LMFDB - Embedded newform 1856.4.a.y.1.3

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:	$5$
Coefficient field:	5.5.13458092.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 $ x^5 - x^4 - 14x^3 + 18x^2 + 20*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.328194$ 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)

$f(q)$	$=$	$q-1.84328 q^{3} -18.3339 q^{5} -16.8583 q^{7} -23.6023 q^{9} +O(q^{10})$	$q-1.84328 q^{3} -18.3339 q^{5} -16.8583 q^{7} -23.6023 q^{9} -52.4385 q^{11} +87.5580 q^{13} +33.7945 q^{15} +15.4072 q^{17} -67.0156 q^{19} +31.0745 q^{21} +132.679 q^{23} +211.133 q^{25} +93.2741 q^{27} +29.0000 q^{29} +90.2221 q^{31} +96.6587 q^{33} +309.078 q^{35} -11.1247 q^{37} -161.394 q^{39} -18.8392 q^{41} +147.756 q^{43} +432.723 q^{45} +21.0963 q^{47} -58.7983 q^{49} -28.3997 q^{51} +290.454 q^{53} +961.404 q^{55} +123.528 q^{57} +337.343 q^{59} -84.0147 q^{61} +397.895 q^{63} -1605.28 q^{65} -330.821 q^{67} -244.564 q^{69} +492.420 q^{71} -347.053 q^{73} -389.176 q^{75} +884.023 q^{77} -986.297 q^{79} +465.333 q^{81} -594.382 q^{83} -282.475 q^{85} -53.4550 q^{87} +1387.04 q^{89} -1476.08 q^{91} -166.304 q^{93} +1228.66 q^{95} -334.003 q^{97} +1237.67 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9}+O(q^{10}) $ 5 * q - 8 * q^3 - 10 * q^5 + 40 * q^7 + 33 * q^9	$ 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9} - 12 q^{11} - 14 q^{13} - 74 q^{15} + 66 q^{17} - 214 q^{19} + 164 q^{23} + 207 q^{25} - 362 q^{27} + 145 q^{29} + 420 q^{31} - 576 q^{33} + 52 q^{35} - 378 q^{37} - 374 q^{39} - 1158 q^{41} + 204 q^{43} + 1506 q^{45} + 248 q^{47} - 283 q^{49} - 228 q^{51} + 554 q^{53} + 546 q^{55} + 44 q^{57} - 440 q^{59} - 618 q^{61} + 804 q^{63} - 1656 q^{65} - 1164 q^{67} + 1968 q^{69} - 692 q^{71} - 1950 q^{73} - 3074 q^{75} + 1616 q^{77} + 272 q^{79} + 1801 q^{81} - 512 q^{83} + 1628 q^{85} - 232 q^{87} + 866 q^{89} - 2580 q^{91} + 40 q^{93} + 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100}) $ 5 * q - 8 * q^3 - 10 * q^5 + 40 * q^7 + 33 * q^9 - 12 * q^11 - 14 * q^13 - 74 * q^15 + 66 * q^17 - 214 * q^19 + 164 * q^23 + 207 * q^25 - 362 * q^27 + 145 * q^29 + 420 * q^31 - 576 * q^33 + 52 * q^35 - 378 * q^37 - 374 * q^39 - 1158 * q^41 + 204 * q^43 + 1506 * q^45 + 248 * q^47 - 283 * q^49 - 228 * q^51 + 554 * q^53 + 546 * q^55 + 44 * q^57 - 440 * q^59 - 618 * q^61 + 804 * q^63 - 1656 * q^65 - 1164 * q^67 + 1968 * q^69 - 692 * q^71 - 1950 * q^73 - 3074 * q^75 + 1616 * q^77 + 272 * q^79 + 1801 * q^81 - 512 * q^83 + 1628 * q^85 - 232 * q^87 + 866 * q^89 - 2580 * q^91 + 40 * q^93 + 2244 * q^95 + 1562 * q^97 + 238 * 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
$3$	−1.84328	−0.354739	−0.177369	−	0.984144i	$-0.556759\pi$
$3$	−1.84328	−0.354739	−0.177369	+	0.984144i	$0.556759\pi$
$4$	0	0
$5$	−18.3339	−1.63984	−0.819918	−	0.572481i	$-0.805981\pi$
$5$	−18.3339	−1.63984	−0.819918	+	0.572481i	$0.805981\pi$
$6$	0	0
$7$	−16.8583	−0.910262	−0.455131	−	0.890425i	$-0.650408\pi$
$7$	−16.8583	−0.910262	−0.455131	+	0.890425i	$0.650408\pi$
$8$	0	0
$9$	−23.6023	−0.874160
$10$	0	0
$11$	−52.4385	−1.43735	−0.718673	−	0.695348i	$-0.755250\pi$
$11$	−52.4385	−1.43735	−0.718673	+	0.695348i	$0.755250\pi$
$12$	0	0
$13$	87.5580	1.86802	0.934008	−	0.357252i	$-0.116286\pi$
$13$	87.5580	1.86802	0.934008	+	0.357252i	$0.116286\pi$
$14$	0	0
$15$	33.7945	0.581713
$16$	0	0
$17$	15.4072	0.219812	0.109906	−	0.993942i	$-0.464945\pi$
$17$	15.4072	0.219812	0.109906	+	0.993942i	$0.464945\pi$
$18$	0	0
$19$	−67.0156	−0.809181	−0.404591	−	0.914498i	$-0.632586\pi$
$19$	−67.0156	−0.809181	−0.404591	+	0.914498i	$0.632586\pi$
$20$	0	0
$21$	31.0745	0.322905
$22$	0	0
$23$	132.679	1.20285	0.601423	−	0.798931i	$-0.294601\pi$
$23$	132.679	1.20285	0.601423	+	0.798931i	$0.294601\pi$
$24$	0	0
$25$	211.133	1.68906
$26$	0	0
$27$	93.2741	0.664837
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	90.2221	0.522721	0.261361	−	0.965241i	$-0.415829\pi$
$31$	90.2221	0.522721	0.261361	+	0.965241i	$0.415829\pi$
$32$	0	0
$33$	96.6587	0.509882
$34$	0	0
$35$	309.078	1.49268
$36$	0	0
$37$	−11.1247	−0.0494296	−0.0247148	−	0.999695i	$-0.507868\pi$
$37$	−11.1247	−0.0494296	−0.0247148	+	0.999695i	$0.507868\pi$
$38$	0	0
$39$	−161.394	−0.662658
$40$	0	0
$41$	−18.8392	−0.0717608	−0.0358804	−	0.999356i	$-0.511424\pi$
$41$	−18.8392	−0.0717608	−0.0358804	+	0.999356i	$0.511424\pi$
$42$	0	0
$43$	147.756	0.524013	0.262007	−	0.965066i	$-0.415616\pi$
$43$	147.756	0.524013	0.262007	+	0.965066i	$0.415616\pi$
$44$	0	0
$45$	432.723	1.43348
$46$	0	0
$47$	21.0963	0.0654726	0.0327363	−	0.999464i	$-0.489578\pi$
$47$	21.0963	0.0654726	0.0327363	+	0.999464i	$0.489578\pi$
$48$	0	0
$49$	−58.7983	−0.171424
$50$	0	0
$51$	−28.3997	−0.0779757
$52$	0	0
$53$	290.454	0.752772	0.376386	−	0.926463i	$-0.377167\pi$
$53$	290.454	0.752772	0.376386	+	0.926463i	$0.377167\pi$
