LMFDB - Embedded newform 504.6.a.m.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [504,6,Mod(1,504)] 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("504.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	504.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-42,0,-98] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$80.8334451857$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{193}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 48 $ x^2 - x - 48
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.44622$ of defining polynomial
Character	$\chi$	$=$	504.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-90.4622 q^{5} -49.0000 q^{7} +552.494 q^{11} -593.172 q^{13} +1422.19 q^{17} +318.997 q^{19} -659.954 q^{23} +5058.41 q^{25} +8185.27 q^{29} -9598.03 q^{31} +4432.65 q^{35} +5180.56 q^{37} +2192.46 q^{41} +7458.29 q^{43} +19561.8 q^{47} +2401.00 q^{49} -36569.6 q^{53} -49979.9 q^{55} -16361.7 q^{59} -10893.1 q^{61} +53659.6 q^{65} +8035.91 q^{67} -55983.3 q^{71} -77752.5 q^{73} -27072.2 q^{77} -3208.43 q^{79} -76626.9 q^{83} -128655. q^{85} +84288.6 q^{89} +29065.4 q^{91} -28857.2 q^{95} +101273. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 42 q^{5} - 98 q^{7} + 716 q^{11} - 714 q^{13} + 1344 q^{17} - 1946 q^{19} + 1792 q^{23} + 4282 q^{25} + 1200 q^{29} - 6804 q^{31} + 2058 q^{35} + 14640 q^{37} - 7896 q^{41} + 524 q^{43} + 18396 q^{47}+ \cdots - 44240 q^{97}+O(q^{100}) $ 2 * q - 42 * q^5 - 98 * q^7 + 716 * q^11 - 714 * q^13 + 1344 * q^17 - 1946 * q^19 + 1792 * q^23 + 4282 * q^25 + 1200 * q^29 - 6804 * q^31 + 2058 * q^35 + 14640 * q^37 - 7896 * q^41 + 524 * q^43 + 18396 * q^47 + 4802 * q^49 - 45132 * q^53 - 42056 * q^55 - 22582 * q^59 - 52822 * q^61 + 47804 * q^65 + 9848 * q^67 + 840 * q^71 - 122052 * q^73 - 35084 * q^77 + 31704 * q^79 - 36974 * q^83 - 132444 * q^85 + 210588 * q^89 + 34986 * q^91 - 138624 * q^95 - 44240 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−90.4622	−1.61824	−0.809119	−	0.587645i	$-0.800055\pi$
$5$	−90.4622	−1.61824	−0.809119	+	0.587645i	$0.800055\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	552.494	1.37672	0.688361	−	0.725369i	$-0.258331\pi$
$11$	552.494	1.37672	0.688361	+	0.725369i	$0.258331\pi$
$12$	0	0
$13$	−593.172	−0.973469	−0.486734	−	0.873550i	$-0.661812\pi$
$13$	−593.172	−0.973469	−0.486734	+	0.873550i	$0.661812\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1422.19	1.19354	0.596769	−	0.802413i	$-0.296451\pi$
$17$	1422.19	1.19354	0.596769	+	0.802413i	$0.296451\pi$
$18$	0	0
$19$	318.997	0.202723	0.101361	−	0.994850i	$-0.467680\pi$
$19$	318.997	0.202723	0.101361	+	0.994850i	$0.467680\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−659.954	−0.260132	−0.130066	−	0.991505i	$-0.541519\pi$
$23$	−659.954	−0.260132	−0.130066	+	0.991505i	$0.541519\pi$
$24$	0	0
$25$	5058.41	1.61869
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8185.27	1.80733	0.903667	−	0.428237i	$-0.140865\pi$
$29$	8185.27	1.80733	0.903667	+	0.428237i	$0.140865\pi$
$30$	0	0
$31$	−9598.03	−1.79382	−0.896908	−	0.442217i	$-0.854192\pi$
$31$	−9598.03	−1.79382	−0.896908	+	0.442217i	$0.854192\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4432.65	0.611636
$36$	0	0
$37$	5180.56	0.622118	0.311059	−	0.950391i	$-0.399316\pi$
$37$	5180.56	0.622118	0.311059	+	0.950391i	$0.399316\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2192.46	0.203691	0.101846	−	0.994800i	$-0.467525\pi$
$41$	2192.46	0.203691	0.101846	+	0.994800i	$0.467525\pi$
$42$	0	0
$43$	7458.29	0.615131	0.307566	−	0.951527i	$-0.400486\pi$
$43$	7458.29	0.615131	0.307566	+	0.951527i	$0.400486\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	19561.8	1.29171	0.645853	−	0.763462i	$-0.276502\pi$
$47$	19561.8	1.29171	0.645853	+	0.763462i	$0.276502\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−36569.6	−1.78826	−0.894129	−	0.447809i	$-0.852205\pi$
$53$	−36569.6	−1.78826	−0.894129	+	0.447809i	$0.852205\pi$
$54$	0	0
$55$	−49979.9	−2.22786
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−16361.7	−0.611927	−0.305963	−	0.952043i	$-0.598979\pi$
$59$	−16361.7	−0.611927	−0.305963	+	0.952043i	$0.598979\pi$
$60$	0	0
$61$	−10893.1	−0.374825	−0.187412	−	0.982281i	$-0.560010\pi$
$61$	−10893.1	−0.374825	−0.187412	+	0.982281i	$0.560010\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	53659.6	1.57530
$66$	0	0
$67$	8035.91	0.218700	0.109350	−	0.994003i	$-0.465123\pi$
$67$	8035.91	0.218700	0.109350	+	0.994003i	$0.465123\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−55983.3	−1.31799	−0.658996	−	0.752146i	$-0.729019\pi$
$71$	−55983.3	−1.31799	−0.658996	+	0.752146i	$0.729019\pi$
$72$	0	0
$73$	−77752.5	−1.70768	−0.853841	−	0.520533i	$-0.825733\pi$
