LMFDB - Embedded newform 1872.4.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,4,Mod(1,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1872.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:	$110.451575531$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1520092.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 40x^{2} - 9x + 81 $ x^4 - x^3 - 40x^2 - 9x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.63814$ of defining polynomial
Character	$\chi$	$=$	1872.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.5342 q^{5} +6.26434 q^{7} +O(q^{10})$	$q-19.5342 q^{5} +6.26434 q^{7} +27.2460 q^{11} -13.0000 q^{13} +30.8471 q^{17} -127.850 q^{19} -84.3198 q^{23} +256.586 q^{25} +272.460 q^{29} +166.264 q^{31} -122.369 q^{35} -198.229 q^{37} +160.385 q^{41} -158.943 q^{43} +305.889 q^{47} -303.758 q^{49} +356.780 q^{53} -532.229 q^{55} +470.384 q^{59} -171.815 q^{61} +253.945 q^{65} -1022.49 q^{67} +188.616 q^{71} +959.975 q^{73} +170.678 q^{77} +1034.80 q^{79} +105.383 q^{83} -602.574 q^{85} -649.860 q^{89} -81.4364 q^{91} +2497.46 q^{95} +707.746 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{7}+O(q^{10}) $ 4 * q - 36 * q^7	$ 4 q - 36 q^{7} - 52 q^{13} - 84 q^{19} + 660 q^{25} + 604 q^{31} + 184 q^{37} - 880 q^{43} - 116 q^{49} - 1152 q^{55} + 656 q^{61} - 3052 q^{67} - 312 q^{73} + 720 q^{79} + 32 q^{85} + 468 q^{91} - 344 q^{97}+O(q^{100}) $ 4 * q - 36 * q^7 - 52 * q^13 - 84 * q^19 + 660 * q^25 + 604 * q^31 + 184 * q^37 - 880 * q^43 - 116 * q^49 - 1152 * q^55 + 656 * q^61 - 3052 * q^67 - 312 * q^73 + 720 * q^79 + 32 * q^85 + 468 * q^91 - 344 * 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
$5$	−19.5342	−1.74719	−0.873597	−	0.486650i	$-0.838219\pi$
$5$	−19.5342	−1.74719	−0.873597	+	0.486650i	$0.838219\pi$
$6$	0	0
$7$	6.26434	0.338242	0.169121	−	0.985595i	$-0.445907\pi$
$7$	6.26434	0.338242	0.169121	+	0.985595i	$0.445907\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	27.2460	0.746816	0.373408	−	0.927667i	$-0.378189\pi$
$11$	27.2460	0.746816	0.373408	+	0.927667i	$0.378189\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	30.8471	0.440089	0.220044	−	0.975490i	$-0.429380\pi$
$17$	30.8471	0.440089	0.220044	+	0.975490i	$0.429380\pi$
$18$	0	0
$19$	−127.850	−1.54373	−0.771865	−	0.635786i	$-0.780676\pi$
$19$	−127.850	−1.54373	−0.771865	+	0.635786i	$0.780676\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−84.3198	−0.764430	−0.382215	−	0.924073i	$-0.624839\pi$
$23$	−84.3198	−0.764430	−0.382215	+	0.924073i	$0.624839\pi$
$24$	0	0
$25$	256.586	2.05269
$26$	0	0
$27$	0	0
$28$	0	0
$29$	272.460	1.74464	0.872320	−	0.488936i	$-0.162615\pi$
$29$	272.460	1.74464	0.872320	+	0.488936i	$0.162615\pi$
$30$	0	0
$31$	166.264	0.963289	0.481644	−	0.876367i	$-0.340040\pi$
$31$	166.264	0.963289	0.481644	+	0.876367i	$0.340040\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−122.369	−0.590975
$36$	0	0
$37$	−198.229	−0.880776	−0.440388	−	0.897808i	$-0.645159\pi$
$37$	−198.229	−0.880776	−0.440388	+	0.897808i	$0.645159\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	160.385	0.610923	0.305462	−	0.952204i	$-0.401189\pi$
$41$	160.385	0.610923	0.305462	+	0.952204i	$0.401189\pi$
$42$	0	0
$43$	−158.943	−0.563687	−0.281843	−	0.959460i	$-0.590946\pi$
$43$	−158.943	−0.563687	−0.281843	+	0.959460i	$0.590946\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	305.889	0.949329	0.474665	−	0.880167i	$-0.342569\pi$
$47$	305.889	0.949329	0.474665	+	0.880167i	$0.342569\pi$
$48$	0	0
$49$	−303.758	−0.885592
$50$	0	0
$51$	0	0
$52$	0	0
$53$	356.780	0.924669	0.462335	−	0.886706i	$-0.347012\pi$
$53$	356.780	0.924669	0.462335	+	0.886706i	$0.347012\pi$
$54$	0	0
$55$	−532.229	−1.30483
$56$	0	0
$57$	0	0
$58$	0	0
$59$	470.384	1.03795	0.518973	−	0.854791i	$-0.326315\pi$
$59$	470.384	1.03795	0.518973	+	0.854791i	$0.326315\pi$
$60$	0	0
$61$	−171.815	−0.360635	−0.180317	−	0.983608i	$-0.557712\pi$
$61$	−171.815	−0.360635	−0.180317	+	0.983608i	$0.557712\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	253.945	0.484585
$66$	0	0
$67$	−1022.49	−1.86444	−0.932220	−	0.361892i	$-0.882131\pi$
$67$	−1022.49	−1.86444	−0.932220	+	0.361892i	$0.882131\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	188.616	0.315276	0.157638	−	0.987497i	$-0.449612\pi$
$71$	188.616	0.315276	0.157638	+	0.987497i	$0.449612\pi$
$72$	0	0
$73$	959.975	1.53913	0.769566	−	0.638568i	$-0.220473\pi$
$73$	959.975	1.53913	0.769566	+	0.638568i	$0.220473\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	170.678	0.252605
