LMFDB - Embedded newform 1856.4.a.be.1.5

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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 19x^{6} + 92x^{4} - 51x^{2} + 5 $ x^8 - 19x^6 + 92x^4 - 51*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 5 $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.354810$ 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)

$f(q)$	$=$	$q+3.50343 q^{3} -6.74822 q^{5} +5.06823 q^{7} -14.7260 q^{9} +O(q^{10})$	$q+3.50343 q^{3} -6.74822 q^{5} +5.06823 q^{7} -14.7260 q^{9} -22.4509 q^{11} +20.4464 q^{13} -23.6419 q^{15} +45.3575 q^{17} +79.8869 q^{19} +17.7562 q^{21} +67.4136 q^{23} -79.4615 q^{25} -146.184 q^{27} -29.0000 q^{29} -133.881 q^{31} -78.6553 q^{33} -34.2015 q^{35} +343.446 q^{37} +71.6325 q^{39} +72.7396 q^{41} -232.616 q^{43} +99.3742 q^{45} +416.631 q^{47} -317.313 q^{49} +158.907 q^{51} +136.783 q^{53} +151.504 q^{55} +279.878 q^{57} -586.105 q^{59} +212.320 q^{61} -74.6346 q^{63} -137.977 q^{65} +138.963 q^{67} +236.179 q^{69} -812.301 q^{71} +163.327 q^{73} -278.388 q^{75} -113.787 q^{77} -926.444 q^{79} -114.544 q^{81} -539.330 q^{83} -306.082 q^{85} -101.599 q^{87} -196.223 q^{89} +103.627 q^{91} -469.043 q^{93} -539.094 q^{95} -1858.85 q^{97} +330.612 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 20 q^{5} + 40 q^{9}+O(q^{10}) $ 8 * q + 20 * q^5 + 40 * q^9	$ 8 q + 20 q^{5} + 40 q^{9} - 4 q^{13} - 140 q^{17} + 28 q^{21} - 256 q^{25} - 232 q^{29} - 344 q^{33} - 280 q^{37} - 700 q^{41} + 56 q^{45} - 256 q^{49} + 604 q^{53} - 2016 q^{57} + 884 q^{61} - 1616 q^{65} + 764 q^{69} - 3504 q^{73} + 1916 q^{77} - 3904 q^{81} + 996 q^{85} - 2924 q^{89} + 2996 q^{93} - 2668 q^{97}+O(q^{100}) $ 8 * q + 20 * q^5 + 40 * q^9 - 4 * q^13 - 140 * q^17 + 28 * q^21 - 256 * q^25 - 232 * q^29 - 344 * q^33 - 280 * q^37 - 700 * q^41 + 56 * q^45 - 256 * q^49 + 604 * q^53 - 2016 * q^57 + 884 * q^61 - 1616 * q^65 + 764 * q^69 - 3504 * q^73 + 1916 * q^77 - 3904 * q^81 + 996 * q^85 - 2924 * q^89 + 2996 * q^93 - 2668 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.50343	0.674235	0.337118	−	0.941463i	$-0.390548\pi$
$3$	3.50343	0.674235	0.337118	+	0.941463i	$0.390548\pi$
$4$	0	0
$5$	−6.74822	−0.603579	−0.301790	−	0.953375i	$-0.597584\pi$
$5$	−6.74822	−0.603579	−0.301790	+	0.953375i	$0.597584\pi$
$6$	0	0
$7$	5.06823	0.273659	0.136829	−	0.990595i	$-0.456309\pi$
$7$	5.06823	0.273659	0.136829	+	0.990595i	$0.456309\pi$
$8$	0	0
$9$	−14.7260	−0.545407
$10$	0	0
$11$	−22.4509	−0.615383	−0.307692	−	0.951486i	$-0.599557\pi$
$11$	−22.4509	−0.615383	−0.307692	+	0.951486i	$0.599557\pi$
$12$	0	0
$13$	20.4464	0.436216	0.218108	−	0.975925i	$-0.430011\pi$
$13$	20.4464	0.436216	0.218108	+	0.975925i	$0.430011\pi$
$14$	0	0
$15$	−23.6419	−0.406954
$16$	0	0
$17$	45.3575	0.647106	0.323553	−	0.946210i	$-0.395123\pi$
$17$	45.3575	0.647106	0.323553	+	0.946210i	$0.395123\pi$
$18$	0	0
$19$	79.8869	0.964595	0.482298	−	0.876007i	$-0.339802\pi$
$19$	79.8869	0.964595	0.482298	+	0.876007i	$0.339802\pi$
$20$	0	0
$21$	17.7562	0.184510
$22$	0	0
$23$	67.4136	0.611161	0.305581	−	0.952166i	$-0.401149\pi$
$23$	67.4136	0.611161	0.305581	+	0.952166i	$0.401149\pi$
$24$	0	0
$25$	−79.4615	−0.635692
$26$	0	0
$27$	−146.184	−1.04197
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−133.881	−0.775669	−0.387834	−	0.921729i	$-0.626777\pi$
$31$	−133.881	−0.775669	−0.387834	+	0.921729i	$0.626777\pi$
$32$	0	0
$33$	−78.6553	−0.414913
$34$	0	0
$35$	−34.2015	−0.165175
$36$	0	0
$37$	343.446	1.52600	0.763002	−	0.646396i	$-0.223725\pi$
$37$	343.446	1.52600	0.763002	+	0.646396i	$0.223725\pi$
$38$	0	0
$39$	71.6325	0.294112
$40$	0	0
$41$	72.7396	0.277074	0.138537	−	0.990357i	$-0.455760\pi$
$41$	72.7396	0.277074	0.138537	+	0.990357i	$0.455760\pi$
$42$	0	0
$43$	−232.616	−0.824968	−0.412484	−	0.910965i	$-0.635339\pi$
$43$	−232.616	−0.824968	−0.412484	+	0.910965i	$0.635339\pi$
$44$	0	0
$45$	99.3742	0.329196
$46$	0	0
$47$	416.631	1.29302	0.646510	−	0.762906i	$-0.276228\pi$
$47$	416.631	1.29302	0.646510	+	0.762906i	$0.276228\pi$
$48$	0	0
$49$	−317.313	−0.925111
$50$	0	0
$51$	158.907	0.436302
$52$	0	0
$53$	136.783	0.354502	0.177251	−	0.984166i	$-0.443280\pi$
$53$	136.783	0.354502	0.177251	+	0.984166i	$0.443280\pi$
$54$	0	0
$55$	151.504	0.371433
$56$	0	0
$57$	279.878	0.650364
$58$	0	0
$59$	−586.105	−1.29329	−0.646647	−	0.762789i	$-0.723829\pi$
$59$	−586.105	−1.29329	−0.646647	+	0.762789i	$0.723829\pi$
$60$	0	0
$61$	212.320	0.445653	0.222826	−	0.974858i	$-0.428472\pi$
$61$	212.320	0.445653	0.222826	+	0.974858i	$0.428472\pi$
$62$	0	0
$63$	−74.6346	−0.149255
$64$	0	0
$65$	−137.977	−0.263291
$66$	0	0
$67$	138.963	0.253389	0.126694	−	0.991942i	$-0.459563\pi$
