LMFDB - Embedded newform 2151.4.a.c.1.1

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:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-5.43685 q^{2} +21.5594 q^{4} +21.1888 q^{5} -20.1660 q^{7} -73.7203 q^{8} +O(q^{10})$	\(q-5.43685 q^{2} +21.5594 q^{4} +21.1888 q^{5} -20.1660 q^{7} -73.7203 q^{8} -115.201 q^{10} +58.1245 q^{11} -34.3940 q^{13} +109.639 q^{14} +228.331 q^{16} +2.49251 q^{17} -34.4858 q^{19} +456.818 q^{20} -316.014 q^{22} +127.252 q^{23} +323.967 q^{25} +186.995 q^{26} -434.765 q^{28} +288.827 q^{29} +286.875 q^{31} -651.641 q^{32} -13.5514 q^{34} -427.294 q^{35} +135.817 q^{37} +187.494 q^{38} -1562.05 q^{40} -360.126 q^{41} -419.768 q^{43} +1253.13 q^{44} -691.850 q^{46} -226.351 q^{47} +63.6663 q^{49} -1761.36 q^{50} -741.514 q^{52} -129.578 q^{53} +1231.59 q^{55} +1486.64 q^{56} -1570.31 q^{58} +579.941 q^{59} +545.156 q^{61} -1559.70 q^{62} +1716.23 q^{64} -728.770 q^{65} -9.17543 q^{67} +53.7369 q^{68} +2323.13 q^{70} +335.879 q^{71} -256.510 q^{73} -738.417 q^{74} -743.493 q^{76} -1172.14 q^{77} -402.337 q^{79} +4838.08 q^{80} +1957.95 q^{82} +364.585 q^{83} +52.8134 q^{85} +2282.22 q^{86} -4284.95 q^{88} +911.616 q^{89} +693.589 q^{91} +2743.47 q^{92} +1230.64 q^{94} -730.715 q^{95} -988.622 q^{97} -346.144 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * 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 $

$2$	−5.43685	−1.92222	−0.961109	−	0.276170i	$-0.910935\pi$
$2$	−5.43685	−1.92222	−0.961109	+	0.276170i	$0.910935\pi$
$3$	0	0
$3$	0	0
$4$	21.5594	2.69492
$5$	21.1888	1.89519	0.947594	−	0.319477i	$-0.103507\pi$
$5$	21.1888	1.89519	0.947594	+	0.319477i	$0.103507\pi$
$6$	0	0
$7$	−20.1660	−1.08886	−0.544430	−	0.838806i	$-0.683254\pi$
$7$	−20.1660	−1.08886	−0.544430	+	0.838806i	$0.683254\pi$
$8$	−73.7203	−3.25801
$9$	0	0
$10$	−115.201	−3.64296
$11$	58.1245	1.59320	0.796600	−	0.604507i	$-0.206630\pi$
$11$	58.1245	1.59320	0.796600	+	0.604507i	$0.206630\pi$
$12$	0	0
$13$	−34.3940	−0.733784	−0.366892	−	0.930264i	$-0.619578\pi$
$13$	−34.3940	−0.733784	−0.366892	+	0.930264i	$0.619578\pi$
$14$	109.639	2.09303
$15$	0	0
$16$	228.331	3.56768
$17$	2.49251	0.0355601	0.0177801	−	0.999842i	$-0.494340\pi$
$17$	2.49251	0.0355601	0.0177801	+	0.999842i	$0.494340\pi$
$18$	0	0
$19$	−34.4858	−0.416400	−0.208200	−	0.978086i	$-0.566760\pi$
$19$	−34.4858	−0.416400	−0.208200	+	0.978086i	$0.566760\pi$
$20$	456.818	5.10738
$21$	0	0
$22$	−316.014	−3.06248
$23$	127.252	1.15365	0.576823	−	0.816869i	$-0.304292\pi$
$23$	127.252	1.15365	0.576823	+	0.816869i	$0.304292\pi$
$24$	0	0
$25$	323.967	2.59174
$26$	186.995	1.41049
$27$	0	0
$28$	−434.765	−2.93439
$29$	288.827	1.84944	0.924721	−	0.380646i	$-0.124298\pi$
$29$	288.827	1.84944	0.924721	+	0.380646i	$0.124298\pi$
$30$	0	0
$31$	286.875	1.66207	0.831037	−	0.556217i	$-0.187748\pi$
$31$	286.875	1.66207	0.831037	+	0.556217i	$0.187748\pi$
$32$	−651.641	−3.59984
$33$	0	0
$34$	−13.5514	−0.0683543
$35$	−427.294	−2.06359
$36$	0	0
$37$	135.817	0.603464	0.301732	−	0.953393i	$-0.402435\pi$
$37$	135.817	0.603464	0.301732	+	0.953393i	$0.402435\pi$
$38$	187.494	0.800411
$39$	0	0
$40$	−1562.05	−6.17453
$41$	−360.126	−1.37176	−0.685882	−	0.727713i	$-0.740583\pi$
$41$	−360.126	−1.37176	−0.685882	+	0.727713i	$0.740583\pi$
$42$	0	0
$43$	−419.768	−1.48870	−0.744349	−	0.667791i	$-0.767240\pi$
$43$	−419.768	−1.48870	−0.744349	+	0.667791i	$0.767240\pi$
$44$	1253.13	4.29355
$45$	0	0
$46$	−691.850	−2.21756
$47$	−226.351	−0.702482	−0.351241	−	0.936285i	$-0.614240\pi$
$47$	−226.351	−0.702482	−0.351241	+	0.936285i	$0.614240\pi$
$48$	0	0
$49$	63.6663	0.185616
$50$	−1761.36	−4.98189
$51$	0	0
$52$	−741.514	−1.97749
$53$	−129.578	−0.335828	−0.167914	−	0.985802i	$-0.553703\pi$
$53$	−129.578	−0.335828	−0.167914	+	0.985802i	$0.553703\pi$
$54$	0	0
$55$	1231.59	3.01941
$56$	1486.64	3.54751
$57$	0	0
$58$	−1570.31	−3.55503
$59$	579.941	1.27969	0.639846	−	0.768503i	$-0.278998\pi$
$59$	579.941	1.27969	0.639846	+	0.768503i	$0.278998\pi$
$60$	0	0
$61$	545.156	1.14426	0.572132	−	0.820162i	$-0.306117\pi$
$61$	545.156	1.14426	0.572132	+	0.820162i	$0.306117\pi$
$62$	−1559.70	−3.19487
$63$	0	0
$64$	1716.23	3.35201
$65$	−728.770	−1.39066
$66$	0	0
$67$	−9.17543	−0.0167307	−0.00836535	−	0.999965i	$-0.502663\pi$
$67$	−9.17543	−0.0167307	−0.00836535	+	0.999965i	$0.502663\pi$
$68$	53.7369	0.0958317
$69$	0	0
$70$	2323.13	3.96668
$71$	335.879	0.561430	0.280715	−	0.959791i	$-0.409428\pi$
$71$	335.879	0.561430	0.280715	+	0.959791i	$0.409428\pi$
$72$	0	0
$73$	−256.510	−0.411264	−0.205632	−	0.978629i	$-0.565925\pi$
$73$	−256.510	−0.411264	−0.205632	+	0.978629i	$0.565925\pi$
$74$	−738.417	−1.15999
$75$	0	0
$76$	−743.493	−1.12216
$77$	−1172.14	−1.73477
$78$	0	0
$79$	−402.337	−0.572993	−0.286497	−	0.958081i	$-0.592491\pi$