$78$	0	0
$79$	1034.80	1.47373	0.736863	−	0.676042i	$-0.236306\pi$
$79$	1034.80	1.47373	0.736863	+	0.676042i	$0.236306\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	105.383	0.139365	0.0696824	−	0.997569i	$-0.477801\pi$
$83$	105.383	0.139365	0.0696824	+	0.997569i	$0.477801\pi$
$84$	0	0
$85$	−602.574	−0.768921
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−649.860	−0.773990	−0.386995	−	0.922082i	$-0.626487\pi$
$89$	−649.860	−0.773990	−0.386995	+	0.922082i	$0.626487\pi$
$90$	0	0
$91$	−81.4364	−0.0938116
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2497.46	2.69720
$96$	0	0
$97$	707.746	0.740832	0.370416	−	0.928866i	$-0.379215\pi$
$97$	707.746	0.740832	0.370416	+	0.928866i	$0.379215\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1409.59	1.38871	0.694353	−	0.719634i	$-0.255690\pi$
$101$	1409.59	1.38871	0.694353	+	0.719634i	$0.255690\pi$
$102$	0	0
$103$	−1519.63	−1.45373	−0.726863	−	0.686783i	$-0.759022\pi$
$103$	−1519.63	−1.45373	−0.726863	+	0.686783i	$0.759022\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1957.57	−1.76865	−0.884323	−	0.466875i	$-0.845380\pi$
$107$	−1957.57	−1.76865	−0.884323	+	0.466875i	$0.845380\pi$
$108$	0	0
$109$	1517.63	1.33360	0.666801	−	0.745236i	$-0.267663\pi$
$109$	1517.63	1.33360	0.666801	+	0.745236i	$0.267663\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2200.27	−1.83171	−0.915857	−	0.401505i	$-0.868487\pi$
$113$	−2200.27	−1.83171	−0.915857	+	0.401505i	$0.868487\pi$
$114$	0	0
$115$	1647.12	1.33561
$116$	0	0
$117$	0	0
$118$	0	0
$119$	193.236	0.148857
$120$	0	0
$121$	−588.656	−0.442266
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2570.43	−1.83925
$126$	0	0
$127$	−1812.16	−1.26617	−0.633083	−	0.774084i	$-0.718211\pi$
$127$	−1812.16	−1.26617	−0.633083	+	0.774084i	$0.718211\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1001.31	−0.667822	−0.333911	−	0.942605i	$-0.608369\pi$
$131$	−1001.31	−0.667822	−0.333911	+	0.942605i	$0.608369\pi$
$132$	0	0
$133$	−800.898	−0.522155
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1244.11	0.775849	0.387925	−	0.921691i	$-0.373192\pi$
$137$	1244.11	0.775849	0.387925	+	0.921691i	$0.373192\pi$
$138$	0	0
$139$	−1348.85	−0.823077	−0.411539	−	0.911392i	$-0.635009\pi$
$139$	−1348.85	−0.823077	−0.411539	+	0.911392i	$0.635009\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−354.198	−0.207129
$144$	0	0
$145$	−5322.29	−3.04822
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1041.63	−0.572710	−0.286355	−	0.958124i	$-0.592444\pi$
$149$	−1041.63	−0.572710	−0.286355	+	0.958124i	$0.592444\pi$
$150$	0	0
$151$	−1361.90	−0.733970	−0.366985	−	0.930227i	$-0.619610\pi$
$151$	−1361.90	−0.733970	−0.366985	+	0.930227i	$0.619610\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3247.85	−1.68305
$156$	0	0
$157$	86.8978	0.0441733	0.0220866	−	0.999756i	$-0.492969\pi$
$157$	86.8978	0.0441733	0.0220866	+	0.999756i	$0.492969\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−528.208	−0.258563
$162$	0	0
$163$	28.5386	0.0137136	0.00685679	−	0.999976i	$-0.497817\pi$
$163$	28.5386	0.0137136	0.00685679	+	0.999976i	$0.497817\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1289.87	−0.597683	−0.298842	−	0.954303i	$-0.596600\pi$
$167$	−1289.87	−0.597683	−0.298842	+	0.954303i	$0.596600\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−603.489	−0.265216	−0.132608	−	0.991169i	$-0.542335\pi$
$173$	−603.489	−0.265216	−0.132608	+	0.991169i	$0.542335\pi$
$174$	0	0
$175$	1607.34	0.694306
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1985.29	−0.828980	−0.414490	−	0.910054i	$-0.636040\pi$
$179$	−1985.29	−0.828980	−0.414490	+	0.910054i	$0.636040\pi$
$180$	0	0
$181$	−3945.79	−1.62038	−0.810189	−	0.586169i	$-0.800635\pi$
$181$	−3945.79	−1.62038	−0.810189	+	0.586169i	$0.800635\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3872.26	1.53889
$186$	0	0
$187$	840.459	0.328665
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1446.48	0.547979	0.273989	−	0.961733i	$-0.411657\pi$
$191$	1446.48	0.547979	0.273989	+	0.961733i	$0.411657\pi$
$192$	0	0
$193$	784.523	0.292597	0.146299	−	0.989240i	$-0.453264\pi$
$193$	784.523	0.292597	0.146299	+	0.989240i	$0.453264\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−494.336	−0.178782	−0.0893909	−	0.995997i	$-0.528492\pi$
$197$	−494.336	−0.178782	−0.0893909	+	0.995997i	$0.528492\pi$
$198$	0	0
$199$	3022.59	1.07671	0.538357	−	0.842717i	$-0.319045\pi$
$199$	3022.59	1.07671	0.538357	+	0.842717i	$0.319045\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1706.78	0.590111
