LMFDB - Embedded newform 84.6.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [84,6,Mod(1,84)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("84.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(84, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	84.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.4722408643$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5569}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1392 $ x^2 - x - 1392
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$37.8129$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} -77.6257 q^{5} -49.0000 q^{7} +81.0000 q^{9} +477.380 q^{11} -63.7544 q^{13} +698.632 q^{15} +1037.63 q^{17} -667.018 q^{19} +441.000 q^{21} +3251.63 q^{23} +2900.75 q^{25} -729.000 q^{27} +2300.97 q^{29} +3717.05 q^{31} -4296.42 q^{33} +3803.66 q^{35} +12245.9 q^{37} +573.790 q^{39} -1829.65 q^{41} -20794.2 q^{43} -6287.68 q^{45} -4283.37 q^{47} +2401.00 q^{49} -9338.63 q^{51} +25718.4 q^{53} -37057.0 q^{55} +6003.16 q^{57} -2838.71 q^{59} +16803.2 q^{61} -3969.00 q^{63} +4948.98 q^{65} -62535.1 q^{67} -29264.6 q^{69} +72301.0 q^{71} -55676.9 q^{73} -26106.8 q^{75} -23391.6 q^{77} -3989.19 q^{79} +6561.00 q^{81} -46092.2 q^{83} -80546.5 q^{85} -20708.7 q^{87} +135385. q^{89} +3123.97 q^{91} -33453.5 q^{93} +51777.7 q^{95} +142878. q^{97} +38667.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} - 6 q^{5} - 98 q^{7} + 162 q^{9} - 90 q^{11} + 768 q^{13} + 54 q^{15} + 1926 q^{17} + 2248 q^{19} + 882 q^{21} + 6354 q^{23} + 4906 q^{25} - 1458 q^{27} + 10572 q^{29} - 3312 q^{31} + 810 q^{33}+ \cdots - 7290 q^{99}+O(q^{100}) $ 2 * q - 18 * q^3 - 6 * q^5 - 98 * q^7 + 162 * q^9 - 90 * q^11 + 768 * q^13 + 54 * q^15 + 1926 * q^17 + 2248 * q^19 + 882 * q^21 + 6354 * q^23 + 4906 * q^25 - 1458 * q^27 + 10572 * q^29 - 3312 * q^31 + 810 * q^33 + 294 * q^35 + 2104 * q^37 - 6912 * q^39 + 1266 * q^41 - 5768 * q^43 - 486 * q^45 + 15612 * q^47 + 4802 * q^49 - 17334 * q^51 + 16512 * q^53 - 77696 * q^55 - 20232 * q^57 - 13140 * q^59 - 5796 * q^61 - 7938 * q^63 + 64524 * q^65 - 56116 * q^67 - 57186 * q^69 + 11022 * q^71 - 85384 * q^73 - 44154 * q^75 + 4410 * q^77 - 19620 * q^79 + 13122 * q^81 - 44424 * q^83 - 16916 * q^85 - 95148 * q^87 + 211218 * q^89 - 37632 * q^91 + 29808 * q^93 + 260568 * q^95 + 44864 * q^97 - 7290 * q^99

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−77.6257	−1.38861	−0.694306	−	0.719680i	$-0.744288\pi$
$5$	−77.6257	−1.38861	−0.694306	+	0.719680i	$0.744288\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	477.380	1.18955	0.594775	−	0.803892i	$-0.297241\pi$
$11$	477.380	1.18955	0.594775	+	0.803892i	$0.297241\pi$
$12$	0	0
$13$	−63.7544	−0.104629	−0.0523145	−	0.998631i	$-0.516660\pi$
$13$	−63.7544	−0.104629	−0.0523145	+	0.998631i	$0.516660\pi$
$14$	0	0
$15$	698.632	0.801715
$16$	0	0
$17$	1037.63	0.870800	0.435400	−	0.900237i	$-0.356607\pi$
$17$	1037.63	0.870800	0.435400	+	0.900237i	$0.356607\pi$
$18$	0	0
$19$	−667.018	−0.423890	−0.211945	−	0.977282i	$-0.567980\pi$
$19$	−667.018	−0.423890	−0.211945	+	0.977282i	$0.567980\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	0	0
$23$	3251.63	1.28168	0.640842	−	0.767673i	$-0.278585\pi$
$23$	3251.63	1.28168	0.640842	+	0.767673i	$0.278585\pi$
$24$	0	0
$25$	2900.75	0.928241
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	2300.97	0.508061	0.254031	−	0.967196i	$-0.418244\pi$
$29$	2300.97	0.508061	0.254031	+	0.967196i	$0.418244\pi$
$30$	0	0
$31$	3717.05	0.694696	0.347348	−	0.937736i	$-0.387082\pi$
$31$	3717.05	0.694696	0.347348	+	0.937736i	$0.387082\pi$
$32$	0	0
$33$	−4296.42	−0.686787
$34$	0	0
$35$	3803.66	0.524846
$36$	0	0
$37$	12245.9	1.47057	0.735284	−	0.677759i	$-0.237049\pi$
$37$	12245.9	1.47057	0.735284	+	0.677759i	$0.237049\pi$
$38$	0	0
$39$	573.790	0.0604075
$40$	0	0
$41$	−1829.65	−0.169984	−0.0849920	−	0.996382i	$-0.527086\pi$
$41$	−1829.65	−0.169984	−0.0849920	+	0.996382i	$0.527086\pi$
$42$	0	0
$43$	−20794.2	−1.71503	−0.857513	−	0.514463i	$-0.827991\pi$
$43$	−20794.2	−1.71503	−0.857513	+	0.514463i	$0.827991\pi$
$44$	0	0
$45$	−6287.68	−0.462870
$46$	0	0
$47$	−4283.37	−0.282840	−0.141420	−	0.989950i	$-0.545167\pi$
$47$	−4283.37	−0.282840	−0.141420	+	0.989950i	$0.545167\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−9338.63	−0.502757
$52$	0	0
$53$	25718.4	1.25764	0.628818	−	0.777553i	$-0.283539\pi$
$53$	25718.4	1.25764	0.628818	+	0.777553i	$0.283539\pi$
$54$	0	0
$55$	−37057.0	−1.65182
$56$	0	0
$57$	6003.16	0.244733
$58$	0	0
$59$	−2838.71	−0.106167	−0.0530837	−	0.998590i	$-0.516905\pi$
$59$	−2838.71	−0.106167	−0.0530837	+	0.998590i	$0.516905\pi$
$60$	0	0
$61$	16803.2	0.578186	0.289093	−	0.957301i	$-0.406646\pi$
$61$	16803.2	0.578186	0.289093	+	0.957301i	$0.406646\pi$
$62$	0	0
$63$	−3969.00	−0.125988
$64$	0	0
$65$	4948.98	0.145289
$66$	0	0