$79$	−402.337	−0.572993	−0.286497	+	0.958081i	$0.592491\pi$
$80$	4838.08	6.76142
$81$	0	0
$82$	1957.95	2.63683
$83$	364.585	0.482149	0.241074	−	0.970507i	$-0.422500\pi$
$83$	364.585	0.482149	0.241074	+	0.970507i	$0.422500\pi$
$84$	0	0
$85$	52.8134	0.0673931
$86$	2282.22	2.86160
$87$	0	0
$88$	−4284.95	−5.19065
$89$	911.616	1.08574	0.542871	−	0.839816i	$-0.317337\pi$
$89$	911.616	1.08574	0.542871	+	0.839816i	$0.317337\pi$
$90$	0	0
$91$	693.589	0.798988
$92$	2743.47	3.10898
$93$	0	0
$94$	1230.64	1.35032
$95$	−730.715	−0.789156
$96$	0	0
$97$	−988.622	−1.03484	−0.517419	−	0.855732i	$-0.673107\pi$
$97$	−988.622	−1.03484	−0.517419	+	0.855732i	$0.673107\pi$
$98$	−346.144	−0.356795
$99$	0	0
$100$	6984.53	6.98453
$101$	−951.895	−0.937793	−0.468896	−	0.883253i	$-0.655348\pi$
$101$	−951.895	−0.937793	−0.468896	+	0.883253i	$0.655348\pi$
$102$	0	0
$103$	−72.7834	−0.0696268	−0.0348134	−	0.999394i	$-0.511084\pi$
$103$	−72.7834	−0.0696268	−0.0348134	+	0.999394i	$0.511084\pi$
$104$	2535.54	2.39067
$105$	0	0
$106$	704.495	0.645534
$107$	1600.49	1.44603	0.723016	−	0.690831i	$-0.242755\pi$
$107$	1600.49	1.44603	0.723016	+	0.690831i	$0.242755\pi$
$108$	0	0
$109$	718.969	0.631787	0.315893	−	0.948795i	$-0.397696\pi$
$109$	718.969	0.631787	0.315893	+	0.948795i	$0.397696\pi$
$110$	−6695.98	−5.80397
$111$	0	0
$112$	−4604.52	−3.88470
$113$	−467.855	−0.389488	−0.194744	−	0.980854i	$-0.562388\pi$
$113$	−467.855	−0.389488	−0.194744	+	0.980854i	$0.562388\pi$
$114$	0	0
$115$	2696.32	2.18638
$116$	6226.93	4.98410
$117$	0	0
$118$	−3153.05	−2.45985
$119$	−50.2638	−0.0387200
$120$	0	0
$121$	2047.46	1.53828
$122$	−2963.93	−2.19952
$123$	0	0
$124$	6184.85	4.47916
$125$	4215.89	3.01664
$126$	0	0
$127$	203.383	0.142105	0.0710524	−	0.997473i	$-0.477364\pi$
$127$	203.383	0.142105	0.0710524	+	0.997473i	$0.477364\pi$
$128$	−4117.74	−2.84344
$129$	0	0
$130$	3962.22	2.67315
$131$	−143.176	−0.0954915	−0.0477457	−	0.998860i	$-0.515204\pi$
$131$	−143.176	−0.0954915	−0.0477457	+	0.998860i	$0.515204\pi$
$132$	0	0
$133$	695.440	0.453401
$134$	49.8854	0.0321600
$135$	0	0
$136$	−183.748	−0.115855
$137$	−560.663	−0.349640	−0.174820	−	0.984600i	$-0.555934\pi$
$137$	−560.663	−0.349640	−0.174820	+	0.984600i	$0.555934\pi$
$138$	0	0
$139$	−922.954	−0.563194	−0.281597	−	0.959533i	$-0.590864\pi$
$139$	−922.954	−0.563194	−0.281597	+	0.959533i	$0.590864\pi$
$140$	−9212.18	−5.56122
$141$	0	0
$142$	−1826.13	−1.07919
$143$	−1999.14	−1.16906
$144$	0	0
$145$	6119.91	3.50504
$146$	1394.61	0.790539
$147$	0	0
$148$	2928.13	1.62629
$149$	−3390.03	−1.86391	−0.931953	−	0.362579i	$-0.881896\pi$
$149$	−3390.03	−1.86391	−0.931953	+	0.362579i	$0.881896\pi$
$150$	0	0
$151$	175.213	0.0944282	0.0472141	−	0.998885i	$-0.484966\pi$
$151$	175.213	0.0944282	0.0472141	+	0.998885i	$0.484966\pi$
$152$	2542.31	1.35663
$153$	0	0
$154$	6372.73	3.33461
$155$	6078.55	3.14994
$156$	0	0
$157$	3170.33	1.61159	0.805796	−	0.592194i	$-0.201738\pi$
$157$	3170.33	1.61159	0.805796	+	0.592194i	$0.201738\pi$
$158$	2187.45	1.10142
$159$	0	0
$160$	−13807.5	−6.82238
$161$	−2566.16	−1.25616
$162$	0	0
$163$	−3857.02	−1.85340	−0.926702	−	0.375797i	$-0.877369\pi$
$163$	−3857.02	−1.85340	−0.926702	+	0.375797i	$0.877369\pi$
$164$	−7764.10	−3.69679
$165$	0	0
$166$	−1982.19	−0.926795
$167$	−2459.26	−1.13954	−0.569771	−	0.821803i	$-0.692968\pi$
$167$	−2459.26	−1.13954	−0.569771	+	0.821803i	$0.692968\pi$
$168$	0	0
$169$	−1014.05	−0.461561
$170$	−287.138	−0.129544
$171$	0	0
$172$	−9049.93	−4.01192
$173$	3962.65	1.74147	0.870737	−	0.491749i	$-0.163642\pi$
$173$	3962.65	1.74147	0.870737	+	0.491749i	$0.163642\pi$
$174$	0	0
$175$	−6533.11	−2.82204
$176$	13271.6	5.68402
$177$	0	0
$178$	−4956.32	−2.08703
$179$	1133.90	0.473472	0.236736	−	0.971574i	$-0.423922\pi$
$179$	1133.90	0.473472	0.236736	+	0.971574i	$0.423922\pi$
$180$	0	0
$181$	−1236.58	−0.507813	−0.253906	−	0.967229i	$-0.581716\pi$
$181$	−1236.58	−0.507813	−0.253906	+	0.967229i	$0.581716\pi$
$182$	−3770.94	−1.53583
$183$	0	0
$184$	−9381.04	−3.75858
$185$	2877.80	1.14368
$186$	0	0
$187$	144.876	0.0566543
$188$	−4879.98	−1.89313
$189$	0	0
$190$	3972.79	1.51693
$191$	−2195.13	−0.831592	−0.415796	−	0.909458i	$-0.636497\pi$
$191$	−2195.13	−0.831592	−0.415796	+	0.909458i	$0.636497\pi$
$192$	0	0
$193$	2806.46	1.04670	0.523351	−	0.852117i	$-0.324682\pi$
$193$	2806.46	1.04670	0.523351	+	0.852117i	$0.324682\pi$
$194$	5374.99	1.98919
$195$	0	0
$196$	1372.61	0.500221
$197$	4583.36	1.65762	0.828809	−	0.559532i	$-0.189019\pi$
$197$	4583.36	1.65762	0.828809	+	0.559532i	$0.189019\pi$
$198$	0	0
$199$	424.495	0.151215	0.0756073	−	0.997138i	$-0.475910\pi$
$199$	424.495	0.151215	0.0756073	+	0.997138i	$0.475910\pi$
$200$	−23883.0	−8.44390
$201$	0	0
$202$	5175.31	1.80264
$203$	−5824.48	−2.01378
