LMFDB - Embedded newform 1080.6.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,6,Mod(1,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, 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("1080.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1080.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:	$173.214525398$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.15881.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 29x - 55 $ x^3 - 29*x - 55
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2}\cdot 11 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.82250$ of defining polynomial
Character	$\chi$	$=$	1080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+25.0000 q^{5} -177.292 q^{7} +O(q^{10})$	$q+25.0000 q^{5} -177.292 q^{7} -778.539 q^{11} +132.539 q^{13} +1634.72 q^{17} +345.079 q^{19} +959.441 q^{23} +625.000 q^{25} +2573.30 q^{29} +4123.80 q^{31} -4432.31 q^{35} +7188.70 q^{37} +19345.5 q^{41} +1526.17 q^{43} -3415.76 q^{47} +14625.6 q^{49} +17770.0 q^{53} -19463.5 q^{55} +17443.5 q^{59} -18471.9 q^{61} +3313.47 q^{65} -54149.5 q^{67} -66498.8 q^{71} -28937.5 q^{73} +138029. q^{77} -88015.0 q^{79} -68424.7 q^{83} +40868.1 q^{85} -116101. q^{89} -23498.1 q^{91} +8626.98 q^{95} -47803.9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 75 q^{5} - 30 q^{7}+O(q^{10}) $ 3 * q + 75 * q^5 - 30 * q^7	$ 3 q + 75 q^{5} - 30 q^{7} - 660 q^{11} - 1278 q^{13} + 1731 q^{17} + 1011 q^{19} + 2433 q^{23} + 1875 q^{25} + 3786 q^{29} + 4131 q^{31} - 750 q^{35} - 7122 q^{37} + 17352 q^{41} + 11556 q^{43} - 19548 q^{47} - 7857 q^{49} - 4965 q^{53} - 16500 q^{55} - 32106 q^{59} - 16317 q^{61} - 31950 q^{65} - 51258 q^{67} - 84072 q^{71} - 161892 q^{73} + 142968 q^{77} + 36441 q^{79} - 64581 q^{83} + 43275 q^{85} - 61584 q^{89} - 123588 q^{91} + 25275 q^{95} + 14760 q^{97}+O(q^{100}) $ 3 * q + 75 * q^5 - 30 * q^7 - 660 * q^11 - 1278 * q^13 + 1731 * q^17 + 1011 * q^19 + 2433 * q^23 + 1875 * q^25 + 3786 * q^29 + 4131 * q^31 - 750 * q^35 - 7122 * q^37 + 17352 * q^41 + 11556 * q^43 - 19548 * q^47 - 7857 * q^49 - 4965 * q^53 - 16500 * q^55 - 32106 * q^59 - 16317 * q^61 - 31950 * q^65 - 51258 * q^67 - 84072 * q^71 - 161892 * q^73 + 142968 * q^77 + 36441 * q^79 - 64581 * q^83 + 43275 * q^85 - 61584 * q^89 - 123588 * q^91 + 25275 * q^95 + 14760 * 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
$7$	−177.292	−1.36756	−0.683778	−	0.729690i	$-0.739664\pi$
$7$	−177.292	−1.36756	−0.683778	+	0.729690i	$0.739664\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−778.539	−1.93999	−0.969993	−	0.243133i	$-0.921825\pi$
$11$	−778.539	−1.93999	−0.969993	+	0.243133i	$0.921825\pi$
$12$	0	0
$13$	132.539	0.217513	0.108756	−	0.994068i	$-0.465313\pi$
$13$	132.539	0.217513	0.108756	+	0.994068i	$0.465313\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1634.72	1.37190	0.685949	−	0.727650i	$-0.259387\pi$
$17$	1634.72	1.37190	0.685949	+	0.727650i	$0.259387\pi$
$18$	0	0
$19$	345.079	0.219298	0.109649	−	0.993970i	$-0.465027\pi$
$19$	345.079	0.219298	0.109649	+	0.993970i	$0.465027\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	959.441	0.378180	0.189090	−	0.981960i	$-0.439446\pi$
$23$	959.441	0.378180	0.189090	+	0.981960i	$0.439446\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2573.30	0.568193	0.284096	−	0.958796i	$-0.408306\pi$
$29$	2573.30	0.568193	0.284096	+	0.958796i	$0.408306\pi$
$30$	0	0
$31$	4123.80	0.770715	0.385358	−	0.922767i	$-0.374078\pi$
$31$	4123.80	0.770715	0.385358	+	0.922767i	$0.374078\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4432.31	−0.611589
$36$	0	0
$37$	7188.70	0.863269	0.431635	−	0.902049i	$-0.357937\pi$
$37$	7188.70	0.863269	0.431635	+	0.902049i	$0.357937\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	19345.5	1.79729	0.898647	−	0.438672i	$-0.144551\pi$
$41$	19345.5	1.79729	0.898647	+	0.438672i	$0.144551\pi$
$42$	0	0
$43$	1526.17	0.125873	0.0629365	−	0.998018i	$-0.479953\pi$
$43$	1526.17	0.125873	0.0629365	+	0.998018i	$0.479953\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3415.76	−0.225550	−0.112775	−	0.993621i	$-0.535974\pi$
$47$	−3415.76	−0.225550	−0.112775	+	0.993621i	$0.535974\pi$
$48$	0	0
$49$	14625.6	0.870208
$50$	0	0
$51$	0	0
$52$	0	0
$53$	17770.0	0.868955	0.434477	−	0.900683i	$-0.356933\pi$
$53$	17770.0	0.868955	0.434477	+	0.900683i	$0.356933\pi$
$54$	0	0
$55$	−19463.5	−0.867588
$56$	0	0
$57$	0	0
$58$	0	0
$59$	17443.5	0.652385	0.326192	−	0.945303i	$-0.394234\pi$
$59$	17443.5	0.652385	0.326192	+	0.945303i	$0.394234\pi$
$60$	0	0
$61$	−18471.9	−0.635603	−0.317802	−	0.948157i	$-0.602945\pi$
$61$	−18471.9	−0.635603	−0.317802	+	0.948157i	$0.602945\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3313.47	0.0972747
