LMFDB - Embedded newform 2151.4.a.h.1.17

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:	$59$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q-2.95118 q^{2} +0.709472 q^{4} -16.6350 q^{5} -30.2397 q^{7} +21.5157 q^{8} +O(q^{10})$	\(q-2.95118 q^{2} +0.709472 q^{4} -16.6350 q^{5} -30.2397 q^{7} +21.5157 q^{8} +49.0928 q^{10} -7.82864 q^{11} +48.9031 q^{13} +89.2428 q^{14} -69.1724 q^{16} +47.2093 q^{17} +48.8394 q^{19} -11.8020 q^{20} +23.1037 q^{22} -130.630 q^{23} +151.722 q^{25} -144.322 q^{26} -21.4542 q^{28} +145.591 q^{29} -122.337 q^{31} +32.0150 q^{32} -139.323 q^{34} +503.036 q^{35} -31.4938 q^{37} -144.134 q^{38} -357.912 q^{40} -123.558 q^{41} -306.194 q^{43} -5.55420 q^{44} +385.514 q^{46} -143.310 q^{47} +571.438 q^{49} -447.758 q^{50} +34.6954 q^{52} +285.587 q^{53} +130.229 q^{55} -650.627 q^{56} -429.666 q^{58} -260.993 q^{59} -604.367 q^{61} +361.038 q^{62} +458.897 q^{64} -813.501 q^{65} -517.550 q^{67} +33.4937 q^{68} -1484.55 q^{70} -246.465 q^{71} -882.001 q^{73} +92.9438 q^{74} +34.6502 q^{76} +236.735 q^{77} -1165.14 q^{79} +1150.68 q^{80} +364.643 q^{82} -34.8380 q^{83} -785.324 q^{85} +903.633 q^{86} -168.438 q^{88} +810.810 q^{89} -1478.81 q^{91} -92.6786 q^{92} +422.934 q^{94} -812.440 q^{95} -192.023 q^{97} -1686.42 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8}+O(q^{10}) $ 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8	$ 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8} - 36 q^{10} + 132 q^{11} + 104 q^{13} + 280 q^{14} + 822 q^{16} + 408 q^{17} + 20 q^{19} + 800 q^{20} - 2 q^{22} + 276 q^{23} + 1477 q^{25} + 780 q^{26} + 224 q^{28} + 696 q^{29} - 380 q^{31} + 896 q^{32} - 72 q^{34} + 700 q^{35} + 224 q^{37} + 988 q^{38} - 258 q^{40} + 2706 q^{41} - 156 q^{43} + 1584 q^{44} + 428 q^{46} + 1316 q^{47} + 2135 q^{49} + 1400 q^{50} + 1092 q^{52} + 1484 q^{53} - 992 q^{55} + 3360 q^{56} - 120 q^{58} + 3186 q^{59} - 254 q^{61} + 1240 q^{62} + 3054 q^{64} + 5120 q^{65} + 288 q^{67} + 9420 q^{68} + 1108 q^{70} + 4468 q^{71} - 1770 q^{73} + 6214 q^{74} + 720 q^{76} + 6352 q^{77} - 746 q^{79} + 7040 q^{80} + 276 q^{82} + 5484 q^{83} + 588 q^{85} + 10152 q^{86} + 1186 q^{88} + 11570 q^{89} + 1768 q^{91} + 15366 q^{92} - 2142 q^{94} + 5736 q^{95} + 2390 q^{97} + 6912 q^{98}+O(q^{100}) $ 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8 - 36 * q^10 + 132 * q^11 + 104 * q^13 + 280 * q^14 + 822 * q^16 + 408 * q^17 + 20 * q^19 + 800 * q^20 - 2 * q^22 + 276 * q^23 + 1477 * q^25 + 780 * q^26 + 224 * q^28 + 696 * q^29 - 380 * q^31 + 896 * q^32 - 72 * q^34 + 700 * q^35 + 224 * q^37 + 988 * q^38 - 258 * q^40 + 2706 * q^41 - 156 * q^43 + 1584 * q^44 + 428 * q^46 + 1316 * q^47 + 2135 * q^49 + 1400 * q^50 + 1092 * q^52 + 1484 * q^53 - 992 * q^55 + 3360 * q^56 - 120 * q^58 + 3186 * q^59 - 254 * q^61 + 1240 * q^62 + 3054 * q^64 + 5120 * q^65 + 288 * q^67 + 9420 * q^68 + 1108 * q^70 + 4468 * q^71 - 1770 * q^73 + 6214 * q^74 + 720 * q^76 + 6352 * q^77 - 746 * q^79 + 7040 * q^80 + 276 * q^82 + 5484 * q^83 + 588 * q^85 + 10152 * q^86 + 1186 * q^88 + 11570 * q^89 + 1768 * q^91 + 15366 * q^92 - 2142 * q^94 + 5736 * q^95 + 2390 * q^97 + 6912 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.95118	−1.04340	−0.521700	−	0.853129i	$-0.674702\pi$
$2$	−2.95118	−1.04340	−0.521700	+	0.853129i	$0.674702\pi$
$3$	0	0
$3$	0	0
$4$	0.709472	0.0886840
$5$	−16.6350	−1.48788	−0.743938	−	0.668249i	$-0.767044\pi$
$5$	−16.6350	−1.48788	−0.743938	+	0.668249i	$0.767044\pi$
$6$	0	0
$7$	−30.2397	−1.63279	−0.816395	−	0.577494i	$-0.804031\pi$
$7$	−30.2397	−1.63279	−0.816395	+	0.577494i	$0.804031\pi$
$8$	21.5157	0.950867
$9$	0	0
$10$	49.0928	1.55245
$11$	−7.82864	−0.214584	−0.107292	−	0.994228i	$-0.534218\pi$
$11$	−7.82864	−0.214584	−0.107292	+	0.994228i	$0.534218\pi$
$12$	0	0
$13$	48.9031	1.04333	0.521665	−	0.853151i	$-0.325311\pi$
$13$	48.9031	1.04333	0.521665	+	0.853151i	$0.325311\pi$
$14$	89.2428	1.70365
$15$	0	0
$16$	−69.1724	−1.08082
$17$	47.2093	0.673526	0.336763	−	0.941590i	$-0.390668\pi$
$17$	47.2093	0.673526	0.336763	+	0.941590i	$0.390668\pi$
$18$	0	0
$19$	48.8394	0.589711	0.294856	−	0.955542i	$-0.404728\pi$
$19$	48.8394	0.589711	0.294856	+	0.955542i	$0.404728\pi$
$20$	−11.8020	−0.131951
$21$	0	0
$22$	23.1037	0.223897
$23$	−130.630	−1.18427	−0.592137	−	0.805837i	$-0.701716\pi$
$23$	−130.630	−1.18427	−0.592137	+	0.805837i	$0.701716\pi$
$24$	0	0
$25$	151.722	1.21377
$26$	−144.322	−1.08861
$27$	0	0
$28$	−21.4542	−0.144802
$29$	145.591	0.932263	0.466131	−	0.884715i	$-0.345647\pi$
$29$	145.591	0.932263	0.466131	+	0.884715i	$0.345647\pi$
$30$	0	0
$31$	−122.337	−0.708784	−0.354392	−	0.935097i	$-0.615312\pi$
$31$	−122.337	−0.708784	−0.354392	+	0.935097i	$0.615312\pi$
$32$	32.0150	0.176860
$33$	0	0
$34$	−139.323	−0.702757
$35$	503.036	2.42939
$36$	0	0
$37$	−31.4938	−0.139934	−0.0699668	−	0.997549i	$-0.522289\pi$
$37$	−31.4938	−0.139934	−0.0699668	+	0.997549i	$0.522289\pi$
$38$	−144.134	−0.615305
$39$	0	0
$40$	−357.912	−1.41477
$41$	−123.558	−0.470648	−0.235324	−	0.971917i	$-0.575615\pi$
$41$	−123.558	−0.470648	−0.235324	+	0.971917i	$0.575615\pi$
$42$	0	0
$43$	−306.194	−1.08591	−0.542955	−	0.839762i	$-0.682694\pi$
$43$	−306.194	−1.08591	−0.542955	+	0.839762i	$0.682694\pi$
