LMFDB - Embedded newform 228.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [228,4,Mod(1,228)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(228, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("228.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 228 = 2^{2} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	228.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:	$13.4524354813$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{897}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 224 $ x^2 - x - 224
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			1.2
Root			$15.4750$ of defining polynomial
Character	$\chi$	$=$	228.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +16.4750 q^{5} +6.47498 q^{7} +9.00000 q^{9} -22.4750 q^{11} +26.0000 q^{13} -49.4249 q^{15} +19.5250 q^{17} +19.0000 q^{19} -19.4249 q^{21} -17.8999 q^{23} +146.425 q^{25} -27.0000 q^{27} +179.900 q^{29} +196.750 q^{31} +67.4249 q^{33} +106.675 q^{35} +166.950 q^{37} -78.0000 q^{39} +356.850 q^{41} +42.4750 q^{43} +148.275 q^{45} +214.475 q^{47} -301.075 q^{49} -58.5751 q^{51} +485.800 q^{53} -370.275 q^{55} -57.0000 q^{57} -461.099 q^{59} -368.075 q^{61} +58.2748 q^{63} +428.349 q^{65} -741.900 q^{67} +53.6997 q^{69} +329.099 q^{71} -831.824 q^{73} -439.275 q^{75} -145.525 q^{77} +109.099 q^{79} +81.0000 q^{81} -353.500 q^{83} +321.674 q^{85} -539.700 q^{87} +215.900 q^{89} +168.349 q^{91} -590.249 q^{93} +313.025 q^{95} -962.499 q^{97} -202.275 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 3 q^{5} - 17 q^{7} + 18 q^{9} - 15 q^{11} + 52 q^{13} - 9 q^{15} + 69 q^{17} + 38 q^{19} + 51 q^{21} + 84 q^{23} + 203 q^{25} - 54 q^{27} + 240 q^{29} + 94 q^{31} + 45 q^{33} + 423 q^{35}+ \cdots - 135 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 3 * q^5 - 17 * q^7 + 18 * q^9 - 15 * q^11 + 52 * q^13 - 9 * q^15 + 69 * q^17 + 38 * q^19 + 51 * q^21 + 84 * q^23 + 203 * q^25 - 54 * q^27 + 240 * q^29 + 94 * q^31 + 45 * q^33 + 423 * q^35 + 274 * q^37 - 156 * q^39 + 534 * q^41 + 55 * q^43 + 27 * q^45 + 399 * q^47 - 93 * q^49 - 207 * q^51 + 732 * q^53 - 471 * q^55 - 114 * q^57 + 156 * q^59 - 227 * q^61 - 153 * q^63 + 78 * q^65 - 1364 * q^67 - 252 * q^69 - 420 * q^71 + 343 * q^73 - 609 * q^75 - 321 * q^77 - 860 * q^79 + 162 * q^81 - 108 * q^83 - 345 * q^85 - 720 * q^87 + 312 * q^89 - 442 * q^91 - 282 * q^93 + 57 * q^95 - 128 * q^97 - 135 * q^99

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	16.4750	1.47357	0.736783	−	0.676129i	$-0.236344\pi$
$5$	16.4750	1.47357	0.736783	+	0.676129i	$0.236344\pi$
$6$	0	0
$7$	6.47498	0.349616	0.174808	−	0.984603i	$-0.444070\pi$
$7$	6.47498	0.349616	0.174808	+	0.984603i	$0.444070\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−22.4750	−0.616042	−0.308021	−	0.951380i	$-0.599667\pi$
$11$	−22.4750	−0.616042	−0.308021	+	0.951380i	$0.599667\pi$
$12$	0	0
$13$	26.0000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	26.0000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−49.4249	−0.850764
$16$	0	0
$17$	19.5250	0.278560	0.139280	−	0.990253i	$-0.455521\pi$
$17$	19.5250	0.278560	0.139280	+	0.990253i	$0.455521\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−19.4249	−0.201851
$22$	0	0
$23$	−17.8999	−0.162278	−0.0811389	−	0.996703i	$-0.525856\pi$
$23$	−17.8999	−0.162278	−0.0811389	+	0.996703i	$0.525856\pi$
$24$	0	0
$25$	146.425	1.17140
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	179.900	1.15195	0.575975	−	0.817467i	$-0.304622\pi$
$29$	179.900	1.15195	0.575975	+	0.817467i	$0.304622\pi$
$30$	0	0
$31$	196.750	1.13991	0.569957	−	0.821675i	$-0.306960\pi$
$31$	196.750	1.13991	0.569957	+	0.821675i	$0.306960\pi$
$32$	0	0
$33$	67.4249	0.355672
$34$	0	0
$35$	106.675	0.515183
$36$	0	0
$37$	166.950	0.741795	0.370897	−	0.928674i	$-0.379050\pi$
$37$	166.950	0.741795	0.370897	+	0.928674i	$0.379050\pi$
$38$	0	0
$39$	−78.0000	−0.320256
$40$	0	0
$41$	356.850	1.35928	0.679641	−	0.733545i	$-0.262135\pi$
$41$	356.850	1.35928	0.679641	+	0.733545i	$0.262135\pi$
$42$	0	0
$43$	42.4750	0.150637	0.0753183	−	0.997160i	$-0.476003\pi$
$43$	42.4750	0.150637	0.0753183	+	0.997160i	$0.476003\pi$
$44$	0	0
$45$	148.275	0.491189
$46$	0	0
$47$	214.475	0.665625	0.332813	−	0.942993i	$-0.392002\pi$
$47$	214.475	0.665625	0.332813	+	0.942993i	$0.392002\pi$
$48$	0	0
$49$	−301.075	−0.877769
$50$	0	0
$51$	−58.5751	−0.160826
$52$	0	0
$53$	485.800	1.25905	0.629526	−	0.776980i	$-0.283249\pi$
$53$	485.800	1.25905	0.629526	+	0.776980i	$0.283249\pi$
$54$	0	0
$55$	−370.275	−0.907779
$56$	0	0
$57$	−57.0000	−0.132453
$58$	0	0
$59$	−461.099	−1.01746	−0.508729	−	0.860927i	$-0.669884\pi$
$59$	−461.099	−1.01746	−0.508729	+	0.860927i	$0.669884\pi$
$60$	0	0