$66$	0	0
$67$	−54149.5	−1.47369	−0.736847	−	0.676059i	$-0.763686\pi$
$67$	−54149.5	−1.47369	−0.736847	+	0.676059i	$0.763686\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−66498.8	−1.56555	−0.782776	−	0.622303i	$-0.786197\pi$
$71$	−66498.8	−1.56555	−0.782776	+	0.622303i	$0.786197\pi$
$72$	0	0
$73$	−28937.5	−0.635556	−0.317778	−	0.948165i	$-0.602937\pi$
$73$	−28937.5	−0.635556	−0.317778	+	0.948165i	$0.602937\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	138029.	2.65304
$78$	0	0
$79$	−88015.0	−1.58668	−0.793339	−	0.608780i	$-0.791659\pi$
$79$	−88015.0	−1.58668	−0.793339	+	0.608780i	$0.791659\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−68424.7	−1.09023	−0.545114	−	0.838362i	$-0.683514\pi$
$83$	−68424.7	−1.09023	−0.545114	+	0.838362i	$0.683514\pi$
$84$	0	0
$85$	40868.1	0.613531
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−116101.	−1.55368	−0.776838	−	0.629701i	$-0.783177\pi$
$89$	−116101.	−1.55368	−0.776838	+	0.629701i	$0.783177\pi$
$90$	0	0
$91$	−23498.1	−0.297461
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8626.98	0.0980731
$96$	0	0
$97$	−47803.9	−0.515863	−0.257931	−	0.966163i	$-0.583041\pi$
$97$	−47803.9	−0.515863	−0.257931	+	0.966163i	$0.583041\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−184461.	−1.79929	−0.899643	−	0.436626i	$-0.856173\pi$
$101$	−184461.	−1.79929	−0.899643	+	0.436626i	$0.856173\pi$
$102$	0	0
$103$	36260.9	0.336780	0.168390	−	0.985720i	$-0.446143\pi$
$103$	36260.9	0.336780	0.168390	+	0.985720i	$0.446143\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	194715.	1.64414	0.822072	−	0.569383i	$-0.192818\pi$
$107$	194715.	1.64414	0.822072	+	0.569383i	$0.192818\pi$
$108$	0	0
$109$	35234.0	0.284051	0.142025	−	0.989863i	$-0.454639\pi$
$109$	35234.0	0.284051	0.142025	+	0.989863i	$0.454639\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−138170.	−1.01793	−0.508965	−	0.860787i	$-0.669972\pi$
$113$	−138170.	−1.01793	−0.508965	+	0.860787i	$0.669972\pi$
$114$	0	0
$115$	23986.0	0.169127
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−289824.	−1.87615
$120$	0	0
$121$	445072.	2.76355
$122$	0	0
$123$	0	0
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	37441.3	0.205988	0.102994	−	0.994682i	$-0.467158\pi$
$127$	37441.3	0.205988	0.102994	+	0.994682i	$0.467158\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−50118.6	−0.255165	−0.127582	−	0.991828i	$-0.540722\pi$
$131$	−50118.6	−0.255165	−0.127582	+	0.991828i	$0.540722\pi$
$132$	0	0
$133$	−61179.9	−0.299902
$134$	0	0
$135$	0	0
$136$	0	0
$137$	73408.5	0.334153	0.167076	−	0.985944i	$-0.446567\pi$
$137$	73408.5	0.334153	0.167076	+	0.985944i	$0.446567\pi$
$138$	0	0
$139$	−32460.7	−0.142502	−0.0712510	−	0.997458i	$-0.522699\pi$
$139$	−32460.7	−0.142502	−0.0712510	+	0.997458i	$0.522699\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−103187.	−0.421972
$144$	0	0
$145$	64332.5	0.254103
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−301999.	−1.11440	−0.557198	−	0.830380i	$-0.688123\pi$
$149$	−301999.	−1.11440	−0.557198	+	0.830380i	$0.688123\pi$
$150$	0	0
$151$	239797.	0.855858	0.427929	−	0.903812i	$-0.359243\pi$
$151$	239797.	0.855858	0.427929	+	0.903812i	$0.359243\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	103095.	0.344674
$156$	0	0
$157$	312913.	1.01315	0.506577	−	0.862195i	$-0.330911\pi$
$157$	312913.	1.01315	0.506577	+	0.862195i	$0.330911\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−170101.	−0.517182
$162$	0	0
$163$	219202.	0.646214	0.323107	−	0.946362i	$-0.395273\pi$
$163$	219202.	0.646214	0.323107	+	0.946362i	$0.395273\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−186815.	−0.518346	−0.259173	−	0.965831i	$-0.583450\pi$
$167$	−186815.	−0.518346	−0.259173	+	0.965831i	$0.583450\pi$
$168$	0	0
$169$	−353726.	−0.952688
$170$	0	0
$171$	0	0
$172$	0	0
$173$	35108.1	0.0891851	0.0445926	−	0.999005i	$-0.485801\pi$
$173$	35108.1	0.0891851	0.0445926	+	0.999005i	$0.485801\pi$
$174$	0	0
$175$	−110808.	−0.273511
$176$	0	0
$177$	0	0
$178$	0	0
$179$	250525.	0.584410	0.292205	−	0.956356i	$-0.405611\pi$
$179$	250525.	0.584410	0.292205	+	0.956356i	$0.405611\pi$
$180$	0	0
$181$	−541712.	−1.22906	−0.614529	−	0.788894i	$-0.710654\pi$
$181$	−541712.	−1.22906	−0.614529	+	0.788894i	$0.710654\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	179718.	0.386066
$186$	0	0
$187$	−1.27270e6	−2.66146
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−180536.	−0.358081	−0.179040	−	0.983842i	$-0.557299\pi$
$191$	−180536.	−0.358081	−0.179040	+	0.983842i	$0.557299\pi$
$192$	0	0
$193$	−531938.	−1.02794	−0.513970	−	0.857808i	$-0.671826\pi$
$193$	−531938.	−1.02794	−0.513970	+	0.857808i	$0.671826\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	488199.	0.896255	0.448127	−	0.893970i	$-0.352091\pi$
