LMFDB - Embedded newform 1911.4.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1911,4,Mod(1,1911)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1911, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1911.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1911 = 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1911.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-3,-12,15,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$112.752650021$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.1038472.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 22x^{2} + 6x + 104 $ x^4 - x^3 - 22x^2 + 6x + 104
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.71034$ of defining polynomial
Character	$\chi$	$=$	1911.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.71034 q^{2} -3.00000 q^{3} +5.76665 q^{4} +15.1555 q^{5} +11.1310 q^{6} +8.28649 q^{8} +9.00000 q^{9} -56.2319 q^{10} +60.6986 q^{11} -17.3000 q^{12} +13.0000 q^{13} -45.4664 q^{15} -76.8789 q^{16} -105.639 q^{17} -33.3931 q^{18} +76.5305 q^{19} +87.3962 q^{20} -225.213 q^{22} -145.001 q^{23} -24.8595 q^{24} +104.688 q^{25} -48.2345 q^{26} -27.0000 q^{27} +140.357 q^{29} +168.696 q^{30} +175.791 q^{31} +218.955 q^{32} -182.096 q^{33} +391.957 q^{34} +51.8999 q^{36} -216.081 q^{37} -283.954 q^{38} -39.0000 q^{39} +125.585 q^{40} +151.463 q^{41} +330.182 q^{43} +350.028 q^{44} +136.399 q^{45} +538.005 q^{46} +344.637 q^{47} +230.637 q^{48} -388.428 q^{50} +316.917 q^{51} +74.9665 q^{52} -669.529 q^{53} +100.179 q^{54} +919.915 q^{55} -229.591 q^{57} -520.773 q^{58} +226.225 q^{59} -262.189 q^{60} +594.623 q^{61} -652.246 q^{62} -197.368 q^{64} +197.021 q^{65} +675.639 q^{66} -556.147 q^{67} -609.183 q^{68} +435.004 q^{69} -171.941 q^{71} +74.5784 q^{72} +97.9290 q^{73} +801.736 q^{74} -314.063 q^{75} +441.325 q^{76} +144.703 q^{78} +1115.36 q^{79} -1165.14 q^{80} +81.0000 q^{81} -561.980 q^{82} +275.245 q^{83} -1601.01 q^{85} -1225.09 q^{86} -421.071 q^{87} +502.978 q^{88} +191.166 q^{89} -506.088 q^{90} -836.172 q^{92} -527.373 q^{93} -1278.72 q^{94} +1159.85 q^{95} -656.866 q^{96} +934.640 q^{97} +546.288 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} - 12 q^{3} + 15 q^{4} + 24 q^{5} + 9 q^{6} - 39 q^{8} + 36 q^{9} + 26 q^{10} + 8 q^{11} - 45 q^{12} + 52 q^{13} - 72 q^{15} - 181 q^{16} + 6 q^{17} - 27 q^{18} + 332 q^{19} + 62 q^{20} - 176 q^{22}+ \cdots + 72 q^{99}+O(q^{100}) $ 4 * q - 3 * q^2 - 12 * q^3 + 15 * q^4 + 24 * q^5 + 9 * q^6 - 39 * q^8 + 36 * q^9 + 26 * q^10 + 8 * q^11 - 45 * q^12 + 52 * q^13 - 72 * q^15 - 181 * q^16 + 6 * q^17 - 27 * q^18 + 332 * q^19 + 62 * q^20 - 176 * q^22 + 6 * q^23 + 117 * q^24 - 12 * q^25 - 39 * q^26 - 108 * q^27 + 8 * q^29 - 78 * q^30 + 184 * q^31 + 485 * q^32 - 24 * q^33 + 326 * q^34 + 135 * q^36 - 244 * q^37 - 516 * q^38 - 156 * q^39 - 22 * q^40 + 296 * q^41 - 224 * q^43 - 336 * q^44 + 216 * q^45 - 404 * q^46 + 750 * q^47 + 543 * q^48 + 35 * q^50 - 18 * q^51 + 195 * q^52 + 680 * q^53 + 81 * q^54 + 600 * q^55 - 996 * q^57 - 334 * q^58 + 334 * q^59 - 186 * q^60 + 1212 * q^61 + 928 * q^62 - 697 * q^64 + 312 * q^65 + 528 * q^66 - 388 * q^67 - 1174 * q^68 - 18 * q^69 - 544 * q^71 - 351 * q^72 + 1572 * q^73 + 1482 * q^74 + 36 * q^75 + 1508 * q^76 + 117 * q^78 + 464 * q^79 - 2066 * q^80 + 324 * q^81 - 1806 * q^82 + 3158 * q^83 - 2228 * q^85 + 324 * q^86 - 24 * q^87 + 984 * q^88 - 1156 * q^89 + 234 * q^90 + 1788 * q^92 - 552 * q^93 - 4604 * q^94 + 1128 * q^95 - 1455 * q^96 + 1928 * q^97 + 72 * q^99

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 $

$2$	−3.71034	−1.31180	−0.655902	−	0.754846i	$-0.727712\pi$
$2$	−3.71034	−1.31180	−0.655902	+	0.754846i	$0.727712\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	5.76665	0.720832
$5$	15.1555	1.35554	0.677772	−	0.735272i	$-0.262945\pi$
$5$	15.1555	1.35554	0.677772	+	0.735272i	$0.262945\pi$
$6$	11.1310	0.757371
$7$	0	0
$7$	0	0
$8$	8.28649	0.366214
$9$	9.00000	0.333333
$10$	−56.2319	−1.77821
$11$	60.6986	1.66376	0.831879	−	0.554958i	$-0.187266\pi$
$11$	60.6986	1.66376	0.831879	+	0.554958i	$0.187266\pi$
$12$	−17.3000	−0.416172
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−45.4664	−0.782624
$16$	−76.8789	−1.20123
$17$	−105.639	−1.50713	−0.753565	−	0.657374i	$-0.771667\pi$
$17$	−105.639	−1.50713	−0.753565	+	0.657374i	$0.771667\pi$
$18$	−33.3931	−0.437268
$19$	76.5305	0.924068	0.462034	−	0.886862i	$-0.347120\pi$
$19$	76.5305	0.924068	0.462034	+	0.886862i	$0.347120\pi$
$20$	87.3962	0.977120
$21$	0	0
$22$	−225.213	−2.18252
$23$	−145.001	−1.31456	−0.657280	−	0.753647i	$-0.728293\pi$
$23$	−145.001	−1.31456	−0.657280	+	0.753647i	$0.728293\pi$
$24$	−24.8595	−0.211434
$25$	104.688	0.837502
$26$	−48.2345	−0.363829
$27$	−27.0000	−0.192450
$28$	0	0
$29$	140.357	0.898747	0.449374	−	0.893344i	$-0.351647\pi$
$29$	140.357	0.898747	0.449374	+	0.893344i	$0.351647\pi$
$30$	168.696	1.02665
$31$	175.791	1.01848	0.509242	−	0.860623i	$-0.329926\pi$
$31$	175.791	1.01848	0.509242	+	0.860623i	$0.329926\pi$
$32$	218.955	1.20957
$33$	−182.096	−0.960571
$34$	391.957	1.97706
$35$	0	0
$36$	51.8999	0.240277
$37$	−216.081	−0.960096	−0.480048	−	0.877242i	$-0.659381\pi$
$37$	−216.081	−0.960096	−0.480048	+	0.877242i	$0.659381\pi$
$38$	−283.954	−1.21220
$39$	−39.0000	−0.160128
$40$	125.585	0.496420
$41$	151.463	0.576941	0.288470	−	0.957489i	$-0.406853\pi$
$41$	151.463	0.576941	0.288470	+	0.957489i	$0.406853\pi$
$42$	0	0
$43$	330.182	1.17098	0.585492	−	0.810678i	$-0.300901\pi$
$43$	330.182	1.17098	0.585492	+	0.810678i	$0.300901\pi$
$44$	350.028	1.19929
$45$	136.399	0.451848