$61$	−368.075	−0.772576	−0.386288	−	0.922378i	$-0.626243\pi$
$61$	−368.075	−0.772576	−0.386288	+	0.922378i	$0.626243\pi$
$62$	0	0
$63$	58.2748	0.116539
$64$	0	0
$65$	428.349	0.817388
$66$	0	0
$67$	−741.900	−1.35280	−0.676399	−	0.736535i	$-0.736461\pi$
$67$	−741.900	−1.35280	−0.676399	+	0.736535i	$0.736461\pi$
$68$	0	0
$69$	53.6997	0.0936912
$70$	0	0
$71$	329.099	0.550097	0.275049	−	0.961430i	$-0.411306\pi$
$71$	329.099	0.550097	0.275049	+	0.961430i	$0.411306\pi$
$72$	0	0
$73$	−831.824	−1.33367	−0.666833	−	0.745207i	$-0.732350\pi$
$73$	−831.824	−1.33367	−0.666833	+	0.745207i	$0.732350\pi$
$74$	0	0
$75$	−439.275	−0.676308
$76$	0	0
$77$	−145.525	−0.215378
$78$	0	0
$79$	109.099	0.155375	0.0776875	−	0.996978i	$-0.475246\pi$
$79$	109.099	0.155375	0.0776875	+	0.996978i	$0.475246\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−353.500	−0.467489	−0.233745	−	0.972298i	$-0.575098\pi$
$83$	−353.500	−0.467489	−0.233745	+	0.972298i	$0.575098\pi$
$84$	0	0
$85$	321.674	0.410476
$86$	0	0
$87$	−539.700	−0.665079
$88$	0	0
$89$	215.900	0.257139	0.128569	−	0.991701i	$-0.458962\pi$
$89$	215.900	0.257139	0.128569	+	0.991701i	$0.458962\pi$
$90$	0	0
$91$	168.349	0.193932
$92$	0	0
$93$	−590.249	−0.658129
$94$	0	0
$95$	313.025	0.338059
$96$	0	0
$97$	−962.499	−1.00749	−0.503747	−	0.863851i	$-0.668046\pi$
$97$	−962.499	−1.00749	−0.503747	+	0.863851i	$0.668046\pi$
$98$	0	0
$99$	−202.275	−0.205347
$100$	0	0
$101$	1097.20	1.08094	0.540472	−	0.841362i	$-0.318246\pi$
$101$	1097.20	1.08094	0.540472	+	0.841362i	$0.318246\pi$
$102$	0	0
$103$	632.848	0.605402	0.302701	−	0.953086i	$-0.402112\pi$
$103$	632.848	0.605402	0.302701	+	0.953086i	$0.402112\pi$
$104$	0	0
$105$	−320.025	−0.297441
$106$	0	0
$107$	1226.45	1.10809	0.554044	−	0.832488i	$-0.313084\pi$
$107$	1226.45	1.10809	0.554044	+	0.832488i	$0.313084\pi$
$108$	0	0
$109$	484.249	0.425529	0.212765	−	0.977103i	$-0.431753\pi$
$109$	484.249	0.425529	0.212765	+	0.977103i	$0.431753\pi$
$110$	0	0
$111$	−500.850	−0.428275
$112$	0	0
$113$	−1305.10	−1.08649	−0.543244	−	0.839575i	$-0.682804\pi$
$113$	−1305.10	−1.08649	−0.543244	+	0.839575i	$0.682804\pi$
$114$	0	0
$115$	−294.901	−0.239127
$116$	0	0
$117$	234.000	0.184900
$118$	0	0
$119$	126.424	0.0973889
$120$	0	0
$121$	−825.875	−0.620492
$122$	0	0
$123$	−1070.55	−0.784782
$124$	0	0
$125$	352.975	0.252569
$126$	0	0
$127$	−1415.35	−0.988912	−0.494456	−	0.869203i	$-0.664633\pi$
$127$	−1415.35	−0.988912	−0.494456	+	0.869203i	$0.664633\pi$
$128$	0	0
$129$	−127.425	−0.0869701
$130$	0	0
$131$	−2913.72	−1.94331	−0.971653	−	0.236412i	$-0.924028\pi$
$131$	−2913.72	−1.94331	−0.971653	+	0.236412i	$0.924028\pi$
$132$	0	0
$133$	123.025	0.0802074
$134$	0	0
$135$	−444.824	−0.283588
$136$	0	0
$137$	1421.72	0.886613	0.443307	−	0.896370i	$-0.353805\pi$
$137$	1421.72	0.886613	0.443307	+	0.896370i	$0.353805\pi$
$138$	0	0
$139$	−2362.02	−1.44133	−0.720663	−	0.693286i	$-0.756162\pi$
$139$	−2362.02	−1.44133	−0.720663	+	0.693286i	$0.756162\pi$
$140$	0	0
$141$	−643.425	−0.384299
$142$	0	0
$143$	−584.349	−0.341719
$144$	0	0
$145$	2963.85	1.69748
$146$	0	0
$147$	903.224	0.506780
$148$	0	0
$149$	−883.172	−0.485586	−0.242793	−	0.970078i	$-0.578064\pi$
$149$	−883.172	−0.485586	−0.242793	+	0.970078i	$0.578064\pi$
$150$	0	0
$151$	325.099	0.175207	0.0876033	−	0.996155i	$-0.472079\pi$
$151$	325.099	0.175207	0.0876033	+	0.996155i	$0.472079\pi$
$152$	0	0
$153$	175.725	0.0928532
$154$	0	0
$155$	3241.45	1.67974
$156$	0	0
$157$	2209.00	1.12291	0.561456	−	0.827506i	$-0.310241\pi$
$157$	2209.00	1.12291	0.561456	+	0.827506i	$0.310241\pi$
$158$	0	0
$159$	−1457.40	−0.726914
$160$	0	0
$161$	−115.902	−0.0567349
$162$	0	0
$163$	−1157.70	−0.556305	−0.278153	−	0.960537i	$-0.589722\pi$
$163$	−1157.70	−0.556305	−0.278153	+	0.960537i	$0.589722\pi$
$164$	0	0
$165$	1110.82	0.524106
$166$	0	0
$167$	−271.949	−0.126012	−0.0630062	−	0.998013i	$-0.520069\pi$
$167$	−271.949	−0.126012	−0.0630062	+	0.998013i	$0.520069\pi$
$168$	0	0
$169$	−1521.00	−0.692308
$170$	0	0
$171$	171.000	0.0764719
$172$	0	0
$173$	1725.70	0.758395	0.379198	−	0.925316i	$-0.376200\pi$
$173$	1725.70	0.758395	0.379198	+	0.925316i	$0.376200\pi$
$174$	0	0
$175$	948.098	0.409540
$176$	0	0
$177$	1383.30	0.587430
$178$	0	0
$179$	1477.85	0.617093	0.308546	−	0.951209i	$-0.400157\pi$
$179$	1477.85	0.617093	0.308546	+	0.951209i	$0.400157\pi$
$180$	0	0
$181$	−1855.00	−0.761773	−0.380886	−	0.924622i	$-0.624381\pi$
$181$	−1855.00	−0.761773	−0.380886	+	0.924622i	$0.624381\pi$
$182$	0	0
$183$	1104.22	0.446047
$184$	0	0
$185$	2750.50	1.09308
$186$	0	0
$187$	−438.824	−0.171604
$188$	0	0
$189$	−174.824	−0.0672836
$190$	0	0
$191$	706.523	0.267655	0.133828	−	0.991005i	$-0.457273\pi$
$191$	706.523	0.267655	0.133828	+	0.991005i	$0.457273\pi$
