LMFDB - Embedded newform 3042.2.b.r.1351.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1351,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("3042.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1014)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.r.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -4.04892i q^{5} -0.692021i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -4.04892i q^{5} -0.692021i q^{7} -1.00000i q^{8} +4.04892 q^{10} -4.85086i q^{11} +0.692021 q^{14} +1.00000 q^{16} +7.38404 q^{17} -1.78017i q^{19} +4.04892i q^{20} +4.85086 q^{22} +5.10992 q^{23} -11.3937 q^{25} +0.692021i q^{28} +3.34481 q^{29} +0.972853i q^{31} +1.00000i q^{32} +7.38404i q^{34} -2.80194 q^{35} -1.28621i q^{37} +1.78017 q^{38} -4.04892 q^{40} -1.50604i q^{41} +8.31767 q^{43} +4.85086i q^{44} +5.10992i q^{46} -7.20775i q^{47} +6.52111 q^{49} -11.3937i q^{50} -13.4765 q^{53} -19.6407 q^{55} -0.692021 q^{56} +3.34481i q^{58} +1.30798i q^{59} -0.396125 q^{61} -0.972853 q^{62} -1.00000 q^{64} +6.05429i q^{67} -7.38404 q^{68} -2.80194i q^{70} +1.32975i q^{71} +7.65279i q^{73} +1.28621 q^{74} +1.78017i q^{76} -3.35690 q^{77} -8.33944 q^{79} -4.04892i q^{80} +1.50604 q^{82} -15.3274i q^{83} -29.8974i q^{85} +8.31767i q^{86} -4.85086 q^{88} +3.10992i q^{89} -5.10992 q^{92} +7.20775 q^{94} -7.20775 q^{95} -8.54288i q^{97} +6.52111i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4}+O(q^{10}) $ 6 * q - 6 * q^4	$ 6 q - 6 q^{4} + 6 q^{10} - 6 q^{14} + 6 q^{16} + 24 q^{17} + 2 q^{22} + 32 q^{23} - 4 q^{25} - 26 q^{29} - 8 q^{35} + 8 q^{38} - 6 q^{40} + 16 q^{43} + 8 q^{49} - 30 q^{53} - 44 q^{55} + 6 q^{56} - 20 q^{61} - 18 q^{62} - 6 q^{64} - 24 q^{68} + 24 q^{74} - 12 q^{77} - 10 q^{79} + 28 q^{82} - 2 q^{88} - 32 q^{92} + 8 q^{94} - 8 q^{95}+O(q^{100}) $ 6 * q - 6 * q^4 + 6 * q^10 - 6 * q^14 + 6 * q^16 + 24 * q^17 + 2 * q^22 + 32 * q^23 - 4 * q^25 - 26 * q^29 - 8 * q^35 + 8 * q^38 - 6 * q^40 + 16 * q^43 + 8 * q^49 - 30 * q^53 - 44 * q^55 + 6 * q^56 - 20 * q^61 - 18 * q^62 - 6 * q^64 - 24 * q^68 + 24 * q^74 - 12 * q^77 - 10 * q^79 + 28 * q^82 - 2 * q^88 - 32 * q^92 + 8 * q^94 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	− 4.04892i	− 1.81073i	−0.424633	−	0.905365i	$-0.639597\pi$
$5$	− 4.04892i	− 1.81073i	0.424633	−	0.905365i	$-0.360403\pi$
$6$	0	0
$7$	− 0.692021i	− 0.261560i	−0.991411	−	0.130780i	$-0.958252\pi$
$7$	− 0.692021i	− 0.261560i	0.991411	−	0.130780i	$-0.0417481\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	4.04892	1.28038
$11$	− 4.85086i	− 1.46259i	−0.682062	−	0.731294i	$-0.738917\pi$
$11$	− 4.85086i	− 1.46259i	0.682062	−	0.731294i	$-0.261083\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0.692021	0.184951
$15$	0	0
$16$	1.00000	0.250000
$17$	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
$19$	− 1.78017i	− 0.408398i	−0.978929	−	0.204199i	$-0.934541\pi$
$19$	− 1.78017i	− 0.408398i	0.978929	−	0.204199i	$-0.0654590\pi$
$20$	4.04892i	0.905365i
$21$	0	0
$22$	4.85086	1.03421
$23$	5.10992	1.06549	0.532746	−	0.846275i	$-0.321160\pi$
$23$	5.10992	1.06549	0.532746	+	0.846275i	$0.321160\pi$
$24$	0	0
$25$	−11.3937	−2.27875
$26$	0	0
$27$	0	0
$28$	0.692021i	0.130780i
$29$	3.34481	0.621116	0.310558	−	0.950554i	$-0.399484\pi$
$29$	3.34481	0.621116	0.310558	+	0.950554i	$0.399484\pi$
$30$	0	0
$31$	0.972853i	0.174730i	0.996176	+	0.0873648i	$0.0278446\pi$
$31$	0.972853i	0.174730i	−0.996176	+	0.0873648i	$0.972155\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	7.38404i	1.26635i
$35$	−2.80194	−0.473614
$36$	0	0
$37$	− 1.28621i	− 0.211451i	−0.994395	−	0.105726i	$-0.966283\pi$
$37$	− 1.28621i	− 0.211451i	0.994395	−	0.105726i	$-0.0337165\pi$
$38$	1.78017	0.288781
$39$	0	0
$40$	−4.04892	−0.640190
$41$	− 1.50604i	− 0.235204i	−0.993061	−	0.117602i	$-0.962479\pi$
$41$	− 1.50604i	− 0.235204i	0.993061	−	0.117602i	$-0.0375207\pi$
$42$	0	0
$43$	8.31767	1.26843	0.634216	−	0.773156i	$-0.281323\pi$
$43$	8.31767	1.26843	0.634216	+	0.773156i	$0.281323\pi$
$44$	4.85086i	0.731294i
$45$	0	0
$46$	5.10992i	0.753416i
$47$	− 7.20775i	− 1.05136i	−0.850683	−	0.525679i	$-0.823811\pi$
$47$	− 7.20775i	− 1.05136i	0.850683	−	0.525679i	$-0.176189\pi$
$48$	0	0
$49$	6.52111	0.931587
$50$	− 11.3937i	− 1.61132i
$51$	0	0
$52$	0	0
$53$	−13.4765	−1.85114	−0.925570	−	0.378577i	$-0.876414\pi$
$53$	−13.4765	−1.85114	−0.925570	+	0.378577i	$0.876414\pi$
$54$	0	0
$55$	−19.6407	−2.64835
$56$	−0.692021	−0.0924753
$57$	0	0
$58$	3.34481i	0.439196i
$59$	1.30798i	0.170284i	0.996369	+	0.0851422i	$0.0271344\pi$
$59$	1.30798i	0.170284i	−0.996369	+	0.0851422i	$0.972866\pi$
$60$	0	0
$61$	−0.396125	−0.0507185	−0.0253593	−	0.999678i	$-0.508073\pi$
$61$	−0.396125	−0.0507185	−0.0253593	+	0.999678i	$0.508073\pi$
$62$	−0.972853	−0.123552
