LMFDB - Embedded newform 2151.4.a.a.1.20

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:	$22$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+4.43192 q^{2} +11.6419 q^{4} +11.7413 q^{5} +6.42321 q^{7} +16.1407 q^{8} +O(q^{10})$	\(q+4.43192 q^{2} +11.6419 q^{4} +11.7413 q^{5} +6.42321 q^{7} +16.1407 q^{8} +52.0364 q^{10} -33.6179 q^{11} -66.2278 q^{13} +28.4672 q^{14} -21.6011 q^{16} +29.6611 q^{17} -27.1748 q^{19} +136.691 q^{20} -148.992 q^{22} -186.640 q^{23} +12.8574 q^{25} -293.516 q^{26} +74.7785 q^{28} -21.8413 q^{29} -193.238 q^{31} -224.860 q^{32} +131.456 q^{34} +75.4166 q^{35} -67.8794 q^{37} -120.437 q^{38} +189.512 q^{40} +74.4545 q^{41} -324.175 q^{43} -391.377 q^{44} -827.173 q^{46} +202.273 q^{47} -301.742 q^{49} +56.9832 q^{50} -771.019 q^{52} +254.801 q^{53} -394.717 q^{55} +103.675 q^{56} -96.7990 q^{58} +474.685 q^{59} +179.092 q^{61} -856.414 q^{62} -823.752 q^{64} -777.599 q^{65} +294.098 q^{67} +345.313 q^{68} +334.241 q^{70} +126.860 q^{71} -920.278 q^{73} -300.836 q^{74} -316.367 q^{76} -215.935 q^{77} +31.6274 q^{79} -253.624 q^{80} +329.977 q^{82} +1354.73 q^{83} +348.259 q^{85} -1436.72 q^{86} -542.617 q^{88} +669.715 q^{89} -425.395 q^{91} -2172.85 q^{92} +896.459 q^{94} -319.067 q^{95} +1611.14 q^{97} -1337.30 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	$ 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) $ 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * 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 $

$2$	4.43192	1.56692	0.783460	−	0.621442i	$-0.213453\pi$
$2$	4.43192	1.56692	0.783460	+	0.621442i	$0.213453\pi$
$3$	0	0
$3$	0	0
$4$	11.6419	1.45524
$5$	11.7413	1.05017	0.525086	−	0.851049i	$-0.324033\pi$
$5$	11.7413	1.05017	0.525086	+	0.851049i	$0.324033\pi$
$6$	0	0
$7$	6.42321	0.346821	0.173410	−	0.984850i	$-0.444521\pi$
$7$	6.42321	0.346821	0.173410	+	0.984850i	$0.444521\pi$
$8$	16.1407	0.713325
$9$	0	0
$10$	52.0364	1.64553
$11$	−33.6179	−0.921472	−0.460736	−	0.887537i	$-0.652415\pi$
$11$	−33.6179	−0.921472	−0.460736	+	0.887537i	$0.652415\pi$
$12$	0	0
$13$	−66.2278	−1.41295	−0.706473	−	0.707740i	$-0.749715\pi$
$13$	−66.2278	−1.41295	−0.706473	+	0.707740i	$0.749715\pi$
$14$	28.4672	0.543440
$15$	0	0
$16$	−21.6011	−0.337517
$17$	29.6611	0.423170	0.211585	−	0.977360i	$-0.432138\pi$
$17$	29.6611	0.423170	0.211585	+	0.977360i	$0.432138\pi$
$18$	0	0
$19$	−27.1748	−0.328123	−0.164061	−	0.986450i	$-0.552460\pi$
$19$	−27.1748	−0.328123	−0.164061	+	0.986450i	$0.552460\pi$
$20$	136.691	1.52825
$21$	0	0
$22$	−148.992	−1.44387
$23$	−186.640	−1.69205	−0.846024	−	0.533145i	$-0.821010\pi$
$23$	−186.640	−1.69205	−0.846024	+	0.533145i	$0.821010\pi$
$24$	0	0
$25$	12.8574	0.102860
$26$	−293.516	−2.21397
$27$	0	0
$28$	74.7785	0.504707
$29$	−21.8413	−0.139856	−0.0699281	−	0.997552i	$-0.522277\pi$
$29$	−21.8413	−0.139856	−0.0699281	+	0.997552i	$0.522277\pi$
$30$	0	0
$31$	−193.238	−1.11956	−0.559782	−	0.828640i	$-0.689115\pi$
$31$	−193.238	−1.11956	−0.559782	+	0.828640i	$0.689115\pi$
$32$	−224.860	−1.24219
$33$	0	0
$34$	131.456	0.663073
$35$	75.4166	0.364221
$36$	0	0
$37$	−67.8794	−0.301603	−0.150801	−	0.988564i	$-0.548185\pi$
$37$	−67.8794	−0.301603	−0.150801	+	0.988564i	$0.548185\pi$
$38$	−120.437	−0.514142
$39$	0	0
$40$	189.512	0.749113
$41$	74.4545	0.283606	0.141803	−	0.989895i	$-0.454710\pi$
$41$	74.4545	0.283606	0.141803	+	0.989895i	$0.454710\pi$
$42$	0	0
$43$	−324.175	−1.14968	−0.574840	−	0.818266i	$-0.694936\pi$
$43$	−324.175	−1.14968	−0.574840	+	0.818266i	$0.694936\pi$
$44$	−391.377	−1.34096
$45$	0	0
$46$	−827.173	−2.65130
$47$	202.273	0.627757	0.313878	−	0.949463i	$-0.398372\pi$
$47$	202.273	0.627757	0.313878	+	0.949463i	$0.398372\pi$
$48$	0	0
$49$	−301.742	−0.879715
$50$	56.9832	0.161173
$51$	0	0
$52$	−771.019	−2.05617
$53$	254.801	0.660371	0.330185	−	0.943916i	$-0.392889\pi$
$53$	254.801	0.660371	0.330185	+	0.943916i	$0.392889\pi$
$54$	0	0
$55$	−394.717	−0.967703
$56$	103.675	0.247396
$57$	0	0
$58$	−96.7990	−0.219144
$59$	474.685	1.04744	0.523718	−	0.851891i	$-0.324544\pi$
$59$	474.685	1.04744	0.523718	+	0.851891i	$0.324544\pi$
$60$	0	0
$61$	179.092	0.375909	0.187954	−	0.982178i	$-0.439814\pi$
$61$	179.092	0.375909	0.187954	+	0.982178i	$0.439814\pi$
$62$	−856.414	−1.75427
$63$	0	0
$64$	−823.752	−1.60889
$65$	−777.599	−1.48383
$66$	0	0
$67$	294.098	0.536265	0.268133	−	0.963382i	$-0.413593\pi$
$67$	294.098	0.536265	0.268133	+	0.963382i	$0.413593\pi$
$68$	345.313	0.615813
$69$	0	0
$70$	334.241	0.570705
$71$	126.860	0.212050	0.106025	−	0.994363i	$-0.466188\pi$
$71$	126.860	0.212050	0.106025	+	0.994363i	$0.466188\pi$
$72$	0	0
$73$	−920.278	−1.47549	−0.737743	−	0.675082i	$-0.764108\pi$
$73$	−920.278	−1.47549	−0.737743	+	0.675082i	$0.764108\pi$
$74$	−300.836	−0.472588
$75$	0	0
$76$	−316.367	−0.477497
$77$	−215.935	−0.319586
$78$	0	0
$79$	31.6274	0.0450425	0.0225213	−	0.999746i	$-0.492831\pi$
$79$	31.6274	0.0450425	0.0225213	+	0.999746i	$0.492831\pi$
$80$	−253.624	−0.354450
$81$	0	0
$82$	329.977	0.444388
