LMFDB - Embedded newform 1815.4.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1815.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:	$107.088466660$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7598722752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 27x^{4} + 156x^{2} - 12 $ x^6 - 27x^4 + 156x^2 - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.86989$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.60194 q^{2} -3.00000 q^{3} +13.1778 q^{4} +5.00000 q^{5} +13.8058 q^{6} -6.69261 q^{7} -23.8281 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.60194 q^{2} -3.00000 q^{3} +13.1778 q^{4} +5.00000 q^{5} +13.8058 q^{6} -6.69261 q^{7} -23.8281 q^{8} +9.00000 q^{9} -23.0097 q^{10} -39.5335 q^{12} +41.4175 q^{13} +30.7990 q^{14} -15.0000 q^{15} +4.23281 q^{16} +49.3883 q^{17} -41.4175 q^{18} +20.3640 q^{19} +65.8892 q^{20} +20.0778 q^{21} -116.277 q^{23} +71.4844 q^{24} +25.0000 q^{25} -190.601 q^{26} -27.0000 q^{27} -88.1942 q^{28} -7.18115 q^{29} +69.0291 q^{30} -72.6203 q^{31} +171.146 q^{32} -227.282 q^{34} -33.4631 q^{35} +118.601 q^{36} -61.8419 q^{37} -93.7140 q^{38} -124.252 q^{39} -119.141 q^{40} +101.888 q^{41} -92.3970 q^{42} +234.205 q^{43} +45.0000 q^{45} +535.098 q^{46} +304.290 q^{47} -12.6984 q^{48} -298.209 q^{49} -115.048 q^{50} -148.165 q^{51} +545.793 q^{52} -90.8986 q^{53} +124.252 q^{54} +159.472 q^{56} -61.0921 q^{57} +33.0472 q^{58} -423.859 q^{59} -197.668 q^{60} -840.145 q^{61} +334.194 q^{62} -60.2335 q^{63} -821.465 q^{64} +207.087 q^{65} -189.117 q^{67} +650.831 q^{68} +348.830 q^{69} +153.995 q^{70} +71.0547 q^{71} -214.453 q^{72} -207.435 q^{73} +284.593 q^{74} -75.0000 q^{75} +268.354 q^{76} +571.802 q^{78} +406.511 q^{79} +21.1640 q^{80} +81.0000 q^{81} -468.883 q^{82} -1034.29 q^{83} +264.583 q^{84} +246.941 q^{85} -1077.80 q^{86} +21.5434 q^{87} -1292.95 q^{89} -207.087 q^{90} -277.191 q^{91} -1532.28 q^{92} +217.861 q^{93} -1400.32 q^{94} +101.820 q^{95} -513.437 q^{96} -1401.68 q^{97} +1372.34 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 18 q^{3} + 12 q^{4} + 30 q^{5} + 54 q^{9}+O(q^{10}) $ 6 * q - 18 * q^3 + 12 * q^4 + 30 * q^5 + 54 * q^9	$ 6 q - 18 q^{3} + 12 q^{4} + 30 q^{5} + 54 q^{9} - 36 q^{12} + 24 q^{14} - 90 q^{15} - 60 q^{16} + 60 q^{20} + 18 q^{23} + 150 q^{25} - 540 q^{26} - 162 q^{27} + 30 q^{31} - 84 q^{34} + 108 q^{36} - 576 q^{37} - 12 q^{38} - 72 q^{42} + 270 q^{45} - 330 q^{47} + 180 q^{48} - 1134 q^{49} - 78 q^{53} + 936 q^{56} - 372 q^{58} - 1488 q^{59} - 180 q^{60} - 252 q^{64} + 120 q^{67} - 54 q^{69} + 120 q^{70} + 156 q^{71} - 450 q^{75} + 1620 q^{78} - 300 q^{80} + 486 q^{81} + 468 q^{82} - 372 q^{86} - 1800 q^{89} - 216 q^{91} - 3972 q^{92} - 90 q^{93} - 180 q^{97}+O(q^{100}) $ 6 * q - 18 * q^3 + 12 * q^4 + 30 * q^5 + 54 * q^9 - 36 * q^12 + 24 * q^14 - 90 * q^15 - 60 * q^16 + 60 * q^20 + 18 * q^23 + 150 * q^25 - 540 * q^26 - 162 * q^27 + 30 * q^31 - 84 * q^34 + 108 * q^36 - 576 * q^37 - 12 * q^38 - 72 * q^42 + 270 * q^45 - 330 * q^47 + 180 * q^48 - 1134 * q^49 - 78 * q^53 + 936 * q^56 - 372 * q^58 - 1488 * q^59 - 180 * q^60 - 252 * q^64 + 120 * q^67 - 54 * q^69 + 120 * q^70 + 156 * q^71 - 450 * q^75 + 1620 * q^78 - 300 * q^80 + 486 * q^81 + 468 * q^82 - 372 * q^86 - 1800 * q^89 - 216 * q^91 - 3972 * q^92 - 90 * q^93 - 180 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.60194	−1.62703	−0.813516	−	0.581543i	$-0.802449\pi$
$2$	−4.60194	−1.62703	−0.813516	+	0.581543i	$0.802449\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	13.1778	1.64723
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	13.8058	0.939367
$7$	−6.69261	−0.361367	−0.180684	−	0.983541i	$-0.557831\pi$
$7$	−6.69261	−0.361367	−0.180684	+	0.983541i	$0.557831\pi$
$8$	−23.8281	−1.05306
$9$	9.00000	0.333333
$10$	−23.0097	−0.727630
$11$	0	0
$11$	0	0
$12$	−39.5335	−0.951029
$13$	41.4175	0.883626	0.441813	−	0.897107i	$-0.354336\pi$
$13$	41.4175	0.883626	0.441813	+	0.897107i	$0.354336\pi$
$14$	30.7990	0.587956
$15$	−15.0000	−0.258199
$16$	4.23281	0.0661376
$17$	49.3883	0.704613	0.352306	−	0.935885i	$-0.385397\pi$
$17$	49.3883	0.704613	0.352306	+	0.935885i	$0.385397\pi$
$18$	−41.4175	−0.542344
$19$	20.3640	0.245886	0.122943	−	0.992414i	$-0.460767\pi$
$19$	20.3640	0.245886	0.122943	+	0.992414i	$0.460767\pi$
$20$	65.8892	0.736664
$21$	20.0778	0.208635
$22$	0	0
$23$	−116.277	−1.05415	−0.527073	−	0.849820i	$-0.676711\pi$
$23$	−116.277	−1.05415	−0.527073	+	0.849820i	$0.676711\pi$
$24$	71.4844	0.607987
$25$	25.0000	0.200000
$26$	−190.601	−1.43769
$27$	−27.0000	−0.192450
$28$	−88.1942	−0.595255
$29$	−7.18115	−0.0459830	−0.0229915	−	0.999736i	$-0.507319\pi$
$29$	−7.18115	−0.0459830	−0.0229915	+	0.999736i	$0.507319\pi$
$30$	69.0291	0.420098
$31$	−72.6203	−0.420742	−0.210371	−	0.977622i	$-0.567467\pi$
$31$	−72.6203	−0.420742	−0.210371	+	0.977622i	$0.567467\pi$
$32$	171.146	0.945456
$33$	0	0
$34$	−227.282	−1.14643
$35$	−33.4631	−0.161608
$36$	118.601	0.549077
$37$	−61.8419	−0.274777	−0.137388	−	0.990517i	$-0.543871\pi$
$37$	−61.8419	−0.274777	−0.137388	+	0.990517i	$0.543871\pi$
$38$	−93.7140	−0.400064
$39$	−124.252	−0.510162
$40$	−119.141	−0.470945
$41$	101.888	0.388103	0.194052	−	0.980991i	$-0.437837\pi$
$41$	101.888	0.388103	0.194052	+	0.980991i	$0.437837\pi$
$42$	−92.3970	−0.339456
$43$	234.205	0.830602	0.415301	−	0.909684i	$-0.363676\pi$
$43$	234.205	0.830602	0.415301	+	0.909684i	$0.363676\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	535.098	1.71513
$47$	304.290	0.944366	0.472183	−	0.881501i	$-0.343466\pi$
$47$	304.290	0.944366	0.472183	+	0.881501i	$0.343466\pi$
$48$	−12.6984	−0.0381846
$49$	−298.209	−0.869414
$50$	−115.048	−0.325406
$51$	−148.165	−0.406808
