LMFDB - Embedded newform 4056.2.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(1,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4056.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.25488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 6x + 6 $ x^4 - 9x^2 - 6x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.28657$ of defining polynomial
Character	$\chi$	$=$	4056.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.55452 q^{5} +2.96046 q^{7} +1.00000 q^{9} -2.24703 q^{11} +1.55452 q^{15} -1.01862 q^{17} -5.35607 q^{19} +2.96046 q^{21} +0.782926 q^{23} -2.58347 q^{25} +1.00000 q^{27} +2.69251 q^{29} +7.28657 q^{31} -2.24703 q^{33} +4.60209 q^{35} +7.80155 q^{37} +5.34474 q^{41} +3.12766 q^{43} +1.55452 q^{45} +7.13799 q^{47} +1.76430 q^{49} -1.01862 q^{51} +13.8388 q^{53} -3.49305 q^{55} -5.35607 q^{57} +4.35506 q^{59} +5.14528 q^{61} +2.96046 q^{63} -12.9977 q^{67} +0.782926 q^{69} +13.8501 q^{71} +8.30519 q^{73} -2.58347 q^{75} -6.65223 q^{77} +1.23570 q^{79} +1.00000 q^{81} +7.67389 q^{83} -1.58347 q^{85} +2.69251 q^{87} -0.494055 q^{89} +7.28657 q^{93} -8.32611 q^{95} -11.1057 q^{97} -2.24703 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} + 2 q^{7} + 4 q^{9} + 10 q^{11} + 4 q^{15} + 12 q^{17} + 2 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 4 q^{27} - 6 q^{29} + 20 q^{31} + 10 q^{33} - 10 q^{35} + 10 q^{37} + 6 q^{41}+ \cdots + 10 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 + 2 * q^7 + 4 * q^9 + 10 * q^11 + 4 * q^15 + 12 * q^17 + 2 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 4 * q^27 - 6 * q^29 + 20 * q^31 + 10 * q^33 - 10 * q^35 + 10 * q^37 + 6 * q^41 - 8 * q^43 + 4 * q^45 + 14 * q^47 + 18 * q^49 + 12 * q^51 + 2 * q^53 + 10 * q^55 + 2 * q^57 + 8 * q^59 - 2 * q^61 + 2 * q^63 - 10 * q^67 - 2 * q^69 - 6 * q^71 + 8 * q^73 + 2 * q^75 - 28 * q^77 - 6 * q^79 + 4 * q^81 + 30 * q^83 + 6 * q^85 - 6 * q^87 + 36 * q^89 + 20 * q^93 - 34 * q^95 - 12 * q^97 + 10 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.55452	0.695202	0.347601	−	0.937642i	$-0.386996\pi$
$5$	1.55452	0.695202	0.347601	+	0.937642i	$0.386996\pi$
$6$	0	0
$7$	2.96046	1.11895	0.559474	−	0.828848i	$-0.311003\pi$
$7$	2.96046	1.11895	0.559474	+	0.828848i	$0.311003\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.24703	−0.677504	−0.338752	−	0.940876i	$-0.610005\pi$
$11$	−2.24703	−0.677504	−0.338752	+	0.940876i	$0.610005\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.55452	0.401375
$16$	0	0
$17$	−1.01862	−0.247052	−0.123526	−	0.992341i	$-0.539420\pi$
$17$	−1.01862	−0.247052	−0.123526	+	0.992341i	$0.539420\pi$
$18$	0	0
$19$	−5.35607	−1.22877	−0.614383	−	0.789008i	$-0.710595\pi$
$19$	−5.35607	−1.22877	−0.614383	+	0.789008i	$0.710595\pi$
$20$	0	0
$21$	2.96046	0.646025
$22$	0	0
$23$	0.782926	0.163251	0.0816257	−	0.996663i	$-0.473989\pi$
$23$	0.782926	0.163251	0.0816257	+	0.996663i	$0.473989\pi$
$24$	0	0
$25$	−2.58347	−0.516694
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.69251	0.499986	0.249993	−	0.968248i	$-0.419572\pi$
$29$	2.69251	0.499986	0.249993	+	0.968248i	$0.419572\pi$
$30$	0	0
$31$	7.28657	1.30871	0.654353	−	0.756189i	$-0.272941\pi$
$31$	7.28657	1.30871	0.654353	+	0.756189i	$0.272941\pi$
$32$	0	0
$33$	−2.24703	−0.391157
$34$	0	0
$35$	4.60209	0.777895
$36$	0	0
$37$	7.80155	1.28257	0.641283	−	0.767304i	$-0.278402\pi$
$37$	7.80155	1.28257	0.641283	+	0.767304i	$0.278402\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.34474	0.834707	0.417354	−	0.908744i	$-0.362958\pi$
$41$	5.34474	0.834707	0.417354	+	0.908744i	$0.362958\pi$
$42$	0	0
$43$	3.12766	0.476964	0.238482	−	0.971147i	$-0.423350\pi$
$43$	3.12766	0.476964	0.238482	+	0.971147i	$0.423350\pi$
$44$	0	0
$45$	1.55452	0.231734
$46$	0	0
$47$	7.13799	1.04118	0.520591	−	0.853806i	$-0.325712\pi$
$47$	7.13799	1.04118	0.520591	+	0.853806i	$0.325712\pi$
$48$	0	0
$49$	1.76430	0.252043
$50$	0	0
$51$	−1.01862	−0.142636
$52$	0	0
$53$	13.8388	1.90090	0.950452	−	0.310871i	$-0.100621\pi$
$53$	13.8388	1.90090	0.950452	+	0.310871i	$0.100621\pi$
$54$	0	0
$55$	−3.49305	−0.471003
$56$	0	0
$57$	−5.35607	−0.709428
$58$	0	0
$59$	4.35506	0.566981	0.283490	−	0.958975i	$-0.408508\pi$
$59$	4.35506	0.566981	0.283490	+	0.958975i	$0.408508\pi$
$60$	0	0
$61$	5.14528	0.658785	0.329393	−	0.944193i	$-0.393156\pi$
$61$	5.14528	0.658785	0.329393	+	0.944193i	$0.393156\pi$
$62$	0	0
$63$	2.96046	0.372982
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12.9977	−1.58792	−0.793961	−	0.607969i	$-0.791985\pi$
$67$	−12.9977	−1.58792	−0.793961	+	0.607969i	$0.791985\pi$
$68$	0	0
$69$	0.782926	0.0942532
$70$	0	0
$71$	13.8501	1.64371	0.821854	−	0.569699i	$-0.192940\pi$
$71$	13.8501	1.64371	0.821854	+	0.569699i	$0.192940\pi$