$54$	0	0
$55$	961.404	2.35701
$56$	0	0
$57$	123.528	0.287048
$58$	0	0
$59$	337.343	0.744379	0.372190	−	0.928157i	$-0.378607\pi$
$59$	337.343	0.744379	0.372190	+	0.928157i	$0.378607\pi$
$60$	0	0
$61$	−84.0147	−0.176344	−0.0881720	−	0.996105i	$-0.528103\pi$
$61$	−84.0147	−0.176344	−0.0881720	+	0.996105i	$0.528103\pi$
$62$	0	0
$63$	397.895	0.795715
$64$	0	0
$65$	−1605.28	−3.06324
$66$	0	0
$67$	−330.821	−0.603228	−0.301614	−	0.953430i	$-0.597525\pi$
$67$	−330.821	−0.603228	−0.301614	+	0.953430i	$0.597525\pi$
$68$	0	0
$69$	−244.564	−0.426696
$70$	0	0
$71$	492.420	0.823092	0.411546	−	0.911389i	$-0.364989\pi$
$71$	492.420	0.823092	0.411546	+	0.911389i	$0.364989\pi$
$72$	0	0
$73$	−347.053	−0.556431	−0.278216	−	0.960519i	$-0.589743\pi$
$73$	−347.053	−0.556431	−0.278216	+	0.960519i	$0.589743\pi$
$74$	0	0
$75$	−389.176	−0.599176
$76$	0	0
$77$	884.023	1.30836
$78$	0	0
$79$	−986.297	−1.40465	−0.702324	−	0.711858i	$-0.747854\pi$
$79$	−986.297	−1.40465	−0.702324	+	0.711858i	$0.747854\pi$
$80$	0	0
$81$	465.333	0.638317
$82$	0	0
$83$	−594.382	−0.786048	−0.393024	−	0.919528i	$-0.628571\pi$
$83$	−594.382	−0.786048	−0.393024	+	0.919528i	$0.628571\pi$
$84$	0	0
$85$	−282.475	−0.360455
$86$	0	0
$87$	−53.4550	−0.0658733
$88$	0	0
$89$	1387.04	1.65197	0.825987	−	0.563689i	$-0.190618\pi$
$89$	1387.04	1.65197	0.825987	+	0.563689i	$0.190618\pi$
$90$	0	0
$91$	−1476.08	−1.70038
$92$	0	0
$93$	−166.304	−0.185430
$94$	0	0
$95$	1228.66	1.32692
$96$	0	0
$97$	−334.003	−0.349617	−0.174808	−	0.984602i	$-0.555931\pi$
$97$	−334.003	−0.349617	−0.174808	+	0.984602i	$0.555931\pi$
$98$	0	0
$99$	1237.67	1.25647
$100$	0	0
$101$	245.919	0.242276	0.121138	−	0.992636i	$-0.461346\pi$
$101$	245.919	0.242276	0.121138	+	0.992636i	$0.461346\pi$
$102$	0	0
$103$	−531.298	−0.508255	−0.254128	−	0.967171i	$-0.581788\pi$
$103$	−531.298	−0.508255	−0.254128	+	0.967171i	$0.581788\pi$
$104$	0	0
$105$	−569.717	−0.529511
$106$	0	0
$107$	429.030	0.387625	0.193812	−	0.981039i	$-0.437915\pi$
$107$	429.030	0.387625	0.193812	+	0.981039i	$0.437915\pi$
$108$	0	0
$109$	967.263	0.849972	0.424986	−	0.905200i	$-0.360279\pi$
$109$	967.263	0.849972	0.424986	+	0.905200i	$0.360279\pi$
$110$	0	0
$111$	20.5060	0.0175346
$112$	0	0
$113$	−1705.23	−1.41960	−0.709798	−	0.704405i	$-0.751214\pi$
$113$	−1705.23	−1.41960	−0.709798	+	0.704405i	$0.751214\pi$
$114$	0	0
$115$	−2432.52	−1.97247
$116$	0	0
$117$	−2066.57	−1.63295
$118$	0	0
$119$	−259.739	−0.200086
$120$	0	0
$121$	1418.80	1.06596
$122$	0	0
$123$	34.7259	0.0254563
$124$	0	0
$125$	−1579.15	−1.12995
$126$	0	0
$127$	−2670.28	−1.86574	−0.932870	−	0.360213i	$-0.882704\pi$
$127$	−2670.28	−1.86574	−0.932870	+	0.360213i	$0.882704\pi$
$128$	0	0
$129$	−272.355	−0.185888
$130$	0	0
$131$	−879.993	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$131$	−879.993	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$132$	0	0
$133$	1129.77	0.736567
$134$	0	0
$135$	−1710.08	−1.09022
$136$	0	0
$137$	2064.15	1.28724	0.643620	−	0.765345i	$-0.277432\pi$
$137$	2064.15	1.28724	0.643620	+	0.765345i	$0.277432\pi$
$138$	0	0
$139$	−605.130	−0.369255	−0.184628	−	0.982809i	$-0.559108\pi$
$139$	−605.130	−0.369255	−0.184628	+	0.982809i	$0.559108\pi$
$140$	0	0
$141$	−38.8863	−0.0232257
$142$	0	0
$143$	−4591.41	−2.68499
$144$	0	0
$145$	−531.684	−0.304510
$146$	0	0
$147$	108.381	0.0608106
$148$	0	0
$149$	775.322	0.426287	0.213144	−	0.977021i	$-0.431630\pi$
$149$	775.322	0.426287	0.213144	+	0.977021i	$0.431630\pi$
$150$	0	0
$151$	427.925	0.230623	0.115311	−	0.993329i	$-0.463213\pi$
$151$	427.925	0.230623	0.115311	+	0.993329i	$0.463213\pi$
$152$	0	0
$153$	−363.646	−0.192151
$154$	0	0
$155$	−1654.12	−0.857177
$156$	0	0
$157$	1680.93	0.854474	0.427237	−	0.904140i	$-0.359487\pi$
$157$	1680.93	0.854474	0.427237	+	0.904140i	$0.359487\pi$
$158$	0	0
$159$	−535.387	−0.267037
$160$	0	0
$161$	−2236.74	−1.09490
$162$	0	0
$163$	−2038.68	−0.979645	−0.489822	−	0.871822i	$-0.662938\pi$
$163$	−2038.68	−0.979645	−0.489822	+	0.871822i	$0.662938\pi$
$164$	0	0
$165$	−1772.13	−0.836123
$166$	0	0
$167$	2543.12	1.17840	0.589199	−	0.807988i	$-0.299443\pi$
$167$	2543.12	1.17840	0.589199	+	0.807988i	$0.299443\pi$
$168$	0	0
$169$	5469.40	2.48949
$170$	0	0
$171$	1581.73	0.707354
$172$	0	0
$173$	306.031	0.134492	0.0672460	−	0.997736i	$-0.478579\pi$
$173$	306.031	0.134492	0.0672460	+	0.997736i	$0.478579\pi$
$174$	0	0
$175$	−3559.34	−1.53749
$176$	0	0
$177$	−621.817	−0.264060
$178$	0	0
$179$	478.797	0.199927	0.0999635	−	0.994991i	$-0.468127\pi$
$179$	478.797	0.199927	0.0999635	+	0.994991i	$0.468127\pi$
$180$	0	0
$181$	478.433	0.196473	0.0982367	−	0.995163i	$-0.468680\pi$
$181$	478.433	0.196473	0.0982367	+	0.995163i	$0.468680\pi$
$182$	0	0
$183$	154.862	0.0625560
$184$	0	0
$185$	203.960	0.0810565
$186$	0	0
$187$	−807.931	−0.315945
$188$	0	0
$189$	−1572.44	−0.605176