$67$	138.963	0.253389	0.126694	+	0.991942i	$0.459563\pi$
$68$	0	0
$69$	236.179	0.412066
$70$	0	0
$71$	−812.301	−1.35778	−0.678890	−	0.734240i	$-0.737539\pi$
$71$	−812.301	−1.35778	−0.678890	+	0.734240i	$0.737539\pi$
$72$	0	0
$73$	163.327	0.261862	0.130931	−	0.991391i	$-0.458203\pi$
$73$	163.327	0.261862	0.130931	+	0.991391i	$0.458203\pi$
$74$	0	0
$75$	−278.388	−0.428606
$76$	0	0
$77$	−113.787	−0.168405
$78$	0	0
$79$	−926.444	−1.31941	−0.659703	−	0.751526i	$-0.729318\pi$
$79$	−926.444	−1.31941	−0.659703	+	0.751526i	$0.729318\pi$
$80$	0	0
$81$	−114.544	−0.157125
$82$	0	0
$83$	−539.330	−0.713243	−0.356621	−	0.934249i	$-0.616071\pi$
$83$	−539.330	−0.713243	−0.356621	+	0.934249i	$0.616071\pi$
$84$	0	0
$85$	−306.082	−0.390580
$86$	0	0
$87$	−101.599	−0.125202
$88$	0	0
$89$	−196.223	−0.233703	−0.116852	−	0.993149i	$-0.537280\pi$
$89$	−196.223	−0.233703	−0.116852	+	0.993149i	$0.537280\pi$
$90$	0	0
$91$	103.627	0.119374
$92$	0	0
$93$	−469.043	−0.522983
$94$	0	0
$95$	−539.094	−0.582209
$96$	0	0
$97$	−1858.85	−1.94575	−0.972874	−	0.231338i	$-0.925690\pi$
$97$	−1858.85	−1.94575	−0.972874	+	0.231338i	$0.925690\pi$
$98$	0	0
$99$	330.612	0.335634
$100$	0	0
$101$	−169.178	−0.166671	−0.0833356	−	0.996522i	$-0.526557\pi$
$101$	−169.178	−0.166671	−0.0833356	+	0.996522i	$0.526557\pi$
$102$	0	0
$103$	601.801	0.575701	0.287850	−	0.957675i	$-0.407059\pi$
$103$	601.801	0.575701	0.287850	+	0.957675i	$0.407059\pi$
$104$	0	0
$105$	−119.823	−0.111367
$106$	0	0
$107$	−599.520	−0.541662	−0.270831	−	0.962627i	$-0.587298\pi$
$107$	−599.520	−0.541662	−0.270831	+	0.962627i	$0.587298\pi$
$108$	0	0
$109$	67.3011	0.0591401	0.0295701	−	0.999563i	$-0.490586\pi$
$109$	67.3011	0.0591401	0.0295701	+	0.999563i	$0.490586\pi$
$110$	0	0
$111$	1203.24	1.02889
$112$	0	0
$113$	−1809.72	−1.50659	−0.753294	−	0.657683i	$-0.771536\pi$
$113$	−1809.72	−1.50659	−0.753294	+	0.657683i	$0.771536\pi$
$114$	0	0
$115$	−454.922	−0.368884
$116$	0	0
$117$	−301.093	−0.237915
$118$	0	0
$119$	229.882	0.177086
$120$	0	0
$121$	−826.955	−0.621303
$122$	0	0
$123$	254.838	0.186813
$124$	0	0
$125$	1379.75	0.987270
$126$	0	0
$127$	−1759.44	−1.22933	−0.614665	−	0.788788i	$-0.710709\pi$
$127$	−1759.44	−1.22933	−0.614665	+	0.788788i	$0.710709\pi$
$128$	0	0
$129$	−814.954	−0.556223
$130$	0	0
$131$	2743.56	1.82981	0.914907	−	0.403664i	$-0.132264\pi$
$131$	2743.56	1.82981	0.914907	+	0.403664i	$0.132264\pi$
$132$	0	0
$133$	404.885	0.263970
$134$	0	0
$135$	986.482	0.628910
$136$	0	0
$137$	−1443.11	−0.899952	−0.449976	−	0.893041i	$-0.648567\pi$
$137$	−1443.11	−0.899952	−0.449976	+	0.893041i	$0.648567\pi$
$138$	0	0
$139$	1737.45	1.06021	0.530103	−	0.847934i	$-0.322153\pi$
$139$	1737.45	1.06021	0.530103	+	0.847934i	$0.322153\pi$
$140$	0	0
$141$	1459.64	0.871800
$142$	0	0
$143$	−459.041	−0.268440
$144$	0	0
$145$	195.698	0.112082
$146$	0	0
$147$	−1111.68	−0.623742
$148$	0	0
$149$	−859.701	−0.472681	−0.236340	−	0.971670i	$-0.575948\pi$
$149$	−859.701	−0.472681	−0.236340	+	0.971670i	$0.575948\pi$
$150$	0	0
$151$	−474.299	−0.255615	−0.127808	−	0.991799i	$-0.540794\pi$
$151$	−474.299	−0.255615	−0.127808	+	0.991799i	$0.540794\pi$
$152$	0	0
$153$	−667.933	−0.352936
$154$	0	0
$155$	903.458	0.468177
$156$	0	0
$157$	338.908	0.172279	0.0861396	−	0.996283i	$-0.472547\pi$
$157$	338.908	0.172279	0.0861396	+	0.996283i	$0.472547\pi$
$158$	0	0
$159$	479.210	0.239018
$160$	0	0
$161$	341.667	0.167249
$162$	0	0
$163$	−1484.38	−0.713287	−0.356643	−	0.934241i	$-0.616079\pi$
$163$	−1484.38	−0.713287	−0.356643	+	0.934241i	$0.616079\pi$
$164$	0	0
$165$	530.783	0.250433
$166$	0	0
$167$	−1790.86	−0.829824	−0.414912	−	0.909862i	$-0.636188\pi$
$167$	−1790.86	−0.829824	−0.414912	+	0.909862i	$0.636188\pi$
$168$	0	0
$169$	−1778.94	−0.809715
$170$	0	0
$171$	−1176.41	−0.526097
$172$	0	0
$173$	−1671.46	−0.734561	−0.367281	−	0.930110i	$-0.619711\pi$
$173$	−1671.46	−0.734561	−0.367281	+	0.930110i	$0.619711\pi$
$174$	0	0
$175$	−402.729	−0.173963
$176$	0	0
$177$	−2053.38	−0.871985
$178$	0	0
$179$	2111.96	0.881874	0.440937	−	0.897538i	$-0.354646\pi$
$179$	2111.96	0.881874	0.440937	+	0.897538i	$0.354646\pi$
$180$	0	0
$181$	1746.36	0.717161	0.358581	−	0.933499i	$-0.383261\pi$
$181$	1746.36	0.717161	0.358581	+	0.933499i	$0.383261\pi$
$182$	0	0
$183$	743.849	0.300475
$184$	0	0
$185$	−2317.65	−0.921064
$186$	0	0
$187$	−1018.32	−0.398218
$188$	0	0
$189$	−740.894	−0.285143
$190$	0	0
$191$	1904.98	0.721672	0.360836	−	0.932629i	$-0.382491\pi$
$191$	1904.98	0.721672	0.360836	+	0.932629i	$0.382491\pi$
$192$	0	0
$193$	−4758.65	−1.77479	−0.887397	−	0.461007i	$-0.847488\pi$
$193$	−4758.65	−1.77479	−0.887397	+	0.461007i	$0.847488\pi$
$194$	0	0
