LMFDB - Embedded newform 8036.2.a.g.1.2

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
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			1.2
Root			$2.30278$ 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)

$f(q)$	$=$	$q+3.30278 q^{3} +3.30278 q^{5} +7.90833 q^{9} +O(q^{10})$	$q+3.30278 q^{3} +3.30278 q^{5} +7.90833 q^{9} +1.00000 q^{11} +5.00000 q^{13} +10.9083 q^{15} -6.21110 q^{17} -3.90833 q^{19} -3.30278 q^{23} +5.90833 q^{25} +16.2111 q^{27} +10.3028 q^{29} -0.0916731 q^{31} +3.30278 q^{33} -0.605551 q^{37} +16.5139 q^{39} +1.00000 q^{41} -11.6056 q^{43} +26.1194 q^{45} +5.39445 q^{47} -20.5139 q^{51} -2.69722 q^{53} +3.30278 q^{55} -12.9083 q^{57} -5.90833 q^{59} +10.2111 q^{61} +16.5139 q^{65} +1.90833 q^{67} -10.9083 q^{69} -12.2111 q^{71} -7.00000 q^{73} +19.5139 q^{75} -16.6056 q^{79} +29.8167 q^{81} +10.8167 q^{83} -20.5139 q^{85} +34.0278 q^{87} +8.51388 q^{89} -0.302776 q^{93} -12.9083 q^{95} -9.11943 q^{97} +7.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 3 q^{5} + 5 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 + 3 * q^5 + 5 * q^9	$ 2 q + 3 q^{3} + 3 q^{5} + 5 q^{9} + 2 q^{11} + 10 q^{13} + 11 q^{15} + 2 q^{17} + 3 q^{19} - 3 q^{23} + q^{25} + 18 q^{27} + 17 q^{29} - 11 q^{31} + 3 q^{33} + 6 q^{37} + 15 q^{39} + 2 q^{41} - 16 q^{43} + 27 q^{45} + 18 q^{47} - 23 q^{51} - 9 q^{53} + 3 q^{55} - 15 q^{57} - q^{59} + 6 q^{61} + 15 q^{65} - 7 q^{67} - 11 q^{69} - 10 q^{71} - 14 q^{73} + 21 q^{75} - 26 q^{79} + 38 q^{81} - 23 q^{85} + 32 q^{87} - q^{89} + 3 q^{93} - 15 q^{95} + 7 q^{97} + 5 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 3 * q^5 + 5 * q^9 + 2 * q^11 + 10 * q^13 + 11 * q^15 + 2 * q^17 + 3 * q^19 - 3 * q^23 + q^25 + 18 * q^27 + 17 * q^29 - 11 * q^31 + 3 * q^33 + 6 * q^37 + 15 * q^39 + 2 * q^41 - 16 * q^43 + 27 * q^45 + 18 * q^47 - 23 * q^51 - 9 * q^53 + 3 * q^55 - 15 * q^57 - q^59 + 6 * q^61 + 15 * q^65 - 7 * q^67 - 11 * q^69 - 10 * q^71 - 14 * q^73 + 21 * q^75 - 26 * q^79 + 38 * q^81 - 23 * q^85 + 32 * q^87 - q^89 + 3 * q^93 - 15 * q^95 + 7 * q^97 + 5 * 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
$3$	3.30278	1.90686	0.953429	−	0.301617i	$-0.0975264\pi$
$3$	3.30278	1.90686	0.953429	+	0.301617i	$0.0975264\pi$
$4$	0	0
$5$	3.30278	1.47705	0.738523	−	0.674228i	$-0.235524\pi$
$5$	3.30278	1.47705	0.738523	+	0.674228i	$0.235524\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.90833	2.63611
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	10.9083	2.81652
$16$	0	0
$17$	−6.21110	−1.50641	−0.753207	−	0.657784i	$-0.771494\pi$
$17$	−6.21110	−1.50641	−0.753207	+	0.657784i	$0.771494\pi$
$18$	0	0
$19$	−3.90833	−0.896632	−0.448316	−	0.893875i	$-0.647976\pi$
$19$	−3.90833	−0.896632	−0.448316	+	0.893875i	$0.647976\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.30278	−0.688676	−0.344338	−	0.938846i	$-0.611897\pi$
$23$	−3.30278	−0.688676	−0.344338	+	0.938846i	$0.611897\pi$
$24$	0	0
$25$	5.90833	1.18167
$26$	0	0
$27$	16.2111	3.11983
$28$	0	0
$29$	10.3028	1.91318	0.956589	−	0.291441i	$-0.0941348\pi$
$29$	10.3028	1.91318	0.956589	+	0.291441i	$0.0941348\pi$
$30$	0	0
$31$	−0.0916731	−0.0164650	−0.00823249	−	0.999966i	$-0.502621\pi$
$31$	−0.0916731	−0.0164650	−0.00823249	+	0.999966i	$0.502621\pi$
$32$	0	0
$33$	3.30278	0.574939
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.605551	−0.0995520	−0.0497760	−	0.998760i	$-0.515851\pi$
$37$	−0.605551	−0.0995520	−0.0497760	+	0.998760i	$0.515851\pi$
$38$	0	0
$39$	16.5139	2.64434
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−11.6056	−1.76983	−0.884915	−	0.465753i	$-0.845784\pi$
$43$	−11.6056	−1.76983	−0.884915	+	0.465753i	$0.845784\pi$
$44$	0	0
$45$	26.1194	3.89365
$46$	0	0
$47$	5.39445	0.786861	0.393431	−	0.919354i	$-0.371288\pi$
$47$	5.39445	0.786861	0.393431	+	0.919354i	$0.371288\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−20.5139	−2.87252
$52$	0	0
$53$	−2.69722	−0.370492	−0.185246	−	0.982692i	$-0.559308\pi$
$53$	−2.69722	−0.370492	−0.185246	+	0.982692i	$0.559308\pi$
$54$	0	0
$55$	3.30278	0.445346
$56$	0	0
$57$	−12.9083	−1.70975
$58$	0	0
$59$	−5.90833	−0.769199	−0.384599	−	0.923084i	$-0.625660\pi$
$59$	−5.90833	−0.769199	−0.384599	+	0.923084i	$0.625660\pi$
$60$	0	0
$61$	10.2111	1.30740	0.653699	−	0.756755i	$-0.273216\pi$
$61$	10.2111	1.30740	0.653699	+	0.756755i	$0.273216\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16.5139	2.04829
$66$	0	0
$67$	1.90833	0.233139	0.116570	−	0.993183i	$-0.462810\pi$
$67$	1.90833	0.233139	0.116570	+	0.993183i	$0.462810\pi$
$68$	0	0
$69$	−10.9083	−1.31321
