LMFDB - Embedded newform 1856.4.a.be.1.1

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			1.1
Root			$2.68601$ 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)

$f(q)$	$=$	$q-7.48152 q^{3} +7.42926 q^{5} +21.6928 q^{7} +28.9731 q^{9} +O(q^{10})$	$q-7.48152 q^{3} +7.42926 q^{5} +21.6928 q^{7} +28.9731 q^{9} +39.4223 q^{11} +37.9793 q^{13} -55.5822 q^{15} -55.1961 q^{17} +40.6394 q^{19} -162.295 q^{21} -130.291 q^{23} -69.8060 q^{25} -14.7618 q^{27} -29.0000 q^{29} -279.845 q^{31} -294.939 q^{33} +161.162 q^{35} -226.565 q^{37} -284.142 q^{39} -77.8132 q^{41} -10.5045 q^{43} +215.249 q^{45} +242.450 q^{47} +127.578 q^{49} +412.951 q^{51} -584.878 q^{53} +292.879 q^{55} -304.044 q^{57} +87.1488 q^{59} -48.1295 q^{61} +628.508 q^{63} +282.158 q^{65} -100.323 q^{67} +974.772 q^{69} -803.695 q^{71} -142.491 q^{73} +522.255 q^{75} +855.182 q^{77} +1141.29 q^{79} -671.833 q^{81} -70.0497 q^{83} -410.067 q^{85} +216.964 q^{87} -823.519 q^{89} +823.877 q^{91} +2093.67 q^{93} +301.921 q^{95} -1660.45 q^{97} +1142.19 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−7.48152	−1.43982	−0.719909	−	0.694068i	$-0.755817\pi$
$3$	−7.48152	−1.43982	−0.719909	+	0.694068i	$0.755817\pi$
$4$	0	0
$5$	7.42926	0.664494	0.332247	−	0.943192i	$-0.392193\pi$
$5$	7.42926	0.664494	0.332247	+	0.943192i	$0.392193\pi$
$6$	0	0
$7$	21.6928	1.17130	0.585651	−	0.810563i	$-0.300839\pi$
$7$	21.6928	1.17130	0.585651	+	0.810563i	$0.300839\pi$
$8$	0	0
$9$	28.9731	1.07308
$10$	0	0
$11$	39.4223	1.08057	0.540286	−	0.841482i	$-0.318316\pi$
$11$	39.4223	1.08057	0.540286	+	0.841482i	$0.318316\pi$
$12$	0	0
$13$	37.9793	0.810273	0.405137	−	0.914256i	$-0.367224\pi$
$13$	37.9793	0.810273	0.405137	+	0.914256i	$0.367224\pi$
$14$	0	0
$15$	−55.5822	−0.956750
$16$	0	0
$17$	−55.1961	−0.787472	−0.393736	−	0.919224i	$-0.628818\pi$
$17$	−55.1961	−0.787472	−0.393736	+	0.919224i	$0.628818\pi$
$18$	0	0
$19$	40.6394	0.490701	0.245350	−	0.969434i	$-0.421097\pi$
$19$	40.6394	0.490701	0.245350	+	0.969434i	$0.421097\pi$
$20$	0	0
$21$	−162.295	−1.68646
$22$	0	0
$23$	−130.291	−1.18119	−0.590597	−	0.806967i	$-0.701108\pi$
$23$	−130.291	−1.18119	−0.590597	+	0.806967i	$0.701108\pi$
$24$	0	0
$25$	−69.8060	−0.558448
$26$	0	0
$27$	−14.7618	−0.105219
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−279.845	−1.62135	−0.810673	−	0.585499i	$-0.800899\pi$
$31$	−279.845	−1.62135	−0.810673	+	0.585499i	$0.800899\pi$
$32$	0	0
$33$	−294.939	−1.55583
$34$	0	0
$35$	161.162	0.778323
$36$	0	0
$37$	−226.565	−1.00668	−0.503339	−	0.864089i	$-0.667895\pi$
$37$	−226.565	−1.00668	−0.503339	+	0.864089i	$0.667895\pi$
$38$	0	0
$39$	−284.142	−1.16665
$40$	0	0
$41$	−77.8132	−0.296399	−0.148200	−	0.988957i	$-0.547348\pi$
$41$	−77.8132	−0.296399	−0.148200	+	0.988957i	$0.547348\pi$
$42$	0	0
$43$	−10.5045	−0.0372539	−0.0186269	−	0.999827i	$-0.505929\pi$
$43$	−10.5045	−0.0372539	−0.0186269	+	0.999827i	$0.505929\pi$
$44$	0	0
$45$	215.249	0.713053
$46$	0	0
$47$	242.450	0.752447	0.376223	−	0.926529i	$-0.377223\pi$
$47$	242.450	0.752447	0.376223	+	0.926529i	$0.377223\pi$
$48$	0	0
$49$	127.578	0.371948
$50$	0	0
$51$	412.951	1.13382
$52$	0	0
$53$	−584.878	−1.51583	−0.757917	−	0.652351i	$-0.773783\pi$
$53$	−584.878	−1.51583	−0.757917	+	0.652351i	$0.773783\pi$
$54$	0	0
$55$	292.879	0.718033
$56$	0	0
$57$	−304.044	−0.706520
$58$	0	0
$59$	87.1488	0.192302	0.0961509	−	0.995367i	$-0.469347\pi$
$59$	87.1488	0.192302	0.0961509	+	0.995367i	$0.469347\pi$
$60$	0	0
$61$	−48.1295	−0.101022	−0.0505111	−	0.998724i	$-0.516085\pi$
$61$	−48.1295	−0.101022	−0.0505111	+	0.998724i	$0.516085\pi$
$62$	0	0
$63$	628.508	1.25690
$64$	0	0
$65$	282.158	0.538421
$66$	0	0
$67$	−100.323	−0.182931	−0.0914655	−	0.995808i	$-0.529155\pi$
$67$	−100.323	−0.182931	−0.0914655	+	0.995808i	$0.529155\pi$
$68$	0	0
$69$	974.772	1.70071
$70$	0	0
$71$	−803.695	−1.34340	−0.671698	−	0.740825i	$-0.734435\pi$
$71$	−803.695	−1.34340	−0.671698	+	0.740825i	$0.734435\pi$
$72$	0	0
$73$	−142.491	−0.228457	−0.114228	−	0.993455i	$-0.536440\pi$
$73$	−142.491	−0.228457	−0.114228	+	0.993455i	$0.536440\pi$
$74$	0	0
$75$	522.255	0.804064
$76$	0	0
$77$	855.182	1.26568
$78$	0	0
$79$	1141.29	1.62539	0.812694	−	0.582691i	$-0.198000\pi$
$79$	1141.29	1.62539	0.812694	+	0.582691i	$0.198000\pi$
$80$	0	0
$81$	−671.833	−0.921582
$82$	0	0
$83$	−70.0497	−0.0926380	−0.0463190	−	0.998927i	$-0.514749\pi$
$83$	−70.0497	−0.0926380	−0.0463190	+	0.998927i	$0.514749\pi$
$84$	0	0
$85$	−410.067	−0.523270