$197$	488199.	0.896255	0.448127	+	0.893970i	$0.352091\pi$
$198$	0	0
$199$	41226.6	0.0737980	0.0368990	−	0.999319i	$-0.488252\pi$
$199$	41226.6	0.0737980	0.0368990	+	0.999319i	$0.488252\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−456227.	−0.777035
$204$	0	0
$205$	483636.	0.803775
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−268658.	−0.425435
$210$	0	0
$211$	727442.	1.12484	0.562422	−	0.826851i	$-0.309870\pi$
$211$	727442.	1.12484	0.562422	+	0.826851i	$0.309870\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	38154.3	0.0562921
$216$	0	0
$217$	−731119.	−1.05400
$218$	0	0
$219$	0	0
$220$	0	0
$221$	216664.	0.298405
$222$	0	0
$223$	1.05221e6	1.41691	0.708453	−	0.705758i	$-0.249394\pi$
$223$	1.05221e6	1.41691	0.708453	+	0.705758i	$0.249394\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−70140.8	−0.0903454	−0.0451727	−	0.998979i	$-0.514384\pi$
$227$	−70140.8	−0.0903454	−0.0451727	+	0.998979i	$0.514384\pi$
$228$	0	0
$229$	331611.	0.417868	0.208934	−	0.977930i	$-0.433001\pi$
$229$	331611.	0.417868	0.208934	+	0.977930i	$0.433001\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.18857e6	−1.43429	−0.717143	−	0.696926i	$-0.754550\pi$
$233$	−1.18857e6	−1.43429	−0.717143	+	0.696926i	$0.754550\pi$
$234$	0	0
$235$	−85394.0	−0.100869
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.65354e6	−1.87249	−0.936246	−	0.351345i	$-0.885724\pi$
$239$	−1.65354e6	−1.87249	−0.936246	+	0.351345i	$0.885724\pi$
$240$	0	0
$241$	1.28873e6	1.42929	0.714643	−	0.699489i	$-0.246589\pi$
$241$	1.28873e6	1.42929	0.714643	+	0.699489i	$0.246589\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	365639.	0.389169
$246$	0	0
$247$	45736.4	0.0477002
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.01541e6	1.01732	0.508662	−	0.860967i	$-0.330140\pi$
$251$	1.01541e6	1.01732	0.508662	+	0.860967i	$0.330140\pi$
$252$	0	0
$253$	−746962.	−0.733664
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−560979.	−0.529802	−0.264901	−	0.964276i	$-0.585339\pi$
$257$	−560979.	−0.529802	−0.264901	+	0.964276i	$0.585339\pi$
$258$	0	0
$259$	−1.27450e6	−1.18057
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−695106.	−0.619672	−0.309836	−	0.950790i	$-0.600274\pi$
$263$	−695106.	−0.619672	−0.309836	+	0.950790i	$0.600274\pi$
$264$	0	0
$265$	444249.	0.388608
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.70042e6	−1.43277	−0.716383	−	0.697707i	$-0.754204\pi$
$269$	−1.70042e6	−1.43277	−0.716383	+	0.697707i	$0.754204\pi$
$270$	0	0
$271$	1.57912e6	1.30615	0.653074	−	0.757294i	$-0.273479\pi$
$271$	1.57912e6	1.30615	0.653074	+	0.757294i	$0.273479\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−486587.	−0.387997
$276$	0	0
$277$	−640171.	−0.501299	−0.250649	−	0.968078i	$-0.580644\pi$
$277$	−640171.	−0.501299	−0.250649	+	0.968078i	$0.580644\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−468521.	−0.353968	−0.176984	−	0.984214i	$-0.556634\pi$
$281$	−468521.	−0.353968	−0.176984	+	0.984214i	$0.556634\pi$
$282$	0	0
$283$	−669897.	−0.497212	−0.248606	−	0.968605i	$-0.579972\pi$
$283$	−669897.	−0.497212	−0.248606	+	0.968605i	$0.579972\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.42980e6	−2.45790
$288$	0	0
$289$	1.25246e6	0.882104
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−223700.	−0.152229	−0.0761145	−	0.997099i	$-0.524251\pi$
$293$	−223700.	−0.152229	−0.0761145	+	0.997099i	$0.524251\pi$
$294$	0	0
$295$	436088.	0.291755
$296$	0	0
$297$	0	0
$298$	0	0
$299$	127163.	0.0822590
$300$	0	0
$301$	−270579.	−0.172138
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−461796.	−0.284250
$306$	0	0
$307$	−1.09907e6	−0.665547	−0.332774	−	0.943007i	$-0.607984\pi$
$307$	−1.09907e6	−0.665547	−0.332774	+	0.943007i	$0.607984\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.84743e6	−1.08310	−0.541548	−	0.840670i	$-0.682162\pi$
$311$	−1.84743e6	−1.08310	−0.541548	+	0.840670i	$0.682162\pi$
$312$	0	0
$313$	84378.0	0.0486820	0.0243410	−	0.999704i	$-0.492251\pi$
$313$	84378.0	0.0486820	0.0243410	+	0.999704i	$0.492251\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.19843e6	−1.78768	−0.893838	−	0.448389i	$-0.851998\pi$
$317$	−3.19843e6	−1.78768	−0.893838	+	0.448389i	$0.851998\pi$
$318$	0	0
$319$	−2.00341e6	−1.10229
$320$	0	0
$321$	0	0
$322$	0	0
$323$	564109.	0.300855
$324$	0	0
$325$	82836.8	0.0435026
$326$	0	0
$327$	0	0
$328$	0	0
$329$	605588.	0.308452
$330$	0	0
$331$	798314.	0.400501	0.200251	−	0.979745i	$-0.435824\pi$
$331$	798314.	0.400501	0.200251	+	0.979745i	$0.435824\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.35374e6	−0.659056
$336$	0	0
$337$	−721479.	−0.346058	−0.173029	−	0.984917i	$-0.555355\pi$