$195$	−483.392	−0.177520
$196$	0	0
$197$	664.799	0.240431	0.120216	−	0.992748i	$-0.461641\pi$
$197$	664.799	0.240431	0.120216	+	0.992748i	$0.461641\pi$
$198$	0	0
$199$	−2902.16	−1.03381	−0.516907	−	0.856042i	$-0.672917\pi$
$199$	−2902.16	−1.03381	−0.516907	+	0.856042i	$0.672917\pi$
$200$	0	0
$201$	486.848	0.170844
$202$	0	0
$203$	−146.979	−0.0508171
$204$	0	0
$205$	−490.863	−0.167236
$206$	0	0
$207$	−992.731	−0.333331
$208$	0	0
$209$	−1793.54	−0.593596
$210$	0	0
$211$	−3698.14	−1.20659	−0.603296	−	0.797518i	$-0.706146\pi$
$211$	−3698.14	−1.20659	−0.603296	+	0.797518i	$0.706146\pi$
$212$	0	0
$213$	−2845.84	−0.915463
$214$	0	0
$215$	1569.74	0.497934
$216$	0	0
$217$	−678.539	−0.212268
$218$	0	0
$219$	572.204	0.176557
$220$	0	0
$221$	927.397	0.282278
$222$	0	0
$223$	1381.34	0.414805	0.207402	−	0.978256i	$-0.433499\pi$
$223$	1381.34	0.414805	0.207402	+	0.978256i	$0.433499\pi$
$224$	0	0
$225$	1170.15	0.346711
$226$	0	0
$227$	2632.75	0.769789	0.384894	−	0.922961i	$-0.374238\pi$
$227$	2632.75	0.769789	0.384894	+	0.922961i	$0.374238\pi$
$228$	0	0
$229$	−2921.66	−0.843094	−0.421547	−	0.906806i	$-0.638513\pi$
$229$	−2921.66	−0.843094	−0.421547	+	0.906806i	$0.638513\pi$
$230$	0	0
$231$	−398.643	−0.113545
$232$	0	0
$233$	2704.54	0.760431	0.380216	−	0.924898i	$-0.375850\pi$
$233$	2704.54	0.760431	0.380216	+	0.924898i	$0.375850\pi$
$234$	0	0
$235$	−2811.52	−0.780440
$236$	0	0
$237$	−3245.73	−0.889590
$238$	0	0
$239$	−320.582	−0.0867645	−0.0433823	−	0.999059i	$-0.513813\pi$
$239$	−320.582	−0.0867645	−0.0433823	+	0.999059i	$0.513813\pi$
$240$	0	0
$241$	1545.98	0.413216	0.206608	−	0.978424i	$-0.433757\pi$
$241$	1545.98	0.413216	0.206608	+	0.978424i	$0.433757\pi$
$242$	0	0
$243$	3545.67	0.936029
$244$	0	0
$245$	2141.30	0.558378
$246$	0	0
$247$	1633.40	0.420772
$248$	0	0
$249$	−1889.50	−0.480893
$250$	0	0
$251$	−2528.06	−0.635737	−0.317868	−	0.948135i	$-0.602967\pi$
$251$	−2528.06	−0.635737	−0.317868	+	0.948135i	$0.602967\pi$
$252$	0	0
$253$	−1513.50	−0.376098
$254$	0	0
$255$	−1072.34	−0.263343
$256$	0	0
$257$	−3050.82	−0.740487	−0.370244	−	0.928935i	$-0.620726\pi$
$257$	−3050.82	−0.740487	−0.370244	+	0.928935i	$0.620726\pi$
$258$	0	0
$259$	1740.66	0.417604
$260$	0	0
$261$	427.054	0.101279
$262$	0	0
$263$	8361.56	1.96044	0.980220	−	0.197910i	$-0.0634152\pi$
$263$	8361.56	1.96044	0.980220	+	0.197910i	$0.0634152\pi$
$264$	0	0
$265$	−923.043	−0.213970
$266$	0	0
$267$	−687.453	−0.157571
$268$	0	0
$269$	−1623.45	−0.367968	−0.183984	−	0.982929i	$-0.558899\pi$
$269$	−1623.45	−0.367968	−0.183984	+	0.982929i	$0.558899\pi$
$270$	0	0
$271$	1624.69	0.364181	0.182091	−	0.983282i	$-0.441714\pi$
$271$	1624.69	0.364181	0.182091	+	0.983282i	$0.441714\pi$
$272$	0	0
$273$	363.050	0.0804864
$274$	0	0
$275$	1783.99	0.391194
$276$	0	0
$277$	−5176.32	−1.12280	−0.561399	−	0.827546i	$-0.689737\pi$
$277$	−5176.32	−1.12280	−0.561399	+	0.827546i	$0.689737\pi$
$278$	0	0
$279$	1971.53	0.423055
$280$	0	0
$281$	−1227.99	−0.260696	−0.130348	−	0.991468i	$-0.541609\pi$
$281$	−1227.99	−0.260696	−0.130348	+	0.991468i	$0.541609\pi$
$282$	0	0
$283$	2760.21	0.579779	0.289889	−	0.957060i	$-0.406381\pi$
$283$	2760.21	0.579779	0.289889	+	0.957060i	$0.406381\pi$
$284$	0	0
$285$	−1888.68	−0.392546
$286$	0	0
$287$	368.661	0.0758235
$288$	0	0
$289$	−2855.70	−0.581254
$290$	0	0
$291$	−6512.34	−1.31189
$292$	0	0
$293$	−7974.31	−1.58998	−0.794990	−	0.606623i	$-0.792524\pi$
$293$	−7974.31	−1.58998	−0.794990	+	0.606623i	$0.792524\pi$
$294$	0	0
$295$	3955.17	0.780605
$296$	0	0
$297$	3281.97	0.641210
$298$	0	0
$299$	1378.37	0.266598
$300$	0	0
$301$	−1178.95	−0.225760
$302$	0	0
$303$	−592.701	−0.112376
$304$	0	0
$305$	−1432.78	−0.268987
$306$	0	0
$307$	5816.97	1.08141	0.540704	−	0.841213i	$-0.318158\pi$
$307$	5816.97	1.08141	0.540704	+	0.841213i	$0.318158\pi$
$308$	0	0
$309$	2108.37	0.388158
$310$	0	0
$311$	−9625.16	−1.75496	−0.877481	−	0.479612i	$-0.840777\pi$
$311$	−9625.16	−1.75496	−0.877481	+	0.479612i	$0.840777\pi$
$312$	0	0
$313$	2115.31	0.381994	0.190997	−	0.981591i	$-0.438828\pi$
$313$	2115.31	0.381994	0.190997	+	0.981591i	$0.438828\pi$
$314$	0	0
$315$	503.651	0.0900873
$316$	0	0
$317$	2421.51	0.429040	0.214520	−	0.976720i	$-0.431181\pi$
$317$	2421.51	0.429040	0.214520	+	0.976720i	$0.431181\pi$
$318$	0	0
$319$	651.078	0.114274
$320$	0	0
$321$	−2100.38	−0.365208
$322$	0	0
$323$	3623.47	0.624195
$324$	0	0
$325$	−1624.70	−0.277299
$326$	0	0
$327$	235.785	0.0398744
$328$	0	0
$329$	2111.58	0.353846
$330$	0	0
$331$	−4071.96	−0.676179	−0.338090	−	0.941114i	$-0.609781\pi$
$331$	−4071.96	−0.676179	−0.338090	+	0.941114i	$0.609781\pi$
$332$	0	0
$333$	−5057.58	−0.832293
$334$	0	0
$335$	−937.754	−0.152940
$336$	0	0