$192$	0	0
$193$	2418.45	0.901989	0.450995	−	0.892527i	$-0.351069\pi$
$193$	2418.45	0.901989	0.450995	+	0.892527i	$0.351069\pi$
$194$	0	0
$195$	−1285.05	−0.471919
$196$	0	0
$197$	86.3019	0.0312120	0.0156060	−	0.999878i	$-0.495032\pi$
$197$	86.3019	0.0312120	0.0156060	+	0.999878i	$0.495032\pi$
$198$	0	0
$199$	−1371.32	−0.488495	−0.244248	−	0.969713i	$-0.578541\pi$
$199$	−1371.32	−0.488495	−0.244248	+	0.969713i	$0.578541\pi$
$200$	0	0
$201$	2225.70	0.781039
$202$	0	0
$203$	1164.85	0.402740
$204$	0	0
$205$	5879.09	2.00299
$206$	0	0
$207$	−161.099	−0.0540926
$208$	0	0
$209$	−427.025	−0.141330
$210$	0	0
$211$	−2099.80	−0.685099	−0.342550	−	0.939500i	$-0.611290\pi$
$211$	−2099.80	−0.685099	−0.342550	+	0.939500i	$0.611290\pi$
$212$	0	0
$213$	−987.298	−0.317599
$214$	0	0
$215$	699.774	0.221973
$216$	0	0
$217$	1273.95	0.398532
$218$	0	0
$219$	2495.47	0.769992
$220$	0	0
$221$	507.651	0.154517
$222$	0	0
$223$	−4160.59	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$223$	−4160.59	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$224$	0	0
$225$	1317.82	0.390466
$226$	0	0
$227$	3743.55	1.09457	0.547287	−	0.836945i	$-0.315661\pi$
$227$	3743.55	1.09457	0.547287	+	0.836945i	$0.315661\pi$
$228$	0	0
$229$	920.777	0.265706	0.132853	−	0.991136i	$-0.457586\pi$
$229$	920.777	0.265706	0.132853	+	0.991136i	$0.457586\pi$
$230$	0	0
$231$	436.575	0.124349
$232$	0	0
$233$	−3756.02	−1.05607	−0.528037	−	0.849221i	$-0.677072\pi$
$233$	−3756.02	−1.05607	−0.528037	+	0.849221i	$0.677072\pi$
$234$	0	0
$235$	3533.47	0.980843
$236$	0	0
$237$	−327.298	−0.0897058
$238$	0	0
$239$	−698.573	−0.189067	−0.0945334	−	0.995522i	$-0.530136\pi$
$239$	−698.573	−0.189067	−0.0945334	+	0.995522i	$0.530136\pi$
$240$	0	0
$241$	−5183.14	−1.38538	−0.692688	−	0.721237i	$-0.743574\pi$
$241$	−5183.14	−1.38538	−0.692688	+	0.721237i	$0.743574\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−4960.20	−1.29345
$246$	0	0
$247$	494.000	0.127257
$248$	0	0
$249$	1060.50	0.269905
$250$	0	0
$251$	−1589.13	−0.399621	−0.199811	−	0.979835i	$-0.564033\pi$
$251$	−1589.13	−0.399621	−0.199811	+	0.979835i	$0.564033\pi$
$252$	0	0
$253$	402.300	0.0999700
$254$	0	0
$255$	−965.023	−0.236989
$256$	0	0
$257$	−4199.95	−1.01940	−0.509699	−	0.860353i	$-0.670243\pi$
$257$	−4199.95	−1.01940	−0.509699	+	0.860353i	$0.670243\pi$
$258$	0	0
$259$	1081.00	0.259343
$260$	0	0
$261$	1619.10	0.383984
$262$	0	0
$263$	3145.07	0.737388	0.368694	−	0.929551i	$-0.379805\pi$
$263$	3145.07	0.737388	0.368694	+	0.929551i	$0.379805\pi$
$264$	0	0
$265$	8003.54	1.85530
$266$	0	0
$267$	−647.700	−0.148459
$268$	0	0
$269$	−1643.79	−0.372580	−0.186290	−	0.982495i	$-0.559646\pi$
$269$	−1643.79	−0.372580	−0.186290	+	0.982495i	$0.559646\pi$
$270$	0	0
$271$	2060.30	0.461823	0.230911	−	0.972975i	$-0.425829\pi$
$271$	2060.30	0.461823	0.230911	+	0.972975i	$0.425829\pi$
$272$	0	0
$273$	−505.048	−0.111967
$274$	0	0
$275$	−3290.90	−0.721631
$276$	0	0
$277$	8093.67	1.75560	0.877800	−	0.479026i	$-0.159010\pi$
$277$	8093.67	1.75560	0.877800	+	0.479026i	$0.159010\pi$
$278$	0	0
$279$	1770.75	0.379971
$280$	0	0
$281$	−4415.30	−0.937348	−0.468674	−	0.883371i	$-0.655268\pi$
$281$	−4415.30	−0.937348	−0.468674	+	0.883371i	$0.655268\pi$
$282$	0	0
$283$	3866.37	0.812126	0.406063	−	0.913845i	$-0.366901\pi$
$283$	3866.37	0.812126	0.406063	+	0.913845i	$0.366901\pi$
$284$	0	0
$285$	−939.074	−0.195179
$286$	0	0
$287$	2310.60	0.475227
$288$	0	0
$289$	−4531.77	−0.922405
$290$	0	0
$291$	2887.50	0.581677
$292$	0	0
$293$	−9959.75	−1.98585	−0.992926	−	0.118735i	$-0.962116\pi$
$293$	−9959.75	−1.98585	−0.992926	+	0.118735i	$0.962116\pi$
$294$	0	0
$295$	−7596.60	−1.49929
$296$	0	0
$297$	606.824	0.118557
$298$	0	0
$299$	−465.398	−0.0900156
$300$	0	0
$301$	275.025	0.0526650
$302$	0	0
$303$	−3291.60	−0.624084
$304$	0	0
$305$	−6064.02	−1.13844
$306$	0	0
$307$	9774.20	1.81708	0.908539	−	0.417800i	$-0.137199\pi$
$307$	9774.20	1.81708	0.908539	+	0.417800i	$0.137199\pi$
$308$	0	0
$309$	−1898.54	−0.349529
$310$	0	0
$311$	−8660.82	−1.57913	−0.789566	−	0.613666i	$-0.789694\pi$
$311$	−8660.82	−1.57913	−0.789566	+	0.613666i	$0.789694\pi$
$312$	0	0
$313$	−2733.10	−0.493560	−0.246780	−	0.969072i	$-0.579372\pi$
$313$	−2733.10	−0.493560	−0.246780	+	0.969072i	$0.579372\pi$
$314$	0	0
$315$	960.076	0.171728
$316$	0	0
$317$	−212.755	−0.0376956	−0.0188478	−	0.999822i	$-0.506000\pi$
$317$	−212.755	−0.0376956	−0.0188478	+	0.999822i	$0.506000\pi$
$318$	0	0
$319$	−4043.25	−0.709650
$320$	0	0
$321$	−3679.35	−0.639755
$322$	0	0
$323$	370.975	0.0639060
$324$	0	0
$325$	3807.05	0.649776
$326$	0	0
$327$	−1452.75	−0.245679
$328$	0	0
$329$	1388.72	0.232713
$330$	0	0
$331$	−7184.19	−1.19299	−0.596494	−	0.802618i	$-0.703440\pi$