$70$	0	0
$71$	−12.2111	−1.44919	−0.724596	−	0.689174i	$-0.757973\pi$
$71$	−12.2111	−1.44919	−0.724596	+	0.689174i	$0.757973\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	19.5139	2.25327
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.6056	−1.86827	−0.934135	−	0.356919i	$-0.883827\pi$
$79$	−16.6056	−1.86827	−0.934135	+	0.356919i	$0.883827\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	0	0
$83$	10.8167	1.18728	0.593641	−	0.804730i	$-0.297690\pi$
$83$	10.8167	1.18728	0.593641	+	0.804730i	$0.297690\pi$
$84$	0	0
$85$	−20.5139	−2.22504
$86$	0	0
$87$	34.0278	3.64816
$88$	0	0
$89$	8.51388	0.902469	0.451235	−	0.892405i	$-0.350984\pi$
$89$	8.51388	0.902469	0.451235	+	0.892405i	$0.350984\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.302776	−0.0313964
$94$	0	0
$95$	−12.9083	−1.32437
$96$	0	0
$97$	−9.11943	−0.925938	−0.462969	−	0.886375i	$-0.653216\pi$
$97$	−9.11943	−0.925938	−0.462969	+	0.886375i	$0.653216\pi$
$98$	0	0
$99$	7.90833	0.794817
$100$	0	0
$101$	−5.60555	−0.557773	−0.278887	−	0.960324i	$-0.589965\pi$
$101$	−5.60555	−0.557773	−0.278887	+	0.960324i	$0.589965\pi$
$102$	0	0
$103$	7.69722	0.758430	0.379215	−	0.925309i	$-0.376194\pi$
$103$	7.69722	0.758430	0.379215	+	0.925309i	$0.376194\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.8167	1.52905	0.764527	−	0.644592i	$-0.222973\pi$
$107$	15.8167	1.52905	0.764527	+	0.644592i	$0.222973\pi$
$108$	0	0
$109$	11.2111	1.07383	0.536914	−	0.843637i	$-0.319590\pi$
$109$	11.2111	1.07383	0.536914	+	0.843637i	$0.319590\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	5.78890	0.544574	0.272287	−	0.962216i	$-0.412220\pi$
$113$	5.78890	0.544574	0.272287	+	0.962216i	$0.412220\pi$
$114$	0	0
$115$	−10.9083	−1.01721
$116$	0	0
$117$	39.5416	3.65563
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	3.30278	0.297801
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	−38.3305	−3.37482
$130$	0	0
$131$	−21.9083	−1.91414	−0.957070	−	0.289858i	$-0.906392\pi$
$131$	−21.9083	−1.91414	−0.957070	+	0.289858i	$0.906392\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	53.5416	4.60813
$136$	0	0
$137$	2.09167	0.178704	0.0893518	−	0.996000i	$-0.471520\pi$
$137$	2.09167	0.178704	0.0893518	+	0.996000i	$0.471520\pi$
$138$	0	0
$139$	15.6056	1.32365	0.661823	−	0.749660i	$-0.269783\pi$
$139$	15.6056	1.32365	0.661823	+	0.749660i	$0.269783\pi$
$140$	0	0
$141$	17.8167	1.50043
$142$	0	0
$143$	5.00000	0.418121
$144$	0	0
$145$	34.0278	2.82585
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.788897	−0.0646290	−0.0323145	−	0.999478i	$-0.510288\pi$
$149$	−0.788897	−0.0646290	−0.0323145	+	0.999478i	$0.510288\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	−49.1194	−3.97107
$154$	0	0
$155$	−0.302776	−0.0243195
$156$	0	0
$157$	18.4222	1.47025	0.735126	−	0.677930i	$-0.237123\pi$
$157$	18.4222	1.47025	0.735126	+	0.677930i	$0.237123\pi$
$158$	0	0
$159$	−8.90833	−0.706476
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−17.2111	−1.34808	−0.674039	−	0.738696i	$-0.735442\pi$
$163$	−17.2111	−1.34808	−0.674039	+	0.738696i	$0.735442\pi$
$164$	0	0
$165$	10.9083	0.849212
$166$	0	0
$167$	−4.21110	−0.325865	−0.162932	−	0.986637i	$-0.552095\pi$
$167$	−4.21110	−0.325865	−0.162932	+	0.986637i	$0.552095\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−30.9083	−2.36362
$172$	0	0
$173$	−10.8167	−0.822375	−0.411187	−	0.911551i	$-0.634886\pi$
$173$	−10.8167	−0.822375	−0.411187	+	0.911551i	$0.634886\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−19.5139	−1.46675
$178$	0	0
$179$	−9.11943	−0.681618	−0.340809	−	0.940133i	$-0.610701\pi$
$179$	−9.11943	−0.681618	−0.340809	+	0.940133i	$0.610701\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	33.7250	2.49302
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−6.21110	−0.454201
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.21110	−0.377062	−0.188531	−	0.982067i	$-0.560373\pi$
$191$	−5.21110	−0.377062	−0.188531	+	0.982067i	$0.560373\pi$
$192$	0	0
$193$	−13.4222	−0.966151	−0.483076	−	0.875579i	$-0.660480\pi$
$193$	−13.4222	−0.966151	−0.483076	+	0.875579i	$0.660480\pi$
$194$	0	0
$195$	54.5416	3.90581
$196$	0	0
$197$	−1.90833	−0.135963	−0.0679813	−	0.997687i	$-0.521656\pi$
$197$	−1.90833	−0.135963	−0.0679813	+	0.997687i	$0.521656\pi$
$198$	0	0
$199$	0.211103	0.0149647	0.00748233	−	0.999972i	$-0.497618\pi$
$199$	0.211103	0.0149647	0.00748233	+	0.999972i	$0.497618\pi$
$200$	0	0
$201$	6.30278	0.444564
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.30278	0.230676
$206$	0	0
$207$	−26.1194	−1.81543
$208$	0	0
$209$	−3.90833	−0.270345