$190$	0	0
$191$	833.106	0.315610	0.157805	−	0.987470i	$-0.449558\pi$
$191$	833.106	0.315610	0.157805	+	0.987470i	$0.449558\pi$
$192$	0	0
$193$	−1449.88	−0.540751	−0.270376	−	0.962755i	$-0.587148\pi$
$193$	−1449.88	−0.540751	−0.270376	+	0.962755i	$0.587148\pi$
$194$	0	0
$195$	2958.98	1.08665
$196$	0	0
$197$	1993.27	0.720886	0.360443	−	0.932781i	$-0.382626\pi$
$197$	1993.27	0.720886	0.360443	+	0.932781i	$0.382626\pi$
$198$	0	0
$199$	−356.359	−0.126943	−0.0634714	−	0.997984i	$-0.520217\pi$
$199$	−356.359	−0.126943	−0.0634714	+	0.997984i	$0.520217\pi$
$200$	0	0
$201$	609.795	0.213988
$202$	0	0
$203$	−488.890	−0.169031
$204$	0	0
$205$	345.397	0.117676
$206$	0	0
$207$	−3131.53	−1.05148
$208$	0	0
$209$	3514.20	1.16307
$210$	0	0
$211$	−4131.66	−1.34803	−0.674017	−	0.738716i	$-0.735433\pi$
$211$	−4131.66	−1.34803	−0.674017	+	0.738716i	$0.735433\pi$
$212$	0	0
$213$	−907.666	−0.291982
$214$	0	0
$215$	−2708.95	−0.859296
$216$	0	0
$217$	−1520.99	−0.475813
$218$	0	0
$219$	639.715	0.197388
$220$	0	0
$221$	1349.02	0.410612
$222$	0	0
$223$	1332.32	0.400086	0.200043	−	0.979787i	$-0.435892\pi$
$223$	1332.32	0.400086	0.200043	+	0.979787i	$0.435892\pi$
$224$	0	0
$225$	−4983.22	−1.47651
$226$	0	0
$227$	1329.33	0.388681	0.194340	−	0.980934i	$-0.437743\pi$
$227$	1329.33	0.388681	0.194340	+	0.980934i	$0.437743\pi$
$228$	0	0
$229$	−5455.47	−1.57427	−0.787135	−	0.616780i	$-0.788437\pi$
$229$	−5455.47	−1.57427	−0.787135	+	0.616780i	$0.788437\pi$
$230$	0	0
$231$	−1629.50	−0.464126
$232$	0	0
$233$	−591.158	−0.166215	−0.0831075	−	0.996541i	$-0.526484\pi$
$233$	−591.158	−0.166215	−0.0831075	+	0.996541i	$0.526484\pi$
$234$	0	0
$235$	−386.778	−0.107364
$236$	0	0
$237$	1818.02	0.498283
$238$	0	0
$239$	6946.01	1.87992	0.939959	−	0.341289i	$-0.110863\pi$
$239$	6946.01	1.87992	0.939959	+	0.341289i	$0.110863\pi$
$240$	0	0
$241$	7105.62	1.89923	0.949613	−	0.313426i	$-0.101477\pi$
$241$	7105.62	1.89923	0.949613	+	0.313426i	$0.101477\pi$
$242$	0	0
$243$	−3376.14	−0.891273
$244$	0	0
$245$	1078.00	0.281106
$246$	0	0
$247$	−5867.75	−1.51156
$248$	0	0
$249$	1095.61	0.278842
$250$	0	0
$251$	4874.53	1.22581	0.612904	−	0.790158i	$-0.290001\pi$
$251$	4874.53	1.22581	0.612904	+	0.790158i	$0.290001\pi$
$252$	0	0
$253$	−6957.49	−1.72891
$254$	0	0
$255$	520.679	0.127867
$256$	0	0
$257$	2488.22	0.603934	0.301967	−	0.953318i	$-0.402357\pi$
$257$	2488.22	0.603934	0.301967	+	0.953318i	$0.402357\pi$
$258$	0	0
$259$	187.544	0.0449939
$260$	0	0
$261$	−684.468	−0.162328
$262$	0	0
$263$	−2812.52	−0.659419	−0.329709	−	0.944082i	$-0.606951\pi$
$263$	−2812.52	−0.659419	−0.329709	+	0.944082i	$0.606951\pi$
$264$	0	0
$265$	−5325.16	−1.23442
$266$	0	0
$267$	−2556.69	−0.586019
$268$	0	0
$269$	5554.63	1.25900	0.629501	−	0.777000i	$-0.283259\pi$
$269$	5554.63	1.25900	0.629501	+	0.777000i	$0.283259\pi$
$270$	0	0
$271$	−3168.41	−0.710211	−0.355105	−	0.934826i	$-0.615555\pi$
$271$	−3168.41	−0.710211	−0.355105	+	0.934826i	$0.615555\pi$
$272$	0	0
$273$	2720.82	0.603192
$274$	0	0
$275$	−11071.5	−2.42777
$276$	0	0
$277$	−3965.64	−0.860189	−0.430095	−	0.902784i	$-0.641520\pi$
$277$	−3965.64	−0.860189	−0.430095	+	0.902784i	$0.641520\pi$
$278$	0	0
$279$	−2129.45	−0.456942
$280$	0	0
$281$	1655.16	0.351383	0.175692	−	0.984445i	$-0.443784\pi$
$281$	1655.16	0.351383	0.175692	+	0.984445i	$0.443784\pi$
$282$	0	0
$283$	−7786.09	−1.63546	−0.817730	−	0.575602i	$-0.804768\pi$
$283$	−7786.09	−1.63546	−0.817730	+	0.575602i	$0.804768\pi$
$284$	0	0
$285$	−2264.76	−0.470711
$286$	0	0
$287$	317.597	0.0653211
$288$	0	0
$289$	−4675.62	−0.951683
$290$	0	0
$291$	615.659	0.124023
$292$	0	0
$293$	−8090.80	−1.61321	−0.806603	−	0.591094i	$-0.798696\pi$
$293$	−8090.80	−1.61321	−0.806603	+	0.591094i	$0.798696\pi$
$294$	0	0
$295$	−6184.83	−1.22066
$296$	0	0
$297$	−4891.16	−0.955601
$298$	0	0
$299$	11617.1	2.24694
$300$	0	0
$301$	−2490.91	−0.476989
$302$	0	0
$303$	−453.297	−0.0859446
$304$	0	0
$305$	1540.32	0.289175
$306$	0	0
$307$	6129.49	1.13951	0.569753	−	0.821816i	$-0.307039\pi$
$307$	6129.49	1.13951	0.569753	+	0.821816i	$0.307039\pi$
$308$	0	0
$309$	979.328	0.180298
$310$	0	0
$311$	−8167.93	−1.48926	−0.744632	−	0.667476i	$-0.767375\pi$
$311$	−8167.93	−1.48926	−0.744632	+	0.667476i	$0.767375\pi$
$312$	0	0
$313$	1877.25	0.339005	0.169502	−	0.985530i	$-0.445784\pi$
$313$	1877.25	0.339005	0.169502	+	0.985530i	$0.445784\pi$
$314$	0	0
$315$	−7294.97	−1.30484
$316$	0	0
$317$	−1222.93	−0.216677	−0.108338	−	0.994114i	$-0.534553\pi$
$317$	−1222.93	−0.216677	−0.108338	+	0.994114i	$0.534553\pi$
$318$	0	0
$319$	−1520.72	−0.266908
$320$	0	0
$321$	−790.820	−0.137506
$322$	0	0
$323$	−1032.52	−0.177867
$324$	0	0
$325$	18486.4	3.15520
$326$	0	0
$327$	−1782.93	−0.301518
$328$	0	0
$329$	−355.648	−0.0595972
$330$	0	0
$331$	−3769.03	−0.625876	−0.312938	−	0.949774i	$-0.601313\pi$
$331$	−3769.03	−0.625876	−0.312938	+	0.949774i	$0.601313\pi$