$83$	1354.73	1.79158	0.895792	−	0.444474i	$-0.146610\pi$
$83$	1354.73	1.79158	0.895792	+	0.444474i	$0.146610\pi$
$84$	0	0
$85$	348.259	0.444401
$86$	−1436.72	−1.80146
$87$	0	0
$88$	−542.617	−0.657309
$89$	669.715	0.797637	0.398818	−	0.917030i	$-0.369420\pi$
$89$	669.715	0.797637	0.398818	+	0.917030i	$0.369420\pi$
$90$	0	0
$91$	−425.395	−0.490039
$92$	−2172.85	−2.46234
$93$	0	0
$94$	896.459	0.983645
$95$	−319.067	−0.344585
$96$	0	0
$97$	1611.14	1.68646	0.843228	−	0.537556i	$-0.180652\pi$
$97$	1611.14	1.68646	0.843228	+	0.537556i	$0.180652\pi$
$98$	−1337.30	−1.37844
$99$	0	0
$100$	149.685	0.149685
$101$	−948.093	−0.934047	−0.467024	−	0.884245i	$-0.654674\pi$
$101$	−948.093	−0.934047	−0.467024	+	0.884245i	$0.654674\pi$
$102$	0	0
$103$	−213.763	−0.204492	−0.102246	−	0.994759i	$-0.532603\pi$
$103$	−213.763	−0.204492	−0.102246	+	0.994759i	$0.532603\pi$
$104$	−1068.96	−1.00789
$105$	0	0
$106$	1129.26	1.03475
$107$	−1318.38	−1.19115	−0.595573	−	0.803301i	$-0.703075\pi$
$107$	−1318.38	−1.19115	−0.595573	+	0.803301i	$0.703075\pi$
$108$	0	0
$109$	437.408	0.384368	0.192184	−	0.981359i	$-0.438443\pi$
$109$	437.408	0.384368	0.192184	+	0.981359i	$0.438443\pi$
$110$	−1749.36	−1.51631
$111$	0	0
$112$	−138.748	−0.117058
$113$	340.699	0.283631	0.141815	−	0.989893i	$-0.454706\pi$
$113$	340.699	0.283631	0.141815	+	0.989893i	$0.454706\pi$
$114$	0	0
$115$	−2191.39	−1.77694
$116$	−254.275	−0.203524
$117$	0	0
$118$	2103.77	1.64125
$119$	190.520	0.146764
$120$	0	0
$121$	−200.834	−0.150889
$122$	793.723	0.589019
$123$	0	0
$124$	−2249.66	−1.62924
$125$	−1316.70	−0.942151
$126$	0	0
$127$	−194.974	−0.136230	−0.0681149	−	0.997677i	$-0.521698\pi$
$127$	−194.974	−0.136230	−0.0681149	+	0.997677i	$0.521698\pi$
$128$	−1851.93	−1.27882
$129$	0	0
$130$	−3446.25	−2.32505
$131$	701.617	0.467943	0.233972	−	0.972243i	$-0.424828\pi$
$131$	701.617	0.467943	0.233972	+	0.972243i	$0.424828\pi$
$132$	0	0
$133$	−174.550	−0.113800
$134$	1303.42	0.840285
$135$	0	0
$136$	478.751	0.301857
$137$	−1352.62	−0.843517	−0.421758	−	0.906708i	$-0.638587\pi$
$137$	−1352.62	−0.843517	−0.421758	+	0.906708i	$0.638587\pi$
$138$	0	0
$139$	2646.49	1.61491	0.807455	−	0.589930i	$-0.200845\pi$
$139$	2646.49	1.61491	0.807455	+	0.589930i	$0.200845\pi$
$140$	877.994	0.530029
$141$	0	0
$142$	562.234	0.332265
$143$	2226.44	1.30199
$144$	0	0
$145$	−256.445	−0.146873
$146$	−4078.60	−2.31197
$147$	0	0
$148$	−790.246	−0.438904
$149$	−1276.10	−0.701623	−0.350811	−	0.936446i	$-0.614094\pi$
$149$	−1276.10	−0.701623	−0.350811	+	0.936446i	$0.614094\pi$
$150$	0	0
$151$	717.895	0.386897	0.193448	−	0.981110i	$-0.438033\pi$
$151$	717.895	0.386897	0.193448	+	0.981110i	$0.438033\pi$
$152$	−438.621	−0.234058
$153$	0	0
$154$	−957.007	−0.500765
$155$	−2268.86	−1.17573
$156$	0	0
$157$	3461.88	1.75980	0.879899	−	0.475160i	$-0.157610\pi$
$157$	3461.88	1.75980	0.879899	+	0.475160i	$0.157610\pi$
$158$	140.170	0.0705781
$159$	0	0
$160$	−2640.14	−1.30451
$161$	−1198.83	−0.586837
$162$	0	0
$163$	−1698.07	−0.815969	−0.407984	−	0.912989i	$-0.633768\pi$
$163$	−1698.07	−0.815969	−0.407984	+	0.912989i	$0.633768\pi$
$164$	866.794	0.412715
$165$	0	0
$166$	6004.07	2.80727
$167$	3459.39	1.60297	0.801484	−	0.598016i	$-0.204044\pi$
$167$	3459.39	1.60297	0.801484	+	0.598016i	$0.204044\pi$
$168$	0	0
$169$	2189.12	0.996414
$170$	1543.46	0.696340
$171$	0	0
$172$	−3774.02	−1.67306
$173$	−2126.41	−0.934499	−0.467249	−	0.884126i	$-0.654755\pi$
$173$	−2126.41	−0.934499	−0.467249	+	0.884126i	$0.654755\pi$
$174$	0	0
$175$	82.5860	0.0356738
$176$	726.184	0.311012
$177$	0	0
$178$	2968.12	1.24983
$179$	4035.27	1.68498	0.842488	−	0.538716i	$-0.181090\pi$
$179$	4035.27	1.68498	0.842488	+	0.538716i	$0.181090\pi$
$180$	0	0
$181$	−465.822	−0.191294	−0.0956471	−	0.995415i	$-0.530492\pi$
$181$	−465.822	−0.191294	−0.0956471	+	0.995415i	$0.530492\pi$
$182$	−1885.32	−0.767852
$183$	0	0
$184$	−3012.50	−1.20698
$185$	−796.990	−0.316735
$186$	0	0
$187$	−997.147	−0.389939
$188$	2354.85	0.913537
$189$	0	0
$190$	−1414.08	−0.539938
$191$	−3254.72	−1.23300	−0.616501	−	0.787354i	$-0.711450\pi$
$191$	−3254.72	−1.23300	−0.616501	+	0.787354i	$0.711450\pi$
$192$	0	0
$193$	−0.201305	−7.50791e−5 0	−3.75395e−5	−	1.00000i	$-0.500012\pi$
$193$	−0.201305	−7.50791e−5 0	−3.75395e−5		1.00000i	$0.500012\pi$
$194$	7140.44	2.64254
$195$	0	0
$196$	−3512.86	−1.28020
$197$	−950.173	−0.343640	−0.171820	−	0.985128i	$-0.554965\pi$
$197$	−950.173	−0.343640	−0.171820	+	0.985128i	$0.554965\pi$
$198$	0	0
$199$	850.338	0.302909	0.151454	−	0.988464i	$-0.451604\pi$
$199$	850.338	0.302909	0.151454	+	0.988464i	$0.451604\pi$
$200$	207.528	0.0733723
$201$	0	0
$202$	−4201.87	−1.46358
$203$	−140.291	−0.0485050
$204$	0	0
$205$	874.191	0.297835
$206$	−947.380	−0.320423
$207$	0	0
$208$	1430.59	0.476893
$209$	913.562	0.302356
$210$	0	0
$211$	958.773	0.312818	0.156409	−	0.987692i	$-0.450008\pi$
$211$	958.773	0.312818	0.156409	+	0.987692i	$0.450008\pi$
$212$	2966.38	0.960998
$213$	0	0
$214$	−5842.95	−1.86643
$215$	−3806.23	−1.20736
$216$	0	0
$217$	−1241.21	−0.388288
$218$	1938.56	0.602274