$52$	545.793	1.45554
$53$	−90.8986	−0.235583	−0.117791	−	0.993038i	$-0.537581\pi$
$53$	−90.8986	−0.235583	−0.117791	+	0.993038i	$0.537581\pi$
$54$	124.252	0.313122
$55$	0	0
$56$	159.472	0.380543
$57$	−61.0921	−0.141962
$58$	33.0472	0.0748157
$59$	−423.859	−0.935283	−0.467641	−	0.883918i	$-0.654896\pi$
$59$	−423.859	−0.935283	−0.467641	+	0.883918i	$0.654896\pi$
$60$	−197.668	−0.425313
$61$	−840.145	−1.76344	−0.881718	−	0.471778i	$-0.843613\pi$
$61$	−840.145	−1.76344	−0.881718	+	0.471778i	$0.843613\pi$
$62$	334.194	0.684560
$63$	−60.2335	−0.120456
$64$	−821.465	−1.60442
$65$	207.087	0.395169
$66$	0	0
$67$	−189.117	−0.344840	−0.172420	−	0.985023i	$-0.555159\pi$
$67$	−189.117	−0.344840	−0.172420	+	0.985023i	$0.555159\pi$
$68$	650.831	1.16066
$69$	348.830	0.608612
$70$	153.995	0.262942
$71$	71.0547	0.118770	0.0593848	−	0.998235i	$-0.481086\pi$
$71$	71.0547	0.118770	0.0593848	+	0.998235i	$0.481086\pi$
$72$	−214.453	−0.351021
$73$	−207.435	−0.332582	−0.166291	−	0.986077i	$-0.553179\pi$
$73$	−207.435	−0.332582	−0.166291	+	0.986077i	$0.553179\pi$
$74$	284.593	0.447070
$75$	−75.0000	−0.115470
$76$	268.354	0.405030
$77$	0	0
$78$	571.802	0.830049
$79$	406.511	0.578938	0.289469	−	0.957187i	$-0.406521\pi$
$79$	406.511	0.578938	0.289469	+	0.957187i	$0.406521\pi$
$80$	21.1640	0.0295776
$81$	81.0000	0.111111
$82$	−468.883	−0.631456
$83$	−1034.29	−1.36780	−0.683902	−	0.729574i	$-0.739718\pi$
$83$	−1034.29	−1.36780	−0.683902	+	0.729574i	$0.739718\pi$
$84$	264.583	0.343671
$85$	246.941	0.315112
$86$	−1077.80	−1.35141
$87$	21.5434	0.0265483
$88$	0	0
$89$	−1292.95	−1.53992	−0.769958	−	0.638094i	$-0.779723\pi$
$89$	−1292.95	−1.53992	−0.769958	+	0.638094i	$0.779723\pi$
$90$	−207.087	−0.242543
$91$	−277.191	−0.319313
$92$	−1532.28	−1.73642
$93$	217.861	0.242915
$94$	−1400.32	−1.53651
$95$	101.820	0.109963
$96$	−513.437	−0.545859
$97$	−1401.68	−1.46721	−0.733604	−	0.679578i	$-0.762163\pi$
$97$	−1401.68	−1.46721	−0.733604	+	0.679578i	$0.762163\pi$
$98$	1372.34	1.41456
$99$	0	0
$100$	329.446	0.329446
$101$	709.987	0.699469	0.349734	−	0.936849i	$-0.386272\pi$
$101$	709.987	0.699469	0.349734	+	0.936849i	$0.386272\pi$
$102$	681.846	0.661890
$103$	1519.29	1.45340	0.726701	−	0.686954i	$-0.241053\pi$
$103$	1519.29	1.45340	0.726701	+	0.686954i	$0.241053\pi$
$104$	−986.900	−0.930514
$105$	100.389	0.0933046
$106$	418.310	0.383300
$107$	1511.95	1.36604	0.683018	−	0.730402i	$-0.260667\pi$
$107$	1511.95	1.36604	0.683018	+	0.730402i	$0.260667\pi$
$108$	−355.802	−0.317010
$109$	−976.109	−0.857746	−0.428873	−	0.903365i	$-0.641089\pi$
$109$	−976.109	−0.857746	−0.428873	+	0.903365i	$0.641089\pi$
$110$	0	0
$111$	185.526	0.158642
$112$	−28.3285	−0.0239000
$113$	1584.44	1.31904	0.659519	−	0.751688i	$-0.270760\pi$
$113$	1584.44	1.31904	0.659519	+	0.751688i	$0.270760\pi$
$114$	281.142	0.230977
$115$	−581.384	−0.471429
$116$	−94.6321	−0.0757445
$117$	372.757	0.294542
$118$	1950.57	1.52173
$119$	−330.537	−0.254624
$120$	357.422	0.271900
$121$	0	0
$122$	3866.30	2.86916
$123$	−305.664	−0.224072
$124$	−956.979	−0.693058
$125$	125.000	0.0894427
$126$	277.191	0.195985
$127$	71.1996	0.0497476	0.0248738	−	0.999691i	$-0.492082\pi$
$127$	71.1996	0.0497476	0.0248738	+	0.999691i	$0.492082\pi$
$128$	2411.17	1.66499
$129$	−702.614	−0.479548
$130$	−953.003	−0.642953
$131$	−75.1630	−0.0501300	−0.0250650	−	0.999686i	$-0.507979\pi$
$131$	−75.1630	−0.0501300	−0.0250650	+	0.999686i	$0.507979\pi$
$132$	0	0
$133$	−136.288	−0.0888550
$134$	870.304	0.561066
$135$	−135.000	−0.0860663
$136$	−1176.83	−0.742003
$137$	1181.45	0.736775	0.368388	−	0.929672i	$-0.379910\pi$
$137$	1181.45	0.736775	0.368388	+	0.929672i	$0.379910\pi$
$138$	−1605.29	−0.990230
$139$	1847.69	1.12748	0.563738	−	0.825954i	$-0.309363\pi$
$139$	1847.69	1.12748	0.563738	+	0.825954i	$0.309363\pi$
$140$	−440.971	−0.266206
$141$	−912.869	−0.545230
$142$	−326.989	−0.193242
$143$	0	0
$144$	38.0953	0.0220459
$145$	−35.9057	−0.0205642
$146$	954.605	0.541121
$147$	894.627	0.501956
$148$	−814.942	−0.452621
$149$	2010.81	1.10558	0.552791	−	0.833320i	$-0.313563\pi$
$149$	2010.81	1.10558	0.552791	+	0.833320i	$0.313563\pi$
$150$	345.145	0.187873
$151$	932.411	0.502507	0.251253	−	0.967921i	$-0.419157\pi$
$151$	932.411	0.502507	0.251253	+	0.967921i	$0.419157\pi$
$152$	−485.236	−0.258933
$153$	444.495	0.234871
$154$	0	0
$155$	−363.102	−0.188161
$156$	−1637.38	−0.840354
$157$	−1705.75	−0.867093	−0.433546	−	0.901131i	$-0.642738\pi$
$157$	−1705.75	−0.867093	−0.433546	+	0.901131i	$0.642738\pi$
$158$	−1870.74	−0.941950
$159$	272.696	0.136014
$160$	855.729	0.422821
$161$	778.195	0.380934
$162$	−372.757	−0.180781
$163$	2812.42	1.35145	0.675723	−	0.737155i	$-0.263831\pi$
$163$	2812.42	1.35145	0.675723	+	0.737155i	$0.263831\pi$
$164$	1342.66	0.639296
$165$	0	0
$166$	4759.72	2.22546
$167$	1098.60	0.509053	0.254527	−	0.967066i	$-0.418080\pi$
$167$	1098.60	0.509053	0.254527	+	0.967066i	$0.418080\pi$
$168$	−478.417	−0.219706
$169$	−481.595	−0.219206
$170$	−1136.41	−0.512698
$171$	183.276	0.0819619
$172$	3086.31	1.36819
$173$	−944.544	−0.415100	−0.207550	−	0.978224i	$-0.566549\pi$
$173$	−944.544	−0.415100	−0.207550	+	0.978224i	$0.566549\pi$
$174$	−99.1416	−0.0431949
$175$	−167.315	−0.0722734
$176$	0	0
$177$	1271.58	0.539986
$178$	5950.08	2.50549
$179$	3129.81	1.30689	0.653445	−	0.756974i	$-0.273323\pi$
$179$	3129.81	1.30689	0.653445	+	0.756974i	$0.273323\pi$
$180$	593.003	0.245555
$181$	−4025.10	−1.65295	−0.826474	−	0.562975i	$-0.809657\pi$
$181$	−4025.10	−1.65295	−0.826474	+	0.562975i	$0.809657\pi$
$182$	1275.62	0.519533
$183$	2520.43	1.01812
$184$	2770.66	1.11008
$185$	−309.209	−0.122884
$186$	−1002.58	−0.395231
$187$	0	0
$188$	4009.88	1.55559
