LMFDB - Embedded newform 3042.2.a.r.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3042,2,Mod(1,3042)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3042.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3042, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,-2,0,0,-2,0,2,-4,0,0,0,0,2,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$24.2904922949$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 234)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.16228$ of defining polynomial
Character	$\chi$	$=$	3042.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -4.16228 q^{5} +3.16228 q^{7} -1.00000 q^{8} +4.16228 q^{10} +1.16228 q^{11} -3.16228 q^{14} +1.00000 q^{16} +3.00000 q^{17} -5.16228 q^{19} -4.16228 q^{20} -1.16228 q^{22} -7.16228 q^{23} +12.3246 q^{25} +3.16228 q^{28} +1.83772 q^{29} -6.32456 q^{31} -1.00000 q^{32} -3.00000 q^{34} -13.1623 q^{35} +3.83772 q^{37} +5.16228 q^{38} +4.16228 q^{40} -3.00000 q^{41} +9.16228 q^{43} +1.16228 q^{44} +7.16228 q^{46} -4.83772 q^{47} +3.00000 q^{49} -12.3246 q^{50} +12.4868 q^{53} -4.83772 q^{55} -3.16228 q^{56} -1.83772 q^{58} +2.32456 q^{59} +0.162278 q^{61} +6.32456 q^{62} +1.00000 q^{64} -2.83772 q^{67} +3.00000 q^{68} +13.1623 q^{70} +7.16228 q^{71} -1.00000 q^{73} -3.83772 q^{74} -5.16228 q^{76} +3.67544 q^{77} -4.00000 q^{79} -4.16228 q^{80} +3.00000 q^{82} +3.48683 q^{83} -12.4868 q^{85} -9.16228 q^{86} -1.16228 q^{88} -12.0000 q^{89} -7.16228 q^{92} +4.83772 q^{94} +21.4868 q^{95} -4.00000 q^{97} -3.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{16} + 6 q^{17} - 4 q^{19} - 2 q^{20} + 4 q^{22} - 8 q^{23} + 12 q^{25} + 10 q^{29} - 2 q^{32} - 6 q^{34} - 20 q^{35} + 14 q^{37}+ \cdots - 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^16 + 6 * q^17 - 4 * q^19 - 2 * q^20 + 4 * q^22 - 8 * q^23 + 12 * q^25 + 10 * q^29 - 2 * q^32 - 6 * q^34 - 20 * q^35 + 14 * q^37 + 4 * q^38 + 2 * q^40 - 6 * q^41 + 12 * q^43 - 4 * q^44 + 8 * q^46 - 16 * q^47 + 6 * q^49 - 12 * q^50 + 6 * q^53 - 16 * q^55 - 10 * q^58 - 8 * q^59 - 6 * q^61 + 2 * q^64 - 12 * q^67 + 6 * q^68 + 20 * q^70 + 8 * q^71 - 2 * q^73 - 14 * q^74 - 4 * q^76 + 20 * q^77 - 8 * q^79 - 2 * q^80 + 6 * q^82 - 12 * q^83 - 6 * q^85 - 12 * q^86 + 4 * q^88 - 24 * q^89 - 8 * q^92 + 16 * q^94 + 24 * q^95 - 8 * q^97 - 6 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−4.16228	−1.86143	−0.930714	−	0.365749i	$-0.880813\pi$
$5$	−4.16228	−1.86143	−0.930714	+	0.365749i	$0.880813\pi$
$6$	0	0
$7$	3.16228	1.19523	0.597614	−	0.801784i	$-0.296115\pi$
$7$	3.16228	1.19523	0.597614	+	0.801784i	$0.296115\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	4.16228	1.31623
$11$	1.16228	0.350440	0.175220	−	0.984529i	$-0.443936\pi$
$11$	1.16228	0.350440	0.175220	+	0.984529i	$0.443936\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−3.16228	−0.845154
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−5.16228	−1.18431	−0.592154	−	0.805825i	$-0.701722\pi$
$19$	−5.16228	−1.18431	−0.592154	+	0.805825i	$0.701722\pi$
$20$	−4.16228	−0.930714
$21$	0	0
$22$	−1.16228	−0.247798
$23$	−7.16228	−1.49344	−0.746719	−	0.665140i	$-0.768372\pi$
$23$	−7.16228	−1.49344	−0.746719	+	0.665140i	$0.768372\pi$
$24$	0	0
$25$	12.3246	2.46491
$26$	0	0
$27$	0	0
$28$	3.16228	0.597614
$29$	1.83772	0.341256	0.170628	−	0.985335i	$-0.445420\pi$
$29$	1.83772	0.341256	0.170628	+	0.985335i	$0.445420\pi$
$30$	0	0
$31$	−6.32456	−1.13592	−0.567962	−	0.823055i	$-0.692268\pi$
$31$	−6.32456	−1.13592	−0.567962	+	0.823055i	$0.692268\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	−13.1623	−2.22483
$36$	0	0
$37$	3.83772	0.630918	0.315459	−	0.948939i	$-0.397842\pi$
$37$	3.83772	0.630918	0.315459	+	0.948939i	$0.397842\pi$
$38$	5.16228	0.837432
$39$	0	0
$40$	4.16228	0.658114
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	9.16228	1.39723	0.698617	−	0.715496i	$-0.253799\pi$
$43$	9.16228	1.39723	0.698617	+	0.715496i	$0.253799\pi$
$44$	1.16228	0.175220
$45$	0	0
$46$	7.16228	1.05602
$47$	−4.83772	−0.705654	−0.352827	−	0.935689i	$-0.614780\pi$
$47$	−4.83772	−0.705654	−0.352827	+	0.935689i	$0.614780\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	−12.3246	−1.74296
$51$	0	0
$52$	0	0
$53$	12.4868	1.71520	0.857599	−	0.514319i	$-0.171955\pi$
$53$	12.4868	1.71520	0.857599	+	0.514319i	$0.171955\pi$
$54$	0	0
$55$	−4.83772	−0.652318
$56$	−3.16228	−0.422577
$57$	0	0
$58$	−1.83772	−0.241305
$59$	2.32456	0.302631	0.151316	−	0.988485i	$-0.451649\pi$
$59$	2.32456	0.302631	0.151316	+	0.988485i	$0.451649\pi$
$60$	0	0
$61$	0.162278	0.0207775	0.0103888	−	0.999946i	$-0.496693\pi$
$61$	0.162278	0.0207775	0.0103888	+	0.999946i	$0.496693\pi$
$62$	6.32456	0.803219
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−2.83772	−0.346683	−0.173341	−	0.984862i	$-0.555456\pi$