$331$	−7184.19	−1.19299	−0.596494	+	0.802618i	$0.703440\pi$
$332$	0	0
$333$	1502.55	0.247265
$334$	0	0
$335$	−12222.8	−1.99344
$336$	0	0
$337$	−11190.8	−1.80891	−0.904453	−	0.426574i	$-0.859720\pi$
$337$	−11190.8	−1.80891	−0.904453	+	0.426574i	$0.859720\pi$
$338$	0	0
$339$	3915.29	0.627285
$340$	0	0
$341$	−4421.95	−0.702234
$342$	0	0
$343$	−4170.37	−0.656498
$344$	0	0
$345$	884.702	0.138060
$346$	0	0
$347$	7770.07	1.20207	0.601036	−	0.799222i	$-0.294755\pi$
$347$	7770.07	1.20207	0.601036	+	0.799222i	$0.294755\pi$
$348$	0	0
$349$	5313.37	0.814951	0.407476	−	0.913216i	$-0.366409\pi$
$349$	5313.37	0.814951	0.407476	+	0.913216i	$0.366409\pi$
$350$	0	0
$351$	−702.000	−0.106752
$352$	0	0
$353$	6873.29	1.03634	0.518170	−	0.855277i	$-0.326613\pi$
$353$	6873.29	1.03634	0.518170	+	0.855277i	$0.326613\pi$
$354$	0	0
$355$	5421.90	0.810605
$356$	0	0
$357$	−379.272	−0.0562275
$358$	0	0
$359$	−3343.38	−0.491523	−0.245761	−	0.969330i	$-0.579038\pi$
$359$	−3343.38	−0.491523	−0.245761	+	0.969330i	$0.579038\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	2477.63	0.358241
$364$	0	0
$365$	−13704.3	−1.96525
$366$	0	0
$367$	−12129.8	−1.72526	−0.862629	−	0.505837i	$-0.831184\pi$
$367$	−12129.8	−1.72526	−0.862629	+	0.505837i	$0.831184\pi$
$368$	0	0
$369$	3211.65	0.453094
$370$	0	0
$371$	3145.54	0.440185
$372$	0	0
$373$	−4017.25	−0.557655	−0.278827	−	0.960341i	$-0.589946\pi$
$373$	−4017.25	−0.557655	−0.278827	+	0.960341i	$0.589946\pi$
$374$	0	0
$375$	−1058.93	−0.145821
$376$	0	0
$377$	4677.40	0.638987
$378$	0	0
$379$	14104.7	1.91164	0.955820	−	0.293951i	$-0.0949704\pi$
$379$	14104.7	1.91164	0.955820	+	0.293951i	$0.0949704\pi$
$380$	0	0
$381$	4246.05	0.570949
$382$	0	0
$383$	10357.3	1.38182	0.690908	−	0.722943i	$-0.257211\pi$
$383$	10357.3	1.38182	0.690908	+	0.722943i	$0.257211\pi$
$384$	0	0
$385$	−2397.52	−0.317374
$386$	0	0
$387$	382.275	0.0502122
$388$	0	0
$389$	−7931.13	−1.03374	−0.516869	−	0.856064i	$-0.672903\pi$
$389$	−7931.13	−1.03374	−0.516869	+	0.856064i	$0.672903\pi$
$390$	0	0
$391$	−349.496	−0.0452041
$392$	0	0
$393$	8741.17	1.12197
$394$	0	0
$395$	1797.41	0.228955
$396$	0	0
$397$	14121.2	1.78519	0.892595	−	0.450859i	$-0.148882\pi$
$397$	14121.2	1.78519	0.892595	+	0.450859i	$0.148882\pi$
$398$	0	0
$399$	−369.074	−0.0463078
$400$	0	0
$401$	7142.94	0.889530	0.444765	−	0.895647i	$-0.353287\pi$
$401$	7142.94	0.889530	0.444765	+	0.895647i	$0.353287\pi$
$402$	0	0
$403$	5115.49	0.632310
$404$	0	0
$405$	1334.47	0.163730
$406$	0	0
$407$	−3752.20	−0.456977
$408$	0	0
$409$	−15385.6	−1.86007	−0.930034	−	0.367473i	$-0.880223\pi$
$409$	−15385.6	−1.86007	−0.930034	+	0.367473i	$0.880223\pi$
$410$	0	0
$411$	−4265.17	−0.511886
$412$	0	0
$413$	−2985.61	−0.355720
$414$	0	0
$415$	−5823.90	−0.688877
$416$	0	0
$417$	7086.07	0.832150
$418$	0	0
$419$	15122.9	1.76325	0.881625	−	0.471950i	$-0.156450\pi$
$419$	15122.9	1.76325	0.881625	+	0.471950i	$0.156450\pi$
$420$	0	0
$421$	−3903.99	−0.451946	−0.225973	−	0.974134i	$-0.572556\pi$
$421$	−3903.99	−0.451946	−0.225973	+	0.974134i	$0.572556\pi$
$422$	0	0
$423$	1930.27	0.221875
$424$	0	0
$425$	2858.95	0.326305
$426$	0	0
$427$	−2383.28	−0.270105
$428$	0	0
$429$	1753.05	0.197291
$430$	0	0
$431$	12165.5	1.35961	0.679807	−	0.733391i	$-0.262064\pi$
$431$	12165.5	1.35961	0.679807	+	0.733391i	$0.262064\pi$
$432$	0	0
$433$	913.905	0.101431	0.0507153	−	0.998713i	$-0.483850\pi$
$433$	913.905	0.101431	0.0507153	+	0.998713i	$0.483850\pi$
$434$	0	0
$435$	−8891.54	−0.980039
$436$	0	0
$437$	−340.098	−0.0372291
$438$	0	0
$439$	12892.5	1.40166	0.700828	−	0.713330i	$-0.252814\pi$
$439$	12892.5	1.40166	0.700828	+	0.713330i	$0.252814\pi$
$440$	0	0
$441$	−2709.67	−0.292590
$442$	0	0
$443$	−1227.01	−0.131596	−0.0657980	−	0.997833i	$-0.520959\pi$
$443$	−1227.01	−0.131596	−0.0657980	+	0.997833i	$0.520959\pi$
$444$	0	0
$445$	3556.95	0.378911
$446$	0	0
$447$	2649.52	0.280353
$448$	0	0
$449$	−10804.2	−1.13560	−0.567799	−	0.823167i	$-0.692205\pi$
$449$	−10804.2	−1.13560	−0.567799	+	0.823167i	$0.692205\pi$
$450$	0	0
$451$	−8020.19	−0.837375
$452$	0	0
$453$	−975.298	−0.101156
$454$	0	0
$455$	2773.55	0.285772
$456$	0	0
$457$	−6196.27	−0.634244	−0.317122	−	0.948385i	$-0.602716\pi$
$457$	−6196.27	−0.634244	−0.317122	+	0.948385i	$0.602716\pi$
$458$	0	0
$459$	−527.176	−0.0536088
$460$	0	0
$461$	−9333.36	−0.942946	−0.471473	−	0.881880i	$-0.656278\pi$
$461$	−9333.36	−0.942946	−0.471473	+	0.881880i	$0.656278\pi$
$462$	0	0
$463$	3356.22	0.336883	0.168441	−	0.985712i	$-0.446127\pi$
$463$	3356.22	0.336883	0.168441	+	0.985712i	$0.446127\pi$
$464$	0	0
$465$	−9724.35	−0.969797
$466$	0	0
$467$	18217.0	1.80510	0.902549	−	0.430587i	$-0.141693\pi$
$467$	18217.0	1.80510	0.902549	+	0.430587i	$0.141693\pi$