$337$	−721479.	−0.346058	−0.173029	+	0.984917i	$0.555355\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.21054e6	−1.49518
$342$	0	0
$343$	386749.	0.177498
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.83135e6	−0.816483	−0.408242	−	0.912874i	$-0.633858\pi$
$347$	−1.83135e6	−0.816483	−0.408242	+	0.912874i	$0.633858\pi$
$348$	0	0
$349$	2.63801e6	1.15935	0.579673	−	0.814849i	$-0.303180\pi$
$349$	2.63801e6	1.15935	0.579673	+	0.814849i	$0.303180\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−640229.	−0.273463	−0.136732	−	0.990608i	$-0.543660\pi$
$353$	−640229.	−0.273463	−0.136732	+	0.990608i	$0.543660\pi$
$354$	0	0
$355$	−1.66247e6	−0.700136
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.88403e6	1.59055	0.795273	−	0.606252i	$-0.207328\pi$
$359$	3.88403e6	1.59055	0.795273	+	0.606252i	$0.207328\pi$
$360$	0	0
$361$	−2.35702e6	−0.951908
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−723437.	−0.284229
$366$	0	0
$367$	−2.89439e6	−1.12174	−0.560870	−	0.827904i	$-0.689533\pi$
$367$	−2.89439e6	−1.12174	−0.560870	+	0.827904i	$0.689533\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.15048e6	−1.18834
$372$	0	0
$373$	−3.55971e6	−1.32478	−0.662389	−	0.749160i	$-0.730457\pi$
$373$	−3.55971e6	−1.32478	−0.662389	+	0.749160i	$0.730457\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	341062.	0.123589
$378$	0	0
$379$	−5.33127e6	−1.90648	−0.953242	−	0.302209i	$-0.902276\pi$
$379$	−5.33127e6	−1.90648	−0.953242	+	0.302209i	$0.902276\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	988098.	0.344194	0.172097	−	0.985080i	$-0.444946\pi$
$383$	988098.	0.344194	0.172097	+	0.985080i	$0.444946\pi$
$384$	0	0
$385$	3.45072e6	1.18647
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.45055e6	−0.821088	−0.410544	−	0.911841i	$-0.634661\pi$
$389$	−2.45055e6	−0.821088	−0.410544	+	0.911841i	$0.634661\pi$
$390$	0	0
$391$	1.56842e6	0.518824
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.20038e6	−0.709584
$396$	0	0
$397$	5.03254e6	1.60255	0.801273	−	0.598299i	$-0.204156\pi$
$397$	5.03254e6	1.60255	0.801273	+	0.598299i	$0.204156\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	421036.	0.130755	0.0653775	−	0.997861i	$-0.479175\pi$
$401$	421036.	0.130755	0.0653775	+	0.997861i	$0.479175\pi$
$402$	0	0
$403$	546564.	0.167640
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.59668e6	−1.67473
$408$	0	0
$409$	5.26037e6	1.55492	0.777459	−	0.628933i	$-0.216508\pi$
$409$	5.26037e6	1.55492	0.777459	+	0.628933i	$0.216508\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.09260e6	−0.892172
$414$	0	0
$415$	−1.71062e6	−0.487565
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5.27635e6	1.46825	0.734123	−	0.679016i	$-0.237593\pi$
$419$	5.27635e6	1.46825	0.734123	+	0.679016i	$0.237593\pi$
$420$	0	0
$421$	6.85695e6	1.88550	0.942748	−	0.333506i	$-0.108232\pi$
$421$	6.85695e6	1.88550	0.942748	+	0.333506i	$0.108232\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.02170e6	0.274380
$426$	0	0
$427$	3.27492e6	0.869222
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.34747e6	1.38661	0.693306	−	0.720643i	$-0.256153\pi$
$431$	5.34747e6	1.38661	0.693306	+	0.720643i	$0.256153\pi$
$432$	0	0
$433$	5.48030e6	1.40470	0.702351	−	0.711831i	$-0.252134\pi$
$433$	5.48030e6	1.40470	0.702351	+	0.711831i	$0.252134\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	331083.	0.0829342
$438$	0	0
$439$	−1.88107e6	−0.465848	−0.232924	−	0.972495i	$-0.574829\pi$
$439$	−1.88107e6	−0.465848	−0.232924	+	0.972495i	$0.574829\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.74311e6	1.63249	0.816245	−	0.577706i	$-0.196052\pi$
$443$	6.74311e6	1.63249	0.816245	+	0.577706i	$0.196052\pi$
$444$	0	0
$445$	−2.90252e6	−0.694825
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.56671e6	−0.366751	−0.183376	−	0.983043i	$-0.558702\pi$
$449$	−1.56671e6	−0.366751	−0.183376	+	0.983043i	$0.558702\pi$
$450$	0	0
$451$	−1.50612e7	−3.48673
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−587453.	−0.133029
$456$	0	0
$457$	−5.65144e6	−1.26581	−0.632905	−	0.774229i	$-0.718138\pi$
$457$	−5.65144e6	−1.26581	−0.632905	+	0.774229i	$0.718138\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.08271e6	−1.55220	−0.776099	−	0.630611i	$-0.782804\pi$
$461$	−7.08271e6	−1.55220	−0.776099	+	0.630611i	$0.782804\pi$
$462$	0	0
$463$	1.96223e6	0.425400	0.212700	−	0.977118i	$-0.431774\pi$
$463$	1.96223e6	0.425400	0.212700	+	0.977118i	$0.431774\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.00128e6	−0.848999	−0.424500	−	0.905428i	$-0.639550\pi$
$467$	−4.00128e6	−0.848999	−0.424500	+	0.905428i	$0.639550\pi$
$468$	0	0
$469$	9.60029e6	2.01536
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.18818e6	−0.244192
