LMFDB - Embedded newform 8036.2.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.470117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 8x^{2} + 7x + 1 $ x^5 - 2x^4 - 6x^3 + 8x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.94177$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.94177 q^{3} -2.68629 q^{5} +0.770473 q^{9} +O(q^{10})$	$q+1.94177 q^{3} -2.68629 q^{5} +0.770473 q^{9} -0.616576 q^{11} +1.70117 q^{13} -5.21616 q^{15} +2.91582 q^{17} -1.49649 q^{19} -9.22765 q^{23} +2.21616 q^{25} -4.32923 q^{27} +10.4723 q^{29} +7.02889 q^{31} -1.19725 q^{33} -4.62554 q^{37} +3.30328 q^{39} -1.00000 q^{41} +4.88909 q^{43} -2.06972 q^{45} +4.96813 q^{47} +5.66185 q^{51} -8.14053 q^{53} +1.65630 q^{55} -2.90584 q^{57} +9.71628 q^{59} +7.04890 q^{61} -4.56983 q^{65} -10.8776 q^{67} -17.9180 q^{69} -2.23698 q^{71} -7.30479 q^{73} +4.30328 q^{75} -12.0081 q^{79} -10.7178 q^{81} +15.9105 q^{83} -7.83274 q^{85} +20.3348 q^{87} -10.3418 q^{89} +13.6485 q^{93} +4.02001 q^{95} -9.32921 q^{97} -0.475055 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} + q^{5} + q^{9}+O(q^{10}) $ 5 * q + 2 * q^3 + q^5 + q^9	$ 5 q + 2 q^{3} + q^{5} + q^{9} - 2 q^{11} + q^{13} - 9 q^{15} + 3 q^{17} + 4 q^{19} - 8 q^{23} - 6 q^{25} + 8 q^{27} - 9 q^{29} + 11 q^{31} - 5 q^{33} - 11 q^{37} - 17 q^{39} - 5 q^{41} - 27 q^{43} + 3 q^{45} + 3 q^{47} - 3 q^{51} - 19 q^{53} + 13 q^{55} - 11 q^{57} + 15 q^{59} + 7 q^{65} - 21 q^{67} - 14 q^{69} - 16 q^{71} + 10 q^{73} - 12 q^{75} - 14 q^{79} - 7 q^{81} + 2 q^{83} - 21 q^{85} + 36 q^{87} - 6 q^{89} + 17 q^{93} + 9 q^{95} - 20 q^{97} - 17 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 + q^5 + q^9 - 2 * q^11 + q^13 - 9 * q^15 + 3 * q^17 + 4 * q^19 - 8 * q^23 - 6 * q^25 + 8 * q^27 - 9 * q^29 + 11 * q^31 - 5 * q^33 - 11 * q^37 - 17 * q^39 - 5 * q^41 - 27 * q^43 + 3 * q^45 + 3 * q^47 - 3 * q^51 - 19 * q^53 + 13 * q^55 - 11 * q^57 + 15 * q^59 + 7 * q^65 - 21 * q^67 - 14 * q^69 - 16 * q^71 + 10 * q^73 - 12 * q^75 - 14 * q^79 - 7 * q^81 + 2 * q^83 - 21 * q^85 + 36 * q^87 - 6 * q^89 + 17 * q^93 + 9 * q^95 - 20 * q^97 - 17 * 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
$3$	1.94177	1.12108	0.560541	−	0.828127i	$-0.310593\pi$
$3$	1.94177	1.12108	0.560541	+	0.828127i	$0.310593\pi$
$4$	0	0
$5$	−2.68629	−1.20135	−0.600673	−	0.799495i	$-0.705101\pi$
$5$	−2.68629	−1.20135	−0.600673	+	0.799495i	$0.705101\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.770473	0.256824
$10$	0	0
$11$	−0.616576	−0.185905	−0.0929524	−	0.995671i	$-0.529630\pi$
$11$	−0.616576	−0.185905	−0.0929524	+	0.995671i	$0.529630\pi$
$12$	0	0
$13$	1.70117	0.471819	0.235910	−	0.971775i	$-0.424193\pi$
$13$	1.70117	0.471819	0.235910	+	0.971775i	$0.424193\pi$
$14$	0	0
$15$	−5.21616	−1.34681
$16$	0	0
$17$	2.91582	0.707190	0.353595	−	0.935399i	$-0.384959\pi$
$17$	2.91582	0.707190	0.353595	+	0.935399i	$0.384959\pi$
$18$	0	0
$19$	−1.49649	−0.343319	−0.171659	−	0.985156i	$-0.554913\pi$
$19$	−1.49649	−0.343319	−0.171659	+	0.985156i	$0.554913\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−9.22765	−1.92410	−0.962049	−	0.272878i	$-0.912025\pi$
$23$	−9.22765	−1.92410	−0.962049	+	0.272878i	$0.912025\pi$
$24$	0	0
$25$	2.21616	0.443232
$26$	0	0
$27$	−4.32923	−0.833161
$28$	0	0
$29$	10.4723	1.94465	0.972327	−	0.233623i	$-0.0750581\pi$
$29$	10.4723	1.94465	0.972327	+	0.233623i	$0.0750581\pi$
$30$	0	0
$31$	7.02889	1.26243	0.631213	−	0.775610i	$-0.282558\pi$
$31$	7.02889	1.26243	0.631213	+	0.775610i	$0.282558\pi$
$32$	0	0
$33$	−1.19725	−0.208414
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.62554	−0.760434	−0.380217	−	0.924897i	$-0.624151\pi$
$37$	−4.62554	−0.760434	−0.380217	+	0.924897i	$0.624151\pi$
$38$	0	0
$39$	3.30328	0.528948
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	4.88909	0.745579	0.372789	−	0.927916i	$-0.378401\pi$
$43$	4.88909	0.745579	0.372789	+	0.927916i	$0.378401\pi$
$44$	0	0
$45$	−2.06972	−0.308535
$46$	0	0
$47$	4.96813	0.724677	0.362338	−	0.932047i	$-0.381979\pi$
$47$	4.96813	0.724677	0.362338	+	0.932047i	$0.381979\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	5.66185	0.792818
$52$	0	0
$53$	−8.14053	−1.11819	−0.559094	−	0.829104i	$-0.688851\pi$
$53$	−8.14053	−1.11819	−0.559094	+	0.829104i	$0.688851\pi$
$54$	0	0
$55$	1.65630	0.223336
$56$	0	0
$57$	−2.90584	−0.384888
$58$	0	0
$59$	9.71628	1.26495	0.632476	−	0.774580i	$-0.282039\pi$
$59$	9.71628	1.26495	0.632476	+	0.774580i	$0.282039\pi$
$60$	0	0
$61$	7.04890	0.902519	0.451260	−	0.892393i	$-0.350975\pi$
$61$	7.04890	0.902519	0.451260	+	0.892393i	$0.350975\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.56983	−0.566818
$66$	0	0
