LMFDB - Embedded newform 84.16.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,16,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, 16, names="a")

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

chi := DirichletCharacter("84.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 16 $
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:	$119.862544284$
Analytic rank:	$0$
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} - 2x^{3} - 47201410x^{2} - 158185874320x - 140304738691800 $ x^4 - 2x^3 - 47201410x^2 - 158185874320*x - 140304738691800
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{2}\cdot 5\cdot 7 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8270.22$ 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)

$f(q)$	$=$	$q+2187.00 q^{3} -84509.8 q^{5} -823543. q^{7} +4.78297e6 q^{9} +O(q^{10})$	$q+2187.00 q^{3} -84509.8 q^{5} -823543. q^{7} +4.78297e6 q^{9} -1.07407e8 q^{11} -3.39726e8 q^{13} -1.84823e8 q^{15} +3.49847e8 q^{17} -5.08916e8 q^{19} -1.80109e9 q^{21} +2.43845e10 q^{23} -2.33757e10 q^{25} +1.04604e10 q^{27} -9.67376e10 q^{29} -6.10655e10 q^{31} -2.34900e11 q^{33} +6.95975e10 q^{35} +8.41861e11 q^{37} -7.42981e11 q^{39} +1.47188e12 q^{41} +1.18276e12 q^{43} -4.04208e11 q^{45} -3.31681e11 q^{47} +6.78223e11 q^{49} +7.65115e11 q^{51} +2.92964e12 q^{53} +9.07697e12 q^{55} -1.11300e12 q^{57} -1.38912e13 q^{59} +1.30651e13 q^{61} -3.93898e12 q^{63} +2.87102e13 q^{65} +2.76757e13 q^{67} +5.33290e13 q^{69} -2.86205e12 q^{71} -9.52347e13 q^{73} -5.11226e13 q^{75} +8.84545e13 q^{77} -2.22994e14 q^{79} +2.28768e13 q^{81} +5.50574e12 q^{83} -2.95655e13 q^{85} -2.11565e14 q^{87} +2.51487e13 q^{89} +2.79779e14 q^{91} -1.33550e14 q^{93} +4.30084e13 q^{95} +1.19559e15 q^{97} -5.13726e14 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8748 q^{3} + 275436 q^{5} - 3294172 q^{7} + 19131876 q^{9}+O(q^{10}) $ 4 * q + 8748 * q^3 + 275436 * q^5 - 3294172 * q^7 + 19131876 * q^9	\( 4 q + 8748 q^{3} + 275436 q^{5} - 3294172 q^{7} + 19131876 q^{9} - 61540668 q^{11} + 369006456 q^{13} + 602378532 q^{15} - 1458569484 q^{17} - 2691246184 q^{19} - 7204354164 q^{21} + 18028282212 q^{23} + 8448695516 q^{25} + 41841412812 q^{27} - 14387019264 q^{29} + 271402618248 q^{31} - 134589440916 q^{33} - 226833389748 q^{35} + 1719450823976 q^{37} + 807017119272 q^{39} + 2784989291868 q^{41} + 1897932676640 q^{43} + 1317401849484 q^{45} + 1872077464008 q^{47} + 2712892291396 q^{49} - 3189891461508 q^{51} + 4740642191736 q^{53} - 21671359654504 q^{55} - 5885755404408 q^{57} + 13984487218296 q^{59} + 32810552663568 q^{61} - 15755922556668 q^{63} + 183566971857672 q^{65} + 4158516498856 q^{67} + 39427853197644 q^{69} + 70250046828444 q^{71} + 27064251481456 q^{73} + 18477297093492 q^{75} + 50681386346724 q^{77} - 238381767283992 q^{79} + 91507169819844 q^{81} - 233193205883808 q^{83} - 893307076687696 q^{85} - 31464411130368 q^{87} + 316154161239228 q^{89} - 303892683793608 q^{91} + 593557526108376 q^{93} - 11\!\cdots\!76 q^{95}+ \cdots - 294347107283292 q^{99}+O(q^{100}) \) 4 * q + 8748 * q^3 + 275436 * q^5 - 3294172 * q^7 + 19131876 * q^9 - 61540668 * q^11 + 369006456 * q^13 + 602378532 * q^15 - 1458569484 * q^17 - 2691246184 * q^19 - 7204354164 * q^21 + 18028282212 * q^23 + 8448695516 * q^25 + 41841412812 * q^27 - 14387019264 * q^29 + 271402618248 * q^31 - 134589440916 * q^33 - 226833389748 * q^35 + 1719450823976 * q^37 + 807017119272 * q^39 + 2784989291868 * q^41 + 1897932676640 * q^43 + 1317401849484 * q^45 + 1872077464008 * q^47 + 2712892291396 * q^49 - 3189891461508 * q^51 + 4740642191736 * q^53 - 21671359654504 * q^55 - 5885755404408 * q^57 + 13984487218296 * q^59 + 32810552663568 * q^61 - 15755922556668 * q^63 + 183566971857672 * q^65 + 4158516498856 * q^67 + 39427853197644 * q^69 + 70250046828444 * q^71 + 27064251481456 * q^73 + 18477297093492 * q^75 + 50681386346724 * q^77 - 238381767283992 * q^79 + 91507169819844 * q^81 - 233193205883808 * q^83 - 893307076687696 * q^85 - 31464411130368 * q^87 + 316154161239228 * q^89 - 303892683793608 * q^91 + 593557526108376 * q^93 - 1197067113655776 * q^95 + 529302182472448 * q^97 - 294347107283292 * 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
$3$	2187.00	0.577350
$3$	2187.00	0.577350
$4$	0	0
$5$	−84509.8	−0.483763	−0.241881	−	0.970306i	$-0.577764\pi$
$5$	−84509.8	−0.483763	−0.241881	+	0.970306i	$0.577764\pi$
$6$	0	0
$7$	−823543.	−0.377964
$7$	−823543.	−0.377964
$8$	0	0
$9$	4.78297e6	0.333333
$10$	0	0
$11$	−1.07407e8	−1.66184	−0.830919	−	0.556393i	$-0.812185\pi$
$11$	−1.07407e8	−1.66184	−0.830919	+	0.556393i	$0.812185\pi$
$12$	0	0
$13$	−3.39726e8	−1.50160	−0.750799	−	0.660531i	$-0.770331\pi$
$13$	−3.39726e8	−1.50160	−0.750799	+	0.660531i	$0.770331\pi$
$14$	0	0
$15$	−1.84823e8	−0.279300
$16$	0	0
$17$	3.49847e8	0.206781	0.103391	−	0.994641i	$-0.467031\pi$
$17$	3.49847e8	0.206781	0.103391	+	0.994641i	$0.467031\pi$
$18$	0	0
$19$	−5.08916e8	−0.130615	−0.0653077	−	0.997865i	$-0.520803\pi$
$19$	−5.08916e8	−0.130615	−0.0653077	+	0.997865i	$0.520803\pi$
$20$	0	0
$21$	−1.80109e9	−0.218218
$22$	0	0
$23$	2.43845e10	1.49333	0.746665	−	0.665200i	$-0.231654\pi$
$23$	2.43845e10	1.49333	0.746665	+	0.665200i	$0.231654\pi$
$24$	0	0
$25$	−2.33757e10	−0.765974
$26$	0	0
$27$	1.04604e10	0.192450
$28$	0	0
$29$	−9.67376e10	−1.04138	−0.520691	−	0.853745i	$-0.674326\pi$
