LMFDB - Embedded newform 84.14.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,14,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, 14, names="a")

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

chi := DirichletCharacter("84.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 14 $
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:	$90.0739803196$
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} - x^{3} - 82385347x^{2} + 61134548293x + 1123635148704894 $ x^4 - x^3 - 82385347x^2 + 61134548293x + 1123635148704894
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{4}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$7252.77$ 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)

$f(q)$	$=$	$q+729.000 q^{3} +48334.6 q^{5} +117649. q^{7} +531441. q^{9} +O(q^{10})$	$q+729.000 q^{3} +48334.6 q^{5} +117649. q^{7} +531441. q^{9} +6.92904e6 q^{11} +1.34436e7 q^{13} +3.52359e7 q^{15} -9.38033e7 q^{17} +9.64052e7 q^{19} +8.57661e7 q^{21} +1.14190e9 q^{23} +1.11553e9 q^{25} +3.87420e8 q^{27} -1.42154e9 q^{29} +3.63957e9 q^{31} +5.05127e9 q^{33} +5.68652e9 q^{35} -2.34867e10 q^{37} +9.80040e9 q^{39} +1.64301e10 q^{41} -3.52441e10 q^{43} +2.56870e10 q^{45} -8.17172e9 q^{47} +1.38413e10 q^{49} -6.83826e10 q^{51} +1.99934e10 q^{53} +3.34912e11 q^{55} +7.02794e10 q^{57} -8.83537e10 q^{59} -3.34130e11 q^{61} +6.25235e10 q^{63} +6.49792e11 q^{65} -2.95848e11 q^{67} +8.32443e11 q^{69} +6.46927e10 q^{71} -1.15752e12 q^{73} +8.13223e11 q^{75} +8.15194e11 q^{77} -1.38129e12 q^{79} +2.82430e11 q^{81} -2.82862e12 q^{83} -4.53395e12 q^{85} -1.03630e12 q^{87} +9.18125e12 q^{89} +1.58163e12 q^{91} +2.65325e12 q^{93} +4.65971e12 q^{95} +1.10479e13 q^{97} +3.68238e12 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2916 q^{3} + 19278 q^{5} + 470596 q^{7} + 2125764 q^{9}+O(q^{10}) $ 4 * q + 2916 * q^3 + 19278 * q^5 + 470596 * q^7 + 2125764 * q^9	$ 4 q + 2916 q^{3} + 19278 q^{5} + 470596 q^{7} + 2125764 q^{9} + 515970 q^{11} + 31470740 q^{13} + 14053662 q^{15} + 40485258 q^{17} + 257760344 q^{19} + 343064484 q^{21} - 492248178 q^{23} + 1141842832 q^{25} + 1549681956 q^{27} + 817801920 q^{29} + 131625608 q^{31} + 376142130 q^{33} + 2268037422 q^{35} + 38485873484 q^{37} + 22942169460 q^{39} + 42425041302 q^{41} + 49587853952 q^{43} + 10245119598 q^{45} - 15206667516 q^{47} + 55365148804 q^{49} + 29513753082 q^{51} + 189707500548 q^{53} - 2269688904 q^{55} + 187907290776 q^{57} + 25327021356 q^{59} + 57555532688 q^{61} + 250094008836 q^{63} - 1243945073628 q^{65} + 459346595468 q^{67} - 358848921762 q^{69} + 510335887698 q^{71} - 957870191236 q^{73} + 832403424528 q^{75} + 60703354530 q^{77} + 21389706452 q^{79} + 1129718145924 q^{81} + 5770833822000 q^{83} + 5429978357868 q^{85} + 596177599680 q^{87} + 7593089475942 q^{89} + 3702501090260 q^{91} + 95955068232 q^{93} + 17821341989208 q^{95} + 5330443230644 q^{97} + 274207612770 q^{99}+O(q^{100}) $ 4 * q + 2916 * q^3 + 19278 * q^5 + 470596 * q^7 + 2125764 * q^9 + 515970 * q^11 + 31470740 * q^13 + 14053662 * q^15 + 40485258 * q^17 + 257760344 * q^19 + 343064484 * q^21 - 492248178 * q^23 + 1141842832 * q^25 + 1549681956 * q^27 + 817801920 * q^29 + 131625608 * q^31 + 376142130 * q^33 + 2268037422 * q^35 + 38485873484 * q^37 + 22942169460 * q^39 + 42425041302 * q^41 + 49587853952 * q^43 + 10245119598 * q^45 - 15206667516 * q^47 + 55365148804 * q^49 + 29513753082 * q^51 + 189707500548 * q^53 - 2269688904 * q^55 + 187907290776 * q^57 + 25327021356 * q^59 + 57555532688 * q^61 + 250094008836 * q^63 - 1243945073628 * q^65 + 459346595468 * q^67 - 358848921762 * q^69 + 510335887698 * q^71 - 957870191236 * q^73 + 832403424528 * q^75 + 60703354530 * q^77 + 21389706452 * q^79 + 1129718145924 * q^81 + 5770833822000 * q^83 + 5429978357868 * q^85 + 596177599680 * q^87 + 7593089475942 * q^89 + 3702501090260 * q^91 + 95955068232 * q^93 + 17821341989208 * q^95 + 5330443230644 * q^97 + 274207612770 * 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
$3$	729.000	0.577350
$3$	729.000	0.577350
$4$	0	0
$5$	48334.6	1.38342	0.691709	−	0.722177i	$-0.256858\pi$
$5$	48334.6	1.38342	0.691709	+	0.722177i	$0.256858\pi$
$6$	0	0
$7$	117649.	0.377964
$7$	117649.	0.377964
$8$	0	0
$9$	531441.	0.333333
$10$	0	0
$11$	6.92904e6	1.17929	0.589645	−	0.807663i	$-0.299268\pi$
$11$	6.92904e6	1.17929	0.589645	+	0.807663i	$0.299268\pi$
$12$	0	0
$13$	1.34436e7	0.772475	0.386238	−	0.922399i	$-0.373774\pi$
$13$	1.34436e7	0.772475	0.386238	+	0.922399i	$0.373774\pi$
$14$	0	0
$15$	3.52359e7	0.798716
$16$	0	0
$17$	−9.38033e7	−0.942541	−0.471270	−	0.881989i	$-0.656204\pi$
$17$	−9.38033e7	−0.942541	−0.471270	+	0.881989i	$0.656204\pi$
$18$	0	0
$19$	9.64052e7	0.470113	0.235056	−	0.971982i	$-0.424472\pi$
$19$	9.64052e7	0.470113	0.235056	+	0.971982i	$0.424472\pi$
$20$	0	0
$21$	8.57661e7	0.218218
$22$	0	0
$23$	1.14190e9	1.60841	0.804204	−	0.594353i	$-0.202592\pi$
$23$	1.14190e9	1.60841	0.804204	+	0.594353i	$0.202592\pi$
$24$	0	0
$25$	1.11553e9	0.913844
$26$	0	0
$27$	3.87420e8	0.192450
$28$	0	0
$29$	−1.42154e9	−0.443784	−0.221892	−	0.975071i	$-0.571223\pi$
$29$	−1.42154e9	−0.443784	−0.221892	+	0.975071i	$0.571223\pi$
$30$	0	0
$31$	3.63957e9	0.736545	0.368273	−	0.929718i	$-0.379949\pi$
$31$	3.63957e9	0.736545	0.368273	+	0.929718i	$0.379949\pi$
$32$	0	0
