LMFDB - Embedded newform 720.6.a.bd.1.2

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{409}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 102 $ x^2 - x - 102
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-9.61187$ 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)

$f(q)$	$=$	$q-25.0000 q^{5} +217.790 q^{7} +O(q^{10})$	$q-25.0000 q^{5} +217.790 q^{7} -199.580 q^{11} +599.790 q^{13} -209.050 q^{17} -2835.27 q^{19} -2093.37 q^{23} +625.000 q^{25} -326.840 q^{29} -3115.69 q^{31} -5444.75 q^{35} -3917.29 q^{37} +9314.86 q^{41} +7572.84 q^{43} -22137.8 q^{47} +30625.5 q^{49} -15796.2 q^{53} +4989.50 q^{55} +32229.6 q^{59} -5745.42 q^{61} -14994.7 q^{65} -6491.20 q^{67} +11870.0 q^{71} +23414.0 q^{73} -43466.5 q^{77} -15779.4 q^{79} -101815. q^{83} +5226.25 q^{85} +22865.0 q^{89} +130628. q^{91} +70881.7 q^{95} +113814. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 50 q^{5} + 112 q^{7}+O(q^{10}) $ 2 * q - 50 * q^5 + 112 * q^7	$ 2 q - 50 q^{5} + 112 q^{7} + 248 q^{11} + 876 q^{13} - 2036 q^{17} - 1464 q^{19} - 3216 q^{23} + 1250 q^{25} - 1948 q^{29} - 2672 q^{31} - 2800 q^{35} + 8668 q^{37} + 7628 q^{41} + 16440 q^{43} - 19360 q^{47} + 25010 q^{49} + 14356 q^{53} - 6200 q^{55} - 904 q^{59} + 20220 q^{61} - 21900 q^{65} + 12904 q^{67} - 40976 q^{71} + 59124 q^{73} - 90816 q^{77} - 107600 q^{79} - 122088 q^{83} + 50900 q^{85} - 103764 q^{89} + 101408 q^{91} + 36600 q^{95} - 24764 q^{97}+O(q^{100}) $ 2 * q - 50 * q^5 + 112 * q^7 + 248 * q^11 + 876 * q^13 - 2036 * q^17 - 1464 * q^19 - 3216 * q^23 + 1250 * q^25 - 1948 * q^29 - 2672 * q^31 - 2800 * q^35 + 8668 * q^37 + 7628 * q^41 + 16440 * q^43 - 19360 * q^47 + 25010 * q^49 + 14356 * q^53 - 6200 * q^55 - 904 * q^59 + 20220 * q^61 - 21900 * q^65 + 12904 * q^67 - 40976 * q^71 + 59124 * q^73 - 90816 * q^77 - 107600 * q^79 - 122088 * q^83 + 50900 * q^85 - 103764 * q^89 + 101408 * q^91 + 36600 * q^95 - 24764 * 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
$7$	217.790	1.67994	0.839968	−	0.542636i	$-0.182574\pi$
$7$	217.790	1.67994	0.839968	+	0.542636i	$0.182574\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−199.580	−0.497319	−0.248660	−	0.968591i	$-0.579990\pi$
$11$	−199.580	−0.497319	−0.248660	+	0.968591i	$0.579990\pi$
$12$	0	0
$13$	599.790	0.984330	0.492165	−	0.870502i	$-0.336206\pi$
$13$	599.790	0.984330	0.492165	+	0.870502i	$0.336206\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−209.050	−0.175440	−0.0877199	−	0.996145i	$-0.527958\pi$
$17$	−209.050	−0.175440	−0.0877199	+	0.996145i	$0.527958\pi$
$18$	0	0
$19$	−2835.27	−1.80182	−0.900908	−	0.434010i	$-0.857098\pi$
$19$	−2835.27	−1.80182	−0.900908	+	0.434010i	$0.857098\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2093.37	−0.825138	−0.412569	−	0.910926i	$-0.635368\pi$
$23$	−2093.37	−0.825138	−0.412569	+	0.910926i	$0.635368\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−326.840	−0.0721673	−0.0360836	−	0.999349i	$-0.511488\pi$
$29$	−326.840	−0.0721673	−0.0360836	+	0.999349i	$0.511488\pi$
$30$	0	0
$31$	−3115.69	−0.582304	−0.291152	−	0.956677i	$-0.594039\pi$
$31$	−3115.69	−0.582304	−0.291152	+	0.956677i	$0.594039\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5444.75	−0.751290
$36$	0	0
$37$	−3917.29	−0.470415	−0.235208	−	0.971945i	$-0.575577\pi$
$37$	−3917.29	−0.470415	−0.235208	+	0.971945i	$0.575577\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9314.86	0.865400	0.432700	−	0.901538i	$-0.357561\pi$
$41$	9314.86	0.865400	0.432700	+	0.901538i	$0.357561\pi$
$42$	0	0
$43$	7572.84	0.624579	0.312290	−	0.949987i	$-0.398904\pi$
$43$	7572.84	0.624579	0.312290	+	0.949987i	$0.398904\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−22137.8	−1.46181	−0.730904	−	0.682480i	$-0.760901\pi$
$47$	−22137.8	−1.46181	−0.730904	+	0.682480i	$0.760901\pi$
$48$	0	0
$49$	30625.5	1.82219
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−15796.2	−0.772436	−0.386218	−	0.922408i	$-0.626219\pi$
$53$	−15796.2	−0.772436	−0.386218	+	0.922408i	$0.626219\pi$
$54$	0	0
$55$	4989.50	0.222408
$56$	0	0
$57$	0	0
$58$	0	0
$59$	32229.6	1.20538	0.602691	−	0.797975i	$-0.294095\pi$
$59$	32229.6	1.20538	0.602691	+	0.797975i	$0.294095\pi$
$60$	0	0
$61$	−5745.42	−0.197696	−0.0988478	−	0.995103i	$-0.531516\pi$
$61$	−5745.42	−0.197696	−0.0988478	+	0.995103i	$0.531516\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14994.7	−0.440206
$66$	0	0
$67$	−6491.20	−0.176660	−0.0883299	−	0.996091i	$-0.528153\pi$
$67$	−6491.20	−0.176660	−0.0883299	+	0.996091i	$0.528153\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11870.0	0.279450	0.139725	−	0.990190i	$-0.455378\pi$
$71$	11870.0	0.279450	0.139725	+	0.990190i	$0.455378\pi$
$72$	0	0
$73$	23414.0	0.514243	0.257121	−	0.966379i	$-0.417226\pi$
