LMFDB - Embedded newform 84.18.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("84.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.3
Root			$35359.1$ of defining polynomial
Character	$\chi$	$=$	84.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6561.00 q^{3} +556376. q^{5} +5.76480e6 q^{7} +4.30467e7 q^{9} +O(q^{10})$	$q-6561.00 q^{3} +556376. q^{5} +5.76480e6 q^{7} +4.30467e7 q^{9} +3.17805e8 q^{11} -2.30148e9 q^{13} -3.65038e9 q^{15} +2.10576e10 q^{17} +2.14549e10 q^{19} -3.78229e10 q^{21} -5.72229e11 q^{23} -4.53385e11 q^{25} -2.82430e11 q^{27} +7.66682e11 q^{29} +2.35994e12 q^{31} -2.08512e12 q^{33} +3.20740e12 q^{35} -2.77407e13 q^{37} +1.51000e13 q^{39} -6.70587e12 q^{41} -7.30774e13 q^{43} +2.39502e13 q^{45} +9.65581e13 q^{47} +3.32329e13 q^{49} -1.38159e14 q^{51} +4.00432e14 q^{53} +1.76819e14 q^{55} -1.40766e14 q^{57} +2.47328e14 q^{59} +1.25955e15 q^{61} +2.48156e14 q^{63} -1.28049e15 q^{65} +8.48157e13 q^{67} +3.75440e15 q^{69} +1.98978e15 q^{71} +4.28825e15 q^{73} +2.97466e15 q^{75} +1.83208e15 q^{77} -1.06756e16 q^{79} +1.85302e15 q^{81} +4.27104e15 q^{83} +1.17159e16 q^{85} -5.03020e15 q^{87} -3.98831e16 q^{89} -1.32676e16 q^{91} -1.54836e16 q^{93} +1.19370e16 q^{95} +3.22278e16 q^{97} +1.36804e16 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 26244 q^{3} + 528276 q^{5} + 23059204 q^{7} + 172186884 q^{9}+O(q^{10}) $ 4 * q - 26244 * q^3 + 528276 * q^5 + 23059204 * q^7 + 172186884 * q^9	$ 4 q - 26244 q^{3} + 528276 q^{5} + 23059204 q^{7} + 172186884 q^{9} - 269632020 q^{11} + 1984752056 q^{13} - 3466018836 q^{15} - 29765040036 q^{17} - 91677835096 q^{19} - 151291437444 q^{21} + 38384818716 q^{23} - 235654160804 q^{25} - 1129718145924 q^{27} - 3815624693904 q^{29} - 4907739231784 q^{31} + 1769055683220 q^{33} + 3045406013076 q^{35} + 13670824759208 q^{37} - 13021958239416 q^{39} + 10307194403604 q^{41} + 125363311182368 q^{43} + 22740549582996 q^{45} - 25278931593000 q^{47} + 132931722278404 q^{49} + 195288427676196 q^{51} - 239725859759304 q^{53} + 694573116031368 q^{55} + 601498276064856 q^{57} + 13\!\cdots\!00 q^{59}+ \cdots - 11\!\cdots\!20 q^{99}+O(q^{100}) $ 4 * q - 26244 * q^3 + 528276 * q^5 + 23059204 * q^7 + 172186884 * q^9 - 269632020 * q^11 + 1984752056 * q^13 - 3466018836 * q^15 - 29765040036 * q^17 - 91677835096 * q^19 - 151291437444 * q^21 + 38384818716 * q^23 - 235654160804 * q^25 - 1129718145924 * q^27 - 3815624693904 * q^29 - 4907739231784 * q^31 + 1769055683220 * q^33 + 3045406013076 * q^35 + 13670824759208 * q^37 - 13021958239416 * q^39 + 10307194403604 * q^41 + 125363311182368 * q^43 + 22740549582996 * q^45 - 25278931593000 * q^47 + 132931722278404 * q^49 + 195288427676196 * q^51 - 239725859759304 * q^53 + 694573116031368 * q^55 + 601498276064856 * q^57 + 1374756208497000 * q^59 + 652646281638080 * q^61 + 992623121070084 * q^63 + 4153293849278904 * q^65 + 6617359123302392 * q^67 - 251842795595676 * q^69 + 2405715947056548 * q^71 + 4250185178265824 * q^73 + 1546126949035044 * q^75 - 1554374938528020 * q^77 + 20052572228473688 * q^79 + 7412080755407364 * q^81 - 5593593799375584 * q^83 - 13012862591692368 * q^85 + 25034313616704144 * q^87 + 1518004216828212 * q^89 + 11441700637180856 * q^91 + 32199677099734824 * q^93 - 149304801904677408 * q^95 - 102152247026326000 * q^97 - 11606774337606420 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−6561.00	−0.577350
$3$	−6561.00	−0.577350
$4$	0	0
$5$	556376.	0.636976	0.318488	−	0.947927i	$-0.396825\pi$
$5$	556376.	0.636976	0.318488	+	0.947927i	$0.396825\pi$
$6$	0	0
$7$	5.76480e6	0.377964
$7$	5.76480e6	0.377964
$8$	0	0
$9$	4.30467e7	0.333333
$10$	0	0
$11$	3.17805e8	0.447015	0.223508	−	0.974702i	$-0.428249\pi$
$11$	3.17805e8	0.447015	0.223508	+	0.974702i	$0.428249\pi$
$12$	0	0
$13$	−2.30148e9	−0.782508	−0.391254	−	0.920283i	$-0.627959\pi$
$13$	−2.30148e9	−0.782508	−0.391254	+	0.920283i	$0.627959\pi$
$14$	0	0
$15$	−3.65038e9	−0.367758
$16$	0	0
$17$	2.10576e10	0.732138	0.366069	−	0.930588i	$-0.380703\pi$
$17$	2.10576e10	0.732138	0.366069	+	0.930588i	$0.380703\pi$
$18$	0	0
$19$	2.14549e10	0.289815	0.144908	−	0.989445i	$-0.453711\pi$
$19$	2.14549e10	0.289815	0.144908	+	0.989445i	$0.453711\pi$
$20$	0	0
$21$	−3.78229e10	−0.218218
$22$	0	0
$23$	−5.72229e11	−1.52364	−0.761822	−	0.647787i	$-0.775695\pi$
$23$	−5.72229e11	−1.52364	−0.761822	+	0.647787i	$0.775695\pi$
$24$	0	0
$25$	−4.53385e11	−0.594261
$26$	0	0
$27$	−2.82430e11	−0.192450
$28$	0	0
$29$	7.66682e11	0.284598	0.142299	−	0.989824i	$-0.454551\pi$
$29$	7.66682e11	0.284598	0.142299	+	0.989824i	$0.454551\pi$
$30$	0	0
$31$	2.35994e12	0.496967	0.248483	−	0.968636i	$-0.420068\pi$
$31$	2.35994e12	0.496967	0.248483	+	0.968636i	$0.420068\pi$
$32$	0	0
$33$	−2.08512e12	−0.258084
$34$	0	0
$35$	3.20740e12	0.240754
$36$	0	0
$37$	−2.77407e13	−1.29838	−0.649191	−	0.760626i	$-0.724892\pi$
$37$	−2.77407e13	−1.29838	−0.649191	+	0.760626i	$0.724892\pi$
$38$	0	0
$39$	1.51000e13	0.451781
$40$	0	0
$41$	−6.70587e12	−0.131157	−0.0655786	−	0.997847i	$-0.520889\pi$
$41$	−6.70587e12	−0.131157	−0.0655786	+	0.997847i	$0.520889\pi$
$42$	0	0
$43$	−7.30774e13	−0.953457	−0.476728	−	0.879051i	$-0.658177\pi$
$43$	−7.30774e13	−0.953457	−0.476728	+	0.879051i	$0.658177\pi$
