LMFDB - Embedded newform 8036.2.a.r.1.3

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + \cdots + 3381 $ x^15 - x^14 - 37x^13 + 39x^12 + 537x^11 - 616x^10 - 3853x^9 + 4929x^8 + 13971x^7 - 20311x^6 - 22309x^5 + 38415x^4 + 8429x^3 - 22584x^2 - 399*x + 3381
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.73742$ 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)

$f(q)$	$=$	$q-2.73742 q^{3} +0.863054 q^{5} +4.49344 q^{9} +O(q^{10})$	$q-2.73742 q^{3} +0.863054 q^{5} +4.49344 q^{9} -5.59068 q^{11} +1.90008 q^{13} -2.36254 q^{15} +4.01928 q^{17} +1.91765 q^{19} +9.06557 q^{23} -4.25514 q^{25} -4.08817 q^{27} +5.61917 q^{29} -9.11195 q^{31} +15.3040 q^{33} +1.45610 q^{37} -5.20129 q^{39} +1.00000 q^{41} +7.83032 q^{43} +3.87808 q^{45} -10.1609 q^{47} -11.0024 q^{51} -1.16468 q^{53} -4.82506 q^{55} -5.24940 q^{57} -6.86817 q^{59} -5.22611 q^{61} +1.63987 q^{65} +14.6443 q^{67} -24.8162 q^{69} +3.14178 q^{71} +4.41735 q^{73} +11.6481 q^{75} -5.30208 q^{79} -2.28930 q^{81} -6.73846 q^{83} +3.46886 q^{85} -15.3820 q^{87} +3.99849 q^{89} +24.9432 q^{93} +1.65503 q^{95} -3.44705 q^{97} -25.1214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * 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
$3$	−2.73742	−1.58045	−0.790224	−	0.612818i	$-0.790036\pi$
$3$	−2.73742	−1.58045	−0.790224	+	0.612818i	$0.790036\pi$
$4$	0	0
$5$	0.863054	0.385970	0.192985	−	0.981202i	$-0.438183\pi$
$5$	0.863054	0.385970	0.192985	+	0.981202i	$0.438183\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	4.49344	1.49781
$10$	0	0
$11$	−5.59068	−1.68565	−0.842826	−	0.538186i	$-0.819110\pi$
$11$	−5.59068	−1.68565	−0.842826	+	0.538186i	$0.819110\pi$
$12$	0	0
$13$	1.90008	0.526986	0.263493	−	0.964661i	$-0.415125\pi$
$13$	1.90008	0.526986	0.263493	+	0.964661i	$0.415125\pi$
$14$	0	0
$15$	−2.36254	−0.610005
$16$	0	0
$17$	4.01928	0.974819	0.487409	−	0.873174i	$-0.337942\pi$
$17$	4.01928	0.974819	0.487409	+	0.873174i	$0.337942\pi$
$18$	0	0
$19$	1.91765	0.439939	0.219969	−	0.975507i	$-0.429404\pi$
$19$	1.91765	0.439939	0.219969	+	0.975507i	$0.429404\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.06557	1.89030	0.945151	−	0.326632i	$-0.105914\pi$
$23$	9.06557	1.89030	0.945151	+	0.326632i	$0.105914\pi$
$24$	0	0
$25$	−4.25514	−0.851027
$26$	0	0
$27$	−4.08817	−0.786769
$28$	0	0
$29$	5.61917	1.04345	0.521727	−	0.853113i	$-0.325288\pi$
$29$	5.61917	1.04345	0.521727	+	0.853113i	$0.325288\pi$
$30$	0	0
$31$	−9.11195	−1.63655	−0.818277	−	0.574824i	$-0.805071\pi$
$31$	−9.11195	−1.63655	−0.818277	+	0.574824i	$0.805071\pi$
$32$	0	0
$33$	15.3040	2.66409
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.45610	0.239382	0.119691	−	0.992811i	$-0.461810\pi$
$37$	1.45610	0.239382	0.119691	+	0.992811i	$0.461810\pi$
$38$	0	0
$39$	−5.20129	−0.832874
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	7.83032	1.19411	0.597056	−	0.802199i	$-0.296337\pi$
$43$	7.83032	1.19411	0.597056	+	0.802199i	$0.296337\pi$
$44$	0	0
$45$	3.87808	0.578111
$46$	0	0
$47$	−10.1609	−1.48212	−0.741059	−	0.671440i	$-0.765676\pi$
$47$	−10.1609	−1.48212	−0.741059	+	0.671440i	$0.765676\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−11.0024	−1.54065
$52$	0	0
$53$	−1.16468	−0.159982	−0.0799909	−	0.996796i	$-0.525489\pi$
$53$	−1.16468	−0.159982	−0.0799909	+	0.996796i	$0.525489\pi$
$54$	0	0
$55$	−4.82506	−0.650611
$56$	0	0
$57$	−5.24940	−0.695300
$58$	0	0
$59$	−6.86817	−0.894159	−0.447080	−	0.894494i	$-0.647536\pi$
$59$	−6.86817	−0.894159	−0.447080	+	0.894494i	$0.647536\pi$
$60$	0	0
$61$	−5.22611	−0.669135	−0.334568	−	0.942372i	$-0.608590\pi$
$61$	−5.22611	−0.669135	−0.334568	+	0.942372i	$0.608590\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.63987	0.203401
$66$	0	0
$67$	14.6443	1.78909	0.894546	−	0.446977i	$-0.147499\pi$
$67$	14.6443	1.78909	0.894546	+	0.446977i	$0.147499\pi$
$68$	0	0
$69$	−24.8162	−2.98752
$70$	0	0
$71$	3.14178	0.372860	0.186430	−	0.982468i	$-0.440308\pi$
$71$	3.14178	0.372860	0.186430	+	0.982468i	$0.440308\pi$
$72$	0	0
$73$	4.41735	0.517012	0.258506	−	0.966010i	$-0.416770\pi$
$73$	4.41735	0.517012	0.258506	+	0.966010i	$0.416770\pi$
$74$	0	0
$75$	11.6481	1.34500
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.30208	−0.596530	−0.298265	−	0.954483i	$-0.596408\pi$
$79$	−5.30208	−0.596530	−0.298265	+	0.954483i	$0.596408\pi$
$80$	0	0
$81$	−2.28930	−0.254367
$82$	0	0
$83$	−6.73846	−0.739642	−0.369821	−	0.929103i	$-0.620581\pi$
