LMFDB - Embedded newform 8112.2.a.ce.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.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:	$64.7746461197$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1014)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.04892 q^{5} -0.692021 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.04892 q^{5} -0.692021 q^{7} +1.00000 q^{9} +4.85086 q^{11} -4.04892 q^{15} +7.38404 q^{17} +1.78017 q^{19} +0.692021 q^{21} -5.10992 q^{23} +11.3937 q^{25} -1.00000 q^{27} -3.34481 q^{29} -0.972853 q^{31} -4.85086 q^{33} -2.80194 q^{35} +1.28621 q^{37} +1.50604 q^{41} +8.31767 q^{43} +4.04892 q^{45} +7.20775 q^{47} -6.52111 q^{49} -7.38404 q^{51} +13.4765 q^{53} +19.6407 q^{55} -1.78017 q^{57} -1.30798 q^{59} -0.396125 q^{61} -0.692021 q^{63} -6.05429 q^{67} +5.10992 q^{69} +1.32975 q^{71} -7.65279 q^{73} -11.3937 q^{75} -3.35690 q^{77} +8.33944 q^{79} +1.00000 q^{81} -15.3274 q^{83} +29.8974 q^{85} +3.34481 q^{87} +3.10992 q^{89} +0.972853 q^{93} +7.20775 q^{95} -8.54288 q^{97} +4.85086 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9} + q^{11} - 3 q^{15} + 12 q^{17} + 4 q^{19} - 3 q^{21} - 16 q^{23} + 2 q^{25} - 3 q^{27} + 13 q^{29} - 9 q^{31} - q^{33} - 4 q^{35} + 12 q^{37} + 14 q^{41} + 8 q^{43} + 3 q^{45} + 4 q^{47} - 4 q^{49} - 12 q^{51} + 15 q^{53} + 22 q^{55} - 4 q^{57} - 9 q^{59} - 10 q^{61} + 3 q^{63} - 6 q^{67} + 16 q^{69} + 6 q^{71} - 5 q^{73} - 2 q^{75} - 6 q^{77} + 5 q^{79} + 3 q^{81} - 7 q^{83} + 26 q^{85} - 13 q^{87} + 10 q^{89} + 9 q^{93} + 4 q^{95} - 7 q^{97} + q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9 + q^11 - 3 * q^15 + 12 * q^17 + 4 * q^19 - 3 * q^21 - 16 * q^23 + 2 * q^25 - 3 * q^27 + 13 * q^29 - 9 * q^31 - q^33 - 4 * q^35 + 12 * q^37 + 14 * q^41 + 8 * q^43 + 3 * q^45 + 4 * q^47 - 4 * q^49 - 12 * q^51 + 15 * q^53 + 22 * q^55 - 4 * q^57 - 9 * q^59 - 10 * q^61 + 3 * q^63 - 6 * q^67 + 16 * q^69 + 6 * q^71 - 5 * q^73 - 2 * q^75 - 6 * q^77 + 5 * q^79 + 3 * q^81 - 7 * q^83 + 26 * q^85 - 13 * q^87 + 10 * q^89 + 9 * q^93 + 4 * q^95 - 7 * q^97 + q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	4.04892	1.81073	0.905365	−	0.424633i	$-0.139597\pi$
$5$	4.04892	1.81073	0.905365	+	0.424633i	$0.139597\pi$
$6$	0	0
$7$	−0.692021	−0.261560	−0.130780	−	0.991411i	$-0.541748\pi$
$7$	−0.692021	−0.261560	−0.130780	+	0.991411i	$0.541748\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.85086	1.46259	0.731294	−	0.682062i	$-0.238917\pi$
$11$	4.85086	1.46259	0.731294	+	0.682062i	$0.238917\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−4.04892	−1.04543
$16$	0	0
$17$	7.38404	1.79089	0.895447	−	0.445169i	$-0.146856\pi$
$17$	7.38404	1.79089	0.895447	+	0.445169i	$0.146856\pi$
$18$	0	0
$19$	1.78017	0.408398	0.204199	−	0.978929i	$-0.434541\pi$
$19$	1.78017	0.408398	0.204199	+	0.978929i	$0.434541\pi$
$20$	0	0
$21$	0.692021	0.151011
$22$	0	0
$23$	−5.10992	−1.06549	−0.532746	−	0.846275i	$-0.678840\pi$
$23$	−5.10992	−1.06549	−0.532746	+	0.846275i	$0.678840\pi$
$24$	0	0
$25$	11.3937	2.27875
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.34481	−0.621116	−0.310558	−	0.950554i	$-0.600516\pi$
$29$	−3.34481	−0.621116	−0.310558	+	0.950554i	$0.600516\pi$
$30$	0	0
$31$	−0.972853	−0.174730	−0.0873648	−	0.996176i	$-0.527845\pi$
$31$	−0.972853	−0.174730	−0.0873648	+	0.996176i	$0.527845\pi$
$32$	0	0
$33$	−4.85086	−0.844425
$34$	0	0
$35$	−2.80194	−0.473614
$36$	0	0
$37$	1.28621	0.211451	0.105726	−	0.994395i	$-0.466283\pi$
$37$	1.28621	0.211451	0.105726	+	0.994395i	$0.466283\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.50604	0.235204	0.117602	−	0.993061i	$-0.462479\pi$
$41$	1.50604	0.235204	0.117602	+	0.993061i	$0.462479\pi$
$42$	0	0
$43$	8.31767	1.26843	0.634216	−	0.773156i	$-0.281323\pi$
$43$	8.31767	1.26843	0.634216	+	0.773156i	$0.281323\pi$
$44$	0	0
$45$	4.04892	0.603577
$46$	0	0
$47$	7.20775	1.05136	0.525679	−	0.850683i	$-0.323811\pi$
$47$	7.20775	1.05136	0.525679	+	0.850683i	$0.323811\pi$
$48$	0	0
$49$	−6.52111	−0.931587
$50$	0	0
$51$	−7.38404	−1.03397
$52$	0	0
$53$	13.4765	1.85114	0.925570	−	0.378577i	$-0.123586\pi$
$53$	13.4765	1.85114	0.925570	+	0.378577i	$0.123586\pi$
$54$	0	0
$55$	19.6407	2.64835
$56$	0	0
$57$	−1.78017	−0.235789
$58$	0	0
$59$	−1.30798	−0.170284	−0.0851422	−	0.996369i	$-0.527134\pi$
$59$	−1.30798	−0.170284	−0.0851422	+	0.996369i	$0.527134\pi$
$60$	0	0
$61$	−0.396125	−0.0507185	−0.0253593	−	0.999678i	$-0.508073\pi$
$61$	−0.396125	−0.0507185	−0.0253593	+	0.999678i	$0.508073\pi$
$62$	0	0
$63$	−0.692021	−0.0871865
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.05429	−0.739650	−0.369825	−	0.929101i	$-0.620582\pi$
$67$	−6.05429	−0.739650	−0.369825	+	0.929101i	$0.620582\pi$
$68$	0	0
$69$	5.10992	0.615162
$70$	0	0
$71$	1.32975	0.157812	0.0789061	−	0.996882i	$-0.474857\pi$
