LMFDB - Embedded newform 840.4.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [840,4,Mod(1,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("840.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-9,0,-15,0,-21,0,27,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$49.5616044048$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.821313.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 117x - 340 $ x^3 - 117*x - 340
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$12.0505$ of defining polynomial
Character	$\chi$	$=$	840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +15.1389 q^{11} -5.13892 q^{13} +15.0000 q^{15} +92.1262 q^{17} -149.669 q^{19} +21.0000 q^{21} +96.4040 q^{23} +25.0000 q^{25} -27.0000 q^{27} +88.2778 q^{29} +117.265 q^{31} -45.4168 q^{33} +35.0000 q^{35} -207.212 q^{37} +15.4168 q^{39} -7.57050 q^{41} +524.550 q^{43} -45.0000 q^{45} -290.682 q^{47} +49.0000 q^{49} -276.379 q^{51} -315.366 q^{53} -75.6946 q^{55} +449.007 q^{57} -500.197 q^{59} +117.642 q^{61} -63.0000 q^{63} +25.6946 q^{65} -805.258 q^{67} -289.212 q^{69} -462.629 q^{71} +39.9766 q^{73} -75.0000 q^{75} -105.972 q^{77} +659.636 q^{79} +81.0000 q^{81} -1228.10 q^{83} -460.631 q^{85} -264.834 q^{87} +451.262 q^{89} +35.9724 q^{91} -351.795 q^{93} +748.346 q^{95} -1.04780 q^{97} +136.250 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 15 q^{5} - 21 q^{7} + 27 q^{9} - 6 q^{11} + 36 q^{13} + 45 q^{15} + 90 q^{17} + 78 q^{19} + 63 q^{21} + 75 q^{25} - 81 q^{27} + 162 q^{29} + 114 q^{31} + 18 q^{33} + 105 q^{35} + 246 q^{37}+ \cdots - 54 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 - 15 * q^5 - 21 * q^7 + 27 * q^9 - 6 * q^11 + 36 * q^13 + 45 * q^15 + 90 * q^17 + 78 * q^19 + 63 * q^21 + 75 * q^25 - 81 * q^27 + 162 * q^29 + 114 * q^31 + 18 * q^33 + 105 * q^35 + 246 * q^37 - 108 * q^39 - 42 * q^41 - 348 * q^43 - 135 * q^45 - 480 * q^47 + 147 * q^49 - 270 * q^51 - 252 * q^53 + 30 * q^55 - 234 * q^57 - 408 * q^59 - 534 * q^61 - 189 * q^63 - 180 * q^65 - 372 * q^67 - 366 * q^71 - 60 * q^73 - 225 * q^75 + 42 * q^77 - 624 * q^79 + 243 * q^81 - 3228 * q^83 - 450 * q^85 - 486 * q^87 - 510 * q^89 - 252 * q^91 - 342 * q^93 - 390 * q^95 - 2304 * q^97 - 54 * 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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	15.1389	0.414960	0.207480	−	0.978239i	$-0.433474\pi$
$11$	15.1389	0.414960	0.207480	+	0.978239i	$0.433474\pi$
$12$	0	0
$13$	−5.13892	−0.109637	−0.0548185	−	0.998496i	$-0.517458\pi$
$13$	−5.13892	−0.109637	−0.0548185	+	0.998496i	$0.517458\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	92.1262	1.31435	0.657173	−	0.753740i	$-0.271752\pi$
$17$	92.1262	1.31435	0.657173	+	0.753740i	$0.271752\pi$
$18$	0	0
$19$	−149.669	−1.80718	−0.903591	−	0.428397i	$-0.859079\pi$
$19$	−149.669	−1.80718	−0.903591	+	0.428397i	$0.859079\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	96.4040	0.873984	0.436992	−	0.899465i	$-0.356044\pi$
$23$	96.4040	0.873984	0.436992	+	0.899465i	$0.356044\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	88.2778	0.565268	0.282634	−	0.959228i	$-0.408792\pi$
$29$	88.2778	0.565268	0.282634	+	0.959228i	$0.408792\pi$
$30$	0	0
$31$	117.265	0.679401	0.339701	−	0.940534i	$-0.389674\pi$
$31$	117.265	0.679401	0.339701	+	0.940534i	$0.389674\pi$
$32$	0	0
$33$	−45.4168	−0.239577
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	−207.212	−0.920688	−0.460344	−	0.887741i	$-0.652274\pi$
$37$	−207.212	−0.920688	−0.460344	+	0.887741i	$0.652274\pi$
$38$	0	0
$39$	15.4168	0.0632989
$40$	0	0
$41$	−7.57050	−0.0288369	−0.0144185	−	0.999896i	$-0.504590\pi$
$41$	−7.57050	−0.0288369	−0.0144185	+	0.999896i	$0.504590\pi$
$42$	0	0
$43$	524.550	1.86031	0.930153	−	0.367172i	$-0.119674\pi$
$43$	524.550	1.86031	0.930153	+	0.367172i	$0.119674\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−290.682	−0.902134	−0.451067	−	0.892490i	$-0.648956\pi$
$47$	−290.682	−0.902134	−0.451067	+	0.892490i	$0.648956\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−276.379	−0.758838
$52$	0	0
$53$	−315.366	−0.817336	−0.408668	−	0.912683i	$-0.634007\pi$
$53$	−315.366	−0.817336	−0.408668	+	0.912683i	$0.634007\pi$
$54$	0	0
$55$	−75.6946	−0.185576
$56$	0	0
$57$	449.007	1.04338
$58$	0	0
$59$	−500.197	−1.10373	−0.551866	−	0.833933i	$-0.686084\pi$
$59$	−500.197	−1.10373	−0.551866	+	0.833933i	$0.686084\pi$
$60$	0	0
$61$	117.642	0.246926	0.123463	−	0.992349i	$-0.460600\pi$
$61$	117.642	0.246926	0.123463	+	0.992349i	$0.460600\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	25.6946	0.0490311
$66$	0	0
$67$	−805.258	−1.46833	−0.734163	−	0.678973i	$-0.762425\pi$