$72$	0	0
$73$	8.30519	0.972049	0.486025	−	0.873945i	$-0.338447\pi$
$73$	8.30519	0.972049	0.486025	+	0.873945i	$0.338447\pi$
$74$	0	0
$75$	−2.58347	−0.298313
$76$	0	0
$77$	−6.65223	−0.758092
$78$	0	0
$79$	1.23570	0.139027	0.0695133	−	0.997581i	$-0.477855\pi$
$79$	1.23570	0.139027	0.0695133	+	0.997581i	$0.477855\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.67389	0.842318	0.421159	−	0.906987i	$-0.361623\pi$
$83$	7.67389	0.842318	0.421159	+	0.906987i	$0.361623\pi$
$84$	0	0
$85$	−1.58347	−0.171751
$86$	0	0
$87$	2.69251	0.288667
$88$	0	0
$89$	−0.494055	−0.0523697	−0.0261849	−	0.999657i	$-0.508336\pi$
$89$	−0.494055	−0.0523697	−0.0261849	+	0.999657i	$0.508336\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.28657	0.755582
$94$	0	0
$95$	−8.32611	−0.854241
$96$	0	0
$97$	−11.1057	−1.12762	−0.563808	−	0.825906i	$-0.690664\pi$
$97$	−11.1057	−1.12762	−0.563808	+	0.825906i	$0.690664\pi$
$98$	0	0
$99$	−2.24703	−0.225835
$100$	0	0
$101$	−15.1866	−1.51112	−0.755560	−	0.655080i	$-0.772635\pi$
$101$	−15.1866	−1.51112	−0.755560	+	0.655080i	$0.772635\pi$
$102$	0	0
$103$	−1.12766	−0.111112	−0.0555559	−	0.998456i	$-0.517693\pi$
$103$	−1.12766	−0.111112	−0.0555559	+	0.998456i	$0.517693\pi$
$104$	0	0
$105$	4.60209	0.449118
$106$	0	0
$107$	−17.0382	−1.64715	−0.823575	−	0.567208i	$-0.808024\pi$
$107$	−17.0382	−1.64715	−0.823575	+	0.567208i	$0.808024\pi$
$108$	0	0
$109$	15.2866	1.46419	0.732094	−	0.681204i	$-0.238543\pi$
$109$	15.2866	1.46419	0.732094	+	0.681204i	$0.238543\pi$
$110$	0	0
$111$	7.80155	0.740490
$112$	0	0
$113$	7.01862	0.660256	0.330128	−	0.943936i	$-0.392908\pi$
$113$	7.01862	0.660256	0.330128	+	0.943936i	$0.392908\pi$
$114$	0	0
$115$	1.21707	0.113493
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.01559	−0.276438
$120$	0	0
$121$	−5.95087	−0.540988
$122$	0	0
$123$	5.34474	0.481919
$124$	0	0
$125$	−11.7887	−1.05441
$126$	0	0
$127$	−14.8015	−1.31342	−0.656712	−	0.754141i	$-0.728053\pi$
$127$	−14.8015	−1.31342	−0.656712	+	0.754141i	$0.728053\pi$
$128$	0	0
$129$	3.12766	0.275375
$130$	0	0
$131$	5.81916	0.508423	0.254211	−	0.967149i	$-0.418184\pi$
$131$	5.81916	0.508423	0.254211	+	0.967149i	$0.418184\pi$
$132$	0	0
$133$	−15.8564	−1.37492
$134$	0	0
$135$	1.55452	0.133792
$136$	0	0
$137$	17.3674	1.48380	0.741899	−	0.670512i	$-0.233926\pi$
$137$	17.3674	1.48380	0.741899	+	0.670512i	$0.233926\pi$
$138$	0	0
$139$	5.88792	0.499407	0.249704	−	0.968322i	$-0.419667\pi$
$139$	5.88792	0.499407	0.249704	+	0.968322i	$0.419667\pi$
$140$	0	0
$141$	7.13799	0.601127
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.18556	0.347592
$146$	0	0
$147$	1.76430	0.145517
$148$	0	0
$149$	−10.5845	−0.867114	−0.433557	−	0.901126i	$-0.642742\pi$
$149$	−10.5845	−0.867114	−0.433557	+	0.901126i	$0.642742\pi$
$150$	0	0
$151$	2.24703	0.182861	0.0914303	−	0.995811i	$-0.470856\pi$
$151$	2.24703	0.182861	0.0914303	+	0.995811i	$0.470856\pi$
$152$	0	0
$153$	−1.01862	−0.0823507
$154$	0	0
$155$	11.3271	0.909816
$156$	0	0
$157$	−8.45782	−0.675007	−0.337504	−	0.941324i	$-0.609583\pi$
$157$	−8.45782	−0.675007	−0.337504	+	0.941324i	$0.609583\pi$
$158$	0	0
$159$	13.8388	1.09749
$160$	0	0
$161$	2.31782	0.182670
$162$	0	0
$163$	−4.20019	−0.328985	−0.164492	−	0.986378i	$-0.552599\pi$
$163$	−4.20019	−0.328985	−0.164492	+	0.986378i	$0.552599\pi$
$164$	0	0
$165$	−3.49305	−0.271934
$166$	0	0
$167$	18.7121	1.44799	0.723994	−	0.689806i	$-0.242304\pi$
$167$	18.7121	1.44799	0.723994	+	0.689806i	$0.242304\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−5.35607	−0.409589
$172$	0	0
$173$	2.89096	0.219796	0.109898	−	0.993943i	$-0.464948\pi$
$173$	2.89096	0.219796	0.109898	+	0.993943i	$0.464948\pi$
$174$	0	0
$175$	−7.64824	−0.578153
$176$	0	0
$177$	4.35506	0.327346
$178$	0	0
$179$	−22.8574	−1.70844	−0.854222	−	0.519909i	$-0.825966\pi$
$179$	−22.8574	−1.70844	−0.854222	+	0.519909i	$0.825966\pi$
$180$	0	0
$181$	−8.09042	−0.601356	−0.300678	−	0.953726i	$-0.597213\pi$
$181$	−8.09042	−0.601356	−0.300678	+	0.953726i	$0.597213\pi$
$182$	0	0
$183$	5.14528	0.380350
$184$	0	0
$185$	12.1277	0.891643
$186$	0	0
$187$	2.28887	0.167379
$188$	0	0
$189$	2.96046	0.215342
$190$	0	0
$191$	−15.3850	−1.11322	−0.556610	−	0.830774i	$-0.687898\pi$
$191$	−15.3850	−1.11322	−0.556610	+	0.830774i	$0.687898\pi$
$192$	0	0
$193$	−0.347036	−0.0249802	−0.0124901	−	0.999922i	$-0.503976\pi$
$193$	−0.347036	−0.0249802	−0.0124901	+	0.999922i	$0.503976\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.8564	1.84219	0.921096	−	0.389335i	$-0.127295\pi$
$197$	25.8564	1.84219	0.921096	+	0.389335i	$0.127295\pi$
$198$	0	0
$199$	−7.34574	−0.520726	−0.260363	−	0.965511i	$-0.583842\pi$
$199$	−7.34574	−0.520726	−0.260363	+	0.965511i	$0.583842\pi$
$200$	0	0
