LMFDB - Embedded newform 2151.4.a.b.1.3

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:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-4.68120 q^{2} +13.9136 q^{4} +9.12054 q^{5} -4.35896 q^{7} -27.6829 q^{8} +O(q^{10})$	\(q-4.68120 q^{2} +13.9136 q^{4} +9.12054 q^{5} -4.35896 q^{7} -27.6829 q^{8} -42.6951 q^{10} -0.264771 q^{11} +32.6234 q^{13} +20.4052 q^{14} +18.2800 q^{16} -37.6237 q^{17} -64.7883 q^{19} +126.900 q^{20} +1.23945 q^{22} +77.1313 q^{23} -41.8157 q^{25} -152.717 q^{26} -60.6490 q^{28} +194.379 q^{29} -176.731 q^{31} +135.891 q^{32} +176.124 q^{34} -39.7561 q^{35} +174.513 q^{37} +303.287 q^{38} -252.483 q^{40} +411.047 q^{41} -487.481 q^{43} -3.68393 q^{44} -361.067 q^{46} +25.3144 q^{47} -323.999 q^{49} +195.748 q^{50} +453.910 q^{52} -635.063 q^{53} -2.41486 q^{55} +120.669 q^{56} -909.928 q^{58} +414.809 q^{59} +521.703 q^{61} +827.314 q^{62} -782.371 q^{64} +297.543 q^{65} +982.659 q^{67} -523.483 q^{68} +186.106 q^{70} +223.780 q^{71} -501.870 q^{73} -816.929 q^{74} -901.440 q^{76} +1.15413 q^{77} -777.767 q^{79} +166.723 q^{80} -1924.19 q^{82} -624.513 q^{83} -343.149 q^{85} +2282.00 q^{86} +7.32963 q^{88} -758.966 q^{89} -142.204 q^{91} +1073.18 q^{92} -118.502 q^{94} -590.904 q^{95} +986.850 q^{97} +1516.71 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) $ 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8	$ 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) $ 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8 - 88 * q^10 + 110 * q^11 - 82 * q^13 - 126 * q^14 + 271 * q^16 - 100 * q^17 - 292 * q^19 + 52 * q^20 - 351 * q^22 + 276 * q^23 + 386 * q^25 - 84 * q^26 - 1010 * q^28 + 38 * q^29 - 432 * q^31 + 452 * q^32 - 524 * q^34 + 166 * q^35 - 936 * q^37 + 41 * q^38 - 1183 * q^40 - 1054 * q^41 - 1804 * q^43 + 341 * q^44 - 888 * q^46 + 560 * q^47 + 1074 * q^49 + 1054 * q^50 - 632 * q^52 + 160 * q^53 - 842 * q^55 - 509 * q^56 - 1266 * q^58 - 846 * q^59 - 2220 * q^61 - 82 * q^62 - 1565 * q^64 - 296 * q^65 - 4752 * q^67 + 1719 * q^68 - 5601 * q^70 + 802 * q^71 - 2732 * q^73 + 4581 * q^74 - 5614 * q^76 + 1008 * q^77 - 3172 * q^79 + 732 * q^80 - 9709 * q^82 + 4780 * q^83 - 4624 * q^85 + 2009 * q^86 - 9331 * q^88 - 4372 * q^89 - 7398 * q^91 + 6138 * q^92 - 7068 * q^94 + 3160 * q^95 - 4846 * q^97 + 3772 * 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 $

$2$	−4.68120	−1.65505	−0.827527	−	0.561426i	$-0.810253\pi$
$2$	−4.68120	−1.65505	−0.827527	+	0.561426i	$0.810253\pi$
$3$	0	0
$3$	0	0
$4$	13.9136	1.73920
$5$	9.12054	0.815766	0.407883	−	0.913034i	$-0.366267\pi$
$5$	9.12054	0.815766	0.407883	+	0.913034i	$0.366267\pi$
$6$	0	0
$7$	−4.35896	−0.235362	−0.117681	−	0.993051i	$-0.537546\pi$
$7$	−4.35896	−0.235362	−0.117681	+	0.993051i	$0.537546\pi$
$8$	−27.6829	−1.22342
$9$	0	0
$10$	−42.6951	−1.35014
$11$	−0.264771	−0.00725741	−0.00362871	−	0.999993i	$-0.501155\pi$
$11$	−0.264771	−0.00725741	−0.00362871	+	0.999993i	$0.501155\pi$
$12$	0	0
$13$	32.6234	0.696008	0.348004	−	0.937493i	$-0.386860\pi$
$13$	32.6234	0.696008	0.348004	+	0.937493i	$0.386860\pi$
$14$	20.4052	0.389537
$15$	0	0
$16$	18.2800	0.285625
$17$	−37.6237	−0.536770	−0.268385	−	0.963312i	$-0.586490\pi$
$17$	−37.6237	−0.536770	−0.268385	+	0.963312i	$0.586490\pi$
$18$	0	0
$19$	−64.7883	−0.782287	−0.391143	−	0.920330i	$-0.627920\pi$
$19$	−64.7883	−0.782287	−0.391143	+	0.920330i	$0.627920\pi$
$20$	126.900	1.41878
$21$	0	0
$22$	1.23945	0.0120114
$23$	77.1313	0.699260	0.349630	−	0.936888i	$-0.386307\pi$
$23$	77.1313	0.699260	0.349630	+	0.936888i	$0.386307\pi$
$24$	0	0
$25$	−41.8157	−0.334526
$26$	−152.717	−1.15193
$27$	0	0
$28$	−60.6490	−0.409342
$29$	194.379	1.24467	0.622333	−	0.782753i	$-0.286185\pi$
$29$	194.379	1.24467	0.622333	+	0.782753i	$0.286185\pi$
$30$	0	0
$31$	−176.731	−1.02393	−0.511966	−	0.859006i	$-0.671083\pi$
$31$	−176.731	−1.02393	−0.511966	+	0.859006i	$0.671083\pi$
$32$	135.891	0.750697
$33$	0	0
$34$	176.124	0.888384
$35$	−39.7561	−0.192000
$36$	0	0
$37$	174.513	0.775398	0.387699	−	0.921786i	$-0.373270\pi$
$37$	174.513	0.775398	0.387699	+	0.921786i	$0.373270\pi$
$38$	303.287	1.29473
$39$	0	0
$40$	−252.483	−0.998026
$41$	411.047	1.56573	0.782863	−	0.622194i	$-0.213759\pi$
$41$	411.047	1.56573	0.782863	+	0.622194i	$0.213759\pi$
$42$	0	0
$43$	−487.481	−1.72884	−0.864421	−	0.502769i	$-0.832315\pi$
$43$	−487.481	−1.72884	−0.864421	+	0.502769i	$0.832315\pi$
$44$	−3.68393	−0.0126221
$45$	0	0
$46$	−361.067	−1.15731
$47$	25.3144	0.0785635	0.0392817	−	0.999228i	$-0.487493\pi$
$47$	25.3144	0.0785635	0.0392817	+	0.999228i	$0.487493\pi$
$48$	0	0
$49$	−323.999	−0.944605
$50$	195.748	0.553658
$51$	0	0
$52$	453.910	1.21050
$53$	−635.063	−1.64590	−0.822949	−	0.568116i	$-0.807673\pi$
$53$	−635.063	−1.64590	−0.822949	+	0.568116i	$0.807673\pi$
$54$	0	0
$55$	−2.41486	−0.00592035
$56$	120.669	0.287947
$57$	0	0
$58$	−909.928	−2.05999
$59$	414.809	0.915315	0.457658	−	0.889129i	$-0.348689\pi$
$59$	414.809	0.915315	0.457658	+	0.889129i	$0.348689\pi$
$60$	0	0
$61$	521.703	1.09504	0.547519	−	0.836794i	$-0.315573\pi$