$67$	−805.258	−1.46833	−0.734163	+	0.678973i	$0.762425\pi$
$68$	0	0
$69$	−289.212	−0.504595
$70$	0	0
$71$	−462.629	−0.773295	−0.386647	−	0.922228i	$-0.626367\pi$
$71$	−462.629	−0.773295	−0.386647	+	0.922228i	$0.626367\pi$
$72$	0	0
$73$	39.9766	0.0640946	0.0320473	−	0.999486i	$-0.489797\pi$
$73$	39.9766	0.0640946	0.0320473	+	0.999486i	$0.489797\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−105.972	−0.156840
$78$	0	0
$79$	659.636	0.939429	0.469714	−	0.882818i	$-0.344357\pi$
$79$	659.636	0.939429	0.469714	+	0.882818i	$0.344357\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1228.10	−1.62412	−0.812058	−	0.583577i	$-0.801653\pi$
$83$	−1228.10	−1.62412	−0.812058	+	0.583577i	$0.801653\pi$
$84$	0	0
$85$	−460.631	−0.587793
$86$	0	0
$87$	−264.834	−0.326358
$88$	0	0
$89$	451.262	0.537457	0.268728	−	0.963216i	$-0.413397\pi$
$89$	451.262	0.537457	0.268728	+	0.963216i	$0.413397\pi$
$90$	0	0
$91$	35.9724	0.0414389
$92$	0	0
$93$	−351.795	−0.392252
$94$	0	0
$95$	748.346	0.808196
$96$	0	0
$97$	−1.04780	−0.00109678	−0.000548392	−	1.00000i	$-0.500175\pi$
$97$	−1.04780	−0.00109678	−0.000548392		1.00000i	$0.500175\pi$
$98$	0	0
$99$	136.250	0.138320
$100$	0	0
$101$	333.439	0.328499	0.164250	−	0.986419i	$-0.447480\pi$
$101$	333.439	0.328499	0.164250	+	0.986419i	$0.447480\pi$
$102$	0	0
$103$	−1877.25	−1.79584	−0.897918	−	0.440164i	$-0.854920\pi$
$103$	−1877.25	−1.79584	−0.897918	+	0.440164i	$0.854920\pi$
$104$	0	0
$105$	−105.000	−0.0975900
$106$	0	0
$107$	−1220.81	−1.10300	−0.551498	−	0.834176i	$-0.685943\pi$
$107$	−1220.81	−1.10300	−0.551498	+	0.834176i	$0.685943\pi$
$108$	0	0
$109$	1210.94	1.06411	0.532053	−	0.846711i	$-0.321421\pi$
$109$	1210.94	1.06411	0.532053	+	0.846711i	$0.321421\pi$
$110$	0	0
$111$	621.636	0.531559
$112$	0	0
$113$	−1970.08	−1.64009	−0.820043	−	0.572302i	$-0.806051\pi$
$113$	−1970.08	−1.64009	−0.820043	+	0.572302i	$0.806051\pi$
$114$	0	0
$115$	−482.020	−0.390857
$116$	0	0
$117$	−46.2503	−0.0365456
$118$	0	0
$119$	−644.883	−0.496776
$120$	0	0
$121$	−1101.81	−0.827809
$122$	0	0
$123$	22.7115	0.0166490
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−667.242	−0.466206	−0.233103	−	0.972452i	$-0.574888\pi$
$127$	−667.242	−0.466206	−0.233103	+	0.972452i	$0.574888\pi$
$128$	0	0
$129$	−1573.65	−1.07405
$130$	0	0
$131$	−248.870	−0.165984	−0.0829918	−	0.996550i	$-0.526448\pi$
$131$	−248.870	−0.165984	−0.0829918	+	0.996550i	$0.526448\pi$
$132$	0	0
$133$	1047.68	0.683050
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−372.532	−0.232318	−0.116159	−	0.993231i	$-0.537058\pi$
$137$	−372.532	−0.232318	−0.116159	+	0.993231i	$0.537058\pi$
$138$	0	0
$139$	−1069.58	−0.652665	−0.326332	−	0.945255i	$-0.605813\pi$
$139$	−1069.58	−0.652665	−0.326332	+	0.945255i	$0.605813\pi$
$140$	0	0
$141$	872.046	0.520847
$142$	0	0
$143$	−77.7977	−0.0454949
$144$	0	0
$145$	−441.389	−0.252796
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	2084.22	1.14595	0.572973	−	0.819574i	$-0.305790\pi$
$149$	2084.22	1.14595	0.572973	+	0.819574i	$0.305790\pi$
$150$	0	0
$151$	770.741	0.415378	0.207689	−	0.978195i	$-0.433406\pi$
$151$	770.741	0.415378	0.207689	+	0.978195i	$0.433406\pi$
$152$	0	0
$153$	829.136	0.438115
$154$	0	0
$155$	−586.325	−0.303837
$156$	0	0
$157$	−1638.86	−0.833090	−0.416545	−	0.909115i	$-0.636759\pi$
$157$	−1638.86	−0.833090	−0.416545	+	0.909115i	$0.636759\pi$
$158$	0	0
$159$	946.097	0.471889
$160$	0	0
$161$	−674.828	−0.330335
$162$	0	0
$163$	387.337	0.186126	0.0930631	−	0.995660i	$-0.470334\pi$
$163$	387.337	0.186126	0.0930631	+	0.995660i	$0.470334\pi$
$164$	0	0
$165$	227.084	0.107142
$166$	0	0
$167$	−3356.70	−1.55539	−0.777693	−	0.628645i	$-0.783610\pi$
$167$	−3356.70	−1.55539	−0.777693	+	0.628645i	$0.783610\pi$
$168$	0	0
$169$	−2170.59	−0.987980
$170$	0	0
$171$	−1347.02	−0.602394
$172$	0	0
$173$	163.309	0.0717695	0.0358847	−	0.999356i	$-0.488575\pi$
$173$	163.309	0.0717695	0.0358847	+	0.999356i	$0.488575\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	1500.59	0.637239
$178$	0	0
$179$	−4375.10	−1.82687	−0.913437	−	0.406980i	$-0.866582\pi$
$179$	−4375.10	−1.82687	−0.913437	+	0.406980i	$0.866582\pi$
$180$	0	0
$181$	746.210	0.306438	0.153219	−	0.988192i	$-0.451036\pi$
$181$	746.210	0.306438	0.153219	+	0.988192i	$0.451036\pi$
$182$	0	0
$183$	−352.925	−0.142563
$184$	0	0
$185$	1036.06	0.411744
$186$	0	0
$187$	1394.69	0.545400
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−4156.89	−1.57478	−0.787388	−	0.616458i	$-0.788567\pi$
$191$	−4156.89	−1.57478	−0.787388	+	0.616458i	$0.788567\pi$
$192$	0	0
$193$	4705.67	1.75504	0.877518	−	0.479544i	$-0.159198\pi$
$193$	4705.67	1.75504	0.877518	+	0.479544i	$0.159198\pi$
$194$	0	0