$29$	−9.67376e10	−1.04138	−0.520691	+	0.853745i	$0.674326\pi$
$30$	0	0
$31$	−6.10655e10	−0.398642	−0.199321	−	0.979934i	$-0.563874\pi$
$31$	−6.10655e10	−0.398642	−0.199321	+	0.979934i	$0.563874\pi$
$32$	0	0
$33$	−2.34900e11	−0.959463
$34$	0	0
$35$	6.95975e10	0.182845
$36$	0	0
$37$	8.41861e11	1.45790	0.728950	−	0.684567i	$-0.240009\pi$
$37$	8.41861e11	1.45790	0.728950	+	0.684567i	$0.240009\pi$
$38$	0	0
$39$	−7.42981e11	−0.866948
$40$	0	0
$41$	1.47188e12	1.18030	0.590151	−	0.807293i	$-0.299068\pi$
$41$	1.47188e12	1.18030	0.590151	+	0.807293i	$0.299068\pi$
$42$	0	0
$43$	1.18276e12	0.663564	0.331782	−	0.943356i	$-0.392350\pi$
$43$	1.18276e12	0.663564	0.331782	+	0.943356i	$0.392350\pi$
$44$	0	0
$45$	−4.04208e11	−0.161254
$46$	0	0
$47$	−3.31681e11	−0.0954965	−0.0477482	−	0.998859i	$-0.515205\pi$
$47$	−3.31681e11	−0.0954965	−0.0477482	+	0.998859i	$0.515205\pi$
$48$	0	0
$49$	6.78223e11	0.142857
$50$	0	0
$51$	7.65115e11	0.119385
$52$	0	0
$53$	2.92964e12	0.342567	0.171284	−	0.985222i	$-0.445209\pi$
$53$	2.92964e12	0.342567	0.171284	+	0.985222i	$0.445209\pi$
$54$	0	0
$55$	9.07697e12	0.803935
$56$	0	0
$57$	−1.11300e12	−0.0754108
$58$	0	0
$59$	−1.38912e13	−0.726689	−0.363344	−	0.931655i	$-0.618365\pi$
$59$	−1.38912e13	−0.726689	−0.363344	+	0.931655i	$0.618365\pi$
$60$	0	0
$61$	1.30651e13	0.532278	0.266139	−	0.963935i	$-0.414252\pi$
$61$	1.30651e13	0.532278	0.266139	+	0.963935i	$0.414252\pi$
$62$	0	0
$63$	−3.93898e12	−0.125988
$64$	0	0
$65$	2.87102e13	0.726417
$66$	0	0
$67$	2.76757e13	0.557875	0.278938	−	0.960309i	$-0.410018\pi$
$67$	2.76757e13	0.557875	0.278938	+	0.960309i	$0.410018\pi$
$68$	0	0
$69$	5.33290e13	0.862174
$70$	0	0
$71$	−2.86205e12	−0.0373457	−0.0186728	−	0.999826i	$-0.505944\pi$
$71$	−2.86205e12	−0.0373457	−0.0186728	+	0.999826i	$0.505944\pi$
$72$	0	0
$73$	−9.52347e13	−1.00896	−0.504480	−	0.863423i	$-0.668316\pi$
$73$	−9.52347e13	−1.00896	−0.504480	+	0.863423i	$0.668316\pi$
$74$	0	0
$75$	−5.11226e13	−0.442235
$76$	0	0
$77$	8.84545e13	0.628116
$78$	0	0
$79$	−2.22994e14	−1.30644	−0.653221	−	0.757167i	$-0.726583\pi$
$79$	−2.22994e14	−1.30644	−0.653221	+	0.757167i	$0.726583\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	0	0
$83$	5.50574e12	0.0222705	0.0111353	−	0.999938i	$-0.496455\pi$
$83$	5.50574e12	0.0222705	0.0111353	+	0.999938i	$0.496455\pi$
$84$	0	0
$85$	−2.95655e13	−0.100033
$86$	0	0
$87$	−2.11565e14	−0.601243
$88$	0	0
$89$	2.51487e13	0.0602685	0.0301342	−	0.999546i	$-0.490407\pi$
$89$	2.51487e13	0.0602685	0.0301342	+	0.999546i	$0.490407\pi$
$90$	0	0
$91$	2.79779e14	0.567551
$92$	0	0
$93$	−1.33550e14	−0.230156
$94$	0	0
$95$	4.30084e13	0.0631868
$96$	0	0
$97$	1.19559e15	1.50243	0.751217	−	0.660055i	$-0.229467\pi$
$97$	1.19559e15	1.50243	0.751217	+	0.660055i	$0.229467\pi$
$98$	0	0
$99$	−5.13726e14	−0.553946
$100$	0	0
$101$	−1.48337e15	−1.37670	−0.688351	−	0.725378i	$-0.741665\pi$
$101$	−1.48337e15	−1.37670	−0.688351	+	0.725378i	$0.741665\pi$
$102$	0	0
$103$	2.24235e15	1.79649	0.898245	−	0.439494i	$-0.144842\pi$
$103$	2.24235e15	1.79649	0.898245	+	0.439494i	$0.144842\pi$
$104$	0	0
$105$	1.52210e14	0.105566
$106$	0	0
$107$	2.53267e15	1.52476	0.762378	−	0.647131i	$-0.224032\pi$
$107$	2.53267e15	1.52476	0.762378	+	0.647131i	$0.224032\pi$
$108$	0	0
$109$	−8.58226e14	−0.449679	−0.224840	−	0.974396i	$-0.572186\pi$
$109$	−8.58226e14	−0.449679	−0.224840	+	0.974396i	$0.572186\pi$
$110$	0	0
$111$	1.84115e15	0.841719
$112$	0	0
$113$	−2.77178e15	−1.10833	−0.554167	−	0.832405i	$-0.686963\pi$
$113$	−2.77178e15	−1.10833	−0.554167	+	0.832405i	$0.686963\pi$
$114$	0	0
$115$	−2.06073e15	−0.722417
$116$	0	0
$117$	−1.62490e15	−0.500533
$118$	0	0
$119$	−2.88114e14	−0.0781560
$120$	0	0
$121$	7.35908e15	1.76171
$122$	0	0
$123$	3.21900e15	0.681447
$124$	0	0
$125$	4.55451e15	0.854312
$126$	0	0
$127$	4.45532e15	0.741909	0.370955	−	0.928651i	$-0.379030\pi$
$127$	4.45532e15	0.741909	0.370955	+	0.928651i	$0.379030\pi$
$128$	0	0
$129$	2.58669e15	0.383109
$130$	0	0
$131$	4.02846e15	0.531624	0.265812	−	0.964025i	$-0.414360\pi$
$131$	4.02846e15	0.531624	0.265812	+	0.964025i	$0.414360\pi$
$132$	0	0
$133$	4.19114e14	0.0493680
$134$	0	0
$135$	−8.84003e14	−0.0931002
$136$	0	0
$137$	6.48047e15	0.611227	0.305613	−	0.952156i	$-0.401138\pi$
$137$	6.48047e15	0.611227	0.305613	+	0.952156i	$0.401138\pi$
$138$	0	0
$139$	−1.02502e16	−0.867208	−0.433604	−	0.901104i	$-0.642758\pi$
$139$	−1.02502e16	−0.867208	−0.433604	+	0.901104i	$0.642758\pi$
$140$	0	0
$141$	−7.25387e14	−0.0551349
$142$	0	0
$143$	3.64891e16	2.49541
$144$	0	0
$145$	8.17528e15	0.503782
$146$	0	0
$147$	1.48327e15	0.0824786
$148$	0	0
$149$	4.30901e15	0.216511	0.108256	−	0.994123i	$-0.465474\pi$
$149$	4.30901e15	0.216511	0.108256	+	0.994123i	$0.465474\pi$
$150$	0	0
$151$	−8.90116e15	−0.404687	−0.202344	−	0.979315i	$-0.564856\pi$
$151$	−8.90116e15	−0.404687	−0.202344	+	0.979315i	$0.564856\pi$
$152$	0	0
$153$	1.67331e15	0.0689271
$154$	0	0
$155$	5.16064e15	0.192848
$156$	0	0
$157$	3.18693e16	1.08175	0.540873	−	0.841104i	$-0.318094\pi$
$157$	3.18693e16	1.08175	0.540873	+	0.841104i	$0.318094\pi$
$158$	0	0
$159$	6.40713e15	0.197781
$160$	0	0
$161$	−2.00817e16	−0.564426
$162$	0	0
$163$	2.96248e16	0.759011	0.379506	−	0.925189i	$-0.376094\pi$
$163$	2.96248e16	0.759011	0.379506	+	0.925189i	$0.376094\pi$
$164$	0	0
$165$	1.98513e16	0.464152
$166$	0	0
$167$	4.90024e16	1.04675	0.523376	−	0.852102i	$-0.324672\pi$
