LMFDB - Embedded newform 3042.2.b.r.1351.1

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			1351.1
Root			$-0.445042i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.r.1351.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -0.692021i q^{5} +0.356896i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -0.692021i q^{5} +0.356896i q^{7} +1.00000i q^{8} -0.692021 q^{10} -2.93900i q^{11} +0.356896 q^{14} +1.00000 q^{16} +6.71379 q^{17} +7.20775i q^{19} +0.692021i q^{20} -2.93900 q^{22} +2.39612 q^{23} +4.52111 q^{25} -0.356896i q^{28} -7.82908 q^{29} +2.76271i q^{31} -1.00000i q^{32} -6.71379i q^{34} +0.246980 q^{35} +10.0978i q^{37} +7.20775 q^{38} +0.692021 q^{40} +4.89008i q^{41} -6.59179 q^{43} +2.93900i q^{44} -2.39612i q^{46} -4.98792i q^{47} +6.87263 q^{49} -4.52111i q^{50} +8.88769 q^{53} -2.03385 q^{55} -0.356896 q^{56} +7.82908i q^{58} -1.64310i q^{59} -6.49396 q^{61} +2.76271 q^{62} -1.00000 q^{64} -13.5254i q^{67} -6.71379 q^{68} -0.246980i q^{70} +6.81163i q^{71} +3.18598i q^{73} +10.0978 q^{74} -7.20775i q^{76} +1.04892 q^{77} +15.0465 q^{79} -0.692021i q^{80} +4.89008 q^{82} -14.8267i q^{83} -4.64609i q^{85} +6.59179i q^{86} +2.93900 q^{88} -0.396125i q^{89} -2.39612 q^{92} -4.98792 q^{94} +4.98792 q^{95} +0.417895i q^{97} -6.87263i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 0.692021i	− 0.309481i	−0.987955	−	0.154741i	$-0.950546\pi$
$5$	− 0.692021i	− 0.309481i	0.987955	−	0.154741i	$-0.0494542\pi$
$6$	0	0
$7$	0.356896i	0.134894i	0.997723	+	0.0674470i	$0.0214854\pi$
$7$	0.356896i	0.134894i	−0.997723	+	0.0674470i	$0.978515\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	−0.692021	−0.218836
$11$	− 2.93900i	− 0.886142i	−0.896486	−	0.443071i	$-0.853889\pi$
$11$	− 2.93900i	− 0.886142i	0.896486	−	0.443071i	$-0.146111\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0.356896	0.0953844
$15$	0	0
$16$	1.00000	0.250000
$17$	6.71379	1.62833	0.814167	−	0.580631i	$-0.197194\pi$
$17$	6.71379	1.62833	0.814167	+	0.580631i	$0.197194\pi$
$18$	0	0
$19$	7.20775i	1.65357i	0.562517	+	0.826786i	$0.309833\pi$
$19$	7.20775i	1.65357i	−0.562517	+	0.826786i	$0.690167\pi$
$20$	0.692021i	0.154741i
$21$	0	0
$22$	−2.93900	−0.626597
$23$	2.39612	0.499627	0.249813	−	0.968294i	$-0.419631\pi$
$23$	2.39612	0.499627	0.249813	+	0.968294i	$0.419631\pi$
$24$	0	0
$25$	4.52111	0.904221
$26$	0	0
$27$	0	0
$28$	− 0.356896i	− 0.0674470i
$29$	−7.82908	−1.45382	−0.726912	−	0.686730i	$-0.759045\pi$
$29$	−7.82908	−1.45382	−0.726912	+	0.686730i	$0.759045\pi$
$30$	0	0
$31$	2.76271i	0.496197i	0.968735	+	0.248099i	$0.0798057\pi$
$31$	2.76271i	0.496197i	−0.968735	+	0.248099i	$0.920194\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 6.71379i	− 1.15141i
$35$	0.246980	0.0417472
$36$	0	0
$37$	10.0978i	1.66007i	0.557708	+	0.830037i	$0.311681\pi$
$37$	10.0978i	1.66007i	−0.557708	+	0.830037i	$0.688319\pi$
$38$	7.20775	1.16925
$39$	0	0
$40$	0.692021	0.109418
$41$	4.89008i	0.763703i	0.924224	+	0.381851i	$0.124713\pi$
$41$	4.89008i	0.763703i	−0.924224	+	0.381851i	$0.875287\pi$
$42$	0	0
$43$	−6.59179	−1.00524	−0.502620	−	0.864508i	$-0.667630\pi$
$43$	−6.59179	−1.00524	−0.502620	+	0.864508i	$0.667630\pi$
$44$	2.93900i	0.443071i
$45$	0	0
$46$	− 2.39612i	− 0.353289i
$47$	− 4.98792i	− 0.727563i	−0.931484	−	0.363781i	$-0.881486\pi$
$47$	− 4.98792i	− 0.727563i	0.931484	−	0.363781i	$-0.118514\pi$
$48$	0	0
$49$	6.87263	0.981804
$50$	− 4.52111i	− 0.639381i
$51$	0	0
$52$	0	0
$53$	8.88769	1.22082	0.610409	−	0.792086i	$-0.291005\pi$
$53$	8.88769	1.22082	0.610409	+	0.792086i	$0.291005\pi$
$54$	0	0
$55$	−2.03385	−0.274245
$56$	−0.356896	−0.0476922
$57$	0	0
$58$	7.82908i	1.02801i
$59$	− 1.64310i	− 0.213914i	−0.994264	−	0.106957i	$-0.965889\pi$
$59$	− 1.64310i	− 0.213914i	0.994264	−	0.106957i	$-0.0341107\pi$
$60$	0	0
$61$	−6.49396	−0.831466	−0.415733	−	0.909487i	$-0.636475\pi$
$61$	−6.49396	−0.831466	−0.415733	+	0.909487i	$0.636475\pi$
$62$	2.76271	0.350864
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 13.5254i	− 1.65239i	−0.563382	−	0.826196i	$-0.690500\pi$
$67$	− 13.5254i	− 1.65239i	0.563382	−	0.826196i	$-0.309500\pi$
$68$	−6.71379	−0.814167
$69$	0	0
$70$	− 0.246980i	− 0.0295197i
$71$	6.81163i	0.808391i	0.914673	+	0.404196i	$0.132449\pi$
$71$	6.81163i	0.808391i	−0.914673	+	0.404196i	$0.867551\pi$
$72$	0	0
$73$	3.18598i	0.372891i	0.982465	+	0.186445i	$0.0596968\pi$
$73$	3.18598i	0.372891i	−0.982465	+	0.186445i	$0.940303\pi$
$74$	10.0978	1.17385
$75$	0	0
$76$	− 7.20775i	− 0.826786i
$77$	1.04892	0.119535
$78$	0	0
$79$	15.0465	1.69287	0.846433	−	0.532495i	$-0.178745\pi$
$79$	15.0465	1.69287	0.846433	+	0.532495i	$0.178745\pi$
$80$	− 0.692021i	− 0.0773704i
$81$	0	0
$82$	4.89008	0.540019
$83$	− 14.8267i	− 1.62744i	−0.581256	−	0.813720i	$-0.697439\pi$
$83$	− 14.8267i	− 1.62744i	0.581256	−	0.813720i	$-0.302561\pi$
$84$	0	0
$85$	− 4.64609i	− 0.503939i
$86$	6.59179i	0.710811i
$87$	0	0
$88$	2.93900	0.313299
$89$	− 0.396125i	− 0.0419891i	−0.999780	−	0.0209946i	$-0.993317\pi$
