LMFDB - Embedded newform 504.6.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,6,Mod(1,504)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	yes
Analytic conductor:	$80.8334451857$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.54138$ 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)

$f(q)$	$=$	$q+84.8276 q^{5} -49.0000 q^{7} +O(q^{10})$	$q+84.8276 q^{5} -49.0000 q^{7} +634.124 q^{11} +895.242 q^{13} -2057.46 q^{17} +2451.95 q^{19} +569.186 q^{23} +4070.73 q^{25} -1471.78 q^{29} -2006.65 q^{31} -4156.55 q^{35} +4860.54 q^{37} +17228.4 q^{41} -15481.3 q^{43} -5006.40 q^{47} +2401.00 q^{49} +19560.3 q^{53} +53791.3 q^{55} -14515.6 q^{59} +3572.67 q^{61} +75941.3 q^{65} -41480.7 q^{67} -9247.05 q^{71} -41350.0 q^{73} -31072.1 q^{77} -37962.2 q^{79} -79211.1 q^{83} -174530. q^{85} +92538.5 q^{89} -43866.9 q^{91} +207993. q^{95} +175419. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 48 q^{5} - 98 q^{7}+O(q^{10}) $ 2 * q + 48 * q^5 - 98 * q^7	$ 2 q + 48 q^{5} - 98 q^{7} + 368 q^{11} - 156 q^{13} - 3312 q^{17} + 3736 q^{19} + 1552 q^{23} + 2302 q^{25} + 1728 q^{29} - 3624 q^{31} - 2352 q^{35} + 6996 q^{37} + 26160 q^{41} - 30184 q^{43} - 11424 q^{47} + 4802 q^{49} - 17376 q^{53} + 63592 q^{55} + 15008 q^{59} + 35564 q^{61} + 114656 q^{65} - 70504 q^{67} + 40752 q^{71} - 53892 q^{73} - 18032 q^{77} - 9744 q^{79} + 31360 q^{83} - 128328 q^{85} + 35952 q^{89} + 7644 q^{91} + 160704 q^{95} + 66652 q^{97}+O(q^{100}) $ 2 * q + 48 * q^5 - 98 * q^7 + 368 * q^11 - 156 * q^13 - 3312 * q^17 + 3736 * q^19 + 1552 * q^23 + 2302 * q^25 + 1728 * q^29 - 3624 * q^31 - 2352 * q^35 + 6996 * q^37 + 26160 * q^41 - 30184 * q^43 - 11424 * q^47 + 4802 * q^49 - 17376 * q^53 + 63592 * q^55 + 15008 * q^59 + 35564 * q^61 + 114656 * q^65 - 70504 * q^67 + 40752 * q^71 - 53892 * q^73 - 18032 * q^77 - 9744 * q^79 + 31360 * q^83 - 128328 * q^85 + 35952 * q^89 + 7644 * q^91 + 160704 * q^95 + 66652 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	84.8276	1.51744	0.758721	−	0.651415i	$-0.225824\pi$
$5$	84.8276	1.51744	0.758721	+	0.651415i	$0.225824\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	634.124	1.58013	0.790065	−	0.613023i	$-0.210047\pi$
$11$	634.124	1.58013	0.790065	+	0.613023i	$0.210047\pi$
$12$	0	0
$13$	895.242	1.46920	0.734602	−	0.678498i	$-0.237369\pi$
$13$	895.242	1.46920	0.734602	+	0.678498i	$0.237369\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2057.46	−1.72667	−0.863335	−	0.504630i	$-0.831629\pi$
$17$	−2057.46	−1.72667	−0.863335	+	0.504630i	$0.831629\pi$
$18$	0	0
$19$	2451.95	1.55821	0.779106	−	0.626892i	$-0.215673\pi$
$19$	2451.95	1.55821	0.779106	+	0.626892i	$0.215673\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	569.186	0.224354	0.112177	−	0.993688i	$-0.464218\pi$
$23$	569.186	0.224354	0.112177	+	0.993688i	$0.464218\pi$
$24$	0	0
$25$	4070.73	1.30263
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1471.78	−0.324974	−0.162487	−	0.986711i	$-0.551951\pi$
$29$	−1471.78	−0.324974	−0.162487	+	0.986711i	$0.551951\pi$
$30$	0	0
$31$	−2006.65	−0.375031	−0.187515	−	0.982262i	$-0.560043\pi$
$31$	−2006.65	−0.375031	−0.187515	+	0.982262i	$0.560043\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4156.55	−0.573539
$36$	0	0
$37$	4860.54	0.583687	0.291844	−	0.956466i	$-0.405731\pi$
$37$	4860.54	0.583687	0.291844	+	0.956466i	$0.405731\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	17228.4	1.60061	0.800307	−	0.599591i	$-0.204670\pi$
$41$	17228.4	1.60061	0.800307	+	0.599591i	$0.204670\pi$
$42$	0	0
$43$	−15481.3	−1.27684	−0.638420	−	0.769689i	$-0.720412\pi$
$43$	−15481.3	−1.27684	−0.638420	+	0.769689i	$0.720412\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5006.40	−0.330583	−0.165292	−	0.986245i	$-0.552857\pi$
$47$	−5006.40	−0.330583	−0.165292	+	0.986245i	$0.552857\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	19560.3	0.956504	0.478252	−	0.878223i	$-0.341271\pi$
$53$	19560.3	0.956504	0.478252	+	0.878223i	$0.341271\pi$
$54$	0	0
$55$	53791.3	2.39776
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−14515.6	−0.542881	−0.271441	−	0.962455i	$-0.587500\pi$
$59$	−14515.6	−0.542881	−0.271441	+	0.962455i	$0.587500\pi$
$60$	0	0
$61$	3572.67	0.122933	0.0614664	−	0.998109i	$-0.480422\pi$
$61$	3572.67	0.122933	0.0614664	+	0.998109i	$0.480422\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	75941.3	2.22943
$66$	0	0
$67$	−41480.7	−1.12891	−0.564455	−	0.825464i	$-0.690914\pi$
$67$	−41480.7	−1.12891	−0.564455	+	0.825464i	$0.690914\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9247.05	−0.217700	−0.108850	−	0.994058i	$-0.534717\pi$
$71$	−9247.05	−0.217700	−0.108850	+	0.994058i	$0.534717\pi$