$219$	0	0
$220$	−4595.27	−1.40824
$221$	−1964.39	−0.597915
$222$	0	0
$223$	1755.18	0.527064	0.263532	−	0.964651i	$-0.415113\pi$
$223$	1755.18	0.527064	0.263532	+	0.964651i	$0.415113\pi$
$224$	−1444.32	−0.430816
$225$	0	0
$226$	1509.95	0.444426
$227$	250.459	0.0732315	0.0366158	−	0.999329i	$-0.488342\pi$
$227$	250.459	0.0732315	0.0366158	+	0.999329i	$0.488342\pi$
$228$	0	0
$229$	−6835.37	−1.97246	−0.986231	−	0.165372i	$-0.947118\pi$
$229$	−6835.37	−1.97246	−0.986231	+	0.165372i	$0.947118\pi$
$230$	−9712.06	−2.78432
$231$	0	0
$232$	−352.534	−0.0997629
$233$	−4731.90	−1.33046	−0.665229	−	0.746639i	$-0.731666\pi$
$233$	−4731.90	−1.33046	−0.665229	+	0.746639i	$0.731666\pi$
$234$	0	0
$235$	2374.94	0.659252
$236$	5526.25	1.52427
$237$	0	0
$238$	844.368	0.229967
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−4235.67	−1.13213	−0.566066	−	0.824360i	$-0.691535\pi$
$241$	−4235.67	−1.13213	−0.566066	+	0.824360i	$0.691535\pi$
$242$	−890.079	−0.236432
$243$	0	0
$244$	2084.98	0.547037
$245$	−3542.84	−0.923852
$246$	0	0
$247$	1799.73	0.463620
$248$	−3118.99	−0.798613
$249$	0	0
$250$	−5835.49	−1.47628
$251$	−2447.32	−0.615433	−0.307717	−	0.951478i	$-0.599565\pi$
$251$	−2447.32	−0.615433	−0.307717	+	0.951478i	$0.599565\pi$
$252$	0	0
$253$	6274.45	1.55917
$254$	−864.111	−0.213461
$255$	0	0
$256$	−1617.57	−0.394915
$257$	−2775.41	−0.673640	−0.336820	−	0.941569i	$-0.609351\pi$
$257$	−2775.41	−0.673640	−0.336820	+	0.941569i	$0.609351\pi$
$258$	0	0
$259$	−436.004	−0.104602
$260$	−9052.74	−2.15933
$261$	0	0
$262$	3109.51	0.733230
$263$	−391.224	−0.0917259	−0.0458629	−	0.998948i	$-0.514604\pi$
$263$	−391.224	−0.0917259	−0.0458629	+	0.998948i	$0.514604\pi$
$264$	0	0
$265$	2991.69	0.693502
$266$	−773.590	−0.178315
$267$	0	0
$268$	3423.86	0.780395
$269$	1160.76	0.263097	0.131548	−	0.991310i	$-0.458005\pi$
$269$	1160.76	0.263097	0.131548	+	0.991310i	$0.458005\pi$
$270$	0	0
$271$	−1446.49	−0.324235	−0.162118	−	0.986771i	$-0.551832\pi$
$271$	−1446.49	−0.324235	−0.162118	+	0.986771i	$0.551832\pi$
$272$	−640.712	−0.142827
$273$	0	0
$274$	−5994.69	−1.32172
$275$	−432.241	−0.0947822
$276$	0	0
$277$	2108.86	0.457435	0.228717	−	0.973493i	$-0.426547\pi$
$277$	2108.86	0.457435	0.228717	+	0.973493i	$0.426547\pi$
$278$	11729.0	2.53043
$279$	0	0
$280$	1217.28	0.259808
$281$	4475.54	0.950137	0.475068	−	0.879949i	$-0.342423\pi$
$281$	4475.54	0.950137	0.475068	+	0.879949i	$0.342423\pi$
$282$	0	0
$283$	1447.46	0.304036	0.152018	−	0.988378i	$-0.451423\pi$
$283$	1447.46	0.304036	0.152018	+	0.988378i	$0.451423\pi$
$284$	1476.90	0.308583
$285$	0	0
$286$	9867.42	2.04011
$287$	478.237	0.0983604
$288$	0	0
$289$	−4033.22	−0.820927
$290$	−1136.54	−0.230138
$291$	0	0
$292$	−10713.8	−2.14718
$293$	−4253.93	−0.848182	−0.424091	−	0.905619i	$-0.639406\pi$
$293$	−4253.93	−0.848182	−0.424091	+	0.905619i	$0.639406\pi$
$294$	0	0
$295$	5573.41	1.09999
$296$	−1095.62	−0.215141
$297$	0	0
$298$	−5655.55	−1.09939
$299$	12360.7	2.39077
$300$	0	0
$301$	−2082.24	−0.398733
$302$	3181.65	0.606237
$303$	0	0
$304$	587.005	0.110747
$305$	2102.77	0.394769
$306$	0	0
$307$	−6324.14	−1.17569	−0.587847	−	0.808972i	$-0.700024\pi$
$307$	−6324.14	−1.17569	−0.587847	+	0.808972i	$0.700024\pi$
$308$	−2513.90	−0.465074
$309$	0	0
$310$	−10055.4	−1.84228
$311$	9730.54	1.77417	0.887087	−	0.461602i	$-0.152725\pi$
$311$	9730.54	1.77417	0.887087	+	0.461602i	$0.152725\pi$
$312$	0	0
$313$	−5888.48	−1.06338	−0.531688	−	0.846941i	$-0.678442\pi$
$313$	−5888.48	−1.06338	−0.531688	+	0.846941i	$0.678442\pi$
$314$	15342.8	2.75746
$315$	0	0
$316$	368.204	0.0655477
$317$	−10954.9	−1.94097	−0.970484	−	0.241167i	$-0.922470\pi$
$317$	−10954.9	−1.94097	−0.970484	+	0.241167i	$0.922470\pi$
$318$	0	0
$319$	734.260	0.128874
$320$	−9671.90	−1.68961
$321$	0	0
$322$	−5313.11	−0.919527
$323$	−806.037	−0.138852
$324$	0	0
$325$	−851.520	−0.145335
$326$	−7525.70	−1.27856
$327$	0	0
$328$	1201.75	0.202303
$329$	1299.24	0.217719
$330$	0	0
$331$	−9221.81	−1.53135	−0.765675	−	0.643228i	$-0.777595\pi$
$331$	−9221.81	−1.53135	−0.765675	+	0.643228i	$0.777595\pi$
$332$	15771.7	2.60718
$333$	0	0
$334$	15331.7	2.51172
$335$	3453.08	0.563170
$336$	0	0
$337$	5536.24	0.894891	0.447445	−	0.894311i	$-0.352334\pi$
$337$	5536.24	0.894891	0.447445	+	0.894311i	$0.352334\pi$
$338$	9702.01	1.56130
$339$	0	0
$340$	4054.41	0.646709
$341$	6496.25	1.03165
$342$	0	0
$343$	−4141.31	−0.651924
$344$	−5232.41	−0.820095
$345$	0	0
$346$	−9424.10	−1.46429
$347$	−9164.04	−1.41773	−0.708864	−	0.705345i	$-0.750792\pi$
$347$	−9164.04	−1.41773	−0.708864	+	0.705345i	$0.750792\pi$
$348$	0	0
$349$	−6045.01	−0.927169	−0.463585	−	0.886053i	$-0.653437\pi$
$349$	−6045.01	−0.927169	−0.463585	+	0.886053i	$0.653437\pi$
$350$	366.015	0.0558980
$351$	0	0
$352$	7559.32	1.14464
$353$	4601.08	0.693742	0.346871	−	0.937913i	$-0.387244\pi$
$353$	4601.08	0.693742	0.346871	+	0.937913i	$0.387244\pi$
$354$	0	0
$355$	1489.50	0.222689
$356$	7796.77	1.16075
$357$	0	0
$358$	17884.0	2.64022
$359$	−11718.8	−1.72283	−0.861415	−	0.507901i	$-0.830421\pi$