$210$	0	0
$211$	−16.1194	−1.10971	−0.554854	−	0.831948i	$-0.687226\pi$
$211$	−16.1194	−1.10971	−0.554854	+	0.831948i	$0.687226\pi$
$212$	0	0
$213$	−40.3305	−2.76340
$214$	0	0
$215$	−38.3305	−2.61412
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−23.1194	−1.56227
$220$	0	0
$221$	−31.0555	−2.08902
$222$	0	0
$223$	4.69722	0.314549	0.157275	−	0.987555i	$-0.449729\pi$
$223$	4.69722	0.314549	0.157275	+	0.987555i	$0.449729\pi$
$224$	0	0
$225$	46.7250	3.11500
$226$	0	0
$227$	−4.81665	−0.319693	−0.159846	−	0.987142i	$-0.551100\pi$
$227$	−4.81665	−0.319693	−0.159846	+	0.987142i	$0.551100\pi$
$228$	0	0
$229$	28.6056	1.89031	0.945154	−	0.326625i	$-0.105911\pi$
$229$	28.6056	1.89031	0.945154	+	0.326625i	$0.105911\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.9083	−1.36975	−0.684875	−	0.728661i	$-0.740143\pi$
$233$	−20.9083	−1.36975	−0.684875	+	0.728661i	$0.740143\pi$
$234$	0	0
$235$	17.8167	1.16223
$236$	0	0
$237$	−54.8444	−3.56253
$238$	0	0
$239$	−21.5139	−1.39162	−0.695809	−	0.718227i	$-0.744954\pi$
$239$	−21.5139	−1.39162	−0.695809	+	0.718227i	$0.744954\pi$
$240$	0	0
$241$	−11.8167	−0.761178	−0.380589	−	0.924744i	$-0.624279\pi$
$241$	−11.8167	−0.761178	−0.380589	+	0.924744i	$0.624279\pi$
$242$	0	0
$243$	49.8444	3.19752
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−19.5416	−1.24340
$248$	0	0
$249$	35.7250	2.26398
$250$	0	0
$251$	4.81665	0.304024	0.152012	−	0.988379i	$-0.451425\pi$
$251$	4.81665	0.304024	0.152012	+	0.988379i	$0.451425\pi$
$252$	0	0
$253$	−3.30278	−0.207644
$254$	0	0
$255$	−67.7527	−4.24284
$256$	0	0
$257$	19.0000	1.18519	0.592594	−	0.805502i	$-0.298104\pi$
$257$	19.0000	1.18519	0.592594	+	0.805502i	$0.298104\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	81.4777	5.04334
$262$	0	0
$263$	8.39445	0.517624	0.258812	−	0.965928i	$-0.416669\pi$
$263$	8.39445	0.517624	0.258812	+	0.965928i	$0.416669\pi$
$264$	0	0
$265$	−8.90833	−0.547234
$266$	0	0
$267$	28.1194	1.72088
$268$	0	0
$269$	−22.3028	−1.35982	−0.679912	−	0.733294i	$-0.737982\pi$
$269$	−22.3028	−1.35982	−0.679912	+	0.733294i	$0.737982\pi$
$270$	0	0
$271$	19.0000	1.15417	0.577084	−	0.816685i	$-0.304191\pi$
$271$	19.0000	1.15417	0.577084	+	0.816685i	$0.304191\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.90833	0.356286
$276$	0	0
$277$	−11.6056	−0.697310	−0.348655	−	0.937251i	$-0.613362\pi$
$277$	−11.6056	−0.697310	−0.348655	+	0.937251i	$0.613362\pi$
$278$	0	0
$279$	−0.724981	−0.0434035
$280$	0	0
$281$	−5.00000	−0.298275	−0.149137	−	0.988816i	$-0.547650\pi$
$281$	−5.00000	−0.298275	−0.149137	+	0.988816i	$0.547650\pi$
$282$	0	0
$283$	−11.6333	−0.691528	−0.345764	−	0.938321i	$-0.612380\pi$
$283$	−11.6333	−0.691528	−0.345764	+	0.938321i	$0.612380\pi$
$284$	0	0
$285$	−42.6333	−2.52538
$286$	0	0
$287$	0	0
$288$	0	0
$289$	21.5778	1.26928
$290$	0	0
$291$	−30.1194	−1.76563
$292$	0	0
$293$	27.8444	1.62669	0.813344	−	0.581783i	$-0.197645\pi$
$293$	27.8444	1.62669	0.813344	+	0.581783i	$0.197645\pi$
$294$	0	0
$295$	−19.5139	−1.13614
$296$	0	0
$297$	16.2111	0.940664
$298$	0	0
$299$	−16.5139	−0.955022
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−18.5139	−1.06359
$304$	0	0
$305$	33.7250	1.93109
$306$	0	0
$307$	−3.21110	−0.183267	−0.0916337	−	0.995793i	$-0.529209\pi$
$307$	−3.21110	−0.183267	−0.0916337	+	0.995793i	$0.529209\pi$
$308$	0	0
$309$	25.4222	1.44622
$310$	0	0
$311$	4.90833	0.278326	0.139163	−	0.990270i	$-0.455559\pi$
$311$	4.90833	0.278326	0.139163	+	0.990270i	$0.455559\pi$
$312$	0	0
$313$	−8.39445	−0.474482	−0.237241	−	0.971451i	$-0.576243\pi$
$313$	−8.39445	−0.474482	−0.237241	+	0.971451i	$0.576243\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.78890	0.100474	0.0502372	−	0.998737i	$-0.484002\pi$
$317$	1.78890	0.100474	0.0502372	+	0.998737i	$0.484002\pi$
$318$	0	0
$319$	10.3028	0.576845
$320$	0	0
$321$	52.2389	2.91569
$322$	0	0
$323$	24.2750	1.35070
$324$	0	0
$325$	29.5416	1.63868
$326$	0	0
$327$	37.0278	2.04764
$328$	0	0
$329$	0	0
$330$	0	0
$331$	15.6333	0.859284	0.429642	−	0.902999i	$-0.358640\pi$
$331$	15.6333	0.859284	0.429642	+	0.902999i	$0.358640\pi$
$332$	0	0
$333$	−4.78890	−0.262430
$334$	0	0
$335$	6.30278	0.344357
$336$	0	0
$337$	0.788897	0.0429740	0.0214870	−	0.999769i	$-0.493160\pi$
$337$	0.788897	0.0429740	0.0214870	+	0.999769i	$0.493160\pi$
$338$	0	0
$339$	19.1194	1.03842
$340$	0	0
$341$	−0.0916731	−0.00496438
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−36.0278	−1.93967
$346$	0	0
$347$	−19.3944	−1.04115	−0.520574	−	0.853816i	$-0.674282\pi$
$347$	−19.3944	−1.04115	−0.520574	+	0.853816i	$0.674282\pi$
