LMFDB - Embedded newform 1944.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1944,2,Mod(1,1944)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1944, 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("1944.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1944 = 2^{3} \cdot 3^{5} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1944.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:	$15.5229181529$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.9926793.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} - x^{3} + 36x^{2} + 9x - 9 $ x^6 - 12x^4 - x^3 + 36x^2 + 9*x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.400207$ of defining polynomial
Character	$\chi$	$=$	1944.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.01336 q^{5} +1.72399 q^{7} +O(q^{10})$	$q-2.01336 q^{5} +1.72399 q^{7} -2.36530 q^{11} +2.55256 q^{13} +1.95067 q^{17} +4.57862 q^{19} +0.627946 q^{23} -0.946382 q^{25} -6.00592 q^{29} -7.74080 q^{31} -3.47101 q^{35} +9.29958 q^{37} -1.44744 q^{41} +4.99753 q^{43} +10.5061 q^{47} -4.02786 q^{49} +8.02110 q^{53} +4.76219 q^{55} +10.1665 q^{59} +14.1554 q^{61} -5.13922 q^{65} -4.22106 q^{67} -3.62601 q^{71} +13.1833 q^{73} -4.07775 q^{77} +5.84622 q^{79} +13.7773 q^{83} -3.92740 q^{85} -15.0325 q^{89} +4.40058 q^{91} -9.21840 q^{95} -3.64723 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{7}+O(q^{10}) $ 6 * q + 3 * q^7	$ 6 q + 3 q^{7} + 3 q^{11} + 3 q^{13} - 9 q^{17} + 9 q^{19} + 6 q^{23} + 18 q^{25} + 3 q^{29} + 24 q^{31} + 15 q^{35} + 15 q^{37} - 21 q^{41} + 18 q^{43} + 6 q^{47} + 21 q^{49} - 12 q^{53} + 18 q^{55} + 12 q^{59} + 18 q^{61} - 33 q^{65} + 24 q^{67} + 15 q^{71} + 6 q^{73} - 3 q^{77} + 21 q^{79} + 18 q^{83} + 24 q^{85} - 18 q^{89} + 30 q^{91} + 39 q^{95} + 27 q^{97}+O(q^{100}) $ 6 * q + 3 * q^7 + 3 * q^11 + 3 * q^13 - 9 * q^17 + 9 * q^19 + 6 * q^23 + 18 * q^25 + 3 * q^29 + 24 * q^31 + 15 * q^35 + 15 * q^37 - 21 * q^41 + 18 * q^43 + 6 * q^47 + 21 * q^49 - 12 * q^53 + 18 * q^55 + 12 * q^59 + 18 * q^61 - 33 * q^65 + 24 * q^67 + 15 * q^71 + 6 * q^73 - 3 * q^77 + 21 * q^79 + 18 * q^83 + 24 * q^85 - 18 * q^89 + 30 * q^91 + 39 * q^95 + 27 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.01336	−0.900402	−0.450201	−	0.892927i	$-0.648648\pi$
$5$	−2.01336	−0.900402	−0.450201	+	0.892927i	$0.648648\pi$
$6$	0	0
$7$	1.72399	0.651607	0.325803	−	0.945438i	$-0.394365\pi$
$7$	1.72399	0.651607	0.325803	+	0.945438i	$0.394365\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.36530	−0.713164	−0.356582	−	0.934264i	$-0.616058\pi$
$11$	−2.36530	−0.713164	−0.356582	+	0.934264i	$0.616058\pi$
$12$	0	0
$13$	2.55256	0.707952	0.353976	−	0.935255i	$-0.384829\pi$
$13$	2.55256	0.707952	0.353976	+	0.935255i	$0.384829\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.95067	0.473107	0.236554	−	0.971618i	$-0.423982\pi$
$17$	1.95067	0.473107	0.236554	+	0.971618i	$0.423982\pi$
$18$	0	0
$19$	4.57862	1.05041	0.525203	−	0.850977i	$-0.323989\pi$
$19$	4.57862	1.05041	0.525203	+	0.850977i	$0.323989\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.627946	0.130936	0.0654679	−	0.997855i	$-0.479146\pi$
$23$	0.627946	0.130936	0.0654679	+	0.997855i	$0.479146\pi$
$24$	0	0
$25$	−0.946382	−0.189276
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.00592	−1.11527	−0.557636	−	0.830086i	$-0.688291\pi$
$29$	−6.00592	−1.11527	−0.557636	+	0.830086i	$0.688291\pi$
$30$	0	0
$31$	−7.74080	−1.39029	−0.695144	−	0.718870i	$-0.744660\pi$
$31$	−7.74080	−1.39029	−0.695144	+	0.718870i	$0.744660\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.47101	−0.586708
$36$	0	0
$37$	9.29958	1.52884	0.764421	−	0.644718i	$-0.223025\pi$
$37$	9.29958	1.52884	0.764421	+	0.644718i	$0.223025\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.44744	−0.226053	−0.113026	−	0.993592i	$-0.536054\pi$
$41$	−1.44744	−0.226053	−0.113026	+	0.993592i	$0.536054\pi$
$42$	0	0
$43$	4.99753	0.762116	0.381058	−	0.924551i	$-0.375560\pi$
$43$	4.99753	0.762116	0.381058	+	0.924551i	$0.375560\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.5061	1.53248	0.766239	−	0.642555i	$-0.222126\pi$
$47$	10.5061	1.53248	0.766239	+	0.642555i	$0.222126\pi$
$48$	0	0
$49$	−4.02786	−0.575408
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.02110	1.10178	0.550891	−	0.834577i	$-0.314288\pi$
$53$	8.02110	1.10178	0.550891	+	0.834577i	$0.314288\pi$
$54$	0	0
$55$	4.76219	0.642134
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.1665	1.32356	0.661781	−	0.749697i	$-0.269801\pi$
$59$	10.1665	1.32356	0.661781	+	0.749697i	$0.269801\pi$
$60$	0	0
$61$	14.1554	1.81242	0.906208	−	0.422832i	$-0.138964\pi$
