LMFDB - Embedded newform 8044.2.a.a.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8044,2,Mod(1,8044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8044 = 2^{2} \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8044.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2316633859$
Analytic rank:	$1$
Dimension:	$80$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64514 q^{3} -1.03664 q^{5} +1.03871 q^{7} +3.99678 q^{9} +O(q^{10})$	$q-2.64514 q^{3} -1.03664 q^{5} +1.03871 q^{7} +3.99678 q^{9} +4.82011 q^{11} -0.130242 q^{13} +2.74206 q^{15} +5.14015 q^{17} -4.17741 q^{19} -2.74753 q^{21} +2.81926 q^{23} -3.92538 q^{25} -2.63663 q^{27} +3.37027 q^{29} -7.55069 q^{31} -12.7499 q^{33} -1.07676 q^{35} -5.03799 q^{37} +0.344508 q^{39} -4.44636 q^{41} -12.8747 q^{43} -4.14323 q^{45} -1.91931 q^{47} -5.92109 q^{49} -13.5964 q^{51} +7.67644 q^{53} -4.99672 q^{55} +11.0498 q^{57} -2.14569 q^{59} +7.38953 q^{61} +4.15148 q^{63} +0.135014 q^{65} +12.6681 q^{67} -7.45736 q^{69} +2.56548 q^{71} -4.58534 q^{73} +10.3832 q^{75} +5.00668 q^{77} +8.83018 q^{79} -5.01608 q^{81} -5.67776 q^{83} -5.32849 q^{85} -8.91485 q^{87} -4.47982 q^{89} -0.135283 q^{91} +19.9727 q^{93} +4.33047 q^{95} +9.41746 q^{97} +19.2649 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.64514	−1.52717	−0.763587	−	0.645705i	$-0.776564\pi$
$3$	−2.64514	−1.52717	−0.763587	+	0.645705i	$0.776564\pi$
$4$	0	0
$5$	−1.03664	−0.463600	−0.231800	−	0.972763i	$-0.574461\pi$
$5$	−1.03664	−0.463600	−0.231800	+	0.972763i	$0.574461\pi$
$6$	0	0
$7$	1.03871	0.392594	0.196297	−	0.980545i	$-0.437108\pi$
$7$	1.03871	0.392594	0.196297	+	0.980545i	$0.437108\pi$
$8$	0	0
$9$	3.99678	1.33226
$10$	0	0
$11$	4.82011	1.45332	0.726659	−	0.686998i	$-0.241072\pi$
$11$	4.82011	1.45332	0.726659	+	0.686998i	$0.241072\pi$
$12$	0	0
$13$	−0.130242	−0.0361226	−0.0180613	−	0.999837i	$-0.505749\pi$
$13$	−0.130242	−0.0361226	−0.0180613	+	0.999837i	$0.505749\pi$
$14$	0	0
$15$	2.74206	0.707998
$16$	0	0
$17$	5.14015	1.24667	0.623335	−	0.781955i	$-0.285777\pi$
$17$	5.14015	1.24667	0.623335	+	0.781955i	$0.285777\pi$
$18$	0	0
$19$	−4.17741	−0.958362	−0.479181	−	0.877716i	$-0.659066\pi$
$19$	−4.17741	−0.958362	−0.479181	+	0.877716i	$0.659066\pi$
$20$	0	0
$21$	−2.74753	−0.599559
$22$	0	0
$23$	2.81926	0.587857	0.293929	−	0.955827i	$-0.405037\pi$
$23$	2.81926	0.587857	0.293929	+	0.955827i	$0.405037\pi$
$24$	0	0
$25$	−3.92538	−0.785075
$26$	0	0
$27$	−2.63663	−0.507420
$28$	0	0
$29$	3.37027	0.625843	0.312922	−	0.949779i	$-0.398692\pi$
$29$	3.37027	0.625843	0.312922	+	0.949779i	$0.398692\pi$
$30$	0	0
$31$	−7.55069	−1.35614	−0.678072	−	0.734995i	$-0.737184\pi$
$31$	−7.55069	−1.35614	−0.678072	+	0.734995i	$0.737184\pi$
$32$	0	0
$33$	−12.7499	−2.21947
$34$	0	0
$35$	−1.07676	−0.182006
$36$	0	0
$37$	−5.03799	−0.828240	−0.414120	−	0.910222i	$-0.635911\pi$
$37$	−5.03799	−0.828240	−0.414120	+	0.910222i	$0.635911\pi$
$38$	0	0
$39$	0.344508	0.0551655
$40$	0	0
$41$	−4.44636	−0.694405	−0.347202	−	0.937790i	$-0.612868\pi$
$41$	−4.44636	−0.694405	−0.347202	+	0.937790i	$0.612868\pi$
$42$	0	0
$43$	−12.8747	−1.96337	−0.981684	−	0.190518i	$-0.938983\pi$
$43$	−12.8747	−1.96337	−0.981684	+	0.190518i	$0.938983\pi$
$44$	0	0
$45$	−4.14323	−0.617636
$46$	0	0
$47$	−1.91931	−0.279960	−0.139980	−	0.990154i	$-0.544704\pi$
$47$	−1.91931	−0.279960	−0.139980	+	0.990154i	$0.544704\pi$
$48$	0	0
$49$	−5.92109	−0.845870
$50$	0	0
$51$	−13.5964	−1.90388
$52$	0	0
$53$	7.67644	1.05444	0.527220	−	0.849729i	$-0.323234\pi$
$53$	7.67644	1.05444	0.527220	+	0.849729i	$0.323234\pi$
$54$	0	0
$55$	−4.99672	−0.673758
$56$	0	0
$57$	11.0498	1.46359
$58$	0	0
$59$	−2.14569	−0.279344	−0.139672	−	0.990198i	$-0.544605\pi$
$59$	−2.14569	−0.279344	−0.139672	+	0.990198i	$0.544605\pi$
$60$	0	0
$61$	7.38953	0.946132	0.473066	−	0.881027i	$-0.343147\pi$
$61$	7.38953	0.946132	0.473066	+	0.881027i	$0.343147\pi$
$62$	0	0
$63$	4.15148	0.523037
$64$	0	0
$65$	0.135014	0.0167464
$66$	0	0
$67$	12.6681	1.54766	0.773829	−	0.633394i	$-0.218339\pi$
$67$	12.6681	1.54766	0.773829	+	0.633394i	$0.218339\pi$
$68$	0	0
$69$	−7.45736	−0.897760
$70$	0	0
$71$	2.56548	0.304466	0.152233	−	0.988345i	$-0.451354\pi$
$71$	2.56548	0.304466	0.152233	+	0.988345i	$0.451354\pi$
$72$	0	0
$73$	−4.58534	−0.536674	−0.268337	−	0.963325i	$-0.586474\pi$
$73$	−4.58534	−0.536674	−0.268337	+	0.963325i	$0.586474\pi$
$74$	0	0
$75$	10.3832	1.19895
$76$	0	0
$77$	5.00668	0.570564
$78$	0	0