$72$	0	0
$73$	−41350.0	−0.908172	−0.454086	−	0.890958i	$-0.650034\pi$
$73$	−41350.0	−0.908172	−0.454086	+	0.890958i	$0.650034\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−31072.1	−0.597233
$78$	0	0
$79$	−37962.2	−0.684359	−0.342179	−	0.939635i	$-0.611165\pi$
$79$	−37962.2	−0.684359	−0.342179	+	0.939635i	$0.611165\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−79211.1	−1.26209	−0.631046	−	0.775746i	$-0.717374\pi$
$83$	−79211.1	−1.26209	−0.631046	+	0.775746i	$0.717374\pi$
$84$	0	0
$85$	−174530.	−2.62012
$86$	0	0
$87$	0	0
$88$	0	0
$89$	92538.5	1.23836	0.619181	−	0.785248i	$-0.287465\pi$
$89$	92538.5	1.23836	0.619181	+	0.785248i	$0.287465\pi$
$90$	0	0
$91$	−43866.9	−0.555307
$92$	0	0
$93$	0	0
$94$	0	0
$95$	207993.	2.36450
$96$	0	0
$97$	175419.	1.89299	0.946495	−	0.322720i	$-0.104597\pi$
$97$	175419.	1.89299	0.946495	+	0.322720i	$0.104597\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−56163.2	−0.547834	−0.273917	−	0.961753i	$-0.588319\pi$
$101$	−56163.2	−0.547834	−0.273917	+	0.961753i	$0.588319\pi$
$102$	0	0
$103$	−158376.	−1.47095	−0.735474	−	0.677553i	$-0.763040\pi$
$103$	−158376.	−1.47095	−0.735474	+	0.677553i	$0.763040\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	23496.4	0.198400	0.0992001	−	0.995068i	$-0.468372\pi$
$107$	23496.4	0.198400	0.0992001	+	0.995068i	$0.468372\pi$
$108$	0	0
$109$	205342.	1.65543	0.827715	−	0.561148i	$-0.189640\pi$
$109$	205342.	1.65543	0.827715	+	0.561148i	$0.189640\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	185993.	1.37025	0.685125	−	0.728426i	$-0.259748\pi$
$113$	185993.	1.37025	0.685125	+	0.728426i	$0.259748\pi$
$114$	0	0
$115$	48282.7	0.340445
$116$	0	0
$117$	0	0
$118$	0	0
$119$	100816.	0.652620
$120$	0	0
$121$	241063.	1.49681
$122$	0	0
$123$	0	0
$124$	0	0
$125$	80223.7	0.459227
$126$	0	0
$127$	348363.	1.91656	0.958281	−	0.285828i	$-0.0922685\pi$
$127$	348363.	1.91656	0.958281	+	0.285828i	$0.0922685\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	294849.	1.50114	0.750571	−	0.660790i	$-0.229779\pi$
$131$	294849.	1.50114	0.750571	+	0.660790i	$0.229779\pi$
$132$	0	0
$133$	−120145.	−0.588949
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−119607.	−0.544446	−0.272223	−	0.962234i	$-0.587759\pi$
$137$	−119607.	−0.544446	−0.272223	+	0.962234i	$0.587759\pi$
$138$	0	0
$139$	145630.	0.639312	0.319656	−	0.947534i	$-0.396433\pi$
$139$	145630.	0.639312	0.319656	+	0.947534i	$0.396433\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	567695.	2.32153
$144$	0	0
$145$	−124848.	−0.493129
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−49750.8	−0.183584	−0.0917919	−	0.995778i	$-0.529259\pi$
$149$	−49750.8	−0.183584	−0.0917919	+	0.995778i	$0.529259\pi$
$150$	0	0
$151$	−214254.	−0.764691	−0.382346	−	0.924019i	$-0.624884\pi$
$151$	−214254.	−0.764691	−0.382346	+	0.924019i	$0.624884\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−170219.	−0.569088
$156$	0	0
$157$	−197114.	−0.638217	−0.319108	−	0.947718i	$-0.603383\pi$
$157$	−197114.	−0.638217	−0.319108	+	0.947718i	$0.603383\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−27890.1	−0.0847980
$162$	0	0
$163$	103013.	0.303686	0.151843	−	0.988405i	$-0.451479\pi$
$163$	103013.	0.303686	0.151843	+	0.988405i	$0.451479\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	231396.	0.642044	0.321022	−	0.947072i	$-0.395974\pi$
$167$	231396.	0.642044	0.321022	+	0.947072i	$0.395974\pi$
$168$	0	0
$169$	430165.	1.15856
$170$	0	0
$171$	0	0
$172$	0	0
$173$	207599.	0.527365	0.263682	−	0.964610i	$-0.415063\pi$
$173$	207599.	0.527365	0.263682	+	0.964610i	$0.415063\pi$
$174$	0	0
$175$	−199466.	−0.492349
$176$	0	0
$177$	0	0
$178$	0	0
$179$	713483.	1.66438	0.832188	−	0.554494i	$-0.187088\pi$
$179$	713483.	1.66438	0.832188	+	0.554494i	$0.187088\pi$
$180$	0	0
$181$	241894.	0.548818	0.274409	−	0.961613i	$-0.411518\pi$
$181$	241894.	0.548818	0.274409	+	0.961613i	$0.411518\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	412308.	0.885712
$186$	0	0
$187$	−1.30469e6	−2.72836
$188$	0	0
$189$	0	0
$190$	0	0
$191$	220340.	0.437029	0.218515	−	0.975834i	$-0.429879\pi$
$191$	220340.	0.437029	0.218515	+	0.975834i	$0.429879\pi$
$192$	0	0
$193$	−736048.	−1.42237	−0.711185	−	0.703005i	$-0.751841\pi$
$193$	−736048.	−1.42237	−0.711185	+	0.703005i	$0.751841\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	586422.	1.07658	0.538288	−	0.842761i	$-0.319071\pi$
$197$	586422.	1.07658	0.538288	+	0.842761i	$0.319071\pi$
$198$	0	0
$199$	−680230.	−1.21765	−0.608826	−	0.793304i	$-0.708359\pi$
$199$	−680230.	−1.21765	−0.608826	+	0.793304i	$0.708359\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	72117.3	0.122828
$204$	0	0