$167$	4.90024e16	1.04675	0.523376	+	0.852102i	$0.324672\pi$
$168$	0	0
$169$	6.42279e16	1.25480
$170$	0	0
$171$	−2.43413e15	−0.0435384
$172$	0	0
$173$	−6.97691e16	−1.14371	−0.571857	−	0.820353i	$-0.693777\pi$
$173$	−6.97691e16	−1.14371	−0.571857	+	0.820353i	$0.693777\pi$
$174$	0	0
$175$	1.92509e16	0.289511
$176$	0	0
$177$	−3.03800e16	−0.419554
$178$	0	0
$179$	7.03205e16	0.892656	0.446328	−	0.894869i	$-0.352731\pi$
$179$	7.03205e16	0.892656	0.446328	+	0.894869i	$0.352731\pi$
$180$	0	0
$181$	−6.60124e16	−0.770968	−0.385484	−	0.922715i	$-0.625965\pi$
$181$	−6.60124e16	−0.770968	−0.385484	+	0.922715i	$0.625965\pi$
$182$	0	0
$183$	2.85733e16	0.307311
$184$	0	0
$185$	−7.11456e16	−0.705277
$186$	0	0
$187$	−3.75761e16	−0.343637
$188$	0	0
$189$	−8.61455e15	−0.0727393
$190$	0	0
$191$	1.36852e17	1.06783	0.533914	−	0.845538i	$-0.320720\pi$
$191$	1.36852e17	1.06783	0.533914	+	0.845538i	$0.320720\pi$
$192$	0	0
$193$	1.57704e17	1.13806	0.569028	−	0.822318i	$-0.307320\pi$
$193$	1.57704e17	1.13806	0.569028	+	0.822318i	$0.307320\pi$
$194$	0	0
$195$	6.27892e16	0.419397
$196$	0	0
$197$	3.05642e17	1.89111	0.945553	−	0.325468i	$-0.105522\pi$
$197$	3.05642e17	1.89111	0.945553	+	0.325468i	$0.105522\pi$
$198$	0	0
$199$	2.40791e17	1.38115	0.690577	−	0.723259i	$-0.257357\pi$
$199$	2.40791e17	1.38115	0.690577	+	0.723259i	$0.257357\pi$
$200$	0	0
$201$	6.05267e16	0.322089
$202$	0	0
$203$	7.96676e16	0.393606
$204$	0	0
$205$	−1.24388e17	−0.570986
$206$	0	0
$207$	1.16630e17	0.497777
$208$	0	0
$209$	5.46613e16	0.217062
$210$	0	0
$211$	−3.59511e17	−1.32921	−0.664605	−	0.747195i	$-0.731400\pi$
$211$	−3.59511e17	−1.32921	−0.664605	+	0.747195i	$0.731400\pi$
$212$	0	0
$213$	−6.25931e15	−0.0215615
$214$	0	0
$215$	−9.99548e16	−0.321008
$216$	0	0
$217$	5.02901e16	0.150673
$218$	0	0
$219$	−2.08278e17	−0.582523
$220$	0	0
$221$	−1.18852e17	−0.310502
$222$	0	0
$223$	3.29694e17	0.805054	0.402527	−	0.915408i	$-0.368132\pi$
$223$	3.29694e17	0.805054	0.402527	+	0.915408i	$0.368132\pi$
$224$	0	0
$225$	−1.11805e17	−0.255325
$226$	0	0
$227$	−8.78052e17	−1.87640	−0.938202	−	0.346088i	$-0.887510\pi$
$227$	−8.78052e17	−1.87640	−0.938202	+	0.346088i	$0.887510\pi$
$228$	0	0
$229$	9.43314e16	0.188751	0.0943756	−	0.995537i	$-0.469915\pi$
$229$	9.43314e16	0.188751	0.0943756	+	0.995537i	$0.469915\pi$
$230$	0	0
$231$	1.93450e17	0.362643
$232$	0	0
$233$	1.39404e17	0.244967	0.122483	−	0.992471i	$-0.460914\pi$
$233$	1.39404e17	0.244967	0.122483	+	0.992471i	$0.460914\pi$
$234$	0	0
$235$	2.80304e16	0.0461976
$236$	0	0
$237$	−4.87688e17	−0.754274
$238$	0	0
$239$	−1.10394e18	−1.60310	−0.801551	−	0.597927i	$-0.795991\pi$
$239$	−1.10394e18	−1.60310	−0.801551	+	0.597927i	$0.795991\pi$
$240$	0	0
$241$	9.13130e17	1.24568	0.622838	−	0.782351i	$-0.285980\pi$
$241$	9.13130e17	1.24568	0.622838	+	0.782351i	$0.285980\pi$
$242$	0	0
$243$	5.00315e16	0.0641500
$244$	0	0
$245$	−5.73165e16	−0.0691089
$246$	0	0
$247$	1.72892e17	0.196132
$248$	0	0
$249$	1.20411e16	0.0128579
$250$	0	0
$251$	−5.12688e16	−0.0515585	−0.0257792	−	0.999668i	$-0.508207\pi$
$251$	−5.12688e16	−0.0515585	−0.0257792	+	0.999668i	$0.508207\pi$
$252$	0	0
$253$	−2.61908e18	−2.48167
$254$	0	0
$255$	−6.46598e16	−0.0577541
$256$	0	0
$257$	−1.32943e18	−1.11987	−0.559933	−	0.828538i	$-0.689173\pi$
$257$	−1.32943e18	−1.11987	−0.559933	+	0.828538i	$0.689173\pi$
$258$	0	0
$259$	−6.93309e17	−0.551034
$260$	0	0
$261$	−4.62693e17	−0.347128
$262$	0	0
$263$	−1.07002e17	−0.0758096	−0.0379048	−	0.999281i	$-0.512068\pi$
$263$	−1.07002e17	−0.0758096	−0.0379048	+	0.999281i	$0.512068\pi$
$264$	0	0
$265$	−2.47584e17	−0.165721
$266$	0	0
$267$	5.50001e16	0.0347960
$268$	0	0
$269$	−1.61523e18	−0.966257	−0.483128	−	0.875550i	$-0.660500\pi$
$269$	−1.61523e18	−0.966257	−0.483128	+	0.875550i	$0.660500\pi$
$270$	0	0
$271$	−4.23363e17	−0.239576	−0.119788	−	0.992799i	$-0.538222\pi$
$271$	−4.23363e17	−0.239576	−0.119788	+	0.992799i	$0.538222\pi$
$272$	0	0
$273$	6.11877e17	0.327676
$274$	0	0
$275$	2.51072e18	1.27292
$276$	0	0
$277$	−3.96615e18	−1.90445	−0.952227	−	0.305391i	$-0.901213\pi$
$277$	−3.96615e18	−1.90445	−0.952227	+	0.305391i	$0.901213\pi$
$278$	0	0
$279$	−2.92074e17	−0.132881
$280$	0	0
$281$	1.31616e18	0.567557	0.283779	−	0.958890i	$-0.408412\pi$
$281$	1.31616e18	0.567557	0.283779	+	0.958890i	$0.408412\pi$
$282$	0	0
$283$	3.45056e18	1.41089	0.705443	−	0.708767i	$-0.250748\pi$
$283$	3.45056e18	1.41089	0.705443	+	0.708767i	$0.250748\pi$
$284$	0	0
$285$	9.40594e16	0.0364809
$286$	0	0
$287$	−1.21215e18	−0.446112
$288$	0	0
$289$	−2.74003e18	−0.957242
$290$	0	0
$291$	2.61476e18	0.867431
$292$	0	0
$293$	4.36840e18	1.37662	0.688312	−	0.725415i	$-0.258352\pi$
$293$	4.36840e18	1.37662	0.688312	+	0.725415i	$0.258352\pi$
$294$	0	0
$295$	1.17394e18	0.351545
$296$	0	0
$297$	−1.12352e18	−0.319821
$298$	0	0
$299$	−8.28406e18	−2.24238
$300$	0	0
$301$	−9.74053e17	−0.250804
$302$	0	0
$303$	−3.24413e18	−0.794839
$304$	0	0
$305$	−1.10413e18	−0.257496
$306$	0	0
$307$	−7.42568e18	−1.64892	−0.824458	−	0.565923i	$-0.808520\pi$
$307$	−7.42568e18	−1.64892	−0.824458	+	0.565923i	$0.808520\pi$
$308$	0	0
$309$	4.90403e18	1.03720
$310$	0	0
$311$	−3.59243e18	−0.723911	−0.361956	−	0.932195i	$-0.617891\pi$
$311$	−3.59243e18	−0.723911	−0.361956	+	0.932195i	$0.617891\pi$
$312$	0	0
$313$	3.55544e18	0.682827	0.341413	−	0.939913i	$-0.389094\pi$
$313$	3.55544e18	0.682827	0.341413	+	0.939913i	$0.389094\pi$
