LMFDB - Embedded newform 928.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [928,4,Mod(1,928)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(928, 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("928.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 928 = 2^{5} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	928.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:	$54.7537724853$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	928.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-7.00000 q^{3} -13.0000 q^{5} +16.0000 q^{7} +22.0000 q^{9} +O(q^{10})$

$q-7.00000 q^{3} -13.0000 q^{5} +16.0000 q^{7} +22.0000 q^{9} -45.0000 q^{11} +61.0000 q^{13} +91.0000 q^{15} -102.000 q^{17} -68.0000 q^{19} -112.000 q^{21} +194.000 q^{23} +44.0000 q^{25} +35.0000 q^{27} -29.0000 q^{29} +149.000 q^{31} +315.000 q^{33} -208.000 q^{35} +400.000 q^{37} -427.000 q^{39} +280.000 q^{41} +263.000 q^{43} -286.000 q^{45} +509.000 q^{47} -87.0000 q^{49} +714.000 q^{51} -605.000 q^{53} +585.000 q^{55} +476.000 q^{57} -578.000 q^{59} -718.000 q^{61} +352.000 q^{63} -793.000 q^{65} -260.000 q^{67} -1358.00 q^{69} +738.000 q^{71} +652.000 q^{73} -308.000 q^{75} -720.000 q^{77} -917.000 q^{79} -839.000 q^{81} +678.000 q^{83} +1326.00 q^{85} +203.000 q^{87} -1008.00 q^{89} +976.000 q^{91} -1043.00 q^{93} +884.000 q^{95} -1764.00 q^{97} -990.000 q^{99} +O(q^{100})$

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
$3$	−7.00000	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−7.00000	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$4$	0	0
$5$	−13.0000	−1.16276	−0.581378	−	0.813634i	$-0.697486\pi$
$5$	−13.0000	−1.16276	−0.581378	+	0.813634i	$0.697486\pi$
$6$	0	0
$7$	16.0000	0.863919	0.431959	−	0.901893i	$-0.357822\pi$
$7$	16.0000	0.863919	0.431959	+	0.901893i	$0.357822\pi$
$8$	0	0
$9$	22.0000	0.814815
$10$	0	0
$11$	−45.0000	−1.23346	−0.616728	−	0.787177i	$-0.711542\pi$
$11$	−45.0000	−1.23346	−0.616728	+	0.787177i	$0.711542\pi$
$12$	0	0
$13$	61.0000	1.30141	0.650706	−	0.759330i	$-0.274473\pi$
$13$	61.0000	1.30141	0.650706	+	0.759330i	$0.274473\pi$
$14$	0	0
$15$	91.0000	1.56641
$16$	0	0
$17$	−102.000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−102.000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	−68.0000	−0.821067	−0.410533	−	0.911846i	$-0.634657\pi$
$19$	−68.0000	−0.821067	−0.410533	+	0.911846i	$0.634657\pi$
$20$	0	0
$21$	−112.000	−1.16383
$22$	0	0
$23$	194.000	1.75877	0.879387	−	0.476108i	$-0.157953\pi$
$23$	194.000	1.75877	0.879387	+	0.476108i	$0.157953\pi$
$24$	0	0
$25$	44.0000	0.352000
$26$	0	0
$27$	35.0000	0.249472
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	149.000	0.863264	0.431632	−	0.902050i	$-0.357938\pi$
$31$	149.000	0.863264	0.431632	+	0.902050i	$0.357938\pi$
$32$	0	0
$33$	315.000	1.66165
$34$	0	0
$35$	−208.000	−1.00453
$36$	0	0
$37$	400.000	1.77729	0.888643	−	0.458599i	$-0.151649\pi$
$37$	400.000	1.77729	0.888643	+	0.458599i	$0.151649\pi$
$38$	0	0
$39$	−427.000	−1.75320
$40$	0	0
$41$	280.000	1.06655	0.533276	−	0.845941i	$-0.320961\pi$
$41$	280.000	1.06655	0.533276	+	0.845941i	$0.320961\pi$
$42$	0	0
$43$	263.000	0.932724	0.466362	−	0.884594i	$-0.345564\pi$
$43$	263.000	0.932724	0.466362	+	0.884594i	$0.345564\pi$
$44$	0	0
$45$	−286.000	−0.947430
$46$	0	0
$47$	509.000	1.57969	0.789843	−	0.613309i	$-0.210162\pi$
$47$	509.000	1.57969	0.789843	+	0.613309i	$0.210162\pi$
$48$	0	0
$49$	−87.0000	−0.253644
$50$	0	0
$51$	714.000	1.96039
$52$	0	0
$53$	−605.000	−1.56798	−0.783992	−	0.620771i	$-0.786820\pi$
$53$	−605.000	−1.56798	−0.783992	+	0.620771i	$0.786820\pi$
$54$	0	0
$55$	585.000	1.43421
$56$	0	0
$57$	476.000	1.10610
$58$	0	0
$59$	−578.000	−1.27541	−0.637705	−	0.770281i	$-0.720116\pi$
$59$	−578.000	−1.27541	−0.637705	+	0.770281i	$0.720116\pi$
$60$	0	0
$61$	−718.000	−1.50706	−0.753529	−	0.657415i	$-0.771650\pi$
$61$	−718.000	−1.50706	−0.753529	+	0.657415i	$0.771650\pi$
$62$	0	0
$63$	352.000	0.703934
$64$	0	0
$65$	−793.000	−1.51322
$66$	0	0
$67$	−260.000	−0.474090	−0.237045	−	0.971499i	$-0.576179\pi$
$67$	−260.000	−0.474090	−0.237045	+	0.971499i	$0.576179\pi$
$68$	0	0
$69$	−1358.00	−2.36933
$70$	0	0
$71$	738.000	1.23358	0.616792	−	0.787126i	$-0.288432\pi$
$71$	738.000	1.23358	0.616792	+	0.787126i	$0.288432\pi$
$72$	0	0
$73$	652.000	1.04535	0.522677	−	0.852531i	$-0.324933\pi$
$73$	652.000	1.04535	0.522677	+	0.852531i	$0.324933\pi$