$89$	− 0.396125i	− 0.0419891i	0.999780	−	0.0209946i	$-0.00668327\pi$
$90$	0	0
$91$	0	0
$92$	−2.39612	−0.249813
$93$	0	0
$94$	−4.98792	−0.514465
$95$	4.98792	0.511750
$96$	0	0
$97$	0.417895i	0.0424308i	0.999775	+	0.0212154i	$0.00675358\pi$
$97$	0.417895i	0.0424308i	−0.999775	+	0.0212154i	$0.993246\pi$
$98$	− 6.87263i	− 0.694240i
$99$	0	0
$100$	−4.52111	−0.452111
$101$	10.0151	0.996536	0.498268	−	0.867023i	$-0.333970\pi$
$101$	10.0151	0.996536	0.498268	+	0.867023i	$0.333970\pi$
$102$	0	0
$103$	9.62565	0.948443	0.474222	−	0.880406i	$-0.342730\pi$
$103$	9.62565	0.948443	0.474222	+	0.880406i	$0.342730\pi$
$104$	0	0
$105$	0	0
$106$	− 8.88769i	− 0.863249i
$107$	6.63102	0.641045	0.320523	−	0.947241i	$-0.396141\pi$
$107$	6.63102	0.641045	0.320523	+	0.947241i	$0.396141\pi$
$108$	0	0
$109$	− 12.9879i	− 1.24402i	−0.783011	−	0.622008i	$-0.786317\pi$
$109$	− 12.9879i	− 1.24402i	0.783011	−	0.622008i	$-0.213683\pi$
$110$	2.03385i	0.193920i
$111$	0	0
$112$	0.356896i	0.0337235i
$113$	−0.792249	−0.0745285	−0.0372643	−	0.999305i	$-0.511864\pi$
$113$	−0.792249	−0.0745285	−0.0372643	+	0.999305i	$0.511864\pi$
$114$	0	0
$115$	− 1.65817i	− 0.154625i
$116$	7.82908	0.726912
$117$	0	0
$118$	−1.64310	−0.151260
$119$	2.39612i	0.219652i
$120$	0	0
$121$	2.36227	0.214752
$122$	6.49396i	0.587935i
$123$	0	0
$124$	− 2.76271i	− 0.248099i
$125$	− 6.58881i	− 0.589321i
$126$	0	0
$127$	18.2174	1.61654	0.808268	−	0.588815i	$-0.200405\pi$
$127$	18.2174	1.61654	0.808268	+	0.588815i	$0.200405\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−2.73556	−0.239007	−0.119504	−	0.992834i	$-0.538130\pi$
$131$	−2.73556	−0.239007	−0.119504	+	0.992834i	$0.538130\pi$
$132$	0	0
$133$	−2.57242	−0.223057
$134$	−13.5254	−1.16842
$135$	0	0
$136$	6.71379i	0.575703i
$137$	7.64742i	0.653363i	0.945135	+	0.326681i	$0.105930\pi$
$137$	7.64742i	0.653363i	−0.945135	+	0.326681i	$0.894070\pi$
$138$	0	0
$139$	3.38404	0.287031	0.143515	−	0.989648i	$-0.454159\pi$
$139$	3.38404	0.287031	0.143515	+	0.989648i	$0.454159\pi$
$140$	−0.246980	−0.0208736
$141$	0	0
$142$	6.81163	0.571619
$143$	0	0
$144$	0	0
$145$	5.41789i	0.449932i
$146$	3.18598	0.263674
$147$	0	0
$148$	− 10.0978i	− 0.830037i
$149$	20.8170i	1.70540i	0.522405	+	0.852698i	$0.325035\pi$
$149$	20.8170i	1.70540i	−0.522405	+	0.852698i	$0.674965\pi$
$150$	0	0
$151$	0.895461i	0.0728715i	0.999336	+	0.0364358i	$0.0116004\pi$
$151$	0.895461i	0.0728715i	−0.999336	+	0.0364358i	$0.988400\pi$
$152$	−7.20775	−0.584626
$153$	0	0
$154$	− 1.04892i	− 0.0845242i
$155$	1.91185	0.153564
$156$	0	0
$157$	8.59179	0.685700	0.342850	−	0.939390i	$-0.388608\pi$
$157$	8.59179	0.685700	0.342850	+	0.939390i	$0.388608\pi$
$158$	− 15.0465i	− 1.19704i
$159$	0	0
$160$	−0.692021	−0.0547091
$161$	0.855167i	0.0673966i
$162$	0	0
$163$	− 1.72587i	− 0.135181i	−0.997713	−	0.0675904i	$-0.978469\pi$
$163$	− 1.72587i	− 0.135181i	0.997713	−	0.0675904i	$-0.0215311\pi$
$164$	− 4.89008i	− 0.381851i
$165$	0	0
$166$	−14.8267	−1.15077
$167$	21.1400i	1.63587i	0.575314	+	0.817933i	$0.304880\pi$
$167$	21.1400i	1.63587i	−0.575314	+	0.817933i	$0.695120\pi$
$168$	0	0
$169$	0	0
$170$	−4.64609	−0.356339
$171$	0	0
$172$	6.59179	0.502620
$173$	9.35450	0.711210	0.355605	−	0.934636i	$-0.384275\pi$
$173$	9.35450	0.711210	0.355605	+	0.934636i	$0.384275\pi$
$174$	0	0
$175$	1.61356i	0.121974i
$176$	− 2.93900i	− 0.221536i
$177$	0	0
$178$	−0.396125	−0.0296908
$179$	3.17523	0.237328	0.118664	−	0.992934i	$-0.462139\pi$
$179$	3.17523	0.237328	0.118664	+	0.992934i	$0.462139\pi$
$180$	0	0
$181$	19.7995	1.47169	0.735844	−	0.677151i	$-0.236786\pi$
$181$	19.7995	1.47169	0.735844	+	0.677151i	$0.236786\pi$
$182$	0	0
$183$	0	0
$184$	2.39612i	0.176645i
$185$	6.98792	0.513762
$186$	0	0
$187$	− 19.7318i	− 1.44294i
$188$	4.98792i	0.363781i
$189$	0	0
$190$	− 4.98792i	− 0.361862i
$191$	−15.2620	−1.10432	−0.552161	−	0.833737i	$-0.686197\pi$
$191$	−15.2620	−1.10432	−0.552161	+	0.833737i	$0.686197\pi$
$192$	0	0
$193$	− 4.76809i	− 0.343214i	−0.985165	−	0.171607i	$-0.945104\pi$
$193$	− 4.76809i	− 0.343214i	0.985165	−	0.171607i	$-0.0548960\pi$
$194$	0.417895	0.0300031
$195$	0	0
$196$	−6.87263	−0.490902
$197$	− 12.2349i	− 0.871700i	−0.900019	−	0.435850i	$-0.856448\pi$
$197$	− 12.2349i	− 0.871700i	0.900019	−	0.435850i	$-0.143552\pi$
$198$	0	0
$199$	11.8485	0.839915	0.419958	−	0.907544i	$-0.362045\pi$
$199$	11.8485	0.839915	0.419958	+	0.907544i	$0.362045\pi$
$200$	4.52111i	0.319690i
$201$	0	0
$202$	− 10.0151i	− 0.704658i
$203$	− 2.79417i	− 0.196112i
$204$	0	0
$205$	3.38404	0.236352
$206$	− 9.62565i	− 0.670651i
$207$	0	0
$208$	0	0
$209$	21.1836	1.46530
$210$	0	0
$211$	−17.2620	−1.18837	−0.594184	−	0.804329i	$-0.702525\pi$
$211$	−17.2620	−1.18837	−0.594184	+	0.804329i	$0.702525\pi$
$212$	−8.88769	−0.610409
$213$	0	0
$214$	− 6.63102i	− 0.453287i
$215$	4.56166i	0.311103i
$216$	0	0
$217$	−0.985999	−0.0669340
$218$	−12.9879	−0.879653
$219$	0	0
$220$	2.03385	0.137122
$221$	0	0
$222$	0	0
$223$	6.76809i	0.453225i	0.973985	+	0.226612i	$0.0727650\pi$