$314$	0	0
$315$	3.32883e17	0.0609484
$316$	0	0
$317$	−7.89866e18	−1.37914	−0.689571	−	0.724218i	$-0.742201\pi$
$317$	−7.89866e18	−1.37914	−0.689571	+	0.724218i	$0.742201\pi$
$318$	0	0
$319$	1.03903e19	1.73061
$320$	0	0
$321$	5.53895e18	0.880319
$322$	0	0
$323$	−1.78043e17	−0.0270088
$324$	0	0
$325$	7.94132e18	1.15018
$326$	0	0
$327$	−1.87694e18	−0.259622
$328$	0	0
$329$	2.73154e17	0.0360943
$330$	0	0
$331$	3.01865e18	0.381157	0.190578	−	0.981672i	$-0.438964\pi$
$331$	3.01865e18	0.381157	0.190578	+	0.981672i	$0.438964\pi$
$332$	0	0
$333$	4.02660e18	0.485967
$334$	0	0
$335$	−2.33887e18	−0.269879
$336$	0	0
$337$	−2.50457e18	−0.276382	−0.138191	−	0.990406i	$-0.544129\pi$
$337$	−2.50457e18	−0.276382	−0.138191	+	0.990406i	$0.544129\pi$
$338$	0	0
$339$	−6.06189e18	−0.639897
$340$	0	0
$341$	6.55888e18	0.662479
$342$	0	0
$343$	−5.58546e17	−0.0539949
$344$	0	0
$345$	−4.50682e18	−0.417088
$346$	0	0
$347$	4.41608e18	0.391351	0.195675	−	0.980669i	$-0.437310\pi$
$347$	4.41608e18	0.391351	0.195675	+	0.980669i	$0.437310\pi$
$348$	0	0
$349$	1.54738e19	1.31343	0.656715	−	0.754139i	$-0.271946\pi$
$349$	1.54738e19	1.31343	0.656715	+	0.754139i	$0.271946\pi$
$350$	0	0
$351$	−3.55365e18	−0.288983
$352$	0	0
$353$	−1.17033e19	−0.912005	−0.456002	−	0.889979i	$-0.650719\pi$
$353$	−1.17033e19	−0.912005	−0.456002	+	0.889979i	$0.650719\pi$
$354$	0	0
$355$	2.41872e17	0.0180664
$356$	0	0
$357$	−6.30105e17	−0.0451234
$358$	0	0
$359$	−4.96931e18	−0.341262	−0.170631	−	0.985335i	$-0.554581\pi$
$359$	−4.96931e18	−0.341262	−0.170631	+	0.985335i	$0.554581\pi$
$360$	0	0
$361$	−1.49221e19	−0.982940
$362$	0	0
$363$	1.60943e19	1.01712
$364$	0	0
$365$	8.04827e18	0.488097
$366$	0	0
$367$	9.75869e18	0.568063	0.284031	−	0.958815i	$-0.408328\pi$
$367$	9.75869e18	0.568063	0.284031	+	0.958815i	$0.408328\pi$
$368$	0	0
$369$	7.03995e18	0.393434
$370$	0	0
$371$	−2.41269e18	−0.129478
$372$	0	0
$373$	2.98456e19	1.53838	0.769191	−	0.639019i	$-0.220660\pi$
$373$	2.98456e19	1.53838	0.769191	+	0.639019i	$0.220660\pi$
$374$	0	0
$375$	9.96071e18	0.493237
$376$	0	0
$377$	3.28643e19	1.56374
$378$	0	0
$379$	−7.64357e18	−0.349544	−0.174772	−	0.984609i	$-0.555919\pi$
$379$	−7.64357e18	−0.349544	−0.174772	+	0.984609i	$0.555919\pi$
$380$	0	0
$381$	9.74378e18	0.428342
$382$	0	0
$383$	1.46109e19	0.617568	0.308784	−	0.951132i	$-0.400078\pi$
$383$	1.46109e19	0.617568	0.308784	+	0.951132i	$0.400078\pi$
$384$	0	0
$385$	−7.47528e18	−0.303859
$386$	0	0
$387$	5.65710e18	0.221188
$388$	0	0
$389$	−5.98929e18	−0.225296	−0.112648	−	0.993635i	$-0.535933\pi$
$389$	−5.98929e18	−0.225296	−0.112648	+	0.993635i	$0.535933\pi$
$390$	0	0
$391$	8.53086e18	0.308793
$392$	0	0
$393$	8.81024e18	0.306933
$394$	0	0
$395$	1.88452e19	0.632008
$396$	0	0
$397$	4.11515e19	1.32879	0.664396	−	0.747380i	$-0.268689\pi$
$397$	4.11515e19	1.32879	0.664396	+	0.747380i	$0.268689\pi$
$398$	0	0
$399$	9.16603e17	0.0285026
$400$	0	0
$401$	−6.02989e19	−1.80604	−0.903019	−	0.429601i	$-0.858654\pi$
$401$	−6.02989e19	−1.80604	−0.903019	+	0.429601i	$0.858654\pi$
$402$	0	0
$403$	2.07456e19	0.598601
$404$	0	0
$405$	−1.93331e18	−0.0537514
$406$	0	0
$407$	−9.04221e19	−2.42279
$408$	0	0
$409$	6.16974e19	1.59346	0.796731	−	0.604334i	$-0.206561\pi$
$409$	6.16974e19	1.59346	0.796731	+	0.604334i	$0.206561\pi$
$410$	0	0
$411$	1.41728e19	0.352892
$412$	0	0
$413$	1.14400e19	0.274663
$414$	0	0
$415$	−4.65290e17	−0.0107736
$416$	0	0
$417$	−2.24173e19	−0.500683
$418$	0	0
$419$	−4.47524e19	−0.964299	−0.482149	−	0.876089i	$-0.660144\pi$
$419$	−4.47524e19	−0.964299	−0.482149	+	0.876089i	$0.660144\pi$
$420$	0	0
$421$	−4.58628e18	−0.0953554	−0.0476777	−	0.998863i	$-0.515182\pi$
$421$	−4.58628e18	−0.0953554	−0.0476777	+	0.998863i	$0.515182\pi$
$422$	0	0
$423$	−1.58642e18	−0.0318322
$424$	0	0
$425$	−8.17790e18	−0.158389
$426$	0	0
$427$	−1.07597e19	−0.201182
$428$	0	0
$429$	7.98016e19	1.44073
$430$	0	0
$431$	3.94926e19	0.688552	0.344276	−	0.938868i	$-0.388124\pi$
$431$	3.94926e19	0.688552	0.344276	+	0.938868i	$0.388124\pi$
$432$	0	0
$433$	1.00198e20	1.68734	0.843668	−	0.536866i	$-0.180392\pi$
$433$	1.00198e20	1.68734	0.843668	+	0.536866i	$0.180392\pi$
$434$	0	0
$435$	1.78793e19	0.290859
$436$	0	0
$437$	−1.24097e19	−0.195052
$438$	0	0
$439$	−2.49377e19	−0.378767	−0.189384	−	0.981903i	$-0.560649\pi$
$439$	−2.49377e19	−0.378767	−0.189384	+	0.981903i	$0.560649\pi$
$440$	0	0
$441$	3.24392e18	0.0476190
$442$	0	0
$443$	−6.22079e19	−0.882709	−0.441354	−	0.897333i	$-0.645502\pi$
$443$	−6.22079e19	−0.882709	−0.441354	+	0.897333i	$0.645502\pi$
$444$	0	0
$445$	−2.12531e18	−0.0291556
$446$	0	0
$447$	9.42380e18	0.125003
$448$	0	0
$449$	2.57075e19	0.329771	0.164886	−	0.986313i	$-0.447274\pi$
$449$	2.57075e19	0.329771	0.164886	+	0.986313i	$0.447274\pi$
$450$	0	0
$451$	−1.58090e20	−1.96147
$452$	0	0
$453$	−1.94668e19	−0.233646
$454$	0	0
$455$	−2.36441e19	−0.274560
$456$	0	0
$457$	−1.27776e20	−1.43574	−0.717871	−	0.696176i	$-0.754883\pi$
$457$	−1.27776e20	−1.43574	−0.717871	+	0.696176i	$0.754883\pi$
$458$	0	0
$459$	3.65952e18	0.0397951
$460$	0	0
$461$	−1.21598e20	−1.27988	−0.639942	−	0.768423i	$-0.721042\pi$
$461$	−1.21598e20	−1.27988	−0.639942	+	0.768423i	$0.721042\pi$
$462$	0	0
$463$	1.41430e20	1.44106	0.720532	−	0.693422i	$-0.243898\pi$
$463$	1.41430e20	1.44106	0.720532	+	0.693422i	$0.243898\pi$
$464$	0	0
$465$	1.12863e19	0.111341