$474$	0	0
$475$	215675.	0.0438596
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.02033e6	0.999755	0.499878	−	0.866096i	$-0.333378\pi$
$479$	5.02033e6	0.999755	0.499878	+	0.866096i	$0.333378\pi$
$480$	0	0
$481$	952782.	0.187772
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.19510e6	−0.230701
$486$	0	0
$487$	4.44446e6	0.849174	0.424587	−	0.905387i	$-0.360419\pi$
$487$	4.44446e6	0.849174	0.424587	+	0.905387i	$0.360419\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.94382e6	−1.48705	−0.743525	−	0.668708i	$-0.766847\pi$
$491$	−7.94382e6	−1.48705	−0.743525	+	0.668708i	$0.766847\pi$
$492$	0	0
$493$	4.20663e6	0.779502
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.17897e7	2.14098
$498$	0	0
$499$	4.40214e6	0.791430	0.395715	−	0.918373i	$-0.370497\pi$
$499$	4.40214e6	0.791430	0.395715	+	0.918373i	$0.370497\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.09339e7	−1.92689	−0.963445	−	0.267907i	$-0.913668\pi$
$503$	−1.09339e7	−1.92689	−0.963445	+	0.267907i	$0.913668\pi$
$504$	0	0
$505$	−4.61152e6	−0.804665
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4.82803e6	−0.825992	−0.412996	−	0.910733i	$-0.635518\pi$
$509$	−4.82803e6	−0.825992	−0.412996	+	0.910733i	$0.635518\pi$
$510$	0	0
$511$	5.13039e6	0.869157
$512$	0	0
$513$	0	0
$514$	0	0
$515$	906524.	0.150613
$516$	0	0
$517$	2.65930e6	0.437564
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.13830e6	0.667924	0.333962	−	0.942586i	$-0.391614\pi$
$521$	4.13830e6	0.667924	0.333962	+	0.942586i	$0.391614\pi$
$522$	0	0
$523$	−9.79480e6	−1.56582	−0.782909	−	0.622136i	$-0.786265\pi$
$523$	−9.79480e6	−1.56582	−0.782909	+	0.622136i	$0.786265\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.74128e6	1.05734
$528$	0	0
$529$	−5.51582e6	−0.856980
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.56402e6	0.390935
$534$	0	0
$535$	4.86788e6	0.735284
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.13866e7	−1.68819
$540$	0	0
$541$	−5.01186e6	−0.736216	−0.368108	−	0.929783i	$-0.619994\pi$
$541$	−5.01186e6	−0.736216	−0.368108	+	0.929783i	$0.619994\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	880850.	0.127031
$546$	0	0
$547$	−3.87207e6	−0.553318	−0.276659	−	0.960968i	$-0.589227\pi$
$547$	−3.87207e6	−0.553318	−0.276659	+	0.960968i	$0.589227\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	887993.	0.124604
$552$	0	0
$553$	1.56044e7	2.16987
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.83098e6	0.932921	0.466461	−	0.884542i	$-0.345529\pi$
$557$	6.83098e6	0.932921	0.466461	+	0.884542i	$0.345529\pi$
$558$	0	0
$559$	202277.	0.0273790
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.10042e7	−1.46314	−0.731571	−	0.681765i	$-0.761213\pi$
$563$	−1.10042e7	−1.46314	−0.731571	+	0.681765i	$0.761213\pi$
$564$	0	0
$565$	−3.45425e6	−0.455232
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.90980e6	−1.02420	−0.512100	−	0.858926i	$-0.671132\pi$
$569$	−7.90980e6	−1.02420	−0.512100	+	0.858926i	$0.671132\pi$
$570$	0	0
$571$	−5.27065e6	−0.676509	−0.338255	−	0.941055i	$-0.609836\pi$
$571$	−5.27065e6	−0.676509	−0.338255	+	0.941055i	$0.609836\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	599650.	0.0756360
$576$	0	0
$577$	16853.3	0.00210740	0.00105370	−	0.999999i	$-0.499665\pi$
$577$	16853.3	0.00210740	0.00105370	+	0.999999i	$0.499665\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.21312e7	1.49095
$582$	0	0
$583$	−1.38346e7	−1.68576
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.46707e7	−1.75733	−0.878667	−	0.477435i	$-0.841567\pi$
$587$	−1.46707e7	−1.75733	−0.878667	+	0.477435i	$0.841567\pi$
$588$	0	0
$589$	1.42304e6	0.169016
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.22349e6	−0.259656	−0.129828	−	0.991537i	$-0.541443\pi$
$593$	−2.22349e6	−0.259656	−0.129828	+	0.991537i	$0.541443\pi$
$594$	0	0
$595$	−7.24560e6	−0.839038
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.24066e7	−1.41281	−0.706405	−	0.707807i	$-0.749684\pi$
$599$	−1.24066e7	−1.41281	−0.706405	+	0.707807i	$0.749684\pi$
$600$	0	0
$601$	−1.44445e6	−0.163124	−0.0815618	−	0.996668i	$-0.525991\pi$
$601$	−1.44445e6	−0.163124	−0.0815618	+	0.996668i	$0.525991\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.11268e7	1.23590
$606$	0	0
$607$	−6.21724e6	−0.684898	−0.342449	−	0.939536i	$-0.611256\pi$
$607$	−6.21724e6	−0.684898	−0.342449	+	0.939536i	$0.611256\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−452721.	−0.0490600
$612$	0	0
$613$	−7.57737e6	−0.814455	−0.407228	−	0.913327i	$-0.633504\pi$
$613$	−7.57737e6	−0.814455	−0.407228	+	0.913327i	$0.633504\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.51893e7	−1.60629	−0.803147	−	0.595782i	$-0.796842\pi$
$617$	−1.51893e7	−1.60629	−0.803147	+	0.595782i	$0.796842\pi$
