LMFDB - Embedded newform 1840.4.a.bb.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + \cdots - 43712 $ x^10 - x^9 - 204x^8 + 42x^7 + 12958x^6 + 5872x^5 - 259871x^4 - 149461x^3 + 1222472x^2 + 627136x - 43712
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$4.78870$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.78870 q^{3} -5.00000 q^{5} -0.156416 q^{7} -4.06831 q^{9} +O(q^{10})$	$q+4.78870 q^{3} -5.00000 q^{5} -0.156416 q^{7} -4.06831 q^{9} +57.4097 q^{11} -17.3244 q^{13} -23.9435 q^{15} -37.0472 q^{17} +121.841 q^{19} -0.749032 q^{21} -23.0000 q^{23} +25.0000 q^{25} -148.777 q^{27} +89.1452 q^{29} +76.6700 q^{31} +274.918 q^{33} +0.782082 q^{35} -183.023 q^{37} -82.9616 q^{39} -237.272 q^{41} +342.074 q^{43} +20.3415 q^{45} +245.602 q^{47} -342.976 q^{49} -177.408 q^{51} +197.588 q^{53} -287.048 q^{55} +583.460 q^{57} +311.314 q^{59} +130.256 q^{61} +0.636350 q^{63} +86.6222 q^{65} +293.742 q^{67} -110.140 q^{69} -765.950 q^{71} +699.714 q^{73} +119.718 q^{75} -8.97981 q^{77} -691.925 q^{79} -602.605 q^{81} +952.614 q^{83} +185.236 q^{85} +426.890 q^{87} -20.9162 q^{89} +2.70983 q^{91} +367.150 q^{93} -609.204 q^{95} +455.261 q^{97} -233.560 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9}+O(q^{10}) $ 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9	$ 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35} + 62 q^{37} - 283 q^{39} + 767 q^{41} - 332 q^{43} - 695 q^{45} + 323 q^{47} + 1162 q^{49} - 176 q^{51} + 558 q^{53} - 70 q^{55} + 46 q^{57} - 822 q^{59} + 318 q^{61} - 2698 q^{63} - 55 q^{65} - 1152 q^{67} - 23 q^{69} - 1247 q^{71} + 1941 q^{73} + 25 q^{75} + 528 q^{77} - 3134 q^{79} + 6210 q^{81} - 482 q^{83} - 340 q^{85} - 1797 q^{87} + 4734 q^{89} - 4992 q^{91} + 4647 q^{93} + 570 q^{95} + 2326 q^{97} - 4356 q^{99}+O(q^{100}) $ 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9 + 14 * q^11 + 11 * q^13 - 5 * q^15 + 68 * q^17 - 114 * q^19 - 232 * q^21 - 230 * q^23 + 250 * q^25 + 433 * q^27 - 273 * q^29 + 129 * q^31 + 98 * q^33 + 140 * q^35 + 62 * q^37 - 283 * q^39 + 767 * q^41 - 332 * q^43 - 695 * q^45 + 323 * q^47 + 1162 * q^49 - 176 * q^51 + 558 * q^53 - 70 * q^55 + 46 * q^57 - 822 * q^59 + 318 * q^61 - 2698 * q^63 - 55 * q^65 - 1152 * q^67 - 23 * q^69 - 1247 * q^71 + 1941 * q^73 + 25 * q^75 + 528 * q^77 - 3134 * q^79 + 6210 * q^81 - 482 * q^83 - 340 * q^85 - 1797 * q^87 + 4734 * q^89 - 4992 * q^91 + 4647 * q^93 + 570 * q^95 + 2326 * q^97 - 4356 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	4.78870	0.921587	0.460793	−	0.887507i	$-0.347565\pi$
$3$	4.78870	0.921587	0.460793	+	0.887507i	$0.347565\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−0.156416	−0.00844569	−0.00422284	−	0.999991i	$-0.501344\pi$
$7$	−0.156416	−0.00844569	−0.00422284	+	0.999991i	$0.501344\pi$
$8$	0	0
$9$	−4.06831	−0.150678
$10$	0	0
$11$	57.4097	1.57361	0.786803	−	0.617204i	$-0.211735\pi$
$11$	57.4097	1.57361	0.786803	+	0.617204i	$0.211735\pi$
$12$	0	0
$13$	−17.3244	−0.369610	−0.184805	−	0.982775i	$-0.559165\pi$
$13$	−17.3244	−0.369610	−0.184805	+	0.982775i	$0.559165\pi$
$14$	0	0
$15$	−23.9435	−0.412146
$16$	0	0
$17$	−37.0472	−0.528545	−0.264272	−	0.964448i	$-0.585132\pi$
$17$	−37.0472	−0.528545	−0.264272	+	0.964448i	$0.585132\pi$
$18$	0	0
$19$	121.841	1.47117	0.735584	−	0.677433i	$-0.236908\pi$
$19$	121.841	1.47117	0.735584	+	0.677433i	$0.236908\pi$
$20$	0	0
$21$	−0.749032	−0.00778343
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−148.777	−1.06045
$28$	0	0
$29$	89.1452	0.570822	0.285411	−	0.958405i	$-0.407870\pi$
$29$	89.1452	0.570822	0.285411	+	0.958405i	$0.407870\pi$
$30$	0	0
$31$	76.6700	0.444205	0.222102	−	0.975023i	$-0.428708\pi$
$31$	76.6700	0.444205	0.222102	+	0.975023i	$0.428708\pi$
$32$	0	0
$33$	274.918	1.45021
$34$	0	0
$35$	0.782082	0.00377703
$36$	0	0
$37$	−183.023	−0.813212	−0.406606	−	0.913604i	$-0.633288\pi$
$37$	−183.023	−0.813212	−0.406606	+	0.913604i	$0.633288\pi$
$38$	0	0
$39$	−82.9616	−0.340628
$40$	0	0
$41$	−237.272	−0.903796	−0.451898	−	0.892070i	$-0.649253\pi$
$41$	−237.272	−0.903796	−0.451898	+	0.892070i	$0.649253\pi$
$42$	0	0
$43$	342.074	1.21316	0.606578	−	0.795024i	$-0.292542\pi$
$43$	342.074	1.21316	0.606578	+	0.795024i	$0.292542\pi$
$44$	0	0
$45$	20.3415	0.0673853
$46$	0	0
$47$	245.602	0.762229	0.381114	−	0.924528i	$-0.375540\pi$
$47$	245.602	0.762229	0.381114	+	0.924528i	$0.375540\pi$
$48$	0	0
$49$	−342.976	−0.999929
$50$	0	0
$51$	−177.408	−0.487100
$52$	0	0
$53$	197.588	0.512091	0.256045	−	0.966665i	$-0.417580\pi$
$53$	197.588	0.512091	0.256045	+	0.966665i	$0.417580\pi$
$54$	0	0
$55$	−287.048	−0.703738
$56$	0	0
$57$	583.460	1.35581
$58$	0	0
$59$	311.314	0.686943	0.343472	−	0.939163i	$-0.388397\pi$
$59$	311.314	0.686943	0.343472	+	0.939163i	$0.388397\pi$