$33$	5.05127e9	0.680863
$34$	0	0
$35$	5.68652e9	0.522883
$36$	0	0
$37$	−2.34867e10	−1.50491	−0.752455	−	0.658643i	$-0.771131\pi$
$37$	−2.34867e10	−1.50491	−0.752455	+	0.658643i	$0.771131\pi$
$38$	0	0
$39$	9.80040e9	0.445989
$40$	0	0
$41$	1.64301e10	0.540189	0.270095	−	0.962834i	$-0.412945\pi$
$41$	1.64301e10	0.540189	0.270095	+	0.962834i	$0.412945\pi$
$42$	0	0
$43$	−3.52441e10	−0.850240	−0.425120	−	0.905137i	$-0.639768\pi$
$43$	−3.52441e10	−0.850240	−0.425120	+	0.905137i	$0.639768\pi$
$44$	0	0
$45$	2.56870e10	0.461139
$46$	0	0
$47$	−8.17172e9	−0.110580	−0.0552901	−	0.998470i	$-0.517608\pi$
$47$	−8.17172e9	−0.110580	−0.0552901	+	0.998470i	$0.517608\pi$
$48$	0	0
$49$	1.38413e10	0.142857
$50$	0	0
$51$	−6.83826e10	−0.544176
$52$	0	0
$53$	1.99934e10	0.123906	0.0619532	−	0.998079i	$-0.480267\pi$
$53$	1.99934e10	0.123906	0.0619532	+	0.998079i	$0.480267\pi$
$54$	0	0
$55$	3.34912e11	1.63145
$56$	0	0
$57$	7.02794e10	0.271420
$58$	0	0
$59$	−8.83537e10	−0.272701	−0.136351	−	0.990661i	$-0.543537\pi$
$59$	−8.83537e10	−0.272701	−0.136351	+	0.990661i	$0.543537\pi$
$60$	0	0
$61$	−3.34130e11	−0.830371	−0.415185	−	0.909737i	$-0.636283\pi$
$61$	−3.34130e11	−0.830371	−0.415185	+	0.909737i	$0.636283\pi$
$62$	0	0
$63$	6.25235e10	0.125988
$64$	0	0
$65$	6.49792e11	1.06866
$66$	0	0
$67$	−2.95848e11	−0.399560	−0.199780	−	0.979841i	$-0.564023\pi$
$67$	−2.95848e11	−0.399560	−0.199780	+	0.979841i	$0.564023\pi$
$68$	0	0
$69$	8.32443e11	0.928615
$70$	0	0
$71$	6.46927e10	0.0599344	0.0299672	−	0.999551i	$-0.490460\pi$
$71$	6.46927e10	0.0599344	0.0299672	+	0.999551i	$0.490460\pi$
$72$	0	0
$73$	−1.15752e12	−0.895221	−0.447611	−	0.894229i	$-0.647725\pi$
$73$	−1.15752e12	−0.895221	−0.447611	+	0.894229i	$0.647725\pi$
$74$	0	0
$75$	8.13223e11	0.527608
$76$	0	0
$77$	8.15194e11	0.445730
$78$	0	0
$79$	−1.38129e12	−0.639306	−0.319653	−	0.947535i	$-0.603566\pi$
$79$	−1.38129e12	−0.639306	−0.319653	+	0.947535i	$0.603566\pi$
$80$	0	0
$81$	2.82430e11	0.111111
$82$	0	0
$83$	−2.82862e12	−0.949658	−0.474829	−	0.880078i	$-0.657490\pi$
$83$	−2.82862e12	−0.949658	−0.474829	+	0.880078i	$0.657490\pi$
$84$	0	0
$85$	−4.53395e12	−1.30393
$86$	0	0
$87$	−1.03630e12	−0.256219
$88$	0	0
$89$	9.18125e12	1.95824	0.979122	−	0.203274i	$-0.0651583\pi$
$89$	9.18125e12	1.95824	0.979122	+	0.203274i	$0.0651583\pi$
$90$	0	0
$91$	1.58163e12	0.291968
$92$	0	0
$93$	2.65325e12	0.425245
$94$	0	0
$95$	4.65971e12	0.650362
$96$	0	0
$97$	1.10479e13	1.34667	0.673336	−	0.739336i	$-0.264861\pi$
$97$	1.10479e13	1.34667	0.673336	+	0.739336i	$0.264861\pi$
$98$	0	0
$99$	3.68238e12	0.393097
$100$	0	0
$101$	8.91736e12	0.835887	0.417943	−	0.908473i	$-0.362751\pi$
$101$	8.91736e12	0.835887	0.417943	+	0.908473i	$0.362751\pi$
$102$	0	0
$103$	−4.15578e12	−0.342934	−0.171467	−	0.985190i	$-0.554851\pi$
$103$	−4.15578e12	−0.342934	−0.171467	+	0.985190i	$0.554851\pi$
$104$	0	0
$105$	4.14547e12	0.301886
$106$	0	0
$107$	2.20476e13	1.42025	0.710127	−	0.704073i	$-0.248637\pi$
$107$	2.20476e13	1.42025	0.710127	+	0.704073i	$0.248637\pi$
$108$	0	0
$109$	2.55303e13	1.45809	0.729045	−	0.684466i	$-0.239965\pi$
$109$	2.55303e13	1.45809	0.729045	+	0.684466i	$0.239965\pi$
$110$	0	0
$111$	−1.71218e13	−0.868861
$112$	0	0
$113$	1.84055e12	0.0831646	0.0415823	−	0.999135i	$-0.486760\pi$
$113$	1.84055e12	0.0831646	0.0415823	+	0.999135i	$0.486760\pi$
$114$	0	0
$115$	5.51932e13	2.22510
$116$	0	0
$117$	7.14449e12	0.257492
$118$	0	0
$119$	−1.10359e13	−0.356247
$120$	0	0
$121$	1.34889e13	0.390724
$122$	0	0
$123$	1.19776e13	0.311879
$124$	0	0
$125$	−5.08340e12	−0.119190
$126$	0	0
$127$	2.32974e13	0.492700	0.246350	−	0.969181i	$-0.420769\pi$
$127$	2.32974e13	0.492700	0.246350	+	0.969181i	$0.420769\pi$
$128$	0	0
$129$	−2.56929e13	−0.490886
$130$	0	0
$131$	−9.69462e13	−1.67597	−0.837987	−	0.545690i	$-0.816268\pi$
$131$	−9.69462e13	−1.67597	−0.837987	+	0.545690i	$0.816268\pi$
$132$	0	0
$133$	1.13420e13	0.177686
$134$	0	0
$135$	1.87258e13	0.266239
$136$	0	0
$137$	6.18134e13	0.798729	0.399364	−	0.916792i	$-0.369231\pi$
$137$	6.18134e13	0.798729	0.399364	+	0.916792i	$0.369231\pi$
$138$	0	0
$139$	1.00307e14	1.17960	0.589798	−	0.807551i	$-0.299207\pi$
$139$	1.00307e14	1.17960	0.589798	+	0.807551i	$0.299207\pi$
$140$	0	0
$141$	−5.95719e12	−0.0638435
$142$	0	0
$143$	9.31514e13	0.910972
$144$	0	0
$145$	−6.87096e13	−0.613939
$146$	0	0
$147$	1.00903e13	0.0824786
$148$	0	0
$149$	8.17468e13	0.612012	0.306006	−	0.952030i	$-0.401007\pi$
$149$	8.17468e13	0.612012	0.306006	+	0.952030i	$0.401007\pi$
$150$	0	0
$151$	1.02755e12	0.00705430	0.00352715	−	0.999994i	$-0.498877\pi$
$151$	1.02755e12	0.00705430	0.00352715	+	0.999994i	$0.498877\pi$
$152$	0	0
$153$	−4.98509e13	−0.314180
$154$	0	0
$155$	1.75917e14	1.01895
$156$	0	0
$157$	2.12861e14	1.13435	0.567177	−	0.823596i	$-0.308035\pi$
$157$	2.12861e14	1.13435	0.567177	+	0.823596i	$0.308035\pi$
$158$	0	0
$159$	1.45752e13	0.0715374
$160$	0	0
$161$	1.34343e14	0.607921
$162$	0	0
$163$	2.08770e14	0.871864	0.435932	−	0.899980i	$-0.356419\pi$
$163$	2.08770e14	0.871864	0.435932	+	0.899980i	$0.356419\pi$
$164$	0	0
$165$	2.44151e14	0.941918
$166$	0	0
$167$	−1.63495e14	−0.583239	−0.291620	−	0.956534i	$-0.594194\pi$
$167$	−1.63495e14	−0.583239	−0.291620	+	0.956534i	$0.594194\pi$
$168$	0	0
$169$	−1.22144e14	−0.403282
$170$	0	0
$171$	5.12337e13	0.156704
$172$	0	0