$67$	−10.8776	−1.32891	−0.664457	−	0.747327i	$-0.731337\pi$
$67$	−10.8776	−1.32891	−0.664457	+	0.747327i	$0.731337\pi$
$68$	0	0
$69$	−17.9180	−2.15707
$70$	0	0
$71$	−2.23698	−0.265480	−0.132740	−	0.991151i	$-0.542378\pi$
$71$	−2.23698	−0.265480	−0.132740	+	0.991151i	$0.542378\pi$
$72$	0	0
$73$	−7.30479	−0.854961	−0.427480	−	0.904025i	$-0.640599\pi$
$73$	−7.30479	−0.854961	−0.427480	+	0.904025i	$0.640599\pi$
$74$	0	0
$75$	4.30328	0.496900
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.0081	−1.35101	−0.675507	−	0.737354i	$-0.736075\pi$
$79$	−12.0081	−1.35101	−0.675507	+	0.737354i	$0.736075\pi$
$80$	0	0
$81$	−10.7178	−1.19087
$82$	0	0
$83$	15.9105	1.74641	0.873203	−	0.487356i	$-0.162039\pi$
$83$	15.9105	1.74641	0.873203	+	0.487356i	$0.162039\pi$
$84$	0	0
$85$	−7.83274	−0.849580
$86$	0	0
$87$	20.3348	2.18012
$88$	0	0
$89$	−10.3418	−1.09623	−0.548115	−	0.836403i	$-0.684654\pi$
$89$	−10.3418	−1.09623	−0.548115	+	0.836403i	$0.684654\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	13.6485	1.41528
$94$	0	0
$95$	4.02001	0.412445
$96$	0	0
$97$	−9.32921	−0.947237	−0.473619	−	0.880730i	$-0.657053\pi$
$97$	−9.32921	−0.947237	−0.473619	+	0.880730i	$0.657053\pi$
$98$	0	0
$99$	−0.475055	−0.0477449
$100$	0	0
$101$	−9.01932	−0.897456	−0.448728	−	0.893668i	$-0.648123\pi$
$101$	−9.01932	−0.897456	−0.448728	+	0.893668i	$0.648123\pi$
$102$	0	0
$103$	−17.3637	−1.71089	−0.855446	−	0.517892i	$-0.826717\pi$
$103$	−17.3637	−1.71089	−0.855446	+	0.517892i	$0.826717\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.1830	−1.08110	−0.540552	−	0.841311i	$-0.681784\pi$
$107$	−11.1830	−1.08110	−0.540552	+	0.841311i	$0.681784\pi$
$108$	0	0
$109$	−8.97808	−0.859944	−0.429972	−	0.902842i	$-0.641477\pi$
$109$	−8.97808	−0.859944	−0.429972	+	0.902842i	$0.641477\pi$
$110$	0	0
$111$	−8.98173	−0.852508
$112$	0	0
$113$	−18.6550	−1.75492	−0.877460	−	0.479651i	$-0.840763\pi$
$113$	−18.6550	−1.75492	−0.877460	+	0.479651i	$0.840763\pi$
$114$	0	0
$115$	24.7882	2.31151
$116$	0	0
$117$	1.31070	0.121175
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.6198	−0.965439
$122$	0	0
$123$	−1.94177	−0.175084
$124$	0	0
$125$	7.47820	0.668871
$126$	0	0
$127$	−3.56174	−0.316053	−0.158027	−	0.987435i	$-0.550513\pi$
$127$	−3.56174	−0.316053	−0.158027	+	0.987435i	$0.550513\pi$
$128$	0	0
$129$	9.49349	0.835855
$130$	0	0
$131$	−8.34370	−0.728992	−0.364496	−	0.931205i	$-0.618759\pi$
$131$	−8.34370	−0.728992	−0.364496	+	0.931205i	$0.618759\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	11.6296	1.00091
$136$	0	0
$137$	6.12563	0.523348	0.261674	−	0.965156i	$-0.415725\pi$
$137$	6.12563	0.523348	0.261674	+	0.965156i	$0.415725\pi$
$138$	0	0
$139$	−18.1886	−1.54273	−0.771367	−	0.636391i	$-0.780427\pi$
$139$	−18.1886	−1.54273	−0.771367	+	0.636391i	$0.780427\pi$
$140$	0	0
$141$	9.64697	0.812422
$142$	0	0
$143$	−1.04890	−0.0877134
$144$	0	0
$145$	−28.1316	−2.33620
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.02297	−0.657267	−0.328634	−	0.944457i	$-0.606588\pi$
$149$	−8.02297	−0.657267	−0.328634	+	0.944457i	$0.606588\pi$
$150$	0	0
$151$	3.66146	0.297966	0.148983	−	0.988840i	$-0.452400\pi$
$151$	3.66146	0.297966	0.148983	+	0.988840i	$0.452400\pi$
$152$	0	0
$153$	2.24656	0.181624
$154$	0	0
$155$	−18.8816	−1.51661
$156$	0	0
$157$	9.63702	0.769118	0.384559	−	0.923100i	$-0.374353\pi$
$157$	9.63702	0.769118	0.384559	+	0.923100i	$0.374353\pi$
$158$	0	0
$159$	−15.8070	−1.25358
$160$	0	0
$161$	0	0
$162$	0	0
$163$	7.71077	0.603954	0.301977	−	0.953315i	$-0.402353\pi$
$163$	7.71077	0.603954	0.301977	+	0.953315i	$0.402353\pi$
$164$	0	0
$165$	3.21616	0.250378
$166$	0	0
$167$	8.37813	0.648319	0.324160	−	0.946002i	$-0.394919\pi$
$167$	8.37813	0.648319	0.324160	+	0.946002i	$0.394919\pi$
$168$	0	0
$169$	−10.1060	−0.777387
$170$	0	0
$171$	−1.15301	−0.0881726
$172$	0	0
$173$	−5.38847	−0.409678	−0.204839	−	0.978796i	$-0.565667\pi$
$173$	−5.38847	−0.409678	−0.204839	+	0.978796i	$0.565667\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	18.8668	1.41811
$178$	0	0
$179$	5.74603	0.429479	0.214739	−	0.976671i	$-0.431110\pi$
$179$	5.74603	0.429479	0.214739	+	0.976671i	$0.431110\pi$
$180$	0	0
$181$	8.17203	0.607422	0.303711	−	0.952764i	$-0.401774\pi$
$181$	8.17203	0.607422	0.303711	+	0.952764i	$0.401774\pi$
$182$	0	0
$183$	13.6873	1.01180
$184$	0	0
$185$	12.4255	0.913544
$186$	0	0
$187$	−1.79782	−0.131470
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.5766	1.41651	0.708255	−	0.705956i	$-0.249483\pi$
$191$	19.5766	1.41651	0.708255	+	0.705956i	$0.249483\pi$
$192$	0	0
$193$	−1.21001	−0.0870985	−0.0435493	−	0.999051i	$-0.513867\pi$
$193$	−1.21001	−0.0870985	−0.0435493	+	0.999051i	$0.513867\pi$
$194$	0	0
$195$	−8.87357	−0.635449
