LMFDB - Embedded newform 2151.4.a.g.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$126.913108422$
Analytic rank:	$1$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.36448 q^{2} +3.31971 q^{4} -9.29992 q^{5} +3.75868 q^{7} +15.7467 q^{8} +O(q^{10})$	\(q-3.36448 q^{2} +3.31971 q^{4} -9.29992 q^{5} +3.75868 q^{7} +15.7467 q^{8} +31.2894 q^{10} -69.7242 q^{11} +86.1163 q^{13} -12.6460 q^{14} -79.5372 q^{16} -91.3877 q^{17} -53.6446 q^{19} -30.8730 q^{20} +234.586 q^{22} +144.473 q^{23} -38.5114 q^{25} -289.736 q^{26} +12.4777 q^{28} +20.4126 q^{29} +303.641 q^{31} +141.627 q^{32} +307.472 q^{34} -34.9554 q^{35} +182.952 q^{37} +180.486 q^{38} -146.443 q^{40} -191.408 q^{41} -309.055 q^{43} -231.464 q^{44} -486.076 q^{46} -7.77095 q^{47} -328.872 q^{49} +129.571 q^{50} +285.881 q^{52} +84.7531 q^{53} +648.430 q^{55} +59.1870 q^{56} -68.6778 q^{58} -37.7112 q^{59} -133.772 q^{61} -1021.59 q^{62} +159.796 q^{64} -800.875 q^{65} +334.841 q^{67} -303.380 q^{68} +117.607 q^{70} +632.410 q^{71} +715.473 q^{73} -615.539 q^{74} -178.085 q^{76} -262.071 q^{77} -1288.80 q^{79} +739.690 q^{80} +643.988 q^{82} -493.990 q^{83} +849.898 q^{85} +1039.81 q^{86} -1097.93 q^{88} +20.7376 q^{89} +323.684 q^{91} +479.608 q^{92} +26.1452 q^{94} +498.891 q^{95} +1185.12 q^{97} +1106.48 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−3.36448	−1.18952	−0.594761	−	0.803902i	$-0.702753\pi$
$2$	−3.36448	−1.18952	−0.594761	+	0.803902i	$0.702753\pi$
$3$	0	0
$3$	0	0
$4$	3.31971	0.414964
$5$	−9.29992	−0.831810	−0.415905	−	0.909408i	$-0.636535\pi$
$5$	−9.29992	−0.831810	−0.415905	+	0.909408i	$0.636535\pi$
$6$	0	0
$7$	3.75868	0.202950	0.101475	−	0.994838i	$-0.467644\pi$
$7$	3.75868	0.202950	0.101475	+	0.994838i	$0.467644\pi$
$8$	15.7467	0.695914
$9$	0	0
$10$	31.2894	0.989457
$11$	−69.7242	−1.91115	−0.955575	−	0.294749i	$-0.904764\pi$
$11$	−69.7242	−1.91115	−0.955575	+	0.294749i	$0.904764\pi$
$12$	0	0
$13$	86.1163	1.83726	0.918629	−	0.395121i	$-0.129297\pi$
$13$	86.1163	1.83726	0.918629	+	0.395121i	$0.129297\pi$
$14$	−12.6460	−0.241413
$15$	0	0
$16$	−79.5372	−1.24277
$17$	−91.3877	−1.30381	−0.651905	−	0.758301i	$-0.726030\pi$
$17$	−91.3877	−1.30381	−0.651905	+	0.758301i	$0.726030\pi$
$18$	0	0
$19$	−53.6446	−0.647733	−0.323866	−	0.946103i	$-0.604983\pi$
$19$	−53.6446	−0.647733	−0.323866	+	0.946103i	$0.604983\pi$
$20$	−30.8730	−0.345171
$21$	0	0
$22$	234.586	2.27336
$23$	144.473	1.30977	0.654885	−	0.755729i	$-0.272717\pi$
$23$	144.473	1.30977	0.654885	+	0.755729i	$0.272717\pi$
$24$	0	0
$25$	−38.5114	−0.308091
$26$	−289.736	−2.18546
$27$	0	0
$28$	12.4777	0.0842167
$29$	20.4126	0.130708	0.0653539	−	0.997862i	$-0.479182\pi$
$29$	20.4126	0.130708	0.0653539	+	0.997862i	$0.479182\pi$
$30$	0	0
$31$	303.641	1.75921	0.879606	−	0.475702i	$-0.157806\pi$
$31$	303.641	1.75921	0.879606	+	0.475702i	$0.157806\pi$
$32$	141.627	0.782387
$33$	0	0
$34$	307.472	1.55091
$35$	−34.9554	−0.168816
$36$	0	0
$37$	182.952	0.812897	0.406449	−	0.913674i	$-0.366767\pi$
$37$	182.952	0.812897	0.406449	+	0.913674i	$0.366767\pi$
$38$	180.486	0.770493
$39$	0	0
$40$	−146.443	−0.578868
$41$	−191.408	−0.729096	−0.364548	−	0.931185i	$-0.618776\pi$
$41$	−191.408	−0.729096	−0.364548	+	0.931185i	$0.618776\pi$
$42$	0	0
$43$	−309.055	−1.09606	−0.548028	−	0.836460i	$-0.684621\pi$
$43$	−309.055	−1.09606	−0.548028	+	0.836460i	$0.684621\pi$
$44$	−231.464	−0.793057
$45$	0	0
$46$	−486.076	−1.55800
$47$	−7.77095	−0.0241172	−0.0120586	−	0.999927i	$-0.503838\pi$
$47$	−7.77095	−0.0241172	−0.0120586	+	0.999927i	$0.503838\pi$
$48$	0	0
$49$	−328.872	−0.958811
$50$	129.571	0.366482
$51$	0	0
$52$	285.881	0.762395
$53$	84.7531	0.219655	0.109828	−	0.993951i	$-0.464970\pi$
$53$	84.7531	0.219655	0.109828	+	0.993951i	$0.464970\pi$
$54$	0	0
$55$	648.430	1.58971
$56$	59.1870	0.141236
$57$	0	0
$58$	−68.6778	−0.155480
$59$	−37.7112	−0.0832133	−0.0416067	−	0.999134i	$-0.513248\pi$
$59$	−37.7112	−0.0832133	−0.0416067	+	0.999134i	$0.513248\pi$
$60$	0	0
$61$	−133.772	−0.280782	−0.140391	−	0.990096i	$-0.544836\pi$
$61$	−133.772	−0.280782	−0.140391	+	0.990096i	$0.544836\pi$
$62$	−1021.59	−2.09262
$63$	0	0
$64$	159.796	0.312101
$65$	−800.875	−1.52825
$66$	0	0
$67$	334.841	0.610558	0.305279	−	0.952263i	$-0.401250\pi$
$67$	334.841	0.610558	0.305279	+	0.952263i	$0.401250\pi$
$68$	−303.380	−0.541034
$69$	0	0
$70$	117.607	0.200810
$71$	632.410	1.05709	0.528544	−	0.848906i	$-0.322738\pi$
$71$	632.410	1.05709	0.528544	+	0.848906i	$0.322738\pi$
$72$	0	0
$73$	715.473	1.14712	0.573560	−	0.819164i	$-0.305562\pi$
$73$	715.473	1.14712	0.573560	+	0.819164i	$0.305562\pi$
$74$	−615.539	−0.966959
$75$	0	0
$76$	−178.085	−0.268786
$77$	−262.071	−0.387867
$78$	0	0
$79$	−1288.80	−1.83546	−0.917729	−	0.397207i	$-0.869979\pi$
$79$	−1288.80	−1.83546	−0.917729	+	0.397207i	$0.869979\pi$