$74$	0	0
$75$	−308.000	−0.474197
$76$	0	0
$77$	−720.000	−1.06561
$78$	0	0
$79$	−917.000	−1.30596	−0.652978	−	0.757377i	$-0.726481\pi$
$79$	−917.000	−1.30596	−0.652978	+	0.757377i	$0.726481\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	0	0
$83$	678.000	0.896629	0.448314	−	0.893876i	$-0.352025\pi$
$83$	678.000	0.896629	0.448314	+	0.893876i	$0.352025\pi$
$84$	0	0
$85$	1326.00	1.69206
$86$	0	0
$87$	203.000	0.250160
$88$	0	0
$89$	−1008.00	−1.20054	−0.600268	−	0.799799i	$-0.704940\pi$
$89$	−1008.00	−1.20054	−0.600268	+	0.799799i	$0.704940\pi$
$90$	0	0
$91$	976.000	1.12431
$92$	0	0
$93$	−1043.00	−1.16295
$94$	0	0
$95$	884.000	0.954700
$96$	0	0
$97$	−1764.00	−1.84646	−0.923232	−	0.384242i	$-0.874463\pi$
$97$	−1764.00	−1.84646	−0.923232	+	0.384242i	$0.874463\pi$
$98$	0	0
$99$	−990.000	−1.00504
$100$	0	0
$101$	−1360.00	−1.33985	−0.669926	−	0.742428i	$-0.733674\pi$
$101$	−1360.00	−1.33985	−0.669926	+	0.742428i	$0.733674\pi$
$102$	0	0
$103$	1498.00	1.43303	0.716516	−	0.697571i	$-0.245736\pi$
$103$	1498.00	1.43303	0.716516	+	0.697571i	$0.245736\pi$
$104$	0	0
$105$	1456.00	1.35325
$106$	0	0
$107$	−772.000	−0.697496	−0.348748	−	0.937217i	$-0.613393\pi$
$107$	−772.000	−0.697496	−0.348748	+	0.937217i	$0.613393\pi$
$108$	0	0
$109$	763.000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	763.000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	−2800.00	−2.39427
$112$	0	0
$113$	−1218.00	−1.01398	−0.506990	−	0.861952i	$-0.669242\pi$
$113$	−1218.00	−1.01398	−0.506990	+	0.861952i	$0.669242\pi$
$114$	0	0
$115$	−2522.00	−2.04502
$116$	0	0
$117$	1342.00	1.06041
$118$	0	0
$119$	−1632.00	−1.25719
$120$	0	0
$121$	694.000	0.521412
$122$	0	0
$123$	−1960.00	−1.43681
$124$	0	0
$125$	1053.00	0.753465
$126$	0	0
$127$	1544.00	1.07880	0.539401	−	0.842049i	$-0.318651\pi$
$127$	1544.00	1.07880	0.539401	+	0.842049i	$0.318651\pi$
$128$	0	0
$129$	−1841.00	−1.25652
$130$	0	0
$131$	−1916.00	−1.27788	−0.638938	−	0.769258i	$-0.720626\pi$
$131$	−1916.00	−1.27788	−0.638938	+	0.769258i	$0.720626\pi$
$132$	0	0
$133$	−1088.00	−0.709335
$134$	0	0
$135$	−455.000	−0.290075
$136$	0	0
$137$	72.0000	0.0449005	0.0224503	−	0.999748i	$-0.492853\pi$
$137$	72.0000	0.0449005	0.0224503	+	0.999748i	$0.492853\pi$
$138$	0	0
$139$	1394.00	0.850630	0.425315	−	0.905045i	$-0.360163\pi$
$139$	1394.00	0.850630	0.425315	+	0.905045i	$0.360163\pi$
$140$	0	0
$141$	−3563.00	−2.12808
$142$	0	0
$143$	−2745.00	−1.60523
$144$	0	0
$145$	377.000	0.215918
$146$	0	0
$147$	609.000	0.341697
$148$	0	0
$149$	−135.000	−0.0742257	−0.0371129	−	0.999311i	$-0.511816\pi$
$149$	−135.000	−0.0742257	−0.0371129	+	0.999311i	$0.511816\pi$
$150$	0	0
$151$	−1708.00	−0.920497	−0.460249	−	0.887790i	$-0.652240\pi$
$151$	−1708.00	−0.920497	−0.460249	+	0.887790i	$0.652240\pi$
$152$	0	0
$153$	−2244.00	−1.18573
$154$	0	0
$155$	−1937.00	−1.00377
$156$	0	0
$157$	128.000	0.0650670	0.0325335	−	0.999471i	$-0.489642\pi$
$157$	128.000	0.0650670	0.0325335	+	0.999471i	$0.489642\pi$
$158$	0	0
$159$	4235.00	2.11231
$160$	0	0
$161$	3104.00	1.51944
$162$	0	0
$163$	1235.00	0.593452	0.296726	−	0.954963i	$-0.404105\pi$
$163$	1235.00	0.593452	0.296726	+	0.954963i	$0.404105\pi$
$164$	0	0
$165$	−4095.00	−1.93209
$166$	0	0
$167$	−2398.00	−1.11115	−0.555577	−	0.831465i	$-0.687503\pi$
$167$	−2398.00	−1.11115	−0.555577	+	0.831465i	$0.687503\pi$
$168$	0	0
$169$	1524.00	0.693673
$170$	0	0
$171$	−1496.00	−0.669017
$172$	0	0
$173$	−1258.00	−0.552855	−0.276428	−	0.961035i	$-0.589151\pi$
$173$	−1258.00	−0.552855	−0.276428	+	0.961035i	$0.589151\pi$
$174$	0	0
$175$	704.000	0.304099
$176$	0	0
$177$	4046.00	1.71817
$178$	0	0
$179$	154.000	0.0643045	0.0321522	−	0.999483i	$-0.489764\pi$
$179$	154.000	0.0643045	0.0321522	+	0.999483i	$0.489764\pi$
$180$	0	0
$181$	3089.00	1.26853	0.634264	−	0.773117i	$-0.281303\pi$
$181$	3089.00	1.26853	0.634264	+	0.773117i	$0.281303\pi$
$182$	0	0
$183$	5026.00	2.03023
$184$	0	0
$185$	−5200.00	−2.06655
$186$	0	0
$187$	4590.00	1.79494
$188$	0	0
$189$	560.000	0.215524
$190$	0	0
$191$	−1088.00	−0.412172	−0.206086	−	0.978534i	$-0.566073\pi$
$191$	−1088.00	−0.412172	−0.206086	+	0.978534i	$0.566073\pi$
$192$	0	0
$193$	−228.000	−0.0850352	−0.0425176	−	0.999096i	$-0.513538\pi$
$193$	−228.000	−0.0850352	−0.0425176	+	0.999096i	$0.513538\pi$
$194$	0	0
$195$	5551.00	2.03854
$196$	0	0
$197$	2226.00	0.805055	0.402528	−	0.915408i	$-0.368132\pi$