$189$	180.701	0.0695451
$190$	−468.570	−0.178914
$191$	−582.534	−0.220684	−0.110342	−	0.993894i	$-0.535195\pi$
$191$	−582.534	−0.220684	−0.110342	+	0.993894i	$0.535195\pi$
$192$	2464.40	0.926315
$193$	−3454.46	−1.28838	−0.644190	−	0.764865i	$-0.722806\pi$
$193$	−3454.46	−1.28838	−0.644190	+	0.764865i	$0.722806\pi$
$194$	6450.45	2.38719
$195$	−621.262	−0.228151
$196$	−3929.75	−1.43212
$197$	2701.82	0.977142	0.488571	−	0.872524i	$-0.337518\pi$
$197$	2701.82	0.977142	0.488571	+	0.872524i	$0.337518\pi$
$198$	0	0
$199$	−4773.59	−1.70046	−0.850229	−	0.526413i	$-0.823537\pi$
$199$	−4773.59	−1.70046	−0.850229	+	0.526413i	$0.823537\pi$
$200$	−595.703	−0.210613
$201$	567.351	0.199094
$202$	−3267.32	−1.13806
$203$	48.0607	0.0166167
$204$	−1952.49	−0.670107
$205$	509.440	0.173565
$206$	−6991.70	−2.36473
$207$	−1046.49	−0.351382
$208$	175.312	0.0584409
$209$	0	0
$210$	−461.985	−0.151809
$211$	2700.03	0.880938	0.440469	−	0.897768i	$-0.354812\pi$
$211$	2700.03	0.880938	0.440469	+	0.897768i	$0.354812\pi$
$212$	−1197.85	−0.388059
$213$	−213.164	−0.0685716
$214$	−6957.91	−2.22258
$215$	1171.02	0.371456
$216$	643.359	0.202662
$217$	486.020	0.152042
$218$	4491.99	1.39558
$219$	622.306	0.192016
$220$	0	0
$221$	2045.54	0.622614
$222$	−853.778	−0.258116
$223$	−1917.44	−0.575791	−0.287895	−	0.957662i	$-0.592956\pi$
$223$	−1917.44	−0.575791	−0.287895	+	0.957662i	$0.592956\pi$
$224$	−1145.41	−0.341657
$225$	225.000	0.0666667
$226$	−7291.48	−2.14612
$227$	4410.61	1.28961	0.644807	−	0.764345i	$-0.276938\pi$
$227$	4410.61	1.28961	0.644807	+	0.764345i	$0.276938\pi$
$228$	−805.062	−0.233844
$229$	1358.00	0.391874	0.195937	−	0.980617i	$-0.437225\pi$
$229$	1358.00	0.391874	0.195937	+	0.980617i	$0.437225\pi$
$230$	2675.49	0.767029
$231$	0	0
$232$	171.113	0.0484230
$233$	3680.72	1.03490	0.517450	−	0.855713i	$-0.326881\pi$
$233$	3680.72	1.03490	0.517450	+	0.855713i	$0.326881\pi$
$234$	−1715.41	−0.479229
$235$	1521.45	0.422333
$236$	−5585.54	−1.54063
$237$	−1219.53	−0.334250
$238$	1521.11	0.414281
$239$	−4121.89	−1.11558	−0.557789	−	0.829983i	$-0.688350\pi$
$239$	−4121.89	−1.11558	−0.557789	+	0.829983i	$0.688350\pi$
$240$	−63.4921	−0.0170767
$241$	−3806.41	−1.01740	−0.508698	−	0.860945i	$-0.669873\pi$
$241$	−3806.41	−1.01740	−0.508698	+	0.860945i	$0.669873\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	−11071.3	−2.90478
$245$	−1491.04	−0.388814
$246$	1406.65	0.364571
$247$	843.426	0.217271
$248$	1730.41	0.443068
$249$	3102.86	0.789702
$250$	−575.242	−0.145526
$251$	7082.86	1.78114	0.890570	−	0.454846i	$-0.150306\pi$
$251$	7082.86	1.78114	0.890570	+	0.454846i	$0.150306\pi$
$252$	−793.748	−0.198418
$253$	0	0
$254$	−327.656	−0.0809409
$255$	−740.824	−0.181930
$256$	−4524.32	−1.10457
$257$	−1504.95	−0.365277	−0.182638	−	0.983180i	$-0.558464\pi$
$257$	−1504.95	−0.365277	−0.182638	+	0.983180i	$0.558464\pi$
$258$	3233.39	0.780240
$259$	413.884	0.0992953
$260$	2728.96	0.650935
$261$	−64.6303	−0.0153277
$262$	345.896	0.0815630
$263$	5077.47	1.19046	0.595229	−	0.803556i	$-0.297061\pi$
$263$	5077.47	1.19046	0.595229	+	0.803556i	$0.297061\pi$
$264$	0	0
$265$	−454.493	−0.105356
$266$	627.191	0.144570
$267$	3878.85	0.889071
$268$	−2492.15	−0.568031
$269$	−7952.90	−1.80259	−0.901295	−	0.433205i	$-0.857383\pi$
$269$	−7952.90	−1.80259	−0.901295	+	0.433205i	$0.857383\pi$
$270$	621.262	0.140033
$271$	−1234.04	−0.276615	−0.138307	−	0.990389i	$-0.544166\pi$
$271$	−1234.04	−0.276615	−0.138307	+	0.990389i	$0.544166\pi$
$272$	209.051	0.0466014
$273$	831.573	0.184356
$274$	−5436.97	−1.19876
$275$	0	0
$276$	4596.83	1.00252
$277$	−5279.41	−1.14516	−0.572580	−	0.819849i	$-0.694057\pi$
$277$	−5279.41	−1.14516	−0.572580	+	0.819849i	$0.694057\pi$
$278$	−8502.96	−1.83444
$279$	−653.583	−0.140247
$280$	797.362	0.170184
$281$	−4283.11	−0.909285	−0.454642	−	0.890674i	$-0.650233\pi$
$281$	−4283.11	−0.909285	−0.454642	+	0.890674i	$0.650233\pi$
$282$	4200.97	0.887106
$283$	−1423.40	−0.298984	−0.149492	−	0.988763i	$-0.547764\pi$
$283$	−1423.40	−0.298984	−0.149492	+	0.988763i	$0.547764\pi$
$284$	936.347	0.195641
$285$	−305.460	−0.0634874
$286$	0	0
$287$	−681.897	−0.140248
$288$	1540.31	0.315152
$289$	−2473.80	−0.503521
$290$	165.236	0.0334586
$291$	4205.04	0.847093
$292$	−2733.55	−0.547839
$293$	−5772.76	−1.15102	−0.575509	−	0.817796i	$-0.695196\pi$
$293$	−5772.76	−1.15102	−0.575509	+	0.817796i	$0.695196\pi$
$294$	−4117.02	−0.816699
$295$	−2119.29	−0.418271
$296$	1473.58	0.289358
$297$	0	0
$298$	−9253.61	−1.79882
$299$	−4815.88	−0.931471
$300$	−988.338	−0.190206
$301$	−1567.44	−0.300152
$302$	−4290.90	−0.817594
$303$	−2129.96	−0.403838
$304$	86.1970	0.0162623
$305$	−4200.72	−0.788632
$306$	−2045.54	−0.382142
$307$	−4205.58	−0.781840	−0.390920	−	0.920425i	$-0.627843\pi$
$307$	−4205.58	−0.781840	−0.390920	+	0.920425i	$0.627843\pi$
$308$	0	0
$309$	−4557.88	−0.839122
$310$	1670.97	0.306144
$311$	2586.28	0.471558	0.235779	−	0.971807i	$-0.424236\pi$
$311$	2586.28	0.471558	0.235779	+	0.971807i	$0.424236\pi$
$312$	2960.70	0.537233
$313$	6936.81	1.25269	0.626345	−	0.779546i	$-0.284550\pi$
$313$	6936.81	1.25269	0.626345	+	0.779546i	$0.284550\pi$
$314$	7849.75	1.41079
$315$	−301.168	−0.0538694
$316$	5356.94	0.953645
$317$	4525.20	0.801768	0.400884	−	0.916129i	$-0.368703\pi$
$317$	4525.20	0.801768	0.400884	+	0.916129i	$0.368703\pi$
$318$	−1254.93	−0.221299
$319$	0	0
$320$	−4107.33	−0.717520
$321$	−4535.85	−0.788681
$322$	−3581.21	−0.619791
$323$	1005.74	0.173254
$324$	1067.41	0.183026
$325$	1035.44	0.176725
$326$	−12942.6	−2.19885
$327$	2928.33	0.495220
$328$	−2427.80	−0.408698
$329$	−2036.49	−0.341263
$330$	0	0
$331$	−1413.65	−0.234748	−0.117374	−	0.993088i	$-0.537448\pi$