$79$	8.83018	0.993473	0.496737	−	0.867901i	$-0.334532\pi$
$79$	8.83018	0.993473	0.496737	+	0.867901i	$0.334532\pi$
$80$	0	0
$81$	−5.01608	−0.557342
$82$	0	0
$83$	−5.67776	−0.623215	−0.311608	−	0.950211i	$-0.600867\pi$
$83$	−5.67776	−0.623215	−0.311608	+	0.950211i	$0.600867\pi$
$84$	0	0
$85$	−5.32849	−0.577955
$86$	0	0
$87$	−8.91485	−0.955772
$88$	0	0
$89$	−4.47982	−0.474860	−0.237430	−	0.971405i	$-0.576305\pi$
$89$	−4.47982	−0.474860	−0.237430	+	0.971405i	$0.576305\pi$
$90$	0	0
$91$	−0.135283	−0.0141815
$92$	0	0
$93$	19.9727	2.07107
$94$	0	0
$95$	4.33047	0.444297
$96$	0	0
$97$	9.41746	0.956199	0.478099	−	0.878306i	$-0.341326\pi$
$97$	9.41746	0.956199	0.478099	+	0.878306i	$0.341326\pi$
$98$	0	0
$99$	19.2649	1.93620
$100$	0	0
$101$	5.64185	0.561385	0.280693	−	0.959798i	$-0.409436\pi$
$101$	5.64185	0.561385	0.280693	+	0.959798i	$0.409436\pi$
$102$	0	0
$103$	−4.21509	−0.415325	−0.207662	−	0.978201i	$-0.566586\pi$
$103$	−4.21509	−0.415325	−0.207662	+	0.978201i	$0.566586\pi$
$104$	0	0
$105$	2.84820	0.277955
$106$	0	0
$107$	−6.19500	−0.598893	−0.299447	−	0.954113i	$-0.596802\pi$
$107$	−6.19500	−0.598893	−0.299447	+	0.954113i	$0.596802\pi$
$108$	0	0
$109$	−12.3407	−1.18203	−0.591013	−	0.806662i	$-0.701272\pi$
$109$	−12.3407	−1.18203	−0.591013	+	0.806662i	$0.701272\pi$
$110$	0	0
$111$	13.3262	1.26487
$112$	0	0
$113$	8.18686	0.770155	0.385077	−	0.922884i	$-0.374175\pi$
$113$	8.18686	0.770155	0.385077	+	0.922884i	$0.374175\pi$
$114$	0	0
$115$	−2.92256	−0.272530
$116$	0	0
$117$	−0.520549	−0.0481247
$118$	0	0
$119$	5.33910	0.489435
$120$	0	0
$121$	12.2335	1.11213
$122$	0	0
$123$	11.7613	1.06048
$124$	0	0
$125$	9.25241	0.827560
$126$	0	0
$127$	−12.4368	−1.10359	−0.551795	−	0.833980i	$-0.686057\pi$
$127$	−12.4368	−1.10359	−0.551795	+	0.833980i	$0.686057\pi$
$128$	0	0
$129$	34.0553	2.99840
$130$	0	0
$131$	16.4766	1.43956	0.719782	−	0.694200i	$-0.244242\pi$
$131$	16.4766	1.43956	0.719782	+	0.694200i	$0.244242\pi$
$132$	0	0
$133$	−4.33909	−0.376247
$134$	0	0
$135$	2.73324	0.235240
$136$	0	0
$137$	−5.15093	−0.440074	−0.220037	−	0.975492i	$-0.570618\pi$
$137$	−5.15093	−0.440074	−0.220037	+	0.975492i	$0.570618\pi$
$138$	0	0
$139$	18.4100	1.56152	0.780758	−	0.624834i	$-0.214833\pi$
$139$	18.4100	1.56152	0.780758	+	0.624834i	$0.214833\pi$
$140$	0	0
$141$	5.07684	0.427547
$142$	0	0
$143$	−0.627780	−0.0524976
$144$	0	0
$145$	−3.49376	−0.290141
$146$	0	0
$147$	15.6621	1.29179
$148$	0	0
$149$	6.51573	0.533789	0.266895	−	0.963726i	$-0.414002\pi$
$149$	6.51573	0.533789	0.266895	+	0.963726i	$0.414002\pi$
$150$	0	0
$151$	−10.3796	−0.844679	−0.422340	−	0.906438i	$-0.638791\pi$
$151$	−10.3796	−0.844679	−0.422340	+	0.906438i	$0.638791\pi$
$152$	0	0
$153$	20.5441	1.66089
$154$	0	0
$155$	7.82735	0.628708
$156$	0	0
$157$	−19.0169	−1.51771	−0.758856	−	0.651259i	$-0.774241\pi$
$157$	−19.0169	−1.51771	−0.758856	+	0.651259i	$0.774241\pi$
$158$	0	0
$159$	−20.3053	−1.61031
$160$	0	0
$161$	2.92838	0.230789
$162$	0	0
$163$	−22.3837	−1.75322	−0.876612	−	0.481198i	$-0.840202\pi$
$163$	−22.3837	−1.75322	−0.876612	+	0.481198i	$0.840202\pi$
$164$	0	0
$165$	13.2170	1.02895
$166$	0	0
$167$	−2.06782	−0.160013	−0.0800064	−	0.996794i	$-0.525494\pi$
$167$	−2.06782	−0.160013	−0.0800064	+	0.996794i	$0.525494\pi$
$168$	0	0
$169$	−12.9830	−0.998695
$170$	0	0
$171$	−16.6962	−1.27679
$172$	0	0
$173$	−16.0682	−1.22164	−0.610821	−	0.791769i	$-0.709161\pi$
$173$	−16.0682	−1.22164	−0.610821	+	0.791769i	$0.709161\pi$
$174$	0	0
$175$	−4.07731	−0.308216
$176$	0	0
$177$	5.67565	0.426608
$178$	0	0
$179$	15.2938	1.14311	0.571557	−	0.820562i	$-0.306339\pi$
$179$	15.2938	1.14311	0.571557	+	0.820562i	$0.306339\pi$
$180$	0	0
$181$	0.446664	0.0332003	0.0166001	−	0.999862i	$-0.494716\pi$
$181$	0.446664	0.0332003	0.0166001	+	0.999862i	$0.494716\pi$
$182$	0	0
$183$	−19.5464	−1.44491
$184$	0	0
$185$	5.22258	0.383972
$186$	0	0
$187$	24.7761	1.81181
$188$	0	0
$189$	−2.73869	−0.199210
$190$	0	0
$191$	−18.5635	−1.34321	−0.671604	−	0.740910i	$-0.734394\pi$
$191$	−18.5635	−1.34321	−0.671604	+	0.740910i	$0.734394\pi$
$192$	0	0
$193$	8.00319	0.576082	0.288041	−	0.957618i	$-0.406996\pi$
$193$	8.00319	0.576082	0.288041	+	0.957618i	$0.406996\pi$
$194$	0	0
$195$	−0.357131	−0.0255747
$196$	0	0
$197$	15.4510	1.10084	0.550420	−	0.834888i	$-0.314468\pi$
$197$	15.4510	1.10084	0.550420	+	0.834888i	$0.314468\pi$
$198$	0	0
$199$	−23.5016	−1.66598	−0.832992	−	0.553285i	$-0.813374\pi$
$199$	−23.5016	−1.66598	−0.832992	+	0.553285i	$0.813374\pi$
$200$	0	0