$67$	−2.83772	−0.346683	−0.173341	+	0.984862i	$0.555456\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	13.1623	1.57319
$71$	7.16228	0.850006	0.425003	−	0.905192i	$-0.360273\pi$
$71$	7.16228	0.850006	0.425003	+	0.905192i	$0.360273\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	−3.83772	−0.446126
$75$	0	0
$76$	−5.16228	−0.592154
$77$	3.67544	0.418856
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−4.16228	−0.465357
$81$	0	0
$82$	3.00000	0.331295
$83$	3.48683	0.382730	0.191365	−	0.981519i	$-0.438709\pi$
$83$	3.48683	0.382730	0.191365	+	0.981519i	$0.438709\pi$
$84$	0	0
$85$	−12.4868	−1.35439
$86$	−9.16228	−0.987994
$87$	0	0
$88$	−1.16228	−0.123899
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	0	0
$92$	−7.16228	−0.746719
$93$	0	0
$94$	4.83772	0.498973
$95$	21.4868	2.20450
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	12.3246	1.23246
$101$	−7.83772	−0.779883	−0.389941	−	0.920840i	$-0.627505\pi$
$101$	−7.83772	−0.779883	−0.389941	+	0.920840i	$0.627505\pi$
$102$	0	0
$103$	−15.8114	−1.55794	−0.778971	−	0.627060i	$-0.784258\pi$
$103$	−15.8114	−1.55794	−0.778971	+	0.627060i	$0.784258\pi$
$104$	0	0
$105$	0	0
$106$	−12.4868	−1.21283
$107$	17.8114	1.72189	0.860946	−	0.508696i	$-0.169872\pi$
$107$	17.8114	1.72189	0.860946	+	0.508696i	$0.169872\pi$
$108$	0	0
$109$	6.64911	0.636869	0.318435	−	0.947945i	$-0.396843\pi$
$109$	6.64911	0.636869	0.318435	+	0.947945i	$0.396843\pi$
$110$	4.83772	0.461259
$111$	0	0
$112$	3.16228	0.298807
$113$	−6.67544	−0.627973	−0.313987	−	0.949427i	$-0.601665\pi$
$113$	−6.67544	−0.627973	−0.313987	+	0.949427i	$0.601665\pi$
$114$	0	0
$115$	29.8114	2.77993
$116$	1.83772	0.170628
$117$	0	0
$118$	−2.32456	−0.213993
$119$	9.48683	0.869657
$120$	0	0
$121$	−9.64911	−0.877192
$122$	−0.162278	−0.0146919
$123$	0	0
$124$	−6.32456	−0.567962
$125$	−30.4868	−2.72683
$126$	0	0
$127$	−18.3246	−1.62604	−0.813021	−	0.582235i	$-0.802178\pi$
$127$	−18.3246	−1.62604	−0.813021	+	0.582235i	$0.802178\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−16.6491	−1.45464	−0.727320	−	0.686299i	$-0.759234\pi$
$131$	−16.6491	−1.45464	−0.727320	+	0.686299i	$0.759234\pi$
$132$	0	0
$133$	−16.3246	−1.41552
$134$	2.83772	0.245142
$135$	0	0
$136$	−3.00000	−0.257248
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	−6.32456	−0.536442	−0.268221	−	0.963357i	$-0.586436\pi$
$139$	−6.32456	−0.536442	−0.268221	+	0.963357i	$0.586436\pi$
$140$	−13.1623	−1.11242
$141$	0	0
$142$	−7.16228	−0.601045
$143$	0	0
$144$	0	0
$145$	−7.64911	−0.635224
$146$	1.00000	0.0827606
$147$	0	0
$148$	3.83772	0.315459
$149$	−0.486833	−0.0398829	−0.0199415	−	0.999801i	$-0.506348\pi$
$149$	−0.486833	−0.0398829	−0.0199415	+	0.999801i	$0.506348\pi$
$150$	0	0
$151$	0.837722	0.0681729	0.0340864	−	0.999419i	$-0.489148\pi$
$151$	0.837722	0.0681729	0.0340864	+	0.999419i	$0.489148\pi$
$152$	5.16228	0.418716
$153$	0	0
$154$	−3.67544	−0.296176
$155$	26.3246	2.11444
$156$	0	0
$157$	−10.4868	−0.836940	−0.418470	−	0.908231i	$-0.637434\pi$
$157$	−10.4868	−0.836940	−0.418470	+	0.908231i	$0.637434\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	4.16228	0.329057
$161$	−22.6491	−1.78500
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−3.48683	−0.270631
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	0	0
$170$	12.4868	0.957696
$171$	0	0
$172$	9.16228	0.698617
$173$	−22.6491	−1.72198	−0.860990	−	0.508622i	$-0.830155\pi$
$173$	−22.6491	−1.72198	−0.860990	+	0.508622i	$0.830155\pi$
$174$	0	0
$175$	38.9737	2.94613
$176$	1.16228	0.0876100
$177$	0	0
$178$	12.0000	0.899438
$179$	−15.4868	−1.15754	−0.578770	−	0.815491i	$-0.696467\pi$
$179$	−15.4868	−1.15754	−0.578770	+	0.815491i	$0.696467\pi$
$180$	0	0
$181$	3.83772	0.285256	0.142628	−	0.989776i	$-0.454445\pi$
$181$	3.83772	0.285256	0.142628	+	0.989776i	$0.454445\pi$
$182$	0	0
$183$	0	0
$184$	7.16228	0.528010
$185$	−15.9737	−1.17441
$186$	0	0
$187$	3.48683	0.254982
$188$	−4.83772	−0.352827
$189$	0	0
$190$	−21.4868	−1.55882
$191$	−14.3246	−1.03649	−0.518244	−	0.855233i	$-0.673414\pi$
$191$	−14.3246	−1.03649	−0.518244	+	0.855233i	$0.673414\pi$
$192$	0	0
$193$	−19.9737	−1.43774	−0.718868	−	0.695147i	$-0.755339\pi$
$193$	−19.9737	−1.43774	−0.718868	+	0.695147i	$0.755339\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	3.00000	0.214286
$197$	−18.9737	−1.35182	−0.675909	−	0.736985i	$-0.736249\pi$
$197$	−18.9737	−1.35182	−0.675909	+	0.736985i	$0.736249\pi$
$198$	0	0
$199$	−6.51317	−0.461706	−0.230853	−	0.972989i	$-0.574152\pi$
$199$	−6.51317	−0.461706	−0.230853	+	0.972989i	$0.574152\pi$
$200$	−12.3246	−0.871478
$201$	0	0
$202$	7.83772	0.551460
$203$	5.81139	0.407879
$204$	0	0
$205$	12.4868	0.872118
$206$	15.8114	1.10163
$207$	0	0