$73$	−77752.5	−1.70768	−0.853841	+	0.520533i	$0.825733\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−27072.2	−0.520352
$78$	0	0
$79$	−3208.43	−0.0578396	−0.0289198	−	0.999582i	$-0.509207\pi$
$79$	−3208.43	−0.0578396	−0.0289198	+	0.999582i	$0.509207\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−76626.9	−1.22092	−0.610458	−	0.792049i	$-0.709015\pi$
$83$	−76626.9	−1.22092	−0.610458	+	0.792049i	$0.709015\pi$
$84$	0	0
$85$	−128655.	−1.93143
$86$	0	0
$87$	0	0
$88$	0	0
$89$	84288.6	1.12796	0.563980	−	0.825788i	$-0.309269\pi$
$89$	84288.6	1.12796	0.563980	+	0.825788i	$0.309269\pi$
$90$	0	0
$91$	29065.4	0.367937
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−28857.2	−0.328054
$96$	0	0
$97$	101273.	1.09286	0.546428	−	0.837506i	$-0.315987\pi$
$97$	101273.	1.09286	0.546428	+	0.837506i	$0.315987\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−77456.5	−0.755535	−0.377767	−	0.925901i	$-0.623308\pi$
$101$	−77456.5	−0.755535	−0.377767	+	0.925901i	$0.623308\pi$
$102$	0	0
$103$	−56716.9	−0.526768	−0.263384	−	0.964691i	$-0.584839\pi$
$103$	−56716.9	−0.526768	−0.263384	+	0.964691i	$0.584839\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	38699.2	0.326770	0.163385	−	0.986562i	$-0.447759\pi$
$107$	38699.2	0.326770	0.163385	+	0.986562i	$0.447759\pi$
$108$	0	0
$109$	−16595.3	−0.133788	−0.0668941	−	0.997760i	$-0.521309\pi$
$109$	−16595.3	−0.133788	−0.0668941	+	0.997760i	$0.521309\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	76723.9	0.565242	0.282621	−	0.959232i	$-0.408796\pi$
$113$	76723.9	0.565242	0.282621	+	0.959232i	$0.408796\pi$
$114$	0	0
$115$	59700.9	0.420955
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−69687.4	−0.451115
$120$	0	0
$121$	144199.	0.895361
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−174901.	−1.00119
$126$	0	0
$127$	68539.3	0.377077	0.188539	−	0.982066i	$-0.439625\pi$
$127$	68539.3	0.377077	0.188539	+	0.982066i	$0.439625\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−95936.2	−0.488432	−0.244216	−	0.969721i	$-0.578531\pi$
$131$	−95936.2	−0.488432	−0.244216	+	0.969721i	$0.578531\pi$
$132$	0	0
$133$	−15630.9	−0.0766221
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−291295.	−1.32597	−0.662983	−	0.748635i	$-0.730710\pi$
$137$	−291295.	−1.32597	−0.662983	+	0.748635i	$0.730710\pi$
$138$	0	0
$139$	−281554.	−1.23602	−0.618009	−	0.786171i	$-0.712061\pi$
$139$	−281554.	−1.23602	−0.618009	+	0.786171i	$0.712061\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−327724.	−1.34019
$144$	0	0
$145$	−740458.	−2.92469
$146$	0	0
$147$	0	0
$148$	0	0
$149$	128283.	0.473372	0.236686	−	0.971586i	$-0.423939\pi$
$149$	128283.	0.473372	0.236686	+	0.971586i	$0.423939\pi$
$150$	0	0
$151$	19017.7	0.0678760	0.0339380	−	0.999424i	$-0.489195\pi$
$151$	19017.7	0.0678760	0.0339380	+	0.999424i	$0.489195\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	868259.	2.90282
$156$	0	0
$157$	−272988.	−0.883883	−0.441941	−	0.897044i	$-0.645710\pi$
$157$	−272988.	−0.883883	−0.441941	+	0.897044i	$0.645710\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	32337.7	0.0983207
$162$	0	0
$163$	315849.	0.931131	0.465565	−	0.885013i	$-0.345851\pi$
$163$	315849.	0.931131	0.465565	+	0.885013i	$0.345851\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	588655.	1.63331	0.816657	−	0.577123i	$-0.195825\pi$
$167$	588655.	1.63331	0.816657	+	0.577123i	$0.195825\pi$
$168$	0	0
$169$	−19440.5	−0.0523590
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−718256.	−1.82458	−0.912292	−	0.409540i	$-0.865689\pi$
$173$	−718256.	−1.82458	−0.912292	+	0.409540i	$0.865689\pi$
$174$	0	0
$175$	−247862.	−0.611808
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−53637.2	−0.125122	−0.0625610	−	0.998041i	$-0.519927\pi$
$179$	−53637.2	−0.125122	−0.0625610	+	0.998041i	$0.519927\pi$
$180$	0	0
$181$	−392213.	−0.889867	−0.444934	−	0.895564i	$-0.646773\pi$
$181$	−392213.	−0.889867	−0.444934	+	0.895564i	$0.646773\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−468645.	−1.00673
$186$	0	0
$187$	785753.	1.64317
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−395509.	−0.784463	−0.392232	−	0.919866i	$-0.628297\pi$
$191$	−395509.	−0.784463	−0.392232	+	0.919866i	$0.628297\pi$
$192$	0	0
$193$	−562892.	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$193$	−562892.	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−402202.	−0.738378	−0.369189	−	0.929354i	$-0.620365\pi$
$197$	−402202.	−0.738378	−0.369189	+	0.929354i	$0.620365\pi$
$198$	0	0
$199$	−455975.	−0.816222	−0.408111	−	0.912932i	$-0.633812\pi$
$199$	−455975.	−0.816222	−0.408111	+	0.912932i	$0.633812\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−401078.	−0.683108