$44$	−5.55420	−0.0190302
$45$	0	0
$46$	385.514	1.23567
$47$	−143.310	−0.444764	−0.222382	−	0.974960i	$-0.571383\pi$
$47$	−143.310	−0.444764	−0.222382	+	0.974960i	$0.571383\pi$
$48$	0	0
$49$	571.438	1.66600
$50$	−447.758	−1.26645
$51$	0	0
$52$	34.6954	0.0925267
$53$	285.587	0.740160	0.370080	−	0.929000i	$-0.379330\pi$
$53$	285.587	0.740160	0.370080	+	0.929000i	$0.379330\pi$
$54$	0	0
$55$	130.229	0.319274
$56$	−650.627	−1.55257
$57$	0	0
$58$	−429.666	−0.972723
$59$	−260.993	−0.575905	−0.287952	−	0.957645i	$-0.592974\pi$
$59$	−260.993	−0.575905	−0.287952	+	0.957645i	$0.592974\pi$
$60$	0	0
$61$	−604.367	−1.26855	−0.634273	−	0.773109i	$-0.718700\pi$
$61$	−604.367	−1.26855	−0.634273	+	0.773109i	$0.718700\pi$
$62$	361.038	0.739545
$63$	0	0
$64$	458.897	0.896284
$65$	−813.501	−1.55234
$66$	0	0
$67$	−517.550	−0.943713	−0.471857	−	0.881675i	$-0.656416\pi$
$67$	−517.550	−0.943713	−0.471857	+	0.881675i	$0.656416\pi$
$68$	33.4937	0.0597310
$69$	0	0
$70$	−1484.55	−2.53482
$71$	−246.465	−0.411972	−0.205986	−	0.978555i	$-0.566040\pi$
$71$	−246.465	−0.411972	−0.205986	+	0.978555i	$0.566040\pi$
$72$	0	0
$73$	−882.001	−1.41412	−0.707058	−	0.707156i	$-0.749978\pi$
$73$	−882.001	−1.41412	−0.707058	+	0.707156i	$0.749978\pi$
$74$	92.9438	0.146007
$75$	0	0
$76$	34.6502	0.0522980
$77$	236.735	0.350370
$78$	0	0
$79$	−1165.14	−1.65935	−0.829676	−	0.558245i	$-0.811475\pi$
$79$	−1165.14	−1.65935	−0.829676	+	0.558245i	$0.811475\pi$
$80$	1150.68	1.60812
$81$	0	0
$82$	364.643	0.491075
$83$	−34.8380	−0.0460719	−0.0230360	−	0.999735i	$-0.507333\pi$
$83$	−34.8380	−0.0460719	−0.0230360	+	0.999735i	$0.507333\pi$
$84$	0	0
$85$	−785.324	−1.00212
$86$	903.633	1.13304
$87$	0	0
$88$	−168.438	−0.204041
$89$	810.810	0.965682	0.482841	−	0.875708i	$-0.339605\pi$
$89$	810.810	0.965682	0.482841	+	0.875708i	$0.339605\pi$
$90$	0	0
$91$	−1478.81	−1.70354
$92$	−92.6786	−0.105026
$93$	0	0
$94$	422.934	0.464067
$95$	−812.440	−0.877417
$96$	0	0
$97$	−192.023	−0.200999	−0.100500	−	0.994937i	$-0.532044\pi$
$97$	−192.023	−0.200999	−0.100500	+	0.994937i	$0.532044\pi$
$98$	−1686.42	−1.73831
$99$	0	0
$100$	107.642	0.107642
$101$	1491.59	1.46949	0.734747	−	0.678341i	$-0.237301\pi$
$101$	1491.59	1.46949	0.734747	+	0.678341i	$0.237301\pi$
$102$	0	0
$103$	−1282.36	−1.22675	−0.613373	−	0.789793i	$-0.710188\pi$
$103$	−1282.36	−1.22675	−0.613373	+	0.789793i	$0.710188\pi$
$104$	1052.18	0.992068
$105$	0	0
$106$	−842.820	−0.772283
$107$	−189.030	−0.170787	−0.0853933	−	0.996347i	$-0.527215\pi$
$107$	−189.030	−0.170787	−0.0853933	+	0.996347i	$0.527215\pi$
$108$	0	0
$109$	−1025.50	−0.901152	−0.450576	−	0.892738i	$-0.648781\pi$
$109$	−1025.50	−0.901152	−0.450576	+	0.892738i	$0.648781\pi$
$110$	−384.329	−0.333131
$111$	0	0
$112$	2091.75	1.76475
$113$	−529.570	−0.440865	−0.220433	−	0.975402i	$-0.570747\pi$
$113$	−529.570	−0.440865	−0.220433	+	0.975402i	$0.570747\pi$
$114$	0	0
$115$	2173.03	1.76205
$116$	103.293	0.0826768
$117$	0	0
$118$	770.237	0.600899
$119$	−1427.59	−1.09973
$120$	0	0
$121$	−1269.71	−0.953954
$122$	1783.60	1.32360
$123$	0	0
$124$	−86.7944	−0.0628578
$125$	−444.514	−0.318069
$126$	0	0
$127$	−231.452	−0.161717	−0.0808584	−	0.996726i	$-0.525766\pi$
$127$	−231.452	−0.161717	−0.0808584	+	0.996726i	$0.525766\pi$
$128$	−1610.41	−1.11204
$129$	0	0
$130$	2400.79	1.61972
$131$	−2676.49	−1.78509	−0.892544	−	0.450961i	$-0.851081\pi$
$131$	−2676.49	−1.78509	−0.892544	+	0.450961i	$0.851081\pi$
$132$	0	0
$133$	−1476.89	−0.962874
$134$	1527.38	0.984670
$135$	0	0
$136$	1015.74	0.640433
$137$	2459.96	1.53408	0.767038	−	0.641602i	$-0.221730\pi$
$137$	2459.96	1.53408	0.767038	+	0.641602i	$0.221730\pi$
$138$	0	0
$139$	−450.639	−0.274984	−0.137492	−	0.990503i	$-0.543904\pi$
$139$	−450.639	−0.274984	−0.137492	+	0.990503i	$0.543904\pi$
$140$	356.890	0.215448
$141$	0	0
$142$	727.362	0.429851
$143$	−382.845	−0.223882
$144$	0	0
$145$	−2421.90	−1.38709
$146$	2602.95	1.47549
$147$	0	0
$148$	−22.3440	−0.0124099
$149$	430.148	0.236504	0.118252	−	0.992984i	$-0.462271\pi$
$149$	430.148	0.236504	0.118252	+	0.992984i	$0.462271\pi$
$150$	0	0
$151$	3413.85	1.83983	0.919917	−	0.392113i	$-0.128256\pi$
$151$	3413.85	1.83983	0.919917	+	0.392113i	$0.128256\pi$
$152$	1050.81	0.560737
$153$	0	0
$154$	−698.649	−0.365576
$155$	2035.06	1.05458
$156$	0	0
$157$	864.393	0.439402	0.219701	−	0.975567i	$-0.429492\pi$
$157$	864.393	0.439402	0.219701	+	0.975567i	$0.429492\pi$
$158$	3438.55	1.73137
$159$	0	0
$160$	−532.568	−0.263145
$161$	3950.22	1.93367
$162$	0	0
$163$	1065.54	0.512024	0.256012	−	0.966674i	$-0.417591\pi$
$163$	1065.54	0.512024	0.256012	+	0.966674i	$0.417591\pi$
$164$	−87.6613	−0.0417390
$165$	0	0
$166$	102.813	0.0480715
$167$	−330.435	−0.153113	−0.0765565	−	0.997065i	$-0.524393\pi$
$167$	−330.435	−0.153113	−0.0765565	+	0.997065i	$0.524393\pi$
$168$	0	0
$169$	194.515	0.0885367
$170$	2317.63	1.04561
$171$	0	0
$172$	−217.236	−0.0963028
$173$	885.439	0.389125	0.194563	−	0.980890i	$-0.437671\pi$
$173$	885.439	0.389125	0.194563	+	0.980890i	$0.437671\pi$
$174$	0	0
$175$	−4588.02	−1.98184
$176$	541.526	0.231926
$177$	0	0
$178$	−2392.85	−1.00759
$179$	3123.34	1.30418	0.652092	−	0.758140i	$-0.273891\pi$
$179$	3123.34	1.30418	0.652092	+	0.758140i	$0.273891\pi$
$180$	0	0
$181$	−612.109	−0.251368	−0.125684	−	0.992070i	$-0.540113\pi$