$201$	−12.9977	−0.916787
$202$	0	0
$203$	7.97105	0.559458
$204$	0	0
$205$	8.30850	0.580291
$206$	0	0
$207$	0.782926	0.0544171
$208$	0	0
$209$	12.0352	0.832494
$210$	0	0
$211$	−28.6952	−1.97546	−0.987729	−	0.156175i	$-0.950084\pi$
$211$	−28.6952	−1.97546	−0.987729	+	0.156175i	$0.950084\pi$
$212$	0	0
$213$	13.8501	0.948995
$214$	0	0
$215$	4.86201	0.331586
$216$	0	0
$217$	21.5716	1.46437
$218$	0	0
$219$	8.30519	0.561213
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2.13698	−0.143103	−0.0715514	−	0.997437i	$-0.522795\pi$
$223$	−2.13698	−0.143103	−0.0715514	+	0.997437i	$0.522795\pi$
$224$	0	0
$225$	−2.58347	−0.172231
$226$	0	0
$227$	−18.3860	−1.22032	−0.610162	−	0.792277i	$-0.708895\pi$
$227$	−18.3860	−1.22032	−0.610162	+	0.792277i	$0.708895\pi$
$228$	0	0
$229$	1.74669	0.115424	0.0577122	−	0.998333i	$-0.481619\pi$
$229$	1.74669	0.115424	0.0577122	+	0.998333i	$0.481619\pi$
$230$	0	0
$231$	−6.65223	−0.437684
$232$	0	0
$233$	−26.3525	−1.72641	−0.863204	−	0.504855i	$-0.831546\pi$
$233$	−26.3525	−1.72641	−0.863204	+	0.504855i	$0.831546\pi$
$234$	0	0
$235$	11.0961	0.723833
$236$	0	0
$237$	1.23570	0.0802671
$238$	0	0
$239$	6.36336	0.411611	0.205806	−	0.978593i	$-0.434019\pi$
$239$	6.36336	0.411611	0.205806	+	0.978593i	$0.434019\pi$
$240$	0	0
$241$	−9.01862	−0.580940	−0.290470	−	0.956884i	$-0.593812\pi$
$241$	−9.01862	−0.580940	−0.290470	+	0.956884i	$0.593812\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.74265	0.175221
$246$	0	0
$247$	0	0
$248$	0	0
$249$	7.67389	0.486313
$250$	0	0
$251$	−18.2760	−1.15357	−0.576785	−	0.816896i	$-0.695693\pi$
$251$	−18.2760	−1.15357	−0.576785	+	0.816896i	$0.695693\pi$
$252$	0	0
$253$	−1.75926	−0.110603
$254$	0	0
$255$	−1.58347	−0.0991606
$256$	0	0
$257$	26.4263	1.64843	0.824214	−	0.566279i	$-0.191617\pi$
$257$	26.4263	1.64843	0.824214	+	0.566279i	$0.191617\pi$
$258$	0	0
$259$	23.0961	1.43512
$260$	0	0
$261$	2.69251	0.166662
$262$	0	0
$263$	4.82017	0.297224	0.148612	−	0.988896i	$-0.452519\pi$
$263$	4.82017	0.297224	0.148612	+	0.988896i	$0.452519\pi$
$264$	0	0
$265$	21.5127	1.32151
$266$	0	0
$267$	−0.494055	−0.0302357
$268$	0	0
$269$	−1.64034	−0.100013	−0.0500066	−	0.998749i	$-0.515924\pi$
$269$	−1.64034	−0.100013	−0.0500066	+	0.998749i	$0.515924\pi$
$270$	0	0
$271$	−12.7300	−0.773294	−0.386647	−	0.922228i	$-0.626367\pi$
$271$	−12.7300	−0.773294	−0.386647	+	0.922228i	$0.626367\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.80512	0.350062
$276$	0	0
$277$	24.5008	1.47211	0.736055	−	0.676922i	$-0.236687\pi$
$277$	24.5008	1.47211	0.736055	+	0.676922i	$0.236687\pi$
$278$	0	0
$279$	7.28657	0.436236
$280$	0	0
$281$	1.93853	0.115643	0.0578215	−	0.998327i	$-0.481585\pi$
$281$	1.93853	0.115643	0.0578215	+	0.998327i	$0.481585\pi$
$282$	0	0
$283$	8.07853	0.480219	0.240109	−	0.970746i	$-0.422817\pi$
$283$	8.07853	0.480219	0.240109	+	0.970746i	$0.422817\pi$
$284$	0	0
$285$	−8.32611	−0.493196
$286$	0	0
$287$	15.8229	0.933994
$288$	0	0
$289$	−15.9624	−0.938965
$290$	0	0
$291$	−11.1057	−0.651030
$292$	0	0
$293$	−28.3411	−1.65571	−0.827854	−	0.560944i	$-0.810438\pi$
$293$	−28.3411	−1.65571	−0.827854	+	0.560944i	$0.810438\pi$
$294$	0	0
$295$	6.77003	0.394166
$296$	0	0
$297$	−2.24703	−0.130386
$298$	0	0
$299$	0	0
$300$	0	0
$301$	9.25931	0.533698
$302$	0	0
$303$	−15.1866	−0.872445
$304$	0	0
$305$	7.99844	0.457989
$306$	0	0
$307$	−30.7025	−1.75229	−0.876143	−	0.482051i	$-0.839892\pi$
$307$	−30.7025	−1.75229	−0.876143	+	0.482051i	$0.839892\pi$
$308$	0	0
$309$	−1.12766	−0.0641504
$310$	0	0
$311$	−13.7338	−0.778772	−0.389386	−	0.921075i	$-0.627313\pi$
$311$	−13.7338	−0.778772	−0.389386	+	0.921075i	$0.627313\pi$
$312$	0	0
$313$	32.1886	1.81941	0.909703	−	0.415260i	$-0.136309\pi$
$313$	32.1886	1.81941	0.909703	+	0.415260i	$0.136309\pi$
$314$	0	0
$315$	4.60209	0.259298
$316$	0	0
$317$	−4.54723	−0.255398	−0.127699	−	0.991813i	$-0.540759\pi$
$317$	−4.54723	−0.255398	−0.127699	+	0.991813i	$0.540759\pi$
$318$	0	0
$319$	−6.05014	−0.338743
$320$	0	0
$321$	−17.0382	−0.950982
$322$	0	0
$323$	5.45581	0.303569
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.2866	0.845349
$328$	0	0
$329$	21.1317	1.16503
$330$	0	0
$331$	14.7899	0.812928	0.406464	−	0.913667i	$-0.366762\pi$
$331$	14.7899	0.812928	0.406464	+	0.913667i	$0.366762\pi$
$332$	0	0
$333$	7.80155	0.427522
$334$	0	0
$335$	−20.2052	−1.10393
$336$	0	0
$337$	3.21808	0.175300	0.0876500	−	0.996151i	$-0.472064\pi$
$337$	3.21808	0.175300	0.0876500	+	0.996151i	$0.472064\pi$
$338$	0	0
$339$	7.01862	0.381199
$340$	0	0
$341$	−16.3731	−0.886654
$342$	0	0
$343$	−15.5001	−0.836924
$344$	0	0
$345$	1.21707	0.0655251
$346$	0	0
$347$	−4.30345	−0.231021	−0.115511	−	0.993306i	$-0.536850\pi$
$347$	−4.30345	−0.231021	−0.115511	+	0.993306i	$0.536850\pi$