$46$	538.005	1.72445
$47$	344.637	1.06959	0.534793	−	0.844983i	$-0.320390\pi$
$47$	344.637	1.06959	0.534793	+	0.844983i	$0.320390\pi$
$48$	230.637	0.693532
$49$	0	0
$50$	−388.428	−1.09864
$51$	316.917	0.870142
$52$	74.9665	0.199923
$53$	−669.529	−1.73522	−0.867612	−	0.497242i	$-0.834346\pi$
$53$	−669.529	−1.73522	−0.867612	+	0.497242i	$0.834346\pi$
$54$	100.179	0.252457
$55$	919.915	2.25530
$56$	0	0
$57$	−229.591	−0.533511
$58$	−520.773	−1.17898
$59$	226.225	0.499186	0.249593	−	0.968351i	$-0.419703\pi$
$59$	226.225	0.499186	0.249593	+	0.968351i	$0.419703\pi$
$60$	−262.189	−0.564140
$61$	594.623	1.24809	0.624046	−	0.781387i	$-0.285488\pi$
$61$	594.623	1.24809	0.624046	+	0.781387i	$0.285488\pi$
$62$	−652.246	−1.33605
$63$	0	0
$64$	−197.368	−0.385485
$65$	197.021	0.375961
$66$	675.639	1.26008
$67$	−556.147	−1.01409	−0.507046	−	0.861919i	$-0.669263\pi$
$67$	−556.147	−1.01409	−0.507046	+	0.861919i	$0.669263\pi$
$68$	−609.183	−1.08639
$69$	435.004	0.758961
$70$	0	0
$71$	−171.941	−0.287403	−0.143701	−	0.989621i	$-0.545901\pi$
$71$	−171.941	−0.287403	−0.143701	+	0.989621i	$0.545901\pi$
$72$	74.5784	0.122071
$73$	97.9290	0.157010	0.0785050	−	0.996914i	$-0.474985\pi$
$73$	97.9290	0.157010	0.0785050	+	0.996914i	$0.474985\pi$
$74$	801.736	1.25946
$75$	−314.063	−0.483532
$76$	441.325	0.666098
$77$	0	0
$78$	144.703	0.210057
$79$	1115.36	1.58845	0.794224	−	0.607626i	$-0.207878\pi$
$79$	1115.36	1.58845	0.794224	+	0.607626i	$0.207878\pi$
$80$	−1165.14	−1.62833
$81$	81.0000	0.111111
$82$	−561.980	−0.756833
$83$	275.245	0.364001	0.182001	−	0.983298i	$-0.441743\pi$
$83$	275.245	0.364001	0.182001	+	0.983298i	$0.441743\pi$
$84$	0	0
$85$	−1601.01	−2.04298
$86$	−1225.09	−1.53610
$87$	−421.071	−0.518892
$88$	502.978	0.609292
$89$	191.166	0.227680	0.113840	−	0.993499i	$-0.463685\pi$
$89$	191.166	0.227680	0.113840	+	0.993499i	$0.463685\pi$
$90$	−506.088	−0.592737
$91$	0	0
$92$	−836.172	−0.947576
$93$	−527.373	−0.588022
$94$	−1278.72	−1.40309
$95$	1159.85	1.25262
$96$	−656.866	−0.698345
$97$	934.640	0.978334	0.489167	−	0.872190i	$-0.337301\pi$
$97$	934.640	0.978334	0.489167	+	0.872190i	$0.337301\pi$
$98$	0	0
$99$	546.288	0.554586
$100$	603.698	0.603698
$101$	134.586	0.132592	0.0662959	−	0.997800i	$-0.478882\pi$
$101$	134.586	0.132592	0.0662959	+	0.997800i	$0.478882\pi$
$102$	−1175.87	−1.14146
$103$	147.902	0.141487	0.0707436	−	0.997495i	$-0.477463\pi$
$103$	147.902	0.141487	0.0707436	+	0.997495i	$0.477463\pi$
$104$	107.724	0.101570
$105$	0	0
$106$	2484.18	2.27628
$107$	1814.42	1.63931	0.819656	−	0.572856i	$-0.194165\pi$
$107$	1814.42	1.63931	0.819656	+	0.572856i	$0.194165\pi$
$108$	−155.700	−0.138724
$109$	−1754.09	−1.54139	−0.770696	−	0.637204i	$-0.780091\pi$
$109$	−1754.09	−1.54139	−0.770696	+	0.637204i	$0.780091\pi$
$110$	−3413.20	−2.95851
$111$	648.244	0.554312
$112$	0	0
$113$	−28.9646	−0.0241129	−0.0120565	−	0.999927i	$-0.503838\pi$
$113$	−28.9646	−0.0241129	−0.0120565	+	0.999927i	$0.503838\pi$
$114$	851.863	0.699862
$115$	−2197.56	−1.78194
$116$	809.391	0.647845
$117$	117.000	0.0924500
$118$	−839.372	−0.654835
$119$	0	0
$120$	−376.756	−0.286608
$121$	2353.32	1.76809
$122$	−2206.26	−1.63725
$123$	−454.389	−0.333097
$124$	1013.73	0.734156
$125$	−307.841	−0.220273
$126$	0	0
$127$	−2130.71	−1.48874	−0.744371	−	0.667767i	$-0.767250\pi$
$127$	−2130.71	−1.48874	−0.744371	+	0.667767i	$0.767250\pi$
$128$	−1019.34	−0.703888
$129$	−990.547	−0.676068
$130$	−731.015	−0.493187
$131$	94.1770	0.0628113	0.0314057	−	0.999507i	$-0.490002\pi$
$131$	94.1770	0.0628113	0.0314057	+	0.999507i	$0.490002\pi$
$132$	−1050.08	−0.692410
$133$	0	0
$134$	2063.50	1.33029
$135$	−409.197	−0.260875
$136$	−875.375	−0.551932
$137$	−244.222	−0.152302	−0.0761508	−	0.997096i	$-0.524263\pi$
$137$	−244.222	−0.152302	−0.0761508	+	0.997096i	$0.524263\pi$
$138$	−1614.01	−0.995609
$139$	1853.58	1.13107	0.565535	−	0.824724i	$-0.308670\pi$
$139$	1853.58	1.13107	0.565535	+	0.824724i	$0.308670\pi$
$140$	0	0
$141$	−1033.91	−0.617525
$142$	637.959	0.377017
$143$	789.082	0.461443
$144$	−691.910	−0.400411
$145$	2127.18	1.21829
$146$	−363.350	−0.205966
$147$	0	0
$148$	−1246.07	−0.692068
$149$	1768.75	0.972492	0.486246	−	0.873822i	$-0.338366\pi$
$149$	1768.75	0.972492	0.486246	+	0.873822i	$0.338366\pi$
$150$	1165.28	0.634300
$151$	1689.75	0.910664	0.455332	−	0.890322i	$-0.349521\pi$
$151$	1689.75	0.910664	0.455332	+	0.890322i	$0.349521\pi$
$152$	634.169	0.338407
$153$	−950.750	−0.502376
$154$	0	0
$155$	2664.19	1.38060
$156$	−224.899	−0.115425
$157$	−1798.51	−0.914247	−0.457124	−	0.889403i	$-0.651120\pi$
$157$	−1798.51	−0.914247	−0.457124	+	0.889403i	$0.651120\pi$
$158$	−4138.35	−2.08373
$159$	2008.59	1.00183
$160$	3318.37	1.63963
$161$	0	0
$162$	−300.538	−0.145756
$163$	−2830.21	−1.35999	−0.679997	−	0.733215i	$-0.738019\pi$
$163$	−2830.21	−1.35999	−0.679997	+	0.733215i	$0.738019\pi$
$164$	873.435	0.415877
$165$	−2759.75	−1.30210
$166$	−1021.26	−0.477499
$167$	−1860.63	−0.862157	−0.431078	−	0.902314i	$-0.641867\pi$
$167$	−1860.63	−0.862157	−0.431078	+	0.902314i	$0.641867\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	5940.28	2.67999
$171$	688.774	0.308023
$172$	1904.05	0.844083
$173$	197.699	0.0868830	0.0434415	−	0.999056i	$-0.486168\pi$
$173$	197.699	0.0868830	0.0434415	+	0.999056i	$0.486168\pi$
$174$	1562.32	0.680685
$175$	0	0
$176$	−4666.45	−1.99856
$177$	−678.675	−0.288205
$178$	−709.290	−0.298672
$179$	−3002.28	−1.25364	−0.626818	−	0.779166i	$-0.715643\pi$
$179$	−3002.28	−1.25364	−0.626818	+	0.779166i	$0.715643\pi$
$180$	786.566	0.325707