$348$	0	0
$349$	−24.0278	−1.28618	−0.643088	−	0.765792i	$-0.722347\pi$
$349$	−24.0278	−1.28618	−0.643088	+	0.765792i	$0.722347\pi$
$350$	0	0
$351$	81.0555	4.32642
$352$	0	0
$353$	30.8444	1.64168	0.820841	−	0.571157i	$-0.193505\pi$
$353$	30.8444	1.64168	0.820841	+	0.571157i	$0.193505\pi$
$354$	0	0
$355$	−40.3305	−2.14052
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.60555	0.454184	0.227092	−	0.973873i	$-0.427078\pi$
$359$	8.60555	0.454184	0.227092	+	0.973873i	$0.427078\pi$
$360$	0	0
$361$	−3.72498	−0.196052
$362$	0	0
$363$	−33.0278	−1.73351
$364$	0	0
$365$	−23.1194	−1.21013
$366$	0	0
$367$	−17.5139	−0.914217	−0.457108	−	0.889411i	$-0.651115\pi$
$367$	−17.5139	−0.914217	−0.457108	+	0.889411i	$0.651115\pi$
$368$	0	0
$369$	7.90833	0.411691
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−14.6972	−0.760993	−0.380497	−	0.924782i	$-0.624247\pi$
$373$	−14.6972	−0.760993	−0.380497	+	0.924782i	$0.624247\pi$
$374$	0	0
$375$	9.90833	0.511664
$376$	0	0
$377$	51.5139	2.65310
$378$	0	0
$379$	−15.8167	−0.812447	−0.406223	−	0.913774i	$-0.633155\pi$
$379$	−15.8167	−0.812447	−0.406223	+	0.913774i	$0.633155\pi$
$380$	0	0
$381$	23.1194	1.18444
$382$	0	0
$383$	−13.6056	−0.695211	−0.347606	−	0.937641i	$-0.613005\pi$
$383$	−13.6056	−0.695211	−0.347606	+	0.937641i	$0.613005\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−91.7805	−4.66546
$388$	0	0
$389$	6.90833	0.350266	0.175133	−	0.984545i	$-0.443964\pi$
$389$	6.90833	0.350266	0.175133	+	0.984545i	$0.443964\pi$
$390$	0	0
$391$	20.5139	1.03743
$392$	0	0
$393$	−72.3583	−3.64999
$394$	0	0
$395$	−54.8444	−2.75952
$396$	0	0
$397$	−13.7250	−0.688837	−0.344419	−	0.938816i	$-0.611924\pi$
$397$	−13.7250	−0.688837	−0.344419	+	0.938816i	$0.611924\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.18335	−0.109031	−0.0545156	−	0.998513i	$-0.517361\pi$
$401$	−2.18335	−0.109031	−0.0545156	+	0.998513i	$0.517361\pi$
$402$	0	0
$403$	−0.458365	−0.0228328
$404$	0	0
$405$	98.4777	4.89340
$406$	0	0
$407$	−0.605551	−0.0300161
$408$	0	0
$409$	−3.60555	−0.178283	−0.0891415	−	0.996019i	$-0.528412\pi$
$409$	−3.60555	−0.178283	−0.0891415	+	0.996019i	$0.528412\pi$
$410$	0	0
$411$	6.90833	0.340763
$412$	0	0
$413$	0	0
$414$	0	0
$415$	35.7250	1.75367
$416$	0	0
$417$	51.5416	2.52400
$418$	0	0
$419$	20.7250	1.01248	0.506241	−	0.862392i	$-0.331035\pi$
$419$	20.7250	1.01248	0.506241	+	0.862392i	$0.331035\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	0	0
$423$	42.6611	2.07425
$424$	0	0
$425$	−36.6972	−1.78008
$426$	0	0
$427$	0	0
$428$	0	0
$429$	16.5139	0.797298
$430$	0	0
$431$	−0.394449	−0.0189999	−0.00949996	−	0.999955i	$-0.503024\pi$
$431$	−0.394449	−0.0189999	−0.00949996	+	0.999955i	$0.503024\pi$
$432$	0	0
$433$	12.0917	0.581089	0.290544	−	0.956862i	$-0.406164\pi$
$433$	12.0917	0.581089	0.290544	+	0.956862i	$0.406164\pi$
$434$	0	0
$435$	112.386	5.38850
$436$	0	0
$437$	12.9083	0.617489
$438$	0	0
$439$	−20.6333	−0.984774	−0.492387	−	0.870376i	$-0.663876\pi$
$439$	−20.6333	−0.984774	−0.492387	+	0.870376i	$0.663876\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.5416	0.928451	0.464226	−	0.885717i	$-0.346333\pi$
$443$	19.5416	0.928451	0.464226	+	0.885717i	$0.346333\pi$
$444$	0	0
$445$	28.1194	1.33299
$446$	0	0
$447$	−2.60555	−0.123238
$448$	0	0
$449$	−23.2111	−1.09540	−0.547700	−	0.836675i	$-0.684496\pi$
$449$	−23.2111	−1.09540	−0.547700	+	0.836675i	$0.684496\pi$
$450$	0	0
$451$	1.00000	0.0470882
$452$	0	0
$453$	−16.5139	−0.775890
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.18335	0.429579	0.214789	−	0.976660i	$-0.431093\pi$
$457$	9.18335	0.429579	0.214789	+	0.976660i	$0.431093\pi$
$458$	0	0
$459$	−100.689	−4.69975
$460$	0	0
$461$	2.81665	0.131185	0.0655923	−	0.997847i	$-0.479106\pi$
$461$	2.81665	0.131185	0.0655923	+	0.997847i	$0.479106\pi$
$462$	0	0
$463$	−27.9361	−1.29830	−0.649150	−	0.760660i	$-0.724875\pi$
$463$	−27.9361	−1.29830	−0.649150	+	0.760660i	$0.724875\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	6.21110	0.287416	0.143708	−	0.989620i	$-0.454097\pi$
$467$	6.21110	0.287416	0.143708	+	0.989620i	$0.454097\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	60.8444	2.80356
$472$	0	0
$473$	−11.6056	−0.533624
$474$	0	0
$475$	−23.0917	−1.05952
$476$	0	0
$477$	−21.3305	−0.976658
$478$	0	0
$479$	−14.8806	−0.679911	−0.339955	−	0.940442i	$-0.610412\pi$
$479$	−14.8806	−0.679911	−0.339955	+	0.940442i	$0.610412\pi$
$480$	0	0
$481$	−3.02776	−0.138054
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−30.1194	−1.36765
$486$	0	0
$487$	−24.6056	−1.11498	−0.557492	−	0.830182i	$-0.688236\pi$