$73$	23414.0	0.514243	0.257121	+	0.966379i	$0.417226\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−43466.5	−0.835465
$78$	0	0
$79$	−15779.4	−0.284460	−0.142230	−	0.989834i	$-0.545427\pi$
$79$	−15779.4	−0.284460	−0.142230	+	0.989834i	$0.545427\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−101815.	−1.62225	−0.811123	−	0.584875i	$-0.801143\pi$
$83$	−101815.	−1.62225	−0.811123	+	0.584875i	$0.801143\pi$
$84$	0	0
$85$	5226.25	0.0784590
$86$	0	0
$87$	0	0
$88$	0	0
$89$	22865.0	0.305982	0.152991	−	0.988228i	$-0.451109\pi$
$89$	22865.0	0.305982	0.152991	+	0.988228i	$0.451109\pi$
$90$	0	0
$91$	130628.	1.65361
$92$	0	0
$93$	0	0
$94$	0	0
$95$	70881.7	0.805797
$96$	0	0
$97$	113814.	1.22819	0.614097	−	0.789230i	$-0.289520\pi$
$97$	113814.	1.22819	0.614097	+	0.789230i	$0.289520\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−88036.9	−0.858739	−0.429370	−	0.903129i	$-0.641264\pi$
$101$	−88036.9	−0.858739	−0.429370	+	0.903129i	$0.641264\pi$
$102$	0	0
$103$	−52836.4	−0.490727	−0.245364	−	0.969431i	$-0.578907\pi$
$103$	−52836.4	−0.490727	−0.245364	+	0.969431i	$0.578907\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−141097.	−1.19141	−0.595703	−	0.803205i	$-0.703126\pi$
$107$	−141097.	−1.19141	−0.595703	+	0.803205i	$0.703126\pi$
$108$	0	0
$109$	−14517.3	−0.117036	−0.0585179	−	0.998286i	$-0.518637\pi$
$109$	−14517.3	−0.117036	−0.0585179	+	0.998286i	$0.518637\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	29499.2	0.217327	0.108664	−	0.994079i	$-0.465343\pi$
$113$	29499.2	0.217327	0.108664	+	0.994079i	$0.465343\pi$
$114$	0	0
$115$	52334.2	0.369013
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−45529.0	−0.294728
$120$	0	0
$121$	−121219.	−0.752674
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−197789.	−1.08816	−0.544081	−	0.839033i	$-0.683121\pi$
$127$	−197789.	−1.08816	−0.544081	+	0.839033i	$0.683121\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−141216.	−0.718959	−0.359480	−	0.933153i	$-0.617046\pi$
$131$	−141216.	−0.718959	−0.359480	+	0.933153i	$0.617046\pi$
$132$	0	0
$133$	−617493.	−3.02694
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−159983.	−0.728235	−0.364117	−	0.931353i	$-0.618629\pi$
$137$	−159983.	−0.728235	−0.364117	+	0.931353i	$0.618629\pi$
$138$	0	0
$139$	−319007.	−1.40044	−0.700219	−	0.713928i	$-0.746914\pi$
$139$	−319007.	−1.40044	−0.700219	+	0.713928i	$0.746914\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−119706.	−0.489526
$144$	0	0
$145$	8171.00	0.0322742
$146$	0	0
$147$	0	0
$148$	0	0
$149$	398964.	1.47220	0.736102	−	0.676870i	$-0.236664\pi$
$149$	398964.	1.47220	0.736102	+	0.676870i	$0.236664\pi$
$150$	0	0
$151$	166591.	0.594577	0.297289	−	0.954788i	$-0.403918\pi$
$151$	166591.	0.594577	0.297289	+	0.954788i	$0.403918\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	77892.2	0.260414
$156$	0	0
$157$	−256463.	−0.830378	−0.415189	−	0.909735i	$-0.636285\pi$
$157$	−256463.	−0.830378	−0.415189	+	0.909735i	$0.636285\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−455915.	−1.38618
$162$	0	0
$163$	−654000.	−1.92801	−0.964004	−	0.265887i	$-0.914335\pi$
$163$	−654000.	−1.92801	−0.964004	+	0.265887i	$0.914335\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	667196.	1.85124	0.925619	−	0.378456i	$-0.123545\pi$
$167$	667196.	1.85124	0.925619	+	0.378456i	$0.123545\pi$
$168$	0	0
$169$	−11545.0	−0.0310940
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−381680.	−0.969582	−0.484791	−	0.874630i	$-0.661104\pi$
$173$	−381680.	−0.969582	−0.484791	+	0.874630i	$0.661104\pi$
$174$	0	0
$175$	136119.	0.335987
$176$	0	0
$177$	0	0
$178$	0	0
$179$	614936.	1.43449	0.717244	−	0.696822i	$-0.245403\pi$
$179$	614936.	1.43449	0.717244	+	0.696822i	$0.245403\pi$
$180$	0	0
$181$	−611099.	−1.38648	−0.693242	−	0.720705i	$-0.743818\pi$
$181$	−611099.	−1.38648	−0.693242	+	0.720705i	$0.743818\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	97932.2	0.210376
$186$	0	0
$187$	41722.2	0.0872496
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−151563.	−0.300615	−0.150307	−	0.988639i	$-0.548026\pi$
$191$	−151563.	−0.300615	−0.150307	+	0.988639i	$0.548026\pi$
$192$	0	0
$193$	669868.	1.29448	0.647241	−	0.762285i	$-0.275923\pi$
$193$	669868.	1.29448	0.647241	+	0.762285i	$0.275923\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	843520.	1.54857	0.774283	−	0.632839i	$-0.218111\pi$
$197$	843520.	1.54857	0.774283	+	0.632839i	$0.218111\pi$
$198$	0	0
$199$	99215.8	0.177602	0.0888010	−	0.996049i	$-0.471696\pi$
$199$	99215.8	0.177602	0.0888010	+	0.996049i	$0.471696\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−71182.5	−0.121236
$204$	0	0
$205$	−232871.	−0.387018
$206$	0	0
$207$	0	0
$208$	0	0
$209$	565863.	0.896078
$210$	0	0
$211$	−954803.	−1.47641	−0.738206	−	0.674575i	$-0.764327\pi$