$205$	1.46145e6	2.42884
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.55484e6	2.46218
$210$	0	0
$211$	142662.	0.220598	0.110299	−	0.993898i	$-0.464819\pi$
$211$	142662.	0.220598	0.110299	+	0.993898i	$0.464819\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.31324e6	−1.93753
$216$	0	0
$217$	98325.8	0.141748
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.84193e6	−2.53683
$222$	0	0
$223$	−1.41803e6	−1.90952	−0.954760	−	0.297377i	$-0.903888\pi$
$223$	−1.41803e6	−1.90952	−0.954760	+	0.297377i	$0.903888\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−75557.1	−0.0973219	−0.0486610	−	0.998815i	$-0.515495\pi$
$227$	−75557.1	−0.0973219	−0.0486610	+	0.998815i	$0.515495\pi$
$228$	0	0
$229$	1.29149e6	1.62743	0.813714	−	0.581265i	$-0.197442\pi$
$229$	1.29149e6	1.62743	0.813714	+	0.581265i	$0.197442\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	112997.	0.136357	0.0681783	−	0.997673i	$-0.478281\pi$
$233$	112997.	0.136357	0.0681783	+	0.997673i	$0.478281\pi$
$234$	0	0
$235$	−424681.	−0.501641
$236$	0	0
$237$	0	0
$238$	0	0
$239$	473209.	0.535869	0.267935	−	0.963437i	$-0.413659\pi$
$239$	473209.	0.535869	0.267935	+	0.963437i	$0.413659\pi$
$240$	0	0
$241$	−1.39122e6	−1.54296	−0.771478	−	0.636256i	$-0.780482\pi$
$241$	−1.39122e6	−1.54296	−0.771478	+	0.636256i	$0.780482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	203671.	0.216778
$246$	0	0
$247$	2.19508e6	2.28933
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−776017.	−0.777476	−0.388738	−	0.921348i	$-0.627089\pi$
$251$	−776017.	−0.777476	−0.388738	+	0.921348i	$0.627089\pi$
$252$	0	0
$253$	360935.	0.354509
$254$	0	0
$255$	0	0
$256$	0	0
$257$	955090.	0.902011	0.451005	−	0.892521i	$-0.351066\pi$
$257$	955090.	0.902011	0.451005	+	0.892521i	$0.351066\pi$
$258$	0	0
$259$	−238166.	−0.220613
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−190821.	−0.170113	−0.0850566	−	0.996376i	$-0.527107\pi$
$263$	−190821.	−0.170113	−0.0850566	+	0.996376i	$0.527107\pi$
$264$	0	0
$265$	1.65926e6	1.45144
$266$	0	0
$267$	0	0
$268$	0	0
$269$	224114.	0.188837	0.0944186	−	0.995533i	$-0.469901\pi$
$269$	224114.	0.188837	0.0944186	+	0.995533i	$0.469901\pi$
$270$	0	0
$271$	687404.	0.568577	0.284288	−	0.958739i	$-0.408243\pi$
$271$	687404.	0.568577	0.284288	+	0.958739i	$0.408243\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.58135e6	2.05833
$276$	0	0
$277$	−1.14588e6	−0.897302	−0.448651	−	0.893707i	$-0.648095\pi$
$277$	−1.14588e6	−0.897302	−0.448651	+	0.893707i	$0.648095\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	755908.	0.571088	0.285544	−	0.958366i	$-0.407826\pi$
$281$	755908.	0.571088	0.285544	+	0.958366i	$0.407826\pi$
$282$	0	0
$283$	−1.92464e6	−1.42851	−0.714257	−	0.699884i	$-0.753235\pi$
$283$	−1.92464e6	−1.42851	−0.714257	+	0.699884i	$0.753235\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−844194.	−0.604975
$288$	0	0
$289$	2.81329e6	1.98139
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.46185e6	−0.994795	−0.497398	−	0.867523i	$-0.665711\pi$
$293$	−1.46185e6	−0.994795	−0.497398	+	0.867523i	$0.665711\pi$
$294$	0	0
$295$	−1.23132e6	−0.823791
$296$	0	0
$297$	0	0
$298$	0	0
$299$	509559.	0.329622
$300$	0	0
$301$	758584.	0.482600
$302$	0	0
$303$	0	0
$304$	0	0
$305$	303061.	0.186544
$306$	0	0
$307$	1.06474e6	0.644756	0.322378	−	0.946611i	$-0.395518\pi$
$307$	1.06474e6	0.644756	0.322378	+	0.946611i	$0.395518\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.59214e6	−1.51970	−0.759850	−	0.650098i	$-0.774728\pi$
$311$	−2.59214e6	−1.51970	−0.759850	+	0.650098i	$0.774728\pi$
$312$	0	0
$313$	−2.14280e6	−1.23629	−0.618147	−	0.786063i	$-0.712116\pi$
$313$	−2.14280e6	−1.23629	−0.618147	+	0.786063i	$0.712116\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.02229e6	−1.68923	−0.844613	−	0.535377i	$-0.820170\pi$
$317$	−3.02229e6	−1.68923	−0.844613	+	0.535377i	$0.820170\pi$
$318$	0	0
$319$	−933292.	−0.513501
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.04478e6	−2.69052
$324$	0	0
$325$	3.64428e6	1.91383
$326$	0	0
$327$	0	0
$328$	0	0
$329$	245314.	0.124949
$330$	0	0
$331$	−1.03801e6	−0.520751	−0.260376	−	0.965507i	$-0.583846\pi$
$331$	−1.03801e6	−0.520751	−0.260376	+	0.965507i	$0.583846\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.51871e6	−1.71306
$336$	0	0
$337$	2.23059e6	1.06991	0.534953	−	0.844882i	$-0.320329\pi$
$337$	2.23059e6	1.06991	0.534953	+	0.844882i	$0.320329\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.27246e6	−0.592598
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.57483e6	1.14795	0.573977	−	0.818872i	$-0.305400\pi$
$347$	2.57483e6	1.14795	0.573977	+	0.818872i	$0.305400\pi$
$348$	0	0
