	0
$907$	6.27586e24	0.455000	0.227500	−	0.973778i	$-0.426945\pi$
$907$	6.27586e24	0.455000	0.227500	+	0.973778i	$0.426945\pi$
$908$	0	0
$909$	4.23146e24	0.301090
$910$	0	0
$911$	−1.10861e25	−0.774234	−0.387117	−	0.922031i	$-0.626529\pi$
$911$	−1.10861e25	−0.774234	−0.387117	+	0.922031i	$0.626529\pi$
$912$	0	0
$913$	−2.51139e24	−0.172152
$914$	0	0
$915$	1.33252e25	0.896589
$916$	0	0
$917$	7.10778e24	0.469456
$918$	0	0
$919$	2.18793e25	1.41857	0.709286	−	0.704920i	$-0.249017\pi$
$919$	2.18793e25	1.41857	0.709286	+	0.704920i	$0.249017\pi$
$920$	0	0
$921$	−1.63527e25	−1.04084
$922$	0	0
$923$	9.39686e24	0.587176
$924$	0	0
$925$	−2.19693e25	−1.34776
$926$	0	0
$927$	−4.31685e24	−0.260009
$928$	0	0
$929$	−9.24125e24	−0.546509	−0.273255	−	0.961942i	$-0.588100\pi$
$929$	−9.24125e24	−0.546509	−0.273255	+	0.961942i	$0.588100\pi$
$930$	0	0
$931$	−4.23541e24	−0.245937
$932$	0	0
$933$	1.07425e25	0.612507
$934$	0	0
$935$	−6.28533e24	−0.351908
$936$	0	0
$937$	−9.59650e24	−0.527626	−0.263813	−	0.964574i	$-0.584980\pi$
$937$	−9.59650e24	−0.527626	−0.263813	+	0.964574i	$0.584980\pi$
$938$	0	0
$939$	−2.06527e25	−1.11511
$940$	0	0
$941$	−2.04215e25	−1.08287	−0.541435	−	0.840743i	$-0.682119\pi$
$941$	−2.04215e25	−1.08287	−0.541435	+	0.840743i	$0.682119\pi$
$942$	0	0
$943$	−1.45129e25	−0.755795
$944$	0	0
$945$	−2.08374e24	−0.106579
$946$	0	0
$947$	−2.83260e25	−1.42302	−0.711508	−	0.702678i	$-0.751987\pi$
$947$	−2.83260e25	−1.42302	−0.711508	+	0.702678i	$0.751987\pi$
$948$	0	0
$949$	3.83875e24	0.189421
$950$	0	0
$951$	3.71642e24	0.180132
$952$	0	0
$953$	3.40065e25	1.61909	0.809546	−	0.587056i	$-0.199713\pi$
$953$	3.40065e25	1.61909	0.809546	+	0.587056i	$0.199713\pi$
$954$	0	0
$955$	−2.73296e25	−1.27822
$956$	0	0
$957$	4.41746e24	0.202965
$958$	0	0
$959$	−1.04868e25	−0.473355
$960$	0	0
$961$	1.49915e24	0.0664808
$962$	0	0
$963$	1.91537e24	0.0834505
$964$	0	0
$965$	−5.81768e25	−2.49039
$966$	0	0
$967$	−1.45084e25	−0.610231	−0.305115	−	0.952315i	$-0.598695\pi$
$967$	−1.45084e25	−0.610231	−0.305115	+	0.952315i	$0.598695\pi$
$968$	0	0
$969$	−2.73791e25	−1.13153
$970$	0	0
$971$	4.90817e25	1.99323	0.996613	−	0.0822374i	$-0.0262066\pi$
$971$	4.90817e25	1.99323	0.996613	+	0.0822374i	$0.0262066\pi$
$972$	0	0
$973$	−2.66316e24	−0.106277
$974$	0	0
$975$	−2.40458e25	−0.942976
$976$	0	0
$977$	−1.65065e25	−0.636140	−0.318070	−	0.948067i	$-0.603035\pi$
$977$	−1.65065e25	−0.636140	−0.318070	+	0.948067i	$0.603035\pi$
$978$	0	0
$979$	3.78497e24	0.143354
$980$	0	0
$981$	−1.11782e25	−0.416087
$982$	0	0
$983$	1.41201e22	0.000516576 0	0.000258288	−	1.00000i	$-0.499918\pi$
$983$	1.41201e22	0.000516576 0	0.000258288		1.00000i	$0.499918\pi$
$984$	0	0
$985$	−4.90391e25	−1.76333
$986$	0	0
$987$	7.71364e24	0.272624
$988$	0	0
$989$	2.60864e25	0.906246
$990$	0	0
$991$	−1.89781e25	−0.648078	−0.324039	−	0.946044i	$-0.605041\pi$
$991$	−1.89781e25	−0.648078	−0.324039	+	0.946044i	$0.605041\pi$
$992$	0	0
$993$	−2.67861e24	−0.0899167
$994$	0	0
$995$	−8.98962e24	−0.296651
$996$	0	0
$997$	6.34337e24	0.205784	0.102892	−	0.994693i	$-0.467190\pi$
$997$	6.34337e24	0.205784	0.102892	+	0.994693i	$0.467190\pi$
$998$	0	0
$999$	7.09122e24	0.226159

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.18.a.a.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.18.a.a.1.4	✓	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)

