LMFDB - Embedded newform 84.14.a.d.1.2

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			1.2
Root			$-3608.22$ 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)

$f(q)$	$=$	$q+729.000 q^{3} -16831.3 q^{5} +117649. q^{7} +531441. q^{9} +O(q^{10})$	$q+729.000 q^{3} -16831.3 q^{5} +117649. q^{7} +531441. q^{9} +6.93891e6 q^{11} -4.45397e6 q^{13} -1.22700e7 q^{15} +1.47755e8 q^{17} -2.83966e8 q^{19} +8.57661e7 q^{21} -1.85997e8 q^{23} -9.37410e8 q^{25} +3.87420e8 q^{27} +2.72179e9 q^{29} -1.54365e9 q^{31} +5.05846e9 q^{33} -1.98019e9 q^{35} +3.53182e9 q^{37} -3.24695e9 q^{39} -1.36893e10 q^{41} +5.67546e10 q^{43} -8.94486e9 q^{45} +1.36999e11 q^{47} +1.38413e10 q^{49} +1.07713e11 q^{51} -1.64179e11 q^{53} -1.16791e11 q^{55} -2.07011e11 q^{57} -2.52920e11 q^{59} +4.46331e11 q^{61} +6.25235e10 q^{63} +7.49663e10 q^{65} +3.09315e11 q^{67} -1.35592e11 q^{69} -1.19400e12 q^{71} +6.11420e11 q^{73} -6.83372e11 q^{75} +8.16356e11 q^{77} +9.09740e11 q^{79} +2.82430e11 q^{81} +4.61316e12 q^{83} -2.48691e12 q^{85} +1.98418e12 q^{87} +6.09449e11 q^{89} -5.24005e11 q^{91} -1.12532e12 q^{93} +4.77953e12 q^{95} +4.78760e12 q^{97} +3.68762e12 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−16831.3	−0.481741	−0.240870	−	0.970557i	$-0.577433\pi$
$5$	−16831.3	−0.481741	−0.240870	+	0.970557i	$0.577433\pi$
$6$	0	0
$7$	117649.	0.377964
$7$	117649.	0.377964
$8$	0	0
$9$	531441.	0.333333
$10$	0	0
$11$	6.93891e6	1.18097	0.590485	−	0.807049i	$-0.298937\pi$
$11$	6.93891e6	1.18097	0.590485	+	0.807049i	$0.298937\pi$
$12$	0	0
$13$	−4.45397e6	−0.255927	−0.127963	−	0.991779i	$-0.540844\pi$
$13$	−4.45397e6	−0.255927	−0.127963	+	0.991779i	$0.540844\pi$
$14$	0	0
$15$	−1.22700e7	−0.278133
$16$	0	0
$17$	1.47755e8	1.48465	0.742325	−	0.670040i	$-0.233723\pi$
$17$	1.47755e8	1.48465	0.742325	+	0.670040i	$0.233723\pi$
$18$	0	0
$19$	−2.83966e8	−1.38474	−0.692370	−	0.721542i	$-0.743433\pi$
$19$	−2.83966e8	−1.38474	−0.692370	+	0.721542i	$0.743433\pi$
$20$	0	0
$21$	8.57661e7	0.218218
$22$	0	0
$23$	−1.85997e8	−0.261984	−0.130992	−	0.991383i	$-0.541816\pi$
$23$	−1.85997e8	−0.261984	−0.130992	+	0.991383i	$0.541816\pi$
$24$	0	0
$25$	−9.37410e8	−0.767926
$26$	0	0
$27$	3.87420e8	0.192450
$28$	0	0
$29$	2.72179e9	0.849704	0.424852	−	0.905263i	$-0.360326\pi$
$29$	2.72179e9	0.849704	0.424852	+	0.905263i	$0.360326\pi$
$30$	0	0
$31$	−1.54365e9	−0.312390	−0.156195	−	0.987726i	$-0.549923\pi$
$31$	−1.54365e9	−0.312390	−0.156195	+	0.987726i	$0.549923\pi$
$32$	0	0
$33$	5.05846e9	0.681833
$34$	0	0
$35$	−1.98019e9	−0.182081
$36$	0	0
$37$	3.53182e9	0.226302	0.113151	−	0.993578i	$-0.463906\pi$
$37$	3.53182e9	0.226302	0.113151	+	0.993578i	$0.463906\pi$
$38$	0	0
$39$	−3.24695e9	−0.147759
$40$	0	0
$41$	−1.36893e10	−0.450075	−0.225037	−	0.974350i	$-0.572250\pi$
$41$	−1.36893e10	−0.450075	−0.225037	+	0.974350i	$0.572250\pi$
$42$	0	0
$43$	5.67546e10	1.36917	0.684583	−	0.728935i	$-0.259984\pi$
$43$	5.67546e10	1.36917	0.684583	+	0.728935i	$0.259984\pi$
$44$	0	0
$45$	−8.94486e9	−0.160580
$46$	0	0
$47$	1.36999e11	1.85388	0.926942	−	0.375206i	$-0.122428\pi$
$47$	1.36999e11	1.85388	0.926942	+	0.375206i	$0.122428\pi$
$48$	0	0
$49$	1.38413e10	0.142857
$50$	0	0
$51$	1.07713e11	0.857163
$52$	0	0
$53$	−1.64179e11	−1.01747	−0.508737	−	0.860922i	$-0.669887\pi$
$53$	−1.64179e11	−1.01747	−0.508737	+	0.860922i	$0.669887\pi$
$54$	0	0
$55$	−1.16791e11	−0.568921
$56$	0	0
$57$	−2.07011e11	−0.799480
$58$	0	0
$59$	−2.52920e11	−0.780629	−0.390315	−	0.920682i	$-0.627634\pi$
$59$	−2.52920e11	−0.780629	−0.390315	+	0.920682i	$0.627634\pi$
$60$	0	0
$61$	4.46331e11	1.10921	0.554604	−	0.832115i	$-0.312870\pi$
$61$	4.46331e11	1.10921	0.554604	+	0.832115i	$0.312870\pi$
$62$	0	0
$63$	6.25235e10	0.125988
$64$	0	0
$65$	7.49663e10	0.123290
$66$	0	0
$67$	3.09315e11	0.417748	0.208874	−	0.977943i	$-0.433020\pi$
$67$	3.09315e11	0.417748	0.208874	+	0.977943i	$0.433020\pi$
$68$	0	0
$69$	−1.35592e11	−0.151257
$70$	0	0
$71$	−1.19400e12	−1.10618	−0.553089	−	0.833122i	$-0.686551\pi$
$71$	−1.19400e12	−1.10618	−0.553089	+	0.833122i	$0.686551\pi$
$72$	0	0
$73$	6.11420e11	0.472870	0.236435	−	0.971647i	$-0.424021\pi$
$73$	6.11420e11	0.472870	0.236435	+	0.971647i	$0.424021\pi$
$74$	0	0
$75$	−6.83372e11	−0.443362
$76$	0	0
$77$	8.16356e11	0.446364
$78$	0	0
$79$	9.09740e11	0.421057	0.210529	−	0.977588i	$-0.432481\pi$
$79$	9.09740e11	0.421057	0.210529	+	0.977588i	$0.432481\pi$
$80$	0	0
$81$	2.82430e11	0.111111
$82$	0	0
$83$	4.61316e12	1.54879	0.774393	−	0.632705i	$-0.218056\pi$
$83$	4.61316e12	1.54879	0.774393	+	0.632705i	$0.218056\pi$
$84$	0	0
$85$	−2.48691e12	−0.715217
$86$	0	0
$87$	1.98418e12	0.490577
$88$	0	0
$89$	6.09449e11	0.129988	0.0649938	−	0.997886i	$-0.479297\pi$