$211$	−954803.	−1.47641	−0.738206	+	0.674575i	$0.764327\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−189321.	−0.279320
$216$	0	0
$217$	−678566.	−0.978234
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−125386.	−0.172691
$222$	0	0
$223$	99120.0	0.133475	0.0667374	−	0.997771i	$-0.478741\pi$
$223$	99120.0	0.133475	0.0667374	+	0.997771i	$0.478741\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−489116.	−0.630009	−0.315005	−	0.949090i	$-0.602006\pi$
$227$	−489116.	−0.630009	−0.315005	+	0.949090i	$0.602006\pi$
$228$	0	0
$229$	−77461.9	−0.0976111	−0.0488056	−	0.998808i	$-0.515541\pi$
$229$	−77461.9	−0.0976111	−0.0488056	+	0.998808i	$0.515541\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	128093.	0.154574	0.0772868	−	0.997009i	$-0.475374\pi$
$233$	128093.	0.154574	0.0772868	+	0.997009i	$0.475374\pi$
$234$	0	0
$235$	553446.	0.653740
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.03342e6	−1.17026	−0.585129	−	0.810940i	$-0.698956\pi$
$239$	−1.03342e6	−1.17026	−0.585129	+	0.810940i	$0.698956\pi$
$240$	0	0
$241$	491546.	0.545157	0.272578	−	0.962134i	$-0.412124\pi$
$241$	491546.	0.545157	0.272578	+	0.962134i	$0.412124\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−765637.	−0.814906
$246$	0	0
$247$	−1.70057e6	−1.77358
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.23517e6	−1.23750	−0.618748	−	0.785590i	$-0.712360\pi$
$251$	−1.23517e6	−1.23750	−0.618748	+	0.785590i	$0.712360\pi$
$252$	0	0
$253$	417795.	0.410357
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.90909e6	−1.80299	−0.901495	−	0.432789i	$-0.857530\pi$
$257$	−1.90909e6	−1.80299	−0.901495	+	0.432789i	$0.857530\pi$
$258$	0	0
$259$	−853146.	−0.790268
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−614059.	−0.547420	−0.273710	−	0.961812i	$-0.588251\pi$
$263$	−614059.	−0.547420	−0.273710	+	0.961812i	$0.588251\pi$
$264$	0	0
$265$	394904.	0.345444
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.05806e6	1.73411	0.867055	−	0.498213i	$-0.166010\pi$
$269$	2.05806e6	1.73411	0.867055	+	0.498213i	$0.166010\pi$
$270$	0	0
$271$	−178987.	−0.148046	−0.0740231	−	0.997257i	$-0.523584\pi$
$271$	−178987.	−0.148046	−0.0740231	+	0.997257i	$0.523584\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−124737.	−0.0994638
$276$	0	0
$277$	−1.89158e6	−1.48124	−0.740619	−	0.671925i	$-0.765468\pi$
$277$	−1.89158e6	−1.48124	−0.740619	+	0.671925i	$0.765468\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.19927e6	−1.66154	−0.830772	−	0.556613i	$-0.812101\pi$
$281$	−2.19927e6	−1.66154	−0.830772	+	0.556613i	$0.812101\pi$
$282$	0	0
$283$	−804296.	−0.596967	−0.298483	−	0.954415i	$-0.596481\pi$
$283$	−804296.	−0.596967	−0.298483	+	0.954415i	$0.596481\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.02868e6	1.45382
$288$	0	0
$289$	−1.37616e6	−0.969221
$290$	0	0
$291$	0	0
$292$	0	0
$293$	435069.	0.296067	0.148033	−	0.988982i	$-0.452706\pi$
$293$	435069.	0.296067	0.148033	+	0.988982i	$0.452706\pi$
$294$	0	0
$295$	−805739.	−0.539063
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.25558e6	−0.812208
$300$	0	0
$301$	1.64929e6	1.04925
$302$	0	0
$303$	0	0
$304$	0	0
$305$	143635.	0.0884122
$306$	0	0
$307$	−379288.	−0.229680	−0.114840	−	0.993384i	$-0.536636\pi$
$307$	−379288.	−0.229680	−0.114840	+	0.993384i	$0.536636\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.71677e6	−1.00649	−0.503246	−	0.864143i	$-0.667861\pi$
$311$	−1.71677e6	−1.00649	−0.503246	+	0.864143i	$0.667861\pi$
$312$	0	0
$313$	−1.46409e6	−0.844711	−0.422356	−	0.906430i	$-0.638797\pi$
$313$	−1.46409e6	−0.844711	−0.422356	+	0.906430i	$0.638797\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−90285.6	−0.0504627	−0.0252313	−	0.999682i	$-0.508032\pi$
$317$	−90285.6	−0.0504627	−0.0252313	+	0.999682i	$0.508032\pi$
$318$	0	0
$319$	65230.7	0.0358902
$320$	0	0
$321$	0	0
$322$	0	0
$323$	592713.	0.316110
$324$	0	0
$325$	374869.	0.196866
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.82140e6	−2.45574
$330$	0	0
$331$	3.37162e6	1.69148	0.845742	−	0.533591i	$-0.179158\pi$
$331$	3.37162e6	1.69148	0.845742	+	0.533591i	$0.179158\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	162280.	0.0790047
$336$	0	0
$337$	−2.09461e6	−1.00468	−0.502341	−	0.864669i	$-0.667528\pi$
$337$	−2.09461e6	−1.00468	−0.502341	+	0.864669i	$0.667528\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	621829.	0.289591
$342$	0	0
$343$	3.00953e6	1.38122
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.42706e6	1.52791	0.763955	−	0.645269i	$-0.223255\pi$
$347$	3.42706e6	1.52791	0.763955	+	0.645269i	$0.223255\pi$
$348$	0	0
$349$	101956.	0.0448074	0.0224037	−	0.999749i	$-0.492868\pi$
$349$	101956.	0.0448074	0.0224037	+	0.999749i	$0.492868\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	704751.	0.301022	0.150511	−	0.988608i	$-0.451908\pi$
