LMFDB - Embedded newform 504.6.a.k.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{106}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 106 $ x^2 - 106
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$10.2956$ 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)

$f(q)$	$=$	$q+23.7738 q^{5} +49.0000 q^{7} +O(q^{10})$	$q+23.7738 q^{5} +49.0000 q^{7} -590.869 q^{11} +505.095 q^{13} -517.964 q^{17} +932.905 q^{19} +3550.73 q^{23} -2559.81 q^{25} -5395.48 q^{29} -8329.00 q^{31} +1164.92 q^{35} -4936.43 q^{37} -5880.27 q^{41} +13540.8 q^{43} +7357.78 q^{47} +2401.00 q^{49} +3341.88 q^{53} -14047.2 q^{55} -42675.2 q^{59} +51327.6 q^{61} +12008.0 q^{65} -33755.0 q^{67} +17848.0 q^{71} -20628.2 q^{73} -28952.6 q^{77} +99202.0 q^{79} -61202.1 q^{83} -12314.0 q^{85} -90153.0 q^{89} +24749.7 q^{91} +22178.7 q^{95} -155739. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 76 q^{5} + 98 q^{7}+O(q^{10}) $ 2 * q - 76 * q^5 + 98 * q^7	$ 2 q - 76 q^{5} + 98 q^{7} - 564 q^{11} + 516 q^{13} + 76 q^{17} + 2360 q^{19} + 2036 q^{23} + 4270 q^{25} - 8320 q^{29} - 6280 q^{31} - 3724 q^{35} - 2460 q^{37} + 14308 q^{41} + 23128 q^{43} - 12712 q^{47} + 4802 q^{49} + 9896 q^{53} - 16728 q^{55} - 60888 q^{59} + 16172 q^{61} + 10920 q^{65} - 62568 q^{67} + 732 q^{71} + 1244 q^{73} - 27636 q^{77} + 11600 q^{79} - 132288 q^{83} - 71576 q^{85} - 93452 q^{89} + 25284 q^{91} - 120208 q^{95} - 272932 q^{97}+O(q^{100}) $ 2 * q - 76 * q^5 + 98 * q^7 - 564 * q^11 + 516 * q^13 + 76 * q^17 + 2360 * q^19 + 2036 * q^23 + 4270 * q^25 - 8320 * q^29 - 6280 * q^31 - 3724 * q^35 - 2460 * q^37 + 14308 * q^41 + 23128 * q^43 - 12712 * q^47 + 4802 * q^49 + 9896 * q^53 - 16728 * q^55 - 60888 * q^59 + 16172 * q^61 + 10920 * q^65 - 62568 * q^67 + 732 * q^71 + 1244 * q^73 - 27636 * q^77 + 11600 * q^79 - 132288 * q^83 - 71576 * q^85 - 93452 * q^89 + 25284 * q^91 - 120208 * q^95 - 272932 * 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
$5$	23.7738	0.425278	0.212639	−	0.977131i	$-0.431794\pi$
$5$	23.7738	0.425278	0.212639	+	0.977131i	$0.431794\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−590.869	−1.47234	−0.736172	−	0.676794i	$-0.763369\pi$
$11$	−590.869	−1.47234	−0.736172	+	0.676794i	$0.763369\pi$
$12$	0	0
$13$	505.095	0.828924	0.414462	−	0.910067i	$-0.363970\pi$
$13$	505.095	0.828924	0.414462	+	0.910067i	$0.363970\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−517.964	−0.434688	−0.217344	−	0.976095i	$-0.569739\pi$
$17$	−517.964	−0.434688	−0.217344	+	0.976095i	$0.569739\pi$
$18$	0	0
$19$	932.905	0.592862	0.296431	−	0.955054i	$-0.404204\pi$
$19$	932.905	0.592862	0.296431	+	0.955054i	$0.404204\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3550.73	1.39958	0.699790	−	0.714349i	$-0.253277\pi$
$23$	3550.73	1.39958	0.699790	+	0.714349i	$0.253277\pi$
$24$	0	0
$25$	−2559.81	−0.819138
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5395.48	−1.19134	−0.595669	−	0.803230i	$-0.703113\pi$
$29$	−5395.48	−1.19134	−0.595669	+	0.803230i	$0.703113\pi$
$30$	0	0
$31$	−8329.00	−1.55664	−0.778321	−	0.627867i	$-0.783928\pi$
$31$	−8329.00	−1.55664	−0.778321	+	0.627867i	$0.783928\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1164.92	0.160740
$36$	0	0
$37$	−4936.43	−0.592800	−0.296400	−	0.955064i	$-0.595786\pi$
$37$	−4936.43	−0.592800	−0.296400	+	0.955064i	$0.595786\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5880.27	−0.546308	−0.273154	−	0.961970i	$-0.588067\pi$
$41$	−5880.27	−0.546308	−0.273154	+	0.961970i	$0.588067\pi$
$42$	0	0
$43$	13540.8	1.11679	0.558396	−	0.829575i	$-0.311417\pi$
$43$	13540.8	1.11679	0.558396	+	0.829575i	$0.311417\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7357.78	0.485850	0.242925	−	0.970045i	$-0.421893\pi$
$47$	7357.78	0.485850	0.242925	+	0.970045i	$0.421893\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3341.88	0.163419	0.0817093	−	0.996656i	$-0.473962\pi$
$53$	3341.88	0.163419	0.0817093	+	0.996656i	$0.473962\pi$
$54$	0	0
$55$	−14047.2	−0.626156
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−42675.2	−1.59605	−0.798023	−	0.602626i	$-0.794121\pi$
$59$	−42675.2	−1.59605	−0.798023	+	0.602626i	$0.794121\pi$
$60$	0	0
$61$	51327.6	1.76615	0.883073	−	0.469235i	$-0.155470\pi$
$61$	51327.6	1.76615	0.883073	+	0.469235i	$0.155470\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	12008.0	0.352523
$66$	0	0
$67$	−33755.0	−0.918651	−0.459325	−	0.888268i	$-0.651909\pi$
$67$	−33755.0	−0.918651	−0.459325	+	0.888268i	$0.651909\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	17848.0	0.420188	0.210094	−	0.977681i	$-0.432623\pi$
$71$	17848.0	0.420188	0.210094	+	0.977681i	$0.432623\pi$
$72$	0	0