$80$	739.690	1.03375
$81$	0	0
$82$	643.988	0.867276
$83$	−493.990	−0.653282	−0.326641	−	0.945148i	$-0.605917\pi$
$83$	−493.990	−0.653282	−0.326641	+	0.945148i	$0.605917\pi$
$84$	0	0
$85$	849.898	1.08452
$86$	1039.81	1.30378
$87$	0	0
$88$	−1097.93	−1.33000
$89$	20.7376	0.0246986	0.0123493	−	0.999924i	$-0.496069\pi$
$89$	20.7376	0.0246986	0.0123493	+	0.999924i	$0.496069\pi$
$90$	0	0
$91$	323.684	0.372871
$92$	479.608	0.543507
$93$	0	0
$94$	26.1452	0.0286880
$95$	498.891	0.538791
$96$	0	0
$97$	1185.12	1.24052	0.620262	−	0.784395i	$-0.287026\pi$
$97$	1185.12	1.24052	0.620262	+	0.784395i	$0.287026\pi$
$98$	1106.48	1.14053
$99$	0	0
$100$	−127.847	−0.127847
$101$	1447.64	1.42619	0.713095	−	0.701068i	$-0.247293\pi$
$101$	1447.64	1.42619	0.713095	+	0.701068i	$0.247293\pi$
$102$	0	0
$103$	213.430	0.204174	0.102087	−	0.994775i	$-0.467448\pi$
$103$	213.430	0.204174	0.102087	+	0.994775i	$0.467448\pi$
$104$	1356.05	1.27857
$105$	0	0
$106$	−285.150	−0.261285
$107$	−593.745	−0.536444	−0.268222	−	0.963357i	$-0.586436\pi$
$107$	−593.745	−0.536444	−0.268222	+	0.963357i	$0.586436\pi$
$108$	0	0
$109$	1436.05	1.26191	0.630957	−	0.775818i	$-0.282662\pi$
$109$	1436.05	1.26191	0.630957	+	0.775818i	$0.282662\pi$
$110$	−2181.63	−1.89100
$111$	0	0
$112$	−298.955	−0.252220
$113$	−763.494	−0.635606	−0.317803	−	0.948157i	$-0.602945\pi$
$113$	−763.494	−0.635606	−0.317803	+	0.948157i	$0.602945\pi$
$114$	0	0
$115$	−1343.59	−1.08948
$116$	67.7639	0.0542390
$117$	0	0
$118$	126.879	0.0989841
$119$	−343.497	−0.264608
$120$	0	0
$121$	3530.47	2.65249
$122$	450.071	0.333996
$123$	0	0
$124$	1008.00	0.730009
$125$	1520.64	1.08808
$126$	0	0
$127$	657.518	0.459412	0.229706	−	0.973260i	$-0.426224\pi$
$127$	657.518	0.459412	0.229706	+	0.973260i	$0.426224\pi$
$128$	−1670.65	−1.15364
$129$	0	0
$130$	2694.52	1.81789
$131$	2555.37	1.70430	0.852151	−	0.523296i	$-0.175298\pi$
$131$	2555.37	1.70430	0.852151	+	0.523296i	$0.175298\pi$
$132$	0	0
$133$	−201.633	−0.131457
$134$	−1126.57	−0.726272
$135$	0	0
$136$	−1439.06	−0.907340
$137$	−2748.77	−1.71418	−0.857092	−	0.515163i	$-0.827732\pi$
$137$	−2748.77	−1.71418	−0.857092	+	0.515163i	$0.827732\pi$
$138$	0	0
$139$	2086.61	1.27326	0.636632	−	0.771168i	$-0.280327\pi$
$139$	2086.61	1.27326	0.636632	+	0.771168i	$0.280327\pi$
$140$	−116.042	−0.0700524
$141$	0	0
$142$	−2127.73	−1.25743
$143$	−6004.39	−3.51128
$144$	0	0
$145$	−189.836	−0.108724
$146$	−2407.19	−1.36452
$147$	0	0
$148$	607.349	0.337323
$149$	−309.085	−0.169941	−0.0849706	−	0.996383i	$-0.527080\pi$
$149$	−309.085	−0.169941	−0.0849706	+	0.996383i	$0.527080\pi$
$150$	0	0
$151$	168.941	0.0910479	0.0455239	−	0.998963i	$-0.485504\pi$
$151$	168.941	0.0910479	0.0455239	+	0.998963i	$0.485504\pi$
$152$	−844.728	−0.450766
$153$	0	0
$154$	881.732	0.461377
$155$	−2823.84	−1.46333
$156$	0	0
$157$	−2971.25	−1.51039	−0.755195	−	0.655500i	$-0.772458\pi$
$157$	−2971.25	−1.51039	−0.755195	+	0.655500i	$0.772458\pi$
$158$	4336.13	2.18332
$159$	0	0
$160$	−1317.12	−0.650798
$161$	543.028	0.265817
$162$	0	0
$163$	3283.31	1.57772	0.788861	−	0.614572i	$-0.210671\pi$
$163$	3283.31	1.57772	0.788861	+	0.614572i	$0.210671\pi$
$164$	−635.419	−0.302548
$165$	0	0
$166$	1662.02	0.777094
$167$	427.973	0.198309	0.0991544	−	0.995072i	$-0.468386\pi$
$167$	427.973	0.198309	0.0991544	+	0.995072i	$0.468386\pi$
$168$	0	0
$169$	5219.01	2.37552
$170$	−2859.46	−1.29006
$171$	0	0
$172$	−1025.97	−0.454824
$173$	2158.39	0.948552	0.474276	−	0.880376i	$-0.342710\pi$
$173$	2158.39	0.948552	0.474276	+	0.880376i	$0.342710\pi$
$174$	0	0
$175$	−144.752	−0.0625271
$176$	5545.67	2.37512
$177$	0	0
$178$	−69.7711	−0.0293796
$179$	−3076.65	−1.28469	−0.642344	−	0.766416i	$-0.722038\pi$
$179$	−3076.65	−1.28469	−0.642344	+	0.766416i	$0.722038\pi$
$180$	0	0
$181$	−3177.54	−1.30489	−0.652445	−	0.757836i	$-0.726257\pi$
$181$	−3177.54	−1.30489	−0.652445	+	0.757836i	$0.726257\pi$
$182$	−1089.03	−0.443538
$183$	0	0
$184$	2274.98	0.911487
$185$	−1701.44	−0.676176
$186$	0	0
$187$	6371.94	2.49178
$188$	−25.7973	−0.0100078
$189$	0	0
$190$	−1678.51	−0.640904
$191$	567.793	0.215100	0.107550	−	0.994200i	$-0.465699\pi$
$191$	567.793	0.215100	0.107550	+	0.994200i	$0.465699\pi$
$192$	0	0
$193$	3223.88	1.20238	0.601192	−	0.799105i	$-0.294693\pi$
$193$	3223.88	1.20238	0.601192	+	0.799105i	$0.294693\pi$
$194$	−3987.31	−1.47563
$195$	0	0
$196$	−1091.76	−0.397872
$197$	−1695.29	−0.613119	−0.306560	−	0.951851i	$-0.599178\pi$
$197$	−1695.29	−0.613119	−0.306560	+	0.951851i	$0.599178\pi$
$198$	0	0
$199$	810.760	0.288810	0.144405	−	0.989519i	$-0.453873\pi$
$199$	810.760	0.288810	0.144405	+	0.989519i	$0.453873\pi$
$200$	−606.429	−0.214405
$201$	0	0
$202$	−4870.54	−1.69648
$203$	76.7245	0.0265271
$204$	0	0
$205$	1780.08	0.606470
$206$	−718.080	−0.242869