$67$	−62535.1	−1.70191	−0.850955	−	0.525238i	$-0.823976\pi$
$67$	−62535.1	−1.70191	−0.850955	+	0.525238i	$0.823976\pi$
$68$	0	0
$69$	−29264.6	−0.739981
$70$	0	0
$71$	72301.0	1.70215	0.851077	−	0.525042i	$-0.175950\pi$
$71$	72301.0	1.70215	0.851077	+	0.525042i	$0.175950\pi$
$72$	0	0
$73$	−55676.9	−1.22283	−0.611417	−	0.791308i	$-0.709400\pi$
$73$	−55676.9	−1.22283	−0.611417	+	0.791308i	$0.709400\pi$
$74$	0	0
$75$	−26106.8	−0.535920
$76$	0	0
$77$	−23391.6	−0.449608
$78$	0	0
$79$	−3989.19	−0.0719146	−0.0359573	−	0.999353i	$-0.511448\pi$
$79$	−3989.19	−0.0719146	−0.0359573	+	0.999353i	$0.511448\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−46092.2	−0.734400	−0.367200	−	0.930142i	$-0.619684\pi$
$83$	−46092.2	−0.734400	−0.367200	+	0.930142i	$0.619684\pi$
$84$	0	0
$85$	−80546.5	−1.20920
$86$	0	0
$87$	−20708.7	−0.293329
$88$	0	0
$89$	135385.	1.81173	0.905867	−	0.423562i	$-0.139220\pi$
$89$	135385.	1.81173	0.905867	+	0.423562i	$0.139220\pi$
$90$	0	0
$91$	3123.97	0.0395460
$92$	0	0
$93$	−33453.5	−0.401083
$94$	0	0
$95$	51777.7	0.588619
$96$	0	0
$97$	142878.	1.54183	0.770914	−	0.636939i	$-0.219800\pi$
$97$	142878.	1.54183	0.770914	+	0.636939i	$0.219800\pi$
$98$	0	0
$99$	38667.8	0.396517
$100$	0	0
$101$	44467.1	0.433746	0.216873	−	0.976200i	$-0.430414\pi$
$101$	44467.1	0.433746	0.216873	+	0.976200i	$0.430414\pi$
$102$	0	0
$103$	202619.	1.88186	0.940931	−	0.338598i	$-0.109953\pi$
$103$	202619.	1.88186	0.940931	+	0.338598i	$0.109953\pi$
$104$	0	0
$105$	−34232.9	−0.303020
$106$	0	0
$107$	99525.9	0.840382	0.420191	−	0.907436i	$-0.361963\pi$
$107$	99525.9	0.840382	0.420191	+	0.907436i	$0.361963\pi$
$108$	0	0
$109$	220930.	1.78110	0.890551	−	0.454883i	$-0.150319\pi$
$109$	220930.	1.78110	0.890551	+	0.454883i	$0.150319\pi$
$110$	0	0
$111$	−110213.	−0.849033
$112$	0	0
$113$	29623.1	0.218240	0.109120	−	0.994029i	$-0.465197\pi$
$113$	29623.1	0.218240	0.109120	+	0.994029i	$0.465197\pi$
$114$	0	0
$115$	−252410.	−1.77976
$116$	0	0
$117$	−5164.11	−0.0348763
$118$	0	0
$119$	−50843.7	−0.329131
$120$	0	0
$121$	66840.8	0.415029
$122$	0	0
$123$	16466.8	0.0981403
$124$	0	0
$125$	17407.2	0.0996448
$126$	0	0
$127$	−264132.	−1.45315	−0.726577	−	0.687086i	$-0.758890\pi$
$127$	−264132.	−1.45315	−0.726577	+	0.687086i	$0.758890\pi$
$128$	0	0
$129$	187148.	0.990170
$130$	0	0
$131$	−77850.1	−0.396352	−0.198176	−	0.980166i	$-0.563502\pi$
$131$	−77850.1	−0.396352	−0.198176	+	0.980166i	$0.563502\pi$
$132$	0	0
$133$	32683.9	0.160215
$134$	0	0
$135$	56589.2	0.267238
$136$	0	0
$137$	372403.	1.69516	0.847581	−	0.530666i	$-0.178058\pi$
$137$	372403.	1.69516	0.847581	+	0.530666i	$0.178058\pi$
$138$	0	0
$139$	−274550.	−1.20527	−0.602636	−	0.798016i	$-0.705883\pi$
$139$	−274550.	−1.20527	−0.602636	+	0.798016i	$0.705883\pi$
$140$	0	0
$141$	38550.3	0.163298
$142$	0	0
$143$	−30435.1	−0.124461
$144$	0	0
$145$	−178615.	−0.705500
$146$	0	0
$147$	−21609.0	−0.0824786
$148$	0	0
$149$	−312982.	−1.15492	−0.577462	−	0.816417i	$-0.695957\pi$
$149$	−312982.	−1.15492	−0.577462	+	0.816417i	$0.695957\pi$
$150$	0	0
$151$	432095.	1.54219	0.771093	−	0.636723i	$-0.219710\pi$
$151$	432095.	1.54219	0.771093	+	0.636723i	$0.219710\pi$
$152$	0	0
$153$	84047.7	0.290267
$154$	0	0
$155$	−288539.	−0.964662
$156$	0	0
$157$	−84603.7	−0.273930	−0.136965	−	0.990576i	$-0.543735\pi$
$157$	−84603.7	−0.273930	−0.136965	+	0.990576i	$0.543735\pi$
$158$	0	0
$159$	−231466.	−0.726096
$160$	0	0
$161$	−159330.	−0.484431
$162$	0	0
$163$	306303.	0.902989	0.451494	−	0.892274i	$-0.350891\pi$
$163$	306303.	0.902989	0.451494	+	0.892274i	$0.350891\pi$
$164$	0	0
$165$	333513.	0.953680
$166$	0	0
$167$	−606514.	−1.68287	−0.841433	−	0.540362i	$-0.818287\pi$
$167$	−606514.	−1.68287	−0.841433	+	0.540362i	$0.818287\pi$
$168$	0	0
$169$	−367228.	−0.989053
$170$	0	0
$171$	−54028.4	−0.141297
$172$	0	0
$173$	−288481.	−0.732828	−0.366414	−	0.930452i	$-0.619415\pi$
$173$	−288481.	−0.732828	−0.366414	+	0.930452i	$0.619415\pi$
$174$	0	0
$175$	−142137.	−0.350842
$176$	0	0
$177$	25548.4	0.0612958
$178$	0	0
$179$	148858.	0.347248	0.173624	−	0.984812i	$-0.444452\pi$
$179$	148858.	0.347248	0.173624	+	0.984812i	$0.444452\pi$
$180$	0	0
$181$	93377.8	0.211859	0.105930	−	0.994374i	$-0.466218\pi$
$181$	93377.8	0.211859	0.105930	+	0.994374i	$0.466218\pi$
$182$	0	0
$183$	−151229.	−0.333816
$184$	0	0
$185$	−950594.	−2.04205
$186$	0	0
$187$	495342.	1.03586
$188$	0	0
$189$	35721.0	0.0727393
$190$	0	0
$191$	246915.	0.489738	0.244869	−	0.969556i	$-0.421255\pi$
$191$	246915.	0.489738	0.244869	+	0.969556i	$0.421255\pi$
$192$	0	0
$193$	481437.	0.930349	0.465175	−	0.885219i	$-0.345992\pi$
$193$	481437.	0.930349	0.465175	+	0.885219i	$0.345992\pi$
$194$	0	0
$195$	−44540.8	−0.0838826
$196$	0	0
$197$	−548236.	−1.00647	−0.503237	−	0.864149i	$-0.667858\pi$
$197$	−548236.	−1.00647	−0.503237	+	0.864149i	$0.667858\pi$
$198$	0	0