$44$	0	0
$45$	2.39502e13	0.212325
$46$	0	0
$47$	9.65581e13	0.591503	0.295752	−	0.955265i	$-0.404430\pi$
$47$	9.65581e13	0.591503	0.295752	+	0.955265i	$0.404430\pi$
$48$	0	0
$49$	3.32329e13	0.142857
$50$	0	0
$51$	−1.38159e14	−0.422700
$52$	0	0
$53$	4.00432e14	0.883453	0.441727	−	0.897150i	$-0.354366\pi$
$53$	4.00432e14	0.883453	0.441727	+	0.897150i	$0.354366\pi$
$54$	0	0
$55$	1.76819e14	0.284738
$56$	0	0
$57$	−1.40766e14	−0.167325
$58$	0	0
$59$	2.47328e14	0.219296	0.109648	−	0.993970i	$-0.465028\pi$
$59$	2.47328e14	0.219296	0.109648	+	0.993970i	$0.465028\pi$
$60$	0	0
$61$	1.25955e15	0.841222	0.420611	−	0.907241i	$-0.361816\pi$
$61$	1.25955e15	0.841222	0.420611	+	0.907241i	$0.361816\pi$
$62$	0	0
$63$	2.48156e14	0.125988
$64$	0	0
$65$	−1.28049e15	−0.498439
$66$	0	0
$67$	8.48157e13	0.0255176	0.0127588	−	0.999919i	$-0.495939\pi$
$67$	8.48157e13	0.0255176	0.0127588	+	0.999919i	$0.495939\pi$
$68$	0	0
$69$	3.75440e15	0.879676
$70$	0	0
$71$	1.98978e15	0.365687	0.182844	−	0.983142i	$-0.441470\pi$
$71$	1.98978e15	0.365687	0.182844	+	0.983142i	$0.441470\pi$
$72$	0	0
$73$	4.28825e15	0.622351	0.311176	−	0.950352i	$-0.399277\pi$
$73$	4.28825e15	0.622351	0.311176	+	0.950352i	$0.399277\pi$
$74$	0	0
$75$	2.97466e15	0.343097
$76$	0	0
$77$	1.83208e15	0.168956
$78$	0	0
$79$	−1.06756e16	−0.791700	−0.395850	−	0.918315i	$-0.629550\pi$
$79$	−1.06756e16	−0.791700	−0.395850	+	0.918315i	$0.629550\pi$
$80$	0	0
$81$	1.85302e15	0.111111
$82$	0	0
$83$	4.27104e15	0.208147	0.104073	−	0.994570i	$-0.466812\pi$
$83$	4.27104e15	0.208147	0.104073	+	0.994570i	$0.466812\pi$
$84$	0	0
$85$	1.17159e16	0.466355
$86$	0	0
$87$	−5.03020e15	−0.164313
$88$	0	0
$89$	−3.98831e16	−1.07392	−0.536962	−	0.843607i	$-0.680428\pi$
$89$	−3.98831e16	−1.07392	−0.536962	+	0.843607i	$0.680428\pi$
$90$	0	0
$91$	−1.32676e16	−0.295760
$92$	0	0
$93$	−1.54836e16	−0.286924
$94$	0	0
$95$	1.19370e16	0.184606
$96$	0	0
$97$	3.22278e16	0.417514	0.208757	−	0.977968i	$-0.433058\pi$
$97$	3.22278e16	0.417514	0.208757	+	0.977968i	$0.433058\pi$
$98$	0	0
$99$	1.36804e16	0.149005
$100$	0	0
$101$	−1.00833e17	−0.926558	−0.463279	−	0.886213i	$-0.653327\pi$
$101$	−1.00833e17	−0.926558	−0.463279	+	0.886213i	$0.653327\pi$
$102$	0	0
$103$	−1.22952e17	−0.956356	−0.478178	−	0.878263i	$-0.658703\pi$
$103$	−1.22952e17	−0.956356	−0.478178	+	0.878263i	$0.658703\pi$
$104$	0	0
$105$	−2.10437e16	−0.139000
$106$	0	0
$107$	1.11649e17	0.628193	0.314097	−	0.949391i	$-0.398298\pi$
$107$	1.11649e17	0.628193	0.314097	+	0.949391i	$0.398298\pi$
$108$	0	0
$109$	−9.34069e16	−0.449008	−0.224504	−	0.974473i	$-0.572076\pi$
$109$	−9.34069e16	−0.449008	−0.224504	+	0.974473i	$0.572076\pi$
$110$	0	0
$111$	1.82007e17	0.749621
$112$	0	0
$113$	1.74099e17	0.616071	0.308035	−	0.951375i	$-0.400328\pi$
$113$	1.74099e17	0.616071	0.308035	+	0.951375i	$0.400328\pi$
$114$	0	0
$115$	−3.18374e17	−0.970525
$116$	0	0
$117$	−9.90711e16	−0.260836
$118$	0	0
$119$	1.21393e17	0.276722
$120$	0	0
$121$	−4.04447e17	−0.800177
$122$	0	0
$123$	4.39972e16	0.0757236
$124$	0	0
$125$	−6.76734e17	−1.01551
$126$	0	0
$127$	2.97979e17	0.390710	0.195355	−	0.980733i	$-0.437414\pi$
$127$	2.97979e17	0.390710	0.195355	+	0.980733i	$0.437414\pi$
$128$	0	0
$129$	4.79461e17	0.550479
$130$	0	0
$131$	2.18977e16	0.0220594	0.0110297	−	0.999939i	$-0.496489\pi$
$131$	2.18977e16	0.0220594	0.0110297	+	0.999939i	$0.496489\pi$
$132$	0	0
$133$	1.23683e17	0.109540
$134$	0	0
$135$	−1.57137e17	−0.122586
$136$	0	0
$137$	8.14435e17	0.560702	0.280351	−	0.959898i	$-0.409549\pi$
$137$	8.14435e17	0.560702	0.280351	+	0.959898i	$0.409549\pi$
$138$	0	0
$139$	−2.01721e18	−1.22779	−0.613895	−	0.789387i	$-0.710398\pi$
$139$	−2.01721e18	−1.22779	−0.613895	+	0.789387i	$0.710398\pi$
$140$	0	0
$141$	−6.33518e17	−0.341504
$142$	0	0
$143$	−7.31421e17	−0.349793
$144$	0	0
$145$	4.26563e17	0.181282
$146$	0	0
$147$	−2.18041e17	−0.0824786
$148$	0	0
$149$	−1.80627e18	−0.609114	−0.304557	−	0.952494i	$-0.598508\pi$
$149$	−1.80627e18	−0.609114	−0.304557	+	0.952494i	$0.598508\pi$
$150$	0	0
$151$	−3.29872e18	−0.993211	−0.496606	−	0.867976i	$-0.665420\pi$
$151$	−3.29872e18	−0.993211	−0.496606	+	0.867976i	$0.665420\pi$
$152$	0	0
$153$	9.06461e17	0.244046
$154$	0	0
$155$	1.31302e18	0.316556
$156$	0	0
$157$	−2.21887e18	−0.479717	−0.239858	−	0.970808i	$-0.577101\pi$
$157$	−2.21887e18	−0.479717	−0.239858	+	0.970808i	$0.577101\pi$
$158$	0	0
$159$	−2.62723e18	−0.510062
$160$	0	0
$161$	−3.29879e18	−0.575883
$162$	0	0
$163$	7.97621e18	1.25372	0.626862	−	0.779131i	$-0.284339\pi$
$163$	7.97621e18	1.25372	0.626862	+	0.779131i	$0.284339\pi$
$164$	0	0
$165$	−1.16011e18	−0.164394
$166$	0	0
$167$	6.58844e18	0.842738	0.421369	−	0.906889i	$-0.361550\pi$
$167$	6.58844e18	0.842738	0.421369	+	0.906889i	$0.361550\pi$
$168$	0	0
$169$	−3.35361e18	−0.387682
$170$	0	0
$171$	9.23564e17	0.0966052
$172$	0	0
$173$	−1.00396e19	−0.951313	−0.475656	−	0.879631i	$-0.657789\pi$
$173$	−1.00396e19	−0.951313	−0.475656	+	0.879631i	$0.657789\pi$
$174$	0	0
$175$	−2.61368e18	−0.224610
$176$	0	0
$177$	−1.62272e18	−0.126611
$178$	0	0
$179$	−8.80949e18	−0.624741	−0.312371	−	0.949960i	$-0.601123\pi$
$179$	−8.80949e18	−0.624741	−0.312371	+	0.949960i	$0.601123\pi$
$180$	0	0
$181$	−2.37517e19	−1.53259	−0.766296	−	0.642488i	$-0.777902\pi$
