LMFDB - Embedded newform 912.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(1,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("912.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.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:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 228)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	912.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.37228 q^{5} +2.37228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.37228 q^{5} +2.37228 q^{7} +1.00000 q^{9} +4.37228 q^{11} +2.00000 q^{13} -4.37228 q^{15} -4.37228 q^{17} -1.00000 q^{19} -2.37228 q^{21} -2.74456 q^{23} +14.1168 q^{25} -1.00000 q^{27} -8.74456 q^{29} -4.74456 q^{31} -4.37228 q^{33} +10.3723 q^{35} -6.74456 q^{37} -2.00000 q^{39} +2.37228 q^{43} +4.37228 q^{45} +7.62772 q^{47} -1.37228 q^{49} +4.37228 q^{51} +8.74456 q^{53} +19.1168 q^{55} +1.00000 q^{57} -8.37228 q^{61} +2.37228 q^{63} +8.74456 q^{65} -13.4891 q^{67} +2.74456 q^{69} +12.0000 q^{71} +0.372281 q^{73} -14.1168 q^{75} +10.3723 q^{77} -8.00000 q^{79} +1.00000 q^{81} +2.74456 q^{83} -19.1168 q^{85} +8.74456 q^{87} +3.25544 q^{89} +4.74456 q^{91} +4.74456 q^{93} -4.37228 q^{95} +14.0000 q^{97} +4.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 3 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 3 * q^5 - q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 3 q^{5} - q^{7} + 2 q^{9} + 3 q^{11} + 4 q^{13} - 3 q^{15} - 3 q^{17} - 2 q^{19} + q^{21} + 6 q^{23} + 11 q^{25} - 2 q^{27} - 6 q^{29} + 2 q^{31} - 3 q^{33} + 15 q^{35} - 2 q^{37} - 4 q^{39} - q^{43} + 3 q^{45} + 21 q^{47} + 3 q^{49} + 3 q^{51} + 6 q^{53} + 21 q^{55} + 2 q^{57} - 11 q^{61} - q^{63} + 6 q^{65} - 4 q^{67} - 6 q^{69} + 24 q^{71} - 5 q^{73} - 11 q^{75} + 15 q^{77} - 16 q^{79} + 2 q^{81} - 6 q^{83} - 21 q^{85} + 6 q^{87} + 18 q^{89} - 2 q^{91} - 2 q^{93} - 3 q^{95} + 28 q^{97} + 3 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 3 * q^5 - q^7 + 2 * q^9 + 3 * q^11 + 4 * q^13 - 3 * q^15 - 3 * q^17 - 2 * q^19 + q^21 + 6 * q^23 + 11 * q^25 - 2 * q^27 - 6 * q^29 + 2 * q^31 - 3 * q^33 + 15 * q^35 - 2 * q^37 - 4 * q^39 - q^43 + 3 * q^45 + 21 * q^47 + 3 * q^49 + 3 * q^51 + 6 * q^53 + 21 * q^55 + 2 * q^57 - 11 * q^61 - q^63 + 6 * q^65 - 4 * q^67 - 6 * q^69 + 24 * q^71 - 5 * q^73 - 11 * q^75 + 15 * q^77 - 16 * q^79 + 2 * q^81 - 6 * q^83 - 21 * q^85 + 6 * q^87 + 18 * q^89 - 2 * q^91 - 2 * q^93 - 3 * q^95 + 28 * q^97 + 3 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	4.37228	1.95534	0.977672	−	0.210138i	$-0.0673912\pi$
$5$	4.37228	1.95534	0.977672	+	0.210138i	$0.0673912\pi$
$6$	0	0
$7$	2.37228	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$7$	2.37228	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.37228	1.31829	0.659146	−	0.752015i	$-0.270918\pi$
$11$	4.37228	1.31829	0.659146	+	0.752015i	$0.270918\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−4.37228	−1.12892
$16$	0	0
$17$	−4.37228	−1.06043	−0.530217	−	0.847862i	$-0.677890\pi$
$17$	−4.37228	−1.06043	−0.530217	+	0.847862i	$0.677890\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.37228	−0.517674
$22$	0	0
$23$	−2.74456	−0.572281	−0.286140	−	0.958188i	$-0.592372\pi$
$23$	−2.74456	−0.572281	−0.286140	+	0.958188i	$0.592372\pi$
$24$	0	0
$25$	14.1168	2.82337
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.74456	−1.62382	−0.811912	−	0.583779i	$-0.801573\pi$
$29$	−8.74456	−1.62382	−0.811912	+	0.583779i	$0.801573\pi$
$30$	0	0
$31$	−4.74456	−0.852149	−0.426074	−	0.904688i	$-0.640104\pi$
$31$	−4.74456	−0.852149	−0.426074	+	0.904688i	$0.640104\pi$
$32$	0	0
$33$	−4.37228	−0.761116
$34$	0	0
$35$	10.3723	1.75324
$36$	0	0
$37$	−6.74456	−1.10880	−0.554400	−	0.832251i	$-0.687052\pi$
$37$	−6.74456	−1.10880	−0.554400	+	0.832251i	$0.687052\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	2.37228	0.361770	0.180885	−	0.983504i	$-0.442104\pi$
$43$	2.37228	0.361770	0.180885	+	0.983504i	$0.442104\pi$
$44$	0	0
$45$	4.37228	0.651781
$46$	0	0
$47$	7.62772	1.11262	0.556309	−	0.830976i	$-0.312217\pi$
$47$	7.62772	1.11262	0.556309	+	0.830976i	$0.312217\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	4.37228	0.612242
$52$	0	0
$53$	8.74456	1.20116	0.600579	−	0.799565i	$-0.294937\pi$
$53$	8.74456	1.20116	0.600579	+	0.799565i	$0.294937\pi$
$54$	0	0
$55$	19.1168	2.57771
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−8.37228	−1.07196	−0.535980	−	0.844230i	$-0.680058\pi$
$61$	−8.37228	−1.07196	−0.535980	+	0.844230i	$0.680058\pi$
$62$	0	0