$353$	704751.	0.301022	0.150511	+	0.988608i	$0.451908\pi$
$354$	0	0
$355$	−296750.	−0.124974
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.50896e6	−1.02744	−0.513721	−	0.857957i	$-0.671733\pi$
$359$	−2.50896e6	−1.02744	−0.513721	+	0.857957i	$0.671733\pi$
$360$	0	0
$361$	5.56266e6	2.24654
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−585350.	−0.229976
$366$	0	0
$367$	4.06900e6	1.57697	0.788483	−	0.615057i	$-0.210867\pi$
$367$	4.06900e6	1.57697	0.788483	+	0.615057i	$0.210867\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.44025e6	−1.29764
$372$	0	0
$373$	−957594.	−0.356377	−0.178188	−	0.983996i	$-0.557024\pi$
$373$	−957594.	−0.356377	−0.178188	+	0.983996i	$0.557024\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−196035.	−0.0710364
$378$	0	0
$379$	−328048.	−0.117311	−0.0586556	−	0.998278i	$-0.518681\pi$
$379$	−328048.	−0.117311	−0.0586556	+	0.998278i	$0.518681\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.22117e6	1.12206	0.561031	−	0.827795i	$-0.310405\pi$
$383$	3.22117e6	1.12206	0.561031	+	0.827795i	$0.310405\pi$
$384$	0	0
$385$	1.08666e6	0.373631
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.38697e6	−0.799785	−0.399893	−	0.916562i	$-0.630953\pi$
$389$	−2.38697e6	−0.799785	−0.399893	+	0.916562i	$0.630953\pi$
$390$	0	0
$391$	437619.	0.144762
$392$	0	0
$393$	0	0
$394$	0	0
$395$	394484.	0.127214
$396$	0	0
$397$	−3.13956e6	−0.999751	−0.499876	−	0.866097i	$-0.666621\pi$
$397$	−3.13956e6	−0.999751	−0.499876	+	0.866097i	$0.666621\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.49379e6	0.774459	0.387229	−	0.921983i	$-0.373432\pi$
$401$	2.49379e6	0.774459	0.387229	+	0.921983i	$0.373432\pi$
$402$	0	0
$403$	−1.86876e6	−0.573180
$404$	0	0
$405$	0	0
$406$	0	0
$407$	781813.	0.233947
$408$	0	0
$409$	114720.	0.0339103	0.0169552	−	0.999856i	$-0.494603\pi$
$409$	114720.	0.0339103	0.0169552	+	0.999856i	$0.494603\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.01928e6	2.02496
$414$	0	0
$415$	2.54538e6	0.725491
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.25805e6	0.628344	0.314172	−	0.949366i	$-0.398273\pi$
$419$	2.25805e6	0.628344	0.314172	+	0.949366i	$0.398273\pi$
$420$	0	0
$421$	−352348.	−0.0968873	−0.0484436	−	0.998826i	$-0.515426\pi$
$421$	−352348.	−0.0968873	−0.0484436	+	0.998826i	$0.515426\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−130656.	−0.0350880
$426$	0	0
$427$	−1.25129e6	−0.332116
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.43169e6	0.630543	0.315272	−	0.949001i	$-0.397904\pi$
$431$	2.43169e6	0.630543	0.315272	+	0.949001i	$0.397904\pi$
$432$	0	0
$433$	5.37990e6	1.37897	0.689485	−	0.724300i	$-0.257837\pi$
$433$	5.37990e6	1.37897	0.689485	+	0.724300i	$0.257837\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.93527e6	1.48675
$438$	0	0
$439$	−6.85919e6	−1.69868	−0.849340	−	0.527847i	$-0.823000\pi$
$439$	−6.85919e6	−1.69868	−0.849340	+	0.527847i	$0.823000\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−568384.	−0.137605	−0.0688023	−	0.997630i	$-0.521918\pi$
$443$	−568384.	−0.137605	−0.0688023	+	0.997630i	$0.521918\pi$
$444$	0	0
$445$	−571624.	−0.136839
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.12355e6	0.965285	0.482642	−	0.875818i	$-0.339677\pi$
$449$	4.12355e6	0.965285	0.482642	+	0.875818i	$0.339677\pi$
$450$	0	0
$451$	−1.85906e6	−0.430380
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.26571e6	−0.739518
$456$	0	0
$457$	1.48425e6	0.332443	0.166221	−	0.986088i	$-0.446843\pi$
$457$	1.48425e6	0.332443	0.166221	+	0.986088i	$0.446843\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.49347e6	−1.42306	−0.711532	−	0.702654i	$-0.751998\pi$
$461$	−6.49347e6	−1.42306	−0.711532	+	0.702654i	$0.751998\pi$
$462$	0	0
$463$	1.85239e6	0.401586	0.200793	−	0.979634i	$-0.435648\pi$
$463$	1.85239e6	0.401586	0.200793	+	0.979634i	$0.435648\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.64352e6	−0.348724	−0.174362	−	0.984682i	$-0.555786\pi$
$467$	−1.64352e6	−0.348724	−0.174362	+	0.984682i	$0.555786\pi$
$468$	0	0
$469$	−1.41372e6	−0.296777
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.51139e6	−0.310615
$474$	0	0
$475$	−1.77204e6	−0.360363
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.72002e6	1.53737	0.768687	−	0.639625i	$-0.220910\pi$
$479$	7.72002e6	1.53737	0.768687	+	0.639625i	$0.220910\pi$
$480$	0	0
$481$	−2.34955e6	−0.463044
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.84535e6	−0.549265
$486$	0	0
$487$	−4.10435e6	−0.784191	−0.392096	−	0.919924i	$-0.628250\pi$
$487$	−4.10435e6	−0.784191	−0.392096	+	0.919924i	$0.628250\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−450180.	−0.0842719	−0.0421360	−	0.999112i	$-0.513416\pi$