$466$	0	0
$467$	−1.60716e20	−1.53526	−0.767630	−	0.640894i	$-0.778564\pi$
$467$	−1.60716e20	−1.53526	−0.767630	+	0.640894i	$0.778564\pi$
$468$	0	0
$469$	−2.27921e19	−0.210857
$470$	0	0
$471$	6.96982e19	0.624546
$472$	0	0
$473$	−1.27037e20	−1.10274
$474$	0	0
$475$	1.18963e19	0.100048
$476$	0	0
$477$	1.40124e19	0.114189
$478$	0	0
$479$	−7.08051e19	−0.559176	−0.279588	−	0.960120i	$-0.590198\pi$
$479$	−7.08051e19	−0.559176	−0.279588	+	0.960120i	$0.590198\pi$
$480$	0	0
$481$	−2.86002e20	−2.18918
$482$	0	0
$483$	−4.39187e19	−0.325871
$484$	0	0
$485$	−1.01039e20	−0.726822
$486$	0	0
$487$	1.82762e20	1.27473	0.637367	−	0.770560i	$-0.280023\pi$
$487$	1.82762e20	1.27473	0.637367	+	0.770560i	$0.280023\pi$
$488$	0	0
$489$	6.47894e19	0.438215
$490$	0	0
$491$	−1.27255e20	−0.834763	−0.417381	−	0.908731i	$-0.637052\pi$
$491$	−1.27255e20	−0.834763	−0.417381	+	0.908731i	$0.637052\pi$
$492$	0	0
$493$	−3.38434e19	−0.215338
$494$	0	0
$495$	4.34149e19	0.267978
$496$	0	0
$497$	2.35702e18	0.0141153
$498$	0	0
$499$	−1.94901e19	−0.113256	−0.0566279	−	0.998395i	$-0.518035\pi$
$499$	−1.94901e19	−0.113256	−0.0566279	+	0.998395i	$0.518035\pi$
$500$	0	0
$501$	1.07168e20	0.604343
$502$	0	0
$503$	−2.31821e20	−1.26880	−0.634399	−	0.773006i	$-0.718752\pi$
$503$	−2.31821e20	−1.26880	−0.634399	+	0.773006i	$0.718752\pi$
$504$	0	0
$505$	1.25360e20	0.665997
$506$	0	0
$507$	1.40466e20	0.724458
$508$	0	0
$509$	3.51035e19	0.175779	0.0878894	−	0.996130i	$-0.471988\pi$
$509$	3.51035e19	0.175779	0.0878894	+	0.996130i	$0.471988\pi$
$510$	0	0
$511$	7.84299e19	0.381351
$512$	0	0
$513$	−5.32344e18	−0.0251369
$514$	0	0
$515$	−1.89501e20	−0.869075
$516$	0	0
$517$	3.56250e19	0.158700
$518$	0	0
$519$	−1.52585e20	−0.660323
$520$	0	0
$521$	9.98913e19	0.419996	0.209998	−	0.977702i	$-0.432654\pi$
$521$	9.98913e19	0.419996	0.209998	+	0.977702i	$0.432654\pi$
$522$	0	0
$523$	1.28542e20	0.525150	0.262575	−	0.964912i	$-0.415428\pi$
$523$	1.28542e20	0.525150	0.262575	+	0.964912i	$0.415428\pi$
$524$	0	0
$525$	4.21016e19	0.167149
$526$	0	0
$527$	−2.13636e19	−0.0824318
$528$	0	0
$529$	3.27971e20	1.23003
$530$	0	0
$531$	−6.64410e19	−0.242230
$532$	0	0
$533$	−5.00035e20	−1.77234
$534$	0	0
$535$	−2.14036e20	−0.737620
$536$	0	0
$537$	1.53791e20	0.515375
$538$	0	0
$539$	−7.28461e19	−0.237405
$540$	0	0
$541$	4.42226e20	1.40173	0.700866	−	0.713293i	$-0.252797\pi$
$541$	4.42226e20	1.40173	0.700866	+	0.713293i	$0.252797\pi$
$542$	0	0
$543$	−1.44369e20	−0.445118
$544$	0	0
$545$	7.25285e19	0.217538
$546$	0	0
$547$	1.92075e20	0.560489	0.280244	−	0.959929i	$-0.409584\pi$
$547$	1.92075e20	0.560489	0.280244	+	0.959929i	$0.409584\pi$
$548$	0	0
$549$	6.24899e19	0.177426
$550$	0	0
$551$	4.92313e19	0.136021
$552$	0	0
$553$	1.83645e20	0.493789
$554$	0	0
$555$	−1.55595e20	−0.407192
$556$	0	0
$557$	−2.96582e20	−0.755494	−0.377747	−	0.925909i	$-0.623301\pi$
$557$	−2.96582e20	−0.755494	−0.377747	+	0.925909i	$0.623301\pi$
$558$	0	0
$559$	−4.01814e20	−0.996407
$560$	0	0
$561$	−8.21790e19	−0.198399
$562$	0	0
$563$	−1.50268e20	−0.353226	−0.176613	−	0.984280i	$-0.556514\pi$
$563$	−1.50268e20	−0.353226	−0.176613	+	0.984280i	$0.556514\pi$
$564$	0	0
$565$	2.34243e20	0.536171
$566$	0	0
$567$	−1.88400e19	−0.0419961
$568$	0	0
$569$	7.13161e20	1.54827	0.774133	−	0.633023i	$-0.218186\pi$
$569$	7.13161e20	1.54827	0.774133	+	0.633023i	$0.218186\pi$
$570$	0	0
$571$	−3.68113e20	−0.778414	−0.389207	−	0.921150i	$-0.627251\pi$
$571$	−3.68113e20	−0.778414	−0.389207	+	0.921150i	$0.627251\pi$
$572$	0	0
$573$	2.99296e20	0.616511
$574$	0	0
$575$	−5.70005e20	−1.14385
$576$	0	0
$577$	5.08725e20	0.994638	0.497319	−	0.867568i	$-0.334318\pi$
$577$	5.08725e20	0.994638	0.497319	+	0.867568i	$0.334318\pi$
$578$	0	0
$579$	3.44899e20	0.657057
$580$	0	0
$581$	−4.53422e18	−0.00841747
$582$	0	0
$583$	−3.14665e20	−0.569291
$584$	0	0
$585$	1.37320e20	0.242139
$586$	0	0
$587$	1.57577e20	0.270836	0.135418	−	0.990789i	$-0.456762\pi$
$587$	1.57577e20	0.270836	0.135418	+	0.990789i	$0.456762\pi$
$588$	0	0
$589$	3.10772e19	0.0520688
$590$	0	0
$591$	6.68438e20	1.09183
$592$	0	0
$593$	−2.97555e20	−0.473868	−0.236934	−	0.971526i	$-0.576142\pi$
$593$	−2.97555e20	−0.473868	−0.236934	+	0.971526i	$0.576142\pi$
$594$	0	0
$595$	2.43485e19	0.0378089
$596$	0	0
$597$	5.26610e20	0.797410
$598$	0	0
$599$	−3.45618e19	−0.0510382	−0.0255191	−	0.999674i	$-0.508124\pi$
$599$	−3.45618e19	−0.0510382	−0.0255191	+	0.999674i	$0.508124\pi$
$600$	0	0
$601$	−4.12621e20	−0.594283	−0.297141	−	0.954834i	$-0.596033\pi$
$601$	−4.12621e20	−0.594283	−0.297141	+	0.954834i	$0.596033\pi$
$602$	0	0
$603$	1.32372e20	0.185958
$604$	0	0
$605$	−6.21915e20	−0.852247
$606$	0	0
$607$	3.97875e20	0.531902	0.265951	−	0.963987i	$-0.414314\pi$
$607$	3.97875e20	0.531902	0.265951	+	0.963987i	$0.414314\pi$
$608$	0	0
$609$	1.74233e20	0.227248
$610$	0	0
$611$	1.12681e20	0.143397
$612$	0	0
$613$	1.06236e21	1.31922	0.659612	−	0.751606i	$-0.270721\pi$
$613$	1.06236e21	1.31922	0.659612	+	0.751606i	$0.270721\pi$
$614$	0	0
$615$	−2.72037e20	−0.329659
$616$	0	0
$617$	1.27309e21	1.50563	0.752817	−	0.658230i	$-0.228694\pi$
$617$	1.27309e21	1.50563	0.752817	+	0.658230i	$0.228694\pi$
$618$	0	0
$619$	−3.75683e20	−0.433652	−0.216826	−	0.976210i	$-0.569570\pi$
$619$	−3.75683e20	−0.433652	−0.216826	+	0.976210i	$0.569570\pi$
$620$	0	0
$621$	2.55071e20	0.287391
$622$	0	0
$623$	−2.07110e19	−0.0227793
$624$	0	0
$625$	3.28468e20	0.352690