$331$	−1413.65	−0.234748	−0.117374	+	0.993088i	$0.537448\pi$
$332$	−13629.7	−2.25309
$333$	−556.577	−0.0915923
$334$	−5055.67	−0.828245
$335$	−945.584	−0.154217
$336$	84.9856	0.0137987
$337$	−7080.38	−1.14449	−0.572244	−	0.820083i	$-0.693927\pi$
$337$	−7080.38	−1.14449	−0.572244	+	0.820083i	$0.693927\pi$
$338$	2216.27	0.356654
$339$	−4753.31	−0.761547
$340$	3254.16	0.519063
$341$	0	0
$342$	−843.426	−0.133355
$343$	4291.36	0.675545
$344$	−5580.66	−0.874677
$345$	1744.15	0.272179
$346$	4346.73	0.675381
$347$	−4467.97	−0.691219	−0.345610	−	0.938378i	$-0.612328\pi$
$347$	−4467.97	−0.691219	−0.345610	+	0.938378i	$0.612328\pi$
$348$	283.896	0.0437311
$349$	−149.249	−0.0228915	−0.0114457	−	0.999934i	$-0.503643\pi$
$349$	−149.249	−0.0228915	−0.0114457	+	0.999934i	$0.503643\pi$
$350$	769.975	0.117591
$351$	−1118.27	−0.170054
$352$	0	0
$353$	−2076.44	−0.313081	−0.156540	−	0.987672i	$-0.550034\pi$
$353$	−2076.44	−0.313081	−0.156540	+	0.987672i	$0.550034\pi$
$354$	−5851.71	−0.878574
$355$	355.273	0.0531154
$356$	−17038.3	−2.53660
$357$	991.610	0.147007
$358$	−14403.2	−2.12635
$359$	−6264.59	−0.920981	−0.460491	−	0.887665i	$-0.652327\pi$
$359$	−6264.59	−0.920981	−0.460491	+	0.887665i	$0.652327\pi$
$360$	−1072.27	−0.156982
$361$	−6444.31	−0.939540
$362$	18523.3	2.68940
$363$	0	0
$364$	−3652.78	−0.525983
$365$	−1037.18	−0.148735
$366$	−11598.9	−1.65651
$367$	72.2149	0.0102714	0.00513568	−	0.999987i	$-0.498365\pi$
$367$	72.2149	0.0102714	0.00513568	+	0.999987i	$0.498365\pi$
$368$	−492.177	−0.0697188
$369$	916.992	0.129368
$370$	1422.96	0.199936
$371$	608.349	0.0851318
$372$	2870.94	0.400137
$373$	5260.85	0.730285	0.365143	−	0.930952i	$-0.381020\pi$
$373$	5260.85	0.730285	0.365143	+	0.930952i	$0.381020\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	−7250.65	−0.994478
$377$	−297.425	−0.0406317
$378$	−831.573	−0.113152
$379$	−4521.17	−0.612763	−0.306381	−	0.951909i	$-0.599118\pi$
$379$	−4521.17	−0.612763	−0.306381	+	0.951909i	$0.599118\pi$
$380$	1341.77	0.181135
$381$	−213.599	−0.0287218
$382$	2680.79	0.359060
$383$	5908.27	0.788246	0.394123	−	0.919058i	$-0.371048\pi$
$383$	5908.27	0.788246	0.394123	+	0.919058i	$0.371048\pi$
$384$	−7233.50	−0.961284
$385$	0	0
$386$	15897.2	2.09624
$387$	2107.84	0.276867
$388$	−18471.1	−2.41683
$389$	−5227.65	−0.681368	−0.340684	−	0.940178i	$-0.610659\pi$
$389$	−5227.65	−0.681368	−0.340684	+	0.940178i	$0.610659\pi$
$390$	2859.01	0.371209
$391$	−5742.71	−0.742765
$392$	7105.76	0.915548
$393$	225.489	0.0289426
$394$	−12433.6	−1.58984
$395$	2032.56	0.258909
$396$	0	0
$397$	−13754.7	−1.73887	−0.869434	−	0.494050i	$-0.835516\pi$
$397$	−13754.7	−1.73887	−0.869434	+	0.494050i	$0.835516\pi$
$398$	21967.8	2.76670
$399$	408.865	0.0513004
$400$	105.820	0.0132275
$401$	−2212.60	−0.275542	−0.137771	−	0.990464i	$-0.543994\pi$
$401$	−2212.60	−0.275542	−0.137771	+	0.990464i	$0.543994\pi$
$402$	−2610.91	−0.323932
$403$	−3007.75	−0.371778
$404$	9356.09	1.15219
$405$	405.000	0.0496904
$406$	−221.172	−0.0270359
$407$	0	0
$408$	3530.49	0.428395
$409$	−15239.4	−1.84239	−0.921196	−	0.389098i	$-0.872787\pi$
$409$	−15239.4	−1.84239	−0.921196	+	0.389098i	$0.872787\pi$
$410$	−2344.41	−0.282396
$411$	−3544.35	−0.425377
$412$	20021.0	2.39409
$413$	2836.72	0.337980
$414$	4815.88	0.571710
$415$	−5171.43	−0.611700
$416$	7088.42	0.835429
$417$	−5543.07	−0.650948
$418$	0	0
$419$	−9907.32	−1.15514	−0.577571	−	0.816341i	$-0.695999\pi$
$419$	−9907.32	−1.15514	−0.577571	+	0.816341i	$0.695999\pi$
$420$	1322.91	0.153694
$421$	10661.9	1.23427	0.617137	−	0.786856i	$-0.288292\pi$
$421$	10661.9	1.23427	0.617137	+	0.786856i	$0.288292\pi$
$422$	−12425.4	−1.43331
$423$	2738.61	0.314789
$424$	2165.94	0.248084
$425$	1234.71	0.140923
$426$	980.968	0.111568
$427$	5622.77	0.637247
$428$	19924.3	2.25018
$429$	0	0
$430$	−5388.98	−0.604371
$431$	−15023.6	−1.67903	−0.839517	−	0.543334i	$-0.817162\pi$
$431$	−15023.6	−1.67903	−0.839517	+	0.543334i	$0.817162\pi$
$432$	−114.286	−0.0127282
$433$	1632.89	0.181228	0.0906142	−	0.995886i	$-0.471117\pi$
$433$	1632.89	0.181228	0.0906142	+	0.995886i	$0.471117\pi$
$434$	−2236.63	−0.247377
$435$	107.717	0.0118728
$436$	−12863.0	−1.41290
$437$	−2367.86	−0.259199
$438$	−2863.81	−0.312416
$439$	10814.5	1.17574	0.587869	−	0.808956i	$-0.299967\pi$
$439$	10814.5	1.17574	0.587869	+	0.808956i	$0.299967\pi$
$440$	0	0
$441$	−2683.88	−0.289805
$442$	−9413.44	−1.01301
$443$	−3582.51	−0.384222	−0.192111	−	0.981373i	$-0.561533\pi$
$443$	−3582.51	−0.384222	−0.192111	+	0.981373i	$0.561533\pi$
$444$	2444.83	0.261321
$445$	−6464.76	−0.688672
$446$	8823.95	0.936829
$447$	−6032.42	−0.638308
$448$	5497.75	0.579786
$449$	12280.5	1.29076	0.645381	−	0.763860i	$-0.276698\pi$
$449$	12280.5	1.29076	0.645381	+	0.763860i	$0.276698\pi$
$450$	−1035.44	−0.108469
$451$	0	0
$452$	20879.5	2.17276
$453$	−2797.23	−0.290122
$454$	−20297.4	−2.09824
$455$	−1385.95	−0.142801
$456$	1455.71	0.149495
$457$	−5432.13	−0.556027	−0.278014	−	0.960577i	$-0.589676\pi$
$457$	−5432.13	−0.556027	−0.278014	+	0.960577i	$0.589676\pi$
$458$	−6249.43	−0.637591
$459$	−1333.48	−0.135603
$460$	−7661.38	−0.776552
$461$	−3539.69	−0.357614	−0.178807	−	0.983884i	$-0.557224\pi$
$461$	−3539.69	−0.357614	−0.178807	+	0.983884i	$0.557224\pi$
$462$	0	0
$463$	−3493.24	−0.350637	−0.175318	−	0.984512i	$-0.556095\pi$
$463$	−3493.24	−0.350637	−0.175318	+	0.984512i	$0.556095\pi$
$464$	−30.3964	−0.00304120
$465$	1089.30	0.108635
$466$	−16938.4	−1.68382
$467$	−13939.3	−1.38123	−0.690613	−	0.723225i	$-0.742659\pi$
$467$	−13939.3	−1.38123	−0.690613	+	0.723225i	$0.742659\pi$
$468$	4912.13	0.485178
$469$	1265.69	0.124614
$470$	−7001.61	−0.687149
$471$	5117.24	0.500616
$472$	10099.8	0.984913
$473$	0	0
$474$	5612.22	0.543835