$181$	−612.109	−0.251368	−0.125684	+	0.992070i	$0.540113\pi$
$182$	4364.25	1.77747
$183$	0	0
$184$	−2810.60	−1.12609
$185$	523.897	0.208204
$186$	0	0
$187$	−369.584	−0.144528
$188$	−101.675	−0.0394435
$189$	0	0
$190$	2397.66	0.915497
$191$	−3649.85	−1.38269	−0.691345	−	0.722525i	$-0.742982\pi$
$191$	−3649.85	−1.38269	−0.691345	+	0.722525i	$0.742982\pi$
$192$	0	0
$193$	76.0810	0.0283753	0.0141876	−	0.999899i	$-0.495484\pi$
$193$	76.0810	0.0283753	0.0141876	+	0.999899i	$0.495484\pi$
$194$	566.694	0.209723
$195$	0	0
$196$	405.419	0.147748
$197$	−3599.52	−1.30180	−0.650902	−	0.759162i	$-0.725609\pi$
$197$	−3599.52	−1.30180	−0.650902	+	0.759162i	$0.725609\pi$
$198$	0	0
$199$	−3965.15	−1.41247	−0.706237	−	0.707976i	$-0.749608\pi$
$199$	−3965.15	−1.41247	−0.706237	+	0.707976i	$0.749608\pi$
$200$	3264.39	1.15414
$201$	0	0
$202$	−4401.96	−1.53327
$203$	−4402.63	−1.52219
$204$	0	0
$205$	2055.39	0.700266
$206$	3784.48	1.27999
$207$	0	0
$208$	−3382.75	−1.12765
$209$	−382.345	−0.126543
$210$	0	0
$211$	315.363	0.102893	0.0514466	−	0.998676i	$-0.483617\pi$
$211$	315.363	0.102893	0.0514466	+	0.998676i	$0.483617\pi$
$212$	202.616	0.0656403
$213$	0	0
$214$	557.861	0.178199
$215$	5093.52	1.61570
$216$	0	0
$217$	3699.42	1.15729
$218$	3026.45	0.940262
$219$	0	0
$220$	92.3938	0.0283145
$221$	2308.68	0.702709
$222$	0	0
$223$	−4273.57	−1.28332	−0.641658	−	0.766991i	$-0.721753\pi$
$223$	−4273.57	−1.28332	−0.641658	+	0.766991i	$0.721753\pi$
$224$	−968.124	−0.288775
$225$	0	0
$226$	1562.86	0.459999
$227$	−4235.90	−1.23853	−0.619265	−	0.785182i	$-0.712569\pi$
$227$	−4235.90	−1.23853	−0.619265	+	0.785182i	$0.712569\pi$
$228$	0	0
$229$	6473.11	1.86793	0.933963	−	0.357370i	$-0.116327\pi$
$229$	6473.11	1.86793	0.933963	+	0.357370i	$0.116327\pi$
$230$	−6413.01	−1.83853
$231$	0	0
$232$	3132.49	0.886458
$233$	2644.21	0.743467	0.371733	−	0.928340i	$-0.378764\pi$
$233$	2644.21	0.743467	0.371733	+	0.928340i	$0.378764\pi$
$234$	0	0
$235$	2383.96	0.661754
$236$	−185.167	−0.0510735
$237$	0	0
$238$	4213.09	1.14745
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−2159.55	−0.577214	−0.288607	−	0.957448i	$-0.593192\pi$
$241$	−2159.55	−0.577214	−0.288607	+	0.957448i	$0.593192\pi$
$242$	3747.15	0.995356
$243$	0	0
$244$	−428.782	−0.112500
$245$	−9505.85	−2.47880
$246$	0	0
$247$	2388.40	0.615263
$248$	−2632.15	−0.673960
$249$	0	0
$250$	1311.84	0.331873
$251$	5514.44	1.38673	0.693364	−	0.720588i	$-0.256128\pi$
$251$	5514.44	1.38673	0.693364	+	0.720588i	$0.256128\pi$
$252$	0	0
$253$	1022.66	0.254126
$254$	683.057	0.168735
$255$	0	0
$256$	1081.43	0.264021
$257$	−3422.74	−0.830758	−0.415379	−	0.909648i	$-0.636351\pi$
$257$	−3422.74	−0.830758	−0.415379	+	0.909648i	$0.636351\pi$
$258$	0	0
$259$	952.362	0.228482
$260$	−577.156	−0.137668
$261$	0	0
$262$	7898.82	1.86256
$263$	−4157.22	−0.974697	−0.487348	−	0.873208i	$-0.662036\pi$
$263$	−4157.22	−0.974697	−0.487348	+	0.873208i	$0.662036\pi$
$264$	0	0
$265$	−4750.73	−1.10127
$266$	4358.56	1.00466
$267$	0	0
$268$	−367.187	−0.0836923
$269$	−74.1339	−0.0168031	−0.00840153	−	0.999965i	$-0.502674\pi$
$269$	−74.1339	−0.0168031	−0.00840153	+	0.999965i	$0.502674\pi$
$270$	0	0
$271$	−3124.03	−0.700263	−0.350131	−	0.936701i	$-0.613863\pi$
$271$	−3124.03	−0.700263	−0.350131	+	0.936701i	$0.613863\pi$
$272$	−3265.58	−0.727959
$273$	0	0
$274$	−7259.78	−1.60066
$275$	−1187.77	−0.260456
$276$	0	0
$277$	−3431.12	−0.744246	−0.372123	−	0.928184i	$-0.621370\pi$
$277$	−3431.12	−0.744246	−0.372123	+	0.928184i	$0.621370\pi$
$278$	1329.92	0.286918
$279$	0	0
$280$	10823.1	2.31002
$281$	−1449.47	−0.307716	−0.153858	−	0.988093i	$-0.549170\pi$
$281$	−1449.47	−0.307716	−0.153858	+	0.988093i	$0.549170\pi$
$282$	0	0
$283$	−3287.48	−0.690532	−0.345266	−	0.938505i	$-0.612211\pi$
$283$	−3287.48	−0.690532	−0.345266	+	0.938505i	$0.612211\pi$
$284$	−174.860	−0.0365353
$285$	0	0
$286$	1129.84	0.233598
$287$	3736.37	0.768470
$288$	0	0
$289$	−2684.28	−0.546363
$290$	7147.48	1.44729
$291$	0	0
$292$	−625.755	−0.125409
$293$	−2698.79	−0.538106	−0.269053	−	0.963125i	$-0.586711\pi$
$293$	−2698.79	−0.538106	−0.269053	+	0.963125i	$0.586711\pi$
$294$	0	0
$295$	4341.60	0.856874
$296$	−677.610	−0.133058
$297$	0	0
$298$	−1269.45	−0.246768
$299$	−6388.23	−1.23559
$300$	0	0
$301$	9259.20	1.77306
$302$	−10074.9	−1.91968
$303$	0	0
$304$	−3378.34	−0.637371
$305$	10053.6	1.88744
$306$	0	0
$307$	10156.3	1.88811	0.944055	−	0.329787i	$-0.106977\pi$
$307$	10156.3	1.88811	0.944055	+	0.329787i	$0.106977\pi$
$308$	167.957	0.0310722
$309$	0	0
$310$	−6005.84	−1.10035
$311$	−5483.97	−0.999895	−0.499948	−	0.866056i	$-0.666647\pi$
$311$	−5483.97	−0.999895	−0.499948	+	0.866056i	$0.666647\pi$
$312$	0	0
$313$	−2535.58	−0.457889	−0.228944	−	0.973439i	$-0.573527\pi$
$313$	−2535.58	−0.457889	−0.228944	+	0.973439i	$0.573527\pi$
$314$	−2550.98	−0.458472
$315$	0	0
$316$	−826.636	−0.147158
$317$	7331.70	1.29902	0.649510	−	0.760353i	$-0.274974\pi$
$317$	7331.70	1.29902	0.649510	+	0.760353i	$0.274974\pi$
$318$	0	0
$319$	−1139.78	−0.200049
$320$	−7633.74	−1.33356
$321$	0	0
$322$	−11657.8	−2.01759
$323$	2305.67	0.397186
$324$	0	0
$325$	7419.66	1.26637
$326$	−3144.61	−0.534245
$327$	0	0
$328$	−2658.44	−0.447524
$329$	4333.65	0.726207
$330$	0	0