$89$	6.09449e11	0.129988	0.0649938	+	0.997886i	$0.479297\pi$
$90$	0	0
$91$	−5.24005e11	−0.0967312
$92$	0	0
$93$	−1.12532e12	−0.180358
$94$	0	0
$95$	4.77953e12	0.667086
$96$	0	0
$97$	4.78760e12	0.583581	0.291791	−	0.956482i	$-0.405749\pi$
$97$	4.78760e12	0.583581	0.291791	+	0.956482i	$0.405749\pi$
$98$	0	0
$99$	3.68762e12	0.393656
$100$	0	0
$101$	2.25982e12	0.211828	0.105914	−	0.994375i	$-0.466223\pi$
$101$	2.25982e12	0.211828	0.105914	+	0.994375i	$0.466223\pi$
$102$	0	0
$103$	2.39271e13	1.97446	0.987228	−	0.159315i	$-0.0509286\pi$
$103$	2.39271e13	1.97446	0.987228	+	0.159315i	$0.0509286\pi$
$104$	0	0
$105$	−1.44356e12	−0.105124
$106$	0	0
$107$	2.26412e12	0.145849	0.0729246	−	0.997337i	$-0.476767\pi$
$107$	2.26412e12	0.145849	0.0729246	+	0.997337i	$0.476767\pi$
$108$	0	0
$109$	6.91471e12	0.394913	0.197457	−	0.980312i	$-0.436732\pi$
$109$	6.91471e12	0.394913	0.197457	+	0.980312i	$0.436732\pi$
$110$	0	0
$111$	2.57470e12	0.130655
$112$	0	0
$113$	9.43824e12	0.426463	0.213231	−	0.977002i	$-0.431601\pi$
$113$	9.43824e12	0.426463	0.213231	+	0.977002i	$0.431601\pi$
$114$	0	0
$115$	3.13058e12	0.126208
$116$	0	0
$117$	−2.36702e12	−0.0853089
$118$	0	0
$119$	1.73832e13	0.561145
$120$	0	0
$121$	1.36257e13	0.394689
$122$	0	0
$123$	−9.97946e12	−0.259851
$124$	0	0
$125$	3.63239e13	0.851682
$126$	0	0
$127$	8.60092e13	1.81895	0.909475	−	0.415759i	$-0.136484\pi$
$127$	8.60092e13	1.81895	0.909475	+	0.415759i	$0.136484\pi$
$128$	0	0
$129$	4.13741e13	0.790489
$130$	0	0
$131$	7.87204e13	1.36089	0.680447	−	0.732798i	$-0.261786\pi$
$131$	7.87204e13	1.36089	0.680447	+	0.732798i	$0.261786\pi$
$132$	0	0
$133$	−3.34083e13	−0.523383
$134$	0	0
$135$	−6.52080e12	−0.0927110
$136$	0	0
$137$	−3.33454e13	−0.430876	−0.215438	−	0.976518i	$-0.569118\pi$
$137$	−3.33454e13	−0.430876	−0.215438	+	0.976518i	$0.569118\pi$
$138$	0	0
$139$	−1.53503e14	−1.80518	−0.902590	−	0.430501i	$-0.858337\pi$
$139$	−1.53503e14	−1.80518	−0.902590	+	0.430501i	$0.858337\pi$
$140$	0	0
$141$	9.98725e13	1.07034
$142$	0	0
$143$	−3.09057e13	−0.302242
$144$	0	0
$145$	−4.58113e13	−0.409337
$146$	0	0
$147$	1.00903e13	0.0824786
$148$	0	0
$149$	2.63817e13	0.197511	0.0987555	−	0.995112i	$-0.468514\pi$
$149$	2.63817e13	0.197511	0.0987555	+	0.995112i	$0.468514\pi$
$150$	0	0
$151$	2.66799e14	1.83161	0.915807	−	0.401619i	$-0.131552\pi$
$151$	2.66799e14	1.83161	0.915807	+	0.401619i	$0.131552\pi$
$152$	0	0
$153$	7.85231e13	0.494883
$154$	0	0
$155$	2.59816e13	0.150491
$156$	0	0
$157$	1.58874e14	0.846651	0.423326	−	0.905978i	$-0.360863\pi$
$157$	1.58874e14	0.846651	0.423326	+	0.905978i	$0.360863\pi$
$158$	0	0
$159$	−1.19686e14	−0.587439
$160$	0	0
$161$	−2.18824e13	−0.0990207
$162$	0	0
$163$	1.93423e14	0.807770	0.403885	−	0.914810i	$-0.367660\pi$
$163$	1.93423e14	0.807770	0.403885	+	0.914810i	$0.367660\pi$
$164$	0	0
$165$	−8.51407e13	−0.328467
$166$	0	0
$167$	2.76411e14	0.986049	0.493024	−	0.870015i	$-0.335891\pi$
$167$	2.76411e14	0.986049	0.493024	+	0.870015i	$0.335891\pi$
$168$	0	0
$169$	−2.83037e14	−0.934501
$170$	0	0
$171$	−1.50911e14	−0.461580
$172$	0	0
$173$	1.59234e14	0.451582	0.225791	−	0.974176i	$-0.427503\pi$
$173$	1.59234e14	0.451582	0.225791	+	0.974176i	$0.427503\pi$
$174$	0	0
$175$	−1.10285e14	−0.290249
$176$	0	0
$177$	−1.84379e14	−0.450696
$178$	0	0
$179$	−4.61147e14	−1.04784	−0.523920	−	0.851767i	$-0.675531\pi$
$179$	−4.61147e14	−1.04784	−0.523920	+	0.851767i	$0.675531\pi$
$180$	0	0
$181$	2.48584e14	0.525489	0.262744	−	0.964866i	$-0.415372\pi$
$181$	2.48584e14	0.525489	0.262744	+	0.964866i	$0.415372\pi$
$182$	0	0
$183$	3.25375e14	0.640401
$184$	0	0
$185$	−5.94452e13	−0.109019
$186$	0	0
$187$	1.02526e15	1.75333
$188$	0	0
$189$	4.55796e13	0.0727393
$190$	0	0
$191$	−5.10794e14	−0.761252	−0.380626	−	0.924729i	$-0.624292\pi$
$191$	−5.10794e14	−0.761252	−0.380626	+	0.924729i	$0.624292\pi$
$192$	0	0
$193$	−1.00056e15	−1.39354	−0.696772	−	0.717293i	$-0.745381\pi$
$193$	−1.00056e15	−1.39354	−0.696772	+	0.717293i	$0.745381\pi$
$194$	0	0
$195$	5.46504e13	0.0711817
$196$	0	0
$197$	−2.67569e14	−0.326141	−0.163071	−	0.986614i	$-0.552140\pi$
$197$	−2.67569e14	−0.326141	−0.163071	+	0.986614i	$0.552140\pi$
$198$	0	0
$199$	−1.05437e15	−1.20350	−0.601752	−	0.798683i	$-0.705530\pi$
$199$	−1.05437e15	−1.20350	−0.601752	+	0.798683i	$0.705530\pi$
$200$	0	0
$201$	2.25490e14	0.241187
$202$	0	0
$203$	3.20216e14	0.321158
$204$	0	0
$205$	2.30408e14	0.216819
$206$	0	0
$207$	−9.88464e13	−0.0873280
$208$	0	0
$209$	−1.97042e15	−1.63534
$210$	0	0
$211$	−5.25130e14	−0.409667	−0.204834	−	0.978797i	$-0.565665\pi$
$211$	−5.25130e14	−0.409667	−0.204834	+	0.978797i	$0.565665\pi$
$212$	0	0
$213$	−8.70425e14	−0.638652
$214$	0	0
$215$	−9.55256e14	−0.659583
$216$	0	0