$332$	0	0
$333$	262.570	0.0432094
$334$	0	0
$335$	6065.25	0.989195
$336$	0	0
$337$	−10900.2	−1.76193	−0.880967	−	0.473179i	$-0.843107\pi$
$337$	−10900.2	−1.76193	−0.880967	+	0.473179i	$0.843107\pi$
$338$	0	0
$339$	3143.21	0.503586
$340$	0	0
$341$	−4731.11	−0.751332
$342$	0	0
$343$	6773.63	1.06630
$344$	0	0
$345$	4483.82	0.699712
$346$	0	0
$347$	8542.30	1.32154	0.660771	−	0.750588i	$-0.270230\pi$
$347$	8542.30	1.32154	0.660771	+	0.750588i	$0.270230\pi$
$348$	0	0
$349$	993.823	0.152430	0.0762151	−	0.997091i	$-0.475716\pi$
$349$	993.823	0.152430	0.0762151	+	0.997091i	$0.475716\pi$
$350$	0	0
$351$	8166.89	1.24193
$352$	0	0
$353$	8191.10	1.23504	0.617519	−	0.786556i	$-0.288138\pi$
$353$	8191.10	1.23504	0.617519	+	0.786556i	$0.288138\pi$
$354$	0	0
$355$	−9027.99	−1.34974
$356$	0	0
$357$	478.771	0.0709783
$358$	0	0
$359$	−4703.71	−0.691510	−0.345755	−	0.938325i	$-0.612377\pi$
$359$	−4703.71	−0.691510	−0.345755	+	0.938325i	$0.612377\pi$
$360$	0	0
$361$	−2367.90	−0.345226
$362$	0	0
$363$	−2615.24	−0.378139
$364$	0	0
$365$	6362.85	0.912456
$366$	0	0
$367$	9431.88	1.34153	0.670763	−	0.741672i	$-0.265967\pi$
$367$	9431.88	1.34153	0.670763	+	0.741672i	$0.265967\pi$
$368$	0	0
$369$	444.650	0.0627305
$370$	0	0
$371$	−4896.55	−0.685219
$372$	0	0
$373$	8281.46	1.14959	0.574796	−	0.818297i	$-0.305081\pi$
$373$	8281.46	1.14959	0.574796	+	0.818297i	$0.305081\pi$
$374$	0	0
$375$	2910.81	0.400836
$376$	0	0
$377$	2539.18	0.346882
$378$	0	0
$379$	−6875.50	−0.931848	−0.465924	−	0.884825i	$-0.654278\pi$
$379$	−6875.50	−0.931848	−0.465924	+	0.884825i	$0.654278\pi$
$380$	0	0
$381$	4922.06	0.661850
$382$	0	0
$383$	−4826.61	−0.643938	−0.321969	−	0.946750i	$-0.604345\pi$
$383$	−4826.61	−0.643938	−0.321969	+	0.946750i	$0.604345\pi$
$384$	0	0
$385$	−16207.6	−2.14550
$386$	0	0
$387$	−3487.38	−0.458072
$388$	0	0
$389$	−4970.57	−0.647861	−0.323930	−	0.946081i	$-0.605004\pi$
$389$	−4970.57	−0.647861	−0.323930	+	0.946081i	$0.605004\pi$
$390$	0	0
$391$	2044.21	0.264400
$392$	0	0
$393$	1622.07	0.208200
$394$	0	0
$395$	18082.7	2.30339
$396$	0	0
$397$	12288.8	1.55355	0.776774	−	0.629779i	$-0.216855\pi$
$397$	12288.8	1.55355	0.776774	+	0.629779i	$0.216855\pi$
$398$	0	0
$399$	−2082.48	−0.261289
$400$	0	0
$401$	−11971.7	−1.49086	−0.745432	−	0.666581i	$-0.767757\pi$
$401$	−11971.7	−1.49086	−0.745432	+	0.666581i	$0.767757\pi$
$402$	0	0
$403$	7899.66	0.976452
$404$	0	0
$405$	−8531.38	−1.04673
$406$	0	0
$407$	583.365	0.0710475
$408$	0	0
$409$	11147.7	1.34772	0.673861	−	0.738858i	$-0.264635\pi$
$409$	11147.7	1.34772	0.673861	+	0.738858i	$0.264635\pi$
$410$	0	0
$411$	−3804.79	−0.456634
$412$	0	0
$413$	−5687.03	−0.677580
$414$	0	0
$415$	10897.4	1.28899
$416$	0	0
$417$	1115.42	0.130989
$418$	0	0
$419$	−11557.3	−1.34752	−0.673762	−	0.738949i	$-0.735323\pi$
$419$	−11557.3	−1.34752	−0.673762	+	0.738949i	$0.735323\pi$
$420$	0	0
$421$	12874.4	1.49040	0.745201	−	0.666840i	$-0.232353\pi$
$421$	12874.4	1.49040	0.745201	+	0.666840i	$0.232353\pi$
$422$	0	0
$423$	−497.922	−0.0572336
$424$	0	0
$425$	3252.97	0.371275
$426$	0	0
$427$	1416.34	0.160519
$428$	0	0
$429$	8463.24	0.952469
$430$	0	0
$431$	−4088.31	−0.456907	−0.228454	−	0.973555i	$-0.573367\pi$
$431$	−4088.31	−0.456907	−0.228454	+	0.973555i	$0.573367\pi$
$432$	0	0
$433$	−3865.90	−0.429060	−0.214530	−	0.976717i	$-0.568822\pi$
$433$	−3865.90	−0.429060	−0.214530	+	0.976717i	$0.568822\pi$
$434$	0	0
$435$	980.040	0.108021
$436$	0	0
$437$	−8891.56	−0.973320
$438$	0	0
$439$	10662.4	1.15920	0.579600	−	0.814901i	$-0.303209\pi$
$439$	10662.4	1.15920	0.579600	+	0.814901i	$0.303209\pi$
$440$	0	0
$441$	1387.78	0.149852
$442$	0	0
$443$	−10288.9	−1.10347	−0.551736	−	0.834019i	$-0.686034\pi$
$443$	−10288.9	−1.10347	−0.551736	+	0.834019i	$0.686034\pi$
$444$	0	0
$445$	−25429.8	−2.70897
$446$	0	0
$447$	−1429.13	−0.151221
$448$	0	0
$449$	12426.2	1.30608	0.653041	−	0.757323i	$-0.273493\pi$
$449$	12426.2	1.30608	0.653041	+	0.757323i	$0.273493\pi$
$450$	0	0
$451$	987.901	0.103145
$452$	0	0
$453$	−788.784	−0.0818108
$454$	0	0
$455$	27062.3	2.78835
$456$	0	0
$457$	−10657.8	−1.09092	−0.545462	−	0.838136i	$-0.683646\pi$
$457$	−10657.8	−1.09092	−0.545462	+	0.838136i	$0.683646\pi$
$458$	0	0
$459$	1437.09	0.146139
$460$	0	0
$461$	−9819.21	−0.992031	−0.496016	−	0.868314i	$-0.665204\pi$
$461$	−9819.21	−0.992031	−0.496016	+	0.868314i	$0.665204\pi$
$462$	0	0
$463$	−19210.5	−1.92827	−0.964135	−	0.265412i	$-0.914492\pi$
$463$	−19210.5	−1.92827	−0.964135	+	0.265412i	$0.914492\pi$
$464$	0	0
$465$	3049.01	0.304074
$466$	0	0
$467$	345.566	0.0342417	0.0171208	−	0.999853i	$-0.494550\pi$
$467$	345.566	0.0342417	0.0171208	+	0.999853i	$0.494550\pi$
$468$	0	0
$469$	5577.08	0.549095
$470$	0	0
$471$	−3098.41	−0.303115
$472$	0	0
$473$	−7748.10	−0.753188
$474$	0	0
$475$	−14149.2	−1.36676
$476$	0	0
$477$	−6855.39	−0.658043