$331$	−9699.52	−1.61068	−0.805338	−	0.592816i	$-0.798016\pi$
$331$	−9699.52	−1.61068	−0.805338	+	0.592816i	$0.798016\pi$
$332$	−24.7166	−0.00408584
$333$	0	0
$334$	975.175	0.159758
$335$	8609.42	1.40413
$336$	0	0
$337$	2893.84	0.467767	0.233883	−	0.972265i	$-0.424857\pi$
$337$	2893.84	0.467767	0.233883	+	0.972265i	$0.424857\pi$
$338$	−574.049	−0.0923792
$339$	0	0
$340$	−557.166	−0.0888722
$341$	957.729	0.152094
$342$	0	0
$343$	−6907.90	−1.08744
$344$	−6587.96	−1.03256
$345$	0	0
$346$	−2613.09	−0.406014
$347$	−3600.31	−0.556987	−0.278494	−	0.960438i	$-0.589835\pi$
$347$	−3600.31	−0.556987	−0.278494	+	0.960438i	$0.589835\pi$
$348$	0	0
$349$	4910.29	0.753129	0.376565	−	0.926390i	$-0.377105\pi$
$349$	4910.29	0.753129	0.376565	+	0.926390i	$0.377105\pi$
$350$	13540.1	2.06785
$351$	0	0
$352$	−250.634	−0.0379512
$353$	5905.36	0.890398	0.445199	−	0.895432i	$-0.353133\pi$
$353$	5905.36	0.890398	0.445199	+	0.895432i	$0.353133\pi$
$354$	0	0
$355$	4099.93	0.612962
$356$	575.247	0.0856406
$357$	0	0
$358$	−9217.53	−1.36079
$359$	−3567.83	−0.524520	−0.262260	−	0.964997i	$-0.584468\pi$
$359$	−3567.83	−0.524520	−0.262260	+	0.964997i	$0.584468\pi$
$360$	0	0
$361$	−4473.72	−0.652240
$362$	1806.44	0.262278
$363$	0	0
$364$	−1049.18	−0.151077
$365$	14672.1	2.10403
$366$	0	0
$367$	2879.49	0.409559	0.204779	−	0.978808i	$-0.434352\pi$
$367$	2879.49	0.409559	0.204779	+	0.978808i	$0.434352\pi$
$368$	9036.02	1.27999
$369$	0	0
$370$	−1546.12	−0.217240
$371$	−8636.07	−1.20852
$372$	0	0
$373$	−766.449	−0.106395	−0.0531974	−	0.998584i	$-0.516941\pi$
$373$	−766.449	−0.106395	−0.0531974	+	0.998584i	$0.516941\pi$
$374$	1090.71	0.150800
$375$	0	0
$376$	−3083.41	−0.422912
$377$	7119.87	0.972658
$378$	0	0
$379$	−5364.00	−0.726993	−0.363496	−	0.931596i	$-0.618417\pi$
$379$	−5364.00	−0.726993	−0.363496	+	0.931596i	$0.618417\pi$
$380$	−576.404	−0.0778129
$381$	0	0
$382$	10771.4	1.44270
$383$	1522.65	0.203143	0.101571	−	0.994828i	$-0.467613\pi$
$383$	1522.65	0.203143	0.101571	+	0.994828i	$0.467613\pi$
$384$	0	0
$385$	−3938.08	−0.521307
$386$	−224.529	−0.0296068
$387$	0	0
$388$	−136.235	−0.0178254
$389$	−2273.74	−0.296358	−0.148179	−	0.988961i	$-0.547341\pi$
$389$	−2273.74	−0.296358	−0.148179	+	0.988961i	$0.547341\pi$
$390$	0	0
$391$	−6166.97	−0.797639
$392$	12294.9	1.58415
$393$	0	0
$394$	10622.8	1.35830
$395$	19382.1	2.46891
$396$	0	0
$397$	−10272.7	−1.29867	−0.649337	−	0.760500i	$-0.724954\pi$
$397$	−10272.7	−1.29867	−0.649337	+	0.760500i	$0.724954\pi$
$398$	11701.9	1.47377
$399$	0	0
$400$	−10495.0	−1.31187
$401$	14406.6	1.79409	0.897047	−	0.441935i	$-0.145708\pi$
$401$	14406.6	1.79409	0.897047	+	0.441935i	$0.145708\pi$
$402$	0	0
$403$	−5982.64	−0.739495
$404$	1058.24	0.130321
$405$	0	0
$406$	12993.0	1.58825
$407$	246.553	0.0300275
$408$	0	0
$409$	5649.95	0.683062	0.341531	−	0.939871i	$-0.389055\pi$
$409$	5649.95	0.683062	0.341531	+	0.939871i	$0.389055\pi$
$410$	−6065.83	−0.730658
$411$	0	0
$412$	−909.800	−0.108793
$413$	7892.34	0.940331
$414$	0	0
$415$	579.529	0.0685493
$416$	1565.63	0.184523
$417$	0	0
$418$	1128.37	0.132035
$419$	−3959.84	−0.461696	−0.230848	−	0.972990i	$-0.574150\pi$
$419$	−3959.84	−0.461696	−0.230848	+	0.972990i	$0.574150\pi$
$420$	0	0
$421$	10821.5	1.25275	0.626375	−	0.779522i	$-0.284538\pi$
$421$	10821.5	1.25275	0.626375	+	0.779522i	$0.284538\pi$
$422$	−930.693	−0.107359
$423$	0	0
$424$	6144.61	0.703793
$425$	7162.67	0.817508
$426$	0	0
$427$	18275.9	2.07127
$428$	−134.111	−0.0151461
$429$	0	0
$430$	−15031.9	−1.68582
$431$	−3895.83	−0.435396	−0.217698	−	0.976016i	$-0.569855\pi$
$431$	−3895.83	−0.435396	−0.217698	+	0.976016i	$0.569855\pi$
$432$	0	0
$433$	453.802	0.0503657	0.0251828	−	0.999683i	$-0.491983\pi$
$433$	453.802	0.0503657	0.0251828	+	0.999683i	$0.491983\pi$
$434$	−10917.7	−1.20752
$435$	0	0
$436$	−727.567	−0.0799177
$437$	−6379.90	−0.698380
$438$	0	0
$439$	−3628.86	−0.394525	−0.197262	−	0.980351i	$-0.563205\pi$
$439$	−3628.86	−0.394525	−0.197262	+	0.980351i	$0.563205\pi$
$440$	2801.96	0.303587
$441$	0	0
$442$	−6813.34	−0.733207
$443$	570.512	0.0611870	0.0305935	−	0.999532i	$-0.490260\pi$
$443$	570.512	0.0611870	0.0305935	+	0.999532i	$0.490260\pi$
$444$	0	0
$445$	−13487.8	−1.43681
$446$	12612.1	1.33901
$447$	0	0
$448$	−13876.9	−1.46344
$449$	−6238.94	−0.655755	−0.327877	−	0.944720i	$-0.606333\pi$
$449$	−6238.94	−0.655755	−0.327877	+	0.944720i	$0.606333\pi$
$450$	0	0
$451$	967.294	0.100994
$452$	−375.715	−0.0390977
$453$	0	0
$454$	12500.9	1.29228
$455$	24600.0	2.53465
$456$	0	0
$457$	−3469.70	−0.355155	−0.177577	−	0.984107i	$-0.556826\pi$
$457$	−3469.70	−0.355155	−0.177577	+	0.984107i	$0.556826\pi$
$458$	−19103.3	−1.94899
$459$	0	0
$460$	1541.70	0.156266
$461$	8651.26	0.874033	0.437016	−	0.899454i	$-0.356035\pi$
$461$	8651.26	0.874033	0.437016	+	0.899454i	$0.356035\pi$
$462$	0	0
$463$	−13077.8	−1.31269	−0.656347	−	0.754459i	$-0.727899\pi$
$463$	−13077.8	−1.31269	−0.656347	+	0.754459i	$0.727899\pi$
$464$	−10070.9	−1.00761
$465$	0	0
$466$	−7803.53	−0.775733
$467$	16473.1	1.63230	0.816151	−	0.577839i	$-0.196104\pi$
$467$	16473.1	1.63230	0.816151	+	0.577839i	$0.196104\pi$
$468$	0	0
$469$	15650.5	1.54088
$470$	−7035.49	−0.690474
$471$	0	0
$472$	−5615.44	−0.547609
$473$	2397.08	0.233019
$474$	0	0
$475$	7409.99	0.715776