$204$	0	0
$205$	−3132.99	−1.06740
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3483.41	−1.15288
$210$	0	0
$211$	−3185.24	−1.03925	−0.519623	−	0.854396i	$-0.673927\pi$
$211$	−3185.24	−1.03925	−0.519623	+	0.854396i	$0.673927\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3104.82	0.984870
$216$	0	0
$217$	1041.54	0.325825
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−401.012	−0.122059
$222$	0	0
$223$	−997.686	−0.299596	−0.149798	−	0.988717i	$-0.547862\pi$
$223$	−997.686	−0.299596	−0.149798	+	0.988717i	$0.547862\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6496.57	−1.89953	−0.949763	−	0.312969i	$-0.898676\pi$
$227$	−6496.57	−1.89953	−0.949763	+	0.312969i	$0.898676\pi$
$228$	0	0
$229$	708.434	0.204431	0.102215	−	0.994762i	$-0.467407\pi$
$229$	708.434	0.204431	0.102215	+	0.994762i	$0.467407\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4919.02	−1.38307	−0.691536	−	0.722342i	$-0.743066\pi$
$233$	−4919.02	−1.38307	−0.691536	+	0.722342i	$0.743066\pi$
$234$	0	0
$235$	−5975.30	−1.65866
$236$	0	0
$237$	0	0
$238$	0	0
$239$	324.370	0.0877898	0.0438949	−	0.999036i	$-0.486023\pi$
$239$	324.370	0.0877898	0.0438949	+	0.999036i	$0.486023\pi$
$240$	0	0
$241$	−1940.43	−0.518649	−0.259324	−	0.965790i	$-0.583500\pi$
$241$	−1940.43	−0.518649	−0.259324	+	0.965790i	$0.583500\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5933.68	1.54730
$246$	0	0
$247$	1662.05	0.428154
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1958.45	−0.492495	−0.246248	−	0.969207i	$-0.579198\pi$
$251$	−1958.45	−0.492495	−0.246248	+	0.969207i	$0.579198\pi$
$252$	0	0
$253$	−2297.38	−0.570889
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4330.35	1.05105	0.525525	−	0.850778i	$-0.323869\pi$
$257$	4330.35	1.05105	0.525525	+	0.850778i	$0.323869\pi$
$258$	0	0
$259$	−1241.78	−0.297916
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2933.60	−0.687807	−0.343904	−	0.939005i	$-0.611749\pi$
$263$	−2933.60	−0.687807	−0.343904	+	0.939005i	$0.611749\pi$
$264$	0	0
$265$	−6969.42	−1.61558
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2458.25	−0.557184	−0.278592	−	0.960410i	$-0.589868\pi$
$269$	−2458.25	−0.557184	−0.278592	+	0.960410i	$0.589868\pi$
$270$	0	0
$271$	−2089.91	−0.468462	−0.234231	−	0.972181i	$-0.575257\pi$
$271$	−2089.91	−0.468462	−0.234231	+	0.972181i	$0.575257\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6990.94	1.53298
$276$	0	0
$277$	−2462.00	−0.534033	−0.267017	−	0.963692i	$-0.586038\pi$
$277$	−2462.00	−0.534033	−0.267017	+	0.963692i	$0.586038\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5775.91	1.22620	0.613100	−	0.790005i	$-0.289922\pi$
$281$	5775.91	1.22620	0.613100	+	0.790005i	$0.289922\pi$
$282$	0	0
$283$	−5093.45	−1.06987	−0.534936	−	0.844892i	$-0.679664\pi$
$283$	−5093.45	−1.06987	−0.534936	+	0.844892i	$0.679664\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1004.70	0.206640
$288$	0	0
$289$	−3961.46	−0.806322
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1503.73	0.299826	0.149913	−	0.988699i	$-0.452101\pi$
$293$	1503.73	0.299826	0.149913	+	0.988699i	$0.452101\pi$
$294$	0	0
$295$	−9188.59	−1.81349
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1096.16	0.212015
$300$	0	0
$301$	−995.670	−0.190663
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3356.28	0.630099
$306$	0	0
$307$	6567.59	1.22095	0.610476	−	0.792035i	$-0.290978\pi$
$307$	6567.59	1.22095	0.610476	+	0.792035i	$0.290978\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5714.66	1.04196	0.520978	−	0.853570i	$-0.325567\pi$
$311$	5714.66	1.04196	0.520978	+	0.853570i	$0.325567\pi$
$312$	0	0
$313$	−4953.59	−0.894548	−0.447274	−	0.894397i	$-0.647605\pi$
$313$	−4953.59	−0.894548	−0.447274	+	0.894397i	$0.647605\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2526.64	0.447666	0.223833	−	0.974628i	$-0.428143\pi$
$317$	2526.64	0.447666	0.223833	+	0.974628i	$0.428143\pi$
$318$	0	0
$319$	7423.44	1.30292
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3943.81	−0.679379
$324$	0	0
$325$	−3335.62	−0.569313
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1916.19	0.321103
$330$	0	0
$331$	−7436.06	−1.23481	−0.617406	−	0.786645i	$-0.711816\pi$
$331$	−7436.06	−1.23481	−0.617406	+	0.786645i	$0.711816\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	19973.6	3.25754
$336$	0	0
$337$	9467.96	1.53042	0.765212	−	0.643778i	$-0.222634\pi$
$337$	9467.96	1.53042	0.765212	+	0.643778i	$0.222634\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4530.04	0.719400
