LMFDB - Embedded newform 1944.2.a.k.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-2.73205 q^{5} -3.46410 q^{7} +O(q^{10})$	$q-2.73205 q^{5} -3.46410 q^{7} +4.19615 q^{11} +4.46410 q^{13} +2.73205 q^{17} +2.46410 q^{19} -5.26795 q^{23} +2.46410 q^{25} -9.46410 q^{29} -4.46410 q^{31} +9.46410 q^{35} +9.46410 q^{37} -2.53590 q^{41} +7.92820 q^{43} -6.92820 q^{47} +5.00000 q^{49} -9.66025 q^{53} -11.4641 q^{55} -12.1962 q^{59} -8.46410 q^{61} -12.1962 q^{65} -10.4641 q^{67} -4.19615 q^{71} -9.39230 q^{73} -14.5359 q^{77} +0.0717968 q^{79} -9.26795 q^{83} -7.46410 q^{85} +12.0000 q^{89} -15.4641 q^{91} -6.73205 q^{95} -1.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^5	$ 2 q - 2 q^{5} - 2 q^{11} + 2 q^{13} + 2 q^{17} - 2 q^{19} - 14 q^{23} - 2 q^{25} - 12 q^{29} - 2 q^{31} + 12 q^{35} + 12 q^{37} - 12 q^{41} + 2 q^{43} + 10 q^{49} - 2 q^{53} - 16 q^{55} - 14 q^{59} - 10 q^{61} - 14 q^{65} - 14 q^{67} + 2 q^{71} + 2 q^{73} - 36 q^{77} + 14 q^{79} - 22 q^{83} - 8 q^{85} + 24 q^{89} - 24 q^{91} - 10 q^{95} + 10 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^11 + 2 * q^13 + 2 * q^17 - 2 * q^19 - 14 * q^23 - 2 * q^25 - 12 * q^29 - 2 * q^31 + 12 * q^35 + 12 * q^37 - 12 * q^41 + 2 * q^43 + 10 * q^49 - 2 * q^53 - 16 * q^55 - 14 * q^59 - 10 * q^61 - 14 * q^65 - 14 * q^67 + 2 * q^71 + 2 * q^73 - 36 * q^77 + 14 * q^79 - 22 * q^83 - 8 * q^85 + 24 * q^89 - 24 * q^91 - 10 * q^95 + 10 * 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
$5$	−2.73205	−1.22181	−0.610905	−	0.791704i	$-0.709194\pi$
$5$	−2.73205	−1.22181	−0.610905	+	0.791704i	$0.709194\pi$
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.19615	1.26519	0.632594	−	0.774484i	$-0.281990\pi$
$11$	4.19615	1.26519	0.632594	+	0.774484i	$0.281990\pi$
$12$	0	0
$13$	4.46410	1.23812	0.619060	−	0.785344i	$-0.287514\pi$
$13$	4.46410	1.23812	0.619060	+	0.785344i	$0.287514\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.73205	0.662620	0.331310	−	0.943522i	$-0.392509\pi$
$17$	2.73205	0.662620	0.331310	+	0.943522i	$0.392509\pi$
$18$	0	0
$19$	2.46410	0.565304	0.282652	−	0.959223i	$-0.408786\pi$
$19$	2.46410	0.565304	0.282652	+	0.959223i	$0.408786\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.26795	−1.09844	−0.549222	−	0.835677i	$-0.685076\pi$
$23$	−5.26795	−1.09844	−0.549222	+	0.835677i	$0.685076\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.46410	−1.75744	−0.878720	−	0.477338i	$-0.841602\pi$
$29$	−9.46410	−1.75744	−0.878720	+	0.477338i	$0.841602\pi$
$30$	0	0
$31$	−4.46410	−0.801776	−0.400888	−	0.916127i	$-0.631298\pi$
$31$	−4.46410	−0.801776	−0.400888	+	0.916127i	$0.631298\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.46410	1.59973
$36$	0	0
$37$	9.46410	1.55589	0.777944	−	0.628333i	$-0.216263\pi$
$37$	9.46410	1.55589	0.777944	+	0.628333i	$0.216263\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.53590	−0.396041	−0.198020	−	0.980198i	$-0.563451\pi$
$41$	−2.53590	−0.396041	−0.198020	+	0.980198i	$0.563451\pi$
$42$	0	0
$43$	7.92820	1.20904	0.604520	−	0.796590i	$-0.293365\pi$
$43$	7.92820	1.20904	0.604520	+	0.796590i	$0.293365\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.66025	−1.32694	−0.663469	−	0.748204i	$-0.730917\pi$
$53$	−9.66025	−1.32694	−0.663469	+	0.748204i	$0.730917\pi$
$54$	0	0
$55$	−11.4641	−1.54582
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.1962	−1.58780	−0.793902	−	0.608046i	$-0.791954\pi$
$59$	−12.1962	−1.58780	−0.793902	+	0.608046i	$0.791954\pi$
$60$	0	0
$61$	−8.46410	−1.08372	−0.541859	−	0.840470i	$-0.682279\pi$
$61$	−8.46410	−1.08372	−0.541859	+	0.840470i	$0.682279\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.1962	−1.51275
$66$	0	0
$67$	−10.4641	−1.27839	−0.639197	−	0.769043i	$-0.720733\pi$
$67$	−10.4641	−1.27839	−0.639197	+	0.769043i	$0.720733\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.19615	−0.497992	−0.248996	−	0.968505i	$-0.580101\pi$
$71$	−4.19615	−0.497992	−0.248996	+	0.968505i	$0.580101\pi$
$72$	0	0
$73$	−9.39230	−1.09929	−0.549643	−	0.835400i	$-0.685236\pi$
$73$	−9.39230	−1.09929	−0.549643	+	0.835400i	$0.685236\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−14.5359	−1.65652
$78$	0	0
$79$	0.0717968	0.00807777	0.00403888	−	0.999992i	$-0.498714\pi$
$79$	0.0717968	0.00807777	0.00403888	+	0.999992i	$0.498714\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.26795	−1.01729	−0.508645	−	0.860976i	$-0.669853\pi$