$337$	−2924.95	−0.472796	−0.236398	−	0.971656i	$-0.575967\pi$
$337$	−2924.95	−0.472796	−0.236398	+	0.971656i	$0.575967\pi$
$338$	0	0
$339$	−6340.24	−1.01580
$340$	0	0
$341$	3005.75	0.477334
$342$	0	0
$343$	−3346.62	−0.526823
$344$	0	0
$345$	−1593.79	−0.248715
$346$	0	0
$347$	−2274.16	−0.351824	−0.175912	−	0.984406i	$-0.556287\pi$
$347$	−2274.16	−0.351824	−0.175912	+	0.984406i	$0.556287\pi$
$348$	0	0
$349$	−1314.05	−0.201546	−0.100773	−	0.994909i	$-0.532132\pi$
$349$	−1314.05	−0.201546	−0.100773	+	0.994909i	$0.532132\pi$
$350$	0	0
$351$	−2988.94	−0.454523
$352$	0	0
$353$	5773.23	0.870476	0.435238	−	0.900315i	$-0.356664\pi$
$353$	5773.23	0.870476	0.435238	+	0.900315i	$0.356664\pi$
$354$	0	0
$355$	5481.59	0.819528
$356$	0	0
$357$	805.375	0.119398
$358$	0	0
$359$	13138.8	1.93159	0.965794	−	0.259311i	$-0.0834954\pi$
$359$	13138.8	1.93159	0.965794	+	0.259311i	$0.0834954\pi$
$360$	0	0
$361$	−477.087	−0.0695563
$362$	0	0
$363$	−2897.18	−0.418905
$364$	0	0
$365$	−1102.17	−0.158055
$366$	0	0
$367$	−2954.04	−0.420162	−0.210081	−	0.977684i	$-0.567373\pi$
$367$	−2954.04	−0.420162	−0.210081	+	0.977684i	$0.567373\pi$
$368$	0	0
$369$	−1071.16	−0.151118
$370$	0	0
$371$	693.248	0.0970126
$372$	0	0
$373$	−2476.76	−0.343812	−0.171906	−	0.985113i	$-0.554993\pi$
$373$	−2476.76	−0.343812	−0.171906	+	0.985113i	$0.554993\pi$
$374$	0	0
$375$	4833.86	0.665652
$376$	0	0
$377$	−592.946	−0.0810033
$378$	0	0
$379$	9969.57	1.35119	0.675597	−	0.737272i	$-0.263886\pi$
$379$	9969.57	1.35119	0.675597	+	0.737272i	$0.263886\pi$
$380$	0	0
$381$	−6164.07	−0.828858
$382$	0	0
$383$	65.3323	0.00871626	0.00435813	−	0.999991i	$-0.498613\pi$
$383$	65.3323	0.00871626	0.00435813	+	0.999991i	$0.498613\pi$
$384$	0	0
$385$	767.856	0.101646
$386$	0	0
$387$	3425.50	0.449943
$388$	0	0
$389$	−958.325	−0.124907	−0.0624537	−	0.998048i	$-0.519893\pi$
$389$	−958.325	−0.124907	−0.0624537	+	0.998048i	$0.519893\pi$
$390$	0	0
$391$	3057.71	0.395486
$392$	0	0
$393$	9611.86	1.23373
$394$	0	0
$395$	6251.85	0.796366
$396$	0	0
$397$	−6593.27	−0.833518	−0.416759	−	0.909017i	$-0.636834\pi$
$397$	−6593.27	−0.833518	−0.416759	+	0.909017i	$0.636834\pi$
$398$	0	0
$399$	1418.49	0.177978
$400$	0	0
$401$	−1027.58	−0.127967	−0.0639834	−	0.997951i	$-0.520380\pi$
$401$	−1027.58	−0.127967	−0.0639834	+	0.997951i	$0.520380\pi$
$402$	0	0
$403$	−2737.38	−0.338359
$404$	0	0
$405$	772.967	0.0948372
$406$	0	0
$407$	−7710.68	−0.939077
$408$	0	0
$409$	1733.46	0.209570	0.104785	−	0.994495i	$-0.466585\pi$
$409$	1733.46	0.209570	0.104785	+	0.994495i	$0.466585\pi$
$410$	0	0
$411$	−5055.84	−0.606779
$412$	0	0
$413$	−2970.51	−0.353921
$414$	0	0
$415$	3639.52	0.430498
$416$	0	0
$417$	6087.03	0.714828
$418$	0	0
$419$	−7022.33	−0.818767	−0.409384	−	0.912362i	$-0.634256\pi$
$419$	−7022.33	−0.818767	−0.409384	+	0.912362i	$0.634256\pi$
$420$	0	0
$421$	−14218.0	−1.64594	−0.822971	−	0.568083i	$-0.807685\pi$
$421$	−14218.0	−1.64594	−0.822971	+	0.568083i	$0.807685\pi$
$422$	0	0
$423$	−6135.31	−0.705222
$424$	0	0
$425$	−3604.17	−0.411360
$426$	0	0
$427$	1076.09	0.121957
$428$	0	0
$429$	−1608.22	−0.180992
$430$	0	0
$431$	−5156.37	−0.576273	−0.288136	−	0.957589i	$-0.593036\pi$
$431$	−5156.37	−0.576273	−0.288136	+	0.957589i	$0.593036\pi$
$432$	0	0
$433$	−8676.62	−0.962983	−0.481492	−	0.876451i	$-0.659905\pi$
$433$	−8676.62	−0.962983	−0.481492	+	0.876451i	$0.659905\pi$
$434$	0	0
$435$	685.615	0.0755695
$436$	0	0
$437$	5385.46	0.589523
$438$	0	0
$439$	5758.81	0.626089	0.313045	−	0.949738i	$-0.398651\pi$
$439$	5758.81	0.626089	0.313045	+	0.949738i	$0.398651\pi$
$440$	0	0
$441$	4672.75	0.504562
$442$	0	0
$443$	1755.56	0.188283	0.0941415	−	0.995559i	$-0.469989\pi$
$443$	1755.56	0.188283	0.0941415	+	0.995559i	$0.469989\pi$
$444$	0	0
$445$	1324.16	0.141058
$446$	0	0
$447$	−3011.90	−0.318698
$448$	0	0
$449$	10362.4	1.08915	0.544577	−	0.838711i	$-0.316690\pi$
$449$	10362.4	1.08915	0.544577	+	0.838711i	$0.316690\pi$
$450$	0	0
$451$	−1633.07	−0.170506
$452$	0	0
$453$	−1661.67	−0.172345
$454$	0	0
$455$	−699.298	−0.0720519
$456$	0	0
$457$	−2767.51	−0.283280	−0.141640	−	0.989918i	$-0.545237\pi$
$457$	−2767.51	−0.283280	−0.141640	+	0.989918i	$0.545237\pi$
$458$	0	0
$459$	−6630.54	−0.674263
$460$	0	0
$461$	−7136.27	−0.720974	−0.360487	−	0.932764i	$-0.617390\pi$
$461$	−7136.27	−0.720974	−0.360487	+	0.932764i	$0.617390\pi$
$462$	0	0
$463$	−2046.79	−0.205448	−0.102724	−	0.994710i	$-0.532756\pi$
$463$	−2046.79	−0.205448	−0.102724	+	0.994710i	$0.532756\pi$
$464$	0	0
$465$	3165.20	0.315662
$466$	0	0
$467$	9820.16	0.973068	0.486534	−	0.873662i	$-0.338261\pi$
$467$	9820.16	0.973068	0.486534	+	0.873662i	$0.338261\pi$
$468$	0	0
$469$	704.297	0.0693420
$470$	0	0