\(f(q)\)	\(=\)	\(q-6561.00 q^{3} +1.27982e6 q^{5} +5.76480e6 q^{7} +4.30467e7 q^{9} +O(q^{10})\)	\(q-6561.00 q^{3} +1.27982e6 q^{5} +5.76480e6 q^{7} +4.30467e7 q^{9} +1.49988e8 q^{11} +4.18855e9 q^{13} -8.39689e9 q^{15} -3.27433e10 q^{17} -1.27446e11 q^{19} -3.78229e10 q^{21} +2.51844e11 q^{23} +8.74996e11 q^{25} -2.82430e11 q^{27} -4.48896e12 q^{29} -4.90401e12 q^{31} -9.84073e11 q^{33} +7.37790e12 q^{35} -2.51079e13 q^{37} -2.74811e13 q^{39} -5.76265e13 q^{41} +1.03582e14 q^{43} +5.50920e13 q^{45} -2.03941e14 q^{47} +3.32329e13 q^{49} +2.14829e14 q^{51} -6.32409e14 q^{53} +1.91958e14 q^{55} +8.36175e14 q^{57} +1.55150e15 q^{59} -1.58692e15 q^{61} +2.48156e14 q^{63} +5.36059e15 q^{65} +4.74535e15 q^{67} -1.65235e15 q^{69} +2.24346e15 q^{71} +9.16486e14 q^{73} -5.74085e15 q^{75} +8.64652e14 q^{77} +5.05681e15 q^{79} +1.85302e15 q^{81} -1.67439e16 q^{83} -4.19055e16 q^{85} +2.94521e16 q^{87} +2.52351e16 q^{89} +2.41462e16 q^{91} +3.21752e16 q^{93} -1.63108e17 q^{95} -8.57721e16 q^{97} +6.45650e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 26244 q^{3} + 528276 q^{5} + 23059204 q^{7} + 172186884 q^{9}+O(q^{10}) \) 4 * q - 26244 * q^3 + 528276 * q^5 + 23059204 * q^7 + 172186884 * q^9	\( 4 q - 26244 q^{3} + 528276 q^{5} + 23059204 q^{7} + 172186884 q^{9} - 269632020 q^{11} + 1984752056 q^{13} - 3466018836 q^{15} - 29765040036 q^{17} - 91677835096 q^{19} - 151291437444 q^{21} + 38384818716 q^{23} - 235654160804 q^{25} - 1129718145924 q^{27} - 3815624693904 q^{29} - 4907739231784 q^{31} + 1769055683220 q^{33} + 3045406013076 q^{35} + 13670824759208 q^{37} - 13021958239416 q^{39} + 10307194403604 q^{41} + 125363311182368 q^{43} + 22740549582996 q^{45} - 25278931593000 q^{47} + 132931722278404 q^{49} + 195288427676196 q^{51} - 239725859759304 q^{53} + 694573116031368 q^{55} + 601498276064856 q^{57} + 13\!\cdots\!00 q^{59}+ \cdots - 11\!\cdots\!20 q^{99}+O(q^{100}) \) 4 * q - 26244 * q^3 + 528276 * q^5 + 23059204 * q^7 + 172186884 * q^9 - 269632020 * q^11 + 1984752056 * q^13 - 3466018836 * q^15 - 29765040036 * q^17 - 91677835096 * q^19 - 151291437444 * q^21 + 38384818716 * q^23 - 235654160804 * q^25 - 1129718145924 * q^27 - 3815624693904 * q^29 - 4907739231784 * q^31 + 1769055683220 * q^33 + 3045406013076 * q^35 + 13670824759208 * q^37 - 13021958239416 * q^39 + 10307194403604 * q^41 + 125363311182368 * q^43 + 22740549582996 * q^45 - 25278931593000 * q^47 + 132931722278404 * q^49 + 195288427676196 * q^51 - 239725859759304 * q^53 + 694573116031368 * q^55 + 601498276064856 * q^57 + 1374756208497000 * q^59 + 652646281638080 * q^61 + 992623121070084 * q^63 + 4153293849278904 * q^65 + 6617359123302392 * q^67 - 251842795595676 * q^69 + 2405715947056548 * q^71 + 4250185178265824 * q^73 + 1546126949035044 * q^75 - 1554374938528020 * q^77 + 20052572228473688 * q^79 + 7412080755407364 * q^81 - 5593593799375584 * q^83 - 13012862591692368 * q^85 + 25034313616704144 * q^87 + 1518004216828212 * q^89 + 11441700637180856 * q^91 + 32199677099734824 * q^93 - 149304801904677408 * q^95 - 102152247026326000 * q^97 - 11606774337606420 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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