$217$	−1.81609e14	−0.118072
$218$	0	0
$219$	4.45726e14	0.273011
$220$	0	0
$221$	−6.58097e14	−0.379962
$222$	0	0
$223$	−2.63699e15	−1.43591	−0.717954	−	0.696091i	$-0.754921\pi$
$223$	−2.63699e15	−1.43591	−0.717954	+	0.696091i	$0.754921\pi$
$224$	0	0
$225$	−4.98178e14	−0.255975
$226$	0	0
$227$	−2.56101e15	−1.24235	−0.621174	−	0.783673i	$-0.713344\pi$
$227$	−2.56101e15	−1.24235	−0.621174	+	0.783673i	$0.713344\pi$
$228$	0	0
$229$	−7.08889e14	−0.324824	−0.162412	−	0.986723i	$-0.551927\pi$
$229$	−7.08889e14	−0.324824	−0.162412	+	0.986723i	$0.551927\pi$
$230$	0	0
$231$	5.95123e14	0.257709
$232$	0	0
$233$	−1.38574e15	−0.567371	−0.283685	−	0.958917i	$-0.591557\pi$
$233$	−1.38574e15	−0.567371	−0.283685	+	0.958917i	$0.591557\pi$
$234$	0	0
$235$	−2.30588e15	−0.893091
$236$	0	0
$237$	6.63200e14	0.243098
$238$	0	0
$239$	2.39521e15	0.831298	0.415649	−	0.909525i	$-0.363554\pi$
$239$	2.39521e15	0.831298	0.415649	+	0.909525i	$0.363554\pi$
$240$	0	0
$241$	5.70784e15	1.87655	0.938277	−	0.345884i	$-0.112421\pi$
$241$	5.70784e15	1.87655	0.938277	+	0.345884i	$0.112421\pi$
$242$	0	0
$243$	2.05891e14	0.0641500
$244$	0	0
$245$	−2.32967e14	−0.0688201
$246$	0	0
$247$	1.26478e15	0.354392
$248$	0	0
$249$	3.36300e15	0.894192
$250$	0	0
$251$	−3.51633e15	−0.887587	−0.443793	−	0.896129i	$-0.646368\pi$
$251$	−3.51633e15	−0.887587	−0.443793	+	0.896129i	$0.646368\pi$
$252$	0	0
$253$	−1.29062e15	−0.309395
$254$	0	0
$255$	−1.81296e15	−0.412930
$256$	0	0
$257$	−3.00438e15	−0.650413	−0.325207	−	0.945643i	$-0.605434\pi$
$257$	−3.00438e15	−0.650413	−0.325207	+	0.945643i	$0.605434\pi$
$258$	0	0
$259$	4.15515e14	0.0855339
$260$	0	0
$261$	1.44647e15	0.283235
$262$	0	0
$263$	5.74031e15	1.06960	0.534802	−	0.844978i	$-0.320386\pi$
$263$	5.74031e15	1.06960	0.534802	+	0.844978i	$0.320386\pi$
$264$	0	0
$265$	2.76334e15	0.490159
$266$	0	0
$267$	4.44288e14	0.0750484
$268$	0	0
$269$	−2.09531e15	−0.337177	−0.168589	−	0.985686i	$-0.553921\pi$
$269$	−2.09531e15	−0.337177	−0.168589	+	0.985686i	$0.553921\pi$
$270$	0	0
$271$	2.34576e15	0.359735	0.179868	−	0.983691i	$-0.442433\pi$
$271$	2.34576e15	0.359735	0.179868	+	0.983691i	$0.442433\pi$
$272$	0	0
$273$	−3.82000e14	−0.0558478
$274$	0	0
$275$	−6.50460e15	−0.906897
$276$	0	0
$277$	−1.04986e16	−1.39641	−0.698207	−	0.715896i	$-0.746018\pi$
$277$	−1.04986e16	−1.39641	−0.698207	+	0.715896i	$0.746018\pi$
$278$	0	0
$279$	−8.20357e14	−0.104130
$280$	0	0
$281$	9.61452e15	1.16503	0.582514	−	0.812820i	$-0.302069\pi$
$281$	9.61452e15	1.16503	0.582514	+	0.812820i	$0.302069\pi$
$282$	0	0
$283$	9.42693e15	1.09083	0.545417	−	0.838165i	$-0.316371\pi$
$283$	9.42693e15	1.09083	0.545417	+	0.838165i	$0.316371\pi$
$284$	0	0
$285$	3.48428e15	0.385142
$286$	0	0
$287$	−1.61053e15	−0.170112
$288$	0	0
$289$	1.19270e16	1.20419
$290$	0	0
$291$	3.49016e15	0.336931
$292$	0	0
$293$	−8.67428e15	−0.800928	−0.400464	−	0.916312i	$-0.631151\pi$
$293$	−8.67428e15	−0.800928	−0.400464	+	0.916312i	$0.631151\pi$
$294$	0	0
$295$	4.25698e15	0.376061
$296$	0	0
$297$	2.68827e15	0.227278
$298$	0	0
$299$	8.28426e14	0.0670488
$300$	0	0
$301$	6.67713e15	0.517496
$302$	0	0
$303$	1.64741e15	0.122299
$304$	0	0
$305$	−7.51234e15	−0.534350
$306$	0	0
$307$	−5.33933e15	−0.363988	−0.181994	−	0.983300i	$-0.558255\pi$
$307$	−5.33933e15	−0.363988	−0.181994	+	0.983300i	$0.558255\pi$
$308$	0	0
$309$	1.74428e16	1.13995
$310$	0	0
$311$	−1.23775e16	−0.775694	−0.387847	−	0.921724i	$-0.626781\pi$
$311$	−1.23775e16	−0.775694	−0.387847	+	0.921724i	$0.626781\pi$
$312$	0	0
$313$	4.38517e15	0.263601	0.131801	−	0.991276i	$-0.457924\pi$
$313$	4.38517e15	0.263601	0.131801	+	0.991276i	$0.457924\pi$
$314$	0	0
$315$	−1.05235e15	−0.0606936
$316$	0	0
$317$	3.07503e16	1.70202	0.851009	−	0.525151i	$-0.175991\pi$
$317$	3.07503e16	1.70202	0.851009	+	0.525151i	$0.175991\pi$
$318$	0	0
$319$	1.88862e16	1.00347
$320$	0	0
$321$	1.65054e15	0.0842061
$322$	0	0
$323$	−4.19574e16	−2.05586
$324$	0	0
$325$	4.17520e15	0.196533
$326$	0	0
$327$	5.04082e15	0.228003
$328$	0	0
$329$	1.61178e16	0.700702
$330$	0	0
$331$	3.42644e16	1.43206	0.716031	−	0.698069i	$-0.245957\pi$
$331$	3.42644e16	1.43206	0.716031	+	0.698069i	$0.245957\pi$
$332$	0	0
$333$	1.87695e15	0.0754338
$334$	0	0
$335$	−5.20618e15	−0.201246
$336$	0	0
$337$	3.09364e16	1.15047	0.575235	−	0.817988i	$-0.304910\pi$
$337$	3.09364e16	1.15047	0.575235	+	0.817988i	$0.304910\pi$
$338$	0	0
$339$	6.88048e15	0.246218
$340$	0	0
$341$	−1.07112e16	−0.368923
$342$	0	0
$343$	1.62841e15	0.0539949
$344$	0	0
$345$	2.28219e15	0.0728665
$346$	0	0
$347$	−4.87720e16	−1.49979	−0.749893	−	0.661560i	$-0.769895\pi$
$347$	−4.87720e16	−1.49979	−0.749893	+	0.661560i	$0.769895\pi$
$348$	0	0
$349$	1.38981e16	0.411709	0.205854	−	0.978583i	$-0.434003\pi$