$491$	−450180.	−0.0842719	−0.0421360	+	0.999112i	$0.513416\pi$
$492$	0	0
$493$	68325.9	0.0126610
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.58517e6	0.469459
$498$	0	0
$499$	2.01251e6	0.361815	0.180907	−	0.983500i	$-0.442097\pi$
$499$	2.01251e6	0.361815	0.180907	+	0.983500i	$0.442097\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.74624e6	0.483970	0.241985	−	0.970280i	$-0.422202\pi$
$503$	2.74624e6	0.483970	0.241985	+	0.970280i	$0.422202\pi$
$504$	0	0
$505$	2.20092e6	0.384040
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.47345e6	0.765328	0.382664	−	0.923888i	$-0.375007\pi$
$509$	4.47345e6	0.765328	0.382664	+	0.923888i	$0.375007\pi$
$510$	0	0
$511$	5.09933e6	0.863895
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.32091e6	0.219460
$516$	0	0
$517$	4.41827e6	0.726985
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.20576e6	0.678814	0.339407	−	0.940640i	$-0.389774\pi$
$521$	4.20576e6	0.678814	0.339407	+	0.940640i	$0.389774\pi$
$522$	0	0
$523$	3.27485e6	0.523524	0.261762	−	0.965132i	$-0.415696\pi$
$523$	3.27485e6	0.523524	0.261762	+	0.965132i	$0.415696\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	651335.	0.102159
$528$	0	0
$529$	−2.05415e6	−0.319148
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.58696e6	0.851839
$534$	0	0
$535$	3.52743e6	0.532813
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.11223e6	−0.906208
$540$	0	0
$541$	−4.55984e6	−0.669817	−0.334909	−	0.942251i	$-0.608705\pi$
$541$	−4.55984e6	−0.669817	−0.334909	+	0.942251i	$0.608705\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	362932.	0.0523400
$546$	0	0
$547$	−1.12208e7	−1.60345	−0.801723	−	0.597695i	$-0.796083\pi$
$547$	−1.12208e7	−1.60345	−0.801723	+	0.597695i	$0.796083\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	926680.	0.130032
$552$	0	0
$553$	−3.43659e6	−0.477875
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.90478e6	−0.260140	−0.130070	−	0.991505i	$-0.541520\pi$
$557$	−1.90478e6	−0.260140	−0.130070	+	0.991505i	$0.541520\pi$
$558$	0	0
$559$	4.54211e6	0.614792
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.81706e6	1.17234	0.586169	−	0.810189i	$-0.300635\pi$
$563$	8.81706e6	1.17234	0.586169	+	0.810189i	$0.300635\pi$
$564$	0	0
$565$	−737481.	−0.0971918
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.58116e6	1.11113	0.555565	−	0.831473i	$-0.312502\pi$
$569$	8.58116e6	1.11113	0.555565	+	0.831473i	$0.312502\pi$
$570$	0	0
$571$	4.01627e6	0.515504	0.257752	−	0.966211i	$-0.417018\pi$
$571$	4.01627e6	0.515504	0.257752	+	0.966211i	$0.417018\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.30836e6	−0.165028
$576$	0	0
$577$	2.73670e6	0.342206	0.171103	−	0.985253i	$-0.445267\pi$
$577$	2.73670e6	0.342206	0.171103	+	0.985253i	$0.445267\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.21743e7	−2.72527
$582$	0	0
$583$	3.15260e6	0.384147
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.12479e6	0.134734	0.0673670	−	0.997728i	$-0.478540\pi$
$587$	1.12479e6	0.134734	0.0673670	+	0.997728i	$0.478540\pi$
$588$	0	0
$589$	8.83382e6	1.04921
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.61107e6	−0.772032	−0.386016	−	0.922492i	$-0.626149\pi$
$593$	−6.61107e6	−0.772032	−0.386016	+	0.922492i	$0.626149\pi$
$594$	0	0
$595$	1.13823e6	0.131806
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.78429e6	0.430941	0.215470	−	0.976510i	$-0.430872\pi$
$599$	3.78429e6	0.430941	0.215470	+	0.976510i	$0.430872\pi$
$600$	0	0
$601$	1.36699e7	1.54376	0.771878	−	0.635771i	$-0.219318\pi$
$601$	1.36699e7	1.54376	0.771878	+	0.635771i	$0.219318\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.03047e6	0.336606
$606$	0	0
$607$	1.05825e7	1.16578	0.582890	−	0.812551i	$-0.301922\pi$
$607$	1.05825e7	1.16578	0.582890	+	0.812551i	$0.301922\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.32780e7	−1.43890
$612$	0	0
$613$	1.49493e7	1.60683	0.803414	−	0.595421i	$-0.203015\pi$
$613$	1.49493e7	1.60683	0.803414	+	0.595421i	$0.203015\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.44272e6	−0.892832	−0.446416	−	0.894826i	$-0.647300\pi$
$617$	−8.44272e6	−0.892832	−0.446416	+	0.894826i	$0.647300\pi$
$618$	0	0
$619$	7.49113e6	0.785815	0.392908	−	0.919578i	$-0.371469\pi$
$619$	7.49113e6	0.785815	0.392908	+	0.919578i	$0.371469\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.97976e6	0.514030
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	818910.	0.0825295
$630$	0	0
$631$	9.74348e6	0.974184	0.487092	−	0.873351i	$-0.338058\pi$
$631$	9.74348e6	0.974184	0.487092	+	0.873351i	$0.338058\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.94473e6	0.486641
$636$	0	0
$637$	1.83689e7	1.79363
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.30814e7	1.25750	0.628750	−	0.777608i	$-0.283567\pi$