$359$	−11718.8	−1.72283	−0.861415	+	0.507901i	$0.830421\pi$
$360$	0	0
$361$	−6120.53	−0.892335
$362$	−2064.48	−0.299743
$363$	0	0
$364$	−4952.41	−0.713124
$365$	−10805.2	−1.54951
$366$	0	0
$367$	2549.66	0.362646	0.181323	−	0.983424i	$-0.441962\pi$
$367$	2549.66	0.362646	0.181323	+	0.983424i	$0.441962\pi$
$368$	4031.62	0.571094
$369$	0	0
$370$	−3532.20	−0.496298
$371$	1636.64	0.229030
$372$	0	0
$373$	1826.08	0.253488	0.126744	−	0.991935i	$-0.459547\pi$
$373$	1826.08	0.253488	0.126744	+	0.991935i	$0.459547\pi$
$374$	−4419.27	−0.611003
$375$	0	0
$376$	3264.83	0.447795
$377$	1446.50	0.197609
$378$	0	0
$379$	−3976.75	−0.538976	−0.269488	−	0.963004i	$-0.586855\pi$
$379$	−3976.75	−0.538976	−0.269488	+	0.963004i	$0.586855\pi$
$380$	−3714.55	−0.501454
$381$	0	0
$382$	−14424.7	−1.93202
$383$	1688.51	0.225271	0.112635	−	0.993636i	$-0.464071\pi$
$383$	1688.51	0.225271	0.112635	+	0.993636i	$0.464071\pi$
$384$	0	0
$385$	−2535.35	−0.335620
$386$	−0.892168	−0.000117643 0
$387$	0	0
$388$	18756.7	2.45420
$389$	9595.89	1.25072	0.625361	−	0.780335i	$-0.284952\pi$
$389$	9595.89	1.25072	0.625361	+	0.780335i	$0.284952\pi$
$390$	0	0
$391$	−5535.95	−0.716023
$392$	−4870.33	−0.627523
$393$	0	0
$394$	−4211.09	−0.538456
$395$	371.346	0.0473024
$396$	0	0
$397$	−2153.46	−0.272240	−0.136120	−	0.990692i	$-0.543463\pi$
$397$	−2153.46	−0.272240	−0.136120	+	0.990692i	$0.543463\pi$
$398$	3768.63	0.474634
$399$	0	0
$400$	−277.735	−0.0347168
$401$	1104.96	0.137604	0.0688020	−	0.997630i	$-0.478082\pi$
$401$	1104.96	0.137604	0.0688020	+	0.997630i	$0.478082\pi$
$402$	0	0
$403$	12797.7	1.58188
$404$	−11037.6	−1.35926
$405$	0	0
$406$	−621.760	−0.0760035
$407$	2281.97	0.277919
$408$	0	0
$409$	−4624.39	−0.559074	−0.279537	−	0.960135i	$-0.590181\pi$
$409$	−4624.39	−0.559074	−0.279537	+	0.960135i	$0.590181\pi$
$410$	3874.34	0.466683
$411$	0	0
$412$	−2488.61	−0.297585
$413$	3049.00	0.363273
$414$	0	0
$415$	15906.3	1.88147
$416$	14892.0	1.75514
$417$	0	0
$418$	4048.83	0.473768
$419$	−15049.7	−1.75472	−0.877361	−	0.479832i	$-0.840698\pi$
$419$	−15049.7	−1.75472	−0.877361	+	0.479832i	$0.840698\pi$
$420$	0	0
$421$	3597.85	0.416505	0.208252	−	0.978075i	$-0.433222\pi$
$421$	3597.85	0.416505	0.208252	+	0.978075i	$0.433222\pi$
$422$	4249.21	0.490161
$423$	0	0
$424$	4112.67	0.471059
$425$	381.366	0.0435270
$426$	0	0
$427$	1150.35	0.130373
$428$	−15348.5	−1.73340
$429$	0	0
$430$	−16868.9	−1.89184
$431$	−8421.15	−0.941143	−0.470571	−	0.882362i	$-0.655952\pi$
$431$	−8421.15	−0.941143	−0.470571	+	0.882362i	$0.655952\pi$
$432$	0	0
$433$	−456.823	−0.0507010	−0.0253505	−	0.999679i	$-0.508070\pi$
$433$	−456.823	−0.0507010	−0.0253505	+	0.999679i	$0.508070\pi$
$434$	−5500.93	−0.608417
$435$	0	0
$436$	5092.27	0.559348
$437$	5071.91	0.555200
$438$	0	0
$439$	−3913.92	−0.425515	−0.212758	−	0.977105i	$-0.568244\pi$
$439$	−3913.92	−0.425515	−0.212758	+	0.977105i	$0.568244\pi$
$440$	−6371.01	−0.690287
$441$	0	0
$442$	−8706.03	−0.936886
$443$	−9716.83	−1.04212	−0.521062	−	0.853519i	$-0.674464\pi$
$443$	−9716.83	−1.04212	−0.521062	+	0.853519i	$0.674464\pi$
$444$	0	0
$445$	7863.31	0.837655
$446$	7778.80	0.825868
$447$	0	0
$448$	−5291.13	−0.557997
$449$	7757.41	0.815356	0.407678	−	0.913126i	$-0.366339\pi$
$449$	7757.41	0.815356	0.407678	+	0.913126i	$0.366339\pi$
$450$	0	0
$451$	−2503.01	−0.261335
$452$	3966.39	0.412750
$453$	0	0
$454$	1110.01	0.114748
$455$	−4994.68	−0.514624
$456$	0	0
$457$	2081.07	0.213017	0.106508	−	0.994312i	$-0.466033\pi$
$457$	2081.07	0.213017	0.106508	+	0.994312i	$0.466033\pi$
$458$	−30293.8	−3.09069
$459$	0	0
$460$	−25512.0	−2.58587
$461$	−15231.8	−1.53886	−0.769432	−	0.638728i	$-0.779461\pi$
$461$	−15231.8	−1.53886	−0.769432	+	0.638728i	$0.779461\pi$
$462$	0	0
$463$	6789.34	0.681485	0.340742	−	0.940157i	$-0.389322\pi$
$463$	6789.34	0.681485	0.340742	+	0.940157i	$0.389322\pi$
$464$	471.796	0.0472038
$465$	0	0
$466$	−20971.4	−2.08472
$467$	−9501.47	−0.941489	−0.470745	−	0.882269i	$-0.656015\pi$
$467$	−9501.47	−0.941489	−0.470745	+	0.882269i	$0.656015\pi$
$468$	0	0
$469$	1889.05	0.185988
$470$	10525.6	1.03300
$471$	0	0
$472$	7661.75	0.747163
$473$	10898.1	1.05940
$474$	0	0
$475$	−349.399	−0.0337506
$476$	2218.01	0.213577
$477$	0	0
$478$	−1059.23	−0.101356
$479$	13193.9	1.25855	0.629273	−	0.777184i	$-0.283353\pi$
$479$	13193.9	1.25855	0.629273	+	0.777184i	$0.283353\pi$
$480$	0	0
$481$	4495.50	0.426148
$482$	−18772.2	−1.77396
$483$	0	0
$484$	−2338.09	−0.219580
$485$	18916.8	1.77107
$486$	0	0
$487$	1521.50	0.141572	0.0707862	−	0.997492i	$-0.477449\pi$
$487$	1521.50	0.141572	0.0707862	+	0.997492i	$0.477449\pi$
$488$	2890.68	0.268145
$489$	0	0
$490$	−15701.6	−1.44760
$491$	−10578.2	−0.972279	−0.486139	−	0.873881i	$-0.661595\pi$
$491$	−10578.2	−0.972279	−0.486139	+	0.873881i	$0.661595\pi$
$492$	0	0
$493$	−647.838	−0.0591829
$494$	7976.26	0.726455
$495$	0	0
$496$	4174.14	0.377872
$497$	814.849	0.0735432
$498$	0	0
$499$	−12634.3	−1.13344	−0.566722	−	0.823909i	$-0.691789\pi$
$499$	−12634.3	−1.13344	−0.566722	+	0.823909i	$0.691789\pi$