$349$	1.38981e16	0.411709	0.205854	+	0.978583i	$0.434003\pi$
$350$	0	0
$351$	−1.72556e15	−0.0492531
$352$	0	0
$353$	−6.92853e15	−0.190593	−0.0952963	−	0.995449i	$-0.530380\pi$
$353$	−6.92853e15	−0.190593	−0.0952963	+	0.995449i	$0.530380\pi$
$354$	0	0
$355$	2.00966e16	0.532891
$356$	0	0
$357$	1.26724e16	0.323977
$358$	0	0
$359$	−1.74394e16	−0.429950	−0.214975	−	0.976620i	$-0.568967\pi$
$359$	−1.74394e16	−0.429950	−0.214975	+	0.976620i	$0.568967\pi$
$360$	0	0
$361$	3.85839e16	0.917506
$362$	0	0
$363$	9.93315e15	0.227874
$364$	0	0
$365$	−1.02910e16	−0.227801
$366$	0	0
$367$	−3.69229e16	−0.788799	−0.394400	−	0.918939i	$-0.629047\pi$
$367$	−3.69229e16	−0.788799	−0.394400	+	0.918939i	$0.629047\pi$
$368$	0	0
$369$	−7.27503e15	−0.150025
$370$	0	0
$371$	−1.93154e16	−0.384569
$372$	0	0
$373$	−5.45077e16	−1.04797	−0.523987	−	0.851726i	$-0.675556\pi$
$373$	−5.45077e16	−1.04797	−0.523987	+	0.851726i	$0.675556\pi$
$374$	0	0
$375$	2.64801e16	0.491719
$376$	0	0
$377$	−1.21228e16	−0.217462
$378$	0	0
$379$	−3.91684e16	−0.678860	−0.339430	−	0.940631i	$-0.610234\pi$
$379$	−3.91684e16	−0.678860	−0.339430	+	0.940631i	$0.610234\pi$
$380$	0	0
$381$	6.27007e16	1.05017
$382$	0	0
$383$	−2.62532e16	−0.425002	−0.212501	−	0.977161i	$-0.568161\pi$
$383$	−2.62532e16	−0.425002	−0.212501	+	0.977161i	$0.568161\pi$
$384$	0	0
$385$	−1.37403e16	−0.215032
$386$	0	0
$387$	3.01617e16	0.456389
$388$	0	0
$389$	−4.96900e16	−0.727104	−0.363552	−	0.931574i	$-0.618436\pi$
$389$	−4.96900e16	−0.727104	−0.363552	+	0.931574i	$0.618436\pi$
$390$	0	0
$391$	−2.74820e16	−0.388955
$392$	0	0
$393$	5.73872e16	0.785712
$394$	0	0
$395$	−1.53121e16	−0.202840
$396$	0	0
$397$	3.27542e16	0.419884	0.209942	−	0.977714i	$-0.432673\pi$
$397$	3.27542e16	0.419884	0.209942	+	0.977714i	$0.432673\pi$
$398$	0	0
$399$	−2.43547e16	−0.302175
$400$	0	0
$401$	−1.23550e17	−1.48390	−0.741952	−	0.670453i	$-0.766100\pi$
$401$	−1.23550e17	−1.48390	−0.741952	+	0.670453i	$0.766100\pi$
$402$	0	0
$403$	6.87536e15	0.0799490
$404$	0	0
$405$	−4.75366e15	−0.0535267
$406$	0	0
$407$	2.45070e16	0.267255
$408$	0	0
$409$	−8.64479e16	−0.913172	−0.456586	−	0.889679i	$-0.650928\pi$
$409$	−8.64479e16	−0.913172	−0.456586	+	0.889679i	$0.650928\pi$
$410$	0	0
$411$	−2.43088e16	−0.248766
$412$	0	0
$413$	−2.97558e16	−0.295050
$414$	0	0
$415$	−7.76457e16	−0.746113
$416$	0	0
$417$	−1.11904e17	−1.04222
$418$	0	0
$419$	−1.56678e17	−1.41454	−0.707270	−	0.706943i	$-0.750074\pi$
$419$	−1.56678e17	−1.41454	−0.707270	+	0.706943i	$0.750074\pi$
$420$	0	0
$421$	5.72013e16	0.500694	0.250347	−	0.968156i	$-0.419455\pi$
$421$	5.72013e16	0.500694	0.250347	+	0.968156i	$0.419455\pi$
$422$	0	0
$423$	7.28071e16	0.617961
$424$	0	0
$425$	−1.38507e17	−1.14010
$426$	0	0
$427$	5.25104e16	0.419241
$428$	0	0
$429$	−2.25303e16	−0.174499
$430$	0	0
$431$	−2.14881e17	−1.61471	−0.807356	−	0.590065i	$-0.799102\pi$
$431$	−2.14881e17	−1.61471	−0.807356	+	0.590065i	$0.799102\pi$
$432$	0	0
$433$	1.21444e17	0.885534	0.442767	−	0.896637i	$-0.353997\pi$
$433$	1.21444e17	0.885534	0.442767	+	0.896637i	$0.353997\pi$
$434$	0	0
$435$	−3.33965e16	−0.236331
$436$	0	0
$437$	5.28169e16	0.362780
$438$	0	0
$439$	3.61040e16	0.240733	0.120367	−	0.992730i	$-0.461593\pi$
$439$	3.61040e16	0.240733	0.120367	+	0.992730i	$0.461593\pi$
$440$	0	0
$441$	7.35583e15	0.0476190
$442$	0	0
$443$	−9.88387e16	−0.621302	−0.310651	−	0.950524i	$-0.600547\pi$
$443$	−9.88387e16	−0.621302	−0.310651	+	0.950524i	$0.600547\pi$
$444$	0	0
$445$	−1.02578e16	−0.0626203
$446$	0	0
$447$	1.92322e16	0.114033
$448$	0	0
$449$	−4.02823e16	−0.232013	−0.116007	−	0.993248i	$-0.537009\pi$
$449$	−4.02823e16	−0.232013	−0.116007	+	0.993248i	$0.537009\pi$
$450$	0	0
$451$	−9.49885e16	−0.531525
$452$	0	0
$453$	1.94496e17	1.05748
$454$	0	0
$455$	8.81971e15	0.0465994
$456$	0	0
$457$	2.76202e17	1.41831	0.709155	−	0.705053i	$-0.249077\pi$
$457$	2.76202e17	1.41831	0.709155	+	0.705053i	$0.249077\pi$
$458$	0	0
$459$	5.72433e16	0.285721
$460$	0	0
$461$	−5.05459e16	−0.245262	−0.122631	−	0.992452i	$-0.539133\pi$
$461$	−5.05459e16	−0.245262	−0.122631	+	0.992452i	$0.539133\pi$
$462$	0	0
$463$	−9.46280e16	−0.446420	−0.223210	−	0.974770i	$-0.571653\pi$
$463$	−9.46280e16	−0.446420	−0.223210	+	0.974770i	$0.571653\pi$
$464$	0	0
$465$	1.89406e16	0.0868860
$466$	0	0
$467$	−2.37692e17	−1.06037	−0.530183	−	0.847883i	$-0.677877\pi$
$467$	−2.37692e17	−1.06037	−0.530183	+	0.847883i	$0.677877\pi$
$468$	0	0
$469$	3.63906e16	0.157894
$470$	0	0
$471$	1.15819e17	0.488814
$472$	0	0
$473$	3.93815e17	1.61694
$474$	0	0
$475$	2.66193e17	1.06338
$476$	0	0
$477$	−8.72512e16	−0.339158
$478$	0	0
$479$	−7.43070e16	−0.281092	−0.140546	−	0.990074i	$-0.544886\pi$
