LMFDB - Embedded newform 720.6.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,6,Mod(1,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, 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("720.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	720.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:	$115.476350265$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{145}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 36 $ x^2 - x - 36
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.52080$ of defining polynomial
Character	$\chi$	$=$	720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+25.0000 q^{5} -160.416 q^{7} +O(q^{10})$	$q+25.0000 q^{5} -160.416 q^{7} -279.584 q^{11} -541.664 q^{13} +777.334 q^{17} +2682.66 q^{19} +3694.48 q^{23} +625.000 q^{25} +8356.62 q^{29} +262.849 q^{31} -4010.40 q^{35} -14949.9 q^{37} -7988.28 q^{41} -5133.38 q^{43} -10567.8 q^{47} +8926.28 q^{49} +21069.4 q^{53} -6989.60 q^{55} -25699.6 q^{59} +8195.14 q^{61} -13541.6 q^{65} -51928.9 q^{67} -23689.2 q^{71} +20933.5 q^{73} +44849.7 q^{77} +39279.7 q^{79} -35911.1 q^{83} +19433.4 q^{85} -94074.7 q^{89} +86891.5 q^{91} +67066.4 q^{95} -12249.9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 50 q^{5} - 80 q^{7}+O(q^{10}) $ 2 * q + 50 * q^5 - 80 * q^7	$ 2 q + 50 q^{5} - 80 q^{7} - 800 q^{11} - 120 q^{13} + 1940 q^{17} + 1512 q^{19} + 1320 q^{23} + 1250 q^{25} + 1300 q^{29} + 5824 q^{31} - 2000 q^{35} - 12560 q^{37} + 400 q^{41} - 25680 q^{43} - 18920 q^{47} - 1414 q^{49} + 49460 q^{53} - 20000 q^{55} - 63200 q^{59} - 49116 q^{61} - 3000 q^{65} - 6080 q^{67} - 65200 q^{71} + 97740 q^{73} + 3000 q^{77} + 46288 q^{79} + 57360 q^{83} + 48500 q^{85} - 87000 q^{89} + 120800 q^{91} + 37800 q^{95} + 10180 q^{97}+O(q^{100}) $ 2 * q + 50 * q^5 - 80 * q^7 - 800 * q^11 - 120 * q^13 + 1940 * q^17 + 1512 * q^19 + 1320 * q^23 + 1250 * q^25 + 1300 * q^29 + 5824 * q^31 - 2000 * q^35 - 12560 * q^37 + 400 * q^41 - 25680 * q^43 - 18920 * q^47 - 1414 * q^49 + 49460 * q^53 - 20000 * q^55 - 63200 * q^59 - 49116 * q^61 - 3000 * q^65 - 6080 * q^67 - 65200 * q^71 + 97740 * q^73 + 3000 * q^77 + 46288 * q^79 + 57360 * q^83 + 48500 * q^85 - 87000 * q^89 + 120800 * q^91 + 37800 * q^95 + 10180 * 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
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−160.416	−1.23738	−0.618689	−	0.785636i	$-0.712336\pi$
$7$	−160.416	−1.23738	−0.618689	+	0.785636i	$0.712336\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−279.584	−0.696676	−0.348338	−	0.937369i	$-0.613254\pi$
$11$	−279.584	−0.696676	−0.348338	+	0.937369i	$0.613254\pi$
$12$	0	0
$13$	−541.664	−0.888938	−0.444469	−	0.895794i	$-0.646608\pi$
$13$	−541.664	−0.888938	−0.444469	+	0.895794i	$0.646608\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	777.334	0.652357	0.326179	−	0.945308i	$-0.394239\pi$
$17$	777.334	0.652357	0.326179	+	0.945308i	$0.394239\pi$
$18$	0	0
$19$	2682.66	1.70483	0.852415	−	0.522867i	$-0.175137\pi$
$19$	2682.66	1.70483	0.852415	+	0.522867i	$0.175137\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3694.48	1.45624	0.728122	−	0.685448i	$-0.240394\pi$
$23$	3694.48	1.45624	0.728122	+	0.685448i	$0.240394\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8356.62	1.84517	0.922584	−	0.385797i	$-0.126074\pi$
$29$	8356.62	1.84517	0.922584	+	0.385797i	$0.126074\pi$
$30$	0	0
$31$	262.849	0.0491250	0.0245625	−	0.999698i	$-0.492181\pi$
$31$	262.849	0.0491250	0.0245625	+	0.999698i	$0.492181\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4010.40	−0.553372
$36$	0	0
$37$	−14949.9	−1.79529	−0.897647	−	0.440716i	$-0.854725\pi$
$37$	−14949.9	−1.79529	−0.897647	+	0.440716i	$0.854725\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7988.28	−0.742154	−0.371077	−	0.928602i	$-0.621011\pi$
$41$	−7988.28	−0.742154	−0.371077	+	0.928602i	$0.621011\pi$
$42$	0	0
$43$	−5133.38	−0.423382	−0.211691	−	0.977337i	$-0.567897\pi$
$43$	−5133.38	−0.423382	−0.211691	+	0.977337i	$0.567897\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10567.8	−0.697816	−0.348908	−	0.937157i	$-0.613447\pi$
$47$	−10567.8	−0.697816	−0.348908	+	0.937157i	$0.613447\pi$
$48$	0	0
$49$	8926.28	0.531105
$50$	0	0
$51$	0	0
$52$	0	0
$53$	21069.4	1.03029	0.515147	−	0.857102i	$-0.327737\pi$
$53$	21069.4	1.03029	0.515147	+	0.857102i	$0.327737\pi$
$54$	0	0
$55$	−6989.60	−0.311563
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−25699.6	−0.961162	−0.480581	−	0.876950i	$-0.659574\pi$
$59$	−25699.6	−0.961162	−0.480581	+	0.876950i	$0.659574\pi$
$60$	0	0
$61$	8195.14	0.281989	0.140994	−	0.990010i	$-0.454970\pi$
$61$	8195.14	0.281989	0.140994	+	0.990010i	$0.454970\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−13541.6	−0.397545
$66$	0	0
$67$	−51928.9	−1.41326	−0.706630	−	0.707584i	$-0.749785\pi$