$641$	1.30814e7	1.25750	0.628750	+	0.777608i	$0.283567\pi$
$642$	0	0
$643$	−6.93389e6	−0.661377	−0.330689	−	0.943740i	$-0.607281\pi$
$643$	−6.93389e6	−0.661377	−0.330689	+	0.943740i	$0.607281\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.98447e7	−1.86373	−0.931866	−	0.362801i	$-0.881820\pi$
$647$	−1.98447e7	−1.86373	−0.931866	+	0.362801i	$0.881820\pi$
$648$	0	0
$649$	−6.43238e6	−0.599459
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.44251e6	0.499478	0.249739	−	0.968313i	$-0.419655\pi$
$653$	5.44251e6	0.499478	0.249739	+	0.968313i	$0.419655\pi$
$654$	0	0
$655$	3.53039e6	0.321528
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.32110e7	1.18501	0.592505	−	0.805567i	$-0.298139\pi$
$659$	1.32110e7	1.18501	0.592505	+	0.805567i	$0.298139\pi$
$660$	0	0
$661$	−1.82594e7	−1.62549	−0.812744	−	0.582621i	$-0.802027\pi$
$661$	−1.82594e7	−1.62549	−0.812744	+	0.582621i	$0.802027\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.54373e7	1.35369
$666$	0	0
$667$	684197.	0.0595479
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.14667e6	0.0983178
$672$	0	0
$673$	1.96595e6	0.167315	0.0836576	−	0.996495i	$-0.473340\pi$
$673$	1.96595e6	0.167315	0.0836576	+	0.996495i	$0.473340\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.56212e7	1.30991	0.654955	−	0.755667i	$-0.272687\pi$
$677$	1.56212e7	1.30991	0.654955	+	0.755667i	$0.272687\pi$
$678$	0	0
$679$	2.47876e7	2.06329
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.86400e6	0.316946	0.158473	−	0.987363i	$-0.449343\pi$
$683$	3.86400e6	0.316946	0.158473	+	0.987363i	$0.449343\pi$
$684$	0	0
$685$	3.99956e6	0.325676
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.47439e6	−0.760332
$690$	0	0
$691$	−8.51624e6	−0.678504	−0.339252	−	0.940695i	$-0.610174\pi$
$691$	−8.51624e6	−0.678504	−0.339252	+	0.940695i	$0.610174\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.97518e6	0.626295
$696$	0	0
$697$	−1.94727e6	−0.151825
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.36126e7	−1.81489	−0.907443	−	0.420176i	$-0.861968\pi$
$701$	−2.36126e7	−1.81489	−0.907443	+	0.420176i	$0.861968\pi$
$702$	0	0
$703$	1.11066e7	0.847602
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.91736e7	−1.44263
$708$	0	0
$709$	1.03409e7	0.772582	0.386291	−	0.922377i	$-0.373756\pi$
$709$	1.03409e7	0.772582	0.386291	+	0.922377i	$0.373756\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.52229e6	0.480481
$714$	0	0
$715$	2.99265e6	0.218923
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.72694e6	0.268862	0.134431	−	0.990923i	$-0.457079\pi$
$719$	3.72694e6	0.268862	0.134431	+	0.990923i	$0.457079\pi$
$720$	0	0
$721$	−1.15072e7	−0.824391
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−204275.	−0.0144335
$726$	0	0
$727$	6.39102e6	0.448471	0.224235	−	0.974535i	$-0.428012\pi$
$727$	6.39102e6	0.448471	0.224235	+	0.974535i	$0.428012\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.58310e6	−0.109576
$732$	0	0
$733$	−1.04155e7	−0.716011	−0.358006	−	0.933719i	$-0.616543\pi$
$733$	−1.04155e7	−0.716011	−0.358006	+	0.933719i	$0.616543\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.29551e6	0.0878564
$738$	0	0
$739$	−2.76262e7	−1.86084	−0.930421	−	0.366492i	$-0.880559\pi$
$739$	−2.76262e7	−1.86084	−0.930421	+	0.366492i	$0.880559\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.79121e6	−0.517765	−0.258882	−	0.965909i	$-0.583354\pi$
$743$	−7.79121e6	−0.517765	−0.258882	+	0.965909i	$0.583354\pi$
$744$	0	0
$745$	−9.97410e6	−0.658390
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.07296e7	−2.00148
$750$	0	0
$751$	−2.52335e7	−1.63259	−0.816296	−	0.577633i	$-0.803976\pi$
$751$	−2.52335e7	−1.63259	−0.816296	+	0.577633i	$0.803976\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.16477e6	−0.265903
$756$	0	0
$757$	−1.13968e7	−0.722844	−0.361422	−	0.932402i	$-0.617709\pi$
$757$	−1.13968e7	−0.722844	−0.361422	+	0.932402i	$0.617709\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.00333e6	−0.313182	−0.156591	−	0.987664i	$-0.550050\pi$
$761$	−5.00333e6	−0.313182	−0.156591	+	0.987664i	$0.550050\pi$
$762$	0	0
$763$	−3.16172e6	−0.196613
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.93310e7	1.18649
$768$	0	0
$769$	−4.48204e6	−0.273313	−0.136656	−	0.990619i	$-0.543636\pi$
$769$	−4.48204e6	−0.273313	−0.136656	+	0.990619i	$0.543636\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.29031e7	−1.37862	−0.689311	−	0.724465i	$-0.742087\pi$
$773$	−2.29031e7	−1.37862	−0.689311	+	0.724465i	$0.742087\pi$
$774$	0	0
$775$	−1.94731e6	−0.116461
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.64101e7	−1.55929
$780$	0	0
$781$	−2.36901e6	−0.138976
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.41158e6	0.371356
$786$	0	0