$342$	0	0
$343$	−4051.51	−0.637787
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2794.85	0.432379	0.216189	−	0.976351i	$-0.430637\pi$
$347$	2794.85	0.432379	0.216189	+	0.976351i	$0.430637\pi$
$348$	0	0
$349$	2602.91	0.399228	0.199614	−	0.979875i	$-0.436031\pi$
$349$	2602.91	0.399228	0.199614	+	0.979875i	$0.436031\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8875.63	−1.33825	−0.669125	−	0.743150i	$-0.733331\pi$
$353$	−8875.63	−1.33825	−0.669125	+	0.743150i	$0.733331\pi$
$354$	0	0
$355$	−3684.47	−0.550849
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8946.74	1.31529	0.657647	−	0.753326i	$-0.271552\pi$
$359$	8946.74	1.31529	0.657647	+	0.753326i	$0.271552\pi$
$360$	0	0
$361$	9486.72	1.38310
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−18752.4	−2.68916
$366$	0	0
$367$	3965.30	0.563997	0.281998	−	0.959415i	$-0.409003\pi$
$367$	3965.30	0.563997	0.281998	+	0.959415i	$0.409003\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2234.99	0.312762
$372$	0	0
$373$	−1457.44	−0.202315	−0.101157	−	0.994870i	$-0.532255\pi$
$373$	−1457.44	−0.202315	−0.101157	+	0.994870i	$0.532255\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3541.98	−0.483876
$378$	0	0
$379$	−1991.75	−0.269945	−0.134973	−	0.990849i	$-0.543095\pi$
$379$	−1991.75	−0.269945	−0.134973	+	0.990849i	$0.543095\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2504.45	−0.334130	−0.167065	−	0.985946i	$-0.553429\pi$
$383$	−2504.45	−0.334130	−0.167065	+	0.985946i	$0.553429\pi$
$384$	0	0
$385$	−3334.06	−0.441350
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1448.12	−0.188747	−0.0943735	−	0.995537i	$-0.530085\pi$
$389$	−1448.12	−0.188747	−0.0943735	+	0.995537i	$0.530085\pi$
$390$	0	0
$391$	−2601.02	−0.336417
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−20214.1	−2.57489
$396$	0	0
$397$	−5634.14	−0.712265	−0.356133	−	0.934435i	$-0.615905\pi$
$397$	−5634.14	−0.712265	−0.356133	+	0.934435i	$0.615905\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3648.07	0.454304	0.227152	−	0.973859i	$-0.427058\pi$
$401$	3648.07	0.454304	0.227152	+	0.973859i	$0.427058\pi$
$402$	0	0
$403$	−2161.44	−0.267168
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5400.96	−0.657778
$408$	0	0
$409$	−4459.91	−0.539190	−0.269595	−	0.962974i	$-0.586890\pi$
$409$	−4459.91	−0.539190	−0.269595	+	0.962974i	$0.586890\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2946.64	0.351077
$414$	0	0
$415$	−2058.57	−0.243497
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4726.74	0.551114	0.275557	−	0.961285i	$-0.411138\pi$
$419$	4726.74	0.551114	0.275557	+	0.961285i	$0.411138\pi$
$420$	0	0
$421$	−3090.51	−0.357772	−0.178886	−	0.983870i	$-0.557249\pi$
$421$	−3090.51	−0.357772	−0.178886	+	0.983870i	$0.557249\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7914.92	0.903365
$426$	0	0
$427$	−1076.31	−0.121982
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11131.9	−1.24409	−0.622045	−	0.782981i	$-0.713698\pi$
$431$	−11131.9	−1.24409	−0.622045	+	0.782981i	$0.713698\pi$
$432$	0	0
$433$	6773.33	0.751745	0.375872	−	0.926671i	$-0.377343\pi$
$433$	6773.33	0.751745	0.375872	+	0.926671i	$0.377343\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10780.3	1.18007
$438$	0	0
$439$	5384.95	0.585443	0.292722	−	0.956198i	$-0.405439\pi$
$439$	5384.95	0.585443	0.292722	+	0.956198i	$0.405439\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	921.806	0.0988631	0.0494315	−	0.998778i	$-0.484259\pi$
$443$	921.806	0.0988631	0.0494315	+	0.998778i	$0.484259\pi$
$444$	0	0
$445$	12694.5	1.35231
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11902.6	1.25104	0.625521	−	0.780207i	$-0.284887\pi$
$449$	11902.6	1.25104	0.625521	+	0.780207i	$0.284887\pi$
$450$	0	0
$451$	4369.84	0.456247
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1590.80	0.163907
$456$	0	0
$457$	1420.66	0.145418	0.0727088	−	0.997353i	$-0.476836\pi$
$457$	1420.66	0.145418	0.0727088	+	0.997353i	$0.476836\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1768.23	0.178644	0.0893218	−	0.996003i	$-0.471530\pi$
$461$	1768.23	0.178644	0.0893218	+	0.996003i	$0.471530\pi$
$462$	0	0
$463$	−18637.4	−1.87074	−0.935370	−	0.353670i	$-0.884933\pi$
$463$	−18637.4	−1.87074	−0.935370	+	0.353670i	$0.884933\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9042.01	−0.895962	−0.447981	−	0.894043i	$-0.647857\pi$
$467$	−9042.01	−0.895962	−0.447981	+	0.894043i	$0.647857\pi$
$468$	0	0
$469$	−6405.25	−0.630633
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4330.55	−0.420970