$67$	−51928.9	−1.41326	−0.706630	+	0.707584i	$0.749785\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−23689.2	−0.557705	−0.278853	−	0.960334i	$-0.589954\pi$
$71$	−23689.2	−0.557705	−0.278853	+	0.960334i	$0.589954\pi$
$72$	0	0
$73$	20933.5	0.459764	0.229882	−	0.973219i	$-0.426166\pi$
$73$	20933.5	0.459764	0.229882	+	0.973219i	$0.426166\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	44849.7	0.862051
$78$	0	0
$79$	39279.7	0.708110	0.354055	−	0.935225i	$-0.384803\pi$
$79$	39279.7	0.708110	0.354055	+	0.935225i	$0.384803\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−35911.1	−0.572181	−0.286091	−	0.958203i	$-0.592356\pi$
$83$	−35911.1	−0.572181	−0.286091	+	0.958203i	$0.592356\pi$
$84$	0	0
$85$	19433.4	0.291743
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−94074.7	−1.25892	−0.629460	−	0.777033i	$-0.716724\pi$
$89$	−94074.7	−1.25892	−0.629460	+	0.777033i	$0.716724\pi$
$90$	0	0
$91$	86891.5	1.09995
$92$	0	0
$93$	0	0
$94$	0	0
$95$	67066.4	0.762423
$96$	0	0
$97$	−12249.9	−0.132191	−0.0660957	−	0.997813i	$-0.521054\pi$
$97$	−12249.9	−0.132191	−0.0660957	+	0.997813i	$0.521054\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−159360.	−1.55445	−0.777223	−	0.629226i	$-0.783372\pi$
$101$	−159360.	−1.55445	−0.777223	+	0.629226i	$0.783372\pi$
$102$	0	0
$103$	−74393.5	−0.690942	−0.345471	−	0.938429i	$-0.612281\pi$
$103$	−74393.5	−0.690942	−0.345471	+	0.938429i	$0.612281\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−170203.	−1.43717	−0.718585	−	0.695439i	$-0.755210\pi$
$107$	−170203.	−1.43717	−0.718585	+	0.695439i	$0.755210\pi$
$108$	0	0
$109$	54355.6	0.438206	0.219103	−	0.975702i	$-0.429687\pi$
$109$	54355.6	0.438206	0.219103	+	0.975702i	$0.429687\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9340.66	−0.0688148	−0.0344074	−	0.999408i	$-0.510954\pi$
$113$	−9340.66	−0.0688148	−0.0344074	+	0.999408i	$0.510954\pi$
$114$	0	0
$115$	92362.0	0.651252
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−124697.	−0.807213
$120$	0	0
$121$	−82883.8	−0.514643
$122$	0	0
$123$	0	0
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	22872.4	0.125835	0.0629176	−	0.998019i	$-0.479959\pi$
$127$	22872.4	0.125835	0.0629176	+	0.998019i	$0.479959\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	345444.	1.75873	0.879366	−	0.476146i	$-0.157967\pi$
$131$	345444.	1.75873	0.879366	+	0.476146i	$0.157967\pi$
$132$	0	0
$133$	−430341.	−2.10952
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−170270.	−0.775064	−0.387532	−	0.921856i	$-0.626672\pi$
$137$	−170270.	−0.775064	−0.387532	+	0.921856i	$0.626672\pi$
$138$	0	0
$139$	268230.	1.17753	0.588763	−	0.808305i	$-0.299615\pi$
$139$	268230.	1.17753	0.588763	+	0.808305i	$0.299615\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	151441.	0.619301
$144$	0	0
$145$	208916.	0.825184
$146$	0	0
$147$	0	0
$148$	0	0
$149$	293424.	1.08276	0.541378	−	0.840779i	$-0.317903\pi$
$149$	293424.	1.08276	0.541378	+	0.840779i	$0.317903\pi$
$150$	0	0
$151$	−365971.	−1.30618	−0.653091	−	0.757279i	$-0.726528\pi$
$151$	−365971.	−1.30618	−0.653091	+	0.757279i	$0.726528\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6571.23	0.0219694
$156$	0	0
$157$	503049.	1.62878	0.814388	−	0.580320i	$-0.197073\pi$
$157$	503049.	1.62878	0.814388	+	0.580320i	$0.197073\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−592654.	−1.80192
$162$	0	0
$163$	−181660.	−0.535538	−0.267769	−	0.963483i	$-0.586286\pi$
$163$	−181660.	−0.535538	−0.267769	+	0.963483i	$0.586286\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−440148.	−1.22126	−0.610629	−	0.791917i	$-0.709083\pi$
$167$	−440148.	−1.22126	−0.610629	+	0.791917i	$0.709083\pi$
$168$	0	0
$169$	−77893.3	−0.209789
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−552730.	−1.40410	−0.702049	−	0.712129i	$-0.747731\pi$
$173$	−552730.	−1.40410	−0.702049	+	0.712129i	$0.747731\pi$
$174$	0	0
$175$	−100260.	−0.247476
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−190220.	−0.443735	−0.221867	−	0.975077i	$-0.571215\pi$
$179$	−190220.	−0.443735	−0.221867	+	0.975077i	$0.571215\pi$
$180$	0	0
$181$	113139.	0.256694	0.128347	−	0.991729i	$-0.459033\pi$
$181$	113139.	0.256694	0.128347	+	0.991729i	$0.459033\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−373749.	−0.802880
$186$	0	0
$187$	−217330.	−0.454482
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−694078.	−1.37665	−0.688327	−	0.725401i	$-0.741654\pi$
$191$	−694078.	−1.37665	−0.688327	+	0.725401i	$0.741654\pi$
$192$	0	0
$193$	−51802.2	−0.100105	−0.0500524	−	0.998747i	$-0.515939\pi$
$193$	−51802.2	−0.100105	−0.0500524	+	0.998747i	$0.515939\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	217963.	0.400145	0.200073	−	0.979781i	$-0.435882\pi$
$197$	217963.	0.400145	0.200073	+	0.979781i	$0.435882\pi$
