LMFDB - Embedded newform 1856.4.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, 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("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 70x + 238 $ x^3 - x^2 - 70*x + 238
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 116)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.21773$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.21773 q^{3} +20.3399 q^{5} +22.2443 q^{7} +25.0956 q^{9} +O(q^{10})$	$q+7.21773 q^{3} +20.3399 q^{5} +22.2443 q^{7} +25.0956 q^{9} +10.3330 q^{11} -62.7753 q^{13} +146.808 q^{15} +70.8570 q^{17} +13.5507 q^{19} +160.553 q^{21} -168.758 q^{23} +288.711 q^{25} -13.7457 q^{27} -29.0000 q^{29} +116.591 q^{31} +74.5804 q^{33} +452.447 q^{35} -270.376 q^{37} -453.095 q^{39} +307.481 q^{41} +377.951 q^{43} +510.441 q^{45} +65.4127 q^{47} +151.810 q^{49} +511.427 q^{51} +28.0053 q^{53} +210.171 q^{55} +97.8051 q^{57} -766.772 q^{59} +565.048 q^{61} +558.234 q^{63} -1276.84 q^{65} +220.567 q^{67} -1218.05 q^{69} +1039.13 q^{71} +815.062 q^{73} +2083.84 q^{75} +229.850 q^{77} +716.722 q^{79} -776.793 q^{81} +785.053 q^{83} +1441.22 q^{85} -209.314 q^{87} +802.772 q^{89} -1396.39 q^{91} +841.523 q^{93} +275.619 q^{95} +1043.98 q^{97} +259.311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 10 q^{3} + 20 q^{5} + 8 q^{7} + 93 q^{9}+O(q^{10}) $ 3 * q + 10 * q^3 + 20 * q^5 + 8 * q^7 + 93 * q^9	$ 3 q + 10 q^{3} + 20 q^{5} + 8 q^{7} + 93 q^{9} - 86 q^{11} - 124 q^{13} + 54 q^{15} + 14 q^{17} - 88 q^{19} - 280 q^{21} + 68 q^{23} + 111 q^{25} + 334 q^{27} - 87 q^{29} + 326 q^{31} + 110 q^{33} + 784 q^{35} - 166 q^{37} - 682 q^{39} + 34 q^{41} + 946 q^{43} + 242 q^{45} + 234 q^{47} + 1067 q^{49} + 1428 q^{51} + 1144 q^{53} + 94 q^{55} + 244 q^{57} - 488 q^{59} - 450 q^{61} - 1096 q^{63} - 1154 q^{65} + 52 q^{67} + 404 q^{69} + 1196 q^{71} + 2434 q^{73} + 1868 q^{75} + 312 q^{77} + 742 q^{79} - 849 q^{81} - 464 q^{83} + 672 q^{85} - 290 q^{87} + 1986 q^{89} - 448 q^{91} + 358 q^{93} + 68 q^{95} - 406 q^{97} - 2540 q^{99}+O(q^{100}) $ 3 * q + 10 * q^3 + 20 * q^5 + 8 * q^7 + 93 * q^9 - 86 * q^11 - 124 * q^13 + 54 * q^15 + 14 * q^17 - 88 * q^19 - 280 * q^21 + 68 * q^23 + 111 * q^25 + 334 * q^27 - 87 * q^29 + 326 * q^31 + 110 * q^33 + 784 * q^35 - 166 * q^37 - 682 * q^39 + 34 * q^41 + 946 * q^43 + 242 * q^45 + 234 * q^47 + 1067 * q^49 + 1428 * q^51 + 1144 * q^53 + 94 * q^55 + 244 * q^57 - 488 * q^59 - 450 * q^61 - 1096 * q^63 - 1154 * q^65 + 52 * q^67 + 404 * q^69 + 1196 * q^71 + 2434 * q^73 + 1868 * q^75 + 312 * q^77 + 742 * q^79 - 849 * q^81 - 464 * q^83 + 672 * q^85 - 290 * q^87 + 1986 * q^89 - 448 * q^91 + 358 * q^93 + 68 * q^95 - 406 * q^97 - 2540 * 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
$3$	7.21773	1.38905	0.694526	−	0.719468i	$-0.255614\pi$
$3$	7.21773	1.38905	0.694526	+	0.719468i	$0.255614\pi$
$4$	0	0
$5$	20.3399	1.81925	0.909627	−	0.415425i	$-0.136367\pi$
$5$	20.3399	1.81925	0.909627	+	0.415425i	$0.136367\pi$
$6$	0	0
$7$	22.2443	1.20108	0.600540	−	0.799595i	$-0.294952\pi$
$7$	22.2443	1.20108	0.600540	+	0.799595i	$0.294952\pi$
$8$	0	0
$9$	25.0956	0.929465
$10$	0	0
$11$	10.3330	0.283227	0.141614	−	0.989922i	$-0.454771\pi$
$11$	10.3330	0.283227	0.141614	+	0.989922i	$0.454771\pi$
$12$	0	0
$13$	−62.7753	−1.33929	−0.669644	−	0.742682i	$-0.733553\pi$
$13$	−62.7753	−1.33929	−0.669644	+	0.742682i	$0.733553\pi$
$14$	0	0
$15$	146.808	2.52704
$16$	0	0
$17$	70.8570	1.01090	0.505452	−	0.862855i	$-0.331326\pi$
$17$	70.8570	1.01090	0.505452	+	0.862855i	$0.331326\pi$
$18$	0	0
$19$	13.5507	0.163618	0.0818089	−	0.996648i	$-0.473930\pi$
$19$	13.5507	0.163618	0.0818089	+	0.996648i	$0.473930\pi$
$20$	0	0
$21$	160.553	1.66836
$22$	0	0
$23$	−168.758	−1.52994	−0.764969	−	0.644067i	$-0.777246\pi$
$23$	−168.758	−1.52994	−0.764969	+	0.644067i	$0.777246\pi$
$24$	0	0
$25$	288.711	2.30969
$26$	0	0
$27$	−13.7457	−0.0979763
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	116.591	0.675496	0.337748	−	0.941237i	$-0.390335\pi$
$31$	116.591	0.675496	0.337748	+	0.941237i	$0.390335\pi$
$32$	0	0
$33$	74.5804	0.393418
$34$	0	0
$35$	452.447	2.18507
$36$	0	0
$37$	−270.376	−1.20134	−0.600670	−	0.799497i	$-0.705099\pi$
$37$	−270.376	−1.20134	−0.600670	+	0.799497i	$0.705099\pi$
$38$	0	0
$39$	−453.095	−1.86034
$40$	0	0
$41$	307.481	1.17123	0.585616	−	0.810589i	$-0.300853\pi$
$41$	307.481	1.17123	0.585616	+	0.810589i	$0.300853\pi$
$42$	0	0
$43$	377.951	1.34039	0.670197	−	0.742183i	$-0.266210\pi$
$43$	377.951	1.34039	0.670197	+	0.742183i	$0.266210\pi$
$44$	0	0
$45$	510.441	1.69093
$46$	0	0
$47$	65.4127	0.203009	0.101504	−	0.994835i	$-0.467634\pi$
$47$	65.4127	0.203009	0.101504	+	0.994835i	$0.467634\pi$
$48$	0	0
$49$	151.810	0.442595
$50$	0	0
$51$	511.427	1.40420
$52$	0	0
$53$	28.0053	0.0725817	0.0362908	−	0.999341i	$-0.488446\pi$