$60$	0	0
$61$	130.256	0.273402	0.136701	−	0.990612i	$-0.456350\pi$
$61$	130.256	0.273402	0.136701	+	0.990612i	$0.456350\pi$
$62$	0	0
$63$	0.636350	0.00127258
$64$	0	0
$65$	86.6222	0.165295
$66$	0	0
$67$	293.742	0.535616	0.267808	−	0.963472i	$-0.413701\pi$
$67$	293.742	0.535616	0.267808	+	0.963472i	$0.413701\pi$
$68$	0	0
$69$	−110.140	−0.192164
$70$	0	0
$71$	−765.950	−1.28030	−0.640152	−	0.768248i	$-0.721129\pi$
$71$	−765.950	−1.28030	−0.640152	+	0.768248i	$0.721129\pi$
$72$	0	0
$73$	699.714	1.12185	0.560927	−	0.827865i	$-0.310445\pi$
$73$	699.714	1.12185	0.560927	+	0.827865i	$0.310445\pi$
$74$	0	0
$75$	119.718	0.184317
$76$	0	0
$77$	−8.97981	−0.0132902
$78$	0	0
$79$	−691.925	−0.985414	−0.492707	−	0.870195i	$-0.663992\pi$
$79$	−691.925	−0.985414	−0.492707	+	0.870195i	$0.663992\pi$
$80$	0	0
$81$	−602.605	−0.826618
$82$	0	0
$83$	952.614	1.25979	0.629897	−	0.776679i	$-0.283097\pi$
$83$	952.614	1.25979	0.629897	+	0.776679i	$0.283097\pi$
$84$	0	0
$85$	185.236	0.236372
$86$	0	0
$87$	426.890	0.526062
$88$	0	0
$89$	−20.9162	−0.0249114	−0.0124557	−	0.999922i	$-0.503965\pi$
$89$	−20.9162	−0.0249114	−0.0124557	+	0.999922i	$0.503965\pi$
$90$	0	0
$91$	2.70983	0.00312161
$92$	0	0
$93$	367.150	0.409373
$94$	0	0
$95$	−609.204	−0.657927
$96$	0	0
$97$	455.261	0.476544	0.238272	−	0.971199i	$-0.423419\pi$
$97$	455.261	0.476544	0.238272	+	0.971199i	$0.423419\pi$
$98$	0	0
$99$	−233.560	−0.237108
$100$	0	0
$101$	679.202	0.669140	0.334570	−	0.942371i	$-0.391409\pi$
$101$	679.202	0.669140	0.334570	+	0.942371i	$0.391409\pi$
$102$	0	0
$103$	−1167.05	−1.11644	−0.558218	−	0.829694i	$-0.688515\pi$
$103$	−1167.05	−1.11644	−0.558218	+	0.829694i	$0.688515\pi$
$104$	0	0
$105$	3.74516	0.00348086
$106$	0	0
$107$	1104.51	0.997917	0.498958	−	0.866626i	$-0.333716\pi$
$107$	1104.51	0.997917	0.498958	+	0.866626i	$0.333716\pi$
$108$	0	0
$109$	1577.51	1.38622	0.693112	−	0.720830i	$-0.256239\pi$
$109$	1577.51	1.38622	0.693112	+	0.720830i	$0.256239\pi$
$110$	0	0
$111$	−876.444	−0.749445
$112$	0	0
$113$	1125.27	0.936780	0.468390	−	0.883522i	$-0.344834\pi$
$113$	1125.27	0.936780	0.468390	+	0.883522i	$0.344834\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	70.4811	0.0556922
$118$	0	0
$119$	5.79479	0.00446393
$120$	0	0
$121$	1964.87	1.47624
$122$	0	0
$123$	−1136.22	−0.832926
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	666.861	0.465939	0.232970	−	0.972484i	$-0.425156\pi$
$127$	666.861	0.465939	0.232970	+	0.972484i	$0.425156\pi$
$128$	0	0
$129$	1638.09	1.11803
$130$	0	0
$131$	856.684	0.571365	0.285682	−	0.958324i	$-0.407780\pi$
$131$	856.684	0.571365	0.285682	+	0.958324i	$0.407780\pi$
$132$	0	0
$133$	−19.0579	−0.0124250
$134$	0	0
$135$	743.885	0.474247
$136$	0	0
$137$	1728.88	1.07816	0.539082	−	0.842253i	$-0.318771\pi$
$137$	1728.88	1.07816	0.539082	+	0.842253i	$0.318771\pi$
$138$	0	0
$139$	−149.716	−0.0913580	−0.0456790	−	0.998956i	$-0.514545\pi$
$139$	−149.716	−0.0913580	−0.0456790	+	0.998956i	$0.514545\pi$
$140$	0	0
$141$	1176.12	0.702460
$142$	0	0
$143$	−994.590	−0.581621
$144$	0	0
$145$	−445.726	−0.255279
$146$	0	0
$147$	−1642.41	−0.921521
$148$	0	0
$149$	1430.37	0.786446	0.393223	−	0.919443i	$-0.371360\pi$
$149$	1430.37	0.786446	0.393223	+	0.919443i	$0.371360\pi$
$150$	0	0
$151$	−3052.73	−1.64521	−0.822607	−	0.568610i	$-0.807481\pi$
$151$	−3052.73	−1.64521	−0.822607	+	0.568610i	$0.807481\pi$
$152$	0	0
$153$	150.719	0.0796401
$154$	0	0
$155$	−383.350	−0.198654
$156$	0	0
$157$	3181.98	1.61751	0.808756	−	0.588144i	$-0.200141\pi$
$157$	3181.98	1.61751	0.808756	+	0.588144i	$0.200141\pi$
$158$	0	0
$159$	946.191	0.471936
$160$	0	0
$161$	3.59758	0.00176105
$162$	0	0
$163$	3054.01	1.46754	0.733769	−	0.679399i	$-0.237759\pi$
$163$	3054.01	1.46754	0.733769	+	0.679399i	$0.237759\pi$
$164$	0	0
$165$	−1374.59	−0.648556
$166$	0	0
$167$	3023.70	1.40108	0.700542	−	0.713612i	$-0.252942\pi$
$167$	3023.70	1.40108	0.700542	+	0.713612i	$0.252942\pi$
$168$	0	0
$169$	−1896.86	−0.863388
$170$	0	0
$171$	−495.686	−0.221673
$172$	0	0
$173$	−2156.05	−0.947523	−0.473761	−	0.880653i	$-0.657104\pi$
$173$	−2156.05	−0.947523	−0.473761	+	0.880653i	$0.657104\pi$
$174$	0	0
$175$	−3.91041	−0.00168914
$176$	0	0
$177$	1490.79	0.633078
$178$	0	0
$179$	3801.46	1.58734	0.793671	−	0.608347i	$-0.208167\pi$
$179$	3801.46	1.58734	0.793671	+	0.608347i	$0.208167\pi$
$180$	0	0
$181$	3562.64	1.46303	0.731517	−	0.681823i	$-0.238813\pi$
$181$	3562.64	1.46303	0.731517	+	0.681823i	$0.238813\pi$
$182$	0	0
$183$	623.755	0.251964
$184$	0	0
$185$	915.116	0.363679
$186$	0	0
$187$	−2126.87	−0.831721
$188$	0	0
$189$	23.2711	0.00895623
$190$	0	0
$191$	−574.403	−0.217604	−0.108802	−	0.994063i	$-0.534701\pi$
$191$	−574.403	−0.217604	−0.108802	+	0.994063i	$0.534701\pi$
$192$	0	0
$193$	608.420	0.226917	0.113459	−	0.993543i	$-0.463807\pi$