$173$	−4.20048e14	−1.19124	−0.595620	−	0.803266i	$-0.703094\pi$
$173$	−4.20048e14	−1.19124	−0.595620	+	0.803266i	$0.703094\pi$
$174$	0	0
$175$	1.31241e14	0.345401
$176$	0	0
$177$	−6.44099e13	−0.157444
$178$	0	0
$179$	5.38324e14	1.22320	0.611602	−	0.791166i	$-0.290525\pi$
$179$	5.38324e14	1.22320	0.611602	+	0.791166i	$0.290525\pi$
$180$	0	0
$181$	−7.52470e14	−1.59066	−0.795332	−	0.606174i	$-0.792704\pi$
$181$	−7.52470e14	−1.59066	−0.795332	+	0.606174i	$0.792704\pi$
$182$	0	0
$183$	−2.43581e14	−0.479415
$184$	0	0
$185$	−1.13522e15	−2.08192
$186$	0	0
$187$	−6.49967e14	−1.11153
$188$	0	0
$189$	4.55796e13	0.0727393
$190$	0	0
$191$	1.14758e15	1.71028	0.855139	−	0.518399i	$-0.173472\pi$
$191$	1.14758e15	1.71028	0.855139	+	0.518399i	$0.173472\pi$
$192$	0	0
$193$	−7.50843e14	−1.04575	−0.522874	−	0.852410i	$-0.675140\pi$
$193$	−7.50843e14	−1.04575	−0.522874	+	0.852410i	$0.675140\pi$
$194$	0	0
$195$	4.73699e14	0.616989
$196$	0	0
$197$	−8.81450e14	−1.07440	−0.537201	−	0.843454i	$-0.680518\pi$
$197$	−8.81450e14	−1.07440	−0.537201	+	0.843454i	$0.680518\pi$
$198$	0	0
$199$	5.65884e14	0.645926	0.322963	−	0.946412i	$-0.395321\pi$
$199$	5.65884e14	0.645926	0.322963	+	0.946412i	$0.395321\pi$
$200$	0	0
$201$	−2.15673e14	−0.230686
$202$	0	0
$203$	−1.67243e14	−0.167735
$204$	0	0
$205$	7.94144e14	0.747308
$206$	0	0
$207$	6.06851e14	0.536136
$208$	0	0
$209$	6.67996e14	0.554399
$210$	0	0
$211$	−2.35085e15	−1.83395	−0.916977	−	0.398939i	$-0.869378\pi$
$211$	−2.35085e15	−1.83395	−0.916977	+	0.398939i	$0.869378\pi$
$212$	0	0
$213$	4.71610e13	0.0346032
$214$	0	0
$215$	−1.70351e15	−1.17624
$216$	0	0
$217$	4.28192e14	0.278388
$218$	0	0
$219$	−8.43833e14	−0.516856
$220$	0	0
$221$	−1.26106e15	−0.728089
$222$	0	0
$223$	−3.46713e15	−1.88794	−0.943972	−	0.330026i	$-0.892943\pi$
$223$	−3.46713e15	−1.88794	−0.943972	+	0.330026i	$0.892943\pi$
$224$	0	0
$225$	5.92839e14	0.304615
$226$	0	0
$227$	−1.07781e15	−0.522848	−0.261424	−	0.965224i	$-0.584192\pi$
$227$	−1.07781e15	−0.522848	−0.261424	+	0.965224i	$0.584192\pi$
$228$	0	0
$229$	1.28468e15	0.588659	0.294330	−	0.955704i	$-0.404904\pi$
$229$	1.28468e15	0.588659	0.294330	+	0.955704i	$0.404904\pi$
$230$	0	0
$231$	5.94277e14	0.257342
$232$	0	0
$233$	−3.78953e15	−1.55157	−0.775785	−	0.630997i	$-0.782646\pi$
$233$	−3.78953e15	−1.55157	−0.775785	+	0.630997i	$0.782646\pi$
$234$	0	0
$235$	−3.94977e14	−0.152979
$236$	0	0
$237$	−1.00696e15	−0.369103
$238$	0	0
$239$	−1.71974e15	−0.596866	−0.298433	−	0.954431i	$-0.596464\pi$
$239$	−1.71974e15	−0.596866	−0.298433	+	0.954431i	$0.596464\pi$
$240$	0	0
$241$	4.73619e15	1.55711	0.778553	−	0.627579i	$-0.215954\pi$
$241$	4.73619e15	1.55711	0.778553	+	0.627579i	$0.215954\pi$
$242$	0	0
$243$	2.05891e14	0.0641500
$244$	0	0
$245$	6.69013e14	0.197631
$246$	0	0
$247$	1.29604e15	0.363151
$248$	0	0
$249$	−2.06207e15	−0.548285
$250$	0	0
$251$	−5.02082e15	−1.26735	−0.633674	−	0.773600i	$-0.718454\pi$
$251$	−5.02082e15	−1.26735	−0.633674	+	0.773600i	$0.718454\pi$
$252$	0	0
$253$	7.91225e15	1.89678
$254$	0	0
$255$	−3.30525e15	−0.752823
$256$	0	0
$257$	−4.15758e14	−0.0900068	−0.0450034	−	0.998987i	$-0.514330\pi$
$257$	−4.15758e14	−0.0900068	−0.0450034	+	0.998987i	$0.514330\pi$
$258$	0	0
$259$	−2.76319e15	−0.568803
$260$	0	0
$261$	−7.55464e14	−0.147928
$262$	0	0
$263$	−6.37979e15	−1.18876	−0.594380	−	0.804184i	$-0.702602\pi$
$263$	−6.37979e15	−1.18876	−0.594380	+	0.804184i	$0.702602\pi$
$264$	0	0
$265$	9.66374e14	0.171414
$266$	0	0
$267$	6.69313e15	1.13059
$268$	0	0
$269$	−3.36332e15	−0.541225	−0.270613	−	0.962688i	$-0.587226\pi$
$269$	−3.36332e15	−0.541225	−0.270613	+	0.962688i	$0.587226\pi$
$270$	0	0
$271$	4.85985e15	0.745285	0.372642	−	0.927975i	$-0.378452\pi$
$271$	4.85985e15	0.745285	0.372642	+	0.927975i	$0.378452\pi$
$272$	0	0
$273$	1.15301e15	0.168568
$274$	0	0
$275$	7.72956e15	1.07769
$276$	0	0
$277$	5.09004e15	0.677022	0.338511	−	0.940962i	$-0.390077\pi$
$277$	5.09004e15	0.677022	0.338511	+	0.940962i	$0.390077\pi$
$278$	0	0
$279$	1.93422e15	0.245515
$280$	0	0
$281$	1.42900e16	1.73157	0.865785	−	0.500417i	$-0.166820\pi$
$281$	1.42900e16	1.73157	0.865785	+	0.500417i	$0.166820\pi$
$282$	0	0
$283$	−3.99299e15	−0.462047	−0.231024	−	0.972948i	$-0.574207\pi$
$283$	−3.99299e15	−0.462047	−0.231024	+	0.972948i	$0.574207\pi$
$284$	0	0
$285$	3.39693e15	0.375487
$286$	0	0
$287$	1.93299e15	0.204172
$288$	0	0
$289$	−1.10552e15	−0.111617
$290$	0	0
$291$	8.05389e15	0.777502
$292$	0	0
$293$	−1.81603e15	−0.167680	−0.0838402	−	0.996479i	$-0.526719\pi$
$293$	−1.81603e15	−0.167680	−0.0838402	+	0.996479i	$0.526719\pi$
$294$	0	0
$295$	−4.27054e15	−0.377259
$296$	0	0
$297$	2.68445e15	0.226954
$298$	0	0
$299$	1.53512e16	1.24246
$300$	0	0
$301$	−4.14643e15	−0.321360
$302$	0	0
$303$	6.50076e15	0.482599
$304$	0	0
$305$	−1.61501e16	−1.14875
$306$	0	0
$307$	−1.00761e14	−0.00686901	−0.00343451	−	0.999994i	$-0.501093\pi$
$307$	−1.00761e14	−0.00686901	−0.00343451	+	0.999994i	$0.501093\pi$
$308$	0	0
$309$	−3.02956e15	−0.197993
$310$	0	0
$311$	−1.89296e16	−1.18631	−0.593157	−	0.805087i	$-0.702119\pi$
$311$	−1.89296e16	−1.18631	−0.593157	+	0.805087i	$0.702119\pi$
$312$	0	0
$313$	2.29247e16	1.37805	0.689027	−	0.724736i	$-0.258038\pi$
$313$	2.29247e16	1.37805	0.689027	+	0.724736i	$0.258038\pi$
$314$	0	0
$315$	3.02205e15	0.174294
$316$	0	0
$317$	−2.00585e16	−1.11023	−0.555116	−	0.831773i	$-0.687326\pi$