$83$	−9.26795	−1.01729	−0.508645	+	0.860976i	$0.669853\pi$
$84$	0	0
$85$	−7.46410	−0.809595
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	−15.4641	−1.62108
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.73205	−0.690694
$96$	0	0
$97$	−1.92820	−0.195779	−0.0978897	−	0.995197i	$-0.531209\pi$
$97$	−1.92820	−0.195779	−0.0978897	+	0.995197i	$0.531209\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.3923	1.63110	0.815548	−	0.578690i	$-0.196436\pi$
$101$	16.3923	1.63110	0.815548	+	0.578690i	$0.196436\pi$
$102$	0	0
$103$	8.85641	0.872648	0.436324	−	0.899790i	$-0.356280\pi$
$103$	8.85641	0.872648	0.436324	+	0.899790i	$0.356280\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.53590	−0.245155	−0.122577	−	0.992459i	$-0.539116\pi$
$107$	−2.53590	−0.245155	−0.122577	+	0.992459i	$0.539116\pi$
$108$	0	0
$109$	9.53590	0.913373	0.456687	−	0.889628i	$-0.349036\pi$
$109$	9.53590	0.913373	0.456687	+	0.889628i	$0.349036\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.92820	0.651751	0.325875	−	0.945413i	$-0.394341\pi$
$113$	6.92820	0.651751	0.325875	+	0.945413i	$0.394341\pi$
$114$	0	0
$115$	14.3923	1.34209
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.46410	−0.867573
$120$	0	0
$121$	6.60770	0.600700
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−11.9282	−1.05846	−0.529228	−	0.848479i	$-0.677519\pi$
$127$	−11.9282	−1.05846	−0.529228	+	0.848479i	$0.677519\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.7321	−1.63663	−0.818313	−	0.574772i	$-0.805091\pi$
$131$	−18.7321	−1.63663	−0.818313	+	0.574772i	$0.805091\pi$
$132$	0	0
$133$	−8.53590	−0.740156
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.07180	−0.433313	−0.216656	−	0.976248i	$-0.569515\pi$
$137$	−5.07180	−0.433313	−0.216656	+	0.976248i	$0.569515\pi$
$138$	0	0
$139$	3.46410	0.293821	0.146911	−	0.989150i	$-0.453067\pi$
$139$	3.46410	0.293821	0.146911	+	0.989150i	$0.453067\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	18.7321	1.56645
$144$	0	0
$145$	25.8564	2.14726
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.0526	−1.47892	−0.739462	−	0.673199i	$-0.764920\pi$
$149$	−18.0526	−1.47892	−0.739462	+	0.673199i	$0.764920\pi$
$150$	0	0
$151$	−10.4641	−0.851557	−0.425778	−	0.904828i	$-0.640000\pi$
$151$	−10.4641	−0.851557	−0.425778	+	0.904828i	$0.640000\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.1962	0.979619
$156$	0	0
$157$	−15.3923	−1.22844	−0.614220	−	0.789135i	$-0.710529\pi$
$157$	−15.3923	−1.22844	−0.614220	+	0.789135i	$0.710529\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	18.2487	1.43820
$162$	0	0
$163$	−1.60770	−0.125924	−0.0629622	−	0.998016i	$-0.520055\pi$
$163$	−1.60770	−0.125924	−0.0629622	+	0.998016i	$0.520055\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.3923	1.26847	0.634237	−	0.773138i	$-0.281314\pi$
$167$	16.3923	1.26847	0.634237	+	0.773138i	$0.281314\pi$
$168$	0	0
$169$	6.92820	0.532939
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	0	0
$175$	−8.53590	−0.645253
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.46410	−0.707380	−0.353690	−	0.935363i	$-0.615073\pi$
$179$	−9.46410	−0.707380	−0.353690	+	0.935363i	$0.615073\pi$
$180$	0	0
$181$	23.3205	1.73340	0.866700	−	0.498830i	$-0.166237\pi$
$181$	23.3205	1.73340	0.866700	+	0.498830i	$0.166237\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−25.8564	−1.90100
$186$	0	0
$187$	11.4641	0.838338
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.07180	−0.366982	−0.183491	−	0.983021i	$-0.558740\pi$
$191$	−5.07180	−0.366982	−0.183491	+	0.983021i	$0.558740\pi$
$192$	0	0
$193$	24.8564	1.78920	0.894602	−	0.446865i	$-0.147459\pi$
$193$	24.8564	1.78920	0.894602	+	0.446865i	$0.147459\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.3205	−0.806553	−0.403276	−	0.915078i	$-0.632129\pi$
$197$	−11.3205	−0.806553	−0.403276	+	0.915078i	$0.632129\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	32.7846	2.30103
$204$	0	0
$205$	6.92820	0.483887
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.3397	0.715215
$210$	0	0
$211$	−12.8564	−0.885072	−0.442536	−	0.896751i	$-0.645921\pi$
$211$	−12.8564	−0.885072	−0.442536	+	0.896751i	$0.645921\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−21.6603	−1.47722
$216$	0	0
$217$	15.4641	1.04977
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.1962	0.820402