$86$	0	0
$87$	216.964	0.267368
$88$	0	0
$89$	−823.519	−0.980819	−0.490409	−	0.871492i	$-0.663153\pi$
$89$	−823.519	−0.980819	−0.490409	+	0.871492i	$0.663153\pi$
$90$	0	0
$91$	823.877	0.949074
$92$	0	0
$93$	2093.67	2.33444
$94$	0	0
$95$	301.921	0.326067
$96$	0	0
$97$	−1660.45	−1.73807	−0.869035	−	0.494751i	$-0.835259\pi$
$97$	−1660.45	−1.73807	−0.869035	+	0.494751i	$0.835259\pi$
$98$	0	0
$99$	1142.19	1.15954
$100$	0	0
$101$	625.051	0.615791	0.307895	−	0.951420i	$-0.400375\pi$
$101$	625.051	0.615791	0.307895	+	0.951420i	$0.400375\pi$
$102$	0	0
$103$	1109.66	1.06154	0.530768	−	0.847517i	$-0.321904\pi$
$103$	1109.66	1.06154	0.530768	+	0.847517i	$0.321904\pi$
$104$	0	0
$105$	−1205.73	−1.12064
$106$	0	0
$107$	813.184	0.734705	0.367353	−	0.930082i	$-0.380264\pi$
$107$	813.184	0.734705	0.367353	+	0.930082i	$0.380264\pi$
$108$	0	0
$109$	−1201.37	−1.05569	−0.527846	−	0.849340i	$-0.677000\pi$
$109$	−1201.37	−1.05569	−0.527846	+	0.849340i	$0.677000\pi$
$110$	0	0
$111$	1695.05	1.44943
$112$	0	0
$113$	1134.64	0.944580	0.472290	−	0.881443i	$-0.343427\pi$
$113$	1134.64	0.944580	0.472290	+	0.881443i	$0.343427\pi$
$114$	0	0
$115$	−967.963	−0.784896
$116$	0	0
$117$	1100.38	0.869486
$118$	0	0
$119$	−1197.36	−0.922368
$120$	0	0
$121$	223.122	0.167635
$122$	0	0
$123$	582.161	0.426762
$124$	0	0
$125$	−1447.27	−1.03558
$126$	0	0
$127$	417.493	0.291705	0.145853	−	0.989306i	$-0.453407\pi$
$127$	417.493	0.291705	0.145853	+	0.989306i	$0.453407\pi$
$128$	0	0
$129$	78.5894	0.0536388
$130$	0	0
$131$	2083.32	1.38947	0.694733	−	0.719268i	$-0.255522\pi$
$131$	2083.32	1.38947	0.694733	+	0.719268i	$0.255522\pi$
$132$	0	0
$133$	881.582	0.574759
$134$	0	0
$135$	−109.670	−0.0699174
$136$	0	0
$137$	−1686.50	−1.05173	−0.525867	−	0.850567i	$-0.676259\pi$
$137$	−1686.50	−1.05173	−0.525867	+	0.850567i	$0.676259\pi$
$138$	0	0
$139$	−1288.81	−0.786444	−0.393222	−	0.919444i	$-0.628640\pi$
$139$	−1288.81	−0.786444	−0.393222	+	0.919444i	$0.628640\pi$
$140$	0	0
$141$	−1813.90	−1.08339
$142$	0	0
$143$	1497.23	0.875558
$144$	0	0
$145$	−215.449	−0.123393
$146$	0	0
$147$	−954.479	−0.535538
$148$	0	0
$149$	−2161.34	−1.18835	−0.594175	−	0.804336i	$-0.702521\pi$
$149$	−2161.34	−1.18835	−0.594175	+	0.804336i	$0.702521\pi$
$150$	0	0
$151$	54.6571	0.0294565	0.0147282	−	0.999892i	$-0.495312\pi$
$151$	54.6571	0.0294565	0.0147282	+	0.999892i	$0.495312\pi$
$152$	0	0
$153$	−1599.20	−0.845019
$154$	0	0
$155$	−2079.05	−1.07737
$156$	0	0
$157$	1422.01	0.722856	0.361428	−	0.932400i	$-0.382289\pi$
$157$	1422.01	0.722856	0.361428	+	0.932400i	$0.382289\pi$
$158$	0	0
$159$	4375.77	2.18253
$160$	0	0
$161$	−2826.37	−1.38354
$162$	0	0
$163$	−2769.38	−1.33076	−0.665382	−	0.746503i	$-0.731731\pi$
$163$	−2769.38	−1.33076	−0.665382	+	0.746503i	$0.731731\pi$
$164$	0	0
$165$	−2191.18	−1.03384
$166$	0	0
$167$	−1181.24	−0.547347	−0.273674	−	0.961823i	$-0.588239\pi$
$167$	−1181.24	−0.547347	−0.273674	+	0.961823i	$0.588239\pi$
$168$	0	0
$169$	−754.576	−0.343458
$170$	0	0
$171$	1177.45	0.526560
$172$	0	0
$173$	2600.85	1.14300	0.571499	−	0.820603i	$-0.306362\pi$
$173$	2600.85	1.14300	0.571499	+	0.820603i	$0.306362\pi$
$174$	0	0
$175$	−1514.29	−0.654112
$176$	0	0
$177$	−652.005	−0.276880
$178$	0	0
$179$	315.474	0.131730	0.0658650	−	0.997829i	$-0.479019\pi$
$179$	315.474	0.131730	0.0658650	+	0.997829i	$0.479019\pi$
$180$	0	0
$181$	−2017.81	−0.828631	−0.414316	−	0.910133i	$-0.635979\pi$
$181$	−2017.81	−0.828631	−0.414316	+	0.910133i	$0.635979\pi$
$182$	0	0
$183$	360.082	0.145454
$184$	0	0
$185$	−1683.21	−0.668930
$186$	0	0
$187$	−2175.96	−0.850920
$188$	0	0
$189$	−320.226	−0.123243
$190$	0	0
$191$	2468.78	0.935259	0.467629	−	0.883925i	$-0.345108\pi$
$191$	2468.78	0.935259	0.467629	+	0.883925i	$0.345108\pi$
$192$	0	0
$193$	3259.67	1.21573	0.607865	−	0.794040i	$-0.292026\pi$
$193$	3259.67	1.21573	0.607865	+	0.794040i	$0.292026\pi$
$194$	0	0
$195$	−2110.97	−0.775229
$196$	0	0
$197$	5504.95	1.99092	0.995461	−	0.0951734i	$-0.0303406\pi$
$197$	5504.95	1.99092	0.995461	+	0.0951734i	$0.0303406\pi$
$198$	0	0
$199$	−3334.30	−1.18775	−0.593876	−	0.804557i	$-0.702403\pi$
$199$	−3334.30	−1.18775	−0.593876	+	0.804557i	$0.702403\pi$
$200$	0	0
$201$	750.567	0.263388
$202$	0	0
$203$	−629.092	−0.217505
$204$	0	0
$205$	−578.095	−0.196956
$206$	0	0
$207$	−3774.92	−1.26751
$208$	0	0
$209$	1602.10	0.530237
$210$	0	0
$211$	−3675.31	−1.19914	−0.599571	−	0.800322i	$-0.704662\pi$
$211$	−3675.31	−1.19914	−0.599571	+	0.800322i	$0.704662\pi$