$317$	−2.00585e16	−1.11023	−0.555116	+	0.831773i	$0.687326\pi$
$318$	0	0
$319$	−9.84990e15	−0.523350
$320$	0	0
$321$	1.60727e16	0.819985
$322$	0	0
$323$	−9.04313e15	−0.443101
$324$	0	0
$325$	1.49968e16	0.705922
$326$	0	0
$327$	1.86116e16	0.841829
$328$	0	0
$329$	−9.61395e14	−0.0417954
$330$	0	0
$331$	9.23373e15	0.385918	0.192959	−	0.981207i	$-0.438192\pi$
$331$	9.23373e15	0.385918	0.192959	+	0.981207i	$0.438192\pi$
$332$	0	0
$333$	−1.24818e16	−0.501637
$334$	0	0
$335$	−1.42997e16	−0.552759
$336$	0	0
$337$	4.15951e16	1.54685	0.773424	−	0.633889i	$-0.218542\pi$
$337$	4.15951e16	1.54685	0.773424	+	0.633889i	$0.218542\pi$
$338$	0	0
$339$	1.34176e15	0.0480151
$340$	0	0
$341$	2.52187e16	0.868600
$342$	0	0
$343$	1.62841e15	0.0539949
$344$	0	0
$345$	4.02358e16	1.28466
$346$	0	0
$347$	5.73745e15	0.176432	0.0882160	−	0.996101i	$-0.471883\pi$
$347$	5.73745e15	0.176432	0.0882160	+	0.996101i	$0.471883\pi$
$348$	0	0
$349$	1.72310e16	0.510440	0.255220	−	0.966883i	$-0.417852\pi$
$349$	1.72310e16	0.510440	0.255220	+	0.966883i	$0.417852\pi$
$350$	0	0
$351$	5.20833e15	0.148663
$352$	0	0
$353$	5.84815e15	0.160873	0.0804365	−	0.996760i	$-0.474369\pi$
$353$	5.84815e15	0.160873	0.0804365	+	0.996760i	$0.474369\pi$
$354$	0	0
$355$	3.12690e15	0.0829143
$356$	0	0
$357$	−8.04514e15	−0.205679
$358$	0	0
$359$	−3.02032e16	−0.744627	−0.372314	−	0.928107i	$-0.621435\pi$
$359$	−3.02032e16	−0.744627	−0.372314	+	0.928107i	$0.621435\pi$
$360$	0	0
$361$	−3.27590e16	−0.778994
$362$	0	0
$363$	9.83338e15	0.225585
$364$	0	0
$365$	−5.59484e16	−1.23846
$366$	0	0
$367$	6.84542e15	0.146242	0.0731208	−	0.997323i	$-0.476704\pi$
$367$	6.84542e15	0.146242	0.0731208	+	0.997323i	$0.476704\pi$
$368$	0	0
$369$	8.73165e15	0.180063
$370$	0	0
$371$	2.35220e15	0.0468322
$372$	0	0
$373$	6.27608e16	1.20665	0.603324	−	0.797496i	$-0.293842\pi$
$373$	6.27608e16	1.20665	0.603324	+	0.797496i	$0.293842\pi$
$374$	0	0
$375$	−3.70580e15	−0.0688143
$376$	0	0
$377$	−1.91106e16	−0.342812
$378$	0	0
$379$	1.11989e17	1.94097	0.970486	−	0.241156i	$-0.0775267\pi$
$379$	1.11989e17	1.94097	0.970486	+	0.241156i	$0.0775267\pi$
$380$	0	0
$381$	1.69838e16	0.284460
$382$	0	0
$383$	−1.00602e17	−1.62859	−0.814297	−	0.580449i	$-0.802877\pi$
$383$	−1.00602e17	−1.62859	−0.814297	+	0.580449i	$0.802877\pi$
$384$	0	0
$385$	3.94021e16	0.616630
$386$	0	0
$387$	−1.87302e16	−0.283413
$388$	0	0
$389$	−5.45888e16	−0.798788	−0.399394	−	0.916779i	$-0.630779\pi$
$389$	−5.45888e16	−0.798788	−0.399394	+	0.916779i	$0.630779\pi$
$390$	0	0
$391$	−1.07114e17	−1.51599
$392$	0	0
$393$	−7.06738e16	−0.967624
$394$	0	0
$395$	−6.67641e16	−0.884426
$396$	0	0
$397$	−1.42610e17	−1.82816	−0.914078	−	0.405539i	$-0.867084\pi$
$397$	−1.42610e17	−1.82816	−0.914078	+	0.405539i	$0.867084\pi$
$398$	0	0
$399$	8.26830e15	0.102587
$400$	0	0
$401$	−4.63342e16	−0.556497	−0.278249	−	0.960509i	$-0.589754\pi$
$401$	−4.63342e16	−0.556497	−0.278249	+	0.960509i	$0.589754\pi$
$402$	0	0
$403$	4.89290e16	0.568963
$404$	0	0
$405$	1.36511e16	0.153713
$406$	0	0
$407$	−1.62740e17	−1.77473
$408$	0	0
$409$	−5.18180e15	−0.0547368	−0.0273684	−	0.999625i	$-0.508713\pi$
$409$	−5.18180e15	−0.0547368	−0.0273684	+	0.999625i	$0.508713\pi$
$410$	0	0
$411$	4.50620e16	0.461146
$412$	0	0
$413$	−1.03947e16	−0.103071
$414$	0	0
$415$	−1.36720e17	−1.31377
$416$	0	0
$417$	7.31235e16	0.681040
$418$	0	0
$419$	−1.70695e17	−1.54110	−0.770548	−	0.637382i	$-0.780017\pi$
$419$	−1.70695e17	−1.54110	−0.770548	+	0.637382i	$0.780017\pi$
$420$	0	0
$421$	1.52791e17	1.33741	0.668704	−	0.743528i	$-0.266849\pi$
$421$	1.52791e17	1.33741	0.668704	+	0.743528i	$0.266849\pi$
$422$	0	0
$423$	−4.34279e15	−0.0368601
$424$	0	0
$425$	−1.04641e17	−0.861335
$426$	0	0
$427$	−3.93101e16	−0.313851
$428$	0	0
$429$	6.79074e16	0.525950
$430$	0	0
$431$	−5.85274e16	−0.439802	−0.219901	−	0.975522i	$-0.570573\pi$
$431$	−5.85274e16	−0.439802	−0.219901	+	0.975522i	$0.570573\pi$
$432$	0	0
$433$	9.53768e16	0.695459	0.347730	−	0.937595i	$-0.386953\pi$
$433$	9.53768e16	0.695459	0.347730	+	0.937595i	$0.386953\pi$
$434$	0	0
$435$	−5.00893e16	−0.354458
$436$	0	0
$437$	1.10085e17	0.756133
$438$	0	0
$439$	−2.23145e17	−1.48788	−0.743938	−	0.668248i	$-0.767044\pi$
$439$	−2.23145e17	−1.48788	−0.743938	+	0.668248i	$0.767044\pi$
$440$	0	0
$441$	7.35583e15	0.0476190
$442$	0	0
$443$	1.86048e16	0.116950	0.0584750	−	0.998289i	$-0.481376\pi$
$443$	1.86048e16	0.116950	0.0584750	+	0.998289i	$0.481376\pi$
$444$	0	0
$445$	4.43772e17	2.70907
$446$	0	0
$447$	5.95934e16	0.353345
$448$	0	0
$449$	−1.26398e17	−0.728015	−0.364007	−	0.931396i	$-0.618592\pi$
$449$	−1.26398e17	−0.728015	−0.364007	+	0.931396i	$0.618592\pi$
$450$	0	0
$451$	1.13845e17	0.637040
$452$	0	0
$453$	7.49086e14	0.00407280
$454$	0	0
$455$	7.64474e16	0.403914
$456$	0	0
$457$	−3.00995e17	−1.54562	−0.772811	−	0.634636i	$-0.781150\pi$
$457$	−3.00995e17	−1.54562	−0.772811	+	0.634636i	$0.781150\pi$
$458$	0	0
$459$	−3.63413e16	−0.181392
$460$	0	0
$461$	1.98935e17	0.965282	0.482641	−	0.875818i	$-0.339678\pi$
$461$	1.98935e17	0.965282	0.482641	+	0.875818i	$0.339678\pi$
$462$	0	0
$463$	−7.31560e16	−0.345123	−0.172561	−	0.984999i	$-0.555204\pi$
$463$	−7.31560e16	−0.345123	−0.172561	+	0.984999i	$0.555204\pi$
$464$	0	0
$465$	1.28244e17	0.588291
$466$	0	0
$467$	2.13655e16	0.0953134	0.0476567	−	0.998864i	$-0.484825\pi$