$197$	2226.00	0.805055	0.402528	+	0.915408i	$0.368132\pi$
$198$	0	0
$199$	−1908.00	−0.679671	−0.339836	−	0.940485i	$-0.610371\pi$
$199$	−1908.00	−0.679671	−0.339836	+	0.940485i	$0.610371\pi$
$200$	0	0
$201$	1820.00	0.638671
$202$	0	0
$203$	−464.000	−0.160426
$204$	0	0
$205$	−3640.00	−1.24014
$206$	0	0
$207$	4268.00	1.43307
$208$	0	0
$209$	3060.00	1.01275
$210$	0	0
$211$	583.000	0.190215	0.0951075	−	0.995467i	$-0.469681\pi$
$211$	583.000	0.190215	0.0951075	+	0.995467i	$0.469681\pi$
$212$	0	0
$213$	−5166.00	−1.66182
$214$	0	0
$215$	−3419.00	−1.08453
$216$	0	0
$217$	2384.00	0.745790
$218$	0	0
$219$	−4564.00	−1.40825
$220$	0	0
$221$	−6222.00	−1.89383
$222$	0	0
$223$	−1594.00	−0.478664	−0.239332	−	0.970938i	$-0.576928\pi$
$223$	−1594.00	−0.478664	−0.239332	+	0.970938i	$0.576928\pi$
$224$	0	0
$225$	968.000	0.286815
$226$	0	0
$227$	2682.00	0.784188	0.392094	−	0.919925i	$-0.371751\pi$
$227$	2682.00	0.784188	0.392094	+	0.919925i	$0.371751\pi$
$228$	0	0
$229$	−960.000	−0.277024	−0.138512	−	0.990361i	$-0.544232\pi$
$229$	−960.000	−0.277024	−0.138512	+	0.990361i	$0.544232\pi$
$230$	0	0
$231$	5040.00	1.43553
$232$	0	0
$233$	−1225.00	−0.344431	−0.172215	−	0.985059i	$-0.555093\pi$
$233$	−1225.00	−0.344431	−0.172215	+	0.985059i	$0.555093\pi$
$234$	0	0
$235$	−6617.00	−1.83679
$236$	0	0
$237$	6419.00	1.75932
$238$	0	0
$239$	−1650.00	−0.446567	−0.223284	−	0.974753i	$-0.571678\pi$
$239$	−1650.00	−0.446567	−0.223284	+	0.974753i	$0.571678\pi$
$240$	0	0
$241$	−3503.00	−0.936299	−0.468150	−	0.883649i	$-0.655079\pi$
$241$	−3503.00	−0.936299	−0.468150	+	0.883649i	$0.655079\pi$
$242$	0	0
$243$	4928.00	1.30095
$244$	0	0
$245$	1131.00	0.294926
$246$	0	0
$247$	−4148.00	−1.06855
$248$	0	0
$249$	−4746.00	−1.20789
$250$	0	0
$251$	−5571.00	−1.40095	−0.700475	−	0.713677i	$-0.747028\pi$
$251$	−5571.00	−1.40095	−0.700475	+	0.713677i	$0.747028\pi$
$252$	0	0
$253$	−8730.00	−2.16937
$254$	0	0
$255$	−9282.00	−2.27946
$256$	0	0
$257$	−4891.00	−1.18713	−0.593565	−	0.804786i	$-0.702280\pi$
$257$	−4891.00	−1.18713	−0.593565	+	0.804786i	$0.702280\pi$
$258$	0	0
$259$	6400.00	1.53543
$260$	0	0
$261$	−638.000	−0.151307
$262$	0	0
$263$	−1287.00	−0.301748	−0.150874	−	0.988553i	$-0.548209\pi$
$263$	−1287.00	−0.301748	−0.150874	+	0.988553i	$0.548209\pi$
$264$	0	0
$265$	7865.00	1.82318
$266$	0	0
$267$	7056.00	1.61730
$268$	0	0
$269$	−3826.00	−0.867195	−0.433597	−	0.901107i	$-0.642756\pi$
$269$	−3826.00	−0.867195	−0.433597	+	0.901107i	$0.642756\pi$
$270$	0	0
$271$	−1713.00	−0.383975	−0.191988	−	0.981397i	$-0.561493\pi$
$271$	−1713.00	−0.383975	−0.191988	+	0.981397i	$0.561493\pi$
$272$	0	0
$273$	−6832.00	−1.51462
$274$	0	0
$275$	−1980.00	−0.434176
$276$	0	0
$277$	7086.00	1.53703	0.768513	−	0.639834i	$-0.220997\pi$
$277$	7086.00	1.53703	0.768513	+	0.639834i	$0.220997\pi$
$278$	0	0
$279$	3278.00	0.703400
$280$	0	0
$281$	7063.00	1.49944	0.749721	−	0.661754i	$-0.230188\pi$
$281$	7063.00	1.49944	0.749721	+	0.661754i	$0.230188\pi$
$282$	0	0
$283$	−5002.00	−1.05066	−0.525332	−	0.850897i	$-0.676059\pi$
$283$	−5002.00	−1.05066	−0.525332	+	0.850897i	$0.676059\pi$
$284$	0	0
$285$	−6188.00	−1.28612
$286$	0	0
$287$	4480.00	0.921415
$288$	0	0
$289$	5491.00	1.11765
$290$	0	0
$291$	12348.0	2.48747
$292$	0	0
$293$	4086.00	0.814699	0.407349	−	0.913272i	$-0.366453\pi$
$293$	4086.00	0.814699	0.407349	+	0.913272i	$0.366453\pi$
$294$	0	0
$295$	7514.00	1.48299
$296$	0	0
$297$	−1575.00	−0.307713
$298$	0	0
$299$	11834.0	2.28889
$300$	0	0
$301$	4208.00	0.805798
$302$	0	0
$303$	9520.00	1.80498
$304$	0	0
$305$	9334.00	1.75234
$306$	0	0
$307$	−1909.00	−0.354894	−0.177447	−	0.984130i	$-0.556784\pi$
$307$	−1909.00	−0.354894	−0.177447	+	0.984130i	$0.556784\pi$
$308$	0	0
$309$	−10486.0	−1.93051
$310$	0	0
$311$	−5808.00	−1.05898	−0.529488	−	0.848317i	$-0.677616\pi$
$311$	−5808.00	−1.05898	−0.529488	+	0.848317i	$0.677616\pi$
$312$	0	0
$313$	−6783.00	−1.22491	−0.612457	−	0.790504i	$-0.709819\pi$
$313$	−6783.00	−1.22491	−0.612457	+	0.790504i	$0.709819\pi$
$314$	0	0
$315$	−4576.00	−0.818503
$316$	0	0
$317$	−6090.00	−1.07902	−0.539509	−	0.841980i	$-0.681390\pi$
$317$	−6090.00	−1.07902	−0.539509	+	0.841980i	$0.681390\pi$
$318$	0	0
$319$	1305.00	0.229047
$320$	0	0
$321$	5404.00	0.939632
$322$	0	0
$323$	6936.00	1.19483
$324$	0	0
$325$	2684.00	0.458097
$326$	0	0
$327$	−5341.00	−0.903235
$328$	0	0
$329$	8144.00	1.36472
$330$	0	0