$468$	0	0
$469$	−4803.79	−0.472960
$470$	0	0
$471$	−6627.00	−0.648314
$472$	0	0
$473$	−954.624	−0.0927985
$474$	0	0
$475$	2782.07	0.268737
$476$	0	0
$477$	4372.20	0.419684
$478$	0	0
$479$	−8736.59	−0.833372	−0.416686	−	0.909051i	$-0.636808\pi$
$479$	−8736.59	−0.833372	−0.416686	+	0.909051i	$0.636808\pi$
$480$	0	0
$481$	4340.70	0.411474
$482$	0	0
$483$	347.705	0.0327559
$484$	0	0
$485$	−15857.1	−1.48461
$486$	0	0
$487$	6001.70	0.558446	0.279223	−	0.960226i	$-0.409923\pi$
$487$	6001.70	0.558446	0.279223	+	0.960226i	$0.409923\pi$
$488$	0	0
$489$	3473.09	0.321183
$490$	0	0
$491$	8182.99	0.752125	0.376062	−	0.926594i	$-0.377278\pi$
$491$	8182.99	0.752125	0.376062	+	0.926594i	$0.377278\pi$
$492$	0	0
$493$	3512.55	0.320887
$494$	0	0
$495$	−3332.47	−0.302593
$496$	0	0
$497$	2130.91	0.192323
$498$	0	0
$499$	17960.8	1.61130	0.805648	−	0.592395i	$-0.201818\pi$
$499$	17960.8	1.61130	0.805648	+	0.592395i	$0.201818\pi$
$500$	0	0
$501$	815.847	0.0727532
$502$	0	0
$503$	−2360.60	−0.209252	−0.104626	−	0.994512i	$-0.533365\pi$
$503$	−2360.60	−0.209252	−0.104626	+	0.994512i	$0.533365\pi$
$504$	0	0
$505$	18076.3	1.59284
$506$	0	0
$507$	4563.00	0.399704
$508$	0	0
$509$	9204.70	0.801554	0.400777	−	0.916176i	$-0.368740\pi$
$509$	9204.70	0.801554	0.400777	+	0.916176i	$0.368740\pi$
$510$	0	0
$511$	−5386.04	−0.466271
$512$	0	0
$513$	−513.000	−0.0441511
$514$	0	0
$515$	10426.2	0.892100
$516$	0	0
$517$	−4820.32	−0.410053
$518$	0	0
$519$	−5177.09	−0.437860
$520$	0	0
$521$	2188.74	0.184051	0.0920254	−	0.995757i	$-0.470666\pi$
$521$	2188.74	0.184051	0.0920254	+	0.995757i	$0.470666\pi$
$522$	0	0
$523$	8302.44	0.694150	0.347075	−	0.937837i	$-0.387175\pi$
$523$	8302.44	0.694150	0.347075	+	0.937837i	$0.387175\pi$
$524$	0	0
$525$	−2844.30	−0.236448
$526$	0	0
$527$	3841.54	0.317534
$528$	0	0
$529$	−11846.6	−0.973666
$530$	0	0
$531$	−4149.89	−0.339153
$532$	0	0
$533$	9278.10	0.753994
$534$	0	0
$535$	20205.7	1.63284
$536$	0	0
$537$	−4433.55	−0.356279
$538$	0	0
$539$	6766.65	0.540742
$540$	0	0
$541$	−17308.0	−1.37547	−0.687734	−	0.725963i	$-0.741394\pi$
$541$	−17308.0	−1.37547	−0.687734	+	0.725963i	$0.741394\pi$
$542$	0	0
$543$	5564.99	0.439810
$544$	0	0
$545$	7978.00	0.627046
$546$	0	0
$547$	879.254	0.0687279	0.0343640	−	0.999409i	$-0.489059\pi$
$547$	879.254	0.0687279	0.0343640	+	0.999409i	$0.489059\pi$
$548$	0	0
$549$	−3312.67	−0.257525
$550$	0	0
$551$	3418.10	0.264276
$552$	0	0
$553$	706.415	0.0543216
$554$	0	0
$555$	−8251.49	−0.631092
$556$	0	0
$557$	18648.1	1.41857	0.709287	−	0.704919i	$-0.249017\pi$
$557$	18648.1	1.41857	0.709287	+	0.704919i	$0.249017\pi$
$558$	0	0
$559$	1104.35	0.0835581
$560$	0	0
$561$	1316.47	0.0990758
$562$	0	0
$563$	9300.78	0.696237	0.348118	−	0.937451i	$-0.386821\pi$
$563$	9300.78	0.696237	0.348118	+	0.937451i	$0.386821\pi$
$564$	0	0
$565$	−21501.5	−1.60101
$566$	0	0
$567$	524.473	0.0388462
$568$	0	0
$569$	18660.2	1.37483	0.687413	−	0.726267i	$-0.258746\pi$
$569$	18660.2	1.37483	0.687413	+	0.726267i	$0.258746\pi$
$570$	0	0
$571$	3844.50	0.281764	0.140882	−	0.990026i	$-0.455006\pi$
$571$	3844.50	0.281764	0.140882	+	0.990026i	$0.455006\pi$
$572$	0	0
$573$	−2119.57	−0.154531
$574$	0	0
$575$	−2620.99	−0.190092
$576$	0	0
$577$	22136.5	1.59715	0.798575	−	0.601896i	$-0.205588\pi$
$577$	22136.5	1.59715	0.798575	+	0.601896i	$0.205588\pi$
$578$	0	0
$579$	−7255.35	−0.520764
$580$	0	0
$581$	−2288.90	−0.163442
$582$	0	0
$583$	−10918.3	−0.775629
$584$	0	0
$585$	3855.15	0.272463
$586$	0	0
$587$	−19056.1	−1.33991	−0.669957	−	0.742400i	$-0.733688\pi$
$587$	−19056.1	−1.33991	−0.669957	+	0.742400i	$0.733688\pi$
$588$	0	0
$589$	3738.25	0.261514
$590$	0	0
$591$	−258.906	−0.0180202
$592$	0	0
$593$	−19393.0	−1.34296	−0.671479	−	0.741023i	$-0.734341\pi$
$593$	−19393.0	−1.34296	−0.671479	+	0.741023i	$0.734341\pi$
$594$	0	0
$595$	2082.83	0.143509
$596$	0	0
$597$	4113.97	0.282033
$598$	0	0
$599$	9171.58	0.625611	0.312805	−	0.949817i	$-0.398731\pi$
$599$	9171.58	0.625611	0.312805	+	0.949817i	$0.398731\pi$
$600$	0	0
$601$	−4335.06	−0.294228	−0.147114	−	0.989120i	$-0.546998\pi$
$601$	−4335.06	−0.294228	−0.147114	+	0.989120i	$0.546998\pi$
$602$	0	0
$603$	−6677.10	−0.450933
$604$	0	0
$605$	−13606.3	−0.914337
$606$	0	0
$607$	−7349.45	−0.491441	−0.245721	−	0.969341i	$-0.579025\pi$
$607$	−7349.45	−0.491441	−0.245721	+	0.969341i	$0.579025\pi$
$608$	0	0
$609$	−3494.54	−0.232522
$610$	0	0
$611$	5576.35	0.369223
$612$	0	0
$613$	−7060.20	−0.465186	−0.232593	−	0.972574i	$-0.574721\pi$
$613$	−7060.20	−0.465186	−0.232593	+	0.972574i	$0.574721\pi$
$614$	0	0
$615$	−17637.3	−1.15643
$616$	0	0
$617$	−400.580	−0.0261374	−0.0130687	−	0.999915i	$-0.504160\pi$
$617$	−400.580	−0.0261374	−0.0130687	+	0.999915i	$0.504160\pi$