$349$	311098.	0.136721	0.0683604	−	0.997661i	$-0.478223\pi$
$349$	311098.	0.136721	0.0683604	+	0.997661i	$0.478223\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.77642e6	1.61303	0.806517	−	0.591212i	$-0.201350\pi$
$353$	3.77642e6	1.61303	0.806517	+	0.591212i	$0.201350\pi$
$354$	0	0
$355$	−784406.	−0.330347
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−996020.	−0.407880	−0.203940	−	0.978983i	$-0.565375\pi$
$359$	−996020.	−0.407880	−0.203940	+	0.978983i	$0.565375\pi$
$360$	0	0
$361$	3.53594e6	1.42803
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.50762e6	−1.37810
$366$	0	0
$367$	−3.83928e6	−1.48794	−0.743969	−	0.668214i	$-0.767059\pi$
$367$	−3.83928e6	−1.48794	−0.743969	+	0.668214i	$0.767059\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−958457.	−0.361525
$372$	0	0
$373$	−5.05517e6	−1.88132	−0.940662	−	0.339344i	$-0.889795\pi$
$373$	−5.05517e6	−1.88132	−0.940662	+	0.339344i	$0.889795\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.31760e6	−0.477453
$378$	0	0
$379$	−3.46825e6	−1.24026	−0.620129	−	0.784500i	$-0.712920\pi$
$379$	−3.46825e6	−1.24026	−0.620129	+	0.784500i	$0.712920\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.58445e6	1.24861	0.624304	−	0.781182i	$-0.285383\pi$
$383$	3.58445e6	1.24861	0.624304	+	0.781182i	$0.285383\pi$
$384$	0	0
$385$	−2.63577e6	−0.906267
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.86098e6	−1.29367	−0.646834	−	0.762630i	$-0.723908\pi$
$389$	−3.86098e6	−1.29367	−0.646834	+	0.762630i	$0.723908\pi$
$390$	0	0
$391$	−1.17108e6	−0.387386
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.22025e6	−1.03848
$396$	0	0
$397$	−1.20136e6	−0.382559	−0.191280	−	0.981536i	$-0.561264\pi$
$397$	−1.20136e6	−0.382559	−0.191280	+	0.981536i	$0.561264\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.88850e6	−1.20760	−0.603798	−	0.797138i	$-0.706347\pi$
$401$	−3.88850e6	−1.20760	−0.603798	+	0.797138i	$0.706347\pi$
$402$	0	0
$403$	−1.79644e6	−0.550997
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.08219e6	0.922301
$408$	0	0
$409$	2.63640e6	0.779296	0.389648	−	0.920964i	$-0.372597\pi$
$409$	2.63640e6	0.779296	0.389648	+	0.920964i	$0.372597\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	711264.	0.205190
$414$	0	0
$415$	−6.71929e6	−1.91515
$416$	0	0
$417$	0	0
$418$	0	0
$419$	432053.	0.120227	0.0601134	−	0.998192i	$-0.480854\pi$
$419$	432053.	0.120227	0.0601134	+	0.998192i	$0.480854\pi$
$420$	0	0
$421$	3.31215e6	0.910763	0.455381	−	0.890296i	$-0.349503\pi$
$421$	3.31215e6	0.910763	0.455381	+	0.890296i	$0.349503\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−8.37537e6	−2.24922
$426$	0	0
$427$	−175061.	−0.0464642
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.30508e6	1.37562	0.687811	−	0.725890i	$-0.258572\pi$
$431$	5.30508e6	1.37562	0.687811	+	0.725890i	$0.258572\pi$
$432$	0	0
$433$	−2.86645e6	−0.734723	−0.367362	−	0.930078i	$-0.619739\pi$
$433$	−2.86645e6	−0.734723	−0.367362	+	0.930078i	$0.619739\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.39561e6	0.349592
$438$	0	0
$439$	758493.	0.187841	0.0939205	−	0.995580i	$-0.470060\pi$
$439$	758493.	0.187841	0.0939205	+	0.995580i	$0.470060\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.02230e6	−0.731693	−0.365846	−	0.930675i	$-0.619220\pi$
$443$	−3.02230e6	−0.731693	−0.365846	+	0.930675i	$0.619220\pi$
$444$	0	0
$445$	7.84982e6	1.87914
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.55957e6	−1.30144	−0.650721	−	0.759317i	$-0.725533\pi$
$449$	−5.55957e6	−1.30144	−0.650721	+	0.759317i	$0.725533\pi$
$450$	0	0
$451$	1.09250e7	2.52918
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.72112e6	−0.842646
$456$	0	0
$457$	1.48117e6	0.331752	0.165876	−	0.986147i	$-0.446955\pi$
$457$	1.48117e6	0.331752	0.165876	+	0.986147i	$0.446955\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12329.0	−0.00270193	−0.00135097	−	0.999999i	$-0.500430\pi$
$461$	−12329.0	−0.00270193	−0.00135097	+	0.999999i	$0.500430\pi$
$462$	0	0
$463$	−1.27201e6	−0.275765	−0.137882	−	0.990449i	$-0.544030\pi$
$463$	−1.27201e6	−0.275765	−0.137882	+	0.990449i	$0.544030\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.15829e6	1.51886	0.759429	−	0.650590i	$-0.225479\pi$
$467$	7.15829e6	1.51886	0.759429	+	0.650590i	$0.225479\pi$
$468$	0	0
$469$	2.03256e6	0.426688
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9.81707e6	−2.01757
$474$	0	0
$475$	9.98120e6	2.02978
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.01478e6	0.998649	0.499325	−	0.866415i	$-0.333582\pi$
$479$	5.01478e6	0.998649	0.499325	+	0.866415i	$0.333582\pi$
$480$	0	0
$481$	4.35136e6	0.857555
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.48804e7	2.87250