$487$	−24.6056	−1.11498	−0.557492	+	0.830182i	$0.688236\pi$
$488$	0	0
$489$	−56.8444	−2.57059
$490$	0	0
$491$	24.7250	1.11582	0.557911	−	0.829901i	$-0.311603\pi$
$491$	24.7250	1.11582	0.557911	+	0.829901i	$0.311603\pi$
$492$	0	0
$493$	−63.9916	−2.88204
$494$	0	0
$495$	26.1194	1.17398
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20.2389	0.906016	0.453008	−	0.891507i	$-0.350351\pi$
$499$	20.2389	0.906016	0.453008	+	0.891507i	$0.350351\pi$
$500$	0	0
$501$	−13.9083	−0.621378
$502$	0	0
$503$	16.3028	0.726905	0.363452	−	0.931613i	$-0.381598\pi$
$503$	16.3028	0.726905	0.363452	+	0.931613i	$0.381598\pi$
$504$	0	0
$505$	−18.5139	−0.823857
$506$	0	0
$507$	39.6333	1.76018
$508$	0	0
$509$	34.9083	1.54728	0.773642	−	0.633623i	$-0.218433\pi$
$509$	34.9083	1.54728	0.773642	+	0.633623i	$0.218433\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−63.3583	−2.79734
$514$	0	0
$515$	25.4222	1.12024
$516$	0	0
$517$	5.39445	0.237248
$518$	0	0
$519$	−35.7250	−1.56815
$520$	0	0
$521$	−6.69722	−0.293411	−0.146705	−	0.989180i	$-0.546867\pi$
$521$	−6.69722	−0.293411	−0.146705	+	0.989180i	$0.546867\pi$
$522$	0	0
$523$	22.3944	0.979241	0.489620	−	0.871936i	$-0.337135\pi$
$523$	22.3944	0.979241	0.489620	+	0.871936i	$0.337135\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.569391	0.0248031
$528$	0	0
$529$	−12.0917	−0.525725
$530$	0	0
$531$	−46.7250	−2.02769
$532$	0	0
$533$	5.00000	0.216574
$534$	0	0
$535$	52.2389	2.25848
$536$	0	0
$537$	−30.1194	−1.29975
$538$	0	0
$539$	0	0
$540$	0	0
$541$	35.3583	1.52017	0.760086	−	0.649823i	$-0.225157\pi$
$541$	35.3583	1.52017	0.760086	+	0.649823i	$0.225157\pi$
$542$	0	0
$543$	33.0278	1.41736
$544$	0	0
$545$	37.0278	1.58609
$546$	0	0
$547$	−31.4222	−1.34352	−0.671758	−	0.740770i	$-0.734461\pi$
$547$	−31.4222	−1.34352	−0.671758	+	0.740770i	$0.734461\pi$
$548$	0	0
$549$	80.7527	3.44644
$550$	0	0
$551$	−40.2666	−1.71542
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−6.60555	−0.280390
$556$	0	0
$557$	−40.5416	−1.71780	−0.858902	−	0.512140i	$-0.828853\pi$
$557$	−40.5416	−1.71780	−0.858902	+	0.512140i	$0.828853\pi$
$558$	0	0
$559$	−58.0278	−2.45431
$560$	0	0
$561$	−20.5139	−0.866097
$562$	0	0
$563$	−8.63331	−0.363851	−0.181925	−	0.983312i	$-0.558233\pi$
$563$	−8.63331	−0.363851	−0.181925	+	0.983312i	$0.558233\pi$
$564$	0	0
$565$	19.1194	0.804360
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−31.5416	−1.32229	−0.661147	−	0.750256i	$-0.729930\pi$
$569$	−31.5416	−1.32229	−0.661147	+	0.750256i	$0.729930\pi$
$570$	0	0
$571$	2.36669	0.0990430	0.0495215	−	0.998773i	$-0.484230\pi$
$571$	2.36669	0.0990430	0.0495215	+	0.998773i	$0.484230\pi$
$572$	0	0
$573$	−17.2111	−0.719004
$574$	0	0
$575$	−19.5139	−0.813785
$576$	0	0
$577$	21.4222	0.891818	0.445909	−	0.895078i	$-0.352880\pi$
$577$	21.4222	0.891818	0.445909	+	0.895078i	$0.352880\pi$
$578$	0	0
$579$	−44.3305	−1.84231
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.69722	−0.111708
$584$	0	0
$585$	130.597	5.39953
$586$	0	0
$587$	−17.2389	−0.711524	−0.355762	−	0.934577i	$-0.615779\pi$
$587$	−17.2389	−0.711524	−0.355762	+	0.934577i	$0.615779\pi$
$588$	0	0
$589$	0.358288	0.0147630
$590$	0	0
$591$	−6.30278	−0.259262
$592$	0	0
$593$	−32.8444	−1.34876	−0.674379	−	0.738385i	$-0.735589\pi$
$593$	−32.8444	−1.34876	−0.674379	+	0.738385i	$0.735589\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.697224	0.0285355
$598$	0	0
$599$	19.8167	0.809687	0.404843	−	0.914386i	$-0.367326\pi$
$599$	19.8167	0.809687	0.404843	+	0.914386i	$0.367326\pi$
$600$	0	0
$601$	−5.69722	−0.232395	−0.116197	−	0.993226i	$-0.537070\pi$
$601$	−5.69722	−0.232395	−0.116197	+	0.993226i	$0.537070\pi$
$602$	0	0
$603$	15.0917	0.614580
$604$	0	0
$605$	−33.0278	−1.34277
$606$	0	0
$607$	24.0278	0.975257	0.487628	−	0.873051i	$-0.337862\pi$
$607$	24.0278	0.975257	0.487628	+	0.873051i	$0.337862\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26.9722	1.09118
$612$	0	0
$613$	−30.1472	−1.21763	−0.608817	−	0.793311i	$-0.708356\pi$
$613$	−30.1472	−1.21763	−0.608817	+	0.793311i	$0.708356\pi$
$614$	0	0
$615$	10.9083	0.439866
$616$	0	0
$617$	44.0278	1.77249	0.886245	−	0.463216i	$-0.153305\pi$
$617$	44.0278	1.77249	0.886245	+	0.463216i	$0.153305\pi$
$618$	0	0
$619$	14.4861	0.582246	0.291123	−	0.956686i	$-0.405971\pi$
$619$	14.4861	0.582246	0.291123	+	0.956686i	$0.405971\pi$
$620$	0	0
$621$	−53.5416	−2.14855
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.6333	−0.785332
$626$	0	0
$627$	−12.9083	−0.515509
$628$	0	0
$629$	3.76114	0.149967
$630$	0	0
$631$	21.8444	0.869612	0.434806	−	0.900524i	$-0.356817\pi$