$204$	0	0
$205$	−198335.	−0.329621
$206$	0	0
$207$	0	0
$208$	0	0
$209$	176244.	0.279093
$210$	0	0
$211$	−1.18264e6	−1.82871	−0.914356	−	0.404911i	$-0.867302\pi$
$211$	−1.18264e6	−1.82871	−0.914356	+	0.404911i	$0.867302\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−674693.	−0.995429
$216$	0	0
$217$	470303.	0.677999
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−843604.	−1.16187
$222$	0	0
$223$	931112.	1.25383	0.626917	−	0.779086i	$-0.284317\pi$
$223$	931112.	1.25383	0.626917	+	0.779086i	$0.284317\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−192653.	−0.248149	−0.124074	−	0.992273i	$-0.539596\pi$
$227$	−192653.	−0.248149	−0.124074	+	0.992273i	$0.539596\pi$
$228$	0	0
$229$	−783934.	−0.987850	−0.493925	−	0.869505i	$-0.664438\pi$
$229$	−783934.	−0.987850	−0.493925	+	0.869505i	$0.664438\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.39976e6	1.68914	0.844568	−	0.535448i	$-0.179857\pi$
$233$	1.39976e6	1.68914	0.844568	+	0.535448i	$0.179857\pi$
$234$	0	0
$235$	−1.76960e6	−2.09029
$236$	0	0
$237$	0	0
$238$	0	0
$239$	643631.	0.728857	0.364429	−	0.931231i	$-0.381264\pi$
$239$	643631.	0.728857	0.364429	+	0.931231i	$0.381264\pi$
$240$	0	0
$241$	−58756.4	−0.0651647	−0.0325824	−	0.999469i	$-0.510373\pi$
$241$	−58756.4	−0.0651647	−0.0325824	+	0.999469i	$0.510373\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−217200.	−0.231177
$246$	0	0
$247$	−189220.	−0.197344
$248$	0	0
$249$	0	0
$250$	0	0
$251$	641680.	0.642886	0.321443	−	0.946929i	$-0.395832\pi$
$251$	641680.	0.642886	0.321443	+	0.946929i	$0.395832\pi$
$252$	0	0
$253$	−364621.	−0.358129
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−810671.	−0.765617	−0.382809	−	0.923828i	$-0.625043\pi$
$257$	−810671.	−0.765617	−0.382809	+	0.923828i	$0.625043\pi$
$258$	0	0
$259$	−253848.	−0.235138
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.04061e6	1.81916	0.909581	−	0.415526i	$-0.136403\pi$
$263$	2.04061e6	1.81916	0.909581	+	0.415526i	$0.136403\pi$
$264$	0	0
$265$	3.30817e6	2.89383
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.28300e6	1.08105	0.540524	−	0.841328i	$-0.318226\pi$
$269$	1.28300e6	1.08105	0.540524	+	0.841328i	$0.318226\pi$
$270$	0	0
$271$	−22511.3	−0.0186199	−0.00930994	−	0.999957i	$-0.502963\pi$
$271$	−22511.3	−0.0186199	−0.00930994	+	0.999957i	$0.502963\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.79474e6	2.22849
$276$	0	0
$277$	368210.	0.288334	0.144167	−	0.989553i	$-0.453950\pi$
$277$	368210.	0.288334	0.144167	+	0.989553i	$0.453950\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.03709e6	−1.53902	−0.769511	−	0.638634i	$-0.779500\pi$
$281$	−2.03709e6	−1.53902	−0.769511	+	0.638634i	$0.779500\pi$
$282$	0	0
$283$	−656410.	−0.487202	−0.243601	−	0.969876i	$-0.578329\pi$
$283$	−656410.	−0.487202	−0.243601	+	0.969876i	$0.578329\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−107431.	−0.0769880
$288$	0	0
$289$	602773.	0.424531
$290$	0	0
$291$	0	0
$292$	0	0
$293$	962295.	0.654846	0.327423	−	0.944878i	$-0.393820\pi$
$293$	962295.	0.654846	0.327423	+	0.944878i	$0.393820\pi$
$294$	0	0
$295$	1.48012e6	0.990243
$296$	0	0
$297$	0	0
$298$	0	0
$299$	391466.	0.253230
$300$	0	0
$301$	−365456.	−0.232498
$302$	0	0
$303$	0	0
$304$	0	0
$305$	985418.	0.606556
$306$	0	0
$307$	−296468.	−0.179528	−0.0897638	−	0.995963i	$-0.528611\pi$
$307$	−296468.	−0.179528	−0.0897638	+	0.995963i	$0.528611\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.12063e6	−0.656996	−0.328498	−	0.944505i	$-0.606542\pi$
$311$	−1.12063e6	−0.656996	−0.328498	+	0.944505i	$0.606542\pi$
$312$	0	0
$313$	1.74908e6	1.00913	0.504567	−	0.863373i	$-0.331652\pi$
$313$	1.74908e6	1.00913	0.504567	+	0.863373i	$0.331652\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.17498e6	−1.77457	−0.887284	−	0.461223i	$-0.847411\pi$
$317$	−3.17498e6	−1.77457	−0.887284	+	0.461223i	$0.847411\pi$
$318$	0	0
$319$	4.52232e6	2.48819
$320$	0	0
$321$	0	0
$322$	0	0
$323$	453675.	0.241957
$324$	0	0
$325$	−3.00051e6	−1.57575
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−958526.	−0.488219
$330$	0	0
$331$	878405.	0.440681	0.220341	−	0.975423i	$-0.429283\pi$
$331$	878405.	0.440681	0.220341	+	0.975423i	$0.429283\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−726946.	−0.353908
$336$	0	0
$337$	2.05261e6	0.984536	0.492268	−	0.870444i	$-0.336168\pi$
$337$	2.05261e6	0.984536	0.492268	+	0.870444i	$0.336168\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.30286e6	−2.46958
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−476182.	−0.212300	−0.106150	−	0.994350i	$-0.533852\pi$
$347$	−476182.	−0.212300	−0.106150	+	0.994350i	$0.533852\pi$
$348$	0	0
$349$	351681.	0.154556	0.0772780	−	0.997010i	$-0.475377\pi$