$486$	0	0
$487$	6.46712e6	1.23563	0.617815	−	0.786323i	$-0.288018\pi$
$487$	6.46712e6	1.23563	0.617815	+	0.786323i	$0.288018\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.89564e6	−1.29084	−0.645418	−	0.763829i	$-0.723317\pi$
$491$	−6.89564e6	−1.29084	−0.645418	+	0.763829i	$0.723317\pi$
$492$	0	0
$493$	3.02813e6	0.561123
$494$	0	0
$495$	0	0
$496$	0	0
$497$	453106.	0.0822827
$498$	0	0
$499$	−1.00230e7	−1.80196	−0.900979	−	0.433862i	$-0.857151\pi$
$499$	−1.00230e7	−1.80196	−0.900979	+	0.433862i	$0.857151\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.02505e7	−1.80644	−0.903220	−	0.429178i	$-0.858803\pi$
$503$	−1.02505e7	−1.80644	−0.903220	+	0.429178i	$0.858803\pi$
$504$	0	0
$505$	−4.76419e6	−0.831306
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.95058e6	−0.333710	−0.166855	−	0.985981i	$-0.553361\pi$
$509$	−1.95058e6	−0.333710	−0.166855	+	0.985981i	$0.553361\pi$
$510$	0	0
$511$	2.02615e6	0.343257
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.34347e7	−2.23208
$516$	0	0
$517$	−3.17468e6	−0.522364
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.02692e6	0.811349	0.405674	−	0.914018i	$-0.367037\pi$
$521$	5.02692e6	0.811349	0.405674	+	0.914018i	$0.367037\pi$
$522$	0	0
$523$	−356811.	−0.0570405	−0.0285203	−	0.999593i	$-0.509080\pi$
$523$	−356811.	−0.0570405	−0.0285203	+	0.999593i	$0.509080\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.12860e6	0.647555
$528$	0	0
$529$	−6.11237e6	−0.949665
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.54236e7	2.35163
$534$	0	0
$535$	1.99314e6	0.301061
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.52253e6	0.225733
$540$	0	0
$541$	−1.16485e7	−1.71111	−0.855555	−	0.517712i	$-0.826784\pi$
$541$	−1.16485e7	−1.71111	−0.855555	+	0.517712i	$0.826784\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.74187e7	2.51202
$546$	0	0
$547$	−2.56657e6	−0.366763	−0.183382	−	0.983042i	$-0.558704\pi$
$547$	−2.56657e6	−0.366763	−0.183382	+	0.983042i	$0.558704\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.60873e6	−0.506378
$552$	0	0
$553$	1.86015e6	0.258663
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.19382e6	−1.25562	−0.627809	−	0.778367i	$-0.716048\pi$
$557$	−9.19382e6	−1.25562	−0.627809	+	0.778367i	$0.716048\pi$
$558$	0	0
$559$	−1.38595e7	−1.87594
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.73412e6	0.363535	0.181768	−	0.983342i	$-0.441818\pi$
$563$	2.73412e6	0.363535	0.181768	+	0.983342i	$0.441818\pi$
$564$	0	0
$565$	1.57773e7	2.07927
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.51619e6	0.714264	0.357132	−	0.934054i	$-0.383755\pi$
$569$	5.51619e6	0.714264	0.357132	+	0.934054i	$0.383755\pi$
$570$	0	0
$571$	−1.15557e7	−1.48322	−0.741612	−	0.670829i	$-0.765938\pi$
$571$	−1.15557e7	−1.48322	−0.741612	+	0.670829i	$0.765938\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.31700e6	0.292251
$576$	0	0
$577$	−9.36955e6	−1.17160	−0.585800	−	0.810456i	$-0.699220\pi$
$577$	−9.36955e6	−1.17160	−0.585800	+	0.810456i	$0.699220\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.88134e6	0.477026
$582$	0	0
$583$	1.24037e7	1.51140
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.61313e6	0.432801	0.216400	−	0.976305i	$-0.430568\pi$
$587$	3.61313e6	0.432801	0.216400	+	0.976305i	$0.430568\pi$
$588$	0	0
$589$	−4.92019e6	−0.584378
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.65554e6	−1.12756	−0.563780	−	0.825925i	$-0.690654\pi$
$593$	−9.65554e6	−1.12756	−0.563780	+	0.825925i	$0.690654\pi$
$594$	0	0
$595$	8.55195e6	0.990314
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.36706e6	0.838932	0.419466	−	0.907771i	$-0.362217\pi$
$599$	7.36706e6	0.838932	0.419466	+	0.907771i	$0.362217\pi$
$600$	0	0
$601$	−9.04647e6	−1.02163	−0.510814	−	0.859691i	$-0.670656\pi$
$601$	−9.04647e6	−1.02163	−0.510814	+	0.859691i	$0.670656\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.04488e7	2.27132
$606$	0	0
$607$	1.07723e7	1.18669	0.593345	−	0.804948i	$-0.297807\pi$
$607$	1.07723e7	1.18669	0.593345	+	0.804948i	$0.297807\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.48194e6	−0.485694
$612$	0	0
$613$	5.64815e6	0.607093	0.303547	−	0.952817i	$-0.401829\pi$
$613$	5.64815e6	0.607093	0.303547	+	0.952817i	$0.401829\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.21062e7	−1.28025	−0.640123	−	0.768272i	$-0.721117\pi$
$617$	−1.21062e7	−1.28025	−0.640123	+	0.768272i	$0.721117\pi$
$618$	0	0
$619$	−3.22164e6	−0.337948	−0.168974	−	0.985620i	$-0.554045\pi$
$619$	−3.22164e6	−0.337948	−0.168974	+	0.985620i	$0.554045\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.53439e6	−0.468057
$624$	0	0
$625$	−5.91583e6	−0.605781