$193$	608.420	0.226917	0.113459	+	0.993543i	$0.463807\pi$
$194$	0	0
$195$	414.808	0.152333
$196$	0	0
$197$	−3593.84	−1.29975	−0.649874	−	0.760042i	$-0.725178\pi$
$197$	−3593.84	−1.29975	−0.649874	+	0.760042i	$0.725178\pi$
$198$	0	0
$199$	3129.11	1.11466	0.557329	−	0.830292i	$-0.311826\pi$
$199$	3129.11	1.11466	0.557329	+	0.830292i	$0.311826\pi$
$200$	0	0
$201$	1406.64	0.493617
$202$	0	0
$203$	−13.9438	−0.00482099
$204$	0	0
$205$	1186.36	0.404190
$206$	0	0
$207$	93.5711	0.0314185
$208$	0	0
$209$	6994.84	2.31504
$210$	0	0
$211$	−4044.47	−1.31959	−0.659794	−	0.751446i	$-0.729357\pi$
$211$	−4044.47	−1.31959	−0.659794	+	0.751446i	$0.729357\pi$
$212$	0	0
$213$	−3667.91	−1.17991
$214$	0	0
$215$	−1710.37	−0.542540
$216$	0	0
$217$	−11.9924	−0.00375161
$218$	0	0
$219$	3350.72	1.03389
$220$	0	0
$221$	641.821	0.195356
$222$	0	0
$223$	3808.00	1.14351	0.571755	−	0.820424i	$-0.306263\pi$
$223$	3808.00	1.14351	0.571755	+	0.820424i	$0.306263\pi$
$224$	0	0
$225$	−101.708	−0.0301356
$226$	0	0
$227$	−3063.54	−0.895745	−0.447872	−	0.894097i	$-0.647818\pi$
$227$	−3063.54	−0.895745	−0.447872	+	0.894097i	$0.647818\pi$
$228$	0	0
$229$	81.4540	0.0235049	0.0117525	−	0.999931i	$-0.496259\pi$
$229$	81.4540	0.0235049	0.0117525	+	0.999931i	$0.496259\pi$
$230$	0	0
$231$	−43.0017	−0.0122481
$232$	0	0
$233$	−415.063	−0.116703	−0.0583513	−	0.998296i	$-0.518584\pi$
$233$	−415.063	−0.116703	−0.0583513	+	0.998296i	$0.518584\pi$
$234$	0	0
$235$	−1228.01	−0.340879
$236$	0	0
$237$	−3313.43	−0.908144
$238$	0	0
$239$	−4180.06	−1.13132	−0.565660	−	0.824638i	$-0.691379\pi$
$239$	−4180.06	−1.13132	−0.565660	+	0.824638i	$0.691379\pi$
$240$	0	0
$241$	3309.29	0.884524	0.442262	−	0.896886i	$-0.354176\pi$
$241$	3309.29	0.884524	0.442262	+	0.896886i	$0.354176\pi$
$242$	0	0
$243$	1131.28	0.298649
$244$	0	0
$245$	1714.88	0.447182
$246$	0	0
$247$	−2110.82	−0.543759
$248$	0	0
$249$	4561.79	1.16101
$250$	0	0
$251$	−2921.05	−0.734561	−0.367281	−	0.930110i	$-0.619711\pi$
$251$	−2921.05	−0.734561	−0.367281	+	0.930110i	$0.619711\pi$
$252$	0	0
$253$	−1320.42	−0.328120
$254$	0	0
$255$	887.040	0.217838
$256$	0	0
$257$	−4285.98	−1.04028	−0.520140	−	0.854081i	$-0.674120\pi$
$257$	−4285.98	−1.04028	−0.520140	+	0.854081i	$0.674120\pi$
$258$	0	0
$259$	28.6278	0.00686813
$260$	0	0
$261$	−362.670	−0.0860104
$262$	0	0
$263$	3728.56	0.874194	0.437097	−	0.899414i	$-0.356007\pi$
$263$	3728.56	0.874194	0.437097	+	0.899414i	$0.356007\pi$
$264$	0	0
$265$	−987.940	−0.229014
$266$	0	0
$267$	−100.162	−0.0229580
$268$	0	0
$269$	−2510.68	−0.569067	−0.284533	−	0.958666i	$-0.591839\pi$
$269$	−2510.68	−0.569067	−0.284533	+	0.958666i	$0.591839\pi$
$270$	0	0
$271$	−1794.35	−0.402210	−0.201105	−	0.979570i	$-0.564453\pi$
$271$	−1794.35	−0.402210	−0.201105	+	0.979570i	$0.564453\pi$
$272$	0	0
$273$	12.9766	0.00287684
$274$	0	0
$275$	1435.24	0.314721
$276$	0	0
$277$	2902.30	0.629539	0.314769	−	0.949168i	$-0.398073\pi$
$277$	2902.30	0.629539	0.314769	+	0.949168i	$0.398073\pi$
$278$	0	0
$279$	−311.917	−0.0669319
$280$	0	0
$281$	−7322.78	−1.55459	−0.777296	−	0.629135i	$-0.783409\pi$
$281$	−7322.78	−1.55459	−0.777296	+	0.629135i	$0.783409\pi$
$282$	0	0
$283$	1397.27	0.293496	0.146748	−	0.989174i	$-0.453119\pi$
$283$	1397.27	0.293496	0.146748	+	0.989174i	$0.453119\pi$
$284$	0	0
$285$	−2917.30	−0.606336
$286$	0	0
$287$	37.1132	0.00763318
$288$	0	0
$289$	−3540.51	−0.720640
$290$	0	0
$291$	2180.11	0.439176
$292$	0	0
$293$	2180.24	0.434713	0.217357	−	0.976092i	$-0.430257\pi$
$293$	2180.24	0.434713	0.217357	+	0.976092i	$0.430257\pi$
$294$	0	0
$295$	−1556.57	−0.307210
$296$	0	0
$297$	−8541.24	−1.66873
$298$	0	0
$299$	398.462	0.0770691
$300$	0	0
$301$	−53.5059	−0.0102459
$302$	0	0
$303$	3252.50	0.616670
$304$	0	0
$305$	−651.278	−0.122269
$306$	0	0
$307$	1664.29	0.309401	0.154700	−	0.987961i	$-0.450559\pi$
$307$	1664.29	0.309401	0.154700	+	0.987961i	$0.450559\pi$
$308$	0	0
$309$	−5588.67	−1.02889
$310$	0	0
$311$	−1909.84	−0.348223	−0.174111	−	0.984726i	$-0.555705\pi$
$311$	−1909.84	−0.348223	−0.174111	+	0.984726i	$0.555705\pi$
$312$	0	0
$313$	−995.149	−0.179710	−0.0898549	−	0.995955i	$-0.528640\pi$
$313$	−995.149	−0.179710	−0.0898549	+	0.995955i	$0.528640\pi$
$314$	0	0
$315$	−3.18175	−0.000569115 0
$316$	0	0
$317$	1418.10	0.251256	0.125628	−	0.992077i	$-0.459905\pi$
$317$	1418.10	0.251256	0.125628	+	0.992077i	$0.459905\pi$
$318$	0	0
$319$	5117.80	0.898249
$320$	0	0
$321$	5289.18	0.919667
$322$	0	0
$323$	−4513.86	−0.777579
$324$	0	0
$325$	−433.111	−0.0739221
$326$	0	0
$327$	7554.25	1.27753
$328$	0	0
$329$	−38.4162	−0.00643755
$330$	0	0
$331$	7631.37	1.26725	0.633623	−	0.773642i	$-0.281567\pi$
$331$	7631.37	1.26725	0.633623	+	0.773642i	$0.281567\pi$
$332$	0	0
$333$	744.595	0.122533
$334$	0	0