$199$	158179.	0.283150	0.141575	−	0.989928i	$-0.454783\pi$
$199$	158179.	0.283150	0.141575	+	0.989928i	$0.454783\pi$
$200$	0	0
$201$	562816.	0.982599
$202$	0	0
$203$	−112748.	−0.192029
$204$	0	0
$205$	142028.	0.236042
$206$	0	0
$207$	263382.	0.427228
$208$	0	0
$209$	−318421.	−0.504238
$210$	0	0
$211$	283510.	0.438392	0.219196	−	0.975681i	$-0.429657\pi$
$211$	283510.	0.438392	0.219196	+	0.975681i	$0.429657\pi$
$212$	0	0
$213$	−650709.	−0.982739
$214$	0	0
$215$	1.61416e6	2.38150
$216$	0	0
$217$	−182136.	−0.262570
$218$	0	0
$219$	501092.	0.706004
$220$	0	0
$221$	−66153.2	−0.0911109
$222$	0	0
$223$	−651135.	−0.876817	−0.438409	−	0.898776i	$-0.644458\pi$
$223$	−651135.	−0.876817	−0.438409	+	0.898776i	$0.644458\pi$
$224$	0	0
$225$	234961.	0.309414
$226$	0	0
$227$	−378294.	−0.487264	−0.243632	−	0.969868i	$-0.578339\pi$
$227$	−378294.	−0.487264	−0.243632	+	0.969868i	$0.578339\pi$
$228$	0	0
$229$	−22332.8	−0.0281420	−0.0140710	−	0.999901i	$-0.504479\pi$
$229$	−22332.8	−0.0281420	−0.0140710	+	0.999901i	$0.504479\pi$
$230$	0	0
$231$	210525.	0.259581
$232$	0	0
$233$	−908940.	−1.09685	−0.548423	−	0.836201i	$-0.684771\pi$
$233$	−908940.	−1.09685	−0.548423	+	0.836201i	$0.684771\pi$
$234$	0	0
$235$	332500.	0.392755
$236$	0	0
$237$	35902.7	0.0415199
$238$	0	0
$239$	1.05363e6	1.19315	0.596573	−	0.802559i	$-0.296529\pi$
$239$	1.05363e6	1.19315	0.596573	+	0.802559i	$0.296529\pi$
$240$	0	0
$241$	1.05233e6	1.16710	0.583550	−	0.812077i	$-0.301663\pi$
$241$	1.05233e6	1.16710	0.583550	+	0.812077i	$0.301663\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−186379.	−0.198373
$246$	0	0
$247$	42525.3	0.0443512
$248$	0	0
$249$	414830.	0.424006
$250$	0	0
$251$	972876.	0.974705	0.487352	−	0.873205i	$-0.337963\pi$
$251$	972876.	0.974705	0.487352	+	0.873205i	$0.337963\pi$
$252$	0	0
$253$	1.55226e6	1.52463
$254$	0	0
$255$	724918.	0.698134
$256$	0	0
$257$	1.77948e6	1.68058	0.840290	−	0.542137i	$-0.182385\pi$
$257$	1.77948e6	1.68058	0.840290	+	0.542137i	$0.182385\pi$
$258$	0	0
$259$	−600047.	−0.555822
$260$	0	0
$261$	186379.	0.169354
$262$	0	0
$263$	21959.6	0.0195765	0.00978827	−	0.999952i	$-0.496884\pi$
$263$	21959.6	0.0195765	0.00978827	+	0.999952i	$0.496884\pi$
$264$	0	0
$265$	−1.99641e6	−1.74637
$266$	0	0
$267$	−1.21846e6	−1.04601
$268$	0	0
$269$	−1.69111e6	−1.42492	−0.712461	−	0.701712i	$-0.752419\pi$
$269$	−1.69111e6	−1.42492	−0.712461	+	0.701712i	$0.752419\pi$
$270$	0	0
$271$	−467863.	−0.386986	−0.193493	−	0.981102i	$-0.561982\pi$
$271$	−467863.	−0.386986	−0.193493	+	0.981102i	$0.561982\pi$
$272$	0	0
$273$	−28115.7	−0.0228319
$274$	0	0
$275$	1.38476e6	1.10419
$276$	0	0
$277$	906169.	0.709594	0.354797	−	0.934943i	$-0.384550\pi$
$277$	906169.	0.709594	0.354797	+	0.934943i	$0.384550\pi$
$278$	0	0
$279$	301081.	0.231565
$280$	0	0
$281$	−781388.	−0.590338	−0.295169	−	0.955445i	$-0.595376\pi$
$281$	−781388.	−0.590338	−0.295169	+	0.955445i	$0.595376\pi$
$282$	0	0
$283$	−1.49843e6	−1.11217	−0.556085	−	0.831125i	$-0.687697\pi$
$283$	−1.49843e6	−1.11217	−0.556085	+	0.831125i	$0.687697\pi$
$284$	0	0
$285$	−466000.	−0.339839
$286$	0	0
$287$	89652.8	0.0642479
$288$	0	0
$289$	−343190.	−0.241707
$290$	0	0
$291$	−1.28590e6	−0.890175
$292$	0	0
$293$	−1.56070e6	−1.06206	−0.531031	−	0.847352i	$-0.678195\pi$
$293$	−1.56070e6	−1.06206	−0.531031	+	0.847352i	$0.678195\pi$
$294$	0	0
$295$	220357.	0.147425
$296$	0	0
$297$	−348010.	−0.228929
$298$	0	0
$299$	−207305.	−0.134101
$300$	0	0
$301$	1.01891e6	0.648219
$302$	0	0
$303$	−400204.	−0.250423
$304$	0	0
$305$	−1.30436e6	−0.802875
$306$	0	0
$307$	−889308.	−0.538525	−0.269263	−	0.963067i	$-0.586780\pi$
$307$	−889308.	−0.538525	−0.269263	+	0.963067i	$0.586780\pi$
$308$	0	0
$309$	−1.82357e6	−1.08649
$310$	0	0
$311$	2.44826e6	1.43535	0.717674	−	0.696380i	$-0.245207\pi$
$311$	2.44826e6	1.43535	0.717674	+	0.696380i	$0.245207\pi$
$312$	0	0
$313$	−2.73102e6	−1.57567	−0.787834	−	0.615887i	$-0.788798\pi$
$313$	−2.73102e6	−1.57567	−0.787834	+	0.615887i	$0.788798\pi$
$314$	0	0
$315$	308097.	0.174949
$316$	0	0
$317$	−249603.	−0.139509	−0.0697545	−	0.997564i	$-0.522222\pi$
$317$	−249603.	−0.139509	−0.0697545	+	0.997564i	$0.522222\pi$
$318$	0	0
$319$	1.09844e6	0.604364
$320$	0	0
$321$	−895733.	−0.485195
$322$	0	0
$323$	−692115.	−0.369124
$324$	0	0
$325$	−184936.	−0.0971209
$326$	0	0
$327$	−1.98837e6	−1.02832
$328$	0	0
$329$	209885.	0.106903
$330$	0	0
$331$	646617.	0.324397	0.162199	−	0.986758i	$-0.448142\pi$
$331$	646617.	0.324397	0.162199	+	0.986758i	$0.448142\pi$
$332$	0	0
$333$	991915.	0.490189
$334$	0	0
$335$	4.85433e6	2.36329
$336$	0	0
$337$	−1.02782e6	−0.492994	−0.246497	−	0.969144i	$-0.579280\pi$
$337$	−1.02782e6	−0.492994	−0.246497	+	0.969144i	$0.579280\pi$
$338$	0	0
$339$	−266608.	−0.126001
$340$	0	0
$341$	1.77445e6	0.826375
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	2.27169e6	1.02755
$346$	0	0