$207$	0	0
$208$	−6849.45	−2.28329
$209$	3740.33	1.23791
$210$	0	0
$211$	−5619.02	−1.83331	−0.916657	−	0.399675i	$-0.869123\pi$
$211$	−5619.02	−1.83331	−0.916657	+	0.399675i	$0.869123\pi$
$212$	281.355	0.0911489
$213$	0	0
$214$	1997.64	0.638112
$215$	2874.19	0.911712
$216$	0	0
$217$	1141.29	0.357032
$218$	−4831.56	−1.50108
$219$	0	0
$220$	2152.60	0.659673
$221$	−7869.97	−2.39544
$222$	0	0
$223$	−147.746	−0.0443667	−0.0221833	−	0.999754i	$-0.507062\pi$
$223$	−147.746	−0.0443667	−0.0221833	+	0.999754i	$0.507062\pi$
$224$	532.332	0.158785
$225$	0	0
$226$	2568.76	0.756068
$227$	−2761.20	−0.807344	−0.403672	−	0.914904i	$-0.632266\pi$
$227$	−2761.20	−0.807344	−0.403672	+	0.914904i	$0.632266\pi$
$228$	0	0
$229$	5111.04	1.47488	0.737439	−	0.675414i	$-0.236035\pi$
$229$	5111.04	1.47488	0.737439	+	0.675414i	$0.236035\pi$
$230$	4520.47	1.29596
$231$	0	0
$232$	321.432	0.0909614
$233$	186.917	0.0525552	0.0262776	−	0.999655i	$-0.491635\pi$
$233$	186.917	0.0525552	0.0262776	+	0.999655i	$0.491635\pi$
$234$	0	0
$235$	72.2692	0.0200609
$236$	−125.190	−0.0345305
$237$	0	0
$238$	1155.69	0.314757
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−5391.23	−1.44100	−0.720498	−	0.693457i	$-0.756087\pi$
$241$	−5391.23	−1.44100	−0.720498	+	0.693457i	$0.756087\pi$
$242$	−11878.2	−3.15520
$243$	0	0
$244$	−444.083	−0.116514
$245$	3058.49	0.797549
$246$	0	0
$247$	−4619.68	−1.19005
$248$	4781.36	1.22426
$249$	0	0
$250$	−5116.17	−1.29430
$251$	6595.87	1.65868	0.829338	−	0.558748i	$-0.188718\pi$
$251$	6595.87	1.65868	0.829338	+	0.558748i	$0.188718\pi$
$252$	0	0
$253$	−10073.3	−2.50317
$254$	−2212.20	−0.546480
$255$	0	0
$256$	4342.49	1.06018
$257$	−4356.08	−1.05730	−0.528648	−	0.848841i	$-0.677301\pi$
$257$	−4356.08	−1.05730	−0.528648	+	0.848841i	$0.677301\pi$
$258$	0	0
$259$	687.660	0.164977
$260$	−2658.67	−0.634168
$261$	0	0
$262$	−8597.48	−2.02731
$263$	2133.91	0.500314	0.250157	−	0.968205i	$-0.419518\pi$
$263$	2133.91	0.500314	0.250157	+	0.968205i	$0.419518\pi$
$264$	0	0
$265$	−788.197	−0.182712
$266$	678.390	0.156371
$267$	0	0
$268$	1111.58	0.253359
$269$	−2757.12	−0.624925	−0.312462	−	0.949930i	$-0.601154\pi$
$269$	−2757.12	−0.624925	−0.312462	+	0.949930i	$0.601154\pi$
$270$	0	0
$271$	617.234	0.138355	0.0691777	−	0.997604i	$-0.477962\pi$
$271$	617.234	0.138355	0.0691777	+	0.997604i	$0.477962\pi$
$272$	7268.72	1.62033
$273$	0	0
$274$	9248.18	2.03906
$275$	2685.18	0.588809
$276$	0	0
$277$	−7811.00	−1.69429	−0.847143	−	0.531364i	$-0.821679\pi$
$277$	−7811.00	−1.69429	−0.847143	+	0.531364i	$0.821679\pi$
$278$	−7020.34	−1.51458
$279$	0	0
$280$	−550.434	−0.117481
$281$	−562.676	−0.119453	−0.0597267	−	0.998215i	$-0.519023\pi$
$281$	−562.676	−0.119453	−0.0597267	+	0.998215i	$0.519023\pi$
$282$	0	0
$283$	−6703.82	−1.40813	−0.704065	−	0.710136i	$-0.748633\pi$
$283$	−6703.82	−1.40813	−0.704065	+	0.710136i	$0.748633\pi$
$284$	2099.42	0.438653
$285$	0	0
$286$	20201.6	4.17674
$287$	−719.442	−0.147970
$288$	0	0
$289$	3438.71	0.699920
$290$	638.698	0.129330
$291$	0	0
$292$	2375.16	0.476013
$293$	5398.26	1.07635	0.538174	−	0.842834i	$-0.319114\pi$
$293$	5398.26	1.07635	0.538174	+	0.842834i	$0.319114\pi$
$294$	0	0
$295$	350.712	0.0692177
$296$	2880.90	0.565707
$297$	0	0
$298$	1039.91	0.202149
$299$	12441.5	2.40639
$300$	0	0
$301$	−1161.64	−0.222444
$302$	−568.398	−0.108303
$303$	0	0
$304$	4266.75	0.804982
$305$	1244.07	0.233557
$306$	0	0
$307$	2717.39	0.505177	0.252589	−	0.967574i	$-0.418718\pi$
$307$	2717.39	0.505177	0.252589	+	0.967574i	$0.418718\pi$
$308$	−870.000	−0.160951
$309$	0	0
$310$	9500.75	1.74067
$311$	3988.66	0.727254	0.363627	−	0.931545i	$-0.381538\pi$
$311$	3988.66	0.727254	0.363627	+	0.931545i	$0.381538\pi$
$312$	0	0
$313$	7412.18	1.33853	0.669267	−	0.743022i	$-0.266608\pi$
$313$	7412.18	1.33853	0.669267	+	0.743022i	$0.266608\pi$
$314$	9996.69	1.79664
$315$	0	0
$316$	−4278.44	−0.761648
$317$	−7563.05	−1.34001	−0.670005	−	0.742357i	$-0.733708\pi$
$317$	−7563.05	−1.34001	−0.670005	+	0.742357i	$0.733708\pi$
$318$	0	0
$319$	−1423.25	−0.249802
$320$	−1486.09	−0.259609
$321$	0	0
$322$	−1827.01	−0.316196
$323$	4902.46	0.844521
$324$	0	0
$325$	−3316.46	−0.566043
$326$	−11046.6	−1.87674
$327$	0	0
$328$	−3014.05	−0.507388
$329$	−29.2085	−0.00489458
$330$	0	0
$331$	−4814.81	−0.799535	−0.399768	−	0.916617i	$-0.630909\pi$
$331$	−4814.81	−0.799535	−0.399768	+	0.916617i	$0.630909\pi$
$332$	−1639.90	−0.271088
$333$	0	0
$334$	−1439.91	−0.235893
$335$	−3114.00	−0.507868
$336$	0	0
$337$	−9243.89	−1.49420	−0.747102	−	0.664709i	$-0.768555\pi$
$337$	−9243.89	−1.49420	−0.747102	+	0.664709i	$0.768555\pi$
$338$	−17559.2	−2.82573
$339$	0	0
$340$	2821.42	0.450037
$341$	−21171.2	−3.36212
$342$	0	0
$343$	−2525.35	−0.397540
$344$	−4866.61	−0.762761
$345$	0	0
$346$	−7261.86	−1.12832