$212$	0	0
$213$	6012.86	1.93425
$214$	0	0
$215$	−78.0405	−0.0247550
$216$	0	0
$217$	−6070.64	−1.89909
$218$	0	0
$219$	1066.05	0.328936
$220$	0	0
$221$	−2096.31	−0.638067
$222$	0	0
$223$	318.163	0.0955416	0.0477708	−	0.998858i	$-0.484788\pi$
$223$	318.163	0.0955416	0.0477708	+	0.998858i	$0.484788\pi$
$224$	0	0
$225$	−2022.50	−0.599259
$226$	0	0
$227$	−213.183	−0.0623323	−0.0311661	−	0.999514i	$-0.509922\pi$
$227$	−213.183	−0.0623323	−0.0311661	+	0.999514i	$0.509922\pi$
$228$	0	0
$229$	−1232.38	−0.355626	−0.177813	−	0.984064i	$-0.556902\pi$
$229$	−1232.38	−0.355626	−0.177813	+	0.984064i	$0.556902\pi$
$230$	0	0
$231$	−6398.06	−1.82234
$232$	0	0
$233$	−3675.32	−1.03338	−0.516691	−	0.856172i	$-0.672836\pi$
$233$	−3675.32	−1.03338	−0.516691	+	0.856172i	$0.672836\pi$
$234$	0	0
$235$	1801.23	0.499996
$236$	0	0
$237$	−8538.61	−2.34026
$238$	0	0
$239$	−3432.96	−0.929120	−0.464560	−	0.885542i	$-0.653787\pi$
$239$	−3432.96	−0.929120	−0.464560	+	0.885542i	$0.653787\pi$
$240$	0	0
$241$	−173.644	−0.0464125	−0.0232063	−	0.999731i	$-0.507387\pi$
$241$	−173.644	−0.0464125	−0.0232063	+	0.999731i	$0.507387\pi$
$242$	0	0
$243$	5424.90	1.43213
$244$	0	0
$245$	947.812	0.247157
$246$	0	0
$247$	1543.45	0.397602
$248$	0	0
$249$	524.078	0.133382
$250$	0	0
$251$	1299.47	0.326779	0.163389	−	0.986562i	$-0.447757\pi$
$251$	1299.47	0.326779	0.163389	+	0.986562i	$0.447757\pi$
$252$	0	0
$253$	−5136.36	−1.27636
$254$	0	0
$255$	3067.92	0.753414
$256$	0	0
$257$	−3130.80	−0.759898	−0.379949	−	0.925007i	$-0.624059\pi$
$257$	−3130.80	−0.759898	−0.379949	+	0.925007i	$0.624059\pi$
$258$	0	0
$259$	−4914.83	−1.17912
$260$	0	0
$261$	−840.220	−0.199266
$262$	0	0
$263$	−3018.47	−0.707707	−0.353854	−	0.935301i	$-0.615129\pi$
$263$	−3018.47	−0.707707	−0.353854	+	0.935301i	$0.615129\pi$
$264$	0	0
$265$	−4345.21	−1.00726
$266$	0	0
$267$	6161.17	1.41220
$268$	0	0
$269$	4052.61	0.918557	0.459279	−	0.888292i	$-0.348108\pi$
$269$	4052.61	0.918557	0.459279	+	0.888292i	$0.348108\pi$
$270$	0	0
$271$	−4384.75	−0.982857	−0.491429	−	0.870918i	$-0.663525\pi$
$271$	−4384.75	−0.982857	−0.491429	+	0.870918i	$0.663525\pi$
$272$	0	0
$273$	−6163.85	−1.36650
$274$	0	0
$275$	−2751.92	−0.603443
$276$	0	0
$277$	−981.943	−0.212994	−0.106497	−	0.994313i	$-0.533963\pi$
$277$	−981.943	−0.212994	−0.106497	+	0.994313i	$0.533963\pi$
$278$	0	0
$279$	−8107.99	−1.73983
$280$	0	0
$281$	7208.06	1.53024	0.765118	−	0.643890i	$-0.222680\pi$
$281$	7208.06	1.53024	0.765118	+	0.643890i	$0.222680\pi$
$282$	0	0
$283$	−1181.73	−0.248222	−0.124111	−	0.992268i	$-0.539608\pi$
$283$	−1181.73	−0.248222	−0.124111	+	0.992268i	$0.539608\pi$
$284$	0	0
$285$	−2258.82	−0.469478
$286$	0	0
$287$	−1687.99	−0.347173
$288$	0	0
$289$	−1866.39	−0.379888
$290$	0	0
$291$	12422.7	2.50250
$292$	0	0
$293$	2991.43	0.596456	0.298228	−	0.954495i	$-0.403605\pi$
$293$	2991.43	0.596456	0.298228	+	0.954495i	$0.403605\pi$
$294$	0	0
$295$	647.451	0.127783
$296$	0	0
$297$	−581.946	−0.113697
$298$	0	0
$299$	−4948.34	−0.957090
$300$	0	0
$301$	−227.871	−0.0436355
$302$	0	0
$303$	−4676.33	−0.886627
$304$	0	0
$305$	−357.567	−0.0671286
$306$	0	0
$307$	−3857.56	−0.717141	−0.358571	−	0.933503i	$-0.616736\pi$
$307$	−3857.56	−0.717141	−0.358571	+	0.933503i	$0.616736\pi$
$308$	0	0
$309$	−8301.95	−1.52842
$310$	0	0
$311$	−3886.42	−0.708614	−0.354307	−	0.935129i	$-0.615283\pi$
$311$	−3886.42	−0.708614	−0.354307	+	0.935129i	$0.615283\pi$
$312$	0	0
$313$	7446.62	1.34475	0.672376	−	0.740210i	$-0.265274\pi$
$313$	7446.62	1.34475	0.672376	+	0.740210i	$0.265274\pi$
$314$	0	0
$315$	4669.35	0.835201
$316$	0	0
$317$	−9286.32	−1.64534	−0.822668	−	0.568522i	$-0.807516\pi$
$317$	−9286.32	−1.64534	−0.822668	+	0.568522i	$0.807516\pi$
$318$	0	0
$319$	−1143.25	−0.200657
$320$	0	0
$321$	−6083.85	−1.05784
$322$	0	0
$323$	−2243.14	−0.386413
$324$	0	0
$325$	−2651.18	−0.452496
$326$	0	0
$327$	8988.07	1.52000
$328$	0	0
$329$	5259.43	0.881342
$330$	0	0
$331$	7078.72	1.17547	0.587737	−	0.809052i	$-0.300019\pi$
$331$	7078.72	1.17547	0.587737	+	0.809052i	$0.300019\pi$
$332$	0	0
$333$	−6564.29	−1.08024
$334$	0	0
$335$	−745.325	−0.121557
$336$	0	0
$337$	9520.27	1.53888	0.769440	−	0.638719i	$-0.220536\pi$
$337$	9520.27	1.53888	0.769440	+	0.638719i	$0.220536\pi$
$338$	0	0
$339$	−8488.80	−1.36002
$340$	0	0
$341$	−11032.2	−1.75198
$342$	0	0
$343$	−4673.10	−0.735638
$344$	0	0
$345$	7241.83	1.13011
$346$	0	0
$347$	−12204.3	−1.88808	−0.944039	−	0.329835i	$-0.893007\pi$