$223$	6.76809i	0.453225i	−0.973985	+	0.226612i	$0.927235\pi$
$224$	0.356896	0.0238461
$225$	0	0
$226$	0.792249i	0.0526996i
$227$	− 23.6799i	− 1.57169i	−0.618422	−	0.785846i	$-0.712228\pi$
$227$	− 23.6799i	− 1.57169i	0.618422	−	0.785846i	$-0.287772\pi$
$228$	0	0
$229$	8.29829i	0.548366i	0.961677	+	0.274183i	$0.0884075\pi$
$229$	8.29829i	0.548366i	−0.961677	+	0.274183i	$0.911593\pi$
$230$	−1.65817	−0.109336
$231$	0	0
$232$	− 7.82908i	− 0.514005i
$233$	−23.9651	−1.57000	−0.785002	−	0.619493i	$-0.787338\pi$
$233$	−23.9651	−1.57000	−0.785002	+	0.619493i	$0.787338\pi$
$234$	0	0
$235$	−3.45175	−0.225167
$236$	1.64310i	0.106957i
$237$	0	0
$238$	2.39612	0.155318
$239$	− 12.6160i	− 0.816058i	−0.912969	−	0.408029i	$-0.866216\pi$
$239$	− 12.6160i	− 0.816058i	0.912969	−	0.408029i	$-0.133784\pi$
$240$	0	0
$241$	− 26.3937i	− 1.70017i	−0.526646	−	0.850085i	$-0.676551\pi$
$241$	− 26.3937i	− 1.70017i	0.526646	−	0.850085i	$-0.323449\pi$
$242$	− 2.36227i	− 0.151853i
$243$	0	0
$244$	6.49396	0.415733
$245$	− 4.75600i	− 0.303850i
$246$	0	0
$247$	0	0
$248$	−2.76271	−0.175432
$249$	0	0
$250$	−6.58881	−0.416713
$251$	−30.0344	−1.89576	−0.947879	−	0.318632i	$-0.896777\pi$
$251$	−30.0344	−1.89576	−0.947879	+	0.318632i	$0.896777\pi$
$252$	0	0
$253$	− 7.04221i	− 0.442740i
$254$	− 18.2174i	− 1.14306i
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.2620	0.702507	0.351254	−	0.936280i	$-0.385756\pi$
$257$	11.2620	0.702507	0.351254	+	0.936280i	$0.385756\pi$
$258$	0	0
$259$	−3.60388	−0.223934
$260$	0	0
$261$	0	0
$262$	2.73556i	0.169004i
$263$	5.54958	0.342202	0.171101	−	0.985254i	$-0.445268\pi$
$263$	5.54958	0.342202	0.171101	+	0.985254i	$0.445268\pi$
$264$	0	0
$265$	− 6.15047i	− 0.377821i
$266$	2.57242i	0.157725i
$267$	0	0
$268$	13.5254i	0.826196i
$269$	−16.6872	−1.01744	−0.508719	−	0.860932i	$-0.669881\pi$
$269$	−16.6872	−1.01744	−0.508719	+	0.860932i	$0.669881\pi$
$270$	0	0
$271$	6.61356i	0.401745i	0.979617	+	0.200873i	$0.0643778\pi$
$271$	6.61356i	0.401745i	−0.979617	+	0.200873i	$0.935622\pi$
$272$	6.71379	0.407083
$273$	0	0
$274$	7.64742	0.461997
$275$	− 13.2875i	− 0.801269i
$276$	0	0
$277$	21.7995	1.30981	0.654904	−	0.755712i	$-0.272709\pi$
$277$	21.7995	1.30981	0.654904	+	0.755712i	$0.272709\pi$
$278$	− 3.38404i	− 0.202961i
$279$	0	0
$280$	0.246980i	0.0147599i
$281$	20.5918i	1.22840i	0.789149	+	0.614202i	$0.210522\pi$
$281$	20.5918i	1.22840i	−0.789149	+	0.614202i	$0.789478\pi$
$282$	0	0
$283$	13.0121	0.773488	0.386744	−	0.922187i	$-0.373600\pi$
$283$	13.0121	0.773488	0.386744	+	0.922187i	$0.373600\pi$
$284$	− 6.81163i	− 0.404196i
$285$	0	0
$286$	0	0
$287$	−1.74525	−0.103019
$288$	0	0
$289$	28.0750	1.65147
$290$	5.41789	0.318150
$291$	0	0
$292$	− 3.18598i	− 0.186445i
$293$	− 14.9390i	− 0.872746i	−0.899766	−	0.436373i	$-0.856263\pi$
$293$	− 14.9390i	− 0.872746i	0.899766	−	0.436373i	$-0.143737\pi$
$294$	0	0
$295$	−1.13706	−0.0662024
$296$	−10.0978	−0.586925
$297$	0	0
$298$	20.8170	1.20590
$299$	0	0
$300$	0	0
$301$	− 2.35258i	− 0.135601i
$302$	0.895461	0.0515280
$303$	0	0
$304$	7.20775i	0.413393i
$305$	4.49396i	0.257323i
$306$	0	0
$307$	− 26.0301i	− 1.48562i	−0.669503	−	0.742809i	$-0.733493\pi$
$307$	− 26.0301i	− 1.48562i	0.669503	−	0.742809i	$-0.266507\pi$
$308$	−1.04892	−0.0597676
$309$	0	0
$310$	− 1.91185i	− 0.108586i
$311$	−4.81163	−0.272842	−0.136421	−	0.990651i	$-0.543560\pi$
$311$	−4.81163	−0.272842	−0.136421	+	0.990651i	$0.543560\pi$
$312$	0	0
$313$	−26.0411	−1.47193	−0.735966	−	0.677018i	$-0.763272\pi$
$313$	−26.0411	−1.47193	−0.735966	+	0.677018i	$0.763272\pi$
$314$	− 8.59179i	− 0.484863i
$315$	0	0
$316$	−15.0465	−0.846433
$317$	11.5211i	0.647090i	0.946213	+	0.323545i	$0.104875\pi$
$317$	11.5211i	0.647090i	−0.946213	+	0.323545i	$0.895125\pi$
$318$	0	0
$319$	23.0097i	1.28830i
$320$	0.692021i	0.0386852i
$321$	0	0
$322$	0.855167	0.0476566
$323$	48.3913i	2.69257i
$324$	0	0
$325$	0	0
$326$	−1.72587	−0.0955873
$327$	0	0
$328$	−4.89008	−0.270010
$329$	1.78017	0.0981438
$330$	0	0
$331$	3.43834i	0.188988i	0.995525	+	0.0944940i	$0.0301233\pi$
$331$	3.43834i	0.188988i	−0.995525	+	0.0944940i	$0.969877\pi$
$332$	14.8267i	0.813720i
$333$	0	0
$334$	21.1400	1.15673
$335$	−9.35988	−0.511385
$336$	0	0
$337$	−8.20105	−0.446739	−0.223370	−	0.974734i	$-0.571706\pi$
$337$	−8.20105	−0.446739	−0.223370	+	0.974734i	$0.571706\pi$
$338$	0	0
$339$	0	0
$340$	4.64609i	0.251970i
$341$	8.11960	0.439701
$342$	0	0
$343$	4.95108i	0.267333i
$344$	− 6.59179i	− 0.355406i
$345$	0	0
$346$	− 9.35450i	− 0.502901i
$347$	−7.86294	−0.422105	−0.211052	−	0.977475i	$-0.567689\pi$
$347$	−7.86294	−0.422105	−0.211052	+	0.977475i	$0.567689\pi$
$348$	0	0
$349$	18.7245i	1.00230i	0.865360	+	0.501151i	$0.167090\pi$
$349$	18.7245i	1.00230i	−0.865360	+	0.501151i	$0.832910\pi$
$350$	1.61356	0.0862486
$351$	0	0
$352$	−2.93900	−0.156649
$353$	31.5448i	1.67896i	0.543391	+	0.839480i	$0.317140\pi$
$353$	31.5448i	1.67896i	−0.543391	+	0.839480i	$0.682860\pi$
$354$	0	0
$355$	4.71379	0.250182
$356$	0.396125i	0.0209946i