$347$	−4200.72	−0.649874	−0.324937	−	0.945736i	$-0.605343\pi$
$347$	−4200.72	−0.649874	−0.324937	+	0.945736i	$0.605343\pi$
$348$	0	0
$349$	7022.27	1.07706	0.538530	−	0.842607i	$-0.318980\pi$
$349$	7022.27	1.07706	0.538530	+	0.842607i	$0.318980\pi$
$350$	487.015	0.0743773
$351$	0	0
$352$	−9874.85	−1.49526
$353$	1287.81	0.194174	0.0970870	−	0.995276i	$-0.469047\pi$
$353$	1287.81	0.194174	0.0970870	+	0.995276i	$0.469047\pi$
$354$	0	0
$355$	−5881.36	−0.879297
$356$	68.8427	0.0102490
$357$	0	0
$358$	10351.3	1.52817
$359$	143.990	0.0211685	0.0105842	−	0.999944i	$-0.496631\pi$
$359$	143.990	0.0211685	0.0105842	+	0.999944i	$0.496631\pi$
$360$	0	0
$361$	−3981.25	−0.580442
$362$	10690.8	1.55220
$363$	0	0
$364$	1074.54	0.154728
$365$	−6653.84	−0.954186
$366$	0	0
$367$	−6277.61	−0.892884	−0.446442	−	0.894812i	$-0.647309\pi$
$367$	−6277.61	−0.892884	−0.446442	+	0.894812i	$0.647309\pi$
$368$	−11491.0	−1.62774
$369$	0	0
$370$	5724.47	0.804327
$371$	318.560	0.0445790
$372$	0	0
$373$	7273.96	1.00974	0.504868	−	0.863197i	$-0.331541\pi$
$373$	7273.96	1.00974	0.504868	+	0.863197i	$0.331541\pi$
$374$	−21438.2	−2.96402
$375$	0	0
$376$	−122.367	−0.0167835
$377$	1757.86	0.240144
$378$	0	0
$379$	−2520.56	−0.341616	−0.170808	−	0.985304i	$-0.554638\pi$
$379$	−2520.56	−0.341616	−0.170808	+	0.985304i	$0.554638\pi$
$380$	1656.17	0.223579
$381$	0	0
$382$	−1910.33	−0.255866
$383$	−2069.24	−0.276065	−0.138033	−	0.990428i	$-0.544078\pi$
$383$	−2069.24	−0.276065	−0.138033	+	0.990428i	$0.544078\pi$
$384$	0	0
$385$	2437.24	0.322632
$386$	−10846.7	−1.43026
$387$	0	0
$388$	3934.26	0.514772
$389$	−11551.9	−1.50566	−0.752831	−	0.658214i	$-0.771312\pi$
$389$	−11551.9	−1.50566	−0.752831	+	0.658214i	$0.771312\pi$
$390$	0	0
$391$	−13203.1	−1.70769
$392$	−5178.67	−0.667250
$393$	0	0
$394$	5703.77	0.729319
$395$	11985.7	1.52675
$396$	0	0
$397$	−7277.79	−0.920055	−0.460028	−	0.887905i	$-0.652160\pi$
$397$	−7277.79	−0.920055	−0.460028	+	0.887905i	$0.652160\pi$
$398$	−2727.78	−0.343546
$399$	0	0
$400$	3063.09	0.382886
$401$	−6002.64	−0.747525	−0.373762	−	0.927525i	$-0.621932\pi$
$401$	−6002.64	−0.747525	−0.373762	+	0.927525i	$0.621932\pi$
$402$	0	0
$403$	26148.5	3.23213
$404$	4805.73	0.591817
$405$	0	0
$406$	−258.138	−0.0315546
$407$	−12756.2	−1.55357
$408$	0	0
$409$	−1210.77	−0.146379	−0.0731894	−	0.997318i	$-0.523318\pi$
$409$	−1210.77	−0.146379	−0.0731894	+	0.997318i	$0.523318\pi$
$410$	−5989.04	−0.721409
$411$	0	0
$412$	708.525	0.0847246
$413$	−141.745	−0.0168881
$414$	0	0
$415$	4594.07	0.543407
$416$	12196.4	1.43745
$417$	0	0
$418$	−12584.3	−1.47253
$419$	−6003.07	−0.699926	−0.349963	−	0.936763i	$-0.613806\pi$
$419$	−6003.07	−0.699926	−0.349963	+	0.936763i	$0.613806\pi$
$420$	0	0
$421$	−5920.42	−0.685376	−0.342688	−	0.939449i	$-0.611337\pi$
$421$	−5920.42	−0.685376	−0.342688	+	0.939449i	$0.611337\pi$
$422$	18905.1	2.18077
$423$	0	0
$424$	1334.58	0.152861
$425$	3519.47	0.401693
$426$	0	0
$427$	−502.805	−0.0569846
$428$	−1971.06	−0.222605
$429$	0	0
$430$	−9670.14	−1.08450
$431$	288.570	0.0322504	0.0161252	−	0.999870i	$-0.494867\pi$
$431$	288.570	0.0322504	0.0161252	+	0.999870i	$0.494867\pi$
$432$	0	0
$433$	−13744.7	−1.52546	−0.762732	−	0.646714i	$-0.776143\pi$
$433$	−13744.7	−1.52546	−0.762732	+	0.646714i	$0.776143\pi$
$434$	−3839.85	−0.424697
$435$	0	0
$436$	4767.27	0.523649
$437$	−7750.21	−0.848381
$438$	0	0
$439$	−2803.41	−0.304782	−0.152391	−	0.988320i	$-0.548697\pi$
$439$	−2803.41	−0.304782	−0.152391	+	0.988320i	$0.548697\pi$
$440$	10210.7	1.10630
$441$	0	0
$442$	26478.3	2.84942
$443$	13686.5	1.46787	0.733936	−	0.679219i	$-0.237681\pi$
$443$	13686.5	1.46787	0.733936	+	0.679219i	$0.237681\pi$
$444$	0	0
$445$	−192.858	−0.0205446
$446$	497.087	0.0527752
$447$	0	0
$448$	600.622	0.0633409
$449$	−1036.51	−0.108944	−0.0544720	−	0.998515i	$-0.517348\pi$
$449$	−1036.51	−0.108944	−0.0544720	+	0.998515i	$0.517348\pi$
$450$	0	0
$451$	13345.8	1.39341
$452$	−2534.58	−0.263753
$453$	0	0
$454$	9289.98	0.960353
$455$	−3010.23	−0.310158
$456$	0	0
$457$	4258.02	0.435846	0.217923	−	0.975966i	$-0.430072\pi$
$457$	4258.02	0.435846	0.217923	+	0.975966i	$0.430072\pi$
$458$	−17196.0	−1.75440
$459$	0	0
$460$	−4460.32	−0.452095
$461$	19721.7	1.99247	0.996237	−	0.0866665i	$-0.0276215\pi$
$461$	19721.7	1.99247	0.996237	+	0.0866665i	$0.0276215\pi$
$462$	0	0
$463$	−11216.9	−1.12590	−0.562951	−	0.826490i	$-0.690334\pi$
$463$	−11216.9	−1.12590	−0.562951	+	0.826490i	$0.690334\pi$
$464$	−1623.56	−0.162440
$465$	0	0
$466$	−628.879	−0.0625156
$467$	−4666.60	−0.462408	−0.231204	−	0.972905i	$-0.574266\pi$
$467$	−4666.60	−0.462408	−0.231204	+	0.972905i	$0.574266\pi$
$468$	0	0
$469$	1258.56	0.123913
$470$	−243.148	−0.0238629
$471$	0	0
$472$	−593.829	−0.0579093
$473$	21548.6	2.09473
$474$	0	0
$475$	2065.93	0.199561