$181$	2208.28	0.906852	0.453426	−	0.891294i	$-0.350202\pi$
$181$	2208.28	0.906852	0.453426	+	0.891294i	$0.350202\pi$
$182$	0	0
$183$	−1783.87	−0.720587
$184$	−1201.55	−0.481411
$185$	−3274.81	−1.30145
$186$	1956.74	0.771370
$187$	−6412.14	−2.50750
$188$	1987.40	0.770991
$189$	0	0
$190$	−4303.46	−1.64319
$191$	−3275.07	−1.24071	−0.620355	−	0.784321i	$-0.713011\pi$
$191$	−3275.07	−1.24071	−0.620355	+	0.784321i	$0.713011\pi$
$192$	592.105	0.222560
$193$	−3591.28	−1.33941	−0.669705	−	0.742627i	$-0.733580\pi$
$193$	−3591.28	−1.33941	−0.669705	+	0.742627i	$0.733580\pi$
$194$	−3467.84	−1.28338
$195$	−591.063	−0.217061
$196$	0	0
$197$	4405.13	1.59316	0.796579	−	0.604534i	$-0.206641\pi$
$197$	4405.13	1.59316	0.796579	+	0.604534i	$0.206641\pi$
$198$	−2026.92	−0.727508
$199$	−4726.98	−1.68385	−0.841927	−	0.539591i	$-0.818579\pi$
$199$	−4726.98	−1.68385	−0.841927	+	0.539591i	$0.818579\pi$
$200$	867.494	0.306705
$201$	1668.44	0.585486
$202$	−499.359	−0.173935
$203$	0	0
$204$	1827.55	0.627226
$205$	2295.49	0.782069
$206$	−548.766	−0.185604
$207$	−1305.01	−0.438186
$208$	−999.426	−0.333162
$209$	4645.30	1.53742
$210$	0	0
$211$	2274.65	0.742147	0.371074	−	0.928603i	$-0.378990\pi$
$211$	2274.65	0.742147	0.371074	+	0.928603i	$0.378990\pi$
$212$	−3860.94	−1.25080
$213$	515.822	0.165932
$214$	−6732.11	−2.15046
$215$	5004.06	1.58732
$216$	−223.735	−0.0704780
$217$	0	0
$218$	6508.29	2.02200
$219$	−293.787	−0.0906497
$220$	5304.83	1.62569
$221$	−1373.31	−0.418002
$222$	−2405.21	−0.727149
$223$	1097.77	0.329650	0.164825	−	0.986323i	$-0.447294\pi$
$223$	1097.77	0.329650	0.164825	+	0.986323i	$0.447294\pi$
$224$	0	0
$225$	942.190	0.279167
$226$	107.469	0.0316314
$227$	3977.47	1.16297	0.581485	−	0.813557i	$-0.302472\pi$
$227$	3977.47	1.16297	0.581485	+	0.813557i	$0.302472\pi$
$228$	−1323.97	−0.384572
$229$	−3552.94	−1.02526	−0.512631	−	0.858609i	$-0.671329\pi$
$229$	−3552.94	−1.02526	−0.512631	+	0.858609i	$0.671329\pi$
$230$	8153.71	2.33756
$231$	0	0
$232$	1163.07	0.329134
$233$	634.762	0.178475	0.0892375	−	0.996010i	$-0.471557\pi$
$233$	634.762	0.178475	0.0892375	+	0.996010i	$0.471557\pi$
$234$	−434.110	−0.121276
$235$	5223.13	1.44987
$236$	1304.56	0.359829
$237$	−3346.07	−0.917090
$238$	0	0
$239$	3389.03	0.917231	0.458616	−	0.888635i	$-0.348345\pi$
$239$	3389.03	0.917231	0.458616	+	0.888635i	$0.348345\pi$
$240$	3495.41	0.940114
$241$	2588.29	0.691810	0.345905	−	0.938270i	$-0.387572\pi$
$241$	2588.29	0.691810	0.345905	+	0.938270i	$0.387572\pi$
$242$	−8731.64	−2.31939
$243$	−243.000	−0.0641500
$244$	3428.98	0.899665
$245$	0	0
$246$	1685.94	0.436958
$247$	994.896	0.256290
$248$	1456.69	0.372984
$249$	−825.736	−0.210156
$250$	1142.20	0.288956
$251$	−2089.73	−0.525507	−0.262754	−	0.964863i	$-0.584631\pi$
$251$	−2089.73	−0.525507	−0.262754	+	0.964863i	$0.584631\pi$
$252$	0	0
$253$	−8801.38	−2.18711
$254$	7905.67	1.95294
$255$	4803.02	1.17952
$256$	5361.04	1.30885
$257$	2521.89	0.612106	0.306053	−	0.952014i	$-0.400991\pi$
$257$	2521.89	0.612106	0.306053	+	0.952014i	$0.400991\pi$
$258$	3675.27	0.886870
$259$	0	0
$260$	1136.15	0.271004
$261$	1263.21	0.299582
$262$	−349.429	−0.0823962
$263$	−5959.98	−1.39737	−0.698685	−	0.715430i	$-0.746231\pi$
$263$	−5959.98	−1.39737	−0.698685	+	0.715430i	$0.746231\pi$
$264$	−1508.94	−0.351775
$265$	−10147.0	−2.35217
$266$	0	0
$267$	−573.497	−0.131451
$268$	−3207.11	−0.730990
$269$	5577.19	1.26412	0.632058	−	0.774921i	$-0.282210\pi$
$269$	5577.19	1.26412	0.632058	+	0.774921i	$0.282210\pi$
$270$	1518.26	0.342217
$271$	−3916.45	−0.877887	−0.438943	−	0.898515i	$-0.644647\pi$
$271$	−3916.45	−0.877887	−0.438943	+	0.898515i	$0.644647\pi$
$272$	8121.41	1.81041
$273$	0	0
$274$	906.149	0.199790
$275$	6354.40	1.39340
$276$	2508.52	0.547083
$277$	−1722.93	−0.373722	−0.186861	−	0.982386i	$-0.559831\pi$
$277$	−1722.93	−0.373722	−0.186861	+	0.982386i	$0.559831\pi$
$278$	−6877.42	−1.48374
$279$	1582.12	0.339495
$280$	0	0
$281$	8339.06	1.77034	0.885172	−	0.465263i	$-0.154040\pi$
$281$	8339.06	1.77034	0.885172	+	0.465263i	$0.154040\pi$
$282$	3836.17	0.810073
$283$	2994.63	0.629018	0.314509	−	0.949254i	$-0.398160\pi$
$283$	2994.63	0.629018	0.314509	+	0.949254i	$0.398160\pi$
$284$	−991.522	−0.207169
$285$	−3479.56	−0.723198
$286$	−2927.77	−0.605323
$287$	0	0
$288$	1970.60	0.403190
$289$	6246.58	1.27144
$290$	−7892.55	−1.59816
$291$	−2803.92	−0.564841
$292$	564.723	0.113178
$293$	8492.60	1.69332	0.846660	−	0.532134i	$-0.178610\pi$
$293$	8492.60	1.69332	0.846660	+	0.532134i	$0.178610\pi$
$294$	0	0
$295$	3428.54	0.676669
$296$	−1790.56	−0.351601
$297$	−1638.86	−0.320190
$298$	−6562.66	−1.27572
$299$	−1885.02	−0.364593
$300$	−1811.09	−0.348545
$301$	0	0
$302$	−6269.57	−1.19461
$303$	−403.757	−0.0765519
$304$	−5883.58	−1.11002
$305$	9011.78	1.69185
$306$	3527.61	0.659020
$307$	−2554.72	−0.474936	−0.237468	−	0.971395i	$-0.576317\pi$
$307$	−2554.72	−0.474936	−0.237468	+	0.971395i	$0.576317\pi$
$308$	0	0
$309$	−443.705	−0.0816877
$310$	−9885.08	−1.81108
$311$	8271.35	1.50812	0.754060	−	0.656806i	$-0.228093\pi$
$311$	8271.35	1.50812	0.754060	+	0.656806i	$0.228093\pi$
$312$	−323.173	−0.0586412
$313$	8332.72	1.50477	0.752385	−	0.658723i	$-0.228903\pi$
$313$	8332.72	1.50477	0.752385	+	0.658723i	$0.228903\pi$
$314$	6673.09	1.19931
$315$	0	0
$316$	6431.87	1.14500
$317$	9836.15	1.74276	0.871378	−	0.490613i	$-0.163227\pi$
$317$	9836.15	1.74276	0.871378	+	0.490613i	$0.163227\pi$
$318$	−7452.55	−1.31421
$319$	8519.49	1.49530
$320$	−2991.21	−0.522543
$321$	−5443.25	−0.946457
$322$	0	0
$323$	−8084.60	−1.39269
$324$	467.099	0.0800924
$325$	1360.94	0.232281
$326$	10501.0	1.78405