$500$	−15328.9	−1.37106
$501$	0	0
$502$	−10846.3	−0.964335
$503$	12052.3	1.06836	0.534182	−	0.845369i	$-0.320620\pi$
$503$	12052.3	1.06836	0.534182	+	0.845369i	$0.320620\pi$
$504$	0	0
$505$	−11131.8	−0.980910
$506$	27807.9	2.44310
$507$	0	0
$508$	−2269.88	−0.198247
$509$	13175.9	1.14737	0.573684	−	0.819076i	$-0.305514\pi$
$509$	13175.9	1.14737	0.573684	+	0.819076i	$0.305514\pi$
$510$	0	0
$511$	−5911.14	−0.511729
$512$	7646.46	0.660018
$513$	0	0
$514$	−12300.4	−1.05554
$515$	−2509.85	−0.214752
$516$	0	0
$517$	−6800.01	−0.578460
$518$	−1932.33	−0.163903
$519$	0	0
$520$	−12551.0	−1.05846
$521$	16729.8	1.40681	0.703404	−	0.710790i	$-0.251662\pi$
$521$	16729.8	1.40681	0.703404	+	0.710790i	$0.251662\pi$
$522$	0	0
$523$	14265.5	1.19271	0.596353	−	0.802722i	$-0.296616\pi$
$523$	14265.5	1.19271	0.596353	+	0.802722i	$0.296616\pi$
$524$	8168.17	0.680970
$525$	0	0
$526$	−1733.87	−0.143727
$527$	−5731.65	−0.473766
$528$	0	0
$529$	22667.4	1.86303
$530$	13258.9	1.08666
$531$	0	0
$532$	−2032.09	−0.165606
$533$	−4930.96	−0.400720
$534$	0	0
$535$	−15479.5	−1.25091
$536$	4746.94	0.382531
$537$	0	0
$538$	5144.41	0.412251
$539$	10144.0	0.810633
$540$	0	0
$541$	−11927.0	−0.947843	−0.473922	−	0.880567i	$-0.657162\pi$
$541$	−11927.0	−0.947843	−0.473922	+	0.880567i	$0.657162\pi$
$542$	−6410.71	−0.508051
$543$	0	0
$544$	−6669.60	−0.525656
$545$	5135.73	0.403652
$546$	0	0
$547$	−21208.1	−1.65776	−0.828880	−	0.559427i	$-0.811021\pi$
$547$	−21208.1	−1.65776	−0.828880	+	0.559427i	$0.811021\pi$
$548$	−15747.0	−1.22752
$549$	0	0
$550$	−1915.66	−0.148516
$551$	593.534	0.0458900
$552$	0	0
$553$	203.149	0.0156217
$554$	9346.32	0.716764
$555$	0	0
$556$	30810.2	2.35008
$557$	−9468.58	−0.720281	−0.360140	−	0.932898i	$-0.617271\pi$
$557$	−9468.58	−0.720281	−0.360140	+	0.932898i	$0.617271\pi$
$558$	0	0
$559$	21469.4	1.62444
$560$	−1629.08	−0.122931
$561$	0	0
$562$	19835.2	1.48879
$563$	−8933.50	−0.668743	−0.334372	−	0.942441i	$-0.608524\pi$
$563$	−8933.50	−0.668743	−0.334372	+	0.942441i	$0.608524\pi$
$564$	0	0
$565$	4000.24	0.297861
$566$	6415.01	0.476401
$567$	0	0
$568$	2047.61	0.151260
$569$	3255.65	0.239866	0.119933	−	0.992782i	$-0.461732\pi$
$569$	3255.65	0.239866	0.119933	+	0.992782i	$0.461732\pi$
$570$	0	0
$571$	23018.3	1.68701	0.843507	−	0.537118i	$-0.180487\pi$
$571$	23018.3	1.68701	0.843507	+	0.537118i	$0.180487\pi$
$572$	25920.1	1.89471
$573$	0	0
$574$	2119.51	0.154123
$575$	−2399.71	−0.174043
$576$	0	0
$577$	−14141.7	−1.02033	−0.510163	−	0.860078i	$-0.670415\pi$
$577$	−14141.7	−1.02033	−0.510163	+	0.860078i	$0.670415\pi$
$578$	−17874.9	−1.28633
$579$	0	0
$580$	−2985.51	−0.213735
$581$	8701.74	0.621358
$582$	0	0
$583$	−8565.90	−0.608513
$584$	−14853.9	−1.05250
$585$	0	0
$586$	−18853.1	−1.32903
$587$	22822.8	1.60477	0.802384	−	0.596808i	$-0.203564\pi$
$587$	22822.8	1.60477	0.802384	+	0.596808i	$0.203564\pi$
$588$	0	0
$589$	5251.20	0.367355
$590$	24700.9	1.72359
$591$	0	0
$592$	1466.27	0.101796
$593$	20090.6	1.39127	0.695633	−	0.718398i	$-0.255124\pi$
$593$	20090.6	1.39127	0.695633	+	0.718398i	$0.255124\pi$
$594$	0	0
$595$	2236.94	0.154127
$596$	−14856.2	−1.02103
$597$	0	0
$598$	54781.9	3.74615
$599$	23493.3	1.60252	0.801259	−	0.598318i	$-0.204164\pi$
$599$	23493.3	1.60252	0.801259	+	0.598318i	$0.204164\pi$
$600$	0	0
$601$	26271.2	1.78307	0.891536	−	0.452951i	$-0.149629\pi$
$601$	26271.2	1.78307	0.891536	+	0.452951i	$0.149629\pi$
$602$	−9228.34	−0.624783
$603$	0	0
$604$	8357.67	0.563028
$605$	−2358.04	−0.158460
$606$	0	0
$607$	14888.7	0.995575	0.497787	−	0.867299i	$-0.334146\pi$
$607$	14888.7	0.995575	0.497787	+	0.867299i	$0.334146\pi$
$608$	6110.53	0.407590
$609$	0	0
$610$	9319.32	0.618571
$611$	−13396.1	−0.886986
$612$	0	0
$613$	8028.48	0.528984	0.264492	−	0.964388i	$-0.414796\pi$
$613$	8028.48	0.528984	0.264492	+	0.964388i	$0.414796\pi$
$614$	−28028.1	−1.84222
$615$	0	0
$616$	−3485.34	−0.227968
$617$	−14781.3	−0.964463	−0.482232	−	0.876044i	$-0.660174\pi$
$617$	−14781.3	−0.964463	−0.482232	+	0.876044i	$0.660174\pi$
$618$	0	0
$619$	−6891.50	−0.447484	−0.223742	−	0.974648i	$-0.571827\pi$
$619$	−6891.50	−0.447484	−0.223742	+	0.974648i	$0.571827\pi$
$620$	−26413.8	−1.71098
$621$	0	0
$622$	43125.0	2.77999
$623$	4301.72	0.276637
$624$	0	0
$625$	−17066.9	−1.09228
$626$	−26097.3	−1.66622
$627$	0	0
$628$	40303.0	2.56093
$629$	−2013.38	−0.127629
$630$	0	0
$631$	−13868.2	−0.874936	−0.437468	−	0.899234i	$-0.644125\pi$
$631$	−13868.2	−0.874936	−0.437468	+	0.899234i	$0.644125\pi$
$632$	510.488	0.0321300
$633$	0	0
$634$	−48551.1	−3.04134
$635$	−2289.25	−0.143065
$636$	0	0
$637$	19983.7	1.24299
$638$	3254.18	0.201935
$639$	0	0
$640$	−21744.0	−1.34298
$641$	−22387.7	−1.37951	−0.689753	−	0.724045i	$-0.742281\pi$
$641$	−22387.7	−1.37951	−0.689753	+	0.724045i	$0.742281\pi$
$642$	0	0
$643$	24966.6	1.53124	0.765619	−	0.643294i	$-0.222433\pi$
$643$	24966.6	1.53124	0.765619	+	0.643294i	$0.222433\pi$
$644$	−13956.6	−0.853989
$645$	0	0
$646$	−3572.29	−0.217569
$647$	4254.36	0.258510	0.129255	−	0.991611i	$-0.458741\pi$