$181$	−2.37517e19	−1.53259	−0.766296	+	0.642488i	$0.777902\pi$
$182$	0	0
$183$	−8.26388e18	−0.485680
$184$	0	0
$185$	−1.54342e19	−0.827038
$186$	0	0
$187$	6.69220e18	0.327277
$188$	0	0
$189$	−1.62815e18	−0.0727393
$190$	0	0
$191$	−3.86575e19	−1.57925	−0.789623	−	0.613592i	$-0.789724\pi$
$191$	−3.86575e19	−1.57925	−0.789623	+	0.613592i	$0.789724\pi$
$192$	0	0
$193$	−2.99609e19	−1.12026	−0.560129	−	0.828406i	$-0.689248\pi$
$193$	−2.99609e19	−1.12026	−0.560129	+	0.828406i	$0.689248\pi$
$194$	0	0
$195$	8.40128e18	0.287774
$196$	0	0
$197$	−5.43933e19	−1.70837	−0.854186	−	0.519967i	$-0.825944\pi$
$197$	−5.43933e19	−1.70837	−0.854186	+	0.519967i	$0.825944\pi$
$198$	0	0
$199$	−2.01556e19	−0.580958	−0.290479	−	0.956881i	$-0.593815\pi$
$199$	−2.01556e19	−0.580958	−0.290479	+	0.956881i	$0.593815\pi$
$200$	0	0
$201$	−5.56476e17	−0.0147326
$202$	0	0
$203$	4.41977e18	0.107568
$204$	0	0
$205$	−3.73098e18	−0.0835440
$206$	0	0
$207$	−2.46326e19	−0.507881
$208$	0	0
$209$	6.81848e18	0.129552
$210$	0	0
$211$	6.42702e18	0.112618	0.0563091	−	0.998413i	$-0.482067\pi$
$211$	6.42702e18	0.112618	0.0563091	+	0.998413i	$0.482067\pi$
$212$	0	0
$213$	−1.30550e19	−0.211130
$214$	0	0
$215$	−4.06585e19	−0.607329
$216$	0	0
$217$	1.36046e19	0.187836
$218$	0	0
$219$	−2.81352e19	−0.359315
$220$	0	0
$221$	−4.84636e19	−0.572904
$222$	0	0
$223$	−9.27886e19	−1.01602	−0.508011	−	0.861351i	$-0.669619\pi$
$223$	−9.27886e19	−1.01602	−0.508011	+	0.861351i	$0.669619\pi$
$224$	0	0
$225$	−1.95168e19	−0.198087
$226$	0	0
$227$	3.73635e19	0.351745	0.175872	−	0.984413i	$-0.443725\pi$
$227$	3.73635e19	0.351745	0.175872	+	0.984413i	$0.443725\pi$
$228$	0	0
$229$	8.39110e19	0.733191	0.366596	−	0.930380i	$-0.380523\pi$
$229$	8.39110e19	0.733191	0.366596	+	0.930380i	$0.380523\pi$
$230$	0	0
$231$	−1.20203e19	−0.0975467
$232$	0	0
$233$	−1.14667e20	−0.864798	−0.432399	−	0.901682i	$-0.642333\pi$
$233$	−1.14667e20	−0.864798	−0.432399	+	0.901682i	$0.642333\pi$
$234$	0	0
$235$	5.37226e19	0.376773
$236$	0	0
$237$	7.00424e19	0.457088
$238$	0	0
$239$	2.45289e20	1.49038	0.745189	−	0.666853i	$-0.232359\pi$
$239$	2.45289e20	1.49038	0.745189	+	0.666853i	$0.232359\pi$
$240$	0	0
$241$	−2.87791e20	−1.62904	−0.814521	−	0.580134i	$-0.803000\pi$
$241$	−2.87791e20	−1.62904	−0.814521	+	0.580134i	$0.803000\pi$
$242$	0	0
$243$	−1.21577e19	−0.0641500
$244$	0	0
$245$	1.84900e19	0.0909966
$246$	0	0
$247$	−4.93781e19	−0.226783
$248$	0	0
$249$	−2.80223e19	−0.120174
$250$	0	0
$251$	−2.50783e20	−1.00478	−0.502390	−	0.864641i	$-0.667546\pi$
$251$	−2.50783e20	−1.00478	−0.502390	+	0.864641i	$0.667546\pi$
$252$	0	0
$253$	−1.81857e20	−0.681092
$254$	0	0
$255$	−7.68683e19	−0.269250
$256$	0	0
$257$	3.38190e20	1.10848	0.554241	−	0.832356i	$-0.313009\pi$
$257$	3.38190e20	1.10848	0.554241	+	0.832356i	$0.313009\pi$
$258$	0	0
$259$	−1.59919e20	−0.490742
$260$	0	0
$261$	3.30031e19	0.0948661
$262$	0	0
$263$	−3.27292e20	−0.881681	−0.440840	−	0.897586i	$-0.645320\pi$
$263$	−3.27292e20	−0.881681	−0.440840	+	0.897586i	$0.645320\pi$
$264$	0	0
$265$	2.22790e20	0.562738
$266$	0	0
$267$	2.61673e20	0.620030
$268$	0	0
$269$	−1.50638e20	−0.334998	−0.167499	−	0.985872i	$-0.553569\pi$
$269$	−1.50638e20	−0.334998	−0.167499	+	0.985872i	$0.553569\pi$
$270$	0	0
$271$	−1.08775e20	−0.227139	−0.113569	−	0.993530i	$-0.536228\pi$
$271$	−1.08775e20	−0.227139	−0.113569	+	0.993530i	$0.536228\pi$
$272$	0	0
$273$	8.70485e19	0.170757
$274$	0	0
$275$	−1.44088e20	−0.265644
$276$	0	0
$277$	6.35624e19	0.110185	0.0550925	−	0.998481i	$-0.482455\pi$
$277$	6.35624e19	0.110185	0.0550925	+	0.998481i	$0.482455\pi$
$278$	0	0
$279$	1.01588e20	0.165656
$280$	0	0
$281$	−8.69402e18	−0.0133419	−0.00667094	−	0.999978i	$-0.502123\pi$
$281$	−8.69402e18	−0.0133419	−0.00667094	+	0.999978i	$0.502123\pi$
$282$	0	0
$283$	5.43359e20	0.785058	0.392529	−	0.919740i	$-0.371600\pi$
$283$	5.43359e20	0.785058	0.392529	+	0.919740i	$0.371600\pi$
$284$	0	0
$285$	−7.83187e19	−0.106582
$286$	0	0
$287$	−3.86580e19	−0.0495728
$288$	0	0
$289$	−3.83818e20	−0.463974
$290$	0	0
$291$	−2.11447e20	−0.241052
$292$	0	0
$293$	−9.33778e19	−0.100431	−0.0502156	−	0.998738i	$-0.515991\pi$
$293$	−9.33778e19	−0.100431	−0.0502156	+	0.998738i	$0.515991\pi$
$294$	0	0
$295$	1.37607e20	0.139686
$296$	0	0
$297$	−8.97574e19	−0.0860281
$298$	0	0
$299$	1.31697e21	1.19226
$300$	0	0
$301$	−4.21276e20	−0.360373
$302$	0	0
$303$	6.61568e20	0.534948
$304$	0	0
$305$	7.00781e20	0.535838
$306$	0	0
$307$	8.01627e20	0.579824	0.289912	−	0.957053i	$-0.406374\pi$
$307$	8.01627e20	0.579824	0.289912	+	0.957053i	$0.406374\pi$
$308$	0	0
$309$	8.06689e20	0.552153
$310$	0	0
$311$	2.14187e21	1.38781	0.693906	−	0.720065i	$-0.255888\pi$
$311$	2.14187e21	1.38781	0.693906	+	0.720065i	$0.255888\pi$
$312$	0	0
$313$	1.96600e21	1.20630	0.603152	−	0.797627i	$-0.293911\pi$
$313$	1.96600e21	1.20630	0.603152	+	0.797627i	$0.293911\pi$
$314$	0	0
$315$	1.38068e20	0.0802514
$316$	0	0
$317$	−1.31329e21	−0.723365	−0.361683	−	0.932301i	$-0.617798\pi$
$317$	−1.31329e21	−0.723365	−0.361683	+	0.932301i	$0.617798\pi$
$318$	0	0
$319$	2.43655e20	0.127220
$320$	0	0
$321$	−7.32530e20	−0.362688
$322$	0	0
$323$	4.51789e20	0.212185
$324$	0	0
$325$	1.04346e21	0.465014
$326$	0	0
$327$	6.12843e20	0.259235
$328$	0	0
