LMFDB - Embedded newform 2151.4.a.g.1.47

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.47
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.13470 q^{2} +1.82636 q^{4} +4.91209 q^{5} +36.3874 q^{7} -19.3525 q^{8} +O(q^{10})$	\(q+3.13470 q^{2} +1.82636 q^{4} +4.91209 q^{5} +36.3874 q^{7} -19.3525 q^{8} +15.3979 q^{10} -0.654849 q^{11} -42.3822 q^{13} +114.064 q^{14} -75.2753 q^{16} +41.8110 q^{17} -79.2470 q^{19} +8.97125 q^{20} -2.05276 q^{22} -103.434 q^{23} -100.871 q^{25} -132.855 q^{26} +66.4566 q^{28} -150.719 q^{29} -43.3756 q^{31} -81.1455 q^{32} +131.065 q^{34} +178.738 q^{35} -328.704 q^{37} -248.416 q^{38} -95.0614 q^{40} -227.006 q^{41} -88.1867 q^{43} -1.19599 q^{44} -324.234 q^{46} -161.362 q^{47} +981.045 q^{49} -316.202 q^{50} -77.4051 q^{52} -206.317 q^{53} -3.21668 q^{55} -704.189 q^{56} -472.461 q^{58} +427.414 q^{59} -38.0290 q^{61} -135.970 q^{62} +347.835 q^{64} -208.185 q^{65} +594.441 q^{67} +76.3620 q^{68} +560.292 q^{70} -905.766 q^{71} +155.597 q^{73} -1030.39 q^{74} -144.734 q^{76} -23.8283 q^{77} +63.7719 q^{79} -369.759 q^{80} -711.597 q^{82} +834.746 q^{83} +205.379 q^{85} -276.439 q^{86} +12.6730 q^{88} +601.888 q^{89} -1542.18 q^{91} -188.907 q^{92} -505.820 q^{94} -389.268 q^{95} -495.710 q^{97} +3075.29 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

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 $

$2$	3.13470	1.10828	0.554142	−	0.832422i	$-0.313046\pi$
$2$	3.13470	1.10828	0.554142	+	0.832422i	$0.313046\pi$
$3$	0	0
$3$	0	0
$4$	1.82636	0.228295
$5$	4.91209	0.439351	0.219675	−	0.975573i	$-0.429500\pi$
$5$	4.91209	0.439351	0.219675	+	0.975573i	$0.429500\pi$
$6$	0	0
$7$	36.3874	1.96474	0.982368	−	0.186956i	$-0.0598621\pi$
$7$	36.3874	1.96474	0.982368	+	0.186956i	$0.0598621\pi$
$8$	−19.3525	−0.855269
$9$	0	0
$10$	15.3979	0.486926
$11$	−0.654849	−0.0179495	−0.00897475	−	0.999960i	$-0.502857\pi$
$11$	−0.654849	−0.0179495	−0.00897475	+	0.999960i	$0.502857\pi$
$12$	0	0
$13$	−42.3822	−0.904207	−0.452104	−	0.891965i	$-0.649326\pi$
$13$	−42.3822	−0.904207	−0.452104	+	0.891965i	$0.649326\pi$
$14$	114.064	2.17749
$15$	0	0
$16$	−75.2753	−1.17618
$17$	41.8110	0.596509	0.298255	−	0.954486i	$-0.403596\pi$
$17$	41.8110	0.596509	0.298255	+	0.954486i	$0.403596\pi$
$18$	0	0
$19$	−79.2470	−0.956869	−0.478434	−	0.878123i	$-0.658796\pi$
$19$	−79.2470	−0.956869	−0.478434	+	0.878123i	$0.658796\pi$
$20$	8.97125	0.100302
$21$	0	0
$22$	−2.05276	−0.0198932
$23$	−103.434	−0.937715	−0.468857	−	0.883274i	$-0.655334\pi$
$23$	−103.434	−0.937715	−0.468857	+	0.883274i	$0.655334\pi$
$24$	0	0
$25$	−100.871	−0.806971
$26$	−132.855	−1.00212
$27$	0	0
$28$	66.4566	0.448540
$29$	−150.719	−0.965100	−0.482550	−	0.875868i	$-0.660289\pi$
$29$	−150.719	−0.965100	−0.482550	+	0.875868i	$0.660289\pi$
$30$	0	0
$31$	−43.3756	−0.251306	−0.125653	−	0.992074i	$-0.540103\pi$
$31$	−43.3756	−0.251306	−0.125653	+	0.992074i	$0.540103\pi$
$32$	−81.1455	−0.448270
$33$	0	0
$34$	131.065	0.661102
$35$	178.738	0.863209
$36$	0	0
$37$	−328.704	−1.46050	−0.730252	−	0.683178i	$-0.760597\pi$
$37$	−328.704	−1.46050	−0.730252	+	0.683178i	$0.760597\pi$
$38$	−248.416	−1.06048
$39$	0	0
$40$	−95.0614	−0.375763
$41$	−227.006	−0.864693	−0.432346	−	0.901708i	$-0.642314\pi$
$41$	−227.006	−0.864693	−0.432346	+	0.901708i	$0.642314\pi$
$42$	0	0
$43$	−88.1867	−0.312752	−0.156376	−	0.987698i	$-0.549981\pi$
$43$	−88.1867	−0.312752	−0.156376	+	0.987698i	$0.549981\pi$
$44$	−1.19599	−0.00409778
$45$	0	0
$46$	−324.234	−1.03925
$47$	−161.362	−0.500787	−0.250394	−	0.968144i	$-0.580560\pi$
$47$	−161.362	−0.500787	−0.250394	+	0.968144i	$0.580560\pi$
$48$	0	0
$49$	981.045	2.86019
$50$	−316.202	−0.894354
$51$	0	0
$52$	−77.4051	−0.206426
$53$	−206.317	−0.534714	−0.267357	−	0.963598i	$-0.586150\pi$
$53$	−206.317	−0.534714	−0.267357	+	0.963598i	$0.586150\pi$
$54$	0	0
$55$	−3.21668	−0.00788613
$56$	−704.189	−1.68038
$57$	0	0
$58$	−472.461	−1.06961
$59$	427.414	0.943128	0.471564	−	0.881832i	$-0.343690\pi$
$59$	427.414	0.943128	0.471564	+	0.881832i	$0.343690\pi$
$60$	0	0
$61$	−38.0290	−0.0798215	−0.0399108	−	0.999203i	$-0.512707\pi$
$61$	−38.0290	−0.0798215	−0.0399108	+	0.999203i	$0.512707\pi$
$62$	−135.970	−0.278519
$63$	0	0
$64$	347.835	0.679366
$65$	−208.185	−0.397264
$66$	0	0
$67$	594.441	1.08392	0.541959	−	0.840405i	$-0.317683\pi$
$67$	594.441	1.08392	0.541959	+	0.840405i	$0.317683\pi$
$68$	76.3620	0.136180
$69$	0	0
$70$	560.292	0.956681
$71$	−905.766	−1.51401	−0.757004	−	0.653410i	$-0.773338\pi$
$71$	−905.766	−1.51401	−0.757004	+	0.653410i	$0.773338\pi$
$72$	0	0
$73$	155.597	0.249469	0.124735	−	0.992190i	$-0.460192\pi$
$73$	155.597	0.249469	0.124735	+	0.992190i	$0.460192\pi$
$74$	−1030.39	−1.61865
$75$	0	0
$76$	−144.734	−0.218448
$77$	−23.8283	−0.0352660
$78$	0	0