$61$	14.1554	1.81242	0.906208	+	0.422832i	$0.138964\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.13922	−0.637441
$66$	0	0
$67$	−4.22106	−0.515685	−0.257842	−	0.966187i	$-0.583012\pi$
$67$	−4.22106	−0.515685	−0.257842	+	0.966187i	$0.583012\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.62601	−0.430329	−0.215164	−	0.976578i	$-0.569029\pi$
$71$	−3.62601	−0.430329	−0.215164	+	0.976578i	$0.569029\pi$
$72$	0	0
$73$	13.1833	1.54299	0.771493	−	0.636238i	$-0.219510\pi$
$73$	13.1833	1.54299	0.771493	+	0.636238i	$0.219510\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.07775	−0.464702
$78$	0	0
$79$	5.84622	0.657751	0.328875	−	0.944373i	$-0.393330\pi$
$79$	5.84622	0.657751	0.328875	+	0.944373i	$0.393330\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.7773	1.51226	0.756128	−	0.654424i	$-0.227089\pi$
$83$	13.7773	1.51226	0.756128	+	0.654424i	$0.227089\pi$
$84$	0	0
$85$	−3.92740	−0.425987
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−15.0325	−1.59344	−0.796722	−	0.604346i	$-0.793434\pi$
$89$	−15.0325	−1.59344	−0.796722	+	0.604346i	$0.793434\pi$
$90$	0	0
$91$	4.40058	0.461306
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−9.21840	−0.945788
$96$	0	0
$97$	−3.64723	−0.370320	−0.185160	−	0.982708i	$-0.559280\pi$
$97$	−3.64723	−0.370320	−0.185160	+	0.982708i	$0.559280\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.98168	0.396192	0.198096	−	0.980183i	$-0.436524\pi$
$101$	3.98168	0.396192	0.198096	+	0.980183i	$0.436524\pi$
$102$	0	0
$103$	2.07839	0.204790	0.102395	−	0.994744i	$-0.467349\pi$
$103$	2.07839	0.204790	0.102395	+	0.994744i	$0.467349\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.00744	0.0973928	0.0486964	−	0.998814i	$-0.484493\pi$
$107$	1.00744	0.0973928	0.0486964	+	0.998814i	$0.484493\pi$
$108$	0	0
$109$	0.124717	0.0119458	0.00597288	−	0.999982i	$-0.498099\pi$
$109$	0.124717	0.0119458	0.00597288	+	0.999982i	$0.498099\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.37842	0.694103	0.347052	−	0.937846i	$-0.387183\pi$
$113$	7.37842	0.694103	0.347052	+	0.937846i	$0.387183\pi$
$114$	0	0
$115$	−1.26428	−0.117895
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.36294	0.308280
$120$	0	0
$121$	−5.40538	−0.491398
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.9722	1.07083
$126$	0	0
$127$	14.9883	1.32999	0.664997	−	0.746846i	$-0.268433\pi$
$127$	14.9883	1.32999	0.664997	+	0.746846i	$0.268433\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.9931	0.960471	0.480236	−	0.877139i	$-0.340551\pi$
$131$	10.9931	0.960471	0.480236	+	0.877139i	$0.340551\pi$
$132$	0	0
$133$	7.89349	0.684452
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.5352	−1.83988	−0.919939	−	0.392063i	$-0.871762\pi$
$137$	−21.5352	−1.83988	−0.919939	+	0.392063i	$0.871762\pi$
$138$	0	0
$139$	9.36693	0.794492	0.397246	−	0.917712i	$-0.369966\pi$
$139$	9.36693	0.794492	0.397246	+	0.917712i	$0.369966\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.03755	−0.504886
$144$	0	0
$145$	12.0921	1.00419
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.0641	−1.31602	−0.658012	−	0.753007i	$-0.728603\pi$
$149$	−16.0641	−1.31602	−0.658012	+	0.753007i	$0.728603\pi$
$150$	0	0
$151$	5.75434	0.468282	0.234141	−	0.972203i	$-0.424772\pi$
$151$	5.75434	0.468282	0.234141	+	0.972203i	$0.424772\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	15.5850	1.25182
$156$	0	0
$157$	−3.60221	−0.287488	−0.143744	−	0.989615i	$-0.545914\pi$
$157$	−3.60221	−0.287488	−0.143744	+	0.989615i	$0.545914\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.08257	0.0853187
$162$	0	0
$163$	21.9582	1.71990	0.859951	−	0.510377i	$-0.170494\pi$
$163$	21.9582	1.71990	0.859951	+	0.510377i	$0.170494\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.2719	1.64607	0.823035	−	0.567991i	$-0.192279\pi$
$167$	21.2719	1.64607	0.823035	+	0.567991i	$0.192279\pi$
$168$	0	0
$169$	−6.48445	−0.498804
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.8524	−1.58538	−0.792691	−	0.609624i	$-0.791320\pi$
$173$	−20.8524	−1.58538	−0.792691	+	0.609624i	$0.791320\pi$
$174$	0	0
$175$	−1.63155	−0.123334
$176$	0	0
$177$	0	0
$178$	0	0
$179$	17.6301	1.31774	0.658868	−	0.752258i	$-0.271035\pi$
$179$	17.6301	1.31774	0.658868	+	0.752258i	$0.271035\pi$
$180$	0	0
$181$	−12.4496	−0.925373	−0.462687	−	0.886522i	$-0.653114\pi$
$181$	−12.4496	−0.925373	−0.462687	+	0.886522i	$0.653114\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−18.7234	−1.37657
$186$	0	0
$187$	−4.61392	−0.337403
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.03476	0.147230	0.0736151	−	0.997287i	$-0.476546\pi$
$191$	2.03476	0.147230	0.0736151	+	0.997287i	$0.476546\pi$
$192$	0	0
$193$	−6.59210	−0.474510	−0.237255	−	0.971447i	$-0.576248\pi$