$467$	2.13655e16	0.0953134	0.0476567	+	0.998864i	$0.484825\pi$
$468$	0	0
$469$	−3.48062e16	−0.151020
$470$	0	0
$471$	1.55176e17	0.654920
$472$	0	0
$473$	−2.44208e17	−1.00268
$474$	0	0
$475$	1.07543e17	0.429610
$476$	0	0
$477$	1.06253e16	0.0413021
$478$	0	0
$479$	−1.80352e17	−0.682245	−0.341122	−	0.940019i	$-0.610807\pi$
$479$	−1.80352e17	−0.682245	−0.341122	+	0.940019i	$0.610807\pi$
$480$	0	0
$481$	−3.15746e17	−1.16251
$482$	0	0
$483$	9.79361e16	0.350983
$484$	0	0
$485$	5.33994e17	1.86301
$486$	0	0
$487$	−4.25474e17	−1.44522	−0.722611	−	0.691255i	$-0.757058\pi$
$487$	−4.25474e17	−1.44522	−0.722611	+	0.691255i	$0.757058\pi$
$488$	0	0
$489$	1.52193e17	0.503371
$490$	0	0
$491$	−3.87893e17	−1.24934	−0.624672	−	0.780887i	$-0.714767\pi$
$491$	−3.87893e17	−1.24934	−0.624672	+	0.780887i	$0.714767\pi$
$492$	0	0
$493$	1.33345e17	0.418285
$494$	0	0
$495$	1.77986e17	0.543817
$496$	0	0
$497$	7.61104e15	0.0226531
$498$	0	0
$499$	3.21368e17	0.931858	0.465929	−	0.884822i	$-0.345720\pi$
$499$	3.21368e17	0.931858	0.465929	+	0.884822i	$0.345720\pi$
$500$	0	0
$501$	−1.19188e17	−0.336733
$502$	0	0
$503$	5.12964e17	1.41220	0.706098	−	0.708114i	$-0.250454\pi$
$503$	5.12964e17	1.41220	0.706098	+	0.708114i	$0.250454\pi$
$504$	0	0
$505$	4.31017e17	1.15638
$506$	0	0
$507$	−8.90431e16	−0.232835
$508$	0	0
$509$	8.10911e16	0.206684	0.103342	−	0.994646i	$-0.467046\pi$
$509$	8.10911e16	0.206684	0.103342	+	0.994646i	$0.467046\pi$
$510$	0	0
$511$	−1.36181e17	−0.338362
$512$	0	0
$513$	3.73494e16	0.0904733
$514$	0	0
$515$	−2.00868e17	−0.474421
$516$	0	0
$517$	−5.66222e16	−0.130406
$518$	0	0
$519$	−3.06215e17	−0.687763
$520$	0	0
$521$	1.93706e17	0.424325	0.212163	−	0.977234i	$-0.431949\pi$
$521$	1.93706e17	0.424325	0.212163	+	0.977234i	$0.431949\pi$
$522$	0	0
$523$	3.95679e17	0.845439	0.422719	−	0.906261i	$-0.361076\pi$
$523$	3.95679e17	0.845439	0.422719	+	0.906261i	$0.361076\pi$
$524$	0	0
$525$	9.56749e16	0.199417
$526$	0	0
$527$	−3.41404e17	−0.694224
$528$	0	0
$529$	7.99894e17	1.58698
$530$	0	0
$531$	−4.69548e16	−0.0909003
$532$	0	0
$533$	2.20881e17	0.417283
$534$	0	0
$535$	1.06566e18	1.96481
$536$	0	0
$537$	3.92438e17	0.706217
$538$	0	0
$539$	9.59068e16	0.168470
$540$	0	0
$541$	−7.99609e17	−1.37118	−0.685592	−	0.727986i	$-0.740456\pi$
$541$	−7.99609e17	−1.37118	−0.685592	+	0.727986i	$0.740456\pi$
$542$	0	0
$543$	−5.48551e17	−0.918370
$544$	0	0
$545$	1.23400e18	2.01715
$546$	0	0
$547$	1.05043e18	1.67668	0.838341	−	0.545147i	$-0.183526\pi$
$547$	1.05043e18	1.67668	0.838341	+	0.545147i	$0.183526\pi$
$548$	0	0
$549$	−1.77571e17	−0.276790
$550$	0	0
$551$	−1.37044e17	−0.208629
$552$	0	0
$553$	−1.62507e17	−0.241635
$554$	0	0
$555$	−8.27576e17	−1.20200
$556$	0	0
$557$	8.24865e17	1.17037	0.585186	−	0.810899i	$-0.301021\pi$
$557$	8.24865e17	1.17037	0.585186	+	0.810899i	$0.301021\pi$
$558$	0	0
$559$	−4.73808e17	−0.656789
$560$	0	0
$561$	−4.73826e17	−0.641741
$562$	0	0
$563$	−8.72012e17	−1.15403	−0.577016	−	0.816733i	$-0.695783\pi$
$563$	−8.72012e17	−1.15403	−0.577016	+	0.816733i	$0.695783\pi$
$564$	0	0
$565$	8.89624e16	0.115051
$566$	0	0
$567$	3.32276e16	0.0419961
$568$	0	0
$569$	−8.24987e17	−1.01910	−0.509550	−	0.860441i	$-0.670188\pi$
$569$	−8.24987e17	−1.01910	−0.509550	+	0.860441i	$0.670188\pi$
$570$	0	0
$571$	−1.32242e18	−1.59674	−0.798372	−	0.602164i	$-0.794305\pi$
$571$	−1.32242e18	−1.59674	−0.798372	+	0.602164i	$0.794305\pi$
$572$	0	0
$573$	8.36587e17	0.987429
$574$	0	0
$575$	1.27382e18	1.46983
$576$	0	0
$577$	6.16377e17	0.695351	0.347675	−	0.937615i	$-0.386971\pi$
$577$	6.16377e17	0.695351	0.347675	+	0.937615i	$0.386971\pi$
$578$	0	0
$579$	−5.47365e17	−0.603763
$580$	0	0
$581$	−3.32785e17	−0.358937
$582$	0	0
$583$	1.38535e17	0.146121
$584$	0	0
$585$	3.45326e17	0.356219
$586$	0	0
$587$	−1.92749e17	−0.194466	−0.0972330	−	0.995262i	$-0.530999\pi$
$587$	−1.92749e17	−0.194466	−0.0972330	+	0.995262i	$0.530999\pi$
$588$	0	0
$589$	3.50874e17	0.346259
$590$	0	0
$591$	−6.42577e17	−0.620307
$592$	0	0
$593$	1.64255e18	1.55119	0.775593	−	0.631234i	$-0.217451\pi$
$593$	1.64255e18	1.55119	0.775593	+	0.631234i	$0.217451\pi$
$594$	0	0
$595$	−5.33414e17	−0.492838
$596$	0	0
$597$	4.12530e17	0.372925
$598$	0	0
$599$	−1.84167e18	−1.62906	−0.814531	−	0.580120i	$-0.803006\pi$
$599$	−1.84167e18	−1.62906	−0.814531	+	0.580120i	$0.803006\pi$
$600$	0	0
$601$	4.73153e16	0.0409560	0.0204780	−	0.999790i	$-0.493481\pi$
$601$	4.73153e16	0.0409560	0.0204780	+	0.999790i	$0.493481\pi$
$602$	0	0
$603$	−1.57226e17	−0.133187
$604$	0	0
$605$	6.51979e17	0.540535
$606$	0	0
$607$	−2.05766e18	−1.66973	−0.834867	−	0.550452i	$-0.814455\pi$
$607$	−2.05766e18	−1.66973	−0.834867	+	0.550452i	$0.814455\pi$
$608$	0	0
$609$	−1.21920e17	−0.0968417
$610$	0	0
$611$	−1.09858e17	−0.0854205
$612$	0	0
$613$	−1.59250e18	−1.21223	−0.606116	−	0.795376i	$-0.707273\pi$
$613$	−1.59250e18	−1.21223	−0.606116	+	0.795376i	$0.707273\pi$
$614$	0	0
$615$	5.78931e17	0.431458
$616$	0	0
$617$	5.79618e16	0.0422949	0.0211475	−	0.999776i	$-0.493268\pi$
$617$	5.79618e16	0.0422949	0.0211475	+	0.999776i	$0.493268\pi$
$618$	0	0
$619$	−6.08326e17	−0.434657	−0.217329	−	0.976098i	$-0.569734\pi$
$619$	−6.08326e17	−0.434657	−0.217329	+	0.976098i	$0.569734\pi$
$620$	0	0
$621$	4.42395e17	0.309538
$622$	0	0
$623$	1.08016e18	0.740147
$624$	0	0
$625$	−1.60744e18	−1.07873
$626$	0	0