$53$	28.0053	0.0725817	0.0362908	+	0.999341i	$0.488446\pi$
$54$	0	0
$55$	210.171	0.515263
$56$	0	0
$57$	97.8051	0.227274
$58$	0	0
$59$	−766.772	−1.69195	−0.845976	−	0.533221i	$-0.820981\pi$
$59$	−766.772	−1.69195	−0.845976	+	0.533221i	$0.820981\pi$
$60$	0	0
$61$	565.048	1.18602	0.593008	−	0.805197i	$-0.297940\pi$
$61$	565.048	1.18602	0.593008	+	0.805197i	$0.297940\pi$
$62$	0	0
$63$	558.234	1.11636
$64$	0	0
$65$	−1276.84	−2.43651
$66$	0	0
$67$	220.567	0.402188	0.201094	−	0.979572i	$-0.435550\pi$
$67$	220.567	0.402188	0.201094	+	0.979572i	$0.435550\pi$
$68$	0	0
$69$	−1218.05	−2.12516
$70$	0	0
$71$	1039.13	1.73693	0.868467	−	0.495747i	$-0.165106\pi$
$71$	1039.13	1.73693	0.868467	+	0.495747i	$0.165106\pi$
$72$	0	0
$73$	815.062	1.30679	0.653396	−	0.757016i	$-0.273344\pi$
$73$	815.062	1.30679	0.653396	+	0.757016i	$0.273344\pi$
$74$	0	0
$75$	2083.84	3.20828
$76$	0	0
$77$	229.850	0.340179
$78$	0	0
$79$	716.722	1.02073	0.510364	−	0.859958i	$-0.329511\pi$
$79$	716.722	1.02073	0.510364	+	0.859958i	$0.329511\pi$
$80$	0	0
$81$	−776.793	−1.06556
$82$	0	0
$83$	785.053	1.03820	0.519101	−	0.854713i	$-0.326267\pi$
$83$	785.053	1.03820	0.519101	+	0.854713i	$0.326267\pi$
$84$	0	0
$85$	1441.22	1.83909
$86$	0	0
$87$	−209.314	−0.257940
$88$	0	0
$89$	802.772	0.956109	0.478054	−	0.878330i	$-0.341342\pi$
$89$	802.772	0.956109	0.478054	+	0.878330i	$0.341342\pi$
$90$	0	0
$91$	−1396.39	−1.60859
$92$	0	0
$93$	841.523	0.938300
$94$	0	0
$95$	275.619	0.297663
$96$	0	0
$97$	1043.98	1.09279	0.546393	−	0.837529i	$-0.316001\pi$
$97$	1043.98	1.09279	0.546393	+	0.837529i	$0.316001\pi$
$98$	0	0
$99$	259.311	0.263250
$100$	0	0
$101$	−776.472	−0.764969	−0.382485	−	0.923962i	$-0.624932\pi$
$101$	−776.472	−0.764969	−0.382485	+	0.923962i	$0.624932\pi$
$102$	0	0
$103$	−956.613	−0.915125	−0.457563	−	0.889177i	$-0.651277\pi$
$103$	−956.613	−0.915125	−0.457563	+	0.889177i	$0.651277\pi$
$104$	0	0
$105$	3265.64	3.03518
$106$	0	0
$107$	−935.122	−0.844875	−0.422437	−	0.906392i	$-0.638825\pi$
$107$	−935.122	−0.844875	−0.422437	+	0.906392i	$0.638825\pi$
$108$	0	0
$109$	−1781.05	−1.56508	−0.782540	−	0.622600i	$-0.786076\pi$
$109$	−1781.05	−1.56508	−0.782540	+	0.622600i	$0.786076\pi$
$110$	0	0
$111$	−1951.50	−1.66872
$112$	0	0
$113$	988.667	0.823061	0.411531	−	0.911396i	$-0.364994\pi$
$113$	988.667	0.823061	0.411531	+	0.911396i	$0.364994\pi$
$114$	0	0
$115$	−3432.53	−2.78335
$116$	0	0
$117$	−1575.38	−1.24482
$118$	0	0
$119$	1576.17	1.21418
$120$	0	0
$121$	−1224.23	−0.919782
$122$	0	0
$123$	2219.32	1.62690
$124$	0	0
$125$	3329.86	2.38266
$126$	0	0
$127$	−2159.98	−1.50919	−0.754595	−	0.656191i	$-0.772167\pi$
$127$	−2159.98	−1.50919	−0.754595	+	0.656191i	$0.772167\pi$
$128$	0	0
$129$	2727.94	1.86188
$130$	0	0
$131$	−284.438	−0.189706	−0.0948529	−	0.995491i	$-0.530238\pi$
$131$	−284.438	−0.189706	−0.0948529	+	0.995491i	$0.530238\pi$
$132$	0	0
$133$	301.426	0.196518
$134$	0	0
$135$	−279.586	−0.178244
$136$	0	0
$137$	−2945.37	−1.83679	−0.918394	−	0.395667i	$-0.870513\pi$
$137$	−2945.37	−1.83679	−0.918394	+	0.395667i	$0.870513\pi$
$138$	0	0
$139$	1314.33	0.802014	0.401007	−	0.916075i	$-0.368660\pi$
$139$	1314.33	0.802014	0.401007	+	0.916075i	$0.368660\pi$
$140$	0	0
$141$	472.131	0.281990
$142$	0	0
$143$	−648.655	−0.379323
$144$	0	0
$145$	−589.857	−0.337827
$146$	0	0
$147$	1095.72	0.614787
$148$	0	0
$149$	279.825	0.153853	0.0769267	−	0.997037i	$-0.475489\pi$
$149$	279.825	0.153853	0.0769267	+	0.997037i	$0.475489\pi$
$150$	0	0
$151$	−785.832	−0.423511	−0.211755	−	0.977323i	$-0.567918\pi$
$151$	−785.832	−0.423511	−0.211755	+	0.977323i	$0.567918\pi$
$152$	0	0
$153$	1778.20	0.939599
$154$	0	0
$155$	2371.45	1.22890
$156$	0	0
$157$	−2496.09	−1.26885	−0.634425	−	0.772984i	$-0.718763\pi$
$157$	−2496.09	−1.26885	−0.634425	+	0.772984i	$0.718763\pi$
$158$	0	0
$159$	202.135	0.100820
$160$	0	0
$161$	−3753.92	−1.83758
$162$	0	0
$163$	−1136.33	−0.546039	−0.273019	−	0.962009i	$-0.588022\pi$
$163$	−1136.33	−0.546039	−0.273019	+	0.962009i	$0.588022\pi$
$164$	0	0
$165$	1516.96	0.715727
$166$	0	0
$167$	−2846.71	−1.31907	−0.659535	−	0.751673i	$-0.729247\pi$
$167$	−2846.71	−1.31907	−0.659535	+	0.751673i	$0.729247\pi$
$168$	0	0
$169$	1743.74	0.793693
$170$	0	0
$171$	340.062	0.152077
$172$	0	0
$173$	18.5058	0.00813276	0.00406638	−	0.999992i	$-0.498706\pi$
$173$	18.5058	0.00813276	0.00406638	+	0.999992i	$0.498706\pi$
$174$	0	0
$175$	6422.18	2.77412
$176$	0	0
$177$	−5534.35	−2.35021
$178$	0	0
$179$	2279.31	0.951752	0.475876	−	0.879512i	$-0.342131\pi$
$179$	2279.31	0.951752	0.475876	+	0.879512i	$0.342131\pi$
$180$	0	0
$181$	−4095.42	−1.68182	−0.840911	−	0.541173i	$-0.817980\pi$
$181$	−4095.42	−1.68182	−0.840911	+	0.541173i	$0.817980\pi$
$182$	0	0
$183$	4078.36	1.64744
$184$	0	0
$185$	−5499.42	−2.18554
$186$	0	0
