LMFDB - Embedded newform 1560.4.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,4,Mod(1,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1560.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1560.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:	$92.0429796090$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 112x^{2} - 28x + 1648 $ x^4 - x^3 - 112x^2 - 28x + 1648
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.91028$ of defining polynomial
Character	$\chi$	$=$	1560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} +23.2203 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -5.00000 q^{5} +23.2203 q^{7} +9.00000 q^{9} +49.6472 q^{11} -13.0000 q^{13} +15.0000 q^{15} -17.0340 q^{17} -129.190 q^{19} -69.6608 q^{21} -149.005 q^{23} +25.0000 q^{25} -27.0000 q^{27} +11.3246 q^{29} -75.4927 q^{31} -148.942 q^{33} -116.101 q^{35} +84.5632 q^{37} +39.0000 q^{39} +136.206 q^{41} -240.808 q^{43} -45.0000 q^{45} +5.99107 q^{47} +196.181 q^{49} +51.1021 q^{51} +25.9866 q^{53} -248.236 q^{55} +387.569 q^{57} +386.054 q^{59} +476.288 q^{61} +208.982 q^{63} +65.0000 q^{65} +298.796 q^{67} +447.016 q^{69} -663.478 q^{71} -99.1380 q^{73} -75.0000 q^{75} +1152.82 q^{77} +480.579 q^{79} +81.0000 q^{81} +397.762 q^{83} +85.1701 q^{85} -33.9739 q^{87} -1319.17 q^{89} -301.863 q^{91} +226.478 q^{93} +645.949 q^{95} -194.579 q^{97} +446.825 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9	$ 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9} + 35 q^{11} - 52 q^{13} + 60 q^{15} - 9 q^{17} + 32 q^{19} + 33 q^{21} - 149 q^{23} + 100 q^{25} - 108 q^{27} + 168 q^{29} + 280 q^{31} - 105 q^{33} + 55 q^{35} + 37 q^{37} + 156 q^{39} + 247 q^{41} + 132 q^{43} - 180 q^{45} + 217 q^{49} + 27 q^{51} + 247 q^{53} - 175 q^{55} - 96 q^{57} + 614 q^{59} + 719 q^{61} - 99 q^{63} + 260 q^{65} + 658 q^{67} + 447 q^{69} + 939 q^{71} - 1452 q^{73} - 300 q^{75} + 199 q^{77} + 289 q^{79} + 324 q^{81} - 118 q^{83} + 45 q^{85} - 504 q^{87} - 1319 q^{89} + 143 q^{91} - 840 q^{93} - 160 q^{95} - 993 q^{97} + 315 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9 + 35 * q^11 - 52 * q^13 + 60 * q^15 - 9 * q^17 + 32 * q^19 + 33 * q^21 - 149 * q^23 + 100 * q^25 - 108 * q^27 + 168 * q^29 + 280 * q^31 - 105 * q^33 + 55 * q^35 + 37 * q^37 + 156 * q^39 + 247 * q^41 + 132 * q^43 - 180 * q^45 + 217 * q^49 + 27 * q^51 + 247 * q^53 - 175 * q^55 - 96 * q^57 + 614 * q^59 + 719 * q^61 - 99 * q^63 + 260 * q^65 + 658 * q^67 + 447 * q^69 + 939 * q^71 - 1452 * q^73 - 300 * q^75 + 199 * q^77 + 289 * q^79 + 324 * q^81 - 118 * q^83 + 45 * q^85 - 504 * q^87 - 1319 * q^89 + 143 * q^91 - 840 * q^93 - 160 * q^95 - 993 * q^97 + 315 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	23.2203	1.25378	0.626888	−	0.779109i	$-0.284328\pi$
$7$	23.2203	1.25378	0.626888	+	0.779109i	$0.284328\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	49.6472	1.36084	0.680418	−	0.732825i	$-0.261798\pi$
$11$	49.6472	1.36084	0.680418	+	0.732825i	$0.261798\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−17.0340	−0.243021	−0.121511	−	0.992590i	$-0.538774\pi$
$17$	−17.0340	−0.243021	−0.121511	+	0.992590i	$0.538774\pi$
$18$	0	0
$19$	−129.190	−1.55990	−0.779952	−	0.625840i	$-0.784756\pi$
$19$	−129.190	−1.55990	−0.779952	+	0.625840i	$0.784756\pi$
$20$	0	0
$21$	−69.6608	−0.723868
$22$	0	0
$23$	−149.005	−1.35086	−0.675429	−	0.737425i	$-0.736042\pi$
$23$	−149.005	−1.35086	−0.675429	+	0.737425i	$0.736042\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	11.3246	0.0725149	0.0362575	−	0.999342i	$-0.488456\pi$
$29$	11.3246	0.0725149	0.0362575	+	0.999342i	$0.488456\pi$
$30$	0	0
$31$	−75.4927	−0.437383	−0.218692	−	0.975794i	$-0.570179\pi$
$31$	−75.4927	−0.437383	−0.218692	+	0.975794i	$0.570179\pi$
$32$	0	0
$33$	−148.942	−0.785679
$34$	0	0
$35$	−116.101	−0.560706
$36$	0	0
$37$	84.5632	0.375733	0.187866	−	0.982195i	$-0.439843\pi$
$37$	84.5632	0.375733	0.187866	+	0.982195i	$0.439843\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	136.206	0.518824	0.259412	−	0.965767i	$-0.416471\pi$
$41$	136.206	0.518824	0.259412	+	0.965767i	$0.416471\pi$
$42$	0	0
$43$	−240.808	−0.854020	−0.427010	−	0.904247i	$-0.640433\pi$
$43$	−240.808	−0.854020	−0.427010	+	0.904247i	$0.640433\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	5.99107	0.0185933	0.00929667	−	0.999957i	$-0.497041\pi$
$47$	5.99107	0.0185933	0.00929667	+	0.999957i	$0.497041\pi$
$48$	0	0
$49$	196.181	0.571955
$50$	0	0
$51$	51.1021	0.140308
$52$	0	0
$53$	25.9866	0.0673496	0.0336748	−	0.999433i	$-0.489279\pi$
$53$	25.9866	0.0673496	0.0336748	+	0.999433i	$0.489279\pi$
$54$	0	0
$55$	−248.236	−0.608584
$56$	0	0
$57$	387.569	0.900611
$58$	0	0
$59$	386.054	0.851863	0.425931	−	0.904755i	$-0.359946\pi$