$335$	−1468.71	−0.239535
$336$	0	0
$337$	2225.65	0.359759	0.179880	−	0.983689i	$-0.442429\pi$
$337$	2225.65	0.359759	0.179880	+	0.983689i	$0.442429\pi$
$338$	0	0
$339$	5388.57	0.863324
$340$	0	0
$341$	4401.60	0.699003
$342$	0	0
$343$	107.298	0.0168908
$344$	0	0
$345$	550.701	0.0859384
$346$	0	0
$347$	8913.21	1.37892	0.689462	−	0.724322i	$-0.257847\pi$
$347$	8913.21	1.37892	0.689462	+	0.724322i	$0.257847\pi$
$348$	0	0
$349$	4776.39	0.732590	0.366295	−	0.930499i	$-0.380626\pi$
$349$	4776.39	0.732590	0.366295	+	0.930499i	$0.380626\pi$
$350$	0	0
$351$	2577.48	0.391953
$352$	0	0
$353$	3864.84	0.582733	0.291366	−	0.956612i	$-0.405890\pi$
$353$	3864.84	0.582733	0.291366	+	0.956612i	$0.405890\pi$
$354$	0	0
$355$	3829.75	0.572569
$356$	0	0
$357$	27.7495	0.00411389
$358$	0	0
$359$	−3113.48	−0.457724	−0.228862	−	0.973459i	$-0.573501\pi$
$359$	−3113.48	−0.457724	−0.228862	+	0.973459i	$0.573501\pi$
$360$	0	0
$361$	7986.19	1.16434
$362$	0	0
$363$	9409.19	1.36048
$364$	0	0
$365$	−3498.57	−0.501708
$366$	0	0
$367$	−980.888	−0.139515	−0.0697574	−	0.997564i	$-0.522223\pi$
$367$	−980.888	−0.139515	−0.0697574	+	0.997564i	$0.522223\pi$
$368$	0	0
$369$	965.295	0.136182
$370$	0	0
$371$	−30.9060	−0.00432496
$372$	0	0
$373$	−775.564	−0.107660	−0.0538300	−	0.998550i	$-0.517143\pi$
$373$	−775.564	−0.107660	−0.0538300	+	0.998550i	$0.517143\pi$
$374$	0	0
$375$	−598.588	−0.0824292
$376$	0	0
$377$	−1544.39	−0.210982
$378$	0	0
$379$	155.428	0.0210654	0.0105327	−	0.999945i	$-0.496647\pi$
$379$	155.428	0.0210654	0.0105327	+	0.999945i	$0.496647\pi$
$380$	0	0
$381$	3193.40	0.429404
$382$	0	0
$383$	1763.82	0.235319	0.117660	−	0.993054i	$-0.462461\pi$
$383$	1763.82	0.235319	0.117660	+	0.993054i	$0.462461\pi$
$384$	0	0
$385$	44.8991	0.00594355
$386$	0	0
$387$	−1391.66	−0.182796
$388$	0	0
$389$	−2095.48	−0.273124	−0.136562	−	0.990632i	$-0.543605\pi$
$389$	−2095.48	−0.273124	−0.136562	+	0.990632i	$0.543605\pi$
$390$	0	0
$391$	852.085	0.110209
$392$	0	0
$393$	4102.41	0.526562
$394$	0	0
$395$	3459.63	0.440690
$396$	0	0
$397$	−6442.77	−0.814492	−0.407246	−	0.913319i	$-0.633511\pi$
$397$	−6442.77	−0.814492	−0.407246	+	0.913319i	$0.633511\pi$
$398$	0	0
$399$	−91.2627	−0.0114507
$400$	0	0
$401$	2941.72	0.366341	0.183170	−	0.983081i	$-0.441364\pi$
$401$	2941.72	0.366341	0.183170	+	0.983081i	$0.441364\pi$
$402$	0	0
$403$	−1328.27	−0.164183
$404$	0	0
$405$	3013.02	0.369675
$406$	0	0
$407$	−10507.3	−1.27968
$408$	0	0
$409$	1444.55	0.174641	0.0873207	−	0.996180i	$-0.472170\pi$
$409$	1444.55	0.174641	0.0873207	+	0.996180i	$0.472170\pi$
$410$	0	0
$411$	8279.11	0.993622
$412$	0	0
$413$	−48.6946	−0.00580171
$414$	0	0
$415$	−4763.07	−0.563397
$416$	0	0
$417$	−716.947	−0.0841943
$418$	0	0
$419$	12100.2	1.41082	0.705412	−	0.708797i	$-0.250762\pi$
$419$	12100.2	1.41082	0.705412	+	0.708797i	$0.250762\pi$
$420$	0	0
$421$	422.517	0.0489127	0.0244563	−	0.999701i	$-0.492215\pi$
$421$	422.517	0.0489127	0.0244563	+	0.999701i	$0.492215\pi$
$422$	0	0
$423$	−999.185	−0.114851
$424$	0	0
$425$	−926.180	−0.105709
$426$	0	0
$427$	−20.3741	−0.00230907
$428$	0	0
$429$	−4762.80	−0.536014
$430$	0	0
$431$	−13720.2	−1.53336	−0.766681	−	0.642028i	$-0.778093\pi$
$431$	−13720.2	−1.53336	−0.766681	+	0.642028i	$0.778093\pi$
$432$	0	0
$433$	−9734.44	−1.08039	−0.540193	−	0.841541i	$-0.681649\pi$
$433$	−9734.44	−1.08039	−0.540193	+	0.841541i	$0.681649\pi$
$434$	0	0
$435$	−2134.45	−0.235262
$436$	0	0
$437$	−2802.34	−0.306760
$438$	0	0
$439$	−11107.3	−1.20757	−0.603783	−	0.797149i	$-0.706340\pi$
$439$	−11107.3	−1.20757	−0.603783	+	0.797149i	$0.706340\pi$
$440$	0	0
$441$	1395.33	0.150667
$442$	0	0
$443$	6148.47	0.659419	0.329710	−	0.944082i	$-0.393049\pi$
$443$	6148.47	0.659419	0.329710	+	0.944082i	$0.393049\pi$
$444$	0	0
$445$	104.581	0.0111407
$446$	0	0
$447$	6849.62	0.724778
$448$	0	0
$449$	1105.16	0.116160	0.0580799	−	0.998312i	$-0.481502\pi$
$449$	1105.16	0.116160	0.0580799	+	0.998312i	$0.481502\pi$
$450$	0	0
$451$	−13621.7	−1.42222
$452$	0	0
$453$	−14618.6	−1.51621
$454$	0	0
$455$	−13.5491	−0.00139603
$456$	0	0
$457$	−3856.17	−0.394714	−0.197357	−	0.980332i	$-0.563236\pi$
$457$	−3856.17	−0.394714	−0.197357	+	0.980332i	$0.563236\pi$
$458$	0	0
$459$	5511.77	0.560495
$460$	0	0
$461$	−15397.7	−1.55562	−0.777810	−	0.628500i	$-0.783669\pi$
$461$	−15397.7	−1.55562	−0.777810	+	0.628500i	$0.783669\pi$
$462$	0	0
$463$	10935.6	1.09767	0.548836	−	0.835930i	$-0.315071\pi$
$463$	10935.6	1.09767	0.548836	+	0.835930i	$0.315071\pi$
$464$	0	0
$465$	−1835.75	−0.183077
$466$	0	0
$467$	−15048.7	−1.49116	−0.745581	−	0.666415i	$-0.767828\pi$
$467$	−15048.7	−1.49116	−0.745581	+	0.666415i	$0.767828\pi$
$468$	0	0
$469$	−45.9460	−0.00452365
$470$	0	0
$471$	15237.6	1.49068
$472$	0	0
$473$	19638.3	1.90903
$474$	0	0
$475$	3046.02	0.294234