$626$	0	0
$627$	1.19544e20	0.125321
$628$	0	0
$629$	2.94523e20	0.301466
$630$	0	0
$631$	1.78227e21	1.78137	0.890685	−	0.454621i	$-0.150225\pi$
$631$	1.78227e21	1.78137	0.890685	+	0.454621i	$0.150225\pi$
$632$	0	0
$633$	−7.86250e20	−0.767419
$634$	0	0
$635$	−3.76518e20	−0.358908
$636$	0	0
$637$	−2.30410e20	−0.214514
$638$	0	0
$639$	−1.36891e19	−0.0124486
$640$	0	0
$641$	1.92517e21	1.71015	0.855074	−	0.518506i	$-0.173512\pi$
$641$	1.92517e21	1.71015	0.855074	+	0.518506i	$0.173512\pi$
$642$	0	0
$643$	−5.14683e20	−0.446640	−0.223320	−	0.974745i	$-0.571689\pi$
$643$	−5.14683e20	−0.446640	−0.223320	+	0.974745i	$0.571689\pi$
$644$	0	0
$645$	−2.18601e20	−0.185334
$646$	0	0
$647$	−1.34408e21	−1.11338	−0.556690	−	0.830720i	$-0.687929\pi$
$647$	−1.34408e21	−1.11338	−0.556690	+	0.830720i	$0.687929\pi$
$648$	0	0
$649$	1.49201e21	1.20764
$650$	0	0
$651$	1.09984e20	0.0869909
$652$	0	0
$653$	9.58468e20	0.740847	0.370424	−	0.928863i	$-0.379213\pi$
$653$	9.58468e20	0.740847	0.370424	+	0.928863i	$0.379213\pi$
$654$	0	0
$655$	−3.40445e20	−0.257180
$656$	0	0
$657$	−4.55505e20	−0.336320
$658$	0	0
$659$	1.71049e21	1.23446	0.617232	−	0.786781i	$-0.288254\pi$
$659$	1.71049e21	1.23446	0.617232	+	0.786781i	$0.288254\pi$
$660$	0	0
$661$	2.20067e21	1.55255	0.776274	−	0.630396i	$-0.217107\pi$
$661$	2.20067e21	1.55255	0.776274	+	0.630396i	$0.217107\pi$
$662$	0	0
$663$	−2.59930e20	−0.179269
$664$	0	0
$665$	−3.54193e19	−0.0238824
$666$	0	0
$667$	−2.35890e21	−1.55513
$668$	0	0
$669$	7.21042e20	0.464798
$670$	0	0
$671$	−1.40329e21	−0.884560
$672$	0	0
$673$	−5.84133e20	−0.360080	−0.180040	−	0.983659i	$-0.557623\pi$
$673$	−5.84133e20	−0.360080	−0.180040	+	0.983659i	$0.557623\pi$
$674$	0	0
$675$	−2.44518e20	−0.147412
$676$	0	0
$677$	2.41595e21	1.42453	0.712267	−	0.701908i	$-0.247668\pi$
$677$	2.41595e21	1.42453	0.712267	+	0.701908i	$0.247668\pi$
$678$	0	0
$679$	−9.84623e20	−0.567867
$680$	0	0
$681$	−1.92030e21	−1.08334
$682$	0	0
$683$	−3.10855e21	−1.71555	−0.857774	−	0.514027i	$-0.828153\pi$
$683$	−3.10855e21	−1.71555	−0.857774	+	0.514027i	$0.828153\pi$
$684$	0	0
$685$	−5.47664e20	−0.295689
$686$	0	0
$687$	2.06303e20	0.108976
$688$	0	0
$689$	−9.95276e20	−0.514398
$690$	0	0
$691$	−3.11559e21	−1.57563	−0.787816	−	0.615910i	$-0.788788\pi$
$691$	−3.11559e21	−1.57563	−0.787816	+	0.615910i	$0.788788\pi$
$692$	0	0
$693$	4.23075e20	0.209372
$694$	0	0
$695$	8.66246e20	0.419523
$696$	0	0
$697$	5.14932e20	0.244064
$698$	0	0
$699$	3.04877e20	0.141432
$700$	0	0
$701$	7.85751e20	0.356779	0.178390	−	0.983960i	$-0.442911\pi$
$701$	7.85751e20	0.356779	0.178390	+	0.983960i	$0.442911\pi$
$702$	0	0
$703$	−4.28437e20	−0.190424
$704$	0	0
$705$	6.13024e19	0.0266722
$706$	0	0
$707$	1.22162e21	0.520344
$708$	0	0
$709$	2.93473e21	1.22383	0.611916	−	0.790923i	$-0.290399\pi$
$709$	2.93473e21	1.22383	0.611916	+	0.790923i	$0.290399\pi$
$710$	0	0
$711$	−1.06657e21	−0.435481
$712$	0	0
$713$	−1.48905e21	−0.595304
$714$	0	0
$715$	−3.08368e21	−1.20719
$716$	0	0
$717$	−2.41432e21	−0.925551
$718$	0	0
$719$	9.59815e19	0.0360347	0.0180174	−	0.999838i	$-0.494265\pi$
$719$	9.59815e19	0.0360347	0.0180174	+	0.999838i	$0.494265\pi$
$720$	0	0
$721$	−1.84667e21	−0.679010
$722$	0	0
$723$	1.99702e21	0.719191
$724$	0	0
$725$	2.26131e21	0.797672
$726$	0	0
$727$	−1.25680e21	−0.434267	−0.217134	−	0.976142i	$-0.569671\pi$
$727$	−1.25680e21	−0.434267	−0.217134	+	0.976142i	$0.569671\pi$
$728$	0	0
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	4.13785e20	0.137213
$732$	0	0
$733$	1.91822e20	0.0623187	0.0311593	−	0.999514i	$-0.490080\pi$
$733$	1.91822e20	0.0623187	0.0311593	+	0.999514i	$0.490080\pi$
$734$	0	0
$735$	−1.25351e20	−0.0399001
$736$	0	0
$737$	−2.97257e21	−0.927098
$738$	0	0
$739$	−4.77946e21	−1.46065	−0.730324	−	0.683100i	$-0.760631\pi$
$739$	−4.77946e21	−1.46065	−0.730324	+	0.683100i	$0.760631\pi$
$740$	0	0
$741$	3.78115e20	0.113237
$742$	0	0
$743$	−6.09817e21	−1.78971	−0.894856	−	0.446354i	$-0.852722\pi$
$743$	−6.09817e21	−1.78971	−0.894856	+	0.446354i	$0.852722\pi$
$744$	0	0
$745$	−3.64154e20	−0.104740
$746$	0	0
$747$	2.63338e19	0.00742351
$748$	0	0
$749$	−2.08576e21	−0.576304
$750$	0	0
$751$	4.22973e21	1.14555	0.572774	−	0.819713i	$-0.305867\pi$
$751$	4.22973e21	1.14555	0.572774	+	0.819713i	$0.305867\pi$
$752$	0	0
$753$	−1.12125e20	−0.0297673
$754$	0	0
$755$	7.52236e20	0.195773
$756$	0	0
$757$	−2.26090e21	−0.576848	−0.288424	−	0.957503i	$-0.593131\pi$
$757$	−2.26090e21	−0.576848	−0.288424	+	0.957503i	$0.593131\pi$
$758$	0	0
$759$	−5.72792e21	−1.43279
$760$	0	0
$761$	2.98405e21	0.731848	0.365924	−	0.930645i	$-0.380753\pi$
$761$	2.98405e21	0.731848	0.365924	+	0.930645i	$0.380753\pi$
$762$	0	0
$763$	7.06786e20	0.169963
$764$	0	0
$765$	−1.41411e20	−0.0333443
$766$	0	0
$767$	4.71919e21	1.09119
$768$	0	0
$769$	−7.07550e21	−1.60439	−0.802194	−	0.597063i	$-0.796334\pi$
$769$	−7.07550e21	−1.60439	−0.802194	+	0.597063i	$0.796334\pi$
$770$	0	0
$771$	−2.90746e21	−0.646555
$772$	0	0
$773$	3.32593e21	0.725383	0.362691	−	0.931909i	$-0.381858\pi$
$773$	3.32593e21	0.725383	0.362691	+	0.931909i	$0.381858\pi$
$774$	0	0
$775$	1.42745e21	0.305350
$776$	0	0
$777$	−1.51627e21	−0.318140
$778$	0	0
$779$	−7.49062e20	−0.154165
$780$	0	0
$781$	3.07405e20	0.0620624
$782$	0	0
$783$	−1.01191e21	−0.200414
$784$	0	0
$785$	−2.69327e21	−0.523308
$786$	0	0
$787$	6.74789e20	0.128635	0.0643173	−	0.997930i	$-0.479513\pi$