$201$	−33.5090	−2.36354
$202$	0	0
$203$	3.50072	0.245702
$204$	0	0
$205$	4.60928	0.321926
$206$	0	0
$207$	11.2680	0.783179
$208$	0	0
$209$	−20.1356	−1.39281
$210$	0	0
$211$	17.0441	1.17336	0.586682	−	0.809817i	$-0.300434\pi$
$211$	17.0441	1.17336	0.586682	+	0.809817i	$0.300434\pi$
$212$	0	0
$213$	−6.78606	−0.464973
$214$	0	0
$215$	13.3464	0.910217
$216$	0	0
$217$	−7.84295	−0.532414
$218$	0	0
$219$	12.1289	0.819595
$220$	0	0
$221$	−0.669463	−0.0450329
$222$	0	0
$223$	−13.8764	−0.929231	−0.464615	−	0.885513i	$-0.653807\pi$
$223$	−13.8764	−0.929231	−0.464615	+	0.885513i	$0.653807\pi$
$224$	0	0
$225$	−15.6889	−1.04593
$226$	0	0
$227$	12.8280	0.851426	0.425713	−	0.904858i	$-0.360023\pi$
$227$	12.8280	0.851426	0.425713	+	0.904858i	$0.360023\pi$
$228$	0	0
$229$	9.91226	0.655020	0.327510	−	0.944848i	$-0.393790\pi$
$229$	9.91226	0.655020	0.327510	+	0.944848i	$0.393790\pi$
$230$	0	0
$231$	−13.2434	−0.871350
$232$	0	0
$233$	−3.36717	−0.220590	−0.110295	−	0.993899i	$-0.535180\pi$
$233$	−3.36717	−0.220590	−0.110295	+	0.993899i	$0.535180\pi$
$234$	0	0
$235$	1.98963	0.129789
$236$	0	0
$237$	−23.3571	−1.51721
$238$	0	0
$239$	−14.0603	−0.909484	−0.454742	−	0.890623i	$-0.650269\pi$
$239$	−14.0603	−0.909484	−0.454742	+	0.890623i	$0.650269\pi$
$240$	0	0
$241$	−3.60154	−0.231996	−0.115998	−	0.993249i	$-0.537007\pi$
$241$	−3.60154	−0.231996	−0.115998	+	0.993249i	$0.537007\pi$
$242$	0	0
$243$	21.1781	1.35858
$244$	0	0
$245$	6.13804	0.392145
$246$	0	0
$247$	0.544073	0.0346185
$248$	0	0
$249$	15.0185	0.951758
$250$	0	0
$251$	−21.1679	−1.33610	−0.668052	−	0.744115i	$-0.732872\pi$
$251$	−21.1679	−1.33610	−0.668052	+	0.744115i	$0.732872\pi$
$252$	0	0
$253$	13.5892	0.854343
$254$	0	0
$255$	14.0946	0.882639
$256$	0	0
$257$	−1.67864	−0.104711	−0.0523555	−	0.998629i	$-0.516673\pi$
$257$	−1.67864	−0.104711	−0.0523555	+	0.998629i	$0.516673\pi$
$258$	0	0
$259$	−5.23299	−0.325162
$260$	0	0
$261$	13.4702	0.833787
$262$	0	0
$263$	27.2634	1.68113	0.840566	−	0.541709i	$-0.182223\pi$
$263$	27.2634	1.68113	0.840566	+	0.541709i	$0.182223\pi$
$264$	0	0
$265$	−7.95771	−0.488838
$266$	0	0
$267$	11.8498	0.725195
$268$	0	0
$269$	13.2364	0.807038	0.403519	−	0.914971i	$-0.367787\pi$
$269$	13.2364	0.807038	0.403519	+	0.914971i	$0.367787\pi$
$270$	0	0
$271$	−15.0830	−0.916229	−0.458115	−	0.888893i	$-0.651475\pi$
$271$	−15.0830	−0.916229	−0.458115	+	0.888893i	$0.651475\pi$
$272$	0	0
$273$	0.357843	0.0216576
$274$	0	0
$275$	−18.9208	−1.14096
$276$	0	0
$277$	21.1939	1.27342	0.636710	−	0.771103i	$-0.280295\pi$
$277$	21.1939	1.27342	0.636710	+	0.771103i	$0.280295\pi$
$278$	0	0
$279$	−30.1785	−1.80674
$280$	0	0
$281$	19.4429	1.15986	0.579932	−	0.814665i	$-0.303079\pi$
$281$	19.4429	1.15986	0.579932	+	0.814665i	$0.303079\pi$
$282$	0	0
$283$	−23.0174	−1.36824	−0.684122	−	0.729367i	$-0.739815\pi$
$283$	−23.0174	−1.36824	−0.684122	+	0.729367i	$0.739815\pi$
$284$	0	0
$285$	−11.4547	−0.678518
$286$	0	0
$287$	−4.61846	−0.272619
$288$	0	0
$289$	9.42112	0.554184
$290$	0	0
$291$	−24.9105	−1.46028
$292$	0	0
$293$	1.19725	0.0699439	0.0349719	−	0.999388i	$-0.488866\pi$
$293$	1.19725	0.0699439	0.0349719	+	0.999388i	$0.488866\pi$
$294$	0	0
$295$	2.22430	0.129504
$296$	0	0
$297$	−12.7089	−0.737443
$298$	0	0
$299$	−0.367186	−0.0212349
$300$	0	0
$301$	−13.3730	−0.770806
$302$	0	0
$303$	−14.9235	−0.857333
$304$	0	0
$305$	−7.66029	−0.438627
$306$	0	0
$307$	−29.2103	−1.66712	−0.833559	−	0.552430i	$-0.813701\pi$
$307$	−29.2103	−1.66712	−0.833559	+	0.552430i	$0.813701\pi$
$308$	0	0
$309$	11.1495	0.634273
$310$	0	0
$311$	−10.1535	−0.575754	−0.287877	−	0.957667i	$-0.592949\pi$
$311$	−10.1535	−0.575754	−0.287877	+	0.957667i	$0.592949\pi$
$312$	0	0
$313$	9.03192	0.510515	0.255257	−	0.966873i	$-0.417840\pi$
$313$	9.03192	0.510515	0.255257	+	0.966873i	$0.417840\pi$
$314$	0	0
$315$	−4.30359	−0.242480
$316$	0	0
$317$	11.0609	0.621240	0.310620	−	0.950534i	$-0.399463\pi$
$317$	11.0609	0.621240	0.310620	+	0.950534i	$0.399463\pi$
$318$	0	0
$319$	16.2451	0.909550
$320$	0	0
$321$	16.3867	0.914614
$322$	0	0
$323$	−21.4725	−1.19476
$324$	0	0
$325$	0.511248	0.0283590
$326$	0	0
$327$	32.6430	1.80516
$328$	0	0
$329$	−1.99360	−0.109910
$330$	0	0
$331$	−28.3423	−1.55783	−0.778917	−	0.627128i	$-0.784230\pi$
$331$	−28.3423	−1.55783	−0.778917	+	0.627128i	$0.784230\pi$
$332$	0	0
$333$	−20.1358	−1.10343
$334$	0	0
$335$	−13.1323	−0.717494
$336$	0	0
$337$	18.9860	1.03423	0.517116	−	0.855915i	$-0.327006\pi$
$337$	18.9860	1.03423	0.517116	+	0.855915i	$0.327006\pi$
