LMFDB - Embedded newform 2151.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$126.913108422$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.57745 q^{2} +23.1079 q^{4} +14.3698 q^{5} -15.9726 q^{7} -84.2636 q^{8} +O(q^{10})$	\(q-5.57745 q^{2} +23.1079 q^{4} +14.3698 q^{5} -15.9726 q^{7} -84.2636 q^{8} -80.1465 q^{10} -67.7046 q^{11} -60.3228 q^{13} +89.0862 q^{14} +285.112 q^{16} -112.003 q^{17} +14.3875 q^{19} +332.055 q^{20} +377.619 q^{22} -148.706 q^{23} +81.4898 q^{25} +336.447 q^{26} -369.093 q^{28} -95.1700 q^{29} -211.404 q^{31} -916.090 q^{32} +624.689 q^{34} -229.522 q^{35} +91.5660 q^{37} -80.2455 q^{38} -1210.85 q^{40} -330.209 q^{41} +116.984 q^{43} -1564.51 q^{44} +829.398 q^{46} -391.297 q^{47} -87.8767 q^{49} -454.505 q^{50} -1393.93 q^{52} -65.7568 q^{53} -972.898 q^{55} +1345.91 q^{56} +530.806 q^{58} -352.345 q^{59} -289.274 q^{61} +1179.09 q^{62} +2828.55 q^{64} -866.824 q^{65} -508.316 q^{67} -2588.15 q^{68} +1280.15 q^{70} +850.297 q^{71} +527.936 q^{73} -510.705 q^{74} +332.465 q^{76} +1081.42 q^{77} -388.395 q^{79} +4096.99 q^{80} +1841.72 q^{82} -274.270 q^{83} -1609.45 q^{85} -652.473 q^{86} +5705.03 q^{88} +1183.09 q^{89} +963.511 q^{91} -3436.28 q^{92} +2182.44 q^{94} +206.745 q^{95} +1624.95 q^{97} +490.127 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−5.57745	−1.97193	−0.985963	−	0.166966i	$-0.946603\pi$
$2$	−5.57745	−1.97193	−0.985963	+	0.166966i	$0.946603\pi$
$3$	0	0
$3$	0	0
$4$	23.1079	2.88849
$5$	14.3698	1.28527	0.642635	−	0.766173i	$-0.277841\pi$
$5$	14.3698	1.28527	0.642635	+	0.766173i	$0.277841\pi$
$6$	0	0
$7$	−15.9726	−0.862438	−0.431219	−	0.902247i	$-0.641916\pi$
$7$	−15.9726	−0.862438	−0.431219	+	0.902247i	$0.641916\pi$
$8$	−84.2636	−3.72396
$9$	0	0
$10$	−80.1465	−2.53446
$11$	−67.7046	−1.85579	−0.927895	−	0.372841i	$-0.878384\pi$
$11$	−67.7046	−1.85579	−0.927895	+	0.372841i	$0.878384\pi$
$12$	0	0
$13$	−60.3228	−1.28696	−0.643482	−	0.765461i	$-0.722511\pi$
$13$	−60.3228	−1.28696	−0.643482	+	0.765461i	$0.722511\pi$
$14$	89.0862	1.70066
$15$	0	0
$16$	285.112	4.45488
$17$	−112.003	−1.59792	−0.798959	−	0.601385i	$-0.794616\pi$
$17$	−112.003	−1.59792	−0.798959	+	0.601385i	$0.794616\pi$
$18$	0	0
$19$	14.3875	0.173722	0.0868610	−	0.996220i	$-0.472316\pi$
$19$	14.3875	0.173722	0.0868610	+	0.996220i	$0.472316\pi$
$20$	332.055	3.71249
$21$	0	0
$22$	377.619	3.65948
$23$	−148.706	−1.34814	−0.674071	−	0.738666i	$-0.735456\pi$
$23$	−148.706	−1.34814	−0.674071	+	0.738666i	$0.735456\pi$
$24$	0	0
$25$	81.4898	0.651918
$26$	336.447	2.53780
$27$	0	0
$28$	−369.093	−2.49114
$29$	−95.1700	−0.609401	−0.304700	−	0.952448i	$-0.598556\pi$
$29$	−95.1700	−0.609401	−0.304700	+	0.952448i	$0.598556\pi$
$30$	0	0
$31$	−211.404	−1.22482	−0.612408	−	0.790542i	$-0.709799\pi$
$31$	−211.404	−1.22482	−0.612408	+	0.790542i	$0.709799\pi$
$32$	−916.090	−5.06073
$33$	0	0
$34$	624.689	3.15098
$35$	−229.522	−1.10847
$36$	0	0
$37$	91.5660	0.406848	0.203424	−	0.979091i	$-0.434793\pi$
$37$	91.5660	0.406848	0.203424	+	0.979091i	$0.434793\pi$
$38$	−80.2455	−0.342567
$39$	0	0
$40$	−1210.85	−4.78629
$41$	−330.209	−1.25780	−0.628902	−	0.777485i	$-0.716495\pi$
$41$	−330.209	−1.25780	−0.628902	+	0.777485i	$0.716495\pi$
$42$	0	0
$43$	116.984	0.414882	0.207441	−	0.978248i	$-0.433487\pi$
$43$	116.984	0.414882	0.207441	+	0.978248i	$0.433487\pi$
$44$	−1564.51	−5.36043
$45$	0	0
$46$	829.398	2.65844
$47$	−391.297	−1.21439	−0.607196	−	0.794552i	$-0.707706\pi$
$47$	−391.297	−1.21439	−0.607196	+	0.794552i	$0.707706\pi$
$48$	0	0
$49$	−87.8767	−0.256200
$50$	−454.505	−1.28553
$51$	0	0
$52$	−1393.93	−3.71738
$53$	−65.7568	−0.170422	−0.0852112	−	0.996363i	$-0.527156\pi$
$53$	−65.7568	−0.170422	−0.0852112	+	0.996363i	$0.527156\pi$
$54$	0	0
$55$	−972.898	−2.38519
$56$	1345.91	3.21169
$57$	0	0
$58$	530.806	1.20169
$59$	−352.345	−0.777482	−0.388741	−	0.921347i	$-0.627090\pi$
$59$	−352.345	−0.777482	−0.388741	+	0.921347i	$0.627090\pi$
$60$	0	0
$61$	−289.274	−0.607176	−0.303588	−	0.952803i	$-0.598185\pi$
$61$	−289.274	−0.607176	−0.303588	+	0.952803i	$0.598185\pi$
$62$	1179.09	2.41524
$63$	0	0
$64$	2828.55	5.52450
$65$	−866.824	−1.65410
$66$	0	0
$67$	−508.316	−0.926875	−0.463438	−	0.886130i	$-0.653384\pi$
$67$	−508.316	−0.926875	−0.463438	+	0.886130i	$0.653384\pi$
$68$	−2588.15	−4.61557
$69$	0	0
$70$	1280.15	2.18581
$71$	850.297	1.42129	0.710646	−	0.703550i	$-0.248403\pi$
$71$	850.297	1.42129	0.710646	+	0.703550i	$0.248403\pi$
$72$	0	0
$73$	527.936	0.846442	0.423221	−	0.906026i	$-0.360899\pi$
$73$	527.936	0.846442	0.423221	+	0.906026i	$0.360899\pi$
$74$	−510.705	−0.802273
$75$	0	0
$76$	332.465	0.501794
$77$	1081.42	1.60051
$78$	0	0
$79$	−388.395	−0.553137	−0.276569	−	0.960994i	$-0.589197\pi$
$79$	−388.395	−0.553137	−0.276569	+	0.960994i	$0.589197\pi$