$83$	−6.73846	−0.739642	−0.369821	+	0.929103i	$0.620581\pi$
$84$	0	0
$85$	3.46886	0.376250
$86$	0	0
$87$	−15.3820	−1.64912
$88$	0	0
$89$	3.99849	0.423839	0.211919	−	0.977287i	$-0.432029\pi$
$89$	3.99849	0.423839	0.211919	+	0.977287i	$0.432029\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	24.9432	2.58649
$94$	0	0
$95$	1.65503	0.169803
$96$	0	0
$97$	−3.44705	−0.349995	−0.174997	−	0.984569i	$-0.555992\pi$
$97$	−3.44705	−0.349995	−0.174997	+	0.984569i	$0.555992\pi$
$98$	0	0
$99$	−25.1214	−2.52479
$100$	0	0
$101$	17.7081	1.76202	0.881011	−	0.473096i	$-0.156864\pi$
$101$	17.7081	1.76202	0.881011	+	0.473096i	$0.156864\pi$
$102$	0	0
$103$	−15.8237	−1.55915	−0.779576	−	0.626308i	$-0.784565\pi$
$103$	−15.8237	−1.55915	−0.779576	+	0.626308i	$0.784565\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.15327	0.884880	0.442440	−	0.896798i	$-0.354113\pi$
$107$	9.15327	0.884880	0.442440	+	0.896798i	$0.354113\pi$
$108$	0	0
$109$	−13.8982	−1.33121	−0.665604	−	0.746305i	$-0.731826\pi$
$109$	−13.8982	−1.33121	−0.665604	+	0.746305i	$0.731826\pi$
$110$	0	0
$111$	−3.98596	−0.378331
$112$	0	0
$113$	9.44966	0.888949	0.444475	−	0.895791i	$-0.353390\pi$
$113$	9.44966	0.888949	0.444475	+	0.895791i	$0.353390\pi$
$114$	0	0
$115$	7.82408	0.729599
$116$	0	0
$117$	8.53788	0.789327
$118$	0	0
$119$	0	0
$120$	0	0
$121$	20.2557	1.84142
$122$	0	0
$123$	−2.73742	−0.246824
$124$	0	0
$125$	−7.98769	−0.714440
$126$	0	0
$127$	−1.14798	−0.101866	−0.0509332	−	0.998702i	$-0.516220\pi$
$127$	−1.14798	−0.101866	−0.0509332	+	0.998702i	$0.516220\pi$
$128$	0	0
$129$	−21.4348	−1.88723
$130$	0	0
$131$	−1.30559	−0.114070	−0.0570348	−	0.998372i	$-0.518165\pi$
$131$	−1.30559	−0.114070	−0.0570348	+	0.998372i	$0.518165\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.52831	−0.303669
$136$	0	0
$137$	12.0254	1.02740	0.513698	−	0.857971i	$-0.328275\pi$
$137$	12.0254	1.02740	0.513698	+	0.857971i	$0.328275\pi$
$138$	0	0
$139$	−6.79439	−0.576293	−0.288146	−	0.957586i	$-0.593039\pi$
$139$	−6.79439	−0.576293	−0.288146	+	0.957586i	$0.593039\pi$
$140$	0	0
$141$	27.8146	2.34241
$142$	0	0
$143$	−10.6227	−0.888315
$144$	0	0
$145$	4.84965	0.402741
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.9624	1.63539	0.817694	−	0.575653i	$-0.195252\pi$
$149$	19.9624	1.63539	0.817694	+	0.575653i	$0.195252\pi$
$150$	0	0
$151$	−7.91750	−0.644317	−0.322158	−	0.946686i	$-0.604408\pi$
$151$	−7.91750	−0.644317	−0.322158	+	0.946686i	$0.604408\pi$
$152$	0	0
$153$	18.0604	1.46010
$154$	0	0
$155$	−7.86411	−0.631660
$156$	0	0
$157$	−7.85253	−0.626700	−0.313350	−	0.949638i	$-0.601451\pi$
$157$	−7.85253	−0.626700	−0.313350	+	0.949638i	$0.601451\pi$
$158$	0	0
$159$	3.18823	0.252843
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.8785	1.40035	0.700174	−	0.713972i	$-0.253106\pi$
$163$	17.8785	1.40035	0.700174	+	0.713972i	$0.253106\pi$
$164$	0	0
$165$	13.2082	1.02826
$166$	0	0
$167$	−5.60745	−0.433918	−0.216959	−	0.976181i	$-0.569614\pi$
$167$	−5.60745	−0.433918	−0.216959	+	0.976181i	$0.569614\pi$
$168$	0	0
$169$	−9.38971	−0.722286
$170$	0	0
$171$	8.61684	0.658946
$172$	0	0
$173$	−7.61552	−0.578997	−0.289499	−	0.957178i	$-0.593489\pi$
$173$	−7.61552	−0.578997	−0.289499	+	0.957178i	$0.593489\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	18.8010	1.41317
$178$	0	0
$179$	−21.5408	−1.61003	−0.805017	−	0.593252i	$-0.797844\pi$
$179$	−21.5408	−1.61003	−0.805017	+	0.593252i	$0.797844\pi$
$180$	0	0
$181$	−4.71251	−0.350278	−0.175139	−	0.984544i	$-0.556038\pi$
$181$	−4.71251	−0.350278	−0.175139	+	0.984544i	$0.556038\pi$
$182$	0	0
$183$	14.3060	1.05753
$184$	0	0
$185$	1.25670	0.0923942
$186$	0	0
$187$	−22.4705	−1.64321
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.1408	−0.806120	−0.403060	−	0.915174i	$-0.632053\pi$
$191$	−11.1408	−0.806120	−0.403060	+	0.915174i	$0.632053\pi$
$192$	0	0
$193$	24.5322	1.76587	0.882933	−	0.469499i	$-0.155566\pi$
$193$	24.5322	1.76587	0.882933	+	0.469499i	$0.155566\pi$
$194$	0	0
$195$	−4.48900	−0.321464
$196$	0	0
$197$	−25.7126	−1.83195	−0.915973	−	0.401241i	$-0.868579\pi$
$197$	−25.7126	−1.83195	−0.915973	+	0.401241i	$0.868579\pi$
$198$	0	0
$199$	0.972796	0.0689597	0.0344798	−	0.999405i	$-0.489023\pi$
$199$	0.972796	0.0689597	0.0344798	+	0.999405i	$0.489023\pi$
$200$	0	0
$201$	−40.0877	−2.82756
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.863054	0.0602783
$206$	0	0
$207$	40.7356	2.83132
$208$	0	0