$79$	63.7719	0.0908216	0.0454108	−	0.998968i	$-0.485540\pi$
$79$	63.7719	0.0908216	0.0454108	+	0.998968i	$0.485540\pi$
$80$	−369.759	−0.516754
$81$	0	0
$82$	−711.597	−0.958326
$83$	834.746	1.10392	0.551960	−	0.833871i	$-0.313880\pi$
$83$	834.746	1.10392	0.551960	+	0.833871i	$0.313880\pi$
$84$	0	0
$85$	205.379	0.262077
$86$	−276.439	−0.346619
$87$	0	0
$88$	12.6730	0.0153516
$89$	601.888	0.716855	0.358427	−	0.933558i	$-0.383313\pi$
$89$	601.888	0.716855	0.358427	+	0.933558i	$0.383313\pi$
$90$	0	0
$91$	−1542.18	−1.77653
$92$	−188.907	−0.214076
$93$	0	0
$94$	−505.820	−0.555015
$95$	−389.268	−0.420401
$96$	0	0
$97$	−495.710	−0.518883	−0.259442	−	0.965759i	$-0.583539\pi$
$97$	−495.710	−0.518883	−0.259442	+	0.965759i	$0.583539\pi$
$98$	3075.29	3.16991
$99$	0	0
$100$	−184.228	−0.184228
$101$	149.594	0.147378	0.0736891	−	0.997281i	$-0.476523\pi$
$101$	149.594	0.147378	0.0736891	+	0.997281i	$0.476523\pi$
$102$	0	0
$103$	564.393	0.539915	0.269958	−	0.962872i	$-0.412990\pi$
$103$	564.393	0.539915	0.269958	+	0.962872i	$0.412990\pi$
$104$	820.202	0.773340
$105$	0	0
$106$	−646.742	−0.592615
$107$	−2111.34	−1.90758	−0.953788	−	0.300480i	$-0.902853\pi$
$107$	−2111.34	−1.90758	−0.953788	+	0.300480i	$0.902853\pi$
$108$	0	0
$109$	240.393	0.211243	0.105621	−	0.994406i	$-0.466317\pi$
$109$	240.393	0.211243	0.105621	+	0.994406i	$0.466317\pi$
$110$	−10.0833	−0.00874008
$111$	0	0
$112$	−2739.07	−2.31088
$113$	548.514	0.456636	0.228318	−	0.973587i	$-0.426677\pi$
$113$	548.514	0.456636	0.228318	+	0.973587i	$0.426677\pi$
$114$	0	0
$115$	−508.076	−0.411986
$116$	−275.268	−0.220328
$117$	0	0
$118$	1339.82	1.04525
$119$	1521.39	1.17198
$120$	0	0
$121$	−1330.57	−0.999678
$122$	−119.210	−0.0884650
$123$	0	0
$124$	−79.2195	−0.0573719
$125$	−1109.50	−0.793894
$126$	0	0
$127$	−2477.98	−1.73138	−0.865690	−	0.500581i	$-0.833120\pi$
$127$	−2477.98	−1.73138	−0.865690	+	0.500581i	$0.833120\pi$
$128$	1739.52	1.20120
$129$	0	0
$130$	−652.598	−0.440282
$131$	240.252	0.160236	0.0801180	−	0.996785i	$-0.474470\pi$
$131$	240.252	0.160236	0.0801180	+	0.996785i	$0.474470\pi$
$132$	0	0
$133$	−2883.59	−1.87999
$134$	1863.39	1.20129
$135$	0	0
$136$	−809.148	−0.510176
$137$	143.473	0.0894728	0.0447364	−	0.998999i	$-0.485755\pi$
$137$	143.473	0.0894728	0.0447364	+	0.998999i	$0.485755\pi$
$138$	0	0
$139$	−340.457	−0.207750	−0.103875	−	0.994590i	$-0.533124\pi$
$139$	−340.457	−0.207750	−0.103875	+	0.994590i	$0.533124\pi$
$140$	326.441	0.197066
$141$	0	0
$142$	−2839.31	−1.67795
$143$	27.7539	0.0162301
$144$	0	0
$145$	−740.348	−0.424017
$146$	487.750	0.276483
$147$	0	0
$148$	−600.333	−0.333426
$149$	−1418.37	−0.779847	−0.389923	−	0.920847i	$-0.627499\pi$
$149$	−1418.37	−0.779847	−0.389923	+	0.920847i	$0.627499\pi$
$150$	0	0
$151$	335.359	0.180736	0.0903679	−	0.995908i	$-0.471196\pi$
$151$	335.359	0.180736	0.0903679	+	0.995908i	$0.471196\pi$
$152$	1533.63	0.818380
$153$	0	0
$154$	−74.6946	−0.0390848
$155$	−213.065	−0.110411
$156$	0	0
$157$	2823.71	1.43539	0.717697	−	0.696355i	$-0.245196\pi$
$157$	2823.71	1.43539	0.717697	+	0.696355i	$0.245196\pi$
$158$	199.906	0.100656
$159$	0	0
$160$	−398.594	−0.196948
$161$	−3763.69	−1.84236
$162$	0	0
$163$	1947.03	0.935603	0.467802	−	0.883833i	$-0.345046\pi$
$163$	1947.03	0.935603	0.467802	+	0.883833i	$0.345046\pi$
$164$	−414.595	−0.197405
$165$	0	0
$166$	2616.68	1.22346
$167$	−2376.81	−1.10134	−0.550669	−	0.834724i	$-0.685627\pi$
$167$	−2376.81	−1.10134	−0.550669	+	0.834724i	$0.685627\pi$
$168$	0	0
$169$	−400.753	−0.182409
$170$	643.803	0.290456
$171$	0	0
$172$	−161.061	−0.0713999
$173$	820.399	0.360542	0.180271	−	0.983617i	$-0.442303\pi$
$173$	820.399	0.360542	0.180271	+	0.983617i	$0.442303\pi$
$174$	0	0
$175$	−3670.45	−1.58549
$176$	49.2940	0.0211118
$177$	0	0
$178$	1886.74	0.794479
$179$	−2688.77	−1.12273	−0.561364	−	0.827569i	$-0.689723\pi$
$179$	−2688.77	−1.12273	−0.561364	+	0.827569i	$0.689723\pi$
$180$	0	0
$181$	2543.72	1.04460	0.522301	−	0.852761i	$-0.325074\pi$
$181$	2543.72	1.04460	0.522301	+	0.852761i	$0.325074\pi$
$182$	−4834.27	−1.96890
$183$	0	0
$184$	2001.70	0.801998
$185$	−1614.62	−0.641673
$186$	0	0
$187$	−27.3799	−0.0107070
$188$	−294.704	−0.114327
$189$	0	0
$190$	−1220.24	−0.465924
$191$	2251.64	0.852999	0.426499	−	0.904488i	$-0.359747\pi$
$191$	2251.64	0.852999	0.426499	+	0.904488i	$0.359747\pi$
$192$	0	0
$193$	889.559	0.331771	0.165886	−	0.986145i	$-0.446952\pi$
$193$	889.559	0.331771	0.165886	+	0.986145i	$0.446952\pi$
$194$	−1553.90	−0.575071
$195$	0	0
$196$	1791.74	0.652968
$197$	3780.75	1.36735	0.683673	−	0.729788i	$-0.260381\pi$
$197$	3780.75	1.36735	0.683673	+	0.729788i	$0.260381\pi$
$198$	0	0
$199$	−118.565	−0.0422355	−0.0211178	−	0.999777i	$-0.506722\pi$
$199$	−118.565	−0.0422355	−0.0211178	+	0.999777i	$0.506722\pi$
$200$	1952.12	0.690177
$201$	0	0
$202$	468.934	0.163337