$479$	−7.43070e16	−0.281092	−0.140546	+	0.990074i	$0.544886\pi$
$480$	0	0
$481$	−1.57306e16	−0.0579166
$482$	0	0
$483$	−1.59522e16	−0.0571696
$484$	0	0
$485$	−8.05816e16	−0.281135
$486$	0	0
$487$	−1.37729e17	−0.467828	−0.233914	−	0.972257i	$-0.575153\pi$
$487$	−1.37729e17	−0.467828	−0.233914	+	0.972257i	$0.575153\pi$
$488$	0	0
$489$	1.41005e17	0.466366
$490$	0	0
$491$	7.05407e16	0.227201	0.113601	−	0.993527i	$-0.463762\pi$
$491$	7.05407e16	0.227201	0.113601	+	0.993527i	$0.463762\pi$
$492$	0	0
$493$	4.02158e17	1.26151
$494$	0	0
$495$	−6.20675e16	−0.189640
$496$	0	0
$497$	−1.40473e17	−0.418096
$498$	0	0
$499$	−4.27709e17	−1.24021	−0.620104	−	0.784519i	$-0.712910\pi$
$499$	−4.27709e17	−1.24021	−0.620104	+	0.784519i	$0.712910\pi$
$500$	0	0
$501$	2.01504e17	0.569296
$502$	0	0
$503$	6.29978e17	1.73434	0.867168	−	0.498015i	$-0.165938\pi$
$503$	6.29978e17	1.73434	0.867168	+	0.498015i	$0.165938\pi$
$504$	0	0
$505$	−3.80357e16	−0.102046
$506$	0	0
$507$	−2.06334e17	−0.539535
$508$	0	0
$509$	6.92905e17	1.76607	0.883036	−	0.469306i	$-0.155496\pi$
$509$	6.92905e17	1.76607	0.883036	+	0.469306i	$0.155496\pi$
$510$	0	0
$511$	7.19330e16	0.178728
$512$	0	0
$513$	−1.10014e17	−0.266493
$514$	0	0
$515$	−4.02724e17	−0.951176
$516$	0	0
$517$	9.50626e17	2.18938
$518$	0	0
$519$	1.16082e17	0.260721
$520$	0	0
$521$	2.21528e17	0.485269	0.242635	−	0.970118i	$-0.421988\pi$
$521$	2.21528e17	0.485269	0.242635	+	0.970118i	$0.421988\pi$
$522$	0	0
$523$	−9.36297e16	−0.200057	−0.100028	−	0.994985i	$-0.531893\pi$
$523$	−9.36297e16	−0.200057	−0.100028	+	0.994985i	$0.531893\pi$
$524$	0	0
$525$	−8.03980e16	−0.167575
$526$	0	0
$527$	−2.28082e17	−0.463790
$528$	0	0
$529$	−4.69441e17	−0.931364
$530$	0	0
$531$	−1.34412e17	−0.260210
$532$	0	0
$533$	6.09716e16	0.115186
$534$	0	0
$535$	−3.81081e16	−0.0702615
$536$	0	0
$537$	−3.36176e17	−0.604971
$538$	0	0
$539$	9.60434e16	0.168710
$540$	0	0
$541$	7.32047e17	1.25533	0.627663	−	0.778485i	$-0.284012\pi$
$541$	7.32047e17	1.25533	0.627663	+	0.778485i	$0.284012\pi$
$542$	0	0
$543$	1.81218e17	0.303391
$544$	0	0
$545$	−1.16384e17	−0.190246
$546$	0	0
$547$	−4.71808e17	−0.753092	−0.376546	−	0.926398i	$-0.622888\pi$
$547$	−4.71808e17	−0.753092	−0.376546	+	0.926398i	$0.622888\pi$
$548$	0	0
$549$	2.37198e17	0.369736
$550$	0	0
$551$	−7.72897e17	−1.17662
$552$	0	0
$553$	1.07030e17	0.159145
$554$	0	0
$555$	−4.33356e16	−0.0629420
$556$	0	0
$557$	6.34214e17	0.899864	0.449932	−	0.893063i	$-0.351448\pi$
$557$	6.34214e17	0.899864	0.449932	+	0.893063i	$0.351448\pi$
$558$	0	0
$559$	−2.52784e17	−0.350406
$560$	0	0
$561$	7.47413e17	1.01228
$562$	0	0
$563$	1.00992e18	1.33654	0.668268	−	0.743920i	$-0.267036\pi$
$563$	1.00992e18	1.33654	0.668268	+	0.743920i	$0.267036\pi$
$564$	0	0
$565$	−1.58858e17	−0.205444
$566$	0	0
$567$	3.32276e16	0.0419961
$568$	0	0
$569$	1.07390e18	1.32659	0.663294	−	0.748359i	$-0.269158\pi$
$569$	1.07390e18	1.32659	0.663294	+	0.748359i	$0.269158\pi$
$570$	0	0
$571$	−1.95807e17	−0.236425	−0.118212	−	0.992988i	$-0.537716\pi$
$571$	−1.95807e17	−0.236425	−0.118212	+	0.992988i	$0.537716\pi$
$572$	0	0
$573$	−3.72369e17	−0.439509
$574$	0	0
$575$	1.74355e17	0.201184
$576$	0	0
$577$	6.07321e17	0.685134	0.342567	−	0.939493i	$-0.388704\pi$
$577$	6.07321e17	0.685134	0.342567	+	0.939493i	$0.388704\pi$
$578$	0	0
$579$	−7.29408e17	−0.804563
$580$	0	0
$581$	5.42734e17	0.585386
$582$	0	0
$583$	−1.13922e18	−1.20161
$584$	0	0
$585$	3.98402e16	0.0410968
$586$	0	0
$587$	8.47840e17	0.855394	0.427697	−	0.903922i	$-0.359325\pi$
$587$	8.47840e17	0.855394	0.427697	+	0.903922i	$0.359325\pi$
$588$	0	0
$589$	4.38344e17	0.432579
$590$	0	0
$591$	−1.95058e17	−0.188298
$592$	0	0
$593$	−2.22479e17	−0.210104	−0.105052	−	0.994467i	$-0.533501\pi$
$593$	−2.22479e17	−0.210104	−0.105052	+	0.994467i	$0.533501\pi$
$594$	0	0
$595$	−2.92583e17	−0.270326
$596$	0	0
$597$	−7.68635e17	−0.694843
$598$	0	0
$599$	−3.76625e17	−0.333146	−0.166573	−	0.986029i	$-0.553270\pi$
$599$	−3.76625e17	−0.333146	−0.166573	+	0.986029i	$0.553270\pi$
$600$	0	0
$601$	−1.20495e18	−1.04300	−0.521502	−	0.853250i	$-0.674628\pi$
$601$	−1.20495e18	−1.04300	−0.521502	+	0.853250i	$0.674628\pi$
$602$	0	0
$603$	1.64383e17	0.139249
$604$	0	0
$605$	−2.29339e17	−0.190138
$606$	0	0
$607$	−2.96571e17	−0.240659	−0.120329	−	0.992734i	$-0.538395\pi$
$607$	−2.96571e17	−0.240659	−0.120329	+	0.992734i	$0.538395\pi$
$608$	0	0
$609$	2.33437e17	0.185421
$610$	0	0
$611$	−6.10191e17	−0.474458
$612$	0	0
$613$	−1.29072e18	−0.982516	−0.491258	−	0.871014i	$-0.663463\pi$
$613$	−1.29072e18	−0.982516	−0.491258	+	0.871014i	$0.663463\pi$
$614$	0	0
$615$	1.67968e17	0.125181
$616$	0	0
$617$	1.10503e18	0.806342	0.403171	−	0.915125i	$-0.367908\pi$
$617$	1.10503e18	0.806342	0.403171	+	0.915125i	$0.367908\pi$