$476$	−1140.31	−0.109803
$477$	0	0
$478$	−804.110	−0.0769438
$479$	−20606.2	−1.96560	−0.982798	−	0.184686i	$-0.940873\pi$
$479$	−20606.2	−1.96560	−0.982798	+	0.184686i	$0.940873\pi$
$480$	0	0
$481$	15755.2	1.49350
$482$	18138.7	1.71410
$483$	0	0
$484$	11720.1	1.10069
$485$	−11021.5	−1.03188
$486$	0	0
$487$	−10570.4	−0.983552	−0.491776	−	0.870722i	$-0.663652\pi$
$487$	−10570.4	−0.983552	−0.491776	+	0.870722i	$0.663652\pi$
$488$	−2106.47	−0.195400
$489$	0	0
$490$	−10290.2	−0.948703
$491$	−4053.86	−0.372603	−0.186301	−	0.982493i	$-0.559650\pi$
$491$	−4053.86	−0.372603	−0.186301	+	0.982493i	$0.559650\pi$
$492$	0	0
$493$	−1865.46	−0.170418
$494$	15542.8	1.41559
$495$	0	0
$496$	−24150.8	−2.18629
$497$	2377.03	0.214536
$498$	0	0
$499$	4010.47	0.359786	0.179893	−	0.983686i	$-0.442425\pi$
$499$	4010.47	0.359786	0.179893	+	0.983686i	$0.442425\pi$
$500$	5048.09	0.451515
$501$	0	0
$502$	−22191.7	−1.97303
$503$	−15500.9	−1.37406	−0.687028	−	0.726631i	$-0.741085\pi$
$503$	−15500.9	−1.37406	−0.687028	+	0.726631i	$0.741085\pi$
$504$	0	0
$505$	−13462.9	−1.18632
$506$	33891.3	2.97757
$507$	0	0
$508$	2182.77	0.190639
$509$	4493.10	0.391264	0.195632	−	0.980677i	$-0.437324\pi$
$509$	4493.10	0.391264	0.195632	+	0.980677i	$0.437324\pi$
$510$	0	0
$511$	2689.23	0.232808
$512$	−1245.03	−0.107467
$513$	0	0
$514$	14655.9	1.25768
$515$	−1984.88	−0.169834
$516$	0	0
$517$	541.823	0.0460916
$518$	−2313.62	−0.196244
$519$	0	0
$520$	−12611.2	−1.06353
$521$	−13135.8	−1.10459	−0.552293	−	0.833650i	$-0.686247\pi$
$521$	−13135.8	−1.10459	−0.552293	+	0.833650i	$0.686247\pi$
$522$	0	0
$523$	−12509.3	−1.04588	−0.522940	−	0.852370i	$-0.675165\pi$
$523$	−12509.3	−1.04588	−0.522940	+	0.852370i	$0.675165\pi$
$524$	8483.08	0.707223
$525$	0	0
$526$	−7179.50	−0.595135
$527$	−27749.1	−2.29368
$528$	0	0
$529$	8705.46	0.715498
$530$	2651.87	0.217339
$531$	0	0
$532$	−669.363	−0.0545500
$533$	−16483.4	−1.33954
$534$	0	0
$535$	5521.78	0.446220
$536$	5272.66	0.424896
$537$	0	0
$538$	9276.28	0.743362
$539$	22930.4	1.83243
$540$	0	0
$541$	−22330.6	−1.77462	−0.887309	−	0.461176i	$-0.847428\pi$
$541$	−22330.6	−1.77462	−0.887309	+	0.461176i	$0.847428\pi$
$542$	−2076.67	−0.164577
$543$	0	0
$544$	−12943.0	−1.02008
$545$	−13355.2	−1.04967
$546$	0	0
$547$	4374.29	0.341922	0.170961	−	0.985278i	$-0.445313\pi$
$547$	4374.29	0.341922	0.170961	+	0.985278i	$0.445313\pi$
$548$	−9125.12	−0.711324
$549$	0	0
$550$	−9034.23	−0.700401
$551$	−1095.03	−0.0846637
$552$	0	0
$553$	−4844.18	−0.372506
$554$	26279.9	2.01539
$555$	0	0
$556$	6926.92	0.528358
$557$	−11192.8	−0.851447	−0.425724	−	0.904853i	$-0.639980\pi$
$557$	−11192.8	−0.851447	−0.425724	+	0.904853i	$0.639980\pi$
$558$	0	0
$559$	−26614.7	−2.01374
$560$	2780.26	0.209799
$561$	0	0
$562$	1893.11	0.142093
$563$	9123.89	0.682995	0.341497	−	0.939883i	$-0.389066\pi$
$563$	9123.89	0.682995	0.341497	+	0.939883i	$0.389066\pi$
$564$	0	0
$565$	7100.44	0.528704
$566$	22554.8	1.67500
$567$	0	0
$568$	9958.39	0.735642
$569$	21130.6	1.55684	0.778419	−	0.627745i	$-0.216022\pi$
$569$	21130.6	1.55684	0.778419	+	0.627745i	$0.216022\pi$
$570$	0	0
$571$	−14945.9	−1.09539	−0.547695	−	0.836678i	$-0.684495\pi$
$571$	−14945.9	−1.09539	−0.547695	+	0.836678i	$0.684495\pi$
$572$	−19932.8	−1.45705
$573$	0	0
$574$	2420.55	0.176013
$575$	−5563.86	−0.403529
$576$	0	0
$577$	−22912.4	−1.65313	−0.826565	−	0.562842i	$-0.809708\pi$
$577$	−22912.4	−1.65313	−0.826565	+	0.562842i	$0.809708\pi$
$578$	−11569.5	−0.832571
$579$	0	0
$580$	−630.199	−0.0451165
$581$	−1856.75	−0.132583
$582$	0	0
$583$	−5909.34	−0.419794
$584$	11266.4	0.798297
$585$	0	0
$586$	−18162.3	−1.28034
$587$	−24633.1	−1.73206	−0.866028	−	0.499996i	$-0.833335\pi$
$587$	−24633.1	−1.73206	−0.866028	+	0.499996i	$0.833335\pi$
$588$	0	0
$589$	−16288.7	−1.13950
$590$	−1179.96	−0.0823360
$591$	0	0
$592$	−14551.5	−1.01024
$593$	−22466.0	−1.55576	−0.777882	−	0.628411i	$-0.783706\pi$
$593$	−22466.0	−1.55576	−0.777882	+	0.628411i	$0.783706\pi$
$594$	0	0
$595$	3194.50	0.220104
$596$	−1026.07	−0.0705194
$597$	0	0
$598$	−41859.1	−2.86245
$599$	17414.9	1.18790	0.593951	−	0.804501i	$-0.297567\pi$
$599$	17414.9	1.18790	0.593951	+	0.804501i	$0.297567\pi$
$600$	0	0
$601$	−7522.83	−0.510587	−0.255293	−	0.966864i	$-0.582172\pi$
$601$	−7522.83	−0.510587	−0.255293	+	0.966864i	$0.582172\pi$
$602$	3908.31	0.264603
$603$	0	0
$604$	560.835	0.0377816
$605$	−32833.1	−2.20637
$606$	0	0
$607$	2872.01	0.192045	0.0960224	−	0.995379i	$-0.469388\pi$
$607$	2872.01	0.192045	0.0960224	+	0.995379i	$0.469388\pi$
$608$	−7597.54	−0.506778
$609$	0	0
$610$	−4185.63	−0.277822
$611$	−669.205	−0.0443095
$612$	0	0
$613$	17050.8	1.12345	0.561726	−	0.827323i	$-0.310137\pi$
$613$	17050.8	1.12345	0.561726	+	0.827323i	$0.310137\pi$
$614$	−9142.58	−0.600919
$615$	0	0
$616$	−4126.77	−0.269922