$222$	0	0
$223$	0.0717968	0.00480787	0.00240393	−	0.999997i	$-0.499235\pi$
$223$	0.0717968	0.00480787	0.00240393	+	0.999997i	$0.499235\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.7846	−1.37952	−0.689761	−	0.724037i	$-0.742285\pi$
$227$	−20.7846	−1.37952	−0.689761	+	0.724037i	$0.742285\pi$
$228$	0	0
$229$	18.3205	1.21065	0.605327	−	0.795977i	$-0.293043\pi$
$229$	18.3205	1.21065	0.605327	+	0.795977i	$0.293043\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.12436	−0.466732	−0.233366	−	0.972389i	$-0.574974\pi$
$233$	−7.12436	−0.466732	−0.233366	+	0.972389i	$0.574974\pi$
$234$	0	0
$235$	18.9282	1.23474
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−23.1244	−1.49579	−0.747895	−	0.663817i	$-0.768935\pi$
$239$	−23.1244	−1.49579	−0.747895	+	0.663817i	$0.768935\pi$
$240$	0	0
$241$	−23.3205	−1.50221	−0.751103	−	0.660185i	$-0.770478\pi$
$241$	−23.3205	−1.50221	−0.751103	+	0.660185i	$0.770478\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−13.6603	−0.872722
$246$	0	0
$247$	11.0000	0.699913
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.58846	−0.542099	−0.271049	−	0.962565i	$-0.587371\pi$
$251$	−8.58846	−0.542099	−0.271049	+	0.962565i	$0.587371\pi$
$252$	0	0
$253$	−22.1051	−1.38974
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.6603	1.35113	0.675565	−	0.737301i	$-0.263900\pi$
$257$	21.6603	1.35113	0.675565	+	0.737301i	$0.263900\pi$
$258$	0	0
$259$	−32.7846	−2.03714
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.1962	0.998698	0.499349	−	0.866401i	$-0.333573\pi$
$263$	16.1962	0.998698	0.499349	+	0.866401i	$0.333573\pi$
$264$	0	0
$265$	26.3923	1.62127
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−18.7321	−1.14211	−0.571057	−	0.820911i	$-0.693466\pi$
$269$	−18.7321	−1.14211	−0.571057	+	0.820911i	$0.693466\pi$
$270$	0	0
$271$	−3.53590	−0.214791	−0.107395	−	0.994216i	$-0.534251\pi$
$271$	−3.53590	−0.214791	−0.107395	+	0.994216i	$0.534251\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.3397	0.623510
$276$	0	0
$277$	−21.3923	−1.28534	−0.642670	−	0.766144i	$-0.722173\pi$
$277$	−21.3923	−1.28534	−0.642670	+	0.766144i	$0.722173\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.26795	0.314260	0.157130	−	0.987578i	$-0.449776\pi$
$281$	5.26795	0.314260	0.157130	+	0.987578i	$0.449776\pi$
$282$	0	0
$283$	−15.5359	−0.923513	−0.461757	−	0.887007i	$-0.652781\pi$
$283$	−15.5359	−0.923513	−0.461757	+	0.887007i	$0.652781\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.78461	0.518539
$288$	0	0
$289$	−9.53590	−0.560935
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.3205	1.36240	0.681199	−	0.732098i	$-0.261459\pi$
$293$	23.3205	1.36240	0.681199	+	0.732098i	$0.261459\pi$
$294$	0	0
$295$	33.3205	1.93999
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−23.5167	−1.36000
$300$	0	0
$301$	−27.4641	−1.58300
$302$	0	0
$303$	0	0
$304$	0	0
$305$	23.1244	1.32410
$306$	0	0
$307$	−27.4641	−1.56746	−0.783730	−	0.621102i	$-0.786685\pi$
$307$	−27.4641	−1.56746	−0.783730	+	0.621102i	$0.786685\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.60770	0.431393	0.215696	−	0.976460i	$-0.430798\pi$
$311$	7.60770	0.431393	0.215696	+	0.976460i	$0.430798\pi$
$312$	0	0
$313$	28.3923	1.60483	0.802414	−	0.596768i	$-0.203549\pi$
$313$	28.3923	1.60483	0.802414	+	0.596768i	$0.203549\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.07180	0.284860	0.142430	−	0.989805i	$-0.454508\pi$
$317$	5.07180	0.284860	0.142430	+	0.989805i	$0.454508\pi$
$318$	0	0
$319$	−39.7128	−2.22349
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.73205	0.374581
$324$	0	0
$325$	11.0000	0.610170
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	13.5359	0.744000	0.372000	−	0.928233i	$-0.378672\pi$
$331$	13.5359	0.744000	0.372000	+	0.928233i	$0.378672\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	28.5885	1.56195
$336$	0	0
$337$	26.5359	1.44550	0.722751	−	0.691108i	$-0.242877\pi$
$337$	26.5359	1.44550	0.722751	+	0.691108i	$0.242877\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−18.7321	−1.01440
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.46410	0.508060	0.254030	−	0.967196i	$-0.418244\pi$
$347$	9.46410	0.508060	0.254030	+	0.967196i	$0.418244\pi$
$348$	0	0
$349$	−7.60770	−0.407231	−0.203615	−	0.979051i	$-0.565269\pi$