$61$	521.703	1.09504	0.547519	+	0.836794i	$0.315573\pi$
$62$	827.314	1.69466
$63$	0	0
$64$	−782.371	−1.52807
$65$	297.543	0.567779
$66$	0	0
$67$	982.659	1.79180	0.895902	−	0.444251i	$-0.146530\pi$
$67$	982.659	1.79180	0.895902	+	0.444251i	$0.146530\pi$
$68$	−523.483	−0.933553
$69$	0	0
$70$	186.106	0.317771
$71$	223.780	0.374054	0.187027	−	0.982355i	$-0.440115\pi$
$71$	223.780	0.374054	0.187027	+	0.982355i	$0.440115\pi$
$72$	0	0
$73$	−501.870	−0.804650	−0.402325	−	0.915497i	$-0.631798\pi$
$73$	−501.870	−0.804650	−0.402325	+	0.915497i	$0.631798\pi$
$74$	−816.929	−1.28333
$75$	0	0
$76$	−901.440	−1.36056
$77$	1.15413	0.00170812
$78$	0	0
$79$	−777.767	−1.10767	−0.553833	−	0.832628i	$-0.686835\pi$
$79$	−777.767	−1.10767	−0.553833	+	0.832628i	$0.686835\pi$
$80$	166.723	0.233003
$81$	0	0
$82$	−1924.19	−2.59136
$83$	−624.513	−0.825894	−0.412947	−	0.910755i	$-0.635501\pi$
$83$	−624.513	−0.825894	−0.412947	+	0.910755i	$0.635501\pi$
$84$	0	0
$85$	−343.149	−0.437879
$86$	2282.00	2.86133
$87$	0	0
$88$	7.32963	0.00887887
$89$	−758.966	−0.903936	−0.451968	−	0.892034i	$-0.649278\pi$
$89$	−758.966	−0.903936	−0.451968	+	0.892034i	$0.649278\pi$
$90$	0	0
$91$	−142.204	−0.163814
$92$	1073.18	1.21616
$93$	0	0
$94$	−118.502	−0.130027
$95$	−590.904	−0.638163
$96$	0	0
$97$	986.850	1.03298	0.516492	−	0.856292i	$-0.327238\pi$
$97$	986.850	1.03298	0.516492	+	0.856292i	$0.327238\pi$
$98$	1516.71	1.56337
$99$	0	0
$100$	−581.808	−0.581808
$101$	−1038.32	−1.02294	−0.511469	−	0.859302i	$-0.670899\pi$
$101$	−1038.32	−1.02294	−0.511469	+	0.859302i	$0.670899\pi$
$102$	0	0
$103$	338.725	0.324034	0.162017	−	0.986788i	$-0.448200\pi$
$103$	338.725	0.324034	0.162017	+	0.986788i	$0.448200\pi$
$104$	−903.109	−0.851510
$105$	0	0
$106$	2972.85	2.72405
$107$	102.259	0.0923903	0.0461951	−	0.998932i	$-0.485290\pi$
$107$	102.259	0.0923903	0.0461951	+	0.998932i	$0.485290\pi$
$108$	0	0
$109$	−1822.39	−1.60141	−0.800704	−	0.599060i	$-0.795541\pi$
$109$	−1822.39	−1.60141	−0.800704	+	0.599060i	$0.795541\pi$
$110$	11.3044	0.00979850
$111$	0	0
$112$	−79.6818	−0.0672252
$113$	−1390.66	−1.15772	−0.578859	−	0.815427i	$-0.696502\pi$
$113$	−1390.66	−1.15772	−0.578859	+	0.815427i	$0.696502\pi$
$114$	0	0
$115$	703.479	0.570432
$116$	2704.52	2.16473
$117$	0	0
$118$	−1941.81	−1.51490
$119$	164.001	0.126335
$120$	0	0
$121$	−1330.93	−0.999947
$122$	−2442.20	−1.81235
$123$	0	0
$124$	−2458.97	−1.78083
$125$	−1521.45	−1.08866
$126$	0	0
$127$	−1373.66	−0.959784	−0.479892	−	0.877328i	$-0.659324\pi$
$127$	−1373.66	−0.959784	−0.479892	+	0.877328i	$0.659324\pi$
$128$	2575.31	1.77834
$129$	0	0
$130$	−1392.86	−0.939705
$131$	−110.361	−0.0736055	−0.0368028	−	0.999323i	$-0.511717\pi$
$131$	−110.361	−0.0736055	−0.0368028	+	0.999323i	$0.511717\pi$
$132$	0	0
$133$	282.410	0.184121
$134$	−4600.02	−2.96553
$135$	0	0
$136$	1041.53	0.656696
$137$	−1359.65	−0.847902	−0.423951	−	0.905685i	$-0.639357\pi$
$137$	−1359.65	−0.847902	−0.423951	+	0.905685i	$0.639357\pi$
$138$	0	0
$139$	1901.87	1.16054	0.580268	−	0.814426i	$-0.302948\pi$
$139$	1901.87	1.16054	0.580268	+	0.814426i	$0.302948\pi$
$140$	−553.152	−0.333927
$141$	0	0
$142$	−1047.56	−0.619080
$143$	−8.63773	−0.00505121
$144$	0	0
$145$	1772.84	1.01536
$146$	2349.35	1.33174
$147$	0	0
$148$	2428.11	1.34857
$149$	669.343	0.368018	0.184009	−	0.982925i	$-0.441092\pi$
$149$	669.343	0.368018	0.184009	+	0.982925i	$0.441092\pi$
$150$	0	0
$151$	1174.47	0.632961	0.316481	−	0.948599i	$-0.397499\pi$
$151$	1174.47	0.632961	0.316481	+	0.948599i	$0.397499\pi$
$152$	1793.53	0.957066
$153$	0	0
$154$	−5.40270	−0.00282703
$155$	−1611.89	−0.835289
$156$	0	0
$157$	−3711.41	−1.88664	−0.943321	−	0.331883i	$-0.892316\pi$
$157$	−3711.41	−1.88664	−0.943321	+	0.331883i	$0.892316\pi$
$158$	3640.88	1.83325
$159$	0	0
$160$	1239.40	0.612393
$161$	−336.212	−0.164579
$162$	0	0
$163$	2096.21	1.00729	0.503645	−	0.863911i	$-0.331992\pi$
$163$	2096.21	1.00729	0.503645	+	0.863911i	$0.331992\pi$
$164$	5719.15	2.72312
$165$	0	0
$166$	2923.47	1.36690
$167$	3826.77	1.77320	0.886600	−	0.462536i	$-0.153061\pi$
$167$	3826.77	1.77320	0.886600	+	0.462536i	$0.153061\pi$
$168$	0	0
$169$	−1132.72	−0.515574
$170$	1606.35	0.724714
$171$	0	0
$172$	−6782.63	−3.00681
$173$	−276.177	−0.121372	−0.0606860	−	0.998157i	$-0.519329\pi$
$173$	−276.177	−0.121372	−0.0606860	+	0.998157i	$0.519329\pi$
$174$	0	0
$175$	182.273	0.0787346
$176$	−4.84002	−0.00207290
$177$	0	0
$178$	3552.87	1.49606
$179$	1641.14	0.685279	0.342639	−	0.939467i	$-0.388679\pi$
$179$	1641.14	0.685279	0.342639	+	0.939467i	$0.388679\pi$
$180$	0	0
$181$	663.031	0.272280	0.136140	−	0.990690i	$-0.456530\pi$
$181$	663.031	0.272280	0.136140	+	0.990690i	$0.456530\pi$
$182$	665.686	0.271120
$183$	0	0
$184$	−2135.21	−0.855490
$185$	1591.65	0.632543
$186$	0	0
$187$	9.96169	0.00389556
$188$	352.215	0.136638
$189$	0	0
$190$	2766.14	1.05619
$191$	5061.36	1.91742	0.958710	−	0.284387i	$-0.0917901\pi$
$191$	5061.36	1.91742	0.958710	+	0.284387i	$0.0917901\pi$
$192$	0	0
$193$	2676.06	0.998066	0.499033	−	0.866583i	$-0.333689\pi$