$631$	21.8444	0.869612	0.434806	+	0.900524i	$0.356817\pi$
$632$	0	0
$633$	−53.2389	−2.11605
$634$	0	0
$635$	23.1194	0.917467
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−96.5694	−3.82023
$640$	0	0
$641$	−0.633308	−0.0250141	−0.0125071	−	0.999922i	$-0.503981\pi$
$641$	−0.633308	−0.0250141	−0.0125071	+	0.999922i	$0.503981\pi$
$642$	0	0
$643$	40.9638	1.61546	0.807728	−	0.589555i	$-0.200697\pi$
$643$	40.9638	1.61546	0.807728	+	0.589555i	$0.200697\pi$
$644$	0	0
$645$	−126.597	−4.98476
$646$	0	0
$647$	6.02776	0.236976	0.118488	−	0.992956i	$-0.462195\pi$
$647$	6.02776	0.236976	0.118488	+	0.992956i	$0.462195\pi$
$648$	0	0
$649$	−5.90833	−0.231922
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−29.7250	−1.16323	−0.581614	−	0.813465i	$-0.697579\pi$
$653$	−29.7250	−1.16323	−0.581614	+	0.813465i	$0.697579\pi$
$654$	0	0
$655$	−72.3583	−2.82727
$656$	0	0
$657$	−55.3583	−2.15973
$658$	0	0
$659$	12.3944	0.482819	0.241410	−	0.970423i	$-0.422390\pi$
$659$	12.3944	0.482819	0.241410	+	0.970423i	$0.422390\pi$
$660$	0	0
$661$	−28.6056	−1.11263	−0.556313	−	0.830972i	$-0.687785\pi$
$661$	−28.6056	−1.11263	−0.556313	+	0.830972i	$0.687785\pi$
$662$	0	0
$663$	−102.569	−3.98347
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−34.0278	−1.31756
$668$	0	0
$669$	15.5139	0.599801
$670$	0	0
$671$	10.2111	0.394195
$672$	0	0
$673$	38.8167	1.49627	0.748136	−	0.663545i	$-0.230949\pi$
$673$	38.8167	1.49627	0.748136	+	0.663545i	$0.230949\pi$
$674$	0	0
$675$	95.7805	3.68659
$676$	0	0
$677$	−45.3583	−1.74326	−0.871630	−	0.490164i	$-0.836937\pi$
$677$	−45.3583	−1.74326	−0.871630	+	0.490164i	$0.836937\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−15.9083	−0.609608
$682$	0	0
$683$	4.63331	0.177289	0.0886443	−	0.996063i	$-0.471747\pi$
$683$	4.63331	0.177289	0.0886443	+	0.996063i	$0.471747\pi$
$684$	0	0
$685$	6.90833	0.263954
$686$	0	0
$687$	94.4777	3.60455
$688$	0	0
$689$	−13.4861	−0.513780
$690$	0	0
$691$	21.6972	0.825401	0.412701	−	0.910867i	$-0.364586\pi$
$691$	21.6972	0.825401	0.412701	+	0.910867i	$0.364586\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	51.5416	1.95509
$696$	0	0
$697$	−6.21110	−0.235262
$698$	0	0
$699$	−69.0555	−2.61192
$700$	0	0
$701$	−12.0917	−0.456696	−0.228348	−	0.973580i	$-0.573332\pi$
$701$	−12.0917	−0.456696	−0.228348	+	0.973580i	$0.573332\pi$
$702$	0	0
$703$	2.36669	0.0892615
$704$	0	0
$705$	58.8444	2.21621
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0.605551	0.0227420	0.0113710	−	0.999935i	$-0.496380\pi$
$709$	0.605551	0.0227420	0.0113710	+	0.999935i	$0.496380\pi$
$710$	0	0
$711$	−131.322	−4.92496
$712$	0	0
$713$	0.302776	0.0113390
$714$	0	0
$715$	16.5139	0.617584
$716$	0	0
$717$	−71.0555	−2.65362
$718$	0	0
$719$	13.2389	0.493726	0.246863	−	0.969050i	$-0.420600\pi$
$719$	13.2389	0.493726	0.246863	+	0.969050i	$0.420600\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−39.0278	−1.45146
$724$	0	0
$725$	60.8722	2.26074
$726$	0	0
$727$	−45.3305	−1.68122	−0.840608	−	0.541644i	$-0.817802\pi$
$727$	−45.3305	−1.68122	−0.840608	+	0.541644i	$0.817802\pi$
$728$	0	0
$729$	75.1749	2.78426
$730$	0	0
$731$	72.0833	2.66610
$732$	0	0
$733$	8.57779	0.316828	0.158414	−	0.987373i	$-0.449362\pi$
$733$	8.57779	0.316828	0.158414	+	0.987373i	$0.449362\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.90833	0.0702941
$738$	0	0
$739$	0.486122	0.0178823	0.00894114	−	0.999960i	$-0.497154\pi$
$739$	0.486122	0.0178823	0.00894114	+	0.999960i	$0.497154\pi$
$740$	0	0
$741$	−64.5416	−2.37100
$742$	0	0
$743$	51.5694	1.89190	0.945949	−	0.324316i	$-0.105134\pi$
$743$	51.5694	1.89190	0.945949	+	0.324316i	$0.105134\pi$
$744$	0	0
$745$	−2.60555	−0.0954600
$746$	0	0
$747$	85.5416	3.12980
$748$	0	0
$749$	0	0
$750$	0	0
$751$	19.2389	0.702036	0.351018	−	0.936369i	$-0.385836\pi$
$751$	19.2389	0.702036	0.351018	+	0.936369i	$0.385836\pi$
$752$	0	0
$753$	15.9083	0.579732
$754$	0	0
$755$	−16.5139	−0.601002
$756$	0	0
$757$	−7.02776	−0.255428	−0.127714	−	0.991811i	$-0.540764\pi$
$757$	−7.02776	−0.255428	−0.127714	+	0.991811i	$0.540764\pi$
$758$	0	0
$759$	−10.9083	−0.395947
$760$	0	0
$761$	−32.0000	−1.16000	−0.580000	−	0.814617i	$-0.696947\pi$
$761$	−32.0000	−1.16000	−0.580000	+	0.814617i	$0.696947\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−162.230	−5.86545
$766$	0	0
$767$	−29.5416	−1.06669
$768$	0	0
$769$	39.0278	1.40738	0.703688	−	0.710509i	$-0.251535\pi$
$769$	39.0278	1.40738	0.703688	+	0.710509i	$0.251535\pi$
$770$	0	0
$771$	62.7527	2.25998
$772$	0	0
$773$	−35.8444	−1.28923	−0.644617	−	0.764506i	$-0.722983\pi$