$474$	0	0
$475$	−32804.6	−3.16880
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9547.64	−0.910737	−0.455368	−	0.890303i	$-0.650492\pi$
$479$	−9547.64	−0.910737	−0.455368	+	0.890303i	$0.650492\pi$
$480$	0	0
$481$	2576.98	0.244283
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13825.3	−1.29438
$486$	0	0
$487$	−3692.81	−0.343608	−0.171804	−	0.985131i	$-0.554960\pi$
$487$	−3692.81	−0.343608	−0.171804	+	0.985131i	$0.554960\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3487.77	0.320572	0.160286	−	0.987071i	$-0.448758\pi$
$491$	3487.77	0.320572	0.160286	+	0.987071i	$0.448758\pi$
$492$	0	0
$493$	8404.59	0.767796
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1181.55	0.106640
$498$	0	0
$499$	−15089.0	−1.35366	−0.676830	−	0.736139i	$-0.736647\pi$
$499$	−15089.0	−1.35366	−0.676830	+	0.736139i	$0.736647\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−8815.48	−0.781437	−0.390718	−	0.920510i	$-0.627773\pi$
$503$	−8815.48	−0.781437	−0.390718	+	0.920510i	$0.627773\pi$
$504$	0	0
$505$	−27535.2	−2.42634
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19491.2	1.69731	0.848655	−	0.528946i	$-0.177413\pi$
$509$	19491.2	1.69731	0.848655	+	0.528946i	$0.177413\pi$
$510$	0	0
$511$	6013.61	0.520599
$512$	0	0
$513$	0	0
$514$	0	0
$515$	29684.8	2.53994
$516$	0	0
$517$	8334.24	0.708974
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6128.73	−0.515364	−0.257682	−	0.966230i	$-0.582959\pi$
$521$	−6128.73	−0.515364	−0.257682	+	0.966230i	$0.582959\pi$
$522$	0	0
$523$	22618.6	1.89109	0.945546	−	0.325489i	$-0.105529\pi$
$523$	22618.6	1.89109	0.945546	+	0.325489i	$0.105529\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5128.77	0.423933
$528$	0	0
$529$	−5057.17	−0.415647
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2085.00	−0.169440
$534$	0	0
$535$	38239.6	3.09017
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−8276.19	−0.661374
$540$	0	0
$541$	−15950.5	−1.26759	−0.633796	−	0.773500i	$-0.718504\pi$
$541$	−15950.5	−1.26759	−0.633796	+	0.773500i	$0.718504\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−29645.7	−2.33006
$546$	0	0
$547$	1972.83	0.154208	0.0771042	−	0.997023i	$-0.475433\pi$
$547$	1972.83	0.154208	0.0771042	+	0.997023i	$0.475433\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−34834.1	−2.69325
$552$	0	0
$553$	6482.35	0.498477
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19974.8	−1.51950	−0.759749	−	0.650217i	$-0.774678\pi$
$557$	−19974.8	−1.51950	−0.759749	+	0.650217i	$0.774678\pi$
$558$	0	0
$559$	2066.25	0.156339
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13153.4	0.984637	0.492319	−	0.870415i	$-0.336149\pi$
$563$	13153.4	0.984637	0.492319	+	0.870415i	$0.336149\pi$
$564$	0	0
$565$	42980.5	3.20036
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4133.86	0.304570	0.152285	−	0.988337i	$-0.451337\pi$
$569$	4133.86	0.304570	0.152285	+	0.988337i	$0.451337\pi$
$570$	0	0
$571$	3394.69	0.248798	0.124399	−	0.992232i	$-0.460300\pi$
$571$	3394.69	0.248798	0.124399	+	0.992232i	$0.460300\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−21635.3	−1.56914
$576$	0	0
$577$	4210.56	0.303792	0.151896	−	0.988396i	$-0.451462\pi$
$577$	4210.56	0.303792	0.151896	+	0.988396i	$0.451462\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	660.154	0.0471391
$582$	0	0
$583$	9720.82	0.690558
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7252.27	0.509937	0.254969	−	0.966949i	$-0.417935\pi$
$587$	7252.27	0.509937	0.254969	+	0.966949i	$0.417935\pi$
$588$	0	0
$589$	−21257.0	−1.48706
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−13899.4	−0.962531	−0.481266	−	0.876575i	$-0.659823\pi$
$593$	−13899.4	−0.962531	−0.481266	+	0.876575i	$0.659823\pi$
$594$	0	0
$595$	−3774.72	−0.260082
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18140.7	−1.23741	−0.618704	−	0.785624i	$-0.712342\pi$
$599$	−18140.7	−1.23741	−0.618704	+	0.785624i	$0.712342\pi$
$600$	0	0
$601$	−26808.3	−1.81952	−0.909761	−	0.415133i	$-0.863735\pi$
$601$	−26808.3	−1.81952	−0.909761	+	0.415133i	$0.863735\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11498.9	0.772724
$606$	0	0
$607$	−3769.98	−0.252091	−0.126045	−	0.992024i	$-0.540228\pi$
$607$	−3769.98	−0.252091	−0.126045	+	0.992024i	$0.540228\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3976.55	−0.263297
$612$	0	0
$613$	2722.99	0.179413	0.0897067	−	0.995968i	$-0.471407\pi$
$613$	2722.99	0.179413	0.0897067	+	0.995968i	$0.471407\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11947.1	0.779533	0.389767	−	0.920914i	$-0.372556\pi$