$71$	1.32975	0.157812	0.0789061	+	0.996882i	$0.474857\pi$
$72$	0	0
$73$	−7.65279	−0.895692	−0.447846	−	0.894111i	$-0.647809\pi$
$73$	−7.65279	−0.895692	−0.447846	+	0.894111i	$0.647809\pi$
$74$	0	0
$75$	−11.3937	−1.31563
$76$	0	0
$77$	−3.35690	−0.382554
$78$	0	0
$79$	8.33944	0.938260	0.469130	−	0.883129i	$-0.344568\pi$
$79$	8.33944	0.938260	0.469130	+	0.883129i	$0.344568\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.3274	−1.68240	−0.841198	−	0.540727i	$-0.818149\pi$
$83$	−15.3274	−1.68240	−0.841198	+	0.540727i	$0.818149\pi$
$84$	0	0
$85$	29.8974	3.24283
$86$	0	0
$87$	3.34481	0.358602
$88$	0	0
$89$	3.10992	0.329650	0.164825	−	0.986323i	$-0.447294\pi$
$89$	3.10992	0.329650	0.164825	+	0.986323i	$0.447294\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.972853	0.100880
$94$	0	0
$95$	7.20775	0.739500
$96$	0	0
$97$	−8.54288	−0.867398	−0.433699	−	0.901058i	$-0.642792\pi$
$97$	−8.54288	−0.867398	−0.433699	+	0.901058i	$0.642792\pi$
$98$	0	0
$99$	4.85086	0.487529
$100$	0	0
$101$	−11.9976	−1.19381	−0.596903	−	0.802313i	$-0.703602\pi$
$101$	−11.9976	−1.19381	−0.596903	+	0.802313i	$0.703602\pi$
$102$	0	0
$103$	12.3230	1.21423	0.607113	−	0.794616i	$-0.292328\pi$
$103$	12.3230	1.21423	0.607113	+	0.794616i	$0.292328\pi$
$104$	0	0
$105$	2.80194	0.273441
$106$	0	0
$107$	−5.89977	−0.570353	−0.285176	−	0.958475i	$-0.592052\pi$
$107$	−5.89977	−0.570353	−0.285176	+	0.958475i	$0.592052\pi$
$108$	0	0
$109$	0.792249	0.0758837	0.0379418	−	0.999280i	$-0.487920\pi$
$109$	0.792249	0.0758837	0.0379418	+	0.999280i	$0.487920\pi$
$110$	0	0
$111$	−1.28621	−0.122081
$112$	0	0
$113$	6.21983	0.585113	0.292556	−	0.956248i	$-0.405494\pi$
$113$	6.21983	0.585113	0.292556	+	0.956248i	$0.405494\pi$
$114$	0	0
$115$	−20.6896	−1.92932
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.10992	−0.468425
$120$	0	0
$121$	12.5308	1.13916
$122$	0	0
$123$	−1.50604	−0.135795
$124$	0	0
$125$	25.8877	2.31547
$126$	0	0
$127$	6.00538	0.532891	0.266446	−	0.963850i	$-0.414151\pi$
$127$	6.00538	0.532891	0.266446	+	0.963850i	$0.414151\pi$
$128$	0	0
$129$	−8.31767	−0.732330
$130$	0	0
$131$	−8.81700	−0.770345	−0.385173	−	0.922845i	$-0.625858\pi$
$131$	−8.81700	−0.770345	−0.385173	+	0.922845i	$0.625858\pi$
$132$	0	0
$133$	−1.23191	−0.106821
$134$	0	0
$135$	−4.04892	−0.348475
$136$	0	0
$137$	−15.7560	−1.34613	−0.673063	−	0.739585i	$-0.735022\pi$
$137$	−15.7560	−1.34613	−0.673063	+	0.739585i	$0.735022\pi$
$138$	0	0
$139$	6.09783	0.517212	0.258606	−	0.965983i	$-0.416737\pi$
$139$	6.09783	0.517212	0.258606	+	0.965983i	$0.416737\pi$
$140$	0	0
$141$	−7.20775	−0.607002
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−13.5429	−1.12467
$146$	0	0
$147$	6.52111	0.537852
$148$	0	0
$149$	−2.55257	−0.209114	−0.104557	−	0.994519i	$-0.533343\pi$
$149$	−2.55257	−0.209114	−0.104557	+	0.994519i	$0.533343\pi$
$150$	0	0
$151$	17.7168	1.44177	0.720885	−	0.693054i	$-0.243735\pi$
$151$	17.7168	1.44177	0.720885	+	0.693054i	$0.243735\pi$
$152$	0	0
$153$	7.38404	0.596964
$154$	0	0
$155$	−3.93900	−0.316388
$156$	0	0
$157$	−6.31767	−0.504205	−0.252102	−	0.967701i	$-0.581122\pi$
$157$	−6.31767	−0.504205	−0.252102	+	0.967701i	$0.581122\pi$
$158$	0	0
$159$	−13.4765	−1.06876
$160$	0	0
$161$	3.53617	0.278689
$162$	0	0
$163$	14.5918	1.14292	0.571459	−	0.820631i	$-0.306378\pi$
$163$	14.5918	1.14292	0.571459	+	0.820631i	$0.306378\pi$
$164$	0	0
$165$	−19.6407	−1.52903
$166$	0	0
$167$	−19.5013	−1.50905	−0.754526	−	0.656270i	$-0.772133\pi$
$167$	−19.5013	−1.50905	−0.754526	+	0.656270i	$0.772133\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1.78017	0.136133
$172$	0	0
$173$	−9.29052	−0.706345	−0.353173	−	0.935558i	$-0.614897\pi$
$173$	−9.29052	−0.706345	−0.353173	+	0.935558i	$0.614897\pi$
$174$	0	0
$175$	−7.88471	−0.596028
$176$	0	0
$177$	1.30798	0.0983137
$178$	0	0
$179$	−22.7928	−1.70362	−0.851808	−	0.523853i	$-0.824494\pi$
$179$	−22.7928	−1.70362	−0.851808	+	0.523853i	$0.824494\pi$
$180$	0	0
$181$	0.537500	0.0399520	0.0199760	−	0.999800i	$-0.493641\pi$
$181$	0.537500	0.0399520	0.0199760	+	0.999800i	$0.493641\pi$
$182$	0	0
$183$	0.396125	0.0292824
$184$	0	0
$185$	5.20775	0.382881
$186$	0	0
$187$	35.8189	2.61934
$188$	0	0
$189$	0.692021	0.0503372
$190$	0	0
$191$	9.79954	0.709070	0.354535	−	0.935043i	$-0.384639\pi$
$191$	9.79954	0.709070	0.354535	+	0.935043i	$0.384639\pi$
$192$	0	0
$193$	14.1957	1.02183	0.510913	−	0.859632i	$-0.329307\pi$
$193$	14.1957	1.02183	0.510913	+	0.859632i	$0.329307\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.00969	0.214431	0.107216	−	0.994236i	$-0.465806\pi$
$197$	3.00969	0.214431	0.107216	+	0.994236i	$0.465806\pi$
$198$	0	0
$199$	−12.8944	−0.914059	−0.457030	−	0.889451i	$-0.651087\pi$
$199$	−12.8944	−0.914059	−0.457030	+	0.889451i	$0.651087\pi$
$200$	0	0