$331$	−3801.00	−0.631184	−0.315592	−	0.948895i	$-0.602203\pi$
$331$	−3801.00	−0.631184	−0.315592	+	0.948895i	$0.602203\pi$
$332$	0	0
$333$	8800.00	1.44816
$334$	0	0
$335$	3380.00	0.551251
$336$	0	0
$337$	−2374.00	−0.383739	−0.191869	−	0.981420i	$-0.561455\pi$
$337$	−2374.00	−0.383739	−0.191869	+	0.981420i	$0.561455\pi$
$338$	0	0
$339$	8526.00	1.36598
$340$	0	0
$341$	−6705.00	−1.06480
$342$	0	0
$343$	−6880.00	−1.08305
$344$	0	0
$345$	17654.0	2.75495
$346$	0	0
$347$	−5346.00	−0.827056	−0.413528	−	0.910491i	$-0.635704\pi$
$347$	−5346.00	−0.827056	−0.413528	+	0.910491i	$0.635704\pi$
$348$	0	0
$349$	9003.00	1.38086	0.690429	−	0.723400i	$-0.257422\pi$
$349$	9003.00	1.38086	0.690429	+	0.723400i	$0.257422\pi$
$350$	0	0
$351$	2135.00	0.324666
$352$	0	0
$353$	10222.0	1.54125	0.770626	−	0.637287i	$-0.219944\pi$
$353$	10222.0	1.54125	0.770626	+	0.637287i	$0.219944\pi$
$354$	0	0
$355$	−9594.00	−1.43436
$356$	0	0
$357$	11424.0	1.69362
$358$	0	0
$359$	719.000	0.105703	0.0528515	−	0.998602i	$-0.483169\pi$
$359$	719.000	0.105703	0.0528515	+	0.998602i	$0.483169\pi$
$360$	0	0
$361$	−2235.00	−0.325849
$362$	0	0
$363$	−4858.00	−0.702421
$364$	0	0
$365$	−8476.00	−1.21549
$366$	0	0
$367$	−6084.00	−0.865347	−0.432673	−	0.901551i	$-0.642430\pi$
$367$	−6084.00	−0.865347	−0.432673	+	0.901551i	$0.642430\pi$
$368$	0	0
$369$	6160.00	0.869043
$370$	0	0
$371$	−9680.00	−1.35461
$372$	0	0
$373$	−1451.00	−0.201421	−0.100710	−	0.994916i	$-0.532112\pi$
$373$	−1451.00	−0.201421	−0.100710	+	0.994916i	$0.532112\pi$
$374$	0	0
$375$	−7371.00	−1.01503
$376$	0	0
$377$	−1769.00	−0.241666
$378$	0	0
$379$	896.000	0.121436	0.0607182	−	0.998155i	$-0.480661\pi$
$379$	896.000	0.121436	0.0607182	+	0.998155i	$0.480661\pi$
$380$	0	0
$381$	−10808.0	−1.45331
$382$	0	0
$383$	13016.0	1.73652	0.868259	−	0.496111i	$-0.165239\pi$
$383$	13016.0	1.73652	0.868259	+	0.496111i	$0.165239\pi$
$384$	0	0
$385$	9360.00	1.23904
$386$	0	0
$387$	5786.00	0.759997
$388$	0	0
$389$	−6560.00	−0.855026	−0.427513	−	0.904009i	$-0.640610\pi$
$389$	−6560.00	−0.855026	−0.427513	+	0.904009i	$0.640610\pi$
$390$	0	0
$391$	−19788.0	−2.55939
$392$	0	0
$393$	13412.0	1.72149
$394$	0	0
$395$	11921.0	1.51851
$396$	0	0
$397$	2757.00	0.348539	0.174269	−	0.984698i	$-0.444244\pi$
$397$	2757.00	0.348539	0.174269	+	0.984698i	$0.444244\pi$
$398$	0	0
$399$	7616.00	0.955581
$400$	0	0
$401$	4995.00	0.622041	0.311020	−	0.950403i	$-0.399329\pi$
$401$	4995.00	0.622041	0.311020	+	0.950403i	$0.399329\pi$
$402$	0	0
$403$	9089.00	1.12346
$404$	0	0
$405$	10907.0	1.33821
$406$	0	0
$407$	−18000.0	−2.19220
$408$	0	0
$409$	3930.00	0.475125	0.237562	−	0.971372i	$-0.423652\pi$
$409$	3930.00	0.475125	0.237562	+	0.971372i	$0.423652\pi$
$410$	0	0
$411$	−504.000	−0.0604878
$412$	0	0
$413$	−9248.00	−1.10185
$414$	0	0
$415$	−8814.00	−1.04256
$416$	0	0
$417$	−9758.00	−1.14593
$418$	0	0
$419$	−3230.00	−0.376601	−0.188301	−	0.982111i	$-0.560298\pi$
$419$	−3230.00	−0.376601	−0.188301	+	0.982111i	$0.560298\pi$
$420$	0	0
$421$	6110.00	0.707323	0.353662	−	0.935373i	$-0.384936\pi$
$421$	6110.00	0.707323	0.353662	+	0.935373i	$0.384936\pi$
$422$	0	0
$423$	11198.0	1.28715
$424$	0	0
$425$	−4488.00	−0.512235
$426$	0	0
$427$	−11488.0	−1.30197
$428$	0	0
$429$	19215.0	2.16249
$430$	0	0
$431$	14218.0	1.58900	0.794498	−	0.607267i	$-0.207734\pi$
$431$	14218.0	1.58900	0.794498	+	0.607267i	$0.207734\pi$
$432$	0	0
$433$	−10008.0	−1.11075	−0.555374	−	0.831601i	$-0.687425\pi$
$433$	−10008.0	−1.11075	−0.555374	+	0.831601i	$0.687425\pi$
$434$	0	0
$435$	−2639.00	−0.290874
$436$	0	0
$437$	−13192.0	−1.44407
$438$	0	0
$439$	−11338.0	−1.23265	−0.616325	−	0.787492i	$-0.711379\pi$
$439$	−11338.0	−1.23265	−0.616325	+	0.787492i	$0.711379\pi$
$440$	0	0
$441$	−1914.00	−0.206673
$442$	0	0
$443$	10180.0	1.09180	0.545899	−	0.837851i	$-0.316188\pi$
$443$	10180.0	1.09180	0.545899	+	0.837851i	$0.316188\pi$
$444$	0	0
$445$	13104.0	1.39593
$446$	0	0
$447$	945.000	0.0999932
$448$	0	0
$449$	−90.0000	−0.00945960	−0.00472980	−	0.999989i	$-0.501506\pi$
$449$	−90.0000	−0.00945960	−0.00472980	+	0.999989i	$0.501506\pi$
$450$	0	0
$451$	−12600.0	−1.31555
$452$	0	0
$453$	11956.0	1.24005
$454$	0	0
$455$	−12688.0	−1.30730
$456$	0	0
$457$	−8906.00	−0.911609	−0.455804	−	0.890080i	$-0.650648\pi$
$457$	−8906.00	−0.911609	−0.455804	+	0.890080i	$0.650648\pi$
$458$	0	0
$459$	−3570.00	−0.363036
$460$	0	0
$461$	−9646.00	−0.974531	−0.487266	−	0.873254i	$-0.662006\pi$