$63$	2.37228	0.298879
$64$	0	0
$65$	8.74456	1.08463
$66$	0	0
$67$	−13.4891	−1.64796	−0.823979	−	0.566620i	$-0.808251\pi$
$67$	−13.4891	−1.64796	−0.823979	+	0.566620i	$0.808251\pi$
$68$	0	0
$69$	2.74456	0.330407
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	0.372281	0.0435722	0.0217861	−	0.999763i	$-0.493065\pi$
$73$	0.372281	0.0435722	0.0217861	+	0.999763i	$0.493065\pi$
$74$	0	0
$75$	−14.1168	−1.63007
$76$	0	0
$77$	10.3723	1.18203
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.74456	0.301255	0.150627	−	0.988591i	$-0.451871\pi$
$83$	2.74456	0.301255	0.150627	+	0.988591i	$0.451871\pi$
$84$	0	0
$85$	−19.1168	−2.07351
$86$	0	0
$87$	8.74456	0.937516
$88$	0	0
$89$	3.25544	0.345076	0.172538	−	0.985003i	$-0.444803\pi$
$89$	3.25544	0.345076	0.172538	+	0.985003i	$0.444803\pi$
$90$	0	0
$91$	4.74456	0.497365
$92$	0	0
$93$	4.74456	0.491988
$94$	0	0
$95$	−4.37228	−0.448587
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	4.37228	0.439431
$100$	0	0
$101$	−11.4891	−1.14321	−0.571605	−	0.820529i	$-0.693679\pi$
$101$	−11.4891	−1.14321	−0.571605	+	0.820529i	$0.693679\pi$
$102$	0	0
$103$	12.7446	1.25576	0.627880	−	0.778311i	$-0.283923\pi$
$103$	12.7446	1.25576	0.627880	+	0.778311i	$0.283923\pi$
$104$	0	0
$105$	−10.3723	−1.01223
$106$	0	0
$107$	−3.25544	−0.314715	−0.157358	−	0.987542i	$-0.550297\pi$
$107$	−3.25544	−0.314715	−0.157358	+	0.987542i	$0.550297\pi$
$108$	0	0
$109$	16.2337	1.55491	0.777453	−	0.628941i	$-0.216511\pi$
$109$	16.2337	1.55491	0.777453	+	0.628941i	$0.216511\pi$
$110$	0	0
$111$	6.74456	0.640166
$112$	0	0
$113$	3.25544	0.306246	0.153123	−	0.988207i	$-0.451067\pi$
$113$	3.25544	0.306246	0.153123	+	0.988207i	$0.451067\pi$
$114$	0	0
$115$	−12.0000	−1.11901
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−10.3723	−0.950825
$120$	0	0
$121$	8.11684	0.737895
$122$	0	0
$123$	0	0
$124$	0	0
$125$	39.8614	3.56531
$126$	0	0
$127$	−22.2337	−1.97292	−0.986460	−	0.164000i	$-0.947560\pi$
$127$	−22.2337	−1.97292	−0.986460	+	0.164000i	$0.947560\pi$
$128$	0	0
$129$	−2.37228	−0.208868
$130$	0	0
$131$	−9.86141	−0.861595	−0.430798	−	0.902449i	$-0.641768\pi$
$131$	−9.86141	−0.861595	−0.430798	+	0.902449i	$0.641768\pi$
$132$	0	0
$133$	−2.37228	−0.205703
$134$	0	0
$135$	−4.37228	−0.376306
$136$	0	0
$137$	−4.37228	−0.373549	−0.186775	−	0.982403i	$-0.559803\pi$
$137$	−4.37228	−0.373549	−0.186775	+	0.982403i	$0.559803\pi$
$138$	0	0
$139$	−9.62772	−0.816612	−0.408306	−	0.912845i	$-0.633880\pi$
$139$	−9.62772	−0.816612	−0.408306	+	0.912845i	$0.633880\pi$
$140$	0	0
$141$	−7.62772	−0.642370
$142$	0	0
$143$	8.74456	0.731257
$144$	0	0
$145$	−38.2337	−3.17513
$146$	0	0
$147$	1.37228	0.113184
$148$	0	0
$149$	−4.37228	−0.358191	−0.179096	−	0.983832i	$-0.557317\pi$
$149$	−4.37228	−0.358191	−0.179096	+	0.983832i	$0.557317\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	−4.37228	−0.353478
$154$	0	0
$155$	−20.7446	−1.66624
$156$	0	0
$157$	−15.4891	−1.23617	−0.618083	−	0.786113i	$-0.712091\pi$
$157$	−15.4891	−1.23617	−0.618083	+	0.786113i	$0.712091\pi$
$158$	0	0
$159$	−8.74456	−0.693489
$160$	0	0
$161$	−6.51087	−0.513129
$162$	0	0
$163$	−13.4891	−1.05655	−0.528275	−	0.849073i	$-0.677161\pi$
$163$	−13.4891	−1.05655	−0.528275	+	0.849073i	$0.677161\pi$
$164$	0	0
$165$	−19.1168	−1.48824
$166$	0	0
$167$	14.2337	1.10144	0.550718	−	0.834691i	$-0.314354\pi$
$167$	14.2337	1.10144	0.550718	+	0.834691i	$0.314354\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	20.7446	1.57718	0.788590	−	0.614919i	$-0.210811\pi$
$173$	20.7446	1.57718	0.788590	+	0.614919i	$0.210811\pi$
$174$	0	0
$175$	33.4891	2.53154
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.25544	−0.243323	−0.121661	−	0.992572i	$-0.538822\pi$
$179$	−3.25544	−0.243323	−0.121661	+	0.992572i	$0.538822\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	8.37228	0.618897
$184$	0	0
$185$	−29.4891	−2.16808
$186$	0	0
$187$	−19.1168	−1.39796
$188$	0	0
$189$	−2.37228	−0.172558
$190$	0	0
$191$	16.3723	1.18466	0.592328	−	0.805697i	$-0.298209\pi$
$191$	16.3723	1.18466	0.592328	+	0.805697i	$0.298209\pi$
$192$	0	0
$193$	5.25544	0.378295	0.189147	−	0.981949i	$-0.439428\pi$
$193$	5.25544	0.378295	0.189147	+	0.981949i	$0.439428\pi$
$194$	0	0
$195$	−8.74456	−0.626211
$196$	0	0
$197$	11.4891	0.818566	0.409283	−	0.912407i	$-0.365779\pi$
$197$	11.4891	0.818566	0.409283	+	0.912407i	$0.365779\pi$
$198$	0	0
$199$	−3.11684	−0.220947	−0.110474	−	0.993879i	$-0.535237\pi$
$199$	−3.11684	−0.220947	−0.110474	+	0.993879i	$0.535237\pi$
$200$	0	0
$201$	13.4891	0.951450
$202$	0	0