$187$	732.162	0.286316
$188$	0	0
$189$	−305.764	−0.117677
$190$	0	0
$191$	1881.70	0.712853	0.356427	−	0.934323i	$-0.383995\pi$
$191$	1881.70	0.712853	0.356427	+	0.934323i	$0.383995\pi$
$192$	0	0
$193$	−2156.17	−0.804169	−0.402084	−	0.915603i	$-0.631714\pi$
$193$	−2156.17	−0.804169	−0.402084	+	0.915603i	$0.631714\pi$
$194$	0	0
$195$	−9215.90	−3.38443
$196$	0	0
$197$	62.9785	0.0227768	0.0113884	−	0.999935i	$-0.496375\pi$
$197$	62.9785	0.0227768	0.0113884	+	0.999935i	$0.496375\pi$
$198$	0	0
$199$	1702.62	0.606509	0.303255	−	0.952910i	$-0.401927\pi$
$199$	1702.62	0.606509	0.303255	+	0.952910i	$0.401927\pi$
$200$	0	0
$201$	1591.99	0.558660
$202$	0	0
$203$	−645.085	−0.223035
$204$	0	0
$205$	6254.14	2.13077
$206$	0	0
$207$	−4235.09	−1.42202
$208$	0	0
$209$	140.018	0.0463411
$210$	0	0
$211$	3012.95	0.983033	0.491516	−	0.870868i	$-0.336443\pi$
$211$	3012.95	0.983033	0.491516	+	0.870868i	$0.336443\pi$
$212$	0	0
$213$	7500.17	2.41269
$214$	0	0
$215$	7687.47	2.43852
$216$	0	0
$217$	2593.49	0.811326
$218$	0	0
$219$	5882.89	1.81520
$220$	0	0
$221$	−4448.07	−1.35389
$222$	0	0
$223$	−1537.84	−0.461799	−0.230899	−	0.972978i	$-0.574167\pi$
$223$	−1537.84	−0.461799	−0.230899	+	0.972978i	$0.574167\pi$
$224$	0	0
$225$	7245.37	2.14678
$226$	0	0
$227$	−1036.94	−0.303189	−0.151594	−	0.988443i	$-0.548441\pi$
$227$	−1036.94	−0.303189	−0.151594	+	0.988443i	$0.548441\pi$
$228$	0	0
$229$	−2008.98	−0.579724	−0.289862	−	0.957068i	$-0.593609\pi$
$229$	−2008.98	−0.579724	−0.289862	+	0.957068i	$0.593609\pi$
$230$	0	0
$231$	1658.99	0.472526
$232$	0	0
$233$	−1864.89	−0.524348	−0.262174	−	0.965021i	$-0.584439\pi$
$233$	−1864.89	−0.524348	−0.262174	+	0.965021i	$0.584439\pi$
$234$	0	0
$235$	1330.49	0.369325
$236$	0	0
$237$	5173.10	1.41784
$238$	0	0
$239$	482.651	0.130628	0.0653141	−	0.997865i	$-0.479195\pi$
$239$	482.651	0.130628	0.0653141	+	0.997865i	$0.479195\pi$
$240$	0	0
$241$	−1057.67	−0.282699	−0.141349	−	0.989960i	$-0.545144\pi$
$241$	−1057.67	−0.282699	−0.141349	+	0.989960i	$0.545144\pi$
$242$	0	0
$243$	−5235.54	−1.38214
$244$	0	0
$245$	3087.80	0.805192
$246$	0	0
$247$	−850.648	−0.219131
$248$	0	0
$249$	5666.30	1.44212
$250$	0	0
$251$	−1546.39	−0.388872	−0.194436	−	0.980915i	$-0.562288\pi$
$251$	−1546.39	−0.388872	−0.194436	+	0.980915i	$0.562288\pi$
$252$	0	0
$253$	−1743.77	−0.433320
$254$	0	0
$255$	10402.4	2.55459
$256$	0	0
$257$	−6918.41	−1.67922	−0.839608	−	0.543193i	$-0.817215\pi$
$257$	−6918.41	−1.67922	−0.839608	+	0.543193i	$0.817215\pi$
$258$	0	0
$259$	−6014.33	−1.44291
$260$	0	0
$261$	−727.771	−0.172597
$262$	0	0
$263$	1421.94	0.333387	0.166693	−	0.986009i	$-0.446691\pi$
$263$	1421.94	0.333387	0.166693	+	0.986009i	$0.446691\pi$
$264$	0	0
$265$	569.625	0.132045
$266$	0	0
$267$	5794.19	1.32808
$268$	0	0
$269$	−5153.64	−1.16812	−0.584058	−	0.811712i	$-0.698536\pi$
$269$	−5153.64	−1.16812	−0.584058	+	0.811712i	$0.698536\pi$
$270$	0	0
$271$	3291.63	0.737830	0.368915	−	0.929463i	$-0.379729\pi$
$271$	3291.63	0.737830	0.368915	+	0.929463i	$0.379729\pi$
$272$	0	0
$273$	−10078.8	−2.23442
$274$	0	0
$275$	2983.24	0.654167
$276$	0	0
$277$	−5913.79	−1.28276	−0.641382	−	0.767222i	$-0.721639\pi$
$277$	−5913.79	−1.28276	−0.641382	+	0.767222i	$0.721639\pi$
$278$	0	0
$279$	2925.92	0.627850
$280$	0	0
$281$	1130.35	0.239969	0.119985	−	0.992776i	$-0.461715\pi$
$281$	1130.35	0.239969	0.119985	+	0.992776i	$0.461715\pi$
$282$	0	0
$283$	76.4994	0.0160686	0.00803431	−	0.999968i	$-0.497443\pi$
$283$	76.4994	0.0160686	0.00803431	+	0.999968i	$0.497443\pi$
$284$	0	0
$285$	1989.34	0.413469
$286$	0	0
$287$	6839.71	1.40674
$288$	0	0
$289$	107.719	0.0219253
$290$	0	0
$291$	7535.17	1.51794
$292$	0	0
$293$	−7005.77	−1.39686	−0.698432	−	0.715676i	$-0.746119\pi$
$293$	−7005.77	−1.39686	−0.698432	+	0.715676i	$0.746119\pi$
$294$	0	0
$295$	−15596.0	−3.07809
$296$	0	0
$297$	−142.034	−0.0277496
$298$	0	0
$299$	10593.9	2.04903
$300$	0	0
$301$	8407.26	1.60992
$302$	0	0
$303$	−5604.37	−1.06258
$304$	0	0
$305$	11493.0	2.15766
$306$	0	0
$307$	−7595.13	−1.41198	−0.705988	−	0.708223i	$-0.749497\pi$
$307$	−7595.13	−1.41198	−0.705988	+	0.708223i	$0.749497\pi$
$308$	0	0
$309$	−6904.57	−1.27116
$310$	0	0
$311$	5895.21	1.07488	0.537439	−	0.843303i	$-0.319392\pi$
$311$	5895.21	1.07488	0.537439	+	0.843303i	$0.319392\pi$
$312$	0	0
$313$	4486.12	0.810130	0.405065	−	0.914288i	$-0.367249\pi$
$313$	4486.12	0.810130	0.405065	+	0.914288i	$0.367249\pi$
$314$	0	0
$315$	11354.4	2.03095
$316$	0	0
$317$	−467.351	−0.0828046	−0.0414023	−	0.999143i	$-0.513183\pi$
$317$	−467.351	−0.0828046	−0.0414023	+	0.999143i	$0.513183\pi$
$318$	0	0
$319$	−299.656	−0.0525940
$320$	0	0
$321$	−6749.45	−1.17358
$322$	0	0
$323$	960.161	0.165402
$324$	0	0
$325$	−18123.9	−3.09334
$326$	0	0
$327$	−12855.1	−2.17398
$328$	0	0
$329$	1455.06	0.243830
$330$	0	0