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.00004e7	−1.00784
$630$	0	0
$631$	−8.59962e6	−0.859816	−0.429908	−	0.902873i	$-0.641454\pi$
$631$	−8.59962e6	−0.859816	−0.429908	+	0.902873i	$0.641454\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.95508e7	2.90827
$636$	0	0
$637$	2.14948e6	0.209886
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.89346e6	0.854920	0.427460	−	0.904034i	$-0.359408\pi$
$641$	8.89346e6	0.854920	0.427460	+	0.904034i	$0.359408\pi$
$642$	0	0
$643$	−6.57628e6	−0.627268	−0.313634	−	0.949544i	$-0.601546\pi$
$643$	−6.57628e6	−0.627268	−0.313634	+	0.949544i	$0.601546\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.16128e7	1.09063	0.545313	−	0.838232i	$-0.316411\pi$
$647$	1.16128e7	1.09063	0.545313	+	0.838232i	$0.316411\pi$
$648$	0	0
$649$	−9.20470e6	−0.857823
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.03391e6	0.645527	0.322763	−	0.946480i	$-0.395388\pi$
$653$	7.03391e6	0.645527	0.322763	+	0.946480i	$0.395388\pi$
$654$	0	0
$655$	2.50114e7	2.27790
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.10193e7	−0.988422	−0.494211	−	0.869342i	$-0.664543\pi$
$659$	−1.10193e7	−0.988422	−0.494211	+	0.869342i	$0.664543\pi$
$660$	0	0
$661$	3.37812e6	0.300726	0.150363	−	0.988631i	$-0.451956\pi$
$661$	3.37812e6	0.300726	0.150363	+	0.988631i	$0.451956\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.01916e7	−0.893696
$666$	0	0
$667$	−837717.	−0.0729093
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.26552e6	0.194250
$672$	0	0
$673$	−3.46735e6	−0.295094	−0.147547	−	0.989055i	$-0.547138\pi$
$673$	−3.46735e6	−0.295094	−0.147547	+	0.989055i	$0.547138\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.81353e7	1.52074	0.760369	−	0.649492i	$-0.225018\pi$
$677$	1.81353e7	1.52074	0.760369	+	0.649492i	$0.225018\pi$
$678$	0	0
$679$	−8.59555e6	−0.715483
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.60726e7	1.31836	0.659182	−	0.751983i	$-0.270903\pi$
$683$	1.60726e7	1.31836	0.659182	+	0.751983i	$0.270903\pi$
$684$	0	0
$685$	−1.01460e7	−0.826166
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.75112e7	1.40530
$690$	0	0
$691$	779272.	0.0620860	0.0310430	−	0.999518i	$-0.490117\pi$
$691$	779272.	0.0620860	0.0310430	+	0.999518i	$0.490117\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.23534e7	0.970120
$696$	0	0
$697$	−3.54469e7	−2.76373
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−616669.	−0.0473977	−0.0236989	−	0.999719i	$-0.507544\pi$
$701$	−616669.	−0.0473977	−0.0236989	+	0.999719i	$0.507544\pi$
$702$	0	0
$703$	1.19178e7	0.909509
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.75200e6	0.207062
$708$	0	0
$709$	1.94867e7	1.45587	0.727935	−	0.685646i	$-0.240480\pi$
$709$	1.94867e7	1.45587	0.727935	+	0.685646i	$0.240480\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.14216e6	−0.0841398
$714$	0	0
$715$	4.81562e7	3.52279
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.07365e6	−0.438155	−0.219078	−	0.975707i	$-0.570305\pi$
$719$	−6.07365e6	−0.438155	−0.219078	+	0.975707i	$0.570305\pi$
$720$	0	0
$721$	7.76044e6	0.555966
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.99122e6	−0.423321
$726$	0	0
$727$	9.38803e6	0.658777	0.329389	−	0.944194i	$-0.393157\pi$
$727$	9.38803e6	0.658777	0.329389	+	0.944194i	$0.393157\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.18522e7	2.20468
$732$	0	0
$733$	6.87253e6	0.472451	0.236226	−	0.971698i	$-0.424090\pi$
$733$	6.87253e6	0.472451	0.236226	+	0.971698i	$0.424090\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.63040e7	−1.78383
$738$	0	0
$739$	3.13208e6	0.210970	0.105485	−	0.994421i	$-0.466360\pi$
$739$	3.13208e6	0.210970	0.105485	+	0.994421i	$0.466360\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.05007e6	−0.534968	−0.267484	−	0.963562i	$-0.586192\pi$
$743$	−8.05007e6	−0.534968	−0.267484	+	0.963562i	$0.586192\pi$
$744$	0	0
$745$	−4.22024e6	−0.278578
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.15132e6	−0.0749882
$750$	0	0
$751$	−1.51621e7	−0.980976	−0.490488	−	0.871448i	$-0.663181\pi$
$751$	−1.51621e7	−0.980976	−0.490488	+	0.871448i	$0.663181\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.81746e7	−1.16037
$756$	0	0
$757$	3.86111e6	0.244891	0.122445	−	0.992475i	$-0.460926\pi$
$757$	3.86111e6	0.244891	0.122445	+	0.992475i	$0.460926\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.74204e7	1.71638	0.858188	−	0.513335i	$-0.171590\pi$
$761$	2.74204e7	1.71638	0.858188	+	0.513335i	$0.171590\pi$
$762$	0	0
$763$	−1.00617e7	−0.625694
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.29950e7	−0.797603
$768$	0	0
$769$	7.96097e6	0.485456	0.242728	−	0.970094i	$-0.421958\pi$