$327$	5262.28	0.889923
$328$	1255.10	0.211284
$329$	0	0
$330$	10239.6	1.70810
$331$	−5544.06	−0.920632	−0.460316	−	0.887755i	$-0.652264\pi$
$331$	−5544.06	−0.920632	−0.460316	+	0.887755i	$0.652264\pi$
$332$	1587.24	0.262384
$333$	−1944.73	−0.320032
$334$	6903.59	1.13098
$335$	−8428.66	−1.37465
$336$	0	0
$337$	−1457.50	−0.235594	−0.117797	−	0.993038i	$-0.537583\pi$
$337$	−1457.50	−0.235594	−0.117797	+	0.993038i	$0.537583\pi$
$338$	−627.048	−0.100908
$339$	86.8937	0.0139216
$340$	−9232.44	−1.47265
$341$	10670.3	1.69451
$342$	−2555.59	−0.404066
$343$	0	0
$344$	2736.05	0.428831
$345$	6592.68	1.02881
$346$	−733.531	−0.113974
$347$	10176.2	1.57431	0.787156	−	0.616754i	$-0.211553\pi$
$347$	10176.2	1.57431	0.787156	+	0.616754i	$0.211553\pi$
$348$	−2428.17	−0.374034
$349$	3374.59	0.517587	0.258793	−	0.965933i	$-0.416675\pi$
$349$	3374.59	0.517587	0.258793	+	0.965933i	$0.416675\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	13290.3	2.01243
$353$	735.069	0.110832	0.0554161	−	0.998463i	$-0.482351\pi$
$353$	735.069	0.110832	0.0554161	+	0.998463i	$0.482351\pi$
$354$	2518.12	0.378069
$355$	−2605.84	−0.389588
$356$	1102.39	0.164119
$357$	0	0
$358$	11139.5	1.64453
$359$	−3669.38	−0.539450	−0.269725	−	0.962937i	$-0.586933\pi$
$359$	−3669.38	−0.539450	−0.269725	+	0.962937i	$0.586933\pi$
$360$	1130.27	0.165473
$361$	−1002.09	−0.146098
$362$	−8193.48	−1.18961
$363$	−7059.97	−1.02081
$364$	0	0
$365$	1484.16	0.212834
$366$	6618.77	0.945269
$367$	1669.40	0.237444	0.118722	−	0.992928i	$-0.462120\pi$
$367$	1669.40	0.237444	0.118722	+	0.992928i	$0.462120\pi$
$368$	11147.5	1.57909
$369$	1363.17	0.192314
$370$	12150.7	1.70725
$371$	0	0
$372$	−3041.18	−0.423865
$373$	−6992.72	−0.970696	−0.485348	−	0.874321i	$-0.661307\pi$
$373$	−6992.72	−0.970696	−0.485348	+	0.874321i	$0.661307\pi$
$374$	23791.2	3.28935
$375$	923.524	0.127175
$376$	2855.83	0.391698
$377$	1824.64	0.249268
$378$	0	0
$379$	−10282.8	−1.39365	−0.696824	−	0.717243i	$-0.745404\pi$
$379$	−10282.8	−1.39365	−0.696824	+	0.717243i	$0.745404\pi$
$380$	6688.48	0.902925
$381$	6392.14	0.859525
$382$	12151.6	1.62757
$383$	7316.05	0.976064	0.488032	−	0.872826i	$-0.337715\pi$
$383$	7316.05	0.976064	0.488032	+	0.872826i	$0.337715\pi$
$384$	3058.02	0.406390
$385$	0	0
$386$	13324.9	1.75704
$387$	2971.64	0.390328
$388$	5389.75	0.705214
$389$	6937.40	0.904216	0.452108	−	0.891963i	$-0.350672\pi$
$389$	6937.40	0.904216	0.452108	+	0.891963i	$0.350672\pi$
$390$	2193.05	0.284742
$391$	15317.8	1.98121
$392$	0	0
$393$	−282.531	−0.0362641
$394$	−16344.5	−2.08991
$395$	16903.7	2.15321
$396$	3150.25	0.399763
$397$	3311.46	0.418633	0.209317	−	0.977848i	$-0.432876\pi$
$397$	3311.46	0.418633	0.209317	+	0.977848i	$0.432876\pi$
$398$	17538.7	2.20889
$399$	0	0
$400$	−8048.28	−1.00604
$401$	−4028.27	−0.501651	−0.250825	−	0.968032i	$-0.580702\pi$
$401$	−4028.27	−0.501651	−0.250825	+	0.968032i	$0.580702\pi$
$402$	−6190.49	−0.768044
$403$	2285.28	0.282477
$404$	776.109	0.0955764
$405$	1227.59	0.150616
$406$	0	0
$407$	−13115.8	−1.59737
$408$	2626.13	0.318658
$409$	−5789.43	−0.699924	−0.349962	−	0.936764i	$-0.613805\pi$
$409$	−5789.43	−0.699924	−0.349962	+	0.936764i	$0.613805\pi$
$410$	−8517.07	−1.02592
$411$	732.667	0.0879314
$412$	852.898	0.101989
$413$	0	0
$414$	4842.04	0.574815
$415$	4171.47	0.493420
$416$	2846.42	0.335474
$417$	−5560.74	−0.653023
$418$	−17235.6	−2.01680
$419$	−11609.2	−1.35357	−0.676784	−	0.736181i	$-0.736627\pi$
$419$	−11609.2	−1.35357	−0.676784	+	0.736181i	$0.736627\pi$
$420$	0	0
$421$	14600.6	1.69023	0.845116	−	0.534583i	$-0.179531\pi$
$421$	14600.6	1.69023	0.845116	+	0.534583i	$0.179531\pi$
$422$	−8439.72	−0.973552
$423$	3101.74	0.356528
$424$	−5548.04	−0.635464
$425$	−11059.1	−1.26222
$426$	−1913.88	−0.217671
$427$	0	0
$428$	10463.1	1.18167
$429$	−2367.25	−0.266414
$430$	−18566.8	−2.08226
$431$	−5252.31	−0.586995	−0.293497	−	0.955960i	$-0.594819\pi$
$431$	−5252.31	−0.586995	−0.293497	+	0.955960i	$0.594819\pi$
$432$	2075.73	0.231177
$433$	11473.7	1.27342	0.636711	−	0.771102i	$-0.280294\pi$
$433$	11473.7	1.27342	0.636711	+	0.771102i	$0.280294\pi$
$434$	0	0
$435$	−6381.53	−0.703381
$436$	−10115.2	−1.11108
$437$	−11097.0	−1.21474
$438$	1090.05	0.118915
$439$	−18002.4	−1.95719	−0.978597	−	0.205788i	$-0.934024\pi$
$439$	−18002.4	−1.95719	−0.978597	+	0.205788i	$0.934024\pi$
$440$	7622.87	0.825922
$441$	0	0
$442$	5095.44	0.548338
$443$	6871.53	0.736967	0.368483	−	0.929634i	$-0.379877\pi$
$443$	6871.53	0.736967	0.368483	+	0.929634i	$0.379877\pi$
$444$	3738.20	0.399566
$445$	2897.20	0.308630
$446$	−4073.10	−0.432437
$447$	−5306.24	−0.561469
$448$	0	0
$449$	−12304.6	−1.29330	−0.646650	−	0.762787i	$-0.723831\pi$
$449$	−12304.6	−1.29330	−0.646650	+	0.762787i	$0.723831\pi$
$450$	−3495.85	−0.366213
$451$	9193.61	0.959889
$452$	−167.029	−0.0173813
$453$	−5069.26	−0.525772
$454$	−14757.8	−1.52559
$455$	0	0
$456$	−1902.51	−0.195379
$457$	−2050.32	−0.209868	−0.104934	−	0.994479i	$-0.533463\pi$
$457$	−2050.32	−0.209868	−0.104934	+	0.994479i	$0.533463\pi$
$458$	13182.6	1.34494
$459$	2852.25	0.290047
$460$	−12672.6	−1.28448
$461$	−16376.2	−1.65448	−0.827239	−	0.561851i	$-0.810089\pi$
$461$	−16376.2	−1.65448	−0.827239	+	0.561851i	$0.810089\pi$
$462$	0	0
$463$	3782.22	0.379643	0.189822	−	0.981819i	$-0.439209\pi$
$463$	3782.22	0.379643	0.189822	+	0.981819i	$0.439209\pi$
$464$	−10790.5	−1.07960
$465$	−7992.58	−0.797091
$466$	−2355.19	−0.234124
$467$	11644.1	1.15380	0.576902	−	0.816813i	$-0.304261\pi$
$467$	11644.1	1.15380	0.576902	+	0.816813i	$0.304261\pi$
$468$	674.698	0.0666409
$469$	0	0