$476$	−1012.84	−0.0975281
$477$	0	0
$478$	705.332	0.0674919
$479$	13102.4	1.24982	0.624910	−	0.780697i	$-0.285136\pi$
$479$	13102.4	1.24982	0.624910	+	0.780697i	$0.285136\pi$
$480$	0	0
$481$	−1540.14	−0.145997
$482$	6373.21	0.602266
$483$	0	0
$484$	−900.826	−0.0846005
$485$	3194.29	0.299062
$486$	0	0
$487$	2802.96	0.260809	0.130405	−	0.991461i	$-0.458372\pi$
$487$	2802.96	0.260809	0.130405	+	0.991461i	$0.458372\pi$
$488$	−13003.4	−1.20622
$489$	0	0
$490$	28053.5	2.58638
$491$	−19817.1	−1.82145	−0.910725	−	0.413013i	$-0.864476\pi$
$491$	−19817.1	−1.82145	−0.910725	+	0.413013i	$0.864476\pi$
$492$	0	0
$493$	6873.26	0.627903
$494$	−7048.59	−0.641966
$495$	0	0
$496$	8462.32	0.766067
$497$	7453.01	0.672663
$498$	0	0
$499$	15423.6	1.38368	0.691839	−	0.722052i	$-0.256801\pi$
$499$	15423.6	1.38368	0.691839	+	0.722052i	$0.256801\pi$
$500$	−315.370	−0.0282076
$501$	0	0
$502$	−16274.1	−1.44691
$503$	12841.3	1.13830	0.569150	−	0.822234i	$-0.307272\pi$
$503$	12841.3	1.13830	0.569150	+	0.822234i	$0.307272\pi$
$504$	0	0
$505$	−24812.6	−2.18642
$506$	−3018.05	−0.265155
$507$	0	0
$508$	−164.209	−0.0143417
$509$	−12494.7	−1.08805	−0.544026	−	0.839068i	$-0.683101\pi$
$509$	−12494.7	−1.08805	−0.544026	+	0.839068i	$0.683101\pi$
$510$	0	0
$511$	26671.4	2.30895
$512$	9691.77	0.836562
$513$	0	0
$514$	10101.1	0.866814
$515$	21332.0	1.82525
$516$	0	0
$517$	1121.92	0.0954393
$518$	−2810.59	−0.238398
$519$	0	0
$520$	−17503.0	−1.47607
$521$	3466.21	0.291473	0.145737	−	0.989323i	$-0.453445\pi$
$521$	3466.21	0.291473	0.145737	+	0.989323i	$0.453445\pi$
$522$	0	0
$523$	9874.89	0.825619	0.412810	−	0.910817i	$-0.364548\pi$
$523$	9874.89	0.825619	0.412810	+	0.910817i	$0.364548\pi$
$524$	−1898.90	−0.158309
$525$	0	0
$526$	12268.7	1.01700
$527$	−5775.42	−0.477384
$528$	0	0
$529$	4897.29	0.402506
$530$	14020.3	1.14906
$531$	0	0
$532$	−1047.81	−0.0853916
$533$	−6042.39	−0.491041
$534$	0	0
$535$	3144.50	0.254109
$536$	−11135.4	−0.897346
$537$	0	0
$538$	218.783	0.0175323
$539$	−4473.58	−0.357497
$540$	0	0
$541$	10796.3	0.857983	0.428992	−	0.903308i	$-0.358869\pi$
$541$	10796.3	0.857983	0.428992	+	0.903308i	$0.358869\pi$
$542$	9219.57	0.730654
$543$	0	0
$544$	1511.41	0.119120
$545$	17059.2	1.34080
$546$	0	0
$547$	8272.86	0.646658	0.323329	−	0.946287i	$-0.395198\pi$
$547$	8272.86	0.646658	0.323329	+	0.946287i	$0.395198\pi$
$548$	1745.27	0.136048
$549$	0	0
$550$	3505.34	0.271760
$551$	7110.59	0.549766
$552$	0	0
$553$	35233.5	2.70937
$554$	10125.9	0.776546
$555$	0	0
$556$	−319.716	−0.0243867
$557$	8859.27	0.673931	0.336965	−	0.941517i	$-0.390599\pi$
$557$	8859.27	0.673931	0.336965	+	0.941517i	$0.390599\pi$
$558$	0	0
$559$	−14973.8	−1.13296
$560$	−34796.2	−2.62573
$561$	0	0
$562$	4277.65	0.321071
$563$	17887.7	1.33903	0.669516	−	0.742797i	$-0.266501\pi$
$563$	17887.7	1.33903	0.669516	+	0.742797i	$0.266501\pi$
$564$	0	0
$565$	8809.38	0.655953
$566$	9701.95	0.720501
$567$	0	0
$568$	−5302.85	−0.391730
$569$	−5355.10	−0.394547	−0.197274	−	0.980348i	$-0.563209\pi$
$569$	−5355.10	−0.394547	−0.197274	+	0.980348i	$0.563209\pi$
$570$	0	0
$571$	16668.1	1.22161	0.610805	−	0.791781i	$-0.290846\pi$
$571$	16668.1	1.22161	0.610805	+	0.791781i	$0.290846\pi$
$572$	−271.618	−0.0198547
$573$	0	0
$574$	−11026.7	−0.801821
$575$	−19819.5	−1.43744
$576$	0	0
$577$	22341.9	1.61197	0.805985	−	0.591937i	$-0.201636\pi$
$577$	22341.9	1.61197	0.805985	+	0.591937i	$0.201636\pi$
$578$	7921.81	0.570076
$579$	0	0
$580$	−1718.27	−0.123013
$581$	1053.49	0.0752257
$582$	0	0
$583$	−2235.76	−0.158826
$584$	−18976.9	−1.34464
$585$	0	0
$586$	7964.62	0.561460
$587$	−13446.6	−0.945487	−0.472744	−	0.881200i	$-0.656736\pi$
$587$	−13446.6	−0.945487	−0.472744	+	0.881200i	$0.656736\pi$
$588$	0	0
$589$	−5974.84	−0.417978
$590$	−12812.9	−0.894063
$591$	0	0
$592$	2178.50	0.151243
$593$	−22921.3	−1.58730	−0.793648	−	0.608377i	$-0.791821\pi$
$593$	−22921.3	−1.58730	−0.793648	+	0.608377i	$0.791821\pi$
$594$	0	0
$595$	23748.0	1.63625
$596$	305.178	0.0209741
$597$	0	0
$598$	18852.8	1.28921
$599$	−25166.7	−1.71667	−0.858334	−	0.513091i	$-0.828500\pi$
$599$	−25166.7	−1.71667	−0.858334	+	0.513091i	$0.828500\pi$
$600$	0	0
$601$	−1390.40	−0.0943688	−0.0471844	−	0.998886i	$-0.515025\pi$
$601$	−1390.40	−0.0943688	−0.0471844	+	0.998886i	$0.515025\pi$
$602$	−27325.6	−1.85001
$603$	0	0
$604$	2422.03	0.163164
$605$	21121.6	1.41936
$606$	0	0
$607$	15650.3	1.04650	0.523249	−	0.852180i	$-0.324720\pi$
$607$	15650.3	1.04650	0.523249	+	0.852180i	$0.324720\pi$
$608$	1563.59	0.104296
$609$	0	0
$610$	−29670.0	−1.96935
$611$	−7008.31	−0.464036
$612$	0	0
$613$	−20202.6	−1.33112	−0.665559	−	0.746345i	$-0.731807\pi$
$613$	−20202.6	−1.33112	−0.665559	+	0.746345i	$0.731807\pi$
$614$	−29973.1	−1.97006
$615$	0	0
$616$	5093.52	0.333156
$617$	3474.03	0.226676	0.113338	−	0.993556i	$-0.463846\pi$
$617$	3474.03	0.226676	0.113338	+	0.993556i	$0.463846\pi$
$618$	0	0
$619$	17527.4	1.13810	0.569051	−	0.822302i	$-0.307311\pi$
$619$	17527.4	1.13810	0.569051	+	0.822302i	$0.307311\pi$
$620$	1443.82	0.0935246
$621$	0	0
$622$	16184.2	1.04329
$623$	−24518.6	−1.57676
$624$	0	0
$625$	−11570.7	−0.740527
$626$	7482.95	0.477761
$627$	0	0
$628$	613.263	0.0389679
$629$	−1486.80	−0.0942489
$630$	0	0
$631$	254.193	0.0160368	0.00801842	−	0.999968i	$-0.497448\pi$