$357$	0	0
$358$	− 3.17523i	− 0.167816i
$359$	2.39612i	0.126463i	0.997999	+	0.0632313i	$0.0201406\pi$
$359$	2.39612i	0.126463i	−0.997999	+	0.0632313i	$0.979859\pi$
$360$	0	0
$361$	−32.9517	−1.73430
$362$	− 19.7995i	− 1.04064i
$363$	0	0
$364$	0	0
$365$	2.20477	0.115403
$366$	0	0
$367$	−0.00431187	−0.000225078 0	−0.000112539	−	1.00000i	$-0.500036\pi$
$367$	−0.00431187	−0.000225078 0	−0.000112539		1.00000i	$0.500036\pi$
$368$	2.39612	0.124907
$369$	0	0
$370$	− 6.98792i	− 0.363285i
$371$	3.17198i	0.164681i
$372$	0	0
$373$	−32.3129	−1.67310	−0.836549	−	0.547892i	$-0.815430\pi$
$373$	−32.3129	−1.67310	−0.836549	+	0.547892i	$0.815430\pi$
$374$	−19.7318	−1.02031
$375$	0	0
$376$	4.98792	0.257232
$377$	0	0
$378$	0	0
$379$	19.7560i	1.01480i	0.861711	+	0.507399i	$0.169393\pi$
$379$	19.7560i	1.01480i	−0.861711	+	0.507399i	$0.830607\pi$
$380$	−4.98792	−0.255875
$381$	0	0
$382$	15.2620i	0.780874i
$383$	− 28.8116i	− 1.47221i	−0.676870	−	0.736103i	$-0.736664\pi$
$383$	− 28.8116i	− 1.47221i	0.676870	−	0.736103i	$-0.263336\pi$
$384$	0	0
$385$	− 0.725873i	− 0.0369939i
$386$	−4.76809	−0.242689
$387$	0	0
$388$	− 0.417895i	− 0.0212154i
$389$	34.7821	1.76352	0.881761	−	0.471697i	$-0.156358\pi$
$389$	34.7821	1.76352	0.881761	+	0.471697i	$0.156358\pi$
$390$	0	0
$391$	16.0871	0.813559
$392$	6.87263i	0.347120i
$393$	0	0
$394$	−12.2349	−0.616385
$395$	− 10.4125i	− 0.523911i
$396$	0	0
$397$	5.15346i	0.258645i	0.991603	+	0.129322i	$0.0412802\pi$
$397$	5.15346i	0.258645i	−0.991603	+	0.129322i	$0.958720\pi$
$398$	− 11.8485i	− 0.593910i
$399$	0	0
$400$	4.52111	0.226055
$401$	13.3250i	0.665417i	0.943030	+	0.332708i	$0.107962\pi$
$401$	13.3250i	0.665417i	−0.943030	+	0.332708i	$0.892038\pi$
$402$	0	0
$403$	0	0
$404$	−10.0151	−0.498268
$405$	0	0
$406$	−2.79417	−0.138672
$407$	29.6775	1.47106
$408$	0	0
$409$	− 24.0237i	− 1.18789i	−0.804504	−	0.593947i	$-0.797569\pi$
$409$	− 24.0237i	− 1.18789i	0.804504	−	0.593947i	$-0.202431\pi$
$410$	− 3.38404i	− 0.167126i
$411$	0	0
$412$	−9.62565	−0.474222
$413$	0.586417	0.0288557
$414$	0	0
$415$	−10.2604	−0.503663
$416$	0	0
$417$	0	0
$418$	− 21.1836i	− 1.03612i
$419$	−13.8049	−0.674415	−0.337207	−	0.941430i	$-0.609482\pi$
$419$	−13.8049	−0.674415	−0.337207	+	0.941430i	$0.609482\pi$
$420$	0	0
$421$	7.72587i	0.376536i	0.982118	+	0.188268i	$0.0602874\pi$
$421$	7.72587i	0.376536i	−0.982118	+	0.188268i	$0.939713\pi$
$422$	17.2620i	0.840303i
$423$	0	0
$424$	8.88769i	0.431624i
$425$	30.3538	1.47237
$426$	0	0
$427$	− 2.31767i	− 0.112160i
$428$	−6.63102	−0.320523
$429$	0	0
$430$	4.56166	0.219983
$431$	− 0.640120i	− 0.0308335i	−0.999881	−	0.0154168i	$-0.995093\pi$
$431$	− 0.640120i	− 0.0308335i	0.999881	−	0.0154168i	$-0.00490750\pi$
$432$	0	0
$433$	−21.2760	−1.02246	−0.511231	−	0.859443i	$-0.670810\pi$
$433$	−21.2760	−1.02246	−0.511231	+	0.859443i	$0.670810\pi$
$434$	0.985999i	0.0473295i
$435$	0	0
$436$	12.9879i	0.622008i
$437$	17.2707i	0.826168i
$438$	0	0
$439$	−12.7181	−0.607002	−0.303501	−	0.952831i	$-0.598156\pi$
$439$	−12.7181	−0.607002	−0.303501	+	0.952831i	$0.598156\pi$
$440$	− 2.03385i	− 0.0969601i
$441$	0	0
$442$	0	0
$443$	−22.5972	−1.07362	−0.536812	−	0.843702i	$-0.680372\pi$
$443$	−22.5972	−1.07362	−0.536812	+	0.843702i	$0.680372\pi$
$444$	0	0
$445$	−0.274127	−0.0129949
$446$	6.76809	0.320478
$447$	0	0
$448$	− 0.356896i	− 0.0168617i
$449$	11.6474i	0.549676i	0.961491	+	0.274838i	$0.0886241\pi$
$449$	11.6474i	0.549676i	−0.961491	+	0.274838i	$0.911376\pi$
$450$	0	0
$451$	14.3720	0.676749
$452$	0.792249	0.0372643
$453$	0	0
$454$	−23.6799	−1.11135
$455$	0	0
$456$	0	0
$457$	− 21.1890i	− 0.991178i	−0.868557	−	0.495589i	$-0.834952\pi$
$457$	− 21.1890i	− 0.991178i	0.868557	−	0.495589i	$-0.165048\pi$
$458$	8.29829	0.387754
$459$	0	0
$460$	1.65817i	0.0773126i
$461$	− 24.0694i	− 1.12102i	−0.828147	−	0.560511i	$-0.810605\pi$
$461$	− 24.0694i	− 1.12102i	0.828147	−	0.560511i	$-0.189395\pi$
$462$	0	0
$463$	18.1715i	0.844502i	0.906479	+	0.422251i	$0.138760\pi$
$463$	18.1715i	0.844502i	−0.906479	+	0.422251i	$0.861240\pi$
$464$	−7.82908	−0.363456
$465$	0	0
$466$	23.9651i	1.11016i
$467$	−2.93123	−0.135641	−0.0678206	−	0.997698i	$-0.521605\pi$
$467$	−2.93123	−0.135641	−0.0678206	+	0.997698i	$0.521605\pi$
$468$	0	0
$469$	4.82717	0.222898
$470$	3.45175i	0.159217i
$471$	0	0
$472$	1.64310	0.0756300
$473$	19.3733i	0.890785i
$474$	0	0
$475$	32.5870i	1.49519i
$476$	− 2.39612i	− 0.109826i
$477$	0	0
$478$	−12.6160	−0.577040
$479$	− 30.7090i	− 1.40313i	−0.712605	−	0.701565i	$-0.752485\pi$
$479$	− 30.7090i	− 1.40313i	0.712605	−	0.701565i	$-0.247515\pi$
$480$	0	0
$481$	0	0
$482$	−26.3937	−1.20220
$483$	0	0
$484$	−2.36227	−0.107376
$485$	0.289192	0.0131315
$486$	0	0
$487$	24.1497i	1.09433i	0.837025	+	0.547164i	$0.184293\pi$
$487$	24.1497i	1.09433i	−0.837025	+	0.547164i	$0.815707\pi$
$488$	− 6.49396i	− 0.293968i
$489$	0	0
$490$	−4.75600	−0.214854
$491$	14.5972	0.658761	0.329381	−	0.944197i	$-0.393160\pi$
$491$	14.5972	0.658761	0.329381	+	0.944197i	$0.393160\pi$