$329$	5.56638e20	0.223567
$330$	0	0
$331$	3.83414e21	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$331$	3.83414e21	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$332$	0	0
$333$	−1.19414e21	−0.432794
$334$	0	0
$335$	4.71894e19	0.0162541
$336$	0	0
$337$	4.25556e21	1.39349	0.696743	−	0.717321i	$-0.254632\pi$
$337$	4.25556e21	1.39349	0.696743	+	0.717321i	$0.254632\pi$
$338$	0	0
$339$	−1.14227e21	−0.355689
$340$	0	0
$341$	7.50001e20	0.222152
$342$	0	0
$343$	1.91581e20	0.0539949
$344$	0	0
$345$	2.08885e21	0.560333
$346$	0	0
$347$	−7.03278e20	−0.179608	−0.0898042	−	0.995959i	$-0.528624\pi$
$347$	−7.03278e20	−0.179608	−0.0898042	+	0.995959i	$0.528624\pi$
$348$	0	0
$349$	1.91207e21	0.465038	0.232519	−	0.972592i	$-0.425303\pi$
$349$	1.91207e21	0.465038	0.232519	+	0.972592i	$0.425303\pi$
$350$	0	0
$351$	6.50006e20	0.150594
$352$	0	0
$353$	−5.23554e21	−1.15578	−0.577892	−	0.816114i	$-0.696124\pi$
$353$	−5.23554e21	−1.15578	−0.577892	+	0.816114i	$0.696124\pi$
$354$	0	0
$355$	1.10707e21	0.232934
$356$	0	0
$357$	−7.96459e20	−0.159766
$358$	0	0
$359$	4.28282e21	0.819269	0.409634	−	0.912250i	$-0.365656\pi$
$359$	4.28282e21	0.819269	0.409634	+	0.912250i	$0.365656\pi$
$360$	0	0
$361$	−5.02007e21	−0.916007
$362$	0	0
$363$	2.65358e21	0.461983
$364$	0	0
$365$	2.38588e21	0.396423
$366$	0	0
$367$	−1.90172e21	−0.301638	−0.150819	−	0.988561i	$-0.548191\pi$
$367$	−1.90172e21	−0.301638	−0.150819	+	0.988561i	$0.548191\pi$
$368$	0	0
$369$	−2.88666e20	−0.0437191
$370$	0	0
$371$	2.30841e21	0.333914
$372$	0	0
$373$	4.78701e21	0.661514	0.330757	−	0.943716i	$-0.392696\pi$
$373$	4.78701e21	0.661514	0.330757	+	0.943716i	$0.392696\pi$
$374$	0	0
$375$	4.44005e21	0.586303
$376$	0	0
$377$	−1.76450e21	−0.222700
$378$	0	0
$379$	−3.98095e21	−0.480345	−0.240173	−	0.970730i	$-0.577204\pi$
$379$	−3.98095e21	−0.480345	−0.240173	+	0.970730i	$0.577204\pi$
$380$	0	0
$381$	−1.95504e21	−0.225577
$382$	0	0
$383$	2.20048e21	0.242844	0.121422	−	0.992601i	$-0.461255\pi$
$383$	2.20048e21	0.242844	0.121422	+	0.992601i	$0.461255\pi$
$384$	0	0
$385$	1.01933e21	0.107621
$386$	0	0
$387$	−3.14574e21	−0.317819
$388$	0	0
$389$	5.26952e21	0.509565	0.254782	−	0.966998i	$-0.417996\pi$
$389$	5.26952e21	0.509565	0.254782	+	0.966998i	$0.417996\pi$
$390$	0	0
$391$	−1.20498e22	−1.11552
$392$	0	0
$393$	−1.43671e20	−0.0127360
$394$	0	0
$395$	−5.93963e21	−0.504294
$396$	0	0
$397$	1.58499e21	0.128916	0.0644582	−	0.997920i	$-0.479468\pi$
$397$	1.58499e21	0.128916	0.0644582	+	0.997920i	$0.479468\pi$
$398$	0	0
$399$	−8.11487e20	−0.0632429
$400$	0	0
$401$	−2.34177e22	−1.74911	−0.874556	−	0.484924i	$-0.838847\pi$
$401$	−2.34177e22	−1.74911	−0.874556	+	0.484924i	$0.838847\pi$
$402$	0	0
$403$	−5.43136e21	−0.388880
$404$	0	0
$405$	1.03098e21	0.0707751
$406$	0	0
$407$	−8.81611e21	−0.580396
$408$	0	0
$409$	1.37482e22	0.868155	0.434077	−	0.900876i	$-0.357074\pi$
$409$	1.37482e22	0.868155	0.434077	+	0.900876i	$0.357074\pi$
$410$	0	0
$411$	−5.34351e21	−0.323721
$412$	0	0
$413$	1.42579e21	0.0828860
$414$	0	0
$415$	2.37630e21	0.132585
$416$	0	0
$417$	1.32349e22	0.708865
$418$	0	0
$419$	−2.24305e22	−1.15350	−0.576752	−	0.816919i	$-0.695680\pi$
$419$	−2.24305e22	−1.15350	−0.576752	+	0.816919i	$0.695680\pi$
$420$	0	0
$421$	3.98714e21	0.196908	0.0984541	−	0.995142i	$-0.468610\pi$
$421$	3.98714e21	0.196908	0.0984541	+	0.995142i	$0.468610\pi$
$422$	0	0
$423$	4.15651e21	0.197168
$424$	0	0
$425$	−9.54721e21	−0.435082
$426$	0	0
$427$	7.26103e21	0.317952
$428$	0	0
$429$	4.79885e21	0.201953
$430$	0	0
$431$	−1.93515e22	−0.782814	−0.391407	−	0.920218i	$-0.628011\pi$
$431$	−1.93515e22	−0.782814	−0.391407	+	0.920218i	$0.628011\pi$
$432$	0	0
$433$	−3.56561e22	−1.38671	−0.693356	−	0.720595i	$-0.743869\pi$
$433$	−3.56561e22	−1.38671	−0.693356	+	0.720595i	$0.743869\pi$
$434$	0	0
$435$	−2.79868e21	−0.104663
$436$	0	0
$437$	−1.22771e22	−0.441576
$438$	0	0
$439$	−4.00001e22	−1.38392	−0.691962	−	0.721934i	$-0.743253\pi$
$439$	−4.00001e22	−1.38392	−0.691962	+	0.721934i	$0.743253\pi$
$440$	0	0
$441$	1.43057e21	0.0476190
$442$	0	0
$443$	2.30685e22	0.738902	0.369451	−	0.929250i	$-0.379546\pi$
$443$	2.30685e22	0.738902	0.369451	+	0.929250i	$0.379546\pi$
$444$	0	0
$445$	−2.21900e22	−0.684064
$446$	0	0
$447$	1.18509e22	0.351672
$448$	0	0
$449$	−1.45829e22	−0.416628	−0.208314	−	0.978062i	$-0.566798\pi$
$449$	−1.45829e22	−0.416628	−0.208314	+	0.978062i	$0.566798\pi$
$450$	0	0
$451$	−2.13116e21	−0.0586293
$452$	0	0
$453$	2.16429e22	0.573431
$454$	0	0
$455$	−7.38175e21	−0.188392
$456$	0	0
$457$	3.40271e22	0.836638	0.418319	−	0.908300i	$-0.362619\pi$
$457$	3.40271e22	0.836638	0.418319	+	0.908300i	$0.362619\pi$
$458$	0	0
$459$	−5.94729e21	−0.140900
$460$	0	0
$461$	−4.85357e22	−1.10816	−0.554081	−	0.832463i	$-0.686930\pi$
$461$	−4.85357e22	−1.10816	−0.554081	+	0.832463i	$0.686930\pi$
$462$	0	0
$463$	2.55987e22	0.563352	0.281676	−	0.959510i	$-0.409110\pi$
$463$	2.55987e22	0.563352	0.281676	+	0.959510i	$0.409110\pi$
$464$	0	0
$465$	−8.61470e21	−0.182764
$466$	0	0
$467$	−6.60855e22	−1.35180	−0.675901	−	0.736993i	$-0.736245\pi$
$467$	−6.60855e22	−1.35180	−0.675901	+	0.736993i	$0.736245\pi$
$468$	0	0
$469$	4.88946e20	0.00964476
$470$	0	0
$471$	1.45580e22	0.276965
$472$	0	0
$473$	−2.32243e22	−0.426210