$196$	0	0
$197$	14.3611	1.02319	0.511594	−	0.859227i	$-0.329055\pi$
$197$	14.3611	1.02319	0.511594	+	0.859227i	$0.329055\pi$
$198$	0	0
$199$	−7.93456	−0.562466	−0.281233	−	0.959640i	$-0.590743\pi$
$199$	−7.93456	−0.562466	−0.281233	+	0.959640i	$0.590743\pi$
$200$	0	0
$201$	−21.1219	−1.48982
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.68629	0.187619
$206$	0	0
$207$	−7.10965	−0.494155
$208$	0	0
$209$	0.922701	0.0638246
$210$	0	0
$211$	−12.6704	−0.872266	−0.436133	−	0.899882i	$-0.643652\pi$
$211$	−12.6704	−0.872266	−0.436133	+	0.899882i	$0.643652\pi$
$212$	0	0
$213$	−4.34370	−0.297625
$214$	0	0
$215$	−13.1335	−0.895698
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−14.1842	−0.958481
$220$	0	0
$221$	4.96030	0.333666
$222$	0	0
$223$	−14.8487	−0.994342	−0.497171	−	0.867653i	$-0.665628\pi$
$223$	−14.8487	−0.994342	−0.497171	+	0.867653i	$0.665628\pi$
$224$	0	0
$225$	1.70749	0.113833
$226$	0	0
$227$	−23.5836	−1.56530	−0.782651	−	0.622461i	$-0.786133\pi$
$227$	−23.5836	−1.56530	−0.782651	+	0.622461i	$0.786133\pi$
$228$	0	0
$229$	5.26135	0.347680	0.173840	−	0.984774i	$-0.444382\pi$
$229$	5.26135	0.347680	0.173840	+	0.984774i	$0.444382\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.0100	0.852312	0.426156	−	0.904650i	$-0.359868\pi$
$233$	13.0100	0.852312	0.426156	+	0.904650i	$0.359868\pi$
$234$	0	0
$235$	−13.3459	−0.870587
$236$	0	0
$237$	−23.3169	−1.51460
$238$	0	0
$239$	22.8401	1.47740	0.738700	−	0.674034i	$-0.235440\pi$
$239$	22.8401	1.47740	0.738700	+	0.674034i	$0.235440\pi$
$240$	0	0
$241$	−24.5969	−1.58443	−0.792214	−	0.610244i	$-0.791071\pi$
$241$	−24.5969	−1.58443	−0.792214	+	0.610244i	$0.791071\pi$
$242$	0	0
$243$	−7.82380	−0.501897
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.54578	−0.161984
$248$	0	0
$249$	30.8946	1.95786
$250$	0	0
$251$	−23.4545	−1.48043	−0.740217	−	0.672368i	$-0.765277\pi$
$251$	−23.4545	−1.48043	−0.740217	+	0.672368i	$0.765277\pi$
$252$	0	0
$253$	5.68955	0.357699
$254$	0	0
$255$	−15.2094	−0.952448
$256$	0	0
$257$	18.5565	1.15752	0.578762	−	0.815497i	$-0.303536\pi$
$257$	18.5565	1.15752	0.578762	+	0.815497i	$0.303536\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.06861	0.499435
$262$	0	0
$263$	13.0340	0.803714	0.401857	−	0.915702i	$-0.368365\pi$
$263$	13.0340	0.803714	0.401857	+	0.915702i	$0.368365\pi$
$264$	0	0
$265$	21.8678	1.34333
$266$	0	0
$267$	−20.0814	−1.22896
$268$	0	0
$269$	8.32781	0.507755	0.253878	−	0.967236i	$-0.418294\pi$
$269$	8.32781	0.507755	0.253878	+	0.967236i	$0.418294\pi$
$270$	0	0
$271$	2.09297	0.127139	0.0635694	−	0.997977i	$-0.479752\pi$
$271$	2.09297	0.127139	0.0635694	+	0.997977i	$0.479752\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.36643	−0.0823990
$276$	0	0
$277$	−8.18490	−0.491783	−0.245891	−	0.969297i	$-0.579081\pi$
$277$	−8.18490	−0.491783	−0.245891	+	0.969297i	$0.579081\pi$
$278$	0	0
$279$	5.41557	0.324222
$280$	0	0
$281$	13.3834	0.798388	0.399194	−	0.916866i	$-0.369290\pi$
$281$	13.3834	0.798388	0.399194	+	0.916866i	$0.369290\pi$
$282$	0	0
$283$	13.1644	0.782541	0.391270	−	0.920276i	$-0.372036\pi$
$283$	13.1644	0.782541	0.391270	+	0.920276i	$0.372036\pi$
$284$	0	0
$285$	7.80594	0.462384
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.49800	−0.499883
$290$	0	0
$291$	−18.1152	−1.06193
$292$	0	0
$293$	−26.7152	−1.56072	−0.780359	−	0.625332i	$-0.784964\pi$
$293$	−26.7152	−1.56072	−0.780359	+	0.625332i	$0.784964\pi$
$294$	0	0
$295$	−26.1008	−1.51964
$296$	0	0
$297$	2.66930	0.154889
$298$	0	0
$299$	−15.6978	−0.907826
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−17.5135	−1.00612
$304$	0	0
$305$	−18.9354	−1.08424
$306$	0	0
$307$	10.9998	0.627790	0.313895	−	0.949458i	$-0.398366\pi$
$307$	10.9998	0.627790	0.313895	+	0.949458i	$0.398366\pi$
$308$	0	0
$309$	−33.7163	−1.91805
$310$	0	0
$311$	−16.1242	−0.914320	−0.457160	−	0.889384i	$-0.651133\pi$
$311$	−16.1242	−0.914320	−0.457160	+	0.889384i	$0.651133\pi$
$312$	0	0
$313$	26.2400	1.48317	0.741586	−	0.670858i	$-0.234074\pi$
$313$	26.2400	1.48317	0.741586	+	0.670858i	$0.234074\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.6496	0.766637	0.383319	−	0.923616i	$-0.374781\pi$
$317$	13.6496	0.766637	0.383319	+	0.923616i	$0.374781\pi$
$318$	0	0
$319$	−6.45696	−0.361521
$320$	0	0
$321$	−21.7149	−1.21201
$322$	0	0
$323$	−4.36350	−0.242792
$324$	0	0
$325$	3.77006	0.209125
$326$	0	0
$327$	−17.4334	−0.964068
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−14.0230	−0.770772	−0.385386	−	0.922755i	$-0.625932\pi$
$331$	−14.0230	−0.770772	−0.385386	+	0.922755i	$0.625932\pi$
$332$	0	0
$333$	−3.56385	−0.195298
$334$	0	0
$335$	29.2205	1.59649
$336$	0	0