$201$	6.05429	0.427037
$202$	0	0
$203$	2.31468	0.162459
$204$	0	0
$205$	6.09783	0.425891
$206$	0	0
$207$	−5.10992	−0.355164
$208$	0	0
$209$	8.63533	0.597319
$210$	0	0
$211$	−7.79954	−0.536943	−0.268471	−	0.963288i	$-0.586518\pi$
$211$	−7.79954	−0.536943	−0.268471	+	0.963288i	$0.586518\pi$
$212$	0	0
$213$	−1.32975	−0.0911129
$214$	0	0
$215$	33.6775	2.29679
$216$	0	0
$217$	0.673235	0.0457022
$218$	0	0
$219$	7.65279	0.517128
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−12.1957	−0.816682	−0.408341	−	0.912829i	$-0.633893\pi$
$223$	−12.1957	−0.816682	−0.408341	+	0.912829i	$0.633893\pi$
$224$	0	0
$225$	11.3937	0.759582
$226$	0	0
$227$	6.74333	0.447571	0.223785	−	0.974638i	$-0.428159\pi$
$227$	6.74333	0.447571	0.223785	+	0.974638i	$0.428159\pi$
$228$	0	0
$229$	19.8237	1.30999	0.654994	−	0.755634i	$-0.272671\pi$
$229$	19.8237	1.30999	0.654994	+	0.755634i	$0.272671\pi$
$230$	0	0
$231$	3.35690	0.220868
$232$	0	0
$233$	−30.0301	−1.96734	−0.983670	−	0.179983i	$-0.942396\pi$
$233$	−30.0301	−1.96734	−0.983670	+	0.179983i	$0.942396\pi$
$234$	0	0
$235$	29.1836	1.90373
$236$	0	0
$237$	−8.33944	−0.541705
$238$	0	0
$239$	22.0978	1.42939	0.714695	−	0.699436i	$-0.246565\pi$
$239$	22.0978	1.42939	0.714695	+	0.699436i	$0.246565\pi$
$240$	0	0
$241$	−10.1274	−0.652362	−0.326181	−	0.945307i	$-0.605762\pi$
$241$	−10.1274	−0.652362	−0.326181	+	0.945307i	$0.605762\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−26.4034	−1.68685
$246$	0	0
$247$	0	0
$248$	0	0
$249$	15.3274	0.971332
$250$	0	0
$251$	−5.54719	−0.350135	−0.175068	−	0.984556i	$-0.556014\pi$
$251$	−5.54719	−0.350135	−0.175068	+	0.984556i	$0.556014\pi$
$252$	0	0
$253$	−24.7875	−1.55837
$254$	0	0
$255$	−29.8974	−1.87225
$256$	0	0
$257$	−13.7995	−0.860792	−0.430396	−	0.902640i	$-0.641626\pi$
$257$	−13.7995	−0.860792	−0.430396	+	0.902640i	$0.641626\pi$
$258$	0	0
$259$	−0.890084	−0.0553071
$260$	0	0
$261$	−3.34481	−0.207039
$262$	0	0
$263$	22.4698	1.38555	0.692773	−	0.721155i	$-0.256389\pi$
$263$	22.4698	1.38555	0.692773	+	0.721155i	$0.256389\pi$
$264$	0	0
$265$	54.5652	3.35192
$266$	0	0
$267$	−3.10992	−0.190324
$268$	0	0
$269$	−26.0140	−1.58610	−0.793051	−	0.609156i	$-0.791509\pi$
$269$	−26.0140	−1.58610	−0.793051	+	0.609156i	$0.791509\pi$
$270$	0	0
$271$	2.88471	0.175233	0.0876167	−	0.996154i	$-0.472075\pi$
$271$	2.88471	0.175233	0.0876167	+	0.996154i	$0.472075\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	55.2693	3.33287
$276$	0	0
$277$	−1.46250	−0.0878731	−0.0439366	−	0.999034i	$-0.513990\pi$
$277$	−1.46250	−0.0878731	−0.0439366	+	0.999034i	$0.513990\pi$
$278$	0	0
$279$	−0.972853	−0.0582432
$280$	0	0
$281$	−5.68233	−0.338980	−0.169490	−	0.985532i	$-0.554212\pi$
$281$	−5.68233	−0.338980	−0.169490	+	0.985532i	$0.554212\pi$
$282$	0	0
$283$	25.2078	1.49845	0.749223	−	0.662318i	$-0.230427\pi$
$283$	25.2078	1.49845	0.749223	+	0.662318i	$0.230427\pi$
$284$	0	0
$285$	−7.20775	−0.426950
$286$	0	0
$287$	−1.04221	−0.0615199
$288$	0	0
$289$	37.5241	2.20730
$290$	0	0
$291$	8.54288	0.500792
$292$	0	0
$293$	7.14914	0.417658	0.208829	−	0.977952i	$-0.433035\pi$
$293$	7.14914	0.417658	0.208829	+	0.977952i	$0.433035\pi$
$294$	0	0
$295$	−5.29590	−0.308339
$296$	0	0
$297$	−4.85086	−0.281475
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−5.75600	−0.331771
$302$	0	0
$303$	11.9976	0.689245
$304$	0	0
$305$	−1.60388	−0.0918376
$306$	0	0
$307$	−17.9952	−1.02704	−0.513521	−	0.858077i	$-0.671659\pi$
$307$	−17.9952	−1.02704	−0.513521	+	0.858077i	$0.671659\pi$
$308$	0	0
$309$	−12.3230	−0.701033
$310$	0	0
$311$	−3.32975	−0.188813	−0.0944064	−	0.995534i	$-0.530095\pi$
$311$	−3.32975	−0.188813	−0.0944064	+	0.995534i	$0.530095\pi$
$312$	0	0
$313$	−17.8834	−1.01083	−0.505414	−	0.862877i	$-0.668660\pi$
$313$	−17.8834	−1.01083	−0.505414	+	0.862877i	$0.668660\pi$
$314$	0	0
$315$	−2.80194	−0.157871
$316$	0	0
$317$	−4.39373	−0.246777	−0.123388	−	0.992358i	$-0.539376\pi$
$317$	−4.39373	−0.246777	−0.123388	+	0.992358i	$0.539376\pi$
$318$	0	0
$319$	−16.2252	−0.908437
$320$	0	0
$321$	5.89977	0.329293
$322$	0	0
$323$	13.1448	0.731398
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.792249	−0.0438115
$328$	0	0
$329$	−4.98792	−0.274993
$330$	0	0
$331$	−25.6775	−1.41137	−0.705683	−	0.708528i	$-0.749360\pi$
$331$	−25.6775	−1.41137	−0.705683	+	0.708528i	$0.749360\pi$
$332$	0	0
$333$	1.28621	0.0704838
$334$	0	0
$335$	−24.5133	−1.33931
$336$	0	0
$337$	−24.6504	−1.34279	−0.671396	−	0.741098i	$-0.734305\pi$
$337$	−24.6504	−1.34279	−0.671396	+	0.741098i	$0.734305\pi$
$338$	0	0
$339$	−6.21983	−0.337815
$340$	0	0
$341$	−4.71917	−0.255557
$342$	0	0
$343$	9.35690	0.505225
$344$	0	0
$345$	20.6896	1.11389
$346$	0	0
$347$	−14.2959	−0.767444	−0.383722	−	0.923449i	$-0.625358\pi$
$347$	−14.2959	−0.767444	−0.383722	+	0.923449i	$0.625358\pi$