$59$	386.054	0.851863	0.425931	+	0.904755i	$0.359946\pi$
$60$	0	0
$61$	476.288	0.999711	0.499856	−	0.866109i	$-0.333386\pi$
$61$	476.288	0.999711	0.499856	+	0.866109i	$0.333386\pi$
$62$	0	0
$63$	208.982	0.417925
$64$	0	0
$65$	65.0000	0.124035
$66$	0	0
$67$	298.796	0.544832	0.272416	−	0.962180i	$-0.412177\pi$
$67$	298.796	0.544832	0.272416	+	0.962180i	$0.412177\pi$
$68$	0	0
$69$	447.016	0.779919
$70$	0	0
$71$	−663.478	−1.10902	−0.554509	−	0.832178i	$-0.687094\pi$
$71$	−663.478	−1.10902	−0.554509	+	0.832178i	$0.687094\pi$
$72$	0	0
$73$	−99.1380	−0.158948	−0.0794741	−	0.996837i	$-0.525324\pi$
$73$	−99.1380	−0.158948	−0.0794741	+	0.996837i	$0.525324\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	1152.82	1.70618
$78$	0	0
$79$	480.579	0.684422	0.342211	−	0.939623i	$-0.388824\pi$
$79$	480.579	0.684422	0.342211	+	0.939623i	$0.388824\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	397.762	0.526025	0.263012	−	0.964792i	$-0.415284\pi$
$83$	397.762	0.526025	0.263012	+	0.964792i	$0.415284\pi$
$84$	0	0
$85$	85.1701	0.108682
$86$	0	0
$87$	−33.9739	−0.0418665
$88$	0	0
$89$	−1319.17	−1.57115	−0.785573	−	0.618769i	$-0.787632\pi$
$89$	−1319.17	−1.57115	−0.785573	+	0.618769i	$0.787632\pi$
$90$	0	0
$91$	−301.863	−0.347735
$92$	0	0
$93$	226.478	0.252523
$94$	0	0
$95$	645.949	0.697610
$96$	0	0
$97$	−194.579	−0.203675	−0.101838	−	0.994801i	$-0.532472\pi$
$97$	−194.579	−0.203675	−0.101838	+	0.994801i	$0.532472\pi$
$98$	0	0
$99$	446.825	0.453612
$100$	0	0
$101$	−788.150	−0.776474	−0.388237	−	0.921560i	$-0.626916\pi$
$101$	−788.150	−0.776474	−0.388237	+	0.921560i	$0.626916\pi$
$102$	0	0
$103$	−250.915	−0.240033	−0.120016	−	0.992772i	$-0.538295\pi$
$103$	−250.915	−0.240033	−0.120016	+	0.992772i	$0.538295\pi$
$104$	0	0
$105$	348.304	0.323724
$106$	0	0
$107$	−628.754	−0.568074	−0.284037	−	0.958813i	$-0.591674\pi$
$107$	−628.754	−0.568074	−0.284037	+	0.958813i	$0.591674\pi$
$108$	0	0
$109$	−1023.54	−0.899426	−0.449713	−	0.893173i	$-0.648474\pi$
$109$	−1023.54	−0.899426	−0.449713	+	0.893173i	$0.648474\pi$
$110$	0	0
$111$	−253.690	−0.216929
$112$	0	0
$113$	−2295.53	−1.91102	−0.955512	−	0.294952i	$-0.904696\pi$
$113$	−2295.53	−1.91102	−0.955512	+	0.294952i	$0.904696\pi$
$114$	0	0
$115$	745.026	0.604122
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−395.535	−0.304694
$120$	0	0
$121$	1133.84	0.851872
$122$	0	0
$123$	−408.618	−0.299543
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−262.837	−0.183646	−0.0918229	−	0.995775i	$-0.529269\pi$
$127$	−262.837	−0.183646	−0.0918229	+	0.995775i	$0.529269\pi$
$128$	0	0
$129$	722.424	0.493069
$130$	0	0
$131$	−928.706	−0.619400	−0.309700	−	0.950834i	$-0.600229\pi$
$131$	−928.706	−0.619400	−0.309700	+	0.950834i	$0.600229\pi$
$132$	0	0
$133$	−2999.82	−1.95577
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−1001.50	−0.624555	−0.312278	−	0.949991i	$-0.601092\pi$
$137$	−1001.50	−0.624555	−0.312278	+	0.949991i	$0.601092\pi$
$138$	0	0
$139$	2351.46	1.43488	0.717440	−	0.696620i	$-0.245314\pi$
$139$	2351.46	1.43488	0.717440	+	0.696620i	$0.245314\pi$
$140$	0	0
$141$	−17.9732	−0.0107349
$142$	0	0
$143$	−645.413	−0.377428
$144$	0	0
$145$	−56.6232	−0.0324297
$146$	0	0
$147$	−588.542	−0.330219
$148$	0	0
$149$	−3146.84	−1.73020	−0.865098	−	0.501602i	$-0.832744\pi$
$149$	−3146.84	−1.73020	−0.865098	+	0.501602i	$0.832744\pi$
$150$	0	0
$151$	703.387	0.379078	0.189539	−	0.981873i	$-0.439301\pi$
$151$	703.387	0.379078	0.189539	+	0.981873i	$0.439301\pi$
$152$	0	0
$153$	−153.306	−0.0810070
$154$	0	0
$155$	377.463	0.195604
$156$	0	0
$157$	−294.653	−0.149783	−0.0748914	−	0.997192i	$-0.523861\pi$
$157$	−294.653	−0.149783	−0.0748914	+	0.997192i	$0.523861\pi$
$158$	0	0
$159$	−77.9597	−0.0388843
$160$	0	0
$161$	−3459.94	−1.69367
$162$	0	0
$163$	537.660	0.258360	0.129180	−	0.991621i	$-0.458765\pi$
$163$	537.660	0.258360	0.129180	+	0.991621i	$0.458765\pi$
$164$	0	0
$165$	744.708	0.351366
$166$	0	0
$167$	1418.70	0.657379	0.328689	−	0.944438i	$-0.393393\pi$
$167$	1418.70	0.657379	0.328689	+	0.944438i	$0.393393\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1162.71	−0.519968
$172$	0	0
$173$	−1763.20	−0.774876	−0.387438	−	0.921896i	$-0.626640\pi$
$173$	−1763.20	−0.774876	−0.387438	+	0.921896i	$0.626640\pi$
$174$	0	0
$175$	580.507	0.250755
$176$	0	0
$177$	−1158.16	−0.491823
$178$	0	0
$179$	−1302.03	−0.543676	−0.271838	−	0.962343i	$-0.587631\pi$
$179$	−1302.03	−0.543676	−0.271838	+	0.962343i	$0.587631\pi$
$180$	0	0
$181$	3252.05	1.33548	0.667742	−	0.744393i	$-0.267261\pi$
$181$	3252.05	1.33548	0.667742	+	0.744393i	$0.267261\pi$
$182$	0	0
$183$	−1428.86	−0.577183
$184$	0	0
$185$	−422.816	−0.168033
$186$	0	0
$187$	−845.691	−0.330712
$188$	0	0
$189$	−626.947	−0.241289
$190$	0	0
$191$	−291.135	−0.110292	−0.0551461	−	0.998478i	$-0.517562\pi$