$787$	6.74789e20	0.128635	0.0643173	+	0.997930i	$0.479513\pi$
$788$	0	0
$789$	−2.34014e20	−0.0437687
$790$	0	0
$791$	2.28268e21	0.418911
$792$	0	0
$793$	−4.43855e21	−0.799268
$794$	0	0
$795$	−5.41465e20	−0.0956791
$796$	0	0
$797$	8.61553e21	1.49398	0.746990	−	0.664835i	$-0.231498\pi$
$797$	8.61553e21	1.49398	0.746990	+	0.664835i	$0.231498\pi$
$798$	0	0
$799$	−1.16038e20	−0.0197469
$800$	0	0
$801$	1.20285e20	0.0200895
$802$	0	0
$803$	1.02289e22	1.67673
$804$	0	0
$805$	1.69710e21	0.273048
$806$	0	0
$807$	−3.53251e21	−0.557869
$808$	0	0
$809$	1.08419e22	1.68070	0.840350	−	0.542045i	$-0.182350\pi$
$809$	1.08419e22	1.68070	0.840350	+	0.542045i	$0.182350\pi$
$810$	0	0
$811$	−3.45917e21	−0.526399	−0.263200	−	0.964741i	$-0.584778\pi$
$811$	−3.45917e21	−0.526399	−0.263200	+	0.964741i	$0.584778\pi$
$812$	0	0
$813$	−9.25896e20	−0.138319
$814$	0	0
$815$	−2.50359e21	−0.367181
$816$	0	0
$817$	−6.01925e20	−0.0866717
$818$	0	0
$819$	1.33817e21	0.189184
$820$	0	0
$821$	−2.74042e21	−0.380403	−0.190201	−	0.981745i	$-0.560914\pi$
$821$	−2.74042e21	−0.380403	−0.190201	+	0.981745i	$0.560914\pi$
$822$	0	0
$823$	1.15599e22	1.57564	0.787820	−	0.615906i	$-0.211210\pi$
$823$	1.15599e22	1.57564	0.787820	+	0.615906i	$0.211210\pi$
$824$	0	0
$825$	5.49094e21	0.734923
$826$	0	0
$827$	−2.83207e21	−0.372232	−0.186116	−	0.982528i	$-0.559590\pi$
$827$	−2.83207e21	−0.372232	−0.186116	+	0.982528i	$0.559590\pi$
$828$	0	0
$829$	−8.42383e21	−1.08730	−0.543651	−	0.839311i	$-0.682959\pi$
$829$	−8.42383e21	−1.08730	−0.543651	+	0.839311i	$0.682959\pi$
$830$	0	0
$831$	−8.67396e21	−1.09954
$832$	0	0
$833$	2.37274e20	0.0295402
$834$	0	0
$835$	−4.14119e21	−0.506379
$836$	0	0
$837$	−6.38767e20	−0.0767187
$838$	0	0
$839$	−4.06996e21	−0.480149	−0.240074	−	0.970754i	$-0.577172\pi$
$839$	−4.06996e21	−0.480149	−0.240074	+	0.970754i	$0.577172\pi$
$840$	0	0
$841$	7.28980e20	0.0844784
$842$	0	0
$843$	2.87843e21	0.327679
$844$	0	0
$845$	−5.42789e21	−0.607024
$846$	0	0
$847$	−6.06052e21	−0.665862
$848$	0	0
$849$	7.54638e21	0.814575
$850$	0	0
$851$	2.05284e22	2.17713
$852$	0	0
$853$	4.96294e21	0.517156	0.258578	−	0.965990i	$-0.416746\pi$
$853$	4.96294e21	0.517156	0.258578	+	0.965990i	$0.416746\pi$
$854$	0	0
$855$	2.05708e20	0.0210623
$856$	0	0
$857$	5.65877e21	0.569332	0.284666	−	0.958627i	$-0.408117\pi$
$857$	5.65877e21	0.569332	0.284666	+	0.958627i	$0.408117\pi$
$858$	0	0
$859$	−1.10602e22	−1.09349	−0.546743	−	0.837300i	$-0.684133\pi$
$859$	−1.10602e22	−1.09349	−0.546743	+	0.837300i	$0.684133\pi$
$860$	0	0
$861$	−2.65098e21	−0.257563
$862$	0	0
$863$	1.19178e22	1.13793	0.568965	−	0.822362i	$-0.307344\pi$
$863$	1.19178e22	1.13793	0.568965	+	0.822362i	$0.307344\pi$
$864$	0	0
$865$	5.89618e21	0.553286
$866$	0	0
$867$	−5.99245e21	−0.552664
$868$	0	0
$869$	2.39512e22	2.17109
$870$	0	0
$871$	−9.40215e21	−0.837705
$872$	0	0
$873$	5.71849e21	0.500812
$874$	0	0
$875$	−3.75083e21	−0.322900
$876$	0	0
$877$	5.08164e21	0.430038	0.215019	−	0.976610i	$-0.431019\pi$
$877$	5.08164e21	0.430038	0.215019	+	0.976610i	$0.431019\pi$
$878$	0	0
$879$	9.55369e21	0.794794
$880$	0	0
$881$	−4.57256e21	−0.373973	−0.186986	−	0.982363i	$-0.559872\pi$
$881$	−4.57256e21	−0.373973	−0.186986	+	0.982363i	$0.559872\pi$
$882$	0	0
$883$	1.59983e22	1.28638	0.643188	−	0.765708i	$-0.277611\pi$
$883$	1.59983e22	1.28638	0.643188	+	0.765708i	$0.277611\pi$
$884$	0	0
$885$	2.56741e21	0.202965
$886$	0	0
$887$	4.60305e21	0.357782	0.178891	−	0.983869i	$-0.442749\pi$
$887$	4.60305e21	0.357782	0.178891	+	0.983869i	$0.442749\pi$
$888$	0	0
$889$	−3.66915e21	−0.280415
$890$	0	0
$891$	−2.45713e21	−0.184649
$892$	0	0
$893$	1.68798e20	0.0124733
$894$	0	0
$895$	−5.94277e21	−0.431834
$896$	0	0
$897$	−1.81172e22	−1.29464
$898$	0	0
$899$	5.90733e21	0.415139
$900$	0	0
$901$	1.02493e21	0.0708364
$902$	0	0
$903$	−2.13025e21	−0.144802
$904$	0	0
$905$	5.57870e21	0.372965
$906$	0	0
$907$	2.53765e22	1.66870	0.834348	−	0.551238i	$-0.185844\pi$
$907$	2.53765e22	1.66870	0.834348	+	0.551238i	$0.185844\pi$
$908$	0	0
$909$	−7.09492e21	−0.458901
$910$	0	0
$911$	−1.48701e22	−0.946076	−0.473038	−	0.881042i	$-0.656843\pi$
$911$	−1.48701e22	−0.946076	−0.473038	+	0.881042i	$0.656843\pi$
$912$	0	0
$913$	−5.91357e20	−0.0370100
$914$	0	0
$915$	−2.41473e21	−0.148665
$916$	0	0
$917$	−3.31761e21	−0.200935
$918$	0	0
$919$	−1.48006e21	−0.0881885	−0.0440942	−	0.999027i	$-0.514040\pi$
$919$	−1.48006e21	−0.0881885	−0.0440942	+	0.999027i	$0.514040\pi$
$920$	0	0
$921$	−1.62400e22	−0.952002
$922$	0	0
$923$	9.72314e20	0.0560782
$924$	0	0
$925$	−1.96791e22	−1.11671
$926$	0	0
$927$	1.07251e22	0.598830
$928$	0	0
$929$	6.53310e21	0.358923	0.179462	−	0.983765i	$-0.442564\pi$
$929$	6.53310e21	0.358923	0.179462	+	0.983765i	$0.442564\pi$
$930$	0	0
$931$	−3.45159e20	−0.0186593
$932$	0	0
$933$	−7.85664e21	−0.417950
$934$	0	0
$935$	3.17555e21	0.166239
$936$	0	0
$937$	2.51992e22	1.29820	0.649098	−	0.760705i	$-0.275147\pi$
$937$	2.51992e22	1.29820	0.649098	+	0.760705i	$0.275147\pi$
$938$	0	0
$939$	7.77574e21	0.394230
$940$	0	0
$941$	7.49410e21	0.373936	0.186968	−	0.982366i	$-0.440134\pi$
$941$	7.49410e21	0.373936	0.186968	+	0.982366i	$0.440134\pi$
$942$	0	0
$943$	3.58911e22	1.76258
$944$	0	0
$945$	7.28014e20	0.0351886
$946$	0	0
$947$	2.27305e22	1.08139	0.540696	−	0.841218i	$-0.318161\pi$
$947$	2.27305e22	1.08139	0.540696	+	0.841218i	$0.318161\pi$
$948$	0	0