$198$	0	0
$199$	385135.	0.689414	0.344707	−	0.938710i	$-0.387978\pi$
$199$	385135.	0.689414	0.344707	+	0.938710i	$0.387978\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.34054e6	−2.28317
$204$	0	0
$205$	−199707.	−0.331901
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−750028.	−1.18771
$210$	0	0
$211$	−300290.	−0.464339	−0.232170	−	0.972675i	$-0.574582\pi$
$211$	−300290.	−0.464339	−0.232170	+	0.972675i	$0.574582\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−128334.	−0.189342
$216$	0	0
$217$	−42165.2	−0.0607862
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−421054.	−0.579905
$222$	0	0
$223$	409580.	0.551539	0.275769	−	0.961224i	$-0.411067\pi$
$223$	409580.	0.551539	0.275769	+	0.961224i	$0.411067\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−386331.	−0.497617	−0.248808	−	0.968553i	$-0.580039\pi$
$227$	−386331.	−0.497617	−0.248808	+	0.968553i	$0.580039\pi$
$228$	0	0
$229$	360206.	0.453902	0.226951	−	0.973906i	$-0.427124\pi$
$229$	360206.	0.453902	0.226951	+	0.973906i	$0.427124\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	376766.	0.454655	0.227327	−	0.973818i	$-0.427001\pi$
$233$	376766.	0.454655	0.227327	+	0.973818i	$0.427001\pi$
$234$	0	0
$235$	−264196.	−0.312073
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−416917.	−0.472123	−0.236061	−	0.971738i	$-0.575857\pi$
$239$	−416917.	−0.472123	−0.236061	+	0.971738i	$0.575857\pi$
$240$	0	0
$241$	−1.15082e6	−1.27634	−0.638168	−	0.769897i	$-0.720308\pi$
$241$	−1.15082e6	−1.27634	−0.638168	+	0.769897i	$0.720308\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	223157.	0.237517
$246$	0	0
$247$	−1.45310e6	−1.51549
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−642245.	−0.643452	−0.321726	−	0.946833i	$-0.604263\pi$
$251$	−642245.	−0.643452	−0.321726	+	0.946833i	$0.604263\pi$
$252$	0	0
$253$	−1.03292e6	−1.01453
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−126474.	−0.119446	−0.0597228	−	0.998215i	$-0.519022\pi$
$257$	−126474.	−0.119446	−0.0597228	+	0.998215i	$0.519022\pi$
$258$	0	0
$259$	2.39821e6	2.22146
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−366879.	−0.327064	−0.163532	−	0.986538i	$-0.552289\pi$
$263$	−366879.	−0.327064	−0.163532	+	0.986538i	$0.552289\pi$
$264$	0	0
$265$	526734.	0.460762
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.21165e6	−1.02093	−0.510465	−	0.859899i	$-0.670527\pi$
$269$	−1.21165e6	−1.02093	−0.510465	+	0.859899i	$0.670527\pi$
$270$	0	0
$271$	1.79322e6	1.48324	0.741619	−	0.670821i	$-0.234058\pi$
$271$	1.79322e6	1.48324	0.741619	+	0.670821i	$0.234058\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−174740.	−0.139335
$276$	0	0
$277$	1.46441e6	1.14674	0.573369	−	0.819297i	$-0.305636\pi$
$277$	1.46441e6	1.14674	0.573369	+	0.819297i	$0.305636\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.12044e6	1.60199	0.800997	−	0.598669i	$-0.204304\pi$
$281$	2.12044e6	1.60199	0.800997	+	0.598669i	$0.204304\pi$
$282$	0	0
$283$	−454281.	−0.337177	−0.168589	−	0.985687i	$-0.553921\pi$
$283$	−454281.	−0.337177	−0.168589	+	0.985687i	$0.553921\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.28145e6	0.918325
$288$	0	0
$289$	−815608.	−0.574430
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.93287e6	−1.31533	−0.657664	−	0.753312i	$-0.728455\pi$
$293$	−1.93287e6	−1.31533	−0.657664	+	0.753312i	$0.728455\pi$
$294$	0	0
$295$	−642490.	−0.429845
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.00117e6	−1.29451
$300$	0	0
$301$	823476.	0.523883
$302$	0	0
$303$	0	0
$304$	0	0
$305$	204878.	0.126109
$306$	0	0
$307$	−224205.	−0.135768	−0.0678842	−	0.997693i	$-0.521625\pi$
$307$	−224205.	−0.135768	−0.0678842	+	0.997693i	$0.521625\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	592091.	0.347126	0.173563	−	0.984823i	$-0.444472\pi$
$311$	592091.	0.347126	0.173563	+	0.984823i	$0.444472\pi$
$312$	0	0
$313$	1.99079e6	1.14859	0.574295	−	0.818648i	$-0.305276\pi$
$313$	1.99079e6	1.14859	0.574295	+	0.818648i	$0.305276\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.89884e6	−1.62023	−0.810113	−	0.586274i	$-0.800594\pi$
$317$	−2.89884e6	−1.62023	−0.810113	+	0.586274i	$0.800594\pi$
$318$	0	0
$319$	−2.33638e6	−1.28548
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.08532e6	1.11216
$324$	0	0
$325$	−338540.	−0.177788
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.69525e6	0.863462
$330$	0	0
$331$	−416722.	−0.209063	−0.104531	−	0.994522i	$-0.533334\pi$
$331$	−416722.	−0.209063	−0.104531	+	0.994522i	$0.533334\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.29822e6	−0.632029
$336$	0	0
$337$	212393.	0.101874	0.0509371	−	0.998702i	$-0.483779\pi$
$337$	212393.	0.101874	0.0509371	+	0.998702i	$0.483779\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−73488.4	−0.0342242
$342$	0	0
$343$	1.26419e6	0.580201