$331$	−2753.95	−0.457314	−0.228657	−	0.973507i	$-0.573433\pi$
$331$	−2753.95	−0.457314	−0.228657	+	0.973507i	$0.573433\pi$
$332$	0	0
$333$	−6785.24	−1.11660
$334$	0	0
$335$	4486.31	0.731682
$336$	0	0
$337$	2917.21	0.471545	0.235773	−	0.971808i	$-0.424238\pi$
$337$	2917.21	0.471545	0.235773	+	0.971808i	$0.424238\pi$
$338$	0	0
$339$	7135.92	1.14327
$340$	0	0
$341$	1204.73	0.191319
$342$	0	0
$343$	−4252.89	−0.669489
$344$	0	0
$345$	−24775.0	−3.86621
$346$	0	0
$347$	1456.25	0.225290	0.112645	−	0.993635i	$-0.464068\pi$
$347$	1456.25	0.225290	0.112645	+	0.993635i	$0.464068\pi$
$348$	0	0
$349$	9045.75	1.38742	0.693708	−	0.720257i	$-0.255976\pi$
$349$	9045.75	1.38742	0.693708	+	0.720257i	$0.255976\pi$
$350$	0	0
$351$	862.891	0.131219
$352$	0	0
$353$	−7644.24	−1.15258	−0.576291	−	0.817244i	$-0.695501\pi$
$353$	−7644.24	−1.15258	−0.576291	+	0.817244i	$0.695501\pi$
$354$	0	0
$355$	21135.8	3.15993
$356$	0	0
$357$	11376.3	1.68655
$358$	0	0
$359$	7037.15	1.03456	0.517279	−	0.855817i	$-0.326945\pi$
$359$	7037.15	1.03456	0.517279	+	0.855817i	$0.326945\pi$
$360$	0	0
$361$	−6675.38	−0.973229
$362$	0	0
$363$	−8836.16	−1.27763
$364$	0	0
$365$	16578.3	2.37739
$366$	0	0
$367$	838.072	0.119202	0.0596008	−	0.998222i	$-0.481017\pi$
$367$	838.072	0.119202	0.0596008	+	0.998222i	$0.481017\pi$
$368$	0	0
$369$	7716.42	1.08862
$370$	0	0
$371$	622.960	0.0871764
$372$	0	0
$373$	−5318.32	−0.738263	−0.369131	−	0.929377i	$-0.620345\pi$
$373$	−5318.32	−0.738263	−0.369131	+	0.929377i	$0.620345\pi$
$374$	0	0
$375$	24034.0	3.30963
$376$	0	0
$377$	1820.48	0.248700
$378$	0	0
$379$	−2543.36	−0.344706	−0.172353	−	0.985035i	$-0.555137\pi$
$379$	−2543.36	−0.344706	−0.172353	+	0.985035i	$0.555137\pi$
$380$	0	0
$381$	−15590.1	−2.09634
$382$	0	0
$383$	10457.6	1.39519	0.697595	−	0.716492i	$-0.254253\pi$
$383$	10457.6	1.39519	0.697595	+	0.716492i	$0.254253\pi$
$384$	0	0
$385$	4675.11	0.618872
$386$	0	0
$387$	9484.89	1.24585
$388$	0	0
$389$	−6237.70	−0.813017	−0.406509	−	0.913647i	$-0.633254\pi$
$389$	−6237.70	−0.813017	−0.406509	+	0.913647i	$0.633254\pi$
$390$	0	0
$391$	−11957.7	−1.54662
$392$	0	0
$393$	−2053.00	−0.263511
$394$	0	0
$395$	14578.0	1.85696
$396$	0	0
$397$	2779.65	0.351402	0.175701	−	0.984444i	$-0.443781\pi$
$397$	2779.65	0.351402	0.175701	+	0.984444i	$0.443781\pi$
$398$	0	0
$399$	2175.61	0.272974
$400$	0	0
$401$	−3175.84	−0.395496	−0.197748	−	0.980253i	$-0.563363\pi$
$401$	−3175.84	−0.395496	−0.197748	+	0.980253i	$0.563363\pi$
$402$	0	0
$403$	−7319.05	−0.904684
$404$	0	0
$405$	−15799.9	−1.93852
$406$	0	0
$407$	−2793.78	−0.340252
$408$	0	0
$409$	−6567.79	−0.794025	−0.397012	−	0.917813i	$-0.629953\pi$
$409$	−6567.79	−0.794025	−0.397012	+	0.917813i	$0.629953\pi$
$410$	0	0
$411$	−21258.9	−2.55139
$412$	0	0
$413$	−17056.3	−2.03217
$414$	0	0
$415$	15967.9	1.88875
$416$	0	0
$417$	9486.47	1.11404
$418$	0	0
$419$	543.863	0.0634115	0.0317058	−	0.999497i	$-0.489906\pi$
$419$	543.863	0.0634115	0.0317058	+	0.999497i	$0.489906\pi$
$420$	0	0
$421$	10540.9	1.22026	0.610132	−	0.792300i	$-0.291116\pi$
$421$	10540.9	1.22026	0.610132	+	0.792300i	$0.291116\pi$
$422$	0	0
$423$	1641.57	0.188690
$424$	0	0
$425$	20457.2	2.33487
$426$	0	0
$427$	12569.1	1.42450
$428$	0	0
$429$	−4681.81	−0.526900
$430$	0	0
$431$	1300.31	0.145322	0.0726608	−	0.997357i	$-0.476851\pi$
$431$	1300.31	0.145322	0.0726608	+	0.997357i	$0.476851\pi$
$432$	0	0
$433$	4554.74	0.505512	0.252756	−	0.967530i	$-0.418663\pi$
$433$	4554.74	0.505512	0.252756	+	0.967530i	$0.418663\pi$
$434$	0	0
$435$	−4257.42	−0.469259
$436$	0	0
$437$	−2286.79	−0.250325
$438$	0	0
$439$	−7033.06	−0.764624	−0.382312	−	0.924033i	$-0.624872\pi$
$439$	−7033.06	−0.764624	−0.382312	+	0.924033i	$0.624872\pi$
$440$	0	0
$441$	3809.76	0.411376
$442$	0	0
$443$	9158.56	0.982249	0.491125	−	0.871089i	$-0.336586\pi$
$443$	9158.56	0.982249	0.491125	+	0.871089i	$0.336586\pi$
$444$	0	0
$445$	16328.3	1.73941
$446$	0	0
$447$	2019.70	0.213710
$448$	0	0
$449$	1258.44	0.132270	0.0661352	−	0.997811i	$-0.478933\pi$
$449$	1258.44	0.132270	0.0661352	+	0.997811i	$0.478933\pi$
$450$	0	0
$451$	3177.19	0.331725
$452$	0	0
$453$	−5671.92	−0.588279
$454$	0	0
$455$	−28402.5	−2.92644
$456$	0	0
$457$	−12652.2	−1.29506	−0.647532	−	0.762038i	$-0.724199\pi$
$457$	−12652.2	−1.29506	−0.647532	+	0.762038i	$0.724199\pi$
$458$	0	0
$459$	−973.979	−0.0990446
$460$	0	0
$461$	15154.3	1.53103	0.765516	−	0.643416i	$-0.222484\pi$
$461$	15154.3	1.53103	0.765516	+	0.643416i	$0.222484\pi$
$462$	0	0
$463$	12734.9	1.27827	0.639137	−	0.769093i	$-0.279292\pi$
$463$	12734.9	1.27827	0.639137	+	0.769093i	$0.279292\pi$
$464$	0	0
$465$	17116.5	1.70701
$466$	0	0
$467$	13830.5	1.37045	0.685226	−	0.728331i	$-0.259703\pi$
$467$	13830.5	1.37045	0.685226	+	0.728331i	$0.259703\pi$
$468$	0	0
$469$	4906.37	0.483060
$470$	0	0
$471$	−18016.1	−1.76250
$472$	0	0