$193$	−6.59210	−0.474510	−0.237255	+	0.971447i	$0.576248\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.03582	0.145046	0.0725232	−	0.997367i	$-0.476895\pi$
$197$	2.03582	0.145046	0.0725232	+	0.997367i	$0.476895\pi$
$198$	0	0
$199$	−13.3565	−0.946819	−0.473410	−	0.880842i	$-0.656977\pi$
$199$	−13.3565	−0.946819	−0.473410	+	0.880842i	$0.656977\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−10.3541	−0.726719
$204$	0	0
$205$	2.91422	0.203538
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−10.8298	−0.749112
$210$	0	0
$211$	4.48027	0.308435	0.154217	−	0.988037i	$-0.450714\pi$
$211$	4.48027	0.308435	0.154217	+	0.988037i	$0.450714\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.0618	−0.686211
$216$	0	0
$217$	−13.3451	−0.905922
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.97920	0.334937
$222$	0	0
$223$	−10.0982	−0.676222	−0.338111	−	0.941106i	$-0.609788\pi$
$223$	−10.0982	−0.676222	−0.338111	+	0.941106i	$0.609788\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.9991	−0.730033	−0.365016	−	0.931001i	$-0.618937\pi$
$227$	−10.9991	−0.730033	−0.365016	+	0.931001i	$0.618937\pi$
$228$	0	0
$229$	12.7767	0.844305	0.422153	−	0.906525i	$-0.361275\pi$
$229$	12.7767	0.844305	0.422153	+	0.906525i	$0.361275\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.13820	0.467639	0.233820	−	0.972280i	$-0.424878\pi$
$233$	7.13820	0.467639	0.233820	+	0.972280i	$0.424878\pi$
$234$	0	0
$235$	−21.1526	−1.37985
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.84428	−0.248666	−0.124333	−	0.992241i	$-0.539679\pi$
$239$	−3.84428	−0.248666	−0.124333	+	0.992241i	$0.539679\pi$
$240$	0	0
$241$	13.7109	0.883197	0.441598	−	0.897213i	$-0.354412\pi$
$241$	13.7109	0.883197	0.441598	+	0.897213i	$0.354412\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	8.10953	0.518099
$246$	0	0
$247$	11.6872	0.743638
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−28.5720	−1.80345	−0.901725	−	0.432310i	$-0.857699\pi$
$251$	−28.5720	−1.80345	−0.901725	+	0.432310i	$0.857699\pi$
$252$	0	0
$253$	−1.48528	−0.0933787
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.40362	−0.274690	−0.137345	−	0.990523i	$-0.543857\pi$
$257$	−4.40362	−0.274690	−0.137345	+	0.990523i	$0.543857\pi$
$258$	0	0
$259$	16.0324	0.996204
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.0263	1.17321	0.586607	−	0.809872i	$-0.300463\pi$
$263$	19.0263	1.17321	0.586607	+	0.809872i	$0.300463\pi$
$264$	0	0
$265$	−16.1494	−0.992047
$266$	0	0
$267$	0	0
$268$	0	0
$269$	27.4196	1.67180	0.835902	−	0.548878i	$-0.184945\pi$
$269$	27.4196	1.67180	0.835902	+	0.548878i	$0.184945\pi$
$270$	0	0
$271$	8.40028	0.510281	0.255140	−	0.966904i	$-0.417878\pi$
$271$	8.40028	0.510281	0.255140	+	0.966904i	$0.417878\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.23847	0.134985
$276$	0	0
$277$	−6.31306	−0.379315	−0.189658	−	0.981850i	$-0.560738\pi$
$277$	−6.31306	−0.379315	−0.189658	+	0.981850i	$0.560738\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−25.8598	−1.54266	−0.771332	−	0.636433i	$-0.780409\pi$
$281$	−25.8598	−1.54266	−0.771332	+	0.636433i	$0.780409\pi$
$282$	0	0
$283$	−31.5345	−1.87453	−0.937266	−	0.348615i	$-0.886652\pi$
$283$	−31.5345	−1.87453	−0.937266	+	0.348615i	$0.886652\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.49538	−0.147297
$288$	0	0
$289$	−13.1949	−0.776169
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−0.381078	−0.0222628	−0.0111314	−	0.999938i	$-0.503543\pi$
$293$	−0.381078	−0.0222628	−0.0111314	+	0.999938i	$0.503543\pi$
$294$	0	0
$295$	−20.4688	−1.19174
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.60287	0.0926963
$300$	0	0
$301$	8.61569	0.496600
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−28.5000	−1.63190
$306$	0	0
$307$	−19.8114	−1.13070	−0.565348	−	0.824853i	$-0.691258\pi$
$307$	−19.8114	−1.13070	−0.565348	+	0.824853i	$0.691258\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.10380	−0.0625906	−0.0312953	−	0.999510i	$-0.509963\pi$
$311$	−1.10380	−0.0625906	−0.0312953	+	0.999510i	$0.509963\pi$
$312$	0	0
$313$	−34.0583	−1.92509	−0.962545	−	0.271121i	$-0.912606\pi$
$313$	−34.0583	−1.92509	−0.962545	+	0.271121i	$0.912606\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.7266	1.89428	0.947138	−	0.320827i	$-0.103961\pi$
$317$	33.7266	1.89428	0.947138	+	0.320827i	$0.103961\pi$
$318$	0	0
$319$	14.2058	0.795371
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.93138	0.496955
$324$	0	0
$325$	−2.41570	−0.133999
$326$	0	0
$327$	0	0
$328$	0	0
$329$	18.1125	0.998574
$330$	0	0
$331$	−12.0041	−0.659806	−0.329903	−	0.944015i	$-0.607016\pi$
$331$	−12.0041	−0.659806	−0.329903	+	0.944015i	$0.607016\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.49852	0.464324