$204$	0	0
$205$	−7630.67	−2.59975
$206$	395.713	0.133838
$207$	0	0
$208$	−7853.24	−2.61790
$209$	−2004.47	−0.663408
$210$	0	0
$211$	−3706.51	−1.20932	−0.604661	−	0.796483i	$-0.706691\pi$
$211$	−3706.51	−1.20932	−0.604661	+	0.796483i	$0.706691\pi$
$212$	−2793.61	−0.905029
$213$	0	0
$214$	−8701.64	−2.77959
$215$	−8894.40	−2.82136
$216$	0	0
$217$	−5785.12	−1.80977
$218$	−3908.93	−1.21443
$219$	0	0
$220$	26552.3	8.13708
$221$	−85.7274	−0.0260934
$222$	0	0
$223$	2594.32	0.779052	0.389526	−	0.921015i	$-0.372639\pi$
$223$	2594.32	0.779052	0.389526	+	0.921015i	$0.372639\pi$
$224$	13141.0	3.91973
$225$	0	0
$226$	2543.66	0.748680
$227$	2885.94	0.843817	0.421908	−	0.906639i	$-0.361360\pi$
$227$	2885.94	0.843817	0.421908	+	0.906639i	$0.361360\pi$
$228$	0	0
$229$	−424.127	−0.122389	−0.0611945	−	0.998126i	$-0.519491\pi$
$229$	−424.127	−0.122389	−0.0611945	+	0.998126i	$0.519491\pi$
$230$	−14659.5	−4.20269
$231$	0	0
$232$	−21292.4	−6.02549
$233$	4328.37	1.21700	0.608499	−	0.793554i	$-0.291772\pi$
$233$	4328.37	1.21700	0.608499	+	0.793554i	$0.291772\pi$
$234$	0	0
$235$	−4796.11	−1.33133
$236$	12503.1	3.44867
$237$	0	0
$238$	273.277	0.0744282
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−713.303	−0.190655	−0.0953276	−	0.995446i	$-0.530390\pi$
$241$	−713.303	−0.190655	−0.0953276	+	0.995446i	$0.530390\pi$
$242$	−11131.7	−2.95692
$243$	0	0
$244$	11753.2	3.08370
$245$	1349.02	0.351777
$246$	0	0
$247$	1186.11	0.305547
$248$	−21148.5	−5.41505
$249$	0	0
$250$	−22921.2	−5.79865
$251$	−1738.25	−0.437121	−0.218561	−	0.975823i	$-0.570136\pi$
$251$	−1738.25	−0.437121	−0.218561	+	0.975823i	$0.570136\pi$
$252$	0	0
$253$	7396.45	1.83799
$254$	−1105.76	−0.273156
$255$	0	0
$256$	8657.75	2.11371
$257$	−1470.20	−0.356843	−0.178422	−	0.983954i	$-0.557099\pi$
$257$	−1470.20	−0.356843	−0.178422	+	0.983954i	$0.557099\pi$
$258$	0	0
$259$	−2738.88	−0.657088
$260$	−15711.8	−3.74771
$261$	0	0
$262$	778.429	0.183555
$263$	3330.67	0.780904	0.390452	−	0.920623i	$-0.372319\pi$
$263$	3330.67	0.780904	0.390452	+	0.920623i	$0.372319\pi$
$264$	0	0
$265$	−2745.60	−0.636457
$266$	−3781.01	−0.871535
$267$	0	0
$268$	−197.816	−0.0450879
$269$	−2471.36	−0.560154	−0.280077	−	0.959978i	$-0.590360\pi$
$269$	−2471.36	−0.560154	−0.280077	+	0.959978i	$0.590360\pi$
$270$	0	0
$271$	3297.34	0.739110	0.369555	−	0.929209i	$-0.379510\pi$
$271$	3297.34	0.739110	0.369555	+	0.929209i	$0.379510\pi$
$272$	569.117	0.126867
$273$	0	0
$274$	3048.24	0.672084
$275$	18830.4	4.12916
$276$	0	0
$277$	3247.02	0.704312	0.352156	−	0.935941i	$-0.385449\pi$
$277$	3247.02	0.704312	0.352156	+	0.935941i	$0.385449\pi$
$278$	5017.97	1.08258
$279$	0	0
$280$	31500.2	6.72320
$281$	6456.24	1.37063	0.685315	−	0.728246i	$-0.259664\pi$
$281$	6456.24	1.37063	0.685315	+	0.728246i	$0.259664\pi$
$282$	0	0
$283$	−5162.81	−1.08444	−0.542221	−	0.840236i	$-0.682416\pi$
$283$	−5162.81	−1.08444	−0.542221	+	0.840236i	$0.682416\pi$
$284$	7241.34	1.51301
$285$	0	0
$286$	10869.0	2.24720
$287$	7262.30	1.49366
$288$	0	0
$289$	−4906.79	−0.998735
$290$	−33273.1	−6.73745
$291$	0	0
$292$	−5530.20	−1.10832
$293$	3359.08	0.669760	0.334880	−	0.942261i	$-0.391304\pi$
$293$	3359.08	0.669760	0.334880	+	0.942261i	$0.391304\pi$
$294$	0	0
$295$	12288.3	2.42526
$296$	−10012.5	−1.96609
$297$	0	0
$298$	18431.1	3.58283
$299$	−4376.71	−0.846527
$300$	0	0
$301$	8465.03	1.62098
$302$	−952.609	−0.181512
$303$	0	0
$304$	−7874.20	−1.48558
$305$	11551.2	2.16859
$306$	0	0
$307$	−1147.42	−0.213312	−0.106656	−	0.994296i	$-0.534014\pi$
$307$	−1147.42	−0.213312	−0.106656	+	0.994296i	$0.534014\pi$
$308$	−25270.5	−4.67507
$309$	0	0
$310$	−33048.2	−6.05488
$311$	−1812.41	−0.330457	−0.165228	−	0.986255i	$-0.552836\pi$
$311$	−1812.41	−0.330457	−0.165228	+	0.986255i	$0.552836\pi$
$312$	0	0
$313$	10852.6	1.95983	0.979914	−	0.199420i	$-0.0639058\pi$
$313$	10852.6	1.95983	0.979914	+	0.199420i	$0.0639058\pi$
$314$	−17236.6	−3.09783
$315$	0	0
$316$	−8674.13	−1.54417
$317$	4758.72	0.843144	0.421572	−	0.906795i	$-0.361479\pi$
$317$	4758.72	0.843144	0.421572	+	0.906795i	$0.361479\pi$
$318$	0	0
$319$	16787.9	2.94653
$320$	36364.9	6.35268
$321$	0	0
$322$	13951.8	2.41461
$323$	−85.9562	−0.0148072
$324$	0	0
$325$	−11142.5	−1.90178
$326$	20970.0	3.56265
$327$	0	0
$328$	26548.6	4.46921
$329$	4564.58	0.764904
$330$	0	0
$331$	−2882.46	−0.478654	−0.239327	−	0.970939i	$-0.576927\pi$
$331$	−2882.46	−0.478654	−0.239327	+	0.970939i	$0.576927\pi$
$332$	7860.21	1.29935
$333$	0	0
$334$	13370.7	2.19045
$335$	−194.417	−0.0317078
$336$	0	0
$337$	−3930.01	−0.635255	−0.317628	−	0.948216i	$-0.602886\pi$
$337$	−3930.01	−0.635255	−0.317628	+	0.948216i	$0.602886\pi$
$338$	5513.24	0.887221
$339$	0	0
$340$	1138.62	0.181619
$341$	16674.5	2.64802
$342$	0	0
$343$	5633.03	0.886750