$195$	−77.0838	−0.0283081
$196$	0	0
$197$	−504.948	−0.182620	−0.0913099	−	0.995823i	$-0.529105\pi$
$197$	−504.948	−0.182620	−0.0913099	+	0.995823i	$0.529105\pi$
$198$	0	0
$199$	335.204	0.119407	0.0597034	−	0.998216i	$-0.480985\pi$
$199$	335.204	0.119407	0.0597034	+	0.998216i	$0.480985\pi$
$200$	0	0
$201$	2415.77	0.847739
$202$	0	0
$203$	−617.945	−0.213651
$204$	0	0
$205$	37.8525	0.0128963
$206$	0	0
$207$	867.636	0.291328
$208$	0	0
$209$	−2265.83	−0.749907
$210$	0	0
$211$	609.859	0.198978	0.0994891	−	0.995039i	$-0.468279\pi$
$211$	609.859	0.198978	0.0994891	+	0.995039i	$0.468279\pi$
$212$	0	0
$213$	1387.89	0.446462
$214$	0	0
$215$	−2622.75	−0.831954
$216$	0	0
$217$	−820.856	−0.256789
$218$	0	0
$219$	−119.930	−0.0370050
$220$	0	0
$221$	−473.429	−0.144101
$222$	0	0
$223$	−4754.27	−1.42766	−0.713832	−	0.700317i	$-0.753042\pi$
$223$	−4754.27	−1.42766	−0.713832	+	0.700317i	$0.753042\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−2623.86	−0.767187	−0.383594	−	0.923502i	$-0.625314\pi$
$227$	−2623.86	−0.767187	−0.383594	+	0.923502i	$0.625314\pi$
$228$	0	0
$229$	−3793.83	−1.09477	−0.547387	−	0.836880i	$-0.684377\pi$
$229$	−3793.83	−1.09477	−0.547387	+	0.836880i	$0.684377\pi$
$230$	0	0
$231$	317.917	0.0905516
$232$	0	0
$233$	4264.30	1.19899	0.599493	−	0.800380i	$-0.295369\pi$
$233$	4264.30	1.19899	0.599493	+	0.800380i	$0.295369\pi$
$234$	0	0
$235$	1453.41	0.403447
$236$	0	0
$237$	−1978.91	−0.542379
$238$	0	0
$239$	4557.87	1.23357	0.616787	−	0.787130i	$-0.288434\pi$
$239$	4557.87	1.23357	0.616787	+	0.787130i	$0.288434\pi$
$240$	0	0
$241$	4643.17	1.24105	0.620524	−	0.784187i	$-0.286920\pi$
$241$	4643.17	1.24105	0.620524	+	0.784187i	$0.286920\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	769.138	0.198134
$248$	0	0
$249$	3684.30	0.937683
$250$	0	0
$251$	2578.36	0.648386	0.324193	−	0.945991i	$-0.394907\pi$
$251$	2578.36	0.648386	0.324193	+	0.945991i	$0.394907\pi$
$252$	0	0
$253$	1459.45	0.362668
$254$	0	0
$255$	1381.89	0.339363
$256$	0	0
$257$	−3339.44	−0.810538	−0.405269	−	0.914197i	$-0.632822\pi$
$257$	−3339.44	−0.810538	−0.405269	+	0.914197i	$0.632822\pi$
$258$	0	0
$259$	1450.48	0.347987
$260$	0	0
$261$	794.501	0.188423
$262$	0	0
$263$	6948.29	1.62909	0.814543	−	0.580103i	$-0.196988\pi$
$263$	6948.29	1.62909	0.814543	+	0.580103i	$0.196988\pi$
$264$	0	0
$265$	1576.83	0.365524
$266$	0	0
$267$	−1353.79	−0.310301
$268$	0	0
$269$	128.376	0.0290975	0.0145487	−	0.999894i	$-0.495369\pi$
$269$	128.376	0.0290975	0.0145487	+	0.999894i	$0.495369\pi$
$270$	0	0
$271$	4003.21	0.897335	0.448668	−	0.893699i	$-0.351899\pi$
$271$	4003.21	0.897335	0.448668	+	0.893699i	$0.351899\pi$
$272$	0	0
$273$	−107.917	−0.0239247
$274$	0	0
$275$	378.473	0.0829919
$276$	0	0
$277$	−2924.87	−0.634434	−0.317217	−	0.948353i	$-0.602748\pi$
$277$	−2924.87	−0.634434	−0.317217	+	0.948353i	$0.602748\pi$
$278$	0	0
$279$	1055.39	0.226467
$280$	0	0
$281$	1837.84	0.390166	0.195083	−	0.980787i	$-0.437502\pi$
$281$	1837.84	0.390166	0.195083	+	0.980787i	$0.437502\pi$
$282$	0	0
$283$	5864.82	1.23190	0.615949	−	0.787786i	$-0.288773\pi$
$283$	5864.82	1.23190	0.615949	+	0.787786i	$0.288773\pi$
$284$	0	0
$285$	−2245.04	−0.466612
$286$	0	0
$287$	52.9935	0.0108993
$288$	0	0
$289$	3574.23	0.727505
$290$	0	0
$291$	3.14340	0.000633228 0
$292$	0	0
$293$	1553.08	0.309665	0.154832	−	0.987941i	$-0.450516\pi$
$293$	1553.08	0.309665	0.154832	+	0.987941i	$0.450516\pi$
$294$	0	0
$295$	2500.99	0.493604
$296$	0	0
$297$	−408.751	−0.0798590
$298$	0	0
$299$	−495.412	−0.0958209
$300$	0	0
$301$	−3671.85	−0.703130
$302$	0	0
$303$	−1000.32	−0.189659
$304$	0	0
$305$	−588.208	−0.110428
$306$	0	0
$307$	−9088.29	−1.68956	−0.844782	−	0.535111i	$-0.820270\pi$
$307$	−9088.29	−1.68956	−0.844782	+	0.535111i	$0.820270\pi$
$308$	0	0
$309$	5631.75	1.03683
$310$	0	0
$311$	607.829	0.110826	0.0554129	−	0.998464i	$-0.482353\pi$
$311$	607.829	0.110826	0.0554129	+	0.998464i	$0.482353\pi$
$312$	0	0
$313$	−4474.40	−0.808012	−0.404006	−	0.914756i	$-0.632383\pi$
$313$	−4474.40	−0.808012	−0.404006	+	0.914756i	$0.632383\pi$
$314$	0	0
$315$	315.000	0.0563436
$316$	0	0
$317$	−4608.87	−0.816593	−0.408297	−	0.912849i	$-0.633877\pi$
$317$	−4608.87	−0.816593	−0.408297	+	0.912849i	$0.633877\pi$
$318$	0	0
$319$	1336.43	0.234564
$320$	0	0
$321$	3662.44	0.636815
$322$	0	0
$323$	−13788.4	−2.37526
$324$	0	0
$325$	−128.473	−0.0219274
$326$	0	0
$327$	−3632.83	−0.614361
$328$	0	0
$329$	2034.77	0.340975
$330$	0	0
$331$	−173.392	−0.0287930	−0.0143965	−	0.999896i	$-0.504583\pi$
$331$	−173.392	−0.0287930	−0.0143965	+	0.999896i	$0.504583\pi$
$332$	0	0
$333$	−1864.91	−0.306896
$334$	0	0
$335$	4026.29	0.656656
$336$	0	0