$461$	−9646.00	−0.974531	−0.487266	+	0.873254i	$0.662006\pi$
$462$	0	0
$463$	11578.0	1.16215	0.581075	−	0.813850i	$-0.302632\pi$
$463$	11578.0	1.16215	0.581075	+	0.813850i	$0.302632\pi$
$464$	0	0
$465$	13559.0	1.35222
$466$	0	0
$467$	4671.00	0.462844	0.231422	−	0.972853i	$-0.425662\pi$
$467$	4671.00	0.462844	0.231422	+	0.972853i	$0.425662\pi$
$468$	0	0
$469$	−4160.00	−0.409576
$470$	0	0
$471$	−896.000	−0.0876550
$472$	0	0
$473$	−11835.0	−1.15047
$474$	0	0
$475$	−2992.00	−0.289016
$476$	0	0
$477$	−13310.0	−1.27762
$478$	0	0
$479$	195.000	0.0186008	0.00930039	−	0.999957i	$-0.497040\pi$
$479$	195.000	0.0186008	0.00930039	+	0.999957i	$0.497040\pi$
$480$	0	0
$481$	24400.0	2.31298
$482$	0	0
$483$	−21728.0	−2.04691
$484$	0	0
$485$	22932.0	2.14699
$486$	0	0
$487$	15604.0	1.45192	0.725960	−	0.687737i	$-0.241396\pi$
$487$	15604.0	1.45192	0.725960	+	0.687737i	$0.241396\pi$
$488$	0	0
$489$	−8645.00	−0.799469
$490$	0	0
$491$	−18507.0	−1.70104	−0.850519	−	0.525945i	$-0.823712\pi$
$491$	−18507.0	−1.70104	−0.850519	+	0.525945i	$0.823712\pi$
$492$	0	0
$493$	2958.00	0.270226
$494$	0	0
$495$	12870.0	1.16861
$496$	0	0
$497$	11808.0	1.06572
$498$	0	0
$499$	−1882.00	−0.168837	−0.0844187	−	0.996430i	$-0.526903\pi$
$499$	−1882.00	−0.168837	−0.0844187	+	0.996430i	$0.526903\pi$
$500$	0	0
$501$	16786.0	1.49689
$502$	0	0
$503$	−7645.00	−0.677681	−0.338841	−	0.940844i	$-0.610035\pi$
$503$	−7645.00	−0.677681	−0.338841	+	0.940844i	$0.610035\pi$
$504$	0	0
$505$	17680.0	1.55792
$506$	0	0
$507$	−10668.0	−0.934482
$508$	0	0
$509$	1255.00	0.109287	0.0546433	−	0.998506i	$-0.482598\pi$
$509$	1255.00	0.109287	0.0546433	+	0.998506i	$0.482598\pi$
$510$	0	0
$511$	10432.0	0.903101
$512$	0	0
$513$	−2380.00	−0.204833
$514$	0	0
$515$	−19474.0	−1.66627
$516$	0	0
$517$	−22905.0	−1.94847
$518$	0	0
$519$	8806.00	0.744779
$520$	0	0
$521$	−873.000	−0.0734104	−0.0367052	−	0.999326i	$-0.511686\pi$
$521$	−873.000	−0.0734104	−0.0367052	+	0.999326i	$0.511686\pi$
$522$	0	0
$523$	14996.0	1.25378	0.626892	−	0.779106i	$-0.284327\pi$
$523$	14996.0	1.25378	0.626892	+	0.779106i	$0.284327\pi$
$524$	0	0
$525$	−4928.00	−0.409668
$526$	0	0
$527$	−15198.0	−1.25623
$528$	0	0
$529$	25469.0	2.09329
$530$	0	0
$531$	−12716.0	−1.03922
$532$	0	0
$533$	17080.0	1.38802
$534$	0	0
$535$	10036.0	0.811017
$536$	0	0
$537$	−1078.00	−0.0866278
$538$	0	0
$539$	3915.00	0.312859
$540$	0	0
$541$	2672.00	0.212344	0.106172	−	0.994348i	$-0.466141\pi$
$541$	2672.00	0.212344	0.106172	+	0.994348i	$0.466141\pi$
$542$	0	0
$543$	−21623.0	−1.70890
$544$	0	0
$545$	−9919.00	−0.779602
$546$	0	0
$547$	−4776.00	−0.373322	−0.186661	−	0.982424i	$-0.559767\pi$
$547$	−4776.00	−0.373322	−0.186661	+	0.982424i	$0.559767\pi$
$548$	0	0
$549$	−15796.0	−1.22797
$550$	0	0
$551$	1972.00	0.152468
$552$	0	0
$553$	−14672.0	−1.12824
$554$	0	0
$555$	36400.0	2.78395
$556$	0	0
$557$	−134.000	−0.0101935	−0.00509673	−	0.999987i	$-0.501622\pi$
$557$	−134.000	−0.0101935	−0.00509673	+	0.999987i	$0.501622\pi$
$558$	0	0
$559$	16043.0	1.21386
$560$	0	0
$561$	−32130.0	−2.41806
$562$	0	0
$563$	−437.000	−0.0327129	−0.0163564	−	0.999866i	$-0.505207\pi$
$563$	−437.000	−0.0327129	−0.0163564	+	0.999866i	$0.505207\pi$
$564$	0	0
$565$	15834.0	1.17901
$566$	0	0
$567$	−13424.0	−0.994277
$568$	0	0
$569$	−23990.0	−1.76751	−0.883755	−	0.467950i	$-0.844993\pi$
$569$	−23990.0	−1.76751	−0.883755	+	0.467950i	$0.844993\pi$
$570$	0	0
$571$	−100.000	−0.00732902	−0.00366451	−	0.999993i	$-0.501166\pi$
$571$	−100.000	−0.00732902	−0.00366451	+	0.999993i	$0.501166\pi$
$572$	0	0
$573$	7616.00	0.555258
$574$	0	0
$575$	8536.00	0.619088
$576$	0	0
$577$	6078.00	0.438528	0.219264	−	0.975666i	$-0.429634\pi$
$577$	6078.00	0.438528	0.219264	+	0.975666i	$0.429634\pi$
$578$	0	0
$579$	1596.00	0.114555
$580$	0	0
$581$	10848.0	0.774614
$582$	0	0
$583$	27225.0	1.93404
$584$	0	0
$585$	−17446.0	−1.23300
$586$	0	0
$587$	−4214.00	−0.296304	−0.148152	−	0.988965i	$-0.547332\pi$
$587$	−4214.00	−0.296304	−0.148152	+	0.988965i	$0.547332\pi$
$588$	0	0
$589$	−10132.0	−0.708798
$590$	0	0
$591$	−15582.0	−1.08453
$592$	0	0
$593$	−28293.0	−1.95928	−0.979641	−	0.200757i	$-0.935660\pi$
$593$	−28293.0	−1.95928	−0.979641	+	0.200757i	$0.935660\pi$
$594$	0	0
$595$	21216.0	1.46180
$596$	0	0
$597$	13356.0	0.915619
$598$	0	0
$599$	8843.00	0.603197	0.301599	−	0.953435i	$-0.402480\pi$
$599$	8843.00	0.603197	0.301599	+	0.953435i	$0.402480\pi$