$618$	0	0
$619$	−19511.6	−1.26694	−0.633470	−	0.773767i	$-0.718370\pi$
$619$	−19511.6	−1.26694	−0.633470	+	0.773767i	$0.718370\pi$
$620$	0	0
$621$	483.298	0.0312304
$622$	0	0
$623$	1397.95	0.0898998
$624$	0	0
$625$	−12487.9	−0.799223
$626$	0	0
$627$	1281.07	0.0815967
$628$	0	0
$629$	3259.70	0.206634
$630$	0	0
$631$	−16140.2	−1.01828	−0.509138	−	0.860685i	$-0.670036\pi$
$631$	−16140.2	−1.01828	−0.509138	+	0.860685i	$0.670036\pi$
$632$	0	0
$633$	6299.39	0.395542
$634$	0	0
$635$	−23317.8	−1.45723
$636$	0	0
$637$	−7827.94	−0.486898
$638$	0	0
$639$	2961.89	0.183366
$640$	0	0
$641$	−1534.50	−0.0945539	−0.0472770	−	0.998882i	$-0.515054\pi$
$641$	−1534.50	−0.0945539	−0.0472770	+	0.998882i	$0.515054\pi$
$642$	0	0
$643$	−24603.9	−1.50899	−0.754497	−	0.656303i	$-0.772119\pi$
$643$	−24603.9	−1.50899	−0.754497	+	0.656303i	$0.772119\pi$
$644$	0	0
$645$	−2099.32	−0.128156
$646$	0	0
$647$	30583.9	1.85839	0.929194	−	0.369591i	$-0.120502\pi$
$647$	30583.9	1.85839	0.929194	+	0.369591i	$0.120502\pi$
$648$	0	0
$649$	10363.2	0.626797
$650$	0	0
$651$	−3821.85	−0.230092
$652$	0	0
$653$	15150.4	0.907935	0.453967	−	0.891018i	$-0.350008\pi$
$653$	15150.4	0.907935	0.453967	+	0.891018i	$0.350008\pi$
$654$	0	0
$655$	−48003.5	−2.86359
$656$	0	0
$657$	−7486.41	−0.444555
$658$	0	0
$659$	−3451.45	−0.204021	−0.102010	−	0.994783i	$-0.532527\pi$
$659$	−3451.45	−0.204021	−0.102010	+	0.994783i	$0.532527\pi$
$660$	0	0
$661$	−1542.56	−0.0907695	−0.0453847	−	0.998970i	$-0.514451\pi$
$661$	−1542.56	−0.0907695	−0.0453847	+	0.998970i	$0.514451\pi$
$662$	0	0
$663$	−1522.95	−0.0892105
$664$	0	0
$665$	2026.83	0.118191
$666$	0	0
$667$	−3220.19	−0.186936
$668$	0	0
$669$	12481.8	0.721336
$670$	0	0
$671$	8272.47	0.475939
$672$	0	0
$673$	−1881.27	−0.107753	−0.0538763	−	0.998548i	$-0.517158\pi$
$673$	−1881.27	−0.107753	−0.0538763	+	0.998548i	$0.517158\pi$
$674$	0	0
$675$	−3953.47	−0.225436
$676$	0	0
$677$	−1483.42	−0.0842135	−0.0421067	−	0.999113i	$-0.513407\pi$
$677$	−1483.42	−0.0842135	−0.0421067	+	0.999113i	$0.513407\pi$
$678$	0	0
$679$	−6232.16	−0.352236
$680$	0	0
$681$	−11230.7	−0.631953
$682$	0	0
$683$	6629.35	0.371398	0.185699	−	0.982607i	$-0.440545\pi$
$683$	6629.35	0.371398	0.185699	+	0.982607i	$0.440545\pi$
$684$	0	0
$685$	23422.9	1.30648
$686$	0	0
$687$	−2762.33	−0.153405
$688$	0	0
$689$	12630.8	0.698396
$690$	0	0
$691$	−21856.7	−1.20328	−0.601641	−	0.798767i	$-0.705486\pi$
$691$	−21856.7	−1.20328	−0.601641	+	0.798767i	$0.705486\pi$
$692$	0	0
$693$	−1309.73	−0.0717927
$694$	0	0
$695$	−38914.3	−2.12389
$696$	0	0
$697$	6967.50	0.378641
$698$	0	0
$699$	11268.1	0.609725
$700$	0	0
$701$	−25538.0	−1.37597	−0.687986	−	0.725724i	$-0.741505\pi$
$701$	−25538.0	−1.37597	−0.687986	+	0.725724i	$0.741505\pi$
$702$	0	0
$703$	3172.05	0.170179
$704$	0	0
$705$	−10600.4	−0.566290
$706$	0	0
$707$	7104.34	0.377916
$708$	0	0
$709$	7419.19	0.392995	0.196498	−	0.980504i	$-0.437043\pi$
$709$	7419.19	0.392995	0.196498	+	0.980504i	$0.437043\pi$
$710$	0	0
$711$	981.893	0.0517917
$712$	0	0
$713$	−3521.80	−0.184983
$714$	0	0
$715$	−9627.15	−0.503545
$716$	0	0
$717$	2095.72	0.109158
$718$	0	0
$719$	−8556.68	−0.443825	−0.221913	−	0.975067i	$-0.571230\pi$
$719$	−8556.68	−0.443825	−0.221913	+	0.975067i	$0.571230\pi$
$720$	0	0
$721$	4097.68	0.211658
$722$	0	0
$723$	15549.4	0.799847
$724$	0	0
$725$	26341.8	1.34939
$726$	0	0
$727$	18454.6	0.941463	0.470731	−	0.882277i	$-0.343990\pi$
$727$	18454.6	0.941463	0.470731	+	0.882277i	$0.343990\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	829.325	0.0419613
$732$	0	0
$733$	33549.5	1.69056	0.845279	−	0.534325i	$-0.179434\pi$
$733$	33549.5	1.69056	0.845279	+	0.534325i	$0.179434\pi$
$734$	0	0
$735$	14880.6	0.746774
$736$	0	0
$737$	16674.2	0.833381
$738$	0	0
$739$	−31214.6	−1.55378	−0.776892	−	0.629634i	$-0.783205\pi$
$739$	−31214.6	−1.55378	−0.776892	+	0.629634i	$0.783205\pi$
$740$	0	0
$741$	−1482.00	−0.0734718
$742$	0	0
$743$	−4365.58	−0.215555	−0.107778	−	0.994175i	$-0.534373\pi$
$743$	−4365.58	−0.215555	−0.107778	+	0.994175i	$0.534373\pi$
$744$	0	0
$745$	−14550.2	−0.715543
$746$	0	0
$747$	−3181.50	−0.155830
$748$	0	0
$749$	7941.24	0.387405
$750$	0	0
$751$	−38096.7	−1.85109	−0.925545	−	0.378639i	$-0.876392\pi$
$751$	−38096.7	−1.85109	−0.925545	+	0.378639i	$0.876392\pi$
$752$	0	0
$753$	4767.38	0.230721
$754$	0	0
$755$	5356.00	0.258179
$756$	0	0
$757$	11366.1	0.545718	0.272859	−	0.962054i	$-0.412031\pi$
$757$	11366.1	0.545718	0.272859	+	0.962054i	$0.412031\pi$
$758$	0	0
$759$	−1206.90	−0.0577177
$760$	0	0
$761$	−15839.4	−0.754506	−0.377253	−	0.926110i	$-0.623131\pi$
$761$	−15839.4	−0.754506	−0.377253	+	0.926110i	$0.623131\pi$
$762$	0	0
$763$	3135.50	0.148772
$764$	0	0