$349$	−7.60770	−0.407231	−0.203615	+	0.979051i	$0.565269\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.33975	−0.124532	−0.0622661	−	0.998060i	$-0.519833\pi$
$353$	−2.33975	−0.124532	−0.0622661	+	0.998060i	$0.519833\pi$
$354$	0	0
$355$	11.4641	0.608451
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.7846	1.09697	0.548485	−	0.836160i	$-0.315205\pi$
$359$	20.7846	1.09697	0.548485	+	0.836160i	$0.315205\pi$
$360$	0	0
$361$	−12.9282	−0.680432
$362$	0	0
$363$	0	0
$364$	0	0
$365$	25.6603	1.34312
$366$	0	0
$367$	−3.14359	−0.164094	−0.0820471	−	0.996628i	$-0.526146\pi$
$367$	−3.14359	−0.164094	−0.0820471	+	0.996628i	$0.526146\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	33.4641	1.73737
$372$	0	0
$373$	−25.0000	−1.29445	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000	−1.29445	−0.647225	+	0.762299i	$0.724071\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−42.2487	−2.17592
$378$	0	0
$379$	−24.7846	−1.27310	−0.636550	−	0.771235i	$-0.719639\pi$
$379$	−24.7846	−1.27310	−0.636550	+	0.771235i	$0.719639\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.60770	0.388735	0.194368	−	0.980929i	$-0.437735\pi$
$383$	7.60770	0.388735	0.194368	+	0.980929i	$0.437735\pi$
$384$	0	0
$385$	39.7128	2.02395
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.66025	0.0841782	0.0420891	−	0.999114i	$-0.486599\pi$
$389$	1.66025	0.0841782	0.0420891	+	0.999114i	$0.486599\pi$
$390$	0	0
$391$	−14.3923	−0.727850
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.196152	−0.00986950
$396$	0	0
$397$	−13.0000	−0.652451	−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000	−0.652451	−0.326226	+	0.945292i	$0.605777\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.39230	0.219341	0.109671	−	0.993968i	$-0.465020\pi$
$401$	4.39230	0.219341	0.109671	+	0.993968i	$0.465020\pi$
$402$	0	0
$403$	−19.9282	−0.992695
$404$	0	0
$405$	0	0
$406$	0	0
$407$	39.7128	1.96849
$408$	0	0
$409$	−11.3205	−0.559763	−0.279882	−	0.960035i	$-0.590295\pi$
$409$	−11.3205	−0.559763	−0.279882	+	0.960035i	$0.590295\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	42.2487	2.07892
$414$	0	0
$415$	25.3205	1.24293
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.2487	−0.891508	−0.445754	−	0.895156i	$-0.647064\pi$
$419$	−18.2487	−0.891508	−0.445754	+	0.895156i	$0.647064\pi$
$420$	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
$423$	0	0
$424$	0	0
$425$	6.73205	0.326552
$426$	0	0
$427$	29.3205	1.41892
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−37.6603	−1.81403	−0.907015	−	0.421098i	$-0.861645\pi$
$431$	−37.6603	−1.81403	−0.907015	+	0.421098i	$0.861645\pi$
$432$	0	0
$433$	5.39230	0.259138	0.129569	−	0.991570i	$-0.458641\pi$
$433$	5.39230	0.259138	0.129569	+	0.991570i	$0.458641\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.9808	−0.620954
$438$	0	0
$439$	−1.60770	−0.0767311	−0.0383656	−	0.999264i	$-0.512215\pi$
$439$	−1.60770	−0.0767311	−0.0383656	+	0.999264i	$0.512215\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.4449	1.44648	0.723240	−	0.690597i	$-0.242652\pi$
$443$	30.4449	1.44648	0.723240	+	0.690597i	$0.242652\pi$
$444$	0	0
$445$	−32.7846	−1.55414
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−24.1962	−1.14189	−0.570944	−	0.820989i	$-0.693422\pi$
$449$	−24.1962	−1.14189	−0.570944	+	0.820989i	$0.693422\pi$
$450$	0	0
$451$	−10.6410	−0.501066
$452$	0	0
$453$	0	0
$454$	0	0
$455$	42.2487	1.98065
$456$	0	0
$457$	−8.85641	−0.414285	−0.207143	−	0.978311i	$-0.566416\pi$
$457$	−8.85641	−0.414285	−0.207143	+	0.978311i	$0.566416\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.8038	0.549760	0.274880	−	0.961479i	$-0.411362\pi$
$461$	11.8038	0.549760	0.274880	+	0.961479i	$0.411362\pi$
$462$	0	0
$463$	27.3923	1.27303	0.636514	−	0.771265i	$-0.280376\pi$
$463$	27.3923	1.27303	0.636514	+	0.771265i	$0.280376\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.53590	−0.117347	−0.0586737	−	0.998277i	$-0.518687\pi$
$467$	−2.53590	−0.117347	−0.0586737	+	0.998277i	$0.518687\pi$
$468$	0	0
$469$	36.2487	1.67381
$470$	0	0
$471$	0	0
$472$	0	0
$473$	33.2679	1.52966
$474$	0	0
$475$	6.07180	0.278593
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.66025	−0.0758589	−0.0379295	−	0.999280i	$-0.512076\pi$
$479$	−1.66025	−0.0758589	−0.0379295	+	0.999280i	$0.512076\pi$
$480$	0	0
$481$	42.2487	1.92638
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.26795	0.239205
$486$	0	0