$80$	4096.99	5.72572
$81$	0	0
$82$	1841.72	2.48030
$83$	−274.270	−0.362712	−0.181356	−	0.983418i	$-0.558049\pi$
$83$	−274.270	−0.362712	−0.181356	+	0.983418i	$0.558049\pi$
$84$	0	0
$85$	−1609.45	−2.05376
$86$	−652.473	−0.818116
$87$	0	0
$88$	5705.03	6.91089
$89$	1183.09	1.40907	0.704535	−	0.709670i	$-0.251156\pi$
$89$	1183.09	1.40907	0.704535	+	0.709670i	$0.251156\pi$
$90$	0	0
$91$	963.511	1.10993
$92$	−3436.28	−3.89409
$93$	0	0
$94$	2182.44	2.39469
$95$	206.745	0.223280
$96$	0	0
$97$	1624.95	1.70091	0.850456	−	0.526047i	$-0.176326\pi$
$97$	1624.95	1.70091	0.850456	+	0.526047i	$0.176326\pi$
$98$	490.127	0.505208
$99$	0	0
$100$	1883.06	1.88306
$101$	−441.705	−0.435161	−0.217580	−	0.976042i	$-0.569816\pi$
$101$	−441.705	−0.435161	−0.217580	+	0.976042i	$0.569816\pi$
$102$	0	0
$103$	−1031.27	−0.986544	−0.493272	−	0.869875i	$-0.664199\pi$
$103$	−1031.27	−0.986544	−0.493272	+	0.869875i	$0.664199\pi$
$104$	5083.02	4.79260
$105$	0	0
$106$	366.755	0.336060
$107$	−814.185	−0.735609	−0.367805	−	0.929903i	$-0.619890\pi$
$107$	−814.185	−0.735609	−0.367805	+	0.929903i	$0.619890\pi$
$108$	0	0
$109$	−347.030	−0.304949	−0.152474	−	0.988307i	$-0.548724\pi$
$109$	−347.030	−0.304949	−0.152474	+	0.988307i	$0.548724\pi$
$110$	5426.29	4.70342
$111$	0	0
$112$	−4553.98	−3.84206
$113$	1883.39	1.56792	0.783960	−	0.620812i	$-0.213197\pi$
$113$	1883.39	1.56792	0.783960	+	0.620812i	$0.213197\pi$
$114$	0	0
$115$	−2136.86	−1.73273
$116$	−2199.18	−1.76025
$117$	0	0
$118$	1965.19	1.53314
$119$	1788.97	1.37811
$120$	0	0
$121$	3252.91	2.44396
$122$	1613.41	1.19731
$123$	0	0
$124$	−4885.10	−3.53787
$125$	−625.231	−0.447379
$126$	0	0
$127$	−2515.47	−1.75758	−0.878789	−	0.477211i	$-0.841648\pi$
$127$	−2515.47	−1.75758	−0.878789	+	0.477211i	$0.841648\pi$
$128$	−8447.34	−5.83318
$129$	0	0
$130$	4834.66	3.26175
$131$	2395.00	1.59734	0.798672	−	0.601766i	$-0.205536\pi$
$131$	2395.00	1.59734	0.798672	+	0.601766i	$0.205536\pi$
$132$	0	0
$133$	−229.806	−0.149825
$134$	2835.10	1.82773
$135$	0	0
$136$	9437.74	5.95058
$137$	1671.81	1.04257	0.521287	−	0.853382i	$-0.325452\pi$
$137$	1671.81	1.04257	0.521287	+	0.853382i	$0.325452\pi$
$138$	0	0
$139$	940.493	0.573896	0.286948	−	0.957946i	$-0.407359\pi$
$139$	940.493	0.573896	0.286948	+	0.957946i	$0.407359\pi$
$140$	−5303.78	−3.20179
$141$	0	0
$142$	−4742.49	−2.80268
$143$	4084.13	2.38834
$144$	0	0
$145$	−1367.57	−0.783245
$146$	−2944.54	−1.66912
$147$	0	0
$148$	2115.90	1.17517
$149$	−1520.42	−0.835958	−0.417979	−	0.908457i	$-0.637261\pi$
$149$	−1520.42	−0.835958	−0.417979	+	0.908457i	$0.637261\pi$
$150$	0	0
$151$	1572.62	0.847539	0.423769	−	0.905770i	$-0.360707\pi$
$151$	1572.62	0.847539	0.423769	+	0.905770i	$0.360707\pi$
$152$	−1212.34	−0.646934
$153$	0	0
$154$	−6031.54	−3.15608
$155$	−3037.82	−1.57422
$156$	0	0
$157$	−2672.08	−1.35831	−0.679156	−	0.733994i	$-0.737654\pi$
$157$	−2672.08	−1.35831	−0.679156	+	0.733994i	$0.737654\pi$
$158$	2166.25	1.09075
$159$	0	0
$160$	−13164.0	−6.50441
$161$	2375.21	1.16269
$162$	0	0
$163$	−536.441	−0.257775	−0.128888	−	0.991659i	$-0.541141\pi$
$163$	−536.441	−0.257775	−0.128888	+	0.991659i	$0.541141\pi$
$164$	−7630.44	−3.63315
$165$	0	0
$166$	1529.73	0.715240
$167$	−3097.39	−1.43523	−0.717615	−	0.696440i	$-0.754766\pi$
$167$	−3097.39	−1.43523	−0.717615	+	0.696440i	$0.754766\pi$
$168$	0	0
$169$	1441.84	0.656278
$170$	8976.62	4.04985
$171$	0	0
$172$	2703.26	1.19838
$173$	−1982.21	−0.871125	−0.435563	−	0.900158i	$-0.643451\pi$
$173$	−1982.21	−0.871125	−0.435563	+	0.900158i	$0.643451\pi$
$174$	0	0
$175$	−1301.60	−0.562239
$176$	−19303.4	−8.26733
$177$	0	0
$178$	−6598.61	−2.77858
$179$	3041.14	1.26986	0.634932	−	0.772568i	$-0.281028\pi$
$179$	3041.14	1.26986	0.634932	+	0.772568i	$0.281028\pi$
$180$	0	0
$181$	95.5415	0.0392351	0.0196175	−	0.999808i	$-0.493755\pi$
$181$	95.5415	0.0392351	0.0196175	+	0.999808i	$0.493755\pi$
$182$	−5373.93	−2.18869
$183$	0	0
$184$	12530.5	5.02043
$185$	1315.78	0.522909
$186$	0	0
$187$	7583.09	2.96540
$188$	−9042.05	−3.50776
$189$	0	0
$190$	−1153.11	−0.440291
$191$	886.005	0.335650	0.167825	−	0.985817i	$-0.446326\pi$
$191$	886.005	0.335650	0.167825	+	0.985817i	$0.446326\pi$
$192$	0	0
$193$	−1024.13	−0.381962	−0.190981	−	0.981594i	$-0.561167\pi$
$193$	−1024.13	−0.381962	−0.190981	+	0.981594i	$0.561167\pi$
$194$	−9063.06	−3.35407
$195$	0	0
$196$	−2030.65	−0.740032
$197$	−519.116	−0.187744	−0.0938718	−	0.995584i	$-0.529924\pi$
$197$	−519.116	−0.187744	−0.0938718	+	0.995584i	$0.529924\pi$
$198$	0	0
$199$	2004.37	0.714001	0.357001	−	0.934104i	$-0.383799\pi$
$199$	2004.37	0.714001	0.357001	+	0.934104i	$0.383799\pi$
$200$	−6866.62	−2.42772
$201$	0	0
$202$	2463.58	0.858105
$203$	1520.11	0.525571
$204$	0	0
$205$	−4745.02	−1.61662