$492$	0	0
$493$	−52.5628	−2.36731
$494$	0	0
$495$	0	0
$496$	2.76271i	0.124049i
$497$	−2.43104	−0.109047
$498$	0	0
$499$	6.85517i	0.306879i	0.988158	+	0.153440i	$0.0490351\pi$
$499$	6.85517i	0.306879i	−0.988158	+	0.153440i	$0.950965\pi$
$500$	6.58881i	0.294661i
$501$	0	0
$502$	30.0344i	1.34050i
$503$	26.7332	1.19197	0.595987	−	0.802994i	$-0.296761\pi$
$503$	26.7332	1.19197	0.595987	+	0.802994i	$0.296761\pi$
$504$	0	0
$505$	− 6.93064i	− 0.308409i
$506$	−7.04221	−0.313065
$507$	0	0
$508$	−18.2174	−0.808268
$509$	20.6595i	0.915716i	0.889025	+	0.457858i	$0.151383\pi$
$509$	20.6595i	0.915716i	−0.889025	+	0.457858i	$0.848617\pi$
$510$	0	0
$511$	−1.13706	−0.0503007
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 11.2620i	− 0.496748i
$515$	− 6.66115i	− 0.293525i
$516$	0	0
$517$	−14.6595	−0.644724
$518$	3.60388i	0.158345i
$519$	0	0
$520$	0	0
$521$	15.0965	0.661390	0.330695	−	0.943738i	$-0.392717\pi$
$521$	15.0965	0.661390	0.330695	+	0.943738i	$0.392717\pi$
$522$	0	0
$523$	0.0349168	0.00152680	0.000763402	−	1.00000i	$-0.499757\pi$
$523$	0.0349168	0.00152680	0.000763402		1.00000i	$0.499757\pi$
$524$	2.73556	0.119504
$525$	0	0
$526$	− 5.54958i	− 0.241973i
$527$	18.5483i	0.807975i
$528$	0	0
$529$	−17.2586	−0.750373
$530$	−6.15047	−0.267159
$531$	0	0
$532$	2.57242	0.111528
$533$	0	0
$534$	0	0
$535$	− 4.58881i	− 0.198392i
$536$	13.5254	0.584209
$537$	0	0
$538$	16.6872i	0.719438i
$539$	− 20.1987i	− 0.870018i
$540$	0	0
$541$	− 13.0858i	− 0.562600i	−0.959620	−	0.281300i	$-0.909234\pi$
$541$	− 13.0858i	− 0.562600i	0.959620	−	0.281300i	$-0.0907657\pi$
$542$	6.61356	0.284077
$543$	0	0
$544$	− 6.71379i	− 0.287851i
$545$	−8.98792	−0.385000
$546$	0	0
$547$	−5.97584	−0.255508	−0.127754	−	0.991806i	$-0.540777\pi$
$547$	−5.97584	−0.255508	−0.127754	+	0.991806i	$0.540777\pi$
$548$	− 7.64742i	− 0.326681i
$549$	0	0
$550$	−13.2875	−0.566582
$551$	− 56.4301i	− 2.40400i
$552$	0	0
$553$	5.37004i	0.228357i
$554$	− 21.7995i	− 0.926174i
$555$	0	0
$556$	−3.38404	−0.143515
$557$	− 10.4397i	− 0.442343i	−0.975235	−	0.221171i	$-0.929012\pi$
$557$	− 10.4397i	− 0.442343i	0.975235	−	0.221171i	$-0.0709880\pi$
$558$	0	0
$559$	0	0
$560$	0.246980	0.0104368
$561$	0	0
$562$	20.5918	0.868612
$563$	−5.66056	−0.238564	−0.119282	−	0.992860i	$-0.538059\pi$
$563$	−5.66056	−0.238564	−0.119282	+	0.992860i	$0.538059\pi$
$564$	0	0
$565$	0.548253i	0.0230652i
$566$	− 13.0121i	− 0.546939i
$567$	0	0
$568$	−6.81163	−0.285809
$569$	20.6568	0.865980	0.432990	−	0.901399i	$-0.357459\pi$
$569$	20.6568	0.865980	0.432990	+	0.901399i	$0.357459\pi$
$570$	0	0
$571$	−44.3672	−1.85671	−0.928354	−	0.371697i	$-0.878776\pi$
$571$	−44.3672	−1.85671	−0.928354	+	0.371697i	$0.878776\pi$
$572$	0	0
$573$	0	0
$574$	1.74525i	0.0728454i
$575$	10.8331	0.451773
$576$	0	0
$577$	29.4426i	1.22571i	0.790194	+	0.612857i	$0.209980\pi$
$577$	29.4426i	1.22571i	−0.790194	+	0.612857i	$0.790020\pi$
$578$	− 28.0750i	− 1.16777i
$579$	0	0
$580$	− 5.41789i	− 0.224966i
$581$	5.29159	0.219532
$582$	0	0
$583$	− 26.1209i	− 1.08182i
$584$	−3.18598	−0.131837
$585$	0	0
$586$	−14.9390	−0.617124
$587$	19.5636i	0.807475i	0.914875	+	0.403738i	$0.132289\pi$
$587$	19.5636i	0.807475i	−0.914875	+	0.403738i	$0.867711\pi$
$588$	0	0
$589$	−19.9129	−0.820498
$590$	1.13706i	0.0468122i
$591$	0	0
$592$	10.0978i	0.415018i
$593$	− 10.8793i	− 0.446761i	−0.974731	−	0.223380i	$-0.928291\pi$
$593$	− 10.8793i	− 0.446761i	0.974731	−	0.223380i	$-0.0717092\pi$
$594$	0	0
$595$	1.65817	0.0679783
$596$	− 20.8170i	− 0.852698i
$597$	0	0
$598$	0	0
$599$	−16.0543	−0.655961	−0.327980	−	0.944685i	$-0.606368\pi$
$599$	−16.0543	−0.655961	−0.327980	+	0.944685i	$0.606368\pi$
$600$	0	0
$601$	12.8955	0.526017	0.263008	−	0.964794i	$-0.415285\pi$
$601$	12.8955	0.526017	0.263008	+	0.964794i	$0.415285\pi$
$602$	−2.35258	−0.0958842
$603$	0	0
$604$	− 0.895461i	− 0.0364358i
$605$	− 1.63474i	− 0.0664618i
$606$	0	0
$607$	44.4741	1.80515	0.902574	−	0.430534i	$-0.141675\pi$
$607$	44.4741	1.80515	0.902574	+	0.430534i	$0.141675\pi$
$608$	7.20775	0.292313
$609$	0	0
$610$	4.49396	0.181955
$611$	0	0
$612$	0	0
$613$	42.0253i	1.69739i	0.528884	+	0.848694i	$0.322610\pi$
$613$	42.0253i	1.69739i	−0.528884	+	0.848694i	$0.677390\pi$
$614$	−26.0301	−1.05049
$615$	0	0
$616$	1.04892i	0.0422621i
$617$	16.6655i	0.670926i	0.942053	+	0.335463i	$0.108893\pi$
$617$	16.6655i	0.670926i	−0.942053	+	0.335463i	$0.891107\pi$
$618$	0	0
$619$	− 39.7512i	− 1.59774i	−0.601506	−	0.798868i	$-0.705432\pi$
$619$	− 39.7512i	− 1.59774i	0.601506	−	0.798868i	$-0.294568\pi$
$620$	−1.91185	−0.0767819
$621$	0	0
$622$	4.81163i	0.192929i
$623$	0.141375	0.00566408
$624$	0	0
$625$	18.0459	0.721837
$626$	26.0411i	1.04081i
$627$	0	0
$628$	−8.59179	−0.342850
$629$	67.7948i	2.70315i
$630$	0	0
$631$	− 14.5767i	− 0.580290i	−0.956983	−	0.290145i	$-0.906296\pi$
$631$	− 14.5767i	− 0.580290i	0.956983	−	0.290145i	$-0.0937036\pi$
$632$	15.0465i	0.598519i
$633$	0	0
$634$	11.5211	0.457562
$635$	− 12.6069i	− 0.500288i
$636$	0	0
$637$	0	0
$638$	23.0097	0.910962