$470$	−19379.6	−1.90195
$471$	5395.53	0.527841
$472$	1874.61	0.182809
$473$	20041.6	1.94823
$474$	12415.1	1.20304
$475$	8011.80	0.773909
$476$	0	0
$477$	−6025.76	−0.578408
$478$	−12574.5	−1.20323
$479$	−3460.09	−0.330053	−0.165027	−	0.986289i	$-0.552771\pi$
$479$	−3460.09	−0.330053	−0.165027	+	0.986289i	$0.552771\pi$
$480$	−9955.11	−0.946638
$481$	−2809.06	−0.266283
$482$	−9603.43	−0.907519
$483$	0	0
$484$	13570.8	1.27449
$485$	14164.9	1.32618
$486$	901.614	0.0841523
$487$	−300.273	−0.0279397	−0.0139699	−	0.999902i	$-0.504447\pi$
$487$	−300.273	−0.0279397	−0.0139699	+	0.999902i	$0.504447\pi$
$488$	4927.33	0.457069
$489$	8490.62	0.785193
$490$	0	0
$491$	7155.24	0.657661	0.328830	−	0.944389i	$-0.393346\pi$
$491$	7155.24	0.657661	0.328830	+	0.944389i	$0.393346\pi$
$492$	−2620.31	−0.240107
$493$	−14827.2	−1.35453
$494$	−3691.41	−0.336203
$495$	8279.24	0.751766
$496$	−13514.6	−1.22344
$497$	0	0
$498$	3063.77	0.275684
$499$	−4180.65	−0.375053	−0.187527	−	0.982260i	$-0.560047\pi$
$499$	−4180.65	−0.375053	−0.187527	+	0.982260i	$0.560047\pi$
$500$	−1775.21	−0.158780
$501$	5581.90	0.497766
$502$	7753.60	0.689363
$503$	−11386.8	−1.00937	−0.504684	−	0.863304i	$-0.668391\pi$
$503$	−11386.8	−1.00937	−0.504684	+	0.863304i	$0.668391\pi$
$504$	0	0
$505$	2039.71	0.179734
$506$	32656.2	2.86906
$507$	−507.000	−0.0444116
$508$	−12287.1	−1.07313
$509$	−19121.7	−1.66514	−0.832569	−	0.553921i	$-0.813131\pi$
$509$	−19121.7	−1.66514	−0.832569	+	0.553921i	$0.813131\pi$
$510$	−17820.8	−1.54729
$511$	0	0
$512$	−11736.6	−1.01307
$513$	−2066.32	−0.177837
$514$	−9357.09	−0.802964
$515$	2241.52	0.191792
$516$	−5712.14	−0.487331
$517$	20919.0	1.77953
$518$	0	0
$519$	−593.096	−0.0501619
$520$	1632.61	0.137682
$521$	−22750.2	−1.91306	−0.956531	−	0.291632i	$-0.905802\pi$
$521$	−22750.2	−1.91306	−0.956531	+	0.291632i	$0.905802\pi$
$522$	−4686.96	−0.392994
$523$	21994.0	1.83887	0.919436	−	0.393241i	$-0.128646\pi$
$523$	21994.0	1.83887	0.919436	+	0.393241i	$0.128646\pi$
$524$	543.086	0.0452764
$525$	0	0
$526$	22113.6	1.83308
$527$	−18570.4	−1.53499
$528$	13999.3	1.15387
$529$	8858.38	0.728066
$530$	37648.9	3.08559
$531$	2036.02	0.166395
$532$	0	0
$533$	1969.02	0.160015
$534$	2127.87	0.172438
$535$	27498.3	2.22216
$536$	−4608.50	−0.371375
$537$	9006.83	0.723787
$538$	−20693.3	−1.65827
$539$	0	0
$540$	−2359.70	−0.188047
$541$	14569.3	1.15782	0.578912	−	0.815390i	$-0.303477\pi$
$541$	14569.3	1.15782	0.578912	+	0.815390i	$0.303477\pi$
$542$	14531.4	1.15162
$543$	−6624.84	−0.523571
$544$	−23130.2	−1.82298
$545$	−26584.1	−2.08942
$546$	0	0
$547$	12056.0	0.942372	0.471186	−	0.882034i	$-0.343826\pi$
$547$	12056.0	0.942372	0.471186	+	0.882034i	$0.343826\pi$
$548$	−1408.35	−0.109784
$549$	5351.60	0.416031
$550$	−23577.0	−1.82787
$551$	10741.6	0.830503
$552$	3604.65	0.277942
$553$	0	0
$554$	6392.68	0.490251
$555$	9824.43	0.751395
$556$	10689.0	0.815311
$557$	−498.089	−0.0378900	−0.0189450	−	0.999821i	$-0.506031\pi$
$557$	−498.089	−0.0378900	−0.0189450	+	0.999821i	$0.506031\pi$
$558$	−5870.21	−0.445351
$559$	4292.37	0.324773
$560$	0	0
$561$	19236.4	1.44770
$562$	−30940.8	−2.32235
$563$	15747.8	1.17884	0.589422	−	0.807825i	$-0.299356\pi$
$563$	15747.8	1.17884	0.589422	+	0.807825i	$0.299356\pi$
$564$	−5962.21	−0.445132
$565$	−438.971	−0.0326861
$566$	−11111.1	−0.825149
$567$	0	0
$568$	−1424.78	−0.105251
$569$	−2834.35	−0.208826	−0.104413	−	0.994534i	$-0.533296\pi$
$569$	−2834.35	−0.208826	−0.104413	+	0.994534i	$0.533296\pi$
$570$	12910.4	0.948695
$571$	−1742.64	−0.127718	−0.0638590	−	0.997959i	$-0.520341\pi$
$571$	−1742.64	−0.127718	−0.0638590	+	0.997959i	$0.520341\pi$
$572$	4550.36	0.332623
$573$	9825.20	0.716324
$574$	0	0
$575$	−15179.9	−1.10095
$576$	−1776.32	−0.128495
$577$	−13655.7	−0.985256	−0.492628	−	0.870240i	$-0.663964\pi$
$577$	−13655.7	−0.985256	−0.492628	+	0.870240i	$0.663964\pi$
$578$	−23177.0	−1.66788
$579$	10773.8	0.773309
$580$	12266.7	0.878183
$581$	0	0
$582$	10403.5	0.740961
$583$	−40639.5	−2.88699
$584$	811.487	0.0574993
$585$	1773.19	0.125320
$586$	−31510.5	−2.22131
$587$	−84.3437	−0.00593056	−0.00296528	−	0.999996i	$-0.500944\pi$
$587$	−84.3437	−0.00593056	−0.00296528	+	0.999996i	$0.500944\pi$
$588$	0	0
$589$	13453.4	0.941149
$590$	−12721.1	−0.887658
$591$	−13215.4	−0.919811
$592$	16612.1	1.15330
$593$	10541.6	0.730003	0.365001	−	0.931007i	$-0.381068\pi$
$593$	10541.6	0.730003	0.365001	+	0.931007i	$0.381068\pi$
$594$	6080.75	0.420027
$595$	0	0
$596$	10199.7	0.701003
$597$	14181.0	0.972174
$598$	6994.06	0.478275
$599$	−9072.15	−0.618828	−0.309414	−	0.950927i	$-0.600133\pi$
$599$	−9072.15	−0.618828	−0.309414	+	0.950927i	$0.600133\pi$
$600$	−2602.48	−0.177076
$601$	−1477.35	−0.100270	−0.0501349	−	0.998742i	$-0.515965\pi$
$601$	−1477.35	−0.100270	−0.0501349	+	0.998742i	$0.515965\pi$
$602$	0	0
$603$	−5005.32	−0.338031
$604$	9744.23	0.656435
$605$	35665.7	2.39672
$606$	1498.08	0.100421
$607$	5069.22	0.338968	0.169484	−	0.985533i	$-0.445790\pi$
$607$	5069.22	0.338968	0.169484	+	0.985533i	$0.445790\pi$
$608$	16756.8	1.11772
$609$	0	0
$610$	−33436.8	−2.21937
$611$	4480.28	0.296650
$612$	−5482.65	−0.362129
$613$	20244.6	1.33388	0.666942	−	0.745109i	$-0.267603\pi$
$613$	20244.6	1.33388	0.666942	+	0.745109i	$0.267603\pi$
$614$	9478.87	0.623023
$615$	−6886.48	−0.451528
$616$	0	0
$617$	−9849.08	−0.642640	−0.321320	−	0.946971i	$-0.604127\pi$
$617$	−9849.08	−0.642640	−0.321320	+	0.946971i	$0.604127\pi$
$618$	1646.30	0.107158
$619$	6625.35	0.430202	0.215101	−	0.976592i	$-0.430992\pi$
$619$	6625.35	0.430202	0.215101	+	0.976592i	$0.430992\pi$
$620$	15363.5	0.995181