$787$	−6.83480e6	−0.393359	−0.196679	−	0.980468i	$-0.563016\pi$
$787$	−6.83480e6	−0.393359	−0.196679	+	0.980468i	$0.563016\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.42464e6	0.365096
$792$	0	0
$793$	−3.44604e6	−0.194598
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.23658e6	−0.0689564	−0.0344782	−	0.999405i	$-0.510977\pi$
$797$	−1.23658e6	−0.0689564	−0.0344782	+	0.999405i	$0.510977\pi$
$798$	0	0
$799$	4.62791e6	0.256459
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.67296e6	−0.255743
$804$	0	0
$805$	1.13979e7	0.619918
$806$	0	0
$807$	0	0
$808$	0	0
$809$	9.63877e6	0.517786	0.258893	−	0.965906i	$-0.416642\pi$
$809$	9.63877e6	0.517786	0.258893	+	0.965906i	$0.416642\pi$
$810$	0	0
$811$	−1.11387e7	−0.594679	−0.297339	−	0.954772i	$-0.596099\pi$
$811$	−1.11387e7	−0.594679	−0.297339	+	0.954772i	$0.596099\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.63500e7	0.862231
$816$	0	0
$817$	−2.14710e7	−1.12538
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.76945e7	1.43396	0.716978	−	0.697096i	$-0.245525\pi$
$821$	2.76945e7	1.43396	0.716978	+	0.697096i	$0.245525\pi$
$822$	0	0
$823$	3.29068e7	1.69351	0.846753	−	0.531987i	$-0.178554\pi$
$823$	3.29068e7	1.69351	0.846753	+	0.531987i	$0.178554\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.46622e7	−1.25392	−0.626958	−	0.779053i	$-0.715700\pi$
$827$	−2.46622e7	−1.25392	−0.626958	+	0.779053i	$0.715700\pi$
$828$	0	0
$829$	2.63681e7	1.33258	0.666288	−	0.745694i	$-0.267882\pi$
$829$	2.63681e7	1.33258	0.666288	+	0.745694i	$0.267882\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.40226e6	−0.319684
$834$	0	0
$835$	−1.66799e7	−0.827899
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.53962e7	1.24556	0.622778	−	0.782399i	$-0.286004\pi$
$839$	2.53962e7	1.24556	0.622778	+	0.782399i	$0.286004\pi$
$840$	0	0
$841$	−2.04043e7	−0.994792
$842$	0	0
$843$	0	0
$844$	0	0
$845$	288624.	0.0139056
$846$	0	0
$847$	−2.64002e7	−1.26444
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.20034e6	0.388157
$852$	0	0
$853$	1.46587e7	0.689799	0.344900	−	0.938640i	$-0.387913\pi$
$853$	1.46587e7	0.689799	0.344900	+	0.938640i	$0.387913\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.44223e7	−1.60099	−0.800493	−	0.599342i	$-0.795429\pi$
$857$	−3.44223e7	−1.60099	−0.800493	+	0.599342i	$0.795429\pi$
$858$	0	0
$859$	−8.53480e6	−0.394648	−0.197324	−	0.980338i	$-0.563225\pi$
$859$	−8.53480e6	−0.394648	−0.197324	+	0.980338i	$0.563225\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.40872e7	0.643870	0.321935	−	0.946762i	$-0.395667\pi$
$863$	1.40872e7	0.643870	0.321935	+	0.946762i	$0.395667\pi$
$864$	0	0
$865$	9.54201e6	0.433610
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.14924e6	0.141468
$870$	0	0
$871$	−3.89336e6	−0.173892
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.40297e6	−0.150258
$876$	0	0
$877$	2.18156e7	0.957786	0.478893	−	0.877873i	$-0.341038\pi$
$877$	2.18156e7	0.957786	0.478893	+	0.877873i	$0.341038\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.92711e7	0.836501	0.418250	−	0.908332i	$-0.362643\pi$
$881$	1.92711e7	0.836501	0.418250	+	0.908332i	$0.362643\pi$
$882$	0	0
$883$	−5.79768e6	−0.250238	−0.125119	−	0.992142i	$-0.539931\pi$
$883$	−5.79768e6	−0.250238	−0.125119	+	0.992142i	$0.539931\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5.72121e6	−0.244163	−0.122081	−	0.992520i	$-0.538957\pi$
$887$	−5.72121e6	−0.244163	−0.122081	+	0.992520i	$0.538957\pi$
$888$	0	0
$889$	−4.30765e7	−1.82804
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.27667e7	2.63391
$894$	0	0
$895$	−1.53734e7	−0.641523
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.01833e6	0.0420233
$900$	0	0
$901$	3.30219e6	0.135516
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.52775e7	0.620055
$906$	0	0
$907$	1.43236e7	0.578139	0.289070	−	0.957308i	$-0.406654\pi$
$907$	1.43236e7	0.578139	0.289070	+	0.957308i	$0.406654\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.60230e7	−1.03887	−0.519434	−	0.854510i	$-0.673857\pi$
$911$	−2.60230e7	−1.03887	−0.519434	+	0.854510i	$0.673857\pi$
$912$	0	0
$913$	2.03203e7	0.806774
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.07553e7	−1.20781
$918$	0	0
$919$	1.51344e7	0.591121	0.295560	−	0.955324i	$-0.404494\pi$
$919$	1.51344e7	0.591121	0.295560	+	0.955324i	$0.404494\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.11951e6	0.275071
$924$	0	0
$925$	−2.44831e6	−0.0940830
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.28489e7	−0.488459	−0.244230	−	0.969717i	$-0.578535\pi$
$929$	−1.28489e7	−0.488459	−0.244230	+	0.969717i	$0.578535\pi$
$930$	0	0
$931$	−8.68315e7	−3.28324
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.04306e6	−0.0390192
$936$	0	0
$937$	−3.08935e7	−1.14952	−0.574762	−	0.818320i	$-0.694905\pi$