$344$	30945.4	4.85019
$345$	0	0
$346$	−21544.4	−3.34749
$347$	−11723.9	−1.81375	−0.906877	−	0.421395i	$-0.861541\pi$
$347$	−11723.9	−1.81375	−0.906877	+	0.421395i	$0.861541\pi$
$348$	0	0
$349$	−4793.87	−0.735271	−0.367636	−	0.929970i	$-0.619833\pi$
$349$	−4793.87	−0.735271	−0.367636	+	0.929970i	$0.619833\pi$
$350$	35519.6	5.42458
$351$	0	0
$352$	−37876.3	−5.73527
$353$	−6196.47	−0.934291	−0.467146	−	0.884180i	$-0.654718\pi$
$353$	−6196.47	−0.934291	−0.467146	+	0.884180i	$0.654718\pi$
$354$	0	0
$355$	7116.89	1.06402
$356$	19653.9	2.92599
$357$	0	0
$358$	−6164.83	−0.910115
$359$	3475.37	0.510927	0.255464	−	0.966819i	$-0.417772\pi$
$359$	3475.37	0.510927	0.255464	+	0.966819i	$0.417772\pi$
$360$	0	0
$361$	−5669.73	−0.826611
$362$	6723.09	0.976126
$363$	0	0
$364$	14953.3	2.15321
$365$	−5435.16	−0.779423
$366$	0	0
$367$	3800.28	0.540526	0.270263	−	0.962787i	$-0.412889\pi$
$367$	3800.28	0.540526	0.270263	+	0.962787i	$0.412889\pi$
$368$	29055.6	4.11583
$369$	0	0
$370$	−15646.2	−2.19840
$371$	2613.06	0.365669
$372$	0	0
$373$	3072.95	0.426571	0.213286	−	0.976990i	$-0.431583\pi$
$373$	3072.95	0.426571	0.213286	+	0.976990i	$0.431583\pi$
$374$	−787.668	−0.108902
$375$	0	0
$376$	16686.6	2.28869
$377$	−9933.93	−1.35709
$378$	0	0
$379$	2211.79	0.299768	0.149884	−	0.988704i	$-0.452110\pi$
$379$	2211.79	0.299768	0.149884	+	0.988704i	$0.452110\pi$
$380$	−15753.8	−2.12671
$381$	0	0
$382$	11934.6	1.59850
$383$	−2103.46	−0.280632	−0.140316	−	0.990107i	$-0.544812\pi$
$383$	−2103.46	−0.280632	−0.140316	+	0.990107i	$0.544812\pi$
$384$	0	0
$385$	−24836.2	−3.28772
$386$	−15258.3	−2.01199
$387$	0	0
$388$	−21314.1	−2.78881
$389$	−6602.56	−0.860574	−0.430287	−	0.902692i	$-0.641588\pi$
$389$	−6602.56	−0.860574	−0.430287	+	0.902692i	$0.641588\pi$
$390$	0	0
$391$	317.176	0.0410238
$392$	−4693.50	−0.604738
$393$	0	0
$394$	−24919.0	−3.18630
$395$	−8525.06	−1.08593
$396$	0	0
$397$	−4180.07	−0.528442	−0.264221	−	0.964462i	$-0.585115\pi$
$397$	−4180.07	−0.528442	−0.264221	+	0.964462i	$0.585115\pi$
$398$	−2307.92	−0.290667
$399$	0	0
$400$	73971.9	9.24648
$401$	13817.7	1.72076	0.860380	−	0.509653i	$-0.170226\pi$
$401$	13817.7	1.72076	0.860380	+	0.509653i	$0.170226\pi$
$402$	0	0
$403$	−9866.80	−1.21960
$404$	−20522.2	−2.52728
$405$	0	0
$406$	31666.8	3.87093
$407$	7894.29	0.961438
$408$	0	0
$409$	9986.49	1.20734	0.603668	−	0.797236i	$-0.293705\pi$
$409$	9986.49	1.20734	0.603668	+	0.797236i	$0.293705\pi$
$410$	41486.8	4.99729
$411$	0	0
$412$	−1569.16	−0.187639
$413$	−11695.1	−1.39341
$414$	0	0
$415$	7725.13	0.913763
$416$	22412.6	2.64151
$417$	0	0
$418$	10898.0	1.27521
$419$	1978.36	0.230667	0.115333	−	0.993327i	$-0.463206\pi$
$419$	1978.36	0.230667	0.115333	+	0.993327i	$0.463206\pi$
$420$	0	0
$421$	3782.60	0.437892	0.218946	−	0.975737i	$-0.429738\pi$
$421$	3782.60	0.437892	0.218946	+	0.975737i	$0.429738\pi$
$422$	20151.8	2.32458
$423$	0	0
$424$	9552.50	1.09413
$425$	807.491	0.0921625
$426$	0	0
$427$	−10993.6	−1.24594
$428$	34505.6	3.89694
$429$	0	0
$430$	48357.5	5.42327
$431$	407.376	0.0455281	0.0227640	−	0.999741i	$-0.492753\pi$
$431$	407.376	0.0455281	0.0227640	+	0.999741i	$0.492753\pi$
$432$	0	0
$433$	−6305.89	−0.699866	−0.349933	−	0.936775i	$-0.613796\pi$
$433$	−6305.89	−0.699866	−0.349933	+	0.936775i	$0.613796\pi$
$434$	31452.8	3.47876
$435$	0	0
$436$	15500.5	1.70262
$437$	−4388.39	−0.480378
$438$	0	0
$439$	15841.6	1.72227	0.861134	−	0.508377i	$-0.169754\pi$
$439$	15841.6	1.72227	0.861134	+	0.508377i	$0.169754\pi$
$440$	−90793.2	−9.83727
$441$	0	0
$442$	466.087	0.0501573
$443$	−8032.06	−0.861433	−0.430717	−	0.902487i	$-0.641739\pi$
$443$	−8032.06	−0.861433	−0.430717	+	0.902487i	$0.641739\pi$
$444$	0	0
$445$	19316.1	2.05769
$446$	−14104.9	−1.49751
$447$	0	0
$448$	−34609.4	−3.64987
$449$	14265.3	1.49938	0.749689	−	0.661791i	$-0.230203\pi$
$449$	14265.3	1.49938	0.749689	+	0.661791i	$0.230203\pi$
$450$	0	0
$451$	−20932.2	−2.18549
$452$	−10086.7	−1.04964
$453$	0	0
$454$	−15690.4	−1.62200
$455$	14696.4	1.51423
$456$	0	0
$457$	3757.21	0.384584	0.192292	−	0.981338i	$-0.438408\pi$
$457$	3757.21	0.384584	0.192292	+	0.981338i	$0.438408\pi$
$458$	2305.92	0.235258
$459$	0	0
$460$	58131.0	5.89211
$461$	−11026.2	−1.11397	−0.556986	−	0.830522i	$-0.688042\pi$
$461$	−11026.2	−1.11397	−0.556986	+	0.830522i	$0.688042\pi$
$462$	0	0
$463$	−6513.94	−0.653842	−0.326921	−	0.945052i	$-0.606011\pi$
$463$	−6513.94	−0.653842	−0.326921	+	0.945052i	$0.606011\pi$
$464$	65948.2	6.59821
$465$	0	0
$466$	−23532.7	−2.33934
$467$	2091.95	0.207289	0.103644	−	0.994614i	$-0.466950\pi$
$467$	2091.95	0.207289	0.103644	+	0.994614i	$0.466950\pi$
$468$	0	0
$469$	185.031	0.0182174
$470$	26075.7	2.55912
$471$	0	0
$472$	−42753.4	−4.16924
$473$	−24398.8	−2.37179
$474$	0	0
$475$	−11172.3	−1.07920
$476$	−1083.66	−0.104347