$347$	−12204.3	−1.88808	−0.944039	+	0.329835i	$0.893007\pi$
$348$	0	0
$349$	3503.83	0.537408	0.268704	−	0.963223i	$-0.413405\pi$
$349$	3503.83	0.537408	0.268704	+	0.963223i	$0.413405\pi$
$350$	0	0
$351$	−560.643	−0.0852562
$352$	0	0
$353$	−10973.0	−1.65449	−0.827246	−	0.561840i	$-0.810094\pi$
$353$	−10973.0	−1.65449	−0.827246	+	0.561840i	$0.810094\pi$
$354$	0	0
$355$	−5970.86	−0.892678
$356$	0	0
$357$	8958.06	1.32804
$358$	0	0
$359$	−1876.05	−0.275806	−0.137903	−	0.990446i	$-0.544036\pi$
$359$	−1876.05	−0.275806	−0.137903	+	0.990446i	$0.544036\pi$
$360$	0	0
$361$	−5207.44	−0.759213
$362$	0	0
$363$	−1669.29	−0.241363
$364$	0	0
$365$	−1058.61	−0.151808
$366$	0	0
$367$	−609.158	−0.0866425	−0.0433212	−	0.999061i	$-0.513794\pi$
$367$	−609.158	−0.0866425	−0.0433212	+	0.999061i	$0.513794\pi$
$368$	0	0
$369$	−2254.49	−0.318060
$370$	0	0
$371$	−12687.6	−1.77550
$372$	0	0
$373$	−5718.61	−0.793830	−0.396915	−	0.917855i	$-0.629919\pi$
$373$	−5718.61	−0.793830	−0.396915	+	0.917855i	$0.629919\pi$
$374$	0	0
$375$	10827.7	1.49105
$376$	0	0
$377$	−1101.40	−0.150464
$378$	0	0
$379$	5140.04	0.696639	0.348320	−	0.937376i	$-0.386752\pi$
$379$	5140.04	0.696639	0.348320	+	0.937376i	$0.386752\pi$
$380$	0	0
$381$	−3123.48	−0.420003
$382$	0	0
$383$	14956.9	1.99547	0.997734	−	0.0672782i	$-0.0214315\pi$
$383$	14956.9	1.99547	0.997734	+	0.0672782i	$0.0214315\pi$
$384$	0	0
$385$	6353.37	0.841033
$386$	0	0
$387$	−304.347	−0.0399763
$388$	0	0
$389$	7361.47	0.959489	0.479745	−	0.877408i	$-0.340729\pi$
$389$	7361.47	0.959489	0.479745	+	0.877408i	$0.340729\pi$
$390$	0	0
$391$	7191.54	0.930158
$392$	0	0
$393$	−15586.4	−2.00058
$394$	0	0
$395$	8478.97	1.08006
$396$	0	0
$397$	−9491.02	−1.19985	−0.599925	−	0.800056i	$-0.704803\pi$
$397$	−9491.02	−1.19985	−0.599925	+	0.800056i	$0.704803\pi$
$398$	0	0
$399$	−6595.57	−0.827548
$400$	0	0
$401$	−13865.8	−1.72675	−0.863375	−	0.504563i	$-0.831654\pi$
$401$	−13865.8	−1.72675	−0.863375	+	0.504563i	$0.831654\pi$
$402$	0	0
$403$	−10628.3	−1.31373
$404$	0	0
$405$	−4991.22	−0.612385
$406$	0	0
$407$	−8931.72	−1.08779
$408$	0	0
$409$	14528.9	1.75650	0.878252	−	0.478198i	$-0.158710\pi$
$409$	14528.9	1.75650	0.878252	+	0.478198i	$0.158710\pi$
$410$	0	0
$411$	12617.6	1.51431
$412$	0	0
$413$	1890.50	0.225243
$414$	0	0
$415$	−520.418	−0.0615574
$416$	0	0
$417$	9642.27	1.13234
$418$	0	0
$419$	7899.89	0.921086	0.460543	−	0.887637i	$-0.347655\pi$
$419$	7899.89	0.921086	0.460543	+	0.887637i	$0.347655\pi$
$420$	0	0
$421$	−2296.02	−0.265799	−0.132899	−	0.991130i	$-0.542429\pi$
$421$	−2296.02	−0.265799	−0.132899	+	0.991130i	$0.542429\pi$
$422$	0	0
$423$	7024.53	0.807434
$424$	0	0
$425$	3853.02	0.439762
$426$	0	0
$427$	−1044.06	−0.118327
$428$	0	0
$429$	−11201.6	−1.26064
$430$	0	0
$431$	6026.75	0.673546	0.336773	−	0.941586i	$-0.390665\pi$
$431$	6026.75	0.673546	0.336773	+	0.941586i	$0.390665\pi$
$432$	0	0
$433$	6821.42	0.757082	0.378541	−	0.925585i	$-0.376426\pi$
$433$	6821.42	0.757082	0.378541	+	0.925585i	$0.376426\pi$
$434$	0	0
$435$	1611.88	0.177664
$436$	0	0
$437$	−5294.93	−0.579613
$438$	0	0
$439$	−8063.71	−0.876674	−0.438337	−	0.898811i	$-0.644432\pi$
$439$	−8063.71	−0.876674	−0.438337	+	0.898811i	$0.644432\pi$
$440$	0	0
$441$	3696.34	0.399129
$442$	0	0
$443$	−5953.69	−0.638529	−0.319264	−	0.947666i	$-0.603436\pi$
$443$	−5953.69	−0.638529	−0.319264	+	0.947666i	$0.603436\pi$
$444$	0	0
$445$	−6118.14	−0.651748
$446$	0	0
$447$	16170.1	1.71101
$448$	0	0
$449$	4812.87	0.505865	0.252932	−	0.967484i	$-0.418605\pi$
$449$	4812.87	0.505865	0.252932	+	0.967484i	$0.418605\pi$
$450$	0	0
$451$	−3067.58	−0.320281
$452$	0	0
$453$	−408.918	−0.0424120
$454$	0	0
$455$	6120.80	0.630654
$456$	0	0
$457$	−4134.97	−0.423252	−0.211626	−	0.977351i	$-0.567876\pi$
$457$	−4134.97	−0.423252	−0.211626	+	0.977351i	$0.567876\pi$
$458$	0	0
$459$	814.796	0.0828571
$460$	0	0
$461$	−950.331	−0.0960115	−0.0480058	−	0.998847i	$-0.515287\pi$
$461$	−950.331	−0.0960115	−0.0480058	+	0.998847i	$0.515287\pi$
$462$	0	0
$463$	11599.2	1.16427	0.582137	−	0.813091i	$-0.302217\pi$
$463$	11599.2	1.16427	0.582137	+	0.813091i	$0.302217\pi$
$464$	0	0
$465$	15554.4	1.55122
$466$	0	0
$467$	−6390.73	−0.633250	−0.316625	−	0.948551i	$-0.602550\pi$
$467$	−6390.73	−0.633250	−0.316625	+	0.948551i	$0.602550\pi$
$468$	0	0
$469$	−2176.28	−0.214268
$470$	0	0
$471$	−10638.8	−1.04078
$472$	0	0
$473$	−414.111	−0.0402555
$474$	0	0
$475$	−2836.87	−0.274031
$476$	0	0
$477$	−16945.7	−1.62661
$478$	0	0