$769$	7.96097e6	0.485456	0.242728	+	0.970094i	$0.421958\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4.09621e6	−0.246566	−0.123283	−	0.992372i	$-0.539342\pi$
$773$	−4.09621e6	−0.246566	−0.123283	+	0.992372i	$0.539342\pi$
$774$	0	0
$775$	−8.16852e6	−0.488527
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.22432e7	2.49410
$780$	0	0
$781$	−5.86378e6	−0.343994
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.67207e7	−0.968457
$786$	0	0
$787$	1.43564e7	0.826244	0.413122	−	0.910676i	$-0.364438\pi$
$787$	1.43564e7	0.826244	0.413122	+	0.910676i	$0.364438\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.11364e6	−0.517906
$792$	0	0
$793$	3.19840e6	0.180613
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.61963e7	−1.46081	−0.730407	−	0.683012i	$-0.760670\pi$
$797$	−2.61963e7	−1.46081	−0.730407	+	0.683012i	$0.760670\pi$
$798$	0	0
$799$	1.03005e7	0.570809
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.62210e7	−1.43503
$804$	0	0
$805$	−2.36585e6	−0.128676
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.31159e6	−0.177896	−0.0889478	−	0.996036i	$-0.528350\pi$
$809$	−3.31159e6	−0.177896	−0.0889478	+	0.996036i	$0.528350\pi$
$810$	0	0
$811$	6.43440e6	0.343523	0.171761	−	0.985139i	$-0.445054\pi$
$811$	6.43440e6	0.343523	0.171761	+	0.985139i	$0.445054\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.73839e6	0.460826
$816$	0	0
$817$	−3.79593e7	−1.98959
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.12076e7	1.09808	0.549039	−	0.835796i	$-0.314994\pi$
$821$	2.12076e7	1.09808	0.549039	+	0.835796i	$0.314994\pi$
$822$	0	0
$823$	1.96206e7	1.00975	0.504875	−	0.863193i	$-0.331539\pi$
$823$	1.96206e7	1.00975	0.504875	+	0.863193i	$0.331539\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.34210e6	−0.474986	−0.237493	−	0.971389i	$-0.576326\pi$
$827$	−9.34210e6	−0.474986	−0.237493	+	0.971389i	$0.576326\pi$
$828$	0	0
$829$	−8.15436e6	−0.412101	−0.206050	−	0.978541i	$-0.566061\pi$
$829$	−8.15436e6	−0.412101	−0.206050	+	0.978541i	$0.566061\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.93997e6	−0.246667
$834$	0	0
$835$	1.96288e7	0.974265
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4.14558e6	0.203320	0.101660	−	0.994819i	$-0.467585\pi$
$839$	4.14558e6	0.203320	0.101660	+	0.994819i	$0.467585\pi$
$840$	0	0
$841$	−1.83450e7	−0.894392
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.64899e7	1.75805
$846$	0	0
$847$	−1.18121e7	−0.565741
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.76655e6	0.130953
$852$	0	0
$853$	3.40559e6	0.160258	0.0801291	−	0.996784i	$-0.474467\pi$
$853$	3.40559e6	0.160258	0.0801291	+	0.996784i	$0.474467\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.26615e6	0.105399	0.0526996	−	0.998610i	$-0.483217\pi$
$857$	2.26615e6	0.105399	0.0526996	+	0.998610i	$0.483217\pi$
$858$	0	0
$859$	−1.45799e7	−0.674174	−0.337087	−	0.941474i	$-0.609442\pi$
$859$	−1.45799e7	−0.674174	−0.337087	+	0.941474i	$0.609442\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.37184e7	1.54113	0.770567	−	0.637359i	$-0.219973\pi$
$863$	3.37184e7	1.54113	0.770567	+	0.637359i	$0.219973\pi$
$864$	0	0
$865$	1.76102e7	0.800245
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.40728e7	−1.08138
$870$	0	0
$871$	−3.71353e7	−1.65860
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.93096e6	−0.173572
$876$	0	0
$877$	1.34267e7	0.589481	0.294740	−	0.955577i	$-0.404767\pi$
$877$	1.34267e7	0.589481	0.294740	+	0.955577i	$0.404767\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−6.01261e6	−0.260990	−0.130495	−	0.991449i	$-0.541657\pi$
$881$	−6.01261e6	−0.260990	−0.130495	+	0.991449i	$0.541657\pi$
$882$	0	0
$883$	−4.30820e7	−1.85949	−0.929747	−	0.368200i	$-0.879974\pi$
$883$	−4.30820e7	−1.85949	−0.929747	+	0.368200i	$0.879974\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.82279e7	−1.20467	−0.602336	−	0.798242i	$-0.705763\pi$
$887$	−2.82279e7	−1.20467	−0.602336	+	0.798242i	$0.705763\pi$
$888$	0	0
$889$	−1.70698e7	−0.724392
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.22754e7	−0.515119
$894$	0	0
$895$	6.05231e7	2.52559
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.95335e6	0.121875
$900$	0	0
$901$	−4.02447e7	−1.65157
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.05193e7	0.832799
$906$	0	0
$907$	−1.56044e7	−0.629836	−0.314918	−	0.949119i	$-0.601977\pi$
$907$	−1.56044e7	−0.629836	−0.314918	+	0.949119i	$0.601977\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.33384e6	0.252855	0.126427	−	0.991976i	$-0.459649\pi$
$911$	6.33384e6	0.252855	0.126427	+	0.991976i	$0.459649\pi$
$912$	0	0
$913$	−5.02297e7	−1.99427
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.44476e7	−0.567378