$618$	0	0
$619$	4.86181e6	0.510001	0.255001	−	0.966941i	$-0.417924\pi$
$619$	4.86181e6	0.510001	0.255001	+	0.966941i	$0.417924\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.05838e7	2.12474
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.17515e7	1.18432
$630$	0	0
$631$	−1.62841e7	−1.62813	−0.814065	−	0.580773i	$-0.802750\pi$
$631$	−1.62841e7	−1.62813	−0.814065	+	0.580773i	$0.802750\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	936033.	0.0921206
$636$	0	0
$637$	1.93846e6	0.189281
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.64531e6	−0.254292	−0.127146	−	0.991884i	$-0.540582\pi$
$641$	−2.64531e6	−0.254292	−0.127146	+	0.991884i	$0.540582\pi$
$642$	0	0
$643$	1.43973e7	1.37326	0.686630	−	0.727007i	$-0.259089\pi$
$643$	1.43973e7	1.37326	0.686630	+	0.727007i	$0.259089\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.26066e6	0.494060	0.247030	−	0.969008i	$-0.420545\pi$
$647$	5.26066e6	0.494060	0.247030	+	0.969008i	$0.420545\pi$
$648$	0	0
$649$	−1.35805e7	−1.26562
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.57874e6	−0.879074	−0.439537	−	0.898225i	$-0.644857\pi$
$653$	−9.57874e6	−0.879074	−0.439537	+	0.898225i	$0.644857\pi$
$654$	0	0
$655$	−1.25297e6	−0.114113
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.87271e6	0.795871	0.397936	−	0.917413i	$-0.369727\pi$
$659$	8.87271e6	0.795871	0.397936	+	0.917413i	$0.369727\pi$
$660$	0	0
$661$	−262438.	−0.0233627	−0.0116814	−	0.999932i	$-0.503718\pi$
$661$	−262438.	−0.0233627	−0.0116814	+	0.999932i	$0.503718\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.52950e6	−0.134120
$666$	0	0
$667$	2.46893e6	0.214879
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.43811e7	1.23306
$672$	0	0
$673$	5.01759e6	0.427029	0.213514	−	0.976940i	$-0.431509\pi$
$673$	5.01759e6	0.427029	0.213514	+	0.976940i	$0.431509\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.24406e7	1.04320	0.521602	−	0.853189i	$-0.325335\pi$
$677$	1.24406e7	1.04320	0.521602	+	0.853189i	$0.325335\pi$
$678$	0	0
$679$	8.47527e6	0.705471
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.34912e6	0.274712	0.137356	−	0.990522i	$-0.456139\pi$
$683$	3.34912e6	0.274712	0.137356	+	0.990522i	$0.456139\pi$
$684$	0	0
$685$	1.83521e6	0.149438
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.35521e6	0.189009
$690$	0	0
$691$	−1.44577e7	−1.15188	−0.575938	−	0.817494i	$-0.695363\pi$
$691$	−1.44577e7	−1.15188	−0.575938	+	0.817494i	$0.695363\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−811517.	−0.0637288
$696$	0	0
$697$	3.16245e7	2.46570
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.17953e6	0.705546	0.352773	−	0.935709i	$-0.385239\pi$
$701$	9.17953e6	0.705546	0.352773	+	0.935709i	$0.385239\pi$
$702$	0	0
$703$	2.48067e6	0.189313
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.27035e7	2.46062
$708$	0	0
$709$	1.55235e7	1.15977	0.579887	−	0.814697i	$-0.303097\pi$
$709$	1.55235e7	1.15977	0.579887	+	0.814697i	$0.303097\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.95655e6	0.291469
$714$	0	0
$715$	−2.57967e6	−0.188712
$716$	0	0
$717$	0	0
$718$	0	0
$719$	5.96476e6	0.430299	0.215150	−	0.976581i	$-0.430976\pi$
$719$	5.96476e6	0.430299	0.215150	+	0.976581i	$0.430976\pi$
$720$	0	0
$721$	−6.42879e6	−0.460565
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.60831e6	0.113639
$726$	0	0
$727$	−6.08552e6	−0.427033	−0.213517	−	0.976939i	$-0.568492\pi$
$727$	−6.08552e6	−0.427033	−0.213517	+	0.976939i	$0.568492\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.49487e6	0.172685
$732$	0	0
$733$	1.96004e7	1.34742	0.673712	−	0.738994i	$-0.264699\pi$
$733$	1.96004e7	1.34742	0.673712	+	0.738994i	$0.264699\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.21575e7	2.85895
$738$	0	0
$739$	1.81747e7	1.22421	0.612106	−	0.790776i	$-0.290323\pi$
$739$	1.81747e7	1.22421	0.612106	+	0.790776i	$0.290323\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.94965e7	1.29564	0.647820	−	0.761793i	$-0.275681\pi$
$743$	1.94965e7	1.29564	0.647820	+	0.761793i	$0.275681\pi$
$744$	0	0
$745$	−7.54996e6	−0.498373
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.45215e7	−2.24846
$750$	0	0
$751$	4.80428e6	0.310834	0.155417	−	0.987849i	$-0.450328\pi$
$751$	4.80428e6	0.310834	0.155417	+	0.987849i	$0.450328\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.99493e6	0.382751
$756$	0	0
$757$	−1.51641e7	−0.961780	−0.480890	−	0.876781i	$-0.659686\pi$
$757$	−1.51641e7	−0.961780	−0.480890	+	0.876781i	$0.659686\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.65799e6	−0.354161	−0.177080	−	0.984196i	$-0.556665\pi$
$761$	−5.65799e6	−0.354161	−0.177080	+	0.984196i	$0.556665\pi$
$762$	0	0
$763$	−6.24672e6	−0.388455