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	6.05429i	0.739650i	0.929101	+	0.369825i	$0.120582\pi$
$67$	6.05429i	0.739650i	−0.929101	+	0.369825i	$0.879418\pi$
$68$	−7.38404	−0.895447
$69$	0	0
$70$	− 2.80194i	− 0.334896i
$71$	1.32975i	0.157812i	0.996882	+	0.0789061i	$0.0251427\pi$
$71$	1.32975i	0.157812i	−0.996882	+	0.0789061i	$0.974857\pi$
$72$	0	0
$73$	7.65279i	0.895692i	0.894111	+	0.447846i	$0.147809\pi$
$73$	7.65279i	0.895692i	−0.894111	+	0.447846i	$0.852191\pi$
$74$	1.28621	0.149519
$75$	0	0
$76$	1.78017i	0.204199i
$77$	−3.35690	−0.382554
$78$	0	0
$79$	−8.33944	−0.938260	−0.469130	−	0.883129i	$-0.655432\pi$
$79$	−8.33944	−0.938260	−0.469130	+	0.883129i	$0.655432\pi$
$80$	− 4.04892i	− 0.452683i
$81$	0	0
$82$	1.50604	0.166314
$83$	− 15.3274i	− 1.68240i	−0.540727	−	0.841198i	$-0.681851\pi$
$83$	− 15.3274i	− 1.68240i	0.540727	−	0.841198i	$-0.318149\pi$
$84$	0	0
$85$	− 29.8974i	− 3.24283i
$86$	8.31767i	0.896917i
$87$	0	0
$88$	−4.85086	−0.517103
$89$	3.10992i	0.329650i	0.986323	+	0.164825i	$0.0527060\pi$
$89$	3.10992i	0.329650i	−0.986323	+	0.164825i	$0.947294\pi$
$90$	0	0
$91$	0	0
$92$	−5.10992	−0.532746
$93$	0	0
$94$	7.20775	0.743423
$95$	−7.20775	−0.739500
$96$	0	0
$97$	− 8.54288i	− 0.867398i	−0.901058	−	0.433699i	$-0.857208\pi$
$97$	− 8.54288i	− 0.867398i	0.901058	−	0.433699i	$-0.142792\pi$
$98$	6.52111i	0.658731i
$99$	0	0
$100$	11.3937	1.13937
$101$	−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
$105$	0	0
$106$	− 13.4765i	− 1.30895i
$107$	−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
$109$	0.792249i	0.0758837i	0.999280	+	0.0379418i	$0.0120802\pi$
$109$	0.792249i	0.0758837i	−0.999280	+	0.0379418i	$0.987920\pi$
$110$	− 19.6407i	− 1.87267i
$111$	0	0
$112$	− 0.692021i	− 0.0653899i
$113$	−6.21983	−0.585113	−0.292556	−	0.956248i	$-0.594506\pi$
$113$	−6.21983	−0.585113	−0.292556	+	0.956248i	$0.594506\pi$
$114$	0	0
$115$	− 20.6896i	− 1.92932i
$116$	−3.34481	−0.310558
$117$	0	0
$118$	−1.30798	−0.120409
$119$	− 5.10992i	− 0.468425i
$120$	0	0
$121$	−12.5308	−1.13916
$122$	− 0.396125i	− 0.0358634i
$123$	0	0
$124$	− 0.972853i	− 0.0873648i
$125$	25.8877i	2.31547i
$126$	0	0
$127$	6.00538	0.532891	0.266446	−	0.963850i	$-0.414151\pi$
$127$	6.00538	0.532891	0.266446	+	0.963850i	$0.414151\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	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
$134$	−6.05429	−0.523011
$135$	0	0
$136$	− 7.38404i	− 0.633176i
$137$	− 15.7560i	− 1.34613i	−0.739585	−	0.673063i	$-0.764978\pi$
$137$	− 15.7560i	− 1.34613i	0.739585	−	0.673063i	$-0.235022\pi$
$138$	0	0
$139$	−6.09783	−0.517212	−0.258606	−	0.965983i	$-0.583263\pi$
$139$	−6.09783	−0.517212	−0.258606	+	0.965983i	$0.583263\pi$
$140$	2.80194	0.236807
$141$	0	0
$142$	−1.32975	−0.111590
$143$	0	0
$144$	0	0
$145$	− 13.5429i	− 1.12467i
$146$	−7.65279	−0.633350
$147$	0	0
$148$	1.28621i	0.105726i
$149$	2.55257i	0.209114i	0.994519	+	0.104557i	$0.0333425\pi$
$149$	2.55257i	0.209114i	−0.994519	+	0.104557i	$0.966657\pi$
$150$	0	0
$151$	17.7168i	1.44177i	0.693054	+	0.720885i	$0.256265\pi$
$151$	17.7168i	1.44177i	−0.693054	+	0.720885i	$0.743735\pi$
$152$	−1.78017	−0.144391
$153$	0	0
$154$	− 3.35690i	− 0.270506i
$155$	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	$0.581122\pi$
$158$	− 8.33944i	− 0.663450i
$159$	0	0
$160$	4.04892	0.320095
$161$	− 3.53617i	− 0.278689i
$162$	0	0
$163$	14.5918i	1.14292i	0.820631	+	0.571459i	$0.193622\pi$
$163$	14.5918i	1.14292i	−0.820631	+	0.571459i	$0.806378\pi$
$164$	1.50604i	0.117602i
$165$	0	0
$166$	15.3274	1.18963
$167$	19.5013i	1.50905i	0.656270	+	0.754526i	$0.272133\pi$
$167$	19.5013i	1.50905i	−0.656270	+	0.754526i	$0.727867\pi$
$168$	0	0
$169$	0	0
$170$	29.8974	2.29302
$171$	0	0
$172$	−8.31767	−0.634216
$173$	−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
$175$	7.88471i	0.596028i
$176$	− 4.85086i	− 0.365647i
$177$	0	0
$178$	−3.10992	−0.233098
$179$	22.7928	1.70362	0.851808	−	0.523853i	$-0.175506\pi$
$179$	22.7928	1.70362	0.851808	+	0.523853i	$0.175506\pi$
$180$	0	0
$181$	−0.537500	−0.0399520	−0.0199760	−	0.999800i	$-0.506359\pi$
$181$	−0.537500	−0.0399520	−0.0199760	+	0.999800i	$0.506359\pi$
$182$	0	0
$183$	0	0
$184$	− 5.10992i	− 0.376708i
$185$	−5.20775	−0.382881
$186$	0	0
$187$	− 35.8189i	− 2.61934i
$188$	7.20775i	0.525679i
$189$	0	0
$190$	− 7.20775i	− 0.522905i
$191$	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
$193$	− 14.1957i	− 1.02183i	−0.859632	−	0.510913i	$-0.829307\pi$
$193$	− 14.1957i	− 1.02183i	0.859632	−	0.510913i	$-0.170693\pi$
$194$	8.54288	0.613343
$195$	0	0
$196$	−6.52111	−0.465793
$197$	− 3.00969i	− 0.214431i	−0.994236	−	0.107216i	$-0.965806\pi$
$197$	− 3.00969i	− 0.214431i	0.994236	−	0.107216i	$-0.0341935\pi$
$198$	0	0
$199$	−12.8944	−0.914059	−0.457030	−	0.889451i	$-0.651087\pi$
$199$	−12.8944	−0.914059	−0.457030	+	0.889451i	$0.651087\pi$