$209$	−10.7209	−0.741584
$210$	0	0
$211$	2.56072	0.176287	0.0881436	−	0.996108i	$-0.471907\pi$
$211$	2.56072	0.176287	0.0881436	+	0.996108i	$0.471907\pi$
$212$	0	0
$213$	−8.60035	−0.589286
$214$	0	0
$215$	6.75799	0.460891
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−12.0921	−0.817110
$220$	0	0
$221$	7.63694	0.513716
$222$	0	0
$223$	21.6493	1.44974	0.724872	−	0.688884i	$-0.241899\pi$
$223$	21.6493	1.44974	0.724872	+	0.688884i	$0.241899\pi$
$224$	0	0
$225$	−19.1202	−1.27468
$226$	0	0
$227$	0.644605	0.0427839	0.0213920	−	0.999771i	$-0.493190\pi$
$227$	0.644605	0.0427839	0.0213920	+	0.999771i	$0.493190\pi$
$228$	0	0
$229$	1.31952	0.0871966	0.0435983	−	0.999049i	$-0.486118\pi$
$229$	1.31952	0.0871966	0.0435983	+	0.999049i	$0.486118\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−28.3002	−1.85401	−0.927003	−	0.375053i	$-0.877624\pi$
$233$	−28.3002	−1.85401	−0.927003	+	0.375053i	$0.877624\pi$
$234$	0	0
$235$	−8.76940	−0.572053
$236$	0	0
$237$	14.5140	0.942785
$238$	0	0
$239$	−9.13220	−0.590713	−0.295356	−	0.955387i	$-0.595438\pi$
$239$	−9.13220	−0.590713	−0.295356	+	0.955387i	$0.595438\pi$
$240$	0	0
$241$	14.9952	0.965924	0.482962	−	0.875641i	$-0.339561\pi$
$241$	14.9952	0.965924	0.482962	+	0.875641i	$0.339561\pi$
$242$	0	0
$243$	18.5313	1.18878
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.64367	0.231841
$248$	0	0
$249$	18.4460	1.16897
$250$	0	0
$251$	19.0748	1.20399	0.601994	−	0.798500i	$-0.294373\pi$
$251$	19.0748	1.20399	0.601994	+	0.798500i	$0.294373\pi$
$252$	0	0
$253$	−50.6827	−3.18639
$254$	0	0
$255$	−9.49570	−0.594644
$256$	0	0
$257$	11.7626	0.733734	0.366867	−	0.930273i	$-0.380430\pi$
$257$	11.7626	0.733734	0.366867	+	0.930273i	$0.380430\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	25.2494	1.56290
$262$	0	0
$263$	−7.87248	−0.485438	−0.242719	−	0.970097i	$-0.578039\pi$
$263$	−7.87248	−0.485438	−0.242719	+	0.970097i	$0.578039\pi$
$264$	0	0
$265$	−1.00519	−0.0617481
$266$	0	0
$267$	−10.9455	−0.669855
$268$	0	0
$269$	11.5817	0.706150	0.353075	−	0.935595i	$-0.385136\pi$
$269$	11.5817	0.706150	0.353075	+	0.935595i	$0.385136\pi$
$270$	0	0
$271$	−12.0932	−0.734609	−0.367305	−	0.930101i	$-0.619719\pi$
$271$	−12.0932	−0.734609	−0.367305	+	0.930101i	$0.619719\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	23.7891	1.43454
$276$	0	0
$277$	21.6075	1.29827	0.649135	−	0.760673i	$-0.275131\pi$
$277$	21.6075	1.29827	0.649135	+	0.760673i	$0.275131\pi$
$278$	0	0
$279$	−40.9440	−2.45125
$280$	0	0
$281$	12.2417	0.730278	0.365139	−	0.930953i	$-0.381022\pi$
$281$	12.2417	0.730278	0.365139	+	0.930953i	$0.381022\pi$
$282$	0	0
$283$	30.5526	1.81616	0.908082	−	0.418791i	$-0.137546\pi$
$283$	30.5526	1.81616	0.908082	+	0.418791i	$0.137546\pi$
$284$	0	0
$285$	−4.53052	−0.268365
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−0.845377	−0.0497281
$290$	0	0
$291$	9.43600	0.553148
$292$	0	0
$293$	−15.2363	−0.890117	−0.445058	−	0.895502i	$-0.646817\pi$
$293$	−15.2363	−0.890117	−0.445058	+	0.895502i	$0.646817\pi$
$294$	0	0
$295$	−5.92760	−0.345118
$296$	0	0
$297$	22.8557	1.32622
$298$	0	0
$299$	17.2253	0.996163
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−48.4744	−2.78478
$304$	0	0
$305$	−4.51042	−0.258266
$306$	0	0
$307$	3.85409	0.219964	0.109982	−	0.993934i	$-0.464921\pi$
$307$	3.85409	0.219964	0.109982	+	0.993934i	$0.464921\pi$
$308$	0	0
$309$	43.3159	2.46416
$310$	0	0
$311$	−5.97634	−0.338887	−0.169444	−	0.985540i	$-0.554197\pi$
$311$	−5.97634	−0.338887	−0.169444	+	0.985540i	$0.554197\pi$
$312$	0	0
$313$	−14.7468	−0.833540	−0.416770	−	0.909012i	$-0.636838\pi$
$313$	−14.7468	−0.833540	−0.416770	+	0.909012i	$0.636838\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.3411	1.70412	0.852062	−	0.523441i	$-0.175352\pi$
$317$	30.3411	1.70412	0.852062	+	0.523441i	$0.175352\pi$
$318$	0	0
$319$	−31.4150	−1.75890
$320$	0	0
$321$	−25.0563	−1.39851
$322$	0	0
$323$	7.70757	0.428860
$324$	0	0
$325$	−8.08508	−0.448480
$326$	0	0
$327$	38.0452	2.10391
$328$	0	0
$329$	0	0
$330$	0	0
$331$	20.2963	1.11559	0.557793	−	0.829980i	$-0.311648\pi$
$331$	20.2963	1.11559	0.557793	+	0.829980i	$0.311648\pi$
$332$	0	0
$333$	6.54292	0.358550
$334$	0	0
$335$	12.6389	0.690535
$336$	0	0
$337$	2.08648	0.113658	0.0568290	−	0.998384i	$-0.481901\pi$
$337$	2.08648	0.113658	0.0568290	+	0.998384i	$0.481901\pi$
$338$	0	0
$339$	−25.8676	−1.40494
$340$	0	0
$341$	50.9420	2.75866