$631$	254.193	0.0160368	0.00801842	+	0.999968i	$0.497448\pi$
$632$	−25068.8	−1.57782
$633$	0	0
$634$	−21637.2	−1.35540
$635$	3850.19	0.240615
$636$	0	0
$637$	27945.1	1.73819
$638$	3363.70	0.208731
$639$	0	0
$640$	26789.1	1.65458
$641$	16952.1	1.04457	0.522284	−	0.852772i	$-0.325080\pi$
$641$	16952.1	1.04457	0.522284	+	0.852772i	$0.325080\pi$
$642$	0	0
$643$	1479.72	0.0907537	0.0453769	−	0.998970i	$-0.485551\pi$
$643$	1479.72	0.0907537	0.0453769	+	0.998970i	$0.485551\pi$
$644$	2802.57	0.171486
$645$	0	0
$646$	−6804.45	−0.414424
$647$	−24399.8	−1.48262	−0.741309	−	0.671164i	$-0.765795\pi$
$647$	−24399.8	−1.48262	−0.741309	+	0.671164i	$0.765795\pi$
$648$	0	0
$649$	2043.22	0.123580
$650$	−21896.8	−1.32133
$651$	0	0
$652$	755.974	0.0454083
$653$	1626.88	0.0974958	0.0487479	−	0.998811i	$-0.484477\pi$
$653$	1626.88	0.0974958	0.0487479	+	0.998811i	$0.484477\pi$
$654$	0	0
$655$	44523.4	2.65599
$656$	8546.84	0.508686
$657$	0	0
$658$	−12789.4	−0.757724
$659$	24392.7	1.44189	0.720946	−	0.692992i	$-0.243708\pi$
$659$	24392.7	1.44189	0.720946	+	0.692992i	$0.243708\pi$
$660$	0	0
$661$	13799.8	0.812029	0.406015	−	0.913867i	$-0.366918\pi$
$661$	13799.8	0.812029	0.406015	+	0.913867i	$0.366918\pi$
$662$	28625.0	1.68058
$663$	0	0
$664$	−749.563	−0.0438083
$665$	24567.9	1.43264
$666$	0	0
$667$	−19018.6	−1.10406
$668$	−234.435	−0.0135787
$669$	0	0
$670$	−25408.0	−1.46507
$671$	4731.37	0.272209
$672$	0	0
$673$	−11475.4	−0.657271	−0.328635	−	0.944457i	$-0.606589\pi$
$673$	−11475.4	−0.657271	−0.328635	+	0.944457i	$0.606589\pi$
$674$	−8540.24	−0.488068
$675$	0	0
$676$	138.003	0.00785179
$677$	−27237.9	−1.54629	−0.773145	−	0.634229i	$-0.781317\pi$
$677$	−27237.9	−1.54629	−0.773145	+	0.634229i	$0.781317\pi$
$678$	0	0
$679$	5806.70	0.328190
$680$	−16896.8	−0.952885
$681$	0	0
$682$	−2826.43	−0.158694
$683$	28388.1	1.59039	0.795197	−	0.606351i	$-0.207367\pi$
$683$	28388.1	1.59039	0.795197	+	0.606351i	$0.207367\pi$
$684$	0	0
$685$	−40921.3	−2.28251
$686$	20386.5	1.13463
$687$	0	0
$688$	21180.2	1.17367
$689$	13966.1	0.772230
$690$	0	0
$691$	−16140.1	−0.888565	−0.444283	−	0.895887i	$-0.646541\pi$
$691$	−16140.1	−0.888565	−0.444283	+	0.895887i	$0.646541\pi$
$692$	628.194	0.0345092
$693$	0	0
$694$	10625.2	0.581161
$695$	7496.36	0.409141
$696$	0	0
$697$	−5833.11	−0.316994
$698$	−14491.2	−0.785815
$699$	0	0
$700$	−3255.07	−0.175757
$701$	16564.9	0.892506	0.446253	−	0.894907i	$-0.352758\pi$
$701$	16564.9	0.892506	0.446253	+	0.894907i	$0.352758\pi$
$702$	0	0
$703$	−1538.14	−0.0825205
$704$	−3592.54	−0.192328
$705$	0	0
$706$	−17427.8	−0.929041
$707$	−45105.3	−2.39937
$708$	0	0
$709$	−31152.4	−1.65014	−0.825072	−	0.565028i	$-0.808865\pi$
$709$	−31152.4	−1.65014	−0.825072	+	0.565028i	$0.808865\pi$
$710$	−12099.6	−0.639565
$711$	0	0
$712$	17445.1	0.918236
$713$	15980.9	0.839395
$714$	0	0
$715$	6368.60	0.333108
$716$	2215.92	0.115660
$717$	0	0
$718$	10529.3	0.547284
$719$	−21689.5	−1.12501	−0.562505	−	0.826794i	$-0.690162\pi$
$719$	−21689.5	−1.12501	−0.562505	+	0.826794i	$0.690162\pi$
$720$	0	0
$721$	38778.2	2.00302
$722$	13202.8	0.680548
$723$	0	0
$724$	−434.274	−0.0222924
$725$	22089.4	1.13156
$726$	0	0
$727$	−37929.0	−1.93495	−0.967475	−	0.252968i	$-0.918593\pi$
$727$	−37929.0	−1.93495	−0.967475	+	0.252968i	$0.918593\pi$
$728$	−31817.7	−1.61984
$729$	0	0
$730$	−43299.9	−2.19534
$731$	−14455.2	−0.731388
$732$	0	0
$733$	30699.1	1.54693	0.773464	−	0.633840i	$-0.218522\pi$
$733$	30699.1	1.54693	0.773464	+	0.633840i	$0.218522\pi$
$734$	−8497.89	−0.427334
$735$	0	0
$736$	−4182.13	−0.209450
$737$	4051.71	0.202506
$738$	0	0
$739$	20883.5	1.03953	0.519764	−	0.854310i	$-0.326020\pi$
$739$	20883.5	1.03953	0.519764	+	0.854310i	$0.326020\pi$
$740$	371.691	0.0184644
$741$	0	0
$742$	25486.6	1.26097
$743$	97.0268	0.00479080	0.00239540	−	0.999997i	$-0.499238\pi$
$743$	97.0268	0.00479080	0.00239540	+	0.999997i	$0.499238\pi$
$744$	0	0
$745$	−7155.49	−0.351889
$746$	2261.93	0.111012
$747$	0	0
$748$	−262.210	−0.0128173
$749$	5716.19	0.278859
$750$	0	0
$751$	13681.8	0.664787	0.332394	−	0.943141i	$-0.392144\pi$
$751$	13681.8	0.664787	0.332394	+	0.943141i	$0.392144\pi$
$752$	9913.11	0.480710
$753$	0	0
$754$	−21012.0	−1.01487
$755$	−56789.2	−2.73744
$756$	0	0
$757$	−26818.7	−1.28764	−0.643819	−	0.765178i	$-0.722651\pi$
$757$	−26818.7	−1.28764	−0.643819	+	0.765178i	$0.722651\pi$
$758$	15830.1	0.758545
$759$	0	0
$760$	−17480.2	−0.834307
$761$	−6888.48	−0.328130	−0.164065	−	0.986450i	$-0.552461\pi$
$761$	−6888.48	−0.328130	−0.164065	+	0.986450i	$0.552461\pi$
$762$	0	0
$763$	31010.9	1.47139
$764$	−2589.47	−0.122623
$765$	0	0
$766$	−4493.61	−0.211959
$767$	−12763.4	−0.600858
$768$	0	0
$769$	−12475.7	−0.585028	−0.292514	−	0.956261i	$-0.594492\pi$
$769$	−12475.7	−0.585028	−0.292514	+	0.956261i	$0.594492\pi$
$770$	11622.0	0.543932
$771$	0	0
$772$	53.9773	0.00251643
$773$	−32256.3	−1.50088	−0.750439	−	0.660940i	$-0.770158\pi$
$773$	−32256.3	−1.50088	−0.750439	+	0.660940i	$0.770158\pi$
$774$	0	0
$775$	−18561.1	−0.860303
$776$	−4131.50	−0.191124
$777$	0	0
$778$	6710.22	0.309220
$779$	−6034.51	−0.277547
$780$	0	0
$781$	1929.48	0.0884024
$782$	18199.8	0.832257
$783$	0	0
$784$	−39527.8	−1.80065
$785$	−14379.1	−0.653775
$786$	0	0
$787$	37930.8	1.71803	0.859014	−	0.511952i	$-0.171078\pi$