$337$	5905.79	0.954627	0.477313	−	0.878733i	$-0.341611\pi$
$337$	5905.79	0.954627	0.477313	+	0.878733i	$0.341611\pi$
$338$	0	0
$339$	5910.25	0.946904
$340$	0	0
$341$	1775.27	0.281924
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	1446.06	0.225662
$346$	0	0
$347$	2991.10	0.462740	0.231370	−	0.972866i	$-0.425679\pi$
$347$	2991.10	0.462740	0.231370	+	0.972866i	$0.425679\pi$
$348$	0	0
$349$	4697.34	0.720466	0.360233	−	0.932862i	$-0.382697\pi$
$349$	4697.34	0.720466	0.360233	+	0.932862i	$0.382697\pi$
$350$	0	0
$351$	138.751	0.0210996
$352$	0	0
$353$	−6315.00	−0.952163	−0.476082	−	0.879401i	$-0.657943\pi$
$353$	−6315.00	−0.952163	−0.476082	+	0.879401i	$0.657943\pi$
$354$	0	0
$355$	2313.14	0.345828
$356$	0	0
$357$	1934.65	0.286814
$358$	0	0
$359$	−6877.46	−1.01108	−0.505541	−	0.862803i	$-0.668707\pi$
$359$	−6877.46	−1.01108	−0.505541	+	0.862803i	$0.668707\pi$
$360$	0	0
$361$	15541.8	2.26590
$362$	0	0
$363$	3305.44	0.477935
$364$	0	0
$365$	−199.883	−0.0286640
$366$	0	0
$367$	11964.7	1.70178	0.850888	−	0.525348i	$-0.176065\pi$
$367$	11964.7	1.70178	0.850888	+	0.525348i	$0.176065\pi$
$368$	0	0
$369$	−68.1345	−0.00961230
$370$	0	0
$371$	2207.56	0.308924
$372$	0	0
$373$	9640.01	1.33818	0.669090	−	0.743182i	$-0.266684\pi$
$373$	9640.01	1.33818	0.669090	+	0.743182i	$0.266684\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−453.653	−0.0619743
$378$	0	0
$379$	−12231.0	−1.65769	−0.828845	−	0.559478i	$-0.811002\pi$
$379$	−12231.0	−1.65769	−0.828845	+	0.559478i	$0.811002\pi$
$380$	0	0
$381$	2001.73	0.269164
$382$	0	0
$383$	−7501.51	−1.00081	−0.500404	−	0.865792i	$-0.666815\pi$
$383$	−7501.51	−1.00081	−0.500404	+	0.865792i	$0.666815\pi$
$384$	0	0
$385$	529.862	0.0701410
$386$	0	0
$387$	4720.95	0.620102
$388$	0	0
$389$	442.505	0.0576758	0.0288379	−	0.999584i	$-0.490819\pi$
$389$	442.505	0.0576758	0.0288379	+	0.999584i	$0.490819\pi$
$390$	0	0
$391$	8881.33	1.14872
$392$	0	0
$393$	746.610	0.0958307
$394$	0	0
$395$	−3298.18	−0.420125
$396$	0	0
$397$	4903.36	0.619880	0.309940	−	0.950756i	$-0.399691\pi$
$397$	4903.36	0.619880	0.309940	+	0.950756i	$0.399691\pi$
$398$	0	0
$399$	−3143.05	−0.394359
$400$	0	0
$401$	12478.7	1.55400	0.777001	−	0.629500i	$-0.216740\pi$
$401$	12478.7	1.55400	0.777001	+	0.629500i	$0.216740\pi$
$402$	0	0
$403$	−602.616	−0.0744874
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−3136.97	−0.382048
$408$	0	0
$409$	−4627.07	−0.559399	−0.279699	−	0.960088i	$-0.590235\pi$
$409$	−4627.07	−0.559399	−0.279699	+	0.960088i	$0.590235\pi$
$410$	0	0
$411$	1117.60	0.134129
$412$	0	0
$413$	3501.38	0.417171
$414$	0	0
$415$	6140.50	0.726326
$416$	0	0
$417$	3208.73	0.376816
$418$	0	0
$419$	−6137.78	−0.715633	−0.357816	−	0.933792i	$-0.616479\pi$
$419$	−6137.78	−0.715633	−0.357816	+	0.933792i	$0.616479\pi$
$420$	0	0
$421$	−11384.4	−1.31791	−0.658956	−	0.752182i	$-0.729002\pi$
$421$	−11384.4	−1.31791	−0.658956	+	0.752182i	$0.729002\pi$
$422$	0	0
$423$	−2616.14	−0.300711
$424$	0	0
$425$	2303.15	0.262869
$426$	0	0
$427$	−823.491	−0.0933291
$428$	0	0
$429$	233.393	0.0262665
$430$	0	0
$431$	5070.20	0.566643	0.283321	−	0.959025i	$-0.408564\pi$
$431$	5070.20	0.566643	0.283321	+	0.959025i	$0.408564\pi$
$432$	0	0
$433$	3694.82	0.410073	0.205037	−	0.978754i	$-0.434269\pi$
$433$	3694.82	0.410073	0.205037	+	0.978754i	$0.434269\pi$
$434$	0	0
$435$	1324.17	0.145952
$436$	0	0
$437$	−14428.7	−1.57945
$438$	0	0
$439$	8787.71	0.955386	0.477693	−	0.878527i	$-0.341473\pi$
$439$	8787.71	0.955386	0.477693	+	0.878527i	$0.341473\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	2129.70	0.228409	0.114205	−	0.993457i	$-0.463568\pi$
$443$	2129.70	0.228409	0.114205	+	0.993457i	$0.463568\pi$
$444$	0	0
$445$	−2256.31	−0.240358
$446$	0	0
$447$	−6252.66	−0.661612
$448$	0	0
$449$	−7821.51	−0.822093	−0.411047	−	0.911614i	$-0.634837\pi$
$449$	−7821.51	−0.822093	−0.411047	+	0.911614i	$0.634837\pi$
$450$	0	0
$451$	−114.609	−0.0119661
$452$	0	0
$453$	−2312.22	−0.239818
$454$	0	0
$455$	−179.862	−0.0185320
$456$	0	0
$457$	−11192.3	−1.14563	−0.572817	−	0.819683i	$-0.694149\pi$
$457$	−11192.3	−1.14563	−0.572817	+	0.819683i	$0.694149\pi$
$458$	0	0
$459$	−2487.41	−0.252946
$460$	0	0
$461$	5287.96	0.534240	0.267120	−	0.963663i	$-0.413928\pi$
$461$	5287.96	0.534240	0.267120	+	0.963663i	$0.413928\pi$
$462$	0	0
$463$	10311.0	1.03497	0.517487	−	0.855691i	$-0.326867\pi$
$463$	10311.0	1.03497	0.517487	+	0.855691i	$0.326867\pi$
$464$	0	0
$465$	1758.98	0.175421
$466$	0	0
$467$	−13752.6	−1.36273	−0.681366	−	0.731943i	$-0.738614\pi$
$467$	−13752.6	−1.36273	−0.681366	+	0.731943i	$0.738614\pi$
$468$	0	0
$469$	5636.80	0.554975
$470$	0	0
$471$	4916.58	0.480985
$472$	0	0
$473$	7941.12	0.771952