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−21.4178	−1.15309
$346$	0	0
$347$	0.664742	0.0356852	0.0178426	−	0.999841i	$-0.494320\pi$
$347$	0.664742	0.0356852	0.0178426	+	0.999841i	$0.494320\pi$
$348$	0	0
$349$	29.1533	1.56054	0.780272	−	0.625441i	$-0.215081\pi$
$349$	29.1533	1.56054	0.780272	+	0.625441i	$0.215081\pi$
$350$	0	0
$351$	−7.76783	−0.414616
$352$	0	0
$353$	30.8199	1.64038	0.820188	−	0.572094i	$-0.193869\pi$
$353$	30.8199	1.64038	0.820188	+	0.572094i	$0.193869\pi$
$354$	0	0
$355$	2.71152	0.143913
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.9341	−0.999304	−0.499652	−	0.866226i	$-0.666539\pi$
$359$	−18.9341	−0.999304	−0.499652	+	0.866226i	$0.666539\pi$
$360$	0	0
$361$	−15.3226	−0.806454
$362$	0	0
$363$	−55.4482	−2.91027
$364$	0	0
$365$	3.81241	0.199551
$366$	0	0
$367$	−8.85703	−0.462333	−0.231167	−	0.972914i	$-0.574254\pi$
$367$	−8.85703	−0.462333	−0.231167	+	0.972914i	$0.574254\pi$
$368$	0	0
$369$	4.49344	0.233919
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3.48349	−0.180368	−0.0901840	−	0.995925i	$-0.528746\pi$
$373$	−3.48349	−0.180368	−0.0901840	+	0.995925i	$0.528746\pi$
$374$	0	0
$375$	21.8656	1.12914
$376$	0	0
$377$	10.6768	0.549885
$378$	0	0
$379$	18.8253	0.966991	0.483496	−	0.875347i	$-0.339367\pi$
$379$	18.8253	0.966991	0.483496	+	0.875347i	$0.339367\pi$
$380$	0	0
$381$	3.14248	0.160994
$382$	0	0
$383$	−0.762543	−0.0389641	−0.0194821	−	0.999810i	$-0.506202\pi$
$383$	−0.762543	−0.0389641	−0.0194821	+	0.999810i	$0.506202\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	35.1851	1.78856
$388$	0	0
$389$	6.98396	0.354101	0.177050	−	0.984202i	$-0.443344\pi$
$389$	6.98396	0.354101	0.177050	+	0.984202i	$0.443344\pi$
$390$	0	0
$391$	36.4371	1.84270
$392$	0	0
$393$	3.57393	0.180281
$394$	0	0
$395$	−4.57598	−0.230243
$396$	0	0
$397$	16.9448	0.850435	0.425218	−	0.905091i	$-0.360198\pi$
$397$	16.9448	0.850435	0.425218	+	0.905091i	$0.360198\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.5586	−0.627147	−0.313573	−	0.949564i	$-0.601526\pi$
$401$	−12.5586	−0.627147	−0.313573	+	0.949564i	$0.601526\pi$
$402$	0	0
$403$	−17.3134	−0.862441
$404$	0	0
$405$	−1.97579	−0.0981779
$406$	0	0
$407$	−8.14061	−0.403515
$408$	0	0
$409$	−19.4920	−0.963817	−0.481909	−	0.876221i	$-0.660056\pi$
$409$	−19.4920	−0.963817	−0.481909	+	0.876221i	$0.660056\pi$
$410$	0	0
$411$	−32.9184	−1.62374
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5.81566	−0.285479
$416$	0	0
$417$	18.5991	0.910801
$418$	0	0
$419$	−34.0935	−1.66558	−0.832788	−	0.553593i	$-0.813256\pi$
$419$	−34.0935	−1.66558	−0.832788	+	0.553593i	$0.813256\pi$
$420$	0	0
$421$	36.1622	1.76244	0.881219	−	0.472708i	$-0.156723\pi$
$421$	36.1622	1.76244	0.881219	+	0.472708i	$0.156723\pi$
$422$	0	0
$423$	−45.6574	−2.21994
$424$	0	0
$425$	−17.1026	−0.829598
$426$	0	0
$427$	0	0
$428$	0	0
$429$	29.0788	1.40394
$430$	0	0
$431$	15.5755	0.750245	0.375122	−	0.926975i	$-0.377601\pi$
$431$	15.5755	0.750245	0.375122	+	0.926975i	$0.377601\pi$
$432$	0	0
$433$	−8.91790	−0.428567	−0.214283	−	0.976772i	$-0.568742\pi$
$433$	−8.91790	−0.428567	−0.214283	+	0.976772i	$0.568742\pi$
$434$	0	0
$435$	−13.2755	−0.636511
$436$	0	0
$437$	17.3846	0.831617
$438$	0	0
$439$	32.4555	1.54901	0.774507	−	0.632565i	$-0.217998\pi$
$439$	32.4555	1.54901	0.774507	+	0.632565i	$0.217998\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.17519	0.293392	0.146696	−	0.989182i	$-0.453136\pi$
$443$	6.17519	0.293392	0.146696	+	0.989182i	$0.453136\pi$
$444$	0	0
$445$	3.45091	0.163589
$446$	0	0
$447$	−54.6455	−2.58464
$448$	0	0
$449$	−33.2976	−1.57141	−0.785706	−	0.618600i	$-0.787700\pi$
$449$	−33.2976	−1.57141	−0.785706	+	0.618600i	$0.787700\pi$
$450$	0	0
$451$	−5.59068	−0.263255
$452$	0	0
$453$	21.6735	1.01831
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.6002	0.495857	0.247929	−	0.968778i	$-0.420250\pi$
$457$	10.6002	0.495857	0.247929	+	0.968778i	$0.420250\pi$
$458$	0	0
$459$	−16.4315	−0.766957
$460$	0	0
$461$	32.3114	1.50489	0.752445	−	0.658655i	$-0.228874\pi$
$461$	32.3114	1.50489	0.752445	+	0.658655i	$0.228874\pi$
$462$	0	0
$463$	24.6791	1.14694	0.573469	−	0.819228i	$-0.305597\pi$
$463$	24.6791	1.14694	0.573469	+	0.819228i	$0.305597\pi$
$464$	0	0
$465$	21.5273	0.998306
$466$	0	0
$467$	2.61652	0.121078	0.0605391	−	0.998166i	$-0.480718\pi$
$467$	2.61652	0.121078	0.0605391	+	0.998166i	$0.480718\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	21.4956	0.990466