$191$	−291.135	−0.110292	−0.0551461	+	0.998478i	$0.517562\pi$
$192$	0	0
$193$	−3266.62	−1.21832	−0.609162	−	0.793046i	$-0.708494\pi$
$193$	−3266.62	−1.21832	−0.609162	+	0.793046i	$0.708494\pi$
$194$	0	0
$195$	−195.000	−0.0716115
$196$	0	0
$197$	3211.09	1.16132	0.580661	−	0.814145i	$-0.302794\pi$
$197$	3211.09	1.16132	0.580661	+	0.814145i	$0.302794\pi$
$198$	0	0
$199$	−359.557	−0.128082	−0.0640410	−	0.997947i	$-0.520399\pi$
$199$	−359.557	−0.128082	−0.0640410	+	0.997947i	$0.520399\pi$
$200$	0	0
$201$	−896.389	−0.314559
$202$	0	0
$203$	262.961	0.0909175
$204$	0	0
$205$	−681.029	−0.232025
$206$	0	0
$207$	−1341.05	−0.450286
$208$	0	0
$209$	−6413.91	−2.12277
$210$	0	0
$211$	−102.062	−0.0332997	−0.0166499	−	0.999861i	$-0.505300\pi$
$211$	−102.062	−0.0332997	−0.0166499	+	0.999861i	$0.505300\pi$
$212$	0	0
$213$	1990.43	0.640292
$214$	0	0
$215$	1204.04	0.381930
$216$	0	0
$217$	−1752.96	−0.548381
$218$	0	0
$219$	297.414	0.0917688
$220$	0	0
$221$	221.442	0.0674019
$222$	0	0
$223$	−1924.30	−0.577849	−0.288925	−	0.957352i	$-0.593298\pi$
$223$	−1924.30	−0.577849	−0.288925	+	0.957352i	$0.593298\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−3269.24	−0.955890	−0.477945	−	0.878390i	$-0.658618\pi$
$227$	−3269.24	−0.955890	−0.477945	+	0.878390i	$0.658618\pi$
$228$	0	0
$229$	−4174.16	−1.20452	−0.602262	−	0.798298i	$-0.705734\pi$
$229$	−4174.16	−1.20452	−0.602262	+	0.798298i	$0.705734\pi$
$230$	0	0
$231$	−3458.46	−0.985065
$232$	0	0
$233$	−1600.84	−0.450106	−0.225053	−	0.974347i	$-0.572256\pi$
$233$	−1600.84	−0.450106	−0.225053	+	0.974347i	$0.572256\pi$
$234$	0	0
$235$	−29.9553	−0.00831520
$236$	0	0
$237$	−1441.74	−0.395151
$238$	0	0
$239$	−1544.22	−0.417939	−0.208969	−	0.977922i	$-0.567011\pi$
$239$	−1544.22	−0.417939	−0.208969	+	0.977922i	$0.567011\pi$
$240$	0	0
$241$	883.106	0.236041	0.118021	−	0.993011i	$-0.462345\pi$
$241$	883.106	0.236041	0.118021	+	0.993011i	$0.462345\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−980.903	−0.255786
$246$	0	0
$247$	1679.47	0.432639
$248$	0	0
$249$	−1193.29	−0.303701
$250$	0	0
$251$	4299.65	1.08124	0.540621	−	0.841267i	$-0.318189\pi$
$251$	4299.65	1.08124	0.540621	+	0.841267i	$0.318189\pi$
$252$	0	0
$253$	−7397.69	−1.83830
$254$	0	0
$255$	−255.510	−0.0627478
$256$	0	0
$257$	1430.76	0.347271	0.173635	−	0.984810i	$-0.444449\pi$
$257$	1430.76	0.347271	0.173635	+	0.984810i	$0.444449\pi$
$258$	0	0
$259$	1963.58	0.471085
$260$	0	0
$261$	101.922	0.0241716
$262$	0	0
$263$	−7683.31	−1.80142	−0.900710	−	0.434421i	$-0.856953\pi$
$263$	−7683.31	−1.80142	−0.900710	+	0.434421i	$0.856953\pi$
$264$	0	0
$265$	−129.933	−0.0301197
$266$	0	0
$267$	3957.52	0.907101
$268$	0	0
$269$	1596.80	0.361927	0.180964	−	0.983490i	$-0.442078\pi$
$269$	1596.80	0.361927	0.180964	+	0.983490i	$0.442078\pi$
$270$	0	0
$271$	4724.67	1.05905	0.529527	−	0.848293i	$-0.322370\pi$
$271$	4724.67	1.05905	0.529527	+	0.848293i	$0.322370\pi$
$272$	0	0
$273$	905.590	0.200765
$274$	0	0
$275$	1241.18	0.272167
$276$	0	0
$277$	612.798	0.132922	0.0664611	−	0.997789i	$-0.478829\pi$
$277$	612.798	0.132922	0.0664611	+	0.997789i	$0.478829\pi$
$278$	0	0
$279$	−679.434	−0.145794
$280$	0	0
$281$	−595.199	−0.126358	−0.0631790	−	0.998002i	$-0.520124\pi$
$281$	−595.199	−0.126358	−0.0631790	+	0.998002i	$0.520124\pi$
$282$	0	0
$283$	6449.89	1.35479	0.677396	−	0.735619i	$-0.263109\pi$
$283$	6449.89	1.35479	0.677396	+	0.735619i	$0.263109\pi$
$284$	0	0
$285$	−1937.85	−0.402765
$286$	0	0
$287$	3162.74	0.650489
$288$	0	0
$289$	−4622.84	−0.940941
$290$	0	0
$291$	583.737	0.117592
$292$	0	0
$293$	8319.92	1.65889	0.829445	−	0.558588i	$-0.188657\pi$
$293$	8319.92	1.65889	0.829445	+	0.558588i	$0.188657\pi$
$294$	0	0
$295$	−1930.27	−0.380965
$296$	0	0
$297$	−1340.47	−0.261893
$298$	0	0
$299$	1937.07	0.374661
$300$	0	0
$301$	−5591.63	−1.07075
$302$	0	0
$303$	2364.45	0.448297
$304$	0	0
$305$	−2381.44	−0.447084
$306$	0	0
$307$	−5382.41	−1.00062	−0.500310	−	0.865846i	$-0.666781\pi$
$307$	−5382.41	−1.00062	−0.500310	+	0.865846i	$0.666781\pi$
$308$	0	0
$309$	752.744	0.138583
$310$	0	0
$311$	−4906.31	−0.894571	−0.447286	−	0.894391i	$-0.647609\pi$
$311$	−4906.31	−0.894571	−0.447286	+	0.894391i	$0.647609\pi$
$312$	0	0
$313$	−1266.89	−0.228782	−0.114391	−	0.993436i	$-0.536492\pi$
$313$	−1266.89	−0.228782	−0.114391	+	0.993436i	$0.536492\pi$
$314$	0	0
$315$	−1044.91	−0.186902
$316$	0	0
$317$	−5575.32	−0.987827	−0.493914	−	0.869511i	$-0.664434\pi$
$317$	−5575.32	−0.987827	−0.493914	+	0.869511i	$0.664434\pi$
$318$	0	0
$319$	562.236	0.0986809
$320$	0	0
$321$	1886.26	0.327978
$322$	0	0
$323$	2200.62	0.379089
$324$	0	0
$325$	−325.000	−0.0554700
$326$	0	0
$327$	3070.62	0.519284
$328$	0	0
$329$	139.114	0.0233119
$330$	0	0
$331$	−4754.45	−0.789512	−0.394756	−	0.918786i	$-0.629171\pi$