$337$	−21.1990	−1.15478	−0.577391	−	0.816467i	$-0.695929\pi$
$337$	−21.1990	−1.15478	−0.577391	+	0.816467i	$0.695929\pi$
$338$	0	0
$339$	−36.2238	−1.96741
$340$	0	0
$341$	−4.33384	−0.234691
$342$	0	0
$343$	0	0
$344$	0	0
$345$	48.1329	2.59139
$346$	0	0
$347$	3.65520	0.196222	0.0981108	−	0.995176i	$-0.468720\pi$
$347$	3.65520	0.196222	0.0981108	+	0.995176i	$0.468720\pi$
$348$	0	0
$349$	29.9786	1.60472	0.802359	−	0.596841i	$-0.203578\pi$
$349$	29.9786	1.60472	0.802359	+	0.596841i	$0.203578\pi$
$350$	0	0
$351$	−7.36475	−0.393101
$352$	0	0
$353$	12.5294	0.666870	0.333435	−	0.942773i	$-0.391792\pi$
$353$	12.5294	0.666870	0.333435	+	0.942773i	$0.391792\pi$
$354$	0	0
$355$	6.00917	0.318934
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.233065	0.0123007	0.00615034	−	0.999981i	$-0.498042\pi$
$359$	0.233065	0.0123007	0.00615034	+	0.999981i	$0.498042\pi$
$360$	0	0
$361$	−16.7605	−0.882132
$362$	0	0
$363$	−20.6213	−1.08234
$364$	0	0
$365$	19.6228	1.02710
$366$	0	0
$367$	23.1545	1.20865	0.604326	−	0.796737i	$-0.293442\pi$
$367$	23.1545	1.20865	0.604326	+	0.796737i	$0.293442\pi$
$368$	0	0
$369$	−0.770473	−0.0401092
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.7029	0.657732	0.328866	−	0.944377i	$-0.393334\pi$
$373$	12.7029	0.657732	0.328866	+	0.944377i	$0.393334\pi$
$374$	0	0
$375$	14.5210	0.749859
$376$	0	0
$377$	17.8151	0.917525
$378$	0	0
$379$	−14.9179	−0.766283	−0.383141	−	0.923690i	$-0.625158\pi$
$379$	−14.9179	−0.766283	−0.383141	+	0.923690i	$0.625158\pi$
$380$	0	0
$381$	−6.91608	−0.354321
$382$	0	0
$383$	−0.921522	−0.0470876	−0.0235438	−	0.999723i	$-0.507495\pi$
$383$	−0.921522	−0.0470876	−0.0235438	+	0.999723i	$0.507495\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.76691	0.191483
$388$	0	0
$389$	−24.0151	−1.21761	−0.608807	−	0.793318i	$-0.708352\pi$
$389$	−24.0151	−1.21761	−0.608807	+	0.793318i	$0.708352\pi$
$390$	0	0
$391$	−26.9061	−1.36070
$392$	0	0
$393$	−16.2015	−0.817260
$394$	0	0
$395$	32.2572	1.62303
$396$	0	0
$397$	−12.9949	−0.652196	−0.326098	−	0.945336i	$-0.605734\pi$
$397$	−12.9949	−0.652196	−0.326098	+	0.945336i	$0.605734\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	19.2074	0.959171	0.479586	−	0.877495i	$-0.340787\pi$
$401$	19.2074	0.959171	0.479586	+	0.877495i	$0.340787\pi$
$402$	0	0
$403$	11.9573	0.595636
$404$	0	0
$405$	28.7911	1.43064
$406$	0	0
$407$	2.85200	0.141368
$408$	0	0
$409$	−17.8957	−0.884884	−0.442442	−	0.896797i	$-0.645888\pi$
$409$	−17.8957	−0.884884	−0.442442	+	0.896797i	$0.645888\pi$
$410$	0	0
$411$	11.8946	0.586716
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−42.7403	−2.09804
$416$	0	0
$417$	−35.3180	−1.72953
$418$	0	0
$419$	−23.1277	−1.12986	−0.564932	−	0.825138i	$-0.691097\pi$
$419$	−23.1277	−1.12986	−0.564932	+	0.825138i	$0.691097\pi$
$420$	0	0
$421$	−19.2191	−0.936683	−0.468341	−	0.883548i	$-0.655148\pi$
$421$	−19.2191	−0.936683	−0.468341	+	0.883548i	$0.655148\pi$
$422$	0	0
$423$	3.82781	0.186115
$424$	0	0
$425$	6.46193	0.313449
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.03672	−0.0983339
$430$	0	0
$431$	−34.8211	−1.67727	−0.838637	−	0.544691i	$-0.816647\pi$
$431$	−34.8211	−1.67727	−0.838637	+	0.544691i	$0.816647\pi$
$432$	0	0
$433$	−20.6604	−0.992875	−0.496437	−	0.868073i	$-0.665359\pi$
$433$	−20.6604	−0.992875	−0.496437	+	0.868073i	$0.665359\pi$
$434$	0	0
$435$	−54.6251	−2.61907
$436$	0	0
$437$	13.8091	0.660579
$438$	0	0
$439$	−17.5338	−0.836841	−0.418421	−	0.908253i	$-0.637416\pi$
$439$	−17.5338	−0.836841	−0.418421	+	0.908253i	$0.637416\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.9057	1.04077	0.520386	−	0.853931i	$-0.325788\pi$
$443$	21.9057	1.04077	0.520386	+	0.853931i	$0.325788\pi$
$444$	0	0
$445$	27.7811	1.31695
$446$	0	0
$447$	−15.5788	−0.736851
$448$	0	0
$449$	−22.4998	−1.06183	−0.530915	−	0.847425i	$-0.678152\pi$
$449$	−22.4998	−1.06183	−0.530915	+	0.847425i	$0.678152\pi$
$450$	0	0
$451$	0.616576	0.0290334
$452$	0	0
$453$	7.10972	0.334044
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.25908	−0.292787	−0.146394	−	0.989226i	$-0.546767\pi$
$457$	−6.25908	−0.292787	−0.146394	+	0.989226i	$0.546767\pi$
$458$	0	0
$459$	−12.6232	−0.589203
$460$	0	0
$461$	−10.8716	−0.506343	−0.253172	−	0.967421i	$-0.581474\pi$
$461$	−10.8716	−0.506343	−0.253172	+	0.967421i	$0.581474\pi$
$462$	0	0
$463$	4.78513	0.222384	0.111192	−	0.993799i	$-0.464533\pi$
$463$	4.78513	0.222384	0.111192	+	0.993799i	$0.464533\pi$
$464$	0	0
$465$	−36.6638	−1.70024
$466$	0	0
$467$	0.974072	0.0450747	0.0225373	−	0.999746i	$-0.492826\pi$
$467$	0.974072	0.0450747	0.0225373	+	0.999746i	$0.492826\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	18.7129	0.862244
$472$	0	0
$473$	−3.01450	−0.138607