$474$	0	0
$475$	−3741.73	−0.361436
$476$	0	0
$477$	−2838.29	−0.272445
$478$	0	0
$479$	−10897.0	−1.03945	−0.519724	−	0.854334i	$-0.673965\pi$
$479$	−10897.0	−1.03945	−0.519724	+	0.854334i	$0.673965\pi$
$480$	0	0
$481$	1064.85	0.100941
$482$	0	0
$483$	2024.48	0.190719
$484$	0	0
$485$	5.23900	0.000490496 0
$486$	0	0
$487$	1423.07	0.132413	0.0662066	−	0.997806i	$-0.478910\pi$
$487$	1423.07	0.132413	0.0662066	+	0.997806i	$0.478910\pi$
$488$	0	0
$489$	−1162.01	−0.107460
$490$	0	0
$491$	−383.757	−0.0352724	−0.0176362	−	0.999844i	$-0.505614\pi$
$491$	−383.757	−0.0352724	−0.0176362	+	0.999844i	$0.505614\pi$
$492$	0	0
$493$	8132.70	0.742958
$494$	0	0
$495$	−681.251	−0.0618585
$496$	0	0
$497$	3238.40	0.292278
$498$	0	0
$499$	13625.0	1.22232	0.611162	−	0.791505i	$-0.290702\pi$
$499$	13625.0	1.22232	0.611162	+	0.791505i	$0.290702\pi$
$500$	0	0
$501$	10070.1	0.898002
$502$	0	0
$503$	−10203.3	−0.904456	−0.452228	−	0.891902i	$-0.649371\pi$
$503$	−10203.3	−0.904456	−0.452228	+	0.891902i	$0.649371\pi$
$504$	0	0
$505$	−1667.19	−0.146909
$506$	0	0
$507$	6511.77	0.570410
$508$	0	0
$509$	−5919.84	−0.515505	−0.257753	−	0.966211i	$-0.582982\pi$
$509$	−5919.84	−0.515505	−0.257753	+	0.966211i	$0.582982\pi$
$510$	0	0
$511$	−279.836	−0.0242255
$512$	0	0
$513$	4041.07	0.347792
$514$	0	0
$515$	9386.25	0.803122
$516$	0	0
$517$	−4400.61	−0.374349
$518$	0	0
$519$	−489.926	−0.0414361
$520$	0	0
$521$	10282.6	0.864661	0.432331	−	0.901715i	$-0.357691\pi$
$521$	10282.6	0.864661	0.432331	+	0.901715i	$0.357691\pi$
$522$	0	0
$523$	−859.562	−0.0718662	−0.0359331	−	0.999354i	$-0.511440\pi$
$523$	−859.562	−0.0718662	−0.0359331	+	0.999354i	$0.511440\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	10803.2	0.892968
$528$	0	0
$529$	−2873.27	−0.236152
$530$	0	0
$531$	−4501.78	−0.367910
$532$	0	0
$533$	38.9042	0.00316159
$534$	0	0
$535$	6104.07	0.493274
$536$	0	0
$537$	13125.3	1.05475
$538$	0	0
$539$	741.807	0.0592799
$540$	0	0
$541$	12145.4	0.965196	0.482598	−	0.875842i	$-0.339693\pi$
$541$	12145.4	0.965196	0.482598	+	0.875842i	$0.339693\pi$
$542$	0	0
$543$	−2238.63	−0.176922
$544$	0	0
$545$	−6054.72	−0.475882
$546$	0	0
$547$	−6958.96	−0.543956	−0.271978	−	0.962304i	$-0.587678\pi$
$547$	−6958.96	−0.543956	−0.271978	+	0.962304i	$0.587678\pi$
$548$	0	0
$549$	1058.77	0.0823085
$550$	0	0
$551$	−13212.5	−1.02154
$552$	0	0
$553$	−4617.45	−0.355071
$554$	0	0
$555$	−3108.18	−0.237721
$556$	0	0
$557$	5172.98	0.393512	0.196756	−	0.980452i	$-0.436959\pi$
$557$	5172.98	0.393512	0.196756	+	0.980452i	$0.436959\pi$
$558$	0	0
$559$	−2695.62	−0.203958
$560$	0	0
$561$	−4184.07	−0.314887
$562$	0	0
$563$	−18219.8	−1.36390	−0.681948	−	0.731401i	$-0.738867\pi$
$563$	−18219.8	−1.36390	−0.681948	+	0.731401i	$0.738867\pi$
$564$	0	0
$565$	9850.41	0.733469
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−824.973	−0.0607815	−0.0303907	−	0.999538i	$-0.509675\pi$
$569$	−824.973	−0.0607815	−0.0303907	+	0.999538i	$0.509675\pi$
$570$	0	0
$571$	16415.6	1.20310	0.601550	−	0.798836i	$-0.294550\pi$
$571$	16415.6	1.20310	0.601550	+	0.798836i	$0.294550\pi$
$572$	0	0
$573$	12470.7	0.909197
$574$	0	0
$575$	2410.10	0.174797
$576$	0	0
$577$	−23308.7	−1.68172	−0.840862	−	0.541250i	$-0.817951\pi$
$577$	−23308.7	−1.68172	−0.840862	+	0.541250i	$0.817951\pi$
$578$	0	0
$579$	−14117.0	−1.01327
$580$	0	0
$581$	8596.70	0.613858
$582$	0	0
$583$	−4774.30	−0.339162
$584$	0	0
$585$	231.251	0.0163437
$586$	0	0
$587$	1713.37	0.120474	0.0602371	−	0.998184i	$-0.480814\pi$
$587$	1713.37	0.120474	0.0602371	+	0.998184i	$0.480814\pi$
$588$	0	0
$589$	−17551.0	−1.22780
$590$	0	0
$591$	1514.85	0.105436
$592$	0	0
$593$	−1729.08	−0.119738	−0.0598690	−	0.998206i	$-0.519068\pi$
$593$	−1729.08	−0.119738	−0.0598690	+	0.998206i	$0.519068\pi$
$594$	0	0
$595$	3224.42	0.222165
$596$	0	0
$597$	−1005.61	−0.0689396
$598$	0	0
$599$	9767.25	0.666242	0.333121	−	0.942884i	$-0.391898\pi$
$599$	9767.25	0.666242	0.333121	+	0.942884i	$0.391898\pi$
$600$	0	0
$601$	−17189.1	−1.16665	−0.583326	−	0.812238i	$-0.698249\pi$
$601$	−17189.1	−1.16665	−0.583326	+	0.812238i	$0.698249\pi$
$602$	0	0
$603$	−7247.32	−0.489442
$604$	0	0
$605$	5509.07	0.370207
$606$	0	0
$607$	−14897.9	−0.996193	−0.498096	−	0.867122i	$-0.665967\pi$
$607$	−14897.9	−0.996193	−0.498096	+	0.867122i	$0.665967\pi$
$608$	0	0
$609$	1853.83	0.123352
$610$	0	0
$611$	1493.79	0.0989072
$612$	0	0
$613$	−24333.8	−1.60332	−0.801658	−	0.597783i	$-0.796048\pi$
$613$	−24333.8	−1.60332	−0.801658	+	0.597783i	$0.796048\pi$
$614$	0	0
$615$	−113.557	−0.00744566
$616$	0	0
$617$	−17445.0	−1.13827	−0.569133	−	0.822246i	$-0.692721\pi$
$617$	−17445.0	−1.13827	−0.569133	+	0.822246i	$0.692721\pi$
$618$	0	0