$647$	4254.36	0.258510	0.129255	+	0.991611i	$0.458741\pi$
$648$	0	0
$649$	−15957.9	−0.965184
$650$	−3773.87	−0.227728
$651$	0	0
$652$	−19768.8	−1.18743
$653$	4413.54	0.264495	0.132247	−	0.991217i	$-0.457781\pi$
$653$	4413.54	0.264495	0.132247	+	0.991217i	$0.457781\pi$
$654$	0	0
$655$	8237.87	0.491420
$656$	−1608.30	−0.0957217
$657$	0	0
$658$	5758.14	0.341149
$659$	18203.8	1.07606	0.538028	−	0.842927i	$-0.319170\pi$
$659$	18203.8	1.07606	0.538028	+	0.842927i	$0.319170\pi$
$660$	0	0
$661$	−10835.4	−0.637589	−0.318795	−	0.947824i	$-0.603278\pi$
$661$	−10835.4	−0.637589	−0.318795	+	0.947824i	$0.603278\pi$
$662$	−40870.3	−2.39950
$663$	0	0
$664$	21866.4	1.27798
$665$	−2049.43	−0.119509
$666$	0	0
$667$	4076.46	0.236643
$668$	40274.0	2.33270
$669$	0	0
$670$	15303.8	0.882443
$671$	−6020.72	−0.346389
$672$	0	0
$673$	9652.28	0.552850	0.276425	−	0.961036i	$-0.410850\pi$
$673$	9652.28	0.552850	0.276425	+	0.961036i	$0.410850\pi$
$674$	24536.2	1.40222
$675$	0	0
$676$	25485.6	1.45002
$677$	28447.0	1.61493	0.807466	−	0.589914i	$-0.200839\pi$
$677$	28447.0	1.61493	0.807466	+	0.589914i	$0.200839\pi$
$678$	0	0
$679$	10348.7	0.584898
$680$	5621.15	0.317002
$681$	0	0
$682$	28790.9	1.61651
$683$	7578.79	0.424589	0.212295	−	0.977206i	$-0.431906\pi$
$683$	7578.79	0.424589	0.212295	+	0.977206i	$0.431906\pi$
$684$	0	0
$685$	−15881.4	−0.885837
$686$	−18354.0	−1.02151
$687$	0	0
$688$	7002.53	0.388036
$689$	−16874.9	−0.933068
$690$	0	0
$691$	−24228.3	−1.33385	−0.666924	−	0.745126i	$-0.732389\pi$
$691$	−24228.3	−1.33385	−0.666924	+	0.745126i	$0.732389\pi$
$692$	−24755.5	−1.35992
$693$	0	0
$694$	−40614.3	−2.22147
$695$	31073.2	1.69593
$696$	0	0
$697$	2208.41	0.120013
$698$	−26791.0	−1.45280
$699$	0	0
$700$	961.460	0.0519140
$701$	19186.0	1.03373	0.516866	−	0.856067i	$-0.327099\pi$
$701$	19186.0	1.03373	0.516866	+	0.856067i	$0.327099\pi$
$702$	0	0
$703$	1844.61	0.0989628
$704$	27692.9	1.48255
$705$	0	0
$706$	20391.6	1.08704
$707$	−6089.80	−0.323947
$708$	0	0
$709$	−3588.22	−0.190069	−0.0950343	−	0.995474i	$-0.530296\pi$
$709$	−3588.22	−0.190069	−0.0950343	+	0.995474i	$0.530296\pi$
$710$	6601.34	0.348935
$711$	0	0
$712$	10809.7	0.568974
$713$	36065.9	1.89436
$714$	0	0
$715$	26141.3	1.36731
$716$	46978.3	2.45204
$717$	0	0
$718$	−51936.9	−2.69954
$719$	33269.0	1.72562	0.862811	−	0.505526i	$-0.168702\pi$
$719$	33269.0	1.72562	0.862811	+	0.505526i	$0.168702\pi$
$720$	0	0
$721$	−1373.04	−0.0709221
$722$	−27125.7	−1.39822
$723$	0	0
$724$	−5423.06	−0.278379
$725$	−280.824	−0.0143856
$726$	0	0
$727$	37966.0	1.93684	0.968419	−	0.249329i	$-0.0802101\pi$
$727$	37966.0	1.93684	0.968419	+	0.249329i	$0.0802101\pi$
$728$	−6866.17	−0.349557
$729$	0	0
$730$	−47887.9	−2.42796
$731$	−9615.41	−0.486510
$732$	0	0
$733$	9538.53	0.480646	0.240323	−	0.970693i	$-0.422747\pi$
$733$	9538.53	0.480646	0.240323	+	0.970693i	$0.422747\pi$
$734$	11299.9	0.568238
$735$	0	0
$736$	41967.8	2.10184
$737$	−9886.97	−0.494153
$738$	0	0
$739$	5437.53	0.270667	0.135333	−	0.990800i	$-0.456789\pi$
$739$	5437.53	0.270667	0.135333	+	0.990800i	$0.456789\pi$
$740$	−9278.50	−0.460925
$741$	0	0
$742$	7253.47	0.358872
$743$	−16270.6	−0.803377	−0.401688	−	0.915776i	$-0.631577\pi$
$743$	−16270.6	−0.803377	−0.401688	+	0.915776i	$0.631577\pi$
$744$	0	0
$745$	−14983.0	−0.736824
$746$	8093.04	0.397195
$747$	0	0
$748$	−11608.7	−0.567455
$749$	−8468.23	−0.413114
$750$	0	0
$751$	14167.4	0.688384	0.344192	−	0.938899i	$-0.388153\pi$
$751$	14167.4	0.688384	0.344192	+	0.938899i	$0.388153\pi$
$752$	−4369.32	−0.211878
$753$	0	0
$754$	6410.78	0.309638
$755$	8429.00	0.406308
$756$	0	0
$757$	−13061.8	−0.627134	−0.313567	−	0.949566i	$-0.601524\pi$
$757$	−13061.8	−0.627134	−0.313567	+	0.949566i	$0.601524\pi$
$758$	−17624.7	−0.844533
$759$	0	0
$760$	−5149.96	−0.245801
$761$	−23975.5	−1.14207	−0.571034	−	0.820927i	$-0.693457\pi$
$761$	−23975.5	−1.14207	−0.571034	+	0.820927i	$0.693457\pi$
$762$	0	0
$763$	2809.56	0.133307
$764$	−37891.2	−1.79431
$765$	0	0
$766$	7483.34	0.352982
$767$	−31437.4	−1.47997
$768$	0	0
$769$	25602.3	1.20058	0.600289	−	0.799783i	$-0.295052\pi$
$769$	25602.3	1.20058	0.600289	+	0.799783i	$0.295052\pi$
$770$	−11236.5	−0.525889
$771$	0	0
$772$	−2.34358	−0.000109258 0
$773$	−5621.51	−0.261568	−0.130784	−	0.991411i	$-0.541749\pi$
$773$	−5621.51	−0.261568	−0.130784	+	0.991411i	$0.541749\pi$
$774$	0	0
$775$	−2484.54	−0.115158
$776$	26004.9	1.20299
$777$	0	0
$778$	42528.2	1.95978
$779$	−2023.29	−0.0930576
$780$	0	0
$781$	−4264.78	−0.195398
$782$	−24534.9	−1.12195
$783$	0	0
$784$	6517.96	0.296919
$785$	40646.9	1.84809
$786$	0	0
$787$	26335.0	1.19281	0.596405	−	0.802684i	$-0.296595\pi$
$787$	26335.0	1.19281	0.596405	+	0.802684i	$0.296595\pi$
$788$	−11061.8	−0.500078
$789$	0	0
$790$	1645.78	0.0741191
$791$	2188.38	0.0983689
$792$	0	0
$793$	−11860.9	−0.531138
$794$	−9543.97	−0.426578
$795$	0	0
$796$	9899.56	0.440805
$797$	−8800.99	−0.391150	−0.195575	−	0.980689i	$-0.562657\pi$
$797$	−8800.99	−0.391150	−0.195575	+	0.980689i	$0.562657\pi$
$798$	0	0
$799$	5999.65	0.265648
$800$	−2891.12	−0.127771