$193$	2676.06	0.998066	0.499033	+	0.866583i	$0.333689\pi$
$194$	−4619.64	−1.70964
$195$	0	0
$196$	−4508.01	−1.64286
$197$	3354.64	1.21324	0.606620	−	0.794992i	$-0.292525\pi$
$197$	3354.64	1.21324	0.606620	+	0.794992i	$0.292525\pi$
$198$	0	0
$199$	1176.04	0.418930	0.209465	−	0.977816i	$-0.432828\pi$
$199$	1176.04	0.418930	0.209465	+	0.977816i	$0.432828\pi$
$200$	1157.58	0.409266
$201$	0	0
$202$	4860.59	1.69302
$203$	−847.292	−0.292947
$204$	0	0
$205$	3748.97	1.27727
$206$	−1585.64	−0.536294
$207$	0	0
$208$	596.355	0.198797
$209$	17.1541	0.00567738
$210$	0	0
$211$	−3752.25	−1.22424	−0.612122	−	0.790763i	$-0.709684\pi$
$211$	−3752.25	−1.22424	−0.612122	+	0.790763i	$0.709684\pi$
$212$	−8836.02	−2.86255
$213$	0	0
$214$	−478.695	−0.152911
$215$	−4446.09	−1.41033
$216$	0	0
$217$	770.365	0.240994
$218$	8530.98	2.65042
$219$	0	0
$220$	−33.5994	−0.0102967
$221$	−1227.41	−0.373596
$222$	0	0
$223$	6092.06	1.82939	0.914696	−	0.404142i	$-0.132430\pi$
$223$	6092.06	1.82939	0.914696	+	0.404142i	$0.132430\pi$
$224$	−592.342	−0.176685
$225$	0	0
$226$	6509.95	1.91609
$227$	462.220	0.135148	0.0675741	−	0.997714i	$-0.478474\pi$
$227$	462.220	0.135148	0.0675741	+	0.997714i	$0.478474\pi$
$228$	0	0
$229$	3109.73	0.897367	0.448683	−	0.893691i	$-0.351893\pi$
$229$	3109.73	0.897367	0.448683	+	0.893691i	$0.351893\pi$
$230$	−3293.12	−0.944097
$231$	0	0
$232$	−5380.97	−1.52275
$233$	−6319.58	−1.77686	−0.888432	−	0.459008i	$-0.848205\pi$
$233$	−6319.58	−1.77686	−0.888432	+	0.459008i	$0.848205\pi$
$234$	0	0
$235$	230.881	0.0640894
$236$	5771.50	1.59192
$237$	0	0
$238$	−767.719	−0.209092
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−1439.87	−0.384854	−0.192427	−	0.981311i	$-0.561636\pi$
$241$	−1439.87	−0.384854	−0.192427	+	0.981311i	$0.561636\pi$
$242$	6230.35	1.65497
$243$	0	0
$244$	7258.78	1.90449
$245$	−2955.05	−0.770576
$246$	0	0
$247$	−2113.61	−0.544478
$248$	4892.43	1.25270
$249$	0	0
$250$	7122.21	1.80179
$251$	−3735.39	−0.939345	−0.469672	−	0.882841i	$-0.655628\pi$
$251$	−3735.39	−0.939345	−0.469672	+	0.882841i	$0.655628\pi$
$252$	0	0
$253$	−20.4221	−0.00507482
$254$	6430.37	1.58849
$255$	0	0
$256$	−5796.57	−1.41518
$257$	−1841.46	−0.446954	−0.223477	−	0.974709i	$-0.571741\pi$
$257$	−1841.46	−0.446954	−0.223477	+	0.974709i	$0.571741\pi$
$258$	0	0
$259$	−760.695	−0.182499
$260$	4139.90	0.987484
$261$	0	0
$262$	516.624	0.121821
$263$	−3783.37	−0.887045	−0.443522	−	0.896263i	$-0.646271\pi$
$263$	−3783.37	−0.887045	−0.443522	+	0.896263i	$0.646271\pi$
$264$	0	0
$265$	−5792.11	−1.34267
$266$	−1322.02	−0.304729
$267$	0	0
$268$	13672.4	3.11631
$269$	1915.55	0.434176	0.217088	−	0.976152i	$-0.430344\pi$
$269$	1915.55	0.434176	0.217088	+	0.976152i	$0.430344\pi$
$270$	0	0
$271$	3558.69	0.797695	0.398847	−	0.917017i	$-0.369410\pi$
$271$	3558.69	0.797695	0.398847	+	0.917017i	$0.369410\pi$
$272$	−687.762	−0.153315
$273$	0	0
$274$	6364.78	1.40332
$275$	11.0716	0.00242779
$276$	0	0
$277$	−1317.45	−0.285770	−0.142885	−	0.989739i	$-0.545638\pi$
$277$	−1317.45	−0.285770	−0.142885	+	0.989739i	$0.545638\pi$
$278$	−8903.03	−1.92075
$279$	0	0
$280$	1100.56	0.234897
$281$	−1490.00	−0.316320	−0.158160	−	0.987413i	$-0.550556\pi$
$281$	−1490.00	−0.316320	−0.158160	+	0.987413i	$0.550556\pi$
$282$	0	0
$283$	−5042.87	−1.05925	−0.529625	−	0.848232i	$-0.677667\pi$
$283$	−5042.87	−1.05925	−0.529625	+	0.848232i	$0.677667\pi$
$284$	3113.60	0.650556
$285$	0	0
$286$	40.4350	0.00836003
$287$	−1791.74	−0.368512
$288$	0	0
$289$	−3497.45	−0.711877
$290$	−8299.03	−1.68047
$291$	0	0
$292$	−6982.83	−1.39945
$293$	1013.97	0.202174	0.101087	−	0.994878i	$-0.467768\pi$
$293$	1013.97	0.202174	0.101087	+	0.994878i	$0.467768\pi$
$294$	0	0
$295$	3783.29	0.746683
$296$	−4831.01	−0.948638
$297$	0	0
$298$	−3133.33	−0.609090
$299$	2516.28	0.486690
$300$	0	0
$301$	2124.91	0.406903
$302$	−5497.94	−1.04759
$303$	0	0
$304$	−1184.33	−0.223441
$305$	4758.22	0.893294
$306$	0	0
$307$	4474.77	0.831886	0.415943	−	0.909391i	$-0.363452\pi$
$307$	4474.77	0.831886	0.415943	+	0.909391i	$0.363452\pi$
$308$	16.0581	0.00297076
$309$	0	0
$310$	7545.56	1.38245
$311$	−3021.46	−0.550905	−0.275452	−	0.961315i	$-0.588828\pi$
$311$	−3021.46	−0.550905	−0.275452	+	0.961315i	$0.588828\pi$
$312$	0	0
$313$	−7111.54	−1.28424	−0.642121	−	0.766603i	$-0.721945\pi$
$313$	−7111.54	−1.28424	−0.642121	+	0.766603i	$0.721945\pi$
$314$	17373.8	3.12249
$315$	0	0
$316$	−10821.6	−1.92646
$317$	6531.10	1.15717	0.578585	−	0.815622i	$-0.303605\pi$
$317$	6531.10	1.15717	0.578585	+	0.815622i	$0.303605\pi$
$318$	0	0
$319$	−51.4660	−0.00903305
$320$	−7135.65	−1.24655
$321$	0	0
$322$	1573.88	0.272387
$323$	2437.58	0.419908
$324$	0	0
$325$	−1364.17	−0.232832
$326$	−9812.80	−1.66712
$327$	0	0
$328$	−11379.0	−1.91554
$329$	−110.345	−0.0184909
$330$	0	0
$331$	−7055.29	−1.17158	−0.585792	−	0.810462i	$-0.699216\pi$
$331$	−7055.29	−1.17158	−0.585792	+	0.810462i	$0.699216\pi$
$332$	−8689.24	−1.43640
$333$	0	0
$334$	−17913.9	−2.93474
$335$	8962.38	1.46169
$336$	0	0
$337$	4291.92	0.693756	0.346878	−	0.937910i	$-0.387242\pi$