$639$	0	0
$640$	0.692021	0.0273546
$641$	38.4349	1.51809	0.759043	−	0.651040i	$-0.225667\pi$
$641$	38.4349	1.51809	0.759043	+	0.651040i	$0.225667\pi$
$642$	0	0
$643$	− 1.04221i	− 0.0411009i	−0.999789	−	0.0205504i	$-0.993458\pi$
$643$	− 1.04221i	− 0.0411009i	0.999789	−	0.0205504i	$-0.00654186\pi$
$644$	− 0.855167i	− 0.0336983i
$645$	0	0
$646$	48.3913	1.90393
$647$	−28.1608	−1.10711	−0.553557	−	0.832811i	$-0.686730\pi$
$647$	−28.1608	−1.10711	−0.553557	+	0.832811i	$0.686730\pi$
$648$	0	0
$649$	−4.82908	−0.189558
$650$	0	0
$651$	0	0
$652$	1.72587i	0.0675904i
$653$	−10.6203	−0.415603	−0.207802	−	0.978171i	$-0.566631\pi$
$653$	−10.6203	−0.415603	−0.207802	+	0.978171i	$0.566631\pi$
$654$	0	0
$655$	1.89307i	0.0739683i
$656$	4.89008i	0.190926i
$657$	0	0
$658$	− 1.78017i	− 0.0693982i
$659$	40.3629	1.57231	0.786157	−	0.618027i	$-0.212068\pi$
$659$	40.3629	1.57231	0.786157	+	0.618027i	$0.212068\pi$
$660$	0	0
$661$	31.9168i	1.24142i	0.784041	+	0.620709i	$0.213155\pi$
$661$	31.9168i	1.24142i	−0.784041	+	0.620709i	$0.786845\pi$
$662$	3.43834	0.133635
$663$	0	0
$664$	14.8267	0.575387
$665$	1.78017i	0.0690319i
$666$	0	0
$667$	−18.7595	−0.726369
$668$	− 21.1400i	− 0.817933i
$669$	0	0
$670$	9.35988i	0.361604i
$671$	19.0858i	0.736797i
$672$	0	0
$673$	−3.82802	−0.147559	−0.0737797	−	0.997275i	$-0.523506\pi$
$673$	−3.82802	−0.147559	−0.0737797	+	0.997275i	$0.523506\pi$
$674$	8.20105i	0.315892i
$675$	0	0
$676$	0	0
$677$	−1.78927	−0.0687670	−0.0343835	−	0.999409i	$-0.510947\pi$
$677$	−1.78927	−0.0687670	−0.0343835	+	0.999409i	$0.510947\pi$
$678$	0	0
$679$	−0.149145	−0.00572366
$680$	4.64609	0.178169
$681$	0	0
$682$	− 8.11960i	− 0.310916i
$683$	0.759725i	0.0290701i	0.999894	+	0.0145350i	$0.00462681\pi$
$683$	0.759725i	0.0290701i	−0.999894	+	0.0145350i	$0.995373\pi$
$684$	0	0
$685$	5.29218	0.202204
$686$	4.95108	0.189033
$687$	0	0
$688$	−6.59179	−0.251310
$689$	0	0
$690$	0	0
$691$	4.65950i	0.177256i	0.996065	+	0.0886278i	$0.0282482\pi$
$691$	4.65950i	0.177256i	−0.996065	+	0.0886278i	$0.971752\pi$
$692$	−9.35450	−0.355605
$693$	0	0
$694$	7.86294i	0.298473i
$695$	− 2.34183i	− 0.0888307i
$696$	0	0
$697$	32.8310i	1.24356i
$698$	18.7245	0.708734
$699$	0	0
$700$	− 1.61356i	− 0.0609870i
$701$	−24.3284	−0.918872	−0.459436	−	0.888211i	$-0.651948\pi$
$701$	−24.3284	−0.918872	−0.459436	+	0.888211i	$0.651948\pi$
$702$	0	0
$703$	−72.7827	−2.74505
$704$	2.93900i	0.110768i
$705$	0	0
$706$	31.5448	1.18720
$707$	3.57434i	0.134427i
$708$	0	0
$709$	29.3927i	1.10386i	0.833889	+	0.551932i	$0.186109\pi$
$709$	29.3927i	1.10386i	−0.833889	+	0.551932i	$0.813891\pi$
$710$	− 4.71379i	− 0.176905i
$711$	0	0
$712$	0.396125	0.0148454
$713$	6.61979i	0.247913i
$714$	0	0
$715$	0	0
$716$	−3.17523	−0.118664
$717$	0	0
$718$	2.39612	0.0894226
$719$	−51.4878	−1.92017	−0.960086	−	0.279704i	$-0.909764\pi$
$719$	−51.4878	−1.92017	−0.960086	+	0.279704i	$0.909764\pi$
$720$	0	0
$721$	3.43535i	0.127939i
$722$	32.9517i	1.22633i
$723$	0	0
$724$	−19.7995	−0.735844
$725$	−35.3961	−1.31458
$726$	0	0
$727$	3.67324	0.136233	0.0681164	−	0.997677i	$-0.478301\pi$
$727$	3.67324	0.136233	0.0681164	+	0.997677i	$0.478301\pi$
$728$	0	0
$729$	0	0
$730$	− 2.20477i	− 0.0816021i
$731$	−44.2559	−1.63686
$732$	0	0
$733$	18.3612i	0.678187i	0.940753	+	0.339093i	$0.110120\pi$
$733$	18.3612i	0.678187i	−0.940753	+	0.339093i	$0.889880\pi$
$734$	0.00431187i	0 0.000159154i
$735$	0	0
$736$	− 2.39612i	− 0.0883223i
$737$	−39.7512	−1.46425
$738$	0	0
$739$	− 1.68233i	− 0.0618856i	−0.999521	−	0.0309428i	$-0.990149\pi$
$739$	− 1.68233i	− 0.0618856i	0.999521	−	0.0309428i	$-0.00985097\pi$
$740$	−6.98792	−0.256881
$741$	0	0
$742$	3.17198	0.116447
$743$	− 48.2935i	− 1.77172i	−0.463956	−	0.885858i	$-0.653570\pi$
$743$	− 48.2935i	− 1.77172i	0.463956	−	0.885858i	$-0.346430\pi$
$744$	0	0
$745$	14.4058	0.527788
$746$	32.3129i	1.18306i
$747$	0	0
$748$	19.7318i	0.721468i
$749$	2.36658i	0.0864731i
$750$	0	0
$751$	−11.7011	−0.426980	−0.213490	−	0.976945i	$-0.568483\pi$
$751$	−11.7011	−0.426980	−0.213490	+	0.976945i	$0.568483\pi$
$752$	− 4.98792i	− 0.181891i
$753$	0	0
$754$	0	0
$755$	0.619678	0.0225524
$756$	0	0
$757$	25.1836	0.915313	0.457657	−	0.889129i	$-0.348689\pi$
$757$	25.1836	0.915313	0.457657	+	0.889129i	$0.348689\pi$
$758$	19.7560	0.717570
$759$	0	0
$760$	4.98792i	0.180931i
$761$	22.4155i	0.812561i	0.913748	+	0.406281i	$0.133174\pi$
$761$	22.4155i	0.812561i	−0.913748	+	0.406281i	$0.866826\pi$
$762$	0	0
$763$	4.63533	0.167810
$764$	15.2620	0.552161
$765$	0	0
$766$	−28.8116	−1.04101
$767$	0	0
$768$	0	0
$769$	− 0.132751i	− 0.00478714i	−0.999997	−	0.00239357i	$-0.999238\pi$
$769$	− 0.132751i	− 0.00478714i	0.999997	−	0.00239357i	$-0.000761898\pi$
$770$	−0.725873	−0.0261587
$771$	0	0
$772$	4.76809i	0.171607i
$773$	48.0694i	1.72893i	0.502689	+	0.864467i	$0.332344\pi$
$773$	48.0694i	1.72893i	−0.502689	+	0.864467i	$0.667656\pi$
$774$	0	0
$775$	12.4905i	0.448672i
$776$	−0.417895	−0.0150015
$777$	0	0
$778$	− 34.7821i	− 1.24700i
$779$	−35.2465	−1.26284