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	12.4868	0.857599
$213$	0	0
$214$	−17.8114	−1.21756
$215$	−38.1359	−2.60085
$216$	0	0
$217$	−20.0000	−1.35769
$218$	−6.64911	−0.450335
$219$	0	0
$220$	−4.83772	−0.326159
$221$	0	0
$222$	0	0
$223$	−1.67544	−0.112196	−0.0560980	−	0.998425i	$-0.517866\pi$
$223$	−1.67544	−0.112196	−0.0560980	+	0.998425i	$0.517866\pi$
$224$	−3.16228	−0.211289
$225$	0	0
$226$	6.67544	0.444044
$227$	15.4868	1.02790	0.513949	−	0.857821i	$-0.328182\pi$
$227$	15.4868	1.02790	0.513949	+	0.857821i	$0.328182\pi$
$228$	0	0
$229$	22.3246	1.47525	0.737624	−	0.675212i	$-0.235948\pi$
$229$	22.3246	1.47525	0.737624	+	0.675212i	$0.235948\pi$
$230$	−29.8114	−1.96570
$231$	0	0
$232$	−1.83772	−0.120652
$233$	16.6491	1.09072	0.545360	−	0.838202i	$-0.316393\pi$
$233$	16.6491	1.09072	0.545360	+	0.838202i	$0.316393\pi$
$234$	0	0
$235$	20.1359	1.31352
$236$	2.32456	0.151316
$237$	0	0
$238$	−9.48683	−0.614940
$239$	21.4868	1.38987	0.694934	−	0.719074i	$-0.255434\pi$
$239$	21.4868	1.38987	0.694934	+	0.719074i	$0.255434\pi$
$240$	0	0
$241$	13.3246	0.858310	0.429155	−	0.903231i	$-0.358811\pi$
$241$	13.3246	0.858310	0.429155	+	0.903231i	$0.358811\pi$
$242$	9.64911	0.620268
$243$	0	0
$244$	0.162278	0.0103888
$245$	−12.4868	−0.797754
$246$	0	0
$247$	0	0
$248$	6.32456	0.401610
$249$	0	0
$250$	30.4868	1.92816
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−8.32456	−0.523360
$254$	18.3246	1.14978
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.9737	1.37068	0.685340	−	0.728223i	$-0.259654\pi$
$257$	21.9737	1.37068	0.685340	+	0.728223i	$0.259654\pi$
$258$	0	0
$259$	12.1359	0.754091
$260$	0	0
$261$	0	0
$262$	16.6491	1.02859
$263$	2.51317	0.154969	0.0774843	−	0.996994i	$-0.475311\pi$
$263$	2.51317	0.154969	0.0774843	+	0.996994i	$0.475311\pi$
$264$	0	0
$265$	−51.9737	−3.19272
$266$	16.3246	1.00092
$267$	0	0
$268$	−2.83772	−0.173341
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−6.32456	−0.384189	−0.192095	−	0.981376i	$-0.561528\pi$
$271$	−6.32456	−0.384189	−0.192095	+	0.981376i	$0.561528\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−3.00000	−0.181237
$275$	14.3246	0.863803
$276$	0	0
$277$	10.8114	0.649593	0.324797	−	0.945784i	$-0.394704\pi$
$277$	10.8114	0.649593	0.324797	+	0.945784i	$0.394704\pi$
$278$	6.32456	0.379322
$279$	0	0
$280$	13.1623	0.786597
$281$	0.675445	0.0402937	0.0201468	−	0.999797i	$-0.493587\pi$
$281$	0.675445	0.0402937	0.0201468	+	0.999797i	$0.493587\pi$
$282$	0	0
$283$	−2.83772	−0.168685	−0.0843425	−	0.996437i	$-0.526879\pi$
$283$	−2.83772	−0.168685	−0.0843425	+	0.996437i	$0.526879\pi$
$284$	7.16228	0.425003
$285$	0	0
$286$	0	0
$287$	−9.48683	−0.559990
$288$	0	0
$289$	−8.00000	−0.470588
$290$	7.64911	0.449171
$291$	0	0
$292$	−1.00000	−0.0585206
$293$	−1.83772	−0.107361	−0.0536804	−	0.998558i	$-0.517095\pi$
$293$	−1.83772	−0.107361	−0.0536804	+	0.998558i	$0.517095\pi$
$294$	0	0
$295$	−9.67544	−0.563326
$296$	−3.83772	−0.223063
$297$	0	0
$298$	0.486833	0.0282015
$299$	0	0
$300$	0	0
$301$	28.9737	1.67001
$302$	−0.837722	−0.0482055
$303$	0	0
$304$	−5.16228	−0.296077
$305$	−0.675445	−0.0386758
$306$	0	0
$307$	11.4868	0.655588	0.327794	−	0.944749i	$-0.393695\pi$
$307$	11.4868	0.655588	0.327794	+	0.944749i	$0.393695\pi$
$308$	3.67544	0.209428
$309$	0	0
$310$	−26.3246	−1.49513
$311$	−21.4868	−1.21841	−0.609203	−	0.793014i	$-0.708511\pi$
$311$	−21.4868	−1.21841	−0.609203	+	0.793014i	$0.708511\pi$
$312$	0	0
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	10.4868	0.591806
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−12.4868	−0.701330	−0.350665	−	0.936501i	$-0.614044\pi$
$317$	−12.4868	−0.701330	−0.350665	+	0.936501i	$0.614044\pi$
$318$	0	0
$319$	2.13594	0.119590
$320$	−4.16228	−0.232678
$321$	0	0
$322$	22.6491	1.26219
$323$	−15.4868	−0.861710
$324$	0	0
$325$	0	0
$326$	16.0000	0.886158
$327$	0	0
$328$	3.00000	0.165647
$329$	−15.2982	−0.843418
$330$	0	0
$331$	−10.9737	−0.603167	−0.301584	−	0.953440i	$-0.597515\pi$
$331$	−10.9737	−0.603167	−0.301584	+	0.953440i	$0.597515\pi$
$332$	3.48683	0.191365
$333$	0	0
$334$	12.0000	0.656611
$335$	11.8114	0.645325
$336$	0	0
$337$	11.0000	0.599208	0.299604	−	0.954064i	$-0.403145\pi$
$337$	11.0000	0.599208	0.299604	+	0.954064i	$0.403145\pi$
$338$	0	0
$339$	0	0
$340$	−12.4868	−0.677194
$341$	−7.35089	−0.398073
$342$	0	0
$343$	−12.6491	−0.682988
$344$	−9.16228	−0.493997
$345$	0	0
$346$	22.6491	1.21762
$347$	15.4868	0.831377	0.415688	−	0.909507i	$-0.363541\pi$
$347$	15.4868	0.831377	0.415688	+	0.909507i	$0.363541\pi$
$348$	0	0
$349$	−5.35089	−0.286427	−0.143213	−	0.989692i	$-0.545743\pi$
$349$	−5.35089	−0.286427	−0.143213	+	0.989692i	$0.545743\pi$
$350$	−38.9737	−2.08323
$351$	0	0
$352$	−1.16228	−0.0619496
$353$	−29.3246	−1.56079	−0.780394	−	0.625288i	$-0.784982\pi$