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.58972e6	−0.708755	−0.354377	−	0.935103i	$-0.615307\pi$
$347$	−1.58972e6	−0.708755	−0.354377	+	0.935103i	$0.615307\pi$
$348$	0	0
$349$	−3.98771e6	−1.75251	−0.876254	−	0.481850i	$-0.839965\pi$
$349$	−3.98771e6	−1.75251	−0.876254	+	0.481850i	$0.839965\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−585594.	−0.250127	−0.125063	−	0.992149i	$-0.539913\pi$
$353$	−585594.	−0.250127	−0.125063	+	0.992149i	$0.539913\pi$
$354$	0	0
$355$	−592231.	−0.249413
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.28470e6	1.75463	0.877313	−	0.479918i	$-0.159334\pi$
$359$	4.28470e6	1.75463	0.877313	+	0.479918i	$0.159334\pi$
$360$	0	0
$361$	4.72054e6	1.90644
$362$	0	0
$363$	0	0
$364$	0	0
$365$	523338.	0.205613
$366$	0	0
$367$	524307.	0.203198	0.101599	−	0.994825i	$-0.467604\pi$
$367$	524307.	0.203198	0.101599	+	0.994825i	$0.467604\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.37986e6	−1.27486
$372$	0	0
$373$	−1.16607e6	−0.433964	−0.216982	−	0.976176i	$-0.569621\pi$
$373$	−1.16607e6	−0.433964	−0.216982	+	0.976176i	$0.569621\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.52648e6	−1.64024
$378$	0	0
$379$	−2.48592e6	−0.888973	−0.444487	−	0.895786i	$-0.646614\pi$
$379$	−2.48592e6	−0.888973	−0.444487	+	0.895786i	$0.646614\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.52361e6	−0.530732	−0.265366	−	0.964148i	$-0.585493\pi$
$383$	−1.52361e6	−0.530732	−0.265366	+	0.964148i	$0.585493\pi$
$384$	0	0
$385$	1.12124e6	0.385521
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.59972e6	−1.54120	−0.770598	−	0.637322i	$-0.780042\pi$
$389$	−4.59972e6	−1.54120	−0.770598	+	0.637322i	$0.780042\pi$
$390$	0	0
$391$	2.87185e6	0.949991
$392$	0	0
$393$	0	0
$394$	0	0
$395$	981993.	0.316677
$396$	0	0
$397$	3.54374e6	1.12846	0.564229	−	0.825618i	$-0.309173\pi$
$397$	3.54374e6	1.12846	0.564229	+	0.825618i	$0.309173\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.01833e6	−0.316248	−0.158124	−	0.987419i	$-0.550545\pi$
$401$	−1.01833e6	−0.316248	−0.158124	+	0.987419i	$0.550545\pi$
$402$	0	0
$403$	−142376.	−0.0436691
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.17977e6	1.25074
$408$	0	0
$409$	−6.32925e6	−1.87087	−0.935436	−	0.353497i	$-0.884992\pi$
$409$	−6.32925e6	−1.87087	−0.935436	+	0.353497i	$0.884992\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.12263e6	1.18932
$414$	0	0
$415$	−897778.	−0.255887
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.41445e6	0.671865	0.335933	−	0.941886i	$-0.390949\pi$
$419$	2.41445e6	0.671865	0.335933	+	0.941886i	$0.390949\pi$
$420$	0	0
$421$	−3.96045e6	−1.08903	−0.544514	−	0.838752i	$-0.683286\pi$
$421$	−3.96045e6	−1.08903	−0.544514	+	0.838752i	$0.683286\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	485834.	0.130471
$426$	0	0
$427$	−1.31463e6	−0.348927
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.48955e6	−0.386245	−0.193123	−	0.981175i	$-0.561861\pi$
$431$	−1.48955e6	−0.386245	−0.193123	+	0.981175i	$0.561861\pi$
$432$	0	0
$433$	−5.66221e6	−1.45133	−0.725665	−	0.688049i	$-0.758468\pi$
$433$	−5.66221e6	−1.45133	−0.725665	+	0.688049i	$0.758468\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.91102e6	2.48265
$438$	0	0
$439$	−8764.09	−0.00217043	−0.00108521	−	0.999999i	$-0.500345\pi$
$439$	−8764.09	−0.00217043	−0.00108521	+	0.999999i	$0.500345\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	962960.	0.233130	0.116565	−	0.993183i	$-0.462812\pi$
$443$	962960.	0.233130	0.116565	+	0.993183i	$0.462812\pi$
$444$	0	0
$445$	−2.35187e6	−0.563006
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.81934e6	0.894071	0.447035	−	0.894516i	$-0.352480\pi$
$449$	3.81934e6	0.894071	0.447035	+	0.894516i	$0.352480\pi$
$450$	0	0
$451$	2.23340e6	0.517040
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.17229e6	0.491914
$456$	0	0
$457$	−2.28095e6	−0.510887	−0.255443	−	0.966824i	$-0.582221\pi$
$457$	−2.28095e6	−0.510887	−0.255443	+	0.966824i	$0.582221\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.56688e6	0.562539	0.281270	−	0.959629i	$-0.409245\pi$
$461$	2.56688e6	0.562539	0.281270	+	0.959629i	$0.409245\pi$
$462$	0	0
$463$	2.16948e6	0.470330	0.235165	−	0.971955i	$-0.424437\pi$
$463$	2.16948e6	0.470330	0.235165	+	0.971955i	$0.424437\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.84499e6	0.391472	0.195736	−	0.980657i	$-0.437290\pi$
$467$	1.84499e6	0.391472	0.195736	+	0.980657i	$0.437290\pi$
$468$	0	0
$469$	8.33022e6	1.74874
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.43521e6	0.294960
$474$	0	0
$475$	1.67666e6	0.340966
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.88467e6	−0.574457	−0.287229	−	0.957862i	$-0.592734\pi$
$479$	−2.88467e6	−0.574457	−0.287229	+	0.957862i	$0.592734\pi$
$480$	0	0
$481$	8.09785e6	1.59590