$475$	509.100	0.0491771
$476$	−4355.76	−0.419424
$477$	−818.087	−0.0785275
$478$	18968.7	1.81508
$479$	17936.3	1.71092	0.855461	−	0.517868i	$-0.173274\pi$
$479$	17936.3	1.71092	0.855461	+	0.517868i	$0.173274\pi$
$480$	−2567.19	−0.244116
$481$	−2561.33	−0.242800
$482$	17516.9	1.65533
$483$	−2334.58	−0.219932
$484$	0	0
$485$	−7008.40	−0.656155
$486$	1118.27	0.104374
$487$	19611.6	1.82482	0.912410	−	0.409277i	$-0.134219\pi$
$487$	19611.6	1.82482	0.912410	+	0.409277i	$0.134219\pi$
$488$	20019.1	1.85701
$489$	−8437.26	−0.780258
$490$	6861.70	0.632612
$491$	−6134.06	−0.563801	−0.281901	−	0.959444i	$-0.590965\pi$
$491$	−6134.06	−0.563801	−0.281901	+	0.959444i	$0.590965\pi$
$492$	−4027.99	−0.369098
$493$	−354.665	−0.0324002
$494$	−3881.39	−0.353506
$495$	0	0
$496$	−307.388	−0.0278269
$497$	−475.541	−0.0429194
$498$	−14279.2	−1.28487
$499$	4691.84	0.420913	0.210456	−	0.977603i	$-0.432505\pi$
$499$	4691.84	0.420913	0.210456	+	0.977603i	$0.432505\pi$
$500$	1647.23	0.147333
$501$	−3295.79	−0.293902
$502$	−32594.9	−2.89797
$503$	−4228.79	−0.374856	−0.187428	−	0.982278i	$-0.560015\pi$
$503$	−4228.79	−0.374856	−0.187428	+	0.982278i	$0.560015\pi$
$504$	1435.25	0.126848
$505$	3549.93	0.312812
$506$	0	0
$507$	1444.78	0.126558
$508$	938.257	0.0819457
$509$	5911.40	0.514770	0.257385	−	0.966309i	$-0.417139\pi$
$509$	5911.40	0.514770	0.257385	+	0.966309i	$0.417139\pi$
$510$	3409.23	0.296006
$511$	1388.28	0.120184
$512$	1531.31	0.132177
$513$	−549.828	−0.0473207
$514$	6925.68	0.594316
$515$	7596.47	0.649981
$516$	−9258.94	−0.789926
$517$	0	0
$518$	−1904.67	−0.161557
$519$	2833.63	0.239658
$520$	−4934.50	−0.416139
$521$	−18861.9	−1.58610	−0.793049	−	0.609158i	$-0.791508\pi$
$521$	−18861.9	−1.58610	−0.793049	+	0.609158i	$0.791508\pi$
$522$	297.425	0.0249386
$523$	−18091.8	−1.51262	−0.756310	−	0.654214i	$-0.773000\pi$
$523$	−18091.8	−1.51262	−0.756310	+	0.654214i	$0.773000\pi$
$524$	−990.487	−0.0825756
$525$	501.946	0.0417271
$526$	−23366.2	−1.93691
$527$	−3586.59	−0.296460
$528$	0	0
$529$	1353.27	0.111225
$530$	2091.55	0.171417
$531$	−3814.73	−0.311761
$532$	−1795.99	−0.146365
$533$	4219.94	0.342938
$534$	−17850.2	−1.44655
$535$	7559.75	0.610910
$536$	4506.30	0.363139
$537$	−9389.44	−0.754533
$538$	36598.8	2.93287
$539$	0	0
$540$	−1779.01	−0.141771
$541$	−1495.55	−0.118851	−0.0594256	−	0.998233i	$-0.518927\pi$
$541$	−1495.55	−0.118851	−0.0594256	+	0.998233i	$0.518927\pi$
$542$	5678.98	0.450061
$543$	12075.3	0.954330
$544$	8452.60	0.666181
$545$	−4880.55	−0.383596
$546$	−3826.85	−0.299952
$547$	−24790.3	−1.93776	−0.968882	−	0.247523i	$-0.920383\pi$
$547$	−24790.3	−1.93776	−0.968882	+	0.247523i	$0.920383\pi$
$548$	15569.0	1.21364
$549$	−7561.30	−0.587812
$550$	0	0
$551$	−146.237	−0.0113065
$552$	−8311.97	−0.640907
$553$	−2720.62	−0.209209
$554$	24295.5	1.86321
$555$	927.628	0.0709471
$556$	24348.6	1.85721
$557$	3954.21	0.300799	0.150400	−	0.988625i	$-0.451944\pi$
$557$	3954.21	0.300799	0.150400	+	0.988625i	$0.451944\pi$
$558$	3007.75	0.228187
$559$	9700.16	0.733941
$560$	−141.643	−0.0106884
$561$	0	0
$562$	19710.6	1.47943
$563$	8058.95	0.603276	0.301638	−	0.953423i	$-0.402467\pi$
$563$	8058.95	0.603276	0.301638	+	0.953423i	$0.402467\pi$
$564$	−12029.6	−0.898119
$565$	7922.19	0.589892
$566$	6550.42	0.486457
$567$	−542.102	−0.0401519
$568$	−1693.10	−0.125072
$569$	10254.9	0.755548	0.377774	−	0.925898i	$-0.376690\pi$
$569$	10254.9	0.755548	0.377774	+	0.925898i	$0.376690\pi$
$570$	1405.71	0.103296
$571$	−11800.1	−0.864834	−0.432417	−	0.901674i	$-0.642339\pi$
$571$	−11800.1	−0.864834	−0.432417	+	0.901674i	$0.642339\pi$
$572$	0	0
$573$	1747.60	0.127412
$574$	3138.05	0.228188
$575$	−2906.92	−0.210829
$576$	−7393.19	−0.534808
$577$	2102.40	0.151688	0.0758441	−	0.997120i	$-0.475835\pi$
$577$	2102.40	0.151688	0.0758441	+	0.997120i	$0.475835\pi$
$578$	11384.3	0.819244
$579$	10363.4	0.743847
$580$	−473.160	−0.0338740
$581$	6922.08	0.494279
$582$	−19351.3	−1.37825
$583$	0	0
$584$	4942.79	0.350230
$585$	1863.79	0.131723
$586$	26565.9	1.87274
$587$	−10342.6	−0.727233	−0.363616	−	0.931549i	$-0.618458\pi$
$587$	−10342.6	−0.727233	−0.363616	+	0.931549i	$0.618458\pi$
$588$	11789.3	0.826838
$589$	−1478.84	−0.103454
$590$	9752.86	0.680540
$591$	−8105.47	−0.564153
$592$	−261.765	−0.0181731
$593$	−16109.4	−1.11557	−0.557786	−	0.829985i	$-0.688349\pi$
$593$	−16109.4	−1.11557	−0.557786	+	0.829985i	$0.688349\pi$
$594$	0	0
$595$	−1652.68	−0.113871
$596$	26498.1	1.82115
$597$	14320.8	0.981760
$598$	22162.4	1.51553
$599$	−7160.18	−0.488409	−0.244205	−	0.969724i	$-0.578527\pi$
$599$	−7160.18	−0.488409	−0.244205	+	0.969724i	$0.578527\pi$
$600$	1787.11	0.121597
$601$	22099.2	1.49991	0.749954	−	0.661490i	$-0.230076\pi$
$601$	22099.2	1.49991	0.749954	+	0.661490i	$0.230076\pi$
$602$	7213.27	0.488357
$603$	−1702.05	−0.114947
$604$	12287.2	0.827744
$605$	0	0
$606$	9801.95	0.657058
$607$	−15583.5	−1.04204	−0.521019	−	0.853545i	$-0.674448\pi$
$607$	−15583.5	−1.04204	−0.521019	+	0.853545i	$0.674448\pi$
$608$	3485.22	0.232474
$609$	−144.182	−0.00959367
$610$	19331.5	1.28313
$611$	12602.9	0.834466
$612$	5857.48	0.386887
$613$	−26636.2	−1.75502	−0.877509	−	0.479561i	$-0.840796\pi$
$613$	−26636.2	−1.75502	−0.877509	+	0.479561i	$0.840796\pi$
$614$	19353.8	1.27208
$615$	−1528.32	−0.100208
$616$	0	0
$617$	793.574	0.0517797	0.0258898	−	0.999665i	$-0.491758\pi$
$617$	793.574	0.0517797	0.0258898	+	0.999665i	$0.491758\pi$
$618$	20975.1	1.36528
$619$	−2306.16	−0.149745	−0.0748727	−	0.997193i	$-0.523855\pi$
$619$	−2306.16	−0.149745	−0.0748727	+	0.997193i	$0.523855\pi$
$620$	−4784.89	−0.309945
$621$	3139.47	0.202871
$622$	−11901.9	−0.767239
$623$	8653.22	0.556475
$624$	−525.936	−0.0337409