$203$	−5484.29	−1.89617
$204$	0	0
$205$	−1115.07	−0.379903
$206$	1769.20	0.598380
$207$	0	0
$208$	3190.33	1.06351
$209$	51.8948	0.0171753
$210$	0	0
$211$	−919.251	−0.299924	−0.149962	−	0.988692i	$-0.547915\pi$
$211$	−919.251	−0.299924	−0.149962	+	0.988692i	$0.547915\pi$
$212$	−376.809	−0.122073
$213$	0	0
$214$	−6618.41	−2.11414
$215$	−433.181	−0.137408
$216$	0	0
$217$	−1578.33	−0.493750
$218$	753.561	0.234117
$219$	0	0
$220$	−5.87482	−0.00180037
$221$	−1772.04	−0.539368
$222$	0	0
$223$	1530.63	0.459635	0.229817	−	0.973234i	$-0.426187\pi$
$223$	1530.63	0.459635	0.229817	+	0.973234i	$0.426187\pi$
$224$	−2952.68	−0.880732
$225$	0	0
$226$	1719.43	0.506082
$227$	−3079.49	−0.900411	−0.450205	−	0.892925i	$-0.648649\pi$
$227$	−3079.49	−0.900411	−0.450205	+	0.892925i	$0.648649\pi$
$228$	0	0
$229$	1241.32	0.358205	0.179103	−	0.983830i	$-0.442681\pi$
$229$	1241.32	0.358205	0.179103	+	0.983830i	$0.442681\pi$
$230$	−1592.67	−0.456597
$231$	0	0
$232$	2916.80	0.825420
$233$	421.662	0.118558	0.0592789	−	0.998241i	$-0.481120\pi$
$233$	421.662	0.118558	0.0592789	+	0.998241i	$0.481120\pi$
$234$	0	0
$235$	−792.623	−0.220021
$236$	780.612	0.215312
$237$	0	0
$238$	4769.12	1.29889
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−990.944	−0.264864	−0.132432	−	0.991192i	$-0.542279\pi$
$241$	−990.944	−0.264864	−0.132432	+	0.991192i	$0.542279\pi$
$242$	−4170.95	−1.10793
$243$	0	0
$244$	−69.4547	−0.0182229
$245$	4818.98	1.25663
$246$	0	0
$247$	3358.66	0.865208
$248$	839.427	0.214934
$249$	0	0
$250$	−3477.95	−0.879861
$251$	−745.327	−0.187429	−0.0937144	−	0.995599i	$-0.529874\pi$
$251$	−745.327	−0.187429	−0.0937144	+	0.995599i	$0.529874\pi$
$252$	0	0
$253$	67.7336	0.0168315
$254$	−7767.73	−1.91886
$255$	0	0
$256$	2670.21	0.651906
$257$	−1146.21	−0.278205	−0.139102	−	0.990278i	$-0.544422\pi$
$257$	−1146.21	−0.278205	−0.139102	+	0.990278i	$0.544422\pi$
$258$	0	0
$259$	−11960.7	−2.86951
$260$	−380.221	−0.0906935
$261$	0	0
$262$	753.119	0.177587
$263$	−6051.57	−1.41884	−0.709421	−	0.704785i	$-0.751044\pi$
$263$	−6051.57	−1.41884	−0.709421	+	0.704785i	$0.751044\pi$
$264$	0	0
$265$	−1013.45	−0.234927
$266$	−9039.21	−2.08357
$267$	0	0
$268$	1085.66	0.247453
$269$	−4009.38	−0.908758	−0.454379	−	0.890808i	$-0.650139\pi$
$269$	−4009.38	−0.908758	−0.454379	+	0.890808i	$0.650139\pi$
$270$	0	0
$271$	6262.91	1.40386	0.701928	−	0.712248i	$-0.252323\pi$
$271$	6262.91	1.40386	0.701928	+	0.712248i	$0.252323\pi$
$272$	−3147.33	−0.701600
$273$	0	0
$274$	449.747	0.0991613
$275$	66.0556	0.0144847
$276$	0	0
$277$	−6892.58	−1.49507	−0.747536	−	0.664222i	$-0.768763\pi$
$277$	−6892.58	−1.49507	−0.747536	+	0.664222i	$0.768763\pi$
$278$	−1067.23	−0.230246
$279$	0	0
$280$	−3459.04	−0.738275
$281$	−1312.21	−0.278577	−0.139288	−	0.990252i	$-0.544482\pi$
$281$	−1312.21	−0.278577	−0.139288	+	0.990252i	$0.544482\pi$
$282$	0	0
$283$	5425.57	1.13963	0.569817	−	0.821771i	$-0.307014\pi$
$283$	5425.57	1.13963	0.569817	+	0.821771i	$0.307014\pi$
$284$	−1654.26	−0.345641
$285$	0	0
$286$	87.0003	0.0179875
$287$	−8260.17	−1.69889
$288$	0	0
$289$	−3164.84	−0.644177
$290$	−2320.77	−0.469932
$291$	0	0
$292$	284.176	0.0569526
$293$	7531.35	1.50166	0.750830	−	0.660496i	$-0.229654\pi$
$293$	7531.35	1.50166	0.750830	+	0.660496i	$0.229654\pi$
$294$	0	0
$295$	2099.50	0.414364
$296$	6361.26	1.24912
$297$	0	0
$298$	−4446.16	−0.864292
$299$	4383.75	0.847888
$300$	0	0
$301$	−3208.89	−0.614476
$302$	1051.25	0.200307
$303$	0	0
$304$	5965.34	1.12545
$305$	−186.802	−0.0350697
$306$	0	0
$307$	591.840	0.110026	0.0550132	−	0.998486i	$-0.482480\pi$
$307$	591.840	0.110026	0.0550132	+	0.998486i	$0.482480\pi$
$308$	−43.5191	−0.00805107
$309$	0	0
$310$	−667.895	−0.122367
$311$	−9484.96	−1.72940	−0.864699	−	0.502290i	$-0.832491\pi$
$311$	−9484.96	−1.72940	−0.864699	+	0.502290i	$0.832491\pi$
$312$	0	0
$313$	−4919.24	−0.888344	−0.444172	−	0.895942i	$-0.646502\pi$
$313$	−4919.24	−0.888344	−0.444172	+	0.895942i	$0.646502\pi$
$314$	8851.51	1.59083
$315$	0	0
$316$	116.471	0.0207341
$317$	496.312	0.0879358	0.0439679	−	0.999033i	$-0.486000\pi$
$317$	496.312	0.0879358	0.0439679	+	0.999033i	$0.486000\pi$
$318$	0	0
$319$	98.6986	0.0173231
$320$	1708.60	0.298480
$321$	0	0
$322$	−11798.1	−2.04186
$323$	−3313.39	−0.570781
$324$	0	0
$325$	4275.15	0.729669
$326$	6103.37	1.03691
$327$	0	0
$328$	4393.14	0.739545
$329$	−5871.53	−0.983915
$330$	0	0
$331$	−2231.61	−0.370575	−0.185287	−	0.982684i	$-0.559322\pi$
$331$	−2231.61	−0.370575	−0.185287	+	0.982684i	$0.559322\pi$
$332$	1524.55	0.252019
$333$	0	0
$334$	−7450.61	−1.22060
$335$	2919.95	0.476220
$336$	0	0
$337$	−3569.71	−0.577016	−0.288508	−	0.957478i	$-0.593159\pi$
$337$	−3569.71	−0.577016	−0.288508	+	0.957478i	$0.593159\pi$
$338$	−1256.24	−0.202161
$339$	0	0
$340$	375.097	0.0598308
$341$	28.4045	0.00451082
$342$	0	0