$200$	11.3937i	0.805658i
$201$	0	0
$202$	− 11.9976i	− 0.844149i
$203$	− 2.31468i	− 0.162459i
$204$	0	0
$205$	−6.09783	−0.425891
$206$	12.3230i	0.858587i
$207$	0	0
$208$	0	0
$209$	−8.63533	−0.597319
$210$	0	0
$211$	7.79954	0.536943	0.268471	−	0.963288i	$-0.413482\pi$
$211$	7.79954	0.536943	0.268471	+	0.963288i	$0.413482\pi$
$212$	13.4765	0.925570
$213$	0	0
$214$	− 5.89977i	− 0.403300i
$215$	− 33.6775i	− 2.29679i
$216$	0	0
$217$	0.673235	0.0457022
$218$	−0.792249	−0.0536579
$219$	0	0
$220$	19.6407	1.32418
$221$	0	0
$222$	0	0
$223$	12.1957i	0.816682i	0.912829	+	0.408341i	$0.133893\pi$
$223$	12.1957i	0.816682i	−0.912829	+	0.408341i	$0.866107\pi$
$224$	0.692021	0.0462376
$225$	0	0
$226$	− 6.21983i	− 0.413737i
$227$	6.74333i	0.447571i	0.974638	+	0.223785i	$0.0718415\pi$
$227$	6.74333i	0.447571i	−0.974638	+	0.223785i	$0.928159\pi$
$228$	0	0
$229$	− 19.8237i	− 1.30999i	−0.755634	−	0.654994i	$-0.772671\pi$
$229$	− 19.8237i	− 1.30999i	0.755634	−	0.654994i	$-0.227329\pi$
$230$	20.6896	1.36423
$231$	0	0
$232$	− 3.34481i	− 0.219598i
$233$	−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
$236$	− 1.30798i	− 0.0851422i
$237$	0	0
$238$	5.10992	0.331227
$239$	22.0978i	1.42939i	0.699436	+	0.714695i	$0.253435\pi$
$239$	22.0978i	1.42939i	−0.699436	+	0.714695i	$0.746565\pi$
$240$	0	0
$241$	10.1274i	0.652362i	0.945307	+	0.326181i	$0.105762\pi$
$241$	10.1274i	0.652362i	−0.945307	+	0.326181i	$0.894238\pi$
$242$	− 12.5308i	− 0.805510i
$243$	0	0
$244$	0.396125	0.0253593
$245$	− 26.4034i	− 1.68685i
$246$	0	0
$247$	0	0
$248$	0.972853	0.0617762
$249$	0	0
$250$	−25.8877	−1.63728
$251$	5.54719	0.350135	0.175068	−	0.984556i	$-0.443986\pi$
$251$	5.54719	0.350135	0.175068	+	0.984556i	$0.443986\pi$
$252$	0	0
$253$	− 24.7875i	− 1.55837i
$254$	6.00538i	0.376811i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−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
$261$	0	0
$262$	− 8.81700i	− 0.544716i
$263$	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
$265$	54.5652i	3.35192i
$266$	− 1.23191i	− 0.0755335i
$267$	0	0
$268$	− 6.05429i	− 0.369825i
$269$	26.0140	1.58610	0.793051	−	0.609156i	$-0.208491\pi$
$269$	26.0140	1.58610	0.793051	+	0.609156i	$0.208491\pi$
$270$	0	0
$271$	2.88471i	0.175233i	0.996154	+	0.0876167i	$0.0279251\pi$
$271$	2.88471i	0.175233i	−0.996154	+	0.0876167i	$0.972075\pi$
$272$	7.38404	0.447723
$273$	0	0
$274$	15.7560	0.951855
$275$	55.2693i	3.33287i
$276$	0	0
$277$	1.46250	0.0878731	0.0439366	−	0.999034i	$-0.486010\pi$
$277$	1.46250	0.0878731	0.0439366	+	0.999034i	$0.486010\pi$
$278$	− 6.09783i	− 0.365724i
$279$	0	0
$280$	2.80194i	0.167448i
$281$	− 5.68233i	− 0.338980i	−0.985532	−	0.169490i	$-0.945788\pi$
$281$	− 5.68233i	− 0.338980i	0.985532	−	0.169490i	$-0.0542120\pi$
$282$	0	0
$283$	25.2078	1.49845	0.749223	−	0.662318i	$-0.230427\pi$
$283$	25.2078	1.49845	0.749223	+	0.662318i	$0.230427\pi$
$284$	− 1.32975i	− 0.0789061i
$285$	0	0
$286$	0	0
$287$	−1.04221	−0.0615199
$288$	0	0
$289$	37.5241	2.20730
$290$	13.5429	0.795265
$291$	0	0
$292$	− 7.65279i	− 0.447846i
$293$	7.14914i	0.417658i	0.977952	+	0.208829i	$0.0669651\pi$
$293$	7.14914i	0.417658i	−0.977952	+	0.208829i	$0.933035\pi$
$294$	0	0
$295$	5.29590	0.308339
$296$	−1.28621	−0.0747593
$297$	0	0
$298$	−2.55257	−0.147866
$299$	0	0
$300$	0	0
$301$	− 5.75600i	− 0.331771i
$302$	−17.7168	−1.01949
$303$	0	0
$304$	− 1.78017i	− 0.102100i
$305$	1.60388i	0.0918376i
$306$	0	0
$307$	− 17.9952i	− 1.02704i	−0.858077	−	0.513521i	$-0.828341\pi$
$307$	− 17.9952i	− 1.02704i	0.858077	−	0.513521i	$-0.171659\pi$
$308$	3.35690	0.191277
$309$	0	0
$310$	3.93900i	0.223720i
$311$	3.32975	0.188813	0.0944064	−	0.995534i	$-0.469905\pi$
$311$	3.32975	0.188813	0.0944064	+	0.995534i	$0.469905\pi$
$312$	0	0
$313$	−17.8834	−1.01083	−0.505414	−	0.862877i	$-0.668660\pi$
$313$	−17.8834	−1.01083	−0.505414	+	0.862877i	$0.668660\pi$
$314$	− 6.31767i	− 0.356527i
$315$	0	0
$316$	8.33944	0.469130
$317$	4.39373i	0.246777i	0.992358	+	0.123388i	$0.0393761\pi$
$317$	4.39373i	0.246777i	−0.992358	+	0.123388i	$0.960624\pi$
$318$	0	0
$319$	− 16.2252i	− 0.908437i
$320$	4.04892i	0.226341i
$321$	0	0
$322$	3.53617	0.197063
$323$	− 13.1448i	− 0.731398i
$324$	0	0
$325$	0	0
$326$	−14.5918	−0.808165
$327$	0	0
$328$	−1.50604	−0.0831572
$329$	−4.98792	−0.274993
$330$	0	0
$331$	25.6775i	1.41137i	0.708528	+	0.705683i	$0.249360\pi$
$331$	25.6775i	1.41137i	−0.708528	+	0.705683i	$0.750640\pi$
$332$	15.3274i	0.841198i
$333$	0	0
$334$	−19.5013	−1.06706
$335$	24.5133	1.33931
$336$	0	0
$337$	24.6504	1.34279	0.671396	−	0.741098i	$-0.265695\pi$
$337$	24.6504	1.34279	0.671396	+	0.741098i	$0.265695\pi$
$338$	0	0
$339$	0	0
$340$	29.8974i	1.62141i
$341$	4.71917	0.255557
$342$	0	0
$343$	− 9.35690i	− 0.505225i
$344$	− 8.31767i	− 0.448459i
$345$	0	0