$353$	−29.3246	−1.56079	−0.780394	+	0.625288i	$0.784982\pi$
$354$	0	0
$355$	−29.8114	−1.58222
$356$	−12.0000	−0.635999
$357$	0	0
$358$	15.4868	0.818505
$359$	−28.4605	−1.50209	−0.751044	−	0.660252i	$-0.770449\pi$
$359$	−28.4605	−1.50209	−0.751044	+	0.660252i	$0.770449\pi$
$360$	0	0
$361$	7.64911	0.402585
$362$	−3.83772	−0.201706
$363$	0	0
$364$	0	0
$365$	4.16228	0.217864
$366$	0	0
$367$	−25.4868	−1.33040	−0.665201	−	0.746664i	$-0.731654\pi$
$367$	−25.4868	−1.33040	−0.665201	+	0.746664i	$0.731654\pi$
$368$	−7.16228	−0.373360
$369$	0	0
$370$	15.9737	0.830431
$371$	39.4868	2.05005
$372$	0	0
$373$	−16.4868	−0.853656	−0.426828	−	0.904333i	$-0.640369\pi$
$373$	−16.4868	−0.853656	−0.426828	+	0.904333i	$0.640369\pi$
$374$	−3.48683	−0.180300
$375$	0	0
$376$	4.83772	0.249486
$377$	0	0
$378$	0	0
$379$	17.6754	0.907927	0.453963	−	0.891020i	$-0.350010\pi$
$379$	17.6754	0.907927	0.453963	+	0.891020i	$0.350010\pi$
$380$	21.4868	1.10225
$381$	0	0
$382$	14.3246	0.732908
$383$	−30.9737	−1.58268	−0.791340	−	0.611376i	$-0.790616\pi$
$383$	−30.9737	−1.58268	−0.791340	+	0.611376i	$0.790616\pi$
$384$	0	0
$385$	−15.2982	−0.779670
$386$	19.9737	1.01663
$387$	0	0
$388$	−4.00000	−0.203069
$389$	−12.4868	−0.633108	−0.316554	−	0.948575i	$-0.602526\pi$
$389$	−12.4868	−0.633108	−0.316554	+	0.948575i	$0.602526\pi$
$390$	0	0
$391$	−21.4868	−1.08664
$392$	−3.00000	−0.151523
$393$	0	0
$394$	18.9737	0.955879
$395$	16.6491	0.837708
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	6.51317	0.326476
$399$	0	0
$400$	12.3246	0.616228
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	0	0
$403$	0	0
$404$	−7.83772	−0.389941
$405$	0	0
$406$	−5.81139	−0.288414
$407$	4.46050	0.221099
$408$	0	0
$409$	14.6754	0.725654	0.362827	−	0.931857i	$-0.381812\pi$
$409$	14.6754	0.725654	0.362827	+	0.931857i	$0.381812\pi$
$410$	−12.4868	−0.616681
$411$	0	0
$412$	−15.8114	−0.778971
$413$	7.35089	0.361714
$414$	0	0
$415$	−14.5132	−0.712423
$416$	0	0
$417$	0	0
$418$	6.00000	0.293470
$419$	30.9737	1.51316	0.756581	−	0.653900i	$-0.226868\pi$
$419$	30.9737	1.51316	0.756581	+	0.653900i	$0.226868\pi$
$420$	0	0
$421$	−23.8377	−1.16178	−0.580890	−	0.813982i	$-0.697295\pi$
$421$	−23.8377	−1.16178	−0.580890	+	0.813982i	$0.697295\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	−12.4868	−0.606414
$425$	36.9737	1.79349
$426$	0	0
$427$	0.513167	0.0248339
$428$	17.8114	0.860946
$429$	0	0
$430$	38.1359	1.83908
$431$	9.48683	0.456965	0.228482	−	0.973548i	$-0.426624\pi$
$431$	9.48683	0.456965	0.228482	+	0.973548i	$0.426624\pi$
$432$	0	0
$433$	−9.32456	−0.448110	−0.224055	−	0.974577i	$-0.571929\pi$
$433$	−9.32456	−0.448110	−0.224055	+	0.974577i	$0.571929\pi$
$434$	20.0000	0.960031
$435$	0	0
$436$	6.64911	0.318435
$437$	36.9737	1.76869
$438$	0	0
$439$	−25.4868	−1.21642	−0.608210	−	0.793776i	$-0.708112\pi$
$439$	−25.4868	−1.21642	−0.608210	+	0.793776i	$0.708112\pi$
$440$	4.83772	0.230629
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	49.9473	2.36773
$446$	1.67544	0.0793346
$447$	0	0
$448$	3.16228	0.149404
$449$	−7.35089	−0.346910	−0.173455	−	0.984842i	$-0.555493\pi$
$449$	−7.35089	−0.346910	−0.173455	+	0.984842i	$0.555493\pi$
$450$	0	0
$451$	−3.48683	−0.164189
$452$	−6.67544	−0.313987
$453$	0	0
$454$	−15.4868	−0.726833
$455$	0	0
$456$	0	0
$457$	−3.32456	−0.155516	−0.0777581	−	0.996972i	$-0.524776\pi$
$457$	−3.32456	−0.155516	−0.0777581	+	0.996972i	$0.524776\pi$
$458$	−22.3246	−1.04316
$459$	0	0
$460$	29.8114	1.38996
$461$	−18.4868	−0.861018	−0.430509	−	0.902586i	$-0.641666\pi$
$461$	−18.4868	−0.861018	−0.430509	+	0.902586i	$0.641666\pi$
$462$	0	0
$463$	15.1623	0.704651	0.352325	−	0.935878i	$-0.385391\pi$
$463$	15.1623	0.704651	0.352325	+	0.935878i	$0.385391\pi$
$464$	1.83772	0.0853141
$465$	0	0
$466$	−16.6491	−0.771255
$467$	−6.18861	−0.286375	−0.143187	−	0.989696i	$-0.545735\pi$
$467$	−6.18861	−0.286375	−0.143187	+	0.989696i	$0.545735\pi$
$468$	0	0
$469$	−8.97367	−0.414365
$470$	−20.1359	−0.928802
$471$	0	0
$472$	−2.32456	−0.106996
$473$	10.6491	0.489647
$474$	0	0
$475$	−63.6228	−2.91921
$476$	9.48683	0.434828
$477$	0	0
$478$	−21.4868	−0.982785
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	0	0
$482$	−13.3246	−0.606917
$483$	0	0
$484$	−9.64911	−0.438596
$485$	16.6491	0.755997
$486$	0	0
$487$	−13.4868	−0.611147	−0.305573	−	0.952169i	$-0.598848\pi$
$487$	−13.4868	−0.611147	−0.305573	+	0.952169i	$0.598848\pi$
$488$	−0.162278	−0.00734596
$489$	0	0
$490$	12.4868	0.564098
$491$	−5.81139	−0.262264	−0.131132	−	0.991365i	$-0.541861\pi$
$491$	−5.81139	−0.262264	−0.131132	+	0.991365i	$0.541861\pi$
$492$	0	0
$493$	5.51317	0.248301
$494$	0	0
$495$	0	0
$496$	−6.32456	−0.283981
$497$	22.6491	1.01595
$498$	0	0
$499$	12.6491	0.566252	0.283126	−	0.959083i	$-0.408629\pi$