$343$	23216.8	3.65478
$344$	1706.64	0.267487
$345$	0	0
$346$	2571.71	0.399583
$347$	5048.57	0.781042	0.390521	−	0.920594i	$-0.372295\pi$
$347$	5048.57	0.781042	0.390521	+	0.920594i	$0.372295\pi$
$348$	0	0
$349$	1860.49	0.285358	0.142679	−	0.989769i	$-0.454428\pi$
$349$	1860.49	0.285358	0.142679	+	0.989769i	$0.454428\pi$
$350$	−11505.8	−1.75717
$351$	0	0
$352$	53.1381	0.00804622
$353$	−11335.4	−1.70912	−0.854562	−	0.519349i	$-0.826175\pi$
$353$	−11335.4	−1.70912	−0.854562	+	0.519349i	$0.826175\pi$
$354$	0	0
$355$	−4449.20	−0.665181
$356$	1099.27	0.163654
$357$	0	0
$358$	−8428.50	−1.24430
$359$	−3148.65	−0.462895	−0.231447	−	0.972847i	$-0.574346\pi$
$359$	−3148.65	−0.462895	−0.231447	+	0.972847i	$0.574346\pi$
$360$	0	0
$361$	−578.917	−0.0844025
$362$	7973.80	1.15772
$363$	0	0
$364$	−2816.57	−0.405573
$365$	764.307	0.109604
$366$	0	0
$367$	6448.54	0.917197	0.458598	−	0.888644i	$-0.348352\pi$
$367$	6448.54	0.917197	0.458598	+	0.888644i	$0.348352\pi$
$368$	7786.01	1.10292
$369$	0	0
$370$	−5061.37	−0.711157
$371$	−7507.35	−1.05057
$372$	0	0
$373$	−1545.86	−0.214588	−0.107294	−	0.994227i	$-0.534219\pi$
$373$	−1545.86	−0.214588	−0.107294	+	0.994227i	$0.534219\pi$
$374$	−85.8279	−0.0118665
$375$	0	0
$376$	3122.75	0.428308
$377$	6387.82	0.872651
$378$	0	0
$379$	−6544.96	−0.887050	−0.443525	−	0.896262i	$-0.646272\pi$
$379$	−6544.96	−0.887050	−0.443525	+	0.896262i	$0.646272\pi$
$380$	−710.945	−0.0959755
$381$	0	0
$382$	7058.21	0.945366
$383$	−14067.3	−1.87677	−0.938386	−	0.345590i	$-0.887679\pi$
$383$	−14067.3	−1.87677	−0.938386	+	0.345590i	$0.887679\pi$
$384$	0	0
$385$	−117.047	−0.0154942
$386$	2788.50	0.367697
$387$	0	0
$388$	−905.345	−0.118459
$389$	−7313.15	−0.953191	−0.476595	−	0.879123i	$-0.658129\pi$
$389$	−7313.15	−0.953191	−0.476595	+	0.879123i	$0.658129\pi$
$390$	0	0
$391$	−4324.67	−0.559355
$392$	−18985.7	−2.44623
$393$	0	0
$394$	11851.5	1.51541
$395$	313.254	0.0399025
$396$	0	0
$397$	3707.91	0.468752	0.234376	−	0.972146i	$-0.424695\pi$
$397$	3707.91	0.468752	0.234376	+	0.972146i	$0.424695\pi$
$398$	−371.667	−0.0468090
$399$	0	0
$400$	7593.12	0.949140
$401$	−6435.24	−0.801398	−0.400699	−	0.916210i	$-0.631233\pi$
$401$	−6435.24	−0.801398	−0.400699	+	0.916210i	$0.631233\pi$
$402$	0	0
$403$	1838.35	0.227233
$404$	273.213	0.0336457
$405$	0	0
$406$	−17191.6	−2.10149
$407$	215.252	0.0262153
$408$	0	0
$409$	−9429.39	−1.13998	−0.569992	−	0.821650i	$-0.693054\pi$
$409$	−9429.39	−1.13998	−0.569992	+	0.821650i	$0.693054\pi$
$410$	−3495.43	−0.421041
$411$	0	0
$412$	1030.78	0.123260
$413$	15552.5	1.85300
$414$	0	0
$415$	4100.35	0.485008
$416$	3439.12	0.405329
$417$	0	0
$418$	162.675	0.0190351
$419$	8421.77	0.981934	0.490967	−	0.871178i	$-0.336644\pi$
$419$	8421.77	0.981934	0.490967	+	0.871178i	$0.336644\pi$
$420$	0	0
$421$	−8329.85	−0.964304	−0.482152	−	0.876088i	$-0.660145\pi$
$421$	−8329.85	−0.964304	−0.482152	+	0.876088i	$0.660145\pi$
$422$	−2881.58	−0.332401
$423$	0	0
$424$	3992.75	0.457324
$425$	−4217.53	−0.481365
$426$	0	0
$427$	−1383.78	−0.156828
$428$	−3856.06	−0.435490
$429$	0	0
$430$	−1357.89	−0.152287
$431$	5846.23	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$431$	5846.23	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$432$	0	0
$433$	11619.0	1.28954	0.644772	−	0.764375i	$-0.276952\pi$
$433$	11619.0	1.28954	0.644772	+	0.764375i	$0.276952\pi$
$434$	−4947.58	−0.547216
$435$	0	0
$436$	439.044	0.0482257
$437$	8196.82	0.897270
$438$	0	0
$439$	10508.6	1.14248	0.571240	−	0.820783i	$-0.306463\pi$
$439$	10508.6	1.14248	0.571240	+	0.820783i	$0.306463\pi$
$440$	62.2509	0.00674476
$441$	0	0
$442$	−5554.82	−0.597773
$443$	−11819.1	−1.26759	−0.633793	−	0.773503i	$-0.718503\pi$
$443$	−11819.1	−1.26759	−0.633793	+	0.773503i	$0.718503\pi$
$444$	0	0
$445$	2956.53	0.314951
$446$	4798.07	0.509406
$447$	0	0
$448$	12656.8	1.33478
$449$	9424.13	0.990539	0.495269	−	0.868739i	$-0.335069\pi$
$449$	9424.13	0.990539	0.495269	+	0.868739i	$0.335069\pi$
$450$	0	0
$451$	148.655	0.0155208
$452$	1001.78	0.104248
$453$	0	0
$454$	−9653.30	−0.997912
$455$	−7575.32	−0.780520
$456$	0	0
$457$	8426.48	0.862526	0.431263	−	0.902226i	$-0.358068\pi$
$457$	8426.48	0.862526	0.431263	+	0.902226i	$0.358068\pi$
$458$	3891.18	0.396993
$459$	0	0
$460$	−927.931	−0.0940543
$461$	11913.7	1.20363	0.601817	−	0.798634i	$-0.294443\pi$
$461$	11913.7	1.20363	0.601817	+	0.798634i	$0.294443\pi$
$462$	0	0
$463$	10300.0	1.03386	0.516932	−	0.856026i	$-0.327074\pi$
$463$	10300.0	1.03386	0.516932	+	0.856026i	$0.327074\pi$
$464$	11345.5	1.13513
$465$	0	0
$466$	1321.78	0.131396
$467$	1463.70	0.145036	0.0725179	−	0.997367i	$-0.476897\pi$
$467$	1463.70	0.145036	0.0725179	+	0.997367i	$0.476897\pi$
$468$	0	0
$469$	21630.2	2.12961
$470$	−2484.64	−0.243846
$471$	0	0
$472$	−8271.54	−0.806628
$473$	57.7490	0.00561375