$627$	4.86969e17	0.320083
$628$	0	0
$629$	2.20313e18	1.41844
$630$	0	0
$631$	2.11280e18	1.33250	0.666251	−	0.745727i	$-0.267898\pi$
$631$	2.11280e18	1.33250	0.666251	+	0.745727i	$0.267898\pi$
$632$	0	0
$633$	−1.71377e18	−1.05883
$634$	0	0
$635$	1.12607e18	0.681610
$636$	0	0
$637$	1.86077e17	0.110354
$638$	0	0
$639$	3.43804e16	0.0199781
$640$	0	0
$641$	−7.67543e17	−0.437044	−0.218522	−	0.975832i	$-0.570124\pi$
$641$	−7.67543e17	−0.437044	−0.218522	+	0.975832i	$0.570124\pi$
$642$	0	0
$643$	−3.02045e18	−1.68539	−0.842694	−	0.538393i	$-0.819031\pi$
$643$	−3.02045e18	−1.68539	−0.842694	+	0.538393i	$0.819031\pi$
$644$	0	0
$645$	−1.24186e18	−0.679100
$646$	0	0
$647$	2.54249e18	1.36264	0.681321	−	0.731985i	$-0.261406\pi$
$647$	2.54249e18	1.36264	0.681321	+	0.731985i	$0.261406\pi$
$648$	0	0
$649$	−6.12206e17	−0.321594
$650$	0	0
$651$	3.12152e17	0.160727
$652$	0	0
$653$	1.88222e17	0.0950024	0.0475012	−	0.998871i	$-0.484874\pi$
$653$	1.88222e17	0.0950024	0.0475012	+	0.998871i	$0.484874\pi$
$654$	0	0
$655$	−4.68586e18	−2.31857
$656$	0	0
$657$	−6.15154e17	−0.298407
$658$	0	0
$659$	−1.95049e18	−0.927658	−0.463829	−	0.885925i	$-0.653525\pi$
$659$	−1.95049e18	−0.927658	−0.463829	+	0.885925i	$0.653525\pi$
$660$	0	0
$661$	2.82493e18	1.31734	0.658671	−	0.752431i	$-0.271119\pi$
$661$	2.82493e18	1.31734	0.658671	+	0.752431i	$0.271119\pi$
$662$	0	0
$663$	−9.19310e17	−0.420362
$664$	0	0
$665$	5.48210e17	0.245814
$666$	0	0
$667$	−1.62325e18	−0.713786
$668$	0	0
$669$	−2.52754e18	−1.09000
$670$	0	0
$671$	−2.31520e18	−0.979247
$672$	0	0
$673$	−3.85388e18	−1.59882	−0.799411	−	0.600784i	$-0.794855\pi$
$673$	−3.85388e18	−1.59882	−0.799411	+	0.600784i	$0.794855\pi$
$674$	0	0
$675$	4.32180e17	0.175869
$676$	0	0
$677$	−3.76901e18	−1.50453	−0.752264	−	0.658861i	$-0.771038\pi$
$677$	−3.76901e18	−1.50453	−0.752264	+	0.658861i	$0.771038\pi$
$678$	0	0
$679$	1.29977e18	0.508995
$680$	0	0
$681$	−7.85726e17	−0.301866
$682$	0	0
$683$	6.31475e16	0.0238025	0.0119012	−	0.999929i	$-0.496212\pi$
$683$	6.31475e16	0.0238025	0.0119012	+	0.999929i	$0.496212\pi$
$684$	0	0
$685$	2.98773e18	1.10498
$686$	0	0
$687$	9.36531e17	0.339863
$688$	0	0
$689$	2.68784e17	0.0957146
$690$	0	0
$691$	−3.63936e18	−1.27180	−0.635899	−	0.771773i	$-0.719370\pi$
$691$	−3.63936e18	−1.27180	−0.635899	+	0.771773i	$0.719370\pi$
$692$	0	0
$693$	4.33228e17	0.148577
$694$	0	0
$695$	4.84828e18	1.63187
$696$	0	0
$697$	−1.54120e18	−0.509150
$698$	0	0
$699$	−2.76256e18	−0.895800
$700$	0	0
$701$	2.17244e18	0.691482	0.345741	−	0.938330i	$-0.387627\pi$
$701$	2.17244e18	0.691482	0.345741	+	0.938330i	$0.387627\pi$
$702$	0	0
$703$	−2.26424e18	−0.707478
$704$	0	0
$705$	−2.87938e17	−0.0883223
$706$	0	0
$707$	1.04912e18	0.315935
$708$	0	0
$709$	1.33940e18	0.396014	0.198007	−	0.980201i	$-0.436553\pi$
$709$	1.33940e18	0.396014	0.198007	+	0.980201i	$0.436553\pi$
$710$	0	0
$711$	−7.34074e17	−0.213102
$712$	0	0
$713$	4.15602e18	1.18467
$714$	0	0
$715$	4.50244e18	1.26025
$716$	0	0
$717$	−1.25369e18	−0.344601
$718$	0	0
$719$	4.53220e18	1.22341	0.611704	−	0.791087i	$-0.290484\pi$
$719$	4.53220e18	1.22341	0.611704	+	0.791087i	$0.290484\pi$
$720$	0	0
$721$	−4.88923e17	−0.129617
$722$	0	0
$723$	3.45268e18	0.898995
$724$	0	0
$725$	−1.58577e18	−0.405550
$726$	0	0
$727$	4.99359e18	1.25441	0.627204	−	0.778855i	$-0.284199\pi$
$727$	4.99359e18	1.25441	0.627204	+	0.778855i	$0.284199\pi$
$728$	0	0
$729$	1.50095e17	0.0370370
$730$	0	0
$731$	3.30601e18	0.801385
$732$	0	0
$733$	−5.95356e18	−1.41775	−0.708877	−	0.705332i	$-0.750798\pi$
$733$	−5.95356e18	−1.41775	−0.708877	+	0.705332i	$0.750798\pi$
$734$	0	0
$735$	4.87711e17	0.114102
$736$	0	0
$737$	−2.04994e18	−0.471197
$738$	0	0
$739$	4.72178e18	1.06639	0.533196	−	0.845992i	$-0.320991\pi$
$739$	4.72178e18	1.06639	0.533196	+	0.845992i	$0.320991\pi$
$740$	0	0
$741$	9.44810e17	0.209665
$742$	0	0
$743$	−2.61373e17	−0.0569946	−0.0284973	−	0.999594i	$-0.509072\pi$
$743$	−2.61373e17	−0.0569946	−0.0284973	+	0.999594i	$0.509072\pi$
$744$	0	0
$745$	3.95120e18	0.846668
$746$	0	0
$747$	−1.50325e18	−0.316553
$748$	0	0
$749$	2.59388e18	0.536806
$750$	0	0
$751$	6.13797e18	1.24843	0.624216	−	0.781252i	$-0.285418\pi$
$751$	6.13797e18	1.24843	0.624216	+	0.781252i	$0.285418\pi$
$752$	0	0
$753$	−3.66018e18	−0.731704
$754$	0	0
$755$	4.96663e16	0.00975904
$756$	0	0
$757$	3.55845e18	0.687286	0.343643	−	0.939100i	$-0.388339\pi$
$757$	3.55845e18	0.687286	0.343643	+	0.939100i	$0.388339\pi$
$758$	0	0
$759$	5.76803e18	1.09511
$760$	0	0
$761$	−6.69140e18	−1.24887	−0.624435	−	0.781077i	$-0.714671\pi$
$761$	−6.69140e18	−1.24887	−0.624435	+	0.781077i	$0.714671\pi$
$762$	0	0
$763$	3.00362e18	0.551106
$764$	0	0
$765$	−2.40952e18	−0.434642
$766$	0	0
$767$	−1.18779e18	−0.210655
$768$	0	0
$769$	2.29623e18	0.400401	0.200200	−	0.979755i	$-0.435841\pi$
$769$	2.29623e18	0.400401	0.200200	+	0.979755i	$0.435841\pi$
$770$	0	0
$771$	−3.03088e17	−0.0519654
$772$	0	0
$773$	3.28937e18	0.554557	0.277278	−	0.960790i	$-0.410568\pi$
$773$	3.28937e18	0.554557	0.277278	+	0.960790i	$0.410568\pi$
$774$	0	0
$775$	4.06006e18	0.673087
$776$	0	0
$777$	−2.01436e18	−0.328398
$778$	0	0
$779$	1.58395e18	0.253950
$780$	0	0
$781$	4.48259e17	0.0706801
$782$	0	0
$783$	−5.50734e17	−0.0854063
$784$	0	0
$785$	1.02886e19	1.56929
$786$	0	0
$787$	−1.27255e18	−0.190915	−0.0954574	−	0.995434i	$-0.530431\pi$