$780$	0	0
$781$	20.0194	0.716350
$782$	− 16.0871i	− 0.575273i
$783$	0	0
$784$	6.87263	0.245451
$785$	− 5.94571i	− 0.212211i
$786$	0	0
$787$	− 6.20908i	− 0.221330i	−0.993858	−	0.110665i	$-0.964702\pi$
$787$	− 6.20908i	− 0.221330i	0.993858	−	0.110665i	$-0.0352980\pi$
$788$	12.2349i	0.435850i
$789$	0	0
$790$	−10.4125	−0.370461
$791$	− 0.282750i	− 0.0100534i
$792$	0	0
$793$	0	0
$794$	5.15346	0.182889
$795$	0	0
$796$	−11.8485	−0.419958
$797$	0.327830	0.0116123	0.00580616	−	0.999983i	$-0.498152\pi$
$797$	0.327830	0.0116123	0.00580616	+	0.999983i	$0.498152\pi$
$798$	0	0
$799$	− 33.4878i	− 1.18471i
$800$	− 4.52111i	− 0.159845i
$801$	0	0
$802$	13.3250	0.470521
$803$	9.36360	0.330434
$804$	0	0
$805$	0.591794	0.0208580
$806$	0	0
$807$	0	0
$808$	10.0151i	0.352329i
$809$	37.4383	1.31626	0.658131	−	0.752904i	$-0.271347\pi$
$809$	37.4383	1.31626	0.658131	+	0.752904i	$0.271347\pi$
$810$	0	0
$811$	17.1448i	0.602037i	0.953618	+	0.301018i	$0.0973265\pi$
$811$	17.1448i	0.602037i	−0.953618	+	0.301018i	$0.902673\pi$
$812$	2.79417i	0.0980561i
$813$	0	0
$814$	− 29.6775i	− 1.04020i
$815$	−1.19434	−0.0418360
$816$	0	0
$817$	− 47.5120i	− 1.66223i
$818$	−24.0237	−0.839969
$819$	0	0
$820$	−3.38404	−0.118176
$821$	17.7885i	0.620824i	0.950602	+	0.310412i	$0.100467\pi$
$821$	17.7885i	0.620824i	−0.950602	+	0.310412i	$0.899533\pi$
$822$	0	0
$823$	−12.2301	−0.426315	−0.213157	−	0.977018i	$-0.568375\pi$
$823$	−12.2301	−0.426315	−0.213157	+	0.977018i	$0.568375\pi$
$824$	9.62565i	0.335325i
$825$	0	0
$826$	− 0.586417i	− 0.0204041i
$827$	20.5623i	0.715020i	0.933909	+	0.357510i	$0.116374\pi$
$827$	20.5623i	0.715020i	−0.933909	+	0.357510i	$0.883626\pi$
$828$	0	0
$829$	25.4470	0.883809	0.441905	−	0.897062i	$-0.354303\pi$
$829$	25.4470	0.883809	0.441905	+	0.897062i	$0.354303\pi$
$830$	10.2604i	0.356143i
$831$	0	0
$832$	0	0
$833$	46.1414	1.59870
$834$	0	0
$835$	14.6294	0.506270
$836$	−21.1836	−0.732650
$837$	0	0
$838$	13.8049i	0.476883i
$839$	5.76676i	0.199091i	0.995033	+	0.0995453i	$0.0317388\pi$
$839$	5.76676i	0.199091i	−0.995033	+	0.0995453i	$0.968261\pi$
$840$	0	0
$841$	32.2946	1.11361
$842$	7.72587	0.266251
$843$	0	0
$844$	17.2620	0.594184
$845$	0	0
$846$	0	0
$847$	0.843085i	0.0289688i
$848$	8.88769	0.305205
$849$	0	0
$850$	− 30.3538i	− 1.04113i
$851$	24.1957i	0.829417i
$852$	0	0
$853$	21.1728i	0.724944i	0.931995	+	0.362472i	$0.118067\pi$
$853$	21.1728i	0.724944i	−0.931995	+	0.362472i	$0.881933\pi$
$854$	−2.31767	−0.0793089
$855$	0	0
$856$	6.63102i	0.226644i
$857$	12.0086	0.410207	0.205103	−	0.978740i	$-0.434247\pi$
$857$	12.0086	0.410207	0.205103	+	0.978740i	$0.434247\pi$
$858$	0	0
$859$	−1.66296	−0.0567393	−0.0283697	−	0.999598i	$-0.509032\pi$
$859$	−1.66296	−0.0567393	−0.0283697	+	0.999598i	$0.509032\pi$
$860$	− 4.56166i	− 0.155551i
$861$	0	0
$862$	−0.640120	−0.0218026
$863$	23.8323i	0.811262i	0.914037	+	0.405631i	$0.132948\pi$
$863$	23.8323i	0.811262i	−0.914037	+	0.405631i	$0.867052\pi$
$864$	0	0
$865$	− 6.47352i	− 0.220106i
$866$	21.2760i	0.722989i
$867$	0	0
$868$	0.985999	0.0334670
$869$	− 44.2218i	− 1.50012i
$870$	0	0
$871$	0	0
$872$	12.9879	0.439826
$873$	0	0
$874$	17.2707	0.584189
$875$	2.35152	0.0794959
$876$	0	0
$877$	− 42.3177i	− 1.42897i	−0.699653	−	0.714483i	$-0.746662\pi$
$877$	− 42.3177i	− 1.42897i	0.699653	−	0.714483i	$-0.253338\pi$
$878$	12.7181i	0.429215i
$879$	0	0
$880$	−2.03385	−0.0685611
$881$	22.2741	0.750434	0.375217	−	0.926937i	$-0.377568\pi$
$881$	22.2741	0.750434	0.375217	+	0.926937i	$0.377568\pi$
$882$	0	0
$883$	8.54229	0.287471	0.143735	−	0.989616i	$-0.454089\pi$
$883$	8.54229	0.287471	0.143735	+	0.989616i	$0.454089\pi$
$884$	0	0
$885$	0	0
$886$	22.5972i	0.759167i
$887$	18.9142	0.635078	0.317539	−	0.948245i	$-0.397144\pi$
$887$	18.9142	0.635078	0.317539	+	0.948245i	$0.397144\pi$
$888$	0	0
$889$	6.50173i	0.218061i
$890$	0.274127i	0.00918875i
$891$	0	0
$892$	− 6.76809i	− 0.226612i
$893$	35.9517	1.20308
$894$	0	0
$895$	− 2.19733i	− 0.0734485i
$896$	−0.356896	−0.0119231
$897$	0	0
$898$	11.6474	0.388679
$899$	− 21.6295i	− 0.721384i
$900$	0	0
$901$	59.6701	1.98790
$902$	− 14.3720i	− 0.478534i
$903$	0	0
$904$	− 0.792249i	− 0.0263498i
$905$	− 13.7017i	− 0.455460i
$906$	0	0
$907$	−13.9517	−0.463258	−0.231629	−	0.972804i	$-0.574405\pi$
$907$	−13.9517	−0.463258	−0.231629	+	0.972804i	$0.574405\pi$
$908$	23.6799i	0.785846i
$909$	0	0
$910$	0	0
$911$	−45.0422	−1.49232	−0.746158	−	0.665769i	$-0.768103\pi$
$911$	−45.0422	−1.49232	−0.746158	+	0.665769i	$0.768103\pi$
$912$	0	0
$913$	−43.5757	−1.44214
$914$	−21.1890	−0.700869
$915$	0	0
$916$	− 8.29829i	− 0.274183i
$917$	− 0.976311i	− 0.0322406i
$918$	0	0
$919$	39.9976	1.31940	0.659700	−	0.751529i	$-0.270683\pi$
$919$	39.9976	1.31940	0.659700	+	0.751529i	$0.270683\pi$
$920$	1.65817	0.0546682
$921$	0	0
$922$	−24.0694	−0.792682
$923$	0	0
$924$	0	0
$925$	45.6534i	1.50107i
$926$	18.1715	0.597153
$927$	0	0
$928$	7.82908i	0.257002i
$929$	− 34.6848i	− 1.13797i	−0.822347	−	0.568986i	$-0.807336\pi$