$73$	−20628.2	−0.453058	−0.226529	−	0.974004i	$-0.572738\pi$
$73$	−20628.2	−0.453058	−0.226529	+	0.974004i	$0.572738\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−28952.6	−0.556494
$78$	0	0
$79$	99202.0	1.78835	0.894175	−	0.447718i	$-0.147763\pi$
$79$	99202.0	1.78835	0.894175	+	0.447718i	$0.147763\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−61202.1	−0.975149	−0.487575	−	0.873081i	$-0.662118\pi$
$83$	−61202.1	−0.975149	−0.487575	+	0.873081i	$0.662118\pi$
$84$	0	0
$85$	−12314.0	−0.184863
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−90153.0	−1.20644	−0.603219	−	0.797576i	$-0.706116\pi$
$89$	−90153.0	−1.20644	−0.603219	+	0.797576i	$0.706116\pi$
$90$	0	0
$91$	24749.7	0.313304
$92$	0	0
$93$	0	0
$94$	0	0
$95$	22178.7	0.252131
$96$	0	0
$97$	−155739.	−1.68062	−0.840309	−	0.542107i	$-0.817627\pi$
$97$	−155739.	−1.68062	−0.840309	+	0.542107i	$0.817627\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	19635.0	0.191526	0.0957628	−	0.995404i	$-0.469471\pi$
$101$	19635.0	0.191526	0.0957628	+	0.995404i	$0.469471\pi$
$102$	0	0
$103$	−88981.8	−0.826434	−0.413217	−	0.910633i	$-0.635595\pi$
$103$	−88981.8	−0.826434	−0.413217	+	0.910633i	$0.635595\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−185903.	−1.56973	−0.784867	−	0.619664i	$-0.787269\pi$
$107$	−185903.	−1.56973	−0.784867	+	0.619664i	$0.787269\pi$
$108$	0	0
$109$	−77652.7	−0.626023	−0.313012	−	0.949749i	$-0.601338\pi$
$109$	−77652.7	−0.626023	−0.313012	+	0.949749i	$0.601338\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−246859.	−1.81867	−0.909334	−	0.416068i	$-0.863408\pi$
$113$	−246859.	−1.81867	−0.909334	+	0.416068i	$0.863408\pi$
$114$	0	0
$115$	84414.2	0.595211
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−25380.2	−0.164296
$120$	0	0
$121$	188075.	1.16780
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−135149.	−0.773640
$126$	0	0
$127$	210563.	1.15844	0.579220	−	0.815171i	$-0.303357\pi$
$127$	210563.	1.15844	0.579220	+	0.815171i	$0.303357\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	85398.9	0.434784	0.217392	−	0.976084i	$-0.430245\pi$
$131$	85398.9	0.434784	0.217392	+	0.976084i	$0.430245\pi$
$132$	0	0
$133$	45712.3	0.224081
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−86716.9	−0.394732	−0.197366	−	0.980330i	$-0.563239\pi$
$137$	−86716.9	−0.394732	−0.197366	+	0.980330i	$0.563239\pi$
$138$	0	0
$139$	−131720.	−0.578247	−0.289124	−	0.957292i	$-0.593364\pi$
$139$	−131720.	−0.578247	−0.289124	+	0.957292i	$0.593364\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−298445.	−1.22046
$144$	0	0
$145$	−128271.	−0.506650
$146$	0	0
$147$	0	0
$148$	0	0
$149$	30659.9	0.113137	0.0565685	−	0.998399i	$-0.481984\pi$
$149$	30659.9	0.113137	0.0565685	+	0.998399i	$0.481984\pi$
$150$	0	0
$151$	125626.	0.448372	0.224186	−	0.974546i	$-0.428028\pi$
$151$	125626.	0.448372	0.224186	+	0.974546i	$0.428028\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−198012.	−0.662006
$156$	0	0
$157$	−516272.	−1.67159	−0.835794	−	0.549043i	$-0.814992\pi$
$157$	−516272.	−1.67159	−0.835794	+	0.549043i	$0.814992\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	173986.	0.528991
$162$	0	0
$163$	−119172.	−0.351321	−0.175660	−	0.984451i	$-0.556206\pi$
$163$	−119172.	−0.351321	−0.175660	+	0.984451i	$0.556206\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−39035.0	−0.108309	−0.0541543	−	0.998533i	$-0.517246\pi$
$167$	−39035.0	−0.108309	−0.0541543	+	0.998533i	$0.517246\pi$
$168$	0	0
$169$	−116172.	−0.312885
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−571616.	−1.45207	−0.726037	−	0.687655i	$-0.758640\pi$
$173$	−571616.	−1.45207	−0.726037	+	0.687655i	$0.758640\pi$
$174$	0	0
$175$	−125431.	−0.309605
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−846291.	−1.97418	−0.987091	−	0.160160i	$-0.948799\pi$
$179$	−846291.	−1.97418	−0.987091	+	0.160160i	$0.948799\pi$
$180$	0	0
$181$	491766.	1.11574	0.557868	−	0.829930i	$-0.311619\pi$
$181$	491766.	1.11574	0.557868	+	0.829930i	$0.311619\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−117358.	−0.252105
$186$	0	0
$187$	306049.	0.640010
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−648886.	−1.28702	−0.643509	−	0.765438i	$-0.722522\pi$
$191$	−648886.	−1.28702	−0.643509	+	0.765438i	$0.722522\pi$
$192$	0	0
$193$	−216436.	−0.418251	−0.209125	−	0.977889i	$-0.567062\pi$
$193$	−216436.	−0.418251	−0.209125	+	0.977889i	$0.567062\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	178935.	0.328497	0.164248	−	0.986419i	$-0.447480\pi$
$197$	178935.	0.328497	0.164248	+	0.986419i	$0.447480\pi$
$198$	0	0
$199$	303836.	0.543885	0.271942	−	0.962314i	$-0.412334\pi$