$621$	3915.04	0.252987
$622$	−30689.5	−1.97836
$623$	0	0
$624$	2998.28	0.192351
$625$	−17751.4	−1.13609
$626$	−30917.3	−1.97397
$627$	−13935.9	−0.887633
$628$	−10371.4	−0.659018
$629$	22826.6	1.44699
$630$	0	0
$631$	15303.0	0.965458	0.482729	−	0.875770i	$-0.339646\pi$
$631$	15303.0	0.965458	0.482729	+	0.875770i	$0.339646\pi$
$632$	9242.38	0.581712
$633$	−6823.94	−0.428479
$634$	−36495.5	−2.28615
$635$	−32291.9	−2.01806
$636$	11582.8	0.722152
$637$	0	0
$638$	−31610.2	−1.96154
$639$	−1547.47	−0.0958010
$640$	−15448.5	−0.954152
$641$	24708.8	1.52253	0.761264	−	0.648442i	$-0.224579\pi$
$641$	24708.8	1.52253	0.761264	+	0.648442i	$0.224579\pi$
$642$	20196.3	1.24157
$643$	17248.5	1.05788	0.528938	−	0.848660i	$-0.322590\pi$
$643$	17248.5	1.05788	0.528938	+	0.848660i	$0.322590\pi$
$644$	0	0
$645$	−15012.2	−0.916441
$646$	29996.6	1.82694
$647$	15232.7	0.925592	0.462796	−	0.886465i	$-0.346846\pi$
$647$	15232.7	0.925592	0.462796	+	0.886465i	$0.346846\pi$
$648$	671.205	0.0406905
$649$	13731.5	0.830524
$650$	−5049.56	−0.304708
$651$	0	0
$652$	−16320.8	−0.980327
$653$	−5254.75	−0.314907	−0.157454	−	0.987526i	$-0.550328\pi$
$653$	−5254.75	−0.314907	−0.157454	+	0.987526i	$0.550328\pi$
$654$	−19524.9	−1.16740
$655$	1427.29	0.0851436
$656$	−11644.3	−0.693040
$657$	881.361	0.0523366
$658$	0	0
$659$	17992.3	1.06355	0.531775	−	0.846885i	$-0.321525\pi$
$659$	17992.3	1.06355	0.531775	+	0.846885i	$0.321525\pi$
$660$	−15914.5	−0.938592
$661$	−22591.6	−1.32937	−0.664684	−	0.747125i	$-0.731434\pi$
$661$	−22591.6	−1.32937	−0.664684	+	0.747125i	$0.731434\pi$
$662$	20570.4	1.20769
$663$	4119.92	0.241334
$664$	2280.82	0.133302
$665$	0	0
$666$	7215.63	0.419820
$667$	−20352.0	−1.18146
$668$	−10729.6	−0.621470
$669$	−3293.31	−0.190324
$670$	31273.2	1.80327
$671$	36092.8	2.07652
$672$	0	0
$673$	11400.1	0.652959	0.326480	−	0.945204i	$-0.394138\pi$
$673$	11400.1	0.652959	0.326480	+	0.945204i	$0.394138\pi$
$674$	5407.83	0.309053
$675$	−2826.57	−0.161177
$676$	974.564	0.0554486
$677$	−623.380	−0.0353892	−0.0176946	−	0.999843i	$-0.505633\pi$
$677$	−623.380	−0.0353892	−0.0176946	+	0.999843i	$0.505633\pi$
$678$	−322.406	−0.0182624
$679$	0	0
$680$	−13266.7	−0.748169
$681$	−11932.4	−0.671441
$682$	−39590.4	−2.22287
$683$	19072.4	1.06850	0.534249	−	0.845327i	$-0.320594\pi$
$683$	19072.4	1.06850	0.534249	+	0.845327i	$0.320594\pi$
$684$	3971.92	0.222033
$685$	−3701.30	−0.206452
$686$	0	0
$687$	10658.8	0.591935
$688$	−25384.1	−1.40663
$689$	−8703.88	−0.481265
$690$	−24461.1	−1.34959
$691$	12350.0	0.679905	0.339953	−	0.940443i	$-0.389589\pi$
$691$	12350.0	0.679905	0.339953	+	0.940443i	$0.389589\pi$
$692$	1140.06	0.0626280
$693$	0	0
$694$	−37757.2	−2.06519
$695$	28091.9	1.53322
$696$	−3489.20	−0.190026
$697$	−16000.4	−0.869524
$698$	−12520.9	−0.678972
$699$	−1904.29	−0.103043
$700$	0	0
$701$	25539.0	1.37603	0.688014	−	0.725697i	$-0.258483\pi$
$701$	25539.0	1.37603	0.688014	+	0.725697i	$0.258483\pi$
$702$	1302.33	0.0700190
$703$	−16536.8	−0.887194
$704$	−11980.0	−0.641354
$705$	−15669.4	−0.837083
$706$	−2727.36	−0.145390
$707$	0	0
$708$	−3913.68	−0.207747
$709$	−233.244	−0.0123549	−0.00617747	−	0.999981i	$-0.501966\pi$
$709$	−233.244	−0.0123549	−0.00617747	+	0.999981i	$0.501966\pi$
$710$	9668.56	0.511063
$711$	10038.2	0.529482
$712$	1584.09	0.0833796
$713$	−25489.9	−1.33886
$714$	0	0
$715$	11958.9	0.625507
$716$	−17313.1	−0.903660
$717$	−10167.1	−0.529564
$718$	13614.7	0.707653
$719$	−14514.9	−0.752871	−0.376435	−	0.926443i	$-0.622850\pi$
$719$	−14514.9	−0.752871	−0.376435	+	0.926443i	$0.622850\pi$
$720$	−10486.2	−0.542775
$721$	0	0
$722$	3718.09	0.191652
$723$	−7764.86	−0.399417
$724$	12734.4	0.653688
$725$	14693.7	0.752702
$726$	26194.9	1.33910
$727$	−29317.8	−1.49565	−0.747824	−	0.663897i	$-0.768901\pi$
$727$	−29317.8	−1.49565	−0.747824	+	0.663897i	$0.768901\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−5506.74	−0.279197
$731$	−34880.1	−1.76483
$732$	−10286.9	−0.519422
$733$	−17283.7	−0.870926	−0.435463	−	0.900207i	$-0.643415\pi$
$733$	−17283.7	−0.870926	−0.435463	+	0.900207i	$0.643415\pi$
$734$	−6194.05	−0.311480
$735$	0	0
$736$	−31748.8	−1.59005
$737$	−33757.4	−1.68720
$738$	−5057.82	−0.252278
$739$	9735.42	0.484605	0.242303	−	0.970201i	$-0.422097\pi$
$739$	9735.42	0.484605	0.242303	+	0.970201i	$0.422097\pi$
$740$	−18884.7	−0.938129
$741$	−2984.69	−0.147969
$742$	0	0
$743$	−27912.2	−1.37819	−0.689097	−	0.724669i	$-0.741993\pi$
$743$	−27912.2	−1.37819	−0.689097	+	0.724669i	$0.741993\pi$
$744$	−4370.07	−0.215342
$745$	26806.2	1.31826
$746$	25945.4	1.27336
$747$	2477.21	0.121334
$748$	−36976.6	−1.80748
$749$	0	0
$750$	−3426.59	−0.166829
$751$	13787.5	0.669923	0.334962	−	0.942232i	$-0.391277\pi$
$751$	13787.5	0.669923	0.334962	+	0.942232i	$0.391277\pi$
$752$	−26495.3	−1.28482
$753$	6269.18	0.303402
$754$	−6770.05	−0.326990
$755$	25609.0	1.23445
$756$	0	0
$757$	−5456.19	−0.261966	−0.130983	−	0.991385i	$-0.541813\pi$
$757$	−5456.19	−0.261966	−0.130983	+	0.991385i	$0.541813\pi$
$758$	38152.8	1.82819
$759$	26404.1	1.26273
$760$	9611.11	0.458726
$761$	−22816.7	−1.08687	−0.543433	−	0.839452i	$-0.682876\pi$
$761$	−22816.7	−1.08687	−0.543433	+	0.839452i	$0.682876\pi$
$762$	−23717.0	−1.12753
$763$	0	0
$764$	−18886.2	−0.894343
$765$	−14409.1	−0.680994
$766$	−27145.1	−1.28041
$767$	2940.92	0.138449
$768$	−16083.1	−0.755664
$769$	39827.7	1.86765	0.933826	−	0.357727i	$-0.116448\pi$
$769$	39827.7	1.86765	0.933826	+	0.357727i	$0.116448\pi$
$770$	0	0
$771$	−7565.68	−0.353400
$772$	−20709.7	−0.965489
$773$	24985.2	1.16256	0.581278	−	0.813705i	$-0.302553\pi$