$474$	0	0
$475$	−9.72735e21	−0.172226
$476$	0	0
$477$	1.72373e22	0.294484
$478$	0	0
$479$	1.24901e22	0.205927	0.102963	−	0.994685i	$-0.467168\pi$
$479$	1.24901e22	0.205927	0.102963	+	0.994685i	$0.467168\pi$
$480$	0	0
$481$	6.38446e22	1.01599
$482$	0	0
$483$	2.16433e22	0.332486
$484$	0	0
$485$	1.79308e22	0.265946
$486$	0	0
$487$	1.25131e22	0.179213	0.0896064	−	0.995977i	$-0.471439\pi$
$487$	1.25131e22	0.179213	0.0896064	+	0.995977i	$0.471439\pi$
$488$	0	0
$489$	−5.23319e22	−0.723838
$490$	0	0
$491$	−3.17848e22	−0.424646	−0.212323	−	0.977200i	$-0.568103\pi$
$491$	−3.17848e22	−0.424646	−0.212323	+	0.977200i	$0.568103\pi$
$492$	0	0
$493$	1.61445e22	0.208365
$494$	0	0
$495$	7.61147e21	0.0949127
$496$	0	0
$497$	1.14707e22	0.138217
$498$	0	0
$499$	−5.34030e22	−0.621887	−0.310943	−	0.950428i	$-0.600645\pi$
$499$	−5.34030e22	−0.621887	−0.310943	+	0.950428i	$0.600645\pi$
$500$	0	0
$501$	−4.32267e22	−0.486555
$502$	0	0
$503$	−7.66144e22	−0.833648	−0.416824	−	0.908987i	$-0.636857\pi$
$503$	−7.66144e22	−0.833648	−0.416824	+	0.908987i	$0.636857\pi$
$504$	0	0
$505$	−5.61012e22	−0.590195
$506$	0	0
$507$	2.20030e22	0.223828
$508$	0	0
$509$	−3.68765e22	−0.362784	−0.181392	−	0.983411i	$-0.558060\pi$
$509$	−3.68765e22	−0.362784	−0.181392	+	0.983411i	$0.558060\pi$
$510$	0	0
$511$	2.47209e22	0.235227
$512$	0	0
$513$	−6.05951e21	−0.0557750
$514$	0	0
$515$	−6.84076e22	−0.609176
$516$	0	0
$517$	3.06866e22	0.264411
$518$	0	0
$519$	6.58696e22	0.549241
$520$	0	0
$521$	−2.06499e23	−1.66647	−0.833235	−	0.552920i	$-0.813514\pi$
$521$	−2.06499e23	−1.66647	−0.833235	+	0.552920i	$0.813514\pi$
$522$	0	0
$523$	5.42076e22	0.423444	0.211722	−	0.977330i	$-0.432093\pi$
$523$	5.42076e22	0.423444	0.211722	+	0.977330i	$0.432093\pi$
$524$	0	0
$525$	1.71483e22	0.129678
$526$	0	0
$527$	4.96948e22	0.363849
$528$	0	0
$529$	1.86396e23	1.32149
$530$	0	0
$531$	1.06466e22	0.0730986
$532$	0	0
$533$	1.54334e22	0.102631
$534$	0	0
$535$	6.21189e22	0.400144
$536$	0	0
$537$	5.77991e22	0.360694
$538$	0	0
$539$	1.05616e22	0.0638593
$540$	0	0
$541$	1.22800e23	0.719486	0.359743	−	0.933051i	$-0.382864\pi$
$541$	1.22800e23	0.719486	0.359743	+	0.933051i	$0.382864\pi$
$542$	0	0
$543$	1.55835e23	0.884842
$544$	0	0
$545$	−5.19694e22	−0.286007
$546$	0	0
$547$	2.32052e23	1.23792	0.618962	−	0.785421i	$-0.287554\pi$
$547$	2.32052e23	1.23792	0.618962	+	0.785421i	$0.287554\pi$
$548$	0	0
$549$	5.42193e22	0.280407
$550$	0	0
$551$	1.64491e22	0.0824810
$552$	0	0
$553$	−6.15425e22	−0.299235
$554$	0	0
$555$	1.01264e23	0.477490
$556$	0	0
$557$	−2.24907e23	−1.02857	−0.514285	−	0.857619i	$-0.671943\pi$
$557$	−2.24907e23	−1.02857	−0.514285	+	0.857619i	$0.671943\pi$
$558$	0	0
$559$	1.68186e23	0.746087
$560$	0	0
$561$	−4.39075e22	−0.188953
$562$	0	0
$563$	−4.74416e22	−0.198079	−0.0990393	−	0.995084i	$-0.531577\pi$
$563$	−4.74416e22	−0.198079	−0.0990393	+	0.995084i	$0.531577\pi$
$564$	0	0
$565$	9.68647e22	0.392422
$566$	0	0
$567$	1.06823e22	0.0419961
$568$	0	0
$569$	1.01817e23	0.388479	0.194239	−	0.980954i	$-0.437776\pi$
$569$	1.01817e23	0.388479	0.194239	+	0.980954i	$0.437776\pi$
$570$	0	0
$571$	3.39879e23	1.25868	0.629342	−	0.777128i	$-0.283324\pi$
$571$	3.39879e23	1.25868	0.629342	+	0.777128i	$0.283324\pi$
$572$	0	0
$573$	2.53632e23	0.911778
$574$	0	0
$575$	2.59440e23	0.905443
$576$	0	0
$577$	4.36062e21	0.0147759	0.00738794	−	0.999973i	$-0.497648\pi$
$577$	4.36062e21	0.0147759	0.00738794	+	0.999973i	$0.497648\pi$
$578$	0	0
$579$	1.96573e23	0.646781
$580$	0	0
$581$	2.46217e22	0.0786721
$582$	0	0
$583$	1.27259e23	0.394917
$584$	0	0
$585$	−5.51208e22	−0.166146
$586$	0	0
$587$	1.58521e23	0.464154	0.232077	−	0.972697i	$-0.425448\pi$
$587$	1.58521e23	0.464154	0.232077	+	0.972697i	$0.425448\pi$
$588$	0	0
$589$	5.06325e22	0.144029
$590$	0	0
$591$	3.56874e23	0.986329
$592$	0	0
$593$	4.34364e22	0.116651	0.0583256	−	0.998298i	$-0.481424\pi$
$593$	4.34364e22	0.116651	0.0583256	+	0.998298i	$0.481424\pi$
$594$	0	0
$595$	6.75401e22	0.176265
$596$	0	0
$597$	1.32241e23	0.335416
$598$	0	0
$599$	−3.13006e23	−0.771657	−0.385828	−	0.922571i	$-0.626084\pi$
$599$	−3.13006e23	−0.771657	−0.385828	+	0.922571i	$0.626084\pi$
$600$	0	0
$601$	−1.85069e23	−0.443507	−0.221754	−	0.975103i	$-0.571178\pi$
$601$	−1.85069e23	−0.443507	−0.221754	+	0.975103i	$0.571178\pi$
$602$	0	0
$603$	3.65104e21	0.00850588
$604$	0	0
$605$	−2.25025e23	−0.509694
$606$	0	0
$607$	2.74088e23	0.603650	0.301825	−	0.953363i	$-0.402404\pi$
$607$	2.74088e23	0.603650	0.301825	+	0.953363i	$0.402404\pi$
$608$	0	0
$609$	−2.89981e22	−0.0621044
$610$	0	0
$611$	−2.22226e23	−0.462856
$612$	0	0
$613$	−4.37801e23	−0.886875	−0.443438	−	0.896305i	$-0.646241\pi$
$613$	−4.37801e23	−0.886875	−0.443438	+	0.896305i	$0.646241\pi$
$614$	0	0
$615$	2.44790e22	0.0482341
$616$	0	0
$617$	−6.99807e23	−1.34139	−0.670694	−	0.741734i	$-0.734004\pi$
$617$	−6.99807e23	−1.34139	−0.670694	+	0.741734i	$0.734004\pi$
$618$	0	0
$619$	−4.76481e22	−0.0888536	−0.0444268	−	0.999013i	$-0.514146\pi$
$619$	−4.76481e22	−0.0888536	−0.0444268	+	0.999013i	$0.514146\pi$
$620$	0	0
$621$	1.61614e23	0.293225
$622$	0	0
$623$	−2.29918e23	−0.405905
$624$	0	0
$625$	−3.06125e22	−0.0525919
$626$	0	0
$627$	−4.47360e22	−0.0747968
$628$	0	0
$629$	−5.84152e23	−0.950595
$630$	0	0