$487$	−17.9282	−0.812404	−0.406202	−	0.913783i	$-0.633147\pi$
$487$	−17.9282	−0.812404	−0.406202	+	0.913783i	$0.633147\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.8564	1.16688	0.583442	−	0.812155i	$-0.301706\pi$
$491$	25.8564	1.16688	0.583442	+	0.812155i	$0.301706\pi$
$492$	0	0
$493$	−25.8564	−1.16451
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.5359	0.652024
$498$	0	0
$499$	5.32051	0.238179	0.119089	−	0.992884i	$-0.462003\pi$
$499$	5.32051	0.238179	0.119089	+	0.992884i	$0.462003\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.5359	0.648124	0.324062	−	0.946036i	$-0.394951\pi$
$503$	14.5359	0.648124	0.324062	+	0.946036i	$0.394951\pi$
$504$	0	0
$505$	−44.7846	−1.99289
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8.78461	−0.389371	−0.194685	−	0.980866i	$-0.562369\pi$
$509$	−8.78461	−0.389371	−0.194685	+	0.980866i	$0.562369\pi$
$510$	0	0
$511$	32.5359	1.43930
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−24.1962	−1.06621
$516$	0	0
$517$	−29.0718	−1.27858
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.1962	−1.23530	−0.617648	−	0.786455i	$-0.711914\pi$
$521$	−28.1962	−1.23530	−0.617648	+	0.786455i	$0.711914\pi$
$522$	0	0
$523$	14.8564	0.649625	0.324813	−	0.945778i	$-0.394699\pi$
$523$	14.8564	0.649625	0.324813	+	0.945778i	$0.394699\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.1962	−0.531273
$528$	0	0
$529$	4.75129	0.206578
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.3205	−0.490346
$534$	0	0
$535$	6.92820	0.299532
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20.9808	0.903705
$540$	0	0
$541$	16.3923	0.704760	0.352380	−	0.935857i	$-0.385372\pi$
$541$	16.3923	0.704760	0.352380	+	0.935857i	$0.385372\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−26.0526	−1.11597
$546$	0	0
$547$	−42.3205	−1.80949	−0.904747	−	0.425949i	$-0.859940\pi$
$547$	−42.3205	−1.80949	−0.904747	+	0.425949i	$0.859940\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−23.3205	−0.993487
$552$	0	0
$553$	−0.248711	−0.0105763
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.19615	0.177797	0.0888983	−	0.996041i	$-0.471665\pi$
$557$	4.19615	0.177797	0.0888983	+	0.996041i	$0.471665\pi$
$558$	0	0
$559$	35.3923	1.49693
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.1244	0.468836	0.234418	−	0.972136i	$-0.424682\pi$
$563$	11.1244	0.468836	0.234418	+	0.972136i	$0.424682\pi$
$564$	0	0
$565$	−18.9282	−0.796315
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.58846	−0.192358	−0.0961791	−	0.995364i	$-0.530662\pi$
$569$	−4.58846	−0.192358	−0.0961791	+	0.995364i	$0.530662\pi$
$570$	0	0
$571$	3.46410	0.144968	0.0724841	−	0.997370i	$-0.476907\pi$
$571$	3.46410	0.144968	0.0724841	+	0.997370i	$0.476907\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.9808	−0.541335
$576$	0	0
$577$	12.8564	0.535219	0.267610	−	0.963527i	$-0.413766\pi$
$577$	12.8564	0.535219	0.267610	+	0.963527i	$0.413766\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	32.1051	1.33194
$582$	0	0
$583$	−40.5359	−1.67883
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.1962	0.668487	0.334243	−	0.942487i	$-0.391519\pi$
$587$	16.1962	0.668487	0.334243	+	0.942487i	$0.391519\pi$
$588$	0	0
$589$	−11.0000	−0.453247
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.58846	−0.188425	−0.0942127	−	0.995552i	$-0.530033\pi$
$593$	−4.58846	−0.188425	−0.0942127	+	0.995552i	$0.530033\pi$
$594$	0	0
$595$	25.8564	1.06001
$596$	0	0
$597$	0	0
$598$	0	0
$599$	39.7128	1.62262	0.811311	−	0.584615i	$-0.198754\pi$
$599$	39.7128	1.62262	0.811311	+	0.584615i	$0.198754\pi$
$600$	0	0
$601$	−22.3205	−0.910473	−0.455236	−	0.890371i	$-0.650445\pi$
$601$	−22.3205	−0.910473	−0.455236	+	0.890371i	$0.650445\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−18.0526	−0.733941
$606$	0	0
$607$	28.7846	1.16833	0.584166	−	0.811634i	$-0.301422\pi$
$607$	28.7846	1.16833	0.584166	+	0.811634i	$0.301422\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−30.9282	−1.25122
$612$	0	0
$613$	−6.07180	−0.245238	−0.122619	−	0.992454i	$-0.539129\pi$
$613$	−6.07180	−0.245238	−0.122619	+	0.992454i	$0.539129\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.53590	0.102091	0.0510457	−	0.998696i	$-0.483745\pi$
$617$	2.53590	0.102091	0.0510457	+	0.998696i	$0.483745\pi$
$618$	0	0
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−41.5692	−1.66544