$199$	303836.	0.543885	0.271942	+	0.962314i	$0.412334\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−264378.	−0.450283
$204$	0	0
$205$	−139796.	−0.232333
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−551224.	−0.872897
$210$	0	0
$211$	634974.	0.981860	0.490930	−	0.871199i	$-0.336657\pi$
$211$	634974.	0.981860	0.490930	+	0.871199i	$0.336657\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	321915.	0.474947
$216$	0	0
$217$	−408121.	−0.588355
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−261621.	−0.360323
$222$	0	0
$223$	1.20650e6	1.62467	0.812337	−	0.583188i	$-0.198195\pi$
$223$	1.20650e6	1.62467	0.812337	+	0.583188i	$0.198195\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	686964.	0.884849	0.442425	−	0.896806i	$-0.354118\pi$
$227$	686964.	0.884849	0.442425	+	0.896806i	$0.354118\pi$
$228$	0	0
$229$	−602459.	−0.759170	−0.379585	−	0.925157i	$-0.623933\pi$
$229$	−602459.	−0.759170	−0.379585	+	0.925157i	$0.623933\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	262764.	0.317085	0.158543	−	0.987352i	$-0.449320\pi$
$233$	262764.	0.317085	0.158543	+	0.987352i	$0.449320\pi$
$234$	0	0
$235$	174922.	0.206621
$236$	0	0
$237$	0	0
$238$	0	0
$239$	194880.	0.220685	0.110342	−	0.993894i	$-0.464805\pi$
$239$	194880.	0.220685	0.110342	+	0.993894i	$0.464805\pi$
$240$	0	0
$241$	352082.	0.390482	0.195241	−	0.980755i	$-0.437451\pi$
$241$	352082.	0.390482	0.195241	+	0.980755i	$0.437451\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	57080.8	0.0607540
$246$	0	0
$247$	471206.	0.491437
$248$	0	0
$249$	0	0
$250$	0	0
$251$	114343.	0.114558	0.0572791	−	0.998358i	$-0.481758\pi$
$251$	114343.	0.114558	0.0572791	+	0.998358i	$0.481758\pi$
$252$	0	0
$253$	−2.09801e6	−2.06066
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.78421e6	−1.68505	−0.842525	−	0.538657i	$-0.818932\pi$
$257$	−1.78421e6	−1.68505	−0.842525	+	0.538657i	$0.818932\pi$
$258$	0	0
$259$	−241885.	−0.224057
$260$	0	0
$261$	0	0
$262$	0	0
$263$	46232.1	0.0412149	0.0206074	−	0.999788i	$-0.493440\pi$
$263$	46232.1	0.0412149	0.0206074	+	0.999788i	$0.493440\pi$
$264$	0	0
$265$	79449.2	0.0694984
$266$	0	0
$267$	0	0
$268$	0	0
$269$	273850.	0.230745	0.115372	−	0.993322i	$-0.463194\pi$
$269$	273850.	0.230745	0.115372	+	0.993322i	$0.463194\pi$
$270$	0	0
$271$	−165105.	−0.136564	−0.0682821	−	0.997666i	$-0.521752\pi$
$271$	−165105.	−0.136564	−0.0682821	+	0.997666i	$0.521752\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.51251e6	1.20605
$276$	0	0
$277$	−605103.	−0.473838	−0.236919	−	0.971529i	$-0.576138\pi$
$277$	−605103.	−0.473838	−0.236919	+	0.971529i	$0.576138\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−618011.	−0.466907	−0.233454	−	0.972368i	$-0.575003\pi$
$281$	−618011.	−0.466907	−0.233454	+	0.972368i	$0.575003\pi$
$282$	0	0
$283$	−218928.	−0.162494	−0.0812468	−	0.996694i	$-0.525890\pi$
$283$	−218928.	−0.162494	−0.0812468	+	0.996694i	$0.525890\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−288133.	−0.206485
$288$	0	0
$289$	−1.15157e6	−0.811047
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.30928e6	1.57147	0.785736	−	0.618562i	$-0.212284\pi$
$293$	2.30928e6	1.57147	0.785736	+	0.618562i	$0.212284\pi$
$294$	0	0
$295$	−1.01455e6	−0.678764
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.79345e6	1.16014
$300$	0	0
$301$	663497.	0.422107
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.22025e6	0.751104
$306$	0	0
$307$	1.61363e6	0.977145	0.488572	−	0.872523i	$-0.337518\pi$
$307$	1.61363e6	0.977145	0.488572	+	0.872523i	$0.337518\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.11596e6	−0.654257	−0.327128	−	0.944980i	$-0.606081\pi$
$311$	−1.11596e6	−0.654257	−0.327128	+	0.944980i	$0.606081\pi$
$312$	0	0
$313$	−2.02307e6	−1.16722	−0.583608	−	0.812036i	$-0.698359\pi$
$313$	−2.02307e6	−1.16722	−0.583608	+	0.812036i	$0.698359\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.04841e6	−0.585982	−0.292991	−	0.956115i	$-0.594651\pi$
$317$	−1.04841e6	−0.585982	−0.292991	+	0.956115i	$0.594651\pi$
$318$	0	0
$319$	3.18802e6	1.75406
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−483211.	−0.257710
$324$	0	0
$325$	−1.29295e6	−0.679004
$326$	0	0
$327$	0	0
$328$	0	0
$329$	360531.	0.183634
$330$	0	0
$331$	3.29848e6	1.65480	0.827398	−	0.561617i	$-0.189820\pi$
$331$	3.29848e6	1.65480	0.827398	+	0.561617i	$0.189820\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−802483.	−0.390682
$336$	0	0
$337$	2.64485e6	1.26861	0.634303	−	0.773084i	$-0.281287\pi$
$337$	2.64485e6	1.26861	0.634303	+	0.773084i	$0.281287\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.92135e6	2.29191
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.09974e6	0.936140	0.468070	−	0.883691i	$-0.344949\pi$