$348$	0	0
$349$	−11.1284	−0.595689	−0.297845	−	0.954614i	$-0.596268\pi$
$349$	−11.1284	−0.595689	−0.297845	+	0.954614i	$0.596268\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.8295	0.576396	0.288198	−	0.957571i	$-0.406944\pi$
$353$	10.8295	0.576396	0.288198	+	0.957571i	$0.406944\pi$
$354$	0	0
$355$	21.5303	1.14271
$356$	0	0
$357$	−3.01559	−0.159602
$358$	0	0
$359$	32.0616	1.69215	0.846073	−	0.533067i	$-0.178961\pi$
$359$	32.0616	1.69215	0.846073	+	0.533067i	$0.178961\pi$
$360$	0	0
$361$	9.68746	0.509866
$362$	0	0
$363$	−5.95087	−0.312340
$364$	0	0
$365$	12.9106	0.675771
$366$	0	0
$367$	0.222122	0.0115947	0.00579734	−	0.999983i	$-0.498155\pi$
$367$	0.222122	0.0115947	0.00579734	+	0.999983i	$0.498155\pi$
$368$	0	0
$369$	5.34474	0.278236
$370$	0	0
$371$	40.9691	2.12701
$372$	0	0
$373$	24.0961	1.24765	0.623826	−	0.781564i	$-0.285578\pi$
$373$	24.0961	1.24765	0.623826	+	0.781564i	$0.285578\pi$
$374$	0	0
$375$	−11.7887	−0.608763
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−16.3518	−0.839933	−0.419967	−	0.907540i	$-0.637958\pi$
$379$	−16.3518	−0.839933	−0.419967	+	0.907540i	$0.637958\pi$
$380$	0	0
$381$	−14.8015	−0.758306
$382$	0	0
$383$	−19.4967	−0.996237	−0.498119	−	0.867109i	$-0.665976\pi$
$383$	−19.4967	−0.996237	−0.498119	+	0.867109i	$0.665976\pi$
$384$	0	0
$385$	−10.3410	−0.527027
$386$	0	0
$387$	3.12766	0.158988
$388$	0	0
$389$	29.6972	1.50571	0.752854	−	0.658187i	$-0.228677\pi$
$389$	29.6972	1.50571	0.752854	+	0.658187i	$0.228677\pi$
$390$	0	0
$391$	−0.797505	−0.0403316
$392$	0	0
$393$	5.81916	0.293538
$394$	0	0
$395$	1.92091	0.0966517
$396$	0	0
$397$	−15.8743	−0.796708	−0.398354	−	0.917232i	$-0.630418\pi$
$397$	−15.8743	−0.796708	−0.398354	+	0.917232i	$0.630418\pi$
$398$	0	0
$399$	−15.8564	−0.793813
$400$	0	0
$401$	−11.5402	−0.576288	−0.288144	−	0.957587i	$-0.593038\pi$
$401$	−11.5402	−0.576288	−0.288144	+	0.957587i	$0.593038\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.55452	0.0772447
$406$	0	0
$407$	−17.5303	−0.868944
$408$	0	0
$409$	−35.0979	−1.73548	−0.867739	−	0.497019i	$-0.834428\pi$
$409$	−35.0979	−1.73548	−0.867739	+	0.497019i	$0.834428\pi$
$410$	0	0
$411$	17.3674	0.856671
$412$	0	0
$413$	12.8930	0.634422
$414$	0	0
$415$	11.9292	0.585582
$416$	0	0
$417$	5.88792	0.288333
$418$	0	0
$419$	22.6284	1.10547	0.552736	−	0.833356i	$-0.313584\pi$
$419$	22.6284	1.10547	0.552736	+	0.833356i	$0.313584\pi$
$420$	0	0
$421$	13.3833	0.652261	0.326130	−	0.945325i	$-0.394255\pi$
$421$	13.3833	0.652261	0.326130	+	0.945325i	$0.394255\pi$
$422$	0	0
$423$	7.13799	0.347061
$424$	0	0
$425$	2.63158	0.127650
$426$	0	0
$427$	15.2324	0.737146
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.0616	−0.966333	−0.483166	−	0.875529i	$-0.660513\pi$
$431$	−20.0616	−0.966333	−0.483166	+	0.875529i	$0.660513\pi$
$432$	0	0
$433$	37.8445	1.81869	0.909346	−	0.416041i	$-0.136583\pi$
$433$	37.8445	1.81869	0.909346	+	0.416041i	$0.136583\pi$
$434$	0	0
$435$	4.18556	0.200682
$436$	0	0
$437$	−4.19340	−0.200598
$438$	0	0
$439$	−25.6796	−1.22562	−0.612811	−	0.790230i	$-0.709961\pi$
$439$	−25.6796	−1.22562	−0.612811	+	0.790230i	$0.709961\pi$
$440$	0	0
$441$	1.76430	0.0840145
$442$	0	0
$443$	−37.4822	−1.78083	−0.890416	−	0.455148i	$-0.849586\pi$
$443$	−37.4822	−1.78083	−0.890416	+	0.455148i	$0.849586\pi$
$444$	0	0
$445$	−0.768019	−0.0364076
$446$	0	0
$447$	−10.5845	−0.500628
$448$	0	0
$449$	−5.92091	−0.279425	−0.139713	−	0.990192i	$-0.544618\pi$
$449$	−5.92091	−0.279425	−0.139713	+	0.990192i	$0.544618\pi$
$450$	0	0
$451$	−12.0098	−0.565518
$452$	0	0
$453$	2.24703	0.105575
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−40.6184	−1.90005	−0.950025	−	0.312175i	$-0.898942\pi$
$457$	−40.6184	−1.90005	−0.950025	+	0.312175i	$0.898942\pi$
$458$	0	0
$459$	−1.01862	−0.0475452
$460$	0	0
$461$	7.86627	0.366369	0.183184	−	0.983079i	$-0.441360\pi$
$461$	7.86627	0.366369	0.183184	+	0.983079i	$0.441360\pi$
$462$	0	0
$463$	−3.96775	−0.184397	−0.0921984	−	0.995741i	$-0.529389\pi$
$463$	−3.96775	−0.184397	−0.0921984	+	0.995741i	$0.529389\pi$
$464$	0	0
$465$	11.3271	0.525283
$466$	0	0
$467$	−1.41249	−0.0653622	−0.0326811	−	0.999466i	$-0.510405\pi$
$467$	−1.41249	−0.0653622	−0.0326811	+	0.999466i	$0.510405\pi$
$468$	0	0
$469$	−38.4791	−1.77680
$470$	0	0
$471$	−8.45782	−0.389716
$472$	0	0
$473$	−7.02794	−0.323145
$474$	0	0
$475$	13.8372	0.634896
$476$	0	0
$477$	13.8388	0.633635
$478$	0	0
$479$	21.4368	0.979474	0.489737	−	0.871870i	$-0.337093\pi$
$479$	21.4368	0.979474	0.489737	+	0.871870i	$0.337093\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	2.31782	0.105464
$484$	0	0
$485$	−17.2641	−0.783922
$486$	0	0
$487$	11.1898	0.507059	0.253529	−	0.967328i	$-0.418409\pi$