$618$	0	0
$619$	1.91510e18	1.36836	0.684182	−	0.729312i	$-0.260160\pi$
$619$	1.91510e18	1.36836	0.684182	+	0.729312i	$0.260160\pi$
$620$	0	0
$621$	−7.20590e16	−0.0504189
$622$	0	0
$623$	7.17010e16	0.0491307
$624$	0	0
$625$	5.32919e17	0.357636
$626$	0	0
$627$	−1.43643e18	−0.944162
$628$	0	0
$629$	5.21844e17	0.335979
$630$	0	0
$631$	−2.01471e18	−1.27064	−0.635319	−	0.772249i	$-0.719132\pi$
$631$	−2.01471e18	−1.27064	−0.635319	+	0.772249i	$0.719132\pi$
$632$	0	0
$633$	−3.82820e17	−0.236522
$634$	0	0
$635$	−1.44765e18	−0.876262
$636$	0	0
$637$	−6.16487e16	−0.0365610
$638$	0	0
$639$	−6.34540e17	−0.368726
$640$	0	0
$641$	−6.59723e17	−0.375651	−0.187825	−	0.982202i	$-0.560144\pi$
$641$	−6.59723e17	−0.375651	−0.187825	+	0.982202i	$0.560144\pi$
$642$	0	0
$643$	1.35094e17	0.0753812	0.0376906	−	0.999289i	$-0.488000\pi$
$643$	1.35094e17	0.0753812	0.0376906	+	0.999289i	$0.488000\pi$
$644$	0	0
$645$	−6.96381e17	−0.380811
$646$	0	0
$647$	−2.90976e17	−0.155948	−0.0779740	−	0.996955i	$-0.524845\pi$
$647$	−2.90976e17	−0.155948	−0.0779740	+	0.996955i	$0.524845\pi$
$648$	0	0
$649$	−1.75499e18	−0.921899
$650$	0	0
$651$	−1.32393e17	−0.0681691
$652$	0	0
$653$	−1.89972e18	−0.958856	−0.479428	−	0.877581i	$-0.659156\pi$
$653$	−1.89972e18	−0.958856	−0.479428	+	0.877581i	$0.659156\pi$
$654$	0	0
$655$	−1.32497e18	−0.655598
$656$	0	0
$657$	3.24934e17	0.157623
$658$	0	0
$659$	1.04869e18	0.498760	0.249380	−	0.968406i	$-0.419773\pi$
$659$	1.04869e18	0.498760	0.249380	+	0.968406i	$0.419773\pi$
$660$	0	0
$661$	−3.01488e18	−1.40592	−0.702960	−	0.711229i	$-0.748139\pi$
$661$	−3.01488e18	−1.40592	−0.702960	+	0.711229i	$0.748139\pi$
$662$	0	0
$663$	−4.79753e17	−0.219371
$664$	0	0
$665$	5.62307e17	0.252135
$666$	0	0
$667$	−5.06245e17	−0.222609
$668$	0	0
$669$	−1.92236e18	−0.829022
$670$	0	0
$671$	3.09705e18	1.30994
$672$	0	0
$673$	−1.84669e18	−0.766118	−0.383059	−	0.923724i	$-0.625129\pi$
$673$	−1.84669e18	−0.766118	−0.383059	+	0.923724i	$0.625129\pi$
$674$	0	0
$675$	−3.63172e17	−0.147787
$676$	0	0
$677$	3.40006e18	1.35725	0.678625	−	0.734485i	$-0.262576\pi$
$677$	3.40006e18	1.35725	0.678625	+	0.734485i	$0.262576\pi$
$678$	0	0
$679$	5.63256e17	0.220573
$680$	0	0
$681$	−1.86698e18	−0.717270
$682$	0	0
$683$	−4.24420e18	−1.59978	−0.799892	−	0.600144i	$-0.795110\pi$
$683$	−4.24420e18	−1.59978	−0.799892	+	0.600144i	$0.795110\pi$
$684$	0	0
$685$	5.61248e17	0.207571
$686$	0	0
$687$	−5.16780e17	−0.187537
$688$	0	0
$689$	7.31247e17	0.260399
$690$	0	0
$691$	2.79035e18	0.975104	0.487552	−	0.873094i	$-0.337890\pi$
$691$	2.79035e18	0.975104	0.487552	+	0.873094i	$0.337890\pi$
$692$	0	0
$693$	4.33845e17	0.148788
$694$	0	0
$695$	2.58366e18	0.869629
$696$	0	0
$697$	−2.02266e18	−0.668204
$698$	0	0
$699$	−1.01020e18	−0.327572
$700$	0	0
$701$	4.02202e18	1.28020	0.640100	−	0.768292i	$-0.278893\pi$
$701$	4.02202e18	1.28020	0.640100	+	0.768292i	$0.278893\pi$
$702$	0	0
$703$	−1.00292e18	−0.313369
$704$	0	0
$705$	−1.68099e18	−0.515626
$706$	0	0
$707$	2.65865e17	0.0800636
$708$	0	0
$709$	2.08042e18	0.615106	0.307553	−	0.951531i	$-0.400490\pi$
$709$	2.08042e18	0.615106	0.307553	+	0.951531i	$0.400490\pi$
$710$	0	0
$711$	4.83473e17	0.140352
$712$	0	0
$713$	2.87114e17	0.0818412
$714$	0	0
$715$	5.20184e17	0.145602
$716$	0	0
$717$	1.74611e18	0.479950
$718$	0	0
$719$	−2.69620e18	−0.727805	−0.363902	−	0.931437i	$-0.618556\pi$
$719$	−2.69620e18	−0.727805	−0.363902	+	0.931437i	$0.618556\pi$
$720$	0	0
$721$	2.81499e18	0.746274
$722$	0	0
$723$	4.16102e18	1.08343
$724$	0	0
$725$	−2.55143e18	−0.652510
$726$	0	0
$727$	2.74015e17	0.0688335	0.0344168	−	0.999408i	$-0.489043\pi$
$727$	2.74015e17	0.0688335	0.0344168	+	0.999408i	$0.489043\pi$
$728$	0	0
$729$	1.50095e17	0.0370370
$730$	0	0
$731$	8.38578e18	2.03273
$732$	0	0
$733$	−3.01229e18	−0.717334	−0.358667	−	0.933466i	$-0.616769\pi$
$733$	−3.01229e18	−0.717334	−0.358667	+	0.933466i	$0.616769\pi$
$734$	0	0
$735$	−1.69833e17	−0.0397333
$736$	0	0
$737$	2.14631e18	0.493348
$738$	0	0
$739$	−6.42884e18	−1.45192	−0.725962	−	0.687734i	$-0.758605\pi$
$739$	−6.42884e18	−1.45192	−0.725962	+	0.687734i	$0.758605\pi$
$740$	0	0
$741$	9.22023e17	0.204608
$742$	0	0
$743$	8.02122e18	1.74909	0.874547	−	0.484941i	$-0.161159\pi$
$743$	8.02122e18	1.74909	0.874547	+	0.484941i	$0.161159\pi$
$744$	0	0
$745$	−4.44038e17	−0.0951491
$746$	0	0
$747$	2.45162e18	0.516262
$748$	0	0
$749$	2.66371e17	0.0551258
$750$	0	0
$751$	−5.93658e18	−1.20747	−0.603735	−	0.797185i	$-0.706322\pi$
$751$	−5.93658e18	−1.20747	−0.603735	+	0.797185i	$0.706322\pi$
$752$	0	0
$753$	−2.56341e18	−0.512448
$754$	0	0
$755$	−4.49058e18	−0.882363
$756$	0	0
$757$	9.79758e18	1.89232	0.946162	−	0.323694i	$-0.104925\pi$