$347$	−1.64916e6	−0.735256	−0.367628	−	0.929973i	$-0.619830\pi$
$347$	−1.64916e6	−0.735256	−0.367628	+	0.929973i	$0.619830\pi$
$348$	0	0
$349$	2.21201e6	0.972128	0.486064	−	0.873923i	$-0.338432\pi$
$349$	2.21201e6	0.972128	0.486064	+	0.873923i	$0.338432\pi$
$350$	0	0
$351$	46477.0	0.0201358
$352$	0	0
$353$	1.10158e6	0.470520	0.235260	−	0.971932i	$-0.424406\pi$
$353$	1.10158e6	0.470520	0.235260	+	0.971932i	$0.424406\pi$
$354$	0	0
$355$	−5.61242e6	−2.36363
$356$	0	0
$357$	457593.	0.190024
$358$	0	0
$359$	−1.29940e6	−0.532116	−0.266058	−	0.963957i	$-0.585721\pi$
$359$	−1.29940e6	−0.532116	−0.266058	+	0.963957i	$0.585721\pi$
$360$	0	0
$361$	−2.03119e6	−0.820317
$362$	0	0
$363$	−601567.	−0.239617
$364$	0	0
$365$	4.32196e6	1.69804
$366$	0	0
$367$	1.55669e6	0.603305	0.301652	−	0.953418i	$-0.402462\pi$
$367$	1.55669e6	0.603305	0.301652	+	0.953418i	$0.402462\pi$
$368$	0	0
$369$	−148202.	−0.0566614
$370$	0	0
$371$	−1.26020e6	−0.475341
$372$	0	0
$373$	−3.12660e6	−1.16359	−0.581796	−	0.813335i	$-0.697650\pi$
$373$	−3.12660e6	−1.16359	−0.581796	+	0.813335i	$0.697650\pi$
$374$	0	0
$375$	−156665.	−0.0575299
$376$	0	0
$377$	−146697.	−0.0531579
$378$	0	0
$379$	−2.96497e6	−1.06029	−0.530143	−	0.847908i	$-0.677862\pi$
$379$	−2.96497e6	−1.06029	−0.530143	+	0.847908i	$0.677862\pi$
$380$	0	0
$381$	2.37719e6	0.838978
$382$	0	0
$383$	2.74774e6	0.957149	0.478574	−	0.878047i	$-0.341154\pi$
$383$	2.74774e6	0.957149	0.478574	+	0.878047i	$0.341154\pi$
$384$	0	0
$385$	1.81579e6	0.624330
$386$	0	0
$387$	−1.68433e6	−0.571675
$388$	0	0
$389$	−611343.	−0.204838	−0.102419	−	0.994741i	$-0.532658\pi$
$389$	−611343.	−0.204838	−0.102419	+	0.994741i	$0.532658\pi$
$390$	0	0
$391$	3.37397e6	1.11609
$392$	0	0
$393$	700651.	0.228834
$394$	0	0
$395$	309664.	0.0998615
$396$	0	0
$397$	2.16794e6	0.690352	0.345176	−	0.938538i	$-0.387819\pi$
$397$	2.16794e6	0.690352	0.345176	+	0.938538i	$0.387819\pi$
$398$	0	0
$399$	−294155.	−0.0925004
$400$	0	0
$401$	−3.14644e6	−0.977143	−0.488572	−	0.872524i	$-0.662482\pi$
$401$	−3.14644e6	−0.977143	−0.488572	+	0.872524i	$0.662482\pi$
$402$	0	0
$403$	−236978.	−0.0726852
$404$	0	0
$405$	−509302.	−0.154290
$406$	0	0
$407$	5.84593e6	1.74931
$408$	0	0
$409$	5.58165e6	1.64989	0.824943	−	0.565215i	$-0.191207\pi$
$409$	5.58165e6	1.64989	0.824943	+	0.565215i	$0.191207\pi$
$410$	0	0
$411$	−3.35162e6	−0.978703
$412$	0	0
$413$	139097.	0.0401275
$414$	0	0
$415$	3.57794e6	1.01980
$416$	0	0
$417$	2.47095e6	0.695864
$418$	0	0
$419$	2.25054e6	0.626257	0.313128	−	0.949711i	$-0.398623\pi$
$419$	2.25054e6	0.626257	0.313128	+	0.949711i	$0.398623\pi$
$420$	0	0
$421$	3.45914e6	0.951180	0.475590	−	0.879667i	$-0.342235\pi$
$421$	3.45914e6	0.951180	0.475590	+	0.879667i	$0.342235\pi$
$422$	0	0
$423$	−346953.	−0.0942800
$424$	0	0
$425$	3.00990e6	0.808313
$426$	0	0
$427$	−823356.	−0.218534
$428$	0	0
$429$	273916.	0.0718578
$430$	0	0
$431$	−6.55926e6	−1.70083	−0.850417	−	0.526109i	$-0.823650\pi$
$431$	−6.55926e6	−1.70083	−0.850417	+	0.526109i	$0.823650\pi$
$432$	0	0
$433$	5.05669e6	1.29612	0.648062	−	0.761587i	$-0.275580\pi$
$433$	5.05669e6	1.29612	0.648062	+	0.761587i	$0.275580\pi$
$434$	0	0
$435$	1.60753e6	0.407320
$436$	0	0
$437$	−2.16889e6	−0.543293
$438$	0	0
$439$	4.23225e6	1.04812	0.524059	−	0.851682i	$-0.324417\pi$
$439$	4.23225e6	1.04812	0.524059	+	0.851682i	$0.324417\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	−6.05525e6	−1.46596	−0.732981	−	0.680249i	$-0.761872\pi$
$443$	−6.05525e6	−1.46596	−0.732981	+	0.680249i	$0.761872\pi$
$444$	0	0
$445$	−1.05093e7	−2.51579
$446$	0	0
$447$	2.81684e6	0.666796
$448$	0	0
$449$	−299186.	−0.0700368	−0.0350184	−	0.999387i	$-0.511149\pi$
$449$	−299186.	−0.0700368	−0.0350184	+	0.999387i	$0.511149\pi$
$450$	0	0
$451$	−873438.	−0.202204
$452$	0	0
$453$	−3.88885e6	−0.890381
$454$	0	0
$455$	−242500.	−0.0549140
$456$	0	0
$457$	3.75177e6	0.840322	0.420161	−	0.907450i	$-0.361974\pi$
$457$	3.75177e6	0.840322	0.420161	+	0.907450i	$0.361974\pi$
$458$	0	0
$459$	−756429.	−0.167586
$460$	0	0
$461$	6.94525e6	1.52207	0.761036	−	0.648709i	$-0.224691\pi$
$461$	6.94525e6	1.52207	0.761036	+	0.648709i	$0.224691\pi$
$462$	0	0
$463$	−9.13226e6	−1.97982	−0.989910	−	0.141697i	$-0.954744\pi$
$463$	−9.13226e6	−1.97982	−0.989910	+	0.141697i	$0.954744\pi$
$464$	0	0
$465$	2.59685e6	0.556948
$466$	0	0
$467$	−424136.	−0.0899940	−0.0449970	−	0.998987i	$-0.514328\pi$
$467$	−424136.	−0.0899940	−0.0449970	+	0.998987i	$0.514328\pi$
$468$	0	0
$469$	3.06422e6	0.643262
$470$	0	0
$471$	761433.	0.158154
$472$	0	0
$473$	−9.92673e6	−2.04011
$474$	0	0
$475$	−1.93485e6	−0.393472
$476$	0	0
$477$	2.08319e6	0.419212
$478$	0	0
$479$	−7.78605e6	−1.55052	−0.775262	−	0.631640i	$-0.782382\pi$
$479$	−7.78605e6	−1.55052	−0.775262	+	0.631640i	$0.782382\pi$
$480$	0	0
$481$	−780727.	−0.153864
$482$	0	0
$483$	1.43397e6	0.279686
$484$	0	0