$474$	0	0
$475$	−3.31647	−0.152170
$476$	0	0
$477$	−6.27206	−0.287178
$478$	0	0
$479$	−9.62638	−0.439840	−0.219920	−	0.975518i	$-0.570580\pi$
$479$	−9.62638	−0.439840	−0.219920	+	0.975518i	$0.570580\pi$
$480$	0	0
$481$	−7.86882	−0.358787
$482$	0	0
$483$	0	0
$484$	0	0
$485$	25.0610	1.13796
$486$	0	0
$487$	−7.00149	−0.317268	−0.158634	−	0.987337i	$-0.550709\pi$
$487$	−7.00149	−0.317268	−0.158634	+	0.987337i	$0.550709\pi$
$488$	0	0
$489$	14.9726	0.677082
$490$	0	0
$491$	−41.6278	−1.87864	−0.939318	−	0.343047i	$-0.888541\pi$
$491$	−41.6278	−1.87864	−0.939318	+	0.343047i	$0.888541\pi$
$492$	0	0
$493$	30.5353	1.37524
$494$	0	0
$495$	1.27614	0.0573581
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12.4714	0.558298	0.279149	−	0.960248i	$-0.409948\pi$
$499$	12.4714	0.558298	0.279149	+	0.960248i	$0.409948\pi$
$500$	0	0
$501$	16.2684	0.726819
$502$	0	0
$503$	−9.28274	−0.413897	−0.206949	−	0.978352i	$-0.566353\pi$
$503$	−9.28274	−0.413897	−0.206949	+	0.978352i	$0.566353\pi$
$504$	0	0
$505$	24.2285	1.07816
$506$	0	0
$507$	−19.6236	−0.871514
$508$	0	0
$509$	−22.5231	−0.998320	−0.499160	−	0.866510i	$-0.666358\pi$
$509$	−22.5231	−0.998320	−0.499160	+	0.866510i	$0.666358\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	6.47866	0.286040
$514$	0	0
$515$	46.6439	2.05537
$516$	0	0
$517$	−3.06323	−0.134721
$518$	0	0
$519$	−10.4632	−0.459283
$520$	0	0
$521$	−23.7247	−1.03940	−0.519699	−	0.854349i	$-0.673956\pi$
$521$	−23.7247	−1.03940	−0.519699	+	0.854349i	$0.673956\pi$
$522$	0	0
$523$	31.9534	1.39722	0.698612	−	0.715500i	$-0.253801\pi$
$523$	31.9534	1.39722	0.698612	+	0.715500i	$0.253801\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.4950	0.892774
$528$	0	0
$529$	62.1495	2.70215
$530$	0	0
$531$	7.48613	0.324870
$532$	0	0
$533$	−1.70117	−0.0736858
$534$	0	0
$535$	30.0408	1.29878
$536$	0	0
$537$	11.1575	0.481481
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.39512	−0.102974	−0.0514871	−	0.998674i	$-0.516396\pi$
$541$	−2.39512	−0.102974	−0.0514871	+	0.998674i	$0.516396\pi$
$542$	0	0
$543$	15.8682	0.680970
$544$	0	0
$545$	24.1177	1.03309
$546$	0	0
$547$	−10.5941	−0.452970	−0.226485	−	0.974015i	$-0.572723\pi$
$547$	−10.5941	−0.452970	−0.226485	+	0.974015i	$0.572723\pi$
$548$	0	0
$549$	5.43099	0.231789
$550$	0	0
$551$	−15.6717	−0.667636
$552$	0	0
$553$	0	0
$554$	0	0
$555$	24.1276	1.02416
$556$	0	0
$557$	−32.4085	−1.37319	−0.686596	−	0.727039i	$-0.740896\pi$
$557$	−32.4085	−1.37319	−0.686596	+	0.727039i	$0.740896\pi$
$558$	0	0
$559$	8.31716	0.351778
$560$	0	0
$561$	−3.49096	−0.147389
$562$	0	0
$563$	−10.5658	−0.445296	−0.222648	−	0.974899i	$-0.571470\pi$
$563$	−10.5658	−0.445296	−0.222648	+	0.974899i	$0.571470\pi$
$564$	0	0
$565$	50.1129	2.10827
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.4511	1.06697	0.533484	−	0.845810i	$-0.320883\pi$
$569$	25.4511	1.06697	0.533484	+	0.845810i	$0.320883\pi$
$570$	0	0
$571$	29.2975	1.22606	0.613031	−	0.790059i	$-0.289950\pi$
$571$	29.2975	1.22606	0.613031	+	0.790059i	$0.289950\pi$
$572$	0	0
$573$	38.0132	1.58802
$574$	0	0
$575$	−20.4500	−0.852822
$576$	0	0
$577$	9.00667	0.374953	0.187476	−	0.982269i	$-0.439969\pi$
$577$	9.00667	0.374953	0.187476	+	0.982269i	$0.439969\pi$
$578$	0	0
$579$	−2.34957	−0.0976446
$580$	0	0
$581$	0	0
$582$	0	0
$583$	5.01926	0.207876
$584$	0	0
$585$	−3.52093	−0.145573
$586$	0	0
$587$	40.3386	1.66495	0.832476	−	0.554061i	$-0.186922\pi$
$587$	40.3386	1.66495	0.832476	+	0.554061i	$0.186922\pi$
$588$	0	0
$589$	−10.5187	−0.433414
$590$	0	0
$591$	27.8860	1.14708
$592$	0	0
$593$	−27.2750	−1.12005	−0.560024	−	0.828476i	$-0.689208\pi$
$593$	−27.2750	−1.12005	−0.560024	+	0.828476i	$0.689208\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.4071	−0.630570
$598$	0	0
$599$	−5.64567	−0.230676	−0.115338	−	0.993326i	$-0.536795\pi$
$599$	−5.64567	−0.230676	−0.115338	+	0.993326i	$0.536795\pi$
$600$	0	0
$601$	7.39390	0.301603	0.150802	−	0.988564i	$-0.451815\pi$
$601$	7.39390	0.301603	0.150802	+	0.988564i	$0.451815\pi$
$602$	0	0
$603$	−8.38092	−0.341297
$604$	0	0
$605$	28.5280	1.15983
$606$	0	0
$607$	2.66935	0.108346	0.0541728	−	0.998532i	$-0.482748\pi$
$607$	2.66935	0.108346	0.0541728	+	0.998532i	$0.482748\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.45163	0.341916
$612$	0	0
$613$	−29.6940	−1.19933	−0.599664	−	0.800252i	$-0.704699\pi$
$613$	−29.6940	−1.19933	−0.599664	+	0.800252i	$0.704699\pi$
$614$	0	0
$615$	5.21616	0.210336
$616$	0	0
$617$	8.47295	0.341108	0.170554	−	0.985348i	$-0.445444\pi$
$617$	8.47295	0.341108	0.170554	+	0.985348i	$0.445444\pi$
$618$	0	0
$619$	−2.44345	−0.0982104	−0.0491052	−	0.998794i	$-0.515637\pi$
$619$	−2.44345	−0.0982104	−0.0491052	+	0.998794i	$0.515637\pi$