$625$	625.000	0.0400000
$626$	−31922.8	−2.03817
$627$	0	0
$628$	−22478.1	−1.42830
$629$	−3054.26	−0.193611
$630$	1385.95	0.0876472
$631$	−15202.8	−0.959136	−0.479568	−	0.877505i	$-0.659207\pi$
$631$	−15202.8	−0.959136	−0.479568	+	0.877505i	$0.659207\pi$
$632$	−9686.40	−0.609659
$633$	−8100.10	−0.508610
$634$	−20824.7	−1.30450
$635$	355.998	0.0222478
$636$	3593.54	0.224046
$637$	−12351.1	−0.768236
$638$	0	0
$639$	639.492	0.0395898
$640$	12055.8	0.744607
$641$	1412.24	0.0870205	0.0435103	−	0.999053i	$-0.486146\pi$
$641$	1412.24	0.0870205	0.0435103	+	0.999053i	$0.486146\pi$
$642$	20873.7	1.28321
$643$	3328.63	0.204150	0.102075	−	0.994777i	$-0.467452\pi$
$643$	3328.63	0.204150	0.102075	+	0.994777i	$0.467452\pi$
$644$	10254.9	0.627486
$645$	−3513.07	−0.214460
$646$	−4628.37	−0.281890
$647$	1286.97	0.0782009	0.0391005	−	0.999235i	$-0.487551\pi$
$647$	1286.97	0.0782009	0.0391005	+	0.999235i	$0.487551\pi$
$648$	−1930.08	−0.117007
$649$	0	0
$650$	−4765.01	−0.287537
$651$	−1458.06	−0.0877816
$652$	37061.6	2.22614
$653$	29020.0	1.73911	0.869557	−	0.493832i	$-0.164404\pi$
$653$	29020.0	1.73911	0.869557	+	0.493832i	$0.164404\pi$
$654$	−13476.0	−0.805738
$655$	−375.815	−0.0224188
$656$	431.273	0.0256682
$657$	−1866.92	−0.110861
$658$	9371.81	0.555245
$659$	−23745.9	−1.40366	−0.701829	−	0.712346i	$-0.747633\pi$
$659$	−23745.9	−1.40366	−0.701829	+	0.712346i	$0.747633\pi$
$660$	0	0
$661$	16783.9	0.987623	0.493811	−	0.869569i	$-0.335603\pi$
$661$	16783.9	0.987623	0.493811	+	0.869569i	$0.335603\pi$
$662$	6505.55	0.381942
$663$	−6136.61	−0.359466
$664$	24645.1	1.44038
$665$	−681.442	−0.0397371
$666$	2561.33	0.149023
$667$	835.000	0.0484728
$668$	14477.1	0.838528
$669$	5752.32	0.332433
$670$	4351.52	0.250916
$671$	0	0
$672$	3436.24	0.197256
$673$	10690.8	0.612330	0.306165	−	0.951978i	$-0.400954\pi$
$673$	10690.8	0.612330	0.306165	+	0.951978i	$0.400954\pi$
$674$	32583.5	1.86212
$675$	−675.000	−0.0384900
$676$	−6346.38	−0.361082
$677$	−30578.8	−1.73595	−0.867975	−	0.496608i	$-0.834579\pi$
$677$	−30578.8	−1.73595	−0.867975	+	0.496608i	$0.834579\pi$
$678$	21874.4	1.23906
$679$	9380.91	0.530200
$680$	−5884.15	−0.331834
$681$	−13231.8	−0.744559
$682$	0	0
$683$	−30905.5	−1.73143	−0.865713	−	0.500540i	$-0.833135\pi$
$683$	−30905.5	−1.73143	−0.865713	+	0.500540i	$0.833135\pi$
$684$	2415.18	0.135010
$685$	5907.26	0.329496
$686$	−19748.6	−1.09913
$687$	−4074.00	−0.226248
$688$	991.343	0.0549340
$689$	−3764.79	−0.208167
$690$	−8026.47	−0.442844
$691$	−18505.2	−1.01877	−0.509386	−	0.860538i	$-0.670127\pi$
$691$	−18505.2	−1.01877	−0.509386	+	0.860538i	$0.670127\pi$
$692$	−12447.1	−0.683766
$693$	0	0
$694$	20561.3	1.12464
$695$	9238.46	0.504222
$696$	−513.340	−0.0279570
$697$	5032.08	0.273463
$698$	686.835	0.0372451
$699$	−11042.2	−0.597500
$700$	−2204.86	−0.119051
$701$	−2288.62	−0.123310	−0.0616549	−	0.998098i	$-0.519638\pi$
$701$	−2288.62	−0.123310	−0.0616549	+	0.998098i	$0.519638\pi$
$702$	5146.22	0.276683
$703$	−1259.35	−0.0675636
$704$	0	0
$705$	−4564.34	−0.243834
$706$	9555.63	0.509392
$707$	−4751.67	−0.252765
$708$	16756.6	0.889481
$709$	2422.69	0.128330	0.0641651	−	0.997939i	$-0.479562\pi$
$709$	2422.69	0.128330	0.0641651	+	0.997939i	$0.479562\pi$
$710$	−1634.95	−0.0864203
$711$	3658.60	0.192979
$712$	30808.6	1.62163
$713$	8444.05	0.443523
$714$	−4563.33	−0.239185
$715$	0	0
$716$	41244.2	2.15275
$717$	12365.7	0.644079
$718$	28829.3	1.49847
$719$	1647.60	0.0854592	0.0427296	−	0.999087i	$-0.486395\pi$
$719$	1647.60	0.0854592	0.0427296	+	0.999087i	$0.486395\pi$
$720$	190.476	0.00985922
$721$	−10168.0	−0.525212
$722$	29656.3	1.52866
$723$	11419.2	0.587393
$724$	−53042.2	−2.72279
$725$	−179.529	−0.00919659
$726$	0	0
$727$	−2539.71	−0.129563	−0.0647817	−	0.997899i	$-0.520635\pi$
$727$	−2539.71	−0.129563	−0.0647817	+	0.997899i	$0.520635\pi$
$728$	6604.94	0.336257
$729$	729.000	0.0370370
$730$	4773.02	0.241997
$731$	11567.0	0.585253
$732$	33213.9	1.67708
$733$	29806.2	1.50194	0.750968	−	0.660339i	$-0.229587\pi$
$733$	29806.2	1.50194	0.750968	+	0.660339i	$0.229587\pi$
$734$	−332.329	−0.0167118
$735$	4473.13	0.224482
$736$	−19900.3	−0.996649
$737$	0	0
$738$	−4219.94	−0.210485
$739$	9399.51	0.467884	0.233942	−	0.972251i	$-0.424837\pi$
$739$	9399.51	0.467884	0.233942	+	0.972251i	$0.424837\pi$
$740$	−4074.71	−0.202418
$741$	−2530.28	−0.125441
$742$	−2799.58	−0.138512
$743$	19915.0	0.983323	0.491661	−	0.870787i	$-0.336390\pi$
$743$	19915.0	0.983323	0.491661	+	0.870787i	$0.336390\pi$
$744$	−5191.22	−0.255805
$745$	10054.0	0.494431
$746$	−24210.1	−1.18820
$747$	−9308.58	−0.455935
$748$	0	0
$749$	−10118.9	−0.493640
$750$	1725.73	0.0840195
$751$	31383.3	1.52489	0.762444	−	0.647054i	$-0.223999\pi$
$751$	31383.3	1.52489	0.762444	+	0.647054i	$0.223999\pi$
$752$	1288.00	0.0624581
$753$	−21248.6	−1.02834
$754$	1368.73	0.0661091
$755$	4662.05	0.224728
$756$	2381.24	0.114557
$757$	−18818.9	−0.903548	−0.451774	−	0.892132i	$-0.649209\pi$
$757$	−18818.9	−0.903548	−0.451774	+	0.892132i	$0.649209\pi$
$758$	20806.2	0.996984
$759$	0	0
$760$	−2426.18	−0.115798
$761$	−4723.62	−0.225008	−0.112504	−	0.993651i	$-0.535887\pi$
$761$	−4723.62	−0.225008	−0.112504	+	0.993651i	$0.535887\pi$
$762$	982.969	0.0467312
$763$	6532.72	0.309961
$764$	−7676.55	−0.363518
$765$	2222.47	0.105037
$766$	−27189.5	−1.28250
$767$	−17555.1	−0.826440
$768$	13573.0	0.637724
$769$	13990.4	0.656056	0.328028	−	0.944668i	$-0.393616\pi$
$769$	13990.4	0.656056	0.328028	+	0.944668i	$0.393616\pi$
$770$	0	0
$771$	4514.84	0.210893
$772$	−45522.3	−2.12226
$773$	−20558.2	−0.956569	−0.478284	−	0.878205i	$-0.658741\pi$
$773$	−20558.2	−0.956569	−0.478284	+	0.878205i	$0.658741\pi$
$774$	−9700.16	−0.450472
$775$	−1815.51	−0.0841483