$338$	0	0
$339$	−21.6554	−1.17616
$340$	0	0
$341$	−36.3952	−1.97091
$342$	0	0
$343$	−13.4212	−0.724677
$344$	0	0
$345$	7.73060	0.416201
$346$	0	0
$347$	−2.46915	−0.132551	−0.0662754	−	0.997801i	$-0.521112\pi$
$347$	−2.46915	−0.132551	−0.0662754	+	0.997801i	$0.521112\pi$
$348$	0	0
$349$	−19.6209	−1.05028	−0.525141	−	0.851015i	$-0.675987\pi$
$349$	−19.6209	−1.05028	−0.525141	+	0.851015i	$0.675987\pi$
$350$	0	0
$351$	0.343400	0.0183293
$352$	0	0
$353$	33.2553	1.77000	0.885000	−	0.465591i	$-0.154158\pi$
$353$	33.2553	1.77000	0.885000	+	0.465591i	$0.154158\pi$
$354$	0	0
$355$	−2.65948	−0.141151
$356$	0	0
$357$	−14.1227	−0.747452
$358$	0	0
$359$	14.5442	0.767615	0.383807	−	0.923413i	$-0.374613\pi$
$359$	14.5442	0.767615	0.383807	+	0.923413i	$0.374613\pi$
$360$	0	0
$361$	−1.54929	−0.0815414
$362$	0	0
$363$	−32.3593	−1.69842
$364$	0	0
$365$	4.75335	0.248802
$366$	0	0
$367$	−3.56897	−0.186299	−0.0931495	−	0.995652i	$-0.529693\pi$
$367$	−3.56897	−0.186299	−0.0931495	+	0.995652i	$0.529693\pi$
$368$	0	0
$369$	−17.7711	−0.925128
$370$	0	0
$371$	7.97357	0.413967
$372$	0	0
$373$	−5.69138	−0.294689	−0.147344	−	0.989085i	$-0.547073\pi$
$373$	−5.69138	−0.294689	−0.147344	+	0.989085i	$0.547073\pi$
$374$	0	0
$375$	−24.4739	−1.26383
$376$	0	0
$377$	−0.438950	−0.0226071
$378$	0	0
$379$	32.4535	1.66702	0.833512	−	0.552502i	$-0.186327\pi$
$379$	32.4535	1.66702	0.833512	+	0.552502i	$0.186327\pi$
$380$	0	0
$381$	32.8972	1.68537
$382$	0	0
$383$	−30.7791	−1.57274	−0.786370	−	0.617756i	$-0.788042\pi$
$383$	−30.7791	−1.57274	−0.786370	+	0.617756i	$0.788042\pi$
$384$	0	0
$385$	−5.19012	−0.264513
$386$	0	0
$387$	−51.4572	−2.61572
$388$	0	0
$389$	21.2263	1.07621	0.538107	−	0.842876i	$-0.319140\pi$
$389$	21.2263	1.07621	0.538107	+	0.842876i	$0.319140\pi$
$390$	0	0
$391$	14.4914	0.732863
$392$	0	0
$393$	−43.5829	−2.19846
$394$	0	0
$395$	−9.15373	−0.460574
$396$	0	0
$397$	10.8491	0.544502	0.272251	−	0.962226i	$-0.412232\pi$
$397$	10.8491	0.544502	0.272251	+	0.962226i	$0.412232\pi$
$398$	0	0
$399$	11.4775	0.574595
$400$	0	0
$401$	−29.5827	−1.47729	−0.738644	−	0.674095i	$-0.764534\pi$
$401$	−29.5827	−1.47729	−0.738644	+	0.674095i	$0.764534\pi$
$402$	0	0
$403$	0.983417	0.0489875
$404$	0	0
$405$	5.19987	0.258384
$406$	0	0
$407$	−24.2837	−1.20370
$408$	0	0
$409$	11.9993	0.593325	0.296662	−	0.954982i	$-0.404126\pi$
$409$	11.9993	0.593325	0.296662	+	0.954982i	$0.404126\pi$
$410$	0	0
$411$	13.6250	0.672070
$412$	0	0
$413$	−2.22874	−0.109669
$414$	0	0
$415$	5.88580	0.288922
$416$	0	0
$417$	−48.6971	−2.38471
$418$	0	0
$419$	−26.4778	−1.29353	−0.646763	−	0.762691i	$-0.723878\pi$
$419$	−26.4778	−1.29353	−0.646763	+	0.762691i	$0.723878\pi$
$420$	0	0
$421$	−20.4735	−0.997817	−0.498909	−	0.866655i	$-0.666266\pi$
$421$	−20.4735	−0.997817	−0.498909	+	0.866655i	$0.666266\pi$
$422$	0	0
$423$	−7.67106	−0.372980
$424$	0	0
$425$	−20.1770	−0.978729
$426$	0	0
$427$	7.67555	0.371446
$428$	0	0
$429$	1.66057	0.0801730
$430$	0	0
$431$	−11.0611	−0.532795	−0.266398	−	0.963863i	$-0.585833\pi$
$431$	−11.0611	−0.532795	−0.266398	+	0.963863i	$0.585833\pi$
$432$	0	0
$433$	2.78589	0.133882	0.0669408	−	0.997757i	$-0.478676\pi$
$433$	2.78589	0.133882	0.0669408	+	0.997757i	$0.478676\pi$
$434$	0	0
$435$	9.24149	0.443096
$436$	0	0
$437$	−11.7772	−0.563380
$438$	0	0
$439$	−5.40047	−0.257751	−0.128875	−	0.991661i	$-0.541137\pi$
$439$	−5.40047	−0.257751	−0.128875	+	0.991661i	$0.541137\pi$
$440$	0	0
$441$	−23.6653	−1.12692
$442$	0	0
$443$	−21.3161	−1.01276	−0.506380	−	0.862311i	$-0.669017\pi$
$443$	−21.3161	−1.01276	−0.506380	+	0.862311i	$0.669017\pi$
$444$	0	0
$445$	4.64397	0.220145
$446$	0	0
$447$	−17.2350	−0.815189
$448$	0	0
$449$	−24.5314	−1.15771	−0.578853	−	0.815432i	$-0.696499\pi$
$449$	−24.5314	−1.15771	−0.578853	+	0.815432i	$0.696499\pi$
$450$	0	0
$451$	−21.4320	−1.00919
$452$	0	0
$453$	27.4555	1.28997
$454$	0	0
$455$	0.140240	0.00657454
$456$	0	0
$457$	10.4517	0.488910	0.244455	−	0.969661i	$-0.421391\pi$
$457$	10.4517	0.488910	0.244455	+	0.969661i	$0.421391\pi$
$458$	0	0
$459$	−13.5527	−0.632585
$460$	0	0
$461$	12.8334	0.597712	0.298856	−	0.954298i	$-0.403395\pi$
$461$	12.8334	0.597712	0.298856	+	0.954298i	$0.403395\pi$
$462$	0	0
$463$	−28.5668	−1.32761	−0.663806	−	0.747905i	$-0.731060\pi$
$463$	−28.5668	−1.32761	−0.663806	+	0.747905i	$0.731060\pi$
$464$	0	0
$465$	−20.7045	−0.960147
$466$	0	0
$467$	−5.85088	−0.270746	−0.135373	−	0.990795i	$-0.543223\pi$