$474$	0	0
$475$	7993.75	0.772165
$476$	2778.62	0.267558
$477$	0	0
$478$	749.194	0.0716890
$479$	−400.076	−0.0381627	−0.0190813	−	0.999818i	$-0.506074\pi$
$479$	−400.076	−0.0381627	−0.0190813	+	0.999818i	$0.506074\pi$
$480$	0	0
$481$	13931.2	1.32060
$482$	−3106.32	−0.293545
$483$	0	0
$484$	−2430.10	−0.228222
$485$	−2434.97	−0.227972
$486$	0	0
$487$	14121.7	1.31400	0.656999	−	0.753891i	$-0.271825\pi$
$487$	14121.7	1.31400	0.656999	+	0.753891i	$0.271825\pi$
$488$	735.957	0.0682689
$489$	0	0
$490$	15106.1	1.39270
$491$	11270.3	1.03589	0.517945	−	0.855414i	$-0.326697\pi$
$491$	11270.3	1.03589	0.517945	+	0.855414i	$0.326697\pi$
$492$	0	0
$493$	−6301.73	−0.575691
$494$	10528.4	0.958896
$495$	0	0
$496$	3265.11	0.295580
$497$	−32958.5	−2.97463
$498$	0	0
$499$	11360.2	1.01915	0.509573	−	0.860428i	$-0.329803\pi$
$499$	11360.2	1.01915	0.509573	+	0.860428i	$0.329803\pi$
$500$	−2026.35	−0.181242
$501$	0	0
$502$	−2336.38	−0.207724
$503$	−16393.5	−1.45318	−0.726589	−	0.687073i	$-0.758895\pi$
$503$	−16393.5	−1.45318	−0.726589	+	0.687073i	$0.758895\pi$
$504$	0	0
$505$	734.821	0.0647507
$506$	212.325	0.0186541
$507$	0	0
$508$	−4525.69	−0.395265
$509$	9306.18	0.810391	0.405196	−	0.914230i	$-0.367203\pi$
$509$	9306.18	0.810391	0.405196	+	0.914230i	$0.367203\pi$
$510$	0	0
$511$	5661.77	0.490141
$512$	−5545.88	−0.478703
$513$	0	0
$514$	−3593.03	−0.308330
$515$	2772.35	0.237212
$516$	0	0
$517$	105.668	0.00898888
$518$	−37493.2	−3.18023
$519$	0	0
$520$	4028.91	0.339768
$521$	19266.7	1.62014	0.810068	−	0.586337i	$-0.199430\pi$
$521$	19266.7	1.62014	0.810068	+	0.586337i	$0.199430\pi$
$522$	0	0
$523$	−11768.3	−0.983920	−0.491960	−	0.870618i	$-0.663719\pi$
$523$	−11768.3	−0.983920	−0.491960	+	0.870618i	$0.663719\pi$
$524$	438.787	0.0365811
$525$	0	0
$526$	−18969.9	−1.57248
$527$	−1813.58	−0.149906
$528$	0	0
$529$	−1468.45	−0.120691
$530$	−3176.86	−0.260366
$531$	0	0
$532$	−5266.48	−0.429194
$533$	9621.01	0.781861
$534$	0	0
$535$	−10371.1	−0.838095
$536$	−11503.9	−0.927041
$537$	0	0
$538$	−12568.2	−1.00716
$539$	−642.437	−0.0513390
$540$	0	0
$541$	17595.6	1.39833	0.699164	−	0.714962i	$-0.253556\pi$
$541$	17595.6	1.39833	0.699164	+	0.714962i	$0.253556\pi$
$542$	19632.4	1.55587
$543$	0	0
$544$	−3392.77	−0.267397
$545$	1180.83	0.0928097
$546$	0	0
$547$	15293.3	1.19542	0.597710	−	0.801713i	$-0.296077\pi$
$547$	15293.3	1.19542	0.597710	+	0.801713i	$0.296077\pi$
$548$	262.034	0.0204262
$549$	0	0
$550$	207.065	0.0160532
$551$	11944.1	0.923474
$552$	0	0
$553$	2320.50	0.178440
$554$	−21606.2	−1.65696
$555$	0	0
$556$	−621.798	−0.0474283
$557$	−7410.55	−0.563725	−0.281863	−	0.959455i	$-0.590952\pi$
$557$	−7410.55	−0.563725	−0.281863	+	0.959455i	$0.590952\pi$
$558$	0	0
$559$	3737.54	0.282793
$560$	−13454.6	−1.01529
$561$	0	0
$562$	−4113.40	−0.308743
$563$	−12331.7	−0.923128	−0.461564	−	0.887107i	$-0.652712\pi$
$563$	−12331.7	−0.923128	−0.461564	+	0.887107i	$0.652712\pi$
$564$	0	0
$565$	2694.35	0.200623
$566$	17007.6	1.26304
$567$	0	0
$568$	17528.9	1.29488
$569$	5561.27	0.409738	0.204869	−	0.978789i	$-0.434323\pi$
$569$	5561.27	0.409738	0.204869	+	0.978789i	$0.434323\pi$
$570$	0	0
$571$	−24919.9	−1.82638	−0.913192	−	0.407529i	$-0.866390\pi$
$571$	−24919.9	−1.82638	−0.913192	+	0.407529i	$0.866390\pi$
$572$	50.6887	0.00370525
$573$	0	0
$574$	−25893.2	−1.88286
$575$	10433.5	0.756708
$576$	0	0
$577$	1269.21	0.0915733	0.0457866	−	0.998951i	$-0.485421\pi$
$577$	1269.21	0.0915733	0.0457866	+	0.998951i	$0.485421\pi$
$578$	−9920.84	−0.713931
$579$	0	0
$580$	−1352.14	−0.0968011
$581$	30374.3	2.16891
$582$	0	0
$583$	135.107	0.00959784
$584$	−3011.19	−0.213363
$585$	0	0
$586$	23608.5	1.66427
$587$	3024.29	0.212651	0.106325	−	0.994331i	$-0.466092\pi$
$587$	3024.29	0.212651	0.106325	+	0.994331i	$0.466092\pi$
$588$	0	0
$589$	3437.38	0.240467
$590$	6581.30	0.459234
$591$	0	0
$592$	24743.3	1.71781
$593$	7958.20	0.551103	0.275552	−	0.961286i	$-0.411140\pi$
$593$	7958.20	0.551103	0.275552	+	0.961286i	$0.411140\pi$
$594$	0	0
$595$	7473.23	0.514912
$596$	−2590.45	−0.178035
$597$	0	0
$598$	13741.7	0.939702
$599$	−15551.4	−1.06079	−0.530395	−	0.847751i	$-0.677956\pi$
$599$	−15551.4	−1.06079	−0.530395	+	0.847751i	$0.677956\pi$
$600$	0	0
$601$	12402.8	0.841798	0.420899	−	0.907108i	$-0.361715\pi$
$601$	12402.8	0.841798	0.420899	+	0.907108i	$0.361715\pi$
$602$	−10058.9	−0.681015
$603$	0	0
$604$	612.486	0.0412611
$605$	−6535.89	−0.439209
$606$	0	0
$607$	−7545.59	−0.504557	−0.252278	−	0.967655i	$-0.581180\pi$
$607$	−7545.59	−0.504557	−0.252278	+	0.967655i	$0.581180\pi$
$608$	6430.53	0.428935
$609$	0	0
$610$	−585.568	−0.0388672
$611$	6838.85	0.452815
$612$	0	0
$613$	3744.35	0.246710	0.123355	−	0.992363i	$-0.460635\pi$
$613$	3744.35	0.246710	0.123355	+	0.992363i	$0.460635\pi$
$614$	1855.24	0.121940
$615$	0	0
$616$	461.138	0.0301619