$476$	0	0
$477$	−803.849	−0.0771608
$478$	0	0
$479$	5703.70	0.544068	0.272034	−	0.962288i	$-0.412304\pi$
$479$	5703.70	0.544068	0.272034	+	0.962288i	$0.412304\pi$
$480$	0	0
$481$	3170.77	0.300571
$482$	0	0
$483$	17.2277	0.00162296
$484$	0	0
$485$	−2276.30	−0.213117
$486$	0	0
$487$	−12039.4	−1.12024	−0.560122	−	0.828410i	$-0.689246\pi$
$487$	−12039.4	−1.12024	−0.560122	+	0.828410i	$0.689246\pi$
$488$	0	0
$489$	14624.8	1.35246
$490$	0	0
$491$	11179.4	1.02753	0.513766	−	0.857930i	$-0.328250\pi$
$491$	11179.4	1.02753	0.513766	+	0.857930i	$0.328250\pi$
$492$	0	0
$493$	−3302.58	−0.301705
$494$	0	0
$495$	1167.80	0.106038
$496$	0	0
$497$	119.807	0.0108130
$498$	0	0
$499$	17502.2	1.57016	0.785078	−	0.619396i	$-0.212623\pi$
$499$	17502.2	1.57016	0.785078	+	0.619396i	$0.212623\pi$
$500$	0	0
$501$	14479.6	1.29122
$502$	0	0
$503$	16936.6	1.50132	0.750662	−	0.660686i	$-0.229735\pi$
$503$	16936.6	1.50132	0.750662	+	0.660686i	$0.229735\pi$
$504$	0	0
$505$	−3396.01	−0.299248
$506$	0	0
$507$	−9083.52	−0.795687
$508$	0	0
$509$	−2758.06	−0.240175	−0.120087	−	0.992763i	$-0.538317\pi$
$509$	−2758.06	−0.240175	−0.120087	+	0.992763i	$0.538317\pi$
$510$	0	0
$511$	−109.447	−0.00947483
$512$	0	0
$513$	−18127.1	−1.56010
$514$	0	0
$515$	5835.26	0.499286
$516$	0	0
$517$	14099.9	1.19945
$518$	0	0
$519$	−10324.7	−0.873224
$520$	0	0
$521$	14450.1	1.21510	0.607552	−	0.794280i	$-0.292152\pi$
$521$	14450.1	1.21510	0.607552	+	0.794280i	$0.292152\pi$
$522$	0	0
$523$	−5421.92	−0.453315	−0.226658	−	0.973974i	$-0.572780\pi$
$523$	−5421.92	−0.453315	−0.226658	+	0.973974i	$0.572780\pi$
$524$	0	0
$525$	−18.7258	−0.00155669
$526$	0	0
$527$	−2840.41	−0.234782
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−1266.52	−0.103507
$532$	0	0
$533$	4110.60	0.334052
$534$	0	0
$535$	−5522.56	−0.446282
$536$	0	0
$537$	18204.1	1.46287
$538$	0	0
$539$	−19690.1	−1.57349
$540$	0	0
$541$	−18761.4	−1.49097	−0.745486	−	0.666521i	$-0.767783\pi$
$541$	−18761.4	−1.49097	−0.745486	+	0.666521i	$0.767783\pi$
$542$	0	0
$543$	17060.4	1.34831
$544$	0	0
$545$	−7887.57	−0.619939
$546$	0	0
$547$	−2415.42	−0.188804	−0.0944022	−	0.995534i	$-0.530094\pi$
$547$	−2415.42	−0.188804	−0.0944022	+	0.995534i	$0.530094\pi$
$548$	0	0
$549$	−529.920	−0.0411957
$550$	0	0
$551$	10861.5	0.839776
$552$	0	0
$553$	108.228	0.00832250
$554$	0	0
$555$	4382.22	0.335162
$556$	0	0
$557$	−2185.97	−0.166288	−0.0831440	−	0.996538i	$-0.526496\pi$
$557$	−2185.97	−0.166288	−0.0831440	+	0.996538i	$0.526496\pi$
$558$	0	0
$559$	−5926.23	−0.448395
$560$	0	0
$561$	−10184.9	−0.766503
$562$	0	0
$563$	23949.7	1.79282	0.896411	−	0.443224i	$-0.146165\pi$
$563$	23949.7	1.79282	0.896411	+	0.443224i	$0.146165\pi$
$564$	0	0
$565$	−5626.33	−0.418941
$566$	0	0
$567$	94.2572	0.00698136
$568$	0	0
$569$	6274.02	0.462251	0.231125	−	0.972924i	$-0.425759\pi$
$569$	6274.02	0.462251	0.231125	+	0.972924i	$0.425759\pi$
$570$	0	0
$571$	−24538.8	−1.79845	−0.899226	−	0.437485i	$-0.855869\pi$
$571$	−24538.8	−1.79845	−0.899226	+	0.437485i	$0.855869\pi$
$572$	0	0
$573$	−2750.65	−0.200541
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	17077.9	1.23217	0.616085	−	0.787680i	$-0.288718\pi$
$577$	17077.9	1.23217	0.616085	+	0.787680i	$0.288718\pi$
$578$	0	0
$579$	2913.55	0.209124
$580$	0	0
$581$	−149.004	−0.0106398
$582$	0	0
$583$	11343.5	0.805829
$584$	0	0
$585$	−352.406	−0.0249063
$586$	0	0
$587$	3705.71	0.260564	0.130282	−	0.991477i	$-0.458412\pi$
$587$	3705.71	0.260564	0.130282	+	0.991477i	$0.458412\pi$
$588$	0	0
$589$	9341.54	0.653500
$590$	0	0
$591$	−17209.8	−1.19783
$592$	0	0
$593$	−21404.7	−1.48227	−0.741136	−	0.671355i	$-0.765712\pi$
$593$	−21404.7	−1.48227	−0.741136	+	0.671355i	$0.765712\pi$
$594$	0	0
$595$	−28.9739	−0.00199633
$596$	0	0
$597$	14984.4	1.02725
$598$	0	0
$599$	13446.8	0.917231	0.458615	−	0.888635i	$-0.348346\pi$
$599$	13446.8	0.917231	0.458615	+	0.888635i	$0.348346\pi$
$600$	0	0
$601$	−20196.9	−1.37080	−0.685398	−	0.728169i	$-0.740372\pi$
$601$	−20196.9	−1.37080	−0.685398	+	0.728169i	$0.740372\pi$
$602$	0	0
$603$	−1195.03	−0.0807056
$604$	0	0
$605$	−9824.36	−0.660193
$606$	0	0
$607$	−19923.0	−1.33221	−0.666103	−	0.745860i	$-0.732039\pi$
$607$	−19923.0	−1.33221	−0.666103	+	0.745860i	$0.732039\pi$
$608$	0	0
$609$	−66.7726	−0.00444296
$610$	0	0
$611$	−4254.92	−0.281728
$612$	0	0
$613$	9116.45	0.600668	0.300334	−	0.953834i	$-0.402902\pi$
$613$	9116.45	0.600668	0.300334	+	0.953834i	$0.402902\pi$
$614$	0	0
$615$	5681.12	0.372496
$616$	0	0
$617$	3513.84	0.229273	0.114637	−	0.993407i	$-0.463430\pi$
$617$	3513.84	0.229273	0.114637	+	0.993407i	$0.463430\pi$
$618$	0	0
$619$	24975.5	1.62173	0.810866	−	0.585232i	$-0.198997\pi$
$619$	24975.5	1.62173	0.810866	+	0.585232i	$0.198997\pi$
$620$	0	0
$621$	3421.87	0.221119
$622$	0	0