$336$	0	0
$337$	24.1501	1.31554	0.657770	−	0.753219i	$-0.271500\pi$
$337$	24.1501	1.31554	0.657770	+	0.753219i	$0.271500\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	18.3093	0.991503
$342$	0	0
$343$	−19.0119	−1.02655
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.98658	0.106645	0.0533227	−	0.998577i	$-0.483019\pi$
$347$	1.98658	0.106645	0.0533227	+	0.998577i	$0.483019\pi$
$348$	0	0
$349$	−7.90764	−0.423286	−0.211643	−	0.977347i	$-0.567881\pi$
$349$	−7.90764	−0.423286	−0.211643	+	0.977347i	$0.567881\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.51374	−0.187017	−0.0935087	−	0.995618i	$-0.529808\pi$
$353$	−3.51374	−0.187017	−0.0935087	+	0.995618i	$0.529808\pi$
$354$	0	0
$355$	7.30047	0.387469
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.47508	0.130630	0.0653148	−	0.997865i	$-0.479195\pi$
$359$	2.47508	0.130630	0.0653148	+	0.997865i	$0.479195\pi$
$360$	0	0
$361$	1.96374	0.103355
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−26.5427	−1.38931
$366$	0	0
$367$	17.1960	0.897622	0.448811	−	0.893627i	$-0.351848\pi$
$367$	17.1960	0.897622	0.448811	+	0.893627i	$0.351848\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.8283	0.717929
$372$	0	0
$373$	−6.74493	−0.349239	−0.174620	−	0.984636i	$-0.555870\pi$
$373$	−6.74493	−0.349239	−0.174620	+	0.984636i	$0.555870\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−15.3305	−0.789559
$378$	0	0
$379$	−30.8627	−1.58531	−0.792655	−	0.609670i	$-0.791302\pi$
$379$	−30.8627	−1.58531	−0.792655	+	0.609670i	$0.791302\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24.1600	1.23452	0.617258	−	0.786761i	$-0.288244\pi$
$383$	24.1600	1.23452	0.617258	+	0.786761i	$0.288244\pi$
$384$	0	0
$385$	8.20997	0.418419
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11.7817	0.597357	0.298678	−	0.954354i	$-0.403454\pi$
$389$	11.7817	0.597357	0.298678	+	0.954354i	$0.403454\pi$
$390$	0	0
$391$	1.22492	0.0619467
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.7705	−0.592240
$396$	0	0
$397$	−6.94137	−0.348377	−0.174189	−	0.984712i	$-0.555730\pi$
$397$	−6.94137	−0.348377	−0.174189	+	0.984712i	$0.555730\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.3162	1.36411	0.682054	−	0.731302i	$-0.261087\pi$
$401$	27.3162	1.36411	0.682054	+	0.731302i	$0.261087\pi$
$402$	0	0
$403$	−19.7588	−0.984258
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21.9963	−1.09031
$408$	0	0
$409$	18.4842	0.913985	0.456993	−	0.889471i	$-0.348927\pi$
$409$	18.4842	0.913985	0.456993	+	0.889471i	$0.348927\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	17.5269	0.862442
$414$	0	0
$415$	−27.7387	−1.36164
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.8648	0.970458	0.485229	−	0.874387i	$-0.338736\pi$
$419$	19.8648	0.970458	0.485229	+	0.874387i	$0.338736\pi$
$420$	0	0
$421$	−18.7200	−0.912358	−0.456179	−	0.889888i	$-0.650782\pi$
$421$	−18.7200	−0.912358	−0.456179	+	0.889888i	$0.650782\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.84608	−0.0895481
$426$	0	0
$427$	24.4038	1.18098
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.9472	1.20166	0.600832	−	0.799375i	$-0.294836\pi$
$431$	24.9472	1.20166	0.600832	+	0.799375i	$0.294836\pi$
$432$	0	0
$433$	−23.2050	−1.11516	−0.557582	−	0.830122i	$-0.688271\pi$
$433$	−23.2050	−1.11516	−0.557582	+	0.830122i	$0.688271\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.87513	0.137536
$438$	0	0
$439$	−23.4764	−1.12047	−0.560234	−	0.828334i	$-0.689289\pi$
$439$	−23.4764	−1.12047	−0.560234	+	0.828334i	$0.689289\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.57940	0.312597	0.156298	−	0.987710i	$-0.450044\pi$
$443$	6.57940	0.312597	0.156298	+	0.987710i	$0.450044\pi$
$444$	0	0
$445$	30.2659	1.43474
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−35.8716	−1.69288	−0.846442	−	0.532480i	$-0.821260\pi$
$449$	−35.8716	−1.69288	−0.846442	+	0.532480i	$0.821260\pi$
$450$	0	0
$451$	3.42363	0.161212
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.85996	−0.415361
$456$	0	0
$457$	9.36215	0.437943	0.218971	−	0.975731i	$-0.429730\pi$
$457$	9.36215	0.437943	0.218971	+	0.975731i	$0.429730\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.35604	−0.109732	−0.0548658	−	0.998494i	$-0.517473\pi$
$461$	−2.35604	−0.109732	−0.0548658	+	0.998494i	$0.517473\pi$
$462$	0	0
$463$	−16.2905	−0.757082	−0.378541	−	0.925584i	$-0.623574\pi$
$463$	−16.2905	−0.757082	−0.378541	+	0.925584i	$0.623574\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.2937	−1.03163	−0.515815	−	0.856700i	$-0.672511\pi$
$467$	−22.2937	−1.03163	−0.515815	+	0.856700i	$0.672511\pi$
$468$	0	0
$469$	−7.27707	−0.336024