$203$	−20.7446	−1.45598
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.74456	−0.190760
$208$	0	0
$209$	−4.37228	−0.302437
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	10.3723	0.707384
$216$	0	0
$217$	−11.2554	−0.764069
$218$	0	0
$219$	−0.372281	−0.0251564
$220$	0	0
$221$	−8.74456	−0.588223
$222$	0	0
$223$	9.48913	0.635439	0.317719	−	0.948185i	$-0.397083\pi$
$223$	9.48913	0.635439	0.317719	+	0.948185i	$0.397083\pi$
$224$	0	0
$225$	14.1168	0.941123
$226$	0	0
$227$	−26.2337	−1.74119	−0.870596	−	0.491999i	$-0.836266\pi$
$227$	−26.2337	−1.74119	−0.870596	+	0.491999i	$0.836266\pi$
$228$	0	0
$229$	0.372281	0.0246010	0.0123005	−	0.999924i	$-0.496085\pi$
$229$	0.372281	0.0246010	0.0123005	+	0.999924i	$0.496085\pi$
$230$	0	0
$231$	−10.3723	−0.682446
$232$	0	0
$233$	4.37228	0.286438	0.143219	−	0.989691i	$-0.454255\pi$
$233$	4.37228	0.286438	0.143219	+	0.989691i	$0.454255\pi$
$234$	0	0
$235$	33.3505	2.17555
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	9.86141	0.637881	0.318941	−	0.947775i	$-0.396673\pi$
$239$	9.86141	0.637881	0.318941	+	0.947775i	$0.396673\pi$
$240$	0	0
$241$	10.7446	0.692118	0.346059	−	0.938213i	$-0.387520\pi$
$241$	10.7446	0.692118	0.346059	+	0.938213i	$0.387520\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.00000	−0.383326
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	−2.74456	−0.173930
$250$	0	0
$251$	−4.37228	−0.275976	−0.137988	−	0.990434i	$-0.544064\pi$
$251$	−4.37228	−0.275976	−0.137988	+	0.990434i	$0.544064\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	19.1168	1.19714
$256$	0	0
$257$	5.48913	0.342402	0.171201	−	0.985236i	$-0.445235\pi$
$257$	5.48913	0.342402	0.171201	+	0.985236i	$0.445235\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	−8.74456	−0.541275
$262$	0	0
$263$	2.13859	0.131871	0.0659357	−	0.997824i	$-0.478997\pi$
$263$	2.13859	0.131871	0.0659357	+	0.997824i	$0.478997\pi$
$264$	0	0
$265$	38.2337	2.34868
$266$	0	0
$267$	−3.25544	−0.199230
$268$	0	0
$269$	3.25544	0.198488	0.0992438	−	0.995063i	$-0.468358\pi$
$269$	3.25544	0.198488	0.0992438	+	0.995063i	$0.468358\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	0	0
$273$	−4.74456	−0.287154
$274$	0	0
$275$	61.7228	3.72203
$276$	0	0
$277$	6.88316	0.413569	0.206784	−	0.978387i	$-0.433700\pi$
$277$	6.88316	0.413569	0.206784	+	0.978387i	$0.433700\pi$
$278$	0	0
$279$	−4.74456	−0.284050
$280$	0	0
$281$	−15.2554	−0.910063	−0.455032	−	0.890475i	$-0.650372\pi$
$281$	−15.2554	−0.910063	−0.455032	+	0.890475i	$0.650372\pi$
$282$	0	0
$283$	8.88316	0.528049	0.264024	−	0.964516i	$-0.414950\pi$
$283$	8.88316	0.528049	0.264024	+	0.964516i	$0.414950\pi$
$284$	0	0
$285$	4.37228	0.258992
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.11684	0.124520
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0	0
$293$	6.51087	0.380369	0.190185	−	0.981748i	$-0.439091\pi$
$293$	6.51087	0.380369	0.190185	+	0.981748i	$0.439091\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.37228	−0.253705
$298$	0	0
$299$	−5.48913	−0.317444
$300$	0	0
$301$	5.62772	0.324376
$302$	0	0
$303$	11.4891	0.660033
$304$	0	0
$305$	−36.6060	−2.09605
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	−12.7446	−0.725013
$310$	0	0
$311$	−9.86141	−0.559189	−0.279595	−	0.960118i	$-0.590200\pi$
$311$	−9.86141	−0.559189	−0.279595	+	0.960118i	$0.590200\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	10.3723	0.584412
$316$	0	0
$317$	6.51087	0.365687	0.182844	−	0.983142i	$-0.441470\pi$
$317$	6.51087	0.365687	0.182844	+	0.983142i	$0.441470\pi$
$318$	0	0
$319$	−38.2337	−2.14068
$320$	0	0
$321$	3.25544	0.181701
$322$	0	0
$323$	4.37228	0.243280
$324$	0	0
$325$	28.2337	1.56612
$326$	0	0
$327$	−16.2337	−0.897725
$328$	0	0
$329$	18.0951	0.997615
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	−6.74456	−0.369600
$334$	0	0
$335$	−58.9783	−3.22233
$336$	0	0
$337$	−3.48913	−0.190065	−0.0950324	−	0.995474i	$-0.530295\pi$
$337$	−3.48913	−0.190065	−0.0950324	+	0.995474i	$0.530295\pi$
$338$	0	0
$339$	−3.25544	−0.176811
$340$	0	0
$341$	−20.7446	−1.12338
$342$	0	0
$343$	−19.8614	−1.07242
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	1.11684	0.0599553	0.0299777	−	0.999551i	$-0.490456\pi$
$347$	1.11684	0.0599553	0.0299777	+	0.999551i	$0.490456\pi$
$348$	0	0
$349$	24.3723	1.30462	0.652309	−	0.757953i	$-0.273800\pi$
$349$	24.3723	1.30462	0.652309	+	0.757953i	$0.273800\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	52.4674	2.78468
$356$	0	0
$357$	10.3723	0.548959
$358$	0	0