$347$	2.09974e6	0.936140	0.468070	+	0.883691i	$0.344949\pi$
$348$	0	0
$349$	−2.91574e6	−1.28140	−0.640701	−	0.767790i	$-0.721356\pi$
$349$	−2.91574e6	−1.28140	−0.640701	+	0.767790i	$0.721356\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.03901e6	1.72520	0.862598	−	0.505891i	$-0.168836\pi$
$353$	4.03901e6	1.72520	0.862598	+	0.505891i	$0.168836\pi$
$354$	0	0
$355$	424314.	0.178697
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.68420e6	0.689696	0.344848	−	0.938658i	$-0.387930\pi$
$359$	1.68420e6	0.689696	0.344848	+	0.938658i	$0.387930\pi$
$360$	0	0
$361$	−1.60579e6	−0.648515
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−490410.	−0.192676
$366$	0	0
$367$	901845.	0.349516	0.174758	−	0.984611i	$-0.444086\pi$
$367$	901845.	0.349516	0.174758	+	0.984611i	$0.444086\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	163752.	0.0617664
$372$	0	0
$373$	−1.48555e6	−0.552861	−0.276431	−	0.961034i	$-0.589152\pi$
$373$	−1.48555e6	−0.552861	−0.276431	+	0.961034i	$0.589152\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.72523e6	−0.987528
$378$	0	0
$379$	2.44302e6	0.873633	0.436817	−	0.899551i	$-0.356106\pi$
$379$	2.44302e6	0.873633	0.436817	+	0.899551i	$0.356106\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−457836.	−0.159482	−0.0797412	−	0.996816i	$-0.525409\pi$
$383$	−457836.	−0.159482	−0.0797412	+	0.996816i	$0.525409\pi$
$384$	0	0
$385$	−688312.	−0.236665
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.35458e6	−0.453868	−0.226934	−	0.973910i	$-0.572870\pi$
$389$	−1.35458e6	−0.453868	−0.226934	+	0.973910i	$0.572870\pi$
$390$	0	0
$391$	−1.83915e6	−0.608380
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.35841e6	0.760546
$396$	0	0
$397$	3.25028e6	1.03501	0.517506	−	0.855680i	$-0.326861\pi$
$397$	3.25028e6	1.03501	0.517506	+	0.855680i	$0.326861\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.99082e6	1.54993	0.774963	−	0.632006i	$-0.217768\pi$
$401$	4.99082e6	1.54993	0.774963	+	0.632006i	$0.217768\pi$
$402$	0	0
$403$	−4.20694e6	−1.29034
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.91678e6	0.872806
$408$	0	0
$409$	−2.16290e6	−0.639334	−0.319667	−	0.947530i	$-0.603571\pi$
$409$	−2.16290e6	−0.639334	−0.319667	+	0.947530i	$0.603571\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.09109e6	−0.603249
$414$	0	0
$415$	−1.45501e6	−0.414710
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.31464e6	−0.365823	−0.182912	−	0.983129i	$-0.558552\pi$
$419$	−1.31464e6	−0.365823	−0.182912	+	0.983129i	$0.558552\pi$
$420$	0	0
$421$	1.14615e6	0.315163	0.157581	−	0.987506i	$-0.449630\pi$
$421$	1.14615e6	0.315163	0.157581	+	0.987506i	$0.449630\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.32589e6	0.356069
$426$	0	0
$427$	2.51505e6	0.667541
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.53987e6	1.43650	0.718251	−	0.695784i	$-0.244943\pi$
$431$	5.53987e6	1.43650	0.718251	+	0.695784i	$0.244943\pi$
$432$	0	0
$433$	−3.52567e6	−0.903696	−0.451848	−	0.892095i	$-0.649235\pi$
$433$	−3.52567e6	−0.903696	−0.451848	+	0.892095i	$0.649235\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.31249e6	0.829757
$438$	0	0
$439$	5.93446e6	1.46967	0.734835	−	0.678246i	$-0.237260\pi$
$439$	5.93446e6	1.46967	0.734835	+	0.678246i	$0.237260\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.19294e6	0.288808	0.144404	−	0.989519i	$-0.453873\pi$
$443$	1.19294e6	0.288808	0.144404	+	0.989519i	$0.453873\pi$
$444$	0	0
$445$	−2.14328e6	−0.513072
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.13944e6	1.20310	0.601548	−	0.798837i	$-0.294551\pi$
$449$	5.13944e6	1.20310	0.601548	+	0.798837i	$0.294551\pi$
$450$	0	0
$451$	3.47447e6	0.804353
$452$	0	0
$453$	0	0
$454$	0	0
$455$	588393.	0.133241
$456$	0	0
$457$	3.93737e6	0.881894	0.440947	−	0.897533i	$-0.354643\pi$
$457$	3.93737e6	0.881894	0.440947	+	0.897533i	$0.354643\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.31710e6	0.946105	0.473053	−	0.881034i	$-0.343152\pi$
$461$	4.31710e6	0.946105	0.473053	+	0.881034i	$0.343152\pi$
$462$	0	0
$463$	−863360.	−0.187171	−0.0935857	−	0.995611i	$-0.529833\pi$
$463$	−863360.	−0.187171	−0.0935857	+	0.995611i	$0.529833\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.84879e6	−0.392280	−0.196140	−	0.980576i	$-0.562841\pi$
$467$	−1.84879e6	−0.392280	−0.196140	+	0.980576i	$0.562841\pi$
$468$	0	0
$469$	−1.65399e6	−0.347217
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.00081e6	−1.64430
$474$	0	0
$475$	−2.38806e6	−0.485636
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.45491e6	1.08630	0.543149	−	0.839637i	$-0.317232\pi$
$479$	5.45491e6	1.08630	0.543149	+	0.839637i	$0.317232\pi$
$480$	0	0
$481$	−2.49337e6	−0.491386