$206$	5751.85	1.94539
$207$	0	0
$208$	−17198.8	−5.73327
$209$	−974.100	−0.322392
$210$	0	0
$211$	673.448	0.219725	0.109863	−	0.993947i	$-0.464959\pi$
$211$	673.448	0.219725	0.109863	+	0.993947i	$0.464959\pi$
$212$	−1519.50	−0.492263
$213$	0	0
$214$	4541.07	1.45057
$215$	1681.03	0.533235
$216$	0	0
$217$	3376.67	1.05633
$218$	1935.54	0.601336
$219$	0	0
$220$	−22481.6	−6.88960
$221$	6756.31	2.05646
$222$	0	0
$223$	−3825.09	−1.14864	−0.574321	−	0.818630i	$-0.694734\pi$
$223$	−3825.09	−1.14864	−0.574321	+	0.818630i	$0.694734\pi$
$224$	14632.3	4.36457
$225$	0	0
$226$	−10504.5	−3.09182
$227$	2927.45	0.855954	0.427977	−	0.903790i	$-0.359226\pi$
$227$	2927.45	0.855954	0.427977	+	0.903790i	$0.359226\pi$
$228$	0	0
$229$	2672.73	0.771262	0.385631	−	0.922653i	$-0.373984\pi$
$229$	2672.73	0.771262	0.385631	+	0.922653i	$0.373984\pi$
$230$	11918.2	3.41681
$231$	0	0
$232$	8019.37	2.26938
$233$	2333.99	0.656243	0.328121	−	0.944636i	$-0.393584\pi$
$233$	2333.99	0.656243	0.328121	+	0.944636i	$0.393584\pi$
$234$	0	0
$235$	−5622.83	−1.56082
$236$	−8141.96	−2.24575
$237$	0	0
$238$	−9977.89	−2.71752
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	2531.77	0.676705	0.338352	−	0.941020i	$-0.390130\pi$
$241$	2531.77	0.676705	0.338352	+	0.941020i	$0.390130\pi$
$242$	−18142.9	−4.81931
$243$	0	0
$244$	−6684.51	−1.75382
$245$	−1262.77	−0.329286
$246$	0	0
$247$	−867.895	−0.223574
$248$	17813.7	4.56116
$249$	0	0
$250$	3487.19	0.882198
$251$	−4699.88	−1.18189	−0.590944	−	0.806712i	$-0.701245\pi$
$251$	−4699.88	−1.18189	−0.590944	+	0.806712i	$0.701245\pi$
$252$	0	0
$253$	10068.1	2.50187
$254$	14029.9	3.46581
$255$	0	0
$256$	24486.2	5.97809
$257$	−3046.48	−0.739433	−0.369717	−	0.929145i	$-0.620545\pi$
$257$	−3046.48	−0.739433	−0.369717	+	0.929145i	$0.620545\pi$
$258$	0	0
$259$	−1462.55	−0.350881
$260$	−20030.5	−4.77784
$261$	0	0
$262$	−13358.0	−3.14984
$263$	−1425.08	−0.334123	−0.167062	−	0.985946i	$-0.553428\pi$
$263$	−1425.08	−0.334123	−0.167062	+	0.985946i	$0.553428\pi$
$264$	0	0
$265$	−944.908	−0.219039
$266$	1281.73	0.295443
$267$	0	0
$268$	−11746.1	−2.67727
$269$	−850.575	−0.192790	−0.0963949	−	0.995343i	$-0.530731\pi$
$269$	−850.575	−0.192790	−0.0963949	+	0.995343i	$0.530731\pi$
$270$	0	0
$271$	−5087.22	−1.14032	−0.570160	−	0.821534i	$-0.693119\pi$
$271$	−5087.22	−1.14032	−0.570160	+	0.821534i	$0.693119\pi$
$272$	−31933.3	−7.11854
$273$	0	0
$274$	−9324.45	−2.05588
$275$	−5517.23	−1.20982
$276$	0	0
$277$	3088.83	0.669999	0.335000	−	0.942218i	$-0.391264\pi$
$277$	3088.83	0.669999	0.335000	+	0.942218i	$0.391264\pi$
$278$	−5245.55	−1.13168
$279$	0	0
$280$	19340.3	4.12788
$281$	−1553.23	−0.329743	−0.164871	−	0.986315i	$-0.552721\pi$
$281$	−1553.23	−0.329743	−0.164871	+	0.986315i	$0.552721\pi$
$282$	0	0
$283$	1236.19	0.259659	0.129830	−	0.991536i	$-0.458557\pi$
$283$	1236.19	0.259659	0.129830	+	0.991536i	$0.458557\pi$
$284$	19648.6	4.10538
$285$	0	0
$286$	−22779.0	−4.70962
$287$	5274.29	1.08478
$288$	0	0
$289$	7631.58	1.55334
$290$	7627.55	1.54450
$291$	0	0
$292$	12199.5	2.44494
$293$	−4404.58	−0.878219	−0.439110	−	0.898433i	$-0.644706\pi$
$293$	−4404.58	−0.878219	−0.439110	+	0.898433i	$0.644706\pi$
$294$	0	0
$295$	−5063.11	−0.999274
$296$	−7715.68	−1.51508
$297$	0	0
$298$	8480.07	1.64845
$299$	8970.34	1.73501
$300$	0	0
$301$	−1868.54	−0.357810
$302$	−8771.23	−1.67128
$303$	0	0
$304$	4102.05	0.773911
$305$	−4156.79	−0.780385
$306$	0	0
$307$	−1774.03	−0.329801	−0.164901	−	0.986310i	$-0.552730\pi$
$307$	−1774.03	−0.329801	−0.164901	+	0.986310i	$0.552730\pi$
$308$	24989.3	4.62304
$309$	0	0
$310$	16943.3	3.10424
$311$	−5383.66	−0.981605	−0.490802	−	0.871271i	$-0.663296\pi$
$311$	−5383.66	−0.981605	−0.490802	+	0.871271i	$0.663296\pi$
$312$	0	0
$313$	−7887.26	−1.42433	−0.712164	−	0.702014i	$-0.752285\pi$
$313$	−7887.26	−1.42433	−0.712164	+	0.702014i	$0.752285\pi$
$314$	14903.4	2.67849
$315$	0	0
$316$	−8974.99	−1.59773
$317$	4098.78	0.726216	0.363108	−	0.931747i	$-0.381716\pi$
$317$	4098.78	0.726216	0.363108	+	0.931747i	$0.381716\pi$
$318$	0	0
$319$	6443.45	1.13092
$320$	40645.5	7.10048
$321$	0	0
$322$	−13247.6	−2.29274
$323$	−1611.44	−0.277594
$324$	0	0
$325$	−4915.69	−0.838996
$326$	2991.97	0.508313
$327$	0	0
$328$	27824.6	4.68401
$329$	6250.02	1.04734
$330$	0	0
$331$	−6892.46	−1.14454	−0.572272	−	0.820064i	$-0.693938\pi$
$331$	−6892.46	−1.14454	−0.572272	+	0.820064i	$0.693938\pi$
$332$	−6337.81	−1.04769
$333$	0	0
$334$	17275.5	2.83017
$335$	−7304.37	−1.19128
$336$	0	0
$337$	−1872.81	−0.302725	−0.151363	−	0.988478i	$-0.548366\pi$
$337$	−1872.81	−0.302725	−0.151363	+	0.988478i	$0.548366\pi$
$338$	−8041.80	−1.29413
$339$	0	0
$340$	−37191.0	−5.93225
$341$	14313.0	2.27300
$342$	0	0
$343$	6882.21	1.08340
$344$	−9857.50	−1.54500
$345$	0	0