$479$	5345.06	0.509858	0.254929	−	0.966960i	$-0.417948\pi$
$479$	5345.06	0.509858	0.254929	+	0.966960i	$0.417948\pi$
$480$	0	0
$481$	−8604.77	−0.815683
$482$	0	0
$483$	21145.5	1.99204
$484$	0	0
$485$	−12335.9	−1.15494
$486$	0	0
$487$	6283.25	0.584644	0.292322	−	0.956320i	$-0.405572\pi$
$487$	6283.25	0.584644	0.292322	+	0.956320i	$0.405572\pi$
$488$	0	0
$489$	20719.2	1.91606
$490$	0	0
$491$	−12492.2	−1.14820	−0.574098	−	0.818786i	$-0.694647\pi$
$491$	−12492.2	−1.14820	−0.574098	+	0.818786i	$0.694647\pi$
$492$	0	0
$493$	1600.69	0.146230
$494$	0	0
$495$	8485.62	0.770505
$496$	0	0
$497$	−17434.4	−1.57352
$498$	0	0
$499$	8918.25	0.800071	0.400036	−	0.916500i	$-0.368998\pi$
$499$	8918.25	0.800071	0.400036	+	0.916500i	$0.368998\pi$
$500$	0	0
$501$	8837.46	0.788081
$502$	0	0
$503$	13508.4	1.19744	0.598720	−	0.800959i	$-0.295676\pi$
$503$	13508.4	1.19744	0.598720	+	0.800959i	$0.295676\pi$
$504$	0	0
$505$	4643.67	0.409189
$506$	0	0
$507$	5645.38	0.494517
$508$	0	0
$509$	15371.9	1.33860	0.669302	−	0.742990i	$-0.266593\pi$
$509$	15371.9	1.33860	0.669302	+	0.742990i	$0.266593\pi$
$510$	0	0
$511$	−3091.04	−0.267592
$512$	0	0
$513$	−599.912	−0.0516311
$514$	0	0
$515$	8243.97	0.705384
$516$	0	0
$517$	9557.96	0.813072
$518$	0	0
$519$	−19458.3	−1.64571
$520$	0	0
$521$	−14818.0	−1.24604	−0.623022	−	0.782204i	$-0.714095\pi$
$521$	−14818.0	−1.24604	−0.623022	+	0.782204i	$0.714095\pi$
$522$	0	0
$523$	18477.5	1.54487	0.772435	−	0.635094i	$-0.219039\pi$
$523$	18477.5	1.54487	0.772435	+	0.635094i	$0.219039\pi$
$524$	0	0
$525$	11329.2	0.941802
$526$	0	0
$527$	15446.4	1.27676
$528$	0	0
$529$	4808.64	0.395220
$530$	0	0
$531$	2524.97	0.206355
$532$	0	0
$533$	−2955.29	−0.240165
$534$	0	0
$535$	6041.36	0.488207
$536$	0	0
$537$	−2360.23	−0.189667
$538$	0	0
$539$	5029.43	0.401917
$540$	0	0
$541$	21020.4	1.67050	0.835249	−	0.549872i	$-0.185323\pi$
$541$	21020.4	1.67050	0.835249	+	0.549872i	$0.185323\pi$
$542$	0	0
$543$	15096.2	1.19308
$544$	0	0
$545$	−8925.29	−0.701500
$546$	0	0
$547$	−14931.3	−1.16712	−0.583562	−	0.812068i	$-0.698342\pi$
$547$	−14931.3	−1.16712	−0.583562	+	0.812068i	$0.698342\pi$
$548$	0	0
$549$	−1394.46	−0.108405
$550$	0	0
$551$	−1178.54	−0.0911208
$552$	0	0
$553$	24757.9	1.90382
$554$	0	0
$555$	12593.0	0.963139
$556$	0	0
$557$	−8309.40	−0.632102	−0.316051	−	0.948742i	$-0.602357\pi$
$557$	−8309.40	−0.632102	−0.316051	+	0.948742i	$0.602357\pi$
$558$	0	0
$559$	−398.952	−0.0301858
$560$	0	0
$561$	16279.5	1.22517
$562$	0	0
$563$	−8522.87	−0.638003	−0.319002	−	0.947754i	$-0.603348\pi$
$563$	−8522.87	−0.638003	−0.319002	+	0.947754i	$0.603348\pi$
$564$	0	0
$565$	8429.51	0.627667
$566$	0	0
$567$	−14573.9	−1.07945
$568$	0	0
$569$	13358.1	0.984182	0.492091	−	0.870544i	$-0.336233\pi$
$569$	13358.1	0.984182	0.492091	+	0.870544i	$0.336233\pi$
$570$	0	0
$571$	15869.2	1.16306	0.581528	−	0.813527i	$-0.302455\pi$
$571$	15869.2	1.16306	0.581528	+	0.813527i	$0.302455\pi$
$572$	0	0
$573$	−18470.2	−1.34660
$574$	0	0
$575$	9095.07	0.659636
$576$	0	0
$577$	−19532.9	−1.40930	−0.704651	−	0.709554i	$-0.748896\pi$
$577$	−19532.9	−1.40930	−0.704651	+	0.709554i	$0.748896\pi$
$578$	0	0
$579$	−24387.3	−1.75043
$580$	0	0
$581$	−1519.58	−0.108507
$582$	0	0
$583$	−23057.3	−1.63797
$584$	0	0
$585$	8174.99	0.577768
$586$	0	0
$587$	7413.80	0.521295	0.260647	−	0.965434i	$-0.416064\pi$
$587$	7413.80	0.521295	0.260647	+	0.965434i	$0.416064\pi$
$588$	0	0
$589$	−11372.7	−0.795596
$590$	0	0
$591$	−41185.4	−2.86657
$592$	0	0
$593$	−6949.89	−0.481278	−0.240639	−	0.970615i	$-0.577357\pi$
$593$	−6949.89	−0.481278	−0.240639	+	0.970615i	$0.577357\pi$
$594$	0	0
$595$	−8895.50	−0.612907
$596$	0	0
$597$	24945.6	1.71015
$598$	0	0
$599$	8126.52	0.554325	0.277162	−	0.960823i	$-0.410606\pi$
$599$	8126.52	0.554325	0.277162	+	0.960823i	$0.410606\pi$
$600$	0	0
$601$	−26777.7	−1.81745	−0.908723	−	0.417399i	$-0.862942\pi$
$601$	−26777.7	−1.81745	−0.908723	+	0.417399i	$0.862942\pi$
$602$	0	0
$603$	−2906.66	−0.196299
$604$	0	0
$605$	1657.63	0.111392
$606$	0	0
$607$	−4214.23	−0.281796	−0.140898	−	0.990024i	$-0.544999\pi$
$607$	−4214.23	−0.281796	−0.140898	+	0.990024i	$0.544999\pi$
$608$	0	0
$609$	4706.56	0.313168
$610$	0	0
$611$	9208.08	0.609687
$612$	0	0
$613$	13411.6	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$613$	13411.6	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$614$	0	0
$615$	4325.03	0.283580
$616$	0	0
$617$	7465.99	0.487147	0.243573	−	0.969882i	$-0.421680\pi$