$499$	12.6491	0.566252	0.283126	+	0.959083i	$0.408629\pi$
$500$	−30.4868	−1.36341
$501$	0	0
$502$	−12.0000	−0.535586
$503$	0.188612	0.00840978	0.00420489	−	0.999991i	$-0.498662\pi$
$503$	0.188612	0.00840978	0.00420489	+	0.999991i	$0.498662\pi$
$504$	0	0
$505$	32.6228	1.45169
$506$	8.32456	0.370072
$507$	0	0
$508$	−18.3246	−0.813021
$509$	28.1623	1.24827	0.624136	−	0.781316i	$-0.285451\pi$
$509$	28.1623	1.24827	0.624136	+	0.781316i	$0.285451\pi$
$510$	0	0
$511$	−3.16228	−0.139891
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−21.9737	−0.969217
$515$	65.8114	2.90000
$516$	0	0
$517$	−5.62278	−0.247289
$518$	−12.1359	−0.533223
$519$	0	0
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	0	0
$523$	−5.16228	−0.225731	−0.112865	−	0.993610i	$-0.536003\pi$
$523$	−5.16228	−0.225731	−0.112865	+	0.993610i	$0.536003\pi$
$524$	−16.6491	−0.727320
$525$	0	0
$526$	−2.51317	−0.109579
$527$	−18.9737	−0.826506
$528$	0	0
$529$	28.2982	1.23036
$530$	51.9737	2.25759
$531$	0	0
$532$	−16.3246	−0.707759
$533$	0	0
$534$	0	0
$535$	−74.1359	−3.20518
$536$	2.83772	0.122571
$537$	0	0
$538$	6.00000	0.258678
$539$	3.48683	0.150189
$540$	0	0
$541$	−15.5132	−0.666963	−0.333482	−	0.942757i	$-0.608223\pi$
$541$	−15.5132	−0.666963	−0.333482	+	0.942757i	$0.608223\pi$
$542$	6.32456	0.271663
$543$	0	0
$544$	−3.00000	−0.128624
$545$	−27.6754	−1.18549
$546$	0	0
$547$	−33.8114	−1.44567	−0.722835	−	0.691020i	$-0.757161\pi$
$547$	−33.8114	−1.44567	−0.722835	+	0.691020i	$0.757161\pi$
$548$	3.00000	0.128154
$549$	0	0
$550$	−14.3246	−0.610801
$551$	−9.48683	−0.404153
$552$	0	0
$553$	−12.6491	−0.537895
$554$	−10.8114	−0.459332
$555$	0	0
$556$	−6.32456	−0.268221
$557$	18.4868	0.783312	0.391656	−	0.920112i	$-0.371902\pi$
$557$	18.4868	0.783312	0.391656	+	0.920112i	$0.371902\pi$
$558$	0	0
$559$	0	0
$560$	−13.1623	−0.556208
$561$	0	0
$562$	−0.675445	−0.0284919
$563$	−40.6491	−1.71316	−0.856578	−	0.516018i	$-0.827414\pi$
$563$	−40.6491	−1.71316	−0.856578	+	0.516018i	$0.827414\pi$
$564$	0	0
$565$	27.7851	1.16893
$566$	2.83772	0.119278
$567$	0	0
$568$	−7.16228	−0.300522
$569$	−27.2982	−1.14440	−0.572200	−	0.820114i	$-0.693910\pi$
$569$	−27.2982	−1.14440	−0.572200	+	0.820114i	$0.693910\pi$
$570$	0	0
$571$	−36.1359	−1.51224	−0.756121	−	0.654432i	$-0.772908\pi$
$571$	−36.1359	−1.51224	−0.756121	+	0.654432i	$0.772908\pi$
$572$	0	0
$573$	0	0
$574$	9.48683	0.395973
$575$	−88.2719	−3.68119
$576$	0	0
$577$	0.350889	0.0146077	0.00730386	−	0.999973i	$-0.497675\pi$
$577$	0.350889	0.0146077	0.00730386	+	0.999973i	$0.497675\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	−7.64911	−0.317612
$581$	11.0263	0.457449
$582$	0	0
$583$	14.5132	0.601074
$584$	1.00000	0.0413803
$585$	0	0
$586$	1.83772	0.0759156
$587$	−28.6491	−1.18248	−0.591238	−	0.806497i	$-0.701360\pi$
$587$	−28.6491	−1.18248	−0.591238	+	0.806497i	$0.701360\pi$
$588$	0	0
$589$	32.6491	1.34528
$590$	9.67544	0.398332
$591$	0	0
$592$	3.83772	0.157729
$593$	−0.675445	−0.0277372	−0.0138686	−	0.999904i	$-0.504415\pi$
$593$	−0.675445	−0.0277372	−0.0138686	+	0.999904i	$0.504415\pi$
$594$	0	0
$595$	−39.4868	−1.61880
$596$	−0.486833	−0.0199415
$597$	0	0
$598$	0	0
$599$	23.6228	0.965200	0.482600	−	0.875841i	$-0.339692\pi$
$599$	23.6228	0.965200	0.482600	+	0.875841i	$0.339692\pi$
$600$	0	0
$601$	−34.2982	−1.39905	−0.699527	−	0.714606i	$-0.746606\pi$
$601$	−34.2982	−1.39905	−0.699527	+	0.714606i	$0.746606\pi$
$602$	−28.9737	−1.18088
$603$	0	0
$604$	0.837722	0.0340864
$605$	40.1623	1.63283
$606$	0	0
$607$	17.2982	0.702113	0.351057	−	0.936354i	$-0.385822\pi$
$607$	17.2982	0.702113	0.351057	+	0.936354i	$0.385822\pi$
$608$	5.16228	0.209358
$609$	0	0
$610$	0.675445	0.0273480
$611$	0	0
$612$	0	0
$613$	20.4868	0.827455	0.413728	−	0.910401i	$-0.364227\pi$
$613$	20.4868	0.827455	0.413728	+	0.910401i	$0.364227\pi$
$614$	−11.4868	−0.463571
$615$	0	0
$616$	−3.67544	−0.148088
$617$	26.6228	1.07179	0.535896	−	0.844284i	$-0.319974\pi$
$617$	26.6228	1.07179	0.535896	+	0.844284i	$0.319974\pi$
$618$	0	0
$619$	24.6491	0.990731	0.495366	−	0.868685i	$-0.335034\pi$
$619$	24.6491	0.990731	0.495366	+	0.868685i	$0.335034\pi$
$620$	26.3246	1.05722
$621$	0	0
$622$	21.4868	0.861544
$623$	−37.9473	−1.52033
$624$	0	0
$625$	65.2719	2.61088
$626$	4.00000	0.159872
$627$	0	0
$628$	−10.4868	−0.418470
$629$	11.5132	0.459060
$630$	0	0
$631$	−25.2982	−1.00711	−0.503553	−	0.863964i	$-0.667974\pi$
$631$	−25.2982	−1.00711	−0.503553	+	0.863964i	$0.667974\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	12.4868	0.495915
$635$	76.2719	3.02676
$636$	0	0
$637$	0	0
$638$	−2.13594	−0.0845628
$639$	0	0
$640$	4.16228	0.164528
$641$	−16.3509	−0.645821	−0.322911	−	0.946429i	$-0.604661\pi$
$641$	−16.3509	−0.645821	−0.322911	+	0.946429i	$0.604661\pi$