$801$	0	0
$802$	4897.11	0.215615
$803$	30937.9	1.35962
$804$	0	0
$805$	−14075.8	−0.616280
$806$	56718.4	2.47869
$807$	0	0
$808$	−15302.9	−0.666279
$809$	−28919.9	−1.25682	−0.628412	−	0.777881i	$-0.716295\pi$
$809$	−28919.9	−1.25682	−0.628412	+	0.777881i	$0.716295\pi$
$810$	0	0
$811$	−2510.47	−0.108699	−0.0543493	−	0.998522i	$-0.517308\pi$
$811$	−2510.47	−0.108699	−0.0543493	+	0.998522i	$0.517308\pi$
$812$	−1633.26	−0.0705865
$813$	0	0
$814$	10113.5	0.435476
$815$	−19937.5	−0.856907
$816$	0	0
$817$	8809.41	0.377236
$818$	−20494.9	−0.876024
$819$	0	0
$820$	10177.3	0.433421
$821$	25922.6	1.10196	0.550978	−	0.834520i	$-0.314255\pi$
$821$	25922.6	1.10196	0.550978	+	0.834520i	$0.314255\pi$
$822$	0	0
$823$	−26762.9	−1.13353	−0.566765	−	0.823879i	$-0.691805\pi$
$823$	−26762.9	−1.13353	−0.566765	+	0.823879i	$0.691805\pi$
$824$	−3450.28	−0.145869
$825$	0	0
$826$	13512.9	0.569220
$827$	45365.0	1.90749	0.953745	−	0.300616i	$-0.0971923\pi$
$827$	45365.0	1.90749	0.953745	+	0.300616i	$0.0971923\pi$
$828$	0	0
$829$	−35434.7	−1.48456	−0.742279	−	0.670091i	$-0.766255\pi$
$829$	−35434.7	−1.48456	−0.742279	+	0.670091i	$0.766255\pi$
$830$	70495.5	2.94811
$831$	0	0
$832$	54555.3	2.27327
$833$	−8950.02	−0.372269
$834$	0	0
$835$	40617.7	1.68339
$836$	10635.6	0.440001
$837$	0	0
$838$	−66699.3	−2.74951
$839$	−12190.2	−0.501610	−0.250805	−	0.968038i	$-0.580695\pi$
$839$	−12190.2	−0.501610	−0.250805	+	0.968038i	$0.580695\pi$
$840$	0	0
$841$	−23912.0	−0.980440
$842$	15945.4	0.652630
$843$	0	0
$844$	11162.0	0.455226
$845$	25703.1	1.04641
$846$	0	0
$847$	−1290.00	−0.0523316
$848$	−5503.98	−0.222886
$849$	0	0
$850$	1690.19	0.0682034
$851$	12669.0	0.510326
$852$	0	0
$853$	−9104.99	−0.365473	−0.182737	−	0.983162i	$-0.558496\pi$
$853$	−9104.99	−0.365473	−0.182737	+	0.983162i	$0.558496\pi$
$854$	5098.25	0.204284
$855$	0	0
$856$	−21279.6	−0.849673
$857$	30408.4	1.21206	0.606028	−	0.795443i	$-0.292762\pi$
$857$	30408.4	1.21206	0.606028	+	0.795443i	$0.292762\pi$
$858$	0	0
$859$	−18161.1	−0.721361	−0.360680	−	0.932689i	$-0.617455\pi$
$859$	−18161.1	−0.721361	−0.360680	+	0.932689i	$0.617455\pi$
$860$	−44311.8	−1.75700
$861$	0	0
$862$	−37321.9	−1.47470
$863$	−20392.3	−0.804358	−0.402179	−	0.915561i	$-0.631747\pi$
$863$	−20392.3	−0.804358	−0.402179	+	0.915561i	$0.631747\pi$
$864$	0	0
$865$	−24966.8	−0.981384
$866$	−2024.61	−0.0794444
$867$	0	0
$868$	−14450.0	−0.565053
$869$	−1063.25	−0.0415054
$870$	0	0
$871$	−19477.5	−0.757713
$872$	7060.07	0.274179
$873$	0	0
$874$	22478.3	0.869954
$875$	−8457.41	−0.326757
$876$	0	0
$877$	41135.1	1.58385	0.791923	−	0.610620i	$-0.209080\pi$
$877$	41135.1	1.58385	0.791923	+	0.610620i	$0.209080\pi$
$878$	−17346.2	−0.666748
$879$	0	0
$880$	8526.32	0.326616
$881$	5084.72	0.194448	0.0972238	−	0.995263i	$-0.469004\pi$
$881$	5084.72	0.194448	0.0972238	+	0.995263i	$0.469004\pi$
$882$	0	0
$883$	−40168.5	−1.53089	−0.765446	−	0.643500i	$-0.777482\pi$
$883$	−40168.5	−1.53089	−0.765446	+	0.643500i	$0.777482\pi$
$884$	−22869.3	−0.870110
$885$	0	0
$886$	−43064.2	−1.63292
$887$	−21339.6	−0.807795	−0.403898	−	0.914804i	$-0.632345\pi$
$887$	−21339.6	−0.807795	−0.403898	+	0.914804i	$0.632345\pi$
$888$	0	0
$889$	−1252.36	−0.0472473
$890$	34849.5	1.31254
$891$	0	0
$892$	20433.6	0.767005
$893$	−5496.74	−0.205981
$894$	0	0
$895$	47379.3	1.76951
$896$	−11895.3	−0.443520
$897$	0	0
$898$	34380.2	1.27760
$899$	4220.57	0.156578
$900$	0	0
$901$	7557.70	0.279449
$902$	−11093.1	−0.409491
$903$	0	0
$904$	5499.12	0.202321
$905$	−5469.34	−0.200892
$906$	0	0
$907$	−50489.5	−1.84838	−0.924188	−	0.381938i	$-0.875257\pi$
$907$	−50489.5	−1.84838	−0.924188	+	0.381938i	$0.875257\pi$
$908$	2915.82	0.106569
$909$	0	0
$910$	−22136.0	−0.806376
$911$	−40812.6	−1.48428	−0.742142	−	0.670243i	$-0.766190\pi$
$911$	−40812.6	−1.48428	−0.742142	+	0.670243i	$0.766190\pi$
$912$	0	0
$913$	−45543.4	−1.65089
$914$	9223.16	0.333780
$915$	0	0
$916$	−79576.8	−2.87041
$917$	4506.63	0.162292
$918$	0	0
$919$	29107.2	1.04478	0.522392	−	0.852705i	$-0.325040\pi$
$919$	29107.2	1.04478	0.522392	+	0.852705i	$0.325040\pi$
$920$	−35370.5	−1.26754
$921$	0	0
$922$	−67506.2	−2.41128
$923$	−8401.67	−0.299615
$924$	0	0
$925$	−872.756	−0.0310227
$926$	30089.8	1.06783
$927$	0	0
$928$	4911.23	0.173728
$929$	−11312.5	−0.399516	−0.199758	−	0.979845i	$-0.564016\pi$
$929$	−11312.5	−0.399516	−0.199758	+	0.979845i	$0.564016\pi$
$930$	0	0
$931$	8199.80	0.288655
$932$	−55088.4	−1.93614
$933$	0	0
$934$	−42109.8	−1.47524
$935$	−11707.8	−0.409503
$936$	0	0
$937$	−40082.3	−1.39747	−0.698736	−	0.715379i	$-0.746254\pi$
$937$	−40082.3	−1.39747	−0.698736	+	0.715379i	$0.746254\pi$
$938$	8372.13	0.291428
$939$	0	0
$940$	27648.9	0.959370
$941$	2708.22	0.0938209	0.0469105	−	0.998899i	$-0.485062\pi$
$941$	2708.22	0.0938209	0.0469105	+	0.998899i	$0.485062\pi$
$942$	0	0
$943$	−13896.2	−0.479875
$944$	−10253.7	−0.353527
$945$	0	0
$946$	48299.5	1.65999
$947$	15460.5	0.530515	0.265257	−	0.964178i	$-0.414543\pi$
$947$	15460.5	0.530515	0.265257	+	0.964178i	$0.414543\pi$
$948$	0	0
$949$	60948.0	2.08478