$617$	7465.99	0.487147	0.243573	+	0.969882i	$0.421680\pi$
$618$	0	0
$619$	−27908.1	−1.81215	−0.906076	−	0.423116i	$-0.860936\pi$
$619$	−27908.1	−1.81215	−0.906076	+	0.423116i	$0.860936\pi$
$620$	0	0
$621$	1923.33	0.124284
$622$	0	0
$623$	−17864.5	−1.14883
$624$	0	0
$625$	−2026.36	−0.129687
$626$	0	0
$627$	−11986.1	−0.763445
$628$	0	0
$629$	12505.5	0.792730
$630$	0	0
$631$	−2565.72	−0.161869	−0.0809347	−	0.996719i	$-0.525791\pi$
$631$	−2565.72	−0.161869	−0.0809347	+	0.996719i	$0.525791\pi$
$632$	0	0
$633$	27496.9	1.72655
$634$	0	0
$635$	3101.67	0.193836
$636$	0	0
$637$	4845.33	0.301380
$638$	0	0
$639$	−23285.5	−1.44157
$640$	0	0
$641$	4751.44	0.292778	0.146389	−	0.989227i	$-0.453235\pi$
$641$	4751.44	0.292778	0.146389	+	0.989227i	$0.453235\pi$
$642$	0	0
$643$	4264.34	0.261538	0.130769	−	0.991413i	$-0.458255\pi$
$643$	4264.34	0.261538	0.130769	+	0.991413i	$0.458255\pi$
$644$	0	0
$645$	583.861	0.0356427
$646$	0	0
$647$	13828.4	0.840263	0.420131	−	0.907463i	$-0.361984\pi$
$647$	13828.4	0.840263	0.420131	+	0.907463i	$0.361984\pi$
$648$	0	0
$649$	3435.61	0.207796
$650$	0	0
$651$	45417.6	2.73434
$652$	0	0
$653$	17151.1	1.02784	0.513918	−	0.857840i	$-0.328194\pi$
$653$	17151.1	1.02784	0.513918	+	0.857840i	$0.328194\pi$
$654$	0	0
$655$	15477.5	0.923291
$656$	0	0
$657$	−4128.42	−0.245152
$658$	0	0
$659$	8518.81	0.503559	0.251780	−	0.967785i	$-0.418984\pi$
$659$	8518.81	0.503559	0.251780	+	0.967785i	$0.418984\pi$
$660$	0	0
$661$	763.507	0.0449273	0.0224637	−	0.999748i	$-0.492849\pi$
$661$	763.507	0.0449273	0.0224637	+	0.999748i	$0.492849\pi$
$662$	0	0
$663$	15683.6	0.918701
$664$	0	0
$665$	6549.51	0.381923
$666$	0	0
$667$	3778.43	0.219342
$668$	0	0
$669$	−2380.34	−0.137563
$670$	0	0
$671$	−1897.38	−0.109162
$672$	0	0
$673$	−156.244	−0.00894913	−0.00447456	−	0.999990i	$-0.501424\pi$
$673$	−156.244	−0.00894913	−0.00447456	+	0.999990i	$0.501424\pi$
$674$	0	0
$675$	1030.47	0.0587594
$676$	0	0
$677$	−1727.96	−0.0980960	−0.0490480	−	0.998796i	$-0.515619\pi$
$677$	−1727.96	−0.0980960	−0.0490480	+	0.998796i	$0.515619\pi$
$678$	0	0
$679$	−36019.7	−2.03580
$680$	0	0
$681$	1594.93	0.0897472
$682$	0	0
$683$	891.383	0.0499382	0.0249691	−	0.999688i	$-0.492051\pi$
$683$	891.383	0.0499382	0.0249691	+	0.999688i	$0.492051\pi$
$684$	0	0
$685$	−12529.5	−0.698871
$686$	0	0
$687$	9220.11	0.512036
$688$	0	0
$689$	−22213.2	−1.22824
$690$	0	0
$691$	−29598.6	−1.62950	−0.814751	−	0.579811i	$-0.803126\pi$
$691$	−29598.6	−1.62950	−0.814751	+	0.579811i	$0.803126\pi$
$692$	0	0
$693$	24777.3	1.35817
$694$	0	0
$695$	−9574.93	−0.522587
$696$	0	0
$697$	4294.99	0.233406
$698$	0	0
$699$	27497.0	1.48788
$700$	0	0
$701$	412.114	0.0222045	0.0111022	−	0.999938i	$-0.496466\pi$
$701$	412.114	0.0222045	0.0111022	+	0.999938i	$0.496466\pi$
$702$	0	0
$703$	−9207.46	−0.493977
$704$	0	0
$705$	−13475.9	−0.719904
$706$	0	0
$707$	13559.1	0.721277
$708$	0	0
$709$	−2219.36	−0.117560	−0.0587799	−	0.998271i	$-0.518721\pi$
$709$	−2219.36	−0.117560	−0.0587799	+	0.998271i	$0.518721\pi$
$710$	0	0
$711$	33066.8	1.74417
$712$	0	0
$713$	36461.2	1.91513
$714$	0	0
$715$	11123.3	0.581803
$716$	0	0
$717$	25683.7	1.33776
$718$	0	0
$719$	3407.15	0.176725	0.0883625	−	0.996088i	$-0.471837\pi$
$719$	3407.15	0.176725	0.0883625	+	0.996088i	$0.471837\pi$
$720$	0	0
$721$	24071.7	1.24338
$722$	0	0
$723$	1299.12	0.0668257
$724$	0	0
$725$	2024.38	0.103701
$726$	0	0
$727$	23405.3	1.19402	0.597011	−	0.802233i	$-0.296355\pi$
$727$	23405.3	1.19402	0.597011	+	0.802233i	$0.296355\pi$
$728$	0	0
$729$	−22447.0	−1.14043
$730$	0	0
$731$	579.806	0.0293364
$732$	0	0
$733$	22249.2	1.12114	0.560568	−	0.828108i	$-0.310583\pi$
$733$	22249.2	1.12114	0.560568	+	0.828108i	$0.310583\pi$
$734$	0	0
$735$	−7091.07	−0.355861
$736$	0	0
$737$	−3954.96	−0.197670
$738$	0	0
$739$	−735.563	−0.0366145	−0.0183073	−	0.999832i	$-0.505828\pi$
$739$	−735.563	−0.0366145	−0.0183073	+	0.999832i	$0.505828\pi$
$740$	0	0
$741$	−11547.4	−0.572474
$742$	0	0
$743$	−12261.5	−0.605427	−0.302713	−	0.953082i	$-0.597892\pi$
$743$	−12261.5	−0.605427	−0.302713	+	0.953082i	$0.597892\pi$
$744$	0	0
$745$	−16057.2	−0.789651
$746$	0	0
$747$	−2029.56	−0.0994079
$748$	0	0
$749$	17640.2	0.860562
$750$	0	0
$751$	13122.2	0.637596	0.318798	−	0.947823i	$-0.396721\pi$
$751$	13122.2	0.637596	0.318798	+	0.947823i	$0.396721\pi$
$752$	0	0
$753$	−9721.97	−0.470502
$754$	0	0
$755$	406.062	0.0195736
$756$	0	0
$757$	−2499.07	−0.119987	−0.0599935	−	0.998199i	$-0.519108\pi$