$787$	37930.8	1.71803	0.859014	+	0.511952i	$0.171078\pi$
$788$	−2553.76	−0.115449
$789$	0	0
$790$	−57200.1	−2.57606
$791$	16014.0	0.719840
$792$	0	0
$793$	−29555.4	−1.32351
$794$	30316.7	1.35504
$795$	0	0
$796$	−2813.16	−0.125264
$797$	5478.85	0.243502	0.121751	−	0.992561i	$-0.461149\pi$
$797$	5478.85	0.243502	0.121751	+	0.992561i	$0.461149\pi$
$798$	0	0
$799$	−6765.57	−0.299560
$800$	4857.37	0.214668
$801$	0	0
$802$	−42516.5	−1.87196
$803$	6904.87	0.303446
$804$	0	0
$805$	−65711.7	−2.87706
$806$	17655.9	0.771590
$807$	0	0
$808$	32092.6	1.39729
$809$	15063.1	0.654623	0.327312	−	0.944916i	$-0.393857\pi$
$809$	15063.1	0.654623	0.327312	+	0.944916i	$0.393857\pi$
$810$	0	0
$811$	5485.13	0.237496	0.118748	−	0.992924i	$-0.462112\pi$
$811$	5485.13	0.237496	0.118748	+	0.992924i	$0.462112\pi$
$812$	−3123.55	−0.134994
$813$	0	0
$814$	−727.623	−0.0313307
$815$	−17725.3	−0.761827
$816$	0	0
$817$	−14954.3	−0.640373
$818$	−16674.0	−0.712707
$819$	0	0
$820$	1458.24	0.0621024
$821$	−1055.15	−0.0448540	−0.0224270	−	0.999748i	$-0.507139\pi$
$821$	−1055.15	−0.0448540	−0.0224270	+	0.999748i	$0.507139\pi$
$822$	0	0
$823$	−17419.3	−0.737785	−0.368893	−	0.929472i	$-0.620263\pi$
$823$	−17419.3	−0.737785	−0.368893	+	0.929472i	$0.620263\pi$
$824$	−27590.9	−1.16647
$825$	0	0
$826$	−23291.7	−0.981141
$827$	1379.47	0.0580033	0.0290016	−	0.999579i	$-0.490767\pi$
$827$	1379.47	0.0580033	0.0290016	+	0.999579i	$0.490767\pi$
$828$	0	0
$829$	−32409.1	−1.35780	−0.678900	−	0.734231i	$-0.737543\pi$
$829$	−32409.1	−1.35780	−0.678900	+	0.734231i	$0.737543\pi$
$830$	−1710.29	−0.0715243
$831$	0	0
$832$	22441.5	0.935119
$833$	26977.2	1.12209
$834$	0	0
$835$	5496.78	0.227813
$836$	−271.263	−0.0112223
$837$	0	0
$838$	11686.2	0.481734
$839$	19286.0	0.793594	0.396797	−	0.917906i	$-0.370122\pi$
$839$	19286.0	0.793594	0.396797	+	0.917906i	$0.370122\pi$
$840$	0	0
$841$	−3192.17	−0.130886
$842$	−31936.2	−1.30712
$843$	0	0
$844$	223.741	0.00912499
$845$	−3235.75	−0.131732
$846$	0	0
$847$	38395.7	1.55761
$848$	−19754.8	−0.799979
$849$	0	0
$850$	−21138.4	−0.852988
$851$	4114.04	0.165720
$852$	0	0
$853$	−9711.21	−0.389807	−0.194904	−	0.980822i	$-0.562439\pi$
$853$	−9711.21	−0.389807	−0.194904	+	0.980822i	$0.562439\pi$
$854$	−53935.4	−2.16116
$855$	0	0
$856$	−4067.10	−0.162395
$857$	−25747.9	−1.02629	−0.513146	−	0.858301i	$-0.671520\pi$
$857$	−25747.9	−1.02629	−0.513146	+	0.858301i	$0.671520\pi$
$858$	0	0
$859$	−1199.86	−0.0476584	−0.0238292	−	0.999716i	$-0.507586\pi$
$859$	−1199.86	−0.0476584	−0.0238292	+	0.999716i	$0.507586\pi$
$860$	3613.71	0.143287
$861$	0	0
$862$	11497.3	0.454292
$863$	1332.02	0.0525407	0.0262704	−	0.999655i	$-0.491637\pi$
$863$	1332.02	0.0525407	0.0262704	+	0.999655i	$0.491637\pi$
$864$	0	0
$865$	−14729.2	−0.578970
$866$	−1339.25	−0.0525515
$867$	0	0
$868$	2624.64	0.102634
$869$	9121.48	0.356070
$870$	0	0
$871$	−25309.8	−0.984604
$872$	−22064.4	−0.856876
$873$	0	0
$874$	18828.2	0.728690
$875$	13442.0	0.519339
$876$	0	0
$877$	20443.6	0.787152	0.393576	−	0.919292i	$-0.371238\pi$
$877$	20443.6	0.787152	0.393576	+	0.919292i	$0.371238\pi$
$878$	10709.4	0.411647
$879$	0	0
$880$	−9008.25	−0.345078
$881$	−30891.8	−1.18135	−0.590676	−	0.806909i	$-0.701139\pi$
$881$	−30891.8	−1.18135	−0.590676	+	0.806909i	$0.701139\pi$
$882$	0	0
$883$	35760.8	1.36291	0.681454	−	0.731861i	$-0.261348\pi$
$883$	35760.8	1.36291	0.681454	+	0.731861i	$0.261348\pi$
$884$	1637.95	0.0623191
$885$	0	0
$886$	−1683.68	−0.0638425
$887$	−42438.8	−1.60649	−0.803245	−	0.595649i	$-0.796895\pi$
$887$	−42438.8	−1.60649	−0.803245	+	0.595649i	$0.796895\pi$
$888$	0	0
$889$	6999.03	0.264050
$890$	39804.9	1.49917
$891$	0	0
$892$	−3031.98	−0.113810
$893$	−6999.17	−0.262283
$894$	0	0
$895$	−51956.5	−1.94046
$896$	48698.3	1.81573
$897$	0	0
$898$	18412.2	0.684215
$899$	−17811.1	−0.660773
$900$	0	0
$901$	13482.4	0.498516
$902$	−2854.66	−0.105377
$903$	0	0
$904$	−11394.1	−0.419205
$905$	10182.4	0.374005
$906$	0	0
$907$	−3380.87	−0.123770	−0.0618852	−	0.998083i	$-0.519711\pi$
$907$	−3380.87	−0.123770	−0.0618852	+	0.998083i	$0.519711\pi$
$908$	−3005.25	−0.109838
$909$	0	0
$910$	−72599.1	−2.64466
$911$	17353.3	0.631111	0.315555	−	0.948907i	$-0.397809\pi$
$911$	17353.3	0.631111	0.315555	+	0.948907i	$0.397809\pi$
$912$	0	0
$913$	272.734	0.00988629
$914$	10239.7	0.370568
$915$	0	0
$916$	4592.49	0.165655
$917$	80936.3	2.91467
$918$	0	0
$919$	−3785.66	−0.135884	−0.0679420	−	0.997689i	$-0.521643\pi$
$919$	−3785.66	−0.135884	−0.0679420	+	0.997689i	$0.521643\pi$
$920$	46754.2	1.67548
$921$	0	0
$922$	−25531.4	−0.911966
$923$	−12052.9	−0.429822
$924$	0	0
$925$	−4778.29	−0.169848
$926$	38595.0	1.36966
$927$	0	0
$928$	4661.11	0.164880
$929$	−3561.15	−0.125767	−0.0628835	−	0.998021i	$-0.520030\pi$
$929$	−3561.15	−0.125767	−0.0628835	+	0.998021i	$0.520030\pi$
$930$	0	0
$931$	27908.7	0.982460
$932$	1875.99	0.0659336
$933$	0	0
$934$	−48615.1	−1.70314
$935$	6148.02	0.215039
$936$	0	0
$937$	7357.98	0.256536	0.128268	−	0.991740i	$-0.459058\pi$
$937$	7357.98	0.256536	0.128268	+	0.991740i	$0.459058\pi$
$938$	−46187.6	−1.60776
$939$	0	0
$940$	1691.35	0.0586870
$941$	8221.73	0.284826	0.142413	−	0.989807i	$-0.454514\pi$
$941$	8221.73	0.284826	0.142413	+	0.989807i	$0.454514\pi$
$942$	0	0