$619$	6733.87	0.437249	0.218625	−	0.975809i	$-0.429843\pi$
$619$	6733.87	0.437249	0.218625	+	0.975809i	$0.429843\pi$
$620$	0	0
$621$	−2602.91	−0.168198
$622$	0	0
$623$	−3158.83	−0.203140
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	6797.49	0.432959
$628$	0	0
$629$	−19089.7	−1.21010
$630$	0	0
$631$	30296.3	1.91137	0.955685	−	0.294391i	$-0.0951167\pi$
$631$	30296.3	1.91137	0.955685	+	0.294391i	$0.0951167\pi$
$632$	0	0
$633$	−1829.58	−0.114880
$634$	0	0
$635$	3336.21	0.208494
$636$	0	0
$637$	−251.807	−0.0156624
$638$	0	0
$639$	−4163.66	−0.257765
$640$	0	0
$641$	−10264.4	−0.632480	−0.316240	−	0.948679i	$-0.602420\pi$
$641$	−10264.4	−0.632480	−0.316240	+	0.948679i	$0.602420\pi$
$642$	0	0
$643$	20436.5	1.25340	0.626702	−	0.779259i	$-0.284404\pi$
$643$	20436.5	1.25340	0.626702	+	0.779259i	$0.284404\pi$
$644$	0	0
$645$	7868.25	0.480329
$646$	0	0
$647$	29516.0	1.79350	0.896749	−	0.442539i	$-0.145922\pi$
$647$	29516.0	1.79350	0.896749	+	0.442539i	$0.145922\pi$
$648$	0	0
$649$	−7572.45	−0.458004
$650$	0	0
$651$	2462.57	0.148257
$652$	0	0
$653$	−14916.1	−0.893896	−0.446948	−	0.894560i	$-0.647489\pi$
$653$	−14916.1	−0.893896	−0.446948	+	0.894560i	$0.647489\pi$
$654$	0	0
$655$	1244.35	0.0742301
$656$	0	0
$657$	359.789	0.0213649
$658$	0	0
$659$	3984.71	0.235542	0.117771	−	0.993041i	$-0.462425\pi$
$659$	3984.71	0.235542	0.117771	+	0.993041i	$0.462425\pi$
$660$	0	0
$661$	7964.74	0.468672	0.234336	−	0.972156i	$-0.424708\pi$
$661$	7964.74	0.468672	0.234336	+	0.972156i	$0.424708\pi$
$662$	0	0
$663$	1420.29	0.0831966
$664$	0	0
$665$	−5238.42	−0.305469
$666$	0	0
$667$	8510.34	0.494035
$668$	0	0
$669$	14262.8	0.824263
$670$	0	0
$671$	1780.97	0.102464
$672$	0	0
$673$	7417.72	0.424862	0.212431	−	0.977176i	$-0.431862\pi$
$673$	7417.72	0.424862	0.212431	+	0.977176i	$0.431862\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−29364.3	−1.66700	−0.833502	−	0.552517i	$-0.813667\pi$
$677$	−29364.3	−1.66700	−0.833502	+	0.552517i	$0.813667\pi$
$678$	0	0
$679$	7.33460	0.000414545 0
$680$	0	0
$681$	7871.57	0.442936
$682$	0	0
$683$	−31390.9	−1.75862	−0.879311	−	0.476248i	$-0.841997\pi$
$683$	−31390.9	−1.75862	−0.879311	+	0.476248i	$0.841997\pi$
$684$	0	0
$685$	1862.66	0.103896
$686$	0	0
$687$	11381.5	0.632068
$688$	0	0
$689$	1620.64	0.0896102
$690$	0	0
$691$	−30693.3	−1.68977	−0.844883	−	0.534952i	$-0.820330\pi$
$691$	−30693.3	−1.68977	−0.844883	+	0.534952i	$0.820330\pi$
$692$	0	0
$693$	−953.752	−0.0522800
$694$	0	0
$695$	5347.89	0.291881
$696$	0	0
$697$	−697.441	−0.0379017
$698$	0	0
$699$	−12792.9	−0.692235
$700$	0	0
$701$	−7091.40	−0.382081	−0.191040	−	0.981582i	$-0.561186\pi$
$701$	−7091.40	−0.382081	−0.191040	+	0.981582i	$0.561186\pi$
$702$	0	0
$703$	31013.2	1.66385
$704$	0	0
$705$	−4360.23	−0.232930
$706$	0	0
$707$	−2334.07	−0.124161
$708$	0	0
$709$	−12330.8	−0.653163	−0.326581	−	0.945169i	$-0.605897\pi$
$709$	−12330.8	−0.653163	−0.326581	+	0.945169i	$0.605897\pi$
$710$	0	0
$711$	5936.73	0.313143
$712$	0	0
$713$	11304.8	0.593785
$714$	0	0
$715$	388.988	0.0203459
$716$	0	0
$717$	−13673.6	−0.712204
$718$	0	0
$719$	−3366.92	−0.174638	−0.0873192	−	0.996180i	$-0.527830\pi$
$719$	−3366.92	−0.174638	−0.0873192	+	0.996180i	$0.527830\pi$
$720$	0	0
$721$	13140.8	0.678762
$722$	0	0
$723$	−13929.5	−0.716520
$724$	0	0
$725$	2206.95	0.113054
$726$	0	0
$727$	3628.17	0.185091	0.0925457	−	0.995708i	$-0.470500\pi$
$727$	3628.17	0.185091	0.0925457	+	0.995708i	$0.470500\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	48324.8	2.44509
$732$	0	0
$733$	−31645.3	−1.59461	−0.797303	−	0.603579i	$-0.793741\pi$
$733$	−31645.3	−1.59461	−0.797303	+	0.603579i	$0.793741\pi$
$734$	0	0
$735$	735.000	0.0368856
$736$	0	0
$737$	−12190.7	−0.609296
$738$	0	0
$739$	−25569.0	−1.27276	−0.636380	−	0.771375i	$-0.719569\pi$
$739$	−25569.0	−1.27276	−0.636380	+	0.771375i	$0.719569\pi$
$740$	0	0
$741$	−2307.41	−0.114393
$742$	0	0
$743$	−22094.7	−1.09095	−0.545475	−	0.838127i	$-0.683651\pi$
$743$	−22094.7	−1.09095	−0.545475	+	0.838127i	$0.683651\pi$
$744$	0	0
$745$	−10421.1	−0.512483
$746$	0	0
$747$	−11052.9	−0.541372
$748$	0	0
$749$	8545.69	0.416893
$750$	0	0
$751$	6666.48	0.323919	0.161960	−	0.986797i	$-0.448219\pi$
$751$	6666.48	0.323919	0.161960	+	0.986797i	$0.448219\pi$
$752$	0	0
$753$	−7735.09	−0.374346
$754$	0	0
$755$	−3853.71	−0.185763
$756$	0	0
$757$	−29275.3	−1.40558	−0.702792	−	0.711395i	$-0.748064\pi$
$757$	−29275.3	−1.40558	−0.702792	+	0.711395i	$0.748064\pi$
$758$	0	0
$759$	−4378.36	−0.209386
$760$	0	0
$761$	−273.757	−0.0130403	−0.00652017	−	0.999979i	$-0.502075\pi$
$761$	−273.757	−0.0130403	−0.00652017	+	0.999979i	$0.502075\pi$
$762$	0	0
$763$	−8476.61	−0.402194