$757$	−2499.07	−0.119987	−0.0599935	+	0.998199i	$0.519108\pi$
$758$	0	0
$759$	38427.8	1.83773
$760$	0	0
$761$	−2058.30	−0.0980463	−0.0490232	−	0.998798i	$-0.515611\pi$
$761$	−2058.30	−0.0980463	−0.0490232	+	0.998798i	$0.515611\pi$
$762$	0	0
$763$	−26061.1	−1.23653
$764$	0	0
$765$	−11880.9	−0.561510
$766$	0	0
$767$	3309.85	0.155817
$768$	0	0
$769$	−35486.0	−1.66405	−0.832026	−	0.554736i	$-0.812819\pi$
$769$	−35486.0	−1.66405	−0.832026	+	0.554736i	$0.812819\pi$
$770$	0	0
$771$	23423.1	1.09412
$772$	0	0
$773$	2048.82	0.0953309	0.0476654	−	0.998863i	$-0.484822\pi$
$773$	2048.82	0.0953309	0.0476654	+	0.998863i	$0.484822\pi$
$774$	0	0
$775$	19534.9	0.905438
$776$	0	0
$777$	36770.4	1.69772
$778$	0	0
$779$	−3162.28	−0.145443
$780$	0	0
$781$	−31683.6	−1.45163
$782$	0	0
$783$	428.093	0.0195387
$784$	0	0
$785$	10564.5	0.480333
$786$	0	0
$787$	−5905.13	−0.267465	−0.133733	−	0.991017i	$-0.542696\pi$
$787$	−5905.13	−0.267465	−0.133733	+	0.991017i	$0.542696\pi$
$788$	0	0
$789$	22582.8	1.01897
$790$	0	0
$791$	24613.4	1.10639
$792$	0	0
$793$	−1827.92	−0.0818555
$794$	0	0
$795$	32508.8	1.45027
$796$	0	0
$797$	7627.74	0.339007	0.169503	−	0.985530i	$-0.445784\pi$
$797$	7627.74	0.339007	0.169503	+	0.985530i	$0.445784\pi$
$798$	0	0
$799$	−13382.3	−0.592531
$800$	0	0
$801$	−23859.9	−1.05250
$802$	0	0
$803$	−5617.34	−0.246864
$804$	0	0
$805$	−20997.8	−0.919350
$806$	0	0
$807$	−30319.7	−1.32256
$808$	0	0
$809$	−16313.5	−0.708963	−0.354482	−	0.935063i	$-0.615343\pi$
$809$	−16313.5	−0.708963	−0.354482	+	0.935063i	$0.615343\pi$
$810$	0	0
$811$	−21467.1	−0.929485	−0.464742	−	0.885446i	$-0.653853\pi$
$811$	−21467.1	−0.929485	−0.464742	+	0.885446i	$0.653853\pi$
$812$	0	0
$813$	32804.6	1.41514
$814$	0	0
$815$	−20574.5	−0.884284
$816$	0	0
$817$	−426.895	−0.0182805
$818$	0	0
$819$	23870.3	1.01843
$820$	0	0
$821$	1288.23	0.0547621	0.0273810	−	0.999625i	$-0.491283\pi$
$821$	1288.23	0.0547621	0.0273810	+	0.999625i	$0.491283\pi$
$822$	0	0
$823$	−45242.7	−1.91624	−0.958118	−	0.286373i	$-0.907550\pi$
$823$	−45242.7	−1.91624	−0.958118	+	0.286373i	$0.907550\pi$
$824$	0	0
$825$	20588.5	0.868849
$826$	0	0
$827$	−24870.2	−1.04573	−0.522867	−	0.852415i	$-0.675137\pi$
$827$	−24870.2	−1.04573	−0.522867	+	0.852415i	$0.675137\pi$
$828$	0	0
$829$	1192.92	0.0499783	0.0249891	−	0.999688i	$-0.492045\pi$
$829$	1192.92	0.0499783	0.0249891	+	0.999688i	$0.492045\pi$
$830$	0	0
$831$	7346.42	0.306672
$832$	0	0
$833$	−7041.82	−0.292899
$834$	0	0
$835$	−8775.73	−0.363709
$836$	0	0
$837$	4131.03	0.170597
$838$	0	0
$839$	−48239.1	−1.98498	−0.992491	−	0.122316i	$-0.960968\pi$
$839$	−48239.1	−1.98498	−0.992491	+	0.122316i	$0.960968\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−53927.2	−2.20326
$844$	0	0
$845$	−5605.95	−0.228225
$846$	0	0
$847$	4840.14	0.196351
$848$	0	0
$849$	8841.17	0.357395
$850$	0	0
$851$	29519.3	1.18908
$852$	0	0
$853$	−30226.9	−1.21330	−0.606652	−	0.794967i	$-0.707488\pi$
$853$	−30226.9	−1.21330	−0.606652	+	0.794967i	$0.707488\pi$
$854$	0	0
$855$	8747.58	0.349896
$856$	0	0
$857$	−20904.0	−0.833216	−0.416608	−	0.909086i	$-0.636781\pi$
$857$	−20904.0	−0.833216	−0.416608	+	0.909086i	$0.636781\pi$
$858$	0	0
$859$	−42115.7	−1.67284	−0.836419	−	0.548090i	$-0.815355\pi$
$859$	−42115.7	−1.67284	−0.836419	+	0.548090i	$0.815355\pi$
$860$	0	0
$861$	12628.7	0.499867
$862$	0	0
$863$	−14974.2	−0.590647	−0.295323	−	0.955397i	$-0.595427\pi$
$863$	−14974.2	−0.590647	−0.295323	+	0.955397i	$0.595427\pi$
$864$	0	0
$865$	19322.4	0.759515
$866$	0	0
$867$	13963.4	0.546970
$868$	0	0
$869$	44992.5	1.75635
$870$	0	0
$871$	−3810.19	−0.148224
$872$	0	0
$873$	−48108.3	−1.86508
$874$	0	0
$875$	−31395.3	−1.21298
$876$	0	0
$877$	18192.8	0.700486	0.350243	−	0.936659i	$-0.386099\pi$
$877$	18192.8	0.700486	0.350243	+	0.936659i	$0.386099\pi$
$878$	0	0
$879$	−22380.5	−0.858788
$880$	0	0
$881$	−47334.6	−1.81015	−0.905076	−	0.425250i	$-0.860186\pi$
$881$	−47334.6	−1.81015	−0.905076	+	0.425250i	$0.860186\pi$
$882$	0	0
$883$	19144.1	0.729614	0.364807	−	0.931083i	$-0.381135\pi$
$883$	19144.1	0.729614	0.364807	+	0.931083i	$0.381135\pi$
$884$	0	0
$885$	−4843.92	−0.183985
$886$	0	0
$887$	37334.6	1.41327	0.706636	−	0.707578i	$-0.250212\pi$
$887$	37334.6	1.41327	0.706636	+	0.707578i	$0.250212\pi$
$888$	0	0
$889$	9056.61	0.341675
$890$	0	0
$891$	−26485.2	−0.995835
$892$	0	0
$893$	9853.02	0.369226
$894$	0	0
$895$	2343.74	0.0875337
$896$	0	0
$897$	37021.1	1.37804
$898$	0	0
$899$	8115.52	0.301076
$900$	0	0
$901$	32283.0	1.19368