$359$	33.8614	1.78714	0.893568	−	0.448927i	$-0.148194\pi$
$359$	33.8614	1.78714	0.893568	+	0.448927i	$0.148194\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−8.11684	−0.426024
$364$	0	0
$365$	1.62772	0.0851987
$366$	0	0
$367$	−2.51087	−0.131067	−0.0655333	−	0.997850i	$-0.520875\pi$
$367$	−2.51087	−0.131067	−0.0655333	+	0.997850i	$0.520875\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	20.7446	1.07700
$372$	0	0
$373$	4.23369	0.219212	0.109606	−	0.993975i	$-0.465041\pi$
$373$	4.23369	0.219212	0.109606	+	0.993975i	$0.465041\pi$
$374$	0	0
$375$	−39.8614	−2.05843
$376$	0	0
$377$	−17.4891	−0.900736
$378$	0	0
$379$	35.7228	1.83496	0.917479	−	0.397785i	$-0.130221\pi$
$379$	35.7228	1.83496	0.917479	+	0.397785i	$0.130221\pi$
$380$	0	0
$381$	22.2337	1.13907
$382$	0	0
$383$	26.2337	1.34048	0.670239	−	0.742145i	$-0.266191\pi$
$383$	26.2337	1.34048	0.670239	+	0.742145i	$0.266191\pi$
$384$	0	0
$385$	45.3505	2.31128
$386$	0	0
$387$	2.37228	0.120590
$388$	0	0
$389$	−19.6277	−0.995165	−0.497582	−	0.867417i	$-0.665779\pi$
$389$	−19.6277	−0.995165	−0.497582	+	0.867417i	$0.665779\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	9.86141	0.497442
$394$	0	0
$395$	−34.9783	−1.75995
$396$	0	0
$397$	−10.6060	−0.532298	−0.266149	−	0.963932i	$-0.585751\pi$
$397$	−10.6060	−0.532298	−0.266149	+	0.963932i	$0.585751\pi$
$398$	0	0
$399$	2.37228	0.118763
$400$	0	0
$401$	−17.4891	−0.873365	−0.436683	−	0.899616i	$-0.643847\pi$
$401$	−17.4891	−0.873365	−0.436683	+	0.899616i	$0.643847\pi$
$402$	0	0
$403$	−9.48913	−0.472687
$404$	0	0
$405$	4.37228	0.217260
$406$	0	0
$407$	−29.4891	−1.46172
$408$	0	0
$409$	−38.4674	−1.90209	−0.951045	−	0.309053i	$-0.899988\pi$
$409$	−38.4674	−1.90209	−0.951045	+	0.309053i	$0.899988\pi$
$410$	0	0
$411$	4.37228	0.215669
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	9.62772	0.471471
$418$	0	0
$419$	−37.7228	−1.84288	−0.921440	−	0.388521i	$-0.872986\pi$
$419$	−37.7228	−1.84288	−0.921440	+	0.388521i	$0.872986\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	7.62772	0.370872
$424$	0	0
$425$	−61.7228	−2.99400
$426$	0	0
$427$	−19.8614	−0.961161
$428$	0	0
$429$	−8.74456	−0.422191
$430$	0	0
$431$	−8.74456	−0.421211	−0.210605	−	0.977571i	$-0.567544\pi$
$431$	−8.74456	−0.421211	−0.210605	+	0.977571i	$0.567544\pi$
$432$	0	0
$433$	−15.4891	−0.744360	−0.372180	−	0.928161i	$-0.621390\pi$
$433$	−15.4891	−0.744360	−0.372180	+	0.928161i	$0.621390\pi$
$434$	0	0
$435$	38.2337	1.83317
$436$	0	0
$437$	2.74456	0.131290
$438$	0	0
$439$	30.2337	1.44298	0.721488	−	0.692427i	$-0.243459\pi$
$439$	30.2337	1.44298	0.721488	+	0.692427i	$0.243459\pi$
$440$	0	0
$441$	−1.37228	−0.0653467
$442$	0	0
$443$	7.62772	0.362404	0.181202	−	0.983446i	$-0.442001\pi$
$443$	7.62772	0.362404	0.181202	+	0.983446i	$0.442001\pi$
$444$	0	0
$445$	14.2337	0.674742
$446$	0	0
$447$	4.37228	0.206802
$448$	0	0
$449$	22.9783	1.08441	0.542205	−	0.840246i	$-0.317589\pi$
$449$	22.9783	1.08441	0.542205	+	0.840246i	$0.317589\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	20.7446	0.972520
$456$	0	0
$457$	−13.8614	−0.648409	−0.324205	−	0.945987i	$-0.605097\pi$
$457$	−13.8614	−0.648409	−0.324205	+	0.945987i	$0.605097\pi$
$458$	0	0
$459$	4.37228	0.204081
$460$	0	0
$461$	−39.3505	−1.83274	−0.916368	−	0.400336i	$-0.868893\pi$
$461$	−39.3505	−1.83274	−0.916368	+	0.400336i	$0.868893\pi$
$462$	0	0
$463$	−17.3505	−0.806348	−0.403174	−	0.915123i	$-0.632093\pi$
$463$	−17.3505	−0.806348	−0.403174	+	0.915123i	$0.632093\pi$
$464$	0	0
$465$	20.7446	0.962006
$466$	0	0
$467$	13.1168	0.606975	0.303488	−	0.952835i	$-0.401849\pi$
$467$	13.1168	0.606975	0.303488	+	0.952835i	$0.401849\pi$
$468$	0	0
$469$	−32.0000	−1.47762
$470$	0	0
$471$	15.4891	0.713701
$472$	0	0
$473$	10.3723	0.476918
$474$	0	0
$475$	−14.1168	−0.647725
$476$	0	0
$477$	8.74456	0.400386
$478$	0	0
$479$	−3.76631	−0.172087	−0.0860436	−	0.996291i	$-0.527422\pi$
$479$	−3.76631	−0.172087	−0.0860436	+	0.996291i	$0.527422\pi$
$480$	0	0
$481$	−13.4891	−0.615051
$482$	0	0
$483$	6.51087	0.296255
$484$	0	0
$485$	61.2119	2.77949
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	13.4891	0.609999
$490$	0	0
$491$	38.7446	1.74852	0.874259	−	0.485460i	$-0.161348\pi$
$491$	38.7446	1.74852	0.874259	+	0.485460i	$0.161348\pi$
$492$	0	0
$493$	38.2337	1.72196
$494$	0	0
$495$	19.1168	0.859238
$496$	0	0
$497$	28.4674	1.27694
$498$	0	0
$499$	−17.3505	−0.776716	−0.388358	−	0.921508i	$-0.626958\pi$
$499$	−17.3505	−0.776716	−0.388358	+	0.921508i	$0.626958\pi$
$500$	0	0
$501$	−14.2337	−0.635914
$502$	0	0