$472$	0	0
$473$	−43.7768	−2.01286
$474$	0	0
$475$	−8.15985	−0.374400
$476$	0	0
$477$	−5.23344	−0.239623
$478$	0	0
$479$	21.8559	0.998621	0.499311	−	0.866423i	$-0.333587\pi$
$479$	21.8559	0.998621	0.499311	+	0.866423i	$0.333587\pi$
$480$	0	0
$481$	2.76671	0.126151
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.97499	−0.135087
$486$	0	0
$487$	23.0302	1.04360	0.521799	−	0.853068i	$-0.325261\pi$
$487$	23.0302	1.04360	0.521799	+	0.853068i	$0.325261\pi$
$488$	0	0
$489$	−48.9408	−2.21318
$490$	0	0
$491$	19.4215	0.876480	0.438240	−	0.898858i	$-0.355602\pi$
$491$	19.4215	0.876480	0.438240	+	0.898858i	$0.355602\pi$
$492$	0	0
$493$	22.5850	1.01718
$494$	0	0
$495$	−21.6811	−0.974494
$496$	0	0
$497$	0	0
$498$	0	0
$499$	14.9571	0.669572	0.334786	−	0.942294i	$-0.391336\pi$
$499$	14.9571	0.669572	0.334786	+	0.942294i	$0.391336\pi$
$500$	0	0
$501$	15.3499	0.685784
$502$	0	0
$503$	12.0347	0.536602	0.268301	−	0.963335i	$-0.413538\pi$
$503$	12.0347	0.536602	0.268301	+	0.963335i	$0.413538\pi$
$504$	0	0
$505$	15.2831	0.680087
$506$	0	0
$507$	25.7035	1.14153
$508$	0	0
$509$	−6.92030	−0.306737	−0.153368	−	0.988169i	$-0.549012\pi$
$509$	−6.92030	−0.306737	−0.153368	+	0.988169i	$0.549012\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−7.83967	−0.346130
$514$	0	0
$515$	−13.6567	−0.601785
$516$	0	0
$517$	56.8063	2.49834
$518$	0	0
$519$	20.8468	0.915075
$520$	0	0
$521$	−2.78492	−0.122009	−0.0610047	−	0.998137i	$-0.519430\pi$
$521$	−2.78492	−0.122009	−0.0610047	+	0.998137i	$0.519430\pi$
$522$	0	0
$523$	−14.8470	−0.649215	−0.324607	−	0.945849i	$-0.605232\pi$
$523$	−14.8470	−0.649215	−0.324607	+	0.945849i	$0.605232\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−36.6235	−1.59534
$528$	0	0
$529$	59.1846	2.57325
$530$	0	0
$531$	−30.8617	−1.33928
$532$	0	0
$533$	1.90008	0.0823014
$534$	0	0
$535$	7.89977	0.341537
$536$	0	0
$537$	58.9661	2.54457
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.23646	0.354113	0.177057	−	0.984201i	$-0.443342\pi$
$541$	8.23646	0.354113	0.177057	+	0.984201i	$0.443342\pi$
$542$	0	0
$543$	12.9001	0.553597
$544$	0	0
$545$	−11.9949	−0.513806
$546$	0	0
$547$	40.5675	1.73454	0.867270	−	0.497838i	$-0.165872\pi$
$547$	40.5675	1.73454	0.867270	+	0.497838i	$0.165872\pi$
$548$	0	0
$549$	−23.4832	−1.00224
$550$	0	0
$551$	10.7756	0.459055
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.44010	−0.146024
$556$	0	0
$557$	22.6289	0.958818	0.479409	−	0.877592i	$-0.340851\pi$
$557$	22.6289	0.958818	0.479409	+	0.877592i	$0.340851\pi$
$558$	0	0
$559$	14.8782	0.629280
$560$	0	0
$561$	61.5111	2.59700
$562$	0	0
$563$	17.3463	0.731061	0.365531	−	0.930799i	$-0.380888\pi$
$563$	17.3463	0.731061	0.365531	+	0.930799i	$0.380888\pi$
$564$	0	0
$565$	8.15557	0.343107
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.5921	−0.444045	−0.222023	−	0.975042i	$-0.571266\pi$
$569$	−10.5921	−0.444045	−0.222023	+	0.975042i	$0.571266\pi$
$570$	0	0
$571$	10.5451	0.441299	0.220649	−	0.975353i	$-0.429182\pi$
$571$	10.5451	0.441299	0.220649	+	0.975353i	$0.429182\pi$
$572$	0	0
$573$	30.4970	1.27403
$574$	0	0
$575$	−38.5753	−1.60870
$576$	0	0
$577$	19.4790	0.810921	0.405460	−	0.914113i	$-0.367111\pi$
$577$	19.4790	0.810921	0.405460	+	0.914113i	$0.367111\pi$
$578$	0	0
$579$	−67.1548	−2.79086
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.51138	0.269674
$584$	0	0
$585$	7.36865	0.304656
$586$	0	0
$587$	−16.5937	−0.684896	−0.342448	−	0.939537i	$-0.611256\pi$
$587$	−16.5937	−0.684896	−0.342448	+	0.939537i	$0.611256\pi$
$588$	0	0
$589$	−17.4735	−0.719983
$590$	0	0
$591$	70.3860	2.89529
$592$	0	0
$593$	22.6551	0.930332	0.465166	−	0.885224i	$-0.345995\pi$
$593$	22.6551	0.930332	0.465166	+	0.885224i	$0.345995\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.66295	−0.108987
$598$	0	0
$599$	−19.2453	−0.786340	−0.393170	−	0.919466i	$-0.628622\pi$
$599$	−19.2453	−0.786340	−0.393170	+	0.919466i	$0.628622\pi$
$600$	0	0
$601$	21.9736	0.896322	0.448161	−	0.893953i	$-0.352079\pi$
$601$	21.9736	0.896322	0.448161	+	0.893953i	$0.352079\pi$
$602$	0	0
$603$	65.8035	2.67973
$604$	0	0
$605$	17.4817	0.710734
$606$	0	0
$607$	−34.3757	−1.39527	−0.697633	−	0.716455i	$-0.745763\pi$
$607$	−34.3757	−1.39527	−0.697633	+	0.716455i	$0.745763\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−19.3065	−0.781056
$612$	0	0
$613$	−24.5084	−0.989885	−0.494943	−	0.868926i	$-0.664811\pi$
$613$	−24.5084	−0.989885	−0.494943	+	0.868926i	$0.664811\pi$