$477$	0	0
$478$	−1299.41	−0.124338
$479$	6892.48	0.657465	0.328732	−	0.944423i	$-0.393379\pi$
$479$	6892.48	0.657465	0.328732	+	0.944423i	$0.393379\pi$
$480$	0	0
$481$	−4671.29	−0.442812
$482$	3878.12	0.366481
$483$	0	0
$484$	44141.9	4.14555
$485$	−20947.8	−1.96121
$486$	0	0
$487$	−11291.5	−1.05065	−0.525327	−	0.850900i	$-0.676057\pi$
$487$	−11291.5	−1.05065	−0.525327	+	0.850900i	$0.676057\pi$
$488$	−40189.0	−3.72802
$489$	0	0
$490$	−7334.40	−0.676193
$491$	−3020.85	−0.277656	−0.138828	−	0.990317i	$-0.544333\pi$
$491$	−3020.85	−0.277656	−0.138828	+	0.990317i	$0.544333\pi$
$492$	0	0
$493$	719.903	0.0657664
$494$	−6448.69	−0.587329
$495$	0	0
$496$	65502.6	5.92974
$497$	−6773.33	−0.611319
$498$	0	0
$499$	−9055.82	−0.812413	−0.406206	−	0.913781i	$-0.633149\pi$
$499$	−9055.82	−0.812413	−0.406206	+	0.913781i	$0.633149\pi$
$500$	90891.9	8.12962
$501$	0	0
$502$	9450.62	0.840242
$503$	−11503.9	−1.01975	−0.509873	−	0.860250i	$-0.670307\pi$
$503$	−11503.9	−1.01975	−0.509873	+	0.860250i	$0.670307\pi$
$504$	0	0
$505$	−20169.6	−1.77729
$506$	−40213.4	−3.53301
$507$	0	0
$508$	4384.81	0.382961
$509$	15953.8	1.38927	0.694636	−	0.719362i	$-0.255566\pi$
$509$	15953.8	1.38927	0.694636	+	0.719362i	$0.255566\pi$
$510$	0	0
$511$	5172.78	0.447809
$512$	−14129.0	−1.21957
$513$	0	0
$514$	7993.27	0.685930
$515$	−1542.20	−0.131956
$516$	0	0
$517$	−13156.5	−1.11919
$518$	14890.9	1.26307
$519$	0	0
$520$	53725.1	4.53077
$521$	−17374.3	−1.46100	−0.730500	−	0.682913i	$-0.760713\pi$
$521$	−17374.3	−1.46100	−0.730500	+	0.682913i	$0.760713\pi$
$522$	0	0
$523$	8796.63	0.735468	0.367734	−	0.929931i	$-0.380134\pi$
$523$	8796.63	0.735468	0.367734	+	0.929931i	$0.380134\pi$
$524$	−3086.79	−0.257342
$525$	0	0
$526$	−18108.3	−1.50107
$527$	715.038	0.0591035
$528$	0	0
$529$	4026.04	0.330898
$530$	14927.4	1.22341
$531$	0	0
$532$	14993.3	1.22188
$533$	12386.2	1.00658
$534$	0	0
$535$	33912.6	2.74050
$536$	676.415	0.0545087
$537$	0	0
$538$	13436.4	1.07674
$539$	3700.57	0.295723
$540$	0	0
$541$	18925.7	1.50403	0.752016	−	0.659145i	$-0.229082\pi$
$541$	18925.7	1.50403	0.752016	+	0.659145i	$0.229082\pi$
$542$	−17927.1	−1.42073
$543$	0	0
$544$	−1624.22	−0.128011
$545$	15234.1	1.19735
$546$	0	0
$547$	9365.93	0.732099	0.366050	−	0.930595i	$-0.380710\pi$
$547$	9365.93	0.732099	0.366050	+	0.930595i	$0.380710\pi$
$548$	−12087.5	−0.942252
$549$	0	0
$550$	−102378.	−7.93714
$551$	−9960.44	−0.770107
$552$	0	0
$553$	8113.52	0.623909
$554$	−17653.6	−1.35384
$555$	0	0
$556$	−19898.3	−1.51776
$557$	6873.60	0.522879	0.261440	−	0.965220i	$-0.415803\pi$
$557$	6873.60	0.522879	0.261440	+	0.965220i	$0.415803\pi$
$558$	0	0
$559$	14437.5	1.09238
$560$	−97564.5	−7.36224
$561$	0	0
$562$	−35101.6	−2.63465
$563$	11181.1	0.836996	0.418498	−	0.908218i	$-0.362557\pi$
$563$	11181.1	0.836996	0.418498	+	0.908218i	$0.362557\pi$
$564$	0	0
$565$	−9913.31	−0.738153
$566$	28069.4	2.08453
$567$	0	0
$568$	−24761.1	−1.82914
$569$	4384.18	0.323013	0.161506	−	0.986872i	$-0.448365\pi$
$569$	4384.18	0.323013	0.161506	+	0.986872i	$0.448365\pi$
$570$	0	0
$571$	19703.0	1.44403	0.722017	−	0.691875i	$-0.243215\pi$
$571$	19703.0	1.44403	0.722017	+	0.691875i	$0.243215\pi$
$572$	−43100.1	−3.15054
$573$	0	0
$574$	−39484.1	−2.87114
$575$	41225.5	2.98995
$576$	0	0
$577$	−16336.5	−1.17868	−0.589339	−	0.807886i	$-0.700612\pi$
$577$	−16336.5	−1.17868	−0.589339	+	0.807886i	$0.700612\pi$
$578$	26677.5	1.91979
$579$	0	0
$580$	131941.	9.44581
$581$	−7352.20	−0.524993
$582$	0	0
$583$	−7531.64	−0.535040
$584$	18910.0	1.33990
$585$	0	0
$586$	−18262.8	−1.28742
$587$	−9751.23	−0.685650	−0.342825	−	0.939399i	$-0.611384\pi$
$587$	−9751.23	−0.685650	−0.342825	+	0.939399i	$0.611384\pi$
$588$	0	0
$589$	−9893.13	−0.692087
$590$	−66809.5	−4.66187
$591$	0	0
$592$	31011.3	2.15296
$593$	22215.3	1.53840	0.769201	−	0.639007i	$-0.220655\pi$
$593$	22215.3	1.53840	0.769201	+	0.639007i	$0.220655\pi$
$594$	0	0
$595$	−1065.03	−0.0733816
$596$	−73086.9	−5.02308
$597$	0	0
$598$	23795.5	1.62721
$599$	12857.0	0.877002	0.438501	−	0.898731i	$-0.355510\pi$
$599$	12857.0	0.877002	0.438501	+	0.898731i	$0.355510\pi$
$600$	0	0
$601$	−12358.8	−0.838811	−0.419405	−	0.907799i	$-0.637761\pi$
$601$	−12358.8	−0.838811	−0.419405	+	0.907799i	$0.637761\pi$
$602$	−46023.1	−3.11588
$603$	0	0
$604$	3777.49	0.254477
$605$	43383.3	2.91534
$606$	0	0
$607$	−1284.44	−0.0858879	−0.0429439	−	0.999077i	$-0.513674\pi$
$607$	−1284.44	−0.0858879	−0.0429439	+	0.999077i	$0.513674\pi$
$608$	22472.4	1.49897
$609$	0	0
$610$	−62802.3	−4.16851
$611$	7785.11	0.515470
$612$	0	0
$613$	−6718.07	−0.442643	−0.221322	−	0.975201i	$-0.571037\pi$
$613$	−6718.07	−0.442643	−0.221322	+	0.975201i	$0.571037\pi$
$614$	6238.37	0.410033
$615$	0	0
$616$	86410.2	5.65190