$346$	11055.7	1.71779
$347$	−11073.5	−1.71313	−0.856565	−	0.516039i	$-0.827406\pi$
$347$	−11073.5	−1.71313	−0.856565	+	0.516039i	$0.827406\pi$
$348$	0	0
$349$	−3738.88	−0.573461	−0.286730	−	0.958011i	$-0.592568\pi$
$349$	−3738.88	−0.573461	−0.286730	+	0.958011i	$0.592568\pi$
$350$	7259.62	1.10869
$351$	0	0
$352$	62023.5	9.39166
$353$	2432.23	0.366727	0.183363	−	0.983045i	$-0.441301\pi$
$353$	2432.23	0.366727	0.183363	+	0.983045i	$0.441301\pi$
$354$	0	0
$355$	12218.6	1.82674
$356$	27338.7	4.07008
$357$	0	0
$358$	−16961.8	−2.50408
$359$	5048.88	0.742255	0.371127	−	0.928582i	$-0.378971\pi$
$359$	5048.88	0.742255	0.371127	+	0.928582i	$0.378971\pi$
$360$	0	0
$361$	−6652.00	−0.969821
$362$	−532.878	−0.0773686
$363$	0	0
$364$	22264.7	3.20601
$365$	7586.32	1.08791
$366$	0	0
$367$	1295.77	0.184302	0.0921509	−	0.995745i	$-0.470626\pi$
$367$	1295.77	0.184302	0.0921509	+	0.995745i	$0.470626\pi$
$368$	−42397.8	−6.00581
$369$	0	0
$370$	−7338.70	−1.03114
$371$	1050.31	0.146979
$372$	0	0
$373$	−11072.0	−1.53696	−0.768478	−	0.639876i	$-0.778986\pi$
$373$	−11072.0	−1.53696	−0.768478	+	0.639876i	$0.778986\pi$
$374$	−42294.3	−5.84755
$375$	0	0
$376$	32972.0	4.52235
$377$	5740.92	0.784277
$378$	0	0
$379$	6086.55	0.824921	0.412461	−	0.910975i	$-0.364669\pi$
$379$	6086.55	0.824921	0.412461	+	0.910975i	$0.364669\pi$
$380$	4777.44	0.644941
$381$	0	0
$382$	−4941.65	−0.661876
$383$	1880.58	0.250896	0.125448	−	0.992100i	$-0.459963\pi$
$383$	1880.58	0.250896	0.125448	+	0.992100i	$0.459963\pi$
$384$	0	0
$385$	15539.7	2.05708
$386$	5712.04	0.753200
$387$	0	0
$388$	37549.1	4.91306
$389$	3610.23	0.470555	0.235277	−	0.971928i	$-0.424400\pi$
$389$	3610.23	0.470555	0.235277	+	0.971928i	$0.424400\pi$
$390$	0	0
$391$	16655.4	2.15422
$392$	7404.80	0.954079
$393$	0	0
$394$	2895.34	0.370217
$395$	−5581.14	−0.710930
$396$	0	0
$397$	−844.801	−0.106799	−0.0533997	−	0.998573i	$-0.517006\pi$
$397$	−844.801	−0.106799	−0.0533997	+	0.998573i	$0.517006\pi$
$398$	−11179.3	−1.40796
$399$	0	0
$400$	23233.7	2.90422
$401$	−344.453	−0.0428956	−0.0214478	−	0.999770i	$-0.506828\pi$
$401$	−344.453	−0.0428956	−0.0214478	+	0.999770i	$0.506828\pi$
$402$	0	0
$403$	12752.5	1.57629
$404$	−10206.9	−1.25696
$405$	0	0
$406$	−8478.34	−1.03639
$407$	−6199.44	−0.755024
$408$	0	0
$409$	4118.33	0.497893	0.248947	−	0.968517i	$-0.419916\pi$
$409$	4118.33	0.497893	0.248947	+	0.968517i	$0.419916\pi$
$410$	26465.1	3.18785
$411$	0	0
$412$	−23830.5	−2.84962
$413$	5627.86	0.670530
$414$	0	0
$415$	−3941.19	−0.466182
$416$	55261.1	6.51298
$417$	0	0
$418$	5432.99	0.635733
$419$	5692.07	0.663666	0.331833	−	0.943338i	$-0.392333\pi$
$419$	5692.07	0.663666	0.331833	+	0.943338i	$0.392333\pi$
$420$	0	0
$421$	−16147.6	−1.86933	−0.934664	−	0.355533i	$-0.884299\pi$
$421$	−16147.6	−1.86933	−0.934664	+	0.355533i	$0.884299\pi$
$422$	−3756.12	−0.433282
$423$	0	0
$424$	5540.90	0.634646
$425$	−9127.07	−1.04171
$426$	0	0
$427$	4620.45	0.523652
$428$	−18814.1	−2.12480
$429$	0	0
$430$	−9375.87	−1.05150
$431$	9046.80	1.01107	0.505533	−	0.862808i	$-0.331296\pi$
$431$	9046.80	1.01107	0.505533	+	0.862808i	$0.331296\pi$
$432$	0	0
$433$	−7938.26	−0.881035	−0.440518	−	0.897744i	$-0.645205\pi$
$433$	−7938.26	−0.881035	−0.440518	+	0.897744i	$0.645205\pi$
$434$	−18833.2	−2.08300
$435$	0	0
$436$	−8019.14	−0.880842
$437$	−2139.50	−0.234202
$438$	0	0
$439$	−8074.87	−0.877888	−0.438944	−	0.898515i	$-0.644647\pi$
$439$	−8074.87	−0.877888	−0.438944	+	0.898515i	$0.644647\pi$
$440$	81979.9	8.88236
$441$	0	0
$442$	−37683.0	−4.05519
$443$	−9070.09	−0.972761	−0.486380	−	0.873747i	$-0.661683\pi$
$443$	−9070.09	−0.972761	−0.486380	+	0.873747i	$0.661683\pi$
$444$	0	0
$445$	17000.7	1.81103
$446$	21334.2	2.26504
$447$	0	0
$448$	−45179.2	−4.76454
$449$	6018.46	0.632580	0.316290	−	0.948663i	$-0.397563\pi$
$449$	6018.46	0.632580	0.316290	+	0.948663i	$0.397563\pi$
$450$	0	0
$451$	22356.7	2.33422
$452$	43521.3	4.52892
$453$	0	0
$454$	−16327.7	−1.68788
$455$	13845.4	1.42656
$456$	0	0
$457$	−17417.6	−1.78285	−0.891426	−	0.453167i	$-0.850294\pi$
$457$	−17417.6	−1.78285	−0.891426	+	0.453167i	$0.850294\pi$
$458$	−14907.0	−1.52087
$459$	0	0
$460$	−49378.5	−5.00496
$461$	−13978.1	−1.41220	−0.706099	−	0.708113i	$-0.749547\pi$
$461$	−13978.1	−1.41220	−0.706099	+	0.708113i	$0.749547\pi$
$462$	0	0
$463$	−8814.45	−0.884756	−0.442378	−	0.896829i	$-0.645865\pi$
$463$	−8814.45	−0.884756	−0.442378	+	0.896829i	$0.645865\pi$
$464$	−27134.1	−2.71481
$465$	0	0
$466$	−13017.7	−1.29406
$467$	−4542.14	−0.450075	−0.225038	−	0.974350i	$-0.572250\pi$
$467$	−4542.14	−0.450075	−0.225038	+	0.974350i	$0.572250\pi$
$468$	0	0
$469$	8119.11	0.799373
$470$	31361.1	3.07783
$471$	0	0
$472$	29689.9	2.89531
$473$	−7920.36	−0.769934
$474$	0	0
$475$	1172.43	0.113253
$476$	41339.4	3.98065