$487$	11.1898	0.507059	0.253529	+	0.967328i	$0.418409\pi$
$488$	0	0
$489$	−4.20019	−0.189939
$490$	0	0
$491$	19.9665	0.901073	0.450537	−	0.892758i	$-0.351233\pi$
$491$	19.9665	0.901073	0.450537	+	0.892758i	$0.351233\pi$
$492$	0	0
$493$	−2.74265	−0.123523
$494$	0	0
$495$	−3.49305	−0.157001
$496$	0	0
$497$	41.0027	1.83922
$498$	0	0
$499$	−4.15156	−0.185849	−0.0929247	−	0.995673i	$-0.529622\pi$
$499$	−4.15156	−0.185849	−0.0929247	+	0.995673i	$0.529622\pi$
$500$	0	0
$501$	18.7121	0.835997
$502$	0	0
$503$	−22.3488	−0.996483	−0.498241	−	0.867038i	$-0.666021\pi$
$503$	−22.3488	−0.996483	−0.498241	+	0.867038i	$0.666021\pi$
$504$	0	0
$505$	−23.6078	−1.05053
$506$	0	0
$507$	0	0
$508$	0	0
$509$	34.6523	1.53594	0.767969	−	0.640487i	$-0.221267\pi$
$509$	34.6523	1.53594	0.767969	+	0.640487i	$0.221267\pi$
$510$	0	0
$511$	24.5872	1.08767
$512$	0	0
$513$	−5.35607	−0.236476
$514$	0	0
$515$	−1.75297	−0.0772452
$516$	0	0
$517$	−16.0393	−0.705406
$518$	0	0
$519$	2.89096	0.126899
$520$	0	0
$521$	25.0186	1.09609	0.548043	−	0.836450i	$-0.315373\pi$
$521$	25.0186	1.09609	0.548043	+	0.836450i	$0.315373\pi$
$522$	0	0
$523$	−27.7484	−1.21335	−0.606676	−	0.794949i	$-0.707497\pi$
$523$	−27.7484	−1.21335	−0.606676	+	0.794949i	$0.707497\pi$
$524$	0	0
$525$	−7.64824	−0.333797
$526$	0	0
$527$	−7.42226	−0.323319
$528$	0	0
$529$	−22.3870	−0.973349
$530$	0	0
$531$	4.35506	0.188994
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−26.4863	−1.14510
$536$	0	0
$537$	−22.8574	−0.986370
$538$	0	0
$539$	−3.96444	−0.170761
$540$	0	0
$541$	−31.4592	−1.35254	−0.676269	−	0.736655i	$-0.736404\pi$
$541$	−31.4592	−1.35254	−0.676269	+	0.736655i	$0.736404\pi$
$542$	0	0
$543$	−8.09042	−0.347193
$544$	0	0
$545$	23.7633	1.01791
$546$	0	0
$547$	−18.1510	−0.776081	−0.388040	−	0.921642i	$-0.626848\pi$
$547$	−18.1510	−0.776081	−0.388040	+	0.921642i	$0.626848\pi$
$548$	0	0
$549$	5.14528	0.219595
$550$	0	0
$551$	−14.4213	−0.614366
$552$	0	0
$553$	3.65822	0.155563
$554$	0	0
$555$	12.1277	0.514791
$556$	0	0
$557$	14.8977	0.631235	0.315618	−	0.948886i	$-0.397788\pi$
$557$	14.8977	0.631235	0.315618	+	0.948886i	$0.397788\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	2.28887	0.0966362
$562$	0	0
$563$	−6.23873	−0.262931	−0.131466	−	0.991321i	$-0.541968\pi$
$563$	−6.23873	−0.262931	−0.131466	+	0.991321i	$0.541968\pi$
$564$	0	0
$565$	10.9106	0.459012
$566$	0	0
$567$	2.96046	0.124327
$568$	0	0
$569$	−15.2268	−0.638342	−0.319171	−	0.947697i	$-0.603405\pi$
$569$	−15.2268	−0.638342	−0.319171	+	0.947697i	$0.603405\pi$
$570$	0	0
$571$	−14.8801	−0.622712	−0.311356	−	0.950293i	$-0.600783\pi$
$571$	−14.8801	−0.622712	−0.311356	+	0.950293i	$0.600783\pi$
$572$	0	0
$573$	−15.3850	−0.642718
$574$	0	0
$575$	−2.02266	−0.0843509
$576$	0	0
$577$	15.6078	0.649762	0.324881	−	0.945755i	$-0.394676\pi$
$577$	15.6078	0.649762	0.324881	+	0.945755i	$0.394676\pi$
$578$	0	0
$579$	−0.347036	−0.0144223
$580$	0	0
$581$	22.7182	0.942510
$582$	0	0
$583$	−31.0961	−1.28787
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−40.2301	−1.66047	−0.830236	−	0.557411i	$-0.811795\pi$
$587$	−40.2301	−1.66047	−0.830236	+	0.557411i	$0.811795\pi$
$588$	0	0
$589$	−39.0274	−1.60809
$590$	0	0
$591$	25.8564	1.06359
$592$	0	0
$593$	37.7225	1.54908	0.774538	−	0.632527i	$-0.217982\pi$
$593$	37.7225	1.54908	0.774538	+	0.632527i	$0.217982\pi$
$594$	0	0
$595$	−4.68779	−0.192181
$596$	0	0
$597$	−7.34574	−0.300641
$598$	0	0
$599$	−30.0518	−1.22788	−0.613942	−	0.789351i	$-0.710417\pi$
$599$	−30.0518	−1.22788	−0.613942	+	0.789351i	$0.710417\pi$
$600$	0	0
$601$	−3.58347	−0.146173	−0.0730863	−	0.997326i	$-0.523285\pi$
$601$	−3.58347	−0.146173	−0.0730863	+	0.997326i	$0.523285\pi$
$602$	0	0
$603$	−12.9977	−0.529307
$604$	0	0
$605$	−9.25074	−0.376096
$606$	0	0
$607$	−23.5679	−0.956590	−0.478295	−	0.878199i	$-0.658745\pi$
$607$	−23.5679	−0.956590	−0.478295	+	0.878199i	$0.658745\pi$
$608$	0	0
$609$	7.97105	0.323003
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−10.2390	−0.413549	−0.206775	−	0.978389i	$-0.566297\pi$
$613$	−10.2390	−0.413549	−0.206775	+	0.978389i	$0.566297\pi$
$614$	0	0
$615$	8.30850	0.335031
$616$	0	0
$617$	37.7804	1.52098	0.760490	−	0.649350i	$-0.224959\pi$
$617$	37.7804	1.52098	0.760490	+	0.649350i	$0.224959\pi$
$618$	0	0
$619$	20.1165	0.808551	0.404275	−	0.914637i	$-0.367524\pi$
$619$	20.1165	0.808551	0.404275	+	0.914637i	$0.367524\pi$
$620$	0	0
$621$	0.782926	0.0314177
$622$	0	0
$623$	−1.46263	−0.0585990
$624$	0	0
$625$	−5.40836	−0.216334
$626$	0	0
$627$	12.0352	0.480641
$628$	0	0
$629$	−7.94682	−0.316861
$630$	0	0
$631$	−23.6225	−0.940395	−0.470198	−	0.882561i	$-0.655817\pi$
$631$	−23.6225	−0.940395	−0.470198	+	0.882561i	$0.655817\pi$
$632$	0	0
$633$	−28.6952	−1.14053
$634$	0	0