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−306247.	−0.0591178
$486$	0	0
$487$	3.37928e6	0.645657	0.322829	−	0.946457i	$-0.395366\pi$
$487$	3.37928e6	0.645657	0.322829	+	0.946457i	$0.395366\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−128325.	−0.0240218	−0.0120109	−	0.999928i	$-0.503823\pi$
$491$	−128325.	−0.0240218	−0.0120109	+	0.999928i	$0.503823\pi$
$492$	0	0
$493$	6.49589e6	1.20371
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.80013e6	0.690093
$498$	0	0
$499$	2.15334e6	0.387135	0.193567	−	0.981087i	$-0.437994\pi$
$499$	2.15334e6	0.387135	0.193567	+	0.981087i	$0.437994\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.42765e6	1.30898	0.654488	−	0.756072i	$-0.272884\pi$
$503$	7.42765e6	1.30898	0.654488	+	0.756072i	$0.272884\pi$
$504$	0	0
$505$	−3.98400e6	−0.695169
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.01756e6	1.37166	0.685832	−	0.727760i	$-0.259439\pi$
$509$	8.01756e6	1.37166	0.685832	+	0.727760i	$0.259439\pi$
$510$	0	0
$511$	−3.35807e6	−0.568902
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.85984e6	−0.308999
$516$	0	0
$517$	2.95460e6	0.486152
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.24266e6	−0.361967	−0.180984	−	0.983486i	$-0.557928\pi$
$521$	−2.24266e6	−0.361967	−0.180984	+	0.983486i	$0.557928\pi$
$522$	0	0
$523$	−7.70554e6	−1.23182	−0.615912	−	0.787815i	$-0.711213\pi$
$523$	−7.70554e6	−1.23182	−0.615912	+	0.787815i	$0.711213\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	204322.	0.0320470
$528$	0	0
$529$	7.21285e6	1.12064
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.32696e6	0.659729
$534$	0	0
$535$	−4.25508e6	−0.642722
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.49564e6	−0.370008
$540$	0	0
$541$	−7.64969e6	−1.12370	−0.561850	−	0.827239i	$-0.689910\pi$
$541$	−7.64969e6	−1.12370	−0.561850	+	0.827239i	$0.689910\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.35889e6	0.195972
$546$	0	0
$547$	−2.76870e6	−0.395647	−0.197823	−	0.980238i	$-0.563387\pi$
$547$	−2.76870e6	−0.395647	−0.197823	+	0.980238i	$0.563387\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.24179e7	3.14569
$552$	0	0
$553$	−6.30110e6	−0.876200
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.21255e6	0.165600	0.0828002	−	0.996566i	$-0.473614\pi$
$557$	1.21255e6	0.165600	0.0828002	+	0.996566i	$0.473614\pi$
$558$	0	0
$559$	2.78057e6	0.376360
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.55586e6	−0.871683	−0.435841	−	0.900024i	$-0.643549\pi$
$563$	−6.55586e6	−0.871683	−0.435841	+	0.900024i	$0.643549\pi$
$564$	0	0
$565$	−233517.	−0.0307749
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.97209e6	1.16175	0.580875	−	0.813993i	$-0.302710\pi$
$569$	8.97209e6	1.16175	0.580875	+	0.813993i	$0.302710\pi$
$570$	0	0
$571$	5.46422e6	0.701354	0.350677	−	0.936496i	$-0.385951\pi$
$571$	5.46422e6	0.701354	0.350677	+	0.936496i	$0.385951\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.30905e6	0.291249
$576$	0	0
$577$	−6.35363e6	−0.794478	−0.397239	−	0.917715i	$-0.630032\pi$
$577$	−6.35363e6	−0.794478	−0.397239	+	0.917715i	$0.630032\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.76072e6	0.708005
$582$	0	0
$583$	−5.89066e6	−0.717781
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.33705e7	−1.60159	−0.800795	−	0.598938i	$-0.795590\pi$
$587$	−1.33705e7	−1.60159	−0.800795	+	0.598938i	$0.795590\pi$
$588$	0	0
$589$	705134.	0.0837497
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.42274e6	−0.166146	−0.0830728	−	0.996543i	$-0.526473\pi$
$593$	−1.42274e6	−0.166146	−0.0830728	+	0.996543i	$0.526473\pi$
$594$	0	0
$595$	−3.11742e6	−0.360997
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.91805e6	0.673925	0.336962	−	0.941518i	$-0.390601\pi$
$599$	5.91805e6	0.673925	0.336962	+	0.941518i	$0.390601\pi$
$600$	0	0
$601$	1.48610e7	1.67827	0.839135	−	0.543924i	$-0.183062\pi$
$601$	1.48610e7	1.67827	0.839135	+	0.543924i	$0.183062\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.07209e6	−0.230155
$606$	0	0
$607$	1.16289e7	1.28105	0.640526	−	0.767937i	$-0.278717\pi$
$607$	1.16289e7	1.28105	0.640526	+	0.767937i	$0.278717\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.72421e6	0.620315
$612$	0	0
$613$	−1.06616e6	−0.114596	−0.0572980	−	0.998357i	$-0.518249\pi$
$613$	−1.06616e6	−0.114596	−0.0572980	+	0.998357i	$0.518249\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	663411.	0.0701568	0.0350784	−	0.999385i	$-0.488832\pi$
$617$	663411.	0.0701568	0.0350784	+	0.999385i	$0.488832\pi$
$618$	0	0
$619$	−2.53444e6	−0.265861	−0.132930	−	0.991125i	$-0.542439\pi$
$619$	−2.53444e6	−0.265861	−0.132930	+	0.991125i	$0.542439\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.50911e7	1.55776
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.16211e7	−1.17117
$630$	0	0