$471$	1187.34	0.116157
$472$	0	0
$473$	5222.45	0.507672
$474$	0	0
$475$	−6347.93	−0.613186
$476$	0	0
$477$	−2014.27	−0.193348
$478$	0	0
$479$	−5722.78	−0.545888	−0.272944	−	0.962030i	$-0.587997\pi$
$479$	−5722.78	−0.545888	−0.272944	+	0.962030i	$0.587997\pi$
$480$	0	0
$481$	7022.23	0.665668
$482$	0	0
$483$	1197.01	0.112765
$484$	0	0
$485$	12543.9	1.17441
$486$	0	0
$487$	−14805.4	−1.37761	−0.688805	−	0.724946i	$-0.741864\pi$
$487$	−14805.4	−1.37761	−0.688805	+	0.724946i	$0.741864\pi$
$488$	0	0
$489$	−5200.43	−0.480923
$490$	0	0
$491$	−2100.56	−0.193069	−0.0965343	−	0.995330i	$-0.530776\pi$
$491$	−2100.56	−0.193069	−0.0965343	+	0.995330i	$0.530776\pi$
$492$	0	0
$493$	−1315.37	−0.120165
$494$	0	0
$495$	−2231.04	−0.202582
$496$	0	0
$497$	−4116.93	−0.371568
$498$	0	0
$499$	17495.0	1.56950	0.784752	−	0.619810i	$-0.212790\pi$
$499$	17495.0	1.56950	0.784752	+	0.619810i	$0.212790\pi$
$500$	0	0
$501$	−6274.13	−0.559496
$502$	0	0
$503$	−657.285	−0.0582642	−0.0291321	−	0.999576i	$-0.509274\pi$
$503$	−657.285	−0.0582642	−0.0291321	+	0.999576i	$0.509274\pi$
$504$	0	0
$505$	1141.65	0.100599
$506$	0	0
$507$	−6232.41	−0.545939
$508$	0	0
$509$	−416.399	−0.0362604	−0.0181302	−	0.999836i	$-0.505771\pi$
$509$	−416.399	−0.0362604	−0.0181302	+	0.999836i	$0.505771\pi$
$510$	0	0
$511$	827.777	0.0716609
$512$	0	0
$513$	−11678.2	−1.00508
$514$	0	0
$515$	−4061.08	−0.347481
$516$	0	0
$517$	−9353.77	−0.795703
$518$	0	0
$519$	−5855.86	−0.495267
$520$	0	0
$521$	12612.5	1.06058	0.530291	−	0.847816i	$-0.322083\pi$
$521$	12612.5	1.06058	0.530291	+	0.847816i	$0.322083\pi$
$522$	0	0
$523$	1894.84	0.158424	0.0792118	−	0.996858i	$-0.474760\pi$
$523$	1894.84	0.158424	0.0792118	+	0.996858i	$0.474760\pi$
$524$	0	0
$525$	−1410.93	−0.117292
$526$	0	0
$527$	−6072.50	−0.501940
$528$	0	0
$529$	−7622.41	−0.626482
$530$	0	0
$531$	8630.97	0.705371
$532$	0	0
$533$	1487.26	0.120864
$534$	0	0
$535$	4045.70	0.326936
$536$	0	0
$537$	7399.11	0.594591
$538$	0	0
$539$	7123.98	0.569298
$540$	0	0
$541$	−16807.5	−1.33569	−0.667846	−	0.744300i	$-0.732783\pi$
$541$	−16807.5	−1.33569	−0.667846	+	0.744300i	$0.732783\pi$
$542$	0	0
$543$	6118.26	0.483535
$544$	0	0
$545$	−454.163	−0.0356958
$546$	0	0
$547$	−12610.7	−0.985733	−0.492866	−	0.870105i	$-0.664051\pi$
$547$	−12610.7	−0.985733	−0.492866	+	0.870105i	$0.664051\pi$
$548$	0	0
$549$	−3126.62	−0.243062
$550$	0	0
$551$	−2316.72	−0.179121
$552$	0	0
$553$	−4695.43	−0.361067
$554$	0	0
$555$	−8119.72	−0.621014
$556$	0	0
$557$	755.062	0.0574381	0.0287190	−	0.999588i	$-0.490857\pi$
$557$	755.062	0.0574381	0.0287190	+	0.999588i	$0.490857\pi$
$558$	0	0
$559$	−4756.16	−0.359865
$560$	0	0
$561$	−3567.60	−0.268493
$562$	0	0
$563$	−16459.1	−1.23209	−0.616045	−	0.787711i	$-0.711266\pi$
$563$	−16459.1	−1.23209	−0.616045	+	0.787711i	$0.711266\pi$
$564$	0	0
$565$	12212.4	0.909346
$566$	0	0
$567$	−580.535	−0.0429985
$568$	0	0
$569$	2566.16	0.189067	0.0945335	−	0.995522i	$-0.469864\pi$
$569$	2566.16	0.189067	0.0945335	+	0.995522i	$0.469864\pi$
$570$	0	0
$571$	2946.30	0.215935	0.107968	−	0.994154i	$-0.465566\pi$
$571$	2946.30	0.215935	0.107968	+	0.994154i	$0.465566\pi$
$572$	0	0
$573$	6673.95	0.486577
$574$	0	0
$575$	−5356.79	−0.388510
$576$	0	0
$577$	16439.8	1.18613	0.593065	−	0.805154i	$-0.297918\pi$
$577$	16439.8	1.18613	0.593065	+	0.805154i	$0.297918\pi$
$578$	0	0
$579$	−16671.6	−1.19663
$580$	0	0
$581$	−2733.45	−0.195185
$582$	0	0
$583$	−3070.91	−0.218155
$584$	0	0
$585$	2031.84	0.143601
$586$	0	0
$587$	−8578.20	−0.603169	−0.301585	−	0.953439i	$-0.597516\pi$
$587$	−8578.20	−0.603169	−0.301585	+	0.953439i	$0.597516\pi$
$588$	0	0
$589$	−10695.3	−0.748206
$590$	0	0
$591$	2329.08	0.162107
$592$	0	0
$593$	−10209.1	−0.706979	−0.353490	−	0.935438i	$-0.615005\pi$
$593$	−10209.1	−0.706979	−0.353490	+	0.935438i	$0.615005\pi$
$594$	0	0
$595$	−1551.29	−0.106885
$596$	0	0
$597$	−10167.5	−0.697033
$598$	0	0
$599$	28912.5	1.97217	0.986087	−	0.166232i	$-0.0531602\pi$
$599$	28912.5	1.97217	0.986087	+	0.166232i	$0.0531602\pi$
$600$	0	0
$601$	18387.2	1.24797	0.623986	−	0.781435i	$-0.285512\pi$
$601$	18387.2	1.24797	0.623986	+	0.781435i	$0.285512\pi$
$602$	0	0
$603$	−2046.37	−0.138200
$604$	0	0
$605$	5580.47	0.375006
$606$	0	0
$607$	−8495.13	−0.568051	−0.284025	−	0.958817i	$-0.591670\pi$
$607$	−8495.13	−0.568051	−0.284025	+	0.958817i	$0.591670\pi$
$608$	0	0
$609$	−514.929	−0.0342627
$610$	0	0
$611$	8518.61	0.564036
$612$	0	0
$613$	2054.08	0.135340	0.0676700	−	0.997708i	$-0.478444\pi$
$613$	2054.08	0.135340	0.0676700	+	0.997708i	$0.478444\pi$
$614$	0	0
$615$	−1719.70	−0.112756
$616$	0	0
$617$	−5368.22	−0.350270	−0.175135	−	0.984544i	$-0.556036\pi$