$642$	0	0
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	−22.6491	−0.892500
$645$	0	0
$646$	15.4868	0.609321
$647$	40.6491	1.59808	0.799041	−	0.601277i	$-0.205341\pi$
$647$	40.6491	1.59808	0.799041	+	0.601277i	$0.205341\pi$
$648$	0	0
$649$	2.70178	0.106054
$650$	0	0
$651$	0	0
$652$	−16.0000	−0.626608
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	69.2982	2.70771
$656$	−3.00000	−0.117130
$657$	0	0
$658$	15.2982	0.596387
$659$	5.02633	0.195798	0.0978991	−	0.995196i	$-0.468788\pi$
$659$	5.02633	0.195798	0.0978991	+	0.995196i	$0.468788\pi$
$660$	0	0
$661$	26.4868	1.03022	0.515109	−	0.857125i	$-0.327751\pi$
$661$	26.4868	1.03022	0.515109	+	0.857125i	$0.327751\pi$
$662$	10.9737	0.426504
$663$	0	0
$664$	−3.48683	−0.135315
$665$	67.9473	2.63488
$666$	0	0
$667$	−13.1623	−0.509645
$668$	−12.0000	−0.464294
$669$	0	0
$670$	−11.8114	−0.456314
$671$	0.188612	0.00728127
$672$	0	0
$673$	14.6754	0.565697	0.282848	−	0.959165i	$-0.408721\pi$
$673$	14.6754	0.565697	0.282848	+	0.959165i	$0.408721\pi$
$674$	−11.0000	−0.423704
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−12.6491	−0.485428
$680$	12.4868	0.478848
$681$	0	0
$682$	7.35089	0.281480
$683$	35.6228	1.36307	0.681534	−	0.731787i	$-0.261313\pi$
$683$	35.6228	1.36307	0.681534	+	0.731787i	$0.261313\pi$
$684$	0	0
$685$	−12.4868	−0.477097
$686$	12.6491	0.482945
$687$	0	0
$688$	9.16228	0.349309
$689$	0	0
$690$	0	0
$691$	9.16228	0.348549	0.174275	−	0.984697i	$-0.444242\pi$
$691$	9.16228	0.348549	0.174275	+	0.984697i	$0.444242\pi$
$692$	−22.6491	−0.860990
$693$	0	0
$694$	−15.4868	−0.587872
$695$	26.3246	0.998547
$696$	0	0
$697$	−9.00000	−0.340899
$698$	5.35089	0.202534
$699$	0	0
$700$	38.9737	1.47307
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−19.8114	−0.747201
$704$	1.16228	0.0438050
$705$	0	0
$706$	29.3246	1.10364
$707$	−24.7851	−0.932138
$708$	0	0
$709$	45.4605	1.70730	0.853652	−	0.520843i	$-0.174382\pi$
$709$	45.4605	1.70730	0.853652	+	0.520843i	$0.174382\pi$
$710$	29.8114	1.11880
$711$	0	0
$712$	12.0000	0.449719
$713$	45.2982	1.69643
$714$	0	0
$715$	0	0
$716$	−15.4868	−0.578770
$717$	0	0
$718$	28.4605	1.06214
$719$	38.3246	1.42926	0.714632	−	0.699500i	$-0.246594\pi$
$719$	38.3246	1.42926	0.714632	+	0.699500i	$0.246594\pi$
$720$	0	0
$721$	−50.0000	−1.86210
$722$	−7.64911	−0.284670
$723$	0	0
$724$	3.83772	0.142628
$725$	22.6491	0.841167
$726$	0	0
$727$	−8.83772	−0.327773	−0.163886	−	0.986479i	$-0.552403\pi$
$727$	−8.83772	−0.327773	−0.163886	+	0.986479i	$0.552403\pi$
$728$	0	0
$729$	0	0
$730$	−4.16228	−0.154053
$731$	27.4868	1.01664
$732$	0	0
$733$	21.4605	0.792662	0.396331	−	0.918108i	$-0.370283\pi$
$733$	21.4605	0.792662	0.396331	+	0.918108i	$0.370283\pi$
$734$	25.4868	0.940736
$735$	0	0
$736$	7.16228	0.264005
$737$	−3.29822	−0.121492
$738$	0	0
$739$	−39.6228	−1.45755	−0.728774	−	0.684755i	$-0.759909\pi$
$739$	−39.6228	−1.45755	−0.728774	+	0.684755i	$0.759909\pi$
$740$	−15.9737	−0.587204
$741$	0	0
$742$	−39.4868	−1.44961
$743$	−2.32456	−0.0852797	−0.0426398	−	0.999091i	$-0.513577\pi$
$743$	−2.32456	−0.0852797	−0.0426398	+	0.999091i	$0.513577\pi$
$744$	0	0
$745$	2.02633	0.0742391
$746$	16.4868	0.603626
$747$	0	0
$748$	3.48683	0.127491
$749$	56.3246	2.05805
$750$	0	0
$751$	38.7851	1.41529	0.707643	−	0.706570i	$-0.249758\pi$
$751$	38.7851	1.41529	0.707643	+	0.706570i	$0.249758\pi$
$752$	−4.83772	−0.176414
$753$	0	0
$754$	0	0
$755$	−3.48683	−0.126899
$756$	0	0
$757$	−26.6491	−0.968578	−0.484289	−	0.874908i	$-0.660922\pi$
$757$	−26.6491	−0.968578	−0.484289	+	0.874908i	$0.660922\pi$
$758$	−17.6754	−0.642001
$759$	0	0
$760$	−21.4868	−0.779409
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	21.0263	0.761204
$764$	−14.3246	−0.518244
$765$	0	0
$766$	30.9737	1.11912
$767$	0	0
$768$	0	0
$769$	3.35089	0.120836	0.0604181	−	0.998173i	$-0.480757\pi$
$769$	3.35089	0.120836	0.0604181	+	0.998173i	$0.480757\pi$
$770$	15.2982	0.551310
$771$	0	0
$772$	−19.9737	−0.718868
$773$	−22.6491	−0.814632	−0.407316	−	0.913287i	$-0.633535\pi$
$773$	−22.6491	−0.814632	−0.407316	+	0.913287i	$0.633535\pi$
$774$	0	0
$775$	−77.9473	−2.79995
$776$	4.00000	0.143592
$777$	0	0
$778$	12.4868	0.447675
$779$	15.4868	0.554873
$780$	0	0
$781$	8.32456	0.297876
$782$	21.4868	0.768368
$783$	0	0
$784$	3.00000	0.107143
$785$	43.6491	1.55790
$786$	0	0
$787$	−9.02633	−0.321754	−0.160877	−	0.986974i	$-0.551432\pi$
$787$	−9.02633	−0.321754	−0.160877	+	0.986974i	$0.551432\pi$
$788$	−18.9737	−0.675909
$789$	0	0
$790$	−16.6491	−0.592349
$791$	−21.1096	−0.750571
$792$	0	0
$793$	0	0
$794$	−26.0000	−0.922705
$795$	0	0
$796$	−6.51317	−0.230853
$797$	18.9737	0.672082	0.336041	−	0.941847i	$-0.390912\pi$
$797$	18.9737	0.672082	0.336041	+	0.941847i	$0.390912\pi$
$798$	0	0
$799$	−14.5132	−0.513439