$348$	0	0
$349$	−11.0616	−0.592113	−0.296057	−	0.955170i	$-0.595672\pi$
$349$	−11.0616	−0.592113	−0.296057	+	0.955170i	$0.595672\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.5047	0.559109	0.279555	−	0.960130i	$-0.409813\pi$
$353$	10.5047	0.559109	0.279555	+	0.960130i	$0.409813\pi$
$354$	0	0
$355$	5.38404	0.285755
$356$	0	0
$357$	5.10992	0.270445
$358$	0	0
$359$	5.10992	0.269691	0.134846	−	0.990867i	$-0.456946\pi$
$359$	5.10992	0.269691	0.134846	+	0.990867i	$0.456946\pi$
$360$	0	0
$361$	−15.8310	−0.833211
$362$	0	0
$363$	−12.5308	−0.657696
$364$	0	0
$365$	−30.9855	−1.62186
$366$	0	0
$367$	8.44803	0.440983	0.220492	−	0.975389i	$-0.429234\pi$
$367$	8.44803	0.440983	0.220492	+	0.975389i	$0.429234\pi$
$368$	0	0
$369$	1.50604	0.0784014
$370$	0	0
$371$	−9.32603	−0.484183
$372$	0	0
$373$	7.69096	0.398223	0.199111	−	0.979977i	$-0.436194\pi$
$373$	7.69096	0.398223	0.199111	+	0.979977i	$0.436194\pi$
$374$	0	0
$375$	−25.8877	−1.33683
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−11.4034	−0.585754	−0.292877	−	0.956150i	$-0.594613\pi$
$379$	−11.4034	−0.585754	−0.292877	+	0.956150i	$0.594613\pi$
$380$	0	0
$381$	−6.00538	−0.307665
$382$	0	0
$383$	20.6703	1.05620	0.528100	−	0.849182i	$-0.322905\pi$
$383$	20.6703	1.05620	0.528100	+	0.849182i	$0.322905\pi$
$384$	0	0
$385$	−13.5918	−0.692702
$386$	0	0
$387$	8.31767	0.422811
$388$	0	0
$389$	17.4776	0.886148	0.443074	−	0.896485i	$-0.353888\pi$
$389$	17.4776	0.886148	0.443074	+	0.896485i	$0.353888\pi$
$390$	0	0
$391$	−37.7318	−1.90818
$392$	0	0
$393$	8.81700	0.444759
$394$	0	0
$395$	33.7657	1.69894
$396$	0	0
$397$	19.3599	0.971645	0.485822	−	0.874058i	$-0.338520\pi$
$397$	19.3599	0.971645	0.485822	+	0.874058i	$0.338520\pi$
$398$	0	0
$399$	1.23191	0.0616728
$400$	0	0
$401$	14.4832	0.723257	0.361628	−	0.932322i	$-0.382221\pi$
$401$	14.4832	0.723257	0.361628	+	0.932322i	$0.382221\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	4.04892	0.201192
$406$	0	0
$407$	6.23921	0.309266
$408$	0	0
$409$	18.8984	0.934468	0.467234	−	0.884134i	$-0.345251\pi$
$409$	18.8984	0.934468	0.467234	+	0.884134i	$0.345251\pi$
$410$	0	0
$411$	15.7560	0.777186
$412$	0	0
$413$	0.905149	0.0445395
$414$	0	0
$415$	−62.0592	−3.04637
$416$	0	0
$417$	−6.09783	−0.298612
$418$	0	0
$419$	21.7603	1.06306	0.531531	−	0.847039i	$-0.321617\pi$
$419$	21.7603	1.06306	0.531531	+	0.847039i	$0.321617\pi$
$420$	0	0
$421$	−20.5918	−1.00358	−0.501791	−	0.864989i	$-0.667325\pi$
$421$	−20.5918	−1.00358	−0.501791	+	0.864989i	$0.667325\pi$
$422$	0	0
$423$	7.20775	0.350453
$424$	0	0
$425$	84.1318	4.08099
$426$	0	0
$427$	0.274127	0.0132659
$428$	0	0
$429$	0	0
$430$	0	0
$431$	34.5133	1.66245	0.831224	−	0.555937i	$-0.187640\pi$
$431$	34.5133	1.66245	0.831224	+	0.555937i	$0.187640\pi$
$432$	0	0
$433$	−2.12631	−0.102184	−0.0510920	−	0.998694i	$-0.516270\pi$
$433$	−2.12631	−0.102184	−0.0510920	+	0.998694i	$0.516270\pi$
$434$	0	0
$435$	13.5429	0.649331
$436$	0	0
$437$	−9.09651	−0.435145
$438$	0	0
$439$	−21.8321	−1.04199	−0.520994	−	0.853560i	$-0.674439\pi$
$439$	−21.8321	−1.04199	−0.520994	+	0.853560i	$0.674439\pi$
$440$	0	0
$441$	−6.52111	−0.310529
$442$	0	0
$443$	7.54048	0.358259	0.179130	−	0.983825i	$-0.442672\pi$
$443$	7.54048	0.358259	0.179130	+	0.983825i	$0.442672\pi$
$444$	0	0
$445$	12.5918	0.596908
$446$	0	0
$447$	2.55257	0.120732
$448$	0	0
$449$	−19.7560	−0.932343	−0.466172	−	0.884694i	$-0.654367\pi$
$449$	−19.7560	−0.932343	−0.466172	+	0.884694i	$0.654367\pi$
$450$	0	0
$451$	7.30559	0.344007
$452$	0	0
$453$	−17.7168	−0.832407
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−23.8582	−1.11604	−0.558019	−	0.829828i	$-0.688438\pi$
$457$	−23.8582	−1.11604	−0.558019	+	0.829828i	$0.688438\pi$
$458$	0	0
$459$	−7.38404	−0.344658
$460$	0	0
$461$	17.5773	0.818657	0.409329	−	0.912387i	$-0.365763\pi$
$461$	17.5773	0.818657	0.409329	+	0.912387i	$0.365763\pi$
$462$	0	0
$463$	23.8431	1.10808	0.554041	−	0.832489i	$-0.313085\pi$
$463$	23.8431	1.10808	0.554041	+	0.832489i	$0.313085\pi$
$464$	0	0
$465$	3.93900	0.182667
$466$	0	0
$467$	−8.61058	−0.398450	−0.199225	−	0.979954i	$-0.563842\pi$
$467$	−8.61058	−0.398450	−0.199225	+	0.979954i	$0.563842\pi$
$468$	0	0
$469$	4.18970	0.193462
$470$	0	0
$471$	6.31767	0.291103
$472$	0	0
$473$	40.3478	1.85519
$474$	0	0
$475$	20.2828	0.930636
$476$	0	0
$477$	13.4765	0.617047
$478$	0	0
$479$	6.58104	0.300695	0.150348	−	0.988633i	$-0.451961\pi$
$479$	6.58104	0.300695	0.150348	+	0.988633i	$0.451961\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−3.53617	−0.160901
$484$	0	0
$485$	−34.5894	−1.57062
$486$	0	0
$487$	−23.2760	−1.05474	−0.527369	−	0.849636i	$-0.676822\pi$
$487$	−23.2760	−1.05474	−0.527369	+	0.849636i	$0.676822\pi$
$488$	0	0
$489$	−14.5918	−0.659864
$490$	0	0
$491$	15.5405	0.701332	0.350666	−	0.936501i	$-0.385955\pi$