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	0	0
$628$	0	0
$629$	25.8564	1.03096
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	32.5885	1.29323
$636$	0	0
$637$	22.3205	0.884371
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19.8038	−0.782205	−0.391102	−	0.920347i	$-0.627906\pi$
$641$	−19.8038	−0.782205	−0.391102	+	0.920347i	$0.627906\pi$
$642$	0	0
$643$	46.3923	1.82953	0.914767	−	0.403982i	$-0.132374\pi$
$643$	46.3923	1.82953	0.914767	+	0.403982i	$0.132374\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.875644	0.0344251	0.0172126	−	0.999852i	$-0.494521\pi$
$647$	0.875644	0.0344251	0.0172126	+	0.999852i	$0.494521\pi$
$648$	0	0
$649$	−51.1769	−2.00887
$650$	0	0
$651$	0	0
$652$	0	0
$653$	32.7846	1.28296	0.641480	−	0.767139i	$-0.278321\pi$
$653$	32.7846	1.28296	0.641480	+	0.767139i	$0.278321\pi$
$654$	0	0
$655$	51.1769	1.99965
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	26.7128	1.03901	0.519504	−	0.854468i	$-0.326117\pi$
$661$	26.7128	1.03901	0.519504	+	0.854468i	$0.326117\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	23.3205	0.904331
$666$	0	0
$667$	49.8564	1.93045
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−35.5167	−1.37111
$672$	0	0
$673$	31.2487	1.20455	0.602275	−	0.798289i	$-0.294261\pi$
$673$	31.2487	1.20455	0.602275	+	0.798289i	$0.294261\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19.9090	0.765164	0.382582	−	0.923922i	$-0.375035\pi$
$677$	19.9090	0.765164	0.382582	+	0.923922i	$0.375035\pi$
$678$	0	0
$679$	6.67949	0.256335
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−38.0526	−1.45604	−0.728020	−	0.685556i	$-0.759559\pi$
$683$	−38.0526	−1.45604	−0.728020	+	0.685556i	$0.759559\pi$
$684$	0	0
$685$	13.8564	0.529426
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−43.1244	−1.64291
$690$	0	0
$691$	30.1769	1.14798	0.573992	−	0.818861i	$-0.305394\pi$
$691$	30.1769	1.14798	0.573992	+	0.818861i	$0.305394\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.46410	−0.358994
$696$	0	0
$697$	−6.92820	−0.262424
$698$	0	0
$699$	0	0
$700$	0	0
$701$	11.3205	0.427570	0.213785	−	0.976881i	$-0.431421\pi$
$701$	11.3205	0.427570	0.213785	+	0.976881i	$0.431421\pi$
$702$	0	0
$703$	23.3205	0.879550
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−56.7846	−2.13561
$708$	0	0
$709$	−13.9282	−0.523085	−0.261542	−	0.965192i	$-0.584231\pi$
$709$	−13.9282	−0.523085	−0.261542	+	0.965192i	$0.584231\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	23.5167	0.880706
$714$	0	0
$715$	−51.1769	−1.91391
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.26795	0.345636	0.172818	−	0.984954i	$-0.444713\pi$
$719$	9.26795	0.345636	0.172818	+	0.984954i	$0.444713\pi$
$720$	0	0
$721$	−30.6795	−1.14256
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−23.3205	−0.866102
$726$	0	0
$727$	−1.60770	−0.0596261	−0.0298131	−	0.999555i	$-0.509491\pi$
$727$	−1.60770	−0.0596261	−0.0298131	+	0.999555i	$0.509491\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	21.6603	0.801133
$732$	0	0
$733$	−24.0718	−0.889112	−0.444556	−	0.895751i	$-0.646639\pi$
$733$	−24.0718	−0.889112	−0.444556	+	0.895751i	$0.646639\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−43.9090	−1.61741
$738$	0	0
$739$	−35.3923	−1.30193	−0.650963	−	0.759109i	$-0.725635\pi$
$739$	−35.3923	−1.30193	−0.650963	+	0.759109i	$0.725635\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26.0526	−0.955776	−0.477888	−	0.878421i	$-0.658597\pi$
$743$	−26.0526	−0.955776	−0.477888	+	0.878421i	$0.658597\pi$
$744$	0	0
$745$	49.3205	1.80696
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.78461	0.320983
$750$	0	0
$751$	−27.4641	−1.00218	−0.501090	−	0.865395i	$-0.667067\pi$
$751$	−27.4641	−1.00218	−0.501090	+	0.865395i	$0.667067\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	28.5885	1.04044
$756$	0	0
$757$	−13.9282	−0.506229	−0.253115	−	0.967436i	$-0.581455\pi$
$757$	−13.9282	−0.506229	−0.253115	+	0.967436i	$0.581455\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23.8038	−0.862889	−0.431444	−	0.902140i	$-0.641996\pi$
$761$	−23.8038	−0.862889	−0.431444	+	0.902140i	$0.641996\pi$
$762$	0	0
$763$	−33.0333	−1.19589
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−54.4449	−1.96589
$768$	0	0
$769$	52.5692	1.89569	0.947847	−	0.318725i	$-0.103255\pi$