$337$	4291.92	0.693756	0.346878	+	0.937910i	$0.387242\pi$
$338$	5302.46	0.853302
$339$	0	0
$340$	−4774.45	−0.761561
$341$	46.7934	0.00743109
$342$	0	0
$343$	2907.43	0.457686
$344$	13494.9	2.11510
$345$	0	0
$346$	1292.84	0.200877
$347$	−375.795	−0.0581376	−0.0290688	−	0.999577i	$-0.509254\pi$
$347$	−375.795	−0.0581376	−0.0290688	+	0.999577i	$0.509254\pi$
$348$	0	0
$349$	−8582.20	−1.31632	−0.658158	−	0.752880i	$-0.728664\pi$
$349$	−8582.20	−1.31632	−0.658158	+	0.752880i	$0.728664\pi$
$350$	−853.257	−0.130310
$351$	0	0
$352$	−35.9799	−0.00544812
$353$	−7051.92	−1.06327	−0.531637	−	0.846972i	$-0.678423\pi$
$353$	−7051.92	−1.06327	−0.531637	+	0.846972i	$0.678423\pi$
$354$	0	0
$355$	2041.00	0.305141
$356$	−10560.0	−1.57213
$357$	0	0
$358$	−7682.52	−1.13417
$359$	−9675.94	−1.42250	−0.711249	−	0.702940i	$-0.751870\pi$
$359$	−9675.94	−1.42250	−0.711249	+	0.702940i	$0.751870\pi$
$360$	0	0
$361$	−2661.48	−0.388027
$362$	−3103.78	−0.450638
$363$	0	0
$364$	−1978.57	−0.284905
$365$	−4577.33	−0.656406
$366$	0	0
$367$	−3635.23	−0.517051	−0.258525	−	0.966004i	$-0.583237\pi$
$367$	−3635.23	−0.517051	−0.258525	+	0.966004i	$0.583237\pi$
$368$	1409.96	0.199726
$369$	0	0
$370$	−7450.84	−1.04689
$371$	2768.21	0.387381
$372$	0	0
$373$	−8596.85	−1.19337	−0.596686	−	0.802475i	$-0.703516\pi$
$373$	−8596.85	−1.19337	−0.596686	+	0.802475i	$0.703516\pi$
$374$	−46.6326	−0.00644737
$375$	0	0
$376$	−700.775	−0.0961163
$377$	6341.31	0.866297
$378$	0	0
$379$	−9086.80	−1.23155	−0.615775	−	0.787922i	$-0.711157\pi$
$379$	−9086.80	−1.23155	−0.615775	+	0.787922i	$0.711157\pi$
$380$	−8221.62	−1.10990
$381$	0	0
$382$	−23693.2	−3.17343
$383$	558.560	0.0745198	0.0372599	−	0.999306i	$-0.488137\pi$
$383$	558.560	0.0745198	0.0372599	+	0.999306i	$0.488137\pi$
$384$	0	0
$385$	10.5263	0.00139342
$386$	−12527.2	−1.65185
$387$	0	0
$388$	13730.7	1.79657
$389$	−2011.49	−0.262176	−0.131088	−	0.991371i	$-0.541847\pi$
$389$	−2011.49	−0.262176	−0.131088	+	0.991371i	$0.541847\pi$
$390$	0	0
$391$	−2901.97	−0.375342
$392$	8969.23	1.15565
$393$	0	0
$394$	−15703.7	−2.00798
$395$	−7093.65	−0.903596
$396$	0	0
$397$	−6832.23	−0.863727	−0.431864	−	0.901939i	$-0.642144\pi$
$397$	−6832.23	−0.863727	−0.431864	+	0.901939i	$0.642144\pi$
$398$	−5505.27	−0.693352
$399$	0	0
$400$	−764.391	−0.0955489
$401$	−6334.54	−0.788857	−0.394429	−	0.918927i	$-0.629057\pi$
$401$	−6334.54	−0.788857	−0.394429	+	0.918927i	$0.629057\pi$
$402$	0	0
$403$	−5765.57	−0.712664
$404$	−14446.8	−1.77910
$405$	0	0
$406$	3966.34	0.484843
$407$	−46.2060	−0.00562738
$408$	0	0
$409$	10967.5	1.32594	0.662969	−	0.748647i	$-0.269296\pi$
$409$	10967.5	1.32594	0.662969	+	0.748647i	$0.269296\pi$
$410$	−17549.7	−2.11394
$411$	0	0
$412$	4712.89	0.563562
$413$	−1808.14	−0.215430
$414$	0	0
$415$	−5695.90	−0.673736
$416$	4433.21	0.522491
$417$	0	0
$418$	−80.3016	−0.00939637
$419$	8586.44	1.00113	0.500567	−	0.865698i	$-0.333125\pi$
$419$	8586.44	1.00113	0.500567	+	0.865698i	$0.333125\pi$
$420$	0	0
$421$	−3535.52	−0.409289	−0.204644	−	0.978836i	$-0.565604\pi$
$421$	−3535.52	−0.409289	−0.204644	+	0.978836i	$0.565604\pi$
$422$	17565.0	2.02619
$423$	0	0
$424$	17580.4	2.01363
$425$	1573.26	0.179564
$426$	0	0
$427$	−2274.09	−0.257730
$428$	1422.80	0.160686
$429$	0	0
$430$	20813.0	2.33417
$431$	−17860.0	−1.99602	−0.998012	−	0.0630170i	$-0.979928\pi$
$431$	−17860.0	−1.99602	−0.998012	+	0.0630170i	$0.979928\pi$
$432$	0	0
$433$	16615.1	1.84404	0.922021	−	0.387141i	$-0.126537\pi$
$433$	16615.1	1.84404	0.922021	+	0.387141i	$0.126537\pi$
$434$	−3606.23	−0.398859
$435$	0	0
$436$	−25356.1	−2.78517
$437$	−4997.20	−0.547022
$438$	0	0
$439$	13100.3	1.42425	0.712123	−	0.702055i	$-0.247734\pi$
$439$	13100.3	1.42425	0.712123	+	0.702055i	$0.247734\pi$
$440$	66.8502	0.00724308
$441$	0	0
$442$	5745.77	0.618322
$443$	−4125.31	−0.442436	−0.221218	−	0.975224i	$-0.571003\pi$
$443$	−4125.31	−0.442436	−0.221218	+	0.975224i	$0.571003\pi$
$444$	0	0
$445$	−6922.18	−0.737400
$446$	−28518.1	−3.02774
$447$	0	0
$448$	3410.33	0.359649
$449$	6483.58	0.681468	0.340734	−	0.940160i	$-0.389324\pi$
$449$	6483.58	0.681468	0.340734	+	0.940160i	$0.389324\pi$
$450$	0	0
$451$	−108.833	−0.0113631
$452$	−19349.1	−2.01351
$453$	0	0
$454$	−2163.75	−0.223678
$455$	−1296.98	−0.133634
$456$	0	0
$457$	10431.6	1.06777	0.533884	−	0.845558i	$-0.320732\pi$
$457$	10431.6	1.06777	0.533884	+	0.845558i	$0.320732\pi$
$458$	−14557.3	−1.48519
$459$	0	0
$460$	9787.94	0.992098
$461$	−10548.2	−1.06568	−0.532840	−	0.846216i	$-0.678875\pi$
$461$	−10548.2	−1.06568	−0.532840	+	0.846216i	$0.678875\pi$
$462$	0	0
$463$	−16878.4	−1.69418	−0.847089	−	0.531451i	$-0.821647\pi$
$463$	−16878.4	−1.69418	−0.847089	+	0.531451i	$0.821647\pi$
$464$	3553.25	0.355508
$465$	0	0
$466$	29583.2	2.94081
$467$	2632.06	0.260808	0.130404	−	0.991461i	$-0.458373\pi$
$467$	2632.06	0.260808	0.130404	+	0.991461i	$0.458373\pi$
$468$	0	0
$469$	−4283.37	−0.421723
$470$	−1080.80	−0.106071
$471$	0	0
$472$	−11483.1	−1.11982
$473$	129.071	0.0125469
$474$	0	0
$475$	2709.17	0.261695
$476$	2281.84	0.219723