$617$	11947.1	0.779533	0.389767	+	0.920914i	$0.372556\pi$
$618$	0	0
$619$	18386.8	1.19391	0.596953	−	0.802276i	$-0.296378\pi$
$619$	18386.8	1.19391	0.596953	+	0.802276i	$0.296378\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4070.95	−0.261796
$624$	0	0
$625$	18138.1	1.16084
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6114.79	−0.387620
$630$	0	0
$631$	25308.3	1.59669	0.798343	−	0.602202i	$-0.205710\pi$
$631$	25308.3	1.59669	0.798343	+	0.602202i	$0.205710\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	35399.1	2.21224
$636$	0	0
$637$	3948.85	0.245619
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−9670.76	−0.595901	−0.297950	−	0.954581i	$-0.596303\pi$
$641$	−9670.76	−0.595901	−0.297950	+	0.954581i	$0.596303\pi$
$642$	0	0
$643$	−19673.0	−1.20657	−0.603286	−	0.797525i	$-0.706142\pi$
$643$	−19673.0	−1.20657	−0.603286	+	0.797525i	$0.706142\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13369.3	0.812367	0.406183	−	0.913792i	$-0.366859\pi$
$647$	13369.3	0.812367	0.406183	+	0.913792i	$0.366859\pi$
$648$	0	0
$649$	12816.1	0.775154
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3088.52	−0.185089	−0.0925446	−	0.995709i	$-0.529500\pi$
$653$	−3088.52	−0.185089	−0.0925446	+	0.995709i	$0.529500\pi$
$654$	0	0
$655$	19559.8	1.16682
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−31326.8	−1.85177	−0.925886	−	0.377802i	$-0.876680\pi$
$659$	−31326.8	−1.85177	−0.925886	+	0.377802i	$0.876680\pi$
$660$	0	0
$661$	146.828	0.00863986	0.00431993	−	0.999991i	$-0.498625\pi$
$661$	146.828	0.00863986	0.00431993	+	0.999991i	$0.498625\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	15644.9	0.912307
$666$	0	0
$667$	−22973.8	−1.33365
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4681.28	−0.269328
$672$	0	0
$673$	−3463.75	−0.198392	−0.0991961	−	0.995068i	$-0.531627\pi$
$673$	−3463.75	−0.198392	−0.0991961	+	0.995068i	$0.531627\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−534.786	−0.0303597	−0.0151798	−	0.999885i	$-0.504832\pi$
$677$	−534.786	−0.0303597	−0.0151798	+	0.999885i	$0.504832\pi$
$678$	0	0
$679$	4433.56	0.250581
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22369.0	1.25318	0.626592	−	0.779348i	$-0.284449\pi$
$683$	22369.0	1.25318	0.626592	+	0.779348i	$0.284449\pi$
$684$	0	0
$685$	−24302.7	−1.35556
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4638.14	−0.256457
$690$	0	0
$691$	−25007.2	−1.37673	−0.688363	−	0.725366i	$-0.741671\pi$
$691$	−25007.2	−1.37673	−0.688363	+	0.725366i	$0.741671\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	26348.7	1.43808
$696$	0	0
$697$	4947.39	0.268861
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−27977.1	−1.50739	−0.753696	−	0.657223i	$-0.771731\pi$
$701$	−27977.1	−1.50739	−0.753696	+	0.657223i	$0.771731\pi$
$702$	0	0
$703$	25343.7	1.35968
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8830.14	0.469720
$708$	0	0
$709$	6374.31	0.337648	0.168824	−	0.985646i	$-0.446003\pi$
$709$	6374.31	0.337648	0.168824	+	0.985646i	$0.446003\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−14019.4	−0.736367
$714$	0	0
$715$	6918.98	0.361895
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25433.3	1.31919	0.659597	−	0.751619i	$-0.270727\pi$
$719$	25433.3	1.31919	0.659597	+	0.751619i	$0.270727\pi$
$720$	0	0
$721$	−9519.48	−0.491711
$722$	0	0
$723$	0	0
$724$	0	0
$725$	69909.4	3.58120
$726$	0	0
$727$	15847.8	0.808476	0.404238	−	0.914654i	$-0.367537\pi$
$727$	15847.8	0.808476	0.404238	+	0.914654i	$0.367537\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4902.91	−0.248072
$732$	0	0
$733$	9378.34	0.472574	0.236287	−	0.971683i	$-0.424069\pi$
$733$	9378.34	0.472574	0.236287	+	0.971683i	$0.424069\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−27858.9	−1.39239
$738$	0	0
$739$	−12956.2	−0.644928	−0.322464	−	0.946582i	$-0.604511\pi$
$739$	−12956.2	−0.644928	−0.322464	+	0.946582i	$0.604511\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16776.6	−0.828363	−0.414181	−	0.910194i	$-0.635932\pi$
$743$	−16776.6	−0.828363	−0.414181	+	0.910194i	$0.635932\pi$
$744$	0	0
$745$	20347.5	1.00064
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−12262.9	−0.598231
$750$	0	0
$751$	−11213.4	−0.544849	−0.272425	−	0.962177i	$-0.587826\pi$
$751$	−11213.4	−0.544849	−0.272425	+	0.962177i	$0.587826\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	26603.6	1.28239
$756$	0	0
$757$	852.253	0.0409190	0.0204595	−	0.999791i	$-0.493487\pi$