$491$	15.5405	0.701332	0.350666	+	0.936501i	$0.385955\pi$
$492$	0	0
$493$	−24.6983	−1.11235
$494$	0	0
$495$	19.6407	0.882784
$496$	0	0
$497$	−0.920215	−0.0412773
$498$	0	0
$499$	9.53617	0.426898	0.213449	−	0.976954i	$-0.431530\pi$
$499$	9.53617	0.426898	0.213449	+	0.976954i	$0.431530\pi$
$500$	0	0
$501$	19.5013	0.871252
$502$	0	0
$503$	13.8345	0.616848	0.308424	−	0.951249i	$-0.400198\pi$
$503$	13.8345	0.616848	0.308424	+	0.951249i	$0.400198\pi$
$504$	0	0
$505$	−48.5773	−2.16166
$506$	0	0
$507$	0	0
$508$	0	0
$509$	40.9638	1.81569	0.907843	−	0.419310i	$-0.137728\pi$
$509$	40.9638	1.81569	0.907843	+	0.419310i	$0.137728\pi$
$510$	0	0
$511$	5.29590	0.234277
$512$	0	0
$513$	−1.78017	−0.0785963
$514$	0	0
$515$	49.8950	2.19864
$516$	0	0
$517$	34.9638	1.53770
$518$	0	0
$519$	9.29052	0.407809
$520$	0	0
$521$	36.3672	1.59327	0.796637	−	0.604457i	$-0.206610\pi$
$521$	36.3672	1.59327	0.796637	+	0.604457i	$0.206610\pi$
$522$	0	0
$523$	6.03013	0.263679	0.131840	−	0.991271i	$-0.457912\pi$
$523$	6.03013	0.263679	0.131840	+	0.991271i	$0.457912\pi$
$524$	0	0
$525$	7.88471	0.344117
$526$	0	0
$527$	−7.18359	−0.312922
$528$	0	0
$529$	3.11124	0.135271
$530$	0	0
$531$	−1.30798	−0.0567614
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−23.8877	−1.03275
$536$	0	0
$537$	22.7928	0.983584
$538$	0	0
$539$	−31.6329	−1.36253
$540$	0	0
$541$	7.92154	0.340574	0.170287	−	0.985395i	$-0.445531\pi$
$541$	7.92154	0.340574	0.170287	+	0.985395i	$0.445531\pi$
$542$	0	0
$543$	−0.537500	−0.0230663
$544$	0	0
$545$	3.20775	0.137405
$546$	0	0
$547$	−18.4155	−0.787390	−0.393695	−	0.919241i	$-0.628803\pi$
$547$	−18.4155	−0.787390	−0.393695	+	0.919241i	$0.628803\pi$
$548$	0	0
$549$	−0.396125	−0.0169062
$550$	0	0
$551$	−5.95433	−0.253663
$552$	0	0
$553$	−5.77107	−0.245411
$554$	0	0
$555$	−5.20775	−0.221057
$556$	0	0
$557$	23.9758	1.01589	0.507944	−	0.861390i	$-0.330406\pi$
$557$	23.9758	1.01589	0.507944	+	0.861390i	$0.330406\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−35.8189	−1.51228
$562$	0	0
$563$	2.29291	0.0966348	0.0483174	−	0.998832i	$-0.484614\pi$
$563$	2.29291	0.0966348	0.0483174	+	0.998832i	$0.484614\pi$
$564$	0	0
$565$	25.1836	1.05948
$566$	0	0
$567$	−0.692021	−0.0290622
$568$	0	0
$569$	−44.3430	−1.85896	−0.929478	−	0.368878i	$-0.879742\pi$
$569$	−44.3430	−1.85896	−0.929478	+	0.368878i	$0.879742\pi$
$570$	0	0
$571$	15.2707	0.639058	0.319529	−	0.947577i	$-0.396475\pi$
$571$	15.2707	0.639058	0.319529	+	0.947577i	$0.396475\pi$
$572$	0	0
$573$	−9.79954	−0.409382
$574$	0	0
$575$	−58.2210	−2.42798
$576$	0	0
$577$	−8.77048	−0.365120	−0.182560	−	0.983195i	$-0.558438\pi$
$577$	−8.77048	−0.365120	−0.182560	+	0.983195i	$0.558438\pi$
$578$	0	0
$579$	−14.1957	−0.589952
$580$	0	0
$581$	10.6069	0.440047
$582$	0	0
$583$	65.3726	2.70745
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−38.1430	−1.57433	−0.787166	−	0.616742i	$-0.788452\pi$
$587$	−38.1430	−1.57433	−0.787166	+	0.616742i	$0.788452\pi$
$588$	0	0
$589$	−1.73184	−0.0713593
$590$	0	0
$591$	−3.00969	−0.123802
$592$	0	0
$593$	37.9517	1.55849	0.779244	−	0.626720i	$-0.215603\pi$
$593$	37.9517	1.55849	0.779244	+	0.626720i	$0.215603\pi$
$594$	0	0
$595$	−20.6896	−0.848192
$596$	0	0
$597$	12.8944	0.527732
$598$	0	0
$599$	3.57971	0.146263	0.0731315	−	0.997322i	$-0.476701\pi$
$599$	3.57971	0.146263	0.0731315	+	0.997322i	$0.476701\pi$
$600$	0	0
$601$	−5.71678	−0.233192	−0.116596	−	0.993179i	$-0.537198\pi$
$601$	−5.71678	−0.233192	−0.116596	+	0.993179i	$0.537198\pi$
$602$	0	0
$603$	−6.05429	−0.246550
$604$	0	0
$605$	50.7362	2.06272
$606$	0	0
$607$	−22.4286	−0.910351	−0.455175	−	0.890402i	$-0.650423\pi$
$607$	−22.4286	−0.910351	−0.455175	+	0.890402i	$0.650423\pi$
$608$	0	0
$609$	−2.31468	−0.0937957
$610$	0	0
$611$	0	0
$612$	0	0
$613$	39.9603	1.61398	0.806991	−	0.590564i	$-0.201095\pi$
$613$	39.9603	1.61398	0.806991	+	0.590564i	$0.201095\pi$
$614$	0	0
$615$	−6.09783	−0.245888
$616$	0	0
$617$	−31.4470	−1.26601	−0.633003	−	0.774149i	$-0.718178\pi$
$617$	−31.4470	−1.26601	−0.633003	+	0.774149i	$0.718178\pi$
$618$	0	0
$619$	−29.3685	−1.18042	−0.590210	−	0.807250i	$-0.700955\pi$
$619$	−29.3685	−1.18042	−0.590210	+	0.807250i	$0.700955\pi$
$620$	0	0
$621$	5.10992	0.205054
$622$	0	0
$623$	−2.15213	−0.0862232
$624$	0	0
$625$	47.8485	1.91394
$626$	0	0
$627$	−8.63533	−0.344862
$628$	0	0
$629$	9.49742	0.378687
$630$	0	0
$631$	21.6799	0.863065	0.431532	−	0.902097i	$-0.357973\pi$
$631$	21.6799	0.863065	0.431532	+	0.902097i	$0.357973\pi$
$632$	0	0
$633$	7.79954	0.310004
$634$	0	0
$635$	24.3153	0.964922
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.32975	0.0526040
$640$	0	0
$641$	14.0108	0.553391	0.276696	−	0.960958i	$-0.410761\pi$
$641$	14.0108	0.553391	0.276696	+	0.960958i	$0.410761\pi$
$642$	0	0