$477$	0	0
$478$	1118.81	0.107057
$479$	12523.9	1.19464	0.597321	−	0.802003i	$-0.296232\pi$
$479$	12523.9	1.19464	0.597321	+	0.802003i	$0.296232\pi$
$480$	0	0
$481$	5693.20	0.539683
$482$	6740.30	0.636955
$483$	0	0
$484$	−18518.1	−1.73911
$485$	9000.60	0.842673
$486$	0	0
$487$	−19818.8	−1.84409	−0.922047	−	0.387077i	$-0.873485\pi$
$487$	−19818.8	−1.84409	−0.922047	+	0.387077i	$0.873485\pi$
$488$	−14442.2	−1.33969
$489$	0	0
$490$	13833.2	1.27535
$491$	−2931.72	−0.269464	−0.134732	−	0.990882i	$-0.543017\pi$
$491$	−2931.72	−0.269464	−0.134732	+	0.990882i	$0.543017\pi$
$492$	0	0
$493$	−7313.27	−0.668100
$494$	9894.24	0.901140
$495$	0	0
$496$	−3230.65	−0.292460
$497$	−975.450	−0.0880381
$498$	0	0
$499$	−9371.84	−0.840764	−0.420382	−	0.907347i	$-0.638104\pi$
$499$	−9371.84	−0.840764	−0.420382	+	0.907347i	$0.638104\pi$
$500$	−21168.9	−1.89340
$501$	0	0
$502$	17486.1	1.55467
$503$	−6844.50	−0.606722	−0.303361	−	0.952876i	$-0.598109\pi$
$503$	−6844.50	−0.606722	−0.303361	+	0.952876i	$0.598109\pi$
$504$	0	0
$505$	−9470.05	−0.834478
$506$	95.6001	0.00839910
$507$	0	0
$508$	−19112.6	−1.66926
$509$	−14791.7	−1.28808	−0.644038	−	0.764993i	$-0.722742\pi$
$509$	−14791.7	−1.28808	−0.644038	+	0.764993i	$0.722742\pi$
$510$	0	0
$511$	2187.63	0.189384
$512$	6532.42	0.563857
$513$	0	0
$514$	8620.25	0.739733
$515$	3089.35	0.264336
$516$	0	0
$517$	−6.70252	−0.000570168 0
$518$	3560.96	0.302046
$519$	0	0
$520$	−8236.84	−0.694633
$521$	7406.49	0.622810	0.311405	−	0.950277i	$-0.399200\pi$
$521$	7406.49	0.622810	0.311405	+	0.950277i	$0.399200\pi$
$522$	0	0
$523$	8139.69	0.680543	0.340271	−	0.940327i	$-0.389481\pi$
$523$	8139.69	0.680543	0.340271	+	0.940327i	$0.389481\pi$
$524$	−1535.53	−0.128015
$525$	0	0
$526$	17710.7	1.46811
$527$	6649.29	0.549616
$528$	0	0
$529$	−6217.77	−0.511036
$530$	27114.0	2.22219
$531$	0	0
$532$	3929.34	0.320223
$533$	13409.7	1.08976
$534$	0	0
$535$	932.658	0.0753689
$536$	−27202.8	−2.19213
$537$	0	0
$538$	−8967.09	−0.718585
$539$	85.7857	0.00685539
$540$	0	0
$541$	−491.245	−0.0390393	−0.0195196	−	0.999809i	$-0.506214\pi$
$541$	−491.245	−0.0390393	−0.0195196	+	0.999809i	$0.506214\pi$
$542$	−16659.0	−1.32023
$543$	0	0
$544$	−5112.71	−0.402952
$545$	−16621.2	−1.30637
$546$	0	0
$547$	−8156.94	−0.637597	−0.318799	−	0.947823i	$-0.603279\pi$
$547$	−8156.94	−0.637597	−0.318799	+	0.947823i	$0.603279\pi$
$548$	−18917.6	−1.47467
$549$	0	0
$550$	−51.8284	−0.00401813
$551$	−12593.5	−0.973686
$552$	0	0
$553$	3390.26	0.260702
$554$	6167.27	0.472964
$555$	0	0
$556$	26461.9	2.01841
$557$	−13967.5	−1.06252	−0.531259	−	0.847210i	$-0.678281\pi$
$557$	−13967.5	−1.06252	−0.531259	+	0.847210i	$0.678281\pi$
$558$	0	0
$559$	−15903.3	−1.20329
$560$	−726.741	−0.0548401
$561$	0	0
$562$	6974.98	0.523527
$563$	2730.23	0.204379	0.102190	−	0.994765i	$-0.467415\pi$
$563$	2730.23	0.204379	0.102190	+	0.994765i	$0.467415\pi$
$564$	0	0
$565$	−12683.6	−0.944427
$566$	23606.7	1.75311
$567$	0	0
$568$	−6194.88	−0.457626
$569$	6459.82	0.475940	0.237970	−	0.971273i	$-0.423518\pi$
$569$	6459.82	0.475940	0.237970	+	0.971273i	$0.423518\pi$
$570$	0	0
$571$	−24766.7	−1.81516	−0.907578	−	0.419884i	$-0.862071\pi$
$571$	−24766.7	−1.81516	−0.907578	+	0.419884i	$0.862071\pi$
$572$	−120.182	−0.00878509
$573$	0	0
$574$	8387.48	0.609907
$575$	−3225.30	−0.233920
$576$	0	0
$577$	−2213.84	−0.159729	−0.0798644	−	0.996806i	$-0.525449\pi$
$577$	−2213.84	−0.159729	−0.0798644	+	0.996806i	$0.525449\pi$
$578$	16372.3	1.17820
$579$	0	0
$580$	24666.7	1.76591
$581$	2722.23	0.194384
$582$	0	0
$583$	168.146	0.0119450
$584$	13893.2	0.984426
$585$	0	0
$586$	−4746.62	−0.334609
$587$	6422.72	0.451608	0.225804	−	0.974173i	$-0.427499\pi$
$587$	6422.72	0.451608	0.225804	+	0.974173i	$0.427499\pi$
$588$	0	0
$589$	11450.1	0.801008
$590$	−17710.3	−1.23580
$591$	0	0
$592$	3190.09	0.221473
$593$	16763.8	1.16089	0.580444	−	0.814300i	$-0.302879\pi$
$593$	16763.8	1.16089	0.580444	+	0.814300i	$0.302879\pi$
$594$	0	0
$595$	1495.77	0.103060
$596$	9312.99	0.640059
$597$	0	0
$598$	−11779.2	−0.805498
$599$	−10500.5	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$599$	−10500.5	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$600$	0	0
$601$	−3600.82	−0.244393	−0.122197	−	0.992506i	$-0.538994\pi$
$601$	−3600.82	−0.244393	−0.122197	+	0.992506i	$0.538994\pi$
$602$	−9947.14	−0.673447
$603$	0	0
$604$	16341.2	1.10085
$605$	−12138.8	−0.815723
$606$	0	0
$607$	9408.77	0.629144	0.314572	−	0.949234i	$-0.398139\pi$
$607$	9408.77	0.629144	0.314572	+	0.949234i	$0.398139\pi$
$608$	−8804.12	−0.587260
$609$	0	0
$610$	−22274.2	−1.47845
$611$	825.841	0.0546808
$612$	0	0
$613$	−245.109	−0.0161498	−0.00807492	−	0.999967i	$-0.502570\pi$
$613$	−245.109	−0.0161498	−0.00807492	+	0.999967i	$0.502570\pi$
$614$	−20947.3	−1.37682
$615$	0	0
$616$	−31.9496	−0.00208975
$617$	−982.946	−0.0641360	−0.0320680	−	0.999486i	$-0.510209\pi$
$617$	−982.946	−0.0641360	−0.0320680	+	0.999486i	$0.510209\pi$
$618$	0	0
$619$	17385.4	1.12888	0.564440	−	0.825474i	$-0.309092\pi$
$619$	17385.4	1.12888	0.564440	+	0.825474i	$0.309092\pi$