$620$	0	0
$621$	39.9486	1.60308
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.1694	−1.24678
$626$	0	0
$627$	1.79167	0.0715526
$628$	0	0
$629$	−13.4872	−0.537771
$630$	0	0
$631$	0.246928	0.00983004	0.00491502	−	0.999988i	$-0.498435\pi$
$631$	0.246928	0.00983004	0.00491502	+	0.999988i	$0.498435\pi$
$632$	0	0
$633$	−24.6030	−0.977882
$634$	0	0
$635$	9.56787	0.379689
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.72353	−0.0681818
$640$	0	0
$641$	−12.6671	−0.500320	−0.250160	−	0.968204i	$-0.580483\pi$
$641$	−12.6671	−0.500320	−0.250160	+	0.968204i	$0.580483\pi$
$642$	0	0
$643$	−44.4641	−1.75349	−0.876746	−	0.480953i	$-0.840291\pi$
$643$	−44.4641	−1.75349	−0.876746	+	0.480953i	$0.840291\pi$
$644$	0	0
$645$	−25.5023	−1.00415
$646$	0	0
$647$	38.0143	1.49450	0.747249	−	0.664544i	$-0.231374\pi$
$647$	38.0143	1.49450	0.747249	+	0.664544i	$0.231374\pi$
$648$	0	0
$649$	−5.99083	−0.235161
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.8090	0.462123	0.231062	−	0.972939i	$-0.425780\pi$
$653$	11.8090	0.462123	0.231062	+	0.972939i	$0.425780\pi$
$654$	0	0
$655$	22.4136	0.875772
$656$	0	0
$657$	−5.62814	−0.219575
$658$	0	0
$659$	44.1657	1.72045	0.860225	−	0.509914i	$-0.170323\pi$
$659$	44.1657	1.72045	0.860225	+	0.509914i	$0.170323\pi$
$660$	0	0
$661$	−18.7184	−0.728061	−0.364030	−	0.931387i	$-0.618600\pi$
$661$	−18.7184	−0.728061	−0.364030	+	0.931387i	$0.618600\pi$
$662$	0	0
$663$	9.63176	0.374066
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−96.6346	−3.74171
$668$	0	0
$669$	−28.8328	−1.11474
$670$	0	0
$671$	−4.34618	−0.167783
$672$	0	0
$673$	−6.79888	−0.262078	−0.131039	−	0.991377i	$-0.541831\pi$
$673$	−6.79888	−0.262078	−0.131039	+	0.991377i	$0.541831\pi$
$674$	0	0
$675$	−9.59427	−0.369284
$676$	0	0
$677$	13.8599	0.532681	0.266340	−	0.963879i	$-0.414185\pi$
$677$	13.8599	0.532681	0.266340	+	0.963879i	$0.414185\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−45.7940	−1.75483
$682$	0	0
$683$	33.9460	1.29891	0.649454	−	0.760401i	$-0.274997\pi$
$683$	33.9460	1.29891	0.649454	+	0.760401i	$0.274997\pi$
$684$	0	0
$685$	−16.4552	−0.628722
$686$	0	0
$687$	10.2163	0.389777
$688$	0	0
$689$	−13.8484	−0.527582
$690$	0	0
$691$	−23.8016	−0.905456	−0.452728	−	0.891649i	$-0.649549\pi$
$691$	−23.8016	−0.905456	−0.452728	+	0.891649i	$0.649549\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	48.8598	1.85336
$696$	0	0
$697$	−2.91582	−0.110444
$698$	0	0
$699$	25.2624	0.955511
$700$	0	0
$701$	48.9994	1.85068	0.925340	−	0.379138i	$-0.123779\pi$
$701$	48.9994	1.85068	0.925340	+	0.379138i	$0.123779\pi$
$702$	0	0
$703$	6.92208	0.261071
$704$	0	0
$705$	−25.9146	−0.976000
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−37.8902	−1.42300	−0.711499	−	0.702687i	$-0.751983\pi$
$709$	−37.8902	−1.42300	−0.711499	+	0.702687i	$0.751983\pi$
$710$	0	0
$711$	−9.25190	−0.346973
$712$	0	0
$713$	−64.8601	−2.42903
$714$	0	0
$715$	2.81765	0.105374
$716$	0	0
$717$	44.3501	1.65629
$718$	0	0
$719$	47.4091	1.76806	0.884031	−	0.467429i	$-0.154820\pi$
$719$	47.4091	1.76806	0.884031	+	0.467429i	$0.154820\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−47.7616	−1.77627
$724$	0	0
$725$	23.2083	0.861934
$726$	0	0
$727$	42.4720	1.57520	0.787600	−	0.616187i	$-0.211323\pi$
$727$	42.4720	1.57520	0.787600	+	0.616187i	$0.211323\pi$
$728$	0	0
$729$	16.9613	0.628198
$730$	0	0
$731$	14.2557	0.527266
$732$	0	0
$733$	4.79759	0.177203	0.0886015	−	0.996067i	$-0.471760\pi$
$733$	4.79759	0.177203	0.0886015	+	0.996067i	$0.471760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.70689	0.247051
$738$	0	0
$739$	−32.6982	−1.20282	−0.601411	−	0.798940i	$-0.705395\pi$
$739$	−32.6982	−1.20282	−0.601411	+	0.798940i	$0.705395\pi$
$740$	0	0
$741$	−4.94333	−0.181598
$742$	0	0
$743$	−16.4685	−0.604172	−0.302086	−	0.953281i	$-0.597683\pi$
$743$	−16.4685	−0.604172	−0.302086	+	0.953281i	$0.597683\pi$
$744$	0	0
$745$	21.5520	0.789606
$746$	0	0
$747$	12.2586	0.448520
$748$	0	0
$749$	0	0
$750$	0	0
$751$	52.4542	1.91408	0.957040	−	0.289956i	$-0.0936408\pi$
$751$	52.4542	1.91408	0.957040	+	0.289956i	$0.0936408\pi$
$752$	0	0
$753$	−45.5432	−1.65969
$754$	0	0
$755$	−9.83576	−0.357960
$756$	0	0
$757$	−35.0448	−1.27372	−0.636862	−	0.770978i	$-0.719768\pi$
$757$	−35.0448	−1.27372	−0.636862	+	0.770978i	$0.719768\pi$
$758$	0	0
$759$	11.0478	0.401010
$760$	0	0
$761$	0.0743581	0.00269548	0.00134774	−	0.999999i	$-0.499571\pi$
$761$	0.0743581	0.00269548	0.00134774	+	0.999999i	$0.499571\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6.03491	−0.218193
$766$	0	0
$767$	16.5290	0.596828
$768$	0	0
$769$	−2.06711	−0.0745418	−0.0372709	−	0.999305i	$-0.511866\pi$