$349$	351681.	0.154556	0.0772780	+	0.997010i	$0.475377\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.73722e6	0.742023	0.371011	−	0.928628i	$-0.379011\pi$
$353$	1.73722e6	0.742023	0.371011	+	0.928628i	$0.379011\pi$
$354$	0	0
$355$	5.06438e6	2.13282
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.69731e6	0.695063	0.347531	−	0.937668i	$-0.387020\pi$
$359$	1.69731e6	0.695063	0.347531	+	0.937668i	$0.387020\pi$
$360$	0	0
$361$	−2.37434e6	−0.958903
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.03366e6	2.76344
$366$	0	0
$367$	−1.11920e6	−0.433752	−0.216876	−	0.976199i	$-0.569587\pi$
$367$	−1.11920e6	−0.433752	−0.216876	+	0.976199i	$0.569587\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.79191e6	0.675898
$372$	0	0
$373$	−837822.	−0.311803	−0.155901	−	0.987773i	$-0.549828\pi$
$373$	−837822.	−0.311803	−0.155901	+	0.987773i	$0.549828\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.85527e6	−1.75938
$378$	0	0
$379$	−1.94713e6	−0.696300	−0.348150	−	0.937439i	$-0.613190\pi$
$379$	−1.94713e6	−0.696300	−0.348150	+	0.937439i	$0.613190\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	202063.	0.0703866	0.0351933	−	0.999381i	$-0.488795\pi$
$383$	202063.	0.0703866	0.0351933	+	0.999381i	$0.488795\pi$
$384$	0	0
$385$	2.44901e6	0.842053
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.23741e6	−1.08473	−0.542367	−	0.840141i	$-0.682472\pi$
$389$	−3.23741e6	−1.08473	−0.542367	+	0.840141i	$0.682472\pi$
$390$	0	0
$391$	−938581.	−0.310477
$392$	0	0
$393$	0	0
$394$	0	0
$395$	290242.	0.0935982
$396$	0	0
$397$	823921.	0.262367	0.131183	−	0.991358i	$-0.458122\pi$
$397$	823921.	0.262367	0.131183	+	0.991358i	$0.458122\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.39337e6	−0.743273	−0.371637	−	0.928378i	$-0.621203\pi$
$401$	−2.39337e6	−0.743273	−0.371637	+	0.928378i	$0.621203\pi$
$402$	0	0
$403$	5.69328e6	1.74622
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.86223e6	0.856483
$408$	0	0
$409$	−581801.	−0.171975	−0.0859876	−	0.996296i	$-0.527405\pi$
$409$	−581801.	−0.171975	−0.0859876	+	0.996296i	$0.527405\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	801725.	0.231287
$414$	0	0
$415$	6.93184e6	1.97573
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.11898e6	−0.867915	−0.433957	−	0.900933i	$-0.642883\pi$
$419$	−3.11898e6	−0.867915	−0.433957	+	0.900933i	$0.642883\pi$
$420$	0	0
$421$	−1.14481e6	−0.314796	−0.157398	−	0.987535i	$-0.550311\pi$
$421$	−1.14481e6	−0.314796	−0.157398	+	0.987535i	$0.550311\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.19403e6	1.93197
$426$	0	0
$427$	533764.	0.141671
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.31012e6	1.11762	0.558812	−	0.829294i	$-0.311257\pi$
$431$	4.31012e6	1.11762	0.558812	+	0.829294i	$0.311257\pi$
$432$	0	0
$433$	−2.79301e6	−0.715901	−0.357950	−	0.933741i	$-0.616524\pi$
$433$	−2.79301e6	−0.715901	−0.357950	+	0.933741i	$0.616524\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−210523.	−0.0527347
$438$	0	0
$439$	252467.	0.0625236	0.0312618	−	0.999511i	$-0.490047\pi$
$439$	252467.	0.0625236	0.0312618	+	0.999511i	$0.490047\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.43282e6	0.831077	0.415539	−	0.909576i	$-0.363593\pi$
$443$	3.43282e6	0.831077	0.415539	+	0.909576i	$0.363593\pi$
$444$	0	0
$445$	−7.62494e6	−1.82531
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.68031e6	0.393345	0.196673	−	0.980469i	$-0.436986\pi$
$449$	1.68031e6	0.393345	0.196673	+	0.980469i	$0.436986\pi$
$450$	0	0
$451$	1.21132e6	0.280426
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.62932e6	−0.595409
$456$	0	0
$457$	−2.94585e6	−0.659812	−0.329906	−	0.944014i	$-0.607017\pi$
$457$	−2.94585e6	−0.659812	−0.329906	+	0.944014i	$0.607017\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.91583e6	0.419860	0.209930	−	0.977716i	$-0.432676\pi$
$461$	1.91583e6	0.419860	0.209930	+	0.977716i	$0.432676\pi$
$462$	0	0
$463$	−3.75401e6	−0.813847	−0.406924	−	0.913462i	$-0.633398\pi$
$463$	−3.75401e6	−0.813847	−0.406924	+	0.913462i	$0.633398\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.10594e6	0.446842	0.223421	−	0.974722i	$-0.428278\pi$
$467$	2.10594e6	0.446842	0.223421	+	0.974722i	$0.428278\pi$
$468$	0	0
$469$	−393759.	−0.0826607
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.12066e6	0.846864
$474$	0	0
$475$	1.61362e6	0.328146
$476$	0	0
$477$	0	0
$478$	0	0
$479$	274926.	0.0547491	0.0273745	−	0.999625i	$-0.491285\pi$
$479$	274926.	0.0547491	0.0273745	+	0.999625i	$0.491285\pi$
$480$	0	0
$481$	−3.07296e6	−0.605612
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.16135e6	−1.76850
$486$	0	0
$487$	7.67128e6	1.46570	0.732851	−	0.680389i	$-0.238189\pi$