$617$	−9982.38	−0.651338	−0.325669	−	0.945484i	$-0.605589\pi$
$617$	−9982.38	−0.651338	−0.325669	+	0.945484i	$0.605589\pi$
$618$	0	0
$619$	−22750.8	−1.47728	−0.738638	−	0.674102i	$-0.764531\pi$
$619$	−22750.8	−1.47728	−0.738638	+	0.674102i	$0.764531\pi$
$620$	131050.	8.48885
$621$	0	0
$622$	9853.78	0.635210
$623$	−18383.6	−1.18222
$624$	0	0
$625$	48833.9	3.12537
$626$	−59004.1	−3.76722
$627$	0	0
$628$	68350.3	4.34311
$629$	338.525	0.0214592
$630$	0	0
$631$	−10285.3	−0.648891	−0.324446	−	0.945904i	$-0.605178\pi$
$631$	−10285.3	−0.648891	−0.324446	+	0.945904i	$0.605178\pi$
$632$	29660.4	1.86682
$633$	0	0
$634$	−25872.5	−1.62071
$635$	4309.45	0.269315
$636$	0	0
$637$	−2189.74	−0.136202
$638$	−91273.4	−5.66387
$639$	0	0
$640$	−87250.3	−5.38886
$641$	−5751.79	−0.354418	−0.177209	−	0.984173i	$-0.556707\pi$
$641$	−5751.79	−0.354418	−0.177209	+	0.984173i	$0.556707\pi$
$642$	0	0
$643$	−19382.4	−1.18875	−0.594375	−	0.804188i	$-0.702600\pi$
$643$	−19382.4	−1.18875	−0.594375	+	0.804188i	$0.702600\pi$
$644$	−55324.7	−3.38525
$645$	0	0
$646$	467.331	0.0284627
$647$	18223.1	1.10730	0.553651	−	0.832749i	$-0.313234\pi$
$647$	18223.1	1.10730	0.553651	+	0.832749i	$0.313234\pi$
$648$	0	0
$649$	33708.7	2.03880
$650$	60580.4	3.65563
$651$	0	0
$652$	−83154.8	−4.99478
$653$	21590.1	1.29385	0.646926	−	0.762552i	$-0.276054\pi$
$653$	21590.1	1.29385	0.646926	+	0.762552i	$0.276054\pi$
$654$	0	0
$655$	−3033.74	−0.180974
$656$	−82228.1	−4.89401
$657$	0	0
$658$	−24817.0	−1.47031
$659$	−18893.0	−1.11679	−0.558397	−	0.829574i	$-0.688583\pi$
$659$	−18893.0	−1.11679	−0.558397	+	0.829574i	$0.688583\pi$
$660$	0	0
$661$	3596.13	0.211608	0.105804	−	0.994387i	$-0.466258\pi$
$661$	3596.13	0.211608	0.105804	+	0.994387i	$0.466258\pi$
$662$	15671.5	0.920076
$663$	0	0
$664$	−26877.3	−1.57084
$665$	14735.6	0.859280
$666$	0	0
$667$	36753.8	2.13360
$668$	−53020.2	−3.07098
$669$	0	0
$670$	1057.01	0.0609493
$671$	31686.9	1.82304
$672$	0	0
$673$	23066.4	1.32117	0.660583	−	0.750753i	$-0.270309\pi$
$673$	23066.4	1.32117	0.660583	+	0.750753i	$0.270309\pi$
$674$	21366.9	1.22110
$675$	0	0
$676$	−21862.3	−1.24387
$677$	29597.5	1.68025	0.840123	−	0.542397i	$-0.182483\pi$
$677$	29597.5	1.68025	0.840123	+	0.542397i	$0.182483\pi$
$678$	0	0
$679$	19936.5	1.12679
$680$	−3893.41	−0.219567
$681$	0	0
$682$	−90656.7	−5.09006
$683$	−313.727	−0.0175761	−0.00878803	−	0.999961i	$-0.502797\pi$
$683$	−313.727	−0.0175761	−0.00878803	+	0.999961i	$0.502797\pi$
$684$	0	0
$685$	−11879.8	−0.662633
$686$	−30626.0	−1.70453
$687$	0	0
$688$	−95846.2	−5.31119
$689$	4456.70	0.246425
$690$	0	0
$691$	7142.52	0.393219	0.196610	−	0.980482i	$-0.437007\pi$
$691$	7142.52	0.393219	0.196610	+	0.980482i	$0.437007\pi$
$692$	85432.3	4.69313
$693$	0	0
$694$	63741.2	3.48643
$695$	−19556.3	−1.06736
$696$	0	0
$697$	−897.618	−0.0487801
$698$	26063.5	1.41335
$699$	0	0
$700$	−140850.	−7.60517
$701$	−8573.27	−0.461923	−0.230961	−	0.972963i	$-0.574187\pi$
$701$	−8573.27	−0.461923	−0.230961	+	0.972963i	$0.574187\pi$
$702$	0	0
$703$	−4683.76	−0.251282
$704$	99754.8	5.34042
$705$	0	0
$706$	33689.3	1.79591
$707$	19195.9	1.02113
$708$	0	0
$709$	−9369.75	−0.496316	−0.248158	−	0.968720i	$-0.579825\pi$
$709$	−9369.75	−0.496316	−0.248158	+	0.968720i	$0.579825\pi$
$710$	−38693.5	−2.04527
$711$	0	0
$712$	−67204.6	−3.53736
$713$	36505.4	1.91744
$714$	0	0
$715$	−42359.4	−2.21560
$716$	24446.1	1.27597
$717$	0	0
$718$	−18895.1	−0.982113
$719$	−5451.24	−0.282750	−0.141375	−	0.989956i	$-0.545152\pi$
$719$	−5451.24	−0.282750	−0.141375	+	0.989956i	$0.545152\pi$
$720$	0	0
$721$	1467.75	0.0758139
$722$	30825.5	1.58893
$723$	0	0
$724$	−26659.8	−1.36851
$725$	93570.5	4.79327
$726$	0	0
$727$	9893.99	0.504742	0.252371	−	0.967631i	$-0.418790\pi$
$727$	9893.99	0.504742	0.252371	+	0.967631i	$0.418790\pi$
$728$	−51131.6	−2.60311
$729$	0	0
$730$	29550.2	1.49822
$731$	−1046.27	−0.0529383
$732$	0	0
$733$	−25285.5	−1.27414	−0.637068	−	0.770808i	$-0.719853\pi$
$733$	−25285.5	−1.27414	−0.637068	+	0.770808i	$0.719853\pi$
$734$	−20661.6	−1.03901
$735$	0	0
$736$	−82922.6	−4.15294
$737$	−533.317	−0.0266553
$738$	0	0
$739$	−8320.96	−0.414197	−0.207098	−	0.978320i	$-0.566402\pi$
$739$	−8320.96	−0.414197	−0.207098	+	0.978320i	$0.566402\pi$
$740$	62043.6	3.08212
$741$	0	0
$742$	−14206.8	−0.702896
$743$	17595.8	0.868812	0.434406	−	0.900717i	$-0.356958\pi$
$743$	17595.8	0.868812	0.434406	+	0.900717i	$0.356958\pi$
$744$	0	0
$745$	−71830.8	−3.53245
$746$	−16707.2	−0.819963
$747$	0	0
$748$	3123.43	0.152679
$749$	−32275.5	−1.57453
$750$	0	0
$751$	−20473.6	−0.994798	−0.497399	−	0.867522i	$-0.665711\pi$
$751$	−20473.6	−0.994798	−0.497399	+	0.867522i	$0.665711\pi$
$752$	−51682.9	−2.50623
$753$	0	0
$754$	54009.3	2.60862
$755$	3712.57	0.178959
$756$	0	0
$757$	11349.7	0.544930	0.272465	−	0.962166i	$-0.412161\pi$