$773$	24985.2	1.16256	0.581278	+	0.813705i	$0.302553\pi$
$774$	−11025.8	−0.512034
$775$	18403.2	0.852983
$776$	7744.88	0.358280
$777$	0	0
$778$	−25740.1	−1.18616
$779$	11591.5	0.533132
$780$	−3408.45	−0.156464
$781$	−10436.6	−0.478169
$782$	−56834.2	−2.59896
$783$	−3789.64	−0.172964
$784$	0	0
$785$	−27257.2	−1.23930
$786$	1048.29	0.0475715
$787$	7568.79	0.342819	0.171409	−	0.985200i	$-0.445168\pi$
$787$	7568.79	0.342819	0.171409	+	0.985200i	$0.445168\pi$
$788$	25402.8	1.14840
$789$	17879.9	0.806772
$790$	−62718.6	−2.82459
$791$	0	0
$792$	4526.81	0.203097
$793$	7730.10	0.346159
$794$	−12286.7	−0.549165
$795$	30441.0	1.35803
$796$	−27258.9	−1.21378
$797$	3362.05	0.149423	0.0747113	−	0.997205i	$-0.476196\pi$
$797$	3362.05	0.149423	0.0747113	+	0.997205i	$0.476196\pi$
$798$	0	0
$799$	−36407.1	−1.61200
$800$	22922.0	1.01302
$801$	1720.49	0.0758933
$802$	14946.3	0.658068
$803$	5944.16	0.261226
$804$	9621.32	0.422037
$805$	0	0
$806$	−8479.19	−0.370554
$807$	−16731.6	−0.729838
$808$	1115.24	0.0485570
$809$	38036.6	1.65302	0.826512	−	0.562920i	$-0.190322\pi$
$809$	38036.6	1.65302	0.826512	+	0.562920i	$0.190322\pi$
$810$	−4554.79	−0.197579
$811$	−939.922	−0.0406968	−0.0203484	−	0.999793i	$-0.506478\pi$
$811$	−939.922	−0.0406968	−0.0203484	+	0.999793i	$0.506478\pi$
$812$	0	0
$813$	11749.3	0.506848
$814$	48664.3	2.09543
$815$	−42893.1	−1.84353
$816$	−24364.2	−1.04524
$817$	25269.0	1.08207
$818$	21480.8	0.918163
$819$	0	0
$820$	13237.3	0.563740
$821$	17499.0	0.743874	0.371937	−	0.928258i	$-0.378694\pi$
$821$	17499.0	0.743874	0.371937	+	0.928258i	$0.378694\pi$
$822$	−2718.45	−0.115349
$823$	−45803.0	−1.93997	−0.969983	−	0.243172i	$-0.921812\pi$
$823$	−45803.0	−1.93997	−0.969983	+	0.243172i	$0.921812\pi$
$824$	1225.59	0.0518147
$825$	−19063.2	−0.804480
$826$	0	0
$827$	−4491.96	−0.188876	−0.0944382	−	0.995531i	$-0.530105\pi$
$827$	−4491.96	−0.188876	−0.0944382	+	0.995531i	$0.530105\pi$
$828$	−7525.55	−0.315859
$829$	−16649.1	−0.697525	−0.348762	−	0.937211i	$-0.613398\pi$
$829$	−16649.1	−0.697525	−0.348762	+	0.937211i	$0.613398\pi$
$830$	−15477.6	−0.647271
$831$	5168.80	0.215769
$832$	−2565.79	−0.106914
$833$	0	0
$834$	20632.3	0.856639
$835$	−28198.8	−1.16869
$836$	26787.8	1.10822
$837$	−4746.36	−0.196007
$838$	43074.0	1.77562
$839$	−18189.8	−0.748487	−0.374243	−	0.927331i	$-0.622098\pi$
$839$	−18189.8	−0.748487	−0.374243	+	0.927331i	$0.622098\pi$
$840$	0	0
$841$	−4688.88	−0.192254
$842$	−54173.1	−2.21726
$843$	−25017.2	−1.02211
$844$	13117.1	0.534963
$845$	2561.27	0.104273
$846$	−11508.5	−0.467696
$847$	0	0
$848$	51472.7	2.08441
$849$	−8983.88	−0.363164
$850$	41033.1	1.65579
$851$	31332.1	1.26210
$852$	2974.57	0.119609
$853$	−16854.3	−0.676530	−0.338265	−	0.941051i	$-0.609840\pi$
$853$	−16854.3	−0.676530	−0.338265	+	0.941051i	$0.609840\pi$
$854$	0	0
$855$	10438.7	0.417539
$856$	15035.1	0.600339
$857$	−13915.7	−0.554670	−0.277335	−	0.960773i	$-0.589451\pi$
$857$	−13915.7	−0.554670	−0.277335	+	0.960773i	$0.589451\pi$
$858$	8783.30	0.349484
$859$	−33490.9	−1.33026	−0.665131	−	0.746727i	$-0.731624\pi$
$859$	−33490.9	−1.33026	−0.665131	+	0.746727i	$0.731624\pi$
$860$	28856.7	1.14419
$861$	0	0
$862$	19487.9	0.770022
$863$	46684.7	1.84144	0.920721	−	0.390222i	$-0.127602\pi$
$863$	46684.7	1.84144	0.920721	+	0.390222i	$0.127602\pi$
$864$	−5911.80	−0.232782
$865$	2996.22	0.117774
$866$	−42571.5	−1.67048
$867$	−18739.7	−0.734066
$868$	0	0
$869$	67700.6	2.64279
$870$	23677.7	0.922699
$871$	−7229.91	−0.281259
$872$	−14535.3	−0.564480
$873$	8411.76	0.326111
$874$	41173.8	1.59350
$875$	0	0
$876$	−1694.17	−0.0653432
$877$	24804.1	0.955045	0.477523	−	0.878619i	$-0.341535\pi$
$877$	24804.1	0.955045	0.477523	+	0.878619i	$0.341535\pi$
$878$	66795.1	2.56746
$879$	−25477.8	−0.977639
$880$	−70722.1	−2.70914
$881$	15752.6	0.602406	0.301203	−	0.953560i	$-0.402612\pi$
$881$	15752.6	0.602406	0.301203	+	0.953560i	$0.402612\pi$
$882$	0	0
$883$	14761.2	0.562575	0.281287	−	0.959624i	$-0.409239\pi$
$883$	14761.2	0.562575	0.281287	+	0.959624i	$0.409239\pi$
$884$	−7919.38	−0.301309
$885$	−10285.6	−0.390675
$886$	−25495.7	−0.966757
$887$	−12017.3	−0.454905	−0.227452	−	0.973789i	$-0.573040\pi$
$887$	−12017.3	−0.454905	−0.227452	+	0.973789i	$0.573040\pi$
$888$	5371.67	0.202997
$889$	0	0
$890$	−10749.6	−0.404863
$891$	4916.59	0.184862
$892$	6330.45	0.237623
$893$	26375.3	0.988370
$894$	19688.0	0.736537
$895$	−45500.9	−1.69936
$896$	0	0
$897$	5655.05	0.210498
$898$	45654.4	1.69656
$899$	24673.5	0.915360
$900$	5433.28	0.201233
$901$	70728.3	2.61521
$902$	−34111.4	−1.25919
$903$	0	0
$904$	−240.015	−0.00883049
$905$	33467.5	1.22928
$906$	18808.7	0.689710
$907$	22681.8	0.830361	0.415181	−	0.909739i	$-0.363718\pi$
$907$	22681.8	0.830361	0.415181	+	0.909739i	$0.363718\pi$
$908$	22936.7	0.838305
$909$	1211.27	0.0441973
$910$	0	0
$911$	−41767.9	−1.51903	−0.759513	−	0.650492i	$-0.774563\pi$
$911$	−41767.9	−1.51903	−0.759513	+	0.650492i	$0.774563\pi$
$912$	17650.7	0.640871
$913$	16707.0	0.605610
$914$	7607.39	0.275306
$915$	−27035.3	−0.976787
$916$	−20488.6	−0.739041
$917$	0	0
$918$	−10582.8	−0.380485
$919$	−19326.8	−0.693726	−0.346863	−	0.937916i	$-0.612753\pi$
$919$	−19326.8	−0.693726	−0.346863	+	0.937916i	$0.612753\pi$
$920$	−18210.1	−0.652574
$921$	7664.15	0.274204
$922$	60761.2	2.17035
$923$	−2235.23	−0.0797112
$924$	0	0
$925$	−22621.1	−0.804083
$926$	−14033.4	−0.498018
$927$	1331.12	0.0471624
$928$	30732.0	1.08710
$929$	14276.7	0.504201	0.252101	−	0.967701i	$-0.418879\pi$
$929$	14276.7	0.504201	0.252101	+	0.967701i	$0.418879\pi$
$930$	29655.2	1.04563
$931$	0	0
$932$	3660.45	0.128650