$620$	−22427.2	−1.45274
$621$	0	0
$622$	14144.1	0.911777
$623$	3308.31	0.212752
$624$	0	0
$625$	−8649.48	−0.553567
$626$	33290.5	2.12549
$627$	0	0
$628$	−51639.2	−3.28125
$629$	−6565.82	−0.416211
$630$	0	0
$631$	−18433.4	−1.16295	−0.581476	−	0.813563i	$-0.697525\pi$
$631$	−18433.4	−1.16295	−0.581476	+	0.813563i	$0.697525\pi$
$632$	21530.8	1.35514
$633$	0	0
$634$	−30573.4	−1.91518
$635$	−12528.5	−0.782959
$636$	0	0
$637$	−10570.0	−0.657452
$638$	240.923	0.0149502
$639$	0	0
$640$	23488.2	1.45071
$641$	−4117.45	−0.253712	−0.126856	−	0.991921i	$-0.540489\pi$
$641$	−4117.45	−0.253712	−0.126856	+	0.991921i	$0.540489\pi$
$642$	0	0
$643$	7095.55	0.435181	0.217590	−	0.976040i	$-0.430180\pi$
$643$	7095.55	0.435181	0.217590	+	0.976040i	$0.430180\pi$
$644$	−4677.93	−0.286237
$645$	0	0
$646$	−11410.8	−0.694971
$647$	−9325.32	−0.566640	−0.283320	−	0.959025i	$-0.591436\pi$
$647$	−9325.32	−0.566640	−0.283320	+	0.959025i	$0.591436\pi$
$648$	0	0
$649$	−109.830	−0.00664282
$650$	6385.95	0.385350
$651$	0	0
$652$	29166.0	1.75188
$653$	20711.9	1.24122	0.620611	−	0.784118i	$-0.286884\pi$
$653$	20711.9	1.24122	0.620611	+	0.784118i	$0.286884\pi$
$654$	0	0
$655$	−1006.56	−0.0600449
$656$	7513.94	0.447210
$657$	0	0
$658$	516.545	0.0306034
$659$	23602.6	1.39518	0.697592	−	0.716495i	$-0.254255\pi$
$659$	23602.6	1.39518	0.697592	+	0.716495i	$0.254255\pi$
$660$	0	0
$661$	−7182.46	−0.422641	−0.211320	−	0.977417i	$-0.567776\pi$
$661$	−7182.46	−0.422641	−0.211320	+	0.977417i	$0.567776\pi$
$662$	33027.2	1.93903
$663$	0	0
$664$	17288.3	1.01042
$665$	2575.73	0.150199
$666$	0	0
$667$	14992.7	0.870345
$668$	53244.3	3.08396
$669$	0	0
$670$	−41954.7	−2.41918
$671$	−138.132	−0.00794713
$672$	0	0
$673$	−18024.1	−1.03236	−0.516180	−	0.856480i	$-0.672647\pi$
$673$	−18024.1	−1.03236	−0.516180	+	0.856480i	$0.672647\pi$
$674$	−20091.3	−1.14820
$675$	0	0
$676$	−15760.2	−0.896687
$677$	−17529.0	−0.995116	−0.497558	−	0.867431i	$-0.665770\pi$
$677$	−17529.0	−0.995116	−0.497558	+	0.867431i	$0.665770\pi$
$678$	0	0
$679$	−4301.64	−0.243125
$680$	9499.34	0.535711
$681$	0	0
$682$	−219.049	−0.0122989
$683$	24535.8	1.37458	0.687288	−	0.726385i	$-0.258801\pi$
$683$	24535.8	1.37458	0.687288	+	0.726385i	$0.258801\pi$
$684$	0	0
$685$	−12400.7	−0.691690
$686$	−13610.2	−0.757495
$687$	0	0
$688$	−8911.15	−0.493800
$689$	−20717.9	−1.14556
$690$	0	0
$691$	−4667.90	−0.256983	−0.128491	−	0.991711i	$-0.541013\pi$
$691$	−4667.90	−0.256983	−0.128491	+	0.991711i	$0.541013\pi$
$692$	−3842.62	−0.211091
$693$	0	0
$694$	1759.17	0.0962208
$695$	17346.1	0.946726
$696$	0	0
$697$	−15465.1	−0.840435
$698$	40175.0	2.17857
$699$	0	0
$700$	2536.08	0.136936
$701$	5060.91	0.272679	0.136339	−	0.990662i	$-0.456466\pi$
$701$	5060.91	0.272679	0.136339	+	0.990662i	$0.456466\pi$
$702$	0	0
$703$	−11306.4	−0.606584
$704$	207.149	0.0110898
$705$	0	0
$706$	33011.4	1.75978
$707$	4526.00	0.240761
$708$	0	0
$709$	−14817.5	−0.784883	−0.392441	−	0.919777i	$-0.628369\pi$
$709$	−14817.5	−0.784883	−0.392441	+	0.919777i	$0.628369\pi$
$710$	−9554.32	−0.505024
$711$	0	0
$712$	21010.4	1.10589
$713$	−13631.5	−0.715994
$714$	0	0
$715$	−78.7808	−0.00412061
$716$	22834.3	1.19184
$717$	0	0
$718$	45295.0	2.35431
$719$	−10031.3	−0.520311	−0.260156	−	0.965567i	$-0.583774\pi$
$719$	−10031.3	−0.520311	−0.260156	+	0.965567i	$0.583774\pi$
$720$	0	0
$721$	−1476.49	−0.0762653
$722$	12458.9	0.642206
$723$	0	0
$724$	9225.17	0.473551
$725$	−8128.11	−0.416373
$726$	0	0
$727$	−27616.6	−1.40886	−0.704430	−	0.709773i	$-0.748797\pi$
$727$	−27616.6	−1.40886	−0.704430	+	0.709773i	$0.748797\pi$
$728$	3936.62	0.200413
$729$	0	0
$730$	21427.4	1.08639
$731$	18340.9	0.927991
$732$	0	0
$733$	9356.75	0.471486	0.235743	−	0.971815i	$-0.424248\pi$
$733$	9356.75	0.471486	0.235743	+	0.971815i	$0.424248\pi$
$734$	17017.3	0.855747
$735$	0	0
$736$	10481.4	0.524932
$737$	−260.180	−0.0130039
$738$	0	0
$739$	879.514	0.0437800	0.0218900	−	0.999760i	$-0.493032\pi$
$739$	879.514	0.0437800	0.0218900	+	0.999760i	$0.493032\pi$
$740$	22145.6	1.10012
$741$	0	0
$742$	−12958.6	−0.641137
$743$	19044.0	0.940319	0.470160	−	0.882581i	$-0.344196\pi$
$743$	19044.0	0.940319	0.470160	+	0.882581i	$0.344196\pi$
$744$	0	0
$745$	6104.77	0.300217
$746$	40243.6	1.97510
$747$	0	0
$748$	138.603	0.00677518
$749$	−445.744	−0.0217452
$750$	0	0
$751$	−7389.73	−0.359061	−0.179531	−	0.983752i	$-0.557458\pi$
$751$	−7389.73	−0.359061	−0.179531	+	0.983752i	$0.557458\pi$
$752$	462.747	0.0224397
$753$	0	0
$754$	−29684.9	−1.43377
$755$	10711.8	0.516348
$756$	0	0
$757$	14359.1	0.689418	0.344709	−	0.938710i	$-0.387978\pi$
$757$	14359.1	0.689418	0.344709	+	0.938710i	$0.387978\pi$
$758$	42537.1	2.03828
$759$	0	0
$760$	16357.9	0.780742
$761$	−40452.8	−1.92696	−0.963478	−	0.267789i	$-0.913707\pi$
$761$	−40452.8	−1.92696	−0.963478	+	0.267789i	$0.913707\pi$
$762$	0	0
$763$	7943.74	0.376910
$764$	70421.9	3.33478
$765$	0	0
$766$	−2614.73	−0.123334
$767$	13532.5	0.637066
$768$	0	0
$769$	−22351.8	−1.04815	−0.524075	−	0.851672i	$-0.675589\pi$
$769$	−22351.8	−1.04815	−0.524075	+	0.851672i	$0.675589\pi$