$346$	− 9.29052i	− 0.499461i
$347$	−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
$349$	11.0616i	0.592113i	0.955170	+	0.296057i	$0.0956717\pi$
$349$	11.0616i	0.592113i	−0.955170	+	0.296057i	$0.904328\pi$
$350$	−7.88471	−0.421455
$351$	0	0
$352$	4.85086	0.258551
$353$	− 10.5047i	− 0.559109i	−0.960130	−	0.279555i	$-0.909813\pi$
$353$	− 10.5047i	− 0.559109i	0.960130	−	0.279555i	$-0.0901868\pi$
$354$	0	0
$355$	5.38404	0.285755
$356$	− 3.10992i	− 0.164825i
$357$	0	0
$358$	22.7928i	1.20464i
$359$	− 5.10992i	− 0.269691i	−0.990867	−	0.134846i	$-0.956946\pi$
$359$	− 5.10992i	− 0.269691i	0.990867	−	0.134846i	$-0.0430538\pi$
$360$	0	0
$361$	15.8310	0.833211
$362$	− 0.537500i	− 0.0282504i
$363$	0	0
$364$	0	0
$365$	30.9855	1.62186
$366$	0	0
$367$	−8.44803	−0.440983	−0.220492	−	0.975389i	$-0.570766\pi$
$367$	−8.44803	−0.440983	−0.220492	+	0.975389i	$0.570766\pi$
$368$	5.10992	0.266373
$369$	0	0
$370$	− 5.20775i	− 0.270738i
$371$	9.32603i	0.484183i
$372$	0	0
$373$	7.69096	0.398223	0.199111	−	0.979977i	$-0.436194\pi$
$373$	7.69096	0.398223	0.199111	+	0.979977i	$0.436194\pi$
$374$	35.8189	1.85215
$375$	0	0
$376$	−7.20775	−0.371711
$377$	0	0
$378$	0	0
$379$	11.4034i	0.585754i	0.956150	+	0.292877i	$0.0946127\pi$
$379$	11.4034i	0.585754i	−0.956150	+	0.292877i	$0.905387\pi$
$380$	7.20775	0.369750
$381$	0	0
$382$	9.79954i	0.501388i
$383$	20.6703i	1.05620i	0.849182	+	0.528100i	$0.177095\pi$
$383$	20.6703i	1.05620i	−0.849182	+	0.528100i	$0.822905\pi$
$384$	0	0
$385$	13.5918i	0.692702i
$386$	14.1957	0.722541
$387$	0	0
$388$	8.54288i	0.433699i
$389$	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
$392$	− 6.52111i	− 0.329366i
$393$	0	0
$394$	3.00969	0.151626
$395$	33.7657i	1.69894i
$396$	0	0
$397$	− 19.3599i	− 0.971645i	−0.874058	−	0.485822i	$-0.838520\pi$
$397$	− 19.3599i	− 0.971645i	0.874058	−	0.485822i	$-0.161480\pi$
$398$	− 12.8944i	− 0.646338i
$399$	0	0
$400$	−11.3937	−0.569687
$401$	14.4832i	0.723257i	0.932322	+	0.361628i	$0.117779\pi$
$401$	14.4832i	0.723257i	−0.932322	+	0.361628i	$0.882221\pi$
$402$	0	0
$403$	0	0
$404$	11.9976	0.596903
$405$	0	0
$406$	2.31468	0.114876
$407$	−6.23921	−0.309266
$408$	0	0
$409$	18.8984i	0.934468i	0.884134	+	0.467234i	$0.154749\pi$
$409$	18.8984i	0.934468i	−0.884134	+	0.467234i	$0.845251\pi$
$410$	− 6.09783i	− 0.301151i
$411$	0	0
$412$	−12.3230	−0.607113
$413$	0.905149	0.0445395
$414$	0	0
$415$	−62.0592	−3.04637
$416$	0	0
$417$	0	0
$418$	− 8.63533i	− 0.422368i
$419$	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
$421$	− 20.5918i	− 1.00358i	−0.864989	−	0.501791i	$-0.832675\pi$
$421$	− 20.5918i	− 1.00358i	0.864989	−	0.501791i	$-0.167325\pi$
$422$	7.79954i	0.379676i
$423$	0	0
$424$	13.4765i	0.654477i
$425$	−84.1318	−4.08099
$426$	0	0
$427$	0.274127i	0.0132659i
$428$	5.89977	0.285176
$429$	0	0
$430$	33.6775	1.62408
$431$	34.5133i	1.66245i	0.555937	+	0.831224i	$0.312360\pi$
$431$	34.5133i	1.66245i	−0.555937	+	0.831224i	$0.687640\pi$
$432$	0	0
$433$	2.12631	0.102184	0.0510920	−	0.998694i	$-0.483730\pi$
$433$	2.12631	0.102184	0.0510920	+	0.998694i	$0.483730\pi$
$434$	0.673235i	0.0323163i
$435$	0	0
$436$	− 0.792249i	− 0.0379418i
$437$	− 9.09651i	− 0.435145i
$438$	0	0
$439$	−21.8321	−1.04199	−0.520994	−	0.853560i	$-0.674439\pi$
$439$	−21.8321	−1.04199	−0.520994	+	0.853560i	$0.674439\pi$
$440$	19.6407i	0.936334i
$441$	0	0
$442$	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
$446$	−12.1957	−0.577482
$447$	0	0
$448$	0.692021i	0.0326949i
$449$	− 19.7560i	− 0.932343i	−0.884694	−	0.466172i	$-0.845633\pi$
$449$	− 19.7560i	− 0.932343i	0.884694	−	0.466172i	$-0.154367\pi$
$450$	0	0
$451$	−7.30559	−0.344007
$452$	6.21983	0.292556
$453$	0	0
$454$	−6.74333	−0.316480
$455$	0	0
$456$	0	0
$457$	− 23.8582i	− 1.11604i	−0.829828	−	0.558019i	$-0.811562\pi$
$457$	− 23.8582i	− 1.11604i	0.829828	−	0.558019i	$-0.188438\pi$
$458$	19.8237	0.926301
$459$	0	0
$460$	20.6896i	0.964659i
$461$	− 17.5773i	− 0.818657i	−0.912387	−	0.409329i	$-0.865763\pi$
$461$	− 17.5773i	− 0.818657i	0.912387	−	0.409329i	$-0.134237\pi$
$462$	0	0
$463$	23.8431i	1.10808i	0.832489	+	0.554041i	$0.186915\pi$
$463$	23.8431i	1.10808i	−0.832489	+	0.554041i	$0.813085\pi$
$464$	3.34481	0.155279
$465$	0	0
$466$	− 30.0301i	− 1.39112i
$467$	8.61058	0.398450	0.199225	−	0.979954i	$-0.436158\pi$
$467$	8.61058	0.398450	0.199225	+	0.979954i	$0.436158\pi$
$468$	0	0
$469$	4.18970	0.193462
$470$	− 29.1836i	− 1.34614i
$471$	0	0
$472$	1.30798	0.0602046
$473$	− 40.3478i	− 1.85519i
$474$	0	0
$475$	20.2828i	0.930636i
$476$	5.10992i	0.234213i
$477$	0	0
$478$	−22.0978	−1.01073
$479$	− 6.58104i	− 0.300695i	−0.988633	−	0.150348i	$-0.951961\pi$
$479$	− 6.58104i	− 0.300695i	0.988633	−	0.150348i	$-0.0480393\pi$
$480$	0	0
$481$	0	0
$482$	−10.1274	−0.461289
$483$	0	0
$484$	12.5308	0.569582
$485$	−34.5894	−1.57062