$617$	−19319.9	−1.26060	−0.630300	−	0.776352i	$-0.717068\pi$
$617$	−19319.9	−1.26060	−0.630300	+	0.776352i	$0.717068\pi$
$618$	0	0
$619$	−4231.21	−0.274744	−0.137372	−	0.990520i	$-0.543866\pi$
$619$	−4231.21	−0.274744	−0.137372	+	0.990520i	$0.543866\pi$
$620$	−389.133	−0.0252064
$621$	0	0
$622$	−29732.5	−1.91667
$623$	21901.2	1.40843
$624$	0	0
$625$	7158.95	0.458173
$626$	−15420.3	−0.984538
$627$	0	0
$628$	5157.12	0.327694
$629$	−13743.4	−0.871204
$630$	0	0
$631$	19238.3	1.21373	0.606866	−	0.794804i	$-0.292426\pi$
$631$	19238.3	1.21373	0.606866	+	0.794804i	$0.292426\pi$
$632$	−1234.15	−0.0776769
$633$	0	0
$634$	1555.79	0.0974579
$635$	−12172.1	−0.760683
$636$	0	0
$637$	−41578.8	−2.58620
$638$	309.391	0.0191989
$639$	0	0
$640$	8544.70	0.527748
$641$	−2485.24	−0.153138	−0.0765688	−	0.997064i	$-0.524396\pi$
$641$	−2485.24	−0.153138	−0.0765688	+	0.997064i	$0.524396\pi$
$642$	0	0
$643$	26682.8	1.63649	0.818247	−	0.574867i	$-0.194946\pi$
$643$	26682.8	1.63649	0.818247	+	0.574867i	$0.194946\pi$
$644$	−6873.86	−0.420602
$645$	0	0
$646$	−10386.5	−0.632588
$647$	−27899.1	−1.69525	−0.847624	−	0.530598i	$-0.821967\pi$
$647$	−27899.1	−1.69525	−0.847624	+	0.530598i	$0.821967\pi$
$648$	0	0
$649$	−279.892	−0.0169287
$650$	13401.3	0.808681
$651$	0	0
$652$	3555.98	0.213594
$653$	2943.64	0.176406	0.0882032	−	0.996103i	$-0.471888\pi$
$653$	2943.64	0.176406	0.0882032	+	0.996103i	$0.471888\pi$
$654$	0	0
$655$	1180.14	0.0703998
$656$	17088.0	1.01703
$657$	0	0
$658$	−18405.5	−1.09046
$659$	−23688.2	−1.40025	−0.700124	−	0.714021i	$-0.746872\pi$
$659$	−23688.2	−1.40025	−0.700124	+	0.714021i	$0.746872\pi$
$660$	0	0
$661$	23515.5	1.38373	0.691867	−	0.722025i	$-0.256788\pi$
$661$	23515.5	1.38373	0.691867	+	0.722025i	$0.256788\pi$
$662$	−6995.43	−0.410702
$663$	0	0
$664$	−16154.4	−0.944148
$665$	−14164.5	−0.825977
$666$	0	0
$667$	15589.5	0.904989
$668$	−4340.92	−0.251430
$669$	0	0
$670$	9153.17	0.527788
$671$	24.9033	0.00143276
$672$	0	0
$673$	6923.43	0.396551	0.198275	−	0.980146i	$-0.436466\pi$
$673$	6923.43	0.396551	0.198275	+	0.980146i	$0.436466\pi$
$674$	−11190.0	−0.639498
$675$	0	0
$676$	−731.920	−0.0416431
$677$	22544.6	1.27985	0.639925	−	0.768437i	$-0.278965\pi$
$677$	22544.6	1.27985	0.639925	+	0.768437i	$0.278965\pi$
$678$	0	0
$679$	−18037.6	−1.01947
$680$	−3974.61	−0.224146
$681$	0	0
$682$	89.0396	0.00499927
$683$	−18136.0	−1.01604	−0.508020	−	0.861345i	$-0.669623\pi$
$683$	−18136.0	−1.01604	−0.508020	+	0.861345i	$0.669623\pi$
$684$	0	0
$685$	704.755	0.0393099
$686$	72777.8	4.05054
$687$	0	0
$688$	6638.28	0.367852
$689$	8744.16	0.483492
$690$	0	0
$691$	−16072.8	−0.884858	−0.442429	−	0.896803i	$-0.645883\pi$
$691$	−16072.8	−0.884858	−0.442429	+	0.896803i	$0.645883\pi$
$692$	1498.34	0.0823100
$693$	0	0
$694$	15825.8	0.865617
$695$	−1672.36	−0.0912750
$696$	0	0
$697$	−9491.35	−0.515797
$698$	5832.10	0.316258
$699$	0	0
$700$	−6703.57	−0.361959
$701$	−10999.4	−0.592640	−0.296320	−	0.955089i	$-0.595760\pi$
$701$	−10999.4	−0.592640	−0.296320	+	0.955089i	$0.595760\pi$
$702$	0	0
$703$	26048.8	1.39751
$704$	−227.780	−0.0121943
$705$	0	0
$706$	−35533.0	−1.89420
$707$	5443.36	0.289559
$708$	0	0
$709$	9279.33	0.491527	0.245763	−	0.969330i	$-0.420961\pi$
$709$	9279.33	0.491527	0.245763	+	0.969330i	$0.420961\pi$
$710$	−13946.9	−0.737210
$711$	0	0
$712$	−11648.1	−0.613103
$713$	4486.50	0.235653
$714$	0	0
$715$	136.330	0.00713070
$716$	−4910.67	−0.256313
$717$	0	0
$718$	−9870.07	−0.513019
$719$	4475.10	0.232119	0.116059	−	0.993242i	$-0.462974\pi$
$719$	4475.10	0.232119	0.116059	+	0.993242i	$0.462974\pi$
$720$	0	0
$721$	20536.8	1.06079
$722$	−1814.73	−0.0935421
$723$	0	0
$724$	4645.75	0.238478
$725$	15203.3	0.778808
$726$	0	0
$727$	24722.2	1.26120	0.630601	−	0.776107i	$-0.282808\pi$
$727$	24722.2	1.26120	0.630601	+	0.776107i	$0.282808\pi$
$728$	29845.0	1.51941
$729$	0	0
$730$	2395.87	0.121473
$731$	−3687.18	−0.186560
$732$	0	0
$733$	2743.32	0.138236	0.0691180	−	0.997608i	$-0.477982\pi$
$733$	2743.32	0.138236	0.0691180	+	0.997608i	$0.477982\pi$
$734$	20214.3	1.01652
$735$	0	0
$736$	8393.19	0.420349
$737$	−389.269	−0.0194558
$738$	0	0
$739$	19302.2	0.960814	0.480407	−	0.877046i	$-0.340489\pi$
$739$	19302.2	0.960814	0.480407	+	0.877046i	$0.340489\pi$
$740$	−2948.89	−0.146491
$741$	0	0
$742$	−23533.3	−1.16433
$743$	33863.2	1.67203	0.836017	−	0.548704i	$-0.184879\pi$
$743$	33863.2	1.67203	0.836017	+	0.548704i	$0.184879\pi$
$744$	0	0
$745$	−6967.15	−0.342626
$746$	−4845.80	−0.237825
$747$	0	0
$748$	−50.0056	−0.00244437
$749$	−76826.1	−3.74789
$750$	0	0
$751$	−30272.2	−1.47090	−0.735451	−	0.677578i	$-0.763029\pi$
$751$	−30272.2	−1.47090	−0.735451	+	0.677578i	$0.763029\pi$
$752$	12146.5	0.589014
$753$	0	0
$754$	20023.9	0.967145
$755$	1647.31	0.0794064
$756$	0	0