$787$	−1.27255e18	−0.190915	−0.0954574	+	0.995434i	$0.530431\pi$
$788$	0	0
$789$	−4.65087e18	−0.686331
$790$	0	0
$791$	2.16539e17	0.0314333
$792$	0	0
$793$	−4.49192e18	−0.641441
$794$	0	0
$795$	7.04486e17	0.0989660
$796$	0	0
$797$	−4.97288e17	−0.0687273	−0.0343636	−	0.999409i	$-0.510940\pi$
$797$	−4.97288e17	−0.0687273	−0.0343636	+	0.999409i	$0.510940\pi$
$798$	0	0
$799$	7.66535e17	0.104226
$800$	0	0
$801$	4.87929e18	0.652748
$802$	0	0
$803$	−8.02051e18	−1.05573
$804$	0	0
$805$	6.49342e18	0.841009
$806$	0	0
$807$	−2.45186e18	−0.312477
$808$	0	0
$809$	−1.27495e19	−1.59892	−0.799461	−	0.600718i	$-0.794881\pi$
$809$	−1.27495e19	−1.59892	−0.799461	+	0.600718i	$0.794881\pi$
$810$	0	0
$811$	−7.39044e18	−0.912084	−0.456042	−	0.889958i	$-0.650733\pi$
$811$	−7.39044e18	−0.912084	−0.456042	+	0.889958i	$0.650733\pi$
$812$	0	0
$813$	3.54283e18	0.430290
$814$	0	0
$815$	1.00908e19	1.20615
$816$	0	0
$817$	−3.39771e18	−0.399709
$818$	0	0
$819$	8.40542e17	0.0973227
$820$	0	0
$821$	−3.08606e18	−0.351701	−0.175851	−	0.984417i	$-0.556268\pi$
$821$	−3.08606e18	−0.351701	−0.175851	+	0.984417i	$0.556268\pi$
$822$	0	0
$823$	−1.35234e19	−1.51700	−0.758500	−	0.651673i	$-0.774068\pi$
$823$	−1.35234e19	−1.51700	−0.758500	+	0.651673i	$0.774068\pi$
$824$	0	0
$825$	5.63485e18	0.622203
$826$	0	0
$827$	1.65535e19	1.79931	0.899654	−	0.436604i	$-0.143819\pi$
$827$	1.65535e19	1.79931	0.899654	+	0.436604i	$0.143819\pi$
$828$	0	0
$829$	2.20527e17	0.0235970	0.0117985	−	0.999930i	$-0.496244\pi$
$829$	2.20527e17	0.0235970	0.0117985	+	0.999930i	$0.496244\pi$
$830$	0	0
$831$	3.71064e18	0.390879
$832$	0	0
$833$	−1.29836e18	−0.134649
$834$	0	0
$835$	−7.90245e18	−0.806863
$836$	0	0
$837$	1.41005e18	0.141748
$838$	0	0
$839$	5.57971e18	0.552280	0.276140	−	0.961117i	$-0.410945\pi$
$839$	5.57971e18	0.552280	0.276140	+	0.961117i	$0.410945\pi$
$840$	0	0
$841$	−8.23985e18	−0.803055
$842$	0	0
$843$	1.04174e19	0.999722
$844$	0	0
$845$	−5.90379e18	−0.557908
$846$	0	0
$847$	1.58695e18	0.147680
$848$	0	0
$849$	−2.91089e18	−0.266763
$850$	0	0
$851$	−2.68194e19	−2.42051
$852$	0	0
$853$	1.64532e19	1.46245	0.731224	−	0.682137i	$-0.238949\pi$
$853$	1.64532e19	1.46245	0.731224	+	0.682137i	$0.238949\pi$
$854$	0	0
$855$	2.47636e18	0.216787
$856$	0	0
$857$	6.75456e18	0.582400	0.291200	−	0.956662i	$-0.405945\pi$
$857$	6.75456e18	0.582400	0.291200	+	0.956662i	$0.405945\pi$
$858$	0	0
$859$	2.22769e19	1.89191	0.945953	−	0.324305i	$-0.105130\pi$
$859$	2.22769e19	1.89191	0.945953	+	0.324305i	$0.105130\pi$
$860$	0	0
$861$	1.40915e18	0.117879
$862$	0	0
$863$	−7.10535e15	−0.000585484 0	−0.000292742	−	1.00000i	$-0.500093\pi$
$863$	−7.10535e15	−0.000585484 0	−0.000292742		1.00000i	$0.500093\pi$
$864$	0	0
$865$	−2.03029e19	−1.64798
$866$	0	0
$867$	−8.05925e17	−0.0644422
$868$	0	0
$869$	−9.57101e18	−0.753926
$870$	0	0
$871$	−3.97727e18	−0.308650
$872$	0	0
$873$	5.87129e18	0.448891
$874$	0	0
$875$	−5.98057e17	−0.0450495
$876$	0	0
$877$	5.21930e18	0.387360	0.193680	−	0.981065i	$-0.437958\pi$
$877$	5.21930e18	0.387360	0.193680	+	0.981065i	$0.437958\pi$
$878$	0	0
$879$	−1.32388e18	−0.0968103
$880$	0	0
$881$	1.85889e19	1.33940	0.669700	−	0.742632i	$-0.266423\pi$
$881$	1.85889e19	1.33940	0.669700	+	0.742632i	$0.266423\pi$
$882$	0	0
$883$	1.97517e18	0.140236	0.0701180	−	0.997539i	$-0.477662\pi$
$883$	1.97517e18	0.140236	0.0701180	+	0.997539i	$0.477662\pi$
$884$	0	0
$885$	−3.11323e18	−0.217811
$886$	0	0
$887$	−4.47316e18	−0.308397	−0.154199	−	0.988040i	$-0.549280\pi$
$887$	−4.47316e18	−0.308397	−0.154199	+	0.988040i	$0.549280\pi$
$888$	0	0
$889$	2.74091e18	0.186223
$890$	0	0
$891$	1.95697e18	0.131032
$892$	0	0
$893$	−7.87797e17	−0.0519852
$894$	0	0
$895$	2.60197e19	1.69220
$896$	0	0
$897$	1.11911e19	0.717332
$898$	0	0
$899$	−5.17380e18	−0.326867
$900$	0	0
$901$	−1.87545e18	−0.116787
$902$	0	0
$903$	−3.02275e18	−0.185538
$904$	0	0
$905$	−3.63704e19	−2.20055
$906$	0	0
$907$	−2.62661e19	−1.56656	−0.783281	−	0.621668i	$-0.786455\pi$
$907$	−2.62661e19	−1.56656	−0.783281	+	0.621668i	$0.786455\pi$
$908$	0	0
$909$	4.73905e18	0.278629
$910$	0	0
$911$	−2.63833e19	−1.52918	−0.764592	−	0.644515i	$-0.777059\pi$
$911$	−2.63833e19	−1.52918	−0.764592	+	0.644515i	$0.777059\pi$
$912$	0	0
$913$	−1.95996e19	−1.11992
$914$	0	0
$915$	−1.17734e19	−0.663231
$916$	0	0
$917$	−1.14056e19	−0.633459
$918$	0	0
$919$	−3.46273e18	−0.189613	−0.0948066	−	0.995496i	$-0.530223\pi$
$919$	−3.46273e18	−0.189613	−0.0948066	+	0.995496i	$0.530223\pi$
$920$	0	0
$921$	−7.34550e16	−0.00396583
$922$	0	0
$923$	8.69705e17	0.0462979
$924$	0	0
$925$	−2.62002e19	−1.37525
$926$	0	0
$927$	−2.20855e18	−0.114311
$928$	0	0
$929$	−2.26128e19	−1.15412	−0.577062	−	0.816700i	$-0.695801\pi$
$929$	−2.26128e19	−1.15412	−0.577062	+	0.816700i	$0.695801\pi$
$930$	0	0
$931$	1.33437e18	0.0671590
$932$	0	0
$933$	−1.37997e19	−0.684919
$934$	0	0
$935$	−3.14159e19	−1.53771
$936$	0	0
$937$	2.93847e19	1.41845	0.709225	−	0.704982i	$-0.249045\pi$
$937$	2.93847e19	1.41845	0.709225	+	0.704982i	$0.249045\pi$
$938$	0	0
$939$	1.67121e19	0.795620
$940$	0	0
$941$	−1.21796e19	−0.571876	−0.285938	−	0.958248i	$-0.592305\pi$
$941$	−1.21796e19	−0.571876	−0.285938	+	0.958248i	$0.592305\pi$
$942$	0	0
$943$	1.87615e19	0.868845
$944$	0	0
$945$	2.20307e18	0.100629
$946$	0	0
$947$	−2.09610e19	−0.944358	−0.472179	−	0.881503i	$-0.656532\pi$