$635$	−23.0093	−0.913096
$636$	0	0
$637$	0	0
$638$	0	0
$639$	13.8501	0.547902
$640$	0	0
$641$	3.57370	0.141153	0.0705763	−	0.997506i	$-0.477516\pi$
$641$	3.57370	0.141153	0.0705763	+	0.997506i	$0.477516\pi$
$642$	0	0
$643$	32.6876	1.28907	0.644537	−	0.764573i	$-0.277050\pi$
$643$	32.6876	1.28907	0.644537	+	0.764573i	$0.277050\pi$
$644$	0	0
$645$	4.86201	0.191442
$646$	0	0
$647$	37.0998	1.45855	0.729273	−	0.684223i	$-0.239859\pi$
$647$	37.0998	1.45855	0.729273	+	0.684223i	$0.239859\pi$
$648$	0	0
$649$	−9.78594	−0.384132
$650$	0	0
$651$	21.5716	0.845457
$652$	0	0
$653$	25.9143	1.01410	0.507052	−	0.861915i	$-0.330735\pi$
$653$	25.9143	1.01410	0.507052	+	0.861915i	$0.330735\pi$
$654$	0	0
$655$	9.04601	0.353457
$656$	0	0
$657$	8.30519	0.324016
$658$	0	0
$659$	−14.9861	−0.583776	−0.291888	−	0.956453i	$-0.594283\pi$
$659$	−14.9861	−0.583776	−0.291888	+	0.956453i	$0.594283\pi$
$660$	0	0
$661$	−12.1765	−0.473611	−0.236806	−	0.971557i	$-0.576100\pi$
$661$	−12.1765	−0.473611	−0.236806	+	0.971557i	$0.576100\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−24.6491	−0.955851
$666$	0	0
$667$	2.10803	0.0816234
$668$	0	0
$669$	−2.13698	−0.0826205
$670$	0	0
$671$	−11.5616	−0.446330
$672$	0	0
$673$	38.8564	1.49780	0.748902	−	0.662681i	$-0.230581\pi$
$673$	38.8564	1.49780	0.748902	+	0.662681i	$0.230581\pi$
$674$	0	0
$675$	−2.58347	−0.0994377
$676$	0	0
$677$	−25.3624	−0.974754	−0.487377	−	0.873192i	$-0.662046\pi$
$677$	−25.3624	−0.974754	−0.487377	+	0.873192i	$0.662046\pi$
$678$	0	0
$679$	−32.8780	−1.26174
$680$	0	0
$681$	−18.3860	−0.704554
$682$	0	0
$683$	15.9189	0.609120	0.304560	−	0.952493i	$-0.401491\pi$
$683$	15.9189	0.609120	0.304560	+	0.952493i	$0.401491\pi$
$684$	0	0
$685$	26.9980	1.03154
$686$	0	0
$687$	1.74669	0.0666403
$688$	0	0
$689$	0	0
$690$	0	0
$691$	3.57084	0.135841	0.0679206	−	0.997691i	$-0.478364\pi$
$691$	3.57084	0.135841	0.0679206	+	0.997691i	$0.478364\pi$
$692$	0	0
$693$	−6.65223	−0.252697
$694$	0	0
$695$	9.15289	0.347189
$696$	0	0
$697$	−5.44426	−0.206216
$698$	0	0
$699$	−26.3525	−0.996742
$700$	0	0
$701$	−1.56585	−0.0591414	−0.0295707	−	0.999563i	$-0.509414\pi$
$701$	−1.56585	−0.0591414	−0.0295707	+	0.999563i	$0.509414\pi$
$702$	0	0
$703$	−41.7856	−1.57597
$704$	0	0
$705$	11.0961	0.417905
$706$	0	0
$707$	−44.9592	−1.69086
$708$	0	0
$709$	−40.7930	−1.53201	−0.766006	−	0.642833i	$-0.777759\pi$
$709$	−40.7930	−1.53201	−0.766006	+	0.642833i	$0.777759\pi$
$710$	0	0
$711$	1.23570	0.0463422
$712$	0	0
$713$	5.70485	0.213648
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.36336	0.237644
$718$	0	0
$719$	−0.831053	−0.0309930	−0.0154965	−	0.999880i	$-0.504933\pi$
$719$	−0.831053	−0.0309930	−0.0154965	+	0.999880i	$0.504933\pi$
$720$	0	0
$721$	−3.33839	−0.124328
$722$	0	0
$723$	−9.01862	−0.335406
$724$	0	0
$725$	−6.95601	−0.258340
$726$	0	0
$727$	−9.85844	−0.365629	−0.182815	−	0.983147i	$-0.558521\pi$
$727$	−9.85844	−0.365629	−0.182815	+	0.983147i	$0.558521\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.18590	−0.117835
$732$	0	0
$733$	−11.6467	−0.430180	−0.215090	−	0.976594i	$-0.569004\pi$
$733$	−11.6467	−0.430180	−0.215090	+	0.976594i	$0.569004\pi$
$734$	0	0
$735$	2.74265	0.101164
$736$	0	0
$737$	29.2062	1.07582
$738$	0	0
$739$	28.1132	1.03416	0.517080	−	0.855937i	$-0.327019\pi$
$739$	28.1132	1.03416	0.517080	+	0.855937i	$0.327019\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.8078	0.616619	0.308309	−	0.951286i	$-0.400237\pi$
$743$	16.8078	0.616619	0.308309	+	0.951286i	$0.400237\pi$
$744$	0	0
$745$	−16.4538	−0.602820
$746$	0	0
$747$	7.67389	0.280773
$748$	0	0
$749$	−50.4410	−1.84307
$750$	0	0
$751$	18.4378	0.672806	0.336403	−	0.941718i	$-0.390789\pi$
$751$	18.4378	0.672806	0.336403	+	0.941718i	$0.390789\pi$
$752$	0	0
$753$	−18.2760	−0.666014
$754$	0	0
$755$	3.49305	0.127125
$756$	0	0
$757$	−1.44805	−0.0526303	−0.0263151	−	0.999654i	$-0.508377\pi$
$757$	−1.44805	−0.0526303	−0.0263151	+	0.999654i	$0.508377\pi$
$758$	0	0
$759$	−1.75926	−0.0638570
$760$	0	0
$761$	−14.7121	−0.533314	−0.266657	−	0.963791i	$-0.585919\pi$
$761$	−14.7121	−0.533314	−0.266657	+	0.963791i	$0.585919\pi$
$762$	0	0
$763$	45.2552	1.63835
$764$	0	0
$765$	−1.58347	−0.0572504
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−9.10703	−0.328408	−0.164204	−	0.986426i	$-0.552506\pi$
$769$	−9.10703	−0.328408	−0.164204	+	0.986426i	$0.552506\pi$
$770$	0	0
$771$	26.4263	0.951720
$772$	0	0
$773$	23.2027	0.834543	0.417272	−	0.908782i	$-0.362986\pi$
$773$	23.2027	0.834543	0.417272	+	0.908782i	$0.362986\pi$
$774$	0	0
$775$	−18.8246	−0.676200
$776$	0	0
$777$	23.0961	0.828570
$778$	0	0
$779$	−28.6268	−1.02566
$780$	0	0
$781$	−31.1216	−1.11362
$782$	0	0
$783$	2.69251	0.0962224
$784$	0	0