$473$	3905.35	0.379636
$474$	0	0
$475$	3912.23	0.377906
$476$	0	0
$477$	702.810	0.0674621
$478$	0	0
$479$	9263.58	0.883641	0.441820	−	0.897104i	$-0.354333\pi$
$479$	9263.58	0.883641	0.441820	+	0.897104i	$0.354333\pi$
$480$	0	0
$481$	16973.0	1.60894
$482$	0	0
$483$	−27094.7	−2.55249
$484$	0	0
$485$	21234.5	1.98806
$486$	0	0
$487$	−1793.51	−0.166882	−0.0834412	−	0.996513i	$-0.526591\pi$
$487$	−1793.51	−0.166882	−0.0834412	+	0.996513i	$0.526591\pi$
$488$	0	0
$489$	−8201.72	−0.758476
$490$	0	0
$491$	−8856.44	−0.814024	−0.407012	−	0.913423i	$-0.633429\pi$
$491$	−8856.44	−0.814024	−0.407012	+	0.913423i	$0.633429\pi$
$492$	0	0
$493$	−2054.85	−0.187720
$494$	0	0
$495$	5274.36	0.478919
$496$	0	0
$497$	23114.8	2.08620
$498$	0	0
$499$	9485.36	0.850948	0.425474	−	0.904971i	$-0.360107\pi$
$499$	9485.36	0.850948	0.425474	+	0.904971i	$0.360107\pi$
$500$	0	0
$501$	−20546.8	−1.83226
$502$	0	0
$503$	1206.30	0.106931	0.0534655	−	0.998570i	$-0.482973\pi$
$503$	1206.30	0.106931	0.0534655	+	0.998570i	$0.482973\pi$
$504$	0	0
$505$	−15793.4	−1.39167
$506$	0	0
$507$	12585.9	1.10248
$508$	0	0
$509$	2047.76	0.178321	0.0891604	−	0.996017i	$-0.471582\pi$
$509$	2047.76	0.178321	0.0891604	+	0.996017i	$0.471582\pi$
$510$	0	0
$511$	18130.5	1.56956
$512$	0	0
$513$	−186.264	−0.0160307
$514$	0	0
$515$	−19457.4	−1.66485
$516$	0	0
$517$	675.906	0.0574977
$518$	0	0
$519$	133.570	0.0112968
$520$	0	0
$521$	−4393.44	−0.369443	−0.184722	−	0.982791i	$-0.559138\pi$
$521$	−4393.44	−0.369443	−0.184722	+	0.982791i	$0.559138\pi$
$522$	0	0
$523$	3692.36	0.308710	0.154355	−	0.988015i	$-0.450670\pi$
$523$	3692.36	0.308710	0.154355	+	0.988015i	$0.450670\pi$
$524$	0	0
$525$	46353.5	3.85340
$526$	0	0
$527$	8261.30	0.682861
$528$	0	0
$529$	16312.4	1.34071
$530$	0	0
$531$	−19242.6	−1.57261
$532$	0	0
$533$	−19302.2	−1.56862
$534$	0	0
$535$	−19020.3	−1.53704
$536$	0	0
$537$	16451.4	1.32203
$538$	0	0
$539$	1568.64	0.125355
$540$	0	0
$541$	−6955.12	−0.552725	−0.276362	−	0.961054i	$-0.589129\pi$
$541$	−6955.12	−0.552725	−0.276362	+	0.961054i	$0.589129\pi$
$542$	0	0
$543$	−29559.6	−2.33614
$544$	0	0
$545$	−36226.4	−2.84728
$546$	0	0
$547$	8300.78	0.648840	0.324420	−	0.945913i	$-0.394831\pi$
$547$	8300.78	0.648840	0.324420	+	0.945913i	$0.394831\pi$
$548$	0	0
$549$	14180.2	1.10236
$550$	0	0
$551$	−392.970	−0.0303831
$552$	0	0
$553$	15943.0	1.22598
$554$	0	0
$555$	−39693.3	−3.03583
$556$	0	0
$557$	−15147.7	−1.15230	−0.576149	−	0.817345i	$-0.695445\pi$
$557$	−15147.7	−1.15230	−0.576149	+	0.817345i	$0.695445\pi$
$558$	0	0
$559$	−23726.0	−1.79517
$560$	0	0
$561$	5284.55	0.397707
$562$	0	0
$563$	−9643.31	−0.721878	−0.360939	−	0.932589i	$-0.617544\pi$
$563$	−9643.31	−0.721878	−0.360939	+	0.932589i	$0.617544\pi$
$564$	0	0
$565$	20109.4	1.49736
$566$	0	0
$567$	−17279.2	−1.27982
$568$	0	0
$569$	18413.5	1.35665	0.678326	−	0.734761i	$-0.262706\pi$
$569$	18413.5	1.35665	0.678326	+	0.734761i	$0.262706\pi$
$570$	0	0
$571$	−17448.5	−1.27881	−0.639403	−	0.768872i	$-0.720818\pi$
$571$	−17448.5	−1.27881	−0.639403	+	0.768872i	$0.720818\pi$
$572$	0	0
$573$	13581.6	0.990190
$574$	0	0
$575$	−48722.4	−3.53368
$576$	0	0
$577$	−15938.7	−1.14998	−0.574988	−	0.818162i	$-0.694993\pi$
$577$	−15938.7	−1.14998	−0.574988	+	0.818162i	$0.694993\pi$
$578$	0	0
$579$	−15562.6	−1.11703
$580$	0	0
$581$	17463.0	1.24696
$582$	0	0
$583$	289.378	0.0205571
$584$	0	0
$585$	−32043.1	−2.26465
$586$	0	0
$587$	−6058.46	−0.425996	−0.212998	−	0.977053i	$-0.568323\pi$
$587$	−6058.46	−0.425996	−0.212998	+	0.977053i	$0.568323\pi$
$588$	0	0
$589$	1579.89	0.110523
$590$	0	0
$591$	454.561	0.0316382
$592$	0	0
$593$	−8479.83	−0.587226	−0.293613	−	0.955924i	$-0.594858\pi$
$593$	−8479.83	−0.587226	−0.293613	+	0.955924i	$0.594858\pi$
$594$	0	0
$595$	32059.1	2.20890
$596$	0	0
$597$	12289.0	0.842473
$598$	0	0
$599$	20498.3	1.39822	0.699112	−	0.715012i	$-0.253579\pi$
$599$	20498.3	1.39822	0.699112	+	0.715012i	$0.253579\pi$
$600$	0	0
$601$	342.226	0.0232275	0.0116137	−	0.999933i	$-0.496303\pi$
$601$	342.226	0.0232275	0.0116137	+	0.999933i	$0.496303\pi$
$602$	0	0
$603$	5535.26	0.373820
$604$	0	0
$605$	−24900.7	−1.67332
$606$	0	0
$607$	−22426.5	−1.49961	−0.749806	−	0.661658i	$-0.769853\pi$
$607$	−22426.5	−1.49961	−0.749806	+	0.661658i	$0.769853\pi$
$608$	0	0
$609$	−4656.05	−0.309807
$610$	0	0
$611$	−4106.30	−0.271887
$612$	0	0
$613$	11621.5	0.765724	0.382862	−	0.923806i	$-0.374939\pi$
$613$	11621.5	0.765724	0.382862	+	0.923806i	$0.374939\pi$
$614$	0	0
$615$	45140.6	2.95975
$616$	0	0
$617$	−21469.5	−1.40086	−0.700429	−	0.713722i	$-0.747008\pi$
$617$	−21469.5	−1.40086	−0.700429	+	0.713722i	$0.747008\pi$
$618$	0	0
$619$	18378.9	1.19339	0.596697	−	0.802467i	$-0.296480\pi$
$619$	18378.9	1.19339	0.596697	+	0.802467i	$0.296480\pi$
$620$	0	0
$621$	2319.70	0.149898
$622$	0	0
$623$	17857.1	1.14836
$624$	0	0