$478$	0	0
$479$	−253.709	−0.0242009	−0.0121005	−	0.999927i	$-0.503852\pi$
$479$	−253.709	−0.0242009	−0.0121005	+	0.999927i	$0.503852\pi$
$480$	0	0
$481$	−974.060	−0.0923354
$482$	0	0
$483$	4122.93	0.388405
$484$	0	0
$485$	6123.58	0.573314
$486$	0	0
$487$	−13255.1	−1.23336	−0.616680	−	0.787214i	$-0.711523\pi$
$487$	−13255.1	−1.23336	−0.616680	+	0.787214i	$0.711523\pi$
$488$	0	0
$489$	3757.86	0.347518
$490$	0	0
$491$	6454.57	0.593260	0.296630	−	0.954993i	$-0.404137\pi$
$491$	6454.57	0.593260	0.296630	+	0.954993i	$0.404137\pi$
$492$	0	0
$493$	446.809	0.0408180
$494$	0	0
$495$	−22691.4	−2.06041
$496$	0	0
$497$	−8301.36	−0.749229
$498$	0	0
$499$	−8090.41	−0.725805	−0.362902	−	0.931827i	$-0.618214\pi$
$499$	−8090.41	−0.725805	−0.362902	+	0.931827i	$0.618214\pi$
$500$	0	0
$501$	−4687.67	−0.418023
$502$	0	0
$503$	−18897.4	−1.67513	−0.837567	−	0.546334i	$-0.816023\pi$
$503$	−18897.4	−1.67513	−0.837567	+	0.546334i	$0.816023\pi$
$504$	0	0
$505$	−4508.66	−0.397293
$506$	0	0
$507$	−10081.6	−0.883117
$508$	0	0
$509$	−4265.15	−0.371413	−0.185707	−	0.982605i	$-0.559457\pi$
$509$	−4265.15	−0.371413	−0.185707	+	0.982605i	$0.559457\pi$
$510$	0	0
$511$	5850.72	0.506498
$512$	0	0
$513$	−6250.82	−0.537974
$514$	0	0
$515$	9740.77	0.833455
$516$	0	0
$517$	−1106.26	−0.0941068
$518$	0	0
$519$	−564.100	−0.0477096
$520$	0	0
$521$	−3324.96	−0.279595	−0.139798	−	0.990180i	$-0.544645\pi$
$521$	−3324.96	−0.279595	−0.139798	+	0.990180i	$0.544645\pi$
$522$	0	0
$523$	−13017.6	−1.08838	−0.544188	−	0.838964i	$-0.683162\pi$
$523$	−13017.6	−1.08838	−0.544188	+	0.838964i	$0.683162\pi$
$524$	0	0
$525$	6560.84	0.545407
$526$	0	0
$527$	1390.07	0.114900
$528$	0	0
$529$	5436.69	0.446839
$530$	0	0
$531$	−7962.09	−0.650707
$532$	0	0
$533$	−1649.52	−0.134050
$534$	0	0
$535$	−7865.80	−0.635641
$536$	0	0
$537$	−882.555	−0.0709219
$538$	0	0
$539$	3083.29	0.246395
$540$	0	0
$541$	17906.8	1.42305	0.711527	−	0.702658i	$-0.248004\pi$
$541$	17906.8	1.42305	0.711527	+	0.702658i	$0.248004\pi$
$542$	0	0
$543$	−881.885	−0.0696967
$544$	0	0
$545$	−17733.7	−1.39382
$546$	0	0
$547$	−1612.94	−0.126078	−0.0630389	−	0.998011i	$-0.520079\pi$
$547$	−1612.94	−0.126078	−0.0630389	+	0.998011i	$0.520079\pi$
$548$	0	0
$549$	1982.94	0.154153
$550$	0	0
$551$	−1943.45	−0.150261
$552$	0	0
$553$	16627.3	1.27860
$554$	0	0
$555$	−375.955	−0.0287539
$556$	0	0
$557$	7803.94	0.593651	0.296826	−	0.954932i	$-0.404072\pi$
$557$	7803.94	0.593651	0.296826	+	0.954932i	$0.404072\pi$
$558$	0	0
$559$	12937.2	0.978865
$560$	0	0
$561$	1489.24	0.112078
$562$	0	0
$563$	12329.5	0.922958	0.461479	−	0.887151i	$-0.347319\pi$
$563$	12329.5	0.922958	0.461479	+	0.887151i	$0.347319\pi$
$564$	0	0
$565$	31263.5	2.32790
$566$	0	0
$567$	−7844.72	−0.581035
$568$	0	0
$569$	1554.46	0.114528	0.0572640	−	0.998359i	$-0.481762\pi$
$569$	1554.46	0.114528	0.0572640	+	0.998359i	$0.481762\pi$
$570$	0	0
$571$	−15951.8	−1.16911	−0.584556	−	0.811353i	$-0.698731\pi$
$571$	−15951.8	−1.16911	−0.584556	+	0.811353i	$0.698731\pi$
$572$	0	0
$573$	−1535.64	−0.111959
$574$	0	0
$575$	28012.9	2.03168
$576$	0	0
$577$	10491.7	0.756975	0.378488	−	0.925606i	$-0.376444\pi$
$577$	10491.7	0.756975	0.378488	+	0.925606i	$0.376444\pi$
$578$	0	0
$579$	2672.54	0.191825
$580$	0	0
$581$	10020.3	0.715509
$582$	0	0
$583$	−15231.0	−1.08199
$584$	0	0
$585$	37888.4	2.67776
$586$	0	0
$587$	−6437.47	−0.452645	−0.226323	−	0.974052i	$-0.572670\pi$
$587$	−6437.47	−0.452645	−0.226323	+	0.974052i	$0.572670\pi$
$588$	0	0
$589$	−6046.29	−0.422976
$590$	0	0
$591$	−3674.15	−0.255726
$592$	0	0
$593$	11240.6	0.778411	0.389205	−	0.921151i	$-0.372750\pi$
$593$	11240.6	0.778411	0.389205	+	0.921151i	$0.372750\pi$
$594$	0	0
$595$	4762.04	0.328108
$596$	0	0
$597$	656.869	0.0450316
$598$	0	0
$599$	15903.5	1.08481	0.542405	−	0.840117i	$-0.317514\pi$
$599$	15903.5	1.08481	0.542405	+	0.840117i	$0.317514\pi$
$600$	0	0
$601$	117.190	0.00795385	0.00397692	−	0.999992i	$-0.498734\pi$
$601$	117.190	0.00795385	0.00397692	+	0.999992i	$0.498734\pi$
$602$	0	0
$603$	7808.16	0.527318
$604$	0	0
$605$	−26012.1	−1.74801
$606$	0	0
$607$	−22047.8	−1.47429	−0.737143	−	0.675737i	$-0.763825\pi$
$607$	−22047.8	−1.47429	−0.737143	+	0.675737i	$0.763825\pi$
$608$	0	0
$609$	901.160	0.0599620
$610$	0	0
$611$	1847.15	0.122304
$612$	0	0
$613$	12719.2	0.838050	0.419025	−	0.907975i	$-0.362372\pi$
$613$	12719.2	0.838050	0.419025	+	0.907975i	$0.362372\pi$
$614$	0	0
$615$	−636.662	−0.0417442
$616$	0	0
$617$	12736.9	0.831064	0.415532	−	0.909578i	$-0.363595\pi$
$617$	12736.9	0.831064	0.415532	+	0.909578i	$0.363595\pi$
$618$	0	0
$619$	−28083.4	−1.82353	−0.911767	−	0.410709i	$-0.865281\pi$
$619$	−28083.4	−1.82353	−0.911767	+	0.410709i	$0.865281\pi$
$620$	0	0
$621$	12375.5	0.799697
$622$	0	0
$623$	−23383.1	−1.50373
$624$	0	0
$625$	2560.44	0.163868
$626$	0	0
$627$	−6477.64	−0.412587
$628$	0	0