$643$	30.7875	1.21414	0.607070	−	0.794649i	$-0.292345\pi$
$643$	30.7875	1.21414	0.607070	+	0.794649i	$0.292345\pi$
$644$	0	0
$645$	−33.6775	−1.32605
$646$	0	0
$647$	16.6025	0.652713	0.326357	−	0.945247i	$-0.394179\pi$
$647$	16.6025	0.652713	0.326357	+	0.945247i	$0.394179\pi$
$648$	0	0
$649$	−6.34481	−0.249056
$650$	0	0
$651$	−0.673235	−0.0263862
$652$	0	0
$653$	28.5459	1.11709	0.558543	−	0.829476i	$-0.311361\pi$
$653$	28.5459	1.11709	0.558543	+	0.829476i	$0.311361\pi$
$654$	0	0
$655$	−35.6993	−1.39489
$656$	0	0
$657$	−7.65279	−0.298564
$658$	0	0
$659$	−27.7187	−1.07977	−0.539884	−	0.841740i	$-0.681532\pi$
$659$	−27.7187	−1.07977	−0.539884	+	0.841740i	$0.681532\pi$
$660$	0	0
$661$	−10.8009	−0.420105	−0.210053	−	0.977690i	$-0.567364\pi$
$661$	−10.8009	−0.420105	−0.210053	+	0.977690i	$0.567364\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.98792	−0.193423
$666$	0	0
$667$	17.0917	0.661794
$668$	0	0
$669$	12.1957	0.471512
$670$	0	0
$671$	−1.92154	−0.0741803
$672$	0	0
$673$	16.3260	0.629322	0.314661	−	0.949204i	$-0.398109\pi$
$673$	16.3260	0.629322	0.314661	+	0.949204i	$0.398109\pi$
$674$	0	0
$675$	−11.3937	−0.438545
$676$	0	0
$677$	41.4252	1.59210	0.796050	−	0.605231i	$-0.206919\pi$
$677$	41.4252	1.59210	0.796050	+	0.605231i	$0.206919\pi$
$678$	0	0
$679$	5.91185	0.226876
$680$	0	0
$681$	−6.74333	−0.258405
$682$	0	0
$683$	31.2325	1.19508	0.597539	−	0.801840i	$-0.296145\pi$
$683$	31.2325	1.19508	0.597539	+	0.801840i	$0.296145\pi$
$684$	0	0
$685$	−63.7948	−2.43747
$686$	0	0
$687$	−19.8237	−0.756322
$688$	0	0
$689$	0	0
$690$	0	0
$691$	24.9638	0.949666	0.474833	−	0.880076i	$-0.342508\pi$
$691$	24.9638	0.949666	0.474833	+	0.880076i	$0.342508\pi$
$692$	0	0
$693$	−3.35690	−0.127518
$694$	0	0
$695$	24.6896	0.936531
$696$	0	0
$697$	11.1207	0.421225
$698$	0	0
$699$	30.0301	1.13584
$700$	0	0
$701$	8.17151	0.308634	0.154317	−	0.988021i	$-0.450682\pi$
$701$	8.17151	0.308634	0.154317	+	0.988021i	$0.450682\pi$
$702$	0	0
$703$	2.28967	0.0863564
$704$	0	0
$705$	−29.1836	−1.09912
$706$	0	0
$707$	8.30260	0.312251
$708$	0	0
$709$	36.7982	1.38199	0.690993	−	0.722861i	$-0.257174\pi$
$709$	36.7982	1.38199	0.690993	+	0.722861i	$0.257174\pi$
$710$	0	0
$711$	8.33944	0.312753
$712$	0	0
$713$	4.97120	0.186173
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−22.0978	−0.825259
$718$	0	0
$719$	−35.2223	−1.31357	−0.656786	−	0.754077i	$-0.728084\pi$
$719$	−35.2223	−1.31357	−0.656786	+	0.754077i	$0.728084\pi$
$720$	0	0
$721$	−8.52781	−0.317592
$722$	0	0
$723$	10.1274	0.376641
$724$	0	0
$725$	−38.1099	−1.41537
$726$	0	0
$727$	−40.6872	−1.50901	−0.754503	−	0.656297i	$-0.772122\pi$
$727$	−40.6872	−1.50901	−0.754503	+	0.656297i	$0.772122\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	61.4180	2.27163
$732$	0	0
$733$	−27.1400	−1.00244	−0.501220	−	0.865320i	$-0.667115\pi$
$733$	−27.1400	−1.00244	−0.501220	+	0.865320i	$0.667115\pi$
$734$	0	0
$735$	26.4034	0.973905
$736$	0	0
$737$	−29.3685	−1.08180
$738$	0	0
$739$	3.72587	0.137058	0.0685292	−	0.997649i	$-0.478169\pi$
$739$	3.72587	0.137058	0.0685292	+	0.997649i	$0.478169\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.8586	0.801915	0.400958	−	0.916097i	$-0.368677\pi$
$743$	21.8586	0.801915	0.400958	+	0.916097i	$0.368677\pi$
$744$	0	0
$745$	−10.3351	−0.378650
$746$	0	0
$747$	−15.3274	−0.560799
$748$	0	0
$749$	4.08277	0.149181
$750$	0	0
$751$	−53.3642	−1.94729	−0.973644	−	0.228075i	$-0.926757\pi$
$751$	−53.3642	−1.94729	−0.973644	+	0.228075i	$0.926757\pi$
$752$	0	0
$753$	5.54719	0.202151
$754$	0	0
$755$	71.7338	2.61066
$756$	0	0
$757$	−4.63533	−0.168474	−0.0842370	−	0.996446i	$-0.526845\pi$
$757$	−4.63533	−0.168474	−0.0842370	+	0.996446i	$0.526845\pi$
$758$	0	0
$759$	24.7875	0.899728
$760$	0	0
$761$	−11.5603	−0.419062	−0.209531	−	0.977802i	$-0.567194\pi$
$761$	−11.5603	−0.419062	−0.209531	+	0.977802i	$0.567194\pi$
$762$	0	0
$763$	−0.548253	−0.0198481
$764$	0	0
$765$	29.8974	1.08094
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−14.7439	−0.531679	−0.265840	−	0.964017i	$-0.585649\pi$
$769$	−14.7439	−0.531679	−0.265840	+	0.964017i	$0.585649\pi$
$770$	0	0
$771$	13.7995	0.496978
$772$	0	0
$773$	6.42268	0.231008	0.115504	−	0.993307i	$-0.463152\pi$
$773$	6.42268	0.231008	0.115504	+	0.993307i	$0.463152\pi$
$774$	0	0
$775$	−11.0844	−0.398164
$776$	0	0
$777$	0.890084	0.0319316
$778$	0	0
$779$	2.68100	0.0960570
$780$	0	0
$781$	6.45042	0.230814
$782$	0	0
$783$	3.34481	0.119534
$784$	0	0
$785$	−25.5797	−0.912979
$786$	0	0
$787$	43.4336	1.54824	0.774119	−	0.633040i	$-0.218193\pi$
$787$	43.4336	1.54824	0.774119	+	0.633040i	$0.218193\pi$
$788$	0	0
$789$	−22.4698	−0.799946
$790$	0	0
$791$	−4.30426	−0.153042
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−54.5652	−1.93523
$796$	0	0
$797$	21.0164	0.744439	0.372219	−	0.928145i	$-0.378597\pi$