$503$	−14.7446	−0.657428	−0.328714	−	0.944430i	$-0.606615\pi$
$503$	−14.7446	−0.657428	−0.328714	+	0.944430i	$0.606615\pi$
$504$	0	0
$505$	−50.2337	−2.23537
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−3.25544	−0.144295	−0.0721474	−	0.997394i	$-0.522985\pi$
$509$	−3.25544	−0.144295	−0.0721474	+	0.997394i	$0.522985\pi$
$510$	0	0
$511$	0.883156	0.0390685
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	55.7228	2.45544
$516$	0	0
$517$	33.3505	1.46675
$518$	0	0
$519$	−20.7446	−0.910585
$520$	0	0
$521$	−17.4891	−0.766212	−0.383106	−	0.923704i	$-0.625146\pi$
$521$	−17.4891	−0.766212	−0.383106	+	0.923704i	$0.625146\pi$
$522$	0	0
$523$	7.25544	0.317258	0.158629	−	0.987338i	$-0.449293\pi$
$523$	7.25544	0.317258	0.158629	+	0.987338i	$0.449293\pi$
$524$	0	0
$525$	−33.4891	−1.46159
$526$	0	0
$527$	20.7446	0.903647
$528$	0	0
$529$	−15.4674	−0.672495
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−14.2337	−0.615376
$536$	0	0
$537$	3.25544	0.140482
$538$	0	0
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−1.86141	−0.0800281	−0.0400141	−	0.999199i	$-0.512740\pi$
$541$	−1.86141	−0.0800281	−0.0400141	+	0.999199i	$0.512740\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	70.9783	3.04037
$546$	0	0
$547$	7.25544	0.310220	0.155110	−	0.987897i	$-0.450427\pi$
$547$	7.25544	0.310220	0.155110	+	0.987897i	$0.450427\pi$
$548$	0	0
$549$	−8.37228	−0.357320
$550$	0	0
$551$	8.74456	0.372531
$552$	0	0
$553$	−18.9783	−0.807037
$554$	0	0
$555$	29.4891	1.25174
$556$	0	0
$557$	−21.8614	−0.926298	−0.463149	−	0.886281i	$-0.653280\pi$
$557$	−21.8614	−0.926298	−0.463149	+	0.886281i	$0.653280\pi$
$558$	0	0
$559$	4.74456	0.200674
$560$	0	0
$561$	19.1168	0.807114
$562$	0	0
$563$	−1.02175	−0.0430616	−0.0215308	−	0.999768i	$-0.506854\pi$
$563$	−1.02175	−0.0430616	−0.0215308	+	0.999768i	$0.506854\pi$
$564$	0	0
$565$	14.2337	0.598816
$566$	0	0
$567$	2.37228	0.0996265
$568$	0	0
$569$	−32.7446	−1.37272	−0.686362	−	0.727260i	$-0.740793\pi$
$569$	−32.7446	−1.37272	−0.686362	+	0.727260i	$0.740793\pi$
$570$	0	0
$571$	26.9783	1.12900	0.564502	−	0.825431i	$-0.309068\pi$
$571$	26.9783	1.12900	0.564502	+	0.825431i	$0.309068\pi$
$572$	0	0
$573$	−16.3723	−0.683962
$574$	0	0
$575$	−38.7446	−1.61576
$576$	0	0
$577$	21.1168	0.879106	0.439553	−	0.898217i	$-0.355137\pi$
$577$	21.1168	0.879106	0.439553	+	0.898217i	$0.355137\pi$
$578$	0	0
$579$	−5.25544	−0.218409
$580$	0	0
$581$	6.51087	0.270117
$582$	0	0
$583$	38.2337	1.58348
$584$	0	0
$585$	8.74456	0.361543
$586$	0	0
$587$	6.60597	0.272658	0.136329	−	0.990664i	$-0.456470\pi$
$587$	6.60597	0.272658	0.136329	+	0.990664i	$0.456470\pi$
$588$	0	0
$589$	4.74456	0.195496
$590$	0	0
$591$	−11.4891	−0.472599
$592$	0	0
$593$	−16.9783	−0.697213	−0.348607	−	0.937269i	$-0.613345\pi$
$593$	−16.9783	−0.697213	−0.348607	+	0.937269i	$0.613345\pi$
$594$	0	0
$595$	−45.3505	−1.85919
$596$	0	0
$597$	3.11684	0.127564
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−1.25544	−0.0512104	−0.0256052	−	0.999672i	$-0.508151\pi$
$601$	−1.25544	−0.0512104	−0.0256052	+	0.999672i	$0.508151\pi$
$602$	0	0
$603$	−13.4891	−0.549320
$604$	0	0
$605$	35.4891	1.44284
$606$	0	0
$607$	19.2554	0.781554	0.390777	−	0.920485i	$-0.372206\pi$
$607$	19.2554	0.781554	0.390777	+	0.920485i	$0.372206\pi$
$608$	0	0
$609$	20.7446	0.840612
$610$	0	0
$611$	15.2554	0.617169
$612$	0	0
$613$	−29.1168	−1.17602	−0.588009	−	0.808854i	$-0.700088\pi$
$613$	−29.1168	−1.17602	−0.588009	+	0.808854i	$0.700088\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10.8832	−0.438139	−0.219070	−	0.975709i	$-0.570302\pi$
$617$	−10.8832	−0.438139	−0.219070	+	0.975709i	$0.570302\pi$
$618$	0	0
$619$	−6.97825	−0.280480	−0.140240	−	0.990118i	$-0.544787\pi$
$619$	−6.97825	−0.280480	−0.140240	+	0.990118i	$0.544787\pi$
$620$	0	0
$621$	2.74456	0.110136
$622$	0	0
$623$	7.72281	0.309408
$624$	0	0
$625$	103.701	4.14804
$626$	0	0
$627$	4.37228	0.174612
$628$	0	0
$629$	29.4891	1.17581
$630$	0	0
$631$	−4.13859	−0.164755	−0.0823774	−	0.996601i	$-0.526251\pi$
$631$	−4.13859	−0.164755	−0.0823774	+	0.996601i	$0.526251\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	−97.2119	−3.85774
$636$	0	0
$637$	−2.74456	−0.108744
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	14.2337	0.562197	0.281098	−	0.959679i	$-0.409301\pi$
$641$	14.2337	0.562197	0.281098	+	0.959679i	$0.409301\pi$
$642$	0	0
$643$	37.3505	1.47296	0.736481	−	0.676459i	$-0.236486\pi$
$643$	37.3505	1.47296	0.736481	+	0.676459i	$0.236486\pi$
$644$	0	0
$645$	−10.3723	−0.408408
$646$	0	0