$623$	3.27164	0.000210394 0
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	33496.2	2.13351
$628$	0	0
$629$	6780.49	0.429819
$630$	0	0
$631$	−30838.5	−1.94558	−0.972789	−	0.231694i	$-0.925573\pi$
$631$	−30838.5	−1.94558	−0.972789	+	0.231694i	$0.925573\pi$
$632$	0	0
$633$	−19367.8	−1.21611
$634$	0	0
$635$	−3334.30	−0.208374
$636$	0	0
$637$	5941.86	0.369584
$638$	0	0
$639$	3116.12	0.192914
$640$	0	0
$641$	28742.3	1.77106	0.885532	−	0.464579i	$-0.153794\pi$
$641$	28742.3	1.77106	0.885532	+	0.464579i	$0.153794\pi$
$642$	0	0
$643$	11124.9	0.682309	0.341154	−	0.940007i	$-0.389182\pi$
$643$	11124.9	0.682309	0.341154	+	0.940007i	$0.389182\pi$
$644$	0	0
$645$	−8190.45	−0.499998
$646$	0	0
$647$	22997.4	1.39740	0.698701	−	0.715413i	$-0.253762\pi$
$647$	22997.4	1.39740	0.698701	+	0.715413i	$0.253762\pi$
$648$	0	0
$649$	17872.5	1.08098
$650$	0	0
$651$	−57.4283	−0.00345744
$652$	0	0
$653$	−5123.03	−0.307013	−0.153507	−	0.988148i	$-0.549057\pi$
$653$	−5123.03	−0.307013	−0.153507	+	0.988148i	$0.549057\pi$
$654$	0	0
$655$	−4283.42	−0.255522
$656$	0	0
$657$	−2846.65	−0.169039
$658$	0	0
$659$	−6578.78	−0.388881	−0.194441	−	0.980914i	$-0.562289\pi$
$659$	−6578.78	−0.388881	−0.194441	+	0.980914i	$0.562289\pi$
$660$	0	0
$661$	−723.212	−0.0425563	−0.0212781	−	0.999774i	$-0.506774\pi$
$661$	−723.212	−0.0425563	−0.0212781	+	0.999774i	$0.506774\pi$
$662$	0	0
$663$	3073.49	0.180037
$664$	0	0
$665$	95.2895	0.00555664
$666$	0	0
$667$	−2050.34	−0.119025
$668$	0	0
$669$	18235.4	1.05384
$670$	0	0
$671$	7477.93	0.430227
$672$	0	0
$673$	18997.3	1.08810	0.544051	−	0.839052i	$-0.316890\pi$
$673$	18997.3	1.08810	0.544051	+	0.839052i	$0.316890\pi$
$674$	0	0
$675$	−3719.42	−0.212090
$676$	0	0
$677$	−25064.9	−1.42293	−0.711465	−	0.702721i	$-0.751968\pi$
$677$	−25064.9	−1.42293	−0.711465	+	0.702721i	$0.751968\pi$
$678$	0	0
$679$	−71.2102	−0.00402474
$680$	0	0
$681$	−14670.4	−0.825506
$682$	0	0
$683$	−26805.5	−1.50173	−0.750867	−	0.660454i	$-0.770364\pi$
$683$	−26805.5	−1.50173	−0.750867	+	0.660454i	$0.770364\pi$
$684$	0	0
$685$	−8644.42	−0.482170
$686$	0	0
$687$	390.059	0.0216618
$688$	0	0
$689$	−3423.10	−0.189274
$690$	0	0
$691$	−30663.7	−1.68814	−0.844068	−	0.536236i	$-0.819846\pi$
$691$	−30663.7	−1.68814	−0.844068	+	0.536236i	$0.819846\pi$
$692$	0	0
$693$	36.5326	0.00200254
$694$	0	0
$695$	748.581	0.0408565
$696$	0	0
$697$	8790.25	0.477697
$698$	0	0
$699$	−1987.62	−0.107552
$700$	0	0
$701$	923.869	0.0497775	0.0248888	−	0.999690i	$-0.492077\pi$
$701$	923.869	0.0497775	0.0248888	+	0.999690i	$0.492077\pi$
$702$	0	0
$703$	−22299.7	−1.19637
$704$	0	0
$705$	−5880.58	−0.314150
$706$	0	0
$707$	−106.238	−0.00565135
$708$	0	0
$709$	−21201.0	−1.12302	−0.561510	−	0.827470i	$-0.689779\pi$
$709$	−21201.0	−1.12302	−0.561510	+	0.827470i	$0.689779\pi$
$710$	0	0
$711$	2814.96	0.148480
$712$	0	0
$713$	−1763.41	−0.0926231
$714$	0	0
$715$	4972.95	0.260109
$716$	0	0
$717$	−20017.1	−1.04261
$718$	0	0
$719$	−9700.52	−0.503155	−0.251577	−	0.967837i	$-0.580949\pi$
$719$	−9700.52	−0.503155	−0.251577	+	0.967837i	$0.580949\pi$
$720$	0	0
$721$	182.546	0.00942908
$722$	0	0
$723$	15847.2	0.815165
$724$	0	0
$725$	2228.63	0.114164
$726$	0	0
$727$	−34644.2	−1.76738	−0.883689	−	0.468074i	$-0.844948\pi$
$727$	−34644.2	−1.76738	−0.883689	+	0.468074i	$0.844948\pi$
$728$	0	0
$729$	21687.7	1.10185
$730$	0	0
$731$	−12672.9	−0.641208
$732$	0	0
$733$	−17256.4	−0.869549	−0.434774	−	0.900539i	$-0.643172\pi$
$733$	−17256.4	−0.869549	−0.434774	+	0.900539i	$0.643172\pi$
$734$	0	0
$735$	8212.04	0.412117
$736$	0	0
$737$	16863.6	0.842849
$738$	0	0
$739$	17467.6	0.869493	0.434747	−	0.900553i	$-0.356838\pi$
$739$	17467.6	0.869493	0.434747	+	0.900553i	$0.356838\pi$
$740$	0	0
$741$	−10108.1	−0.501121
$742$	0	0
$743$	−3084.21	−0.152286	−0.0761432	−	0.997097i	$-0.524261\pi$
$743$	−3084.21	−0.152286	−0.0761432	+	0.997097i	$0.524261\pi$
$744$	0	0
$745$	−7151.85	−0.351709
$746$	0	0
$747$	−3875.52	−0.189823
$748$	0	0
$749$	−172.764	−0.00842810
$750$	0	0
$751$	−15395.3	−0.748046	−0.374023	−	0.927420i	$-0.622022\pi$
$751$	−15395.3	−0.748046	−0.374023	+	0.927420i	$0.622022\pi$
$752$	0	0
$753$	−13988.0	−0.676962
$754$	0	0
$755$	15263.6	0.735762
$756$	0	0
$757$	1499.60	0.0719998	0.0359999	−	0.999352i	$-0.488538\pi$
$757$	1499.60	0.0719998	0.0359999	+	0.999352i	$0.488538\pi$
$758$	0	0
$759$	−6323.11	−0.302391
$760$	0	0
$761$	−25871.4	−1.23237	−0.616187	−	0.787600i	$-0.711323\pi$
$761$	−25871.4	−1.23237	−0.616187	+	0.787600i	$0.711323\pi$
$762$	0	0
$763$	−246.749	−0.0117076
$764$	0	0
$765$	−753.597	−0.0356161
$766$	0	0
$767$	−5393.34	−0.253901
$768$	0	0
$769$	−25772.7	−1.20857	−0.604283	−	0.796770i	$-0.706540\pi$
$769$	−25772.7	−1.20857	−0.604283	+	0.796770i	$0.706540\pi$
$770$	0	0
$771$	−20524.3	−0.958708
$772$	0	0