$757$	852.253	0.0409190	0.0204595	+	0.999791i	$0.493487\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35706.1	1.70085	0.850425	−	0.526097i	$-0.176345\pi$
$761$	35706.1	1.70085	0.850425	+	0.526097i	$0.176345\pi$
$762$	0	0
$763$	9506.95	0.451081
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6114.99	−0.287874
$768$	0	0
$769$	−3663.63	−0.171800	−0.0858999	−	0.996304i	$-0.527377\pi$
$769$	−3663.63	−0.171800	−0.0858999	+	0.996304i	$0.527377\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	17985.6	0.836865	0.418432	−	0.908248i	$-0.362580\pi$
$773$	17985.6	0.836865	0.418432	+	0.908248i	$0.362580\pi$
$774$	0	0
$775$	42661.1	1.97733
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−20505.2	−0.943101
$780$	0	0
$781$	5139.03	0.235453
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1697.48	−0.0771793
$786$	0	0
$787$	19322.0	0.875167	0.437584	−	0.899178i	$-0.355834\pi$
$787$	19322.0	0.875167	0.437584	+	0.899178i	$0.355834\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13783.2	−0.619563
$792$	0	0
$793$	2233.60	0.100022
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10689.5	−0.475082	−0.237541	−	0.971378i	$-0.576341\pi$
$797$	−10689.5	−0.475082	−0.237541	+	0.971378i	$0.576341\pi$
$798$	0	0
$799$	9435.77	0.417789
$800$	0	0
$801$	0	0
$802$	0	0
$803$	26155.5	1.14945
$804$	0	0
$805$	10318.1	0.451759
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16756.4	0.728213	0.364107	−	0.931357i	$-0.381374\pi$
$809$	16756.4	0.728213	0.364107	+	0.931357i	$0.381374\pi$
$810$	0	0
$811$	33333.7	1.44329	0.721643	−	0.692266i	$-0.243387\pi$
$811$	33333.7	1.44329	0.721643	+	0.692266i	$0.243387\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−557.479	−0.0239603
$816$	0	0
$817$	20320.9	0.870180
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21242.1	0.902988	0.451494	−	0.892274i	$-0.350891\pi$
$821$	21242.1	0.902988	0.451494	+	0.892274i	$0.350891\pi$
$822$	0	0
$823$	−3994.06	−0.169167	−0.0845834	−	0.996416i	$-0.526956\pi$
$823$	−3994.06	−0.169167	−0.0845834	+	0.996416i	$0.526956\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16952.5	0.712814	0.356407	−	0.934331i	$-0.384002\pi$
$827$	16952.5	0.712814	0.356407	+	0.934331i	$0.384002\pi$
$828$	0	0
$829$	38917.5	1.63047	0.815237	−	0.579128i	$-0.196607\pi$
$829$	38917.5	1.63047	0.815237	+	0.579128i	$0.196607\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9370.04	−0.389739
$834$	0	0
$835$	25196.6	1.04427
$836$	0	0
$837$	0	0
$838$	0	0
$839$	25671.3	1.05634	0.528172	−	0.849138i	$-0.322878\pi$
$839$	25671.3	1.05634	0.528172	+	0.849138i	$0.322878\pi$
$840$	0	0
$841$	49845.4	2.04377
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3301.28	−0.134400
$846$	0	0
$847$	−3687.54	−0.149593
$848$	0	0
$849$	0	0
$850$	0	0
$851$	16714.7	0.673292
$852$	0	0
$853$	−37444.6	−1.50302	−0.751511	−	0.659720i	$-0.770675\pi$
$853$	−37444.6	−1.50302	−0.751511	+	0.659720i	$0.770675\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	36610.6	1.45927	0.729636	−	0.683836i	$-0.239690\pi$
$857$	36610.6	1.45927	0.729636	+	0.683836i	$0.239690\pi$
$858$	0	0
$859$	−18043.5	−0.716688	−0.358344	−	0.933590i	$-0.616659\pi$
$859$	−18043.5	−0.716688	−0.358344	+	0.933590i	$0.616659\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8697.94	−0.343084	−0.171542	−	0.985177i	$-0.554875\pi$
$863$	−8697.94	−0.343084	−0.171542	+	0.985177i	$0.554875\pi$
$864$	0	0
$865$	11788.7	0.463384
$866$	0	0
$867$	0	0
$868$	0	0
$869$	28194.2	1.10060
$870$	0	0
$871$	13292.4	0.517103
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−16102.0	−0.622113
$876$	0	0
$877$	26657.4	1.02640	0.513202	−	0.858268i	$-0.328459\pi$
$877$	26657.4	1.02640	0.513202	+	0.858268i	$0.328459\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−13896.5	−0.531424	−0.265712	−	0.964053i	$-0.585607\pi$
$881$	−13896.5	−0.531424	−0.265712	+	0.964053i	$0.585607\pi$
$882$	0	0
$883$	−24343.3	−0.927767	−0.463884	−	0.885896i	$-0.653544\pi$
$883$	−24343.3	−0.927767	−0.463884	+	0.885896i	$0.653544\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4592.33	0.173839	0.0869196	−	0.996215i	$-0.472298\pi$
$887$	4592.33	0.173839	0.0869196	+	0.996215i	$0.472298\pi$
$888$	0	0
$889$	−11352.0	−0.428271
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−39108.0	−1.46551
$894$	0	0
$895$	38781.1	1.44839
$896$	0	0
$897$	0	0
$898$	0	0
$899$	45300.4	1.68059
$900$	0	0
$901$	11005.6	0.406937
$902$	0	0
$903$	0	0
$904$	0	0
$905$	77078.0	2.83111
$906$	0	0
$907$	−31406.7	−1.14977	−0.574885	−	0.818234i	$-0.694953\pi$