$331$	−4754.45	−0.789512	−0.394756	+	0.918786i	$0.629171\pi$
$332$	0	0
$333$	761.069	0.125244
$334$	0	0
$335$	−1493.98	−0.243656
$336$	0	0
$337$	3407.39	0.550779	0.275389	−	0.961333i	$-0.411193\pi$
$337$	3407.39	0.550779	0.275389	+	0.961333i	$0.411193\pi$
$338$	0	0
$339$	6886.60	1.10333
$340$	0	0
$341$	−3748.00	−0.595207
$342$	0	0
$343$	−3409.18	−0.536672
$344$	0	0
$345$	−2235.08	−0.348790
$346$	0	0
$347$	8395.30	1.29880	0.649400	−	0.760447i	$-0.275020\pi$
$347$	8395.30	1.29880	0.649400	+	0.760447i	$0.275020\pi$
$348$	0	0
$349$	−3399.02	−0.521333	−0.260667	−	0.965429i	$-0.583942\pi$
$349$	−3399.02	−0.521333	−0.260667	+	0.965429i	$0.583942\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	−10179.0	−1.53477	−0.767383	−	0.641189i	$-0.778441\pi$
$353$	−10179.0	−1.53477	−0.767383	+	0.641189i	$0.778441\pi$
$354$	0	0
$355$	3317.39	0.495968
$356$	0	0
$357$	1186.60	0.175915
$358$	0	0
$359$	4145.91	0.609506	0.304753	−	0.952431i	$-0.401426\pi$
$359$	4145.91	0.609506	0.304753	+	0.952431i	$0.401426\pi$
$360$	0	0
$361$	9831.00	1.43330
$362$	0	0
$363$	−3401.53	−0.491829
$364$	0	0
$365$	495.690	0.0710838
$366$	0	0
$367$	−1100.58	−0.156539	−0.0782695	−	0.996932i	$-0.524939\pi$
$367$	−1100.58	−0.156539	−0.0782695	+	0.996932i	$0.524939\pi$
$368$	0	0
$369$	1225.85	0.172941
$370$	0	0
$371$	603.415	0.0844414
$372$	0	0
$373$	−2144.52	−0.297691	−0.148846	−	0.988860i	$-0.547556\pi$
$373$	−2144.52	−0.297691	−0.148846	+	0.988860i	$0.547556\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−147.220	−0.0201120
$378$	0	0
$379$	−12011.6	−1.62795	−0.813976	−	0.580898i	$-0.802701\pi$
$379$	−12011.6	−1.62795	−0.813976	+	0.580898i	$0.802701\pi$
$380$	0	0
$381$	788.511	0.106028
$382$	0	0
$383$	−7653.88	−1.02114	−0.510568	−	0.859838i	$-0.670565\pi$
$383$	−7653.88	−1.02114	−0.510568	+	0.859838i	$0.670565\pi$
$384$	0	0
$385$	−5764.10	−0.763028
$386$	0	0
$387$	−2167.27	−0.284673
$388$	0	0
$389$	−2839.13	−0.370050	−0.185025	−	0.982734i	$-0.559237\pi$
$389$	−2839.13	−0.370050	−0.185025	+	0.982734i	$0.559237\pi$
$390$	0	0
$391$	2538.16	0.328287
$392$	0	0
$393$	2786.12	0.357611
$394$	0	0
$395$	−2402.89	−0.306083
$396$	0	0
$397$	−11721.1	−1.48177	−0.740887	−	0.671630i	$-0.765594\pi$
$397$	−11721.1	−1.48177	−0.740887	+	0.671630i	$0.765594\pi$
$398$	0	0
$399$	8999.46	1.12916
$400$	0	0
$401$	−12477.7	−1.55388	−0.776939	−	0.629575i	$-0.783229\pi$
$401$	−12477.7	−1.55388	−0.776939	+	0.629575i	$0.783229\pi$
$402$	0	0
$403$	981.405	0.121308
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	4198.32	0.511310
$408$	0	0
$409$	15313.1	1.85131	0.925654	−	0.378372i	$-0.123516\pi$
$409$	15313.1	1.85131	0.925654	+	0.378372i	$0.123516\pi$
$410$	0	0
$411$	3004.50	0.360587
$412$	0	0
$413$	8964.27	1.06805
$414$	0	0
$415$	−1988.81	−0.235246
$416$	0	0
$417$	−7054.39	−0.828429
$418$	0	0
$419$	4618.31	0.538471	0.269235	−	0.963074i	$-0.413229\pi$
$419$	4618.31	0.538471	0.269235	+	0.963074i	$0.413229\pi$
$420$	0	0
$421$	2759.12	0.319410	0.159705	−	0.987165i	$-0.448946\pi$
$421$	2759.12	0.319410	0.159705	+	0.987165i	$0.448946\pi$
$422$	0	0
$423$	53.9196	0.00619778
$424$	0	0
$425$	−425.851	−0.0486042
$426$	0	0
$427$	11059.5	1.25341
$428$	0	0
$429$	1936.24	0.217908
$430$	0	0
$431$	−7281.63	−0.813790	−0.406895	−	0.913475i	$-0.633389\pi$
$431$	−7281.63	−0.813790	−0.406895	+	0.913475i	$0.633389\pi$
$432$	0	0
$433$	13179.3	1.46272	0.731361	−	0.681991i	$-0.238885\pi$
$433$	13179.3	1.46272	0.731361	+	0.681991i	$0.238885\pi$
$434$	0	0
$435$	169.870	0.0187233
$436$	0	0
$437$	19250.0	2.10721
$438$	0	0
$439$	−4500.39	−0.489275	−0.244637	−	0.969615i	$-0.578669\pi$
$439$	−4500.39	−0.489275	−0.244637	+	0.969615i	$0.578669\pi$
$440$	0	0
$441$	1765.63	0.190652
$442$	0	0
$443$	−6316.54	−0.677444	−0.338722	−	0.940886i	$-0.609995\pi$
$443$	−6316.54	−0.677444	−0.338722	+	0.940886i	$0.609995\pi$
$444$	0	0
$445$	6595.86	0.702638
$446$	0	0
$447$	9440.53	0.998930
$448$	0	0
$449$	−2245.51	−0.236018	−0.118009	−	0.993013i	$-0.537651\pi$
$449$	−2245.51	−0.236018	−0.118009	+	0.993013i	$0.537651\pi$
$450$	0	0
$451$	6762.24	0.706034
$452$	0	0
$453$	−2110.16	−0.218861
$454$	0	0
$455$	1509.32	0.155512
$456$	0	0
$457$	−14385.4	−1.47248	−0.736238	−	0.676723i	$-0.763400\pi$
$457$	−14385.4	−1.47248	−0.736238	+	0.676723i	$0.763400\pi$
$458$	0	0
$459$	459.919	0.0467694
$460$	0	0
$461$	−3968.69	−0.400955	−0.200478	−	0.979698i	$-0.564249\pi$
$461$	−3968.69	−0.400955	−0.200478	+	0.979698i	$0.564249\pi$
$462$	0	0
$463$	−7545.90	−0.757425	−0.378713	−	0.925514i	$-0.623633\pi$
$463$	−7545.90	−0.757425	−0.378713	+	0.925514i	$0.623633\pi$
$464$	0	0
$465$	−1132.39	−0.112932
$466$	0	0
$467$	−19129.0	−1.89547	−0.947736	−	0.319056i	$-0.896634\pi$
$467$	−19129.0	−1.89547	−0.947736	+	0.319056i	$0.896634\pi$
$468$	0	0
$469$	6938.13	0.683098