$764$	0	0
$765$	−4145.68	−0.195931
$766$	0	0
$767$	2570.47	0.121010
$768$	0	0
$769$	−2086.13	−0.0978256	−0.0489128	−	0.998803i	$-0.515576\pi$
$769$	−2086.13	−0.0978256	−0.0489128	+	0.998803i	$0.515576\pi$
$770$	0	0
$771$	10018.3	0.467965
$772$	0	0
$773$	−14578.1	−0.678317	−0.339159	−	0.940729i	$-0.610142\pi$
$773$	−14578.1	−0.678317	−0.339159	+	0.940729i	$0.610142\pi$
$774$	0	0
$775$	2931.63	0.135880
$776$	0	0
$777$	−4351.45	−0.200911
$778$	0	0
$779$	1133.07	0.0521135
$780$	0	0
$781$	−7003.70	−0.320886
$782$	0	0
$783$	−2383.50	−0.108786
$784$	0	0
$785$	8194.29	0.372569
$786$	0	0
$787$	15742.9	0.713054	0.356527	−	0.934285i	$-0.383961\pi$
$787$	15742.9	0.713054	0.356527	+	0.934285i	$0.383961\pi$
$788$	0	0
$789$	−20844.9	−0.940553
$790$	0	0
$791$	13790.6	0.619894
$792$	0	0
$793$	−604.550	−0.0270722
$794$	0	0
$795$	−4730.49	−0.211035
$796$	0	0
$797$	20245.1	0.899771	0.449886	−	0.893086i	$-0.351465\pi$
$797$	20245.1	0.899771	0.449886	+	0.893086i	$0.351465\pi$
$798$	0	0
$799$	−26779.4	−1.18572
$800$	0	0
$801$	4061.36	0.179152
$802$	0	0
$803$	605.202	0.0265967
$804$	0	0
$805$	3374.14	0.147730
$806$	0	0
$807$	−385.128	−0.0167994
$808$	0	0
$809$	−11670.4	−0.507179	−0.253590	−	0.967312i	$-0.581611\pi$
$809$	−11670.4	−0.507179	−0.253590	+	0.967312i	$0.581611\pi$
$810$	0	0
$811$	−30477.0	−1.31959	−0.659797	−	0.751444i	$-0.729358\pi$
$811$	−30477.0	−1.31959	−0.659797	+	0.751444i	$0.729358\pi$
$812$	0	0
$813$	−12009.6	−0.518077
$814$	0	0
$815$	−1936.68	−0.0832382
$816$	0	0
$817$	−78509.0	−3.36191
$818$	0	0
$819$	323.752	0.0138130
$820$	0	0
$821$	26161.8	1.11212	0.556061	−	0.831141i	$-0.312312\pi$
$821$	26161.8	1.11212	0.556061	+	0.831141i	$0.312312\pi$
$822$	0	0
$823$	29339.3	1.24265	0.621327	−	0.783551i	$-0.286594\pi$
$823$	29339.3	1.24265	0.621327	+	0.783551i	$0.286594\pi$
$824$	0	0
$825$	−1135.42	−0.0479154
$826$	0	0
$827$	−43018.7	−1.80884	−0.904418	−	0.426648i	$-0.859694\pi$
$827$	−43018.7	−1.80884	−0.904418	+	0.426648i	$0.859694\pi$
$828$	0	0
$829$	−9244.41	−0.387300	−0.193650	−	0.981071i	$-0.562033\pi$
$829$	−9244.41	−0.387300	−0.193650	+	0.981071i	$0.562033\pi$
$830$	0	0
$831$	8774.60	0.366290
$832$	0	0
$833$	4514.18	0.187764
$834$	0	0
$835$	16783.5	0.695590
$836$	0	0
$837$	−3166.16	−0.130751
$838$	0	0
$839$	−24042.0	−0.989299	−0.494649	−	0.869093i	$-0.664704\pi$
$839$	−24042.0	−0.989299	−0.494649	+	0.869093i	$0.664704\pi$
$840$	0	0
$841$	−16596.0	−0.680472
$842$	0	0
$843$	−5513.53	−0.225262
$844$	0	0
$845$	10853.0	0.441838
$846$	0	0
$847$	7712.69	0.312882
$848$	0	0
$849$	−17594.5	−0.711237
$850$	0	0
$851$	−19976.1	−0.804666
$852$	0	0
$853$	−8523.38	−0.342128	−0.171064	−	0.985260i	$-0.554720\pi$
$853$	−8523.38	−0.342128	−0.171064	+	0.985260i	$0.554720\pi$
$854$	0	0
$855$	6735.11	0.269399
$856$	0	0
$857$	−9207.05	−0.366986	−0.183493	−	0.983021i	$-0.558740\pi$
$857$	−9207.05	−0.366986	−0.183493	+	0.983021i	$0.558740\pi$
$858$	0	0
$859$	−38938.2	−1.54663	−0.773315	−	0.634022i	$-0.781403\pi$
$859$	−38938.2	−1.54663	−0.773315	+	0.634022i	$0.781403\pi$
$860$	0	0
$861$	−158.980	−0.00629273
$862$	0	0
$863$	44665.8	1.76181	0.880906	−	0.473292i	$-0.156934\pi$
$863$	44665.8	1.76181	0.880906	+	0.473292i	$0.156934\pi$
$864$	0	0
$865$	−816.543	−0.0320963
$866$	0	0
$867$	−10722.7	−0.420025
$868$	0	0
$869$	9986.18	0.389825
$870$	0	0
$871$	4138.15	0.160983
$872$	0	0
$873$	−9.43020	−0.000365594 0
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	7517.99	0.289469	0.144735	−	0.989471i	$-0.453767\pi$
$877$	7517.99	0.289469	0.144735	+	0.989471i	$0.453767\pi$
$878$	0	0
$879$	−4659.23	−0.178785
$880$	0	0
$881$	36609.0	1.39999	0.699993	−	0.714149i	$-0.253186\pi$
$881$	36609.0	1.39999	0.699993	+	0.714149i	$0.253186\pi$
$882$	0	0
$883$	12894.3	0.491426	0.245713	−	0.969343i	$-0.420978\pi$
$883$	12894.3	0.491426	0.245713	+	0.969343i	$0.420978\pi$
$884$	0	0
$885$	−7502.96	−0.284982
$886$	0	0
$887$	22287.0	0.843656	0.421828	−	0.906676i	$-0.361389\pi$
$887$	22287.0	0.843656	0.421828	+	0.906676i	$0.361389\pi$
$888$	0	0
$889$	4670.69	0.176209
$890$	0	0
$891$	1226.25	0.0461066
$892$	0	0
$893$	43506.1	1.63032
$894$	0	0
$895$	21875.5	0.817003
$896$	0	0
$897$	1486.24	0.0553222
$898$	0	0
$899$	10351.9	0.384044
$900$	0	0
$901$	−29053.4	−1.07426
$902$	0	0
$903$	11015.6	0.405952
$904$	0	0
$905$	−3731.05	−0.137043
$906$	0	0
$907$	41841.2	1.53177	0.765884	−	0.642979i	$-0.222302\pi$
$907$	41841.2	1.53177	0.765884	+	0.642979i	$0.222302\pi$
$908$	0	0
$909$	3000.95	0.109500
$910$	0	0
$911$	−45265.4	−1.64622	−0.823112	−	0.567879i	$-0.807764\pi$
$911$	−45265.4	−1.64622	−0.823112	+	0.567879i	$0.807764\pi$
$912$	0	0
$913$	−18592.1	−0.673942
$914$	0	0
$915$	1764.62	0.0637559