$949$	3.23537e22	1.51505
$950$	0	0
$951$	−1.72744e22	−0.796248
$952$	0	0
$953$	−2.96702e22	−1.34624	−0.673121	−	0.739532i	$-0.735047\pi$
$953$	−2.96702e22	−1.34624	−0.673121	+	0.739532i	$0.735047\pi$
$954$	0	0
$955$	−1.15654e22	−0.516576
$956$	0	0
$957$	2.27236e22	0.999168
$958$	0	0
$959$	−5.33695e21	−0.231022
$960$	0	0
$961$	−1.97363e22	−0.841084
$962$	0	0
$963$	1.21137e22	0.508252
$964$	0	0
$965$	−1.33276e22	−0.550549
$966$	0	0
$967$	−1.66711e22	−0.678057	−0.339028	−	0.940776i	$-0.610098\pi$
$967$	−1.66711e22	−0.678057	−0.339028	+	0.940776i	$0.610098\pi$
$968$	0	0
$969$	−3.89379e20	−0.0155935
$970$	0	0
$971$	1.23464e22	0.486849	0.243425	−	0.969920i	$-0.421729\pi$
$971$	1.23464e22	0.486849	0.243425	+	0.969920i	$0.421729\pi$
$972$	0	0
$973$	8.44151e21	0.327774
$974$	0	0
$975$	1.73677e22	0.664060
$976$	0	0
$977$	4.41539e21	0.166249	0.0831246	−	0.996539i	$-0.473510\pi$
$977$	4.41539e21	0.166249	0.0831246	+	0.996539i	$0.473510\pi$
$978$	0	0
$979$	−2.70115e21	−0.100156
$980$	0	0
$981$	−4.10487e21	−0.149893
$982$	0	0
$983$	−4.01082e22	−1.44239	−0.721194	−	0.692733i	$-0.756406\pi$
$983$	−4.01082e22	−1.44239	−0.721194	+	0.692733i	$0.756406\pi$
$984$	0	0
$985$	−2.58297e22	−0.914846
$986$	0	0
$987$	5.97388e20	0.0208390
$988$	0	0
$989$	2.88410e22	0.990920
$990$	0	0
$991$	1.40735e22	0.476266	0.238133	−	0.971233i	$-0.423465\pi$
$991$	1.40735e22	0.476266	0.238133	+	0.971233i	$0.423465\pi$
$992$	0	0
$993$	6.60180e21	0.220061
$994$	0	0
$995$	−2.03492e22	−0.668151
$996$	0	0
$997$	−3.67223e22	−1.18772	−0.593862	−	0.804567i	$-0.702398\pi$
$997$	−3.67223e22	−1.18772	−0.593862	+	0.804567i	$0.702398\pi$
$998$	0	0
$999$	8.80617e21	0.280573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.16.a.d.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.16.a.d.1.1	✓	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:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)

\(f(q)\)	\(=\)	\(q+2187.00 q^{3} -84509.8 q^{5} -823543. q^{7} +4.78297e6 q^{9} +O(q^{10})\)	\(q+2187.00 q^{3} -84509.8 q^{5} -823543. q^{7} +4.78297e6 q^{9} -1.07407e8 q^{11} -3.39726e8 q^{13} -1.84823e8 q^{15} +3.49847e8 q^{17} -5.08916e8 q^{19} -1.80109e9 q^{21} +2.43845e10 q^{23} -2.33757e10 q^{25} +1.04604e10 q^{27} -9.67376e10 q^{29} -6.10655e10 q^{31} -2.34900e11 q^{33} +6.95975e10 q^{35} +8.41861e11 q^{37} -7.42981e11 q^{39} +1.47188e12 q^{41} +1.18276e12 q^{43} -4.04208e11 q^{45} -3.31681e11 q^{47} +6.78223e11 q^{49} +7.65115e11 q^{51} +2.92964e12 q^{53} +9.07697e12 q^{55} -1.11300e12 q^{57} -1.38912e13 q^{59} +1.30651e13 q^{61} -3.93898e12 q^{63} +2.87102e13 q^{65} +2.76757e13 q^{67} +5.33290e13 q^{69} -2.86205e12 q^{71} -9.52347e13 q^{73} -5.11226e13 q^{75} +8.84545e13 q^{77} -2.22994e14 q^{79} +2.28768e13 q^{81} +5.50574e12 q^{83} -2.95655e13 q^{85} -2.11565e14 q^{87} +2.51487e13 q^{89} +2.79779e14 q^{91} -1.33550e14 q^{93} +4.30084e13 q^{95} +1.19559e15 q^{97} -5.13726e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8748 q^{3} + 275436 q^{5} - 3294172 q^{7} + 19131876 q^{9}+O(q^{10}) \) 4 * q + 8748 * q^3 + 275436 * q^5 - 3294172 * q^7 + 19131876 * q^9	\( 4 q + 8748 q^{3} + 275436 q^{5} - 3294172 q^{7} + 19131876 q^{9} - 61540668 q^{11} + 369006456 q^{13} + 602378532 q^{15} - 1458569484 q^{17} - 2691246184 q^{19} - 7204354164 q^{21} + 18028282212 q^{23} + 8448695516 q^{25} + 41841412812 q^{27} - 14387019264 q^{29} + 271402618248 q^{31} - 134589440916 q^{33} - 226833389748 q^{35} + 1719450823976 q^{37} + 807017119272 q^{39} + 2784989291868 q^{41} + 1897932676640 q^{43} + 1317401849484 q^{45} + 1872077464008 q^{47} + 2712892291396 q^{49} - 3189891461508 q^{51} + 4740642191736 q^{53} - 21671359654504 q^{55} - 5885755404408 q^{57} + 13984487218296 q^{59} + 32810552663568 q^{61} - 15755922556668 q^{63} + 183566971857672 q^{65} + 4158516498856 q^{67} + 39427853197644 q^{69} + 70250046828444 q^{71} + 27064251481456 q^{73} + 18477297093492 q^{75} + 50681386346724 q^{77} - 238381767283992 q^{79} + 91507169819844 q^{81} - 233193205883808 q^{83} - 893307076687696 q^{85} - 31464411130368 q^{87} + 316154161239228 q^{89} - 303892683793608 q^{91} + 593557526108376 q^{93} - 11\!\cdots\!76 q^{95}+ \cdots - 294347107283292 q^{99}+O(q^{100}) \) 4 * q + 8748 * q^3 + 275436 * q^5 - 3294172 * q^7 + 19131876 * q^9 - 61540668 * q^11 + 369006456 * q^13 + 602378532 * q^15 - 1458569484 * q^17 - 2691246184 * q^19 - 7204354164 * q^21 + 18028282212 * q^23 + 8448695516 * q^25 + 41841412812 * q^27 - 14387019264 * q^29 + 271402618248 * q^31 - 134589440916 * q^33 - 226833389748 * q^35 + 1719450823976 * q^37 + 807017119272 * q^39 + 2784989291868 * q^41 + 1897932676640 * q^43 + 1317401849484 * q^45 + 1872077464008 * q^47 + 2712892291396 * q^49 - 3189891461508 * q^51 + 4740642191736 * q^53 - 21671359654504 * q^55 - 5885755404408 * q^57 + 13984487218296 * q^59 + 32810552663568 * q^61 - 15755922556668 * q^63 + 183566971857672 * q^65 + 4158516498856 * q^67 + 39427853197644 * q^69 + 70250046828444 * q^71 + 27064251481456 * q^73 + 18477297093492 * q^75 + 50681386346724 * q^77 - 238381767283992 * q^79 + 91507169819844 * q^81 - 233193205883808 * q^83 - 893307076687696 * q^85 - 31464411130368 * q^87 + 316154161239228 * q^89 - 303892683793608 * q^91 + 593557526108376 * q^93 - 1197067113655776 * q^95 + 529302182472448 * q^97 - 294347107283292 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more