$467$	−5.85088	−0.270746	−0.135373	+	0.990795i	$0.543223\pi$
$468$	0	0
$469$	13.1585	0.607601
$470$	0	0
$471$	50.3023	2.31781
$472$	0	0
$473$	−62.0573	−2.85340
$474$	0	0
$475$	16.3979	0.752387
$476$	0	0
$477$	30.6811	1.40479
$478$	0	0
$479$	−18.6832	−0.853658	−0.426829	−	0.904332i	$-0.640369\pi$
$479$	−18.6832	−0.853658	−0.426829	+	0.904332i	$0.640369\pi$
$480$	0	0
$481$	0.656157	0.0299182
$482$	0	0
$483$	−7.74600	−0.352455
$484$	0	0
$485$	−9.76252	−0.443293
$486$	0	0
$487$	27.4446	1.24364	0.621818	−	0.783162i	$-0.286394\pi$
$487$	27.4446	1.24364	0.621818	+	0.783162i	$0.286394\pi$
$488$	0	0
$489$	59.2080	2.67748
$490$	0	0
$491$	5.71864	0.258079	0.129039	−	0.991639i	$-0.458811\pi$
$491$	5.71864	0.258079	0.129039	+	0.991639i	$0.458811\pi$
$492$	0	0
$493$	17.3237	0.780220
$494$	0	0
$495$	−19.9708	−0.897621
$496$	0	0
$497$	2.66478	0.119532
$498$	0	0
$499$	12.8880	0.576945	0.288472	−	0.957488i	$-0.406853\pi$
$499$	12.8880	0.576945	0.288472	+	0.957488i	$0.406853\pi$
$500$	0	0
$501$	5.46968	0.244367
$502$	0	0
$503$	−6.52922	−0.291123	−0.145562	−	0.989349i	$-0.546499\pi$
$503$	−6.52922	−0.291123	−0.145562	+	0.989349i	$0.546499\pi$
$504$	0	0
$505$	−5.84857	−0.260258
$506$	0	0
$507$	34.3420	1.52518
$508$	0	0
$509$	−22.9398	−1.01679	−0.508395	−	0.861124i	$-0.669761\pi$
$509$	−22.9398	−1.01679	−0.508395	+	0.861124i	$0.669761\pi$
$510$	0	0
$511$	−4.76282	−0.210695
$512$	0	0
$513$	11.0143	0.486293
$514$	0	0
$515$	4.36953	0.192544
$516$	0	0
$517$	−9.25128	−0.406871
$518$	0	0
$519$	42.5027	1.86566
$520$	0	0
$521$	21.3856	0.936918	0.468459	−	0.883485i	$-0.344809\pi$
$521$	21.3856	0.936918	0.468459	+	0.883485i	$0.344809\pi$
$522$	0	0
$523$	−7.66015	−0.334955	−0.167478	−	0.985876i	$-0.553562\pi$
$523$	−7.66015	−0.334955	−0.167478	+	0.985876i	$0.553562\pi$
$524$	0	0
$525$	10.7851	0.470699
$526$	0	0
$527$	−38.8117	−1.69066
$528$	0	0
$529$	−15.0518	−0.654424
$530$	0	0
$531$	−8.57584	−0.372160
$532$	0	0
$533$	0.579102	0.0250837
$534$	0	0
$535$	6.42199	0.277647
$536$	0	0
$537$	−40.4544	−1.74574
$538$	0	0
$539$	−28.5403	−1.22932
$540$	0	0
$541$	−2.36149	−0.101528	−0.0507641	−	0.998711i	$-0.516166\pi$
$541$	−2.36149	−0.101528	−0.0507641	+	0.998711i	$0.516166\pi$
$542$	0	0
$543$	−1.18149	−0.0507026
$544$	0	0
$545$	12.7929	0.547987
$546$	0	0
$547$	10.2201	0.436978	0.218489	−	0.975839i	$-0.429887\pi$
$547$	10.2201	0.436978	0.218489	+	0.975839i	$0.429887\pi$
$548$	0	0
$549$	29.5343	1.26050
$550$	0	0
$551$	−14.0790	−0.599785
$552$	0	0
$553$	9.17196	0.390031
$554$	0	0
$555$	−13.8145	−0.586392
$556$	0	0
$557$	23.8574	1.01087	0.505436	−	0.862864i	$-0.331332\pi$
$557$	23.8574	1.01087	0.505436	+	0.862864i	$0.331332\pi$
$558$	0	0
$559$	1.67682	0.0709219
$560$	0	0
$561$	−65.5363	−2.76694
$562$	0	0
$563$	−36.7093	−1.54711	−0.773557	−	0.633727i	$-0.781524\pi$
$563$	−36.7093	−1.54711	−0.773557	+	0.633727i	$0.781524\pi$
$564$	0	0
$565$	−8.48683	−0.357044
$566$	0	0
$567$	−5.21023	−0.218809
$568$	0	0
$569$	−29.5502	−1.23881	−0.619404	−	0.785073i	$-0.712626\pi$
$569$	−29.5502	−1.23881	−0.619404	+	0.785073i	$0.712626\pi$
$570$	0	0
$571$	2.21362	0.0926370	0.0463185	−	0.998927i	$-0.485251\pi$
$571$	2.21362	0.0926370	0.0463185	+	0.998927i	$0.485251\pi$
$572$	0	0
$573$	49.1032	2.05131
$574$	0	0
$575$	−11.0667	−0.461512
$576$	0	0
$577$	1.02017	0.0424703	0.0212352	−	0.999775i	$-0.493240\pi$
$577$	1.02017	0.0424703	0.0212352	+	0.999775i	$0.493240\pi$
$578$	0	0
$579$	−21.1696	−0.879778
$580$	0	0
$581$	−5.89752	−0.244670
$582$	0	0
$583$	37.0013	1.53244
$584$	0	0
$585$	0.539622	0.0223106
$586$	0	0
$587$	−23.2056	−0.957796	−0.478898	−	0.877870i	$-0.658964\pi$
$587$	−23.2056	−0.957796	−0.478898	+	0.877870i	$0.658964\pi$
$588$	0	0
$589$	31.5423	1.29968
$590$	0	0
$591$	−40.8701	−1.68117
$592$	0	0
$593$	−25.6284	−1.05243	−0.526217	−	0.850350i	$-0.676390\pi$
$593$	−25.6284	−1.05243	−0.526217	+	0.850350i	$0.676390\pi$
$594$	0	0
$595$	−5.53473	−0.226902
$596$	0	0
$597$	62.1651	2.54425
$598$	0	0
$599$	−32.3967	−1.32369	−0.661847	−	0.749639i	$-0.730227\pi$
$599$	−32.3967	−1.32369	−0.661847	+	0.749639i	$0.730227\pi$
$600$	0	0
$601$	−27.9265	−1.13915	−0.569573	−	0.821941i	$-0.692891\pi$
$601$	−27.9265	−1.13915	−0.569573	+	0.821941i	$0.692891\pi$
$602$	0	0
$603$	50.6318	2.06188
$604$	0	0
$605$	−12.6817	−0.515585
$606$	0	0
$607$	−43.4389	−1.76313	−0.881566	−	0.472061i	$-0.843510\pi$
$607$	−43.4389	−1.76313	−0.881566	+	0.472061i	$0.843510\pi$
$608$	0	0
$609$	−9.25990	−0.375230
$610$	0	0
$611$	0.249974	0.0101129
$612$	0	0