$765$	2895.07	0.136825
$766$	0	0
$767$	−11988.6	−0.564384
$768$	0	0
$769$	22054.9	1.03423	0.517113	−	0.855917i	$-0.327007\pi$
$769$	22054.9	1.03423	0.517113	+	0.855917i	$0.327007\pi$
$770$	0	0
$771$	12599.8	0.588550
$772$	0	0
$773$	−23479.1	−1.09248	−0.546239	−	0.837629i	$-0.683941\pi$
$773$	−23479.1	−1.09248	−0.546239	+	0.837629i	$0.683941\pi$
$774$	0	0
$775$	28809.1	1.33529
$776$	0	0
$777$	−3242.99	−0.149732
$778$	0	0
$779$	6780.15	0.311841
$780$	0	0
$781$	−7396.50	−0.338883
$782$	0	0
$783$	−4857.30	−0.221693
$784$	0	0
$785$	36393.2	1.65469
$786$	0	0
$787$	13540.3	0.613291	0.306645	−	0.951824i	$-0.400793\pi$
$787$	13540.3	0.613291	0.306645	+	0.951824i	$0.400793\pi$
$788$	0	0
$789$	−9435.20	−0.425731
$790$	0	0
$791$	−8450.48	−0.379854
$792$	0	0
$793$	−9569.94	−0.428548
$794$	0	0
$795$	−24010.6	−1.07116
$796$	0	0
$797$	24586.9	1.09274	0.546368	−	0.837545i	$-0.316010\pi$
$797$	24586.9	1.09274	0.546368	+	0.837545i	$0.316010\pi$
$798$	0	0
$799$	4187.63	0.185416
$800$	0	0
$801$	1943.10	0.0857129
$802$	0	0
$803$	18695.2	0.821594
$804$	0	0
$805$	−1909.48	−0.0836027
$806$	0	0
$807$	4931.38	0.215109
$808$	0	0
$809$	−10158.0	−0.441454	−0.220727	−	0.975336i	$-0.570843\pi$
$809$	−10158.0	−0.441454	−0.220727	+	0.975336i	$0.570843\pi$
$810$	0	0
$811$	39398.7	1.70589	0.852945	−	0.522001i	$-0.174814\pi$
$811$	39398.7	1.70589	0.852945	+	0.522001i	$0.174814\pi$
$812$	0	0
$813$	−6180.89	−0.266634
$814$	0	0
$815$	−19073.0	−0.819753
$816$	0	0
$817$	807.025	0.0345584
$818$	0	0
$819$	1515.15	0.0646440
$820$	0	0
$821$	−23969.8	−1.01894	−0.509471	−	0.860488i	$-0.670159\pi$
$821$	−23969.8	−1.01894	−0.509471	+	0.860488i	$0.670159\pi$
$822$	0	0
$823$	25637.7	1.08587	0.542936	−	0.839774i	$-0.317313\pi$
$823$	25637.7	1.08587	0.542936	+	0.839774i	$0.317313\pi$
$824$	0	0
$825$	9872.69	0.416634
$826$	0	0
$827$	−27913.4	−1.17369	−0.586846	−	0.809699i	$-0.699631\pi$
$827$	−27913.4	−1.17369	−0.586846	+	0.809699i	$0.699631\pi$
$828$	0	0
$829$	41489.4	1.73822	0.869112	−	0.494615i	$-0.164691\pi$
$829$	41489.4	1.73822	0.869112	+	0.494615i	$0.164691\pi$
$830$	0	0
$831$	−24281.0	−1.01360
$832$	0	0
$833$	−5878.49	−0.244511
$834$	0	0
$835$	−4480.36	−0.185688
$836$	0	0
$837$	−5312.24	−0.219376
$838$	0	0
$839$	19409.4	0.798672	0.399336	−	0.916805i	$-0.369241\pi$
$839$	19409.4	0.798672	0.399336	+	0.916805i	$0.369241\pi$
$840$	0	0
$841$	7974.98	0.326991
$842$	0	0
$843$	13245.9	0.541178
$844$	0	0
$845$	−25058.4	−1.02016
$846$	0	0
$847$	−5347.53	−0.216934
$848$	0	0
$849$	−11599.1	−0.468881
$850$	0	0
$851$	−2988.39	−0.120377
$852$	0	0
$853$	−23801.7	−0.955397	−0.477698	−	0.878524i	$-0.658529\pi$
$853$	−23801.7	−0.955397	−0.477698	+	0.878524i	$0.658529\pi$
$854$	0	0
$855$	2817.22	0.112686
$856$	0	0
$857$	22763.1	0.907317	0.453659	−	0.891176i	$-0.350119\pi$
$857$	22763.1	0.907317	0.453659	+	0.891176i	$0.350119\pi$
$858$	0	0
$859$	36138.3	1.43542	0.717709	−	0.696343i	$-0.245191\pi$
$859$	36138.3	1.43542	0.717709	+	0.696343i	$0.245191\pi$
$860$	0	0
$861$	−6931.79	−0.274372
$862$	0	0
$863$	−3079.32	−0.121462	−0.0607308	−	0.998154i	$-0.519343\pi$
$863$	−3079.32	−0.121462	−0.0607308	+	0.998154i	$0.519343\pi$
$864$	0	0
$865$	28430.8	1.11755
$866$	0	0
$867$	13595.3	0.532551
$868$	0	0
$869$	−2452.00	−0.0957175
$870$	0	0
$871$	−19289.4	−0.750398
$872$	0	0
$873$	−8662.49	−0.335831
$874$	0	0
$875$	2285.51	0.0883020
$876$	0	0
$877$	−592.185	−0.0228012	−0.0114006	−	0.999935i	$-0.503629\pi$
$877$	−592.185	−0.0228012	−0.0114006	+	0.999935i	$0.503629\pi$
$878$	0	0
$879$	29879.2	1.14653
$880$	0	0
$881$	24642.0	0.942350	0.471175	−	0.882040i	$-0.343830\pi$
$881$	24642.0	0.942350	0.471175	+	0.882040i	$0.343830\pi$
$882$	0	0
$883$	−2212.23	−0.0843119	−0.0421560	−	0.999111i	$-0.513423\pi$
$883$	−2212.23	−0.0843119	−0.0421560	+	0.999111i	$0.513423\pi$
$884$	0	0
$885$	22789.8	0.865617
$886$	0	0
$887$	−21990.5	−0.832433	−0.416216	−	0.909266i	$-0.636644\pi$
$887$	−21990.5	−0.832433	−0.416216	+	0.909266i	$0.636644\pi$
$888$	0	0
$889$	−9164.35	−0.345740
$890$	0	0
$891$	−1820.47	−0.0684491
$892$	0	0
$893$	4075.02	0.152705
$894$	0	0
$895$	24347.5	0.909328
$896$	0	0
$897$	1396.19	0.0519705
$898$	0	0
$899$	35395.3	1.31312
$900$	0	0
$901$	9485.25	0.350721
$902$	0	0
$903$	−825.074	−0.0304061
$904$	0	0
$905$	−30561.0	−1.12252
$906$	0	0
$907$	30234.0	1.10684	0.553420	−	0.832902i	$-0.313322\pi$
$907$	30234.0	1.10684	0.553420	+	0.832902i	$0.313322\pi$
$908$	0	0
$909$	9874.79	0.360315
$910$	0	0
$911$	−17697.8	−0.643637	−0.321818	−	0.946801i	$-0.604294\pi$
$911$	−17697.8	−0.643637	−0.321818	+	0.946801i	$0.604294\pi$
$912$	0	0
$913$	7944.90	0.287993
$914$	0	0
$915$	18192.1	0.657280
$916$	0	0
$917$	−18866.3	−0.679411
$918$	0	0