$477$	0	0
$478$	−1333.01	−0.127553
$479$	−6423.37	−0.612717	−0.306358	−	0.951916i	$-0.599111\pi$
$479$	−6423.37	−0.612717	−0.306358	+	0.951916i	$0.599111\pi$
$480$	0	0
$481$	−5523.52	−0.523598
$482$	−14120.8	−1.33441
$483$	0	0
$484$	75168.0	7.05935
$485$	23350.1	2.18613
$486$	0	0
$487$	−15432.9	−1.43600	−0.717998	−	0.696045i	$-0.754941\pi$
$487$	−15432.9	−1.43600	−0.717998	+	0.696045i	$0.754941\pi$
$488$	24375.2	2.26110
$489$	0	0
$490$	7043.01	0.649328
$491$	5832.90	0.536120	0.268060	−	0.963402i	$-0.413617\pi$
$491$	5832.90	0.536120	0.268060	+	0.963402i	$0.413617\pi$
$492$	0	0
$493$	10659.3	0.973773
$494$	4840.64	0.440871
$495$	0	0
$496$	−60273.9	−5.45641
$497$	−13581.4	−1.22578
$498$	0	0
$499$	3854.90	0.345830	0.172915	−	0.984937i	$-0.444681\pi$
$499$	3854.90	0.345830	0.172915	+	0.984937i	$0.444681\pi$
$500$	−14447.8	−1.29225
$501$	0	0
$502$	26213.3	2.33060
$503$	−10380.5	−0.920165	−0.460083	−	0.887876i	$-0.652180\pi$
$503$	−10380.5	−0.920165	−0.460083	+	0.887876i	$0.652180\pi$
$504$	0	0
$505$	−6347.19	−0.559299
$506$	−56154.0	−4.93350
$507$	0	0
$508$	−58127.4	−5.07674
$509$	19981.3	1.73999	0.869997	−	0.493058i	$-0.164121\pi$
$509$	19981.3	1.73999	0.869997	+	0.493058i	$0.164121\pi$
$510$	0	0
$511$	−8432.51	−0.730004
$512$	−68992.0	−5.95516
$513$	0	0
$514$	16991.6	1.45811
$515$	−14819.1	−1.26798
$516$	0	0
$517$	26492.6	2.25366
$518$	8157.27	0.691911
$519$	0	0
$520$	73041.7	6.15979
$521$	−10235.1	−0.860672	−0.430336	−	0.902669i	$-0.641605\pi$
$521$	−10235.1	−0.860672	−0.430336	+	0.902669i	$0.641605\pi$
$522$	0	0
$523$	10132.2	0.847136	0.423568	−	0.905864i	$-0.360778\pi$
$523$	10132.2	0.847136	0.423568	+	0.905864i	$0.360778\pi$
$524$	55343.5	4.61391
$525$	0	0
$526$	7948.33	0.658866
$527$	23677.8	1.95716
$528$	0	0
$529$	9946.38	0.817488
$530$	5270.18	0.431928
$531$	0	0
$532$	−5310.33	−0.432767
$533$	19919.1	1.61875
$534$	0	0
$535$	−11699.6	−0.945456
$536$	42832.5	3.45165
$537$	0	0
$538$	4744.03	0.380167
$539$	5949.65	0.475454
$540$	0	0
$541$	−15419.9	−1.22542	−0.612712	−	0.790306i	$-0.709921\pi$
$541$	−15419.9	−1.22542	−0.612712	+	0.790306i	$0.709921\pi$
$542$	28373.7	2.24863
$543$	0	0
$544$	102604.	8.08664
$545$	−4986.73	−0.391942
$546$	0	0
$547$	21732.5	1.69875	0.849374	−	0.527791i	$-0.176979\pi$
$547$	21732.5	1.69875	0.849374	+	0.527791i	$0.176979\pi$
$548$	38632.1	3.01146
$549$	0	0
$550$	30772.1	2.38568
$551$	−1369.26	−0.105866
$552$	0	0
$553$	6203.67	0.477047
$554$	−17227.8	−1.32119
$555$	0	0
$556$	21732.8	1.65769
$557$	−770.209	−0.0585903	−0.0292951	−	0.999571i	$-0.509326\pi$
$557$	−770.209	−0.0585903	−0.0292951	+	0.999571i	$0.509326\pi$
$558$	0	0
$559$	−7056.81	−0.533938
$560$	−65439.6	−4.93808
$561$	0	0
$562$	8663.04	0.650228
$563$	−11660.7	−0.872895	−0.436448	−	0.899730i	$-0.643764\pi$
$563$	−11660.7	−0.872895	−0.436448	+	0.899730i	$0.643764\pi$
$564$	0	0
$565$	27063.9	2.01520
$566$	−6894.76	−0.512029
$567$	0	0
$568$	−71649.1	−5.29283
$569$	−19237.4	−1.41735	−0.708675	−	0.705535i	$-0.750707\pi$
$569$	−19237.4	−1.41735	−0.708675	+	0.705535i	$0.750707\pi$
$570$	0	0
$571$	−11812.8	−0.865765	−0.432883	−	0.901450i	$-0.642504\pi$
$571$	−11812.8	−0.865765	−0.432883	+	0.901450i	$0.642504\pi$
$572$	94375.7	6.89868
$573$	0	0
$574$	−29417.1	−2.13910
$575$	−12118.0	−0.878879
$576$	0	0
$577$	9039.46	0.652197	0.326098	−	0.945336i	$-0.394266\pi$
$577$	9039.46	0.652197	0.326098	+	0.945336i	$0.394266\pi$
$578$	−42564.7	−3.06308
$579$	0	0
$580$	−31601.7	−2.26239
$581$	4380.80	0.312816
$582$	0	0
$583$	4452.03	0.316268
$584$	−44485.8	−3.15212
$585$	0	0
$586$	24566.3	1.73178
$587$	−15916.4	−1.11915	−0.559574	−	0.828781i	$-0.689035\pi$
$587$	−15916.4	−1.11915	−0.559574	+	0.828781i	$0.689035\pi$
$588$	0	0
$589$	−3041.57	−0.212777
$590$	28239.2	1.97049
$591$	0	0
$592$	26106.6	1.81246
$593$	13088.7	0.906387	0.453194	−	0.891412i	$-0.350285\pi$
$593$	13088.7	0.906387	0.453194	+	0.891412i	$0.350285\pi$
$594$	0	0
$595$	25707.1	1.77124
$596$	−35133.8	−2.41466
$597$	0	0
$598$	−50031.6	−3.42131
$599$	−23912.9	−1.63114	−0.815571	−	0.578658i	$-0.803577\pi$
$599$	−23912.9	−1.63114	−0.815571	+	0.578658i	$0.803577\pi$
$600$	0	0
$601$	−7423.50	−0.503845	−0.251923	−	0.967747i	$-0.581063\pi$
$601$	−7423.50	−0.503845	−0.251923	+	0.967747i	$0.581063\pi$
$602$	10421.7	0.705574
$603$	0	0
$604$	36340.1	2.44811
$605$	46743.5	3.14115
$606$	0	0
$607$	6424.37	0.429584	0.214792	−	0.976660i	$-0.431093\pi$
$607$	6424.37	0.429584	0.214792	+	0.976660i	$0.431093\pi$
$608$	−13180.3	−0.879161
$609$	0	0
$610$	23184.3	1.53886
$611$	23604.1	1.56288
$612$	0	0
$613$	12568.3	0.828109	0.414054	−	0.910252i	$-0.364112\pi$
$613$	12568.3	0.828109	0.414054	+	0.910252i	$0.364112\pi$
$614$	9894.53	0.650343
$615$	0	0
$616$	−91124.1	−5.96022