$617$	−2203.39	−0.143768	−0.0718842	−	0.997413i	$-0.522901\pi$
$617$	−2203.39	−0.143768	−0.0718842	+	0.997413i	$0.522901\pi$
$618$	0	0
$619$	−5831.73	−0.378671	−0.189335	−	0.981912i	$-0.560633\pi$
$619$	−5831.73	−0.378671	−0.189335	+	0.981912i	$0.560633\pi$
$620$	−9374.33	−0.607229
$621$	0	0
$622$	−13419.8	−0.865085
$623$	77.9459	0.00501258
$624$	0	0
$625$	−9327.94	−0.596988
$626$	−24938.1	−1.59222
$627$	0	0
$628$	−9863.67	−0.626757
$629$	−16719.6	−1.05986
$630$	0	0
$631$	9155.36	0.577605	0.288803	−	0.957389i	$-0.406743\pi$
$631$	9155.36	0.577605	0.288803	+	0.957389i	$0.406743\pi$
$632$	−20294.4	−1.27732
$633$	0	0
$634$	25445.7	1.59397
$635$	−6114.87	−0.382143
$636$	0	0
$637$	−28321.3	−1.76158
$638$	4788.50	0.297145
$639$	0	0
$640$	15536.9	0.959609
$641$	−15883.3	−0.978711	−0.489355	−	0.872085i	$-0.662768\pi$
$641$	−15883.3	−0.978711	−0.489355	+	0.872085i	$0.662768\pi$
$642$	0	0
$643$	3981.15	0.244170	0.122085	−	0.992520i	$-0.461042\pi$
$643$	3981.15	0.244170	0.122085	+	0.992520i	$0.461042\pi$
$644$	1802.70	0.110305
$645$	0	0
$646$	−16494.2	−1.00458
$647$	14585.7	0.886279	0.443139	−	0.896453i	$-0.353865\pi$
$647$	14585.7	0.886279	0.443139	+	0.896453i	$0.353865\pi$
$648$	0	0
$649$	2629.39	0.159033
$650$	11158.2	0.673321
$651$	0	0
$652$	10899.6	0.654697
$653$	−6997.75	−0.419361	−0.209681	−	0.977770i	$-0.567242\pi$
$653$	−6997.75	−0.419361	−0.209681	+	0.977770i	$0.567242\pi$
$654$	0	0
$655$	−23764.7	−1.41766
$656$	15224.1	0.906098
$657$	0	0
$658$	98.2714	0.00582221
$659$	29294.4	1.73163	0.865817	−	0.500361i	$-0.166800\pi$
$659$	29294.4	1.73163	0.865817	+	0.500361i	$0.166800\pi$
$660$	0	0
$661$	−22701.0	−1.33581	−0.667903	−	0.744249i	$-0.732808\pi$
$661$	−22701.0	−1.33581	−0.667903	+	0.744249i	$0.732808\pi$
$662$	16199.3	0.951065
$663$	0	0
$664$	−7778.73	−0.454628
$665$	1875.17	0.109347
$666$	0	0
$667$	2949.07	0.171197
$668$	1420.75	0.0822909
$669$	0	0
$670$	10477.0	0.604121
$671$	9327.12	0.536616
$672$	0	0
$673$	−28620.0	−1.63926	−0.819629	−	0.572894i	$-0.805820\pi$
$673$	−28620.0	−1.63926	−0.819629	+	0.572894i	$0.805820\pi$
$674$	31100.9	1.77739
$675$	0	0
$676$	17325.6	0.985753
$677$	17363.9	0.985746	0.492873	−	0.870101i	$-0.335947\pi$
$677$	17363.9	0.985746	0.492873	+	0.870101i	$0.335947\pi$
$678$	0	0
$679$	4454.49	0.251764
$680$	13383.1	0.754734
$681$	0	0
$682$	71229.9	3.99932
$683$	−18795.3	−1.05298	−0.526488	−	0.850183i	$-0.676491\pi$
$683$	−18795.3	−1.05298	−0.526488	+	0.850183i	$0.676491\pi$
$684$	0	0
$685$	25563.4	1.42588
$686$	8496.50	0.472883
$687$	0	0
$688$	24581.4	1.36215
$689$	7298.62	0.403563
$690$	0	0
$691$	−3088.74	−0.170045	−0.0850225	−	0.996379i	$-0.527096\pi$
$691$	−3088.74	−0.170045	−0.0850225	+	0.996379i	$0.527096\pi$
$692$	7165.23	0.393615
$693$	0	0
$694$	14133.2	0.773040
$695$	−19405.3	−1.05911
$696$	0	0
$697$	17492.3	0.950603
$698$	−23626.3	−1.28119
$699$	0	0
$700$	−480.535	−0.0259465
$701$	23674.8	1.27558	0.637792	−	0.770209i	$-0.279848\pi$
$701$	23674.8	1.27558	0.637792	+	0.770209i	$0.279848\pi$
$702$	0	0
$703$	−9814.42	−0.526540
$704$	−11141.6	−0.596473
$705$	0	0
$706$	−4332.82	−0.230974
$707$	5441.20	0.289445
$708$	0	0
$709$	22101.6	1.17072	0.585362	−	0.810772i	$-0.300952\pi$
$709$	22101.6	1.17072	0.585362	+	0.810772i	$0.300952\pi$
$710$	19787.7	1.04594
$711$	0	0
$712$	326.549	0.0171881
$713$	43868.0	2.30416
$714$	0	0
$715$	55840.4	2.92072
$716$	−10213.6	−0.533099
$717$	0	0
$718$	−484.450	−0.0251804
$719$	−3159.64	−0.163887	−0.0819435	−	0.996637i	$-0.526113\pi$
$719$	−3159.64	−0.163887	−0.0819435	+	0.996637i	$0.526113\pi$
$720$	0	0
$721$	802.215	0.0414370
$722$	13394.8	0.690449
$723$	0	0
$724$	−10548.5	−0.541482
$725$	−786.119	−0.0402699
$726$	0	0
$727$	11599.7	0.591760	0.295880	−	0.955225i	$-0.404387\pi$
$727$	11599.7	0.591760	0.295880	+	0.955225i	$0.404387\pi$
$728$	5096.96	0.259486
$729$	0	0
$730$	22386.7	1.13503
$731$	28243.8	1.42905
$732$	0	0
$733$	−30956.4	−1.55989	−0.779946	−	0.625846i	$-0.784754\pi$
$733$	−30956.4	−1.55989	−0.779946	+	0.625846i	$0.784754\pi$
$734$	21120.9	1.06211
$735$	0	0
$736$	20461.3	1.02475
$737$	−23346.5	−1.16687
$738$	0	0
$739$	−17815.8	−0.886826	−0.443413	−	0.896317i	$-0.646232\pi$
$739$	−17815.8	−0.886826	−0.443413	+	0.896317i	$0.646232\pi$
$740$	−5648.30	−0.280589
$741$	0	0
$742$	−1071.79	−0.0530277
$743$	32457.3	1.60261	0.801307	−	0.598253i	$-0.204138\pi$
$743$	32457.3	1.60261	0.801307	+	0.598253i	$0.204138\pi$
$744$	0	0
$745$	2874.47	0.141359
$746$	−24473.1	−1.20110
$747$	0	0
$748$	21153.0	1.03400
$749$	−2231.70	−0.108871
$750$	0	0
$751$	15687.8	0.762260	0.381130	−	0.924521i	$-0.375535\pi$
$751$	15687.8	0.762260	0.381130	+	0.924521i	$0.375535\pi$
$752$	618.079	0.0299721
$753$	0	0
$754$	−5914.27	−0.285657
$755$	−1571.14	−0.0757346
$756$	0	0
$757$	−8795.67	−0.422304	−0.211152	−	0.977453i	$-0.567721\pi$