$629$	−171.401	−0.0108652
$630$	0	0
$631$	281.496	0.0177594	0.00887969	−	0.999961i	$-0.497173\pi$
$631$	281.496	0.0177594	0.00887969	+	0.999961i	$0.497173\pi$
$632$	0	0
$633$	7615.79	0.478200
$634$	0	0
$635$	48956.7	3.05951
$636$	0	0
$637$	−5148.26	−0.320222
$638$	0	0
$639$	−11622.3	−0.719514
$640$	0	0
$641$	−8440.98	−0.520123	−0.260061	−	0.965592i	$-0.583743\pi$
$641$	−8440.98	−0.520123	−0.260061	+	0.965592i	$0.583743\pi$
$642$	0	0
$643$	1173.61	0.0719792	0.0359896	−	0.999352i	$-0.488542\pi$
$643$	1173.61	0.0719792	0.0359896	+	0.999352i	$0.488542\pi$
$644$	0	0
$645$	4993.34	0.304825
$646$	0	0
$647$	−10845.7	−0.659025	−0.329513	−	0.944151i	$-0.606884\pi$
$647$	−10845.7	−0.659025	−0.329513	+	0.944151i	$0.606884\pi$
$648$	0	0
$649$	−17689.8	−1.06993
$650$	0	0
$651$	2803.60	0.168789
$652$	0	0
$653$	5282.40	0.316564	0.158282	−	0.987394i	$-0.449404\pi$
$653$	5282.40	0.316564	0.158282	+	0.987394i	$0.449404\pi$
$654$	0	0
$655$	16133.7	0.962437
$656$	0	0
$657$	8191.26	0.486410
$658$	0	0
$659$	19243.7	1.13752	0.568761	−	0.822503i	$-0.307423\pi$
$659$	19243.7	1.13752	0.568761	+	0.822503i	$0.307423\pi$
$660$	0	0
$661$	−29196.9	−1.71804	−0.859021	−	0.511940i	$-0.828927\pi$
$661$	−29196.9	−1.71804	−0.859021	+	0.511940i	$0.828927\pi$
$662$	0	0
$663$	−2486.62	−0.145660
$664$	0	0
$665$	−20713.1	−1.20785
$666$	0	0
$667$	3847.69	0.223363
$668$	0	0
$669$	−2455.84	−0.141926
$670$	0	0
$671$	4405.61	0.253467
$672$	0	0
$673$	19924.5	1.14121	0.570605	−	0.821224i	$-0.306709\pi$
$673$	19924.5	1.14121	0.570605	+	0.821224i	$0.306709\pi$
$674$	0	0
$675$	19693.2	1.12295
$676$	0	0
$677$	4980.43	0.282738	0.141369	−	0.989957i	$-0.454850\pi$
$677$	4980.43	0.282738	0.141369	+	0.989957i	$0.454850\pi$
$678$	0	0
$679$	5630.71	0.318243
$680$	0	0
$681$	−2450.32	−0.137880
$682$	0	0
$683$	−29295.8	−1.64125	−0.820624	−	0.571468i	$-0.806374\pi$
$683$	−29295.8	−1.64125	−0.820624	+	0.571468i	$0.806374\pi$
$684$	0	0
$685$	−37843.9	−2.11086
$686$	0	0
$687$	10055.9	0.558455
$688$	0	0
$689$	25431.5	1.40619
$690$	0	0
$691$	32759.8	1.80353	0.901766	−	0.432225i	$-0.142271\pi$
$691$	32759.8	1.80353	0.901766	+	0.432225i	$0.142271\pi$
$692$	0	0
$693$	−20865.0	−1.14372
$694$	0	0
$695$	11094.4	0.605518
$696$	0	0
$697$	−290.260	−0.0157739
$698$	0	0
$699$	1089.67	0.0589629
$700$	0	0
$701$	−27958.5	−1.50639	−0.753195	−	0.657797i	$-0.771488\pi$
$701$	−27958.5	−1.50639	−0.753195	+	0.657797i	$0.771488\pi$
$702$	0	0
$703$	745.532	0.0399975
$704$	0	0
$705$	712.939	0.0380863
$706$	0	0
$707$	−4145.77	−0.220534
$708$	0	0
$709$	31863.5	1.68781	0.843906	−	0.536492i	$-0.180251\pi$
$709$	31863.5	1.68781	0.843906	+	0.536492i	$0.180251\pi$
$710$	0	0
$711$	23278.9	1.22789
$712$	0	0
$713$	11970.6	0.628753
$714$	0	0
$715$	84178.6	4.40294
$716$	0	0
$717$	−12803.4	−0.666879
$718$	0	0
$719$	7944.76	0.412085	0.206043	−	0.978543i	$-0.433941\pi$
$719$	7944.76	0.412085	0.206043	+	0.978543i	$0.433941\pi$
$720$	0	0
$721$	8956.76	0.462645
$722$	0	0
$723$	−13097.6	−0.673729
$724$	0	0
$725$	6122.85	0.313651
$726$	0	0
$727$	28640.3	1.46109	0.730543	−	0.682866i	$-0.239267\pi$
$727$	28640.3	1.46109	0.730543	+	0.682866i	$0.239267\pi$
$728$	0	0
$729$	−6340.84	−0.322148
$730$	0	0
$731$	2276.51	0.115184
$732$	0	0
$733$	−11852.7	−0.597258	−0.298629	−	0.954369i	$-0.596529\pi$
$733$	−11852.7	−0.597258	−0.298629	+	0.954369i	$0.596529\pi$
$734$	0	0
$735$	−1987.06	−0.0997194
$736$	0	0
$737$	17347.8	0.867047
$738$	0	0
$739$	24052.5	1.19727	0.598636	−	0.801021i	$-0.295710\pi$
$739$	24052.5	1.19727	0.598636	+	0.801021i	$0.295710\pi$
$740$	0	0
$741$	10815.9	0.536210
$742$	0	0
$743$	−12530.4	−0.618704	−0.309352	−	0.950948i	$-0.600112\pi$
$743$	−12530.4	−0.618704	−0.309352	+	0.950948i	$0.600112\pi$
$744$	0	0
$745$	−14214.7	−0.699042
$746$	0	0
$747$	14028.8	0.687132
$748$	0	0
$749$	−7232.71	−0.352840
$750$	0	0
$751$	30921.6	1.50246	0.751229	−	0.660042i	$-0.229461\pi$
$751$	30921.6	1.50246	0.751229	+	0.660042i	$0.229461\pi$
$752$	0	0
$753$	−8985.11	−0.434841
$754$	0	0
$755$	−7845.54	−0.378184
$756$	0	0
$757$	6257.40	0.300435	0.150217	−	0.988653i	$-0.452003\pi$
$757$	6257.40	0.300435	0.150217	+	0.988653i	$0.452003\pi$
$758$	0	0
$759$	12824.6	0.613310
$760$	0	0
$761$	−20094.8	−0.957211	−0.478605	−	0.878030i	$-0.658858\pi$
$761$	−20094.8	−0.957211	−0.478605	+	0.878030i	$0.658858\pi$
$762$	0	0
$763$	−16306.4	−0.773697
$764$	0	0
$765$	6667.06	0.315095
$766$	0	0
$767$	29537.1	1.39051
$768$	0	0
$769$	−26647.8	−1.24960	−0.624801	−	0.780784i	$-0.714820\pi$
$769$	−26647.8	−1.24960	−0.624801	+	0.780784i	$0.714820\pi$
$770$	0	0
$771$	−4586.48	−0.214239
$772$	0	0
$773$	18513.2	0.861414	0.430707	−	0.902492i	$-0.358264\pi$
$773$	18513.2	0.861414	0.430707	+	0.902492i	$0.358264\pi$
$774$	0	0
$775$	19048.8	0.882909
$776$	0	0
$777$	−345.696	−0.0159611
$778$	0	0
$779$	1262.52	0.0580675
$780$	0	0
$781$	−25821.8	−1.18307
$782$	0	0