$614$	0	0
$615$	−2.36254	−0.0952667
$616$	0	0
$617$	−11.7986	−0.474993	−0.237496	−	0.971388i	$-0.576327\pi$
$617$	−11.7986	−0.474993	−0.237496	+	0.971388i	$0.576327\pi$
$618$	0	0
$619$	32.1597	1.29261	0.646304	−	0.763080i	$-0.276314\pi$
$619$	32.1597	1.29261	0.646304	+	0.763080i	$0.276314\pi$
$620$	0	0
$621$	−37.0616	−1.48723
$622$	0	0
$623$	0	0
$624$	0	0
$625$	14.3819	0.575275
$626$	0	0
$627$	29.3477	1.17203
$628$	0	0
$629$	5.85249	0.233354
$630$	0	0
$631$	37.4091	1.48923	0.744617	−	0.667492i	$-0.232632\pi$
$631$	37.4091	1.48923	0.744617	+	0.667492i	$0.232632\pi$
$632$	0	0
$633$	−7.00975	−0.278613
$634$	0	0
$635$	−0.990765	−0.0393173
$636$	0	0
$637$	0	0
$638$	0	0
$639$	14.1174	0.558475
$640$	0	0
$641$	4.05878	0.160312	0.0801560	−	0.996782i	$-0.474458\pi$
$641$	4.05878	0.160312	0.0801560	+	0.996782i	$0.474458\pi$
$642$	0	0
$643$	26.5083	1.04539	0.522693	−	0.852521i	$-0.324927\pi$
$643$	26.5083	1.04539	0.522693	+	0.852521i	$0.324927\pi$
$644$	0	0
$645$	−18.4994	−0.728414
$646$	0	0
$647$	−9.38614	−0.369007	−0.184504	−	0.982832i	$-0.559068\pi$
$647$	−9.38614	−0.369007	−0.184504	+	0.982832i	$0.559068\pi$
$648$	0	0
$649$	38.3977	1.50724
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.32782	−0.286760	−0.143380	−	0.989668i	$-0.545797\pi$
$653$	−7.32782	−0.286760	−0.143380	+	0.989668i	$0.545797\pi$
$654$	0	0
$655$	−1.12679	−0.0440274
$656$	0	0
$657$	19.8491	0.774387
$658$	0	0
$659$	5.69956	0.222023	0.111012	−	0.993819i	$-0.464591\pi$
$659$	5.69956	0.222023	0.111012	+	0.993819i	$0.464591\pi$
$660$	0	0
$661$	−44.0578	−1.71365	−0.856824	−	0.515608i	$-0.827566\pi$
$661$	−44.0578	−1.71365	−0.856824	+	0.515608i	$0.827566\pi$
$662$	0	0
$663$	−20.9055	−0.811901
$664$	0	0
$665$	0	0
$666$	0	0
$667$	50.9410	1.97244
$668$	0	0
$669$	−59.2631	−2.29124
$670$	0	0
$671$	29.2175	1.12793
$672$	0	0
$673$	5.92353	0.228335	0.114168	−	0.993461i	$-0.463580\pi$
$673$	5.92353	0.228335	0.114168	+	0.993461i	$0.463580\pi$
$674$	0	0
$675$	17.3957	0.669562
$676$	0	0
$677$	20.0586	0.770914	0.385457	−	0.922726i	$-0.374044\pi$
$677$	20.0586	0.770914	0.385457	+	0.922726i	$0.374044\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.76455	−0.0676178
$682$	0	0
$683$	41.5746	1.59081	0.795404	−	0.606080i	$-0.207259\pi$
$683$	41.5746	1.59081	0.795404	+	0.606080i	$0.207259\pi$
$684$	0	0
$685$	10.3785	0.396543
$686$	0	0
$687$	−3.61209	−0.137810
$688$	0	0
$689$	−2.21299	−0.0843082
$690$	0	0
$691$	2.77305	0.105492	0.0527459	−	0.998608i	$-0.483203\pi$
$691$	2.77305	0.105492	0.0527459	+	0.998608i	$0.483203\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.86393	−0.222432
$696$	0	0
$697$	4.01928	0.152241
$698$	0	0
$699$	77.4693	2.93016
$700$	0	0
$701$	32.4197	1.22448	0.612238	−	0.790674i	$-0.290269\pi$
$701$	32.4197	1.22448	0.612238	+	0.790674i	$0.290269\pi$
$702$	0	0
$703$	2.79229	0.105313
$704$	0	0
$705$	24.0055	0.904099
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10.5705	−0.396984	−0.198492	−	0.980103i	$-0.563604\pi$
$709$	−10.5705	−0.396984	−0.198492	+	0.980103i	$0.563604\pi$
$710$	0	0
$711$	−23.8246	−0.893492
$712$	0	0
$713$	−82.6051	−3.09358
$714$	0	0
$715$	−9.16797	−0.342863
$716$	0	0
$717$	24.9986	0.933590
$718$	0	0
$719$	33.0291	1.23178	0.615889	−	0.787833i	$-0.288797\pi$
$719$	33.0291	1.23178	0.615889	+	0.787833i	$0.288797\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−41.0480	−1.52659
$724$	0	0
$725$	−23.9103	−0.888007
$726$	0	0
$727$	−27.4483	−1.01800	−0.509001	−	0.860766i	$-0.669985\pi$
$727$	−27.4483	−1.01800	−0.509001	+	0.860766i	$0.669985\pi$
$728$	0	0
$729$	−43.8599	−1.62444
$730$	0	0
$731$	31.4722	1.16404
$732$	0	0
$733$	−9.03135	−0.333581	−0.166790	−	0.985992i	$-0.553340\pi$
$733$	−9.03135	−0.333581	−0.166790	+	0.985992i	$0.553340\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−81.8718	−3.01579
$738$	0	0
$739$	40.2255	1.47972	0.739859	−	0.672762i	$-0.234892\pi$
$739$	40.2255	1.47972	0.739859	+	0.672762i	$0.234892\pi$
$740$	0	0
$741$	−9.97425	−0.366413
$742$	0	0
$743$	2.30265	0.0844761	0.0422380	−	0.999108i	$-0.486551\pi$
$743$	2.30265	0.0844761	0.0422380	+	0.999108i	$0.486551\pi$
$744$	0	0
$745$	17.2287	0.631210
$746$	0	0
$747$	−30.2789	−1.10785
$748$	0	0
$749$	0	0
$750$	0	0
$751$	19.5371	0.712921	0.356460	−	0.934310i	$-0.383984\pi$
$751$	19.5371	0.712921	0.356460	+	0.934310i	$0.383984\pi$
$752$	0	0
$753$	−52.2156	−1.90284
$754$	0	0
$755$	−6.83323	−0.248687