$625$	31640.2	2.02497
$626$	0	0
$627$	1010.62	0.0643701
$628$	0	0
$629$	−19158.1	−1.21444
$630$	0	0
$631$	−874.199	−0.0551526	−0.0275763	−	0.999620i	$-0.508779\pi$
$631$	−874.199	−0.0551526	−0.0275763	+	0.999620i	$0.508779\pi$
$632$	0	0
$633$	21746.6	1.36548
$634$	0	0
$635$	−43933.7	−2.74560
$636$	0	0
$637$	−9529.92	−0.592762
$638$	0	0
$639$	26077.6	1.61442
$640$	0	0
$641$	−11449.8	−0.705525	−0.352762	−	0.935713i	$-0.614758\pi$
$641$	−11449.8	−0.705525	−0.352762	+	0.935713i	$0.614758\pi$
$642$	0	0
$643$	2190.33	0.134336	0.0671681	−	0.997742i	$-0.478604\pi$
$643$	2190.33	0.134336	0.0671681	+	0.997742i	$0.478604\pi$
$644$	0	0
$645$	55486.1	3.38723
$646$	0	0
$647$	−22530.5	−1.36903	−0.684516	−	0.728998i	$-0.739986\pi$
$647$	−22530.5	−1.36903	−0.684516	+	0.728998i	$0.739986\pi$
$648$	0	0
$649$	−7923.01	−0.479207
$650$	0	0
$651$	18719.1	1.12697
$652$	0	0
$653$	9695.42	0.581028	0.290514	−	0.956871i	$-0.406174\pi$
$653$	9695.42	0.581028	0.290514	+	0.956871i	$0.406174\pi$
$654$	0	0
$655$	−5785.44	−0.345123
$656$	0	0
$657$	20454.4	1.21462
$658$	0	0
$659$	−7940.74	−0.469389	−0.234695	−	0.972069i	$-0.575409\pi$
$659$	−7940.74	−0.469389	−0.234695	+	0.972069i	$0.575409\pi$
$660$	0	0
$661$	−30424.6	−1.79029	−0.895144	−	0.445778i	$-0.852927\pi$
$661$	−30424.6	−1.79029	−0.895144	+	0.445778i	$0.852927\pi$
$662$	0	0
$663$	−32105.0	−1.88062
$664$	0	0
$665$	6130.96	0.357517
$666$	0	0
$667$	4893.99	0.284102
$668$	0	0
$669$	−11099.7	−0.641462
$670$	0	0
$671$	5838.61	0.335912
$672$	0	0
$673$	25780.2	1.47660	0.738301	−	0.674472i	$-0.235628\pi$
$673$	25780.2	1.47660	0.738301	+	0.674472i	$0.235628\pi$
$674$	0	0
$675$	−3968.53	−0.226295
$676$	0	0
$677$	14742.0	0.836897	0.418449	−	0.908240i	$-0.362574\pi$
$677$	14742.0	0.836897	0.418449	+	0.908240i	$0.362574\pi$
$678$	0	0
$679$	23222.7	1.31252
$680$	0	0
$681$	−7484.32	−0.421145
$682$	0	0
$683$	9739.56	0.545643	0.272821	−	0.962065i	$-0.412043\pi$
$683$	9739.56	0.545643	0.272821	+	0.962065i	$0.412043\pi$
$684$	0	0
$685$	−59908.5	−3.34159
$686$	0	0
$687$	−14500.2	−0.805267
$688$	0	0
$689$	−1758.04	−0.0972078
$690$	0	0
$691$	24386.5	1.34256	0.671279	−	0.741205i	$-0.265745\pi$
$691$	24386.5	1.34256	0.671279	+	0.741205i	$0.265745\pi$
$692$	0	0
$693$	5768.20	0.316185
$694$	0	0
$695$	26733.3	1.45907
$696$	0	0
$697$	21787.2	1.18400
$698$	0	0
$699$	−13460.3	−0.728347
$700$	0	0
$701$	−19464.0	−1.04871	−0.524354	−	0.851500i	$-0.675693\pi$
$701$	−19464.0	−1.04871	−0.524354	+	0.851500i	$0.675693\pi$
$702$	0	0
$703$	−3663.78	−0.196561
$704$	0	0
$705$	9603.08	0.513011
$706$	0	0
$707$	−17272.1	−0.918790
$708$	0	0
$709$	12647.6	0.669946	0.334973	−	0.942228i	$-0.391273\pi$
$709$	12647.6	0.669946	0.334973	+	0.942228i	$0.391273\pi$
$710$	0	0
$711$	17986.5	0.948731
$712$	0	0
$713$	−19675.7	−1.03347
$714$	0	0
$715$	−13193.6	−0.690086
$716$	0	0
$717$	3483.65	0.181449
$718$	0	0
$719$	21453.6	1.11277	0.556387	−	0.830923i	$-0.312187\pi$
$719$	21453.6	1.11277	0.556387	+	0.830923i	$0.312187\pi$
$720$	0	0
$721$	−21279.2	−1.09914
$722$	0	0
$723$	−7633.96	−0.392683
$724$	0	0
$725$	−8372.62	−0.428898
$726$	0	0
$727$	24254.7	1.23735	0.618677	−	0.785645i	$-0.287669\pi$
$727$	24254.7	1.23735	0.618677	+	0.785645i	$0.287669\pi$
$728$	0	0
$729$	−16815.3	−0.854306
$730$	0	0
$731$	26780.5	1.35501
$732$	0	0
$733$	27449.1	1.38316	0.691580	−	0.722300i	$-0.256915\pi$
$733$	27449.1	1.38316	0.691580	+	0.722300i	$0.256915\pi$
$734$	0	0
$735$	22286.9	1.11845
$736$	0	0
$737$	2279.11	0.113911
$738$	0	0
$739$	2066.13	0.102847	0.0514234	−	0.998677i	$-0.483624\pi$
$739$	2066.13	0.102847	0.0514234	+	0.998677i	$0.483624\pi$
$740$	0	0
$741$	−6139.75	−0.304385
$742$	0	0
$743$	−23960.4	−1.18307	−0.591537	−	0.806278i	$-0.701479\pi$
$743$	−23960.4	−1.18307	−0.591537	+	0.806278i	$0.701479\pi$
$744$	0	0
$745$	5691.61	0.279898
$746$	0	0
$747$	19701.3	0.964973
$748$	0	0
$749$	−20801.1	−1.01476
$750$	0	0
$751$	−31339.1	−1.52275	−0.761373	−	0.648315i	$-0.775474\pi$
$751$	−31339.1	−1.52275	−0.761373	+	0.648315i	$0.775474\pi$
$752$	0	0
$753$	−11161.4	−0.540164
$754$	0	0
$755$	−15983.7	−0.770474
$756$	0	0
$757$	−9882.75	−0.474498	−0.237249	−	0.971449i	$-0.576246\pi$
$757$	−9882.75	−0.474498	−0.237249	+	0.971449i	$0.576246\pi$
$758$	0	0
$759$	−12586.1	−0.601904
$760$	0	0
$761$	−27344.7	−1.30256	−0.651278	−	0.758839i	$-0.725767\pi$
$761$	−27344.7	−1.30256	−0.651278	+	0.758839i	$0.725767\pi$
$762$	0	0
$763$	−39618.3	−1.87979
$764$	0	0
$765$	36168.3	1.70937
$766$	0	0
$767$	48134.3	2.26601
$768$	0	0
$769$	14093.4	0.660885	0.330442	−	0.943826i	$-0.392802\pi$
$769$	14093.4	0.660885	0.330442	+	0.943826i	$0.392802\pi$
$770$	0	0
$771$	−49935.2	−2.33252
$772$	0	0
$773$	23275.4	1.08300	0.541498	−	0.840702i	$-0.317857\pi$
$773$	23275.4	1.08300	0.541498	+	0.840702i	$0.317857\pi$
$774$	0	0
$775$	33661.1	1.56019
$776$	0	0