$773$	−35.8444	−1.28923	−0.644617	+	0.764506i	$0.722983\pi$
$774$	0	0
$775$	−0.541635	−0.0194561
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.90833	−0.140030
$780$	0	0
$781$	−12.2111	−0.436948
$782$	0	0
$783$	167.019	5.96878
$784$	0	0
$785$	60.8444	2.17163
$786$	0	0
$787$	3.39445	0.120999	0.0604995	−	0.998168i	$-0.480731\pi$
$787$	3.39445	0.120999	0.0604995	+	0.998168i	$0.480731\pi$
$788$	0	0
$789$	27.7250	0.987035
$790$	0	0
$791$	0	0
$792$	0	0
$793$	51.0555	1.81303
$794$	0	0
$795$	−29.4222	−1.04350
$796$	0	0
$797$	−14.6333	−0.518338	−0.259169	−	0.965832i	$-0.583449\pi$
$797$	−14.6333	−0.518338	−0.259169	+	0.965832i	$0.583449\pi$
$798$	0	0
$799$	−33.5055	−1.18534
$800$	0	0
$801$	67.3305	2.37901
$802$	0	0
$803$	−7.00000	−0.247025
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−73.6611	−2.59299
$808$	0	0
$809$	−5.66947	−0.199328	−0.0996639	−	0.995021i	$-0.531777\pi$
$809$	−5.66947	−0.199328	−0.0996639	+	0.995021i	$0.531777\pi$
$810$	0	0
$811$	3.00000	0.105344	0.0526721	−	0.998612i	$-0.483226\pi$
$811$	3.00000	0.105344	0.0526721	+	0.998612i	$0.483226\pi$
$812$	0	0
$813$	62.7527	2.20083
$814$	0	0
$815$	−56.8444	−1.99117
$816$	0	0
$817$	45.3583	1.58689
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17.8444	−0.622774	−0.311387	−	0.950283i	$-0.600794\pi$
$821$	−17.8444	−0.622774	−0.311387	+	0.950283i	$0.600794\pi$
$822$	0	0
$823$	−49.5139	−1.72595	−0.862973	−	0.505251i	$-0.831400\pi$
$823$	−49.5139	−1.72595	−0.862973	+	0.505251i	$0.831400\pi$
$824$	0	0
$825$	19.5139	0.679386
$826$	0	0
$827$	24.2111	0.841903	0.420951	−	0.907083i	$-0.361696\pi$
$827$	24.2111	0.841903	0.420951	+	0.907083i	$0.361696\pi$
$828$	0	0
$829$	−0.633308	−0.0219957	−0.0109978	−	0.999940i	$-0.503501\pi$
$829$	−0.633308	−0.0219957	−0.0109978	+	0.999940i	$0.503501\pi$
$830$	0	0
$831$	−38.3305	−1.32967
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−13.9083	−0.481318
$836$	0	0
$837$	−1.48612	−0.0513679
$838$	0	0
$839$	26.6056	0.918526	0.459263	−	0.888300i	$-0.348114\pi$
$839$	26.6056	0.918526	0.459263	+	0.888300i	$0.348114\pi$
$840$	0	0
$841$	77.1472	2.66025
$842$	0	0
$843$	−16.5139	−0.568768
$844$	0	0
$845$	39.6333	1.36343
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−38.4222	−1.31865
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	10.3667	0.354949	0.177474	−	0.984125i	$-0.443207\pi$
$853$	10.3667	0.354949	0.177474	+	0.984125i	$0.443207\pi$
$854$	0	0
$855$	−102.083	−3.49117
$856$	0	0
$857$	45.5139	1.55472	0.777362	−	0.629053i	$-0.216557\pi$
$857$	45.5139	1.55472	0.777362	+	0.629053i	$0.216557\pi$
$858$	0	0
$859$	6.51388	0.222251	0.111125	−	0.993806i	$-0.464555\pi$
$859$	6.51388	0.222251	0.111125	+	0.993806i	$0.464555\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15.8444	−0.539350	−0.269675	−	0.962951i	$-0.586916\pi$
$863$	−15.8444	−0.539350	−0.269675	+	0.962951i	$0.586916\pi$
$864$	0	0
$865$	−35.7250	−1.21469
$866$	0	0
$867$	71.2666	2.42034
$868$	0	0
$869$	−16.6056	−0.563305
$870$	0	0
$871$	9.54163	0.323306
$872$	0	0
$873$	−72.1194	−2.44087
$874$	0	0
$875$	0	0
$876$	0	0
$877$	54.9638	1.85600	0.927998	−	0.372584i	$-0.121528\pi$
$877$	54.9638	1.85600	0.927998	+	0.372584i	$0.121528\pi$
$878$	0	0
$879$	91.9638	3.10186
$880$	0	0
$881$	23.2111	0.782002	0.391001	−	0.920390i	$-0.372129\pi$
$881$	23.2111	0.782002	0.391001	+	0.920390i	$0.372129\pi$
$882$	0	0
$883$	46.5694	1.56718	0.783592	−	0.621275i	$-0.213385\pi$
$883$	46.5694	1.56718	0.783592	+	0.621275i	$0.213385\pi$
$884$	0	0
$885$	−64.4500	−2.16646
$886$	0	0
$887$	16.8444	0.565580	0.282790	−	0.959182i	$-0.408740\pi$
$887$	16.8444	0.565580	0.282790	+	0.959182i	$0.408740\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	29.8167	0.998895
$892$	0	0
$893$	−21.0833	−0.705525
$894$	0	0
$895$	−30.1194	−1.00678
$896$	0	0
$897$	−54.5416	−1.82109
$898$	0	0
$899$	−0.944487	−0.0315004
$900$	0	0
$901$	16.7527	0.558115
$902$	0	0
$903$	0	0
$904$	0	0
$905$	33.0278	1.09788
$906$	0	0
$907$	30.1194	1.00010	0.500050	−	0.865997i	$-0.333315\pi$
$907$	30.1194	1.00010	0.500050	+	0.865997i	$0.333315\pi$
$908$	0	0
$909$	−44.3305	−1.47035
$910$	0	0
$911$	−28.8444	−0.955658	−0.477829	−	0.878453i	$-0.658576\pi$
$911$	−28.8444	−0.955658	−0.477829	+	0.878453i	$0.658576\pi$
$912$	0	0
$913$	10.8167	0.357979
$914$	0	0
$915$	111.386	3.68231
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.51388	0.0499382	0.0249691	−	0.999688i	$-0.492051\pi$
$919$	1.51388	0.0499382	0.0249691	+	0.999688i	$0.492051\pi$
$920$	0	0
$921$	−10.6056	−0.349465
$922$	0	0
$923$	−61.0555	−2.00967
$924$	0	0
$925$	−3.57779	−0.117637
$926$	0	0