$631$	−102057.	−0.0102040	−0.00510199	−	0.999987i	$-0.501624\pi$
$631$	−102057.	−0.0102040	−0.00510199	+	0.999987i	$0.501624\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	571809.	0.0562752
$636$	0	0
$637$	−4.83504e6	−0.472119
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.67995e6	−0.353750	−0.176875	−	0.984233i	$-0.556599\pi$
$641$	−3.67995e6	−0.353750	−0.176875	+	0.984233i	$0.556599\pi$
$642$	0	0
$643$	−8.95581e6	−0.854235	−0.427118	−	0.904196i	$-0.640471\pi$
$643$	−8.95581e6	−0.854235	−0.427118	+	0.904196i	$0.640471\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.36816e6	−0.128492	−0.0642458	−	0.997934i	$-0.520464\pi$
$647$	−1.36816e6	−0.128492	−0.0642458	+	0.997934i	$0.520464\pi$
$648$	0	0
$649$	7.18520e6	0.669618
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.23673e6	0.297046	0.148523	−	0.988909i	$-0.452548\pi$
$653$	3.23673e6	0.297046	0.148523	+	0.988909i	$0.452548\pi$
$654$	0	0
$655$	8.63611e6	0.786529
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5.17492e6	−0.464184	−0.232092	−	0.972694i	$-0.574557\pi$
$659$	−5.17492e6	−0.464184	−0.232092	+	0.972694i	$0.574557\pi$
$660$	0	0
$661$	−1.07060e7	−0.953064	−0.476532	−	0.879157i	$-0.658106\pi$
$661$	−1.07060e7	−0.953064	−0.476532	+	0.879157i	$0.658106\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.07585e7	−0.943405
$666$	0	0
$667$	3.08734e7	2.68701
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.29123e6	−0.196455
$672$	0	0
$673$	−2.15653e7	−1.83535	−0.917673	−	0.397338i	$-0.869934\pi$
$673$	−2.15653e7	−1.83535	−0.917673	+	0.397338i	$0.869934\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.09762e6	0.511315	0.255658	−	0.966767i	$-0.417708\pi$
$677$	6.09762e6	0.511315	0.255658	+	0.966767i	$0.417708\pi$
$678$	0	0
$679$	1.96508e6	0.163571
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2.04240e7	−1.67528	−0.837642	−	0.546220i	$-0.816066\pi$
$683$	−2.04240e7	−1.67528	−0.837642	+	0.546220i	$0.816066\pi$
$684$	0	0
$685$	−4.25676e6	−0.346619
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.14125e7	−0.915868
$690$	0	0
$691$	2.91819e6	0.232497	0.116249	−	0.993220i	$-0.462913\pi$
$691$	2.91819e6	0.232497	0.116249	+	0.993220i	$0.462913\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.70576e6	0.526606
$696$	0	0
$697$	−6.20957e6	−0.484150
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.78394e6	−0.675141	−0.337570	−	0.941300i	$-0.609605\pi$
$701$	−8.78394e6	−0.675141	−0.337570	+	0.941300i	$0.609605\pi$
$702$	0	0
$703$	−4.01056e7	−3.06067
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.55639e7	1.92344
$708$	0	0
$709$	1.66475e7	1.24375	0.621875	−	0.783116i	$-0.286371\pi$
$709$	1.66475e7	1.24375	0.621875	+	0.783116i	$0.286371\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	971092.	0.0715379
$714$	0	0
$715$	3.78601e6	0.276960
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.55062e7	1.84003	0.920013	−	0.391888i	$-0.128178\pi$
$719$	2.55062e7	1.84003	0.920013	+	0.391888i	$0.128178\pi$
$720$	0	0
$721$	1.19339e7	0.854957
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.22289e6	0.369033
$726$	0	0
$727$	−1.36506e7	−0.957890	−0.478945	−	0.877845i	$-0.658981\pi$
$727$	−1.36506e7	−0.957890	−0.478945	+	0.877845i	$0.658981\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3.99035e6	−0.276196
$732$	0	0
$733$	−1.38922e7	−0.955018	−0.477509	−	0.878627i	$-0.658460\pi$
$733$	−1.38922e7	−0.955018	−0.477509	+	0.878627i	$0.658460\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.45185e7	0.984584
$738$	0	0
$739$	−837888.	−0.0564384	−0.0282192	−	0.999602i	$-0.508984\pi$
$739$	−837888.	−0.0564384	−0.0282192	+	0.999602i	$0.508984\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.07258e6	0.536464	0.268232	−	0.963354i	$-0.413561\pi$
$743$	8.07258e6	0.536464	0.268232	+	0.963354i	$0.413561\pi$
$744$	0	0
$745$	7.33561e6	0.484223
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.73033e7	1.77832
$750$	0	0
$751$	1.74892e7	1.13154	0.565770	−	0.824563i	$-0.308579\pi$
$751$	1.74892e7	1.13154	0.565770	+	0.824563i	$0.308579\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9.14927e6	−0.584143
$756$	0	0
$757$	2.19231e7	1.39047	0.695236	−	0.718781i	$-0.255300\pi$
$757$	2.19231e7	1.39047	0.695236	+	0.718781i	$0.255300\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.18585e7	0.742278	0.371139	−	0.928577i	$-0.378967\pi$
$761$	1.18585e7	0.742278	0.371139	+	0.928577i	$0.378967\pi$
$762$	0	0
$763$	−8.71951e6	−0.542227
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.39206e7	0.854413
$768$	0	0
$769$	−2.94181e6	−0.179390	−0.0896950	−	0.995969i	$-0.528589\pi$
$769$	−2.94181e6	−0.179390	−0.0896950	+	0.995969i	$0.528589\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.01745e7	−1.21438	−0.607188	−	0.794558i	$-0.707703\pi$