$950$	−1548.51	−0.0528845
$951$	0	0
$952$	3075.12	0.104690
$953$	12109.6	0.411614	0.205807	−	0.978593i	$-0.434018\pi$
$953$	12109.6	0.411614	0.205807	+	0.978593i	$0.434018\pi$
$954$	0	0
$955$	−38214.6	−1.29486
$956$	−2782.42	−0.0941316
$957$	0	0
$958$	58474.2	1.97204
$959$	−8688.14	−0.292549
$960$	0	0
$961$	7549.81	0.253426
$962$	19923.7	0.667740
$963$	0	0
$964$	−49311.3	−1.64752
$965$	−2.36358	−7.88459e−5 0
$966$	0	0
$967$	−39971.5	−1.32926	−0.664632	−	0.747171i	$-0.731411\pi$
$967$	−39971.5	−1.32926	−0.664632	+	0.747171i	$0.731411\pi$
$968$	−3241.60	−0.107633
$969$	0	0
$970$	83837.8	2.77512
$971$	−9447.30	−0.312233	−0.156117	−	0.987739i	$-0.549898\pi$
$971$	−9447.30	−0.312233	−0.156117	+	0.987739i	$0.549898\pi$
$972$	0	0
$973$	16999.0	0.560084
$974$	6743.17	0.221833
$975$	0	0
$976$	−3868.59	−0.126875
$977$	−36175.0	−1.18459	−0.592293	−	0.805722i	$-0.701777\pi$
$977$	−36175.0	−1.18459	−0.592293	+	0.805722i	$0.701777\pi$
$978$	0	0
$979$	−22514.4	−0.735000
$980$	−41245.4	−1.34443
$981$	0	0
$982$	−46881.9	−1.52348
$983$	36082.6	1.17076	0.585380	−	0.810759i	$-0.300945\pi$
$983$	36082.6	1.17076	0.585380	+	0.810759i	$0.300945\pi$
$984$	0	0
$985$	−11156.2	−0.360880
$986$	−2871.17	−0.0927349
$987$	0	0
$988$	20952.3	0.674678
$989$	60504.0	1.94531
$990$	0	0
$991$	−54791.5	−1.75632	−0.878158	−	0.478371i	$-0.841227\pi$
$991$	−54791.5	−1.75632	−0.878158	+	0.478371i	$0.841227\pi$
$992$	43451.4	1.39071
$993$	0	0
$994$	3611.35	0.115236
$995$	9984.05	0.318106
$996$	0	0
$997$	3815.64	0.121206	0.0606030	−	0.998162i	$-0.480698\pi$
$997$	3815.64	0.121206	0.0606030	+	0.998162i	$0.480698\pi$
$998$	−55994.2	−1.77602
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.a.1.20		22
3.2	odd	2		239.4.a.a.1.3	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.a.1.3	✓	22	3.2	odd	2
2151.4.a.a.1.20		22	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)

\(f(q)\)	\(=\)	\(q+4.43192 q^{2} +11.6419 q^{4} +11.7413 q^{5} +6.42321 q^{7} +16.1407 q^{8} +O(q^{10})\)	\(q+4.43192 q^{2} +11.6419 q^{4} +11.7413 q^{5} +6.42321 q^{7} +16.1407 q^{8} +52.0364 q^{10} -33.6179 q^{11} -66.2278 q^{13} +28.4672 q^{14} -21.6011 q^{16} +29.6611 q^{17} -27.1748 q^{19} +136.691 q^{20} -148.992 q^{22} -186.640 q^{23} +12.8574 q^{25} -293.516 q^{26} +74.7785 q^{28} -21.8413 q^{29} -193.238 q^{31} -224.860 q^{32} +131.456 q^{34} +75.4166 q^{35} -67.8794 q^{37} -120.437 q^{38} +189.512 q^{40} +74.4545 q^{41} -324.175 q^{43} -391.377 q^{44} -827.173 q^{46} +202.273 q^{47} -301.742 q^{49} +56.9832 q^{50} -771.019 q^{52} +254.801 q^{53} -394.717 q^{55} +103.675 q^{56} -96.7990 q^{58} +474.685 q^{59} +179.092 q^{61} -856.414 q^{62} -823.752 q^{64} -777.599 q^{65} +294.098 q^{67} +345.313 q^{68} +334.241 q^{70} +126.860 q^{71} -920.278 q^{73} -300.836 q^{74} -316.367 q^{76} -215.935 q^{77} +31.6274 q^{79} -253.624 q^{80} +329.977 q^{82} +1354.73 q^{83} +348.259 q^{85} -1436.72 q^{86} -542.617 q^{88} +669.715 q^{89} -425.395 q^{91} -2172.85 q^{92} +896.459 q^{94} -319.067 q^{95} +1611.14 q^{97} -1337.30 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8}+O(q^{10}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8	\( 22 q + 4 q^{2} + 50 q^{4} + 37 q^{5} - 52 q^{7} + 69 q^{8} - 93 q^{10} + 77 q^{11} - 218 q^{13} + 111 q^{14} - 42 q^{16} + 219 q^{17} - 476 q^{19} + 314 q^{20} - 390 q^{22} + 202 q^{23} - 271 q^{25} + 220 q^{26} - 515 q^{28} + 307 q^{29} - 1001 q^{31} + 771 q^{32} - 1297 q^{34} + 430 q^{35} - 922 q^{37} - 49 q^{38} - 1344 q^{40} + 1188 q^{41} - 192 q^{43} + 547 q^{44} - 1178 q^{46} + 102 q^{47} - 1952 q^{49} + 471 q^{50} - 1785 q^{52} + 580 q^{53} - 1730 q^{55} + 804 q^{56} - 1156 q^{58} + 1528 q^{59} - 1631 q^{61} - 2206 q^{62} + 327 q^{64} - 44 q^{65} - 689 q^{67} - 2522 q^{68} + 1175 q^{70} - 341 q^{71} - 2260 q^{73} - 4027 q^{74} - 1855 q^{76} - 1578 q^{77} + 396 q^{79} - 6183 q^{80} + 4936 q^{82} - 1065 q^{83} + 144 q^{85} - 2915 q^{86} + 1068 q^{88} + 1984 q^{89} - 2186 q^{91} - 6720 q^{92} + 174 q^{94} - 2804 q^{95} - 4946 q^{97} - 7149 q^{98}+O(q^{100}) \) 22 * q + 4 * q^2 + 50 * q^4 + 37 * q^5 - 52 * q^7 + 69 * q^8 - 93 * q^10 + 77 * q^11 - 218 * q^13 + 111 * q^14 - 42 * q^16 + 219 * q^17 - 476 * q^19 + 314 * q^20 - 390 * q^22 + 202 * q^23 - 271 * q^25 + 220 * q^26 - 515 * q^28 + 307 * q^29 - 1001 * q^31 + 771 * q^32 - 1297 * q^34 + 430 * q^35 - 922 * q^37 - 49 * q^38 - 1344 * q^40 + 1188 * q^41 - 192 * q^43 + 547 * q^44 - 1178 * q^46 + 102 * q^47 - 1952 * q^49 + 471 * q^50 - 1785 * q^52 + 580 * q^53 - 1730 * q^55 + 804 * q^56 - 1156 * q^58 + 1528 * q^59 - 1631 * q^61 - 2206 * q^62 + 327 * q^64 - 44 * q^65 - 689 * q^67 - 2522 * q^68 + 1175 * q^70 - 341 * q^71 - 2260 * q^73 - 4027 * q^74 - 1855 * q^76 - 1578 * q^77 + 396 * q^79 - 6183 * q^80 + 4936 * q^82 - 1065 * q^83 + 144 * q^85 - 2915 * q^86 + 1068 * q^88 + 1984 * q^89 - 2186 * q^91 - 6720 * q^92 + 174 * q^94 - 2804 * q^95 - 4946 * q^97 - 7149 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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