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.70251e6	−0.714731
$486$	0	0
$487$	344795.	0.0658777	0.0329388	−	0.999457i	$-0.489513\pi$
$487$	344795.	0.0658777	0.0329388	+	0.999457i	$0.489513\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.06521e7	−1.99404	−0.997018	−	0.0771721i	$-0.975411\pi$
$491$	−1.06521e7	−1.99404	−0.997018	+	0.0771721i	$0.975411\pi$
$492$	0	0
$493$	2.79466e6	0.517860
$494$	0	0
$495$	0	0
$496$	0	0
$497$	874551.	0.158816
$498$	0	0
$499$	−2.85842e6	−0.513895	−0.256948	−	0.966425i	$-0.582717\pi$
$499$	−2.85842e6	−0.513895	−0.256948	+	0.966425i	$0.582717\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.14241e6	0.553786	0.276893	−	0.960901i	$-0.410695\pi$
$503$	3.14241e6	0.553786	0.276893	+	0.960901i	$0.410695\pi$
$504$	0	0
$505$	466797.	0.0814517
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.66192e6	0.284325	0.142163	−	0.989843i	$-0.454594\pi$
$509$	1.66192e6	0.284325	0.142163	+	0.989843i	$0.454594\pi$
$510$	0	0
$511$	−1.01078e6	−0.171240
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.11543e6	−0.351464
$516$	0	0
$517$	−4.34748e6	−0.715338
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.18324e6	−0.190976	−0.0954881	−	0.995431i	$-0.530441\pi$
$521$	−1.18324e6	−0.190976	−0.0954881	+	0.995431i	$0.530441\pi$
$522$	0	0
$523$	479133.	0.0765953	0.0382976	−	0.999266i	$-0.487806\pi$
$523$	479133.	0.0765953	0.0382976	+	0.999266i	$0.487806\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.31412e6	0.676653
$528$	0	0
$529$	6.17131e6	0.958822
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.97009e6	−0.452848
$534$	0	0
$535$	−4.41961e6	−0.667574
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.41868e6	−0.210335
$540$	0	0
$541$	−5.40143e6	−0.793442	−0.396721	−	0.917939i	$-0.629852\pi$
$541$	−5.40143e6	−0.793442	−0.396721	+	0.917939i	$0.629852\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.84610e6	−0.266234
$546$	0	0
$547$	−3.61094e6	−0.516003	−0.258001	−	0.966145i	$-0.583064\pi$
$547$	−3.61094e6	−0.516003	−0.258001	+	0.966145i	$0.583064\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.03347e6	−0.706298
$552$	0	0
$553$	4.86090e6	0.675933
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.04321e6	−1.23505	−0.617525	−	0.786551i	$-0.711865\pi$
$557$	−9.04321e6	−1.23505	−0.617525	+	0.786551i	$0.711865\pi$
$558$	0	0
$559$	6.83937e6	0.925735
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.40826e6	0.453170	0.226585	−	0.973991i	$-0.427244\pi$
$563$	3.40826e6	0.453170	0.226585	+	0.973991i	$0.427244\pi$
$564$	0	0
$565$	−5.86878e6	−0.773440
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.29430e6	0.426562	0.213281	−	0.976991i	$-0.431585\pi$
$569$	3.29430e6	0.426562	0.213281	+	0.976991i	$0.431585\pi$
$570$	0	0
$571$	5.86813e6	0.753199	0.376599	−	0.926376i	$-0.377093\pi$
$571$	5.86813e6	0.753199	0.376599	+	0.926376i	$0.377093\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.08917e6	−1.14645
$576$	0	0
$577$	1.23173e6	0.154019	0.0770096	−	0.997030i	$-0.475463\pi$
$577$	1.23173e6	0.154019	0.0770096	+	0.997030i	$0.475463\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.99890e6	−0.368572
$582$	0	0
$583$	−1.97461e6	−0.240608
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.58196e6	−0.788424	−0.394212	−	0.919020i	$-0.628982\pi$
$587$	−6.58196e6	−0.788424	−0.394212	+	0.919020i	$0.628982\pi$
$588$	0	0
$589$	−7.77016e6	−0.922873
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.93053e6	0.459002	0.229501	−	0.973308i	$-0.426291\pi$
$593$	3.93053e6	0.459002	0.229501	+	0.973308i	$0.426291\pi$
$594$	0	0
$595$	−603384.	−0.0698717
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.32868e7	−1.51305	−0.756524	−	0.653966i	$-0.773104\pi$
$599$	−1.32868e7	−1.51305	−0.756524	+	0.653966i	$0.773104\pi$
$600$	0	0
$601$	1.25036e7	1.41205	0.706025	−	0.708187i	$-0.250487\pi$
$601$	1.25036e7	1.41205	0.706025	+	0.708187i	$0.250487\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.47126e6	0.496639
$606$	0	0
$607$	−1.24254e7	−1.36880	−0.684400	−	0.729107i	$-0.739936\pi$
$607$	−1.24254e7	−1.36880	−0.684400	+	0.729107i	$0.739936\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.71638e6	0.402733
$612$	0	0
$613$	3.63638e6	0.390857	0.195429	−	0.980718i	$-0.437390\pi$
$613$	3.63638e6	0.390857	0.195429	+	0.980718i	$0.437390\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.50994e6	−0.794188	−0.397094	−	0.917778i	$-0.629981\pi$
$617$	−7.50994e6	−0.794188	−0.397094	+	0.917778i	$0.629981\pi$
$618$	0	0
$619$	3.35237e6	0.351662	0.175831	−	0.984420i	$-0.443739\pi$
$619$	3.35237e6	0.351662	0.175831	+	0.984420i	$0.443739\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.41750e6	−0.455991