$617$	−19534.1	−1.27457	−0.637287	−	0.770626i	$-0.719944\pi$
$617$	−19534.1	−1.27457	−0.637287	+	0.770626i	$0.719944\pi$
$618$	0	0
$619$	−17015.2	−1.10485	−0.552423	−	0.833564i	$-0.686297\pi$
$619$	−17015.2	−1.10485	−0.552423	+	0.833564i	$0.686297\pi$
$620$	−70197.7	−4.54711
$621$	0	0
$622$	30027.1	1.93565
$623$	−18897.0	−1.21524
$624$	0	0
$625$	−19170.6	−1.22692
$626$	43990.8	2.80867
$627$	0	0
$628$	−61746.1	−3.92347
$629$	−10255.6	−0.650109
$630$	0	0
$631$	−23196.5	−1.46345	−0.731727	−	0.681598i	$-0.761285\pi$
$631$	−23196.5	−1.46345	−0.731727	+	0.681598i	$0.761285\pi$
$632$	32727.5	2.05986
$633$	0	0
$634$	−22860.7	−1.43204
$635$	−36146.7	−2.25896
$636$	0	0
$637$	5300.97	0.329721
$638$	−35938.0	−2.23009
$639$	0	0
$640$	−121386.	−7.49721
$641$	−756.951	−0.0466424	−0.0233212	−	0.999728i	$-0.507424\pi$
$641$	−756.951	−0.0466424	−0.0233212	+	0.999728i	$0.507424\pi$
$642$	0	0
$643$	6502.91	0.398833	0.199417	−	0.979915i	$-0.436095\pi$
$643$	6502.91	0.398833	0.199417	+	0.979915i	$0.436095\pi$
$644$	54886.2	3.35842
$645$	0	0
$646$	8987.71	0.547394
$647$	−1653.32	−0.100462	−0.0502310	−	0.998738i	$-0.515996\pi$
$647$	−1653.32	−0.100462	−0.0502310	+	0.998738i	$0.515996\pi$
$648$	0	0
$649$	23855.4	1.44284
$650$	27417.0	1.65444
$651$	0	0
$652$	−12396.0	−0.744581
$653$	−18474.2	−1.10712	−0.553561	−	0.832809i	$-0.686731\pi$
$653$	−18474.2	−1.10712	−0.553561	+	0.832809i	$0.686731\pi$
$654$	0	0
$655$	34415.6	2.05302
$656$	−94146.6	−5.60337
$657$	0	0
$658$	−34859.1	−2.06527
$659$	−23387.7	−1.38248	−0.691240	−	0.722625i	$-0.742935\pi$
$659$	−23387.7	−1.38248	−0.691240	+	0.722625i	$0.742935\pi$
$660$	0	0
$661$	−18677.2	−1.09903	−0.549515	−	0.835484i	$-0.685188\pi$
$661$	−18677.2	−1.09903	−0.549515	+	0.835484i	$0.685188\pi$
$662$	38442.3	2.25695
$663$	0	0
$664$	23111.0	1.35072
$665$	−3302.25	−0.192565
$666$	0	0
$667$	14152.3	0.821559
$668$	−71574.3	−4.14565
$669$	0	0
$670$	40739.7	2.34912
$671$	19585.2	1.12679
$672$	0	0
$673$	27159.9	1.55563	0.777814	−	0.628494i	$-0.216328\pi$
$673$	27159.9	1.55563	0.777814	+	0.628494i	$0.216328\pi$
$674$	10445.5	0.596952
$675$	0	0
$676$	33318.0	1.89565
$677$	−9668.12	−0.548857	−0.274428	−	0.961608i	$-0.588489\pi$
$677$	−9668.12	−0.548857	−0.274428	+	0.961608i	$0.588489\pi$
$678$	0	0
$679$	−25954.6	−1.46693
$680$	135618.	7.64811
$681$	0	0
$682$	−79830.1	−4.48219
$683$	22837.5	1.27943	0.639717	−	0.768611i	$-0.279052\pi$
$683$	22837.5	1.27943	0.639717	+	0.768611i	$0.279052\pi$
$684$	0	0
$685$	24023.5	1.33999
$686$	−38385.2	−2.13637
$687$	0	0
$688$	33353.6	1.84825
$689$	3966.63	0.219328
$690$	0	0
$691$	3814.19	0.209984	0.104992	−	0.994473i	$-0.466518\pi$
$691$	3814.19	0.209984	0.104992	+	0.994473i	$0.466518\pi$
$692$	−45804.7	−2.51624
$693$	0	0
$694$	61761.8	3.37817
$695$	13514.6	0.737611
$696$	0	0
$697$	36984.2	2.00987
$698$	20853.4	1.13082
$699$	0	0
$700$	−30077.3	−1.62402
$701$	−8033.74	−0.432854	−0.216427	−	0.976299i	$-0.569440\pi$
$701$	−8033.74	−0.432854	−0.216427	+	0.976299i	$0.569440\pi$
$702$	0	0
$703$	1317.41	0.0706784
$704$	−191506.	−10.2523
$705$	0	0
$706$	−13565.6	−0.723158
$707$	7055.16	0.375299
$708$	0	0
$709$	13699.6	0.725668	0.362834	−	0.931854i	$-0.381809\pi$
$709$	13699.6	0.725668	0.362834	+	0.931854i	$0.381809\pi$
$710$	−68148.4	−3.60220
$711$	0	0
$712$	−99691.3	−5.24732
$713$	31437.0	1.65123
$714$	0	0
$715$	58687.9	3.06966
$716$	70274.4	3.66799
$717$	0	0
$718$	−28159.8	−1.46367
$719$	19109.5	0.991186	0.495593	−	0.868555i	$-0.334951\pi$
$719$	19109.5	0.991186	0.495593	+	0.868555i	$0.334951\pi$
$720$	0	0
$721$	16472.0	0.850834
$722$	37101.2	1.91241
$723$	0	0
$724$	2207.77	0.113330
$725$	−7755.38	−0.397280
$726$	0	0
$727$	−16594.7	−0.846578	−0.423289	−	0.905995i	$-0.639124\pi$
$727$	−16594.7	−0.846578	−0.423289	+	0.905995i	$0.639124\pi$
$728$	−81188.9	−4.13332
$729$	0	0
$730$	−42312.3	−2.14527
$731$	−13102.5	−0.662947
$732$	0	0
$733$	−36811.1	−1.85491	−0.927454	−	0.373936i	$-0.878008\pi$
$733$	−36811.1	−1.85491	−0.927454	+	0.373936i	$0.878008\pi$
$734$	−7227.10	−0.363430
$735$	0	0
$736$	136228.	6.82259
$737$	34415.3	1.72009
$738$	0	0
$739$	21945.4	1.09239	0.546193	−	0.837659i	$-0.316077\pi$
$739$	21945.4	1.09239	0.546193	+	0.837659i	$0.316077\pi$
$740$	30405.0	1.51042
$741$	0	0
$742$	−5858.02	−0.289831
$743$	−19115.2	−0.943834	−0.471917	−	0.881643i	$-0.656438\pi$
$743$	−19115.2	−0.943834	−0.471917	+	0.881643i	$0.656438\pi$
$744$	0	0
$745$	−21848.1	−1.07443
$746$	61753.3	3.03076
$747$	0	0
$748$	175229.	8.56553
$749$	13004.6	0.634418
$750$	0	0
$751$	−27030.6	−1.31339	−0.656697	−	0.754154i	$-0.728047\pi$
$751$	−27030.6	−1.31339	−0.656697	+	0.754154i	$0.728047\pi$
$752$	−111563.	−5.40998
$753$	0	0
$754$	−32019.7	−1.54654
$755$	22598.2	1.08932
$756$	0	0
$757$	7649.19	0.367258	0.183629	−	0.982996i	$-0.441215\pi$