$800$	−12.3246	−0.435739
$801$	0	0
$802$	−15.0000	−0.529668
$803$	−1.16228	−0.0410159
$804$	0	0
$805$	94.2719	3.32265
$806$	0	0
$807$	0	0
$808$	7.83772	0.275730
$809$	−3.00000	−0.105474	−0.0527372	−	0.998608i	$-0.516795\pi$
$809$	−3.00000	−0.105474	−0.0527372	+	0.998608i	$0.516795\pi$
$810$	0	0
$811$	1.02633	0.0360395	0.0180197	−	0.999838i	$-0.494264\pi$
$811$	1.02633	0.0360395	0.0180197	+	0.999838i	$0.494264\pi$
$812$	5.81139	0.203940
$813$	0	0
$814$	−4.46050	−0.156340
$815$	66.5964	2.33277
$816$	0	0
$817$	−47.2982	−1.65476
$818$	−14.6754	−0.513115
$819$	0	0
$820$	12.4868	0.436059
$821$	−10.6491	−0.371657	−0.185828	−	0.982582i	$-0.559497\pi$
$821$	−10.6491	−0.371657	−0.185828	+	0.982582i	$0.559497\pi$
$822$	0	0
$823$	53.2982	1.85786	0.928930	−	0.370256i	$-0.120730\pi$
$823$	53.2982	1.85786	0.928930	+	0.370256i	$0.120730\pi$
$824$	15.8114	0.550816
$825$	0	0
$826$	−7.35089	−0.255770
$827$	6.97367	0.242498	0.121249	−	0.992622i	$-0.461310\pi$
$827$	6.97367	0.242498	0.121249	+	0.992622i	$0.461310\pi$
$828$	0	0
$829$	12.1623	0.422413	0.211207	−	0.977441i	$-0.432261\pi$
$829$	12.1623	0.422413	0.211207	+	0.977441i	$0.432261\pi$
$830$	14.5132	0.503759
$831$	0	0
$832$	0	0
$833$	9.00000	0.311832
$834$	0	0
$835$	49.9473	1.72850
$836$	−6.00000	−0.207514
$837$	0	0
$838$	−30.9737	−1.06997
$839$	−4.64911	−0.160505	−0.0802526	−	0.996775i	$-0.525573\pi$
$839$	−4.64911	−0.160505	−0.0802526	+	0.996775i	$0.525573\pi$
$840$	0	0
$841$	−25.6228	−0.883544
$842$	23.8377	0.821502
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	−30.5132	−1.04844
$848$	12.4868	0.428800
$849$	0	0
$850$	−36.9737	−1.26819
$851$	−27.4868	−0.942236
$852$	0	0
$853$	55.1359	1.88782	0.943909	−	0.330205i	$-0.107118\pi$
$853$	55.1359	1.88782	0.943909	+	0.330205i	$0.107118\pi$
$854$	−0.513167	−0.0175602
$855$	0	0
$856$	−17.8114	−0.608781
$857$	25.6491	0.876157	0.438078	−	0.898937i	$-0.355659\pi$
$857$	25.6491	0.876157	0.438078	+	0.898937i	$0.355659\pi$
$858$	0	0
$859$	28.5132	0.972857	0.486428	−	0.873720i	$-0.338299\pi$
$859$	28.5132	0.972857	0.486428	+	0.873720i	$0.338299\pi$
$860$	−38.1359	−1.30042
$861$	0	0
$862$	−9.48683	−0.323123
$863$	−47.8114	−1.62752	−0.813759	−	0.581202i	$-0.802583\pi$
$863$	−47.8114	−1.62752	−0.813759	+	0.581202i	$0.802583\pi$
$864$	0	0
$865$	94.2719	3.20534
$866$	9.32456	0.316861
$867$	0	0
$868$	−20.0000	−0.678844
$869$	−4.64911	−0.157710
$870$	0	0
$871$	0	0
$872$	−6.64911	−0.225167
$873$	0	0
$874$	−36.9737	−1.25065
$875$	−96.4078	−3.25918
$876$	0	0
$877$	−45.1359	−1.52413	−0.762066	−	0.647499i	$-0.775815\pi$
$877$	−45.1359	−1.52413	−0.762066	+	0.647499i	$0.775815\pi$
$878$	25.4868	0.860139
$879$	0	0
$880$	−4.83772	−0.163080
$881$	−9.97367	−0.336021	−0.168011	−	0.985785i	$-0.553734\pi$
$881$	−9.97367	−0.336021	−0.168011	+	0.985785i	$0.553734\pi$
$882$	0	0
$883$	−11.3509	−0.381988	−0.190994	−	0.981591i	$-0.561171\pi$
$883$	−11.3509	−0.381988	−0.190994	+	0.981591i	$0.561171\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38.3246	−1.28681	−0.643406	−	0.765525i	$-0.722479\pi$
$887$	−38.3246	−1.28681	−0.643406	+	0.765525i	$0.722479\pi$
$888$	0	0
$889$	−57.9473	−1.94349
$890$	−49.9473	−1.67424
$891$	0	0
$892$	−1.67544	−0.0560980
$893$	24.9737	0.835712
$894$	0	0
$895$	64.4605	2.15468
$896$	−3.16228	−0.105644
$897$	0	0
$898$	7.35089	0.245302
$899$	−11.6228	−0.387641
$900$	0	0
$901$	37.4605	1.24799
$902$	3.48683	0.116099
$903$	0	0
$904$	6.67544	0.222022
$905$	−15.9737	−0.530983
$906$	0	0
$907$	36.2719	1.20439	0.602194	−	0.798350i	$-0.294293\pi$
$907$	36.2719	1.20439	0.602194	+	0.798350i	$0.294293\pi$
$908$	15.4868	0.513949
$909$	0	0
$910$	0	0
$911$	49.9473	1.65483	0.827414	−	0.561592i	$-0.189811\pi$
$911$	49.9473	1.65483	0.827414	+	0.561592i	$0.189811\pi$
$912$	0	0
$913$	4.05267	0.134124
$914$	3.32456	0.109967
$915$	0	0
$916$	22.3246	0.737624
$917$	−52.6491	−1.73863
$918$	0	0
$919$	3.35089	0.110536	0.0552678	−	0.998472i	$-0.482399\pi$
$919$	3.35089	0.110536	0.0552678	+	0.998472i	$0.482399\pi$
$920$	−29.8114	−0.982852
$921$	0	0
$922$	18.4868	0.608831
$923$	0	0
$924$	0	0
$925$	47.2982	1.55516
$926$	−15.1623	−0.498263
$927$	0	0
$928$	−1.83772	−0.0603262
$929$	−28.9473	−0.949731	−0.474866	−	0.880058i	$-0.657503\pi$
$929$	−28.9473	−0.949731	−0.474866	+	0.880058i	$0.657503\pi$
$930$	0	0
$931$	−15.4868	−0.507560
$932$	16.6491	0.545360
$933$	0	0
$934$	6.18861	0.202498
$935$	−14.5132	−0.474631
$936$	0	0
$937$	14.2982	0.467103	0.233551	−	0.972344i	$-0.424965\pi$
$937$	14.2982	0.467103	0.233551	+	0.972344i	$0.424965\pi$
$938$	8.97367	0.293001
$939$	0	0
$940$	20.1359	0.656762
$941$	48.5964	1.58420	0.792099	−	0.610392i	$-0.208988\pi$
$941$	48.5964	1.58420	0.792099	+	0.610392i	$0.208988\pi$
$942$	0	0
$943$	21.4868	0.699708
$944$	2.32456	0.0756578