$783$	2704.95	0.123457
$784$	0	0
$785$	−30818.0	−1.40120
$786$	0	0
$787$	29524.5	1.33727	0.668636	−	0.743590i	$-0.266878\pi$
$787$	29524.5	1.33727	0.668636	+	0.743590i	$0.266878\pi$
$788$	0	0
$789$	5184.25	0.233921
$790$	0	0
$791$	28747.2	1.29220
$792$	0	0
$793$	−7356.16	−0.329413
$794$	0	0
$795$	9815.74	0.437897
$796$	0	0
$797$	−38789.4	−1.72395	−0.861976	−	0.506949i	$-0.830773\pi$
$797$	−38789.4	−1.72395	−0.861976	+	0.506949i	$0.830773\pi$
$798$	0	0
$799$	325.035	0.0143916
$800$	0	0
$801$	−32737.3	−1.44409
$802$	0	0
$803$	18199.0	0.799785
$804$	0	0
$805$	41008.2	1.79546
$806$	0	0
$807$	−10238.7	−0.446617
$808$	0	0
$809$	−11552.0	−0.502037	−0.251018	−	0.967982i	$-0.580765\pi$
$809$	−11552.0	−0.502037	−0.251018	+	0.967982i	$0.580765\pi$
$810$	0	0
$811$	−26939.1	−1.16641	−0.583205	−	0.812325i	$-0.698202\pi$
$811$	−26939.1	−1.16641	−0.583205	+	0.812325i	$0.698202\pi$
$812$	0	0
$813$	5840.25	0.251939
$814$	0	0
$815$	37377.1	1.60646
$816$	0	0
$817$	−9901.96	−0.424022
$818$	0	0
$819$	34838.9	1.48641
$820$	0	0
$821$	7558.67	0.321315	0.160657	−	0.987010i	$-0.448639\pi$
$821$	7558.67	0.321315	0.160657	+	0.987010i	$0.448639\pi$
$822$	0	0
$823$	−3201.90	−0.135615	−0.0678076	−	0.997698i	$-0.521600\pi$
$823$	−3201.90	−0.135615	−0.0678076	+	0.997698i	$0.521600\pi$
$824$	0	0
$825$	20407.8	0.861223
$826$	0	0
$827$	11479.1	0.482670	0.241335	−	0.970442i	$-0.422415\pi$
$827$	11479.1	0.482670	0.241335	+	0.970442i	$0.422415\pi$
$828$	0	0
$829$	1667.94	0.0698794	0.0349397	−	0.999389i	$-0.488876\pi$
$829$	1667.94	0.0698794	0.0349397	+	0.999389i	$0.488876\pi$
$830$	0	0
$831$	7309.78	0.305142
$832$	0	0
$833$	−905.917	−0.0376809
$834$	0	0
$835$	−46625.3	−1.93238
$836$	0	0
$837$	8415.38	0.347525
$838$	0	0
$839$	15210.3	0.625884	0.312942	−	0.949772i	$-0.398685\pi$
$839$	15210.3	0.625884	0.312942	+	0.949772i	$0.398685\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−3050.92	−0.124649
$844$	0	0
$845$	−100276.	−4.08235
$846$	0	0
$847$	−23918.5	−0.970306
$848$	0	0
$849$	14351.9	0.580161
$850$	0	0
$851$	−1476.02	−0.0594563
$852$	0	0
$853$	18281.2	0.733805	0.366902	−	0.930259i	$-0.380418\pi$
$853$	18281.2	0.733805	0.366902	+	0.930259i	$0.380418\pi$
$854$	0	0
$855$	−28999.2	−1.15994
$856$	0	0
$857$	585.415	0.0233342	0.0116671	−	0.999932i	$-0.496286\pi$
$857$	585.415	0.0233342	0.0116671	+	0.999932i	$0.496286\pi$
$858$	0	0
$859$	935.611	0.0371626	0.0185813	−	0.999827i	$-0.494085\pi$
$859$	935.611	0.0371626	0.0185813	+	0.999827i	$0.494085\pi$
$860$	0	0
$861$	−585.419	−0.0231719
$862$	0	0
$863$	−12110.7	−0.477696	−0.238848	−	0.971057i	$-0.576770\pi$
$863$	−12110.7	−0.477696	−0.238848	+	0.971057i	$0.576770\pi$
$864$	0	0
$865$	−5610.75	−0.220545
$866$	0	0
$867$	8618.46	0.337599
$868$	0	0
$869$	51720.0	2.01896
$870$	0	0
$871$	−28966.1	−1.12684
$872$	0	0
$873$	7883.24	0.305621
$874$	0	0
$875$	26621.8	1.02855
$876$	0	0
$877$	−5841.41	−0.224915	−0.112458	−	0.993657i	$-0.535872\pi$
$877$	−5841.41	−0.224915	−0.112458	+	0.993657i	$0.535872\pi$
$878$	0	0
$879$	14913.6	0.572267
$880$	0	0
$881$	−47826.2	−1.82895	−0.914476	−	0.404641i	$-0.867397\pi$
$881$	−47826.2	−1.82895	−0.914476	+	0.404641i	$0.867397\pi$
$882$	0	0
$883$	−17337.5	−0.660761	−0.330381	−	0.943848i	$-0.607177\pi$
$883$	−17337.5	−0.660761	−0.330381	+	0.943848i	$0.607177\pi$
$884$	0	0
$885$	11400.4	0.433015
$886$	0	0
$887$	35181.5	1.33177	0.665885	−	0.746055i	$-0.268054\pi$
$887$	35181.5	1.33177	0.665885	+	0.746055i	$0.268054\pi$
$888$	0	0
$889$	45016.3	1.69831
$890$	0	0
$891$	−24401.4	−0.917482
$892$	0	0
$893$	−1413.78	−0.0529792
$894$	0	0
$895$	−8778.22	−0.327848
$896$	0	0
$897$	−21413.5	−0.797075
$898$	0	0
$899$	2616.44	0.0970669
$900$	0	0
$901$	4475.08	0.165468
$902$	0	0
$903$	4591.44	0.169207
$904$	0	0
$905$	−8771.56	−0.322184
$906$	0	0
$907$	28184.4	1.03180	0.515902	−	0.856648i	$-0.327457\pi$
$907$	28184.4	1.03180	0.515902	+	0.856648i	$0.327457\pi$
$908$	0	0
$909$	−5804.26	−0.211788
$910$	0	0
$911$	−34136.6	−1.24149	−0.620745	−	0.784013i	$-0.713170\pi$
$911$	−34136.6	−1.24149	−0.620745	+	0.784013i	$0.713170\pi$
$912$	0	0
$913$	31168.5	1.12982
$914$	0	0
$915$	−2839.23	−0.102582
$916$	0	0
$917$	14835.2	0.534242
$918$	0	0
$919$	15512.0	0.556794	0.278397	−	0.960466i	$-0.410197\pi$
$919$	15512.0	0.556794	0.278397	+	0.960466i	$0.410197\pi$
$920$	0	0
$921$	−11298.3	−0.404227
$922$	0	0
$923$	43115.3	1.53755
$924$	0	0
$925$	−2348.80	−0.0834897
$926$	0	0
$927$	12539.9	0.444297
$928$	0	0
$929$	−3100.72	−0.109506	−0.0547531	−	0.998500i	$-0.517437\pi$
$929$	−3100.72	−0.109506	−0.0547531	+	0.998500i	$0.517437\pi$
$930$	0	0
$931$	3940.40	0.138713
$932$	0	0
$933$	15055.8	0.528299
$934$	0	0
$935$	14812.5	0.518098
$936$	0	0
$937$	−19638.8	−0.684708	−0.342354	−	0.939571i	$-0.611224\pi$
$937$	−19638.8	−0.684708	−0.342354	+	0.939571i	$0.611224\pi$
$938$	0	0
$939$	−3460.29	−0.120258