$757$	7649.19	0.367258	0.183629	+	0.982996i	$0.441215\pi$
$758$	−33947.4	−1.62668
$759$	0	0
$760$	−17421.1	−0.831485
$761$	40543.7	1.93129	0.965643	−	0.259872i	$-0.0836802\pi$
$761$	40543.7	1.93129	0.965643	+	0.259872i	$0.0836802\pi$
$762$	0	0
$763$	5542.96	0.263000
$764$	20473.7	0.969520
$765$	0	0
$766$	−10488.8	−0.494747
$767$	21254.4	1.00059
$768$	0	0
$769$	28218.7	1.32327	0.661634	−	0.749827i	$-0.269863\pi$
$769$	28218.7	1.32327	0.661634	+	0.749827i	$0.269863\pi$
$770$	−86671.8	−4.05641
$771$	0	0
$772$	−23665.6	−1.10329
$773$	30546.6	1.42132	0.710662	−	0.703533i	$-0.248395\pi$
$773$	30546.6	1.42132	0.710662	+	0.703533i	$0.248395\pi$
$774$	0	0
$775$	−17227.3	−0.798480
$776$	−136924.	−6.33413
$777$	0	0
$778$	−20135.8	−0.927898
$779$	−4750.88	−0.218508
$780$	0	0
$781$	−57569.0	−2.63762
$782$	−92894.7	−4.24797
$783$	0	0
$784$	−25054.7	−1.14134
$785$	−38397.1	−1.74580
$786$	0	0
$787$	9440.72	0.427605	0.213803	−	0.976877i	$-0.431415\pi$
$787$	9440.72	0.427605	0.213803	+	0.976877i	$0.431415\pi$
$788$	−11995.7	−0.542296
$789$	0	0
$790$	31128.5	1.40190
$791$	−30082.7	−1.35223
$792$	0	0
$793$	17449.8	0.781414
$794$	4711.84	0.210600
$795$	0	0
$796$	46316.9	2.06239
$797$	18359.7	0.815977	0.407988	−	0.912987i	$-0.366230\pi$
$797$	18359.7	0.815977	0.407988	+	0.912987i	$0.366230\pi$
$798$	0	0
$799$	43826.2	1.94050
$800$	−74652.0	−3.29918
$801$	0	0
$802$	1921.17	0.0845869
$803$	−35743.7	−1.57082
$804$	0	0
$805$	34131.2	1.49437
$806$	−71126.3	−3.10833
$807$	0	0
$808$	37219.6	1.62052
$809$	37498.7	1.62965	0.814823	−	0.579709i	$-0.196834\pi$
$809$	37498.7	1.62965	0.814823	+	0.579709i	$0.196834\pi$
$810$	0	0
$811$	−33350.0	−1.44399	−0.721995	−	0.691898i	$-0.756775\pi$
$811$	−33350.0	−1.44399	−0.721995	+	0.691898i	$0.756775\pi$
$812$	35126.6	1.51811
$813$	0	0
$814$	34577.0	1.48885
$815$	−7708.53	−0.331311
$816$	0	0
$817$	1683.11	0.0720741
$818$	−22969.8	−0.981808
$819$	0	0
$820$	−109648.	−4.66958
$821$	−17321.0	−0.736304	−0.368152	−	0.929766i	$-0.620009\pi$
$821$	−17321.0	−0.736304	−0.368152	+	0.929766i	$0.620009\pi$
$822$	0	0
$823$	−19991.5	−0.846730	−0.423365	−	0.905959i	$-0.639151\pi$
$823$	−19991.5	−0.846730	−0.423365	+	0.905959i	$0.639151\pi$
$824$	86898.5	3.67385
$825$	0	0
$826$	−31389.1	−1.32224
$827$	−7462.52	−0.313781	−0.156891	−	0.987616i	$-0.550147\pi$
$827$	−7462.52	−0.313781	−0.156891	+	0.987616i	$0.550147\pi$
$828$	0	0
$829$	−40382.8	−1.69186	−0.845930	−	0.533294i	$-0.820954\pi$
$829$	−40382.8	−1.69186	−0.845930	+	0.533294i	$0.820954\pi$
$830$	21981.8	0.919276
$831$	0	0
$832$	−170626.	−7.10984
$833$	9842.42	0.409387
$834$	0	0
$835$	−44508.8	−1.84466
$836$	−22509.4	−0.931225
$837$	0	0
$838$	−31747.2	−1.30870
$839$	29456.3	1.21209	0.606045	−	0.795430i	$-0.292755\pi$
$839$	29456.3	1.21209	0.606045	+	0.795430i	$0.292755\pi$
$840$	0	0
$841$	−15331.7	−0.628630
$842$	90062.5	3.68617
$843$	0	0
$844$	15562.0	0.634674
$845$	20718.9	0.843494
$846$	0	0
$847$	−51957.4	−2.10776
$848$	−18748.1	−0.759211
$849$	0	0
$850$	50905.7	2.05418
$851$	−13616.4	−0.548488
$852$	0	0
$853$	47797.4	1.91858	0.959291	−	0.282419i	$-0.0911368\pi$
$853$	47797.4	1.91858	0.959291	+	0.282419i	$0.0911368\pi$
$854$	−25770.3	−1.03260
$855$	0	0
$856$	68606.1	2.73938
$857$	−19750.1	−0.787223	−0.393612	−	0.919277i	$-0.628775\pi$
$857$	−19750.1	−0.787223	−0.393612	+	0.919277i	$0.628775\pi$
$858$	0	0
$859$	−15036.6	−0.597256	−0.298628	−	0.954370i	$-0.596529\pi$
$859$	−15036.6	−0.597256	−0.298628	+	0.954370i	$0.596529\pi$
$860$	38845.2	1.54024
$861$	0	0
$862$	−50458.0	−1.99374
$863$	−13702.8	−0.540496	−0.270248	−	0.962791i	$-0.587106\pi$
$863$	−13702.8	−0.540496	−0.270248	+	0.962791i	$0.587106\pi$
$864$	0	0
$865$	−28483.9	−1.11963
$866$	44275.2	1.73734
$867$	0	0
$868$	78027.7	3.05119
$869$	26296.1	1.02651
$870$	0	0
$871$	30663.0	1.19286
$872$	29242.0	1.13562
$873$	0	0
$874$	11933.0	0.461829
$875$	9986.55	0.385837
$876$	0	0
$877$	28808.8	1.10924	0.554621	−	0.832103i	$-0.312863\pi$
$877$	28808.8	1.10924	0.554621	+	0.832103i	$0.312863\pi$
$878$	45037.2	1.73113
$879$	0	0
$880$	−277385.	−10.6257
$881$	12840.3	0.491035	0.245518	−	0.969392i	$-0.421042\pi$
$881$	12840.3	0.491035	0.245518	+	0.969392i	$0.421042\pi$
$882$	0	0
$883$	−21452.9	−0.817608	−0.408804	−	0.912622i	$-0.634054\pi$
$883$	−21452.9	−0.817608	−0.408804	+	0.912622i	$0.634054\pi$
$884$	156124.	5.94008
$885$	0	0
$886$	50588.0	1.91821
$887$	1146.26	0.0433907	0.0216953	−	0.999765i	$-0.493094\pi$
$887$	1146.26	0.0433907	0.0216953	+	0.999765i	$0.493094\pi$
$888$	0	0
$889$	40178.6	1.51580
$890$	−94820.5	−3.57122
$891$	0	0
$892$	−88389.9	−3.31784
$893$	−5629.78	−0.210967
$894$	0	0
$895$	43700.5	1.63212
$896$	134926.	5.03076
$897$	0	0
$898$	−33567.6	−1.24740
$899$	20119.3	0.746404
$900$	0	0
$901$	7364.93	0.272321