$613$	−27.4282	−1.10781	−0.553907	−	0.832579i	$-0.686864\pi$
$613$	−27.4282	−1.10781	−0.553907	+	0.832579i	$0.686864\pi$
$614$	0	0
$615$	−12.1922	−0.491637
$616$	0	0
$617$	47.9049	1.92858	0.964290	−	0.264848i	$-0.0853219\pi$
$617$	47.9049	1.92858	0.964290	+	0.264848i	$0.0853219\pi$
$618$	0	0
$619$	−19.5116	−0.784239	−0.392119	−	0.919914i	$-0.628258\pi$
$619$	−19.5116	−0.784239	−0.392119	+	0.919914i	$0.628258\pi$
$620$	0	0
$621$	−7.43336	−0.298291
$622$	0	0
$623$	−4.65322	−0.186427
$624$	0	0
$625$	10.0355	0.401418
$626$	0	0
$627$	53.2614	2.12706
$628$	0	0
$629$	−25.8960	−1.03254
$630$	0	0
$631$	−29.9655	−1.19291	−0.596454	−	0.802647i	$-0.703424\pi$
$631$	−29.9655	−1.19291	−0.596454	+	0.802647i	$0.703424\pi$
$632$	0	0
$633$	−45.0841	−1.79193
$634$	0	0
$635$	12.8925	0.511624
$636$	0	0
$637$	0.771174	0.0305550
$638$	0	0
$639$	10.2537	0.405629
$640$	0	0
$641$	−8.66517	−0.342254	−0.171127	−	0.985249i	$-0.554741\pi$
$641$	−8.66517	−0.342254	−0.171127	+	0.985249i	$0.554741\pi$
$642$	0	0
$643$	32.9016	1.29751	0.648757	−	0.760996i	$-0.275289\pi$
$643$	32.9016	1.29751	0.648757	+	0.760996i	$0.275289\pi$
$644$	0	0
$645$	−35.3031	−1.39006
$646$	0	0
$647$	−7.49277	−0.294571	−0.147286	−	0.989094i	$-0.547054\pi$
$647$	−7.49277	−0.294571	−0.147286	+	0.989094i	$0.547054\pi$
$648$	0	0
$649$	−10.3424	−0.405976
$650$	0	0
$651$	20.7457	0.813089
$652$	0	0
$653$	39.5323	1.54702	0.773510	−	0.633784i	$-0.218499\pi$
$653$	39.5323	1.54702	0.773510	+	0.633784i	$0.218499\pi$
$654$	0	0
$655$	−17.0803	−0.667381
$656$	0	0
$657$	−18.3266	−0.714990
$658$	0	0
$659$	−36.9158	−1.43804	−0.719018	−	0.694991i	$-0.755408\pi$
$659$	−36.9158	−1.43804	−0.719018	+	0.694991i	$0.755408\pi$
$660$	0	0
$661$	−40.1667	−1.56230	−0.781152	−	0.624341i	$-0.785367\pi$
$661$	−40.1667	−1.56230	−0.781152	+	0.624341i	$0.785367\pi$
$662$	0	0
$663$	1.77082	0.0687731
$664$	0	0
$665$	4.49808	0.174428
$666$	0	0
$667$	9.50168	0.367907
$668$	0	0
$669$	36.7050	1.41910
$670$	0	0
$671$	35.6184	1.37503
$672$	0	0
$673$	19.1071	0.736526	0.368263	−	0.929722i	$-0.379953\pi$
$673$	19.1071	0.736526	0.368263	+	0.929722i	$0.379953\pi$
$674$	0	0
$675$	10.3498	0.398363
$676$	0	0
$677$	−10.9120	−0.419383	−0.209692	−	0.977768i	$-0.567246\pi$
$677$	−10.9120	−0.419383	−0.209692	+	0.977768i	$0.567246\pi$
$678$	0	0
$679$	9.78197	0.375398
$680$	0	0
$681$	−33.9320	−1.30028
$682$	0	0
$683$	12.8100	0.490160	0.245080	−	0.969503i	$-0.421186\pi$
$683$	12.8100	0.490160	0.245080	+	0.969503i	$0.421186\pi$
$684$	0	0
$685$	5.33967	0.204018
$686$	0	0
$687$	−26.2193	−1.00033
$688$	0	0
$689$	−0.999795	−0.0380891
$690$	0	0
$691$	0.194689	0.00740631	0.00370316	−	0.999993i	$-0.498821\pi$
$691$	0.194689	0.00740631	0.00370316	+	0.999993i	$0.498821\pi$
$692$	0	0
$693$	20.0106	0.760140
$694$	0	0
$695$	−19.0845	−0.723918
$696$	0	0
$697$	−22.8550	−0.865693
$698$	0	0
$699$	8.90664	0.336880
$700$	0	0
$701$	−9.13883	−0.345169	−0.172584	−	0.984995i	$-0.555212\pi$
$701$	−9.13883	−0.345169	−0.172584	+	0.984995i	$0.555212\pi$
$702$	0	0
$703$	21.0457	0.793755
$704$	0	0
$705$	−5.26286	−0.198211
$706$	0	0
$707$	5.86023	0.220396
$708$	0	0
$709$	23.7788	0.893032	0.446516	−	0.894776i	$-0.352665\pi$
$709$	23.7788	0.893032	0.446516	+	0.894776i	$0.352665\pi$
$710$	0	0
$711$	35.2923	1.32357
$712$	0	0
$713$	−21.2874	−0.797219
$714$	0	0
$715$	0.650783	0.0243379
$716$	0	0
$717$	37.1915	1.38894
$718$	0	0
$719$	−8.75462	−0.326492	−0.163246	−	0.986585i	$-0.552196\pi$
$719$	−8.75462	−0.326492	−0.163246	+	0.986585i	$0.552196\pi$
$720$	0	0
$721$	−4.37823	−0.163054
$722$	0	0
$723$	9.52659	0.354298
$724$	0	0
$725$	−13.2296	−0.491334
$726$	0	0
$727$	−3.59933	−0.133492	−0.0667459	−	0.997770i	$-0.521262\pi$
$727$	−3.59933	−0.133492	−0.0667459	+	0.997770i	$0.521262\pi$
$728$	0	0
$729$	−40.9710	−1.51744
$730$	0	0
$731$	−66.1777	−2.44767
$732$	0	0
$733$	22.1972	0.819873	0.409936	−	0.912114i	$-0.365551\pi$
$733$	22.1972	0.819873	0.409936	+	0.912114i	$0.365551\pi$
$734$	0	0
$735$	−16.2360	−0.598874
$736$	0	0
$737$	61.0618	2.24924
$738$	0	0
$739$	31.4451	1.15673	0.578364	−	0.815779i	$-0.303691\pi$
$739$	31.4451	1.15673	0.578364	+	0.815779i	$0.303691\pi$
$740$	0	0
$741$	−1.43915	−0.0528685
$742$	0	0
$743$	50.0669	1.83678	0.918388	−	0.395681i	$-0.129492\pi$
$743$	50.0669	1.83678	0.918388	+	0.395681i	$0.129492\pi$
$744$	0	0
$745$	−6.75447	−0.247465
$746$	0	0
$747$	−22.6928	−0.830285
$748$	0	0
$749$	−6.43478	−0.235122
$750$	0	0
$751$	25.9990	0.948716	0.474358	−	0.880332i	$-0.342680\pi$