$757$	11349.7	0.544930	0.272465	+	0.962166i	$0.412161\pi$
$758$	−12025.2	−0.576219
$759$	0	0
$760$	53868.5	2.57107
$761$	−33383.6	−1.59022	−0.795108	−	0.606468i	$-0.792586\pi$
$761$	−33383.6	−1.59022	−0.795108	+	0.606468i	$0.792586\pi$
$762$	0	0
$763$	−14498.7	−0.687927
$764$	−47325.6	−2.24107
$765$	0	0
$766$	11436.2	0.539435
$767$	−19946.5	−0.939017
$768$	0	0
$769$	33150.7	1.55454	0.777272	−	0.629164i	$-0.216603\pi$
$769$	33150.7	1.55454	0.777272	+	0.629164i	$0.216603\pi$
$770$	135031.	6.31971
$771$	0	0
$772$	60505.5	2.82078
$773$	26587.2	1.23709	0.618547	−	0.785748i	$-0.287722\pi$
$773$	26587.2	1.23709	0.618547	+	0.785748i	$0.287722\pi$
$774$	0	0
$775$	92938.2	4.30766
$776$	72881.5	3.37151
$777$	0	0
$778$	35897.2	1.65421
$779$	12419.3	0.571202
$780$	0	0
$781$	19522.8	0.894470
$782$	−1724.44	−0.0788566
$783$	0	0
$784$	14537.0	0.662218
$785$	67175.6	3.05427
$786$	0	0
$787$	−377.173	−0.0170836	−0.00854178	−	0.999964i	$-0.502719\pi$
$787$	−377.173	−0.0170836	−0.00854178	+	0.999964i	$0.502719\pi$
$788$	98814.2	4.46715
$789$	0	0
$790$	46349.5	2.08739
$791$	9434.75	0.424098
$792$	0	0
$793$	−18750.1	−0.839642
$794$	22726.4	1.01578
$795$	0	0
$796$	9151.85	0.407511
$797$	16345.8	0.726474	0.363237	−	0.931697i	$-0.381672\pi$
$797$	16345.8	0.726474	0.363237	+	0.931697i	$0.381672\pi$
$798$	0	0
$799$	−564.181	−0.0249803
$800$	−211110.	−9.32985
$801$	0	0
$802$	−75125.0	−3.30768
$803$	−14909.5	−0.655226
$804$	0	0
$805$	−54373.9	−2.38066
$806$	53644.3	2.34434
$807$	0	0
$808$	70173.9	3.05533
$809$	−43097.6	−1.87297	−0.936485	−	0.350708i	$-0.885941\pi$
$809$	−43097.6	−1.87297	−0.936485	+	0.350708i	$0.885941\pi$
$810$	0	0
$811$	10251.9	0.443888	0.221944	−	0.975059i	$-0.428760\pi$
$811$	10251.9	0.443888	0.221944	+	0.975059i	$0.428760\pi$
$812$	−125572.	−5.42699
$813$	0	0
$814$	−42920.1	−1.84809
$815$	−81725.7	−3.51255
$816$	0	0
$817$	14476.1	0.619894
$818$	−54295.1	−2.32076
$819$	0	0
$820$	−164512.	−7.00612
$821$	23807.1	1.01203	0.506013	−	0.862526i	$-0.331119\pi$
$821$	23807.1	1.01203	0.506013	+	0.862526i	$0.331119\pi$
$822$	0	0
$823$	−12364.6	−0.523699	−0.261850	−	0.965109i	$-0.584332\pi$
$823$	−12364.6	−0.523699	−0.261850	+	0.965109i	$0.584332\pi$
$824$	5365.61	0.226845
$825$	0	0
$826$	63584.3	2.67843
$827$	22253.2	0.935696	0.467848	−	0.883809i	$-0.345030\pi$
$827$	22253.2	0.935696	0.467848	+	0.883809i	$0.345030\pi$
$828$	0	0
$829$	−5832.30	−0.244348	−0.122174	−	0.992509i	$-0.538987\pi$
$829$	−5832.30	−0.244348	−0.122174	+	0.992509i	$0.538987\pi$
$830$	−42000.4	−1.75645
$831$	0	0
$832$	−59028.0	−2.45965
$833$	158.689	0.00660053
$834$	0	0
$835$	−52109.0	−2.15965
$836$	−43215.1	−1.78783
$837$	0	0
$838$	−10756.1	−0.443392
$839$	15875.2	0.653246	0.326623	−	0.945155i	$-0.394089\pi$
$839$	15875.2	0.653246	0.326623	+	0.945155i	$0.394089\pi$
$840$	0	0
$841$	59032.0	2.42044
$842$	−20565.4	−0.841724
$843$	0	0
$844$	−79910.1	−3.25903
$845$	−21486.5	−0.874745
$846$	0	0
$847$	−41289.0	−1.67498
$848$	−29586.6	−1.19812
$849$	0	0
$850$	−4390.21	−0.177156
$851$	17283.0	0.696184
$852$	0	0
$853$	26409.2	1.06006	0.530032	−	0.847978i	$-0.322180\pi$
$853$	26409.2	1.06006	0.530032	+	0.847978i	$0.322180\pi$
$854$	59770.6	2.39497
$855$	0	0
$856$	−117989.	−4.71118
$857$	−14760.2	−0.588331	−0.294165	−	0.955755i	$-0.595042\pi$
$857$	−14760.2	−0.588331	−0.294165	+	0.955755i	$0.595042\pi$
$858$	0	0
$859$	6730.92	0.267353	0.133676	−	0.991025i	$-0.457322\pi$
$859$	6730.92	0.267353	0.133676	+	0.991025i	$0.457322\pi$
$860$	−191758.	−7.60335
$861$	0	0
$862$	−2214.84	−0.0875149
$863$	−18557.6	−0.731991	−0.365995	−	0.930617i	$-0.619271\pi$
$863$	−18557.6	−0.731991	−0.365995	+	0.930617i	$0.619271\pi$
$864$	0	0
$865$	83964.1	3.30042
$866$	34284.2	1.34529
$867$	0	0
$868$	−124723.	−4.87718
$869$	−23385.6	−0.912892
$870$	0	0
$871$	315.580	0.0122767
$872$	−53002.6	−2.05836
$873$	0	0
$874$	23859.0	0.923391
$875$	−85017.5	−3.28470
$876$	0	0
$877$	39238.3	1.51082	0.755408	−	0.655255i	$-0.227439\pi$
$877$	39238.3	1.51082	0.755408	+	0.655255i	$0.227439\pi$
$878$	−86128.2	−3.31058
$879$	0	0
$880$	281211.	10.7723
$881$	22060.4	0.843626	0.421813	−	0.906683i	$-0.361394\pi$
$881$	22060.4	0.843626	0.421813	+	0.906683i	$0.361394\pi$
$882$	0	0
$883$	34767.4	1.32505	0.662523	−	0.749042i	$-0.269486\pi$
$883$	34767.4	1.32505	0.662523	+	0.749042i	$0.269486\pi$
$884$	−1848.23	−0.0703197
$885$	0	0
$886$	43669.1	1.65586
$887$	−33883.9	−1.28265	−0.641325	−	0.767270i	$-0.721615\pi$
$887$	−33883.9	−1.28265	−0.641325	+	0.767270i	$0.721615\pi$
$888$	0	0
$889$	−4101.41	−0.154732
$890$	−105019.	−3.95532
$891$	0	0
$892$	55931.9	2.09948
$893$	7805.89	0.292513
$894$	0	0
$895$	24026.0	0.897318
$896$	83038.3	3.09611
$897$	0	0
$898$	−77558.2	−2.88213
$899$	82857.3	3.07391
$900$	0	0