$631$	1.17277e24	1.85765	0.928823	−	0.370523i	$-0.120822\pi$
$631$	1.17277e24	1.85765	0.928823	+	0.370523i	$0.120822\pi$
$632$	0	0
$633$	−4.21677e22	−0.0650202
$634$	0	0
$635$	1.65789e23	0.248873
$636$	0	0
$637$	−7.64849e22	−0.111787
$638$	0	0
$639$	8.56537e22	0.121896
$640$	0	0
$641$	−9.73584e22	−0.134921	−0.0674606	−	0.997722i	$-0.521490\pi$
$641$	−9.73584e22	−0.134921	−0.0674606	+	0.997722i	$0.521490\pi$
$642$	0	0
$643$	−7.57616e23	−1.02248	−0.511241	−	0.859437i	$-0.670814\pi$
$643$	−7.57616e23	−1.02248	−0.511241	+	0.859437i	$0.670814\pi$
$644$	0	0
$645$	2.66760e23	0.350642
$646$	0	0
$647$	1.06301e23	0.136098	0.0680491	−	0.997682i	$-0.478323\pi$
$647$	1.06301e23	0.136098	0.0680491	+	0.997682i	$0.478323\pi$
$648$	0	0
$649$	7.86018e22	0.0980286
$650$	0	0
$651$	−8.92599e22	−0.108447
$652$	0	0
$653$	−1.43513e24	−1.69875	−0.849377	−	0.527786i	$-0.823022\pi$
$653$	−1.43513e24	−1.69875	−0.849377	+	0.527786i	$0.823022\pi$
$654$	0	0
$655$	1.21834e22	0.0140513
$656$	0	0
$657$	1.84595e23	0.207450
$658$	0	0
$659$	−1.19627e24	−1.31010	−0.655048	−	0.755587i	$-0.727352\pi$
$659$	−1.19627e24	−1.31010	−0.655048	+	0.755587i	$0.727352\pi$
$660$	0	0
$661$	5.93495e23	0.633439	0.316720	−	0.948519i	$-0.397419\pi$
$661$	5.93495e23	0.633439	0.316720	+	0.948519i	$0.397419\pi$
$662$	0	0
$663$	3.17970e23	0.330766
$664$	0	0
$665$	6.88145e22	0.0697743
$666$	0	0
$667$	−4.38718e23	−0.433626
$668$	0	0
$669$	6.08786e23	0.586601
$670$	0	0
$671$	4.00290e23	0.376039
$672$	0	0
$673$	4.37003e23	0.400273	0.200136	−	0.979768i	$-0.435861\pi$
$673$	4.37003e23	0.400273	0.200136	+	0.979768i	$0.435861\pi$
$674$	0	0
$675$	1.28049e23	0.114366
$676$	0	0
$677$	−1.35725e24	−1.18210	−0.591051	−	0.806634i	$-0.701287\pi$
$677$	−1.35725e24	−1.18210	−0.591051	+	0.806634i	$0.701287\pi$
$678$	0	0
$679$	1.85787e23	0.157805
$680$	0	0
$681$	−2.45142e23	−0.203080
$682$	0	0
$683$	2.92426e23	0.236287	0.118144	−	0.992997i	$-0.462306\pi$
$683$	2.92426e23	0.236287	0.118144	+	0.992997i	$0.462306\pi$
$684$	0	0
$685$	4.53132e23	0.357154
$686$	0	0
$687$	−5.50540e23	−0.423308
$688$	0	0
$689$	−9.21585e23	−0.691309
$690$	0	0
$691$	−1.56675e23	−0.114667	−0.0573333	−	0.998355i	$-0.518260\pi$
$691$	−1.56675e23	−0.114667	−0.0573333	+	0.998355i	$0.518260\pi$
$692$	0	0
$693$	7.88650e22	0.0563186
$694$	0	0
$695$	−1.12232e24	−0.782073
$696$	0	0
$697$	−1.41209e23	−0.0960252
$698$	0	0
$699$	7.52333e23	0.499291
$700$	0	0
$701$	−1.84448e24	−1.19473	−0.597366	−	0.801969i	$-0.703786\pi$
$701$	−1.84448e24	−1.19473	−0.597366	+	0.801969i	$0.703786\pi$
$702$	0	0
$703$	−5.95174e23	−0.376291
$704$	0	0
$705$	−3.52474e23	−0.217530
$706$	0	0
$707$	−5.81284e23	−0.350206
$708$	0	0
$709$	−1.11997e23	−0.0658739	−0.0329369	−	0.999457i	$-0.510486\pi$
$709$	−1.11997e23	−0.0658739	−0.0329369	+	0.999457i	$0.510486\pi$
$710$	0	0
$711$	−4.59548e23	−0.263900
$712$	0	0
$713$	−1.35043e24	−0.757201
$714$	0	0
$715$	−4.06945e23	−0.222810
$716$	0	0
$717$	−1.60934e24	−0.860470
$718$	0	0
$719$	1.08878e24	0.568519	0.284259	−	0.958747i	$-0.408252\pi$
$719$	1.08878e24	0.568519	0.284259	+	0.958747i	$0.408252\pi$
$720$	0	0
$721$	−7.08795e23	−0.361469
$722$	0	0
$723$	1.88820e24	0.940528
$724$	0	0
$725$	−3.47602e23	−0.169126
$726$	0	0
$727$	3.95909e24	1.88171	0.940855	−	0.338810i	$-0.110024\pi$
$727$	3.95909e24	1.88171	0.940855	+	0.338810i	$0.110024\pi$
$728$	0	0
$729$	7.97664e22	0.0370370
$730$	0	0
$731$	−1.53883e24	−0.698062
$732$	0	0
$733$	−3.50724e24	−1.55447	−0.777234	−	0.629211i	$-0.783378\pi$
$733$	−3.50724e24	−1.55447	−0.777234	+	0.629211i	$0.783378\pi$
$734$	0	0
$735$	−1.21313e23	−0.0525369
$736$	0	0
$737$	2.69548e22	0.0114068
$738$	0	0
$739$	2.55054e24	1.05476	0.527381	−	0.849629i	$-0.323174\pi$
$739$	2.55054e24	1.05476	0.527381	+	0.849629i	$0.323174\pi$
$740$	0	0
$741$	3.23970e23	0.130933
$742$	0	0
$743$	−9.04947e23	−0.357452	−0.178726	−	0.983899i	$-0.557198\pi$
$743$	−9.04947e23	−0.357452	−0.178726	+	0.983899i	$0.557198\pi$
$744$	0	0
$745$	−1.00496e24	−0.387991
$746$	0	0
$747$	1.83854e23	0.0693823
$748$	0	0
$749$	6.43635e23	0.237435
$750$	0	0
$751$	−5.24477e23	−0.189141	−0.0945707	−	0.995518i	$-0.530148\pi$
$751$	−5.24477e23	−0.189141	−0.0945707	+	0.995518i	$0.530148\pi$
$752$	0	0
$753$	1.64538e24	0.580109
$754$	0	0
$755$	−1.83533e24	−0.632652
$756$	0	0
$757$	−6.78743e23	−0.228765	−0.114383	−	0.993437i	$-0.536489\pi$
$757$	−6.78743e23	−0.228765	−0.114383	+	0.993437i	$0.536489\pi$
$758$	0	0
$759$	1.19316e24	0.393229
$760$	0	0
$761$	−1.08592e24	−0.349966	−0.174983	−	0.984571i	$-0.555987\pi$
$761$	−1.08592e24	−0.349966	−0.174983	+	0.984571i	$0.555987\pi$
$762$	0	0
$763$	−5.38472e23	−0.169709
$764$	0	0
$765$	5.04333e23	0.155452
$766$	0	0
$767$	−5.69219e23	−0.171601
$768$	0	0
$769$	−1.62887e24	−0.480301	−0.240151	−	0.970736i	$-0.577197\pi$
$769$	−1.62887e24	−0.480301	−0.240151	+	0.970736i	$0.577197\pi$
$770$	0	0
$771$	−2.21886e24	−0.639983
$772$	0	0
$773$	6.93808e23	0.195755	0.0978776	−	0.995198i	$-0.468795\pi$
$773$	6.93808e23	0.195755	0.0978776	+	0.995198i	$0.468795\pi$
$774$	0	0
$775$	−1.06996e24	−0.295328
$776$	0	0
$777$	1.04923e24	0.283330
$778$	0	0
$779$	−1.43874e23	−0.0380114
$780$	0	0
$781$	6.32363e23	0.163468
$782$	0	0
$783$	−2.16534e23	−0.0547710
$784$	0	0
$785$	−1.23453e24	−0.305568