$756$	0	0
$757$	8.28174	0.301005	0.150502	−	0.988610i	$-0.451911\pi$
$757$	8.28174	0.301005	0.150502	+	0.988610i	$0.451911\pi$
$758$	0	0
$759$	138.740	5.03593
$760$	0	0
$761$	−18.7920	−0.681210	−0.340605	−	0.940206i	$-0.610632\pi$
$761$	−18.7920	−0.681210	−0.340605	+	0.940206i	$0.610632\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	15.5871	0.563553
$766$	0	0
$767$	−13.0500	−0.471210
$768$	0	0
$769$	15.1327	0.545700	0.272850	−	0.962057i	$-0.412034\pi$
$769$	15.1327	0.545700	0.272850	+	0.962057i	$0.412034\pi$
$770$	0	0
$771$	−32.1992	−1.15963
$772$	0	0
$773$	−26.9296	−0.968591	−0.484295	−	0.874905i	$-0.660924\pi$
$773$	−26.9296	−0.968591	−0.484295	+	0.874905i	$0.660924\pi$
$774$	0	0
$775$	38.7726	1.39275
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.91765	0.0687069
$780$	0	0
$781$	−17.5647	−0.628513
$782$	0	0
$783$	−22.9721	−0.820957
$784$	0	0
$785$	−6.77716	−0.241887
$786$	0	0
$787$	2.91892	0.104048	0.0520241	−	0.998646i	$-0.483433\pi$
$787$	2.91892	0.104048	0.0520241	+	0.998646i	$0.483433\pi$
$788$	0	0
$789$	21.5502	0.767209
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9.93001	−0.352625
$794$	0	0
$795$	2.75161	0.0975896
$796$	0	0
$797$	16.4159	0.581481	0.290740	−	0.956802i	$-0.406098\pi$
$797$	16.4159	0.581481	0.290740	+	0.956802i	$0.406098\pi$
$798$	0	0
$799$	−40.8395	−1.44480
$800$	0	0
$801$	17.9670	0.634832
$802$	0	0
$803$	−24.6960	−0.871502
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−31.7040	−1.11603
$808$	0	0
$809$	49.3677	1.73568	0.867838	−	0.496846i	$-0.165509\pi$
$809$	49.3677	1.73568	0.867838	+	0.496846i	$0.165509\pi$
$810$	0	0
$811$	23.4315	0.822793	0.411396	−	0.911457i	$-0.365041\pi$
$811$	23.4315	0.822793	0.411396	+	0.911457i	$0.365041\pi$
$812$	0	0
$813$	33.1041	1.16101
$814$	0	0
$815$	15.4301	0.540492
$816$	0	0
$817$	15.0158	0.525336
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.32744	−0.151029	−0.0755143	−	0.997145i	$-0.524060\pi$
$821$	−4.32744	−0.151029	−0.0755143	+	0.997145i	$0.524060\pi$
$822$	0	0
$823$	−13.0805	−0.455956	−0.227978	−	0.973666i	$-0.573211\pi$
$823$	−13.0805	−0.455956	−0.227978	+	0.973666i	$0.573211\pi$
$824$	0	0
$825$	−65.1206	−2.26721
$826$	0	0
$827$	6.08156	0.211477	0.105738	−	0.994394i	$-0.466279\pi$
$827$	6.08156	0.211477	0.105738	+	0.994394i	$0.466279\pi$
$828$	0	0
$829$	−20.9946	−0.729172	−0.364586	−	0.931170i	$-0.618789\pi$
$829$	−20.9946	−0.729172	−0.364586	+	0.931170i	$0.618789\pi$
$830$	0	0
$831$	−59.1487	−2.05185
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4.83953	−0.167479
$836$	0	0
$837$	37.2512	1.28759
$838$	0	0
$839$	50.0675	1.72852	0.864261	−	0.503044i	$-0.167787\pi$
$839$	50.0675	1.72852	0.864261	+	0.503044i	$0.167787\pi$
$840$	0	0
$841$	2.57505	0.0887948
$842$	0	0
$843$	−33.5106	−1.15417
$844$	0	0
$845$	−8.10383	−0.278780
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−83.6352	−2.87035
$850$	0	0
$851$	13.2004	0.452504
$852$	0	0
$853$	−31.8851	−1.09173	−0.545863	−	0.837875i	$-0.683798\pi$
$853$	−31.8851	−1.09173	−0.545863	+	0.837875i	$0.683798\pi$
$854$	0	0
$855$	7.43680	0.254333
$856$	0	0
$857$	−50.4764	−1.72424	−0.862121	−	0.506703i	$-0.830864\pi$
$857$	−50.4764	−1.72424	−0.862121	+	0.506703i	$0.830864\pi$
$858$	0	0
$859$	−54.6739	−1.86545	−0.932724	−	0.360591i	$-0.882575\pi$
$859$	−54.6739	−1.86545	−0.932724	+	0.360591i	$0.882575\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.0631	1.05740	0.528700	−	0.848809i	$-0.322680\pi$
$863$	31.0631	1.05740	0.528700	+	0.848809i	$0.322680\pi$
$864$	0	0
$865$	−6.57261	−0.223475
$866$	0	0
$867$	2.31415	0.0785926
$868$	0	0
$869$	29.6422	1.00554
$870$	0	0
$871$	27.8254	0.942826
$872$	0	0
$873$	−15.4891	−0.524227
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.75325	−0.0592031	−0.0296016	−	0.999562i	$-0.509424\pi$
$877$	−1.75325	−0.0592031	−0.0296016	+	0.999562i	$0.509424\pi$
$878$	0	0
$879$	41.7082	1.40678
$880$	0	0
$881$	7.18675	0.242128	0.121064	−	0.992645i	$-0.461369\pi$
$881$	7.18675	0.242128	0.121064	+	0.992645i	$0.461369\pi$
$882$	0	0
$883$	−22.2021	−0.747159	−0.373579	−	0.927598i	$-0.621870\pi$
$883$	−22.2021	−0.747159	−0.373579	+	0.927598i	$0.621870\pi$
$884$	0	0
$885$	16.2263	0.545441
$886$	0	0
$887$	7.26838	0.244048	0.122024	−	0.992527i	$-0.461061\pi$
$887$	7.26838	0.244048	0.122024	+	0.992527i	$0.461061\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	12.7988	0.428774
$892$	0	0
$893$	−19.4850	−0.652041