$773$	−2.01745e7	−1.21438	−0.607188	+	0.794558i	$0.707703\pi$
$774$	0	0
$775$	164281.	0.00982500
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.14298e7	−1.26525
$780$	0	0
$781$	6.62313e6	0.388540
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.25762e7	0.728411
$786$	0	0
$787$	−8.17269e6	−0.470358	−0.235179	−	0.971952i	$-0.575568\pi$
$787$	−8.17269e6	−0.470358	−0.235179	+	0.971952i	$0.575568\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.49839e6	0.0851499
$792$	0	0
$793$	−4.43901e6	−0.250670
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.66457e7	−0.928234	−0.464117	−	0.885774i	$-0.653628\pi$
$797$	−1.66457e7	−0.928234	−0.464117	+	0.885774i	$0.653628\pi$
$798$	0	0
$799$	−8.21474e6	−0.455226
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.85267e6	−0.320306
$804$	0	0
$805$	−1.48163e7	−0.805845
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7.91242e6	−0.425048	−0.212524	−	0.977156i	$-0.568168\pi$
$809$	−7.91242e6	−0.425048	−0.212524	+	0.977156i	$0.568168\pi$
$810$	0	0
$811$	4.07133e6	0.217362	0.108681	−	0.994077i	$-0.465337\pi$
$811$	4.07133e6	0.217362	0.108681	+	0.994077i	$0.465337\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.54150e6	−0.239500
$816$	0	0
$817$	−1.37711e7	−0.721794
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.86139e7	−0.963782	−0.481891	−	0.876231i	$-0.660050\pi$
$821$	−1.86139e7	−0.963782	−0.481891	+	0.876231i	$0.660050\pi$
$822$	0	0
$823$	3.54074e6	0.182220	0.0911098	−	0.995841i	$-0.470959\pi$
$823$	3.54074e6	0.182220	0.0911098	+	0.995841i	$0.470959\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.76580e7	−1.91467	−0.957335	−	0.288981i	$-0.906684\pi$
$827$	−3.76580e7	−1.91467	−0.957335	+	0.288981i	$0.906684\pi$
$828$	0	0
$829$	1.16650e7	0.589518	0.294759	−	0.955572i	$-0.404761\pi$
$829$	1.16650e7	0.589518	0.294759	+	0.955572i	$0.404761\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.93870e6	0.346470
$834$	0	0
$835$	−1.10037e7	−0.546163
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.07091e7	−1.01568	−0.507839	−	0.861452i	$-0.669556\pi$
$839$	−2.07091e7	−1.01568	−0.507839	+	0.861452i	$0.669556\pi$
$840$	0	0
$841$	4.93220e7	2.40464
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.94733e6	−0.0938207
$846$	0	0
$847$	1.32959e7	0.636808
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.52323e7	−2.61438
$852$	0	0
$853$	−1.94473e7	−0.915137	−0.457569	−	0.889174i	$-0.651280\pi$
$853$	−1.94473e7	−0.915137	−0.457569	+	0.889174i	$0.651280\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.10458e7	−0.978843	−0.489421	−	0.872047i	$-0.662792\pi$
$857$	−2.10458e7	−0.978843	−0.489421	+	0.872047i	$0.662792\pi$
$858$	0	0
$859$	−2.40179e7	−1.11058	−0.555292	−	0.831656i	$-0.687393\pi$
$859$	−2.40179e7	−1.11058	−0.555292	+	0.831656i	$0.687393\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.25620e6	−0.240240	−0.120120	−	0.992759i	$-0.538328\pi$
$863$	−5.25620e6	−0.240240	−0.120120	+	0.992759i	$0.538328\pi$
$864$	0	0
$865$	−1.38182e7	−0.627932
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.09820e7	−0.493323
$870$	0	0
$871$	2.81280e7	1.25630
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.50650e6	−0.110674
$876$	0	0
$877$	−2.99490e6	−0.131487	−0.0657436	−	0.997837i	$-0.520942\pi$
$877$	−2.99490e6	−0.131487	−0.0657436	+	0.997837i	$0.520942\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.19714e7	0.953714	0.476857	−	0.878981i	$-0.341776\pi$
$881$	2.19714e7	0.953714	0.476857	+	0.878981i	$0.341776\pi$
$882$	0	0
$883$	1.05962e7	0.457348	0.228674	−	0.973503i	$-0.426561\pi$
$883$	1.05962e7	0.457348	0.228674	+	0.973503i	$0.426561\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.57778e6	0.0673343	0.0336672	−	0.999433i	$-0.489281\pi$
$887$	1.57778e6	0.0673343	0.0336672	+	0.999433i	$0.489281\pi$
$888$	0	0
$889$	−3.66909e6	−0.155706
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.83498e7	−1.18966
$894$	0	0
$895$	−4.75550e6	−0.198444
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.19653e6	0.0906438
$900$	0	0
$901$	1.63779e7	0.672121
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.82847e6	0.114797
$906$	0	0
$907$	3.03048e7	1.22319	0.611594	−	0.791171i	$-0.290528\pi$
$907$	3.03048e7	1.22319	0.611594	+	0.791171i	$0.290528\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.14226e7	−0.456005	−0.228003	−	0.973661i	$-0.573219\pi$
$911$	−1.14226e7	−0.456005	−0.228003	+	0.973661i	$0.573219\pi$
$912$	0	0
$913$	1.00402e7	0.398625
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.54148e7	−2.17622
$918$	0	0
$919$	4.43959e7	1.73402	0.867010	−	0.498291i	$-0.166039\pi$
$919$	4.43959e7	1.73402	0.867010	+	0.498291i	$0.166039\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.28316e7	0.495766
$924$	0	0
$925$	−9.34372e6	−0.359059