$470$	0	0
$471$	883.960	0.0864771
$472$	0	0
$473$	−11955.4	−1.16218
$474$	0	0
$475$	−3229.74	−0.311981
$476$	0	0
$477$	233.879	0.0224499
$478$	0	0
$479$	13797.8	1.31615	0.658077	−	0.752951i	$-0.271370\pi$
$479$	13797.8	1.31615	0.658077	+	0.752951i	$0.271370\pi$
$480$	0	0
$481$	−1099.32	−0.104209
$482$	0	0
$483$	10379.8	0.977844
$484$	0	0
$485$	972.896	0.0910864
$486$	0	0
$487$	−6382.88	−0.593913	−0.296957	−	0.954891i	$-0.595972\pi$
$487$	−6382.88	−0.593913	−0.296957	+	0.954891i	$0.595972\pi$
$488$	0	0
$489$	−1612.98	−0.149164
$490$	0	0
$491$	10192.6	0.936838	0.468419	−	0.883506i	$-0.344824\pi$
$491$	10192.6	0.936838	0.468419	+	0.883506i	$0.344824\pi$
$492$	0	0
$493$	−192.904	−0.0176227
$494$	0	0
$495$	−2234.12	−0.202861
$496$	0	0
$497$	−15406.1	−1.39046
$498$	0	0
$499$	14894.6	1.33622	0.668109	−	0.744063i	$-0.267104\pi$
$499$	14894.6	1.33622	0.668109	+	0.744063i	$0.267104\pi$
$500$	0	0
$501$	−4256.10	−0.379538
$502$	0	0
$503$	−21343.0	−1.89192	−0.945962	−	0.324277i	$-0.894879\pi$
$503$	−21343.0	−1.89192	−0.945962	+	0.324277i	$0.894879\pi$
$504$	0	0
$505$	3940.75	0.347250
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−377.349	−0.0328599	−0.0164300	−	0.999865i	$-0.505230\pi$
$509$	−377.349	−0.0328599	−0.0164300	+	0.999865i	$0.505230\pi$
$510$	0	0
$511$	−2302.01	−0.199286
$512$	0	0
$513$	3488.12	0.300204
$514$	0	0
$515$	1254.57	0.107346
$516$	0	0
$517$	297.440	0.0253025
$518$	0	0
$519$	5289.60	0.447375
$520$	0	0
$521$	−8872.27	−0.746068	−0.373034	−	0.927818i	$-0.621682\pi$
$521$	−8872.27	−0.746068	−0.373034	+	0.927818i	$0.621682\pi$
$522$	0	0
$523$	3758.24	0.314219	0.157109	−	0.987581i	$-0.449782\pi$
$523$	3758.24	0.314219	0.157109	+	0.987581i	$0.449782\pi$
$524$	0	0
$525$	−1741.52	−0.144774
$526$	0	0
$527$	1285.94	0.106293
$528$	0	0
$529$	10035.6	0.824819
$530$	0	0
$531$	3474.48	0.283954
$532$	0	0
$533$	−1770.68	−0.143896
$534$	0	0
$535$	3143.77	0.254050
$536$	0	0
$537$	3906.08	0.313891
$538$	0	0
$539$	9739.81	0.778337
$540$	0	0
$541$	4013.18	0.318928	0.159464	−	0.987204i	$-0.449023\pi$
$541$	4013.18	0.318928	0.159464	+	0.987204i	$0.449023\pi$
$542$	0	0
$543$	−9756.14	−0.771042
$544$	0	0
$545$	5117.71	0.402236
$546$	0	0
$547$	−12219.2	−0.955125	−0.477562	−	0.878598i	$-0.658480\pi$
$547$	−12219.2	−0.955125	−0.477562	+	0.878598i	$0.658480\pi$
$548$	0	0
$549$	4286.59	0.333237
$550$	0	0
$551$	−1463.03	−0.113116
$552$	0	0
$553$	11159.2	0.858112
$554$	0	0
$555$	1268.45	0.0970137
$556$	0	0
$557$	5058.95	0.384838	0.192419	−	0.981313i	$-0.438367\pi$
$557$	5058.95	0.384838	0.192419	+	0.981313i	$0.438367\pi$
$558$	0	0
$559$	3130.50	0.236863
$560$	0	0
$561$	2537.07	0.190936
$562$	0	0
$563$	2866.27	0.214563	0.107281	−	0.994229i	$-0.465785\pi$
$563$	2866.27	0.214563	0.107281	+	0.994229i	$0.465785\pi$
$564$	0	0
$565$	11477.7	0.854636
$566$	0	0
$567$	1880.84	0.139308
$568$	0	0
$569$	−6474.64	−0.477032	−0.238516	−	0.971139i	$-0.576661\pi$
$569$	−6474.64	−0.477032	−0.238516	+	0.971139i	$0.576661\pi$
$570$	0	0
$571$	−3151.94	−0.231007	−0.115503	−	0.993307i	$-0.536848\pi$
$571$	−3151.94	−0.231007	−0.115503	+	0.993307i	$0.536848\pi$
$572$	0	0
$573$	873.406	0.0636772
$574$	0	0
$575$	−3725.13	−0.270172
$576$	0	0
$577$	−12780.9	−0.922139	−0.461069	−	0.887364i	$-0.652534\pi$
$577$	−12780.9	−0.922139	−0.461069	+	0.887364i	$0.652534\pi$
$578$	0	0
$579$	9799.86	0.703400
$580$	0	0
$581$	9236.14	0.659518
$582$	0	0
$583$	1290.16	0.0916518
$584$	0	0
$585$	585.000	0.0413449
$586$	0	0
$587$	6197.74	0.435789	0.217895	−	0.975972i	$-0.430081\pi$
$587$	6197.74	0.435789	0.217895	+	0.975972i	$0.430081\pi$
$588$	0	0
$589$	9752.88	0.682276
$590$	0	0
$591$	−9633.26	−0.670490
$592$	0	0
$593$	−21102.6	−1.46135	−0.730674	−	0.682727i	$-0.760794\pi$
$593$	−21102.6	−1.46135	−0.730674	+	0.682727i	$0.760794\pi$
$594$	0	0
$595$	1977.67	0.136263
$596$	0	0
$597$	1078.67	0.0739482
$598$	0	0
$599$	16931.0	1.15490	0.577448	−	0.816427i	$-0.304049\pi$
$599$	16931.0	1.15490	0.577448	+	0.816427i	$0.304049\pi$
$600$	0	0
$601$	26929.3	1.82774	0.913868	−	0.406012i	$-0.133081\pi$
$601$	26929.3	1.82774	0.913868	+	0.406012i	$0.133081\pi$
$602$	0	0
$603$	2689.17	0.181611
$604$	0	0
$605$	−5669.21	−0.380969
$606$	0	0
$607$	14848.3	0.992875	0.496437	−	0.868073i	$-0.334641\pi$
$607$	14848.3	0.992875	0.496437	+	0.868073i	$0.334641\pi$
$608$	0	0
$609$	−788.883	−0.0524912
$610$	0	0
$611$	−77.8839	−0.00515687
$612$	0	0
$613$	379.793	0.0250240	0.0125120	−	0.999922i	$-0.496017\pi$
$613$	379.793	0.0250240	0.0125120	+	0.999922i	$0.496017\pi$
$614$	0	0
$615$	2043.09	0.133960
$616$	0	0
$617$	−21935.2	−1.43124	−0.715621	−	0.698489i	$-0.753856\pi$
$617$	−21935.2	−1.43124	−0.715621	+	0.698489i	$0.753856\pi$
$618$	0	0
$619$	9575.78	0.621782	0.310891	−	0.950446i	$-0.399373\pi$