$487$	7.67128e6	1.46570	0.732851	+	0.680389i	$0.238189\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.42352e6	0.453673	0.226837	−	0.973933i	$-0.427162\pi$
$491$	2.42352e6	0.453673	0.226837	+	0.973933i	$0.427162\pi$
$492$	0	0
$493$	1.16410e7	2.15712
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.74318e6	0.498154
$498$	0	0
$499$	−8.22317e6	−1.47839	−0.739193	−	0.673494i	$-0.764793\pi$
$499$	−8.22317e6	−1.47839	−0.739193	+	0.673494i	$0.764793\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.98186e6	−0.525493	−0.262746	−	0.964865i	$-0.584628\pi$
$503$	−2.98186e6	−0.525493	−0.262746	+	0.964865i	$0.584628\pi$
$504$	0	0
$505$	7.00689e6	1.22263
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.26867e6	1.07246	0.536230	−	0.844072i	$-0.319848\pi$
$509$	6.26867e6	1.07246	0.536230	+	0.844072i	$0.319848\pi$
$510$	0	0
$511$	3.80987e6	0.645443
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.13073e6	0.852435
$516$	0	0
$517$	1.08078e7	1.77832
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.12134e6	0.665188	0.332594	−	0.943070i	$-0.392076\pi$
$521$	4.12134e6	0.665188	0.332594	+	0.943070i	$0.392076\pi$
$522$	0	0
$523$	−4.70469e6	−0.752102	−0.376051	−	0.926599i	$-0.622718\pi$
$523$	−4.70469e6	−0.752102	−0.376051	+	0.926599i	$0.622718\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.36502e7	−2.14099
$528$	0	0
$529$	−6.00080e6	−0.932331
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.30051e6	−0.198287
$534$	0	0
$535$	−3.50081e6	−0.528791
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.32654e6	0.196674
$540$	0	0
$541$	4.47840e6	0.657855	0.328927	−	0.944355i	$-0.393313\pi$
$541$	4.47840e6	0.657855	0.328927	+	0.944355i	$0.393313\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.50124e6	0.216501
$546$	0	0
$547$	−6.49187e6	−0.927687	−0.463843	−	0.885917i	$-0.653530\pi$
$547$	−6.49187e6	−0.927687	−0.463843	+	0.885917i	$0.653530\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.61108e6	0.366388
$552$	0	0
$553$	157213.	0.0218613
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.50858e6	0.342602	0.171301	−	0.985219i	$-0.445203\pi$
$557$	2.50858e6	0.342602	0.171301	+	0.985219i	$0.445203\pi$
$558$	0	0
$559$	−4.42404e6	−0.598811
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.52702e6	0.468961	0.234480	−	0.972121i	$-0.424661\pi$
$563$	3.52702e6	0.468961	0.234480	+	0.972121i	$0.424661\pi$
$564$	0	0
$565$	−6.94061e6	−0.914696
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9.75283e6	−1.26284	−0.631422	−	0.775439i	$-0.717529\pi$
$569$	−9.75283e6	−1.26284	−0.631422	+	0.775439i	$0.717529\pi$
$570$	0	0
$571$	1.70085e6	0.218312	0.109156	−	0.994025i	$-0.465185\pi$
$571$	1.70085e6	0.218312	0.109156	+	0.994025i	$0.465185\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.33832e6	−0.421074
$576$	0	0
$577$	−1.17066e7	−1.46383	−0.731914	−	0.681397i	$-0.761373\pi$
$577$	−1.17066e7	−1.46383	−0.731914	+	0.681397i	$0.761373\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.75472e6	0.461463
$582$	0	0
$583$	−2.02045e7	−2.46193
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.75683e6	0.450015	0.225007	−	0.974357i	$-0.427759\pi$
$587$	3.75683e6	0.450015	0.225007	+	0.974357i	$0.427759\pi$
$588$	0	0
$589$	−3.06175e6	−0.363648
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.64338e6	−1.00936	−0.504681	−	0.863306i	$-0.668390\pi$
$593$	−8.64338e6	−1.00936	−0.504681	+	0.863306i	$0.668390\pi$
$594$	0	0
$595$	6.30408e6	0.730011
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−9.31930e6	−1.06125	−0.530623	−	0.847608i	$-0.678042\pi$
$599$	−9.31930e6	−1.06125	−0.530623	+	0.847608i	$0.678042\pi$
$600$	0	0
$601$	910438.	0.102817	0.0514084	−	0.998678i	$-0.483629\pi$
$601$	910438.	0.102817	0.0514084	+	0.998678i	$0.483629\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.30445e7	−1.44891
$606$	0	0
$607$	5.70173e6	0.628109	0.314054	−	0.949405i	$-0.398313\pi$
$607$	5.70173e6	0.628109	0.314054	+	0.949405i	$0.398313\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.16035e7	−1.25743
$612$	0	0
$613$	1.61327e7	1.73403	0.867016	−	0.498280i	$-0.166035\pi$
$613$	1.61327e7	1.73403	0.867016	+	0.498280i	$0.166035\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.53575e6	0.585415	0.292708	−	0.956202i	$-0.405444\pi$
$617$	5.53575e6	0.585415	0.292708	+	0.956202i	$0.405444\pi$
$618$	0	0
$619$	−1.88448e7	−1.97681	−0.988407	−	0.151828i	$-0.951484\pi$
$619$	−1.88448e7	−1.97681	−0.988407	+	0.151828i	$0.951484\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.13014e6	−0.426329
$624$	0	0
$625$	14378.1	0.00147232
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.36776e6	0.742521
$630$	0	0
$631$	2.63269e6	0.263224	0.131612	−	0.991301i	$-0.457985\pi$