$486$	0	0
$487$	23.2760i	1.05474i	0.849636	+	0.527369i	$0.176822\pi$
$487$	23.2760i	1.05474i	−0.849636	+	0.527369i	$0.823178\pi$
$488$	0.396125i	0.0179317i
$489$	0	0
$490$	26.4034	1.19278
$491$	−15.5405	−0.701332	−0.350666	−	0.936501i	$-0.614045\pi$
$491$	−15.5405	−0.701332	−0.350666	+	0.936501i	$0.614045\pi$
$492$	0	0
$493$	24.6983	1.11235
$494$	0	0
$495$	0	0
$496$	0.972853i	0.0436824i
$497$	0.920215	0.0412773
$498$	0	0
$499$	− 9.53617i	− 0.426898i	−0.976954	−	0.213449i	$-0.931530\pi$
$499$	− 9.53617i	− 0.426898i	0.976954	−	0.213449i	$-0.0684697\pi$
$500$	− 25.8877i	− 1.15773i
$501$	0	0
$502$	5.54719i	0.247583i
$503$	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
$505$	48.5773i	2.16166i
$506$	24.7875	1.10194
$507$	0	0
$508$	−6.00538	−0.266446
$509$	− 40.9638i	− 1.81569i	−0.419310	−	0.907843i	$-0.637728\pi$
$509$	− 40.9638i	− 1.81569i	0.419310	−	0.907843i	$-0.362272\pi$
$510$	0	0
$511$	5.29590	0.234277
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 13.7995i	− 0.608672i
$515$	− 49.8950i	− 2.19864i
$516$	0	0
$517$	−34.9638	−1.53770
$518$	− 0.890084i	− 0.0391080i
$519$	0	0
$520$	0	0
$521$	−36.3672	−1.59327	−0.796637	−	0.604457i	$-0.793390\pi$
$521$	−36.3672	−1.59327	−0.796637	+	0.604457i	$0.793390\pi$
$522$	0	0
$523$	−6.03013	−0.263679	−0.131840	−	0.991271i	$-0.542088\pi$
$523$	−6.03013	−0.263679	−0.131840	+	0.991271i	$0.542088\pi$
$524$	8.81700	0.385173
$525$	0	0
$526$	22.4698i	0.979730i
$527$	7.18359i	0.312922i
$528$	0	0
$529$	3.11124	0.135271
$530$	−54.5652	−2.37016
$531$	0	0
$532$	1.23191	0.0534103
$533$	0	0
$534$	0	0
$535$	23.8877i	1.03275i
$536$	6.05429	0.261506
$537$	0	0
$538$	26.0140i	1.12154i
$539$	− 31.6329i	− 1.36253i
$540$	0	0
$541$	− 7.92154i	− 0.340574i	−0.985395	−	0.170287i	$-0.945531\pi$
$541$	− 7.92154i	− 0.340574i	0.985395	−	0.170287i	$-0.0544694\pi$
$542$	−2.88471	−0.123909
$543$	0	0
$544$	7.38404i	0.316588i
$545$	3.20775	0.137405
$546$	0	0
$547$	18.4155	0.787390	0.393695	−	0.919241i	$-0.371197\pi$
$547$	18.4155	0.787390	0.393695	+	0.919241i	$0.371197\pi$
$548$	15.7560i	0.673063i
$549$	0	0
$550$	−55.2693	−2.35669
$551$	− 5.95433i	− 0.253663i
$552$	0	0
$553$	5.77107i	0.245411i
$554$	1.46250i	0.0621357i
$555$	0	0
$556$	6.09783	0.258606
$557$	23.9758i	1.01589i	0.861390	+	0.507944i	$0.169594\pi$
$557$	23.9758i	1.01589i	−0.861390	+	0.507944i	$0.830406\pi$
$558$	0	0
$559$	0	0
$560$	−2.80194	−0.118403
$561$	0	0
$562$	5.68233	0.239695
$563$	−2.29291	−0.0966348	−0.0483174	−	0.998832i	$-0.515386\pi$
$563$	−2.29291	−0.0966348	−0.0483174	+	0.998832i	$0.515386\pi$
$564$	0	0
$565$	25.1836i	1.05948i
$566$	25.2078i	1.05956i
$567$	0	0
$568$	1.32975	0.0557950
$569$	−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
$573$	0	0
$574$	− 1.04221i	− 0.0435011i
$575$	−58.2210	−2.42798
$576$	0	0
$577$	− 8.77048i	− 0.365120i	−0.983195	−	0.182560i	$-0.941562\pi$
$577$	− 8.77048i	− 0.365120i	0.983195	−	0.182560i	$-0.0584383\pi$
$578$	37.5241i	1.56080i
$579$	0	0
$580$	13.5429i	0.562337i
$581$	−10.6069	−0.440047
$582$	0	0
$583$	65.3726i	2.70745i
$584$	7.65279	0.316675
$585$	0	0
$586$	−7.14914	−0.295328
$587$	− 38.1430i	− 1.57433i	−0.616742	−	0.787166i	$-0.711548\pi$
$587$	− 38.1430i	− 1.57433i	0.616742	−	0.787166i	$-0.288452\pi$
$588$	0	0
$589$	1.73184	0.0713593
$590$	5.29590i	0.218029i
$591$	0	0
$592$	− 1.28621i	− 0.0528628i
$593$	37.9517i	1.55849i	0.626720	+	0.779244i	$0.284397\pi$
$593$	37.9517i	1.55849i	−0.626720	+	0.779244i	$0.715603\pi$
$594$	0	0
$595$	−20.6896	−0.848192
$596$	− 2.55257i	− 0.104557i
$597$	0	0
$598$	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	$0.537198\pi$
$602$	5.75600	0.234597
$603$	0	0
$604$	− 17.7168i	− 0.720885i
$605$	50.7362i	2.06272i
$606$	0	0
$607$	22.4286	0.910351	0.455175	−	0.890402i	$-0.349577\pi$
$607$	22.4286	0.910351	0.455175	+	0.890402i	$0.349577\pi$
$608$	1.78017	0.0721953
$609$	0	0
$610$	−1.60388	−0.0649390
$611$	0	0
$612$	0	0
$613$	39.9603i	1.61398i	0.590564	+	0.806991i	$0.298905\pi$
$613$	39.9603i	1.61398i	−0.590564	+	0.806991i	$0.701095\pi$
$614$	17.9952	0.726228
$615$	0	0
$616$	3.35690i	0.135253i
$617$	31.4470i	1.26601i	0.774149	+	0.633003i	$0.218178\pi$
$617$	31.4470i	1.26601i	−0.774149	+	0.633003i	$0.781822\pi$
$618$	0	0
$619$	− 29.3685i	− 1.18042i	−0.807250	−	0.590210i	$-0.799045\pi$
$619$	− 29.3685i	− 1.18042i	0.807250	−	0.590210i	$-0.200955\pi$
$620$	−3.93900	−0.158194
$621$	0	0
$622$	3.32975i	0.133511i
$623$	2.15213	0.0862232
$624$	0	0
$625$	47.8485	1.91394
$626$	− 17.8834i	− 0.714764i
$627$	0	0
$628$	6.31767	0.252102
$629$	− 9.49742i	− 0.378687i
$630$	0	0
$631$	21.6799i	0.863065i	0.902097	+	0.431532i	$0.142027\pi$
$631$	21.6799i	0.863065i	−0.902097	+	0.431532i	$0.857973\pi$
$632$	8.33944i	0.331725i
$633$	0	0
$634$	−4.39373	−0.174497
$635$	− 24.3153i	− 0.964922i
$636$	0	0
$637$	0	0