$757$	9.79758e18	1.89232	0.946162	+	0.323694i	$0.104925\pi$
$758$	0	0
$759$	−9.40859e17	−0.178629
$760$	0	0
$761$	−2.70979e18	−0.505750	−0.252875	−	0.967499i	$-0.581376\pi$
$761$	−2.70979e18	−0.505750	−0.252875	+	0.967499i	$0.581376\pi$
$762$	0	0
$763$	8.13508e17	0.149263
$764$	0	0
$765$	−1.32165e18	−0.238406
$766$	0	0
$767$	1.12650e18	0.199784
$768$	0	0
$769$	4.54004e18	0.791660	0.395830	−	0.918324i	$-0.370457\pi$
$769$	4.54004e18	0.791660	0.395830	+	0.918324i	$0.370457\pi$
$770$	0	0
$771$	−2.19019e18	−0.375516
$772$	0	0
$773$	5.58635e18	0.941805	0.470903	−	0.882185i	$-0.343928\pi$
$773$	5.58635e18	0.941805	0.470903	+	0.882185i	$0.343928\pi$
$774$	0	0
$775$	1.44703e18	0.239892
$776$	0	0
$777$	3.02911e17	0.0493830
$778$	0	0
$779$	3.88729e18	0.623237
$780$	0	0
$781$	−8.28505e18	−1.30636
$782$	0	0
$783$	1.05448e18	0.163526
$784$	0	0
$785$	−2.67406e18	−0.407866
$786$	0	0
$787$	1.03749e19	1.55649	0.778245	−	0.627961i	$-0.216110\pi$
$787$	1.03749e19	1.55649	0.778245	+	0.627961i	$0.216110\pi$
$788$	0	0
$789$	4.18468e18	0.617536
$790$	0	0
$791$	1.11040e18	0.161188
$792$	0	0
$793$	−1.98794e18	−0.283876
$794$	0	0
$795$	2.01448e18	0.282993
$796$	0	0
$797$	9.33254e18	1.28980	0.644898	−	0.764269i	$-0.276900\pi$
$797$	9.33254e18	1.28980	0.644898	+	0.764269i	$0.276900\pi$
$798$	0	0
$799$	2.02423e19	2.75237
$800$	0	0
$801$	3.23886e17	0.0433292
$802$	0	0
$803$	4.24259e18	0.558445
$804$	0	0
$805$	3.68309e17	0.0477023
$806$	0	0
$807$	−1.52748e18	−0.194670
$808$	0	0
$809$	−1.39047e19	−1.74380	−0.871901	−	0.489683i	$-0.837112\pi$
$809$	−1.39047e19	−1.74380	−0.871901	+	0.489683i	$0.837112\pi$
$810$	0	0
$811$	1.49833e19	1.84915	0.924575	−	0.381000i	$-0.124420\pi$
$811$	1.49833e19	1.84915	0.924575	+	0.381000i	$0.124420\pi$
$812$	0	0
$813$	1.71006e18	0.207693
$814$	0	0
$815$	−3.25556e18	−0.389136
$816$	0	0
$817$	−1.61164e19	−1.89594
$818$	0	0
$819$	−2.78478e17	−0.0322437
$820$	0	0
$821$	−1.20542e19	−1.37375	−0.686875	−	0.726776i	$-0.741018\pi$
$821$	−1.20542e19	−1.37375	−0.686875	+	0.726776i	$0.741018\pi$
$822$	0	0
$823$	−1.02391e18	−0.114859	−0.0574293	−	0.998350i	$-0.518290\pi$
$823$	−1.02391e18	−0.114859	−0.0574293	+	0.998350i	$0.518290\pi$
$824$	0	0
$825$	−4.74185e18	−0.523597
$826$	0	0
$827$	−1.67445e19	−1.82006	−0.910031	−	0.414541i	$-0.863942\pi$
$827$	−1.67445e19	−1.82006	−0.910031	+	0.414541i	$0.863942\pi$
$828$	0	0
$829$	2.52152e18	0.269809	0.134905	−	0.990859i	$-0.456927\pi$
$829$	2.52152e18	0.269809	0.134905	+	0.990859i	$0.456927\pi$
$830$	0	0
$831$	−7.65350e18	−0.806220
$832$	0	0
$833$	2.04512e18	0.212093
$834$	0	0
$835$	−4.65237e18	−0.475020
$836$	0	0
$837$	−5.98040e17	−0.0601195
$838$	0	0
$839$	3.56609e18	0.352971	0.176486	−	0.984303i	$-0.443527\pi$
$839$	3.56609e18	0.352971	0.176486	+	0.984303i	$0.443527\pi$
$840$	0	0
$841$	−2.85249e18	−0.278003
$842$	0	0
$843$	7.00899e18	0.672630
$844$	0	0
$845$	4.76389e18	0.450187
$846$	0	0
$847$	1.60305e18	0.149178
$848$	0	0
$849$	6.87223e18	0.629793
$850$	0	0
$851$	−6.56908e17	−0.0592874
$852$	0	0
$853$	−2.52869e18	−0.224764	−0.112382	−	0.993665i	$-0.535848\pi$
$853$	−2.52869e18	−0.224764	−0.112382	+	0.993665i	$0.535848\pi$
$854$	0	0
$855$	2.54004e18	0.222362
$856$	0	0
$857$	3.19030e18	0.275078	0.137539	−	0.990496i	$-0.456081\pi$
$857$	3.19030e18	0.275078	0.137539	+	0.990496i	$0.456081\pi$
$858$	0	0
$859$	4.28369e16	0.00363800	0.00181900	−	0.999998i	$-0.499421\pi$
$859$	4.28369e16	0.00363800	0.00181900	+	0.999998i	$0.499421\pi$
$860$	0	0
$861$	−1.17407e18	−0.0982144
$862$	0	0
$863$	−1.85927e18	−0.153205	−0.0766024	−	0.997062i	$-0.524407\pi$
$863$	−1.85927e18	−0.153205	−0.0766024	+	0.997062i	$0.524407\pi$
$864$	0	0
$865$	−2.68012e18	−0.217546
$866$	0	0
$867$	8.69476e18	0.695238
$868$	0	0
$869$	6.31260e18	0.497256
$870$	0	0
$871$	−1.37768e18	−0.106913
$872$	0	0
$873$	2.54433e18	0.194527
$874$	0	0
$875$	4.27347e18	0.321905
$876$	0	0
$877$	−9.55182e18	−0.708906	−0.354453	−	0.935074i	$-0.615333\pi$
$877$	−9.55182e18	−0.708906	−0.354453	+	0.935074i	$0.615333\pi$
$878$	0	0
$879$	−6.32355e18	−0.462416
$880$	0	0
$881$	4.90258e18	0.353249	0.176625	−	0.984278i	$-0.443482\pi$
$881$	4.90258e18	0.353249	0.176625	+	0.984278i	$0.443482\pi$
$882$	0	0
$883$	−4.98079e18	−0.353634	−0.176817	−	0.984244i	$-0.556580\pi$
$883$	−4.98079e18	−0.353634	−0.176817	+	0.984244i	$0.556580\pi$
$884$	0	0
$885$	3.10334e18	0.217119
$886$	0	0
$887$	−2.07352e19	−1.42957	−0.714783	−	0.699346i	$-0.753475\pi$
$887$	−2.07352e19	−1.42957	−0.714783	+	0.699346i	$0.753475\pi$
$888$	0	0
$889$	1.01189e19	0.687498
$890$	0	0
$891$	1.95975e18	0.131219
$892$	0	0
$893$	−3.89032e19	−2.56715
$894$	0	0
$895$	7.76172e18	0.504787
$896$	0	0
$897$	6.03922e17	0.0387106
$898$	0	0
$899$	−4.20148e18	−0.265439