$769$	−2.06711	−0.0745418	−0.0372709	+	0.999305i	$0.511866\pi$
$770$	0	0
$771$	36.0325	1.29768
$772$	0	0
$773$	33.1292	1.19157	0.595787	−	0.803143i	$-0.296840\pi$
$773$	33.1292	1.19157	0.595787	+	0.803143i	$0.296840\pi$
$774$	0	0
$775$	15.5771	0.559548
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.49649	0.0536174
$780$	0	0
$781$	1.37927	0.0493541
$782$	0	0
$783$	−45.3369	−1.62021
$784$	0	0
$785$	−25.8879	−0.923977
$786$	0	0
$787$	−22.7134	−0.809646	−0.404823	−	0.914395i	$-0.632667\pi$
$787$	−22.7134	−0.809646	−0.404823	+	0.914395i	$0.632667\pi$
$788$	0	0
$789$	25.3091	0.901029
$790$	0	0
$791$	0	0
$792$	0	0
$793$	11.9914	0.425826
$794$	0	0
$795$	42.4623	1.50598
$796$	0	0
$797$	10.3885	0.367978	0.183989	−	0.982928i	$-0.441099\pi$
$797$	10.3885	0.367978	0.183989	+	0.982928i	$0.441099\pi$
$798$	0	0
$799$	14.4862	0.512484
$800$	0	0
$801$	−7.96809	−0.281539
$802$	0	0
$803$	4.50396	0.158941
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.1707	0.569235
$808$	0	0
$809$	−40.1639	−1.41209	−0.706044	−	0.708168i	$-0.749522\pi$
$809$	−40.1639	−1.41209	−0.706044	+	0.708168i	$0.749522\pi$
$810$	0	0
$811$	34.0648	1.19618	0.598089	−	0.801430i	$-0.295927\pi$
$811$	34.0648	1.19618	0.598089	+	0.801430i	$0.295927\pi$
$812$	0	0
$813$	4.06406	0.142533
$814$	0	0
$815$	−20.7134	−0.725558
$816$	0	0
$817$	−7.31648	−0.255971
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.84366	−0.169045	−0.0845225	−	0.996422i	$-0.526936\pi$
$821$	−4.84366	−0.169045	−0.0845225	+	0.996422i	$0.526936\pi$
$822$	0	0
$823$	17.9308	0.625027	0.312513	−	0.949913i	$-0.398829\pi$
$823$	17.9308	0.625027	0.312513	+	0.949913i	$0.398829\pi$
$824$	0	0
$825$	−2.65330	−0.0923760
$826$	0	0
$827$	−47.5813	−1.65456	−0.827281	−	0.561788i	$-0.810114\pi$
$827$	−47.5813	−1.65456	−0.827281	+	0.561788i	$0.810114\pi$
$828$	0	0
$829$	−17.0618	−0.592580	−0.296290	−	0.955098i	$-0.595749\pi$
$829$	−17.0618	−0.592580	−0.296290	+	0.955098i	$0.595749\pi$
$830$	0	0
$831$	−15.8932	−0.551329
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−22.5061	−0.778856
$836$	0	0
$837$	−30.4297	−1.05180
$838$	0	0
$839$	−8.06960	−0.278594	−0.139297	−	0.990251i	$-0.544484\pi$
$839$	−8.06960	−0.278594	−0.139297	+	0.990251i	$0.544484\pi$
$840$	0	0
$841$	80.6688	2.78168
$842$	0	0
$843$	25.9875	0.895058
$844$	0	0
$845$	27.1477	0.933911
$846$	0	0
$847$	0	0
$848$	0	0
$849$	25.5622	0.877292
$850$	0	0
$851$	42.6828	1.46315
$852$	0	0
$853$	32.8221	1.12381	0.561904	−	0.827203i	$-0.310069\pi$
$853$	32.8221	1.12381	0.561904	+	0.827203i	$0.310069\pi$
$854$	0	0
$855$	3.09731	0.105926
$856$	0	0
$857$	−9.35655	−0.319614	−0.159807	−	0.987148i	$-0.551087\pi$
$857$	−9.35655	−0.319614	−0.159807	+	0.987148i	$0.551087\pi$
$858$	0	0
$859$	25.2909	0.862913	0.431457	−	0.902134i	$-0.358000\pi$
$859$	25.2909	0.862913	0.431457	+	0.902134i	$0.358000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	44.5622	1.51692	0.758458	−	0.651722i	$-0.225953\pi$
$863$	44.5622	1.51692	0.758458	+	0.651722i	$0.225953\pi$
$864$	0	0
$865$	14.4750	0.492165
$866$	0	0
$867$	−16.5012	−0.560409
$868$	0	0
$869$	7.40389	0.251160
$870$	0	0
$871$	−18.5047	−0.627007
$872$	0	0
$873$	−7.18790	−0.243274
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9.83308	−0.332040	−0.166020	−	0.986122i	$-0.553092\pi$
$877$	−9.83308	−0.332040	−0.166020	+	0.986122i	$0.553092\pi$
$878$	0	0
$879$	−51.8748	−1.74969
$880$	0	0
$881$	23.6685	0.797412	0.398706	−	0.917079i	$-0.369459\pi$
$881$	23.6685	0.797412	0.398706	+	0.917079i	$0.369459\pi$
$882$	0	0
$883$	−23.7461	−0.799120	−0.399560	−	0.916707i	$-0.630837\pi$
$883$	−23.7461	−0.799120	−0.399560	+	0.916707i	$0.630837\pi$
$884$	0	0
$885$	−50.6817	−1.70365
$886$	0	0
$887$	5.13770	0.172507	0.0862536	−	0.996273i	$-0.472510\pi$
$887$	5.13770	0.172507	0.0862536	+	0.996273i	$0.472510\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.60834	0.221388
$892$	0	0
$893$	−7.43477	−0.248795
$894$	0	0
$895$	−15.4355	−0.515952
$896$	0	0
$897$	−30.4815	−1.01775
$898$	0	0
$899$	73.6085	2.45498
$900$	0	0
$901$	−23.7363	−0.790771
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21.9525	−0.729724
$906$	0	0
$907$	47.8817	1.58989	0.794943	−	0.606685i	$-0.207501\pi$
$907$	47.8817	1.58989	0.794943	+	0.606685i	$0.207501\pi$
$908$	0	0
$909$	−6.94914	−0.230489
$910$	0	0
$911$	34.3791	1.13903	0.569516	−	0.821981i	$-0.307131\pi$
$911$	34.3791	1.13903	0.569516	+	0.821981i	$0.307131\pi$
$912$	0	0
$913$	−9.81005	−0.324665
$914$	0	0
$915$	−36.7682	−1.21552
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0.0563714	0.00185952	0.000929760	−	1.00000i	$-0.499704\pi$
$919$	0.0563714	0.00185952	0.000929760		1.00000i	$0.499704\pi$
$920$	0	0
$921$	21.3590	0.703804
$922$	0	0