$916$	0	0
$917$	1742.09	0.0627359
$918$	0	0
$919$	18890.5	0.678063	0.339031	−	0.940775i	$-0.389901\pi$
$919$	18890.5	0.678063	0.339031	+	0.940775i	$0.389901\pi$
$920$	0	0
$921$	27264.9	0.975470
$922$	0	0
$923$	2377.41	0.0847817
$924$	0	0
$925$	−5180.30	−0.184138
$926$	0	0
$927$	−16895.3	−0.598612
$928$	0	0
$929$	−29185.3	−1.03072	−0.515360	−	0.856974i	$-0.672342\pi$
$929$	−29185.3	−1.03072	−0.515360	+	0.856974i	$0.672342\pi$
$930$	0	0
$931$	−7333.79	−0.258169
$932$	0	0
$933$	−1823.49	−0.0639853
$934$	0	0
$935$	−6973.45	−0.243911
$936$	0	0
$937$	−46388.2	−1.61733	−0.808663	−	0.588272i	$-0.799809\pi$
$937$	−46388.2	−1.61733	−0.808663	+	0.588272i	$0.799809\pi$
$938$	0	0
$939$	13423.2	0.466506
$940$	0	0
$941$	−25332.0	−0.877578	−0.438789	−	0.898590i	$-0.644592\pi$
$941$	−25332.0	−0.877578	−0.438789	+	0.898590i	$0.644592\pi$
$942$	0	0
$943$	−729.826	−0.0252030
$944$	0	0
$945$	−945.000	−0.0325300
$946$	0	0
$947$	−19350.3	−0.663992	−0.331996	−	0.943281i	$-0.607722\pi$
$947$	−19350.3	−0.663992	−0.331996	+	0.943281i	$0.607722\pi$
$948$	0	0
$949$	−205.436	−0.00702713
$950$	0	0
$951$	13826.6	0.471460
$952$	0	0
$953$	−31814.6	−1.08140	−0.540701	−	0.841215i	$-0.681841\pi$
$953$	−31814.6	−1.08140	−0.540701	+	0.841215i	$0.681841\pi$
$954$	0	0
$955$	20784.5	0.704261
$956$	0	0
$957$	−4009.29	−0.135425
$958$	0	0
$959$	2607.73	0.0878080
$960$	0	0
$961$	−16039.9	−0.538414
$962$	0	0
$963$	−10987.3	−0.367665
$964$	0	0
$965$	−23528.4	−0.784876
$966$	0	0
$967$	−14698.2	−0.488793	−0.244396	−	0.969675i	$-0.578590\pi$
$967$	−14698.2	−0.488793	−0.244396	+	0.969675i	$0.578590\pi$
$968$	0	0
$969$	41365.3	1.37136
$970$	0	0
$971$	30064.1	0.993617	0.496809	−	0.867860i	$-0.334505\pi$
$971$	30064.1	0.993617	0.496809	+	0.867860i	$0.334505\pi$
$972$	0	0
$973$	7487.05	0.246684
$974$	0	0
$975$	385.419	0.0126598
$976$	0	0
$977$	−10575.9	−0.346319	−0.173159	−	0.984894i	$-0.555398\pi$
$977$	−10575.9	−0.346319	−0.173159	+	0.984894i	$0.555398\pi$
$978$	0	0
$979$	6831.62	0.223023
$980$	0	0
$981$	10898.5	0.354702
$982$	0	0
$983$	21051.3	0.683044	0.341522	−	0.939874i	$-0.389058\pi$
$983$	21051.3	0.683044	0.341522	+	0.939874i	$0.389058\pi$
$984$	0	0
$985$	2524.74	0.0816700
$986$	0	0
$987$	−6104.32	−0.196862
$988$	0	0
$989$	50568.8	1.62588
$990$	0	0
$991$	8846.57	0.283573	0.141786	−	0.989897i	$-0.454715\pi$
$991$	8846.57	0.283573	0.141786	+	0.989897i	$0.454715\pi$
$992$	0	0
$993$	520.175	0.0166236
$994$	0	0
$995$	−1676.02	−0.0534003
$996$	0	0
$997$	30958.8	0.983424	0.491712	−	0.870758i	$-0.336371\pi$
$997$	30958.8	0.983424	0.491712	+	0.870758i	$0.336371\pi$
$998$	0	0
$999$	5594.73	0.177186

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.4.a.p.1.2	✓	3
4.3	odd	2		1680.4.a.bs.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.4.a.p.1.2	✓	3	1.1	even	1	trivial
1680.4.a.bs.1.2		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	840.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +15.1389 q^{11} -5.13892 q^{13} +15.0000 q^{15} +92.1262 q^{17} -149.669 q^{19} +21.0000 q^{21} +96.4040 q^{23} +25.0000 q^{25} -27.0000 q^{27} +88.2778 q^{29} +117.265 q^{31} -45.4168 q^{33} +35.0000 q^{35} -207.212 q^{37} +15.4168 q^{39} -7.57050 q^{41} +524.550 q^{43} -45.0000 q^{45} -290.682 q^{47} +49.0000 q^{49} -276.379 q^{51} -315.366 q^{53} -75.6946 q^{55} +449.007 q^{57} -500.197 q^{59} +117.642 q^{61} -63.0000 q^{63} +25.6946 q^{65} -805.258 q^{67} -289.212 q^{69} -462.629 q^{71} +39.9766 q^{73} -75.0000 q^{75} -105.972 q^{77} +659.636 q^{79} +81.0000 q^{81} -1228.10 q^{83} -460.631 q^{85} -264.834 q^{87} +451.262 q^{89} +35.9724 q^{91} -351.795 q^{93} +748.346 q^{95} -1.04780 q^{97} +136.250 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 15 q^{5} - 21 q^{7} + 27 q^{9} - 6 q^{11} + 36 q^{13} + 45 q^{15} + 90 q^{17} + 78 q^{19} + 63 q^{21} + 75 q^{25} - 81 q^{27} + 162 q^{29} + 114 q^{31} + 18 q^{33} + 105 q^{35} + 246 q^{37}+ \cdots - 54 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 - 15 * q^5 - 21 * q^7 + 27 * q^9 - 6 * q^11 + 36 * q^13 + 45 * q^15 + 90 * q^17 + 78 * q^19 + 63 * q^21 + 75 * q^25 - 81 * q^27 + 162 * q^29 + 114 * q^31 + 18 * q^33 + 105 * q^35 + 246 * q^37 - 108 * q^39 - 42 * q^41 - 348 * q^43 - 135 * q^45 - 480 * q^47 + 147 * q^49 - 270 * q^51 - 252 * q^53 + 30 * q^55 - 234 * q^57 - 408 * q^59 - 534 * q^61 - 189 * q^63 - 180 * q^65 - 372 * q^67 - 366 * q^71 - 60 * q^73 - 225 * q^75 + 42 * q^77 - 624 * q^79 + 243 * q^81 - 3228 * q^83 - 450 * q^85 - 486 * q^87 - 510 * q^89 - 252 * q^91 - 342 * q^93 - 390 * q^95 - 2304 * q^97 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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