$769$	52.5692	1.89569	0.947847	+	0.318725i	$0.103255\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.5359	0.954430	0.477215	−	0.878787i	$-0.341646\pi$
$773$	26.5359	0.954430	0.477215	+	0.878787i	$0.341646\pi$
$774$	0	0
$775$	−11.0000	−0.395132
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.24871	−0.223883
$780$	0	0
$781$	−17.6077	−0.630053
$782$	0	0
$783$	0	0
$784$	0	0
$785$	42.0526	1.50092
$786$	0	0
$787$	−21.6410	−0.771419	−0.385709	−	0.922620i	$-0.626043\pi$
$787$	−21.6410	−0.771419	−0.385709	+	0.922620i	$0.626043\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	−37.7846	−1.34177
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.26795	−0.186600	−0.0933002	−	0.995638i	$-0.529742\pi$
$797$	−5.26795	−0.186600	−0.0933002	+	0.995638i	$0.529742\pi$
$798$	0	0
$799$	−18.9282	−0.669632
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−39.4115	−1.39080
$804$	0	0
$805$	−49.8564	−1.75721
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.6603	0.761534	0.380767	−	0.924671i	$-0.375660\pi$
$809$	21.6603	0.761534	0.380767	+	0.924671i	$0.375660\pi$
$810$	0	0
$811$	4.21539	0.148022	0.0740112	−	0.997257i	$-0.476420\pi$
$811$	4.21539	0.148022	0.0740112	+	0.997257i	$0.476420\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.39230	0.153856
$816$	0	0
$817$	19.5359	0.683475
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.85641	−0.0647890	−0.0323945	−	0.999475i	$-0.510313\pi$
$821$	−1.85641	−0.0647890	−0.0323945	+	0.999475i	$0.510313\pi$
$822$	0	0
$823$	−20.1769	−0.703323	−0.351662	−	0.936127i	$-0.614383\pi$
$823$	−20.1769	−0.703323	−0.351662	+	0.936127i	$0.614383\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.5167	0.817754	0.408877	−	0.912589i	$-0.365920\pi$
$827$	23.5167	0.817754	0.408877	+	0.912589i	$0.365920\pi$
$828$	0	0
$829$	−35.2487	−1.22424	−0.612119	−	0.790766i	$-0.709683\pi$
$829$	−35.2487	−1.22424	−0.612119	+	0.790766i	$0.709683\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	13.6603	0.473300
$834$	0	0
$835$	−44.7846	−1.54984
$836$	0	0
$837$	0	0
$838$	0	0
$839$	23.3205	0.805113	0.402557	−	0.915395i	$-0.368122\pi$
$839$	23.3205	0.805113	0.402557	+	0.915395i	$0.368122\pi$
$840$	0	0
$841$	60.5692	2.08859
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−18.9282	−0.651150
$846$	0	0
$847$	−22.8897	−0.786500
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−49.8564	−1.70906
$852$	0	0
$853$	−9.78461	−0.335019	−0.167509	−	0.985870i	$-0.553572\pi$
$853$	−9.78461	−0.335019	−0.167509	+	0.985870i	$0.553572\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49.1769	1.67985	0.839926	−	0.542701i	$-0.182598\pi$
$857$	49.1769	1.67985	0.839926	+	0.542701i	$0.182598\pi$
$858$	0	0
$859$	39.4641	1.34650	0.673249	−	0.739416i	$-0.264898\pi$
$859$	39.4641	1.34650	0.673249	+	0.739416i	$0.264898\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	54.0526	1.83997	0.919985	−	0.391953i	$-0.128200\pi$
$863$	54.0526	1.83997	0.919985	+	0.391953i	$0.128200\pi$
$864$	0	0
$865$	−18.9282	−0.643578
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.301270	0.0102199
$870$	0	0
$871$	−46.7128	−1.58280
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−38.5359	−1.30126	−0.650632	−	0.759393i	$-0.725496\pi$
$877$	−38.5359	−1.30126	−0.650632	+	0.759393i	$0.725496\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.66025	−0.0559354	−0.0279677	−	0.999609i	$-0.508904\pi$
$881$	−1.66025	−0.0559354	−0.0279677	+	0.999609i	$0.508904\pi$
$882$	0	0
$883$	46.3205	1.55881	0.779405	−	0.626521i	$-0.215522\pi$
$883$	46.3205	1.55881	0.779405	+	0.626521i	$0.215522\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.87564	−0.163708	−0.0818541	−	0.996644i	$-0.526084\pi$
$887$	−4.87564	−0.163708	−0.0818541	+	0.996644i	$0.526084\pi$
$888$	0	0
$889$	41.3205	1.38585
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−17.0718	−0.571286
$894$	0	0
$895$	25.8564	0.864284
$896$	0	0
$897$	0	0
$898$	0	0
$899$	42.2487	1.40907
$900$	0	0
$901$	−26.3923	−0.879255
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−63.7128	−2.11789
$906$	0	0
$907$	7.00000	0.232431	0.116216	−	0.993224i	$-0.462924\pi$
$907$	7.00000	0.232431	0.116216	+	0.993224i	$0.462924\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−24.1962	−0.801654	−0.400827	−	0.916154i	$-0.631277\pi$
$911$	−24.1962	−0.801654	−0.400827	+	0.916154i	$0.631277\pi$
$912$	0	0
$913$	−38.8897	−1.28706
$914$	0	0
$915$	0	0
$916$	0	0
$917$	64.8897	2.14285