$900$	0	0
$901$	−2.42582e19	−1.51059
$902$	0	0
$903$	4.86762e18	0.298777
$904$	0	0
$905$	−4.18401e18	−0.253149
$906$	0	0
$907$	−1.23785e19	−0.738278	−0.369139	−	0.929374i	$-0.620347\pi$
$907$	−1.23785e19	−0.738278	−0.369139	+	0.929374i	$0.620347\pi$
$908$	0	0
$909$	1.20096e18	0.0706095
$910$	0	0
$911$	−7.80820e18	−0.452565	−0.226283	−	0.974062i	$-0.572657\pi$
$911$	−7.80820e18	−0.452565	−0.226283	+	0.974062i	$0.572657\pi$
$912$	0	0
$913$	3.20103e19	1.82907
$914$	0	0
$915$	−5.47649e18	−0.308507
$916$	0	0
$917$	9.26138e18	0.514369
$918$	0	0
$919$	3.07890e19	1.68595	0.842976	−	0.537951i	$-0.180802\pi$
$919$	3.07890e19	1.68595	0.842976	+	0.537951i	$0.180802\pi$
$920$	0	0
$921$	−3.89237e18	−0.210149
$922$	0	0
$923$	5.31804e18	0.283100
$924$	0	0
$925$	−3.31076e18	−0.173783
$926$	0	0
$927$	1.27158e19	0.658152
$928$	0	0
$929$	−3.09891e19	−1.58164	−0.790818	−	0.612052i	$-0.790344\pi$
$929$	−3.09891e19	−1.58164	−0.790818	+	0.612052i	$0.790344\pi$
$930$	0	0
$931$	−3.93046e18	−0.197820
$932$	0	0
$933$	−9.02320e18	−0.447847
$934$	0	0
$935$	−1.72565e19	−0.844649
$936$	0	0
$937$	−7.88022e17	−0.0380392	−0.0190196	−	0.999819i	$-0.506054\pi$
$937$	−7.88022e17	−0.0380392	−0.0190196	+	0.999819i	$0.506054\pi$
$938$	0	0
$939$	3.19679e18	0.152190
$940$	0	0
$941$	−2.68836e19	−1.26228	−0.631138	−	0.775670i	$-0.717412\pi$
$941$	−2.68836e19	−1.26228	−0.631138	+	0.775670i	$0.717412\pi$
$942$	0	0
$943$	2.54616e18	0.117912
$944$	0	0
$945$	−7.67166e17	−0.0350415
$946$	0	0
$947$	−3.38940e19	−1.52703	−0.763517	−	0.645788i	$-0.776529\pi$
$947$	−3.38940e19	−1.52703	−0.763517	+	0.645788i	$0.776529\pi$
$948$	0	0
$949$	−2.72325e18	−0.121020
$950$	0	0
$951$	2.24170e19	0.982660
$952$	0	0
$953$	−1.38566e19	−0.599173	−0.299586	−	0.954069i	$-0.596849\pi$
$953$	−1.38566e19	−0.599173	−0.299586	+	0.954069i	$0.596849\pi$
$954$	0	0
$955$	8.59734e18	0.366726
$956$	0	0
$957$	1.37681e19	0.579356
$958$	0	0
$959$	−3.92305e18	−0.162856
$960$	0	0
$961$	−2.20347e19	−0.902413
$962$	0	0
$963$	1.20324e18	0.0486164
$964$	0	0
$965$	1.68408e19	0.671327
$966$	0	0
$967$	−9.82197e17	−0.0386302	−0.0193151	−	0.999813i	$-0.506149\pi$
$967$	−9.82197e17	−0.0386302	−0.0193151	+	0.999813i	$0.506149\pi$
$968$	0	0
$969$	−3.05870e19	−1.18695
$970$	0	0
$971$	−1.82100e19	−0.697244	−0.348622	−	0.937263i	$-0.613350\pi$
$971$	−1.82100e19	−0.697244	−0.348622	+	0.937263i	$0.613350\pi$
$972$	0	0
$973$	−1.80595e19	−0.682294
$974$	0	0
$975$	3.04372e18	0.113468
$976$	0	0
$977$	−1.75890e19	−0.647035	−0.323517	−	0.946222i	$-0.604865\pi$
$977$	−1.75890e19	−0.647035	−0.323517	+	0.946222i	$0.604865\pi$
$978$	0	0
$979$	4.22891e18	0.153511
$980$	0	0
$981$	3.67476e18	0.131638
$982$	0	0
$983$	4.61488e19	1.63141	0.815705	−	0.578469i	$-0.196349\pi$
$983$	4.61488e19	1.63141	0.815705	+	0.578469i	$0.196349\pi$
$984$	0	0
$985$	4.50354e18	0.157115
$986$	0	0
$987$	1.17499e19	0.404550
$988$	0	0
$989$	−1.05562e19	−0.358700
$990$	0	0
$991$	−3.99245e19	−1.33894	−0.669470	−	0.742839i	$-0.733479\pi$
$991$	−3.99245e19	−1.33894	−0.669470	+	0.742839i	$0.733479\pi$
$992$	0	0
$993$	2.49788e19	0.826801
$994$	0	0
$995$	1.77464e19	0.579777
$996$	0	0
$997$	3.94762e18	0.127296	0.0636482	−	0.997972i	$-0.479726\pi$
$997$	3.94762e18	0.127296	0.0636482	+	0.997972i	$0.479726\pi$
$998$	0	0
$999$	1.36830e18	0.0435518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.14.a.d.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.14.a.d.1.2	✓	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)

\(f(q)\)	\(=\)	\(q+729.000 q^{3} -16831.3 q^{5} +117649. q^{7} +531441. q^{9} +O(q^{10})\)	\(q+729.000 q^{3} -16831.3 q^{5} +117649. q^{7} +531441. q^{9} +6.93891e6 q^{11} -4.45397e6 q^{13} -1.22700e7 q^{15} +1.47755e8 q^{17} -2.83966e8 q^{19} +8.57661e7 q^{21} -1.85997e8 q^{23} -9.37410e8 q^{25} +3.87420e8 q^{27} +2.72179e9 q^{29} -1.54365e9 q^{31} +5.05846e9 q^{33} -1.98019e9 q^{35} +3.53182e9 q^{37} -3.24695e9 q^{39} -1.36893e10 q^{41} +5.67546e10 q^{43} -8.94486e9 q^{45} +1.36999e11 q^{47} +1.38413e10 q^{49} +1.07713e11 q^{51} -1.64179e11 q^{53} -1.16791e11 q^{55} -2.07011e11 q^{57} -2.52920e11 q^{59} +4.46331e11 q^{61} +6.25235e10 q^{63} +7.49663e10 q^{65} +3.09315e11 q^{67} -1.35592e11 q^{69} -1.19400e12 q^{71} +6.11420e11 q^{73} -6.83372e11 q^{75} +8.16356e11 q^{77} +9.09740e11 q^{79} +2.82430e11 q^{81} +4.61316e12 q^{83} -2.48691e12 q^{85} +1.98418e12 q^{87} +6.09449e11 q^{89} -5.24005e11 q^{91} -1.12532e12 q^{93} +4.77953e12 q^{95} +4.78760e12 q^{97} +3.68762e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2916 q^{3} + 19278 q^{5} + 470596 q^{7} + 2125764 q^{9}+O(q^{10}) \) 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

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