$923$	−3.80547	−0.125259
$924$	0	0
$925$	−10.2509	−0.337049
$926$	0	0
$927$	−13.3782	−0.439399
$928$	0	0
$929$	−8.49686	−0.278773	−0.139387	−	0.990238i	$-0.544513\pi$
$929$	−8.49686	−0.278773	−0.139387	+	0.990238i	$0.544513\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−31.3095	−1.02503
$934$	0	0
$935$	4.82948	0.157941
$936$	0	0
$937$	−14.4166	−0.470969	−0.235485	−	0.971878i	$-0.575668\pi$
$937$	−14.4166	−0.470969	−0.235485	+	0.971878i	$0.575668\pi$
$938$	0	0
$939$	50.9520	1.66276
$940$	0	0
$941$	49.9747	1.62913	0.814565	−	0.580072i	$-0.196976\pi$
$941$	49.9747	1.62913	0.814565	+	0.580072i	$0.196976\pi$
$942$	0	0
$943$	9.22765	0.300494
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.2007	−0.428965	−0.214483	−	0.976728i	$-0.568807\pi$
$947$	−13.2007	−0.428965	−0.214483	+	0.976728i	$0.568807\pi$
$948$	0	0
$949$	−12.4267	−0.403387
$950$	0	0
$951$	26.5044	0.859463
$952$	0	0
$953$	27.3461	0.885826	0.442913	−	0.896565i	$-0.353945\pi$
$953$	27.3461	0.885826	0.442913	+	0.896565i	$0.353945\pi$
$954$	0	0
$955$	−52.5884	−1.70172
$956$	0	0
$957$	−12.5379	−0.405294
$958$	0	0
$959$	0	0
$960$	0	0
$961$	18.4052	0.593718
$962$	0	0
$963$	−8.61621	−0.277654
$964$	0	0
$965$	3.25044	0.104635
$966$	0	0
$967$	−39.6323	−1.27449	−0.637245	−	0.770662i	$-0.719926\pi$
$967$	−39.6323	−1.27449	−0.637245	+	0.770662i	$0.719926\pi$
$968$	0	0
$969$	−8.47291	−0.272189
$970$	0	0
$971$	−28.7757	−0.923456	−0.461728	−	0.887022i	$-0.652770\pi$
$971$	−28.7757	−0.923456	−0.461728	+	0.887022i	$0.652770\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	7.32060	0.234447
$976$	0	0
$977$	16.0596	0.513793	0.256896	−	0.966439i	$-0.417300\pi$
$977$	16.0596	0.513793	0.256896	+	0.966439i	$0.417300\pi$
$978$	0	0
$979$	6.37652	0.203794
$980$	0	0
$981$	−6.91737	−0.220855
$982$	0	0
$983$	31.8476	1.01578	0.507890	−	0.861422i	$-0.330425\pi$
$983$	31.8476	1.01578	0.507890	+	0.861422i	$0.330425\pi$
$984$	0	0
$985$	−38.5782	−1.22920
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−45.1148	−1.43457
$990$	0	0
$991$	10.7518	0.341541	0.170771	−	0.985311i	$-0.445374\pi$
$991$	10.7518	0.341541	0.170771	+	0.985311i	$0.445374\pi$
$992$	0	0
$993$	−27.2294	−0.864099
$994$	0	0
$995$	21.3145	0.675716
$996$	0	0
$997$	27.6437	0.875486	0.437743	−	0.899100i	$-0.355778\pi$
$997$	27.6437	0.875486	0.437743	+	0.899100i	$0.355778\pi$
$998$	0	0
$999$	20.0250	0.633563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.l.1.4		5
7.6	odd	2		1148.2.a.c.1.2	✓	5
28.27	even	2		4592.2.a.be.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.c.1.2	✓	5	7.6	odd	2
4592.2.a.be.1.4		5	28.27	even	2
8036.2.a.l.1.4		5	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.94177 q^{3} -2.68629 q^{5} +0.770473 q^{9} +O(q^{10})\)	\(q+1.94177 q^{3} -2.68629 q^{5} +0.770473 q^{9} -0.616576 q^{11} +1.70117 q^{13} -5.21616 q^{15} +2.91582 q^{17} -1.49649 q^{19} -9.22765 q^{23} +2.21616 q^{25} -4.32923 q^{27} +10.4723 q^{29} +7.02889 q^{31} -1.19725 q^{33} -4.62554 q^{37} +3.30328 q^{39} -1.00000 q^{41} +4.88909 q^{43} -2.06972 q^{45} +4.96813 q^{47} +5.66185 q^{51} -8.14053 q^{53} +1.65630 q^{55} -2.90584 q^{57} +9.71628 q^{59} +7.04890 q^{61} -4.56983 q^{65} -10.8776 q^{67} -17.9180 q^{69} -2.23698 q^{71} -7.30479 q^{73} +4.30328 q^{75} -12.0081 q^{79} -10.7178 q^{81} +15.9105 q^{83} -7.83274 q^{85} +20.3348 q^{87} -10.3418 q^{89} +13.6485 q^{93} +4.02001 q^{95} -9.32921 q^{97} -0.475055 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} + q^{5} + q^{9}+O(q^{10}) \) 5 * q + 2 * q^3 + q^5 + q^9	\( 5 q + 2 q^{3} + q^{5} + q^{9} - 2 q^{11} + q^{13} - 9 q^{15} + 3 q^{17} + 4 q^{19} - 8 q^{23} - 6 q^{25} + 8 q^{27} - 9 q^{29} + 11 q^{31} - 5 q^{33} - 11 q^{37} - 17 q^{39} - 5 q^{41} - 27 q^{43} + 3 q^{45} + 3 q^{47} - 3 q^{51} - 19 q^{53} + 13 q^{55} - 11 q^{57} + 15 q^{59} + 7 q^{65} - 21 q^{67} - 14 q^{69} - 16 q^{71} + 10 q^{73} - 12 q^{75} - 14 q^{79} - 7 q^{81} + 2 q^{83} - 21 q^{85} + 36 q^{87} - 6 q^{89} + 17 q^{93} + 9 q^{95} - 20 q^{97} - 17 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 + q^5 + q^9 - 2 * q^11 + q^13 - 9 * q^15 + 3 * q^17 + 4 * q^19 - 8 * q^23 - 6 * q^25 + 8 * q^27 - 9 * q^29 + 11 * q^31 - 5 * q^33 - 11 * q^37 - 17 * q^39 - 5 * q^41 - 27 * q^43 + 3 * q^45 + 3 * q^47 - 3 * q^51 - 19 * q^53 + 13 * q^55 - 11 * q^57 + 15 * q^59 + 7 * q^65 - 21 * q^67 - 14 * q^69 - 16 * q^71 + 10 * q^73 - 12 * q^75 - 14 * q^79 - 7 * q^81 + 2 * q^83 - 21 * q^85 + 36 * q^87 - 6 * q^89 + 17 * q^93 + 9 * q^95 - 20 * q^97 - 17 * q^99

Introduction

L-functions

Modular forms

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Groups

Database

Properties

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