$918$	0	0
$919$	2.11662e7	0.826713	0.413357	−	0.910569i	$-0.364356\pi$
$919$	2.11662e7	0.826713	0.413357	+	0.910569i	$0.364356\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.27835e6	−0.319845
$924$	0	0
$925$	1.97859e7	0.760330
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.25377e6	0.123694	0.0618468	−	0.998086i	$-0.480301\pi$
$929$	3.25377e6	0.123694	0.0618468	+	0.998086i	$0.480301\pi$
$930$	0	0
$931$	5.88712e6	0.222602
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.10674e8	−4.14014
$936$	0	0
$937$	−1.00930e7	−0.375555	−0.187777	−	0.982212i	$-0.560128\pi$
$937$	−1.00930e7	−0.375555	−0.187777	+	0.982212i	$0.560128\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.04029e7	1.11929	0.559643	−	0.828734i	$-0.310938\pi$
$941$	3.04029e7	1.11929	0.559643	+	0.828734i	$0.310938\pi$
$942$	0	0
$943$	9.80619e6	0.359105
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.37126e7	−1.58391	−0.791957	−	0.610577i	$-0.790938\pi$
$947$	−4.37126e7	−1.58391	−0.791957	+	0.610577i	$0.790938\pi$
$948$	0	0
$949$	−3.70182e7	−1.33429
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.31900e7	−0.827120	−0.413560	−	0.910477i	$-0.635715\pi$
$953$	−2.31900e7	−0.827120	−0.413560	+	0.910477i	$0.635715\pi$
$954$	0	0
$955$	1.86909e7	0.663167
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.86074e6	0.205781
$960$	0	0
$961$	−2.46025e7	−0.859352
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.24372e7	−2.15837
$966$	0	0
$967$	−2.78623e7	−0.958188	−0.479094	−	0.877764i	$-0.659035\pi$
$967$	−2.78623e7	−0.958188	−0.479094	+	0.877764i	$0.659035\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.65933e7	−1.24553	−0.622763	−	0.782410i	$-0.713990\pi$
$971$	−3.65933e7	−1.24553	−0.622763	+	0.782410i	$0.713990\pi$
$972$	0	0
$973$	−7.13586e6	−0.241637
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.83769e7	−0.615938	−0.307969	−	0.951396i	$-0.599649\pi$
$977$	−1.83769e7	−0.615938	−0.307969	+	0.951396i	$0.599649\pi$
$978$	0	0
$979$	5.86809e7	1.95677
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.09387e7	1.68137	0.840687	−	0.541522i	$-0.182152\pi$
$983$	5.09387e7	1.68137	0.840687	+	0.541522i	$0.182152\pi$
$984$	0	0
$985$	4.97448e7	1.63364
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.81174e6	−0.286465
$990$	0	0
$991$	−946169.	−0.0306044	−0.0153022	−	0.999883i	$-0.504871\pi$
$991$	−946169.	−0.0306044	−0.0153022	+	0.999883i	$0.504871\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.77023e7	−1.84772
$996$	0	0
$997$	8.61505e6	0.274486	0.137243	−	0.990537i	$-0.456176\pi$
$997$	8.61505e6	0.274486	0.137243	+	0.990537i	$0.456176\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.6.a.r.1.2	yes	2
3.2	odd	2		504.6.a.l.1.1	✓	2
4.3	odd	2		1008.6.a.bs.1.2		2
12.11	even	2		1008.6.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.6.a.l.1.1	✓	2	3.2	odd	2
504.6.a.r.1.2	yes	2	1.1	even	1	trivial
1008.6.a.bh.1.1		2	12.11	even	2
1008.6.a.bs.1.2		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+84.8276 q^{5} -49.0000 q^{7} +O(q^{10})\)	\(q+84.8276 q^{5} -49.0000 q^{7} +634.124 q^{11} +895.242 q^{13} -2057.46 q^{17} +2451.95 q^{19} +569.186 q^{23} +4070.73 q^{25} -1471.78 q^{29} -2006.65 q^{31} -4156.55 q^{35} +4860.54 q^{37} +17228.4 q^{41} -15481.3 q^{43} -5006.40 q^{47} +2401.00 q^{49} +19560.3 q^{53} +53791.3 q^{55} -14515.6 q^{59} +3572.67 q^{61} +75941.3 q^{65} -41480.7 q^{67} -9247.05 q^{71} -41350.0 q^{73} -31072.1 q^{77} -37962.2 q^{79} -79211.1 q^{83} -174530. q^{85} +92538.5 q^{89} -43866.9 q^{91} +207993. q^{95} +175419. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 48 q^{5} - 98 q^{7}+O(q^{10}) \) 2 * q + 48 * q^5 - 98 * q^7	\( 2 q + 48 q^{5} - 98 q^{7} + 368 q^{11} - 156 q^{13} - 3312 q^{17} + 3736 q^{19} + 1552 q^{23} + 2302 q^{25} + 1728 q^{29} - 3624 q^{31} - 2352 q^{35} + 6996 q^{37} + 26160 q^{41} - 30184 q^{43} - 11424 q^{47} + 4802 q^{49} - 17376 q^{53} + 63592 q^{55} + 15008 q^{59} + 35564 q^{61} + 114656 q^{65} - 70504 q^{67} + 40752 q^{71} - 53892 q^{73} - 18032 q^{77} - 9744 q^{79} + 31360 q^{83} - 128328 q^{85} + 35952 q^{89} + 7644 q^{91} + 160704 q^{95} + 66652 q^{97}+O(q^{100}) \) 2 * q + 48 * q^5 - 98 * q^7 + 368 * q^11 - 156 * q^13 - 3312 * q^17 + 3736 * q^19 + 1552 * q^23 + 2302 * q^25 + 1728 * q^29 - 3624 * q^31 - 2352 * q^35 + 6996 * q^37 + 26160 * q^41 - 30184 * q^43 - 11424 * q^47 + 4802 * q^49 - 17376 * q^53 + 63592 * q^55 + 15008 * q^59 + 35564 * q^61 + 114656 * q^65 - 70504 * q^67 + 40752 * q^71 - 53892 * q^73 - 18032 * q^77 - 9744 * q^79 + 31360 * q^83 - 128328 * q^85 + 35952 * q^89 + 7644 * q^91 + 160704 * q^95 + 66652 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more