$751$	25.9990	0.948716	0.474358	+	0.880332i	$0.342680\pi$
$752$	0	0
$753$	55.9920	2.04046
$754$	0	0
$755$	10.7599	0.391593
$756$	0	0
$757$	39.9301	1.45128	0.725642	−	0.688073i	$-0.241543\pi$
$757$	39.9301	1.45128	0.725642	+	0.688073i	$0.241543\pi$
$758$	0	0
$759$	−35.9453	−1.30473
$760$	0	0
$761$	23.0525	0.835652	0.417826	−	0.908527i	$-0.362792\pi$
$761$	23.0525	0.835652	0.417826	+	0.908527i	$0.362792\pi$
$762$	0	0
$763$	−12.8184	−0.464056
$764$	0	0
$765$	−21.2968	−0.769987
$766$	0	0
$767$	0.279458	0.0100906
$768$	0	0
$769$	5.32809	0.192136	0.0960680	−	0.995375i	$-0.469373\pi$
$769$	5.32809	0.192136	0.0960680	+	0.995375i	$0.469373\pi$
$770$	0	0
$771$	4.44025	0.159912
$772$	0	0
$773$	20.4770	0.736505	0.368253	−	0.929726i	$-0.379956\pi$
$773$	20.4770	0.736505	0.368253	+	0.929726i	$0.379956\pi$
$774$	0	0
$775$	29.6393	1.06468
$776$	0	0
$777$	13.8420	0.496579
$778$	0	0
$779$	18.5742	0.665492
$780$	0	0
$781$	12.3659	0.442487
$782$	0	0
$783$	−8.88617	−0.317566
$784$	0	0
$785$	19.7137	0.703610
$786$	0	0
$787$	−5.62502	−0.200510	−0.100255	−	0.994962i	$-0.531966\pi$
$787$	−5.62502	−0.200510	−0.100255	+	0.994962i	$0.531966\pi$
$788$	0	0
$789$	−72.1155	−2.56738
$790$	0	0
$791$	8.50374	0.302358
$792$	0	0
$793$	−0.962426	−0.0341768
$794$	0	0
$795$	21.0493	0.746541
$796$	0	0
$797$	9.00011	0.318800	0.159400	−	0.987214i	$-0.449044\pi$
$797$	9.00011	0.318800	0.159400	+	0.987214i	$0.449044\pi$
$798$	0	0
$799$	−9.86553	−0.349017
$800$	0	0
$801$	−17.9049	−0.632638
$802$	0	0
$803$	−22.1019	−0.779958
$804$	0	0
$805$	−3.03568	−0.106994
$806$	0	0
$807$	−35.0122	−1.23249
$808$	0	0
$809$	−17.1979	−0.604647	−0.302323	−	0.953205i	$-0.597762\pi$
$809$	−17.1979	−0.604647	−0.302323	+	0.953205i	$0.597762\pi$
$810$	0	0
$811$	−18.3768	−0.645297	−0.322649	−	0.946519i	$-0.604573\pi$
$811$	−18.3768	−0.645297	−0.322649	+	0.946519i	$0.604573\pi$
$812$	0	0
$813$	39.8968	1.39924
$814$	0	0
$815$	23.2038	0.812794
$816$	0	0
$817$	53.7827	1.88162
$818$	0	0
$819$	−0.540697	−0.0188935
$820$	0	0
$821$	−29.1448	−1.01716	−0.508581	−	0.861014i	$-0.669830\pi$
$821$	−29.1448	−1.01716	−0.508581	+	0.861014i	$0.669830\pi$
$822$	0	0
$823$	−28.3758	−0.989118	−0.494559	−	0.869144i	$-0.664670\pi$
$823$	−28.3758	−0.989118	−0.494559	+	0.869144i	$0.664670\pi$
$824$	0	0
$825$	50.0481	1.74245
$826$	0	0
$827$	10.7519	0.373879	0.186940	−	0.982371i	$-0.440143\pi$
$827$	10.7519	0.373879	0.186940	+	0.982371i	$0.440143\pi$
$828$	0	0
$829$	−55.2891	−1.92027	−0.960135	−	0.279536i	$-0.909819\pi$
$829$	−55.2891	−1.92027	−0.960135	+	0.279536i	$0.909819\pi$
$830$	0	0
$831$	−56.0610	−1.94473
$832$	0	0
$833$	−30.4353	−1.05452
$834$	0	0
$835$	2.14359	0.0741819
$836$	0	0
$837$	19.9084	0.688135
$838$	0	0
$839$	−36.8790	−1.27320	−0.636602	−	0.771193i	$-0.719661\pi$
$839$	−36.8790	−1.27320	−0.636602	+	0.771193i	$0.719661\pi$
$840$	0	0
$841$	−17.6413	−0.608320
$842$	0	0
$843$	−51.4292	−1.77131
$844$	0	0
$845$	13.4587	0.462995
$846$	0	0
$847$	12.7070	0.436617
$848$	0	0
$849$	60.8844	2.08955
$850$	0	0
$851$	−14.2034	−0.486887
$852$	0	0
$853$	21.3099	0.729637	0.364818	−	0.931079i	$-0.381131\pi$
$853$	21.3099	0.729637	0.364818	+	0.931079i	$0.381131\pi$
$854$	0	0
$855$	17.3079	0.591919
$856$	0	0
$857$	30.7122	1.04911	0.524554	−	0.851377i	$-0.324232\pi$
$857$	30.7122	1.04911	0.524554	+	0.851377i	$0.324232\pi$
$858$	0	0
$859$	−22.7427	−0.775972	−0.387986	−	0.921665i	$-0.626829\pi$
$859$	−22.7427	−0.775972	−0.387986	+	0.921665i	$0.626829\pi$
$860$	0	0
$861$	12.2165	0.416337
$862$	0	0
$863$	14.5961	0.496857	0.248429	−	0.968650i	$-0.420086\pi$
$863$	14.5961	0.496857	0.248429	+	0.968650i	$0.420086\pi$
$864$	0	0
$865$	16.6569	0.566353
$866$	0	0
$867$	−24.9202	−0.846335
$868$	0	0
$869$	42.5625	1.44383
$870$	0	0
$871$	−1.64992	−0.0559054
$872$	0	0
$873$	37.6396	1.27391
$874$	0	0
$875$	9.61053	0.324895
$876$	0	0
$877$	−39.5247	−1.33465	−0.667327	−	0.744765i	$-0.732562\pi$
$877$	−39.5247	−1.33465	−0.667327	+	0.744765i	$0.732562\pi$
$878$	0	0
$879$	−3.16689	−0.106816
$880$	0	0
$881$	−44.0151	−1.48291	−0.741454	−	0.671004i	$-0.765863\pi$
$881$	−44.0151	−1.48291	−0.741454	+	0.671004i	$0.765863\pi$
$882$	0	0
$883$	39.2637	1.32133	0.660664	−	0.750682i	$-0.270275\pi$
$883$	39.2637	1.32133	0.660664	+	0.750682i	$0.270275\pi$
$884$	0	0
$885$	−5.88360	−0.197775
$886$	0	0
$887$	−33.5930	−1.12794	−0.563971	−	0.825794i	$-0.690727\pi$
$887$	−33.5930	−1.12794	−0.563971	+	0.825794i	$0.690727\pi$
$888$	0	0
$889$	−12.9182	−0.433263
$890$	0	0