$776$	33399.4	1.54506
$777$	−1241.65	−0.0573282
$778$	24057.3	1.10861
$779$	2074.85	0.0954290
$780$	−8186.89	−0.375818
$781$	0	0
$782$	26427.6	1.20850
$783$	193.891	0.00884943
$784$	−1262.26	−0.0575010
$785$	−8528.74	−0.387776
$786$	−1037.69	−0.0470904
$787$	38497.3	1.74368	0.871842	−	0.489787i	$-0.162925\pi$
$787$	38497.3	1.74368	0.871842	+	0.489787i	$0.162925\pi$
$788$	35604.2	1.60958
$789$	−15232.4	−0.687311
$790$	−9353.71	−0.421253
$791$	−10604.0	−0.476657
$792$	0	0
$793$	−34796.7	−1.55822
$794$	63298.4	2.82919
$795$	1363.48	0.0608272
$796$	−62905.7	−2.80105
$797$	−36994.7	−1.64419	−0.822094	−	0.569352i	$-0.807194\pi$
$797$	−36994.7	−1.64419	−0.822094	+	0.569352i	$0.807194\pi$
$798$	−1881.57	−0.0834674
$799$	15028.3	0.665412
$800$	4278.65	0.189091
$801$	−11636.6	−0.513305
$802$	10182.3	0.448315
$803$	0	0
$804$	7476.46	0.327953
$805$	3890.97	0.170359
$806$	13841.5	0.604895
$807$	23858.7	1.04073
$808$	−16917.6	−0.736585
$809$	16439.2	0.714426	0.357213	−	0.934023i	$-0.383727\pi$
$809$	16439.2	0.714426	0.357213	+	0.934023i	$0.383727\pi$
$810$	−1863.79	−0.0808478
$811$	−10436.3	−0.451871	−0.225935	−	0.974142i	$-0.572544\pi$
$811$	−10436.3	−0.451871	−0.225935	+	0.974142i	$0.572544\pi$
$812$	633.336	0.0273716
$813$	3702.12	0.159704
$814$	0	0
$815$	14062.1	0.604385
$816$	−627.154	−0.0269053
$817$	4769.35	0.204233
$818$	70130.7	2.99763
$819$	−2494.72	−0.106438
$820$	6713.32	0.285902
$821$	−21461.9	−0.912335	−0.456168	−	0.889894i	$-0.650778\pi$
$821$	−21461.9	−0.912335	−0.456168	+	0.889894i	$0.650778\pi$
$822$	16310.9	0.692102
$823$	−21752.0	−0.921296	−0.460648	−	0.887583i	$-0.652383\pi$
$823$	−21752.0	−0.921296	−0.460648	+	0.887583i	$0.652383\pi$
$824$	−36201.9	−1.53053
$825$	0	0
$826$	−13054.4	−0.549905
$827$	−34671.5	−1.45785	−0.728927	−	0.684592i	$-0.759980\pi$
$827$	−34671.5	−1.45785	−0.728927	+	0.684592i	$0.759980\pi$
$828$	−13790.5	−0.578807
$829$	−28626.9	−1.19934	−0.599670	−	0.800247i	$-0.704702\pi$
$829$	−28626.9	−1.19934	−0.599670	+	0.800247i	$0.704702\pi$
$830$	23798.6	0.995256
$831$	15838.2	0.661158
$832$	−34023.0	−1.41771
$833$	−14728.0	−0.612600
$834$	25508.9	1.05911
$835$	5492.98	0.227655
$836$	0	0
$837$	1960.75	0.0809718
$838$	45592.9	1.87945
$839$	41272.7	1.69832	0.849161	−	0.528135i	$-0.177108\pi$
$839$	41272.7	1.69832	0.849161	+	0.528135i	$0.177108\pi$
$840$	−2392.09	−0.0982557
$841$	−24337.4	−0.997886
$842$	−49065.4	−2.00820
$843$	12849.3	0.524976
$844$	35580.6	1.45111
$845$	−2407.97	−0.0980317
$846$	−12602.9	−0.512171
$847$	0	0
$848$	−384.756	−0.0155809
$849$	4270.21	0.172619
$850$	−5682.05	−0.229285
$851$	7190.77	0.289655
$852$	−2809.04	−0.112953
$853$	20565.0	0.825479	0.412740	−	0.910849i	$-0.364572\pi$
$853$	20565.0	0.825479	0.412740	+	0.910849i	$0.364572\pi$
$854$	−25875.6	−1.03682
$855$	916.381	0.0366545
$856$	−36026.9	−1.43852
$857$	20386.4	0.812586	0.406293	−	0.913743i	$-0.366821\pi$
$857$	20386.4	0.812586	0.406293	+	0.913743i	$0.366821\pi$
$858$	0	0
$859$	18369.1	0.729622	0.364811	−	0.931082i	$-0.381134\pi$
$859$	18369.1	0.729622	0.364811	+	0.931082i	$0.381134\pi$
$860$	15431.6	0.611874
$861$	2045.69	0.0809721
$862$	69137.9	2.73184
$863$	−8320.64	−0.328201	−0.164101	−	0.986444i	$-0.552472\pi$
$863$	−8320.64	−0.328201	−0.164101	+	0.986444i	$0.552472\pi$
$864$	−4620.94	−0.181953
$865$	−4722.72	−0.185639
$866$	−7514.48	−0.294864
$867$	7421.39	0.290708
$868$	6404.69	0.250449
$869$	0	0
$870$	−495.708	−0.0193173
$871$	−7832.74	−0.304710
$872$	23258.8	0.903261
$873$	−12615.1	−0.489069
$874$	10896.8	0.421726
$875$	−836.577	−0.0323217
$876$	8200.65	0.316295
$877$	36524.6	1.40633	0.703164	−	0.711028i	$-0.251770\pi$
$877$	36524.6	1.40633	0.703164	+	0.711028i	$0.251770\pi$
$878$	−49767.8	−1.91296
$879$	17318.3	0.664540
$880$	0	0
$881$	−44255.1	−1.69239	−0.846193	−	0.532877i	$-0.821111\pi$
$881$	−44255.1	−1.69239	−0.846193	+	0.532877i	$0.821111\pi$
$882$	12351.1	0.471521
$883$	35224.7	1.34248	0.671238	−	0.741242i	$-0.265763\pi$
$883$	35224.7	1.34248	0.671238	+	0.741242i	$0.265763\pi$
$884$	26955.8	1.02559
$885$	6357.88	0.241489
$886$	16486.5	0.625141
$887$	−15629.8	−0.591653	−0.295826	−	0.955242i	$-0.595595\pi$
$887$	−15629.8	−0.591653	−0.295826	+	0.955242i	$0.595595\pi$
$888$	−4420.73	−0.167061
$889$	−476.511	−0.0179771
$890$	29750.4	1.12049
$891$	0	0
$892$	−25267.7	−0.948460
$893$	6196.56	0.232206
$894$	27760.8	1.03855
$895$	15649.1	0.584459
$896$	−16137.0	−0.601673
$897$	14447.7	0.537785
$898$	−56514.1	−2.10011
$899$	521.497	0.0193469
$900$	2965.01	0.109815
$901$	−4489.33	−0.165995
$902$	0	0
$903$	4702.32	0.173293
$904$	−37754.2	−1.38903
$905$	−20125.5	−0.739221
$906$	12872.7	0.472038
$907$	−31746.6	−1.16221	−0.581107	−	0.813827i	$-0.697380\pi$
$907$	−31746.6	−1.16221	−0.581107	+	0.813827i	$0.697380\pi$
$908$	58122.3	2.12429
$909$	6389.88	0.233156
$910$	6378.08	0.232342
$911$	−16896.0	−0.614477	−0.307238	−	0.951633i	$-0.599405\pi$
$911$	−16896.0	−0.614477	−0.307238	+	0.951633i	$0.599405\pi$
$912$	−258.591	−0.00938904
$913$	0	0
$914$	24998.3	0.904674
$915$	12602.2	0.455317
$916$	17895.5	0.645506
$917$	503.037	0.0181153
$918$	6136.61	0.220630
$919$	−25110.4	−0.901323	−0.450662	−	0.892695i	$-0.648812\pi$
$919$	−25110.4	−0.901323	−0.450662	+	0.892695i	$0.648812\pi$
$920$	13853.3	0.496445
$921$	12616.7	0.451396
$922$	16289.4	0.581848
$923$	2942.90	0.104948
$924$	0	0
$925$	−1546.05	−0.0549554
$926$	16075.7	0.570497
$927$	13673.6	0.484467
$928$	−1229.02	−0.0434749
$929$	37432.3	1.32197	0.660987	−	0.750398i	$-0.270138\pi$
$929$	37432.3	1.32197	0.660987	+	0.750398i	$0.270138\pi$
$930$	−5012.91	−0.176753
$931$	−6072.73	−0.213776
$932$	48503.9	1.70472
$933$	−7758.84	−0.272254
$934$	64147.6	2.24730
$935$	0	0