$786$	0	0
$787$	6.16458e24	1.49320	0.746600	−	0.665273i	$-0.231685\pi$
$787$	6.16458e24	1.49320	0.746600	+	0.665273i	$0.231685\pi$
$788$	0	0
$789$	2.14736e24	0.509039
$790$	0	0
$791$	1.00365e24	0.232853
$792$	0	0
$793$	−2.89882e24	−0.658262
$794$	0	0
$795$	−1.46173e24	−0.324897
$796$	0	0
$797$	1.40720e24	0.306168	0.153084	−	0.988213i	$-0.451079\pi$
$797$	1.40720e24	0.306168	0.153084	+	0.988213i	$0.451079\pi$
$798$	0	0
$799$	2.03328e24	0.433062
$800$	0	0
$801$	−1.71683e24	−0.357975
$802$	0	0
$803$	1.36283e24	0.278201
$804$	0	0
$805$	−1.83537e24	−0.366824
$806$	0	0
$807$	9.88339e23	0.193411
$808$	0	0
$809$	1.04899e24	0.201006	0.100503	−	0.994937i	$-0.467955\pi$
$809$	1.04899e24	0.201006	0.100503	+	0.994937i	$0.467955\pi$
$810$	0	0
$811$	6.97510e22	0.0130880	0.00654400	−	0.999979i	$-0.497917\pi$
$811$	6.97510e22	0.0130880	0.00654400	+	0.999979i	$0.497917\pi$
$812$	0	0
$813$	7.13675e23	0.131139
$814$	0	0
$815$	4.43777e24	0.798592
$816$	0	0
$817$	−1.56787e24	−0.276327
$818$	0	0
$819$	−5.71125e23	−0.0985867
$820$	0	0
$821$	3.10859e24	0.525590	0.262795	−	0.964852i	$-0.415356\pi$
$821$	3.10859e24	0.525590	0.262795	+	0.964852i	$0.415356\pi$
$822$	0	0
$823$	7.50912e24	1.24363	0.621814	−	0.783165i	$-0.286396\pi$
$823$	7.50912e24	1.24363	0.621814	+	0.783165i	$0.286396\pi$
$824$	0	0
$825$	9.45361e23	0.153370
$826$	0	0
$827$	4.33416e24	0.688824	0.344412	−	0.938819i	$-0.388078\pi$
$827$	4.33416e24	0.688824	0.344412	+	0.938819i	$0.388078\pi$
$828$	0	0
$829$	3.45527e24	0.537983	0.268992	−	0.963143i	$-0.413310\pi$
$829$	3.45527e24	0.537983	0.268992	+	0.963143i	$0.413310\pi$
$830$	0	0
$831$	−4.17033e23	−0.0636153
$832$	0	0
$833$	6.99806e23	0.104591
$834$	0	0
$835$	3.66565e24	0.536804
$836$	0	0
$837$	−6.66518e23	−0.0956413
$838$	0	0
$839$	1.03957e25	1.46176	0.730882	−	0.682504i	$-0.239109\pi$
$839$	1.03957e25	1.46176	0.730882	+	0.682504i	$0.239109\pi$
$840$	0	0
$841$	−6.66935e24	−0.919004
$842$	0	0
$843$	5.70415e22	0.00770294
$844$	0	0
$845$	−1.86587e24	−0.246944
$846$	0	0
$847$	−2.33156e24	−0.302439
$848$	0	0
$849$	−3.56498e24	−0.453254
$850$	0	0
$851$	1.58740e25	1.97827
$852$	0	0
$853$	−9.66859e24	−1.18113	−0.590563	−	0.806991i	$-0.701094\pi$
$853$	−9.66859e24	−1.18113	−0.590563	+	0.806991i	$0.701094\pi$
$854$	0	0
$855$	5.13849e23	0.0615352
$856$	0	0
$857$	−3.97481e24	−0.466637	−0.233318	−	0.972400i	$-0.574958\pi$
$857$	−3.97481e24	−0.466637	−0.233318	+	0.972400i	$0.574958\pi$
$858$	0	0
$859$	1.13340e24	0.130449	0.0652247	−	0.997871i	$-0.479224\pi$
$859$	1.13340e24	0.130449	0.0652247	+	0.997871i	$0.479224\pi$
$860$	0	0
$861$	2.53635e23	0.0286208
$862$	0	0
$863$	1.15492e25	1.27779	0.638895	−	0.769294i	$-0.279392\pi$
$863$	1.15492e25	1.27779	0.638895	+	0.769294i	$0.279392\pi$
$864$	0	0
$865$	−5.58577e24	−0.605963
$866$	0	0
$867$	2.51823e24	0.267875
$868$	0	0
$869$	−3.39275e24	−0.353902
$870$	0	0
$871$	−1.95202e23	−0.0199677
$872$	0	0
$873$	1.38730e24	0.139171
$874$	0	0
$875$	−3.90124e24	−0.383825
$876$	0	0
$877$	1.26471e25	1.22037	0.610187	−	0.792258i	$-0.291094\pi$
$877$	1.26471e25	1.22037	0.610187	+	0.792258i	$0.291094\pi$
$878$	0	0
$879$	6.12652e23	0.0579840
$880$	0	0
$881$	1.20121e25	1.11512	0.557561	−	0.830136i	$-0.311737\pi$
$881$	1.20121e25	1.11512	0.557561	+	0.830136i	$0.311737\pi$
$882$	0	0
$883$	1.35514e25	1.23401	0.617005	−	0.786959i	$-0.288346\pi$
$883$	1.35514e25	1.23401	0.617005	+	0.786959i	$0.288346\pi$
$884$	0	0
$885$	−9.02840e23	−0.0806479
$886$	0	0
$887$	1.24050e24	0.108704	0.0543521	−	0.998522i	$-0.482691\pi$
$887$	1.24050e24	0.108704	0.0543521	+	0.998522i	$0.482691\pi$
$888$	0	0
$889$	1.71779e24	0.147675
$890$	0	0
$891$	5.88898e23	0.0496684
$892$	0	0
$893$	2.07165e24	0.171427
$894$	0	0
$895$	−4.90139e24	−0.397945
$896$	0	0
$897$	−8.64066e24	−0.688353
$898$	0	0
$899$	1.80933e24	0.141436
$900$	0	0
$901$	8.43213e24	0.646810
$902$	0	0
$903$	2.76400e24	0.208061
$904$	0	0
$905$	−1.32149e25	−0.976224
$906$	0	0
$907$	2.23717e25	1.62194	0.810972	−	0.585084i	$-0.198939\pi$
$907$	2.23717e25	1.62194	0.810972	+	0.585084i	$0.198939\pi$
$908$	0	0
$909$	−4.34055e24	−0.308853
$910$	0	0
$911$	−5.84358e23	−0.0408106	−0.0204053	−	0.999792i	$-0.506496\pi$
$911$	−5.84358e23	−0.0408106	−0.0204053	+	0.999792i	$0.506496\pi$
$912$	0	0
$913$	1.35736e24	0.0930449
$914$	0	0
$915$	−4.59782e24	−0.309366
$916$	0	0
$917$	1.26236e23	0.00833765
$918$	0	0
$919$	−2.40891e25	−1.56185	−0.780925	−	0.624624i	$-0.785252\pi$
$919$	−2.40891e25	−1.56185	−0.780925	+	0.624624i	$0.785252\pi$
$920$	0	0
$921$	−5.25947e24	−0.334761
$922$	0	0
$923$	−4.57945e24	−0.286153
$924$	0	0
$925$	1.25772e25	0.771578
$926$	0	0
$927$	−5.29269e24	−0.318785
$928$	0	0
$929$	2.40982e25	1.42512	0.712560	−	0.701611i	$-0.247536\pi$
$929$	2.40982e25	1.42512	0.712560	+	0.701611i	$0.247536\pi$
$930$	0	0
$931$	7.13010e23	0.0414022
$932$	0	0
$933$	−1.40528e25	−0.801254
$934$	0	0
$935$	3.72338e24	0.208468
$936$	0	0
$937$	−3.21344e25	−1.76678	−0.883391	−	0.468637i	$-0.844745\pi$
$937$	−3.21344e25	−1.76678	−0.883391	+	0.468637i	$0.844745\pi$
$938$	0	0
$939$	−1.28989e25	−0.696459
$940$	0	0
$941$	−1.96282e25	−1.04080	−0.520401	−	0.853922i	$-0.674218\pi$
$941$	−1.96282e25	−1.04080	−0.520401	+	0.853922i	$0.674218\pi$
$942$	0	0
$943$	3.83729e24	0.199837