$617$	−5368.22	−0.350270	−0.175135	+	0.984544i	$0.556036\pi$
$618$	0	0
$619$	−12026.5	−0.780916	−0.390458	−	0.920621i	$-0.627683\pi$
$619$	−12026.5	−0.780916	−0.390458	+	0.920621i	$0.627683\pi$
$620$	0	0
$621$	−9854.79	−0.636810
$622$	0	0
$623$	−994.502	−0.0639549
$624$	0	0
$625$	621.824	0.0397967
$626$	0	0
$627$	−6283.53	−0.400223
$628$	0	0
$629$	15577.8	0.987486
$630$	0	0
$631$	5964.45	0.376293	0.188147	−	0.982141i	$-0.439752\pi$
$631$	5964.45	0.376293	0.188147	+	0.982141i	$0.439752\pi$
$632$	0	0
$633$	−12956.2	−0.813526
$634$	0	0
$635$	11873.1	0.741999
$636$	0	0
$637$	−6487.91	−0.403549
$638$	0	0
$639$	11961.9	0.740543
$640$	0	0
$641$	−18376.9	−1.13236	−0.566181	−	0.824281i	$-0.691580\pi$
$641$	−18376.9	−1.13236	−0.566181	+	0.824281i	$0.691580\pi$
$642$	0	0
$643$	−1716.99	−0.105306	−0.0526529	−	0.998613i	$-0.516768\pi$
$643$	−1716.99	−0.105306	−0.0526529	+	0.998613i	$0.516768\pi$
$644$	0	0
$645$	5499.49	0.335724
$646$	0	0
$647$	15572.9	0.946266	0.473133	−	0.880991i	$-0.343123\pi$
$647$	15572.9	0.946266	0.473133	+	0.880991i	$0.343123\pi$
$648$	0	0
$649$	13158.6	0.795872
$650$	0	0
$651$	−2377.21	−0.143119
$652$	0	0
$653$	408.474	0.0244790	0.0122395	−	0.999925i	$-0.496104\pi$
$653$	408.474	0.0244790	0.0122395	+	0.999925i	$0.496104\pi$
$654$	0	0
$655$	−18514.1	−1.10444
$656$	0	0
$657$	−2405.15	−0.142822
$658$	0	0
$659$	−2482.85	−0.146765	−0.0733824	−	0.997304i	$-0.523379\pi$
$659$	−2482.85	−0.146765	−0.0733824	+	0.997304i	$0.523379\pi$
$660$	0	0
$661$	18389.9	1.08212	0.541061	−	0.840983i	$-0.318023\pi$
$661$	18389.9	1.08212	0.541061	+	0.840983i	$0.318023\pi$
$662$	0	0
$663$	3249.07	0.190322
$664$	0	0
$665$	−2732.25	−0.159327
$666$	0	0
$667$	−1954.99	−0.113490
$668$	0	0
$669$	4839.43	0.279676
$670$	0	0
$671$	−4766.79	−0.274247
$672$	0	0
$673$	7094.41	0.406344	0.203172	−	0.979143i	$-0.434875\pi$
$673$	7094.41	0.406344	0.203172	+	0.979143i	$0.434875\pi$
$674$	0	0
$675$	11616.0	0.662371
$676$	0	0
$677$	34603.8	1.96445	0.982224	−	0.187715i	$-0.0601080\pi$
$677$	34603.8	1.96445	0.982224	+	0.187715i	$0.0601080\pi$
$678$	0	0
$679$	−9421.07	−0.532470
$680$	0	0
$681$	9223.67	0.519019
$682$	0	0
$683$	−9849.61	−0.551808	−0.275904	−	0.961185i	$-0.588977\pi$
$683$	−9849.61	−0.551808	−0.275904	+	0.961185i	$0.588977\pi$
$684$	0	0
$685$	9738.44	0.543192
$686$	0	0
$687$	−10235.8	−0.568444
$688$	0	0
$689$	2796.72	0.154640
$690$	0	0
$691$	25353.4	1.39579	0.697894	−	0.716201i	$-0.254121\pi$
$691$	25353.4	1.39579	0.697894	+	0.716201i	$0.254121\pi$
$692$	0	0
$693$	1675.62	0.0918492
$694$	0	0
$695$	−11724.7	−0.639918
$696$	0	0
$697$	3299.28	0.179296
$698$	0	0
$699$	9475.17	0.512710
$700$	0	0
$701$	25751.8	1.38749	0.693746	−	0.720220i	$-0.255959\pi$
$701$	25751.8	1.38749	0.693746	+	0.720220i	$0.255959\pi$
$702$	0	0
$703$	27436.8	1.47198
$704$	0	0
$705$	−9849.96	−0.526200
$706$	0	0
$707$	−857.430	−0.0456110
$708$	0	0
$709$	36751.0	1.94670	0.973352	−	0.229317i	$-0.0736492\pi$
$709$	36751.0	1.94670	0.973352	+	0.229317i	$0.0736492\pi$
$710$	0	0
$711$	13642.8	0.719613
$712$	0	0
$713$	−9025.40	−0.474059
$714$	0	0
$715$	3097.71	0.162025
$716$	0	0
$717$	−1123.14	−0.0584997
$718$	0	0
$719$	6541.00	0.339274	0.169637	−	0.985507i	$-0.445740\pi$
$719$	6541.00	0.339274	0.169637	+	0.985507i	$0.445740\pi$
$720$	0	0
$721$	3050.06	0.157545
$722$	0	0
$723$	5416.22	0.278605
$724$	0	0
$725$	2304.38	0.118045
$726$	0	0
$727$	17838.1	0.910014	0.455007	−	0.890488i	$-0.349637\pi$
$727$	17838.1	0.910014	0.455007	+	0.890488i	$0.349637\pi$
$728$	0	0
$729$	15514.7	0.788228
$730$	0	0
$731$	−10550.9	−0.533842
$732$	0	0
$733$	−24295.4	−1.22424	−0.612121	−	0.790764i	$-0.709684\pi$
$733$	−24295.4	−1.22424	−0.612121	+	0.790764i	$0.709684\pi$
$734$	0	0
$735$	7501.89	0.376478
$736$	0	0
$737$	−3119.85	−0.155931
$738$	0	0
$739$	−38041.0	−1.89359	−0.946794	−	0.321840i	$-0.895699\pi$
$739$	−38041.0	−1.89359	−0.946794	+	0.321840i	$0.895699\pi$
$740$	0	0
$741$	5722.50	0.283699
$742$	0	0
$743$	−21176.9	−1.04563	−0.522816	−	0.852446i	$-0.675118\pi$
$743$	−21176.9	−1.04563	−0.522816	+	0.852446i	$0.675118\pi$
$744$	0	0
$745$	5801.45	0.285300
$746$	0	0
$747$	7942.16	0.389007
$748$	0	0
$749$	−3038.51	−0.148230
$750$	0	0
$751$	11864.8	0.576500	0.288250	−	0.957555i	$-0.406927\pi$
$751$	11864.8	0.576500	0.288250	+	0.957555i	$0.406927\pi$
$752$	0	0
$753$	−8856.89	−0.428636
$754$	0	0
$755$	3200.68	0.154284
$756$	0	0
$757$	27080.6	1.30021	0.650105	−	0.759844i	$-0.274725\pi$
$757$	27080.6	1.30021	0.650105	+	0.759844i	$0.274725\pi$
$758$	0	0
$759$	−5302.44	−0.253579
$760$	0	0
$761$	−7232.38	−0.344512	−0.172256	−	0.985052i	$-0.555106\pi$
$761$	−7232.38	−0.344512	−0.172256	+	0.985052i	$0.555106\pi$
$762$	0	0
$763$	341.097	0.0161842