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.8206	−0.543514
$474$	0	0
$475$	−4.33312	−0.198817
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.80255	0.310817	0.155408	−	0.987850i	$-0.450331\pi$
$479$	6.80255	0.310817	0.155408	+	0.987850i	$0.450331\pi$
$480$	0	0
$481$	23.7377	1.08235
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.34318	0.333437
$486$	0	0
$487$	1.42376	0.0645167	0.0322583	−	0.999480i	$-0.489730\pi$
$487$	1.42376	0.0645167	0.0322583	+	0.999480i	$0.489730\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−40.2610	−1.81695	−0.908476	−	0.417937i	$-0.862753\pi$
$491$	−40.2610	−1.81695	−0.908476	+	0.417937i	$0.862753\pi$
$492$	0	0
$493$	−11.7156	−0.527643
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.25121	−0.280405
$498$	0	0
$499$	−23.1021	−1.03419	−0.517097	−	0.855927i	$-0.672987\pi$
$499$	−23.1021	−1.03419	−0.517097	+	0.855927i	$0.672987\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.3822	0.997971	0.498985	−	0.866610i	$-0.333706\pi$
$503$	22.3822	0.997971	0.498985	+	0.866610i	$0.333706\pi$
$504$	0	0
$505$	−8.01655	−0.356732
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.3659	−0.592432	−0.296216	−	0.955121i	$-0.595725\pi$
$509$	−13.3659	−0.592432	−0.296216	+	0.955121i	$0.595725\pi$
$510$	0	0
$511$	22.7278	1.00542
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.18456	−0.184394
$516$	0	0
$517$	−24.8501	−1.09291
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.4993	−0.459983	−0.229991	−	0.973193i	$-0.573870\pi$
$521$	−10.4993	−0.459983	−0.229991	+	0.973193i	$0.573870\pi$
$522$	0	0
$523$	5.91415	0.258608	0.129304	−	0.991605i	$-0.458726\pi$
$523$	5.91415	0.258608	0.129304	+	0.991605i	$0.458726\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.0998	−0.657756
$528$	0	0
$529$	−22.6057	−0.982856
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.69468	−0.160034
$534$	0	0
$535$	−2.02834	−0.0876927
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.52708	0.410360
$540$	0	0
$541$	−30.8650	−1.32699	−0.663494	−	0.748181i	$-0.730927\pi$
$541$	−30.8650	−1.32699	−0.663494	+	0.748181i	$0.730927\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.251101	−0.0107560
$546$	0	0
$547$	25.5268	1.09145	0.545723	−	0.837966i	$-0.316255\pi$
$547$	25.5268	1.09145	0.545723	+	0.837966i	$0.316255\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−27.4988	−1.17149
$552$	0	0
$553$	10.0788	0.428595
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.9156	0.886223	0.443112	−	0.896466i	$-0.353875\pi$
$557$	20.9156	0.886223	0.443112	+	0.896466i	$0.353875\pi$
$558$	0	0
$559$	12.7565	0.539542
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.83075	−0.245737	−0.122868	−	0.992423i	$-0.539209\pi$
$563$	−5.83075	−0.245737	−0.122868	+	0.992423i	$0.539209\pi$
$564$	0	0
$565$	−14.8554	−0.624972
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.86246	0.371534	0.185767	−	0.982594i	$-0.440523\pi$
$569$	8.86246	0.371534	0.185767	+	0.982594i	$0.440523\pi$
$570$	0	0
$571$	22.5260	0.942685	0.471342	−	0.881950i	$-0.343770\pi$
$571$	22.5260	0.942685	0.471342	+	0.881950i	$0.343770\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.594277	−0.0247831
$576$	0	0
$577$	−11.4375	−0.476148	−0.238074	−	0.971247i	$-0.576516\pi$
$577$	−11.4375	−0.476148	−0.238074	+	0.971247i	$0.576516\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	23.7519	0.985397
$582$	0	0
$583$	−18.9723	−0.785751
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.0526	−1.07531	−0.537654	−	0.843166i	$-0.680689\pi$
$587$	−26.0526	−1.07531	−0.537654	+	0.843166i	$0.680689\pi$
$588$	0	0
$589$	−35.4422	−1.46037
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−39.8172	−1.63509	−0.817547	−	0.575862i	$-0.804667\pi$
$593$	−39.8172	−1.63509	−0.817547	+	0.575862i	$0.804667\pi$
$594$	0	0
$595$	−6.77080	−0.277576
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.1443	0.496202	0.248101	−	0.968734i	$-0.420193\pi$
$599$	12.1443	0.496202	0.248101	+	0.968734i	$0.420193\pi$
$600$	0	0
$601$	38.5072	1.57074	0.785372	−	0.619025i	$-0.212472\pi$
$601$	38.5072	1.57074	0.785372	+	0.619025i	$0.212472\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.8830	0.442455
$606$	0	0
$607$	0.784388	0.0318373	0.0159187	−	0.999873i	$-0.494933\pi$
$607$	0.784388	0.0318373	0.0159187	+	0.999873i	$0.494933\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26.8175	1.08492
$612$	0	0
$613$	16.3354	0.659782	0.329891	−	0.944019i	$-0.392988\pi$
$613$	16.3354	0.659782	0.329891	+	0.944019i	$0.392988\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.6455	0.951931	0.475966	−	0.879464i	$-0.342099\pi$
$617$	23.6455	0.951931	0.475966	+	0.879464i	$0.342099\pi$
$618$	0	0