$757$	−2202.96	−0.105770	−0.0528849	−	0.998601i	$-0.516842\pi$
$757$	−2202.96	−0.105770	−0.0528849	+	0.998601i	$0.516842\pi$
$758$	−20516.5	−0.983104
$759$	0	0
$760$	7533.32	0.359556
$761$	4867.98	0.231885	0.115942	−	0.993256i	$-0.463011\pi$
$761$	4867.98	0.231885	0.115942	+	0.993256i	$0.463011\pi$
$762$	0	0
$763$	8747.28	0.415037
$764$	4112.30	0.194736
$765$	0	0
$766$	−44096.7	−2.08000
$767$	−18114.7	−0.852784
$768$	0	0
$769$	−1744.23	−0.0817928	−0.0408964	−	0.999163i	$-0.513021\pi$
$769$	−1744.23	−0.0817928	−0.0408964	+	0.999163i	$0.513021\pi$
$770$	−366.907	−0.0171719
$771$	0	0
$772$	1624.66	0.0757418
$773$	31838.9	1.48145	0.740727	−	0.671806i	$-0.234481\pi$
$773$	31838.9	1.48145	0.740727	+	0.671806i	$0.234481\pi$
$774$	0	0
$775$	4375.35	0.202797
$776$	9593.23	0.443785
$777$	0	0
$778$	−22924.5	−1.05641
$779$	17989.5	0.827397
$780$	0	0
$781$	593.140	0.0271757
$782$	−13556.6	−0.619925
$783$	0	0
$784$	−73848.5	−3.36409
$785$	13870.3	0.630642
$786$	0	0
$787$	−6132.91	−0.277782	−0.138891	−	0.990308i	$-0.544354\pi$
$787$	−6132.91	−0.277782	−0.138891	+	0.990308i	$0.544354\pi$
$788$	6905.02	0.312159
$789$	0	0
$790$	981.957	0.0442234
$791$	19959.0	0.897169
$792$	0	0
$793$	1611.75	0.0721752
$794$	11623.2	0.519510
$795$	0	0
$796$	−216.543	−0.00964217
$797$	40316.5	1.79182	0.895911	−	0.444233i	$-0.146524\pi$
$797$	40316.5	1.79182	0.895911	+	0.444233i	$0.146524\pi$
$798$	0	0
$799$	−6746.68	−0.298724
$800$	8185.26	0.361741
$801$	0	0
$802$	−20172.6	−0.888178
$803$	−101.893	−0.00447785
$804$	0	0
$805$	−18487.6	−0.809443
$806$	5762.68	0.251838
$807$	0	0
$808$	−2895.03	−0.126048
$809$	−1070.47	−0.0465211	−0.0232605	−	0.999729i	$-0.507405\pi$
$809$	−1070.47	−0.0465211	−0.0232605	+	0.999729i	$0.507405\pi$
$810$	0	0
$811$	31220.1	1.35177	0.675885	−	0.737007i	$-0.263761\pi$
$811$	31220.1	1.35177	0.675885	+	0.737007i	$0.263761\pi$
$812$	−10016.3	−0.432886
$813$	0	0
$814$	674.750	0.0290540
$815$	9564.00	0.411058
$816$	0	0
$817$	6988.53	0.299263
$818$	−29558.4	−1.26343
$819$	0	0
$820$	−2036.53	−0.0867301
$821$	42159.9	1.79219	0.896096	−	0.443860i	$-0.146391\pi$
$821$	42159.9	1.79219	0.896096	+	0.443860i	$0.146391\pi$
$822$	0	0
$823$	1888.54	0.0799884	0.0399942	−	0.999200i	$-0.487266\pi$
$823$	1888.54	0.0799884	0.0399942	+	0.999200i	$0.487266\pi$
$824$	−10922.4	−0.461773
$825$	0	0
$826$	48752.5	2.05365
$827$	8491.02	0.357027	0.178514	−	0.983937i	$-0.442871\pi$
$827$	8491.02	0.357027	0.178514	+	0.983937i	$0.442871\pi$
$828$	0	0
$829$	30959.9	1.29708	0.648541	−	0.761180i	$-0.275380\pi$
$829$	30959.9	1.29708	0.648541	+	0.761180i	$0.275380\pi$
$830$	12853.4	0.537527
$831$	0	0
$832$	−14742.0	−0.614288
$833$	41018.5	1.70613
$834$	0	0
$835$	−11675.1	−0.483874
$836$	94.7787	0.00392104
$837$	0	0
$838$	26399.7	1.08826
$839$	−40700.5	−1.67478	−0.837389	−	0.546608i	$-0.815919\pi$
$839$	−40700.5	−1.67478	−0.837389	+	0.546608i	$0.815919\pi$
$840$	0	0
$841$	−1672.64	−0.0685818
$842$	−26111.6	−1.06872
$843$	0	0
$844$	−1678.88	−0.0684711
$845$	−1968.54	−0.0801416
$846$	0	0
$847$	−48416.1	−1.96410
$848$	15530.6	0.628918
$849$	0	0
$850$	−13220.7	−0.533490
$851$	33999.1	1.36954
$852$	0	0
$853$	38855.2	1.55964	0.779822	−	0.626001i	$-0.215309\pi$
$853$	38855.2	1.55964	0.779822	+	0.626001i	$0.215309\pi$
$854$	−4337.73	−0.173810
$855$	0	0
$856$	40859.7	1.63149
$857$	−44433.6	−1.77109	−0.885544	−	0.464555i	$-0.846214\pi$
$857$	−44433.6	−1.77109	−0.885544	+	0.464555i	$0.846214\pi$
$858$	0	0
$859$	−7511.88	−0.298372	−0.149186	−	0.988809i	$-0.547665\pi$
$859$	−7511.88	−0.298372	−0.149186	+	0.988809i	$0.547665\pi$
$860$	−791.146	−0.0313696
$861$	0	0
$862$	18326.2	0.724122
$863$	−8869.88	−0.349866	−0.174933	−	0.984580i	$-0.555971\pi$
$863$	−8869.88	−0.349866	−0.174933	+	0.984580i	$0.555971\pi$
$864$	0	0
$865$	4029.87	0.158404
$866$	36422.0	1.42918
$867$	0	0
$868$	−2882.59	−0.112721
$869$	−41.7610	−0.00163020
$870$	0	0
$871$	−25193.7	−0.980086
$872$	−4652.21	−0.180669
$873$	0	0
$874$	25694.6	0.994430
$875$	−40371.9	−1.55979
$876$	0	0
$877$	−42946.9	−1.65361	−0.826804	−	0.562491i	$-0.809843\pi$
$877$	−42946.9	−1.65361	−0.826804	+	0.562491i	$0.809843\pi$
$878$	32941.4	1.26619
$879$	0	0
$880$	242.137	0.00927548
$881$	10522.8	0.402409	0.201205	−	0.979549i	$-0.435514\pi$
$881$	10522.8	0.402409	0.201205	+	0.979549i	$0.435514\pi$
$882$	0	0
$883$	−33736.0	−1.28574	−0.642870	−	0.765975i	$-0.722256\pi$
$883$	−33736.0	−1.28574	−0.642870	+	0.765975i	$0.722256\pi$
$884$	−3236.38	−0.123135
$885$	0	0
$886$	−37049.3	−1.40485
$887$	−13266.6	−0.502196	−0.251098	−	0.967962i	$-0.580792\pi$
$887$	−13266.6	−0.502196	−0.251098	+	0.967962i	$0.580792\pi$
$888$	0	0
$889$	−90167.3	−3.40170
$890$	9267.85	0.349055
$891$	0	0
$892$	2795.48	0.104932
$893$	12787.4	0.479187
$894$	0	0
$895$	−13207.5	−0.493271
$896$	63296.8	2.36004
$897$	0	0
$898$	29541.8	1.09780