$647$	21.8614	0.859461	0.429730	−	0.902957i	$-0.358609\pi$
$647$	21.8614	0.859461	0.429730	+	0.902957i	$0.358609\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	11.2554	0.441135
$652$	0	0
$653$	30.6060	1.19770	0.598852	−	0.800860i	$-0.295624\pi$
$653$	30.6060	1.19770	0.598852	+	0.800860i	$0.295624\pi$
$654$	0	0
$655$	−43.1168	−1.68471
$656$	0	0
$657$	0.372281	0.0145241
$658$	0	0
$659$	8.74456	0.340640	0.170320	−	0.985389i	$-0.445520\pi$
$659$	8.74456	0.340640	0.170320	+	0.985389i	$0.445520\pi$
$660$	0	0
$661$	−24.2337	−0.942581	−0.471291	−	0.881978i	$-0.656212\pi$
$661$	−24.2337	−0.942581	−0.471291	+	0.881978i	$0.656212\pi$
$662$	0	0
$663$	8.74456	0.339611
$664$	0	0
$665$	−10.3723	−0.402220
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	−9.48913	−0.366871
$670$	0	0
$671$	−36.6060	−1.41316
$672$	0	0
$673$	−36.2337	−1.39671	−0.698353	−	0.715753i	$-0.746083\pi$
$673$	−36.2337	−1.39671	−0.698353	+	0.715753i	$0.746083\pi$
$674$	0	0
$675$	−14.1168	−0.543358
$676$	0	0
$677$	8.74456	0.336081	0.168040	−	0.985780i	$-0.446256\pi$
$677$	8.74456	0.336081	0.168040	+	0.985780i	$0.446256\pi$
$678$	0	0
$679$	33.2119	1.27456
$680$	0	0
$681$	26.2337	1.00528
$682$	0	0
$683$	−20.7446	−0.793769	−0.396884	−	0.917869i	$-0.629909\pi$
$683$	−20.7446	−0.793769	−0.396884	+	0.917869i	$0.629909\pi$
$684$	0	0
$685$	−19.1168	−0.730417
$686$	0	0
$687$	−0.372281	−0.0142034
$688$	0	0
$689$	17.4891	0.666283
$690$	0	0
$691$	−29.3505	−1.11655	−0.558273	−	0.829657i	$-0.688536\pi$
$691$	−29.3505	−1.11655	−0.558273	+	0.829657i	$0.688536\pi$
$692$	0	0
$693$	10.3723	0.394010
$694$	0	0
$695$	−42.0951	−1.59676
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−4.37228	−0.165375
$700$	0	0
$701$	12.5109	0.472529	0.236265	−	0.971689i	$-0.424077\pi$
$701$	12.5109	0.472529	0.236265	+	0.971689i	$0.424077\pi$
$702$	0	0
$703$	6.74456	0.254376
$704$	0	0
$705$	−33.3505	−1.25605
$706$	0	0
$707$	−27.2554	−1.02505
$708$	0	0
$709$	−50.4674	−1.89534	−0.947671	−	0.319248i	$-0.896570\pi$
$709$	−50.4674	−1.89534	−0.947671	+	0.319248i	$0.896570\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	13.0217	0.487668
$714$	0	0
$715$	38.2337	1.42986
$716$	0	0
$717$	−9.86141	−0.368281
$718$	0	0
$719$	10.8832	0.405873	0.202937	−	0.979192i	$-0.434951\pi$
$719$	10.8832	0.405873	0.202937	+	0.979192i	$0.434951\pi$
$720$	0	0
$721$	30.2337	1.12596
$722$	0	0
$723$	−10.7446	−0.399595
$724$	0	0
$725$	−123.446	−4.58466
$726$	0	0
$727$	39.5842	1.46810	0.734049	−	0.679097i	$-0.237628\pi$
$727$	39.5842	1.46810	0.734049	+	0.679097i	$0.237628\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.3723	−0.383633
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	−58.9783	−2.17249
$738$	0	0
$739$	−44.6060	−1.64086	−0.820429	−	0.571749i	$-0.806265\pi$
$739$	−44.6060	−1.64086	−0.820429	+	0.571749i	$0.806265\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	−40.4674	−1.48460	−0.742302	−	0.670065i	$-0.766266\pi$
$743$	−40.4674	−1.48460	−0.742302	+	0.670065i	$0.766266\pi$
$744$	0	0
$745$	−19.1168	−0.700387
$746$	0	0
$747$	2.74456	0.100418
$748$	0	0
$749$	−7.72281	−0.282185
$750$	0	0
$751$	2.97825	0.108678	0.0543390	−	0.998523i	$-0.482695\pi$
$751$	2.97825	0.108678	0.0543390	+	0.998523i	$0.482695\pi$
$752$	0	0
$753$	4.37228	0.159335
$754$	0	0
$755$	−34.9783	−1.27299
$756$	0	0
$757$	32.0951	1.16652	0.583258	−	0.812287i	$-0.301778\pi$
$757$	32.0951	1.16652	0.583258	+	0.812287i	$0.301778\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	41.5842	1.50743	0.753713	−	0.657203i	$-0.228261\pi$
$761$	41.5842	1.50743	0.753713	+	0.657203i	$0.228261\pi$
$762$	0	0
$763$	38.5109	1.39419
$764$	0	0
$765$	−19.1168	−0.691171
$766$	0	0
$767$	0	0
$768$	0	0
$769$	21.1168	0.761493	0.380746	−	0.924679i	$-0.375667\pi$
$769$	21.1168	0.761493	0.380746	+	0.924679i	$0.375667\pi$
$770$	0	0
$771$	−5.48913	−0.197686
$772$	0	0
$773$	41.4891	1.49226	0.746130	−	0.665800i	$-0.231910\pi$
$773$	41.4891	1.49226	0.746130	+	0.665800i	$0.231910\pi$
$774$	0	0
$775$	−66.9783	−2.40593
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	0	0
$780$	0	0
$781$	52.4674	1.87743
$782$	0	0
$783$	8.74456	0.312505
$784$	0	0
$785$	−67.7228	−2.41713
$786$	0	0
$787$	21.4891	0.766005	0.383002	−	0.923747i	$-0.374890\pi$
$787$	21.4891	0.766005	0.383002	+	0.923747i	$0.374890\pi$
$788$	0	0
$789$	−2.13859	−0.0761359
$790$	0	0
$791$	7.72281	0.274592
$792$	0	0
$793$	−16.7446	−0.594617
$794$	0	0
$795$	−38.2337	−1.35601
$796$	0	0
$797$	−6.51087	−0.230627	−0.115314	−	0.993329i	$-0.536787\pi$