$937$	−3.08935e7	−1.14952	−0.574762	+	0.818320i	$0.694905\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.48240e7	0.913897	0.456948	−	0.889493i	$-0.348942\pi$
$941$	2.48240e7	0.913897	0.456948	+	0.889493i	$0.348942\pi$
$942$	0	0
$943$	−1.94994e7	−0.714074
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.75170e7	−0.997071	−0.498535	−	0.866869i	$-0.666129\pi$
$947$	−2.75170e7	−0.997071	−0.498535	+	0.866869i	$0.666129\pi$
$948$	0	0
$949$	1.40435e7	0.506185
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.04677e7	1.80004	0.900019	−	0.435851i	$-0.143553\pi$
$953$	5.04677e7	1.80004	0.900019	+	0.435851i	$0.143553\pi$
$954$	0	0
$955$	3.78908e6	0.134439
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.48426e7	−1.22339
$960$	0	0
$961$	−1.89216e7	−0.660922
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.67467e7	−0.578910
$966$	0	0
$967$	2.51529e7	0.865010	0.432505	−	0.901632i	$-0.357630\pi$
$967$	2.51529e7	0.865010	0.432505	+	0.901632i	$0.357630\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.68530e6	−0.227548	−0.113774	−	0.993507i	$-0.536294\pi$
$971$	−6.68530e6	−0.227548	−0.113774	+	0.993507i	$0.536294\pi$
$972$	0	0
$973$	−6.94766e7	−2.35265
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.48404e7	−0.832573	−0.416286	−	0.909234i	$-0.636669\pi$
$977$	−2.48404e7	−0.832573	−0.416286	+	0.909234i	$0.636669\pi$
$978$	0	0
$979$	−4.56339e6	−0.152171
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.77957e7	1.24755	0.623776	−	0.781603i	$-0.285598\pi$
$983$	3.77957e7	1.24755	0.623776	+	0.781603i	$0.285598\pi$
$984$	0	0
$985$	−2.10880e7	−0.692540
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.58528e7	−0.515364
$990$	0	0
$991$	7.63130e6	0.246839	0.123420	−	0.992355i	$-0.460614\pi$
$991$	7.63130e6	0.246839	0.123420	+	0.992355i	$0.460614\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.48039e6	−0.0794261
$996$	0	0
$997$	−6.87906e6	−0.219175	−0.109588	−	0.993977i	$-0.534953\pi$
$997$	−6.87906e6	−0.219175	−0.109588	+	0.993977i	$0.534953\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.6.a.bd.1.2		2
3.2	odd	2		240.6.a.q.1.2		2
4.3	odd	2		45.6.a.e.1.1		2
12.11	even	2		15.6.a.c.1.2	✓	2
20.3	even	4		225.6.b.g.199.3		4
20.7	even	4		225.6.b.g.199.2		4
20.19	odd	2		225.6.a.m.1.2		2
24.5	odd	2		960.6.a.bf.1.2		2
24.11	even	2		960.6.a.bj.1.1		2
60.23	odd	4		75.6.b.e.49.2		4
60.47	odd	4		75.6.b.e.49.3		4
60.59	even	2		75.6.a.h.1.1		2
84.83	odd	2		735.6.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.6.a.c.1.2	✓	2	12.11	even	2
45.6.a.e.1.1		2	4.3	odd	2
75.6.a.h.1.1		2	60.59	even	2
75.6.b.e.49.2		4	60.23	odd	4
75.6.b.e.49.3		4	60.47	odd	4
225.6.a.m.1.2		2	20.19	odd	2
225.6.b.g.199.2		4	20.7	even	4
225.6.b.g.199.3		4	20.3	even	4
240.6.a.q.1.2		2	3.2	odd	2
720.6.a.bd.1.2		2	1.1	even	1	trivial
735.6.a.g.1.2		2	84.83	odd	2
960.6.a.bf.1.2		2	24.5	odd	2
960.6.a.bj.1.1		2	24.11	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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.a (trivial)

\(f(q)\)	\(=\)	\(q-25.0000 q^{5} +217.790 q^{7} +O(q^{10})\)	\(q-25.0000 q^{5} +217.790 q^{7} -199.580 q^{11} +599.790 q^{13} -209.050 q^{17} -2835.27 q^{19} -2093.37 q^{23} +625.000 q^{25} -326.840 q^{29} -3115.69 q^{31} -5444.75 q^{35} -3917.29 q^{37} +9314.86 q^{41} +7572.84 q^{43} -22137.8 q^{47} +30625.5 q^{49} -15796.2 q^{53} +4989.50 q^{55} +32229.6 q^{59} -5745.42 q^{61} -14994.7 q^{65} -6491.20 q^{67} +11870.0 q^{71} +23414.0 q^{73} -43466.5 q^{77} -15779.4 q^{79} -101815. q^{83} +5226.25 q^{85} +22865.0 q^{89} +130628. q^{91} +70881.7 q^{95} +113814. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 50 q^{5} + 112 q^{7}+O(q^{10}) \) 2 * q - 50 * q^5 + 112 * q^7	\( 2 q - 50 q^{5} + 112 q^{7} + 248 q^{11} + 876 q^{13} - 2036 q^{17} - 1464 q^{19} - 3216 q^{23} + 1250 q^{25} - 1948 q^{29} - 2672 q^{31} - 2800 q^{35} + 8668 q^{37} + 7628 q^{41} + 16440 q^{43} - 19360 q^{47} + 25010 q^{49} + 14356 q^{53} - 6200 q^{55} - 904 q^{59} + 20220 q^{61} - 21900 q^{65} + 12904 q^{67} - 40976 q^{71} + 59124 q^{73} - 90816 q^{77} - 107600 q^{79} - 122088 q^{83} + 50900 q^{85} - 103764 q^{89} + 101408 q^{91} + 36600 q^{95} - 24764 q^{97}+O(q^{100}) \) 2 * q - 50 * q^5 + 112 * q^7 + 248 * q^11 + 876 * q^13 - 2036 * q^17 - 1464 * q^19 - 3216 * q^23 + 1250 * q^25 - 1948 * q^29 - 2672 * q^31 - 2800 * q^35 + 8668 * q^37 + 7628 * q^41 + 16440 * q^43 - 19360 * q^47 + 25010 * q^49 + 14356 * q^53 - 6200 * q^55 - 904 * q^59 + 20220 * q^61 - 21900 * q^65 + 12904 * q^67 - 40976 * q^71 + 59124 * q^73 - 90816 * q^77 - 107600 * q^79 - 122088 * q^83 + 50900 * q^85 - 103764 * q^89 + 101408 * q^91 + 36600 * q^95 - 24764 * q^97

Introduction

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