$638$	16.2252	0.642362
$639$	0	0
$640$	−4.04892	−0.160048
$641$	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
$643$	− 30.7875i	− 1.21414i	−0.794649	−	0.607070i	$-0.792345\pi$
$643$	− 30.7875i	− 1.21414i	0.794649	−	0.607070i	$-0.207655\pi$
$644$	3.53617i	0.139345i
$645$	0	0
$646$	13.1448	0.517177
$647$	−16.6025	−0.652713	−0.326357	−	0.945247i	$-0.605821\pi$
$647$	−16.6025	−0.652713	−0.326357	+	0.945247i	$0.605821\pi$
$648$	0	0
$649$	6.34481	0.249056
$650$	0	0
$651$	0	0
$652$	− 14.5918i	− 0.571459i
$653$	−28.5459	−1.11709	−0.558543	−	0.829476i	$-0.688639\pi$
$653$	−28.5459	−1.11709	−0.558543	+	0.829476i	$0.688639\pi$
$654$	0	0
$655$	35.6993i	1.39489i
$656$	− 1.50604i	− 0.0588010i
$657$	0	0
$658$	− 4.98792i	− 0.194449i
$659$	−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
$661$	10.8009i	0.420105i	0.977690	+	0.210053i	$0.0673635\pi$
$661$	10.8009i	0.420105i	−0.977690	+	0.210053i	$0.932636\pi$
$662$	−25.6775	−0.997986
$663$	0	0
$664$	−15.3274	−0.594817
$665$	4.98792i	0.193423i
$666$	0	0
$667$	17.0917	0.661794
$668$	− 19.5013i	− 0.754526i
$669$	0	0
$670$	24.5133i	0.947033i
$671$	1.92154i	0.0741803i
$672$	0	0
$673$	−16.3260	−0.629322	−0.314661	−	0.949204i	$-0.601891\pi$
$673$	−16.3260	−0.629322	−0.314661	+	0.949204i	$0.601891\pi$
$674$	24.6504i	0.949498i
$675$	0	0
$676$	0	0
$677$	−41.4252	−1.59210	−0.796050	−	0.605231i	$-0.793081\pi$
$677$	−41.4252	−1.59210	−0.796050	+	0.605231i	$0.793081\pi$
$678$	0	0
$679$	−5.91185	−0.226876
$680$	−29.8974	−1.14651
$681$	0	0
$682$	4.71917i	0.180706i
$683$	− 31.2325i	− 1.19508i	−0.801840	−	0.597539i	$-0.796145\pi$
$683$	− 31.2325i	− 1.19508i	0.801840	−	0.597539i	$-0.203855\pi$
$684$	0	0
$685$	−63.7948	−2.43747
$686$	9.35690	0.357248
$687$	0	0
$688$	8.31767	0.317108
$689$	0	0
$690$	0	0
$691$	− 24.9638i	− 0.949666i	−0.880076	−	0.474833i	$-0.842508\pi$
$691$	− 24.9638i	− 0.949666i	0.880076	−	0.474833i	$-0.157492\pi$
$692$	9.29052	0.353173
$693$	0	0
$694$	− 14.2959i	− 0.542665i
$695$	24.6896i	0.936531i
$696$	0	0
$697$	− 11.1207i	− 0.421225i
$698$	−11.0616	−0.418687
$699$	0	0
$700$	− 7.88471i	− 0.298014i
$701$	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
$704$	4.85086i	0.182823i
$705$	0	0
$706$	10.5047	0.395350
$707$	8.30260i	0.312251i
$708$	0	0
$709$	− 36.7982i	− 1.38199i	−0.722861	−	0.690993i	$-0.757174\pi$
$709$	− 36.7982i	− 1.38199i	0.722861	−	0.690993i	$-0.242826\pi$
$710$	5.38404i	0.202060i
$711$	0	0
$712$	3.10992	0.116549
$713$	4.97120i	0.186173i
$714$	0	0
$715$	0	0
$716$	−22.7928	−0.851808
$717$	0	0
$718$	5.10992	0.190700
$719$	35.2223	1.31357	0.656786	−	0.754077i	$-0.271916\pi$
$719$	35.2223	1.31357	0.656786	+	0.754077i	$0.271916\pi$
$720$	0	0
$721$	− 8.52781i	− 0.317592i
$722$	15.8310i	0.589169i
$723$	0	0
$724$	0.537500	0.0199760
$725$	−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
$729$	0	0
$730$	30.9855i	1.14683i
$731$	61.4180	2.27163
$732$	0	0
$733$	− 27.1400i	− 1.00244i	−0.865320	−	0.501220i	$-0.832885\pi$
$733$	− 27.1400i	− 1.00244i	0.865320	−	0.501220i	$-0.167115\pi$
$734$	− 8.44803i	− 0.311822i
$735$	0	0
$736$	5.10992i	0.188354i
$737$	29.3685	1.08180
$738$	0	0
$739$	3.72587i	0.137058i	0.997649	+	0.0685292i	$0.0218306\pi$
$739$	3.72587i	0.137058i	−0.997649	+	0.0685292i	$0.978169\pi$
$740$	5.20775	0.191441
$741$	0	0
$742$	−9.32603	−0.342369
$743$	21.8586i	0.801915i	0.916097	+	0.400958i	$0.131323\pi$
$743$	21.8586i	0.801915i	−0.916097	+	0.400958i	$0.868677\pi$
$744$	0	0
$745$	10.3351	0.378650
$746$	7.69096i	0.281586i
$747$	0	0
$748$	35.8189i	1.30967i
$749$	4.08277i	0.149181i
$750$	0	0
$751$	−53.3642	−1.94729	−0.973644	−	0.228075i	$-0.926757\pi$
$751$	−53.3642	−1.94729	−0.973644	+	0.228075i	$0.926757\pi$
$752$	− 7.20775i	− 0.262840i
$753$	0	0
$754$	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	$0.526845\pi$
$758$	−11.4034	−0.414191
$759$	0	0
$760$	7.20775i	0.261453i
$761$	− 11.5603i	− 0.419062i	−0.977802	−	0.209531i	$-0.932806\pi$
$761$	− 11.5603i	− 0.419062i	0.977802	−	0.209531i	$-0.0671937\pi$
$762$	0	0
$763$	0.548253	0.0198481
$764$	−9.79954	−0.354535
$765$	0	0
$766$	−20.6703	−0.746847
$767$	0	0
$768$	0	0
$769$	− 14.7439i	− 0.531679i	−0.964017	−	0.265840i	$-0.914351\pi$
$769$	− 14.7439i	− 0.531679i	0.964017	−	0.265840i	$-0.0856492\pi$
$770$	−13.5918	−0.489814
$771$	0	0
$772$	14.1957i	0.510913i
$773$	− 6.42268i	− 0.231008i	−0.993307	−	0.115504i	$-0.963152\pi$
$773$	− 6.42268i	− 0.231008i	0.993307	−	0.115504i	$-0.0368483\pi$
$774$	0	0
$775$	− 11.0844i	− 0.398164i
$776$	−8.54288	−0.306671
$777$	0	0
$778$	17.4776i	0.626601i
$779$	−2.68100	−0.0960570
$780$	0	0
$781$	6.45042	0.230814
$782$	37.7318i	1.34929i
$783$	0	0
$784$	6.52111	0.232897
$785$	25.5797i	0.912979i
$786$	0	0
$787$	43.4336i	1.54824i	0.633040	+	0.774119i	$0.281807\pi$
$787$	43.4336i	1.54824i	−0.633040	+	0.774119i	$0.718193\pi$