$901$	−322.973	−0.0119421
$902$	113805.	4.20099
$903$	0	0
$904$	34490.4	1.26895
$905$	−26201.7	−0.962401
$906$	0	0
$907$	−9262.28	−0.339084	−0.169542	−	0.985523i	$-0.554229\pi$
$907$	−9262.28	−0.339084	−0.169542	+	0.985523i	$0.554229\pi$
$908$	62219.0	2.27402
$909$	0	0
$910$	−79901.9	−2.91068
$911$	43952.7	1.59848	0.799241	−	0.601011i	$-0.205235\pi$
$911$	43952.7	1.59848	0.799241	+	0.601011i	$0.205235\pi$
$912$	0	0
$913$	21191.3	0.768159
$914$	−20427.4	−0.739255
$915$	0	0
$916$	−9143.91	−0.329829
$917$	2887.29	0.103977
$918$	0	0
$919$	27081.8	0.972085	0.486042	−	0.873935i	$-0.338440\pi$
$919$	27081.8	0.972085	0.486042	+	0.873935i	$0.338440\pi$
$920$	−198773.	−7.12323
$921$	0	0
$922$	59947.8	2.14130
$923$	−11552.2	−0.411968
$924$	0	0
$925$	44000.2	1.56402
$926$	35415.4	1.25683
$927$	0	0
$928$	−188212.	−6.65770
$929$	54050.7	1.90888	0.954438	−	0.298410i	$-0.0964561\pi$
$929$	54050.7	1.90888	0.954438	+	0.298410i	$0.0964561\pi$
$930$	0	0
$931$	−2195.59	−0.0772905
$932$	93316.8	3.27972
$933$	0	0
$934$	−11373.6	−0.398454
$935$	3069.75	0.107371
$936$	0	0
$937$	27909.9	0.973081	0.486541	−	0.873658i	$-0.338259\pi$
$937$	27909.9	0.973081	0.486541	+	0.873658i	$0.338259\pi$
$938$	−1005.99	−0.0350178
$939$	0	0
$940$	−103401.	−3.58784
$941$	−18805.5	−0.651477	−0.325739	−	0.945460i	$-0.605613\pi$
$941$	−18805.5	−0.651477	−0.325739	+	0.945460i	$0.605613\pi$
$942$	0	0
$943$	−45826.8	−1.58253
$944$	132419.	4.56553
$945$	0	0
$946$	132653.	4.55910
$947$	440.658	0.0151209	0.00756044	−	0.999971i	$-0.497593\pi$
$947$	440.658	0.0151209	0.00756044	+	0.999971i	$0.497593\pi$
$948$	0	0
$949$	8822.43	0.301779
$950$	60742.1	2.07446
$951$	0	0
$952$	3705.46	0.126150
$953$	−22127.2	−0.752120	−0.376060	−	0.926595i	$-0.622721\pi$
$953$	−22127.2	−0.752120	−0.376060	+	0.926595i	$0.622721\pi$
$954$	0	0
$955$	−46512.3	−1.57602
$956$	5152.69	0.174320
$957$	0	0
$958$	−37473.4	−1.26379
$959$	11306.3	0.380709
$960$	0	0
$961$	52506.4	1.76249
$962$	25397.1	0.851181
$963$	0	0
$964$	−15378.4	−0.513800
$965$	59465.7	1.98370
$966$	0	0
$967$	−40244.7	−1.33835	−0.669173	−	0.743106i	$-0.733352\pi$
$967$	−40244.7	−1.33835	−0.669173	+	0.743106i	$0.733352\pi$
$968$	−150939.	−5.01174
$969$	0	0
$970$	113890.	3.76988
$971$	−31150.6	−1.02953	−0.514763	−	0.857332i	$-0.672120\pi$
$971$	−31150.6	−1.02953	−0.514763	+	0.857332i	$0.672120\pi$
$972$	0	0
$973$	18612.3	0.613239
$974$	61390.5	2.01959
$975$	0	0
$976$	124476.	4.08236
$977$	−2021.98	−0.0662116	−0.0331058	−	0.999452i	$-0.510540\pi$
$977$	−2021.98	−0.0662116	−0.0331058	+	0.999452i	$0.510540\pi$
$978$	0	0
$979$	52987.2	1.72980
$980$	29083.9	0.948012
$981$	0	0
$982$	16423.9	0.533715
$983$	−9840.09	−0.319278	−0.159639	−	0.987175i	$-0.551033\pi$
$983$	−9840.09	−0.319278	−0.159639	+	0.987175i	$0.551033\pi$
$984$	0	0
$985$	97116.0	3.14150
$986$	−3914.01	−0.126417
$987$	0	0
$988$	25571.7	0.823426
$989$	−53416.3	−1.71743
$990$	0	0
$991$	−48686.2	−1.56061	−0.780307	−	0.625397i	$-0.784937\pi$
$991$	−48686.2	−1.56061	−0.780307	+	0.625397i	$0.784937\pi$
$992$	−186940.	−5.98321
$993$	0	0
$994$	36825.6	1.17509
$995$	8994.57	0.286580
$996$	0	0
$997$	47521.5	1.50955	0.754774	−	0.655985i	$-0.227746\pi$
$997$	47521.5	1.50955	0.754774	+	0.655985i	$0.227746\pi$
$998$	49235.1	1.56163
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.c.1.1		28
3.2	odd	2		717.4.a.a.1.28	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.a.1.28	✓	28	3.2	odd	2
2151.4.a.c.1.1		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)

\(f(q)\)	\(=\)	\(q-5.43685 q^{2} +21.5594 q^{4} +21.1888 q^{5} -20.1660 q^{7} -73.7203 q^{8} +O(q^{10})\)	\(q-5.43685 q^{2} +21.5594 q^{4} +21.1888 q^{5} -20.1660 q^{7} -73.7203 q^{8} -115.201 q^{10} +58.1245 q^{11} -34.3940 q^{13} +109.639 q^{14} +228.331 q^{16} +2.49251 q^{17} -34.4858 q^{19} +456.818 q^{20} -316.014 q^{22} +127.252 q^{23} +323.967 q^{25} +186.995 q^{26} -434.765 q^{28} +288.827 q^{29} +286.875 q^{31} -651.641 q^{32} -13.5514 q^{34} -427.294 q^{35} +135.817 q^{37} +187.494 q^{38} -1562.05 q^{40} -360.126 q^{41} -419.768 q^{43} +1253.13 q^{44} -691.850 q^{46} -226.351 q^{47} +63.6663 q^{49} -1761.36 q^{50} -741.514 q^{52} -129.578 q^{53} +1231.59 q^{55} +1486.64 q^{56} -1570.31 q^{58} +579.941 q^{59} +545.156 q^{61} -1559.70 q^{62} +1716.23 q^{64} -728.770 q^{65} -9.17543 q^{67} +53.7369 q^{68} +2323.13 q^{70} +335.879 q^{71} -256.510 q^{73} -738.417 q^{74} -743.493 q^{76} -1172.14 q^{77} -402.337 q^{79} +4838.08 q^{80} +1957.95 q^{82} +364.585 q^{83} +52.8134 q^{85} +2282.22 q^{86} -4284.95 q^{88} +911.616 q^{89} +693.589 q^{91} +2743.47 q^{92} +1230.64 q^{94} -730.715 q^{95} -988.622 q^{97} -346.144 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * q^98

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