$940$	0	0
$941$	50033.6	1.73332	0.866658	−	0.498903i	$-0.166264\pi$
$941$	50033.6	1.73332	0.866658	+	0.498903i	$0.166264\pi$
$942$	0	0
$943$	−2499.57	−0.0863172
$944$	0	0
$945$	28829.0	0.992389
$946$	0	0
$947$	−19758.4	−0.677994	−0.338997	−	0.940787i	$-0.610088\pi$
$947$	−19758.4	−0.677994	−0.338997	+	0.940787i	$0.610088\pi$
$948$	0	0
$949$	−30387.3	−1.03942
$950$	0	0
$951$	2254.20	0.0768636
$952$	0	0
$953$	33843.4	1.15036	0.575180	−	0.818027i	$-0.304932\pi$
$953$	33843.4	1.15036	0.575180	+	0.818027i	$0.304932\pi$
$954$	0	0
$955$	−15274.1	−0.517548
$956$	0	0
$957$	2803.10	0.0946828
$958$	0	0
$959$	−34798.0	−1.17173
$960$	0	0
$961$	−21651.0	−0.726762
$962$	0	0
$963$	−10126.1	−0.338846
$964$	0	0
$965$	26582.1	0.886743
$966$	0	0
$967$	35236.9	1.17181	0.585906	−	0.810379i	$-0.300739\pi$
$967$	35236.9	1.17181	0.585906	+	0.810379i	$0.300739\pi$
$968$	0	0
$969$	1903.23	0.0630964
$970$	0	0
$971$	4505.40	0.148903	0.0744517	−	0.997225i	$-0.476279\pi$
$971$	4505.40	0.148903	0.0744517	+	0.997225i	$0.476279\pi$
$972$	0	0
$973$	10201.5	0.336119
$974$	0	0
$975$	−34075.5	−1.11927
$976$	0	0
$977$	−15167.4	−0.496671	−0.248336	−	0.968674i	$-0.579884\pi$
$977$	−15167.4	−0.496671	−0.248336	+	0.968674i	$0.579884\pi$
$978$	0	0
$979$	−72734.2	−2.37446
$980$	0	0
$981$	−22829.7	−0.743012
$982$	0	0
$983$	25342.5	0.822278	0.411139	−	0.911573i	$-0.365131\pi$
$983$	25342.5	0.822278	0.411139	+	0.911573i	$0.365131\pi$
$984$	0	0
$985$	−36544.4	−1.18213
$986$	0	0
$987$	655.557	0.0211414
$988$	0	0
$989$	19604.1	0.630307
$990$	0	0
$991$	40576.1	1.30065	0.650324	−	0.759657i	$-0.274633\pi$
$991$	40576.1	1.30065	0.650324	+	0.759657i	$0.274633\pi$
$992$	0	0
$993$	6947.37	0.222022
$994$	0	0
$995$	6533.46	0.208166
$996$	0	0
$997$	−7442.49	−0.236415	−0.118208	−	0.992989i	$-0.537715\pi$
$997$	−7442.49	−0.236415	−0.118208	+	0.992989i	$0.537715\pi$
$998$	0	0
$999$	−1037.65	−0.0328627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.y.1.3		5
4.3	odd	2		1856.4.a.bb.1.3		5
8.3	odd	2		464.4.a.l.1.3		5
8.5	even	2		29.4.a.b.1.4	✓	5
24.5	odd	2		261.4.a.f.1.2		5
40.29	even	2		725.4.a.c.1.2		5
56.13	odd	2		1421.4.a.e.1.4		5
232.173	even	2		841.4.a.b.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.4.a.b.1.4	✓	5	8.5	even	2
261.4.a.f.1.2		5	24.5	odd	2
464.4.a.l.1.3		5	8.3	odd	2
725.4.a.c.1.2		5	40.29	even	2
841.4.a.b.1.2		5	232.173	even	2
1421.4.a.e.1.4		5	56.13	odd	2
1856.4.a.y.1.3		5	1.1	even	1	trivial
1856.4.a.bb.1.3		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q-1.84328 q^{3} -18.3339 q^{5} -16.8583 q^{7} -23.6023 q^{9} +O(q^{10})\)	\(q-1.84328 q^{3} -18.3339 q^{5} -16.8583 q^{7} -23.6023 q^{9} -52.4385 q^{11} +87.5580 q^{13} +33.7945 q^{15} +15.4072 q^{17} -67.0156 q^{19} +31.0745 q^{21} +132.679 q^{23} +211.133 q^{25} +93.2741 q^{27} +29.0000 q^{29} +90.2221 q^{31} +96.6587 q^{33} +309.078 q^{35} -11.1247 q^{37} -161.394 q^{39} -18.8392 q^{41} +147.756 q^{43} +432.723 q^{45} +21.0963 q^{47} -58.7983 q^{49} -28.3997 q^{51} +290.454 q^{53} +961.404 q^{55} +123.528 q^{57} +337.343 q^{59} -84.0147 q^{61} +397.895 q^{63} -1605.28 q^{65} -330.821 q^{67} -244.564 q^{69} +492.420 q^{71} -347.053 q^{73} -389.176 q^{75} +884.023 q^{77} -986.297 q^{79} +465.333 q^{81} -594.382 q^{83} -282.475 q^{85} -53.4550 q^{87} +1387.04 q^{89} -1476.08 q^{91} -166.304 q^{93} +1228.66 q^{95} -334.003 q^{97} +1237.67 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9}+O(q^{10}) \) 5 * q - 8 * q^3 - 10 * q^5 + 40 * q^7 + 33 * q^9	\( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9} - 12 q^{11} - 14 q^{13} - 74 q^{15} + 66 q^{17} - 214 q^{19} + 164 q^{23} + 207 q^{25} - 362 q^{27} + 145 q^{29} + 420 q^{31} - 576 q^{33} + 52 q^{35} - 378 q^{37} - 374 q^{39} - 1158 q^{41} + 204 q^{43} + 1506 q^{45} + 248 q^{47} - 283 q^{49} - 228 q^{51} + 554 q^{53} + 546 q^{55} + 44 q^{57} - 440 q^{59} - 618 q^{61} + 804 q^{63} - 1656 q^{65} - 1164 q^{67} + 1968 q^{69} - 692 q^{71} - 1950 q^{73} - 3074 q^{75} + 1616 q^{77} + 272 q^{79} + 1801 q^{81} - 512 q^{83} + 1628 q^{85} - 232 q^{87} + 866 q^{89} - 2580 q^{91} + 40 q^{93} + 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100}) \) 5 * q - 8 * q^3 - 10 * q^5 + 40 * q^7 + 33 * q^9 - 12 * q^11 - 14 * q^13 - 74 * q^15 + 66 * q^17 - 214 * q^19 + 164 * q^23 + 207 * q^25 - 362 * q^27 + 145 * q^29 + 420 * q^31 - 576 * q^33 + 52 * q^35 - 378 * q^37 - 374 * q^39 - 1158 * q^41 + 204 * q^43 + 1506 * q^45 + 248 * q^47 - 283 * q^49 - 228 * q^51 + 554 * q^53 + 546 * q^55 + 44 * q^57 - 440 * q^59 - 618 * q^61 + 804 * q^63 - 1656 * q^65 - 1164 * q^67 + 1968 * q^69 - 692 * q^71 - 1950 * q^73 - 3074 * q^75 + 1616 * q^77 + 272 * q^79 + 1801 * q^81 - 512 * q^83 + 1628 * q^85 - 232 * q^87 + 866 * q^89 - 2580 * q^91 + 40 * q^93 + 2244 * q^95 + 1562 * q^97 + 238 * q^99

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