$631$	2.63269e6	0.263224	0.131612	+	0.991301i	$0.457985\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.20021e6	−0.610200
$636$	0	0
$637$	−1.42420e6	−0.139067
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.42939e7	−1.37406	−0.687028	−	0.726631i	$-0.741085\pi$
$641$	−1.42939e7	−1.37406	−0.687028	+	0.726631i	$0.741085\pi$
$642$	0	0
$643$	−1.63874e7	−1.56308	−0.781542	−	0.623853i	$-0.785566\pi$
$643$	−1.63874e7	−1.56308	−0.781542	+	0.623853i	$0.785566\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.19303e7	−1.12045	−0.560224	−	0.828341i	$-0.689285\pi$
$647$	−1.19303e7	−1.12045	−0.560224	+	0.828341i	$0.689285\pi$
$648$	0	0
$649$	−9.03977e6	−0.842453
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.47862e6	−0.319245	−0.159622	−	0.987178i	$-0.551028\pi$
$653$	−3.47862e6	−0.319245	−0.159622	+	0.987178i	$0.551028\pi$
$654$	0	0
$655$	8.67860e6	0.790399
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2.09397e7	−1.87826	−0.939131	−	0.343560i	$-0.888367\pi$
$659$	−2.09397e7	−1.87826	−0.939131	+	0.343560i	$0.888367\pi$
$660$	0	0
$661$	−1.02566e7	−0.913059	−0.456530	−	0.889708i	$-0.650908\pi$
$661$	−1.02566e7	−0.913059	−0.456530	+	0.889708i	$0.650908\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.41400e6	0.123993
$666$	0	0
$667$	−5.40190e6	−0.470145
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.01840e6	−0.516029
$672$	0	0
$673$	716398.	0.0609701	0.0304851	−	0.999535i	$-0.490295\pi$
$673$	716398.	0.0609701	0.0304851	+	0.999535i	$0.490295\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−945625.	−0.0792953	−0.0396476	−	0.999214i	$-0.512624\pi$
$677$	−945625.	−0.0792953	−0.0396476	+	0.999214i	$0.512624\pi$
$678$	0	0
$679$	−4.96236e6	−0.413061
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.56992e6	−0.292824	−0.146412	−	0.989224i	$-0.546773\pi$
$683$	−3.56992e6	−0.292824	−0.146412	+	0.989224i	$0.546773\pi$
$684$	0	0
$685$	2.63512e7	2.14573
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.16920e7	1.74081
$690$	0	0
$691$	9.67811e6	0.771073	0.385536	−	0.922693i	$-0.374016\pi$
$691$	9.67811e6	0.771073	0.385536	+	0.922693i	$0.374016\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.54700e7	2.00017
$696$	0	0
$697$	3.11810e6	0.243113
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.52404e7	−1.94000	−0.970000	−	0.243105i	$-0.921834\pi$
$701$	−2.52404e7	−1.94000	−0.970000	+	0.243105i	$0.921834\pi$
$702$	0	0
$703$	1.65259e6	0.126118
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.79537e6	0.285565
$708$	0	0
$709$	−1.74744e7	−1.30553	−0.652765	−	0.757560i	$-0.726391\pi$
$709$	−1.74744e7	−1.30553	−0.652765	+	0.757560i	$0.726391\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.33426e6	0.466629
$714$	0	0
$715$	2.96466e7	2.16875
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.73361e6	0.702185	0.351093	−	0.936341i	$-0.385810\pi$
$719$	9.73361e6	0.702185	0.351093	+	0.936341i	$0.385810\pi$
$720$	0	0
$721$	2.77913e6	0.199100
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.14045e7	2.92552
$726$	0	0
$727$	−1.60454e7	−1.12594	−0.562971	−	0.826477i	$-0.690342\pi$
$727$	−1.60454e7	−1.12594	−0.562971	+	0.826477i	$0.690342\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.06071e7	0.734182
$732$	0	0
$733$	1.04534e7	0.718615	0.359307	−	0.933219i	$-0.383013\pi$
$733$	1.04534e7	0.718615	0.359307	+	0.933219i	$0.383013\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.43979e6	0.301088
$738$	0	0
$739$	−7.47638e6	−0.503594	−0.251797	−	0.967780i	$-0.581022\pi$
$739$	−7.47638e6	−0.503594	−0.251797	+	0.967780i	$0.581022\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.33998e7	0.890483	0.445241	−	0.895411i	$-0.353118\pi$
$743$	1.33998e7	0.890483	0.445241	+	0.895411i	$0.353118\pi$
$744$	0	0
$745$	−1.16047e7	−0.766028
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.89626e6	−0.123507
$750$	0	0
$751$	9.13903e6	0.591290	0.295645	−	0.955298i	$-0.404465\pi$
$751$	9.13903e6	0.591290	0.295645	+	0.955298i	$0.404465\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.72039e6	−0.109840
$756$	0	0
$757$	−1.16376e7	−0.738117	−0.369059	−	0.929406i	$-0.620320\pi$
$757$	−1.16376e7	−0.738117	−0.369059	+	0.929406i	$0.620320\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.07072e6	0.505186	0.252593	−	0.967573i	$-0.418717\pi$
$761$	8.07072e6	0.505186	0.252593	+	0.967573i	$0.418717\pi$
$762$	0	0
$763$	813167.	0.0505672
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.70532e6	0.595692
$768$	0	0
$769$	−1.30085e7	−0.793255	−0.396628	−	0.917980i	$-0.629820\pi$
$769$	−1.30085e7	−0.793255	−0.396628	+	0.917980i	$0.629820\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.37637e7	−1.43043	−0.715214	−	0.698906i	$-0.753671\pi$