$624$	0	0
$625$	4.78639e6	0.490126
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.55689e6	0.257683
$630$	0	0
$631$	−2.66773e6	−0.266728	−0.133364	−	0.991067i	$-0.542578\pi$
$631$	−2.66773e6	−0.266728	−0.133364	+	0.991067i	$0.542578\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.00589e6	0.492659
$636$	0	0
$637$	1.21273e6	0.118418
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.30265e6	−0.798126	−0.399063	−	0.916923i	$-0.630665\pi$
$641$	−8.30265e6	−0.798126	−0.399063	+	0.916923i	$0.630665\pi$
$642$	0	0
$643$	1.16120e7	1.10759	0.553796	−	0.832652i	$-0.313179\pi$
$643$	1.16120e7	1.10759	0.553796	+	0.832652i	$0.313179\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.91709e7	1.80045	0.900227	−	0.435421i	$-0.143400\pi$
$647$	1.91709e7	1.80045	0.900227	+	0.435421i	$0.143400\pi$
$648$	0	0
$649$	2.52155e7	2.34993
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.36350e7	−1.25133	−0.625665	−	0.780092i	$-0.715172\pi$
$653$	−1.36350e7	−1.25133	−0.625665	+	0.780092i	$0.715172\pi$
$654$	0	0
$655$	2.03025e6	0.184904
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.71937e6	0.333623	0.166812	−	0.985989i	$-0.446653\pi$
$659$	3.71937e6	0.333623	0.166812	+	0.985989i	$0.446653\pi$
$660$	0	0
$661$	9.69481e6	0.863050	0.431525	−	0.902101i	$-0.357976\pi$
$661$	9.69481e6	0.863050	0.431525	+	0.902101i	$0.357976\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.08676e6	0.0952966
$666$	0	0
$667$	−1.91579e7	−1.66737
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.03279e7	−2.60038
$672$	0	0
$673$	−1.51973e6	−0.129339	−0.0646695	−	0.997907i	$-0.520599\pi$
$673$	−1.51973e6	−0.129339	−0.0646695	+	0.997907i	$0.520599\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.26234e7	−1.89709	−0.948543	−	0.316648i	$-0.897443\pi$
$677$	−2.26234e7	−1.89709	−0.948543	+	0.316648i	$0.897443\pi$
$678$	0	0
$679$	−7.63123e6	−0.635214
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.21373e6	−0.0995567	−0.0497784	−	0.998760i	$-0.515851\pi$
$683$	−1.21373e6	−0.0995567	−0.0497784	+	0.998760i	$0.515851\pi$
$684$	0	0
$685$	−2.06159e6	−0.167871
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.68797e6	0.135462
$690$	0	0
$691$	−1.40481e7	−1.11924	−0.559619	−	0.828750i	$-0.689053\pi$
$691$	−1.40481e7	−1.11924	−0.559619	+	0.828750i	$0.689053\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.13147e6	−0.245916
$696$	0	0
$697$	3.04577e6	0.237473
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.42260e7	−1.09342	−0.546711	−	0.837321i	$-0.684120\pi$
$701$	−1.42260e7	−1.09342	−0.546711	+	0.837321i	$0.684120\pi$
$702$	0	0
$703$	−4.60522e6	−0.351449
$704$	0	0
$705$	0	0
$706$	0	0
$707$	962113.	0.0723898
$708$	0	0
$709$	−8.49859e6	−0.634938	−0.317469	−	0.948269i	$-0.602833\pi$
$709$	−8.49859e6	−0.634938	−0.317469	+	0.948269i	$0.602833\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.95740e7	−2.17864
$714$	0	0
$715$	−7.09517e6	−0.519036
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7.77769e6	0.561085	0.280542	−	0.959842i	$-0.409486\pi$
$719$	7.77769e6	0.561085	0.280542	+	0.959842i	$0.409486\pi$
$720$	0	0
$721$	−4.36011e6	−0.312363
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.38114e7	0.975870
$726$	0	0
$727$	−2.03361e7	−1.42703	−0.713513	−	0.700642i	$-0.752897\pi$
$727$	−2.03361e7	−1.42703	−0.713513	+	0.700642i	$0.752897\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.01363e6	−0.485455
$732$	0	0
$733$	−6.91670e6	−0.475488	−0.237744	−	0.971328i	$-0.576408\pi$
$733$	−6.91670e6	−0.475488	−0.237744	+	0.971328i	$0.576408\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.99448e7	1.35257
$738$	0	0
$739$	−2.12775e7	−1.43321	−0.716605	−	0.697479i	$-0.754305\pi$
$739$	−2.12775e7	−1.43321	−0.716605	+	0.697479i	$0.754305\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.99548e7	−1.32610	−0.663049	−	0.748576i	$-0.730738\pi$
$743$	−1.99548e7	−1.32610	−0.663049	+	0.748576i	$0.730738\pi$
$744$	0	0
$745$	728902.	0.0481148
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.10923e6	−0.593304
$750$	0	0
$751$	5.84360e6	0.378077	0.189039	−	0.981970i	$-0.439463\pi$
$751$	5.84360e6	0.378077	0.189039	+	0.981970i	$0.439463\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.98661e6	0.190683
$756$	0	0
$757$	2.32278e7	1.47322	0.736612	−	0.676315i	$-0.236424\pi$
$757$	2.32278e7	1.47322	0.736612	+	0.676315i	$0.236424\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.93636e7	1.21206	0.606030	−	0.795441i	$-0.292761\pi$
$761$	1.93636e7	1.21206	0.606030	+	0.795441i	$0.292761\pi$
$762$	0	0
$763$	−3.80498e6	−0.236614
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.15550e7	−1.32300
$768$	0	0