$797$	21.0164	0.744439	0.372219	+	0.928145i	$0.378597\pi$
$798$	0	0
$799$	53.2223	1.88287
$800$	0	0
$801$	3.10992	0.109883
$802$	0	0
$803$	−37.1226	−1.31003
$804$	0	0
$805$	14.3177	0.504631
$806$	0	0
$807$	26.0140	0.915736
$808$	0	0
$809$	−8.32245	−0.292602	−0.146301	−	0.989240i	$-0.546737\pi$
$809$	−8.32245	−0.292602	−0.146301	+	0.989240i	$0.546737\pi$
$810$	0	0
$811$	14.4638	0.507894	0.253947	−	0.967218i	$-0.418271\pi$
$811$	14.4638	0.507894	0.253947	+	0.967218i	$0.418271\pi$
$812$	0	0
$813$	−2.88471	−0.101171
$814$	0	0
$815$	59.0810	2.06952
$816$	0	0
$817$	14.8068	0.518026
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38.4161	−1.34073	−0.670365	−	0.742031i	$-0.733863\pi$
$821$	−38.4161	−1.34073	−0.670365	+	0.742031i	$0.733863\pi$
$822$	0	0
$823$	40.9748	1.42829	0.714145	−	0.699997i	$-0.246816\pi$
$823$	40.9748	1.42829	0.714145	+	0.699997i	$0.246816\pi$
$824$	0	0
$825$	−55.2693	−1.92423
$826$	0	0
$827$	−3.51035	−0.122067	−0.0610335	−	0.998136i	$-0.519440\pi$
$827$	−3.51035	−0.122067	−0.0610335	+	0.998136i	$0.519440\pi$
$828$	0	0
$829$	−13.2185	−0.459098	−0.229549	−	0.973297i	$-0.573725\pi$
$829$	−13.2185	−0.459098	−0.229549	+	0.973297i	$0.573725\pi$
$830$	0	0
$831$	1.46250	0.0507336
$832$	0	0
$833$	−48.1521	−1.66837
$834$	0	0
$835$	−78.9590	−2.73249
$836$	0	0
$837$	0.972853	0.0336267
$838$	0	0
$839$	−55.8491	−1.92812	−0.964062	−	0.265678i	$-0.914404\pi$
$839$	−55.8491	−1.92812	−0.964062	+	0.265678i	$0.914404\pi$
$840$	0	0
$841$	−17.8122	−0.614214
$842$	0	0
$843$	5.68233	0.195710
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8.67158	−0.297959
$848$	0	0
$849$	−25.2078	−0.865128
$850$	0	0
$851$	−6.57242	−0.225300
$852$	0	0
$853$	21.8103	0.746770	0.373385	−	0.927676i	$-0.378197\pi$
$853$	21.8103	0.746770	0.373385	+	0.927676i	$0.378197\pi$
$854$	0	0
$855$	7.20775	0.246500
$856$	0	0
$857$	28.8961	0.987070	0.493535	−	0.869726i	$-0.335704\pi$
$857$	28.8961	0.987070	0.493535	+	0.869726i	$0.335704\pi$
$858$	0	0
$859$	17.2755	0.589431	0.294715	−	0.955585i	$-0.404775\pi$
$859$	17.2755	0.589431	0.294715	+	0.955585i	$0.404775\pi$
$860$	0	0
$861$	1.04221	0.0355185
$862$	0	0
$863$	−44.7741	−1.52413	−0.762063	−	0.647503i	$-0.775813\pi$
$863$	−44.7741	−1.52413	−0.762063	+	0.647503i	$0.775813\pi$
$864$	0	0
$865$	−37.6165	−1.27900
$866$	0	0
$867$	−37.5241	−1.27438
$868$	0	0
$869$	40.4534	1.37229
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−8.54288	−0.289133
$874$	0	0
$875$	−17.9148	−0.605632
$876$	0	0
$877$	40.2741	1.35996	0.679980	−	0.733230i	$-0.261988\pi$
$877$	40.2741	1.35996	0.679980	+	0.733230i	$0.261988\pi$
$878$	0	0
$879$	−7.14914	−0.241135
$880$	0	0
$881$	9.40821	0.316971	0.158485	−	0.987361i	$-0.449339\pi$
$881$	9.40821	0.316971	0.158485	+	0.987361i	$0.449339\pi$
$882$	0	0
$883$	51.2271	1.72393	0.861965	−	0.506968i	$-0.169234\pi$
$883$	51.2271	1.72393	0.861965	+	0.506968i	$0.169234\pi$
$884$	0	0
$885$	5.29590	0.178020
$886$	0	0
$887$	39.9215	1.34043	0.670217	−	0.742165i	$-0.266201\pi$
$887$	39.9215	1.34043	0.670217	+	0.742165i	$0.266201\pi$
$888$	0	0
$889$	−4.15585	−0.139383
$890$	0	0
$891$	4.85086	0.162510
$892$	0	0
$893$	12.8310	0.429373
$894$	0	0
$895$	−92.2863	−3.08479
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.25401	0.108527
$900$	0	0
$901$	99.5111	3.31519
$902$	0	0
$903$	5.75600	0.191548
$904$	0	0
$905$	2.17629	0.0723424
$906$	0	0
$907$	34.8310	1.15654	0.578272	−	0.815844i	$-0.303727\pi$
$907$	34.8310	1.15654	0.578272	+	0.815844i	$0.303727\pi$
$908$	0	0
$909$	−11.9976	−0.397936
$910$	0	0
$911$	−13.2125	−0.437751	−0.218875	−	0.975753i	$-0.570239\pi$
$911$	−13.2125	−0.437751	−0.218875	+	0.975753i	$0.570239\pi$
$912$	0	0
$913$	−74.3508	−2.46065
$914$	0	0
$915$	1.60388	0.0530225
$916$	0	0
$917$	6.10156	0.201491
$918$	0	0
$919$	−21.0175	−0.693302	−0.346651	−	0.937994i	$-0.612681\pi$
$919$	−21.0175	−0.693302	−0.346651	+	0.937994i	$0.612681\pi$
$920$	0	0
$921$	17.9952	0.592962
$922$	0	0
$923$	0	0
$924$	0	0
$925$	14.6547	0.481844
$926$	0	0
$927$	12.3230	0.404742
$928$	0	0
$929$	26.9965	0.885728	0.442864	−	0.896589i	$-0.353962\pi$
$929$	26.9965	0.885728	0.442864	+	0.896589i	$0.353962\pi$
$930$	0	0
$931$	−11.6087	−0.380459
$932$	0	0
$933$	3.32975	0.109011
$934$	0	0
$935$	145.028	4.74292
$936$	0	0
$937$	23.1745	0.757078	0.378539	−	0.925585i	$-0.376427\pi$
$937$	23.1745	0.757078	0.378539	+	0.925585i	$0.376427\pi$
$938$	0	0
$939$	17.8834	0.583602
$940$	0	0
$941$	−7.54048	−0.245813	−0.122906	−	0.992418i	$-0.539221\pi$
$941$	−7.54048	−0.245813	−0.122906	+	0.992418i	$0.539221\pi$
$942$	0	0
$943$	−7.69574	−0.250608
$944$	0	0
$945$	2.80194	0.0911470
$946$	0	0
$947$	20.6601	0.671363	0.335681	−	0.941976i	$-0.391033\pi$
$947$	20.6601	0.671363	0.335681	+	0.941976i	$0.391033\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	4.39373	0.142477