$773$	−2.37637e7	−1.43043	−0.715214	+	0.698906i	$0.753671\pi$
$774$	0	0
$775$	−4.85508e7	−2.90364
$776$	0	0
$777$	0	0
$778$	0	0
$779$	699389.	0.0412929
$780$	0	0
$781$	−3.09305e7	−1.81451
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.46951e7	1.43033
$786$	0	0
$787$	−2.84940e7	−1.63989	−0.819947	−	0.572439i	$-0.805997\pi$
$787$	−2.84940e7	−1.63989	−0.819947	+	0.572439i	$0.805997\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.75947e6	−0.213641
$792$	0	0
$793$	6.46150e6	0.364880
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.43450e7	−0.799938	−0.399969	−	0.916529i	$-0.630979\pi$
$797$	−1.43450e7	−0.799938	−0.399969	+	0.916529i	$0.630979\pi$
$798$	0	0
$799$	2.78206e7	1.54170
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.29578e7	−2.35100
$804$	0	0
$805$	−2.92534e6	−0.159106
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.59284e6	0.193004	0.0965020	−	0.995333i	$-0.469235\pi$
$809$	3.59284e6	0.193004	0.0965020	+	0.995333i	$0.469235\pi$
$810$	0	0
$811$	3.52968e7	1.88445	0.942223	−	0.334988i	$-0.108732\pi$
$811$	3.52968e7	1.88445	0.942223	+	0.334988i	$0.108732\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.85724e7	−1.50679
$816$	0	0
$817$	2.37917e6	0.124701
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.04558e7	−0.541378	−0.270689	−	0.962667i	$-0.587252\pi$
$821$	−1.04558e7	−0.541378	−0.270689	+	0.962667i	$0.587252\pi$
$822$	0	0
$823$	4.99190e6	0.256901	0.128451	−	0.991716i	$-0.459000\pi$
$823$	4.99190e6	0.256901	0.128451	+	0.991716i	$0.459000\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.06198e7	1.04839	0.524193	−	0.851599i	$-0.324367\pi$
$827$	2.06198e7	1.04839	0.524193	+	0.851599i	$0.324367\pi$
$828$	0	0
$829$	−506843.	−0.0256146	−0.0128073	−	0.999918i	$-0.504077\pi$
$829$	−506843.	−0.0256146	−0.0128073	+	0.999918i	$0.504077\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.41468e6	0.170505
$834$	0	0
$835$	−5.32511e7	−2.64309
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4.41851e6	0.216706	0.108353	−	0.994112i	$-0.465442\pi$
$839$	4.41851e6	0.216706	0.108353	+	0.994112i	$0.465442\pi$
$840$	0	0
$841$	4.64876e7	2.26645
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.75863e6	0.0847292
$846$	0	0
$847$	−7.06574e6	−0.338415
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.41893e6	−0.161833
$852$	0	0
$853$	−1.11737e7	−0.525806	−0.262903	−	0.964822i	$-0.584680\pi$
$853$	−1.11737e7	−0.525806	−0.262903	+	0.964822i	$0.584680\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.12308e7	−0.987448	−0.493724	−	0.869619i	$-0.664365\pi$
$857$	−2.12308e7	−0.987448	−0.493724	+	0.869619i	$0.664365\pi$
$858$	0	0
$859$	3.64229e7	1.68419	0.842096	−	0.539328i	$-0.181322\pi$
$859$	3.64229e7	1.68419	0.842096	+	0.539328i	$0.181322\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.53298e7	1.15772	0.578861	−	0.815426i	$-0.303497\pi$
$863$	2.53298e7	1.15772	0.578861	+	0.815426i	$0.303497\pi$
$864$	0	0
$865$	6.49750e7	2.95261
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.77264e6	−0.0796290
$870$	0	0
$871$	−4.76667e6	−0.212897
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.57014e6	0.378415
$876$	0	0
$877$	−577138.	−0.0253385	−0.0126692	−	0.999920i	$-0.504033\pi$
$877$	−577138.	−0.0253385	−0.0126692	+	0.999920i	$0.504033\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.15071e7	0.933561	0.466781	−	0.884373i	$-0.345414\pi$
$881$	2.15071e7	0.933561	0.466781	+	0.884373i	$0.345414\pi$
$882$	0	0
$883$	−3.71948e6	−0.160539	−0.0802695	−	0.996773i	$-0.525578\pi$
$883$	−3.71948e6	−0.160539	−0.0802695	+	0.996773i	$0.525578\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.81760e7	−0.775694	−0.387847	−	0.921724i	$-0.626781\pi$
$887$	−1.81760e7	−0.775694	−0.387847	+	0.921724i	$0.626781\pi$
$888$	0	0
$889$	−3.35842e6	−0.142522
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.24015e6	0.261858
$894$	0	0
$895$	4.85214e6	0.202477
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.85625e7	−3.24202
$900$	0	0
$901$	−5.20090e7	−2.13435
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.54804e7	1.44002
$906$	0	0
$907$	−1.35430e7	−0.546635	−0.273317	−	0.961924i	$-0.588121\pi$
$907$	−1.35430e7	−0.546635	−0.273317	+	0.961924i	$0.588121\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	412746.	0.0164773	0.00823866	−	0.999966i	$-0.497378\pi$
$911$	412746.	0.0164773	0.00823866	+	0.999966i	$0.497378\pi$
$912$	0	0
$913$	−4.23359e7	−1.68086
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.70087e6	0.184610
$918$	0	0
$919$	2.79059e7	1.08995	0.544976	−	0.838452i	$-0.316539\pi$
$919$	2.79059e7	1.08995	0.544976	+	0.838452i	$0.316539\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.32077e7	1.28302
$924$	0	0