$769$	1.83678e7	1.12006	0.560031	−	0.828472i	$-0.310789\pi$
$769$	1.83678e7	1.12006	0.560031	+	0.828472i	$0.310789\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.91439e7	1.75428	0.877140	−	0.480234i	$-0.159448\pi$
$773$	2.91439e7	1.75428	0.877140	+	0.480234i	$0.159448\pi$
$774$	0	0
$775$	2.13206e7	1.27510
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.48573e6	−0.323885
$780$	0	0
$781$	−1.05458e7	−0.618661
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.22737e7	−0.710890
$786$	0	0
$787$	4.53087e6	0.260762	0.130381	−	0.991464i	$-0.458380\pi$
$787$	4.53087e6	0.260762	0.130381	+	0.991464i	$0.458380\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.20961e7	−0.687392
$792$	0	0
$793$	2.59253e7	1.46400
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.19253e7	0.665005	0.332503	−	0.943102i	$-0.392107\pi$
$797$	1.19253e7	0.665005	0.332503	+	0.943102i	$0.392107\pi$
$798$	0	0
$799$	−3.81107e6	−0.211193
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.21886e7	0.667057
$804$	0	0
$805$	4.13629e6	0.224968
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.25486e7	−0.674099	−0.337049	−	0.941487i	$-0.609429\pi$
$809$	−1.25486e7	−0.674099	−0.337049	+	0.941487i	$0.609429\pi$
$810$	0	0
$811$	−5.87349e6	−0.313577	−0.156788	−	0.987632i	$-0.550114\pi$
$811$	−5.87349e6	−0.313577	−0.156788	+	0.987632i	$0.550114\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.83316e6	−0.149409
$816$	0	0
$817$	1.26322e7	0.662103
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.44021e7	1.26348	0.631742	−	0.775178i	$-0.282340\pi$
$821$	2.44021e7	1.26348	0.631742	+	0.775178i	$0.282340\pi$
$822$	0	0
$823$	2.45208e7	1.26193	0.630965	−	0.775812i	$-0.282659\pi$
$823$	2.45208e7	1.26193	0.630965	+	0.775812i	$0.282659\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.15215e7	−0.585792	−0.292896	−	0.956144i	$-0.594619\pi$
$827$	−1.15215e7	−0.585792	−0.292896	+	0.956144i	$0.594619\pi$
$828$	0	0
$829$	−2.67057e7	−1.34964	−0.674820	−	0.737982i	$-0.735779\pi$
$829$	−2.67057e7	−1.34964	−0.674820	+	0.737982i	$0.735779\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.24363e6	−0.0620982
$834$	0	0
$835$	−928009.	−0.0460613
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.76624e7	1.35671	0.678353	−	0.734737i	$-0.262694\pi$
$839$	2.76624e7	1.35671	0.678353	+	0.734737i	$0.262694\pi$
$840$	0	0
$841$	8.60001e6	0.419285
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.76185e6	−0.133063
$846$	0	0
$847$	9.21568e6	0.441386
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.75279e7	−0.829671
$852$	0	0
$853$	−1.46417e7	−0.688999	−0.344499	−	0.938787i	$-0.611951\pi$
$853$	−1.46417e7	−0.688999	−0.344499	+	0.938787i	$0.611951\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.18815e7	1.94792	0.973959	−	0.226725i	$-0.0728019\pi$
$857$	4.18815e7	1.94792	0.973959	+	0.226725i	$0.0728019\pi$
$858$	0	0
$859$	−3.52752e7	−1.63112	−0.815562	−	0.578670i	$-0.803572\pi$
$859$	−3.52752e7	−1.63112	−0.815562	+	0.578670i	$0.803572\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.07135e7	1.40379	0.701895	−	0.712280i	$-0.252337\pi$
$863$	3.07135e7	1.40379	0.701895	+	0.712280i	$0.252337\pi$
$864$	0	0
$865$	−1.35895e7	−0.617536
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5.86154e7	−2.63307
$870$	0	0
$871$	−1.70495e7	−0.761492
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.62232e6	−0.292408
$876$	0	0
$877$	2.97875e7	1.30778	0.653891	−	0.756588i	$-0.273135\pi$
$877$	2.97875e7	1.30778	0.653891	+	0.756588i	$0.273135\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−3.74145e7	−1.62405	−0.812026	−	0.583621i	$-0.801635\pi$
$881$	−3.74145e7	−1.62405	−0.812026	+	0.583621i	$0.801635\pi$
$882$	0	0
$883$	1.30401e7	0.562832	0.281416	−	0.959586i	$-0.409196\pi$
$883$	1.30401e7	0.562832	0.281416	+	0.959586i	$0.409196\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.76838e7	0.754687	0.377343	−	0.926073i	$-0.376838\pi$
$887$	1.76838e7	0.754687	0.377343	+	0.926073i	$0.376838\pi$
$888$	0	0
$889$	1.03176e7	0.437849
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.86411e6	0.288042
$894$	0	0
$895$	−2.01195e7	−0.839577
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.49389e7	1.85448
$900$	0	0
$901$	−1.73097e6	−0.0710360
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.16911e7	0.474499
$906$	0	0
$907$	4.31052e6	0.173985	0.0869925	−	0.996209i	$-0.472274\pi$
$907$	4.31052e6	0.173985	0.0869925	+	0.996209i	$0.472274\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.16954e7	0.866106	0.433053	−	0.901368i	$-0.357436\pi$
$911$	2.16954e7	0.866106	0.433053	+	0.901368i	$0.357436\pi$
$912$	0	0
$913$	3.61624e7	1.43576
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.18454e6	0.164333