$902$	−124693.	−4.60291
$903$	0	0
$904$	−158702.	−5.83887
$905$	1372.91	0.0504276
$906$	0	0
$907$	34566.7	1.26546	0.632728	−	0.774374i	$-0.281935\pi$
$907$	34566.7	1.26546	0.632728	+	0.774374i	$0.281935\pi$
$908$	67647.2	2.47241
$909$	0	0
$910$	−77222.1	−2.81306
$911$	−42850.7	−1.55840	−0.779202	−	0.626773i	$-0.784375\pi$
$911$	−42850.7	−1.55840	−0.779202	+	0.626773i	$0.784375\pi$
$912$	0	0
$913$	18569.3	0.673117
$914$	97146.0	3.51565
$915$	0	0
$916$	61761.2	2.22778
$917$	−38254.3	−1.37761
$918$	0	0
$919$	6886.69	0.247194	0.123597	−	0.992333i	$-0.460557\pi$
$919$	6886.69	0.247194	0.123597	+	0.992333i	$0.460557\pi$
$920$	180060.	6.45260
$921$	0	0
$922$	77961.9	2.78475
$923$	−51292.3	−1.82915
$924$	0	0
$925$	7461.69	0.265231
$926$	49162.1	1.74467
$927$	0	0
$928$	87184.3	3.08401
$929$	28108.1	0.992677	0.496338	−	0.868129i	$-0.334678\pi$
$929$	28108.1	0.992677	0.496338	+	0.868129i	$0.334678\pi$
$930$	0	0
$931$	−1264.33	−0.0445076
$932$	53933.6	1.89555
$933$	0	0
$934$	25333.5	0.887515
$935$	108967.	3.81134
$936$	0	0
$937$	27099.9	0.944839	0.472419	−	0.881374i	$-0.343381\pi$
$937$	27099.9	0.944839	0.472419	+	0.881374i	$0.343381\pi$
$938$	−45283.9	−1.57630
$939$	0	0
$940$	−129932.	−4.50842
$941$	47104.9	1.63186	0.815928	−	0.578153i	$-0.196226\pi$
$941$	47104.9	1.63186	0.815928	+	0.578153i	$0.196226\pi$
$942$	0	0
$943$	49103.9	1.69570
$944$	−100458.	−3.46359
$945$	0	0
$946$	44175.4	1.51825
$947$	−14957.8	−0.513265	−0.256632	−	0.966509i	$-0.582613\pi$
$947$	−14957.8	−0.513265	−0.256632	+	0.966509i	$0.582613\pi$
$948$	0	0
$949$	−31846.6	−1.08934
$950$	−6539.19	−0.223326
$951$	0	0
$952$	−150745.	−5.13201
$953$	30282.6	1.02933	0.514664	−	0.857392i	$-0.327917\pi$
$953$	30282.6	1.02933	0.514664	+	0.857392i	$0.327917\pi$
$954$	0	0
$955$	12731.7	0.431400
$956$	5522.79	0.186841
$957$	0	0
$958$	35826.0	1.20823
$959$	−26703.2	−0.899155
$960$	0	0
$961$	14900.6	0.500172
$962$	30807.1	1.03250
$963$	0	0
$964$	58504.0	1.95465
$965$	−14716.5	−0.490924
$966$	0	0
$967$	−2455.70	−0.0816648	−0.0408324	−	0.999166i	$-0.513001\pi$
$967$	−2455.70	−0.0816648	−0.0408324	+	0.999166i	$0.513001\pi$
$968$	−274102.	−9.10121
$969$	0	0
$970$	−130234.	−4.31089
$971$	26842.8	0.887154	0.443577	−	0.896236i	$-0.353709\pi$
$971$	26842.8	0.887154	0.443577	+	0.896236i	$0.353709\pi$
$972$	0	0
$973$	−15022.1	−0.494950
$974$	86076.0	2.83168
$975$	0	0
$976$	−82475.5	−2.70489
$977$	−13850.0	−0.453533	−0.226767	−	0.973949i	$-0.572815\pi$
$977$	−13850.0	−0.453533	−0.226767	+	0.973949i	$0.572815\pi$
$978$	0	0
$979$	−80100.5	−2.61494
$980$	−29179.9	−0.951140
$981$	0	0
$982$	−32532.7	−1.05719
$983$	−46046.2	−1.49405	−0.747023	−	0.664798i	$-0.768518\pi$
$983$	−46046.2	−1.49405	−0.747023	+	0.664798i	$0.768518\pi$
$984$	0	0
$985$	−7459.57	−0.241301
$986$	−59451.6	−1.92021
$987$	0	0
$988$	−20055.2	−0.645791
$989$	−17396.2	−0.559320
$990$	0	0
$991$	−36312.5	−1.16398	−0.581990	−	0.813196i	$-0.697726\pi$
$991$	−36312.5	−1.16398	−0.581990	+	0.813196i	$0.697726\pi$
$992$	193665.	6.19846
$993$	0	0
$994$	75749.7	2.41714
$995$	28802.3	0.917684
$996$	0	0
$997$	−32437.8	−1.03041	−0.515204	−	0.857068i	$-0.672284\pi$
$997$	−32437.8	−1.03041	−0.515204	+	0.857068i	$0.672284\pi$
$998$	−21500.5	−0.681951
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.e.1.1		32
3.2	odd	2		717.4.a.c.1.32	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.c.1.32	✓	32	3.2	odd	2
2151.4.a.e.1.1		32	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-5.57745 q^{2} +23.1079 q^{4} +14.3698 q^{5} -15.9726 q^{7} -84.2636 q^{8} +O(q^{10})\)	\(q-5.57745 q^{2} +23.1079 q^{4} +14.3698 q^{5} -15.9726 q^{7} -84.2636 q^{8} -80.1465 q^{10} -67.7046 q^{11} -60.3228 q^{13} +89.0862 q^{14} +285.112 q^{16} -112.003 q^{17} +14.3875 q^{19} +332.055 q^{20} +377.619 q^{22} -148.706 q^{23} +81.4898 q^{25} +336.447 q^{26} -369.093 q^{28} -95.1700 q^{29} -211.404 q^{31} -916.090 q^{32} +624.689 q^{34} -229.522 q^{35} +91.5660 q^{37} -80.2455 q^{38} -1210.85 q^{40} -330.209 q^{41} +116.984 q^{43} -1564.51 q^{44} +829.398 q^{46} -391.297 q^{47} -87.8767 q^{49} -454.505 q^{50} -1393.93 q^{52} -65.7568 q^{53} -972.898 q^{55} +1345.91 q^{56} +530.806 q^{58} -352.345 q^{59} -289.274 q^{61} +1179.09 q^{62} +2828.55 q^{64} -866.824 q^{65} -508.316 q^{67} -2588.15 q^{68} +1280.15 q^{70} +850.297 q^{71} +527.936 q^{73} -510.705 q^{74} +332.465 q^{76} +1081.42 q^{77} -388.395 q^{79} +4096.99 q^{80} +1841.72 q^{82} -274.270 q^{83} -1609.45 q^{85} -652.473 q^{86} +5705.03 q^{88} +1183.09 q^{89} +963.511 q^{91} -3436.28 q^{92} +2182.44 q^{94} +206.745 q^{95} +1624.95 q^{97} +490.127 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * q^98

Introduction

L-functions

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