$764$	0	0
$765$	4507.36	0.213025
$766$	0	0
$767$	−11983.7	−0.564156
$768$	0	0
$769$	10444.7	0.489787	0.244894	−	0.969550i	$-0.421247\pi$
$769$	10444.7	0.489787	0.244894	+	0.969550i	$0.421247\pi$
$770$	0	0
$771$	−10688.3	−0.499263
$772$	0	0
$773$	22573.6	1.05035	0.525173	−	0.850995i	$-0.324001\pi$
$773$	22573.6	1.05035	0.525173	+	0.850995i	$0.324001\pi$
$774$	0	0
$775$	10638.4	0.493087
$776$	0	0
$777$	6098.28	0.281563
$778$	0	0
$779$	5810.94	0.267264
$780$	0	0
$781$	18236.9	0.835555
$782$	0	0
$783$	4239.34	0.193489
$784$	0	0
$785$	−2287.03	−0.103984
$786$	0	0
$787$	33268.0	1.50683	0.753415	−	0.657545i	$-0.228405\pi$
$787$	33268.0	1.50683	0.753415	+	0.657545i	$0.228405\pi$
$788$	0	0
$789$	29294.1	1.32180
$790$	0	0
$791$	−9172.09	−0.412291
$792$	0	0
$793$	4341.18	0.194401
$794$	0	0
$795$	−3233.82	−0.144266
$796$	0	0
$797$	17000.3	0.755561	0.377781	−	0.925895i	$-0.376687\pi$
$797$	17000.3	0.755561	0.377781	+	0.925895i	$0.376687\pi$
$798$	0	0
$799$	18897.3	0.836721
$800$	0	0
$801$	2889.58	0.127463
$802$	0	0
$803$	−3666.84	−0.161146
$804$	0	0
$805$	−2305.65	−0.100948
$806$	0	0
$807$	−5687.64	−0.248097
$808$	0	0
$809$	−19036.8	−0.827317	−0.413658	−	0.910432i	$-0.635749\pi$
$809$	−19036.8	−0.827317	−0.413658	+	0.910432i	$0.635749\pi$
$810$	0	0
$811$	43037.9	1.86346	0.931729	−	0.363155i	$-0.118300\pi$
$811$	43037.9	1.86346	0.931729	+	0.363155i	$0.118300\pi$
$812$	0	0
$813$	5692.00	0.245544
$814$	0	0
$815$	10016.9	0.430525
$816$	0	0
$817$	−18583.0	−0.795760
$818$	0	0
$819$	−1526.01	−0.0651076
$820$	0	0
$821$	18755.1	0.797268	0.398634	−	0.917110i	$-0.369484\pi$
$821$	18755.1	0.797268	0.398634	+	0.917110i	$0.369484\pi$
$822$	0	0
$823$	−34237.8	−1.45013	−0.725064	−	0.688682i	$-0.758190\pi$
$823$	−34237.8	−1.45013	−0.725064	+	0.688682i	$0.758190\pi$
$824$	0	0
$825$	6250.07	0.263757
$826$	0	0
$827$	−7145.93	−0.300470	−0.150235	−	0.988650i	$-0.548003\pi$
$827$	−7145.93	−0.300470	−0.150235	+	0.988650i	$0.548003\pi$
$828$	0	0
$829$	−31620.6	−1.32476	−0.662382	−	0.749166i	$-0.730455\pi$
$829$	−31620.6	−1.32476	−0.662382	+	0.749166i	$0.730455\pi$
$830$	0	0
$831$	−18134.9	−0.757030
$832$	0	0
$833$	−14392.5	−0.598645
$834$	0	0
$835$	12085.1	0.500864
$836$	0	0
$837$	19571.3	0.808222
$838$	0	0
$839$	335.782	0.0138170	0.00690851	−	0.999976i	$-0.497801\pi$
$839$	335.782	0.0138170	0.00690851	+	0.999976i	$0.497801\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−4302.16	−0.175770
$844$	0	0
$845$	12004.7	0.488727
$846$	0	0
$847$	−4191.20	−0.170025
$848$	0	0
$849$	9670.20	0.390907
$850$	0	0
$851$	23152.9	0.932634
$852$	0	0
$853$	2615.21	0.104974	0.0524872	−	0.998622i	$-0.483285\pi$
$853$	2615.21	0.104974	0.0524872	+	0.998622i	$0.483285\pi$
$854$	0	0
$855$	7938.69	0.317541
$856$	0	0
$857$	24072.4	0.959506	0.479753	−	0.877404i	$-0.340726\pi$
$857$	24072.4	0.959506	0.479753	+	0.877404i	$0.340726\pi$
$858$	0	0
$859$	−21753.0	−0.864032	−0.432016	−	0.901866i	$-0.642198\pi$
$859$	−21753.0	−0.864032	−0.432016	+	0.901866i	$0.642198\pi$
$860$	0	0
$861$	1291.58	0.0511229
$862$	0	0
$863$	26023.5	1.02648	0.513239	−	0.858246i	$-0.328445\pi$
$863$	26023.5	1.02648	0.513239	+	0.858246i	$0.328445\pi$
$864$	0	0
$865$	11279.4	0.443366
$866$	0	0
$867$	−10004.7	−0.391902
$868$	0	0
$869$	20799.5	0.811940
$870$	0	0
$871$	2841.30	0.110532
$872$	0	0
$873$	27373.4	1.06122
$874$	0	0
$875$	6992.89	0.270175
$876$	0	0
$877$	−26143.6	−1.00662	−0.503310	−	0.864106i	$-0.667885\pi$
$877$	−26143.6	−1.00662	−0.503310	+	0.864106i	$0.667885\pi$
$878$	0	0
$879$	−27937.4	−1.07202
$880$	0	0
$881$	24084.2	0.921019	0.460510	−	0.887655i	$-0.347667\pi$
$881$	24084.2	0.921019	0.460510	+	0.887655i	$0.347667\pi$
$882$	0	0
$883$	861.933	0.0328498	0.0164249	−	0.999865i	$-0.494772\pi$
$883$	861.933	0.0328498	0.0164249	+	0.999865i	$0.494772\pi$
$884$	0	0
$885$	13856.6	0.526312
$886$	0	0
$887$	−24546.4	−0.929187	−0.464594	−	0.885524i	$-0.653800\pi$
$887$	−24546.4	−0.929187	−0.464594	+	0.885524i	$0.653800\pi$
$888$	0	0
$889$	−8917.24	−0.336417
$890$	0	0
$891$	2571.62	0.0966919
$892$	0	0
$893$	33283.4	1.24724
$894$	0	0
$895$	−14252.0	−0.532281
$896$	0	0
$897$	4829.01	0.179750
$898$	0	0
$899$	3882.55	0.144038
$900$	0	0
$901$	6204.14	0.229400
$902$	0	0
$903$	−4130.37	−0.152215
$904$	0	0
$905$	−11784.8	−0.432863
$906$	0	0
$907$	38997.7	1.42767	0.713835	−	0.700313i	$-0.246956\pi$
$907$	38997.7	1.42767	0.713835	+	0.700313i	$0.246956\pi$
$908$	0	0
$909$	2491.31	0.0909036
$910$	0	0
$911$	49929.6	1.81585	0.907927	−	0.419129i	$-0.137664\pi$
$911$	49929.6	1.81585	0.907927	+	0.419129i	$0.137664\pi$
$912$	0	0
$913$	12108.5	0.438918
$914$	0	0
$915$	−5019.66	−0.181360
$916$	0	0
$917$	13905.0	0.500744
$918$	0	0