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.31194e6	0.141902
$768$	0	0
$769$	3.26224e7	1.98930	0.994650	−	0.103299i	$-0.0329398\pi$
$769$	3.26224e7	1.98930	0.994650	+	0.103299i	$0.0329398\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.93469e6	−0.417425	−0.208712	−	0.977977i	$-0.566927\pi$
$773$	−6.93469e6	−0.417425	−0.208712	+	0.977977i	$0.566927\pi$
$774$	0	0
$775$	2.57738e6	0.154143
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.67572e6	0.394143
$780$	0	0
$781$	5.17719e7	3.03715
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.82284e6	0.453096
$786$	0	0
$787$	−2.48568e7	−1.43057	−0.715285	−	0.698833i	$-0.753703\pi$
$787$	−2.48568e7	−1.43057	−0.715285	+	0.698833i	$0.753703\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.44965e7	1.39208
$792$	0	0
$793$	−2.44824e6	−0.138252
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.45001e6	0.0808583	0.0404292	−	0.999182i	$-0.487127\pi$
$797$	1.45001e6	0.0808583	0.0404292	+	0.999182i	$0.487127\pi$
$798$	0	0
$799$	−5.58382e6	−0.309431
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.25289e7	1.23297
$804$	0	0
$805$	−4.25254e6	−0.231291
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.88981e6	0.477552	0.238776	−	0.971075i	$-0.423254\pi$
$809$	8.88981e6	0.477552	0.238776	+	0.971075i	$0.423254\pi$
$810$	0	0
$811$	8.16694e6	0.436021	0.218011	−	0.975946i	$-0.430043\pi$
$811$	8.16694e6	0.436021	0.218011	+	0.975946i	$0.430043\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.48006e6	0.288996
$816$	0	0
$817$	526651.	0.0276037
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.76397e7	−0.913339	−0.456670	−	0.889636i	$-0.650958\pi$
$821$	−1.76397e7	−0.913339	−0.456670	+	0.889636i	$0.650958\pi$
$822$	0	0
$823$	−3.01825e7	−1.55330	−0.776651	−	0.629931i	$-0.783083\pi$
$823$	−3.01825e7	−1.55330	−0.776651	+	0.629931i	$0.783083\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.94875e7	−0.990817	−0.495408	−	0.868660i	$-0.664982\pi$
$827$	−1.94875e7	−0.990817	−0.495408	+	0.868660i	$0.664982\pi$
$828$	0	0
$829$	−2.29876e7	−1.16174	−0.580868	−	0.813998i	$-0.697287\pi$
$829$	−2.29876e7	−1.16174	−0.580868	+	0.813998i	$0.697287\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.39088e7	1.19384
$834$	0	0
$835$	−4.67037e6	−0.231811
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.59410e7	0.781827	0.390913	−	0.920427i	$-0.372159\pi$
$839$	1.59410e7	0.781827	0.390913	+	0.920427i	$0.372159\pi$
$840$	0	0
$841$	−1.38893e7	−0.677157
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8.84316e6	−0.426055
$846$	0	0
$847$	−7.89078e7	−3.77930
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.89713e6	0.326471
$852$	0	0
$853$	1.10065e7	0.517938	0.258969	−	0.965886i	$-0.416617\pi$
$853$	1.10065e7	0.517938	0.258969	+	0.965886i	$0.416617\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.02680e7	−0.942667	−0.471334	−	0.881955i	$-0.656227\pi$
$857$	−2.02680e7	−0.942667	−0.471334	+	0.881955i	$0.656227\pi$
$858$	0	0
$859$	9.35167e6	0.432420	0.216210	−	0.976347i	$-0.430630\pi$
$859$	9.35167e6	0.432420	0.216210	+	0.976347i	$0.430630\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.37371e7	−1.54199	−0.770993	−	0.636843i	$-0.780240\pi$
$863$	−3.37371e7	−1.54199	−0.770993	+	0.636843i	$0.780240\pi$
$864$	0	0
$865$	877703.	0.0398848
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.85231e7	3.07813
$870$	0	0
$871$	−7.17691e6	−0.320547
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.77019e6	−0.122318
$876$	0	0
$877$	5.27863e6	0.231751	0.115876	−	0.993264i	$-0.463033\pi$
$877$	5.27863e6	0.231751	0.115876	+	0.993264i	$0.463033\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.46500e6	0.410848	0.205424	−	0.978673i	$-0.434143\pi$
$881$	9.46500e6	0.410848	0.205424	+	0.978673i	$0.434143\pi$
$882$	0	0
$883$	−3.19136e7	−1.37744	−0.688722	−	0.725026i	$-0.741828\pi$
$883$	−3.19136e7	−1.37744	−0.688722	+	0.725026i	$0.741828\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.25237e7	0.534469	0.267235	−	0.963631i	$-0.413890\pi$
$887$	1.25237e7	0.534469	0.267235	+	0.963631i	$0.413890\pi$
$888$	0	0
$889$	−6.63806e6	−0.281700
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.17871e6	−0.0494627
$894$	0	0
$895$	6.26311e6	0.261356
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.06118e7	0.437915
$900$	0	0
$901$	2.90490e7	1.19212
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.35428e7	−0.549651
$906$	0	0
$907$	2.30028e7	0.928458	0.464229	−	0.885715i	$-0.346331\pi$
$907$	2.30028e7	0.928458	0.464229	+	0.885715i	$0.346331\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.53237e7	−1.41017	−0.705084	−	0.709124i	$-0.749091\pi$
$911$	−3.53237e7	−1.41017	−0.705084	+	0.709124i	$0.749091\pi$