$619$	9575.78	0.621782	0.310891	+	0.950446i	$0.399373\pi$
$620$	0	0
$621$	4023.14	0.259973
$622$	0	0
$623$	−30631.5	−1.96987
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	19241.7	1.22558
$628$	0	0
$629$	−1440.45	−0.0913109
$630$	0	0
$631$	−5910.73	−0.372904	−0.186452	−	0.982464i	$-0.559699\pi$
$631$	−5910.73	−0.372904	−0.186452	+	0.982464i	$0.559699\pi$
$632$	0	0
$633$	306.186	0.0192256
$634$	0	0
$635$	1314.19	0.0821289
$636$	0	0
$637$	−2550.35	−0.158632
$638$	0	0
$639$	−5971.30	−0.369673
$640$	0	0
$641$	−8916.88	−0.549447	−0.274724	−	0.961523i	$-0.588586\pi$
$641$	−8916.88	−0.549447	−0.274724	+	0.961523i	$0.588586\pi$
$642$	0	0
$643$	28527.5	1.74963	0.874817	−	0.484454i	$-0.160982\pi$
$643$	28527.5	1.74963	0.874817	+	0.484454i	$0.160982\pi$
$644$	0	0
$645$	−3612.12	−0.220507
$646$	0	0
$647$	8185.55	0.497383	0.248692	−	0.968583i	$-0.419999\pi$
$647$	8185.55	0.497383	0.248692	+	0.968583i	$0.419999\pi$
$648$	0	0
$649$	19166.5	1.15925
$650$	0	0
$651$	5258.88	0.316608
$652$	0	0
$653$	−29817.7	−1.78692	−0.893458	−	0.449147i	$-0.851728\pi$
$653$	−29817.7	−1.78692	−0.893458	+	0.449147i	$0.851728\pi$
$654$	0	0
$655$	4643.53	0.277004
$656$	0	0
$657$	−892.242	−0.0529827
$658$	0	0
$659$	24829.1	1.46769	0.733843	−	0.679319i	$-0.237725\pi$
$659$	24829.1	1.46769	0.733843	+	0.679319i	$0.237725\pi$
$660$	0	0
$661$	25208.7	1.48337	0.741684	−	0.670750i	$-0.234027\pi$
$661$	25208.7	1.48337	0.741684	+	0.670750i	$0.234027\pi$
$662$	0	0
$663$	−664.327	−0.0389145
$664$	0	0
$665$	14999.1	0.874647
$666$	0	0
$667$	−1687.43	−0.0979574
$668$	0	0
$669$	5772.89	0.333621
$670$	0	0
$671$	23646.3	1.36044
$672$	0	0
$673$	−663.063	−0.0379780	−0.0189890	−	0.999820i	$-0.506045\pi$
$673$	−663.063	−0.0379780	−0.0189890	+	0.999820i	$0.506045\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−26514.9	−1.50524	−0.752621	−	0.658454i	$-0.771211\pi$
$677$	−26514.9	−1.50524	−0.752621	+	0.658454i	$0.771211\pi$
$678$	0	0
$679$	−4518.18	−0.255363
$680$	0	0
$681$	9807.72	0.551883
$682$	0	0
$683$	10672.0	0.597883	0.298942	−	0.954271i	$-0.403366\pi$
$683$	10672.0	0.597883	0.298942	+	0.954271i	$0.403366\pi$
$684$	0	0
$685$	5007.51	0.279310
$686$	0	0
$687$	12522.5	0.695432
$688$	0	0
$689$	−337.826	−0.0186794
$690$	0	0
$691$	4110.95	0.226321	0.113161	−	0.993577i	$-0.463903\pi$
$691$	4110.95	0.226321	0.113161	+	0.993577i	$0.463903\pi$
$692$	0	0
$693$	10375.4	0.568728
$694$	0	0
$695$	−11757.3	−0.641698
$696$	0	0
$697$	−2320.13	−0.126085
$698$	0	0
$699$	4802.53	0.259869
$700$	0	0
$701$	−2003.40	−0.107942	−0.0539711	−	0.998542i	$-0.517188\pi$
$701$	−2003.40	−0.107942	−0.0539711	+	0.998542i	$0.517188\pi$
$702$	0	0
$703$	−10924.7	−0.586107
$704$	0	0
$705$	89.8660	0.00480078
$706$	0	0
$707$	−18301.1	−0.973525
$708$	0	0
$709$	10991.7	0.582234	0.291117	−	0.956688i	$-0.405973\pi$
$709$	10991.7	0.582234	0.291117	+	0.956688i	$0.405973\pi$
$710$	0	0
$711$	4325.21	0.228141
$712$	0	0
$713$	11248.8	0.590843
$714$	0	0
$715$	3227.07	0.168791
$716$	0	0
$717$	4632.66	0.241297
$718$	0	0
$719$	24009.2	1.24533	0.622665	−	0.782489i	$-0.286050\pi$
$719$	24009.2	1.24533	0.622665	+	0.782489i	$0.286050\pi$
$720$	0	0
$721$	−5826.30	−0.300947
$722$	0	0
$723$	−2649.32	−0.136278
$724$	0	0
$725$	283.116	0.0145030
$726$	0	0
$727$	30166.0	1.53892	0.769460	−	0.638694i	$-0.220525\pi$
$727$	30166.0	1.53892	0.769460	+	0.638694i	$0.220525\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	4101.93	0.207545
$732$	0	0
$733$	−21467.6	−1.08175	−0.540875	−	0.841103i	$-0.681907\pi$
$733$	−21467.6	−1.08175	−0.540875	+	0.841103i	$0.681907\pi$
$734$	0	0
$735$	2942.71	0.147678
$736$	0	0
$737$	14834.4	0.741427
$738$	0	0
$739$	28578.3	1.42256	0.711279	−	0.702910i	$-0.248116\pi$
$739$	28578.3	1.42256	0.711279	+	0.702910i	$0.248116\pi$
$740$	0	0
$741$	−5038.40	−0.249784
$742$	0	0
$743$	13095.0	0.646581	0.323290	−	0.946300i	$-0.395211\pi$
$743$	13095.0	0.646581	0.323290	+	0.946300i	$0.395211\pi$
$744$	0	0
$745$	15734.2	0.773768
$746$	0	0
$747$	3579.86	0.175342
$748$	0	0
$749$	−14599.8	−0.712238
$750$	0	0
$751$	24389.9	1.18509	0.592543	−	0.805539i	$-0.298124\pi$
$751$	24389.9	1.18509	0.592543	+	0.805539i	$0.298124\pi$
$752$	0	0
$753$	−12899.0	−0.624255
$754$	0	0
$755$	−3516.93	−0.169529
$756$	0	0
$757$	24785.6	1.19002	0.595012	−	0.803717i	$-0.297147\pi$
$757$	24785.6	1.19002	0.595012	+	0.803717i	$0.297147\pi$
$758$	0	0
$759$	22193.1	1.06134
$760$	0	0
$761$	−18590.7	−0.885563	−0.442781	−	0.896630i	$-0.646008\pi$
$761$	−18590.7	−0.885563	−0.442781	+	0.896630i	$0.646008\pi$
$762$	0	0
$763$	−23766.9	−1.12768
$764$	0	0
$765$	766.531	0.0362274
$766$	0	0
$767$	−5018.70	−0.236264
$768$	0	0
$769$	−7587.31	−0.355794	−0.177897	−	0.984049i	$-0.556929\pi$
$769$	−7587.31	−0.355794	−0.177897	+	0.984049i	$0.556929\pi$
$770$	0	0
$771$	−4292.29	−0.200497