$773$	−23172.5	−1.07821	−0.539106	−	0.842238i	$-0.681238\pi$
$773$	−23172.5	−1.07821	−0.539106	+	0.842238i	$0.681238\pi$
$774$	0	0
$775$	1916.75	0.0888409
$776$	0	0
$777$	137.090	0.00632958
$778$	0	0
$779$	−28909.4	−1.32964
$780$	0	0
$781$	−43973.0	−2.01469
$782$	0	0
$783$	−13262.7	−0.605328
$784$	0	0
$785$	−15909.9	−0.723374
$786$	0	0
$787$	−24900.0	−1.12781	−0.563906	−	0.825839i	$-0.690702\pi$
$787$	−24900.0	−1.12781	−0.563906	+	0.825839i	$0.690702\pi$
$788$	0	0
$789$	17855.0	0.805646
$790$	0	0
$791$	−176.010	−0.00791175
$792$	0	0
$793$	−2256.60	−0.101052
$794$	0	0
$795$	−4730.96	−0.211056
$796$	0	0
$797$	36170.1	1.60754	0.803771	−	0.594939i	$-0.202824\pi$
$797$	36170.1	1.60754	0.803771	+	0.594939i	$0.202824\pi$
$798$	0	0
$799$	−9098.87	−0.402872
$800$	0	0
$801$	85.0937	0.00375360
$802$	0	0
$803$	40170.4	1.76536
$804$	0	0
$805$	−17.9879	−0.000787565 0
$806$	0	0
$807$	−12022.9	−0.524444
$808$	0	0
$809$	18615.1	0.808991	0.404495	−	0.914540i	$-0.367447\pi$
$809$	18615.1	0.808991	0.404495	+	0.914540i	$0.367447\pi$
$810$	0	0
$811$	−42242.1	−1.82900	−0.914501	−	0.404584i	$-0.867416\pi$
$811$	−42242.1	−1.82900	−0.914501	+	0.404584i	$0.867416\pi$
$812$	0	0
$813$	−8592.60	−0.370671
$814$	0	0
$815$	−15270.1	−0.656303
$816$	0	0
$817$	41678.5	1.78476
$818$	0	0
$819$	−11.0244	−0.000470359 0
$820$	0	0
$821$	14337.6	0.609484	0.304742	−	0.952435i	$-0.401430\pi$
$821$	14337.6	0.609484	0.304742	+	0.952435i	$0.401430\pi$
$822$	0	0
$823$	−42536.5	−1.80162	−0.900808	−	0.434218i	$-0.857025\pi$
$823$	−42536.5	−1.80162	−0.900808	+	0.434218i	$0.857025\pi$
$824$	0	0
$825$	6872.95	0.290043
$826$	0	0
$827$	9881.30	0.415485	0.207743	−	0.978183i	$-0.433388\pi$
$827$	9881.30	0.415485	0.207743	+	0.978183i	$0.433388\pi$
$828$	0	0
$829$	26382.5	1.10531	0.552654	−	0.833411i	$-0.313615\pi$
$829$	26382.5	1.10531	0.552654	+	0.833411i	$0.313615\pi$
$830$	0	0
$831$	13898.3	0.580174
$832$	0	0
$833$	12706.3	0.528507
$834$	0	0
$835$	−15118.5	−0.626583
$836$	0	0
$837$	−11406.7	−0.471057
$838$	0	0
$839$	−32817.3	−1.35039	−0.675197	−	0.737638i	$-0.735941\pi$
$839$	−32817.3	−1.35039	−0.675197	+	0.737638i	$0.735941\pi$
$840$	0	0
$841$	−16442.1	−0.674162
$842$	0	0
$843$	−35066.6	−1.43269
$844$	0	0
$845$	9484.32	0.386119
$846$	0	0
$847$	−307.338	−0.0124678
$848$	0	0
$849$	6691.13	0.270482
$850$	0	0
$851$	4209.53	0.169566
$852$	0	0
$853$	−38507.6	−1.54569	−0.772846	−	0.634593i	$-0.781168\pi$
$853$	−38507.6	−1.54569	−0.772846	+	0.634593i	$0.781168\pi$
$854$	0	0
$855$	2478.43	0.0991351
$856$	0	0
$857$	30326.4	1.20878	0.604392	−	0.796687i	$-0.293416\pi$
$857$	30326.4	1.20878	0.604392	+	0.796687i	$0.293416\pi$
$858$	0	0
$859$	−33637.3	−1.33608	−0.668039	−	0.744126i	$-0.732866\pi$
$859$	−33637.3	−1.33608	−0.668039	+	0.744126i	$0.732866\pi$
$860$	0	0
$861$	177.724	0.00703464
$862$	0	0
$863$	−6236.47	−0.245993	−0.122996	−	0.992407i	$-0.539250\pi$
$863$	−6236.47	−0.245993	−0.122996	+	0.992407i	$0.539250\pi$
$864$	0	0
$865$	10780.2	0.423745
$866$	0	0
$867$	−16954.4	−0.664133
$868$	0	0
$869$	−39723.2	−1.55065
$870$	0	0
$871$	−5088.91	−0.197969
$872$	0	0
$873$	−1852.14	−0.0718046
$874$	0	0
$875$	19.5520	0.000755405 0
$876$	0	0
$877$	6601.16	0.254168	0.127084	−	0.991892i	$-0.459438\pi$
$877$	6601.16	0.254168	0.127084	+	0.991892i	$0.459438\pi$
$878$	0	0
$879$	10440.5	0.400626
$880$	0	0
$881$	3981.08	0.152243	0.0761214	−	0.997099i	$-0.475746\pi$
$881$	3981.08	0.152243	0.0761214	+	0.997099i	$0.475746\pi$
$882$	0	0
$883$	−5915.16	−0.225437	−0.112719	−	0.993627i	$-0.535956\pi$
$883$	−5915.16	−0.225437	−0.112719	+	0.993627i	$0.535956\pi$
$884$	0	0
$885$	−7453.96	−0.283121
$886$	0	0
$887$	35592.2	1.34731	0.673657	−	0.739044i	$-0.264722\pi$
$887$	35592.2	1.34731	0.673657	+	0.739044i	$0.264722\pi$
$888$	0	0
$889$	−104.308	−0.00393518
$890$	0	0
$891$	−34595.3	−1.30077
$892$	0	0
$893$	29924.4	1.12137
$894$	0	0
$895$	−19007.3	−0.709881
$896$	0	0
$897$	1908.12	0.0710258
$898$	0	0
$899$	6834.76	0.253562
$900$	0	0
$901$	−7320.08	−0.270663
$902$	0	0
$903$	−256.224	−0.00944252
$904$	0	0
$905$	−17813.2	−0.654289
$906$	0	0
$907$	−11252.0	−0.411925	−0.205962	−	0.978560i	$-0.566032\pi$
$907$	−11252.0	−0.411925	−0.205962	+	0.978560i	$0.566032\pi$
$908$	0	0
$909$	−2763.20	−0.100825
$910$	0	0
$911$	11927.1	0.433769	0.216885	−	0.976197i	$-0.430410\pi$
$911$	11927.1	0.433769	0.216885	+	0.976197i	$0.430410\pi$
$912$	0	0
$913$	54689.2	1.98242
$914$	0	0
$915$	−3118.78	−0.112682
$916$	0	0
$917$	−133.999	−0.00482557
$918$	0	0
$919$	14494.3	0.520265	0.260132	−	0.965573i	$-0.416234\pi$
$919$	14494.3	0.520265	0.260132	+	0.965573i	$0.416234\pi$
$920$	0	0
$921$	7969.79	0.285140
$922$	0	0
$923$	13269.7	0.473213
$924$	0	0
$925$	−4575.58	−0.162642
$926$	0	0
$927$	4747.92	0.168223
$928$	0	0
$929$	11080.2	0.391314	0.195657	−	0.980672i	$-0.437316\pi$