$788$	3.00969i	0.107216i
$789$	0	0
$790$	−33.7657	−1.20133
$791$	4.30426i	0.153042i
$792$	0	0
$793$	0	0
$794$	19.3599	0.687056
$795$	0	0
$796$	12.8944	0.457030
$797$	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
$799$	− 53.2223i	− 1.88287i
$800$	− 11.3937i	− 0.402829i
$801$	0	0
$802$	−14.4832	−0.511420
$803$	37.1226	1.31003
$804$	0	0
$805$	−14.3177	−0.504631
$806$	0	0
$807$	0	0
$808$	11.9976i	0.422074i
$809$	8.32245	0.292602	0.146301	−	0.989240i	$-0.453263\pi$
$809$	8.32245	0.292602	0.146301	+	0.989240i	$0.453263\pi$
$810$	0	0
$811$	− 14.4638i	− 0.507894i	−0.967218	−	0.253947i	$-0.918271\pi$
$811$	− 14.4638i	− 0.507894i	0.967218	−	0.253947i	$-0.0817288\pi$
$812$	2.31468i	0.0812295i
$813$	0	0
$814$	− 6.23921i	− 0.218684i
$815$	59.0810	2.06952
$816$	0	0
$817$	− 14.8068i	− 0.518026i
$818$	−18.8984	−0.660769
$819$	0	0
$820$	6.09783	0.212946
$821$	38.4161i	1.34073i	0.742031	+	0.670365i	$0.233863\pi$
$821$	38.4161i	1.34073i	−0.742031	+	0.670365i	$0.766137\pi$
$822$	0	0
$823$	40.9748	1.42829	0.714145	−	0.699997i	$-0.246816\pi$
$823$	40.9748	1.42829	0.714145	+	0.699997i	$0.246816\pi$
$824$	− 12.3230i	− 0.429294i
$825$	0	0
$826$	0.905149i	0.0314942i
$827$	3.51035i	0.122067i	0.998136	+	0.0610335i	$0.0194396\pi$
$827$	3.51035i	0.122067i	−0.998136	+	0.0610335i	$0.980560\pi$
$828$	0	0
$829$	13.2185	0.459098	0.229549	−	0.973297i	$-0.426275\pi$
$829$	13.2185	0.459098	0.229549	+	0.973297i	$0.426275\pi$
$830$	− 62.0592i	− 2.15411i
$831$	0	0
$832$	0	0
$833$	48.1521	1.66837
$834$	0	0
$835$	78.9590	2.73249
$836$	8.63533	0.298659
$837$	0	0
$838$	21.7603i	0.751698i
$839$	55.8491i	1.92812i	0.265678	+	0.964062i	$0.414404\pi$
$839$	55.8491i	1.92812i	−0.265678	+	0.964062i	$0.585596\pi$
$840$	0	0
$841$	−17.8122	−0.614214
$842$	20.5918	0.709640
$843$	0	0
$844$	−7.79954	−0.268471
$845$	0	0
$846$	0	0
$847$	8.67158i	0.297959i
$848$	−13.4765	−0.462785
$849$	0	0
$850$	− 84.1318i	− 2.88570i
$851$	− 6.57242i	− 0.225300i
$852$	0	0
$853$	− 21.8103i	− 0.746770i	−0.927676	−	0.373385i	$-0.878197\pi$
$853$	− 21.8103i	− 0.746770i	0.927676	−	0.373385i	$-0.121803\pi$
$854$	−0.274127	−0.00938042
$855$	0	0
$856$	5.89977i	0.201650i
$857$	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	$-0.595225\pi$
$859$	−17.2755	−0.589431	−0.294715	+	0.955585i	$0.595225\pi$
$860$	33.6775i	1.14839i
$861$	0	0
$862$	−34.5133	−1.17553
$863$	− 44.7741i	− 1.52413i	−0.647503	−	0.762063i	$-0.724187\pi$
$863$	− 44.7741i	− 1.52413i	0.647503	−	0.762063i	$-0.275813\pi$
$864$	0	0
$865$	37.6165i	1.27900i
$866$	2.12631i	0.0722549i
$867$	0	0
$868$	−0.673235	−0.0228511
$869$	40.4534i	1.37229i
$870$	0	0
$871$	0	0
$872$	0.792249	0.0268289
$873$	0	0
$874$	9.09651	0.307694
$875$	17.9148	0.605632
$876$	0	0
$877$	40.2741i	1.35996i	0.733230	+	0.679980i	$0.238012\pi$
$877$	40.2741i	1.35996i	−0.733230	+	0.679980i	$0.761988\pi$
$878$	− 21.8321i	− 0.736797i
$879$	0	0
$880$	−19.6407	−0.662088
$881$	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
$885$	0	0
$886$	7.54048i	0.253328i
$887$	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
$889$	− 4.15585i	− 0.139383i
$890$	12.5918i	0.422078i
$891$	0	0
$892$	− 12.1957i	− 0.408341i
$893$	−12.8310	−0.429373
$894$	0	0
$895$	− 92.2863i	− 3.08479i
$896$	−0.692021	−0.0231188
$897$	0	0
$898$	19.7560	0.659266
$899$	3.25401i	0.108527i
$900$	0	0
$901$	−99.5111	−3.31519
$902$	− 7.30559i	− 0.243249i
$903$	0	0
$904$	6.21983i	0.206869i
$905$	2.17629i	0.0723424i
$906$	0	0
$907$	34.8310	1.15654	0.578272	−	0.815844i	$-0.303727\pi$
$907$	34.8310	1.15654	0.578272	+	0.815844i	$0.303727\pi$
$908$	− 6.74333i	− 0.223785i
$909$	0	0
$910$	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
$914$	23.8582	0.789157
$915$	0	0
$916$	19.8237i	0.654994i
$917$	6.10156i	0.201491i
$918$	0	0
$919$	21.0175	0.693302	0.346651	−	0.937994i	$-0.387319\pi$
$919$	21.0175	0.693302	0.346651	+	0.937994i	$0.387319\pi$
$920$	−20.6896	−0.682117
$921$	0	0
$922$	17.5773	0.578878
$923$	0	0
$924$	0	0
$925$	14.6547i	0.481844i
$926$	−23.8431	−0.783532
$927$	0	0
$928$	3.34481i	0.109799i
$929$	− 26.9965i	− 0.885728i	−0.896589	−	0.442864i	$-0.853962\pi$
$929$	− 26.9965i	− 0.885728i	0.896589	−	0.442864i	$-0.146038\pi$
$930$	0	0
$931$	− 11.6087i	− 0.380459i
$932$	30.0301	0.983670
$933$	0	0
$934$	8.61058i	0.281747i
$935$	−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	$0.376427\pi$
$938$	4.18970i	0.136799i
$939$	0	0
$940$	29.1836	0.951864
$941$	7.54048i	0.245813i	0.992418	+	0.122906i	$0.0392215\pi$
$941$	7.54048i	0.245813i	−0.992418	+	0.122906i	$0.960779\pi$
$942$	0	0
$943$	− 7.69574i	− 0.250608i
$944$	1.30798i	0.0425711i
$945$	0	0
$946$	40.3478	1.31182
$947$	− 20.6601i	− 0.671363i	−0.941976	−	0.335681i	$-0.891033\pi$
$947$	− 20.6601i	− 0.671363i	0.941976	−	0.335681i	$-0.108967\pi$
$948$	0	0
$949$	0	0