$619$	−40.8315	−1.64116	−0.820578	−	0.571535i	$-0.806348\pi$
$619$	−40.8315	−1.64116	−0.820578	+	0.571535i	$0.806348\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−25.9159	−1.03830
$624$	0	0
$625$	−19.3724	−0.774898
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.1404	0.723306
$630$	0	0
$631$	−46.3424	−1.84486	−0.922430	−	0.386164i	$-0.873800\pi$
$631$	−46.3424	−1.84486	−0.922430	+	0.386164i	$0.873800\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−30.1768	−1.19753
$636$	0	0
$637$	−10.2813	−0.407361
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17.6395	0.696717	0.348358	−	0.937361i	$-0.386739\pi$
$641$	17.6395	0.696717	0.348358	+	0.937361i	$0.386739\pi$
$642$	0	0
$643$	4.31097	0.170008	0.0850041	−	0.996381i	$-0.472910\pi$
$643$	4.31097	0.170008	0.0850041	+	0.996381i	$0.472910\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.0982	0.711515	0.355758	−	0.934578i	$-0.384223\pi$
$647$	18.0982	0.711515	0.355758	+	0.934578i	$0.384223\pi$
$648$	0	0
$649$	−24.0467	−0.943916
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.9054	−0.426762	−0.213381	−	0.976969i	$-0.568448\pi$
$653$	−10.9054	−0.426762	−0.213381	+	0.976969i	$0.568448\pi$
$654$	0	0
$655$	−22.1331	−0.864810
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−29.2001	−1.13747	−0.568736	−	0.822520i	$-0.692568\pi$
$659$	−29.2001	−1.13747	−0.568736	+	0.822520i	$0.692568\pi$
$660$	0	0
$661$	8.53495	0.331971	0.165986	−	0.986128i	$-0.446919\pi$
$661$	8.53495	0.331971	0.165986	+	0.986128i	$0.446919\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−15.8924	−0.616282
$666$	0	0
$667$	−3.77139	−0.146029
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−33.4818	−1.29255
$672$	0	0
$673$	−4.05596	−0.156346	−0.0781729	−	0.996940i	$-0.524909\pi$
$673$	−4.05596	−0.156346	−0.0781729	+	0.996940i	$0.524909\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.1988	1.39123	0.695616	−	0.718414i	$-0.255132\pi$
$677$	36.1988	1.39123	0.695616	+	0.718414i	$0.255132\pi$
$678$	0	0
$679$	−6.28778	−0.241303
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.02223	0.306962	0.153481	−	0.988152i	$-0.450952\pi$
$683$	8.02223	0.306962	0.153481	+	0.988152i	$0.450952\pi$
$684$	0	0
$685$	43.3581	1.65663
$686$	0	0
$687$	0	0
$688$	0	0
$689$	20.4743	0.780009
$690$	0	0
$691$	35.0520	1.33344	0.666720	−	0.745308i	$-0.267698\pi$
$691$	35.0520	1.33344	0.666720	+	0.745308i	$0.267698\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−18.8590	−0.715362
$696$	0	0
$697$	−2.82349	−0.106947
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.84094	0.0695315	0.0347658	−	0.999395i	$-0.488931\pi$
$701$	1.84094	0.0695315	0.0347658	+	0.999395i	$0.488931\pi$
$702$	0	0
$703$	42.5792	1.60591
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.86438	0.258161
$708$	0	0
$709$	14.6368	0.549695	0.274847	−	0.961488i	$-0.411373\pi$
$709$	14.6368	0.549695	0.274847	+	0.961488i	$0.411373\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.86081	−0.182039
$714$	0	0
$715$	12.1558	0.454600
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−49.2729	−1.83757	−0.918785	−	0.394759i	$-0.870828\pi$
$719$	−49.2729	−1.83757	−0.918785	+	0.394759i	$0.870828\pi$
$720$	0	0
$721$	3.58313	0.133443
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.68390	0.211095
$726$	0	0
$727$	36.8194	1.36556	0.682778	−	0.730626i	$-0.260772\pi$
$727$	36.8194	1.36556	0.682778	+	0.730626i	$0.260772\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.74854	0.360563
$732$	0	0
$733$	−16.6179	−0.613796	−0.306898	−	0.951742i	$-0.599291\pi$
$733$	−16.6179	−0.613796	−0.306898	+	0.951742i	$0.599291\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.98406	0.367768
$738$	0	0
$739$	39.2198	1.44272	0.721362	−	0.692559i	$-0.243517\pi$
$739$	39.2198	1.44272	0.721362	+	0.692559i	$0.243517\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	47.0213	1.72505	0.862523	−	0.506018i	$-0.168883\pi$
$743$	47.0213	1.72505	0.862523	+	0.506018i	$0.168883\pi$
$744$	0	0
$745$	32.3429	1.18495
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.73682	0.0634619
$750$	0	0
$751$	26.2168	0.956663	0.478331	−	0.878179i	$-0.341242\pi$
$751$	26.2168	0.956663	0.478331	+	0.878179i	$0.341242\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11.5856	−0.421642
$756$	0	0
$757$	−6.83630	−0.248469	−0.124235	−	0.992253i	$-0.539648\pi$
$757$	−6.83630	−0.248469	−0.124235	+	0.992253i	$0.539648\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.3777	1.39119	0.695595	−	0.718434i	$-0.255141\pi$
$761$	38.3777	1.39119	0.695595	+	0.718434i	$0.255141\pi$
$762$	0	0
$763$	0.215011	0.00778394
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.9505	0.937018
$768$	0	0