$944$	0	0
$945$	−9.05863e23	−0.0463332
$946$	0	0
$947$	1.83724e25	0.922976	0.461488	−	0.887147i	$-0.347316\pi$
$947$	1.83724e25	0.922976	0.461488	+	0.887147i	$0.347316\pi$
$948$	0	0
$949$	−9.86931e24	−0.486995
$950$	0	0
$951$	8.61651e24	0.417635
$952$	0	0
$953$	4.84933e24	0.230883	0.115442	−	0.993314i	$-0.463172\pi$
$953$	4.84933e24	0.230883	0.115442	+	0.993314i	$0.463172\pi$
$954$	0	0
$955$	−2.15081e25	−1.00594
$956$	0	0
$957$	−1.59862e24	−0.0734504
$958$	0	0
$959$	4.69506e24	0.211925
$960$	0	0
$961$	−1.69808e25	−0.753024
$962$	0	0
$963$	4.80613e24	0.209398
$964$	0	0
$965$	−1.66695e25	−0.713577
$966$	0	0
$967$	−1.42722e25	−0.600299	−0.300149	−	0.953892i	$-0.597037\pi$
$967$	−1.42722e25	−0.600299	−0.300149	+	0.953892i	$0.597037\pi$
$968$	0	0
$969$	−2.96419e24	−0.122505
$970$	0	0
$971$	2.49371e25	1.01270	0.506351	−	0.862327i	$-0.330994\pi$
$971$	2.49371e25	1.01270	0.506351	+	0.862327i	$0.330994\pi$
$972$	0	0
$973$	−1.16288e25	−0.464061
$974$	0	0
$975$	−6.84612e24	−0.268476
$976$	0	0
$977$	−2.29961e25	−0.886237	−0.443119	−	0.896463i	$-0.646128\pi$
$977$	−2.29961e25	−0.886237	−0.443119	+	0.896463i	$0.646128\pi$
$978$	0	0
$979$	−1.26750e25	−0.480060
$980$	0	0
$981$	−4.02086e24	−0.149669
$982$	0	0
$983$	1.23875e22	0.000453189 0	0.000226595	−	1.00000i	$-0.499928\pi$
$983$	1.23875e22	0.000453189 0	0.000226595		1.00000i	$0.499928\pi$
$984$	0	0
$985$	−3.02631e25	−1.08819
$986$	0	0
$987$	−3.65210e24	−0.129077
$988$	0	0
$989$	4.18170e25	1.45273
$990$	0	0
$991$	−3.27707e25	−1.11908	−0.559539	−	0.828804i	$-0.689022\pi$
$991$	−3.27707e25	−1.11908	−0.559539	+	0.828804i	$0.689022\pi$
$992$	0	0
$993$	−2.51558e25	−0.844441
$994$	0	0
$995$	−1.12141e25	−0.370056
$996$	0	0
$997$	−3.33854e25	−1.08305	−0.541525	−	0.840685i	$-0.682153\pi$
$997$	−3.33854e25	−1.08305	−0.541525	+	0.840685i	$0.682153\pi$
$998$	0	0
$999$	7.83478e24	0.249874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.18.a.a.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.18.a.a.1.3	✓	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-6561.00 q^{3} +556376. q^{5} +5.76480e6 q^{7} +4.30467e7 q^{9} +O(q^{10})\)	\(q-6561.00 q^{3} +556376. q^{5} +5.76480e6 q^{7} +4.30467e7 q^{9} +3.17805e8 q^{11} -2.30148e9 q^{13} -3.65038e9 q^{15} +2.10576e10 q^{17} +2.14549e10 q^{19} -3.78229e10 q^{21} -5.72229e11 q^{23} -4.53385e11 q^{25} -2.82430e11 q^{27} +7.66682e11 q^{29} +2.35994e12 q^{31} -2.08512e12 q^{33} +3.20740e12 q^{35} -2.77407e13 q^{37} +1.51000e13 q^{39} -6.70587e12 q^{41} -7.30774e13 q^{43} +2.39502e13 q^{45} +9.65581e13 q^{47} +3.32329e13 q^{49} -1.38159e14 q^{51} +4.00432e14 q^{53} +1.76819e14 q^{55} -1.40766e14 q^{57} +2.47328e14 q^{59} +1.25955e15 q^{61} +2.48156e14 q^{63} -1.28049e15 q^{65} +8.48157e13 q^{67} +3.75440e15 q^{69} +1.98978e15 q^{71} +4.28825e15 q^{73} +2.97466e15 q^{75} +1.83208e15 q^{77} -1.06756e16 q^{79} +1.85302e15 q^{81} +4.27104e15 q^{83} +1.17159e16 q^{85} -5.03020e15 q^{87} -3.98831e16 q^{89} -1.32676e16 q^{91} -1.54836e16 q^{93} +1.19370e16 q^{95} +3.22278e16 q^{97} +1.36804e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 26244 q^{3} + 528276 q^{5} + 23059204 q^{7} + 172186884 q^{9}+O(q^{10}) \) 4 * q - 26244 * q^3 + 528276 * q^5 + 23059204 * q^7 + 172186884 * q^9	\( 4 q - 26244 q^{3} + 528276 q^{5} + 23059204 q^{7} + 172186884 q^{9} - 269632020 q^{11} + 1984752056 q^{13} - 3466018836 q^{15} - 29765040036 q^{17} - 91677835096 q^{19} - 151291437444 q^{21} + 38384818716 q^{23} - 235654160804 q^{25} - 1129718145924 q^{27} - 3815624693904 q^{29} - 4907739231784 q^{31} + 1769055683220 q^{33} + 3045406013076 q^{35} + 13670824759208 q^{37} - 13021958239416 q^{39} + 10307194403604 q^{41} + 125363311182368 q^{43} + 22740549582996 q^{45} - 25278931593000 q^{47} + 132931722278404 q^{49} + 195288427676196 q^{51} - 239725859759304 q^{53} + 694573116031368 q^{55} + 601498276064856 q^{57} + 13\!\cdots\!00 q^{59}+ \cdots - 11\!\cdots\!20 q^{99}+O(q^{100}) \) 4 * q - 26244 * q^3 + 528276 * q^5 + 23059204 * q^7 + 172186884 * q^9 - 269632020 * q^11 + 1984752056 * q^13 - 3466018836 * q^15 - 29765040036 * q^17 - 91677835096 * q^19 - 151291437444 * q^21 + 38384818716 * q^23 - 235654160804 * q^25 - 1129718145924 * q^27 - 3815624693904 * q^29 - 4907739231784 * q^31 + 1769055683220 * q^33 + 3045406013076 * q^35 + 13670824759208 * q^37 - 13021958239416 * q^39 + 10307194403604 * q^41 + 125363311182368 * q^43 + 22740549582996 * q^45 - 25278931593000 * q^47 + 132931722278404 * q^49 + 195288427676196 * q^51 - 239725859759304 * q^53 + 694573116031368 * q^55 + 601498276064856 * q^57 + 1374756208497000 * q^59 + 652646281638080 * q^61 + 992623121070084 * q^63 + 4153293849278904 * q^65 + 6617359123302392 * q^67 - 251842795595676 * q^69 + 2405715947056548 * q^71 + 4250185178265824 * q^73 + 1546126949035044 * q^75 - 1554374938528020 * q^77 + 20052572228473688 * q^79 + 7412080755407364 * q^81 - 5593593799375584 * q^83 - 13012862591692368 * q^85 + 25034313616704144 * q^87 + 1518004216828212 * q^89 + 11441700637180856 * q^91 + 32199677099734824 * q^93 - 149304801904677408 * q^95 - 102152247026326000 * q^97 - 11606774337606420 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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