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.73604e7	−1.04012	−0.520059	−	0.854130i	$-0.674090\pi$
$929$	−2.73604e7	−1.04012	−0.520059	+	0.854130i	$0.674090\pi$
$930$	0	0
$931$	2.39461e7	0.905443
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.43326e6	−0.203250
$936$	0	0
$937$	−1.23707e7	−0.460306	−0.230153	−	0.973154i	$-0.573923\pi$
$937$	−1.23707e7	−0.460306	−0.230153	+	0.973154i	$0.573923\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.77596e7	0.653822	0.326911	−	0.945055i	$-0.393992\pi$
$941$	1.77596e7	0.653822	0.326911	+	0.945055i	$0.393992\pi$
$942$	0	0
$943$	−2.95126e7	−1.08076
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.50046e7	−1.63073	−0.815364	−	0.578949i	$-0.803463\pi$
$947$	−4.50046e7	−1.63073	−0.815364	+	0.578949i	$0.803463\pi$
$948$	0	0
$949$	−1.13389e7	−0.408701
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.10110e7	1.46274	0.731372	−	0.681979i	$-0.238881\pi$
$953$	4.10110e7	1.46274	0.731372	+	0.681979i	$0.238881\pi$
$954$	0	0
$955$	−1.73519e7	−0.615658
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.73141e7	0.959048
$960$	0	0
$961$	−2.85601e7	−0.997587
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.29505e6	−0.0447682
$966$	0	0
$967$	−2.23360e7	−0.768137	−0.384069	−	0.923305i	$-0.625477\pi$
$967$	−2.23360e7	−0.768137	−0.384069	+	0.923305i	$0.625477\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.34728e7	−1.13932	−0.569658	−	0.821882i	$-0.692924\pi$
$971$	−3.34728e7	−1.13932	−0.569658	+	0.821882i	$0.692924\pi$
$972$	0	0
$973$	−4.30284e7	−1.45705
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8.52076e6	−0.285589	−0.142795	−	0.989752i	$-0.545609\pi$
$977$	−8.52076e6	−0.285589	−0.142795	+	0.989752i	$0.545609\pi$
$978$	0	0
$979$	2.63018e7	0.877058
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.01155e7	−0.663969	−0.331985	−	0.943285i	$-0.607718\pi$
$983$	−2.01155e7	−0.663969	−0.331985	+	0.943285i	$0.607718\pi$
$984$	0	0
$985$	5.44908e6	0.178950
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.89652e7	−0.616547
$990$	0	0
$991$	−3.24195e7	−1.04863	−0.524315	−	0.851524i	$-0.675679\pi$
$991$	−3.24195e7	−1.04863	−0.524315	+	0.851524i	$0.675679\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.62836e6	0.308315
$996$	0	0
$997$	4.47995e7	1.42737	0.713683	−	0.700469i	$-0.247026\pi$
$997$	4.47995e7	1.42737	0.713683	+	0.700469i	$0.247026\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.6.a.be.1.1		2
3.2	odd	2		720.6.a.y.1.1		2
4.3	odd	2		45.6.a.f.1.2	yes	2
12.11	even	2		45.6.a.d.1.1	✓	2
20.3	even	4		225.6.b.k.199.1		4
20.7	even	4		225.6.b.k.199.4		4
20.19	odd	2		225.6.a.k.1.1		2
60.23	odd	4		225.6.b.j.199.4		4
60.47	odd	4		225.6.b.j.199.1		4
60.59	even	2		225.6.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.6.a.d.1.1	✓	2	12.11	even	2
45.6.a.f.1.2	yes	2	4.3	odd	2
225.6.a.k.1.1		2	20.19	odd	2
225.6.a.r.1.2		2	60.59	even	2
225.6.b.j.199.1		4	60.47	odd	4
225.6.b.j.199.4		4	60.23	odd	4
225.6.b.k.199.1		4	20.3	even	4
225.6.b.k.199.4		4	20.7	even	4
720.6.a.y.1.1		2	3.2	odd	2
720.6.a.be.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.a (trivial)

\(f(q)\)	\(=\)	\(q+25.0000 q^{5} -160.416 q^{7} +O(q^{10})\)	\(q+25.0000 q^{5} -160.416 q^{7} -279.584 q^{11} -541.664 q^{13} +777.334 q^{17} +2682.66 q^{19} +3694.48 q^{23} +625.000 q^{25} +8356.62 q^{29} +262.849 q^{31} -4010.40 q^{35} -14949.9 q^{37} -7988.28 q^{41} -5133.38 q^{43} -10567.8 q^{47} +8926.28 q^{49} +21069.4 q^{53} -6989.60 q^{55} -25699.6 q^{59} +8195.14 q^{61} -13541.6 q^{65} -51928.9 q^{67} -23689.2 q^{71} +20933.5 q^{73} +44849.7 q^{77} +39279.7 q^{79} -35911.1 q^{83} +19433.4 q^{85} -94074.7 q^{89} +86891.5 q^{91} +67066.4 q^{95} -12249.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 50 q^{5} - 80 q^{7}+O(q^{10}) \) 2 * q + 50 * q^5 - 80 * q^7	\( 2 q + 50 q^{5} - 80 q^{7} - 800 q^{11} - 120 q^{13} + 1940 q^{17} + 1512 q^{19} + 1320 q^{23} + 1250 q^{25} + 1300 q^{29} + 5824 q^{31} - 2000 q^{35} - 12560 q^{37} + 400 q^{41} - 25680 q^{43} - 18920 q^{47} - 1414 q^{49} + 49460 q^{53} - 20000 q^{55} - 63200 q^{59} - 49116 q^{61} - 3000 q^{65} - 6080 q^{67} - 65200 q^{71} + 97740 q^{73} + 3000 q^{77} + 46288 q^{79} + 57360 q^{83} + 48500 q^{85} - 87000 q^{89} + 120800 q^{91} + 37800 q^{95} + 10180 q^{97}+O(q^{100}) \) 2 * q + 50 * q^5 - 80 * q^7 - 800 * q^11 - 120 * q^13 + 1940 * q^17 + 1512 * q^19 + 1320 * q^23 + 1250 * q^25 + 1300 * q^29 + 5824 * q^31 - 2000 * q^35 - 12560 * q^37 + 400 * q^41 - 25680 * q^43 - 18920 * q^47 - 1414 * q^49 + 49460 * q^53 - 20000 * q^55 - 63200 * q^59 - 49116 * q^61 - 3000 * q^65 - 6080 * q^67 - 65200 * q^71 + 97740 * q^73 + 3000 * q^77 + 46288 * q^79 + 57360 * q^83 + 48500 * q^85 - 87000 * q^89 + 120800 * q^91 + 37800 * q^95 + 10180 * q^97

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