$600$	0	0
$601$	−22548.0	−1.53037	−0.765185	−	0.643811i	$-0.777352\pi$
$601$	−22548.0	−1.53037	−0.765185	+	0.643811i	$0.777352\pi$
$602$	0	0
$603$	−5720.00	−0.386296
$604$	0	0
$605$	−9022.00	−0.606275
$606$	0	0
$607$	−17061.0	−1.14083	−0.570416	−	0.821356i	$-0.693218\pi$
$607$	−17061.0	−1.14083	−0.570416	+	0.821356i	$0.693218\pi$
$608$	0	0
$609$	3248.00	0.216118
$610$	0	0
$611$	31049.0	2.05582
$612$	0	0
$613$	21251.0	1.40020	0.700098	−	0.714047i	$-0.253140\pi$
$613$	21251.0	1.40020	0.700098	+	0.714047i	$0.253140\pi$
$614$	0	0
$615$	25480.0	1.67065
$616$	0	0
$617$	200.000	0.0130498	0.00652488	−	0.999979i	$-0.497923\pi$
$617$	200.000	0.0130498	0.00652488	+	0.999979i	$0.497923\pi$
$618$	0	0
$619$	−19533.0	−1.26833	−0.634166	−	0.773197i	$-0.718656\pi$
$619$	−19533.0	−1.26833	−0.634166	+	0.773197i	$0.718656\pi$
$620$	0	0
$621$	6790.00	0.438765
$622$	0	0
$623$	−16128.0	−1.03717
$624$	0	0
$625$	−19189.0	−1.22810
$626$	0	0
$627$	−21420.0	−1.36433
$628$	0	0
$629$	−40800.0	−2.58633
$630$	0	0
$631$	12570.0	0.793033	0.396516	−	0.918028i	$-0.370219\pi$
$631$	12570.0	0.793033	0.396516	+	0.918028i	$0.370219\pi$
$632$	0	0
$633$	−4081.00	−0.256248
$634$	0	0
$635$	−20072.0	−1.25438
$636$	0	0
$637$	−5307.00	−0.330096
$638$	0	0
$639$	16236.0	1.00514
$640$	0	0
$641$	17446.0	1.07500	0.537500	−	0.843263i	$-0.319369\pi$
$641$	17446.0	1.07500	0.537500	+	0.843263i	$0.319369\pi$
$642$	0	0
$643$	−11324.0	−0.694518	−0.347259	−	0.937769i	$-0.612887\pi$
$643$	−11324.0	−0.694518	−0.347259	+	0.937769i	$0.612887\pi$
$644$	0	0
$645$	23933.0	1.46102
$646$	0	0
$647$	2922.00	0.177551	0.0887756	−	0.996052i	$-0.471705\pi$
$647$	2922.00	0.177551	0.0887756	+	0.996052i	$0.471705\pi$
$648$	0	0
$649$	26010.0	1.57316
$650$	0	0
$651$	−16688.0	−1.00469
$652$	0	0
$653$	23120.0	1.38554	0.692768	−	0.721160i	$-0.256391\pi$
$653$	23120.0	1.38554	0.692768	+	0.721160i	$0.256391\pi$
$654$	0	0
$655$	24908.0	1.48586
$656$	0	0
$657$	14344.0	0.851770
$658$	0	0
$659$	32901.0	1.94483	0.972414	−	0.233264i	$-0.0749406\pi$
$659$	32901.0	1.94483	0.972414	+	0.233264i	$0.0749406\pi$
$660$	0	0
$661$	−18494.0	−1.08825	−0.544125	−	0.839004i	$-0.683138\pi$
$661$	−18494.0	−1.08825	−0.544125	+	0.839004i	$0.683138\pi$
$662$	0	0
$663$	43554.0	2.55128
$664$	0	0
$665$	14144.0	0.824783
$666$	0	0
$667$	−5626.00	−0.326596
$668$	0	0
$669$	11158.0	0.644833
$670$	0	0
$671$	32310.0	1.85889
$672$	0	0
$673$	−15119.0	−0.865965	−0.432983	−	0.901402i	$-0.642539\pi$
$673$	−15119.0	−0.865965	−0.432983	+	0.901402i	$0.642539\pi$
$674$	0	0
$675$	1540.00	0.0878143
$676$	0	0
$677$	6946.00	0.394323	0.197161	−	0.980371i	$-0.436828\pi$
$677$	6946.00	0.394323	0.197161	+	0.980371i	$0.436828\pi$
$678$	0	0
$679$	−28224.0	−1.59520
$680$	0	0
$681$	−18774.0	−1.05642
$682$	0	0
$683$	31124.0	1.74367	0.871835	−	0.489799i	$-0.162930\pi$
$683$	31124.0	1.74367	0.871835	+	0.489799i	$0.162930\pi$
$684$	0	0
$685$	−936.000	−0.0522084
$686$	0	0
$687$	6720.00	0.373194
$688$	0	0
$689$	−36905.0	−2.04059
$690$	0	0
$691$	12664.0	0.697194	0.348597	−	0.937273i	$-0.386658\pi$
$691$	12664.0	0.697194	0.348597	+	0.937273i	$0.386658\pi$
$692$	0	0
$693$	−15840.0	−0.868271
$694$	0	0
$695$	−18122.0	−0.989074
$696$	0	0
$697$	−28560.0	−1.55206
$698$	0	0
$699$	8575.00	0.464000
$700$	0	0
$701$	−18715.0	−1.00835	−0.504177	−	0.863600i	$-0.668204\pi$
$701$	−18715.0	−1.00835	−0.504177	+	0.863600i	$0.668204\pi$
$702$	0	0
$703$	−27200.0	−1.45927
$704$	0	0
$705$	46319.0	2.47443
$706$	0	0
$707$	−21760.0	−1.15752
$708$	0	0
$709$	−7531.00	−0.398918	−0.199459	−	0.979906i	$-0.563918\pi$
$709$	−7531.00	−0.398918	−0.199459	+	0.979906i	$0.563918\pi$
$710$	0	0
$711$	−20174.0	−1.06411
$712$	0	0
$713$	28906.0	1.51829
$714$	0	0
$715$	35685.0	1.86649
$716$	0	0
$717$	11550.0	0.601594
$718$	0	0
$719$	35338.0	1.83294	0.916471	−	0.400102i	$-0.131025\pi$
$719$	35338.0	1.83294	0.916471	+	0.400102i	$0.131025\pi$
$720$	0	0
$721$	23968.0	1.23802
$722$	0	0
$723$	24521.0	1.26134
$724$	0	0
$725$	−1276.00	−0.0653648
$726$	0	0
$727$	−13976.0	−0.712986	−0.356493	−	0.934298i	$-0.616028\pi$
$727$	−13976.0	−0.712986	−0.356493	+	0.934298i	$0.616028\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	0	0
$731$	−26826.0	−1.35731
$732$	0	0
$733$	7172.00	0.361397	0.180698	−	0.983539i	$-0.442164\pi$
$733$	7172.00	0.361397	0.180698	+	0.983539i	$0.442164\pi$
$734$	0	0
$735$	−7917.00	−0.397310
$736$	0	0
$737$	11700.0	0.584769