$769$	−15.9329	−0.574554	−0.287277	−	0.957848i	$-0.592750\pi$
$769$	−15.9329	−0.574554	−0.287277	+	0.957848i	$0.592750\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	20.1580	0.725033	0.362517	−	0.931977i	$-0.381918\pi$
$773$	20.1580	0.725033	0.362517	+	0.931977i	$0.381918\pi$
$774$	0	0
$775$	7.32576	0.263149
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.62729	−0.237447
$780$	0	0
$781$	8.57660	0.306895
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.25254	0.258854
$786$	0	0
$787$	1.98113	0.0706197	0.0353099	−	0.999376i	$-0.488758\pi$
$787$	1.98113	0.0706197	0.0353099	+	0.999376i	$0.488758\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.7203	0.452283
$792$	0	0
$793$	36.1325	1.28310
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.4536	−0.547396	−0.273698	−	0.961816i	$-0.588247\pi$
$797$	−15.4536	−0.547396	−0.273698	+	0.961816i	$0.588247\pi$
$798$	0	0
$799$	20.4940	0.725027
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−31.1823	−1.10040
$804$	0	0
$805$	−2.17961	−0.0768211
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−21.8551	−0.768385	−0.384193	−	0.923253i	$-0.625520\pi$
$809$	−21.8551	−0.768385	−0.384193	+	0.923253i	$0.625520\pi$
$810$	0	0
$811$	17.3874	0.610554	0.305277	−	0.952264i	$-0.401251\pi$
$811$	17.3874	0.610554	0.305277	+	0.952264i	$0.401251\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−44.2098	−1.54860
$816$	0	0
$817$	22.8818	0.800532
$818$	0	0
$819$	0	0
$820$	0	0
$821$	55.7669	1.94628	0.973140	−	0.230216i	$-0.0739431\pi$
$821$	55.7669	1.94628	0.973140	+	0.230216i	$0.0739431\pi$
$822$	0	0
$823$	12.9595	0.451739	0.225869	−	0.974158i	$-0.427478\pi$
$823$	12.9595	0.451739	0.225869	+	0.974158i	$0.427478\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.2552	0.773889	0.386944	−	0.922103i	$-0.373531\pi$
$827$	22.2552	0.773889	0.386944	+	0.922103i	$0.373531\pi$
$828$	0	0
$829$	−0.807488	−0.0280452	−0.0140226	−	0.999902i	$-0.504464\pi$
$829$	−0.807488	−0.0280452	−0.0140226	+	0.999902i	$0.504464\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.85703	−0.272230
$834$	0	0
$835$	−42.8280	−1.48212
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7.26142	0.250692	0.125346	−	0.992113i	$-0.459996\pi$
$839$	7.26142	0.250692	0.125346	+	0.992113i	$0.459996\pi$
$840$	0	0
$841$	7.07108	0.243830
$842$	0	0
$843$	0	0
$844$	0	0
$845$	13.0555	0.449124
$846$	0	0
$847$	−9.31881	−0.320198
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.83964	0.200180
$852$	0	0
$853$	−44.2453	−1.51493	−0.757464	−	0.652876i	$-0.773562\pi$
$853$	−44.2453	−1.51493	−0.757464	+	0.652876i	$0.773562\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.12852	0.0727088	0.0363544	−	0.999339i	$-0.488425\pi$
$857$	2.12852	0.0727088	0.0363544	+	0.999339i	$0.488425\pi$
$858$	0	0
$859$	2.94066	0.100334	0.0501671	−	0.998741i	$-0.484025\pi$
$859$	2.94066	0.100334	0.0501671	+	0.998741i	$0.484025\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47.3492	−1.61178	−0.805892	−	0.592063i	$-0.798314\pi$
$863$	−47.3492	−1.61178	−0.805892	+	0.592063i	$0.798314\pi$
$864$	0	0
$865$	41.9835	1.42748
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13.8280	−0.469084
$870$	0	0
$871$	−10.7745	−0.365080
$872$	0	0
$873$	0	0
$874$	0	0
$875$	20.6400	0.697758
$876$	0	0
$877$	1.29148	0.0436102	0.0218051	−	0.999762i	$-0.493059\pi$
$877$	1.29148	0.0436102	0.0218051	+	0.999762i	$0.493059\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−43.5582	−1.46751	−0.733757	−	0.679412i	$-0.762235\pi$
$881$	−43.5582	−1.46751	−0.733757	+	0.679412i	$0.762235\pi$
$882$	0	0
$883$	25.8892	0.871241	0.435620	−	0.900131i	$-0.356529\pi$
$883$	25.8892	0.871241	0.435620	+	0.900131i	$0.356529\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	35.6886	1.19831	0.599153	−	0.800634i	$-0.295504\pi$
$887$	35.6886	1.19831	0.599153	+	0.800634i	$0.295504\pi$
$888$	0	0
$889$	25.8396	0.866634
$890$	0	0
$891$	0	0
$892$	0	0
$893$	48.1036	1.60973
$894$	0	0
$895$	−35.4958	−1.18649
$896$	0	0
$897$	0	0
$898$	0	0
$899$	46.4906	1.55055
$900$	0	0
$901$	15.6465	0.521262
$902$	0	0
$903$	0	0
$904$	0	0
$905$	25.0656	0.833208
$906$	0	0
$907$	13.4383	0.446212	0.223106	−	0.974794i	$-0.428380\pi$
$907$	13.4383	0.446212	0.223106	+	0.974794i	$0.428380\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−41.9366	−1.38942	−0.694710	−	0.719290i	$-0.744467\pi$
$911$	−41.9366	−1.38942	−0.694710	+	0.719290i	$0.744467\pi$
$912$	0	0
$913$	−32.5874	−1.07849
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.9520	0.625850
$918$	0	0
$919$	41.4196	1.36631	0.683154	−	0.730275i	$-0.260608\pi$
$919$	41.4196	1.36631	0.683154	+	0.730275i	$0.260608\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.25561	−0.304652