$925$	2.62054e7	1.00702
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.94840e7	0.740695	0.370348	−	0.928893i	$-0.379239\pi$
$929$	1.94840e7	0.740695	0.370348	+	0.928893i	$0.379239\pi$
$930$	0	0
$931$	765912.	0.0289604
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.10809e7	−2.65904
$936$	0	0
$937$	−1.89631e7	−0.705602	−0.352801	−	0.935698i	$-0.614771\pi$
$937$	−1.89631e7	−0.705602	−0.352801	+	0.935698i	$0.614771\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.30498e7	0.848579	0.424290	−	0.905526i	$-0.360524\pi$
$941$	2.30498e7	0.848579	0.424290	+	0.905526i	$0.360524\pi$
$942$	0	0
$943$	−1.44692e6	−0.0529866
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.49310e7	1.26572	0.632858	−	0.774268i	$-0.281882\pi$
$947$	3.49310e7	1.26572	0.632858	+	0.774268i	$0.281882\pi$
$948$	0	0
$949$	4.61206e7	1.66238
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.74501e7	−0.979066	−0.489533	−	0.871985i	$-0.662833\pi$
$953$	−2.74501e7	−0.979066	−0.489533	+	0.871985i	$0.662833\pi$
$954$	0	0
$955$	3.57786e7	1.26945
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.42735e7	0.501168
$960$	0	0
$961$	6.34930e7	2.21778
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.09204e7	1.76025
$966$	0	0
$967$	−9.37608e6	−0.322445	−0.161222	−	0.986918i	$-0.551544\pi$
$967$	−9.37608e6	−0.322445	−0.161222	+	0.986918i	$0.551544\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7.23359e6	−0.246210	−0.123105	−	0.992394i	$-0.539285\pi$
$971$	−7.23359e6	−0.246210	−0.123105	+	0.992394i	$0.539285\pi$
$972$	0	0
$973$	1.37962e7	0.467171
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.88022e7	0.965360	0.482680	−	0.875797i	$-0.339663\pi$
$977$	2.88022e7	0.965360	0.482680	+	0.875797i	$0.339663\pi$
$978$	0	0
$979$	4.65690e7	1.55289
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.21848e7	−1.72250	−0.861252	−	0.508178i	$-0.830319\pi$
$983$	−5.21848e7	−1.72250	−0.861252	+	0.508178i	$0.830319\pi$
$984$	0	0
$985$	3.63841e7	1.19487
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.92212e6	−0.160015
$990$	0	0
$991$	−3.89578e7	−1.26012	−0.630058	−	0.776548i	$-0.716969\pi$
$991$	−3.89578e7	−1.26012	−0.630058	+	0.776548i	$0.716969\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.12485e7	1.32084
$996$	0	0
$997$	−3.01716e7	−0.961305	−0.480652	−	0.876911i	$-0.659600\pi$
$997$	−3.01716e7	−0.961305	−0.480652	+	0.876911i	$0.659600\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.6.a.m.1.1		2
3.2	odd	2		56.6.a.d.1.1	✓	2
4.3	odd	2		1008.6.a.bi.1.1		2
12.11	even	2		112.6.a.j.1.2		2
21.2	odd	6		392.6.i.k.361.2		4
21.5	even	6		392.6.i.h.361.1		4
21.11	odd	6		392.6.i.k.177.2		4
21.17	even	6		392.6.i.h.177.1		4
21.20	even	2		392.6.a.e.1.2		2
24.5	odd	2		448.6.a.x.1.2		2
24.11	even	2		448.6.a.r.1.1		2
84.83	odd	2		784.6.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.a.d.1.1	✓	2	3.2	odd	2
112.6.a.j.1.2		2	12.11	even	2
392.6.a.e.1.2		2	21.20	even	2
392.6.i.h.177.1		4	21.17	even	6
392.6.i.h.361.1		4	21.5	even	6
392.6.i.k.177.2		4	21.11	odd	6
392.6.i.k.361.2		4	21.2	odd	6
448.6.a.r.1.1		2	24.11	even	2
448.6.a.x.1.2		2	24.5	odd	2
504.6.a.m.1.1		2	1.1	even	1	trivial
784.6.a.q.1.1		2	84.83	odd	2
1008.6.a.bi.1.1		2	4.3	odd	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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	504.a (trivial)

\(f(q)\)	\(=\)	\(q-90.4622 q^{5} -49.0000 q^{7} +552.494 q^{11} -593.172 q^{13} +1422.19 q^{17} +318.997 q^{19} -659.954 q^{23} +5058.41 q^{25} +8185.27 q^{29} -9598.03 q^{31} +4432.65 q^{35} +5180.56 q^{37} +2192.46 q^{41} +7458.29 q^{43} +19561.8 q^{47} +2401.00 q^{49} -36569.6 q^{53} -49979.9 q^{55} -16361.7 q^{59} -10893.1 q^{61} +53659.6 q^{65} +8035.91 q^{67} -55983.3 q^{71} -77752.5 q^{73} -27072.2 q^{77} -3208.43 q^{79} -76626.9 q^{83} -128655. q^{85} +84288.6 q^{89} +29065.4 q^{91} -28857.2 q^{95} +101273. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 42 q^{5} - 98 q^{7} + 716 q^{11} - 714 q^{13} + 1344 q^{17} - 1946 q^{19} + 1792 q^{23} + 4282 q^{25} + 1200 q^{29} - 6804 q^{31} + 2058 q^{35} + 14640 q^{37} - 7896 q^{41} + 524 q^{43} + 18396 q^{47}+ \cdots - 44240 q^{97}+O(q^{100}) \) 2 * q - 42 * q^5 - 98 * q^7 + 716 * q^11 - 714 * q^13 + 1344 * q^17 - 1946 * q^19 + 1792 * q^23 + 4282 * q^25 + 1200 * q^29 - 6804 * q^31 + 2058 * q^35 + 14640 * q^37 - 7896 * q^41 + 524 * q^43 + 18396 * q^47 + 4802 * q^49 - 45132 * q^53 - 42056 * q^55 - 22582 * q^59 - 52822 * q^61 + 47804 * q^65 + 9848 * q^67 + 840 * q^71 - 122052 * q^73 - 35084 * q^77 + 31704 * q^79 - 36974 * q^83 - 132444 * q^85 + 210588 * q^89 + 34986 * q^91 - 138624 * q^95 - 44240 * q^97

Introduction

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