$738$	0	0
$739$	−35091.0	−1.74674	−0.873372	−	0.487054i	$-0.838071\pi$
$739$	−35091.0	−1.74674	−0.873372	+	0.487054i	$0.838071\pi$
$740$	0	0
$741$	29036.0	1.43949
$742$	0	0
$743$	−10992.0	−0.542742	−0.271371	−	0.962475i	$-0.587477\pi$
$743$	−10992.0	−0.542742	−0.271371	+	0.962475i	$0.587477\pi$
$744$	0	0
$745$	1755.00	0.0863063
$746$	0	0
$747$	14916.0	0.730586
$748$	0	0
$749$	−12352.0	−0.602580
$750$	0	0
$751$	−20988.0	−1.01979	−0.509895	−	0.860236i	$-0.670316\pi$
$751$	−20988.0	−1.01979	−0.509895	+	0.860236i	$0.670316\pi$
$752$	0	0
$753$	38997.0	1.88729
$754$	0	0
$755$	22204.0	1.07031
$756$	0	0
$757$	−39094.0	−1.87701	−0.938504	−	0.345267i	$-0.887788\pi$
$757$	−39094.0	−1.87701	−0.938504	+	0.345267i	$0.887788\pi$
$758$	0	0
$759$	61110.0	2.92247
$760$	0	0
$761$	−27270.0	−1.29900	−0.649499	−	0.760363i	$-0.725021\pi$
$761$	−27270.0	−1.29900	−0.649499	+	0.760363i	$0.725021\pi$
$762$	0	0
$763$	12208.0	0.579239
$764$	0	0
$765$	29172.0	1.37871
$766$	0	0
$767$	−35258.0	−1.65983
$768$	0	0
$769$	3462.00	0.162344	0.0811722	−	0.996700i	$-0.474134\pi$
$769$	3462.00	0.162344	0.0811722	+	0.996700i	$0.474134\pi$
$770$	0	0
$771$	34237.0	1.59924
$772$	0	0
$773$	−10632.0	−0.494704	−0.247352	−	0.968926i	$-0.579560\pi$
$773$	−10632.0	−0.494704	−0.247352	+	0.968926i	$0.579560\pi$
$774$	0	0
$775$	6556.00	0.303869
$776$	0	0
$777$	−44800.0	−2.06846
$778$	0	0
$779$	−19040.0	−0.875711
$780$	0	0
$781$	−33210.0	−1.52157
$782$	0	0
$783$	−1015.00	−0.0463259
$784$	0	0
$785$	−1664.00	−0.0756570
$786$	0	0
$787$	−3754.00	−0.170033	−0.0850163	−	0.996380i	$-0.527094\pi$
$787$	−3754.00	−0.170033	−0.0850163	+	0.996380i	$0.527094\pi$
$788$	0	0
$789$	9009.00	0.406500
$790$	0	0
$791$	−19488.0	−0.875997
$792$	0	0
$793$	−43798.0	−1.96130
$794$	0	0
$795$	−55055.0	−2.45610
$796$	0	0
$797$	9378.00	0.416795	0.208398	−	0.978044i	$-0.433175\pi$
$797$	9378.00	0.416795	0.208398	+	0.978044i	$0.433175\pi$
$798$	0	0
$799$	−51918.0	−2.29878
$800$	0	0
$801$	−22176.0	−0.978215
$802$	0	0
$803$	−29340.0	−1.28940
$804$	0	0
$805$	−40352.0	−1.76673
$806$	0	0
$807$	26782.0	1.16824
$808$	0	0
$809$	−3708.00	−0.161145	−0.0805725	−	0.996749i	$-0.525675\pi$
$809$	−3708.00	−0.161145	−0.0805725	+	0.996749i	$0.525675\pi$
$810$	0	0
$811$	13232.0	0.572920	0.286460	−	0.958092i	$-0.407521\pi$
$811$	13232.0	0.572920	0.286460	+	0.958092i	$0.407521\pi$
$812$	0	0
$813$	11991.0	0.517273
$814$	0	0
$815$	−16055.0	−0.690039
$816$	0	0
$817$	−17884.0	−0.765829
$818$	0	0
$819$	21472.0	0.916108
$820$	0	0
$821$	5445.00	0.231464	0.115732	−	0.993280i	$-0.463079\pi$
$821$	5445.00	0.231464	0.115732	+	0.993280i	$0.463079\pi$
$822$	0	0
$823$	−17576.0	−0.744424	−0.372212	−	0.928148i	$-0.621401\pi$
$823$	−17576.0	−0.744424	−0.372212	+	0.928148i	$0.621401\pi$
$824$	0	0
$825$	13860.0	0.584901
$826$	0	0
$827$	10143.0	0.426489	0.213245	−	0.976999i	$-0.431597\pi$
$827$	10143.0	0.426489	0.213245	+	0.976999i	$0.431597\pi$
$828$	0	0
$829$	−8558.00	−0.358542	−0.179271	−	0.983800i	$-0.557374\pi$
$829$	−8558.00	−0.358542	−0.179271	+	0.983800i	$0.557374\pi$
$830$	0	0
$831$	−49602.0	−2.07061
$832$	0	0
$833$	8874.00	0.369107
$834$	0	0
$835$	31174.0	1.29200
$836$	0	0
$837$	5215.00	0.215361
$838$	0	0
$839$	−30467.0	−1.25368	−0.626840	−	0.779148i	$-0.715652\pi$
$839$	−30467.0	−1.25368	−0.626840	+	0.779148i	$0.715652\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−49441.0	−2.01997
$844$	0	0
$845$	−19812.0	−0.806572
$846$	0	0
$847$	11104.0	0.450458
$848$	0	0
$849$	35014.0	1.41540
$850$	0	0
$851$	77600.0	3.12584
$852$	0	0
$853$	−154.000	−0.00618155	−0.00309077	−	0.999995i	$-0.500984\pi$
$853$	−154.000	−0.00618155	−0.00309077	+	0.999995i	$0.500984\pi$
$854$	0	0
$855$	19448.0	0.777904
$856$	0	0
$857$	−10463.0	−0.417047	−0.208523	−	0.978017i	$-0.566866\pi$
$857$	−10463.0	−0.417047	−0.208523	+	0.978017i	$0.566866\pi$
$858$	0	0
$859$	24749.0	0.983033	0.491516	−	0.870868i	$-0.336443\pi$
$859$	24749.0	0.983033	0.491516	+	0.870868i	$0.336443\pi$
$860$	0	0
$861$	−31360.0	−1.24128
$862$	0	0
$863$	−20586.0	−0.812000	−0.406000	−	0.913873i	$-0.633077\pi$
$863$	−20586.0	−0.812000	−0.406000	+	0.913873i	$0.633077\pi$
$864$	0	0
$865$	16354.0	0.642835
$866$	0	0
$867$	−38437.0	−1.50564
$868$	0	0
$869$	41265.0	1.61084
$870$	0	0
$871$	−15860.0	−0.616987
$872$	0	0
$873$	−38808.0	−1.50453
$874$	0	0
$875$	16848.0	0.650933
$876$	0	0
$877$	5051.00	0.194481	0.0972407	−	0.995261i	$-0.468998\pi$