$785$	−13.1478	−0.469267
$786$	0	0
$787$	−12.7879	−0.455840	−0.227920	−	0.973680i	$-0.573192\pi$
$787$	−12.7879	−0.455840	−0.227920	+	0.973680i	$0.573192\pi$
$788$	0	0
$789$	4.82017	0.171603
$790$	0	0
$791$	20.7783	0.738792
$792$	0	0
$793$	0	0
$794$	0	0
$795$	21.5127	0.762976
$796$	0	0
$797$	40.4330	1.43221	0.716106	−	0.697992i	$-0.245923\pi$
$797$	40.4330	1.43221	0.716106	+	0.697992i	$0.245923\pi$
$798$	0	0
$799$	−7.27091	−0.257226
$800$	0	0
$801$	−0.494055	−0.0174566
$802$	0	0
$803$	−18.6620	−0.658568
$804$	0	0
$805$	3.60310	0.126992
$806$	0	0
$807$	−1.64034	−0.0577426
$808$	0	0
$809$	44.5607	1.56667	0.783335	−	0.621599i	$-0.213517\pi$
$809$	44.5607	1.56667	0.783335	+	0.621599i	$0.213517\pi$
$810$	0	0
$811$	−1.90102	−0.0667537	−0.0333769	−	0.999443i	$-0.510626\pi$
$811$	−1.90102	−0.0667537	−0.0333769	+	0.999443i	$0.510626\pi$
$812$	0	0
$813$	−12.7300	−0.446461
$814$	0	0
$815$	−6.52929	−0.228711
$816$	0	0
$817$	−16.7520	−0.586077
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43.0062	1.50093	0.750463	−	0.660913i	$-0.229831\pi$
$821$	43.0062	1.50093	0.750463	+	0.660913i	$0.229831\pi$
$822$	0	0
$823$	32.1490	1.12064	0.560321	−	0.828275i	$-0.310678\pi$
$823$	32.1490	1.12064	0.560321	+	0.828275i	$0.310678\pi$
$824$	0	0
$825$	5.80512	0.202108
$826$	0	0
$827$	−2.20149	−0.0765533	−0.0382766	−	0.999267i	$-0.512187\pi$
$827$	−2.20149	−0.0765533	−0.0382766	+	0.999267i	$0.512187\pi$
$828$	0	0
$829$	37.7013	1.30942	0.654709	−	0.755881i	$-0.272791\pi$
$829$	37.7013	1.30942	0.654709	+	0.755881i	$0.272791\pi$
$830$	0	0
$831$	24.5008	0.849923
$832$	0	0
$833$	−1.79716	−0.0622679
$834$	0	0
$835$	29.0884	1.00665
$836$	0	0
$837$	7.28657	0.251861
$838$	0	0
$839$	−27.8776	−0.962442	−0.481221	−	0.876599i	$-0.659806\pi$
$839$	−27.8776	−0.962442	−0.481221	+	0.876599i	$0.659806\pi$
$840$	0	0
$841$	−21.7504	−0.750014
$842$	0	0
$843$	1.93853	0.0667665
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−17.6173	−0.605337
$848$	0	0
$849$	8.07853	0.277254
$850$	0	0
$851$	6.10803	0.209381
$852$	0	0
$853$	3.38475	0.115891	0.0579457	−	0.998320i	$-0.481545\pi$
$853$	3.38475	0.115891	0.0579457	+	0.998320i	$0.481545\pi$
$854$	0	0
$855$	−8.32611	−0.284747
$856$	0	0
$857$	0.567882	0.0193985	0.00969925	−	0.999953i	$-0.496913\pi$
$857$	0.567882	0.0193985	0.00969925	+	0.999953i	$0.496913\pi$
$858$	0	0
$859$	−43.1391	−1.47189	−0.735944	−	0.677043i	$-0.763261\pi$
$859$	−43.1391	−1.47189	−0.735944	+	0.677043i	$0.763261\pi$
$860$	0	0
$861$	15.8229	0.539242
$862$	0	0
$863$	21.6466	0.736860	0.368430	−	0.929656i	$-0.379895\pi$
$863$	21.6466	0.736860	0.368430	+	0.929656i	$0.379895\pi$
$864$	0	0
$865$	4.49406	0.152802
$866$	0	0
$867$	−15.9624	−0.542112
$868$	0	0
$869$	−2.77664	−0.0941911
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−11.1057	−0.375872
$874$	0	0
$875$	−34.8998	−1.17983
$876$	0	0
$877$	−42.3494	−1.43004	−0.715019	−	0.699105i	$-0.753582\pi$
$877$	−42.3494	−1.43004	−0.715019	+	0.699105i	$0.753582\pi$
$878$	0	0
$879$	−28.3411	−0.955923
$880$	0	0
$881$	−29.1903	−0.983445	−0.491722	−	0.870752i	$-0.663633\pi$
$881$	−29.1903	−0.983445	−0.491722	+	0.870752i	$0.663633\pi$
$882$	0	0
$883$	41.6461	1.40150	0.700751	−	0.713406i	$-0.252848\pi$
$883$	41.6461	1.40150	0.700751	+	0.713406i	$0.252848\pi$
$884$	0	0
$885$	6.77003	0.227572
$886$	0	0
$887$	40.6471	1.36480	0.682398	−	0.730981i	$-0.260937\pi$
$887$	40.6471	1.36480	0.682398	+	0.730981i	$0.260937\pi$
$888$	0	0
$889$	−43.8193	−1.46965
$890$	0	0
$891$	−2.24703	−0.0752783
$892$	0	0
$893$	−38.2315	−1.27937
$894$	0	0
$895$	−35.5323	−1.18771
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19.6191	0.654335
$900$	0	0
$901$	−14.0965	−0.469622
$902$	0	0
$903$	9.25931	0.308130
$904$	0	0
$905$	−12.5767	−0.418064
$906$	0	0
$907$	37.7561	1.25367	0.626836	−	0.779151i	$-0.284350\pi$
$907$	37.7561	1.25367	0.626836	+	0.779151i	$0.284350\pi$
$908$	0	0
$909$	−15.1866	−0.503706
$910$	0	0
$911$	−42.4249	−1.40560	−0.702801	−	0.711387i	$-0.748067\pi$
$911$	−42.4249	−1.40560	−0.702801	+	0.711387i	$0.748067\pi$
$912$	0	0
$913$	−17.2434	−0.570674
$914$	0	0
$915$	7.99844	0.264420
$916$	0	0
$917$	17.2274	0.568898
$918$	0	0
$919$	49.8346	1.64389	0.821947	−	0.569565i	$-0.192888\pi$
$919$	49.8346	1.64389	0.821947	+	0.569565i	$0.192888\pi$
$920$	0	0
$921$	−30.7025	−1.01168
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−20.1550	−0.662694
$926$	0	0
$927$	−1.12766	−0.0370373
$928$	0	0
$929$	44.1893	1.44980	0.724901	−	0.688853i	$-0.241885\pi$
$929$	44.1893	1.44980	0.724901	+	0.688853i	$0.241885\pi$
$930$	0	0
$931$	−9.44973	−0.309703
$932$	0	0
$933$	−13.7338	−0.449624
$934$	0	0
$935$	3.55810	0.116362
$936$	0	0
$937$	−22.9644	−0.750215	−0.375107	−	0.926981i	$-0.622394\pi$