$797$	−6.51087	−0.230627	−0.115314	+	0.993329i	$0.536787\pi$
$798$	0	0
$799$	−33.3505	−1.17986
$800$	0	0
$801$	3.25544	0.115025
$802$	0	0
$803$	1.62772	0.0574409
$804$	0	0
$805$	−28.4674	−1.00334
$806$	0	0
$807$	−3.25544	−0.114597
$808$	0	0
$809$	−19.6277	−0.690074	−0.345037	−	0.938589i	$-0.612134\pi$
$809$	−19.6277	−0.690074	−0.345037	+	0.938589i	$0.612134\pi$
$810$	0	0
$811$	−34.2337	−1.20211	−0.601054	−	0.799209i	$-0.705252\pi$
$811$	−34.2337	−1.20211	−0.601054	+	0.799209i	$0.705252\pi$
$812$	0	0
$813$	−4.00000	−0.140286
$814$	0	0
$815$	−58.9783	−2.06592
$816$	0	0
$817$	−2.37228	−0.0829956
$818$	0	0
$819$	4.74456	0.165788
$820$	0	0
$821$	41.5842	1.45130	0.725650	−	0.688064i	$-0.241539\pi$
$821$	41.5842	1.45130	0.725650	+	0.688064i	$0.241539\pi$
$822$	0	0
$823$	37.3505	1.30196	0.650979	−	0.759096i	$-0.274359\pi$
$823$	37.3505	1.30196	0.650979	+	0.759096i	$0.274359\pi$
$824$	0	0
$825$	−61.7228	−2.14891
$826$	0	0
$827$	17.4891	0.608156	0.304078	−	0.952647i	$-0.401652\pi$
$827$	17.4891	0.608156	0.304078	+	0.952647i	$0.401652\pi$
$828$	0	0
$829$	−7.76631	−0.269735	−0.134868	−	0.990864i	$-0.543061\pi$
$829$	−7.76631	−0.269735	−0.134868	+	0.990864i	$0.543061\pi$
$830$	0	0
$831$	−6.88316	−0.238774
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	62.2337	2.15369
$836$	0	0
$837$	4.74456	0.163996
$838$	0	0
$839$	−8.74456	−0.301896	−0.150948	−	0.988542i	$-0.548233\pi$
$839$	−8.74456	−0.301896	−0.150948	+	0.988542i	$0.548233\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	0	0
$843$	15.2554	0.525425
$844$	0	0
$845$	−39.3505	−1.35370
$846$	0	0
$847$	19.2554	0.661625
$848$	0	0
$849$	−8.88316	−0.304869
$850$	0	0
$851$	18.5109	0.634545
$852$	0	0
$853$	31.4891	1.07817	0.539084	−	0.842252i	$-0.318771\pi$
$853$	31.4891	1.07817	0.539084	+	0.842252i	$0.318771\pi$
$854$	0	0
$855$	−4.37228	−0.149529
$856$	0	0
$857$	10.9783	0.375010	0.187505	−	0.982264i	$-0.439960\pi$
$857$	10.9783	0.375010	0.187505	+	0.982264i	$0.439960\pi$
$858$	0	0
$859$	−44.6060	−1.52194	−0.760968	−	0.648789i	$-0.775276\pi$
$859$	−44.6060	−1.52194	−0.760968	+	0.648789i	$0.775276\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.02175	−0.0347808	−0.0173904	−	0.999849i	$-0.505536\pi$
$863$	−1.02175	−0.0347808	−0.0173904	+	0.999849i	$0.505536\pi$
$864$	0	0
$865$	90.7011	3.08393
$866$	0	0
$867$	−2.11684	−0.0718918
$868$	0	0
$869$	−34.9783	−1.18656
$870$	0	0
$871$	−26.9783	−0.914123
$872$	0	0
$873$	14.0000	0.473828
$874$	0	0
$875$	94.5625	3.19679
$876$	0	0
$877$	−16.5109	−0.557533	−0.278766	−	0.960359i	$-0.589925\pi$
$877$	−16.5109	−0.557533	−0.278766	+	0.960359i	$0.589925\pi$
$878$	0	0
$879$	−6.51087	−0.219606
$880$	0	0
$881$	−56.8397	−1.91498	−0.957488	−	0.288472i	$-0.906853\pi$
$881$	−56.8397	−1.91498	−0.957488	+	0.288472i	$0.906853\pi$
$882$	0	0
$883$	1.35053	0.0454490	0.0227245	−	0.999742i	$-0.492766\pi$
$883$	1.35053	0.0454490	0.0227245	+	0.999742i	$0.492766\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41.4891	1.39307	0.696534	−	0.717524i	$-0.254724\pi$
$887$	41.4891	1.39307	0.696534	+	0.717524i	$0.254724\pi$
$888$	0	0
$889$	−52.7446	−1.76900
$890$	0	0
$891$	4.37228	0.146477
$892$	0	0
$893$	−7.62772	−0.255252
$894$	0	0
$895$	−14.2337	−0.475780
$896$	0	0
$897$	5.48913	0.183277
$898$	0	0
$899$	41.4891	1.38374
$900$	0	0
$901$	−38.2337	−1.27375
$902$	0	0
$903$	−5.62772	−0.187279
$904$	0	0
$905$	−96.1902	−3.19747
$906$	0	0
$907$	−5.76631	−0.191467	−0.0957336	−	0.995407i	$-0.530520\pi$
$907$	−5.76631	−0.191467	−0.0957336	+	0.995407i	$0.530520\pi$
$908$	0	0
$909$	−11.4891	−0.381070
$910$	0	0
$911$	38.2337	1.26674	0.633369	−	0.773850i	$-0.281671\pi$
$911$	38.2337	1.26674	0.633369	+	0.773850i	$0.281671\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	36.6060	1.21016
$916$	0	0
$917$	−23.3940	−0.772539
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	−95.2119	−3.13055
$926$	0	0
$927$	12.7446	0.418586
$928$	0	0
$929$	11.4891	0.376946	0.188473	−	0.982078i	$-0.439646\pi$
$929$	11.4891	0.376946	0.188473	+	0.982078i	$0.439646\pi$
$930$	0	0
$931$	1.37228	0.0449747
$932$	0	0
$933$	9.86141	0.322848
$934$	0	0
$935$	−83.5842	−2.73350
$936$	0	0
$937$	16.8397	0.550128	0.275064	−	0.961426i	$-0.411301\pi$
$937$	16.8397	0.550128	0.275064	+	0.961426i	$0.411301\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	27.2554	0.888502	0.444251	−	0.895902i	$-0.353470\pi$
$941$	27.2554	0.888502	0.444251	+	0.895902i	$0.353470\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−10.3723	−0.337410
$946$	0	0
$947$	−13.7228	−0.445932	−0.222966	−	0.974826i	$-0.571574\pi$