$924$	0	0
$925$	−8.80096	−0.289374
$926$	0	0
$927$	0	0
$928$	0	0
$929$	41.0825	1.34787	0.673936	−	0.738789i	$-0.264602\pi$
$929$	41.0825	1.34787	0.673936	+	0.738789i	$0.264602\pi$
$930$	0	0
$931$	−18.4420	−0.604413
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9.28947	0.303798
$936$	0	0
$937$	−5.81518	−0.189974	−0.0949868	−	0.995479i	$-0.530281\pi$
$937$	−5.81518	−0.189974	−0.0949868	+	0.995479i	$0.530281\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−0.555676	−0.0181145	−0.00905725	−	0.999959i	$-0.502883\pi$
$941$	−0.555676	−0.0181145	−0.00905725	+	0.999959i	$0.502883\pi$
$942$	0	0
$943$	−0.908916	−0.0295984
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−35.3229	−1.14784	−0.573921	−	0.818911i	$-0.694578\pi$
$947$	−35.3229	−1.14784	−0.573921	+	0.818911i	$0.694578\pi$
$948$	0	0
$949$	33.6511	1.09236
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−51.2159	−1.65904	−0.829522	−	0.558474i	$-0.811387\pi$
$953$	−51.2159	−1.65904	−0.829522	+	0.558474i	$0.811387\pi$
$954$	0	0
$955$	−4.09671	−0.132566
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−37.1265	−1.19888
$960$	0	0
$961$	28.9200	0.932903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.2723	0.427249
$966$	0	0
$967$	−33.8235	−1.08769	−0.543845	−	0.839185i	$-0.683032\pi$
$967$	−33.8235	−1.08769	−0.543845	+	0.839185i	$0.683032\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−25.7100	−0.825074	−0.412537	−	0.910941i	$-0.635357\pi$
$971$	−25.7100	−0.825074	−0.412537	+	0.910941i	$0.635357\pi$
$972$	0	0
$973$	16.1485	0.517697
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−38.5144	−1.23219	−0.616093	−	0.787673i	$-0.711285\pi$
$977$	−38.5144	−1.23219	−0.616093	+	0.787673i	$0.711285\pi$
$978$	0	0
$979$	35.5564	1.13639
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27.0170	−0.861708	−0.430854	−	0.902422i	$-0.641788\pi$
$983$	−27.0170	−0.861708	−0.430854	+	0.902422i	$0.641788\pi$
$984$	0	0
$985$	−4.09885	−0.130600
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.13818	0.0997883
$990$	0	0
$991$	37.0010	1.17538	0.587688	−	0.809088i	$-0.300038\pi$
$991$	37.0010	1.17538	0.587688	+	0.809088i	$0.300038\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	26.8915	0.852518
$996$	0	0
$997$	45.3947	1.43766	0.718832	−	0.695184i	$-0.244677\pi$
$997$	45.3947	1.43766	0.718832	+	0.695184i	$0.244677\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1944.2.a.r.1.2	yes	6
3.2	odd	2		1944.2.a.q.1.5	✓	6
4.3	odd	2		3888.2.a.bl.1.2		6
9.2	odd	6		1944.2.i.r.1297.2		12
9.4	even	3		1944.2.i.q.649.5		12
9.5	odd	6		1944.2.i.r.649.2		12
9.7	even	3		1944.2.i.q.1297.5		12
12.11	even	2		3888.2.a.bm.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1944.2.a.q.1.5	✓	6	3.2	odd	2
1944.2.a.r.1.2	yes	6	1.1	even	1	trivial
1944.2.i.q.649.5		12	9.4	even	3
1944.2.i.q.1297.5		12	9.7	even	3
1944.2.i.r.649.2		12	9.5	odd	6
1944.2.i.r.1297.2		12	9.2	odd	6
3888.2.a.bl.1.2		6	4.3	odd	2
3888.2.a.bm.1.5		6	12.11	even	2

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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:	\( N \)	\(=\)	\( 1944 = 2^{3} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1944.a (trivial)

\(f(q)\)	\(=\)	\(q-2.01336 q^{5} +1.72399 q^{7} +O(q^{10})\)	\(q-2.01336 q^{5} +1.72399 q^{7} -2.36530 q^{11} +2.55256 q^{13} +1.95067 q^{17} +4.57862 q^{19} +0.627946 q^{23} -0.946382 q^{25} -6.00592 q^{29} -7.74080 q^{31} -3.47101 q^{35} +9.29958 q^{37} -1.44744 q^{41} +4.99753 q^{43} +10.5061 q^{47} -4.02786 q^{49} +8.02110 q^{53} +4.76219 q^{55} +10.1665 q^{59} +14.1554 q^{61} -5.13922 q^{65} -4.22106 q^{67} -3.62601 q^{71} +13.1833 q^{73} -4.07775 q^{77} +5.84622 q^{79} +13.7773 q^{83} -3.92740 q^{85} -15.0325 q^{89} +4.40058 q^{91} -9.21840 q^{95} -3.64723 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{7}+O(q^{10}) \) 6 * q + 3 * q^7	\( 6 q + 3 q^{7} + 3 q^{11} + 3 q^{13} - 9 q^{17} + 9 q^{19} + 6 q^{23} + 18 q^{25} + 3 q^{29} + 24 q^{31} + 15 q^{35} + 15 q^{37} - 21 q^{41} + 18 q^{43} + 6 q^{47} + 21 q^{49} - 12 q^{53} + 18 q^{55} + 12 q^{59} + 18 q^{61} - 33 q^{65} + 24 q^{67} + 15 q^{71} + 6 q^{73} - 3 q^{77} + 21 q^{79} + 18 q^{83} + 24 q^{85} - 18 q^{89} + 30 q^{91} + 39 q^{95} + 27 q^{97}+O(q^{100}) \) 6 * q + 3 * q^7 + 3 * q^11 + 3 * q^13 - 9 * q^17 + 9 * q^19 + 6 * q^23 + 18 * q^25 + 3 * q^29 + 24 * q^31 + 15 * q^35 + 15 * q^37 - 21 * q^41 + 18 * q^43 + 6 * q^47 + 21 * q^49 - 12 * q^53 + 18 * q^55 + 12 * q^59 + 18 * q^61 - 33 * q^65 + 24 * q^67 + 15 * q^71 + 6 * q^73 - 3 * q^77 + 21 * q^79 + 18 * q^83 + 24 * q^85 - 18 * q^89 + 30 * q^91 + 39 * q^95 + 27 * q^97

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