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−18.7321	−0.616573
$924$	0	0
$925$	23.3205	0.766774
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−19.4115	−0.636872	−0.318436	−	0.947944i	$-0.603158\pi$
$929$	−19.4115	−0.636872	−0.318436	+	0.947944i	$0.603158\pi$
$930$	0	0
$931$	12.3205	0.403788
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−31.3205	−1.02429
$936$	0	0
$937$	4.39230	0.143490	0.0717452	−	0.997423i	$-0.477143\pi$
$937$	4.39230	0.143490	0.0717452	+	0.997423i	$0.477143\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6.92820	−0.225853	−0.112926	−	0.993603i	$-0.536022\pi$
$941$	−6.92820	−0.225853	−0.112926	+	0.993603i	$0.536022\pi$
$942$	0	0
$943$	13.3590	0.435028
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.2487	0.982951	0.491476	−	0.870891i	$-0.336458\pi$
$947$	30.2487	0.982951	0.491476	+	0.870891i	$0.336458\pi$
$948$	0	0
$949$	−41.9282	−1.36105
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.7846	1.45072	0.725358	−	0.688372i	$-0.241674\pi$
$953$	44.7846	1.45072	0.725358	+	0.688372i	$0.241674\pi$
$954$	0	0
$955$	13.8564	0.448383
$956$	0	0
$957$	0	0
$958$	0	0
$959$	17.5692	0.567340
$960$	0	0
$961$	−11.0718	−0.357155
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−67.9090	−2.18607
$966$	0	0
$967$	5.24871	0.168787	0.0843936	−	0.996432i	$-0.473105\pi$
$967$	5.24871	0.168787	0.0843936	+	0.996432i	$0.473105\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.6410	−0.726585	−0.363292	−	0.931675i	$-0.618347\pi$
$971$	−22.6410	−0.726585	−0.363292	+	0.931675i	$0.618347\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23.3205	−0.746089	−0.373045	−	0.927813i	$-0.621686\pi$
$977$	−23.3205	−0.746089	−0.373045	+	0.927813i	$0.621686\pi$
$978$	0	0
$979$	50.3538	1.60932
$980$	0	0
$981$	0	0
$982$	0	0
$983$	19.6077	0.625388	0.312694	−	0.949854i	$-0.398769\pi$
$983$	19.6077	0.625388	0.312694	+	0.949854i	$0.398769\pi$
$984$	0	0
$985$	30.9282	0.985454
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−41.7654	−1.32806
$990$	0	0
$991$	0.0717968	0.00228070	0.00114035	−	0.999999i	$-0.499637\pi$
$991$	0.0717968	0.00228070	0.00114035	+	0.999999i	$0.499637\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	30.0526	0.952730
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1944.2.a.k.1.1	✓	2
3.2	odd	2		1944.2.a.n.1.2	yes	2
4.3	odd	2		3888.2.a.w.1.1		2
9.2	odd	6		1944.2.i.k.1297.1		4
9.4	even	3		1944.2.i.n.649.2		4
9.5	odd	6		1944.2.i.k.649.1		4
9.7	even	3		1944.2.i.n.1297.2		4
12.11	even	2		3888.2.a.bc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1944.2.a.k.1.1	✓	2	1.1	even	1	trivial
1944.2.a.n.1.2	yes	2	3.2	odd	2
1944.2.i.k.649.1		4	9.5	odd	6
1944.2.i.k.1297.1		4	9.2	odd	6
1944.2.i.n.649.2		4	9.4	even	3
1944.2.i.n.1297.2		4	9.7	even	3
3888.2.a.w.1.1		2	4.3	odd	2
3888.2.a.bc.1.2		2	12.11	even	2

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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)

\(f(q)\)	\(=\)	\(q-2.73205 q^{5} -3.46410 q^{7} +O(q^{10})\)	\(q-2.73205 q^{5} -3.46410 q^{7} +4.19615 q^{11} +4.46410 q^{13} +2.73205 q^{17} +2.46410 q^{19} -5.26795 q^{23} +2.46410 q^{25} -9.46410 q^{29} -4.46410 q^{31} +9.46410 q^{35} +9.46410 q^{37} -2.53590 q^{41} +7.92820 q^{43} -6.92820 q^{47} +5.00000 q^{49} -9.66025 q^{53} -11.4641 q^{55} -12.1962 q^{59} -8.46410 q^{61} -12.1962 q^{65} -10.4641 q^{67} -4.19615 q^{71} -9.39230 q^{73} -14.5359 q^{77} +0.0717968 q^{79} -9.26795 q^{83} -7.46410 q^{85} +12.0000 q^{89} -15.4641 q^{91} -6.73205 q^{95} -1.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} - 2 q^{11} + 2 q^{13} + 2 q^{17} - 2 q^{19} - 14 q^{23} - 2 q^{25} - 12 q^{29} - 2 q^{31} + 12 q^{35} + 12 q^{37} - 12 q^{41} + 2 q^{43} + 10 q^{49} - 2 q^{53} - 16 q^{55} - 14 q^{59} - 10 q^{61} - 14 q^{65} - 14 q^{67} + 2 q^{71} + 2 q^{73} - 36 q^{77} + 14 q^{79} - 22 q^{83} - 8 q^{85} + 24 q^{89} - 24 q^{91} - 10 q^{95} + 10 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^11 + 2 * q^13 + 2 * q^17 - 2 * q^19 - 14 * q^23 - 2 * q^25 - 12 * q^29 - 2 * q^31 + 12 * q^35 + 12 * q^37 - 12 * q^41 + 2 * q^43 + 10 * q^49 - 2 * q^53 - 16 * q^55 - 14 * q^59 - 10 * q^61 - 14 * q^65 - 14 * q^67 + 2 * q^71 + 2 * q^73 - 36 * q^77 + 14 * q^79 - 22 * q^83 - 8 * q^85 + 24 * q^89 - 24 * q^91 - 10 * q^95 + 10 * q^97

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