$919$	20227.7	0.726062	0.363031	−	0.931777i	$-0.381742\pi$
$919$	20227.7	0.726062	0.363031	+	0.931777i	$0.381742\pi$
$920$	0	0
$921$	20379.3	0.729123
$922$	0	0
$923$	−16608.6	−0.592286
$924$	0	0
$925$	−27290.7	−0.970069
$926$	0	0
$927$	−8862.11	−0.313991
$928$	0	0
$929$	−26106.4	−0.921983	−0.460992	−	0.887405i	$-0.652506\pi$
$929$	−26106.4	−0.921983	−0.460992	+	0.887405i	$0.652506\pi$
$930$	0	0
$931$	−25349.2	−0.892358
$932$	0	0
$933$	−33721.1	−1.18326
$934$	0	0
$935$	6871.83	0.240356
$936$	0	0
$937$	3449.56	0.120269	0.0601346	−	0.998190i	$-0.480847\pi$
$937$	3449.56	0.120269	0.0601346	+	0.998190i	$0.480847\pi$
$938$	0	0
$939$	7410.83	0.257554
$940$	0	0
$941$	14748.8	0.510941	0.255471	−	0.966817i	$-0.417770\pi$
$941$	14748.8	0.510941	0.255471	+	0.966817i	$0.417770\pi$
$942$	0	0
$943$	4903.64	0.169337
$944$	0	0
$945$	4999.72	0.172107
$946$	0	0
$947$	−35707.9	−1.22529	−0.612646	−	0.790358i	$-0.709895\pi$
$947$	−35707.9	−1.22529	−0.612646	+	0.790358i	$0.709895\pi$
$948$	0	0
$949$	3339.45	0.114229
$950$	0	0
$951$	8483.60	0.289274
$952$	0	0
$953$	12441.5	0.422896	0.211448	−	0.977389i	$-0.432182\pi$
$953$	12441.5	0.422896	0.211448	+	0.977389i	$0.432182\pi$
$954$	0	0
$955$	−12855.2	−0.435586
$956$	0	0
$957$	2281.00	0.0770474
$958$	0	0
$959$	−7314.02	−0.246280
$960$	0	0
$961$	−11866.9	−0.398338
$962$	0	0
$963$	8828.53	0.295426
$964$	0	0
$965$	32112.4	1.07123
$966$	0	0
$967$	2697.84	0.0897172	0.0448586	−	0.998993i	$-0.485716\pi$
$967$	2697.84	0.0897172	0.0448586	+	0.998993i	$0.485716\pi$
$968$	0	0
$969$	12694.6	0.420854
$970$	0	0
$971$	51090.8	1.68855	0.844275	−	0.535910i	$-0.180031\pi$
$971$	51090.8	1.68855	0.844275	+	0.535910i	$0.180031\pi$
$972$	0	0
$973$	8805.79	0.290134
$974$	0	0
$975$	−5692.03	−0.186965
$976$	0	0
$977$	−36334.4	−1.18981	−0.594903	−	0.803797i	$-0.702810\pi$
$977$	−36334.4	−1.18981	−0.594903	+	0.803797i	$0.702810\pi$
$978$	0	0
$979$	4405.39	0.143817
$980$	0	0
$981$	−991.075	−0.0322554
$982$	0	0
$983$	11595.3	0.376227	0.188114	−	0.982147i	$-0.439763\pi$
$983$	11595.3	0.376227	0.188114	+	0.982147i	$0.439763\pi$
$984$	0	0
$985$	−4486.21	−0.145119
$986$	0	0
$987$	7397.78	0.238575
$988$	0	0
$989$	−15681.5	−0.504188
$990$	0	0
$991$	−37459.3	−1.20074	−0.600369	−	0.799723i	$-0.704980\pi$
$991$	−37459.3	−1.20074	−0.600369	+	0.799723i	$0.704980\pi$
$992$	0	0
$993$	−14265.8	−0.455904
$994$	0	0
$995$	19584.4	0.623988
$996$	0	0
$997$	−35586.7	−1.13043	−0.565216	−	0.824943i	$-0.691207\pi$
$997$	−35586.7	−1.13043	−0.565216	+	0.824943i	$0.691207\pi$
$998$	0	0
$999$	−50206.3	−1.59005

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.be.1.5		8
4.3	odd	2	inner	1856.4.a.be.1.4		8
8.3	odd	2		928.4.a.c.1.5	yes	8
8.5	even	2		928.4.a.c.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.c.1.4	✓	8	8.5	even	2
928.4.a.c.1.5	yes	8	8.3	odd	2
1856.4.a.be.1.4		8	4.3	odd	2	inner
1856.4.a.be.1.5		8	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)

\(f(q)\)	\(=\)	\(q+3.50343 q^{3} -6.74822 q^{5} +5.06823 q^{7} -14.7260 q^{9} +O(q^{10})\)	\(q+3.50343 q^{3} -6.74822 q^{5} +5.06823 q^{7} -14.7260 q^{9} -22.4509 q^{11} +20.4464 q^{13} -23.6419 q^{15} +45.3575 q^{17} +79.8869 q^{19} +17.7562 q^{21} +67.4136 q^{23} -79.4615 q^{25} -146.184 q^{27} -29.0000 q^{29} -133.881 q^{31} -78.6553 q^{33} -34.2015 q^{35} +343.446 q^{37} +71.6325 q^{39} +72.7396 q^{41} -232.616 q^{43} +99.3742 q^{45} +416.631 q^{47} -317.313 q^{49} +158.907 q^{51} +136.783 q^{53} +151.504 q^{55} +279.878 q^{57} -586.105 q^{59} +212.320 q^{61} -74.6346 q^{63} -137.977 q^{65} +138.963 q^{67} +236.179 q^{69} -812.301 q^{71} +163.327 q^{73} -278.388 q^{75} -113.787 q^{77} -926.444 q^{79} -114.544 q^{81} -539.330 q^{83} -306.082 q^{85} -101.599 q^{87} -196.223 q^{89} +103.627 q^{91} -469.043 q^{93} -539.094 q^{95} -1858.85 q^{97} +330.612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 20 q^{5} + 40 q^{9}+O(q^{10}) \) 8 * q + 20 * q^5 + 40 * q^9	\( 8 q + 20 q^{5} + 40 q^{9} - 4 q^{13} - 140 q^{17} + 28 q^{21} - 256 q^{25} - 232 q^{29} - 344 q^{33} - 280 q^{37} - 700 q^{41} + 56 q^{45} - 256 q^{49} + 604 q^{53} - 2016 q^{57} + 884 q^{61} - 1616 q^{65} + 764 q^{69} - 3504 q^{73} + 1916 q^{77} - 3904 q^{81} + 996 q^{85} - 2924 q^{89} + 2996 q^{93} - 2668 q^{97}+O(q^{100}) \) 8 * q + 20 * q^5 + 40 * q^9 - 4 * q^13 - 140 * q^17 + 28 * q^21 - 256 * q^25 - 232 * q^29 - 344 * q^33 - 280 * q^37 - 700 * q^41 + 56 * q^45 - 256 * q^49 + 604 * q^53 - 2016 * q^57 + 884 * q^61 - 1616 * q^65 + 764 * q^69 - 3504 * q^73 + 1916 * q^77 - 3904 * q^81 + 996 * q^85 - 2924 * q^89 + 2996 * q^93 - 2668 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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