$952$	0	0
$953$	−30.2935	−0.981303	−0.490651	−	0.871356i	$-0.663241\pi$
$953$	−30.2935	−0.981303	−0.490651	+	0.871356i	$0.663241\pi$
$954$	0	0
$955$	39.6775	1.28394
$956$	0	0
$957$	16.2252	0.524487
$958$	0	0
$959$	10.9035	0.352092
$960$	0	0
$961$	−30.0536	−0.969470
$962$	0	0
$963$	−5.89977	−0.190118
$964$	0	0
$965$	57.4771	1.85025
$966$	0	0
$967$	−20.7289	−0.666595	−0.333298	−	0.942822i	$-0.608161\pi$
$967$	−20.7289	−0.666595	−0.333298	+	0.942822i	$0.608161\pi$
$968$	0	0
$969$	−13.1448	−0.422273
$970$	0	0
$971$	39.1094	1.25508	0.627541	−	0.778584i	$-0.284062\pi$
$971$	39.1094	1.25508	0.627541	+	0.778584i	$0.284062\pi$
$972$	0	0
$973$	−4.21983	−0.135282
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.27545	−0.232762	−0.116381	−	0.993205i	$-0.537129\pi$
$977$	−7.27545	−0.232762	−0.116381	+	0.993205i	$0.537129\pi$
$978$	0	0
$979$	15.0858	0.482143
$980$	0	0
$981$	0.792249	0.0252946
$982$	0	0
$983$	53.4857	1.70593	0.852965	−	0.521969i	$-0.174802\pi$
$983$	53.4857	1.70593	0.852965	+	0.521969i	$0.174802\pi$
$984$	0	0
$985$	12.1860	0.388278
$986$	0	0
$987$	4.98792	0.158767
$988$	0	0
$989$	−42.5026	−1.35150
$990$	0	0
$991$	−16.0575	−0.510085	−0.255042	−	0.966930i	$-0.582089\pi$
$991$	−16.0575	−0.510085	−0.255042	+	0.966930i	$0.582089\pi$
$992$	0	0
$993$	25.6775	0.814852
$994$	0	0
$995$	−52.2083	−1.65512
$996$	0	0
$997$	22.4263	0.710247	0.355123	−	0.934819i	$-0.384439\pi$
$997$	22.4263	0.710247	0.355123	+	0.934819i	$0.384439\pi$
$998$	0	0
$999$	−1.28621	−0.0406938

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.ce.1.3		3
4.3	odd	2		1014.2.a.m.1.3	✓	3
12.11	even	2		3042.2.a.be.1.1		3
13.12	even	2		8112.2.a.bz.1.1		3
52.3	odd	6		1014.2.e.m.529.3		6
52.7	even	12		1014.2.i.g.361.3		12
52.11	even	12		1014.2.i.g.823.6		12
52.15	even	12		1014.2.i.g.823.1		12
52.19	even	12		1014.2.i.g.361.4		12
52.23	odd	6		1014.2.e.k.529.1		6
52.31	even	4		1014.2.b.g.337.4		6
52.35	odd	6		1014.2.e.m.991.3		6
52.43	odd	6		1014.2.e.k.991.1		6
52.47	even	4		1014.2.b.g.337.3		6
52.51	odd	2		1014.2.a.o.1.1	yes	3
156.47	odd	4		3042.2.b.r.1351.4		6
156.83	odd	4		3042.2.b.r.1351.3		6
156.155	even	2		3042.2.a.bd.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.2.a.m.1.3	✓	3	4.3	odd	2
1014.2.a.o.1.1	yes	3	52.51	odd	2
1014.2.b.g.337.3		6	52.47	even	4
1014.2.b.g.337.4		6	52.31	even	4
1014.2.e.k.529.1		6	52.23	odd	6
1014.2.e.k.991.1		6	52.43	odd	6
1014.2.e.m.529.3		6	52.3	odd	6
1014.2.e.m.991.3		6	52.35	odd	6
1014.2.i.g.361.3		12	52.7	even	12
1014.2.i.g.361.4		12	52.19	even	12
1014.2.i.g.823.1		12	52.15	even	12
1014.2.i.g.823.6		12	52.11	even	12
3042.2.a.bd.1.3		3	156.155	even	2
3042.2.a.be.1.1		3	12.11	even	2
3042.2.b.r.1351.3		6	156.83	odd	4
3042.2.b.r.1351.4		6	156.47	odd	4
8112.2.a.bz.1.1		3	13.12	even	2
8112.2.a.ce.1.3		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:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.04892 q^{5} -0.692021 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.04892 q^{5} -0.692021 q^{7} +1.00000 q^{9} +4.85086 q^{11} -4.04892 q^{15} +7.38404 q^{17} +1.78017 q^{19} +0.692021 q^{21} -5.10992 q^{23} +11.3937 q^{25} -1.00000 q^{27} -3.34481 q^{29} -0.972853 q^{31} -4.85086 q^{33} -2.80194 q^{35} +1.28621 q^{37} +1.50604 q^{41} +8.31767 q^{43} +4.04892 q^{45} +7.20775 q^{47} -6.52111 q^{49} -7.38404 q^{51} +13.4765 q^{53} +19.6407 q^{55} -1.78017 q^{57} -1.30798 q^{59} -0.396125 q^{61} -0.692021 q^{63} -6.05429 q^{67} +5.10992 q^{69} +1.32975 q^{71} -7.65279 q^{73} -11.3937 q^{75} -3.35690 q^{77} +8.33944 q^{79} +1.00000 q^{81} -15.3274 q^{83} +29.8974 q^{85} +3.34481 q^{87} +3.10992 q^{89} +0.972853 q^{93} +7.20775 q^{95} -8.54288 q^{97} +4.85086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9} + q^{11} - 3 q^{15} + 12 q^{17} + 4 q^{19} - 3 q^{21} - 16 q^{23} + 2 q^{25} - 3 q^{27} + 13 q^{29} - 9 q^{31} - q^{33} - 4 q^{35} + 12 q^{37} + 14 q^{41} + 8 q^{43} + 3 q^{45} + 4 q^{47} - 4 q^{49} - 12 q^{51} + 15 q^{53} + 22 q^{55} - 4 q^{57} - 9 q^{59} - 10 q^{61} + 3 q^{63} - 6 q^{67} + 16 q^{69} + 6 q^{71} - 5 q^{73} - 2 q^{75} - 6 q^{77} + 5 q^{79} + 3 q^{81} - 7 q^{83} + 26 q^{85} - 13 q^{87} + 10 q^{89} + 9 q^{93} + 4 q^{95} - 7 q^{97} + q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9 + q^11 - 3 * q^15 + 12 * q^17 + 4 * q^19 - 3 * q^21 - 16 * q^23 + 2 * q^25 - 3 * q^27 + 13 * q^29 - 9 * q^31 - q^33 - 4 * q^35 + 12 * q^37 + 14 * q^41 + 8 * q^43 + 3 * q^45 + 4 * q^47 - 4 * q^49 - 12 * q^51 + 15 * q^53 + 22 * q^55 - 4 * q^57 - 9 * q^59 - 10 * q^61 + 3 * q^63 - 6 * q^67 + 16 * q^69 + 6 * q^71 - 5 * q^73 - 2 * q^75 - 6 * q^77 + 5 * q^79 + 3 * q^81 - 7 * q^83 + 26 * q^85 - 13 * q^87 + 10 * q^89 + 9 * q^93 + 4 * q^95 - 7 * q^97 + q^99

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