$777$	−43409.8	−2.00427
$778$	0	0
$779$	4166.58	0.191634
$780$	0	0
$781$	10737.3	0.491947
$782$	0	0
$783$	398.625	0.0181937
$784$	0	0
$785$	−50770.2	−2.30836
$786$	0	0
$787$	32710.9	1.48160	0.740799	−	0.671726i	$-0.234447\pi$
$787$	32710.9	1.48160	0.740799	+	0.671726i	$0.234447\pi$
$788$	0	0
$789$	10263.2	0.463091
$790$	0	0
$791$	21992.2	0.988563
$792$	0	0
$793$	−35471.1	−1.58842
$794$	0	0
$795$	4111.40	0.183417
$796$	0	0
$797$	2107.47	0.0936645	0.0468322	−	0.998903i	$-0.485087\pi$
$797$	2107.47	0.0936645	0.0468322	+	0.998903i	$0.485087\pi$
$798$	0	0
$799$	4634.95	0.205222
$800$	0	0
$801$	20146.0	0.888670
$802$	0	0
$803$	8422.00	0.370119
$804$	0	0
$805$	−76354.2	−3.34302
$806$	0	0
$807$	−37197.6	−1.62257
$808$	0	0
$809$	−29404.8	−1.27790	−0.638948	−	0.769250i	$-0.720630\pi$
$809$	−29404.8	−1.27790	−0.638948	+	0.769250i	$0.720630\pi$
$810$	0	0
$811$	−2041.30	−0.0883844	−0.0441922	−	0.999023i	$-0.514071\pi$
$811$	−2041.30	−0.0883844	−0.0441922	+	0.999023i	$0.514071\pi$
$812$	0	0
$813$	23758.0	1.02488
$814$	0	0
$815$	−23112.8	−0.993383
$816$	0	0
$817$	5121.49	0.219312
$818$	0	0
$819$	−35043.3	−1.49513
$820$	0	0
$821$	11592.9	0.492808	0.246404	−	0.969167i	$-0.420751\pi$
$821$	11592.9	0.492808	0.246404	+	0.969167i	$0.420751\pi$
$822$	0	0
$823$	−35294.1	−1.49487	−0.747434	−	0.664336i	$-0.768714\pi$
$823$	−35294.1	−1.49487	−0.747434	+	0.664336i	$0.768714\pi$
$824$	0	0
$825$	21532.2	0.908672
$826$	0	0
$827$	20275.4	0.852533	0.426266	−	0.904598i	$-0.359829\pi$
$827$	20275.4	0.852533	0.426266	+	0.904598i	$0.359829\pi$
$828$	0	0
$829$	−11610.6	−0.486432	−0.243216	−	0.969972i	$-0.578202\pi$
$829$	−11610.6	−0.486432	−0.243216	+	0.969972i	$0.578202\pi$
$830$	0	0
$831$	−42684.1	−1.78183
$832$	0	0
$833$	10756.8	0.447420
$834$	0	0
$835$	−57901.7	−2.39973
$836$	0	0
$837$	−1602.63	−0.0661826
$838$	0	0
$839$	31941.1	1.31434	0.657170	−	0.753743i	$-0.271753\pi$
$839$	31941.1	1.31434	0.657170	+	0.753743i	$0.271753\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	8158.59	0.333330
$844$	0	0
$845$	35467.5	1.44393
$846$	0	0
$847$	−27232.2	−1.10473
$848$	0	0
$849$	552.152	0.0223201
$850$	0	0
$851$	45628.3	1.83797
$852$	0	0
$853$	−2934.94	−0.117808	−0.0589042	−	0.998264i	$-0.518761\pi$
$853$	−2934.94	−0.117808	−0.0589042	+	0.998264i	$0.518761\pi$
$854$	0	0
$855$	6916.82	0.276667
$856$	0	0
$857$	17341.5	0.691219	0.345609	−	0.938378i	$-0.387672\pi$
$857$	17341.5	0.691219	0.345609	+	0.938378i	$0.387672\pi$
$858$	0	0
$859$	4513.41	0.179273	0.0896366	−	0.995975i	$-0.471429\pi$
$859$	4513.41	0.179273	0.0896366	+	0.995975i	$0.471429\pi$
$860$	0	0
$861$	49367.2	1.95404
$862$	0	0
$863$	5821.54	0.229626	0.114813	−	0.993387i	$-0.463373\pi$
$863$	5821.54	0.229626	0.114813	+	0.993387i	$0.463373\pi$
$864$	0	0
$865$	376.405	0.0147956
$866$	0	0
$867$	777.486	0.0304554
$868$	0	0
$869$	7405.85	0.289098
$870$	0	0
$871$	−13846.2	−0.538645
$872$	0	0
$873$	26199.3	1.01571
$874$	0	0
$875$	74070.6	2.86176
$876$	0	0
$877$	−51188.0	−1.97092	−0.985460	−	0.169906i	$-0.945653\pi$
$877$	−51188.0	−1.97092	−0.985460	+	0.169906i	$0.945653\pi$
$878$	0	0
$879$	−50565.7	−1.94032
$880$	0	0
$881$	32900.6	1.25817	0.629085	−	0.777336i	$-0.283430\pi$
$881$	32900.6	1.25817	0.629085	+	0.777336i	$0.283430\pi$
$882$	0	0
$883$	−833.893	−0.0317811	−0.0158906	−	0.999874i	$-0.505058\pi$
$883$	−833.893	−0.0317811	−0.0158906	+	0.999874i	$0.505058\pi$
$884$	0	0
$885$	−112568.	−4.27563
$886$	0	0
$887$	−35232.7	−1.33371	−0.666854	−	0.745189i	$-0.732359\pi$
$887$	−35232.7	−1.33371	−0.666854	+	0.745189i	$0.732359\pi$
$888$	0	0
$889$	−48047.3	−1.81266
$890$	0	0
$891$	−8026.56	−0.301796
$892$	0	0
$893$	886.386	0.0332159
$894$	0	0
$895$	46360.9	1.73148
$896$	0	0
$897$	76463.6	2.84621
$898$	0	0
$899$	−3381.14	−0.125437
$900$	0	0
$901$	1984.37	0.0733730
$902$	0	0
$903$	60681.3	2.23626
$904$	0	0
$905$	−83300.3	−3.05966
$906$	0	0
$907$	−42333.5	−1.54979	−0.774895	−	0.632090i	$-0.782197\pi$
$907$	−42333.5	−1.54979	−0.774895	+	0.632090i	$0.782197\pi$
$908$	0	0
$909$	−19486.0	−0.711012
$910$	0	0
$911$	9917.58	0.360685	0.180342	−	0.983604i	$-0.442279\pi$
$911$	9917.58	0.360685	0.180342	+	0.983604i	$0.442279\pi$
$912$	0	0
$913$	8111.91	0.294047
$914$	0	0
$915$	82953.3	2.99711
$916$	0	0
$917$	−6327.13	−0.227852
$918$	0	0
$919$	−23537.2	−0.844852	−0.422426	−	0.906397i	$-0.638821\pi$
$919$	−23537.2	−0.844852	−0.422426	+	0.906397i	$0.638821\pi$
$920$	0	0
$921$	−54819.6	−1.96131
$922$	0	0
$923$	−65231.9	−2.32626
$924$	0	0
$925$	−78060.6	−2.77472
$926$	0	0
$927$	−24006.8	−0.850577
$928$	0	0
$929$	9433.24	0.333148	0.166574	−	0.986029i	$-0.446730\pi$
$929$	9433.24	0.333148	0.166574	+	0.986029i	$0.446730\pi$
$930$	0	0
$931$	2057.13	0.0724164
$932$	0	0
$933$	42550.0	1.49306
$934$	0	0
$935$	14892.1	0.520881
$936$	0	0
$937$	−21154.4	−0.737551	−0.368776	−	0.929518i	$-0.620223\pi$