$485$	−1.10910e7	−2.14100
$486$	0	0
$487$	2.32259e6	0.443763	0.221881	−	0.975074i	$-0.428780\pi$
$487$	2.32259e6	0.443763	0.221881	+	0.975074i	$0.428780\pi$
$488$	0	0
$489$	−2.75673e6	−0.521341
$490$	0	0
$491$	6.01036e6	1.12512	0.562558	−	0.826758i	$-0.309817\pi$
$491$	6.01036e6	1.12512	0.562558	+	0.826758i	$0.309817\pi$
$492$	0	0
$493$	2.38755e6	0.442420
$494$	0	0
$495$	−3.00162e6	−0.550607
$496$	0	0
$497$	−3.54275e6	−0.643353
$498$	0	0
$499$	3.37385e6	0.606562	0.303281	−	0.952901i	$-0.401918\pi$
$499$	3.37385e6	0.606562	0.303281	+	0.952901i	$0.401918\pi$
$500$	0	0
$501$	5.45862e6	0.971603
$502$	0	0
$503$	−1.22068e6	−0.215120	−0.107560	−	0.994199i	$-0.534304\pi$
$503$	−1.22068e6	−0.215120	−0.107560	+	0.994199i	$0.534304\pi$
$504$	0	0
$505$	−3.45179e6	−0.602304
$506$	0	0
$507$	3.30506e6	0.571030
$508$	0	0
$509$	1.56897e6	0.268423	0.134212	−	0.990953i	$-0.457150\pi$
$509$	1.56897e6	0.268423	0.134212	+	0.990953i	$0.457150\pi$
$510$	0	0
$511$	2.72817e6	0.462188
$512$	0	0
$513$	486256.	0.0815777
$514$	0	0
$515$	−1.57285e7	−2.61318
$516$	0	0
$517$	−2.04480e6	−0.336452
$518$	0	0
$519$	2.59633e6	0.423098
$520$	0	0
$521$	1.06779e7	1.72342	0.861708	−	0.507404i	$-0.169395\pi$
$521$	1.06779e7	1.72342	0.861708	+	0.507404i	$0.169395\pi$
$522$	0	0
$523$	−1.21007e7	−1.93444	−0.967219	−	0.253943i	$-0.918272\pi$
$523$	−1.21007e7	−1.93444	−0.967219	+	0.253943i	$0.918272\pi$
$524$	0	0
$525$	1.27923e6	0.202559
$526$	0	0
$527$	3.85691e6	0.604941
$528$	0	0
$529$	4.13673e6	0.642714
$530$	0	0
$531$	−229936.	−0.0353892
$532$	0	0
$533$	116648.	0.0177852
$534$	0	0
$535$	−7.72577e6	−1.16696
$536$	0	0
$537$	−1.33972e6	−0.200484
$538$	0	0
$539$	1.14619e6	0.169936
$540$	0	0
$541$	−3.19805e6	−0.469778	−0.234889	−	0.972022i	$-0.575473\pi$
$541$	−3.19805e6	−0.469778	−0.234889	+	0.972022i	$0.575473\pi$
$542$	0	0
$543$	−840400.	−0.122317
$544$	0	0
$545$	−1.71499e7	−2.47326
$546$	0	0
$547$	3.97811e6	0.568471	0.284235	−	0.958755i	$-0.408260\pi$
$547$	3.97811e6	0.568471	0.284235	+	0.958755i	$0.408260\pi$
$548$	0	0
$549$	1.36106e6	0.192729
$550$	0	0
$551$	−1.53479e6	−0.215362
$552$	0	0
$553$	195470.	0.0271812
$554$	0	0
$555$	8.55534e6	1.17898
$556$	0	0
$557$	1.14113e7	1.55847	0.779236	−	0.626731i	$-0.215608\pi$
$557$	1.14113e7	1.55847	0.779236	+	0.626731i	$0.215608\pi$
$558$	0	0
$559$	1.32572e6	0.179441
$560$	0	0
$561$	−4.45808e6	−0.598054
$562$	0	0
$563$	−6.46973e6	−0.860231	−0.430115	−	0.902774i	$-0.641527\pi$
$563$	−6.46973e6	−0.860231	−0.430115	+	0.902774i	$0.641527\pi$
$564$	0	0
$565$	−2.29952e6	−0.303051
$566$	0	0
$567$	−321489.	−0.0419961
$568$	0	0
$569$	717028.	0.0928444	0.0464222	−	0.998922i	$-0.485218\pi$
$569$	717028.	0.0928444	0.0464222	+	0.998922i	$0.485218\pi$
$570$	0	0
$571$	6.00286e6	0.770491	0.385246	−	0.922814i	$-0.374117\pi$
$571$	6.00286e6	0.770491	0.385246	+	0.922814i	$0.374117\pi$
$572$	0	0
$573$	−2.22223e6	−0.282750
$574$	0	0
$575$	9.43217e6	1.18971
$576$	0	0
$577$	1.55712e7	1.94707	0.973537	−	0.228531i	$-0.0733922\pi$
$577$	1.55712e7	1.94707	0.973537	+	0.228531i	$0.0733922\pi$
$578$	0	0
$579$	−4.33293e6	−0.537137
$580$	0	0
$581$	2.25852e6	0.277577
$582$	0	0
$583$	1.22775e7	1.49602
$584$	0	0
$585$	400868.	0.0484296
$586$	0	0
$587$	7.84621e6	0.939863	0.469931	−	0.882703i	$-0.344279\pi$
$587$	7.84621e6	0.939863	0.469931	+	0.882703i	$0.344279\pi$
$588$	0	0
$589$	−2.47934e6	−0.294475
$590$	0	0
$591$	4.93413e6	0.581088
$592$	0	0
$593$	1.33248e7	1.55605	0.778026	−	0.628232i	$-0.216221\pi$
$593$	1.33248e7	1.55605	0.778026	+	0.628232i	$0.216221\pi$
$594$	0	0
$595$	3.94678e6	0.457036
$596$	0	0
$597$	−1.42361e6	−0.163477
$598$	0	0
$599$	−7.77916e6	−0.885861	−0.442930	−	0.896556i	$-0.646061\pi$
$599$	−7.77916e6	−0.885861	−0.442930	+	0.896556i	$0.646061\pi$
$600$	0	0
$601$	8.62898e6	0.974480	0.487240	−	0.873268i	$-0.338004\pi$
$601$	8.62898e6	0.974480	0.487240	+	0.873268i	$0.338004\pi$
$602$	0	0
$603$	−5.06534e6	−0.567304
$604$	0	0
$605$	−5.18857e6	−0.576314
$606$	0	0
$607$	−7.09353e6	−0.781431	−0.390715	−	0.920512i	$-0.627772\pi$
$607$	−7.09353e6	−0.781431	−0.390715	+	0.920512i	$0.627772\pi$
$608$	0	0
$609$	1.01473e6	0.110868
$610$	0	0
$611$	273084.	0.0295932
$612$	0	0
$613$	4.52640e6	0.486521	0.243261	−	0.969961i	$-0.421783\pi$
$613$	4.52640e6	0.486521	0.243261	+	0.969961i	$0.421783\pi$
$614$	0	0
$615$	−1.27825e6	−0.136279
$616$	0	0
$617$	−8.38009e6	−0.886208	−0.443104	−	0.896470i	$-0.646123\pi$
$617$	−8.38009e6	−0.886208	−0.443104	+	0.896470i	$0.646123\pi$
$618$	0	0
$619$	−168342.	−0.0176590	−0.00882949	−	0.999961i	$-0.502811\pi$
$619$	−168342.	−0.0176590	−0.00882949	+	0.999961i	$0.502811\pi$
$620$	0	0
$621$	−2.37044e6	−0.246660
$622$	0	0
$623$	−6.63385e6	−0.684771
$624$	0	0
$625$	−1.04161e7	−1.06661
$626$	0	0
$627$	2.86579e6	0.291122
$628$	0	0
$629$	1.27066e7	1.28057
$630$	0	0
$631$	1.66208e7	1.66180	0.830899	−	0.556423i	$-0.187826\pi$