$929$	− 34.6848i	− 1.13797i	0.822347	−	0.568986i	$-0.192664\pi$
$930$	0	0
$931$	49.5362i	1.62348i
$932$	23.9651	0.785002
$933$	0	0
$934$	2.93123i	0.0959128i
$935$	−13.6549	−0.446562
$936$	0	0
$937$	−19.1260	−0.624821	−0.312410	−	0.949947i	$-0.601136\pi$
$937$	−19.1260	−0.624821	−0.312410	+	0.949947i	$0.601136\pi$
$938$	− 4.82717i	− 0.157613i
$939$	0	0
$940$	3.45175	0.112584
$941$	22.5972i	0.736647i	0.929698	+	0.368323i	$0.120068\pi$
$941$	22.5972i	0.736647i	−0.929698	+	0.368323i	$0.879932\pi$
$942$	0	0
$943$	11.7172i	0.381566i
$944$	− 1.64310i	− 0.0534785i
$945$	0	0
$946$	19.3733	0.629880
$947$	− 27.4359i	− 0.891548i	−0.895145	−	0.445774i	$-0.852928\pi$
$947$	− 27.4359i	− 0.891548i	0.895145	−	0.445774i	$-0.147072\pi$
$948$	0	0
$949$	0	0
$950$	32.5870	1.05726
$951$	0	0
$952$	−2.39612	−0.0776588
$953$	−1.84787	−0.0598584	−0.0299292	−	0.999552i	$-0.509528\pi$
$953$	−1.84787	−0.0598584	−0.0299292	+	0.999552i	$0.509528\pi$
$954$	0	0
$955$	10.5617i	0.341767i
$956$	12.6160i	0.408029i
$957$	0	0
$958$	−30.7090	−0.992163
$959$	−2.72933	−0.0881347
$960$	0	0
$961$	23.3674	0.753788
$962$	0	0
$963$	0	0
$964$	26.3937i	0.850085i
$965$	−3.29962	−0.106218
$966$	0	0
$967$	− 8.88471i	− 0.285713i	−0.989743	−	0.142856i	$-0.954371\pi$
$967$	− 8.88471i	− 0.285713i	0.989743	−	0.142856i	$-0.0456287\pi$
$968$	2.36227i	0.0759263i
$969$	0	0
$970$	− 0.289192i	− 0.00928540i
$971$	−35.0863	−1.12597	−0.562987	−	0.826466i	$-0.690348\pi$
$971$	−35.0863	−1.12597	−0.562987	+	0.826466i	$0.690348\pi$
$972$	0	0
$973$	1.20775i	0.0387187i
$974$	24.1497	0.773807
$975$	0	0
$976$	−6.49396	−0.207867
$977$	8.33704i	0.266726i	0.991067	+	0.133363i	$0.0425776\pi$
$977$	8.33704i	0.266726i	−0.991067	+	0.133363i	$0.957422\pi$
$978$	0	0
$979$	−1.16421	−0.0372083
$980$	4.75600i	0.151925i
$981$	0	0
$982$	− 14.5972i	− 0.465814i
$983$	− 55.6051i	− 1.77353i	−0.462224	−	0.886763i	$-0.652949\pi$
$983$	− 55.6051i	− 1.77353i	0.462224	−	0.886763i	$-0.347051\pi$
$984$	0	0
$985$	−8.46681	−0.269775
$986$	52.5628i	1.67394i
$987$	0	0
$988$	0	0
$989$	−15.7948	−0.502244
$990$	0	0
$991$	−43.5967	−1.38489	−0.692447	−	0.721468i	$-0.743468\pi$
$991$	−43.5967	−1.38489	−0.692447	+	0.721468i	$0.743468\pi$
$992$	2.76271	0.0877161
$993$	0	0
$994$	2.43104i	0.0771079i
$995$	− 8.19939i	− 0.259938i
$996$	0	0
$997$	22.4590	0.711285	0.355643	−	0.934622i	$-0.384262\pi$
$997$	22.4590	0.711285	0.355643	+	0.934622i	$0.384262\pi$
$998$	6.85517	0.216997
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.r.1351.1		6
3.2	odd	2		1014.2.b.g.337.6		6
13.5	odd	4		3042.2.a.bd.1.1		3
13.8	odd	4		3042.2.a.be.1.3		3
13.12	even	2	inner	3042.2.b.r.1351.6		6
39.2	even	12		1014.2.e.k.529.3		6
39.5	even	4		1014.2.a.o.1.3	yes	3
39.8	even	4		1014.2.a.m.1.1	✓	3
39.11	even	12		1014.2.e.m.529.1		6
39.17	odd	6		1014.2.i.g.361.1		12
39.20	even	12		1014.2.e.m.991.1		6
39.23	odd	6		1014.2.i.g.823.4		12
39.29	odd	6		1014.2.i.g.823.3		12
39.32	even	12		1014.2.e.k.991.3		6
39.35	odd	6		1014.2.i.g.361.6		12
39.38	odd	2		1014.2.b.g.337.1		6
156.47	odd	4		8112.2.a.ce.1.1		3
156.83	odd	4		8112.2.a.bz.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.2.a.m.1.1	✓	3	39.8	even	4
1014.2.a.o.1.3	yes	3	39.5	even	4
1014.2.b.g.337.1		6	39.38	odd	2
1014.2.b.g.337.6		6	3.2	odd	2
1014.2.e.k.529.3		6	39.2	even	12
1014.2.e.k.991.3		6	39.32	even	12
1014.2.e.m.529.1		6	39.11	even	12
1014.2.e.m.991.1		6	39.20	even	12
1014.2.i.g.361.1		12	39.17	odd	6
1014.2.i.g.361.6		12	39.35	odd	6
1014.2.i.g.823.3		12	39.29	odd	6
1014.2.i.g.823.4		12	39.23	odd	6
3042.2.a.bd.1.1		3	13.5	odd	4
3042.2.a.be.1.3		3	13.8	odd	4
3042.2.b.r.1351.1		6	1.1	even	1	trivial
3042.2.b.r.1351.6		6	13.12	even	2	inner
8112.2.a.bz.1.3		3	156.83	odd	4
8112.2.a.ce.1.1		3	156.47	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -0.692021i q^{5} +0.356896i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -0.692021i q^{5} +0.356896i q^{7} +1.00000i q^{8} -0.692021 q^{10} -2.93900i q^{11} +0.356896 q^{14} +1.00000 q^{16} +6.71379 q^{17} +7.20775i q^{19} +0.692021i q^{20} -2.93900 q^{22} +2.39612 q^{23} +4.52111 q^{25} -0.356896i q^{28} -7.82908 q^{29} +2.76271i q^{31} -1.00000i q^{32} -6.71379i q^{34} +0.246980 q^{35} +10.0978i q^{37} +7.20775 q^{38} +0.692021 q^{40} +4.89008i q^{41} -6.59179 q^{43} +2.93900i q^{44} -2.39612i q^{46} -4.98792i q^{47} +6.87263 q^{49} -4.52111i q^{50} +8.88769 q^{53} -2.03385 q^{55} -0.356896 q^{56} +7.82908i q^{58} -1.64310i q^{59} -6.49396 q^{61} +2.76271 q^{62} -1.00000 q^{64} -13.5254i q^{67} -6.71379 q^{68} -0.246980i q^{70} +6.81163i q^{71} +3.18598i q^{73} +10.0978 q^{74} -7.20775i q^{76} +1.04892 q^{77} +15.0465 q^{79} -0.692021i q^{80} +4.89008 q^{82} -14.8267i q^{83} -4.64609i q^{85} +6.59179i q^{86} +2.93900 q^{88} -0.396125i q^{89} -2.39612 q^{92} -4.98792 q^{94} +4.98792 q^{95} +0.417895i q^{97} -6.87263i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4}+O(q^{10}) \) 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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