$907$	−31406.7	−1.14977	−0.574885	+	0.818234i	$0.694953\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44825.0	−1.63021	−0.815103	−	0.579317i	$-0.803319\pi$
$911$	−44825.0	−1.63021	−0.815103	+	0.579317i	$0.803319\pi$
$912$	0	0
$913$	2871.26	0.104080
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6272.53	−0.225886
$918$	0	0
$919$	−22241.1	−0.798332	−0.399166	−	0.916879i	$-0.630700\pi$
$919$	−22241.1	−0.798332	−0.399166	+	0.916879i	$0.630700\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2452.01	−0.0874419
$924$	0	0
$925$	−50862.9	−1.80796
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16292.5	−0.575393	−0.287697	−	0.957722i	$-0.592889\pi$
$929$	−16292.5	−0.575393	−0.287697	+	0.957722i	$0.592889\pi$
$930$	0	0
$931$	38835.6	1.36712
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−16417.7	−0.574242
$936$	0	0
$937$	24400.8	0.850734	0.425367	−	0.905021i	$-0.360145\pi$
$937$	24400.8	0.850734	0.425367	+	0.905021i	$0.360145\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−41529.9	−1.43872	−0.719360	−	0.694638i	$-0.755565\pi$
$941$	−41529.9	−1.43872	−0.719360	+	0.694638i	$0.755565\pi$
$942$	0	0
$943$	−13523.6	−0.467008
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29981.8	−1.02880	−0.514402	−	0.857549i	$-0.671986\pi$
$947$	−29981.8	−1.02880	−0.514402	+	0.857549i	$0.671986\pi$
$948$	0	0
$949$	−12479.7	−0.426878
$950$	0	0
$951$	0	0
$952$	0	0
$953$	28594.4	0.971943	0.485972	−	0.873975i	$-0.338466\pi$
$953$	28594.4	0.971943	0.485972	+	0.873975i	$0.338466\pi$
$954$	0	0
$955$	−28256.0	−0.957426
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7793.52	0.262425
$960$	0	0
$961$	−2147.17	−0.0720745
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15325.1	−0.511224
$966$	0	0
$967$	28729.1	0.955393	0.477696	−	0.878525i	$-0.341472\pi$
$967$	28729.1	0.955393	0.477696	+	0.878525i	$0.341472\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−27550.7	−0.910549	−0.455274	−	0.890351i	$-0.650459\pi$
$971$	−27550.7	−0.910549	−0.455274	+	0.890351i	$0.650459\pi$
$972$	0	0
$973$	−8449.64	−0.278400
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23812.0	−0.779746	−0.389873	−	0.920869i	$-0.627481\pi$
$977$	−23812.0	−0.779746	−0.389873	+	0.920869i	$0.627481\pi$
$978$	0	0
$979$	−17706.1	−0.578028
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32668.3	1.05998	0.529988	−	0.848005i	$-0.322196\pi$
$983$	32668.3	1.05998	0.529988	+	0.848005i	$0.322196\pi$
$984$	0	0
$985$	9656.48	0.312367
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13402.0	0.430899
$990$	0	0
$991$	19654.5	0.630015	0.315008	−	0.949089i	$-0.397993\pi$
$991$	19654.5	0.630015	0.315008	+	0.949089i	$0.397993\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−59044.0	−1.88123
$996$	0	0
$997$	17666.5	0.561188	0.280594	−	0.959827i	$-0.409469\pi$
$997$	17666.5	0.561188	0.280594	+	0.959827i	$0.409469\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.bo.1.1		4
3.2	odd	2	inner	1872.4.a.bo.1.4		4
4.3	odd	2		117.4.a.g.1.2	✓	4
12.11	even	2		117.4.a.g.1.3	yes	4
52.51	odd	2		1521.4.a.ba.1.3		4
156.155	even	2		1521.4.a.ba.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.a.g.1.2	✓	4	4.3	odd	2
117.4.a.g.1.3	yes	4	12.11	even	2
1521.4.a.ba.1.2		4	156.155	even	2
1521.4.a.ba.1.3		4	52.51	odd	2
1872.4.a.bo.1.1		4	1.1	even	1	trivial
1872.4.a.bo.1.4		4	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

\(f(q)\)	\(=\)	\(q-19.5342 q^{5} +6.26434 q^{7} +O(q^{10})\)	\(q-19.5342 q^{5} +6.26434 q^{7} +27.2460 q^{11} -13.0000 q^{13} +30.8471 q^{17} -127.850 q^{19} -84.3198 q^{23} +256.586 q^{25} +272.460 q^{29} +166.264 q^{31} -122.369 q^{35} -198.229 q^{37} +160.385 q^{41} -158.943 q^{43} +305.889 q^{47} -303.758 q^{49} +356.780 q^{53} -532.229 q^{55} +470.384 q^{59} -171.815 q^{61} +253.945 q^{65} -1022.49 q^{67} +188.616 q^{71} +959.975 q^{73} +170.678 q^{77} +1034.80 q^{79} +105.383 q^{83} -602.574 q^{85} -649.860 q^{89} -81.4364 q^{91} +2497.46 q^{95} +707.746 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{7}+O(q^{10}) \) 4 * q - 36 * q^7	\( 4 q - 36 q^{7} - 52 q^{13} - 84 q^{19} + 660 q^{25} + 604 q^{31} + 184 q^{37} - 880 q^{43} - 116 q^{49} - 1152 q^{55} + 656 q^{61} - 3052 q^{67} - 312 q^{73} + 720 q^{79} + 32 q^{85} + 468 q^{91} - 344 q^{97}+O(q^{100}) \) 4 * q - 36 * q^7 - 52 * q^13 - 84 * q^19 + 660 * q^25 + 604 * q^31 + 184 * q^37 - 880 * q^43 - 116 * q^49 - 1152 * q^55 + 656 * q^61 - 3052 * q^67 - 312 * q^73 + 720 * q^79 + 32 * q^85 + 468 * q^91 - 344 * q^97

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