$950$	−20.2828	−0.658059
$951$	0	0
$952$	−5.10992	−0.165613
$953$	−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
$955$	− 39.6775i	− 1.28394i
$956$	− 22.0978i	− 0.714695i
$957$	0	0
$958$	6.58104	0.212624
$959$	−10.9035	−0.352092
$960$	0	0
$961$	30.0536	0.969470
$962$	0	0
$963$	0	0
$964$	− 10.1274i	− 0.326181i
$965$	−57.4771	−1.85025
$966$	0	0
$967$	20.7289i	0.666595i	0.942822	+	0.333298i	$0.108161\pi$
$967$	20.7289i	0.666595i	−0.942822	+	0.333298i	$0.891839\pi$
$968$	12.5308i	0.402755i
$969$	0	0
$970$	− 34.5894i	− 1.11060i
$971$	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
$973$	4.21983i	0.135282i
$974$	−23.2760	−0.745813
$975$	0	0
$976$	−0.396125	−0.0126796
$977$	7.27545i	0.232762i	0.993205	+	0.116381i	$0.0371294\pi$
$977$	7.27545i	0.232762i	−0.993205	+	0.116381i	$0.962871\pi$
$978$	0	0
$979$	15.0858	0.482143
$980$	26.4034i	0.843426i
$981$	0	0
$982$	− 15.5405i	− 0.495917i
$983$	− 53.4857i	− 1.70593i	−0.521969	−	0.852965i	$-0.674802\pi$
$983$	− 53.4857i	− 1.70593i	0.521969	−	0.852965i	$-0.325198\pi$
$984$	0	0
$985$	−12.1860	−0.388278
$986$	24.6983i	0.786553i
$987$	0	0
$988$	0	0
$989$	42.5026	1.35150
$990$	0	0
$991$	16.0575	0.510085	0.255042	−	0.966930i	$-0.417911\pi$
$991$	16.0575	0.510085	0.255042	+	0.966930i	$0.417911\pi$
$992$	−0.972853	−0.0308881
$993$	0	0
$994$	0.920215i	0.0291874i
$995$	52.2083i	1.65512i
$996$	0	0
$997$	22.4263	0.710247	0.355123	−	0.934819i	$-0.384439\pi$
$997$	22.4263	0.710247	0.355123	+	0.934819i	$0.384439\pi$
$998$	9.53617	0.301862
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.r.1351.4		6
3.2	odd	2		1014.2.b.g.337.3		6
13.5	odd	4		3042.2.a.be.1.1		3
13.8	odd	4		3042.2.a.bd.1.3		3
13.12	even	2	inner	3042.2.b.r.1351.3		6
39.2	even	12		1014.2.e.m.529.3		6
39.5	even	4		1014.2.a.m.1.3	✓	3
39.8	even	4		1014.2.a.o.1.1	yes	3
39.11	even	12		1014.2.e.k.529.1		6
39.17	odd	6		1014.2.i.g.361.4		12
39.20	even	12		1014.2.e.k.991.1		6
39.23	odd	6		1014.2.i.g.823.1		12
39.29	odd	6		1014.2.i.g.823.6		12
39.32	even	12		1014.2.e.m.991.3		6
39.35	odd	6		1014.2.i.g.361.3		12
39.38	odd	2		1014.2.b.g.337.4		6
156.47	odd	4		8112.2.a.bz.1.1		3
156.83	odd	4		8112.2.a.ce.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.2.a.m.1.3	✓	3	39.5	even	4
1014.2.a.o.1.1	yes	3	39.8	even	4
1014.2.b.g.337.3		6	3.2	odd	2
1014.2.b.g.337.4		6	39.38	odd	2
1014.2.e.k.529.1		6	39.11	even	12
1014.2.e.k.991.1		6	39.20	even	12
1014.2.e.m.529.3		6	39.2	even	12
1014.2.e.m.991.3		6	39.32	even	12
1014.2.i.g.361.3		12	39.35	odd	6
1014.2.i.g.361.4		12	39.17	odd	6
1014.2.i.g.823.1		12	39.23	odd	6
1014.2.i.g.823.6		12	39.29	odd	6
3042.2.a.bd.1.3		3	13.8	odd	4
3042.2.a.be.1.1		3	13.5	odd	4
3042.2.b.r.1351.3		6	13.12	even	2	inner
3042.2.b.r.1351.4		6	1.1	even	1	trivial
8112.2.a.bz.1.1		3	156.47	odd	4
8112.2.a.ce.1.3		3	156.83	odd	4

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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -4.04892i q^{5} -0.692021i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -4.04892i q^{5} -0.692021i q^{7} -1.00000i q^{8} +4.04892 q^{10} -4.85086i q^{11} +0.692021 q^{14} +1.00000 q^{16} +7.38404 q^{17} -1.78017i q^{19} +4.04892i q^{20} +4.85086 q^{22} +5.10992 q^{23} -11.3937 q^{25} +0.692021i q^{28} +3.34481 q^{29} +0.972853i q^{31} +1.00000i q^{32} +7.38404i q^{34} -2.80194 q^{35} -1.28621i q^{37} +1.78017 q^{38} -4.04892 q^{40} -1.50604i q^{41} +8.31767 q^{43} +4.85086i q^{44} +5.10992i q^{46} -7.20775i q^{47} +6.52111 q^{49} -11.3937i q^{50} -13.4765 q^{53} -19.6407 q^{55} -0.692021 q^{56} +3.34481i q^{58} +1.30798i q^{59} -0.396125 q^{61} -0.972853 q^{62} -1.00000 q^{64} +6.05429i q^{67} -7.38404 q^{68} -2.80194i q^{70} +1.32975i q^{71} +7.65279i q^{73} +1.28621 q^{74} +1.78017i q^{76} -3.35690 q^{77} -8.33944 q^{79} -4.04892i q^{80} +1.50604 q^{82} -15.3274i q^{83} -29.8974i q^{85} +8.31767i q^{86} -4.85086 q^{88} +3.10992i q^{89} -5.10992 q^{92} +7.20775 q^{94} -7.20775 q^{95} -8.54288i q^{97} +6.52111i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4}+O(q^{10}) \) 6 * q - 6 * q^4	\( 6 q - 6 q^{4} + 6 q^{10} - 6 q^{14} + 6 q^{16} + 24 q^{17} + 2 q^{22} + 32 q^{23} - 4 q^{25} - 26 q^{29} - 8 q^{35} + 8 q^{38} - 6 q^{40} + 16 q^{43} + 8 q^{49} - 30 q^{53} - 44 q^{55} + 6 q^{56} - 20 q^{61} - 18 q^{62} - 6 q^{64} - 24 q^{68} + 24 q^{74} - 12 q^{77} - 10 q^{79} + 28 q^{82} - 2 q^{88} - 32 q^{92} + 8 q^{94} - 8 q^{95}+O(q^{100}) \) 6 * q - 6 * q^4 + 6 * q^10 - 6 * q^14 + 6 * q^16 + 24 * q^17 + 2 * q^22 + 32 * q^23 - 4 * q^25 - 26 * q^29 - 8 * q^35 + 8 * q^38 - 6 * q^40 + 16 * q^43 + 8 * q^49 - 30 * q^53 - 44 * q^55 + 6 * q^56 - 20 * q^61 - 18 * q^62 - 6 * q^64 - 24 * q^68 + 24 * q^74 - 12 * q^77 - 10 * q^79 + 28 * q^82 - 2 * q^88 - 32 * q^92 + 8 * q^94 - 8 * q^95

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