$757$	−8795.67	−0.422304	−0.211152	+	0.977453i	$0.567721\pi$
$758$	8480.36	0.406359
$759$	0	0
$760$	7855.91	0.374952
$761$	14704.2	0.700428	0.350214	−	0.936670i	$-0.386109\pi$
$761$	14704.2	0.700428	0.350214	+	0.936670i	$0.386109\pi$
$762$	0	0
$763$	5397.66	0.256105
$764$	1884.91	0.0892586
$765$	0	0
$766$	6961.90	0.328386
$767$	−3247.55	−0.152884
$768$	0	0
$769$	7451.53	0.349426	0.174713	−	0.984619i	$-0.444100\pi$
$769$	7451.53	0.349426	0.174713	+	0.984619i	$0.444100\pi$
$770$	−8200.04	−0.383778
$771$	0	0
$772$	10702.3	0.498945
$773$	4152.37	0.193209	0.0966045	−	0.995323i	$-0.469202\pi$
$773$	4152.37	0.193209	0.0966045	+	0.995323i	$0.469202\pi$
$774$	0	0
$775$	−11693.7	−0.541998
$776$	18661.8	0.863298
$777$	0	0
$778$	38866.0	1.79102
$779$	10268.0	0.472259
$780$	0	0
$781$	−44094.3	−2.02025
$782$	44421.4	2.03134
$783$	0	0
$784$	26157.6	1.19158
$785$	27632.4	1.25636
$786$	0	0
$787$	11972.9	0.542298	0.271149	−	0.962537i	$-0.412596\pi$
$787$	11972.9	0.542298	0.271149	+	0.962537i	$0.412596\pi$
$788$	−5627.87	−0.254422
$789$	0	0
$790$	−40325.7	−1.81611
$791$	−2869.73	−0.128996
$792$	0	0
$793$	−11519.9	−0.515869
$794$	24486.0	1.09443
$795$	0	0
$796$	2691.49	0.119846
$797$	22819.5	1.01419	0.507094	−	0.861891i	$-0.330720\pi$
$797$	22819.5	1.01419	0.507094	+	0.861891i	$0.330720\pi$
$798$	0	0
$799$	710.169	0.0314443
$800$	−5454.27	−0.241047
$801$	0	0
$802$	20195.7	0.889197
$803$	−49885.8	−2.19232
$804$	0	0
$805$	−5050.12	−0.221110
$806$	−87975.9	−3.84469
$807$	0	0
$808$	22795.5	0.992505
$809$	17267.6	0.750427	0.375213	−	0.926938i	$-0.377569\pi$
$809$	17267.6	0.750427	0.375213	+	0.926938i	$0.377569\pi$
$810$	0	0
$811$	36812.9	1.59393	0.796965	−	0.604025i	$-0.206437\pi$
$811$	36812.9	1.59393	0.796965	+	0.604025i	$0.206437\pi$
$812$	254.703	0.0110078
$813$	0	0
$814$	42918.0	1.84800
$815$	−30534.5	−1.31237
$816$	0	0
$817$	16579.1	0.709952
$818$	4073.62	0.174121
$819$	0	0
$820$	5909.35	0.251663
$821$	−23379.4	−0.993846	−0.496923	−	0.867795i	$-0.665537\pi$
$821$	−23379.4	−0.993846	−0.496923	+	0.867795i	$0.665537\pi$
$822$	0	0
$823$	−19489.3	−0.825459	−0.412730	−	0.910854i	$-0.635425\pi$
$823$	−19489.3	−0.825459	−0.412730	+	0.910854i	$0.635425\pi$
$824$	3360.82	0.142087
$825$	0	0
$826$	476.896	0.0200888
$827$	42718.9	1.79623	0.898114	−	0.439763i	$-0.144938\pi$
$827$	42718.9	1.79623	0.898114	+	0.439763i	$0.144938\pi$
$828$	0	0
$829$	16125.5	0.675587	0.337793	−	0.941220i	$-0.390320\pi$
$829$	16125.5	0.675587	0.337793	+	0.941220i	$0.390320\pi$
$830$	−15456.6	−0.646395
$831$	0	0
$832$	13761.0	0.573411
$833$	30054.9	1.25011
$834$	0	0
$835$	−3980.12	−0.164955
$836$	12416.8	0.513689
$837$	0	0
$838$	20197.2	0.832578
$839$	38875.7	1.59969	0.799844	−	0.600207i	$-0.204915\pi$
$839$	38875.7	1.59969	0.799844	+	0.600207i	$0.204915\pi$
$840$	0	0
$841$	−23972.3	−0.982915
$842$	19919.1	0.815270
$843$	0	0
$844$	−18653.5	−0.760759
$845$	−48536.4	−1.97598
$846$	0	0
$847$	13269.9	0.538323
$848$	−6741.02	−0.272981
$849$	0	0
$850$	−11841.2	−0.477822
$851$	26431.7	1.06471
$852$	0	0
$853$	24210.3	0.971801	0.485901	−	0.874014i	$-0.338492\pi$
$853$	24210.3	0.971801	0.485901	+	0.874014i	$0.338492\pi$
$854$	1691.68	0.0677845
$855$	0	0
$856$	−9349.55	−0.373319
$857$	−45734.3	−1.82293	−0.911466	−	0.411375i	$-0.865049\pi$
$857$	−45734.3	−1.82293	−0.911466	+	0.411375i	$0.865049\pi$
$858$	0	0
$859$	−43000.2	−1.70797	−0.853986	−	0.520295i	$-0.825822\pi$
$859$	−43000.2	−1.70797	−0.853986	+	0.520295i	$0.825822\pi$
$860$	9541.47	0.378327
$861$	0	0
$862$	−970.886	−0.0383626
$863$	−8356.00	−0.329596	−0.164798	−	0.986327i	$-0.552697\pi$
$863$	−8356.00	−0.329596	−0.164798	+	0.986327i	$0.552697\pi$
$864$	0	0
$865$	−20072.9	−0.789016
$866$	46243.6	1.81457
$867$	0	0
$868$	3788.75	0.148155
$869$	89860.5	3.50783
$870$	0	0
$871$	28835.3	1.12175
$872$	22613.1	0.878184
$873$	0	0
$874$	26075.4	1.00917
$875$	5715.62	0.220826
$876$	0	0
$877$	−11075.6	−0.426450	−0.213225	−	0.977003i	$-0.568397\pi$
$877$	−11075.6	−0.426450	−0.213225	+	0.977003i	$0.568397\pi$
$878$	9432.00	0.362545
$879$	0	0
$880$	−51574.3	−1.97565
$881$	−34952.4	−1.33663	−0.668317	−	0.743876i	$-0.732985\pi$
$881$	−34952.4	−1.33663	−0.668317	+	0.743876i	$0.732985\pi$
$882$	0	0
$883$	−15570.7	−0.593428	−0.296714	−	0.954966i	$-0.595891\pi$
$883$	−15570.7	−0.593428	−0.296714	+	0.954966i	$0.595891\pi$
$884$	−26126.0	−0.994018
$885$	0	0
$886$	−46048.1	−1.74607
$887$	−6033.06	−0.228377	−0.114188	−	0.993459i	$-0.536427\pi$
$887$	−6033.06	−0.228377	−0.114188	+	0.993459i	$0.536427\pi$
$888$	0	0
$889$	2471.40	0.0932375
$890$	648.865	0.0244382
$891$	0	0
$892$	−490.472	−0.0184106
$893$	416.870	0.0156215
$894$	0	0
$895$	28612.6	1.06862
$896$	−6279.43	−0.234131
$897$	0	0
$898$	3487.31	0.129591
$899$	6198.11	0.229943
$900$	0	0