$631$	1.66208e7	1.66180	0.830899	+	0.556423i	$0.187826\pi$
$632$	0	0
$633$	−2.55159e6	−0.253106
$634$	0	0
$635$	2.05034e7	2.01786
$636$	0	0
$637$	−153074.	−0.0149470
$638$	0	0
$639$	5.85638e6	0.567384
$640$	0	0
$641$	1.02473e6	0.0985068	0.0492534	−	0.998786i	$-0.484316\pi$
$641$	1.02473e6	0.0985068	0.0492534	+	0.998786i	$0.484316\pi$
$642$	0	0
$643$	−1.22962e7	−1.17286	−0.586428	−	0.810001i	$-0.699466\pi$
$643$	−1.22962e7	−1.17286	−0.586428	+	0.810001i	$0.699466\pi$
$644$	0	0
$645$	−1.45275e7	−1.37496
$646$	0	0
$647$	−2.16537e6	−0.203363	−0.101682	−	0.994817i	$-0.532422\pi$
$647$	−2.16537e6	−0.203363	−0.101682	+	0.994817i	$0.532422\pi$
$648$	0	0
$649$	−1.35515e6	−0.126292
$650$	0	0
$651$	1.63922e6	0.151595
$652$	0	0
$653$	−1.22501e7	−1.12424	−0.562119	−	0.827056i	$-0.690014\pi$
$653$	−1.22501e7	−1.12424	−0.562119	+	0.827056i	$0.690014\pi$
$654$	0	0
$655$	6.04317e6	0.550379
$656$	0	0
$657$	−4.50983e6	−0.407612
$658$	0	0
$659$	1.18607e6	0.106389	0.0531944	−	0.998584i	$-0.483060\pi$
$659$	1.18607e6	0.106389	0.0531944	+	0.998584i	$0.483060\pi$
$660$	0	0
$661$	−1.45382e7	−1.29421	−0.647107	−	0.762399i	$-0.724021\pi$
$661$	−1.45382e7	−1.29421	−0.647107	+	0.762399i	$0.724021\pi$
$662$	0	0
$663$	595379.	0.0526029
$664$	0	0
$665$	−2.53711e6	−0.222477
$666$	0	0
$667$	7.48190e6	0.651174
$668$	0	0
$669$	5.86022e6	0.506231
$670$	0	0
$671$	8.02151e6	0.687781
$672$	0	0
$673$	−5.99405e6	−0.510132	−0.255066	−	0.966924i	$-0.582097\pi$
$673$	−5.99405e6	−0.510132	−0.255066	+	0.966924i	$0.582097\pi$
$674$	0	0
$675$	−2.11465e6	−0.178640
$676$	0	0
$677$	2.08389e7	1.74744	0.873721	−	0.486428i	$-0.161700\pi$
$677$	2.08389e7	1.74744	0.873721	+	0.486428i	$0.161700\pi$
$678$	0	0
$679$	−7.00102e6	−0.582756
$680$	0	0
$681$	3.40464e6	0.281322
$682$	0	0
$683$	4.55565e6	0.373679	0.186840	−	0.982390i	$-0.440176\pi$
$683$	4.55565e6	0.373679	0.186840	+	0.982390i	$0.440176\pi$
$684$	0	0
$685$	−2.89080e7	−2.35392
$686$	0	0
$687$	200995.	0.0162478
$688$	0	0
$689$	−1.63966e6	−0.131585
$690$	0	0
$691$	1.68542e6	0.134281	0.0671404	−	0.997744i	$-0.478612\pi$
$691$	1.68542e6	0.134281	0.0671404	+	0.997744i	$0.478612\pi$
$692$	0	0
$693$	−1.89472e6	−0.149869
$694$	0	0
$695$	2.13122e7	1.67365
$696$	0	0
$697$	−1.89849e6	−0.148022
$698$	0	0
$699$	8.18046e6	0.633264
$700$	0	0
$701$	2.45349e6	0.188577	0.0942887	−	0.995545i	$-0.469942\pi$
$701$	2.45349e6	0.188577	0.0942887	+	0.995545i	$0.469942\pi$
$702$	0	0
$703$	−8.16820e6	−0.623359
$704$	0	0
$705$	−2.99250e6	−0.226757
$706$	0	0
$707$	−2.17889e6	−0.163940
$708$	0	0
$709$	−1.32314e7	−0.988530	−0.494265	−	0.869311i	$-0.664563\pi$
$709$	−1.32314e7	−0.988530	−0.494265	+	0.869311i	$0.664563\pi$
$710$	0	0
$711$	−323125.	−0.0239715
$712$	0	0
$713$	1.20865e7	0.890380
$714$	0	0
$715$	2.36255e6	0.172828
$716$	0	0
$717$	−9.48267e6	−0.688863
$718$	0	0
$719$	1.35337e7	0.976324	0.488162	−	0.872753i	$-0.337667\pi$
$719$	1.35337e7	0.976324	0.488162	+	0.872753i	$0.337667\pi$
$720$	0	0
$721$	−9.92835e6	−0.711277
$722$	0	0
$723$	−9.47094e6	−0.673826
$724$	0	0
$725$	6.67455e6	0.471604
$726$	0	0
$727$	5.29416e6	0.371502	0.185751	−	0.982597i	$-0.440528\pi$
$727$	5.29416e6	0.371502	0.185751	+	0.982597i	$0.440528\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.15766e7	−1.49344
$732$	0	0
$733$	−2.25014e7	−1.54685	−0.773426	−	0.633886i	$-0.781459\pi$
$733$	−2.25014e7	−1.54685	−0.773426	+	0.633886i	$0.781459\pi$
$734$	0	0
$735$	1.67741e6	0.114531
$736$	0	0
$737$	−2.98530e7	−2.02451
$738$	0	0
$739$	1.60739e7	1.08271	0.541353	−	0.840795i	$-0.317912\pi$
$739$	1.60739e7	1.08271	0.541353	+	0.840795i	$0.317912\pi$
$740$	0	0
$741$	−382728.	−0.0256062
$742$	0	0
$743$	−2.31604e6	−0.153913	−0.0769563	−	0.997034i	$-0.524520\pi$
$743$	−2.31604e6	−0.153913	−0.0769563	+	0.997034i	$0.524520\pi$
$744$	0	0
$745$	2.42955e7	1.60374
$746$	0	0
$747$	−3.73347e6	−0.244800
$748$	0	0
$749$	−4.87677e6	−0.317634
$750$	0	0
$751$	−7.77263e6	−0.502885	−0.251442	−	0.967872i	$-0.580905\pi$
$751$	−7.77263e6	−0.502885	−0.251442	+	0.967872i	$0.580905\pi$
$752$	0	0
$753$	−8.75588e6	−0.562746
$754$	0	0
$755$	−3.35417e7	−2.14150
$756$	0	0
$757$	−1.59716e7	−1.01300	−0.506498	−	0.862241i	$-0.669060\pi$
$757$	−1.59716e7	−1.01300	−0.506498	+	0.862241i	$0.669060\pi$
$758$	0	0
$759$	−1.39704e7	−0.880244
$760$	0	0
$761$	1.84940e7	1.15763	0.578815	−	0.815459i	$-0.303515\pi$
$761$	1.84940e7	1.15763	0.578815	+	0.815459i	$0.303515\pi$
$762$	0	0
$763$	−1.08256e7	−0.673193
$764$	0	0
$765$	−6.52426e6	−0.403068
$766$	0	0
$767$	180980.	0.0111082
$768$	0	0
$769$	−2.33524e7	−1.42402	−0.712009	−	0.702170i	$-0.752214\pi$
$769$	−2.33524e7	−1.42402	−0.712009	+	0.702170i	$0.752214\pi$
$770$	0	0
$771$	−1.60153e7	−0.970283
$772$	0	0
$773$	7.25262e6	0.436562	0.218281	−	0.975886i	$-0.429955\pi$
$773$	7.25262e6	0.436562	0.218281	+	0.975886i	$0.429955\pi$
$774$	0	0