$937$	−22.9644	−0.750215	−0.375107	+	0.926981i	$0.622394\pi$
$938$	0	0
$939$	32.1886	1.05043
$940$	0	0
$941$	6.09255	0.198611	0.0993057	−	0.995057i	$-0.468338\pi$
$941$	6.09255	0.198611	0.0993057	+	0.995057i	$0.468338\pi$
$942$	0	0
$943$	4.18453	0.136267
$944$	0	0
$945$	4.60209	0.149706
$946$	0	0
$947$	−41.6982	−1.35501	−0.677505	−	0.735518i	$-0.736939\pi$
$947$	−41.6982	−1.35501	−0.677505	+	0.735518i	$0.736939\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−4.54723	−0.147454
$952$	0	0
$953$	35.3850	1.14623	0.573117	−	0.819474i	$-0.305734\pi$
$953$	35.3850	1.14623	0.573117	+	0.819474i	$0.305734\pi$
$954$	0	0
$955$	−23.9163	−0.773914
$956$	0	0
$957$	−6.05014	−0.195573
$958$	0	0
$959$	51.4154	1.66029
$960$	0	0
$961$	22.0941	0.712713
$962$	0	0
$963$	−17.0382	−0.549050
$964$	0	0
$965$	−0.539474	−0.0173663
$966$	0	0
$967$	−4.08684	−0.131424	−0.0657120	−	0.997839i	$-0.520932\pi$
$967$	−4.08684	−0.131424	−0.0657120	+	0.997839i	$0.520932\pi$
$968$	0	0
$969$	5.45581	0.175266
$970$	0	0
$971$	−5.72402	−0.183693	−0.0918463	−	0.995773i	$-0.529277\pi$
$971$	−5.72402	−0.183693	−0.0918463	+	0.995773i	$0.529277\pi$
$972$	0	0
$973$	17.4309	0.558810
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.04498	0.289375	0.144687	−	0.989477i	$-0.453782\pi$
$977$	9.04498	0.289375	0.144687	+	0.989477i	$0.453782\pi$
$978$	0	0
$979$	1.11016	0.0354807
$980$	0	0
$981$	15.2866	0.488063
$982$	0	0
$983$	−7.05061	−0.224879	−0.112440	−	0.993659i	$-0.535867\pi$
$983$	−7.05061	−0.224879	−0.112440	+	0.993659i	$0.535867\pi$
$984$	0	0
$985$	40.1943	1.28070
$986$	0	0
$987$	21.1317	0.672630
$988$	0	0
$989$	2.44873	0.0778650
$990$	0	0
$991$	−19.9292	−0.633072	−0.316536	−	0.948580i	$-0.602520\pi$
$991$	−19.9292	−0.633072	−0.316536	+	0.948580i	$0.602520\pi$
$992$	0	0
$993$	14.7899	0.469344
$994$	0	0
$995$	−11.4191	−0.362010
$996$	0	0
$997$	−22.1395	−0.701164	−0.350582	−	0.936532i	$-0.614016\pi$
$997$	−22.1395	−0.701164	−0.350582	+	0.936532i	$0.614016\pi$
$998$	0	0
$999$	7.80155	0.246830

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.be.1.3		4
4.3	odd	2		8112.2.a.cs.1.3		4
13.5	odd	4		4056.2.c.p.337.2		8
13.6	odd	12		312.2.bf.b.49.1	✓	8
13.8	odd	4		4056.2.c.p.337.7		8
13.11	odd	12		312.2.bf.b.121.4	yes	8
13.12	even	2		4056.2.a.bd.1.2		4
39.11	even	12		936.2.bi.c.433.1		8
39.32	even	12		936.2.bi.c.361.4		8
52.11	even	12		624.2.bv.g.433.4		8
52.19	even	12		624.2.bv.g.49.1		8
52.51	odd	2		8112.2.a.cq.1.2		4
156.11	odd	12		1872.2.by.m.433.1		8
156.71	odd	12		1872.2.by.m.1297.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.bf.b.49.1	✓	8	13.6	odd	12
312.2.bf.b.121.4	yes	8	13.11	odd	12
624.2.bv.g.49.1		8	52.19	even	12
624.2.bv.g.433.4		8	52.11	even	12
936.2.bi.c.361.4		8	39.32	even	12
936.2.bi.c.433.1		8	39.11	even	12
1872.2.by.m.433.1		8	156.11	odd	12
1872.2.by.m.1297.4		8	156.71	odd	12
4056.2.a.bd.1.2		4	13.12	even	2
4056.2.a.be.1.3		4	1.1	even	1	trivial
4056.2.c.p.337.2		8	13.5	odd	4
4056.2.c.p.337.7		8	13.8	odd	4
8112.2.a.cq.1.2		4	52.51	odd	2
8112.2.a.cs.1.3		4	4.3	odd	2

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:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.55452 q^{5} +2.96046 q^{7} +1.00000 q^{9} -2.24703 q^{11} +1.55452 q^{15} -1.01862 q^{17} -5.35607 q^{19} +2.96046 q^{21} +0.782926 q^{23} -2.58347 q^{25} +1.00000 q^{27} +2.69251 q^{29} +7.28657 q^{31} -2.24703 q^{33} +4.60209 q^{35} +7.80155 q^{37} +5.34474 q^{41} +3.12766 q^{43} +1.55452 q^{45} +7.13799 q^{47} +1.76430 q^{49} -1.01862 q^{51} +13.8388 q^{53} -3.49305 q^{55} -5.35607 q^{57} +4.35506 q^{59} +5.14528 q^{61} +2.96046 q^{63} -12.9977 q^{67} +0.782926 q^{69} +13.8501 q^{71} +8.30519 q^{73} -2.58347 q^{75} -6.65223 q^{77} +1.23570 q^{79} +1.00000 q^{81} +7.67389 q^{83} -1.58347 q^{85} +2.69251 q^{87} -0.494055 q^{89} +7.28657 q^{93} -8.32611 q^{95} -11.1057 q^{97} -2.24703 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} + 2 q^{7} + 4 q^{9} + 10 q^{11} + 4 q^{15} + 12 q^{17} + 2 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 4 q^{27} - 6 q^{29} + 20 q^{31} + 10 q^{33} - 10 q^{35} + 10 q^{37} + 6 q^{41}+ \cdots + 10 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 + 2 * q^7 + 4 * q^9 + 10 * q^11 + 4 * q^15 + 12 * q^17 + 2 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 + 4 * q^27 - 6 * q^29 + 20 * q^31 + 10 * q^33 - 10 * q^35 + 10 * q^37 + 6 * q^41 - 8 * q^43 + 4 * q^45 + 14 * q^47 + 18 * q^49 + 12 * q^51 + 2 * q^53 + 10 * q^55 + 2 * q^57 + 8 * q^59 - 2 * q^61 + 2 * q^63 - 10 * q^67 - 2 * q^69 - 6 * q^71 + 8 * q^73 + 2 * q^75 - 28 * q^77 - 6 * q^79 + 4 * q^81 + 30 * q^83 + 6 * q^85 - 6 * q^87 + 36 * q^89 + 20 * q^93 - 34 * q^95 - 12 * q^97 + 10 * q^99

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