$945$	0	0
$946$	−10.6491	−0.346232
$947$	−18.9737	−0.616561	−0.308281	−	0.951295i	$-0.599754\pi$
$947$	−18.9737	−0.616561	−0.308281	+	0.951295i	$0.599754\pi$
$948$	0	0
$949$	0	0
$950$	63.6228	2.06420
$951$	0	0
$952$	−9.48683	−0.307470
$953$	27.2982	0.884276	0.442138	−	0.896947i	$-0.354220\pi$
$953$	27.2982	0.884276	0.442138	+	0.896947i	$0.354220\pi$
$954$	0	0
$955$	59.6228	1.92935
$956$	21.4868	0.694934
$957$	0	0
$958$	−24.0000	−0.775405
$959$	9.48683	0.306346
$960$	0	0
$961$	9.00000	0.290323
$962$	0	0
$963$	0	0
$964$	13.3246	0.429155
$965$	83.1359	2.67624
$966$	0	0
$967$	34.5132	1.10987	0.554934	−	0.831894i	$-0.312743\pi$
$967$	34.5132	1.10987	0.554934	+	0.831894i	$0.312743\pi$
$968$	9.64911	0.310134
$969$	0	0
$970$	−16.6491	−0.534571
$971$	18.9737	0.608894	0.304447	−	0.952529i	$-0.401528\pi$
$971$	18.9737	0.608894	0.304447	+	0.952529i	$0.401528\pi$
$972$	0	0
$973$	−20.0000	−0.641171
$974$	13.4868	0.432146
$975$	0	0
$976$	0.162278	0.00519438
$977$	−21.0000	−0.671850	−0.335925	−	0.941889i	$-0.609049\pi$
$977$	−21.0000	−0.671850	−0.335925	+	0.941889i	$0.609049\pi$
$978$	0	0
$979$	−13.9473	−0.445759
$980$	−12.4868	−0.398877
$981$	0	0
$982$	5.81139	0.185449
$983$	−23.6228	−0.753450	−0.376725	−	0.926325i	$-0.622950\pi$
$983$	−23.6228	−0.753450	−0.376725	+	0.926325i	$0.622950\pi$
$984$	0	0
$985$	78.9737	2.51631
$986$	−5.51317	−0.175575
$987$	0	0
$988$	0	0
$989$	−65.6228	−2.08668
$990$	0	0
$991$	26.7851	0.850855	0.425428	−	0.904992i	$-0.360124\pi$
$991$	26.7851	0.850855	0.425428	+	0.904992i	$0.360124\pi$
$992$	6.32456	0.200805
$993$	0	0
$994$	−22.6491	−0.718386
$995$	27.1096	0.859432
$996$	0	0
$997$	−59.4605	−1.88313	−0.941566	−	0.336827i	$-0.890646\pi$
$997$	−59.4605	−1.88313	−0.941566	+	0.336827i	$0.890646\pi$
$998$	−12.6491	−0.400401
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.r.1.1		2
3.2	odd	2		3042.2.a.w.1.2		2
13.3	even	3		234.2.h.e.217.1	yes	4
13.5	odd	4		3042.2.b.j.1351.4		4
13.8	odd	4		3042.2.b.j.1351.1		4
13.9	even	3		234.2.h.e.55.1	yes	4
13.12	even	2		3042.2.a.x.1.2		2
39.5	even	4		3042.2.b.k.1351.1		4
39.8	even	4		3042.2.b.k.1351.4		4
39.29	odd	6		234.2.h.d.217.2	yes	4
39.35	odd	6		234.2.h.d.55.2	✓	4
39.38	odd	2		3042.2.a.q.1.1		2
52.3	odd	6		1872.2.t.n.1153.1		4
52.35	odd	6		1872.2.t.n.289.1		4
156.35	even	6		1872.2.t.p.289.2		4
156.107	even	6		1872.2.t.p.1153.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.h.d.55.2	✓	4	39.35	odd	6
234.2.h.d.217.2	yes	4	39.29	odd	6
234.2.h.e.55.1	yes	4	13.9	even	3
234.2.h.e.217.1	yes	4	13.3	even	3
1872.2.t.n.289.1		4	52.35	odd	6
1872.2.t.n.1153.1		4	52.3	odd	6
1872.2.t.p.289.2		4	156.35	even	6
1872.2.t.p.1153.2		4	156.107	even	6
3042.2.a.q.1.1		2	39.38	odd	2
3042.2.a.r.1.1		2	1.1	even	1	trivial
3042.2.a.w.1.2		2	3.2	odd	2
3042.2.a.x.1.2		2	13.12	even	2
3042.2.b.j.1351.1		4	13.8	odd	4
3042.2.b.j.1351.4		4	13.5	odd	4
3042.2.b.k.1351.1		4	39.5	even	4
3042.2.b.k.1351.4		4	39.8	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.16228 q^{5} +3.16228 q^{7} -1.00000 q^{8} +4.16228 q^{10} +1.16228 q^{11} -3.16228 q^{14} +1.00000 q^{16} +3.00000 q^{17} -5.16228 q^{19} -4.16228 q^{20} -1.16228 q^{22} -7.16228 q^{23} +12.3246 q^{25} +3.16228 q^{28} +1.83772 q^{29} -6.32456 q^{31} -1.00000 q^{32} -3.00000 q^{34} -13.1623 q^{35} +3.83772 q^{37} +5.16228 q^{38} +4.16228 q^{40} -3.00000 q^{41} +9.16228 q^{43} +1.16228 q^{44} +7.16228 q^{46} -4.83772 q^{47} +3.00000 q^{49} -12.3246 q^{50} +12.4868 q^{53} -4.83772 q^{55} -3.16228 q^{56} -1.83772 q^{58} +2.32456 q^{59} +0.162278 q^{61} +6.32456 q^{62} +1.00000 q^{64} -2.83772 q^{67} +3.00000 q^{68} +13.1623 q^{70} +7.16228 q^{71} -1.00000 q^{73} -3.83772 q^{74} -5.16228 q^{76} +3.67544 q^{77} -4.00000 q^{79} -4.16228 q^{80} +3.00000 q^{82} +3.48683 q^{83} -12.4868 q^{85} -9.16228 q^{86} -1.16228 q^{88} -12.0000 q^{89} -7.16228 q^{92} +4.83772 q^{94} +21.4868 q^{95} -4.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{16} + 6 q^{17} - 4 q^{19} - 2 q^{20} + 4 q^{22} - 8 q^{23} + 12 q^{25} + 10 q^{29} - 2 q^{32} - 6 q^{34} - 20 q^{35} + 14 q^{37}+ \cdots - 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^16 + 6 * q^17 - 4 * q^19 - 2 * q^20 + 4 * q^22 - 8 * q^23 + 12 * q^25 + 10 * q^29 - 2 * q^32 - 6 * q^34 - 20 * q^35 + 14 * q^37 + 4 * q^38 + 2 * q^40 - 6 * q^41 + 12 * q^43 - 4 * q^44 + 8 * q^46 - 16 * q^47 + 6 * q^49 - 12 * q^50 + 6 * q^53 - 16 * q^55 - 10 * q^58 - 8 * q^59 - 6 * q^61 + 2 * q^64 - 12 * q^67 + 6 * q^68 + 20 * q^70 + 8 * q^71 - 2 * q^73 - 14 * q^74 - 4 * q^76 + 20 * q^77 - 8 * q^79 - 2 * q^80 + 6 * q^82 - 12 * q^83 - 6 * q^85 - 12 * q^86 + 4 * q^88 - 24 * q^89 - 8 * q^92 + 16 * q^94 + 24 * q^95 - 8 * q^97 - 6 * q^98

Introduction

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