$899$	6537.54	0.242535
$900$	0	0
$901$	−8626.32	−0.318961
$902$	465.989	0.0172015
$903$	0	0
$904$	−10615.1	−0.390546
$905$	12495.0	0.458947
$906$	0	0
$907$	−39503.9	−1.44620	−0.723100	−	0.690743i	$-0.757284\pi$
$907$	−39503.9	−1.44620	−0.723100	+	0.690743i	$0.757284\pi$
$908$	−5624.27	−0.205559
$909$	0	0
$910$	−23746.4	−0.865038
$911$	−45573.3	−1.65742	−0.828711	−	0.559676i	$-0.810925\pi$
$911$	−45573.3	−1.65742	−0.828711	+	0.559676i	$0.810925\pi$
$912$	0	0
$913$	−546.633	−0.0198148
$914$	26414.5	0.955924
$915$	0	0
$916$	2267.10	0.0817765
$917$	8742.16	0.314822
$918$	0	0
$919$	−43624.1	−1.56586	−0.782930	−	0.622110i	$-0.786276\pi$
$919$	−43624.1	−1.56586	−0.782930	+	0.622110i	$0.786276\pi$
$920$	9832.56	0.352358
$921$	0	0
$922$	37345.9	1.33397
$923$	38388.3	1.36898
$924$	0	0
$925$	33156.8	1.17858
$926$	32287.3	1.14582
$927$	0	0
$928$	12230.2	0.432625
$929$	40824.6	1.44178	0.720889	−	0.693050i	$-0.243734\pi$
$929$	40824.6	1.44178	0.720889	+	0.693050i	$0.243734\pi$
$930$	0	0
$931$	−77744.9	−2.73683
$932$	770.107	0.0270662
$933$	0	0
$934$	4588.25	0.160741
$935$	−134.493	−0.00470415
$936$	0	0
$937$	2247.16	0.0783474	0.0391737	−	0.999232i	$-0.487527\pi$
$937$	2247.16	0.0783474	0.0391737	+	0.999232i	$0.487527\pi$
$938$	67804.2	2.36022
$939$	0	0
$940$	−1447.62	−0.0502298
$941$	−52797.3	−1.82906	−0.914528	−	0.404522i	$-0.867438\pi$
$941$	−52797.3	−1.82906	−0.914528	+	0.404522i	$0.867438\pi$
$942$	0	0
$943$	23480.1	0.810835
$944$	−32173.7	−1.10929
$945$	0	0
$946$	181.026	0.00622163
$947$	−8350.80	−0.286552	−0.143276	−	0.989683i	$-0.545764\pi$
$947$	−8350.80	−0.286552	−0.143276	+	0.989683i	$0.545764\pi$
$948$	0	0
$949$	−6594.53	−0.225572
$950$	25058.0	0.855779
$951$	0	0
$952$	−29442.8	−1.00236
$953$	18670.3	0.634616	0.317308	−	0.948323i	$-0.397221\pi$
$953$	18670.3	0.634616	0.317308	+	0.948323i	$0.397221\pi$
$954$	0	0
$955$	11060.2	0.374766
$956$	436.500	0.0147672
$957$	0	0
$958$	−1254.12	−0.0422951
$959$	5220.63	0.175790
$960$	0	0
$961$	−27909.6	−0.936845
$962$	43670.1	1.46360
$963$	0	0
$964$	−1809.82	−0.0604673
$965$	4369.60	0.145764
$966$	0	0
$967$	34012.8	1.13110	0.565551	−	0.824713i	$-0.308663\pi$
$967$	34012.8	1.13110	0.565551	+	0.824713i	$0.308663\pi$
$968$	25749.9	0.854993
$969$	0	0
$970$	−7632.91	−0.252658
$971$	−45188.5	−1.49348	−0.746739	−	0.665117i	$-0.768381\pi$
$971$	−45188.5	−1.49348	−0.746739	+	0.665117i	$0.768381\pi$
$972$	0	0
$973$	−12388.4	−0.408174
$974$	44267.5	1.45628
$975$	0	0
$976$	2862.64	0.0938842
$977$	21244.1	0.695658	0.347829	−	0.937558i	$-0.386919\pi$
$977$	21244.1	0.695658	0.347829	+	0.937558i	$0.386919\pi$
$978$	0	0
$979$	−394.146	−0.0128672
$980$	8801.20	0.286882
$981$	0	0
$982$	35329.1	1.14806
$983$	−35357.8	−1.14724	−0.573621	−	0.819121i	$-0.694462\pi$
$983$	−35357.8	−1.14724	−0.573621	+	0.819121i	$0.694462\pi$
$984$	0	0
$985$	18571.4	0.600745
$986$	−19754.0	−0.638030
$987$	0	0
$988$	6134.12	0.197523
$989$	9121.49	0.293273
$990$	0	0
$991$	21457.7	0.687815	0.343908	−	0.939003i	$-0.388249\pi$
$991$	21457.7	0.687815	0.343908	+	0.939003i	$0.388249\pi$
$992$	3519.73	0.112653
$993$	0	0
$994$	−103315.	−3.29674
$995$	−582.404	−0.0185562
$996$	0	0
$997$	−26463.2	−0.840621	−0.420310	−	0.907380i	$-0.638079\pi$
$997$	−26463.2	−0.840621	−0.420310	+	0.907380i	$0.638079\pi$
$998$	35610.9	1.12950
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.47	✓	59
3.2	odd	2		2151.4.a.h.1.13	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.47	✓	59	1.1	even	1	trivial
2151.4.a.h.1.13	yes	59	3.2	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+3.13470 q^{2} +1.82636 q^{4} +4.91209 q^{5} +36.3874 q^{7} -19.3525 q^{8} +O(q^{10})\)	\(q+3.13470 q^{2} +1.82636 q^{4} +4.91209 q^{5} +36.3874 q^{7} -19.3525 q^{8} +15.3979 q^{10} -0.654849 q^{11} -42.3822 q^{13} +114.064 q^{14} -75.2753 q^{16} +41.8110 q^{17} -79.2470 q^{19} +8.97125 q^{20} -2.05276 q^{22} -103.434 q^{23} -100.871 q^{25} -132.855 q^{26} +66.4566 q^{28} -150.719 q^{29} -43.3756 q^{31} -81.1455 q^{32} +131.065 q^{34} +178.738 q^{35} -328.704 q^{37} -248.416 q^{38} -95.0614 q^{40} -227.006 q^{41} -88.1867 q^{43} -1.19599 q^{44} -324.234 q^{46} -161.362 q^{47} +981.045 q^{49} -316.202 q^{50} -77.4051 q^{52} -206.317 q^{53} -3.21668 q^{55} -704.189 q^{56} -472.461 q^{58} +427.414 q^{59} -38.0290 q^{61} -135.970 q^{62} +347.835 q^{64} -208.185 q^{65} +594.441 q^{67} +76.3620 q^{68} +560.292 q^{70} -905.766 q^{71} +155.597 q^{73} -1030.39 q^{74} -144.734 q^{76} -23.8283 q^{77} +63.7719 q^{79} -369.759 q^{80} -711.597 q^{82} +834.746 q^{83} +205.379 q^{85} -276.439 q^{86} +12.6730 q^{88} +601.888 q^{89} -1542.18 q^{91} -188.907 q^{92} -505.820 q^{94} -389.268 q^{95} -495.710 q^{97} +3075.29 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

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