$918$	0	0
$919$	−1.38075e7	−0.539296	−0.269648	−	0.962959i	$-0.586907\pi$
$919$	−1.38075e7	−0.539296	−0.269648	+	0.962959i	$0.586907\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.01493e6	0.348304
$924$	0	0
$925$	1.26363e7	0.485585
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.28005e6	0.314770	0.157385	−	0.987537i	$-0.449694\pi$
$929$	8.28005e6	0.314770	0.157385	+	0.987537i	$0.449694\pi$
$930$	0	0
$931$	2.23990e6	0.0846945
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.27594e6	0.272182
$936$	0	0
$937$	−3.03112e7	−1.12786	−0.563928	−	0.825824i	$-0.690711\pi$
$937$	−3.03112e7	−1.12786	−0.563928	+	0.825824i	$0.690711\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.15564e7	1.52990	0.764952	−	0.644087i	$-0.222763\pi$
$941$	4.15564e7	1.52990	0.764952	+	0.644087i	$0.222763\pi$
$942$	0	0
$943$	−2.08792e7	−0.764601
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.94364e6	−0.179132	−0.0895658	−	0.995981i	$-0.528548\pi$
$947$	−4.94364e6	−0.179132	−0.0895658	+	0.995981i	$0.528548\pi$
$948$	0	0
$949$	−1.04192e7	−0.375551
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.18359e7	1.84884	0.924418	−	0.381380i	$-0.124551\pi$
$953$	5.18359e7	1.84884	0.924418	+	0.381380i	$0.124551\pi$
$954$	0	0
$955$	−1.54265e7	−0.547341
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.24913e6	−0.149195
$960$	0	0
$961$	4.07430e7	1.42313
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.14551e6	−0.177873
$966$	0	0
$967$	−1.92627e7	−0.662446	−0.331223	−	0.943553i	$-0.607461\pi$
$967$	−1.92627e7	−0.662446	−0.331223	+	0.943553i	$0.607461\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.97229e6	0.237316	0.118658	−	0.992935i	$-0.462141\pi$
$971$	6.97229e6	0.237316	0.118658	+	0.992935i	$0.462141\pi$
$972$	0	0
$973$	−6.45426e6	−0.218557
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.65588e7	−1.56051	−0.780253	−	0.625465i	$-0.784910\pi$
$977$	−4.65588e7	−1.56051	−0.780253	+	0.625465i	$0.784910\pi$
$978$	0	0
$979$	5.32686e7	1.77629
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.10229e7	1.35408	0.677038	−	0.735948i	$-0.263263\pi$
$983$	4.10229e7	1.35408	0.677038	+	0.735948i	$0.263263\pi$
$984$	0	0
$985$	4.25397e6	0.139702
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.80795e7	1.56304
$990$	0	0
$991$	−3.31717e7	−1.07296	−0.536480	−	0.843913i	$-0.680247\pi$
$991$	−3.31717e7	−1.07296	−0.536480	+	0.843913i	$0.680247\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.22334e6	0.231302
$996$	0	0
$997$	−5.20233e7	−1.65753	−0.828763	−	0.559600i	$-0.810955\pi$
$997$	−5.20233e7	−1.65753	−0.828763	+	0.559600i	$0.810955\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.6.a.k.1.2	✓	2
3.2	odd	2		504.6.a.u.1.1	yes	2
4.3	odd	2		1008.6.a.bg.1.2		2
12.11	even	2		1008.6.a.bv.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.6.a.k.1.2	✓	2	1.1	even	1	trivial
504.6.a.u.1.1	yes	2	3.2	odd	2
1008.6.a.bg.1.2		2	4.3	odd	2
1008.6.a.bv.1.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+23.7738 q^{5} +49.0000 q^{7} +O(q^{10})\)	\(q+23.7738 q^{5} +49.0000 q^{7} -590.869 q^{11} +505.095 q^{13} -517.964 q^{17} +932.905 q^{19} +3550.73 q^{23} -2559.81 q^{25} -5395.48 q^{29} -8329.00 q^{31} +1164.92 q^{35} -4936.43 q^{37} -5880.27 q^{41} +13540.8 q^{43} +7357.78 q^{47} +2401.00 q^{49} +3341.88 q^{53} -14047.2 q^{55} -42675.2 q^{59} +51327.6 q^{61} +12008.0 q^{65} -33755.0 q^{67} +17848.0 q^{71} -20628.2 q^{73} -28952.6 q^{77} +99202.0 q^{79} -61202.1 q^{83} -12314.0 q^{85} -90153.0 q^{89} +24749.7 q^{91} +22178.7 q^{95} -155739. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 76 q^{5} + 98 q^{7}+O(q^{10}) \) 2 * q - 76 * q^5 + 98 * q^7	\( 2 q - 76 q^{5} + 98 q^{7} - 564 q^{11} + 516 q^{13} + 76 q^{17} + 2360 q^{19} + 2036 q^{23} + 4270 q^{25} - 8320 q^{29} - 6280 q^{31} - 3724 q^{35} - 2460 q^{37} + 14308 q^{41} + 23128 q^{43} - 12712 q^{47} + 4802 q^{49} + 9896 q^{53} - 16728 q^{55} - 60888 q^{59} + 16172 q^{61} + 10920 q^{65} - 62568 q^{67} + 732 q^{71} + 1244 q^{73} - 27636 q^{77} + 11600 q^{79} - 132288 q^{83} - 71576 q^{85} - 93452 q^{89} + 25284 q^{91} - 120208 q^{95} - 272932 q^{97}+O(q^{100}) \) 2 * q - 76 * q^5 + 98 * q^7 - 564 * q^11 + 516 * q^13 + 76 * q^17 + 2360 * q^19 + 2036 * q^23 + 4270 * q^25 - 8320 * q^29 - 6280 * q^31 - 3724 * q^35 - 2460 * q^37 + 14308 * q^41 + 23128 * q^43 - 12712 * q^47 + 4802 * q^49 + 9896 * q^53 - 16728 * q^55 - 60888 * q^59 + 16172 * q^61 + 10920 * q^65 - 62568 * q^67 + 732 * q^71 + 1244 * q^73 - 27636 * q^77 + 11600 * q^79 - 132288 * q^83 - 71576 * q^85 - 93452 * q^89 + 25284 * q^91 - 120208 * q^95 - 272932 * q^97

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