$894$	0	0
$895$	−18.5909	−0.621424
$896$	0	0
$897$	−47.1527	−1.57438
$898$	0	0
$899$	−51.2016	−1.70767
$900$	0	0
$901$	−4.68120	−0.155953
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.06715	−0.135197
$906$	0	0
$907$	53.2523	1.76821	0.884107	−	0.467285i	$-0.154768\pi$
$907$	53.2523	1.76821	0.884107	+	0.467285i	$0.154768\pi$
$908$	0	0
$909$	79.5703	2.63918
$910$	0	0
$911$	−30.2046	−1.00072	−0.500361	−	0.865817i	$-0.666799\pi$
$911$	−30.2046	−1.00072	−0.500361	+	0.865817i	$0.666799\pi$
$912$	0	0
$913$	37.6725	1.24678
$914$	0	0
$915$	12.3469	0.408176
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−37.1650	−1.22596	−0.612980	−	0.790099i	$-0.710029\pi$
$919$	−37.1650	−1.22596	−0.612980	+	0.790099i	$0.710029\pi$
$920$	0	0
$921$	−10.5502	−0.347642
$922$	0	0
$923$	5.96961	0.196492
$924$	0	0
$925$	−6.19592	−0.203721
$926$	0	0
$927$	−71.1027	−2.33532
$928$	0	0
$929$	46.2523	1.51749	0.758744	−	0.651389i	$-0.225813\pi$
$929$	46.2523	1.51749	0.758744	+	0.651389i	$0.225813\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	16.3597	0.535593
$934$	0	0
$935$	−19.3933	−0.634228
$936$	0	0
$937$	−29.1366	−0.951850	−0.475925	−	0.879486i	$-0.657887\pi$
$937$	−29.1366	−0.951850	−0.475925	+	0.879486i	$0.657887\pi$
$938$	0	0
$939$	40.3682	1.31737
$940$	0	0
$941$	−32.7856	−1.06878	−0.534391	−	0.845238i	$-0.679459\pi$
$941$	−32.7856	−1.06878	−0.534391	+	0.845238i	$0.679459\pi$
$942$	0	0
$943$	9.06557	0.295216
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23.0635	−0.749465	−0.374732	−	0.927133i	$-0.622265\pi$
$947$	−23.0635	−0.749465	−0.374732	+	0.927133i	$0.622265\pi$
$948$	0	0
$949$	8.39330	0.272458
$950$	0	0
$951$	−83.0561	−2.69328
$952$	0	0
$953$	16.8472	0.545734	0.272867	−	0.962052i	$-0.412028\pi$
$953$	16.8472	0.545734	0.272867	+	0.962052i	$0.412028\pi$
$954$	0	0
$955$	−9.61512	−0.311138
$956$	0	0
$957$	85.9958	2.77985
$958$	0	0
$959$	0	0
$960$	0	0
$961$	52.0276	1.67831
$962$	0	0
$963$	41.1297	1.32539
$964$	0	0
$965$	21.1726	0.681570
$966$	0	0
$967$	12.4420	0.400107	0.200053	−	0.979785i	$-0.435888\pi$
$967$	12.4420	0.400107	0.200053	+	0.979785i	$0.435888\pi$
$968$	0	0
$969$	−21.0988	−0.677791
$970$	0	0
$971$	10.8254	0.347404	0.173702	−	0.984798i	$-0.444427\pi$
$971$	10.8254	0.347404	0.173702	+	0.984798i	$0.444427\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	22.1322	0.708798
$976$	0	0
$977$	−26.1505	−0.836628	−0.418314	−	0.908303i	$-0.637379\pi$
$977$	−26.1505	−0.836628	−0.418314	+	0.908303i	$0.637379\pi$
$978$	0	0
$979$	−22.3543	−0.714445
$980$	0	0
$981$	−62.4509	−1.99390
$982$	0	0
$983$	−44.7712	−1.42798	−0.713990	−	0.700156i	$-0.753114\pi$
$983$	−44.7712	−1.42798	−0.713990	+	0.700156i	$0.753114\pi$
$984$	0	0
$985$	−22.1913	−0.707075
$986$	0	0
$987$	0	0
$988$	0	0
$989$	70.9863	2.25723
$990$	0	0
$991$	−22.5346	−0.715834	−0.357917	−	0.933753i	$-0.616513\pi$
$991$	−22.5346	−0.715834	−0.357917	+	0.933753i	$0.616513\pi$
$992$	0	0
$993$	−55.5594	−1.76312
$994$	0	0
$995$	0.839576	0.0266163
$996$	0	0
$997$	0.655575	0.0207623	0.0103811	−	0.999946i	$-0.496696\pi$
$997$	0.655575	0.0207623	0.0103811	+	0.999946i	$0.496696\pi$
$998$	0	0
$999$	−5.95280	−0.188338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.r.1.3		15
7.3	odd	6		1148.2.i.e.821.3	yes	30
7.5	odd	6		1148.2.i.e.165.3	✓	30
7.6	odd	2		8036.2.a.q.1.13		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.e.165.3	✓	30	7.5	odd	6
1148.2.i.e.821.3	yes	30	7.3	odd	6
8036.2.a.q.1.13		15	7.6	odd	2
8036.2.a.r.1.3		15	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)

\(f(q)\)	\(=\)	\(q-2.73742 q^{3} +0.863054 q^{5} +4.49344 q^{9} +O(q^{10})\)	\(q-2.73742 q^{3} +0.863054 q^{5} +4.49344 q^{9} -5.59068 q^{11} +1.90008 q^{13} -2.36254 q^{15} +4.01928 q^{17} +1.91765 q^{19} +9.06557 q^{23} -4.25514 q^{25} -4.08817 q^{27} +5.61917 q^{29} -9.11195 q^{31} +15.3040 q^{33} +1.45610 q^{37} -5.20129 q^{39} +1.00000 q^{41} +7.83032 q^{43} +3.87808 q^{45} -10.1609 q^{47} -11.0024 q^{51} -1.16468 q^{53} -4.82506 q^{55} -5.24940 q^{57} -6.86817 q^{59} -5.22611 q^{61} +1.63987 q^{65} +14.6443 q^{67} -24.8162 q^{69} +3.14178 q^{71} +4.41735 q^{73} +11.6481 q^{75} -5.30208 q^{79} -2.28930 q^{81} -6.73846 q^{83} +3.46886 q^{85} -15.3820 q^{87} +3.99849 q^{89} +24.9432 q^{93} +1.65503 q^{95} -3.44705 q^{97} -25.1214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * q^99

Introduction

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