$936$	−8882.10	−0.310171
$937$	11908.1	0.415177	0.207589	−	0.978216i	$-0.433438\pi$
$937$	11908.1	0.415177	0.207589	+	0.978216i	$0.433438\pi$
$938$	−5824.61	−0.202751
$939$	−20810.4	−0.723241
$940$	20049.4	0.695680
$941$	31455.4	1.08971	0.544855	−	0.838530i	$-0.316585\pi$
$941$	31455.4	1.08971	0.544855	+	0.838530i	$0.316585\pi$
$942$	−23549.2	−0.814518
$943$	−11847.2	−0.409118
$944$	−1794.11	−0.0618574
$945$	903.503	0.0311015
$946$	0	0
$947$	35195.7	1.20772	0.603858	−	0.797092i	$-0.293629\pi$
$947$	35195.7	1.20772	0.603858	+	0.797092i	$0.293629\pi$
$948$	−16070.8	−0.550587
$949$	−8591.44	−0.293878
$950$	−2342.85	−0.0800127
$951$	−13575.6	−0.462901
$952$	7876.07	0.268135
$953$	19026.1	0.646711	0.323355	−	0.946278i	$-0.395189\pi$
$953$	19026.1	0.646711	0.323355	+	0.946278i	$0.395189\pi$
$954$	3764.79	0.127767
$955$	−2912.67	−0.0986931
$956$	−54317.7	−1.83761
$957$	0	0
$958$	−82541.9	−2.78372
$959$	−7907.00	−0.266246
$960$	12322.0	0.414261
$961$	−24517.3	−0.822976
$962$	11787.1	0.395043
$963$	13607.6	0.455345
$964$	−50160.2	−1.67588
$965$	−17272.3	−0.576181
$966$	10743.6	0.357837
$967$	22769.6	0.757209	0.378605	−	0.925558i	$-0.376404\pi$
$967$	22769.6	0.757209	0.378605	+	0.925558i	$0.376404\pi$
$968$	0	0
$969$	−3017.23	−0.100028
$970$	32252.2	1.06758
$971$	−12911.1	−0.426713	−0.213357	−	0.976974i	$-0.568440\pi$
$971$	−12911.1	−0.426713	−0.213357	+	0.976974i	$0.568440\pi$
$972$	−3202.22	−0.105670
$973$	−12365.9	−0.407433
$974$	−90251.5	−2.96904
$975$	−3106.31	−0.102032
$976$	−3556.17	−0.116629
$977$	−38839.1	−1.27182	−0.635912	−	0.771761i	$-0.719376\pi$
$977$	−38839.1	−1.27182	−0.635912	+	0.771761i	$0.719376\pi$
$978$	38827.8	1.26950
$979$	0	0
$980$	−19648.8	−0.640466
$981$	−8784.98	−0.285915
$982$	28228.6	0.917322
$983$	31594.4	1.02513	0.512566	−	0.858648i	$-0.328695\pi$
$983$	31594.4	1.02513	0.512566	+	0.858648i	$0.328695\pi$
$984$	7283.40	0.235962
$985$	13509.1	0.436991
$986$	1632.15	0.0527161
$987$	6109.48	0.197028
$988$	11114.5	0.357895
$989$	−27232.5	−0.875576
$990$	0	0
$991$	−45708.5	−1.46516	−0.732582	−	0.680679i	$-0.761685\pi$
$991$	−45708.5	−1.46516	−0.732582	+	0.680679i	$0.761685\pi$
$992$	−12428.7	−0.397793
$993$	4240.96	0.135532
$994$	2188.41	0.0698312
$995$	−23868.0	−0.760468
$996$	40889.0	1.30082
$997$	6951.24	0.220810	0.110405	−	0.993887i	$-0.464785\pi$
$997$	6951.24	0.220810	0.110405	+	0.993887i	$0.464785\pi$
$998$	−21591.6	−0.684838
$999$	1669.73	0.0528808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.z.1.1	✓	6
11.10	odd	2	inner	1815.4.a.z.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.4.a.z.1.1	✓	6	1.1	even	1	trivial
1815.4.a.z.1.6	yes	6	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q-4.60194 q^{2} -3.00000 q^{3} +13.1778 q^{4} +5.00000 q^{5} +13.8058 q^{6} -6.69261 q^{7} -23.8281 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.60194 q^{2} -3.00000 q^{3} +13.1778 q^{4} +5.00000 q^{5} +13.8058 q^{6} -6.69261 q^{7} -23.8281 q^{8} +9.00000 q^{9} -23.0097 q^{10} -39.5335 q^{12} +41.4175 q^{13} +30.7990 q^{14} -15.0000 q^{15} +4.23281 q^{16} +49.3883 q^{17} -41.4175 q^{18} +20.3640 q^{19} +65.8892 q^{20} +20.0778 q^{21} -116.277 q^{23} +71.4844 q^{24} +25.0000 q^{25} -190.601 q^{26} -27.0000 q^{27} -88.1942 q^{28} -7.18115 q^{29} +69.0291 q^{30} -72.6203 q^{31} +171.146 q^{32} -227.282 q^{34} -33.4631 q^{35} +118.601 q^{36} -61.8419 q^{37} -93.7140 q^{38} -124.252 q^{39} -119.141 q^{40} +101.888 q^{41} -92.3970 q^{42} +234.205 q^{43} +45.0000 q^{45} +535.098 q^{46} +304.290 q^{47} -12.6984 q^{48} -298.209 q^{49} -115.048 q^{50} -148.165 q^{51} +545.793 q^{52} -90.8986 q^{53} +124.252 q^{54} +159.472 q^{56} -61.0921 q^{57} +33.0472 q^{58} -423.859 q^{59} -197.668 q^{60} -840.145 q^{61} +334.194 q^{62} -60.2335 q^{63} -821.465 q^{64} +207.087 q^{65} -189.117 q^{67} +650.831 q^{68} +348.830 q^{69} +153.995 q^{70} +71.0547 q^{71} -214.453 q^{72} -207.435 q^{73} +284.593 q^{74} -75.0000 q^{75} +268.354 q^{76} +571.802 q^{78} +406.511 q^{79} +21.1640 q^{80} +81.0000 q^{81} -468.883 q^{82} -1034.29 q^{83} +264.583 q^{84} +246.941 q^{85} -1077.80 q^{86} +21.5434 q^{87} -1292.95 q^{89} -207.087 q^{90} -277.191 q^{91} -1532.28 q^{92} +217.861 q^{93} -1400.32 q^{94} +101.820 q^{95} -513.437 q^{96} -1401.68 q^{97} +1372.34 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 18 q^{3} + 12 q^{4} + 30 q^{5} + 54 q^{9}+O(q^{10}) \) 6 * q - 18 * q^3 + 12 * q^4 + 30 * q^5 + 54 * q^9	\( 6 q - 18 q^{3} + 12 q^{4} + 30 q^{5} + 54 q^{9} - 36 q^{12} + 24 q^{14} - 90 q^{15} - 60 q^{16} + 60 q^{20} + 18 q^{23} + 150 q^{25} - 540 q^{26} - 162 q^{27} + 30 q^{31} - 84 q^{34} + 108 q^{36} - 576 q^{37} - 12 q^{38} - 72 q^{42} + 270 q^{45} - 330 q^{47} + 180 q^{48} - 1134 q^{49} - 78 q^{53} + 936 q^{56} - 372 q^{58} - 1488 q^{59} - 180 q^{60} - 252 q^{64} + 120 q^{67} - 54 q^{69} + 120 q^{70} + 156 q^{71} - 450 q^{75} + 1620 q^{78} - 300 q^{80} + 486 q^{81} + 468 q^{82} - 372 q^{86} - 1800 q^{89} - 216 q^{91} - 3972 q^{92} - 90 q^{93} - 180 q^{97}+O(q^{100}) \) 6 * q - 18 * q^3 + 12 * q^4 + 30 * q^5 + 54 * q^9 - 36 * q^12 + 24 * q^14 - 90 * q^15 - 60 * q^16 + 60 * q^20 + 18 * q^23 + 150 * q^25 - 540 * q^26 - 162 * q^27 + 30 * q^31 - 84 * q^34 + 108 * q^36 - 576 * q^37 - 12 * q^38 - 72 * q^42 + 270 * q^45 - 330 * q^47 + 180 * q^48 - 1134 * q^49 - 78 * q^53 + 936 * q^56 - 372 * q^58 - 1488 * q^59 - 180 * q^60 - 252 * q^64 + 120 * q^67 - 54 * q^69 + 120 * q^70 + 156 * q^71 - 450 * q^75 + 1620 * q^78 - 300 * q^80 + 486 * q^81 + 468 * q^82 - 372 * q^86 - 1800 * q^89 - 216 * q^91 - 3972 * q^92 - 90 * q^93 - 180 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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