$943$	16140.5	0.557377
$944$	18053.5	0.622449
$945$	0	0
$946$	−7074.21	−0.243132
$947$	−22232.7	−0.762899	−0.381449	−	0.924390i	$-0.624575\pi$
$947$	−22232.7	−0.762899	−0.381449	+	0.924390i	$0.624575\pi$
$948$	0	0
$949$	−43132.6	−1.47539
$950$	−21868.2	−0.746841
$951$	0	0
$952$	−30715.6	−1.04569
$953$	5855.90	0.199047	0.0995233	−	0.995035i	$-0.468268\pi$
$953$	5855.90	0.199047	0.0995233	+	0.995035i	$0.468268\pi$
$954$	0	0
$955$	60715.1	2.05727
$956$	−169.564	−0.00573649
$957$	0	0
$958$	−38667.5	−1.30406
$959$	−74388.3	−2.50482
$960$	0	0
$961$	−14824.8	−0.497625
$962$	4545.24	0.152333
$963$	0	0
$964$	−1532.14	−0.0511897
$965$	−1265.60	−0.0422189
$966$	0	0
$967$	26083.7	0.867421	0.433710	−	0.901052i	$-0.357204\pi$
$967$	26083.7	0.867421	0.433710	+	0.901052i	$0.357204\pi$
$968$	−27318.7	−0.907083
$969$	0	0
$970$	−9426.92	−0.312041
$971$	34276.3	1.13283	0.566415	−	0.824120i	$-0.308330\pi$
$971$	34276.3	1.13283	0.566415	+	0.824120i	$0.308330\pi$
$972$	0	0
$973$	13627.2	0.448990
$974$	−8272.04	−0.272129
$975$	0	0
$976$	41805.5	1.37107
$977$	−11390.8	−0.373002	−0.186501	−	0.982455i	$-0.559715\pi$
$977$	−11390.8	−0.373002	−0.186501	+	0.982455i	$0.559715\pi$
$978$	0	0
$979$	−6347.54	−0.207220
$980$	−6744.13	−0.219830
$981$	0	0
$982$	58483.8	1.90050
$983$	−49563.1	−1.60815	−0.804077	−	0.594525i	$-0.797340\pi$
$983$	−49563.1	−1.60815	−0.804077	+	0.594525i	$0.797340\pi$
$984$	0	0
$985$	59877.9	1.93692
$986$	−20284.2	−0.655154
$987$	0	0
$988$	1694.50	0.0545640
$989$	39998.2	1.28601
$990$	0	0
$991$	31276.5	1.00255	0.501277	−	0.865287i	$-0.332864\pi$
$991$	31276.5	1.00255	0.501277	+	0.865287i	$0.332864\pi$
$992$	−3916.61	−0.125355
$993$	0	0
$994$	−21995.2	−0.701856
$995$	65960.1	2.10158
$996$	0	0
$997$	22856.0	0.726035	0.363018	−	0.931782i	$-0.381747\pi$
$997$	22856.0	0.726035	0.363018	+	0.931782i	$0.381747\pi$
$998$	−45517.9	−1.44373
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.h.1.17	yes	59
3.2	odd	2		2151.4.a.g.1.43	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.43	✓	59	3.2	odd	2
2151.4.a.h.1.17	yes	59	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)

\(f(q)\)	\(=\)	\(q-2.95118 q^{2} +0.709472 q^{4} -16.6350 q^{5} -30.2397 q^{7} +21.5157 q^{8} +O(q^{10})\)	\(q-2.95118 q^{2} +0.709472 q^{4} -16.6350 q^{5} -30.2397 q^{7} +21.5157 q^{8} +49.0928 q^{10} -7.82864 q^{11} +48.9031 q^{13} +89.2428 q^{14} -69.1724 q^{16} +47.2093 q^{17} +48.8394 q^{19} -11.8020 q^{20} +23.1037 q^{22} -130.630 q^{23} +151.722 q^{25} -144.322 q^{26} -21.4542 q^{28} +145.591 q^{29} -122.337 q^{31} +32.0150 q^{32} -139.323 q^{34} +503.036 q^{35} -31.4938 q^{37} -144.134 q^{38} -357.912 q^{40} -123.558 q^{41} -306.194 q^{43} -5.55420 q^{44} +385.514 q^{46} -143.310 q^{47} +571.438 q^{49} -447.758 q^{50} +34.6954 q^{52} +285.587 q^{53} +130.229 q^{55} -650.627 q^{56} -429.666 q^{58} -260.993 q^{59} -604.367 q^{61} +361.038 q^{62} +458.897 q^{64} -813.501 q^{65} -517.550 q^{67} +33.4937 q^{68} -1484.55 q^{70} -246.465 q^{71} -882.001 q^{73} +92.9438 q^{74} +34.6502 q^{76} +236.735 q^{77} -1165.14 q^{79} +1150.68 q^{80} +364.643 q^{82} -34.8380 q^{83} -785.324 q^{85} +903.633 q^{86} -168.438 q^{88} +810.810 q^{89} -1478.81 q^{91} -92.6786 q^{92} +422.934 q^{94} -812.440 q^{95} -192.023 q^{97} -1686.42 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8}+O(q^{10}) \) 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8	\( 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8} - 36 q^{10} + 132 q^{11} + 104 q^{13} + 280 q^{14} + 822 q^{16} + 408 q^{17} + 20 q^{19} + 800 q^{20} - 2 q^{22} + 276 q^{23} + 1477 q^{25} + 780 q^{26} + 224 q^{28} + 696 q^{29} - 380 q^{31} + 896 q^{32} - 72 q^{34} + 700 q^{35} + 224 q^{37} + 988 q^{38} - 258 q^{40} + 2706 q^{41} - 156 q^{43} + 1584 q^{44} + 428 q^{46} + 1316 q^{47} + 2135 q^{49} + 1400 q^{50} + 1092 q^{52} + 1484 q^{53} - 992 q^{55} + 3360 q^{56} - 120 q^{58} + 3186 q^{59} - 254 q^{61} + 1240 q^{62} + 3054 q^{64} + 5120 q^{65} + 288 q^{67} + 9420 q^{68} + 1108 q^{70} + 4468 q^{71} - 1770 q^{73} + 6214 q^{74} + 720 q^{76} + 6352 q^{77} - 746 q^{79} + 7040 q^{80} + 276 q^{82} + 5484 q^{83} + 588 q^{85} + 10152 q^{86} + 1186 q^{88} + 11570 q^{89} + 1768 q^{91} + 15366 q^{92} - 2142 q^{94} + 5736 q^{95} + 2390 q^{97} + 6912 q^{98}+O(q^{100}) \) 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8 - 36 * q^10 + 132 * q^11 + 104 * q^13 + 280 * q^14 + 822 * q^16 + 408 * q^17 + 20 * q^19 + 800 * q^20 - 2 * q^22 + 276 * q^23 + 1477 * q^25 + 780 * q^26 + 224 * q^28 + 696 * q^29 - 380 * q^31 + 896 * q^32 - 72 * q^34 + 700 * q^35 + 224 * q^37 + 988 * q^38 - 258 * q^40 + 2706 * q^41 - 156 * q^43 + 1584 * q^44 + 428 * q^46 + 1316 * q^47 + 2135 * q^49 + 1400 * q^50 + 1092 * q^52 + 1484 * q^53 - 992 * q^55 + 3360 * q^56 - 120 * q^58 + 3186 * q^59 - 254 * q^61 + 1240 * q^62 + 3054 * q^64 + 5120 * q^65 + 288 * q^67 + 9420 * q^68 + 1108 * q^70 + 4468 * q^71 - 1770 * q^73 + 6214 * q^74 + 720 * q^76 + 6352 * q^77 - 746 * q^79 + 7040 * q^80 + 276 * q^82 + 5484 * q^83 + 588 * q^85 + 10152 * q^86 + 1186 * q^88 + 11570 * q^89 + 1768 * q^91 + 15366 * q^92 - 2142 * q^94 + 5736 * q^95 + 2390 * q^97 + 6912 * q^98

Introduction

L-functions

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