$772$	0	0
$773$	25128.8	1.16923	0.584617	−	0.811309i	$-0.301245\pi$
$773$	25128.8	1.16923	0.584617	+	0.811309i	$0.301245\pi$
$774$	0	0
$775$	−1887.32	−0.0874767
$776$	0	0
$777$	−5890.74	−0.271981
$778$	0	0
$779$	−17596.4	−0.809315
$780$	0	0
$781$	−32939.8	−1.50919
$782$	0	0
$783$	−305.765	−0.0139555
$784$	0	0
$785$	1473.27	0.0669849
$786$	0	0
$787$	13800.7	0.625084	0.312542	−	0.949904i	$-0.398819\pi$
$787$	13800.7	0.625084	0.312542	+	0.949904i	$0.398819\pi$
$788$	0	0
$789$	23049.9	1.04005
$790$	0	0
$791$	−53302.9	−2.39600
$792$	0	0
$793$	−6191.74	−0.277270
$794$	0	0
$795$	389.799	0.0173896
$796$	0	0
$797$	−12692.1	−0.564085	−0.282042	−	0.959402i	$-0.591012\pi$
$797$	−12692.1	−0.564085	−0.282042	+	0.959402i	$0.591012\pi$
$798$	0	0
$799$	−102.052	−0.00451857
$800$	0	0
$801$	−11872.5	−0.523715
$802$	0	0
$803$	−4921.92	−0.216302
$804$	0	0
$805$	17299.7	0.757434
$806$	0	0
$807$	−4790.39	−0.208959
$808$	0	0
$809$	−29998.7	−1.30371	−0.651854	−	0.758345i	$-0.726008\pi$
$809$	−29998.7	−1.30371	−0.651854	+	0.758345i	$0.726008\pi$
$810$	0	0
$811$	23498.4	1.01744	0.508718	−	0.860933i	$-0.330120\pi$
$811$	23498.4	1.01744	0.508718	+	0.860933i	$0.330120\pi$
$812$	0	0
$813$	−14174.0	−0.611445
$814$	0	0
$815$	−2688.30	−0.115542
$816$	0	0
$817$	31109.9	1.33219
$818$	0	0
$819$	−2716.77	−0.115912
$820$	0	0
$821$	−23543.7	−1.00083	−0.500415	−	0.865786i	$-0.666819\pi$
$821$	−23543.7	−1.00083	−0.500415	+	0.865786i	$0.666819\pi$
$822$	0	0
$823$	−27705.8	−1.17347	−0.586734	−	0.809780i	$-0.699587\pi$
$823$	−27705.8	−1.17347	−0.586734	+	0.809780i	$0.699587\pi$
$824$	0	0
$825$	−3723.54	−0.157136
$826$	0	0
$827$	−30947.0	−1.30125	−0.650625	−	0.759399i	$-0.725493\pi$
$827$	−30947.0	−1.30125	−0.650625	+	0.759399i	$0.725493\pi$
$828$	0	0
$829$	24338.4	1.01967	0.509836	−	0.860271i	$-0.329706\pi$
$829$	24338.4	1.01967	0.509836	+	0.860271i	$0.329706\pi$
$830$	0	0
$831$	−1838.39	−0.0767427
$832$	0	0
$833$	−3341.75	−0.138997
$834$	0	0
$835$	−7093.50	−0.293989
$836$	0	0
$837$	2038.30	0.0841745
$838$	0	0
$839$	17887.5	0.736048	0.368024	−	0.929816i	$-0.380034\pi$
$839$	17887.5	0.736048	0.368024	+	0.929816i	$0.380034\pi$
$840$	0	0
$841$	−24260.8	−0.994742
$842$	0	0
$843$	1785.60	0.0729528
$844$	0	0
$845$	−845.000	−0.0344010
$846$	0	0
$847$	26328.1	1.06806
$848$	0	0
$849$	−19349.7	−0.782189
$850$	0	0
$851$	−12600.4	−0.507562
$852$	0	0
$853$	−25173.5	−1.01046	−0.505231	−	0.862984i	$-0.668593\pi$
$853$	−25173.5	−1.01046	−0.505231	+	0.862984i	$0.668593\pi$
$854$	0	0
$855$	5813.54	0.232537
$856$	0	0
$857$	5689.32	0.226772	0.113386	−	0.993551i	$-0.463830\pi$
$857$	5689.32	0.226772	0.113386	+	0.993551i	$0.463830\pi$
$858$	0	0
$859$	−4295.98	−0.170637	−0.0853185	−	0.996354i	$-0.527191\pi$
$859$	−4295.98	−0.170637	−0.0853185	+	0.996354i	$0.527191\pi$
$860$	0	0
$861$	−9488.21	−0.375560
$862$	0	0
$863$	35970.1	1.41881	0.709407	−	0.704799i	$-0.248963\pi$
$863$	35970.1	1.41881	0.709407	+	0.704799i	$0.248963\pi$
$864$	0	0
$865$	8815.99	0.346535
$866$	0	0
$867$	13868.5	0.543252
$868$	0	0
$869$	23859.4	0.931386
$870$	0	0
$871$	−3884.35	−0.151109
$872$	0	0
$873$	−1751.21	−0.0678918
$874$	0	0
$875$	−2902.53	−0.112141
$876$	0	0
$877$	−25910.4	−0.997641	−0.498821	−	0.866705i	$-0.666233\pi$
$877$	−25910.4	−0.997641	−0.498821	+	0.866705i	$0.666233\pi$
$878$	0	0
$879$	−24959.8	−0.957761
$880$	0	0
$881$	17916.9	0.685171	0.342585	−	0.939487i	$-0.388697\pi$
$881$	17916.9	0.685171	0.342585	+	0.939487i	$0.388697\pi$
$882$	0	0
$883$	−44338.8	−1.68983	−0.844915	−	0.534901i	$-0.820349\pi$
$883$	−44338.8	−1.68983	−0.844915	+	0.534901i	$0.820349\pi$
$884$	0	0
$885$	5790.81	0.219950
$886$	0	0
$887$	38292.9	1.44955	0.724774	−	0.688986i	$-0.241944\pi$
$887$	38292.9	1.44955	0.724774	+	0.688986i	$0.241944\pi$
$888$	0	0
$889$	−6103.15	−0.230251
$890$	0	0
$891$	4021.42	0.151204
$892$	0	0
$893$	−773.985	−0.0290038
$894$	0	0
$895$	6510.13	0.243139
$896$	0	0
$897$	−5811.21	−0.216310
$898$	0	0
$899$	−854.927	−0.0317168
$900$	0	0
$901$	−442.656	−0.0163674
$902$	0	0
$903$	16774.9	0.618198
$904$	0	0
$905$	−16260.2	−0.597247
$906$	0	0
$907$	−30113.8	−1.10244	−0.551220	−	0.834360i	$-0.685837\pi$
$907$	−30113.8	−1.10244	−0.551220	+	0.834360i	$0.685837\pi$
$908$	0	0
$909$	−7093.35	−0.258825
$910$	0	0
$911$	20016.2	0.727953	0.363977	−	0.931408i	$-0.381419\pi$
$911$	20016.2	0.727953	0.363977	+	0.931408i	$0.381419\pi$
$912$	0	0
$913$	19747.8	0.715833
$914$	0	0
$915$	7144.31	0.258124
$916$	0	0
$917$	−21564.8	−0.776590
$918$	0	0
$919$	−46456.3	−1.66752	−0.833760	−	0.552127i	$-0.813816\pi$
$919$	−46456.3	−1.66752	−0.833760	+	0.552127i	$0.813816\pi$
$920$	0	0
$921$	16147.2	0.577708
$922$	0	0
$923$	8625.21	0.307586
$924$	0	0
$925$	2114.08	0.0751465
$926$	0	0
$927$	−2258.23	−0.0800108
$928$	0	0