$775$	1.07823e7	0.644845
$776$	0	0
$777$	5.40042e6	0.320904
$778$	0	0
$779$	1.22041e6	0.0720546
$780$	0	0
$781$	3.45151e7	2.02480
$782$	0	0
$783$	−1.67741e6	−0.0977764
$784$	0	0
$785$	6.56742e6	0.380383
$786$	0	0
$787$	1.18935e6	0.0684497	0.0342248	−	0.999414i	$-0.489104\pi$
$787$	1.18935e6	0.0684497	0.0342248	+	0.999414i	$0.489104\pi$
$788$	0	0
$789$	−197637.	−0.0113025
$790$	0	0
$791$	−1.45153e6	−0.0824871
$792$	0	0
$793$	−1.07128e6	−0.0604949
$794$	0	0
$795$	1.79677e7	1.00827
$796$	0	0
$797$	2.14041e7	1.19358	0.596790	−	0.802397i	$-0.296442\pi$
$797$	2.14041e7	1.19358	0.596790	+	0.802397i	$0.296442\pi$
$798$	0	0
$799$	−4.44453e6	−0.246297
$800$	0	0
$801$	1.09662e7	0.603911
$802$	0	0
$803$	−2.65790e7	−1.45462
$804$	0	0
$805$	1.23681e7	0.672686
$806$	0	0
$807$	1.52200e7	0.822679
$808$	0	0
$809$	9.60341e6	0.515886	0.257943	−	0.966160i	$-0.416955\pi$
$809$	9.60341e6	0.515886	0.257943	+	0.966160i	$0.416955\pi$
$810$	0	0
$811$	2.62263e7	1.40018	0.700091	−	0.714054i	$-0.253143\pi$
$811$	2.62263e7	1.40018	0.700091	+	0.714054i	$0.253143\pi$
$812$	0	0
$813$	4.21077e6	0.223427
$814$	0	0
$815$	−2.37770e7	−1.25390
$816$	0	0
$817$	1.38701e7	0.726982
$818$	0	0
$819$	253041.	0.0131820
$820$	0	0
$821$	3.65685e7	1.89343	0.946715	−	0.322073i	$-0.104380\pi$
$821$	3.65685e7	1.89343	0.946715	+	0.322073i	$0.104380\pi$
$822$	0	0
$823$	−7.73488e6	−0.398065	−0.199032	−	0.979993i	$-0.563780\pi$
$823$	−7.73488e6	−0.398065	−0.199032	+	0.979993i	$0.563780\pi$
$824$	0	0
$825$	−1.24629e7	−0.637504
$826$	0	0
$827$	−1.47172e7	−0.748277	−0.374138	−	0.927373i	$-0.622061\pi$
$827$	−1.47172e7	−0.748277	−0.374138	+	0.927373i	$0.622061\pi$
$828$	0	0
$829$	−7.41886e6	−0.374931	−0.187465	−	0.982271i	$-0.560027\pi$
$829$	−7.41886e6	−0.374931	−0.187465	+	0.982271i	$0.560027\pi$
$830$	0	0
$831$	−8.15552e6	−0.409684
$832$	0	0
$833$	2.49134e6	0.124400
$834$	0	0
$835$	4.70811e7	2.33685
$836$	0	0
$837$	−2.70973e6	−0.133694
$838$	0	0
$839$	116538.	0.00571559	0.00285780	−	0.999996i	$-0.499090\pi$
$839$	116538.	0.00571559	0.00285780	+	0.999996i	$0.499090\pi$
$840$	0	0
$841$	−1.52167e7	−0.741874
$842$	0	0
$843$	7.03249e6	0.340832
$844$	0	0
$845$	2.85064e7	1.37341
$846$	0	0
$847$	−3.27520e6	−0.156866
$848$	0	0
$849$	1.34859e7	0.642112
$850$	0	0
$851$	3.98190e7	1.88480
$852$	0	0
$853$	−1.91763e7	−0.902387	−0.451193	−	0.892426i	$-0.649002\pi$
$853$	−1.91763e7	−0.902387	−0.451193	+	0.892426i	$0.649002\pi$
$854$	0	0
$855$	4.19400e6	0.196206
$856$	0	0
$857$	−1.62548e6	−0.0756012	−0.0378006	−	0.999285i	$-0.512035\pi$
$857$	−1.62548e6	−0.0756012	−0.0378006	+	0.999285i	$0.512035\pi$
$858$	0	0
$859$	−1.65931e7	−0.767265	−0.383632	−	0.923486i	$-0.625327\pi$
$859$	−1.65931e7	−0.767265	−0.383632	+	0.923486i	$0.625327\pi$
$860$	0	0
$861$	−806875.	−0.0370936
$862$	0	0
$863$	−2.30415e7	−1.05314	−0.526568	−	0.850133i	$-0.676521\pi$
$863$	−2.30415e7	−1.05314	−0.526568	+	0.850133i	$0.676521\pi$
$864$	0	0
$865$	2.23935e7	1.01761
$866$	0	0
$867$	3.08871e6	0.139550
$868$	0	0
$869$	−1.90436e6	−0.0855460
$870$	0	0
$871$	3.98689e6	0.178069
$872$	0	0
$873$	1.15731e7	0.513943
$874$	0	0
$875$	−852954.	−0.0376622
$876$	0	0
$877$	1.84493e7	0.809993	0.404996	−	0.914318i	$-0.367273\pi$
$877$	1.84493e7	0.809993	0.404996	+	0.914318i	$0.367273\pi$
$878$	0	0
$879$	1.40463e7	0.613182
$880$	0	0
$881$	−3.70548e7	−1.60844	−0.804219	−	0.594333i	$-0.797416\pi$
$881$	−3.70548e7	−1.60844	−0.804219	+	0.594333i	$0.797416\pi$
$882$	0	0
$883$	−5.28466e6	−0.228095	−0.114047	−	0.993475i	$-0.536382\pi$
$883$	−5.28466e6	−0.228095	−0.114047	+	0.993475i	$0.536382\pi$
$884$	0	0
$885$	−1.98321e6	−0.0851161
$886$	0	0
$887$	−1.98545e7	−0.847326	−0.423663	−	0.905820i	$-0.639256\pi$
$887$	−1.98545e7	−0.847326	−0.423663	+	0.905820i	$0.639256\pi$
$888$	0	0
$889$	1.29425e7	0.549240
$890$	0	0
$891$	3.13209e6	0.132172
$892$	0	0
$893$	2.85708e6	0.119893
$894$	0	0
$895$	−1.15552e7	−0.482193
$896$	0	0
$897$	1.86575e6	0.0774234
$898$	0	0
$899$	8.55283e6	0.352948
$900$	0	0
$901$	2.66861e7	1.09515
$902$	0	0
$903$	−9.17023e6	−0.374249
$904$	0	0
$905$	−7.24852e6	−0.294190
$906$	0	0
$907$	−2.00483e7	−0.809207	−0.404603	−	0.914492i	$-0.632590\pi$
$907$	−2.00483e7	−0.809207	−0.404603	+	0.914492i	$0.632590\pi$
$908$	0	0
$909$	3.60183e6	0.144582
$910$	0	0
$911$	−765753.	−0.0305698	−0.0152849	−	0.999883i	$-0.504866\pi$
$911$	−765753.	−0.0305698	−0.0152849	+	0.999883i	$0.504866\pi$
$912$	0	0
$913$	−2.20035e7	−0.873605
$914$	0	0
$915$	1.17392e7	0.463540
$916$	0	0
$917$	3.81465e6	0.149807
$918$	0	0
$919$	−1.87846e7	−0.733692	−0.366846	−	0.930282i	$-0.619562\pi$
$919$	−1.87846e7	−0.733692	−0.366846	+	0.930282i	$0.619562\pi$
$920$	0	0
$921$	8.00377e6	0.310918
$922$	0	0
$923$	−4.60951e6	−0.178094
$924$	0	0
$925$	3.55222e7	1.36504
$926$	0	0
$927$	1.64122e7	0.627288
$928$	0	0
$929$	−4.56290e7	−1.73461	−0.867304	−	0.497778i	$-0.834149\pi$