$929$	11080.2	0.391314	0.195657	+	0.980672i	$0.437316\pi$
$930$	0	0
$931$	−41788.4	−1.47106
$932$	0	0
$933$	−9145.67	−0.320917
$934$	0	0
$935$	10634.3	0.371957
$936$	0	0
$937$	−35395.5	−1.23407	−0.617034	−	0.786937i	$-0.711666\pi$
$937$	−35395.5	−1.23407	−0.617034	+	0.786937i	$0.711666\pi$
$938$	0	0
$939$	−4765.47	−0.165618
$940$	0	0
$941$	857.274	0.0296985	0.0148493	−	0.999890i	$-0.495273\pi$
$941$	857.274	0.0296985	0.0148493	+	0.999890i	$0.495273\pi$
$942$	0	0
$943$	5457.25	0.188455
$944$	0	0
$945$	−116.356	−0.00400535
$946$	0	0
$947$	−50904.8	−1.74676	−0.873382	−	0.487036i	$-0.838078\pi$
$947$	−50904.8	−1.74676	−0.873382	+	0.487036i	$0.838078\pi$
$948$	0	0
$949$	−12122.2	−0.414649
$950$	0	0
$951$	6790.85	0.231555
$952$	0	0
$953$	−18498.5	−0.628779	−0.314390	−	0.949294i	$-0.601800\pi$
$953$	−18498.5	−0.628779	−0.314390	+	0.949294i	$0.601800\pi$
$954$	0	0
$955$	2872.01	0.0973154
$956$	0	0
$957$	24507.6	0.827815
$958$	0	0
$959$	−270.426	−0.00910584
$960$	0	0
$961$	−23912.7	−0.802682
$962$	0	0
$963$	−4493.49	−0.150364
$964$	0	0
$965$	−3042.10	−0.101481
$966$	0	0
$967$	−2338.84	−0.0777787	−0.0388894	−	0.999244i	$-0.512382\pi$
$967$	−2338.84	−0.0777787	−0.0388894	+	0.999244i	$0.512382\pi$
$968$	0	0
$969$	−21615.5	−0.716606
$970$	0	0
$971$	−35751.0	−1.18157	−0.590785	−	0.806829i	$-0.701182\pi$
$971$	−35751.0	−1.18157	−0.590785	+	0.806829i	$0.701182\pi$
$972$	0	0
$973$	23.4181	0.000771581 0
$974$	0	0
$975$	−2074.04	−0.0681256
$976$	0	0
$977$	16985.7	0.556214	0.278107	−	0.960550i	$-0.410293\pi$
$977$	16985.7	0.556214	0.278107	+	0.960550i	$0.410293\pi$
$978$	0	0
$979$	−1200.79	−0.0392008
$980$	0	0
$981$	−6417.81	−0.208874
$982$	0	0
$983$	−9817.16	−0.318534	−0.159267	−	0.987236i	$-0.550913\pi$
$983$	−9817.16	−0.318534	−0.159267	+	0.987236i	$0.550913\pi$
$984$	0	0
$985$	17969.2	0.581265
$986$	0	0
$987$	−183.964	−0.00593276
$988$	0	0
$989$	−7867.69	−0.252961
$990$	0	0
$991$	17724.8	0.568159	0.284079	−	0.958801i	$-0.408312\pi$
$991$	17724.8	0.568159	0.284079	+	0.958801i	$0.408312\pi$
$992$	0	0
$993$	36544.4	1.16788
$994$	0	0
$995$	−15645.6	−0.498490
$996$	0	0
$997$	2137.03	0.0678842	0.0339421	−	0.999424i	$-0.489194\pi$
$997$	2137.03	0.0678842	0.0339421	+	0.999424i	$0.489194\pi$
$998$	0	0
$999$	27229.6	0.862370

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.bb.1.8		10
4.3	odd	2		920.4.a.g.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.g.1.3	✓	10	4.3	odd	2
1840.4.a.bb.1.8		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q+4.78870 q^{3} -5.00000 q^{5} -0.156416 q^{7} -4.06831 q^{9} +O(q^{10})\)	\(q+4.78870 q^{3} -5.00000 q^{5} -0.156416 q^{7} -4.06831 q^{9} +57.4097 q^{11} -17.3244 q^{13} -23.9435 q^{15} -37.0472 q^{17} +121.841 q^{19} -0.749032 q^{21} -23.0000 q^{23} +25.0000 q^{25} -148.777 q^{27} +89.1452 q^{29} +76.6700 q^{31} +274.918 q^{33} +0.782082 q^{35} -183.023 q^{37} -82.9616 q^{39} -237.272 q^{41} +342.074 q^{43} +20.3415 q^{45} +245.602 q^{47} -342.976 q^{49} -177.408 q^{51} +197.588 q^{53} -287.048 q^{55} +583.460 q^{57} +311.314 q^{59} +130.256 q^{61} +0.636350 q^{63} +86.6222 q^{65} +293.742 q^{67} -110.140 q^{69} -765.950 q^{71} +699.714 q^{73} +119.718 q^{75} -8.97981 q^{77} -691.925 q^{79} -602.605 q^{81} +952.614 q^{83} +185.236 q^{85} +426.890 q^{87} -20.9162 q^{89} +2.70983 q^{91} +367.150 q^{93} -609.204 q^{95} +455.261 q^{97} -233.560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9}+O(q^{10}) \) 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9	\( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35} + 62 q^{37} - 283 q^{39} + 767 q^{41} - 332 q^{43} - 695 q^{45} + 323 q^{47} + 1162 q^{49} - 176 q^{51} + 558 q^{53} - 70 q^{55} + 46 q^{57} - 822 q^{59} + 318 q^{61} - 2698 q^{63} - 55 q^{65} - 1152 q^{67} - 23 q^{69} - 1247 q^{71} + 1941 q^{73} + 25 q^{75} + 528 q^{77} - 3134 q^{79} + 6210 q^{81} - 482 q^{83} - 340 q^{85} - 1797 q^{87} + 4734 q^{89} - 4992 q^{91} + 4647 q^{93} + 570 q^{95} + 2326 q^{97} - 4356 q^{99}+O(q^{100}) \) 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9 + 14 * q^11 + 11 * q^13 - 5 * q^15 + 68 * q^17 - 114 * q^19 - 232 * q^21 - 230 * q^23 + 250 * q^25 + 433 * q^27 - 273 * q^29 + 129 * q^31 + 98 * q^33 + 140 * q^35 + 62 * q^37 - 283 * q^39 + 767 * q^41 - 332 * q^43 - 695 * q^45 + 323 * q^47 + 1162 * q^49 - 176 * q^51 + 558 * q^53 - 70 * q^55 + 46 * q^57 - 822 * q^59 + 318 * q^61 - 2698 * q^63 - 55 * q^65 - 1152 * q^67 - 23 * q^69 - 1247 * q^71 + 1941 * q^73 + 25 * q^75 + 528 * q^77 - 3134 * q^79 + 6210 * q^81 - 482 * q^83 - 340 * q^85 - 1797 * q^87 + 4734 * q^89 - 4992 * q^91 + 4647 * q^93 + 570 * q^95 + 2326 * q^97 - 4356 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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