$947$	−2.09610e19	−0.944358	−0.472179	+	0.881503i	$0.656532\pi$
$948$	0	0
$949$	−1.55613e19	−0.691536
$950$	0	0
$951$	−1.46227e19	−0.640993
$952$	0	0
$953$	3.72845e19	1.61222	0.806111	−	0.591764i	$-0.201568\pi$
$953$	3.72845e19	1.61222	0.806111	+	0.591764i	$0.201568\pi$
$954$	0	0
$955$	5.54679e19	2.36603
$956$	0	0
$957$	−7.18058e18	−0.302156
$958$	0	0
$959$	7.27229e18	0.301891
$960$	0	0
$961$	−1.11711e19	−0.457501
$962$	0	0
$963$	1.17170e19	0.473418
$964$	0	0
$965$	−3.62917e19	−1.44671
$966$	0	0
$967$	2.28570e19	0.898976	0.449488	−	0.893286i	$-0.351607\pi$
$967$	2.28570e19	0.898976	0.449488	+	0.893286i	$0.351607\pi$
$968$	0	0
$969$	−6.59244e18	−0.255824
$970$	0	0
$971$	−1.24219e19	−0.475625	−0.237812	−	0.971311i	$-0.576430\pi$
$971$	−1.24219e19	−0.475625	−0.237812	+	0.971311i	$0.576430\pi$
$972$	0	0
$973$	1.18010e19	0.445845
$974$	0	0
$975$	1.09327e19	0.407564
$976$	0	0
$977$	−1.88012e19	−0.691626	−0.345813	−	0.938303i	$-0.612397\pi$
$977$	−1.88012e19	−0.691626	−0.345813	+	0.938303i	$0.612397\pi$
$978$	0	0
$979$	6.36172e19	2.30934
$980$	0	0
$981$	1.35679e19	0.486030
$982$	0	0
$983$	−1.58981e19	−0.562015	−0.281007	−	0.959706i	$-0.590669\pi$
$983$	−1.58981e19	−0.562015	−0.281007	+	0.959706i	$0.590669\pi$
$984$	0	0
$985$	−4.26045e19	−1.48635
$986$	0	0
$987$	−7.00857e17	−0.0241306
$988$	0	0
$989$	−4.02451e19	−1.36753
$990$	0	0
$991$	−1.95086e19	−0.654255	−0.327128	−	0.944980i	$-0.606081\pi$
$991$	−1.95086e19	−0.654255	−0.327128	+	0.944980i	$0.606081\pi$
$992$	0	0
$993$	6.73139e18	0.222810
$994$	0	0
$995$	2.73518e19	0.893585
$996$	0	0
$997$	3.36866e19	1.08627	0.543136	−	0.839645i	$-0.317237\pi$
$997$	3.36866e19	1.08627	0.543136	+	0.839645i	$0.317237\pi$
$998$	0	0
$999$	−9.09923e18	−0.289620

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.14.a.d.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.14.a.d.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:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)

\(f(q)\)	\(=\)	\(q+729.000 q^{3} +48334.6 q^{5} +117649. q^{7} +531441. q^{9} +O(q^{10})\)	\(q+729.000 q^{3} +48334.6 q^{5} +117649. q^{7} +531441. q^{9} +6.92904e6 q^{11} +1.34436e7 q^{13} +3.52359e7 q^{15} -9.38033e7 q^{17} +9.64052e7 q^{19} +8.57661e7 q^{21} +1.14190e9 q^{23} +1.11553e9 q^{25} +3.87420e8 q^{27} -1.42154e9 q^{29} +3.63957e9 q^{31} +5.05127e9 q^{33} +5.68652e9 q^{35} -2.34867e10 q^{37} +9.80040e9 q^{39} +1.64301e10 q^{41} -3.52441e10 q^{43} +2.56870e10 q^{45} -8.17172e9 q^{47} +1.38413e10 q^{49} -6.83826e10 q^{51} +1.99934e10 q^{53} +3.34912e11 q^{55} +7.02794e10 q^{57} -8.83537e10 q^{59} -3.34130e11 q^{61} +6.25235e10 q^{63} +6.49792e11 q^{65} -2.95848e11 q^{67} +8.32443e11 q^{69} +6.46927e10 q^{71} -1.15752e12 q^{73} +8.13223e11 q^{75} +8.15194e11 q^{77} -1.38129e12 q^{79} +2.82430e11 q^{81} -2.82862e12 q^{83} -4.53395e12 q^{85} -1.03630e12 q^{87} +9.18125e12 q^{89} +1.58163e12 q^{91} +2.65325e12 q^{93} +4.65971e12 q^{95} +1.10479e13 q^{97} +3.68238e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2916 q^{3} + 19278 q^{5} + 470596 q^{7} + 2125764 q^{9}+O(q^{10}) \) 4 * q + 2916 * q^3 + 19278 * q^5 + 470596 * q^7 + 2125764 * q^9	\( 4 q + 2916 q^{3} + 19278 q^{5} + 470596 q^{7} + 2125764 q^{9} + 515970 q^{11} + 31470740 q^{13} + 14053662 q^{15} + 40485258 q^{17} + 257760344 q^{19} + 343064484 q^{21} - 492248178 q^{23} + 1141842832 q^{25} + 1549681956 q^{27} + 817801920 q^{29} + 131625608 q^{31} + 376142130 q^{33} + 2268037422 q^{35} + 38485873484 q^{37} + 22942169460 q^{39} + 42425041302 q^{41} + 49587853952 q^{43} + 10245119598 q^{45} - 15206667516 q^{47} + 55365148804 q^{49} + 29513753082 q^{51} + 189707500548 q^{53} - 2269688904 q^{55} + 187907290776 q^{57} + 25327021356 q^{59} + 57555532688 q^{61} + 250094008836 q^{63} - 1243945073628 q^{65} + 459346595468 q^{67} - 358848921762 q^{69} + 510335887698 q^{71} - 957870191236 q^{73} + 832403424528 q^{75} + 60703354530 q^{77} + 21389706452 q^{79} + 1129718145924 q^{81} + 5770833822000 q^{83} + 5429978357868 q^{85} + 596177599680 q^{87} + 7593089475942 q^{89} + 3702501090260 q^{91} + 95955068232 q^{93} + 17821341989208 q^{95} + 5330443230644 q^{97} + 274207612770 q^{99}+O(q^{100}) \) 4 * q + 2916 * q^3 + 19278 * q^5 + 470596 * q^7 + 2125764 * q^9 + 515970 * q^11 + 31470740 * q^13 + 14053662 * q^15 + 40485258 * q^17 + 257760344 * q^19 + 343064484 * q^21 - 492248178 * q^23 + 1141842832 * q^25 + 1549681956 * q^27 + 817801920 * q^29 + 131625608 * q^31 + 376142130 * q^33 + 2268037422 * q^35 + 38485873484 * q^37 + 22942169460 * q^39 + 42425041302 * q^41 + 49587853952 * q^43 + 10245119598 * q^45 - 15206667516 * q^47 + 55365148804 * q^49 + 29513753082 * q^51 + 189707500548 * q^53 - 2269688904 * q^55 + 187907290776 * q^57 + 25327021356 * q^59 + 57555532688 * q^61 + 250094008836 * q^63 - 1243945073628 * q^65 + 459346595468 * q^67 - 358848921762 * q^69 + 510335887698 * q^71 - 957870191236 * q^73 + 832403424528 * q^75 + 60703354530 * q^77 + 21389706452 * q^79 + 1129718145924 * q^81 + 5770833822000 * q^83 + 5429978357868 * q^85 + 596177599680 * q^87 + 7593089475942 * q^89 + 3702501090260 * q^91 + 95955068232 * q^93 + 17821341989208 * q^95 + 5330443230644 * q^97 + 274207612770 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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