$929$	−4.56290e7	−1.73461	−0.867304	+	0.497778i	$0.834149\pi$
$930$	0	0
$931$	−1.60151e6	−0.0605557
$932$	0	0
$933$	−2.20344e7	−0.828698
$934$	0	0
$935$	−3.84513e7	−1.43841
$936$	0	0
$937$	6.67800e6	0.248484	0.124242	−	0.992252i	$-0.460350\pi$
$937$	6.67800e6	0.248484	0.124242	+	0.992252i	$0.460350\pi$
$938$	0	0
$939$	2.45792e7	0.909713
$940$	0	0
$941$	3.42716e7	1.26171	0.630857	−	0.775899i	$-0.282703\pi$
$941$	3.42716e7	1.26171	0.630857	+	0.775899i	$0.282703\pi$
$942$	0	0
$943$	−5.94933e6	−0.217866
$944$	0	0
$945$	−2.77287e6	−0.101007
$946$	0	0
$947$	5.28766e7	1.91597	0.957984	−	0.286821i	$-0.0925984\pi$
$947$	5.28766e7	1.91597	0.957984	+	0.286821i	$0.0925984\pi$
$948$	0	0
$949$	3.54965e6	0.127944
$950$	0	0
$951$	2.24643e6	0.0805455
$952$	0	0
$953$	5.33439e6	0.190262	0.0951311	−	0.995465i	$-0.469673\pi$
$953$	5.33439e6	0.190262	0.0951311	+	0.995465i	$0.469673\pi$
$954$	0	0
$955$	−1.91669e7	−0.680056
$956$	0	0
$957$	−9.88594e6	−0.348930
$958$	0	0
$959$	−1.82477e7	−0.640711
$960$	0	0
$961$	−1.48127e7	−0.517398
$962$	0	0
$963$	8.06160e6	0.280127
$964$	0	0
$965$	−3.73719e7	−1.29189
$966$	0	0
$967$	−1.93877e7	−0.666744	−0.333372	−	0.942795i	$-0.608187\pi$
$967$	−1.93877e7	−0.666744	−0.333372	+	0.942795i	$0.608187\pi$
$968$	0	0
$969$	6.22903e6	0.213114
$970$	0	0
$971$	1.07463e7	0.365772	0.182886	−	0.983134i	$-0.441456\pi$
$971$	1.07463e7	0.365772	0.182886	+	0.983134i	$0.441456\pi$
$972$	0	0
$973$	1.34530e7	0.455550
$974$	0	0
$975$	1.66442e6	0.0560728
$976$	0	0
$977$	−2.41051e7	−0.807926	−0.403963	−	0.914775i	$-0.632368\pi$
$977$	−2.41051e7	−0.807926	−0.403963	+	0.914775i	$0.632368\pi$
$978$	0	0
$979$	6.46300e7	2.15515
$980$	0	0
$981$	1.78953e7	0.593701
$982$	0	0
$983$	−4.05038e7	−1.33694	−0.668470	−	0.743739i	$-0.733050\pi$
$983$	−4.05038e7	−1.33694	−0.668470	+	0.743739i	$0.733050\pi$
$984$	0	0
$985$	4.25573e7	1.39760
$986$	0	0
$987$	−1.88897e6	−0.0617207
$988$	0	0
$989$	−6.76149e7	−2.19812
$990$	0	0
$991$	−4.01588e7	−1.29896	−0.649482	−	0.760377i	$-0.725014\pi$
$991$	−4.01588e7	−1.29896	−0.649482	+	0.760377i	$0.725014\pi$
$992$	0	0
$993$	−5.81955e6	−0.187291
$994$	0	0
$995$	−1.22788e7	−0.393185
$996$	0	0
$997$	−2.72276e7	−0.867505	−0.433753	−	0.901032i	$-0.642811\pi$
$997$	−2.72276e7	−0.867505	−0.433753	+	0.901032i	$0.642811\pi$
$998$	0	0
$999$	−8.92723e6	−0.283011

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.6.a.c.1.1	✓	2
3.2	odd	2		252.6.a.h.1.2		2
4.3	odd	2		336.6.a.x.1.1		2
7.2	even	3		588.6.i.l.361.2		4
7.3	odd	6		588.6.i.i.373.1		4
7.4	even	3		588.6.i.l.373.2		4
7.5	odd	6		588.6.i.i.361.1		4
7.6	odd	2		588.6.a.k.1.2		2
12.11	even	2		1008.6.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.a.c.1.1	✓	2	1.1	even	1	trivial
252.6.a.h.1.2		2	3.2	odd	2
336.6.a.x.1.1		2	4.3	odd	2
588.6.a.k.1.2		2	7.6	odd	2
588.6.i.i.361.1		4	7.5	odd	6
588.6.i.i.373.1		4	7.3	odd	6
588.6.i.l.361.2		4	7.2	even	3
588.6.i.l.373.2		4	7.4	even	3
1008.6.a.bo.1.2		2	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -77.6257 q^{5} -49.0000 q^{7} +81.0000 q^{9} +477.380 q^{11} -63.7544 q^{13} +698.632 q^{15} +1037.63 q^{17} -667.018 q^{19} +441.000 q^{21} +3251.63 q^{23} +2900.75 q^{25} -729.000 q^{27} +2300.97 q^{29} +3717.05 q^{31} -4296.42 q^{33} +3803.66 q^{35} +12245.9 q^{37} +573.790 q^{39} -1829.65 q^{41} -20794.2 q^{43} -6287.68 q^{45} -4283.37 q^{47} +2401.00 q^{49} -9338.63 q^{51} +25718.4 q^{53} -37057.0 q^{55} +6003.16 q^{57} -2838.71 q^{59} +16803.2 q^{61} -3969.00 q^{63} +4948.98 q^{65} -62535.1 q^{67} -29264.6 q^{69} +72301.0 q^{71} -55676.9 q^{73} -26106.8 q^{75} -23391.6 q^{77} -3989.19 q^{79} +6561.00 q^{81} -46092.2 q^{83} -80546.5 q^{85} -20708.7 q^{87} +135385. q^{89} +3123.97 q^{91} -33453.5 q^{93} +51777.7 q^{95} +142878. q^{97} +38667.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} - 6 q^{5} - 98 q^{7} + 162 q^{9} - 90 q^{11} + 768 q^{13} + 54 q^{15} + 1926 q^{17} + 2248 q^{19} + 882 q^{21} + 6354 q^{23} + 4906 q^{25} - 1458 q^{27} + 10572 q^{29} - 3312 q^{31} + 810 q^{33}+ \cdots - 7290 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 - 6 * q^5 - 98 * q^7 + 162 * q^9 - 90 * q^11 + 768 * q^13 + 54 * q^15 + 1926 * q^17 + 2248 * q^19 + 882 * q^21 + 6354 * q^23 + 4906 * q^25 - 1458 * q^27 + 10572 * q^29 - 3312 * q^31 + 810 * q^33 + 294 * q^35 + 2104 * q^37 - 6912 * q^39 + 1266 * q^41 - 5768 * q^43 - 486 * q^45 + 15612 * q^47 + 4802 * q^49 - 17334 * q^51 + 16512 * q^53 - 77696 * q^55 - 20232 * q^57 - 13140 * q^59 - 5796 * q^61 - 7938 * q^63 + 64524 * q^65 - 56116 * q^67 - 57186 * q^69 + 11022 * q^71 - 85384 * q^73 - 44154 * q^75 + 4410 * q^77 - 19620 * q^79 + 13122 * q^81 - 44424 * q^83 - 16916 * q^85 - 95148 * q^87 + 211218 * q^89 - 37632 * q^91 + 29808 * q^93 + 260568 * q^95 + 44864 * q^97 - 7290 * q^99

Introduction

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