$891$	−24.1780	−0.809995
$892$	0	0
$893$	8.01773	0.268303
$894$	0	0
$895$	−15.8542	−0.529948
$896$	0	0
$897$	0.971260	0.0324294
$898$	0	0
$899$	−25.4479	−0.848734
$900$	0	0
$901$	39.4581	1.31454
$902$	0	0
$903$	35.3735	1.17715
$904$	0	0
$905$	−0.463030	−0.0153916
$906$	0	0
$907$	−32.5091	−1.07945	−0.539724	−	0.841842i	$-0.681471\pi$
$907$	−32.5091	−1.07945	−0.539724	+	0.841842i	$0.681471\pi$
$908$	0	0
$909$	22.5493	0.747912
$910$	0	0
$911$	9.18546	0.304328	0.152164	−	0.988355i	$-0.451376\pi$
$911$	9.18546	0.304328	0.152164	+	0.988355i	$0.451376\pi$
$912$	0	0
$913$	−27.3674	−0.905730
$914$	0	0
$915$	20.2626	0.669859
$916$	0	0
$917$	17.1143	0.565164
$918$	0	0
$919$	14.9530	0.493256	0.246628	−	0.969110i	$-0.420677\pi$
$919$	14.9530	0.493256	0.246628	+	0.969110i	$0.420677\pi$
$920$	0	0
$921$	77.2653	2.54598
$922$	0	0
$923$	−0.334133	−0.0109981
$924$	0	0
$925$	19.7760	0.650231
$926$	0	0
$927$	−16.8468	−0.553321
$928$	0	0
$929$	−25.3191	−0.830694	−0.415347	−	0.909663i	$-0.636340\pi$
$929$	−25.3191	−0.830694	−0.415347	+	0.909663i	$0.636340\pi$
$930$	0	0
$931$	24.7348	0.810650
$932$	0	0
$933$	26.8575	0.879276
$934$	0	0
$935$	−25.6839	−0.839953
$936$	0	0
$937$	45.4166	1.48370	0.741848	−	0.670568i	$-0.233950\pi$
$937$	45.4166	1.48370	0.741848	+	0.670568i	$0.233950\pi$
$938$	0	0
$939$	−23.8907	−0.779645
$940$	0	0
$941$	19.8213	0.646156	0.323078	−	0.946372i	$-0.395282\pi$
$941$	19.8213	0.646156	0.323078	+	0.946372i	$0.395282\pi$
$942$	0	0
$943$	−12.5355	−0.408211
$944$	0	0
$945$	2.83903	0.0923537
$946$	0	0
$947$	−7.84850	−0.255042	−0.127521	−	0.991836i	$-0.540702\pi$
$947$	−7.84850	−0.255042	−0.127521	+	0.991836i	$0.540702\pi$
$948$	0	0
$949$	0.597204	0.0193861
$950$	0	0
$951$	−29.2576	−0.948741
$952$	0	0
$953$	6.12908	0.198540	0.0992701	−	0.995061i	$-0.468349\pi$
$953$	6.12908	0.198540	0.0992701	+	0.995061i	$0.468349\pi$
$954$	0	0
$955$	19.2437	0.622711
$956$	0	0
$957$	−42.9706	−1.38904
$958$	0	0
$959$	−5.35030	−0.172770
$960$	0	0
$961$	26.0130	0.839128
$962$	0	0
$963$	−24.7601	−0.797882
$964$	0	0
$965$	−8.29643	−0.267072
$966$	0	0
$967$	6.58200	0.211663	0.105831	−	0.994384i	$-0.466250\pi$
$967$	6.58200	0.211663	0.105831	+	0.994384i	$0.466250\pi$
$968$	0	0
$969$	56.7978	1.82461
$970$	0	0
$971$	22.7713	0.730766	0.365383	−	0.930857i	$-0.380938\pi$
$971$	22.7713	0.730766	0.365383	+	0.930857i	$0.380938\pi$
$972$	0	0
$973$	19.1226	0.613041
$974$	0	0
$975$	−1.35233	−0.0433091
$976$	0	0
$977$	−25.8096	−0.825721	−0.412861	−	0.910794i	$-0.635470\pi$
$977$	−25.8096	−0.825721	−0.412861	+	0.910794i	$0.635470\pi$
$978$	0	0
$979$	−21.5932	−0.690123
$980$	0	0
$981$	−49.3232	−1.57477
$982$	0	0
$983$	−20.4128	−0.651066	−0.325533	−	0.945531i	$-0.605544\pi$
$983$	−20.4128	−0.651066	−0.325533	+	0.945531i	$0.605544\pi$
$984$	0	0
$985$	−16.0171	−0.510349
$986$	0	0
$987$	5.27335	0.167852
$988$	0	0
$989$	−36.2971	−1.15418
$990$	0	0
$991$	−38.9887	−1.23852	−0.619258	−	0.785188i	$-0.712566\pi$
$991$	−38.9887	−1.23852	−0.619258	+	0.785188i	$0.712566\pi$
$992$	0	0
$993$	74.9694	2.37908
$994$	0	0
$995$	24.3627	0.772350
$996$	0	0
$997$	−34.9448	−1.10671	−0.553357	−	0.832944i	$-0.686653\pi$
$997$	−34.9448	−1.10671	−0.553357	+	0.832944i	$0.686653\pi$
$998$	0	0
$999$	13.2833	0.420266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.10	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.10	✓	80	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8044 = 2^{2} \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64514 q^{3} -1.03664 q^{5} +1.03871 q^{7} +3.99678 q^{9} +O(q^{10})\)	\(q-2.64514 q^{3} -1.03664 q^{5} +1.03871 q^{7} +3.99678 q^{9} +4.82011 q^{11} -0.130242 q^{13} +2.74206 q^{15} +5.14015 q^{17} -4.17741 q^{19} -2.74753 q^{21} +2.81926 q^{23} -3.92538 q^{25} -2.63663 q^{27} +3.37027 q^{29} -7.55069 q^{31} -12.7499 q^{33} -1.07676 q^{35} -5.03799 q^{37} +0.344508 q^{39} -4.44636 q^{41} -12.8747 q^{43} -4.14323 q^{45} -1.91931 q^{47} -5.92109 q^{49} -13.5964 q^{51} +7.67644 q^{53} -4.99672 q^{55} +11.0498 q^{57} -2.14569 q^{59} +7.38953 q^{61} +4.15148 q^{63} +0.135014 q^{65} +12.6681 q^{67} -7.45736 q^{69} +2.56548 q^{71} -4.58534 q^{73} +10.3832 q^{75} +5.00668 q^{77} +8.83018 q^{79} -5.01608 q^{81} -5.67776 q^{83} -5.32849 q^{85} -8.91485 q^{87} -4.47982 q^{89} -0.135283 q^{91} +19.9727 q^{93} +4.33047 q^{95} +9.41746 q^{97} +19.2649 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * q^99

Introduction

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