$947$	−13.7228	−0.445932	−0.222966	+	0.974826i	$0.571574\pi$
$948$	0	0
$949$	0.744563	0.0241695
$950$	0	0
$951$	−6.51087	−0.211130
$952$	0	0
$953$	16.4674	0.533431	0.266715	−	0.963775i	$-0.414062\pi$
$953$	16.4674	0.533431	0.266715	+	0.963775i	$0.414062\pi$
$954$	0	0
$955$	71.5842	2.31641
$956$	0	0
$957$	38.2337	1.23592
$958$	0	0
$959$	−10.3723	−0.334938
$960$	0	0
$961$	−8.48913	−0.273843
$962$	0	0
$963$	−3.25544	−0.104905
$964$	0	0
$965$	22.9783	0.739696
$966$	0	0
$967$	50.9783	1.63935	0.819675	−	0.572829i	$-0.194154\pi$
$967$	50.9783	1.63935	0.819675	+	0.572829i	$0.194154\pi$
$968$	0	0
$969$	−4.37228	−0.140458
$970$	0	0
$971$	1.02175	0.0327895	0.0163947	−	0.999866i	$-0.494781\pi$
$971$	1.02175	0.0327895	0.0163947	+	0.999866i	$0.494781\pi$
$972$	0	0
$973$	−22.8397	−0.732206
$974$	0	0
$975$	−28.2337	−0.904202
$976$	0	0
$977$	7.72281	0.247075	0.123537	−	0.992340i	$-0.460576\pi$
$977$	7.72281	0.247075	0.123537	+	0.992340i	$0.460576\pi$
$978$	0	0
$979$	14.2337	0.454911
$980$	0	0
$981$	16.2337	0.518302
$982$	0	0
$983$	6.51087	0.207665	0.103832	−	0.994595i	$-0.466889\pi$
$983$	6.51087	0.207665	0.103832	+	0.994595i	$0.466889\pi$
$984$	0	0
$985$	50.2337	1.60058
$986$	0	0
$987$	−18.0951	−0.575973
$988$	0	0
$989$	−6.51087	−0.207034
$990$	0	0
$991$	37.9565	1.20573	0.602864	−	0.797844i	$-0.294026\pi$
$991$	37.9565	1.20573	0.602864	+	0.797844i	$0.294026\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	−13.6277	−0.432028
$996$	0	0
$997$	18.8832	0.598036	0.299018	−	0.954248i	$-0.403341\pi$
$997$	18.8832	0.598036	0.299018	+	0.954248i	$0.403341\pi$
$998$	0	0
$999$	6.74456	0.213389

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.a.n.1.2		2
3.2	odd	2		2736.2.a.y.1.1		2
4.3	odd	2		228.2.a.c.1.2	✓	2
8.3	odd	2		3648.2.a.bk.1.1		2
8.5	even	2		3648.2.a.bq.1.1		2
12.11	even	2		684.2.a.d.1.1		2
20.3	even	4		5700.2.f.m.3649.4		4
20.7	even	4		5700.2.f.m.3649.1		4
20.19	odd	2		5700.2.a.t.1.2		2
76.75	even	2		4332.2.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.a.c.1.2	✓	2	4.3	odd	2
684.2.a.d.1.1		2	12.11	even	2
912.2.a.n.1.2		2	1.1	even	1	trivial
2736.2.a.y.1.1		2	3.2	odd	2
3648.2.a.bk.1.1		2	8.3	odd	2
3648.2.a.bq.1.1		2	8.5	even	2
4332.2.a.i.1.2		2	76.75	even	2
5700.2.a.t.1.2		2	20.19	odd	2
5700.2.f.m.3649.1		4	20.7	even	4
5700.2.f.m.3649.4		4	20.3	even	4

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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.37228 q^{5} +2.37228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.37228 q^{5} +2.37228 q^{7} +1.00000 q^{9} +4.37228 q^{11} +2.00000 q^{13} -4.37228 q^{15} -4.37228 q^{17} -1.00000 q^{19} -2.37228 q^{21} -2.74456 q^{23} +14.1168 q^{25} -1.00000 q^{27} -8.74456 q^{29} -4.74456 q^{31} -4.37228 q^{33} +10.3723 q^{35} -6.74456 q^{37} -2.00000 q^{39} +2.37228 q^{43} +4.37228 q^{45} +7.62772 q^{47} -1.37228 q^{49} +4.37228 q^{51} +8.74456 q^{53} +19.1168 q^{55} +1.00000 q^{57} -8.37228 q^{61} +2.37228 q^{63} +8.74456 q^{65} -13.4891 q^{67} +2.74456 q^{69} +12.0000 q^{71} +0.372281 q^{73} -14.1168 q^{75} +10.3723 q^{77} -8.00000 q^{79} +1.00000 q^{81} +2.74456 q^{83} -19.1168 q^{85} +8.74456 q^{87} +3.25544 q^{89} +4.74456 q^{91} +4.74456 q^{93} -4.37228 q^{95} +14.0000 q^{97} +4.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 3 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 3 * q^5 - q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 3 q^{5} - q^{7} + 2 q^{9} + 3 q^{11} + 4 q^{13} - 3 q^{15} - 3 q^{17} - 2 q^{19} + q^{21} + 6 q^{23} + 11 q^{25} - 2 q^{27} - 6 q^{29} + 2 q^{31} - 3 q^{33} + 15 q^{35} - 2 q^{37} - 4 q^{39} - q^{43} + 3 q^{45} + 21 q^{47} + 3 q^{49} + 3 q^{51} + 6 q^{53} + 21 q^{55} + 2 q^{57} - 11 q^{61} - q^{63} + 6 q^{65} - 4 q^{67} - 6 q^{69} + 24 q^{71} - 5 q^{73} - 11 q^{75} + 15 q^{77} - 16 q^{79} + 2 q^{81} - 6 q^{83} - 21 q^{85} + 6 q^{87} + 18 q^{89} - 2 q^{91} - 2 q^{93} - 3 q^{95} + 28 q^{97} + 3 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 3 * q^5 - q^7 + 2 * q^9 + 3 * q^11 + 4 * q^13 - 3 * q^15 - 3 * q^17 - 2 * q^19 + q^21 + 6 * q^23 + 11 * q^25 - 2 * q^27 - 6 * q^29 + 2 * q^31 - 3 * q^33 + 15 * q^35 - 2 * q^37 - 4 * q^39 - q^43 + 3 * q^45 + 21 * q^47 + 3 * q^49 + 3 * q^51 + 6 * q^53 + 21 * q^55 + 2 * q^57 - 11 * q^61 - q^63 + 6 * q^65 - 4 * q^67 - 6 * q^69 + 24 * q^71 - 5 * q^73 - 11 * q^75 + 15 * q^77 - 16 * q^79 + 2 * q^81 - 6 * q^83 - 21 * q^85 + 6 * q^87 + 18 * q^89 - 2 * q^91 - 2 * q^93 - 3 * q^95 + 28 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more