$912$	0	0
$913$	5.32713e7	2.11503
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.88565e6	0.348952
$918$	0	0
$919$	−1.41122e7	−0.551195	−0.275598	−	0.961273i	$-0.588876\pi$
$919$	−1.41122e7	−0.551195	−0.275598	+	0.961273i	$0.588876\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.81367e6	−0.340528
$924$	0	0
$925$	4.49294e6	0.172654
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.42561e7	1.68242	0.841209	−	0.540709i	$-0.181844\pi$
$929$	4.42561e7	1.68242	0.841209	+	0.540709i	$0.181844\pi$
$930$	0	0
$931$	5.04698e6	0.190835
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.18174e7	−1.19024
$936$	0	0
$937$	−4.43783e7	−1.65128	−0.825642	−	0.564194i	$-0.809187\pi$
$937$	−4.43783e7	−1.65128	−0.825642	+	0.564194i	$0.809187\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.55733e6	0.130964	0.0654818	−	0.997854i	$-0.479142\pi$
$941$	3.55733e6	0.130964	0.0654818	+	0.997854i	$0.479142\pi$
$942$	0	0
$943$	1.85608e7	0.679701
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.69862e6	−0.242723	−0.121361	−	0.992608i	$-0.538726\pi$
$947$	−6.69862e6	−0.242723	−0.121361	+	0.992608i	$0.538726\pi$
$948$	0	0
$949$	−3.83534e6	−0.138241
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.31129e7	0.467698	0.233849	−	0.972273i	$-0.424868\pi$
$953$	1.31129e7	0.467698	0.233849	+	0.972273i	$0.424868\pi$
$954$	0	0
$955$	−4.51341e6	−0.160139
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.30148e7	−0.456972
$960$	0	0
$961$	−1.16234e7	−0.405998
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.32984e7	−0.459708
$966$	0	0
$967$	−5.15407e7	−1.77249	−0.886246	−	0.463215i	$-0.846696\pi$
$967$	−5.15407e7	−1.77249	−0.886246	+	0.463215i	$0.846696\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.13728e6	0.242932	0.121466	−	0.992596i	$-0.461240\pi$
$971$	7.13728e6	0.242932	0.121466	+	0.992596i	$0.461240\pi$
$972$	0	0
$973$	5.75503e6	0.194879
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.94348e7	−1.99207	−0.996034	−	0.0889707i	$-0.971642\pi$
$977$	−5.94348e7	−1.99207	−0.996034	+	0.0889707i	$0.971642\pi$
$978$	0	0
$979$	9.03890e7	3.01411
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.29387e7	0.427078	0.213539	−	0.976934i	$-0.431501\pi$
$983$	1.29387e7	0.427078	0.213539	+	0.976934i	$0.431501\pi$
$984$	0	0
$985$	1.22050e7	0.400817
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.46427e6	0.0476026
$990$	0	0
$991$	3.39160e7	1.09704	0.548518	−	0.836139i	$-0.315192\pi$
$991$	3.39160e7	1.09704	0.548518	+	0.836139i	$0.315192\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.03066e6	0.0330035
$996$	0	0
$997$	1.25463e7	0.399739	0.199869	−	0.979823i	$-0.435948\pi$
$997$	1.25463e7	0.399739	0.199869	+	0.979823i	$0.435948\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.6.a.g.1.1	yes	3
3.2	odd	2		1080.6.a.e.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.6.a.e.1.1	✓	3	3.2	odd	2
1080.6.a.g.1.1	yes	3	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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q+25.0000 q^{5} -177.292 q^{7} +O(q^{10})\)	\(q+25.0000 q^{5} -177.292 q^{7} -778.539 q^{11} +132.539 q^{13} +1634.72 q^{17} +345.079 q^{19} +959.441 q^{23} +625.000 q^{25} +2573.30 q^{29} +4123.80 q^{31} -4432.31 q^{35} +7188.70 q^{37} +19345.5 q^{41} +1526.17 q^{43} -3415.76 q^{47} +14625.6 q^{49} +17770.0 q^{53} -19463.5 q^{55} +17443.5 q^{59} -18471.9 q^{61} +3313.47 q^{65} -54149.5 q^{67} -66498.8 q^{71} -28937.5 q^{73} +138029. q^{77} -88015.0 q^{79} -68424.7 q^{83} +40868.1 q^{85} -116101. q^{89} -23498.1 q^{91} +8626.98 q^{95} -47803.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 75 q^{5} - 30 q^{7}+O(q^{10}) \) 3 * q + 75 * q^5 - 30 * q^7	\( 3 q + 75 q^{5} - 30 q^{7} - 660 q^{11} - 1278 q^{13} + 1731 q^{17} + 1011 q^{19} + 2433 q^{23} + 1875 q^{25} + 3786 q^{29} + 4131 q^{31} - 750 q^{35} - 7122 q^{37} + 17352 q^{41} + 11556 q^{43} - 19548 q^{47} - 7857 q^{49} - 4965 q^{53} - 16500 q^{55} - 32106 q^{59} - 16317 q^{61} - 31950 q^{65} - 51258 q^{67} - 84072 q^{71} - 161892 q^{73} + 142968 q^{77} + 36441 q^{79} - 64581 q^{83} + 43275 q^{85} - 61584 q^{89} - 123588 q^{91} + 25275 q^{95} + 14760 q^{97}+O(q^{100}) \) 3 * q + 75 * q^5 - 30 * q^7 - 660 * q^11 - 1278 * q^13 + 1731 * q^17 + 1011 * q^19 + 2433 * q^23 + 1875 * q^25 + 3786 * q^29 + 4131 * q^31 - 750 * q^35 - 7122 * q^37 + 17352 * q^41 + 11556 * q^43 - 19548 * q^47 - 7857 * q^49 - 4965 * q^53 - 16500 * q^55 - 32106 * q^59 - 16317 * q^61 - 31950 * q^65 - 51258 * q^67 - 84072 * q^71 - 161892 * q^73 + 142968 * q^77 + 36441 * q^79 - 64581 * q^83 + 43275 * q^85 - 61584 * q^89 - 123588 * q^91 + 25275 * q^95 + 14760 * q^97

Introduction

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