$770$	−49.2756	−0.00230619
$771$	0	0
$772$	37233.7	1.73584
$773$	20207.8	0.940262	0.470131	−	0.882597i	$-0.344207\pi$
$773$	20207.8	0.940262	0.470131	+	0.882597i	$0.344207\pi$
$774$	0	0
$775$	7390.15	0.342532
$776$	−27318.8	−1.26377
$777$	0	0
$778$	9416.18	0.433916
$779$	−26631.0	−1.22485
$780$	0	0
$781$	−59.2506	−0.00271467
$782$	13584.7	0.621211
$783$	0	0
$784$	−5922.71	−0.269803
$785$	−33850.1	−1.53906
$786$	0	0
$787$	−32312.5	−1.46355	−0.731777	−	0.681544i	$-0.761309\pi$
$787$	−32312.5	−1.46355	−0.731777	+	0.681544i	$0.761309\pi$
$788$	46675.2	2.11007
$789$	0	0
$790$	33206.8	1.49550
$791$	6061.83	0.272483
$792$	0	0
$793$	17019.7	0.762154
$794$	31983.0	1.42951
$795$	0	0
$796$	16362.9	0.728605
$797$	−34708.6	−1.54259	−0.771293	−	0.636480i	$-0.780390\pi$
$797$	−34708.6	−1.54259	−0.771293	+	0.636480i	$0.780390\pi$
$798$	0	0
$799$	−952.422	−0.0421706
$800$	−5682.36	−0.251127
$801$	0	0
$802$	29653.2	1.30560
$803$	132.881	0.00583968
$804$	0	0
$805$	−3066.44	−0.134258
$806$	26989.8	1.17950
$807$	0	0
$808$	28743.7	1.25148
$809$	30434.6	1.32265	0.661325	−	0.750100i	$-0.269995\pi$
$809$	30434.6	1.32265	0.661325	+	0.750100i	$0.269995\pi$
$810$	0	0
$811$	−19981.0	−0.865139	−0.432569	−	0.901601i	$-0.642393\pi$
$811$	−19981.0	−0.865139	−0.432569	+	0.901601i	$0.642393\pi$
$812$	−11788.9	−0.509494
$813$	0	0
$814$	216.299	0.00931362
$815$	19118.6	0.821713
$816$	0	0
$817$	31583.1	1.35245
$818$	−51341.1	−2.19450
$819$	0	0
$820$	52161.8	2.22142
$821$	−30639.4	−1.30247	−0.651233	−	0.758878i	$-0.725748\pi$
$821$	−30639.4	−1.30247	−0.651233	+	0.758878i	$0.725748\pi$
$822$	0	0
$823$	−11260.1	−0.476916	−0.238458	−	0.971153i	$-0.576642\pi$
$823$	−11260.1	−0.476916	−0.238458	+	0.971153i	$0.576642\pi$
$824$	−9376.87	−0.396431
$825$	0	0
$826$	8464.26	0.356549
$827$	187.472	0.00788277	0.00394138	−	0.999992i	$-0.498745\pi$
$827$	187.472	0.00788277	0.00394138	+	0.999992i	$0.498745\pi$
$828$	0	0
$829$	−33688.3	−1.41139	−0.705696	−	0.708515i	$-0.749366\pi$
$829$	−33688.3	−1.41139	−0.705696	+	0.708515i	$0.749366\pi$
$830$	26663.6	1.11507
$831$	0	0
$832$	−25523.6	−1.06355
$833$	12190.1	0.507036
$834$	0	0
$835$	34902.2	1.44652
$836$	238.675	0.00987412
$837$	0	0
$838$	−40194.9	−1.65693
$839$	25781.0	1.06086	0.530428	−	0.847730i	$-0.322031\pi$
$839$	25781.0	1.06086	0.530428	+	0.847730i	$0.322031\pi$
$840$	0	0
$841$	13394.3	0.549194
$842$	16550.5	0.677395
$843$	0	0
$844$	−52207.4	−2.12921
$845$	−10331.0	−0.420587
$846$	0	0
$847$	5801.47	0.235349
$848$	−11608.9	−0.470109
$849$	0	0
$850$	−7364.76	−0.297187
$851$	13460.4	0.542205
$852$	0	0
$853$	−3821.08	−0.153378	−0.0766889	−	0.997055i	$-0.524435\pi$
$853$	−3821.08	−0.153378	−0.0766889	+	0.997055i	$0.524435\pi$
$854$	10645.4	0.426557
$855$	0	0
$856$	−2830.83	−0.113032
$857$	18507.9	0.737709	0.368854	−	0.929487i	$-0.379750\pi$
$857$	18507.9	0.737709	0.368854	+	0.929487i	$0.379750\pi$
$858$	0	0
$859$	−12742.7	−0.506142	−0.253071	−	0.967448i	$-0.581441\pi$
$859$	−12742.7	−0.506142	−0.253071	+	0.967448i	$0.581441\pi$
$860$	−61861.3	−2.45285
$861$	0	0
$862$	83606.3	3.30353
$863$	4991.67	0.196893	0.0984464	−	0.995142i	$-0.468613\pi$
$863$	4991.67	0.196893	0.0984464	+	0.995142i	$0.468613\pi$
$864$	0	0
$865$	−2518.88	−0.0990111
$866$	−77778.5	−3.05199
$867$	0	0
$868$	10718.6	0.419138
$869$	205.930	0.00803879
$870$	0	0
$871$	32057.7	1.24711
$872$	50449.0	1.95920
$873$	0	0
$874$	23392.9	0.905351
$875$	6631.94	0.256229
$876$	0	0
$877$	−42740.7	−1.64567	−0.822835	−	0.568281i	$-0.807609\pi$
$877$	−42740.7	−1.64567	−0.822835	+	0.568281i	$0.807609\pi$
$878$	−61325.2	−2.35720
$879$	0	0
$880$	−44.1436	−0.00169100
$881$	−33732.0	−1.28997	−0.644983	−	0.764197i	$-0.723136\pi$
$881$	−33732.0	−1.28997	−0.644983	+	0.764197i	$0.723136\pi$
$882$	0	0
$883$	33426.0	1.27392	0.636962	−	0.770895i	$-0.280191\pi$
$883$	33426.0	1.27392	0.636962	+	0.770895i	$0.280191\pi$
$884$	−17077.8	−0.649760
$885$	0	0
$886$	19311.4	0.732255
$887$	1170.82	0.0443204	0.0221602	−	0.999754i	$-0.492946\pi$
$887$	1170.82	0.0443204	0.0221602	+	0.999754i	$0.492946\pi$
$888$	0	0
$889$	5987.73	0.225897
$890$	32404.1	1.22044
$891$	0	0
$892$	84762.6	3.18169
$893$	−1640.08	−0.0614592
$894$	0	0
$895$	14968.1	0.559027
$896$	−11225.7	−0.418553
$897$	0	0
$898$	−30350.9	−1.12787
$899$	−34352.9	−1.27445
$900$	0	0
$901$	23893.4	0.883469
$902$	509.471	0.0188066
$903$	0	0
$904$	38497.4	1.41638
$905$	6047.20	0.222117
$906$	0	0
$907$	−2560.30	−0.0937302	−0.0468651	−	0.998901i	$-0.514923\pi$
$907$	−2560.30	−0.0937302	−0.0468651	+	0.998901i	$0.514923\pi$
$908$	6431.16	0.235050
$909$	0	0
$910$	6071.41	0.221171
$911$	−52904.5	−1.92404	−0.962022	−	0.272974i	$-0.911993\pi$
$911$	−52904.5	−1.92404	−0.962022	+	0.272974i	$0.911993\pi$
$912$	0	0
$913$	165.353	0.00599385
$914$	−48832.4	−1.76721
$915$	0	0
$916$	43267.7	1.56070
$917$	481.061	0.0173239
$918$	0	0
$919$	−21301.2	−0.764593	−0.382297	−	0.924040i	$-0.624867\pi$
$919$	−21301.2	−0.764593	−0.382297	+	0.924040i	$0.624867\pi$
$920$	−19474.3	−0.697879
$921$	0	0
$922$	49378.2	1.76376