$927$	60.8722	1.99930
$928$	0	0
$929$	32.5416	1.06766	0.533828	−	0.845593i	$-0.320753\pi$
$929$	32.5416	1.06766	0.533828	+	0.845593i	$0.320753\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	16.2111	0.530728
$934$	0	0
$935$	−20.5139	−0.670876
$936$	0	0
$937$	38.0278	1.24231	0.621156	−	0.783687i	$-0.286663\pi$
$937$	38.0278	1.24231	0.621156	+	0.783687i	$0.286663\pi$
$938$	0	0
$939$	−27.7250	−0.904771
$940$	0	0
$941$	5.30278	0.172866	0.0864328	−	0.996258i	$-0.472453\pi$
$941$	5.30278	0.172866	0.0864328	+	0.996258i	$0.472453\pi$
$942$	0	0
$943$	−3.30278	−0.107553
$944$	0	0
$945$	0	0
$946$	0	0
$947$	54.9083	1.78428	0.892140	−	0.451758i	$-0.149203\pi$
$947$	54.9083	1.78428	0.892140	+	0.451758i	$0.149203\pi$
$948$	0	0
$949$	−35.0000	−1.13615
$950$	0	0
$951$	5.90833	0.191591
$952$	0	0
$953$	51.1472	1.65682	0.828410	−	0.560122i	$-0.189246\pi$
$953$	51.1472	1.65682	0.828410	+	0.560122i	$0.189246\pi$
$954$	0	0
$955$	−17.2111	−0.556938
$956$	0	0
$957$	34.0278	1.09996
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.9916	−0.999729
$962$	0	0
$963$	125.083	4.03075
$964$	0	0
$965$	−44.3305	−1.42705
$966$	0	0
$967$	−38.4222	−1.23557	−0.617787	−	0.786345i	$-0.711971\pi$
$967$	−38.4222	−1.23557	−0.617787	+	0.786345i	$0.711971\pi$
$968$	0	0
$969$	80.1749	2.57559
$970$	0	0
$971$	52.0278	1.66965	0.834825	−	0.550515i	$-0.185569\pi$
$971$	52.0278	1.66965	0.834825	+	0.550515i	$0.185569\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	97.5694	3.12472
$976$	0	0
$977$	−19.6972	−0.630170	−0.315085	−	0.949063i	$-0.602033\pi$
$977$	−19.6972	−0.630170	−0.315085	+	0.949063i	$0.602033\pi$
$978$	0	0
$979$	8.51388	0.272105
$980$	0	0
$981$	88.6611	2.83073
$982$	0	0
$983$	4.02776	0.128465	0.0642327	−	0.997935i	$-0.479540\pi$
$983$	4.02776	0.128465	0.0642327	+	0.997935i	$0.479540\pi$
$984$	0	0
$985$	−6.30278	−0.200823
$986$	0	0
$987$	0	0
$988$	0	0
$989$	38.3305	1.21884
$990$	0	0
$991$	36.8167	1.16952	0.584760	−	0.811207i	$-0.301189\pi$
$991$	36.8167	1.16952	0.584760	+	0.811207i	$0.301189\pi$
$992$	0	0
$993$	51.6333	1.63853
$994$	0	0
$995$	0.697224	0.0221035
$996$	0	0
$997$	5.27502	0.167062	0.0835308	−	0.996505i	$-0.473380\pi$
$997$	5.27502	0.167062	0.0835308	+	0.996505i	$0.473380\pi$
$998$	0	0
$999$	−9.81665	−0.310585

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.g.1.2		2
7.6	odd	2		1148.2.a.a.1.1	✓	2
28.27	even	2		4592.2.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.a.1.1	✓	2	7.6	odd	2
4592.2.a.o.1.2		2	28.27	even	2
8036.2.a.g.1.2		2	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)

\(f(q)\)	\(=\)	\(q+3.30278 q^{3} +3.30278 q^{5} +7.90833 q^{9} +O(q^{10})\)	\(q+3.30278 q^{3} +3.30278 q^{5} +7.90833 q^{9} +1.00000 q^{11} +5.00000 q^{13} +10.9083 q^{15} -6.21110 q^{17} -3.90833 q^{19} -3.30278 q^{23} +5.90833 q^{25} +16.2111 q^{27} +10.3028 q^{29} -0.0916731 q^{31} +3.30278 q^{33} -0.605551 q^{37} +16.5139 q^{39} +1.00000 q^{41} -11.6056 q^{43} +26.1194 q^{45} +5.39445 q^{47} -20.5139 q^{51} -2.69722 q^{53} +3.30278 q^{55} -12.9083 q^{57} -5.90833 q^{59} +10.2111 q^{61} +16.5139 q^{65} +1.90833 q^{67} -10.9083 q^{69} -12.2111 q^{71} -7.00000 q^{73} +19.5139 q^{75} -16.6056 q^{79} +29.8167 q^{81} +10.8167 q^{83} -20.5139 q^{85} +34.0278 q^{87} +8.51388 q^{89} -0.302776 q^{93} -12.9083 q^{95} -9.11943 q^{97} +7.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 3 q^{5} + 5 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 + 3 * q^5 + 5 * q^9	\( 2 q + 3 q^{3} + 3 q^{5} + 5 q^{9} + 2 q^{11} + 10 q^{13} + 11 q^{15} + 2 q^{17} + 3 q^{19} - 3 q^{23} + q^{25} + 18 q^{27} + 17 q^{29} - 11 q^{31} + 3 q^{33} + 6 q^{37} + 15 q^{39} + 2 q^{41} - 16 q^{43} + 27 q^{45} + 18 q^{47} - 23 q^{51} - 9 q^{53} + 3 q^{55} - 15 q^{57} - q^{59} + 6 q^{61} + 15 q^{65} - 7 q^{67} - 11 q^{69} - 10 q^{71} - 14 q^{73} + 21 q^{75} - 26 q^{79} + 38 q^{81} - 23 q^{85} + 32 q^{87} - q^{89} + 3 q^{93} - 15 q^{95} + 7 q^{97} + 5 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 3 * q^5 + 5 * q^9 + 2 * q^11 + 10 * q^13 + 11 * q^15 + 2 * q^17 + 3 * q^19 - 3 * q^23 + q^25 + 18 * q^27 + 17 * q^29 - 11 * q^31 + 3 * q^33 + 6 * q^37 + 15 * q^39 + 2 * q^41 - 16 * q^43 + 27 * q^45 + 18 * q^47 - 23 * q^51 - 9 * q^53 + 3 * q^55 - 15 * q^57 - q^59 + 6 * q^61 + 15 * q^65 - 7 * q^67 - 11 * q^69 - 10 * q^71 - 14 * q^73 + 21 * q^75 - 26 * q^79 + 38 * q^81 - 23 * q^85 + 32 * q^87 - q^89 + 3 * q^93 - 15 * q^95 + 7 * q^97 + 5 * q^99

Introduction

L-functions

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Database

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