$901$	−7745.39	−0.286389
$902$	−44901.6	−1.65749
$903$	0	0
$904$	−12022.5	−0.442327
$905$	29550.9	1.08542
$906$	0	0
$907$	28077.5	1.02789	0.513946	−	0.857822i	$-0.328183\pi$
$907$	28077.5	1.02789	0.513946	+	0.857822i	$0.328183\pi$
$908$	−9166.37	−0.335018
$909$	0	0
$910$	10127.9	0.368940
$911$	−34056.0	−1.23856	−0.619279	−	0.785171i	$-0.712575\pi$
$911$	−34056.0	−1.23856	−0.619279	+	0.785171i	$0.712575\pi$
$912$	0	0
$913$	34443.1	1.24852
$914$	−14326.0	−0.518449
$915$	0	0
$916$	16967.2	0.612021
$917$	9604.82	0.345888
$918$	0	0
$919$	−17320.1	−0.621695	−0.310848	−	0.950460i	$-0.600613\pi$
$919$	−17320.1	−0.621695	−0.310848	+	0.950460i	$0.600613\pi$
$920$	−21157.1	−0.758185
$921$	0	0
$922$	−66353.2	−2.37009
$923$	54460.8	1.94214
$924$	0	0
$925$	−7045.76	−0.250447
$926$	37739.0	1.33929
$927$	0	0
$928$	2890.98	0.102264
$929$	31529.5	1.11351	0.556753	−	0.830678i	$-0.312047\pi$
$929$	31529.5	1.11351	0.556753	+	0.830678i	$0.312047\pi$
$930$	0	0
$931$	17642.2	0.621054
$932$	620.511	0.0218085
$933$	0	0
$934$	15700.7	0.550045
$935$	−59258.5	−2.07269
$936$	0	0
$937$	2938.10	0.102437	0.0512185	−	0.998687i	$-0.483690\pi$
$937$	2938.10	0.102437	0.0512185	+	0.998687i	$0.483690\pi$
$938$	−4234.40	−0.147397
$939$	0	0
$940$	239.913	0.00832456
$941$	−17965.1	−0.622364	−0.311182	−	0.950350i	$-0.600725\pi$
$941$	−17965.1	−0.622364	−0.311182	+	0.950350i	$0.600725\pi$
$942$	0	0
$943$	−27653.3	−0.954948
$944$	2999.45	0.103415
$945$	0	0
$946$	−72499.8	−2.49173
$947$	14780.5	0.507183	0.253592	−	0.967311i	$-0.418388\pi$
$947$	14780.5	0.507183	0.253592	+	0.967311i	$0.418388\pi$
$948$	0	0
$949$	61613.8	2.10756
$950$	−6950.78	−0.237382
$951$	0	0
$952$	−5408.96	−0.184144
$953$	44472.3	1.51165	0.755824	−	0.654775i	$-0.227237\pi$
$953$	44472.3	1.51165	0.755824	+	0.654775i	$0.227237\pi$
$954$	0	0
$955$	−5280.43	−0.178922
$956$	793.410	0.0268418
$957$	0	0
$958$	69329.0	2.33812
$959$	−10331.8	−0.347893
$960$	0	0
$961$	62407.1	2.09483
$962$	−53008.0	−1.77655
$963$	0	0
$964$	−17897.3	−0.597960
$965$	−29981.8	−1.00015
$966$	0	0
$967$	17344.7	0.576804	0.288402	−	0.957509i	$-0.406876\pi$
$967$	17344.7	0.576804	0.288402	+	0.957509i	$0.406876\pi$
$968$	55593.3	1.84591
$969$	0	0
$970$	37081.7	1.22745
$971$	43715.3	1.44479	0.722394	−	0.691481i	$-0.243042\pi$
$971$	43715.3	1.44479	0.722394	+	0.691481i	$0.243042\pi$
$972$	0	0
$973$	7842.89	0.258408
$974$	35563.8	1.16996
$975$	0	0
$976$	10639.8	0.348947
$977$	−6339.04	−0.207578	−0.103789	−	0.994599i	$-0.533097\pi$
$977$	−6339.04	−0.207578	−0.103789	+	0.994599i	$0.533097\pi$
$978$	0	0
$979$	−1445.91	−0.0472028
$980$	10153.3	0.330954
$981$	0	0
$982$	13639.1	0.443219
$983$	15307.9	0.496691	0.248345	−	0.968672i	$-0.420113\pi$
$983$	15307.9	0.496691	0.248345	+	0.968672i	$0.420113\pi$
$984$	0	0
$985$	15766.1	0.509999
$986$	6276.30	0.202716
$987$	0	0
$988$	−15336.0	−0.493828
$989$	−44650.1	−1.43558
$990$	0	0
$991$	28952.3	0.928052	0.464026	−	0.885822i	$-0.346404\pi$
$991$	28952.3	0.928052	0.464026	+	0.885822i	$0.346404\pi$
$992$	43003.9	1.37639
$993$	0	0
$994$	−7997.46	−0.255195
$995$	−7540.01	−0.240235
$996$	0	0
$997$	19274.7	0.612272	0.306136	−	0.951988i	$-0.400964\pi$
$997$	19274.7	0.612272	0.306136	+	0.951988i	$0.400964\pi$
$998$	−13493.1	−0.427974
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.14	✓	59
3.2	odd	2		2151.4.a.h.1.46	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.14	✓	59	1.1	even	1	trivial
2151.4.a.h.1.46	yes	59	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.36448 q^{2} +3.31971 q^{4} -9.29992 q^{5} +3.75868 q^{7} +15.7467 q^{8} +O(q^{10})\)	\(q-3.36448 q^{2} +3.31971 q^{4} -9.29992 q^{5} +3.75868 q^{7} +15.7467 q^{8} +31.2894 q^{10} -69.7242 q^{11} +86.1163 q^{13} -12.6460 q^{14} -79.5372 q^{16} -91.3877 q^{17} -53.6446 q^{19} -30.8730 q^{20} +234.586 q^{22} +144.473 q^{23} -38.5114 q^{25} -289.736 q^{26} +12.4777 q^{28} +20.4126 q^{29} +303.641 q^{31} +141.627 q^{32} +307.472 q^{34} -34.9554 q^{35} +182.952 q^{37} +180.486 q^{38} -146.443 q^{40} -191.408 q^{41} -309.055 q^{43} -231.464 q^{44} -486.076 q^{46} -7.77095 q^{47} -328.872 q^{49} +129.571 q^{50} +285.881 q^{52} +84.7531 q^{53} +648.430 q^{55} +59.1870 q^{56} -68.6778 q^{58} -37.7112 q^{59} -133.772 q^{61} -1021.59 q^{62} +159.796 q^{64} -800.875 q^{65} +334.841 q^{67} -303.380 q^{68} +117.607 q^{70} +632.410 q^{71} +715.473 q^{73} -615.539 q^{74} -178.085 q^{76} -262.071 q^{77} -1288.80 q^{79} +739.690 q^{80} +643.988 q^{82} -493.990 q^{83} +849.898 q^{85} +1039.81 q^{86} -1097.93 q^{88} +20.7376 q^{89} +323.684 q^{91} +479.608 q^{92} +26.1452 q^{94} +498.891 q^{95} +1185.12 q^{97} +1106.48 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 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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