$902$	0	0
$903$	1704.82	0.0628273
$904$	0	0
$905$	−14990.8	−0.550620
$906$	0	0
$907$	−17138.9	−0.627439	−0.313719	−	0.949516i	$-0.601575\pi$
$907$	−17138.9	−0.627439	−0.313719	+	0.949516i	$0.601575\pi$
$908$	0	0
$909$	18109.7	0.660792
$910$	0	0
$911$	2972.24	0.108095	0.0540475	−	0.998538i	$-0.482788\pi$
$911$	2972.24	0.108095	0.0540475	+	0.998538i	$0.482788\pi$
$912$	0	0
$913$	−2761.53	−0.100102
$914$	0	0
$915$	2675.14	0.0966530
$916$	0	0
$917$	45193.0	1.62748
$918$	0	0
$919$	−23257.1	−0.834800	−0.417400	−	0.908723i	$-0.637059\pi$
$919$	−23257.1	−0.834800	−0.417400	+	0.908723i	$0.637059\pi$
$920$	0	0
$921$	28860.4	1.03255
$922$	0	0
$923$	−30523.7	−1.08852
$924$	0	0
$925$	15815.6	0.562177
$926$	0	0
$927$	32150.3	1.13911
$928$	0	0
$929$	−8966.22	−0.316655	−0.158327	−	0.987387i	$-0.550610\pi$
$929$	−8966.22	−0.316655	−0.158327	+	0.987387i	$0.550610\pi$
$930$	0	0
$931$	5184.70	0.182515
$932$	0	0
$933$	29076.3	1.02028
$934$	0	0
$935$	−16165.8	−0.565431
$936$	0	0
$937$	−38107.4	−1.32862	−0.664309	−	0.747458i	$-0.731274\pi$
$937$	−38107.4	−1.32862	−0.664309	+	0.747458i	$0.731274\pi$
$938$	0	0
$939$	−55712.0	−1.93620
$940$	0	0
$941$	−11452.2	−0.396738	−0.198369	−	0.980127i	$-0.563564\pi$
$941$	−11452.2	−0.396738	−0.198369	+	0.980127i	$0.563564\pi$
$942$	0	0
$943$	10138.3	0.350105
$944$	0	0
$945$	−2379.04	−0.0818944
$946$	0	0
$947$	13450.2	0.461533	0.230767	−	0.973009i	$-0.425877\pi$
$947$	13450.2	0.461533	0.230767	+	0.973009i	$0.425877\pi$
$948$	0	0
$949$	−5411.72	−0.185112
$950$	0	0
$951$	69475.8	2.36899
$952$	0	0
$953$	−37721.2	−1.28217	−0.641086	−	0.767469i	$-0.721516\pi$
$953$	−37721.2	−1.28217	−0.641086	+	0.767469i	$0.721516\pi$
$954$	0	0
$955$	18341.2	0.621473
$956$	0	0
$957$	8553.23	0.288910
$958$	0	0
$959$	−36585.0	−1.23190
$960$	0	0
$961$	48522.5	1.62876
$962$	0	0
$963$	23560.5	0.788396
$964$	0	0
$965$	24216.9	0.807845
$966$	0	0
$967$	35139.1	1.16856	0.584280	−	0.811552i	$-0.301377\pi$
$967$	35139.1	1.16856	0.584280	+	0.811552i	$0.301377\pi$
$968$	0	0
$969$	16782.1	0.556365
$970$	0	0
$971$	−4508.15	−0.148994	−0.0744972	−	0.997221i	$-0.523735\pi$
$971$	−4508.15	−0.148994	−0.0744972	+	0.997221i	$0.523735\pi$
$972$	0	0
$973$	−27958.0	−0.921163
$974$	0	0
$975$	19834.9	0.651512
$976$	0	0
$977$	32839.4	1.07536	0.537679	−	0.843150i	$-0.319301\pi$
$977$	32839.4	1.07536	0.537679	+	0.843150i	$0.319301\pi$
$978$	0	0
$979$	−32465.1	−1.05984
$980$	0	0
$981$	−34807.4	−1.13284
$982$	0	0
$983$	−44340.0	−1.43868	−0.719342	−	0.694656i	$-0.755557\pi$
$983$	−44340.0	−1.43868	−0.719342	+	0.694656i	$0.755557\pi$
$984$	0	0
$985$	40897.7	1.32295
$986$	0	0
$987$	−39348.5	−1.26897
$988$	0	0
$989$	1368.63	0.0440041
$990$	0	0
$991$	37665.0	1.20733	0.603667	−	0.797237i	$-0.293706\pi$
$991$	37665.0	1.20733	0.603667	+	0.797237i	$0.293706\pi$
$992$	0	0
$993$	−52959.6	−1.69247
$994$	0	0
$995$	−24771.4	−0.789253
$996$	0	0
$997$	45617.8	1.44908	0.724539	−	0.689234i	$-0.242053\pi$
$997$	45617.8	1.44908	0.724539	+	0.689234i	$0.242053\pi$
$998$	0	0
$999$	3344.51	0.105922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.be.1.1		8
4.3	odd	2	inner	1856.4.a.be.1.8		8
8.3	odd	2		928.4.a.c.1.1	✓	8
8.5	even	2		928.4.a.c.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.c.1.1	✓	8	8.3	odd	2
928.4.a.c.1.8	yes	8	8.5	even	2
1856.4.a.be.1.1		8	1.1	even	1	trivial
1856.4.a.be.1.8		8	4.3	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-7.48152 q^{3} +7.42926 q^{5} +21.6928 q^{7} +28.9731 q^{9} +O(q^{10})\)	\(q-7.48152 q^{3} +7.42926 q^{5} +21.6928 q^{7} +28.9731 q^{9} +39.4223 q^{11} +37.9793 q^{13} -55.5822 q^{15} -55.1961 q^{17} +40.6394 q^{19} -162.295 q^{21} -130.291 q^{23} -69.8060 q^{25} -14.7618 q^{27} -29.0000 q^{29} -279.845 q^{31} -294.939 q^{33} +161.162 q^{35} -226.565 q^{37} -284.142 q^{39} -77.8132 q^{41} -10.5045 q^{43} +215.249 q^{45} +242.450 q^{47} +127.578 q^{49} +412.951 q^{51} -584.878 q^{53} +292.879 q^{55} -304.044 q^{57} +87.1488 q^{59} -48.1295 q^{61} +628.508 q^{63} +282.158 q^{65} -100.323 q^{67} +974.772 q^{69} -803.695 q^{71} -142.491 q^{73} +522.255 q^{75} +855.182 q^{77} +1141.29 q^{79} -671.833 q^{81} -70.0497 q^{83} -410.067 q^{85} +216.964 q^{87} -823.519 q^{89} +823.877 q^{91} +2093.67 q^{93} +301.921 q^{95} -1660.45 q^{97} +1142.19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 20 q^{5} + 40 q^{9}+O(q^{10}) \) 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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