$877$	5051.00	0.194481	0.0972407	+	0.995261i	$0.468998\pi$
$878$	0	0
$879$	−28602.0	−1.09752
$880$	0	0
$881$	−38990.0	−1.49104	−0.745520	−	0.666483i	$-0.767799\pi$
$881$	−38990.0	−1.49104	−0.745520	+	0.666483i	$0.767799\pi$
$882$	0	0
$883$	−27180.0	−1.03588	−0.517939	−	0.855418i	$-0.673300\pi$
$883$	−27180.0	−1.03588	−0.517939	+	0.855418i	$0.673300\pi$
$884$	0	0
$885$	−52598.0	−1.99781
$886$	0	0
$887$	−17435.0	−0.659989	−0.329994	−	0.943983i	$-0.607047\pi$
$887$	−17435.0	−0.659989	−0.329994	+	0.943983i	$0.607047\pi$
$888$	0	0
$889$	24704.0	0.931997
$890$	0	0
$891$	37755.0	1.41957
$892$	0	0
$893$	−34612.0	−1.29703
$894$	0	0
$895$	−2002.00	−0.0747704
$896$	0	0
$897$	−82838.0	−3.08348
$898$	0	0
$899$	−4321.00	−0.160304
$900$	0	0
$901$	61710.0	2.28175
$902$	0	0
$903$	−29456.0	−1.08553
$904$	0	0
$905$	−40157.0	−1.47499
$906$	0	0
$907$	−11500.0	−0.421005	−0.210502	−	0.977593i	$-0.567510\pi$
$907$	−11500.0	−0.421005	−0.210502	+	0.977593i	$0.567510\pi$
$908$	0	0
$909$	−29920.0	−1.09173
$910$	0	0
$911$	−7969.00	−0.289819	−0.144909	−	0.989445i	$-0.546289\pi$
$911$	−7969.00	−0.289819	−0.144909	+	0.989445i	$0.546289\pi$
$912$	0	0
$913$	−30510.0	−1.10595
$914$	0	0
$915$	−65338.0	−2.36066
$916$	0	0
$917$	−30656.0	−1.10398
$918$	0	0
$919$	−24414.0	−0.876326	−0.438163	−	0.898896i	$-0.644371\pi$
$919$	−24414.0	−0.876326	−0.438163	+	0.898896i	$0.644371\pi$
$920$	0	0
$921$	13363.0	0.478095
$922$	0	0
$923$	45018.0	1.60540
$924$	0	0
$925$	17600.0	0.625605
$926$	0	0
$927$	32956.0	1.16766
$928$	0	0
$929$	−18694.0	−0.660205	−0.330102	−	0.943945i	$-0.607083\pi$
$929$	−18694.0	−0.660205	−0.330102	+	0.943945i	$0.607083\pi$
$930$	0	0
$931$	5916.00	0.208259
$932$	0	0
$933$	40656.0	1.42660
$934$	0	0
$935$	−59670.0	−2.08708
$936$	0	0
$937$	14394.0	0.501848	0.250924	−	0.968007i	$-0.419266\pi$
$937$	14394.0	0.501848	0.250924	+	0.968007i	$0.419266\pi$
$938$	0	0
$939$	47481.0	1.65014
$940$	0	0
$941$	−11925.0	−0.413118	−0.206559	−	0.978434i	$-0.566227\pi$
$941$	−11925.0	−0.413118	−0.206559	+	0.978434i	$0.566227\pi$
$942$	0	0
$943$	54320.0	1.87582
$944$	0	0
$945$	−7280.00	−0.250602
$946$	0	0
$947$	−53321.0	−1.82967	−0.914836	−	0.403825i	$-0.867680\pi$
$947$	−53321.0	−1.82967	−0.914836	+	0.403825i	$0.867680\pi$
$948$	0	0
$949$	39772.0	1.36044
$950$	0	0
$951$	42630.0	1.45360
$952$	0	0
$953$	−38301.0	−1.30188	−0.650940	−	0.759129i	$-0.725625\pi$
$953$	−38301.0	−1.30188	−0.650940	+	0.759129i	$0.725625\pi$
$954$	0	0
$955$	14144.0	0.479256
$956$	0	0
$957$	−9135.00	−0.308561
$958$	0	0
$959$	1152.00	0.0387904
$960$	0	0
$961$	−7590.00	−0.254775
$962$	0	0
$963$	−16984.0	−0.568330
$964$	0	0
$965$	2964.00	0.0988752
$966$	0	0
$967$	−40011.0	−1.33058	−0.665288	−	0.746587i	$-0.731691\pi$
$967$	−40011.0	−1.33058	−0.665288	+	0.746587i	$0.731691\pi$
$968$	0	0
$969$	−48552.0	−1.60961
$970$	0	0
$971$	−53500.0	−1.76817	−0.884087	−	0.467323i	$-0.845219\pi$
$971$	−53500.0	−1.76817	−0.884087	+	0.467323i	$0.845219\pi$
$972$	0	0
$973$	22304.0	0.734875
$974$	0	0
$975$	−18788.0	−0.617126
$976$	0	0
$977$	28449.0	0.931591	0.465795	−	0.884892i	$-0.345768\pi$
$977$	28449.0	0.931591	0.465795	+	0.884892i	$0.345768\pi$
$978$	0	0
$979$	45360.0	1.48081
$980$	0	0
$981$	16786.0	0.546316
$982$	0	0
$983$	48529.0	1.57460	0.787301	−	0.616568i	$-0.211478\pi$
$983$	48529.0	1.57460	0.787301	+	0.616568i	$0.211478\pi$
$984$	0	0
$985$	−28938.0	−0.936083
$986$	0	0
$987$	−57008.0	−1.83848
$988$	0	0
$989$	51022.0	1.64045
$990$	0	0
$991$	8142.00	0.260988	0.130494	−	0.991449i	$-0.458344\pi$
$991$	8142.00	0.260988	0.130494	+	0.991449i	$0.458344\pi$
$992$	0	0
$993$	26607.0	0.850300
$994$	0	0
$995$	24804.0	0.790291
$996$	0	0
$997$	−30572.0	−0.971138	−0.485569	−	0.874198i	$-0.661388\pi$
$997$	−30572.0	−0.971138	−0.485569	+	0.874198i	$0.661388\pi$
$998$	0	0
$999$	14000.0	0.443384

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	928.4.a.a.1.1	✓	1
4.3	odd	2		928.4.a.b.1.1	yes	1
8.3	odd	2		1856.4.a.b.1.1		1
8.5	even	2		1856.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.a.1.1	✓	1	1.1	even	1	trivial
928.4.a.b.1.1	yes	1	4.3	odd	2
1856.4.a.b.1.1		1	8.3	odd	2
1856.4.a.e.1.1		1	8.5	even	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:	\( N \)	\(=\)	\( 928 = 2^{5} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	928.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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