$919$	−14971.6	−0.537395	−0.268698	−	0.963225i	$-0.586593\pi$
$919$	−14971.6	−0.537395	−0.268698	+	0.963225i	$0.586593\pi$
$920$	0	0
$921$	−29322.6	−1.04909
$922$	0	0
$923$	8556.58	0.305139
$924$	0	0
$925$	24445.6	0.868938
$926$	0	0
$927$	5695.63	0.201801
$928$	0	0
$929$	11711.3	0.413601	0.206800	−	0.978383i	$-0.433695\pi$
$929$	11711.3	0.413601	0.206800	+	0.978383i	$0.433695\pi$
$930$	0	0
$931$	−5720.42	−0.201374
$932$	0	0
$933$	25982.4	0.911712
$934$	0	0
$935$	−7229.62	−0.252871
$936$	0	0
$937$	21703.5	0.756696	0.378348	−	0.925663i	$-0.376492\pi$
$937$	21703.5	0.756696	0.378348	+	0.925663i	$0.376492\pi$
$938$	0	0
$939$	8199.31	0.284957
$940$	0	0
$941$	−4472.21	−0.154931	−0.0774654	−	0.996995i	$-0.524683\pi$
$941$	−4472.21	−0.154931	−0.0774654	+	0.996995i	$0.524683\pi$
$942$	0	0
$943$	−6387.58	−0.220581
$944$	0	0
$945$	−2880.23	−0.0991469
$946$	0	0
$947$	2492.89	0.0855418	0.0427709	−	0.999085i	$-0.486381\pi$
$947$	2492.89	0.0855418	0.0427709	+	0.999085i	$0.486381\pi$
$948$	0	0
$949$	−21627.4	−0.739785
$950$	0	0
$951$	638.264	0.0217636
$952$	0	0
$953$	12389.2	0.421119	0.210560	−	0.977581i	$-0.432471\pi$
$953$	12389.2	0.421119	0.210560	+	0.977581i	$0.432471\pi$
$954$	0	0
$955$	11639.9	0.394408
$956$	0	0
$957$	12129.7	0.409717
$958$	0	0
$959$	9205.63	0.309974
$960$	0	0
$961$	8919.48	0.299402
$962$	0	0
$963$	11038.0	0.369362
$964$	0	0
$965$	39843.9	1.32914
$966$	0	0
$967$	−24561.6	−0.816802	−0.408401	−	0.912803i	$-0.633914\pi$
$967$	−24561.6	−0.816802	−0.408401	+	0.912803i	$0.633914\pi$
$968$	0	0
$969$	−1112.93	−0.0368961
$970$	0	0
$971$	−34972.4	−1.15584	−0.577918	−	0.816095i	$-0.696135\pi$
$971$	−34972.4	−1.15584	−0.577918	+	0.816095i	$0.696135\pi$
$972$	0	0
$973$	−15294.1	−0.503910
$974$	0	0
$975$	−11421.1	−0.375148
$976$	0	0
$977$	−34669.6	−1.13529	−0.567645	−	0.823274i	$-0.692145\pi$
$977$	−34669.6	−1.13529	−0.567645	+	0.823274i	$0.692145\pi$
$978$	0	0
$979$	−4852.35	−0.158408
$980$	0	0
$981$	4358.24	0.141843
$982$	0	0
$983$	−36962.5	−1.19931	−0.599654	−	0.800260i	$-0.704695\pi$
$983$	−36962.5	−1.19931	−0.599654	+	0.800260i	$0.704695\pi$
$984$	0	0
$985$	1421.82	0.0459929
$986$	0	0
$987$	−4166.16	−0.134357
$988$	0	0
$989$	−760.299	−0.0244450
$990$	0	0
$991$	−15655.1	−0.501816	−0.250908	−	0.968011i	$-0.580729\pi$
$991$	−15655.1	−0.501816	−0.250908	+	0.968011i	$0.580729\pi$
$992$	0	0
$993$	21552.6	0.688772
$994$	0	0
$995$	−22592.5	−0.719830
$996$	0	0
$997$	−39083.4	−1.24151	−0.620754	−	0.784005i	$-0.713174\pi$
$997$	−39083.4	−1.24151	−0.620754	+	0.784005i	$0.713174\pi$
$998$	0	0
$999$	−4507.65	−0.142758

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	228.4.a.d.1.2	✓	2
3.2	odd	2		684.4.a.f.1.1		2
4.3	odd	2		912.4.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.4.a.d.1.2	✓	2	1.1	even	1	trivial
684.4.a.f.1.1		2	3.2	odd	2
912.4.a.l.1.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	228.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +16.4750 q^{5} +6.47498 q^{7} +9.00000 q^{9} -22.4750 q^{11} +26.0000 q^{13} -49.4249 q^{15} +19.5250 q^{17} +19.0000 q^{19} -19.4249 q^{21} -17.8999 q^{23} +146.425 q^{25} -27.0000 q^{27} +179.900 q^{29} +196.750 q^{31} +67.4249 q^{33} +106.675 q^{35} +166.950 q^{37} -78.0000 q^{39} +356.850 q^{41} +42.4750 q^{43} +148.275 q^{45} +214.475 q^{47} -301.075 q^{49} -58.5751 q^{51} +485.800 q^{53} -370.275 q^{55} -57.0000 q^{57} -461.099 q^{59} -368.075 q^{61} +58.2748 q^{63} +428.349 q^{65} -741.900 q^{67} +53.6997 q^{69} +329.099 q^{71} -831.824 q^{73} -439.275 q^{75} -145.525 q^{77} +109.099 q^{79} +81.0000 q^{81} -353.500 q^{83} +321.674 q^{85} -539.700 q^{87} +215.900 q^{89} +168.349 q^{91} -590.249 q^{93} +313.025 q^{95} -962.499 q^{97} -202.275 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 3 q^{5} - 17 q^{7} + 18 q^{9} - 15 q^{11} + 52 q^{13} - 9 q^{15} + 69 q^{17} + 38 q^{19} + 51 q^{21} + 84 q^{23} + 203 q^{25} - 54 q^{27} + 240 q^{29} + 94 q^{31} + 45 q^{33} + 423 q^{35}+ \cdots - 135 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 3 * q^5 - 17 * q^7 + 18 * q^9 - 15 * q^11 + 52 * q^13 - 9 * q^15 + 69 * q^17 + 38 * q^19 + 51 * q^21 + 84 * q^23 + 203 * q^25 - 54 * q^27 + 240 * q^29 + 94 * q^31 + 45 * q^33 + 423 * q^35 + 274 * q^37 - 156 * q^39 + 534 * q^41 + 55 * q^43 + 27 * q^45 + 399 * q^47 - 93 * q^49 - 207 * q^51 + 732 * q^53 - 471 * q^55 - 114 * q^57 + 156 * q^59 - 227 * q^61 - 153 * q^63 + 78 * q^65 - 1364 * q^67 - 252 * q^69 - 420 * q^71 + 343 * q^73 - 609 * q^75 - 321 * q^77 - 860 * q^79 + 162 * q^81 - 108 * q^83 - 345 * q^85 - 720 * q^87 + 312 * q^89 - 442 * q^91 - 282 * q^93 + 57 * q^95 - 128 * q^97 - 135 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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