$937$	−21154.4	−0.737551	−0.368776	+	0.929518i	$0.620223\pi$
$938$	0	0
$939$	32379.6	1.12531
$940$	0	0
$941$	22459.3	0.778057	0.389028	−	0.921226i	$-0.372811\pi$
$941$	22459.3	0.778057	0.389028	+	0.921226i	$0.372811\pi$
$942$	0	0
$943$	−51890.1	−1.79191
$944$	0	0
$945$	−6219.20	−0.214085
$946$	0	0
$947$	−53773.3	−1.84519	−0.922596	−	0.385767i	$-0.873937\pi$
$947$	−53773.3	−1.84519	−0.922596	+	0.385767i	$0.873937\pi$
$948$	0	0
$949$	−51165.8	−1.75017
$950$	0	0
$951$	−3373.21	−0.115020
$952$	0	0
$953$	−3891.53	−0.132276	−0.0661379	−	0.997810i	$-0.521068\pi$
$953$	−3891.53	−0.132276	−0.0661379	+	0.997810i	$0.521068\pi$
$954$	0	0
$955$	38273.5	1.29686
$956$	0	0
$957$	−2162.83	−0.0730558
$958$	0	0
$959$	−65517.8	−2.20613
$960$	0	0
$961$	−16197.5	−0.543705
$962$	0	0
$963$	−23467.4	−0.785282
$964$	0	0
$965$	−43856.3	−1.46299
$966$	0	0
$967$	−7715.55	−0.256583	−0.128291	−	0.991737i	$-0.540949\pi$
$967$	−7715.55	−0.256583	−0.128291	+	0.991737i	$0.540949\pi$
$968$	0	0
$969$	6930.18	0.229752
$970$	0	0
$971$	−11607.4	−0.383623	−0.191812	−	0.981432i	$-0.561436\pi$
$971$	−11607.4	−0.383623	−0.191812	+	0.981432i	$0.561436\pi$
$972$	0	0
$973$	29236.4	0.963283
$974$	0	0
$975$	−130814.	−4.29681
$976$	0	0
$977$	36000.6	1.17888	0.589438	−	0.807813i	$-0.299349\pi$
$977$	36000.6	1.17888	0.589438	+	0.807813i	$0.299349\pi$
$978$	0	0
$979$	8295.01	0.270796
$980$	0	0
$981$	−44696.5	−1.45469
$982$	0	0
$983$	−20581.0	−0.667783	−0.333891	−	0.942612i	$-0.608362\pi$
$983$	−20581.0	−0.667783	−0.333891	+	0.942612i	$0.608362\pi$
$984$	0	0
$985$	1280.98	0.0414368
$986$	0	0
$987$	10502.2	0.338693
$988$	0	0
$989$	−63782.4	−2.05072
$990$	0	0
$991$	4576.25	0.146689	0.0733447	−	0.997307i	$-0.476633\pi$
$991$	4576.25	0.146689	0.0733447	+	0.997307i	$0.476633\pi$
$992$	0	0
$993$	−19877.3	−0.635233
$994$	0	0
$995$	34631.0	1.10339
$996$	0	0
$997$	18403.8	0.584608	0.292304	−	0.956325i	$-0.405578\pi$
$997$	18403.8	0.584608	0.292304	+	0.956325i	$0.405578\pi$
$998$	0	0
$999$	3716.51	0.117703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.v.1.2		3
4.3	odd	2		1856.4.a.o.1.2		3
8.3	odd	2		464.4.a.j.1.2		3
8.5	even	2		116.4.a.c.1.2	✓	3
24.5	odd	2		1044.4.a.f.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.4.a.c.1.2	✓	3	8.5	even	2
464.4.a.j.1.2		3	8.3	odd	2
1044.4.a.f.1.3		3	24.5	odd	2
1856.4.a.o.1.2		3	4.3	odd	2
1856.4.a.v.1.2		3	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+7.21773 q^{3} +20.3399 q^{5} +22.2443 q^{7} +25.0956 q^{9} +O(q^{10})\)	\(q+7.21773 q^{3} +20.3399 q^{5} +22.2443 q^{7} +25.0956 q^{9} +10.3330 q^{11} -62.7753 q^{13} +146.808 q^{15} +70.8570 q^{17} +13.5507 q^{19} +160.553 q^{21} -168.758 q^{23} +288.711 q^{25} -13.7457 q^{27} -29.0000 q^{29} +116.591 q^{31} +74.5804 q^{33} +452.447 q^{35} -270.376 q^{37} -453.095 q^{39} +307.481 q^{41} +377.951 q^{43} +510.441 q^{45} +65.4127 q^{47} +151.810 q^{49} +511.427 q^{51} +28.0053 q^{53} +210.171 q^{55} +97.8051 q^{57} -766.772 q^{59} +565.048 q^{61} +558.234 q^{63} -1276.84 q^{65} +220.567 q^{67} -1218.05 q^{69} +1039.13 q^{71} +815.062 q^{73} +2083.84 q^{75} +229.850 q^{77} +716.722 q^{79} -776.793 q^{81} +785.053 q^{83} +1441.22 q^{85} -209.314 q^{87} +802.772 q^{89} -1396.39 q^{91} +841.523 q^{93} +275.619 q^{95} +1043.98 q^{97} +259.311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 10 q^{3} + 20 q^{5} + 8 q^{7} + 93 q^{9}+O(q^{10}) \) 3 * q + 10 * q^3 + 20 * q^5 + 8 * q^7 + 93 * q^9	\( 3 q + 10 q^{3} + 20 q^{5} + 8 q^{7} + 93 q^{9} - 86 q^{11} - 124 q^{13} + 54 q^{15} + 14 q^{17} - 88 q^{19} - 280 q^{21} + 68 q^{23} + 111 q^{25} + 334 q^{27} - 87 q^{29} + 326 q^{31} + 110 q^{33} + 784 q^{35} - 166 q^{37} - 682 q^{39} + 34 q^{41} + 946 q^{43} + 242 q^{45} + 234 q^{47} + 1067 q^{49} + 1428 q^{51} + 1144 q^{53} + 94 q^{55} + 244 q^{57} - 488 q^{59} - 450 q^{61} - 1096 q^{63} - 1154 q^{65} + 52 q^{67} + 404 q^{69} + 1196 q^{71} + 2434 q^{73} + 1868 q^{75} + 312 q^{77} + 742 q^{79} - 849 q^{81} - 464 q^{83} + 672 q^{85} - 290 q^{87} + 1986 q^{89} - 448 q^{91} + 358 q^{93} + 68 q^{95} - 406 q^{97} - 2540 q^{99}+O(q^{100}) \) 3 * q + 10 * q^3 + 20 * q^5 + 8 * q^7 + 93 * q^9 - 86 * q^11 - 124 * q^13 + 54 * q^15 + 14 * q^17 - 88 * q^19 - 280 * q^21 + 68 * q^23 + 111 * q^25 + 334 * q^27 - 87 * q^29 + 326 * q^31 + 110 * q^33 + 784 * q^35 - 166 * q^37 - 682 * q^39 + 34 * q^41 + 946 * q^43 + 242 * q^45 + 234 * q^47 + 1067 * q^49 + 1428 * q^51 + 1144 * q^53 + 94 * q^55 + 244 * q^57 - 488 * q^59 - 450 * q^61 - 1096 * q^63 - 1154 * q^65 + 52 * q^67 + 404 * q^69 + 1196 * q^71 + 2434 * q^73 + 1868 * q^75 + 312 * q^77 + 742 * q^79 - 849 * q^81 - 464 * q^83 + 672 * q^85 - 290 * q^87 + 1986 * q^89 - 448 * q^91 + 358 * q^93 + 68 * q^95 - 406 * q^97 - 2540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more