$933$	−24814.0	−0.870713
$934$	−43203.8	−1.51357
$935$	−97178.9	−3.39903
$936$	969.519	0.0338565
$937$	15265.0	0.532214	0.266107	−	0.963943i	$-0.414262\pi$
$937$	15265.0	0.532214	0.266107	+	0.963943i	$0.414262\pi$
$938$	0	0
$939$	−24998.2	−0.868780
$940$	30120.0	1.04511
$941$	22508.7	0.779768	0.389884	−	0.920864i	$-0.372515\pi$
$941$	22508.7	0.779768	0.389884	+	0.920864i	$0.372515\pi$
$942$	−20019.3	−0.692424
$943$	−21962.4	−0.758423
$944$	−17391.9	−0.599639
$945$	0	0
$946$	−74361.3	−2.55570
$947$	17943.5	0.615720	0.307860	−	0.951432i	$-0.400387\pi$
$947$	17943.5	0.615720	0.307860	+	0.951432i	$0.400387\pi$
$948$	−19295.6	−0.661068
$949$	1273.08	0.0435467
$950$	−29726.5	−1.01522
$951$	−29508.5	−1.00618
$952$	0	0
$953$	−5504.20	−0.187092	−0.0935460	−	0.995615i	$-0.529820\pi$
$953$	−5504.20	−0.187092	−0.0935460	+	0.995615i	$0.529820\pi$
$954$	22357.6	0.758758
$955$	−49635.1	−1.68184
$956$	19543.4	0.661169
$957$	−25558.5	−0.863310
$958$	12838.1	0.432965
$959$	0	0
$960$	8973.63	0.301690
$961$	1111.51	0.0373104
$962$	10422.6	0.349311
$963$	16329.8	0.546437
$964$	14925.7	0.498678
$965$	−54427.5	−1.81563
$966$	0	0
$967$	−27534.1	−0.915652	−0.457826	−	0.889042i	$-0.651372\pi$
$967$	−27534.1	−0.915652	−0.457826	+	0.889042i	$0.651372\pi$
$968$	19500.8	0.647499
$969$	24253.8	0.804070
$970$	−52556.6	−1.73968
$971$	26329.7	0.870194	0.435097	−	0.900383i	$-0.356714\pi$
$971$	26329.7	0.870194	0.435097	+	0.900383i	$0.356714\pi$
$972$	−1401.30	−0.0462414
$973$	0	0
$974$	1114.11	0.0366515
$975$	−4082.82	−0.134108
$976$	−45714.0	−1.49925
$977$	55209.8	1.80790	0.903950	−	0.427638i	$-0.140654\pi$
$977$	55209.8	1.80790	0.903950	+	0.427638i	$0.140654\pi$
$978$	−31503.1	−1.03002
$979$	11603.5	0.378804
$980$	0	0
$981$	−15786.8	−0.513797
$982$	−26548.4	−0.862722
$983$	−16795.9	−0.544971	−0.272486	−	0.962160i	$-0.587846\pi$
$983$	−16795.9	−0.544971	−0.272486	+	0.962160i	$0.587846\pi$
$984$	−3765.29	−0.121985
$985$	66761.7	2.15960
$986$	55013.9	1.77688
$987$	0	0
$988$	5737.22	0.184742
$989$	−47876.9	−1.53933
$990$	−30718.8	−0.986170
$991$	−61142.2	−1.95988	−0.979942	−	0.199282i	$-0.936139\pi$
$991$	−61142.2	−1.95988	−0.979942	+	0.199282i	$0.936139\pi$
$992$	38490.4	1.23193
$993$	16632.2	0.531527
$994$	0	0
$995$	−71639.6	−2.28254
$996$	−4761.73	−0.151487
$997$	−7852.57	−0.249442	−0.124721	−	0.992192i	$-0.539804\pi$
$997$	−7852.57	−0.249442	−0.124721	+	0.992192i	$0.539804\pi$
$998$	15511.7	0.491997
$999$	5834.20	0.184771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1911.4.a.l.1.2		4
7.6	odd	2		273.4.a.f.1.2	✓	4
21.20	even	2		819.4.a.g.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.4.a.f.1.2	✓	4	7.6	odd	2
819.4.a.g.1.3		4	21.20	even	2
1911.4.a.l.1.2		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1911.a (trivial)

\(f(q)\)	\(=\)	\(q-3.71034 q^{2} -3.00000 q^{3} +5.76665 q^{4} +15.1555 q^{5} +11.1310 q^{6} +8.28649 q^{8} +9.00000 q^{9} -56.2319 q^{10} +60.6986 q^{11} -17.3000 q^{12} +13.0000 q^{13} -45.4664 q^{15} -76.8789 q^{16} -105.639 q^{17} -33.3931 q^{18} +76.5305 q^{19} +87.3962 q^{20} -225.213 q^{22} -145.001 q^{23} -24.8595 q^{24} +104.688 q^{25} -48.2345 q^{26} -27.0000 q^{27} +140.357 q^{29} +168.696 q^{30} +175.791 q^{31} +218.955 q^{32} -182.096 q^{33} +391.957 q^{34} +51.8999 q^{36} -216.081 q^{37} -283.954 q^{38} -39.0000 q^{39} +125.585 q^{40} +151.463 q^{41} +330.182 q^{43} +350.028 q^{44} +136.399 q^{45} +538.005 q^{46} +344.637 q^{47} +230.637 q^{48} -388.428 q^{50} +316.917 q^{51} +74.9665 q^{52} -669.529 q^{53} +100.179 q^{54} +919.915 q^{55} -229.591 q^{57} -520.773 q^{58} +226.225 q^{59} -262.189 q^{60} +594.623 q^{61} -652.246 q^{62} -197.368 q^{64} +197.021 q^{65} +675.639 q^{66} -556.147 q^{67} -609.183 q^{68} +435.004 q^{69} -171.941 q^{71} +74.5784 q^{72} +97.9290 q^{73} +801.736 q^{74} -314.063 q^{75} +441.325 q^{76} +144.703 q^{78} +1115.36 q^{79} -1165.14 q^{80} +81.0000 q^{81} -561.980 q^{82} +275.245 q^{83} -1601.01 q^{85} -1225.09 q^{86} -421.071 q^{87} +502.978 q^{88} +191.166 q^{89} -506.088 q^{90} -836.172 q^{92} -527.373 q^{93} -1278.72 q^{94} +1159.85 q^{95} -656.866 q^{96} +934.640 q^{97} +546.288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} - 12 q^{3} + 15 q^{4} + 24 q^{5} + 9 q^{6} - 39 q^{8} + 36 q^{9} + 26 q^{10} + 8 q^{11} - 45 q^{12} + 52 q^{13} - 72 q^{15} - 181 q^{16} + 6 q^{17} - 27 q^{18} + 332 q^{19} + 62 q^{20} - 176 q^{22}+ \cdots + 72 q^{99}+O(q^{100}) \) 4 * q - 3 * q^2 - 12 * q^3 + 15 * q^4 + 24 * q^5 + 9 * q^6 - 39 * q^8 + 36 * q^9 + 26 * q^10 + 8 * q^11 - 45 * q^12 + 52 * q^13 - 72 * q^15 - 181 * q^16 + 6 * q^17 - 27 * q^18 + 332 * q^19 + 62 * q^20 - 176 * q^22 + 6 * q^23 + 117 * q^24 - 12 * q^25 - 39 * q^26 - 108 * q^27 + 8 * q^29 - 78 * q^30 + 184 * q^31 + 485 * q^32 - 24 * q^33 + 326 * q^34 + 135 * q^36 - 244 * q^37 - 516 * q^38 - 156 * q^39 - 22 * q^40 + 296 * q^41 - 224 * q^43 - 336 * q^44 + 216 * q^45 - 404 * q^46 + 750 * q^47 + 543 * q^48 + 35 * q^50 - 18 * q^51 + 195 * q^52 + 680 * q^53 + 81 * q^54 + 600 * q^55 - 996 * q^57 - 334 * q^58 + 334 * q^59 - 186 * q^60 + 1212 * q^61 + 928 * q^62 - 697 * q^64 + 312 * q^65 + 528 * q^66 - 388 * q^67 - 1174 * q^68 - 18 * q^69 - 544 * q^71 - 351 * q^72 + 1572 * q^73 + 1482 * q^74 + 36 * q^75 + 1508 * q^76 + 117 * q^78 + 464 * q^79 - 2066 * q^80 + 324 * q^81 - 1806 * q^82 + 3158 * q^83 - 2228 * q^85 + 324 * q^86 - 24 * q^87 + 984 * q^88 - 1156 * q^89 + 234 * q^90 + 1788 * q^92 - 552 * q^93 - 4604 * q^94 + 1128 * q^95 - 1455 * q^96 + 1928 * q^97 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more