$923$	7300.47	0.260344
$924$	0	0
$925$	−7297.38	−0.259391
$926$	79011.0	2.80396
$927$	0	0
$928$	26414.3	0.934367
$929$	−56298.8	−1.98827	−0.994136	−	0.108134i	$-0.965512\pi$
$929$	−56298.8	−1.98827	−0.994136	+	0.108134i	$0.965512\pi$
$930$	0	0
$931$	20991.4	0.738952
$932$	−87928.3	−3.09033
$933$	0	0
$934$	−12321.2	−0.431652
$935$	90.8560	0.00317787
$936$	0	0
$937$	−5930.96	−0.206783	−0.103392	−	0.994641i	$-0.532970\pi$
$937$	−5930.96	−0.206783	−0.103392	+	0.994641i	$0.532970\pi$
$938$	20051.3	0.697974
$939$	0	0
$940$	3212.39	0.111465
$941$	−16512.8	−0.572052	−0.286026	−	0.958222i	$-0.592334\pi$
$941$	−16512.8	−0.572052	−0.286026	+	0.958222i	$0.592334\pi$
$942$	0	0
$943$	31704.6	1.09485
$944$	7582.72	0.261437
$945$	0	0
$946$	−604.207	−0.0207658
$947$	50831.4	1.74424	0.872122	−	0.489288i	$-0.162743\pi$
$947$	50831.4	1.74424	0.872122	+	0.489288i	$0.162743\pi$
$948$	0	0
$949$	−16372.7	−0.560042
$950$	−12682.2	−0.433120
$951$	0	0
$952$	−4540.00	−0.154561
$953$	−1247.25	−0.0423949	−0.0211975	−	0.999775i	$-0.506748\pi$
$953$	−1247.25	−0.0423949	−0.0211975	+	0.999775i	$0.506748\pi$
$954$	0	0
$955$	46162.3	1.56417
$956$	−3325.36	−0.112500
$957$	0	0
$958$	−58627.0	−1.97720
$959$	5926.66	0.199564
$960$	0	0
$961$	1442.96	0.0484359
$962$	−26651.0	−0.893204
$963$	0	0
$964$	−20033.8	−0.669340
$965$	24407.1	0.814189
$966$	0	0
$967$	−18874.2	−0.627667	−0.313833	−	0.949478i	$-0.601613\pi$
$967$	−18874.2	−0.627667	−0.313833	+	0.949478i	$0.601613\pi$
$968$	36844.0	1.22336
$969$	0	0
$970$	−42133.6	−1.39467
$971$	49892.5	1.64895	0.824473	−	0.565902i	$-0.191472\pi$
$971$	49892.5	1.64895	0.824473	+	0.565902i	$0.191472\pi$
$972$	0	0
$973$	−8290.18	−0.273146
$974$	92775.6	3.05208
$975$	0	0
$976$	9536.73	0.312770
$977$	1866.45	0.0611186	0.0305593	−	0.999533i	$-0.490271\pi$
$977$	1866.45	0.0611186	0.0305593	+	0.999533i	$0.490271\pi$
$978$	0	0
$979$	200.952	0.00656023
$980$	−41115.5	−1.34019
$981$	0	0
$982$	13724.0	0.445978
$983$	39636.9	1.28608	0.643042	−	0.765831i	$-0.277672\pi$
$983$	39636.9	1.28608	0.643042	+	0.765831i	$0.277672\pi$
$984$	0	0
$985$	30596.1	0.989720
$986$	34234.9	1.10574
$987$	0	0
$988$	−29408.0	−0.946957
$989$	−37600.0	−1.20891
$990$	0	0
$991$	46902.1	1.50343	0.751713	−	0.659490i	$-0.229228\pi$
$991$	46902.1	1.50343	0.751713	+	0.659490i	$0.229228\pi$
$992$	−24016.1	−0.768662
$993$	0	0
$994$	4566.28	0.145708
$995$	10726.1	0.341749
$996$	0	0
$997$	12513.5	0.397500	0.198750	−	0.980050i	$-0.436312\pi$
$997$	12513.5	0.397500	0.198750	+	0.980050i	$0.436312\pi$
$998$	43871.4	1.39151
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.b.1.3		28
3.2	odd	2		717.4.a.b.1.26	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.b.1.26	✓	28	3.2	odd	2
2151.4.a.b.1.3		28	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-4.68120 q^{2} +13.9136 q^{4} +9.12054 q^{5} -4.35896 q^{7} -27.6829 q^{8} +O(q^{10})\)	\(q-4.68120 q^{2} +13.9136 q^{4} +9.12054 q^{5} -4.35896 q^{7} -27.6829 q^{8} -42.6951 q^{10} -0.264771 q^{11} +32.6234 q^{13} +20.4052 q^{14} +18.2800 q^{16} -37.6237 q^{17} -64.7883 q^{19} +126.900 q^{20} +1.23945 q^{22} +77.1313 q^{23} -41.8157 q^{25} -152.717 q^{26} -60.6490 q^{28} +194.379 q^{29} -176.731 q^{31} +135.891 q^{32} +176.124 q^{34} -39.7561 q^{35} +174.513 q^{37} +303.287 q^{38} -252.483 q^{40} +411.047 q^{41} -487.481 q^{43} -3.68393 q^{44} -361.067 q^{46} +25.3144 q^{47} -323.999 q^{49} +195.748 q^{50} +453.910 q^{52} -635.063 q^{53} -2.41486 q^{55} +120.669 q^{56} -909.928 q^{58} +414.809 q^{59} +521.703 q^{61} +827.314 q^{62} -782.371 q^{64} +297.543 q^{65} +982.659 q^{67} -523.483 q^{68} +186.106 q^{70} +223.780 q^{71} -501.870 q^{73} -816.929 q^{74} -901.440 q^{76} +1.15413 q^{77} -777.767 q^{79} +166.723 q^{80} -1924.19 q^{82} -624.513 q^{83} -343.149 q^{85} +2282.00 q^{86} +7.32963 q^{88} -758.966 q^{89} -142.204 q^{91} +1073.18 q^{92} -118.502 q^{94} -590.904 q^{95} +986.850 q^{97} +1516.71 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) \) 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8	\( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) \) 28 * q + 5 * q^2 + 103 * q^4 - 6 * q^5 - 68 * q^7 + 39 * q^8 - 88 * q^10 + 110 * q^11 - 82 * q^13 - 126 * q^14 + 271 * q^16 - 100 * q^17 - 292 * q^19 + 52 * q^20 - 351 * q^22 + 276 * q^23 + 386 * q^25 - 84 * q^26 - 1010 * q^28 + 38 * q^29 - 432 * q^31 + 452 * q^32 - 524 * q^34 + 166 * q^35 - 936 * q^37 + 41 * q^38 - 1183 * q^40 - 1054 * q^41 - 1804 * q^43 + 341 * q^44 - 888 * q^46 + 560 * q^47 + 1074 * q^49 + 1054 * q^50 - 632 * q^52 + 160 * q^53 - 842 * q^55 - 509 * q^56 - 1266 * q^58 - 846 * q^59 - 2220 * q^61 - 82 * q^62 - 1565 * q^64 - 296 * q^65 - 4752 * q^67 + 1719 * q^68 - 5601 * q^70 + 802 * q^71 - 2732 * q^73 + 4581 * q^74 - 5614 * q^76 + 1008 * q^77 - 3172 * q^79 + 732 * q^80 - 9709 * q^82 + 4780 * q^83 - 4624 * q^85 + 2009 * q^86 - 9331 * q^88 - 4372 * q^89 - 7398 * q^91 + 6138 * q^92 - 7068 * q^94 + 3160 * q^95 - 4846 * q^97 + 3772 * q^98

Introduction

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