$929$	−15074.1	−0.532363	−0.266181	−	0.963923i	$-0.585762\pi$
$929$	−15074.1	−0.532363	−0.266181	+	0.963923i	$0.585762\pi$
$930$	0	0
$931$	−25344.5	−0.892195
$932$	0	0
$933$	14718.9	0.516481
$934$	0	0
$935$	4228.46	0.147899
$936$	0	0
$937$	−4632.84	−0.161524	−0.0807621	−	0.996733i	$-0.525735\pi$
$937$	−4632.84	−0.161524	−0.0807621	+	0.996733i	$0.525735\pi$
$938$	0	0
$939$	3800.66	0.132087
$940$	0	0
$941$	−4489.76	−0.155539	−0.0777695	−	0.996971i	$-0.524780\pi$
$941$	−4489.76	−0.155539	−0.0777695	+	0.996971i	$0.524780\pi$
$942$	0	0
$943$	−20295.4	−0.700858
$944$	0	0
$945$	3134.74	0.107908
$946$	0	0
$947$	−7931.39	−0.272160	−0.136080	−	0.990698i	$-0.543450\pi$
$947$	−7931.39	−0.272160	−0.136080	+	0.990698i	$0.543450\pi$
$948$	0	0
$949$	1288.79	0.0440843
$950$	0	0
$951$	16726.0	0.570322
$952$	0	0
$953$	−5819.48	−0.197808	−0.0989042	−	0.995097i	$-0.531534\pi$
$953$	−5819.48	−0.197808	−0.0989042	+	0.995097i	$0.531534\pi$
$954$	0	0
$955$	1455.68	0.0493242
$956$	0	0
$957$	−1686.71	−0.0569734
$958$	0	0
$959$	−23255.1	−0.783052
$960$	0	0
$961$	−24091.9	−0.808696
$962$	0	0
$963$	−5658.78	−0.189358
$964$	0	0
$965$	16333.1	0.544851
$966$	0	0
$967$	26094.8	0.867789	0.433895	−	0.900964i	$-0.357139\pi$
$967$	26094.8	0.867789	0.433895	+	0.900964i	$0.357139\pi$
$968$	0	0
$969$	−6601.87	−0.218867
$970$	0	0
$971$	−4233.31	−0.139911	−0.0699554	−	0.997550i	$-0.522286\pi$
$971$	−4233.31	−0.139911	−0.0699554	+	0.997550i	$0.522286\pi$
$972$	0	0
$973$	54601.6	1.79902
$974$	0	0
$975$	975.000	0.0320256
$976$	0	0
$977$	53025.5	1.73637	0.868185	−	0.496240i	$-0.165286\pi$
$977$	53025.5	1.73637	0.868185	+	0.496240i	$0.165286\pi$
$978$	0	0
$979$	−65493.2	−2.13807
$980$	0	0
$981$	−9211.87	−0.299809
$982$	0	0
$983$	46282.5	1.50171	0.750856	−	0.660466i	$-0.229641\pi$
$983$	46282.5	1.50171	0.750856	+	0.660466i	$0.229641\pi$
$984$	0	0
$985$	−16055.4	−0.519359
$986$	0	0
$987$	−417.343	−0.0134591
$988$	0	0
$989$	35881.7	1.15366
$990$	0	0
$991$	−12335.4	−0.395405	−0.197703	−	0.980262i	$-0.563348\pi$
$991$	−12335.4	−0.395405	−0.197703	+	0.980262i	$0.563348\pi$
$992$	0	0
$993$	14263.4	0.455825
$994$	0	0
$995$	1797.79	0.0572800
$996$	0	0
$997$	21791.1	0.692207	0.346103	−	0.938196i	$-0.387505\pi$
$997$	21791.1	0.692207	0.346103	+	0.938196i	$0.387505\pi$
$998$	0	0
$999$	−2283.21	−0.0723098

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.4.a.l.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.4.a.l.1.4	✓	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1560.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} +23.2203 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -5.00000 q^{5} +23.2203 q^{7} +9.00000 q^{9} +49.6472 q^{11} -13.0000 q^{13} +15.0000 q^{15} -17.0340 q^{17} -129.190 q^{19} -69.6608 q^{21} -149.005 q^{23} +25.0000 q^{25} -27.0000 q^{27} +11.3246 q^{29} -75.4927 q^{31} -148.942 q^{33} -116.101 q^{35} +84.5632 q^{37} +39.0000 q^{39} +136.206 q^{41} -240.808 q^{43} -45.0000 q^{45} +5.99107 q^{47} +196.181 q^{49} +51.1021 q^{51} +25.9866 q^{53} -248.236 q^{55} +387.569 q^{57} +386.054 q^{59} +476.288 q^{61} +208.982 q^{63} +65.0000 q^{65} +298.796 q^{67} +447.016 q^{69} -663.478 q^{71} -99.1380 q^{73} -75.0000 q^{75} +1152.82 q^{77} +480.579 q^{79} +81.0000 q^{81} +397.762 q^{83} +85.1701 q^{85} -33.9739 q^{87} -1319.17 q^{89} -301.863 q^{91} +226.478 q^{93} +645.949 q^{95} -194.579 q^{97} +446.825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9	\( 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9} + 35 q^{11} - 52 q^{13} + 60 q^{15} - 9 q^{17} + 32 q^{19} + 33 q^{21} - 149 q^{23} + 100 q^{25} - 108 q^{27} + 168 q^{29} + 280 q^{31} - 105 q^{33} + 55 q^{35} + 37 q^{37} + 156 q^{39} + 247 q^{41} + 132 q^{43} - 180 q^{45} + 217 q^{49} + 27 q^{51} + 247 q^{53} - 175 q^{55} - 96 q^{57} + 614 q^{59} + 719 q^{61} - 99 q^{63} + 260 q^{65} + 658 q^{67} + 447 q^{69} + 939 q^{71} - 1452 q^{73} - 300 q^{75} + 199 q^{77} + 289 q^{79} + 324 q^{81} - 118 q^{83} + 45 q^{85} - 504 q^{87} - 1319 q^{89} + 143 q^{91} - 840 q^{93} - 160 q^{95} - 993 q^{97} + 315 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9 + 35 * q^11 - 52 * q^13 + 60 * q^15 - 9 * q^17 + 32 * q^19 + 33 * q^21 - 149 * q^23 + 100 * q^25 - 108 * q^27 + 168 * q^29 + 280 * q^31 - 105 * q^33 + 55 * q^35 + 37 * q^37 + 156 * q^39 + 247 * q^41 + 132 * q^43 - 180 * q^45 + 217 * q^49 + 27 * q^51 + 247 * q^53 - 175 * q^55 - 96 * q^57 + 614 * q^59 + 719 * q^61 - 99 * q^63 + 260 * q^65 + 658 * q^67 + 447 * q^69 + 939 * q^71 - 1452 * q^73 - 300 * q^75 + 199 * q^77 + 289 * q^79 + 324 * q^81 - 118 * q^83 + 45 * q^85 - 504 * q^87 - 1319 * q^89 + 143 * q^91 - 840 * q^93 - 160 * q^95 - 993 * q^97 + 315 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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