LMFDB - Embedded newform 1840.4.a.u.1.4

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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 80x^{4} + 105x^{3} + 1328x^{2} - 816x - 4032 $ x^6 - 2x^5 - 80x^4 + 105x^3 + 1328x^2 - 816*x - 4032
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.67963$ 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)

$f(q)$	$=$	$q+2.67963 q^{3} +5.00000 q^{5} -24.9473 q^{7} -19.8196 q^{9} +O(q^{10})$	$q+2.67963 q^{3} +5.00000 q^{5} -24.9473 q^{7} -19.8196 q^{9} +26.0279 q^{11} +54.5603 q^{13} +13.3981 q^{15} -70.5481 q^{17} +86.5444 q^{19} -66.8495 q^{21} -23.0000 q^{23} +25.0000 q^{25} -125.459 q^{27} -150.814 q^{29} +197.127 q^{31} +69.7451 q^{33} -124.737 q^{35} +159.729 q^{37} +146.201 q^{39} -139.725 q^{41} +191.106 q^{43} -99.0980 q^{45} -411.237 q^{47} +279.369 q^{49} -189.043 q^{51} +100.118 q^{53} +130.140 q^{55} +231.907 q^{57} -295.192 q^{59} -789.233 q^{61} +494.446 q^{63} +272.802 q^{65} +670.195 q^{67} -61.6314 q^{69} -598.061 q^{71} +535.992 q^{73} +66.9907 q^{75} -649.326 q^{77} +1133.10 q^{79} +198.945 q^{81} -806.388 q^{83} -352.740 q^{85} -404.125 q^{87} -874.184 q^{89} -1361.13 q^{91} +528.227 q^{93} +432.722 q^{95} -892.297 q^{97} -515.863 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9	$ 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9} + 23 q^{11} - 42 q^{13} + 20 q^{15} - 124 q^{17} - 53 q^{19} - 114 q^{21} - 138 q^{23} + 150 q^{25} + 103 q^{27} - 320 q^{29} + 229 q^{31} - 583 q^{33} + 70 q^{35} - 377 q^{37} - 37 q^{39} - 683 q^{41} + 168 q^{43} + 20 q^{45} - 211 q^{47} - 374 q^{49} - 777 q^{51} - 613 q^{53} + 115 q^{55} - 316 q^{57} - 1029 q^{59} - 1169 q^{61} - 183 q^{63} - 210 q^{65} + 1227 q^{67} - 92 q^{69} - 237 q^{71} - 1001 q^{73} + 100 q^{75} - 1498 q^{77} + 898 q^{79} - 838 q^{81} + 1281 q^{83} - 620 q^{85} + 695 q^{87} - 2780 q^{89} - 857 q^{91} - 1569 q^{93} - 265 q^{95} + 91 q^{97} + 1015 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9 + 23 * q^11 - 42 * q^13 + 20 * q^15 - 124 * q^17 - 53 * q^19 - 114 * q^21 - 138 * q^23 + 150 * q^25 + 103 * q^27 - 320 * q^29 + 229 * q^31 - 583 * q^33 + 70 * q^35 - 377 * q^37 - 37 * q^39 - 683 * q^41 + 168 * q^43 + 20 * q^45 - 211 * q^47 - 374 * q^49 - 777 * q^51 - 613 * q^53 + 115 * q^55 - 316 * q^57 - 1029 * q^59 - 1169 * q^61 - 183 * q^63 - 210 * q^65 + 1227 * q^67 - 92 * q^69 - 237 * q^71 - 1001 * q^73 + 100 * q^75 - 1498 * q^77 + 898 * q^79 - 838 * q^81 + 1281 * q^83 - 620 * q^85 + 695 * q^87 - 2780 * q^89 - 857 * q^91 - 1569 * q^93 - 265 * q^95 + 91 * q^97 + 1015 * 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
$3$	2.67963	0.515695	0.257847	−	0.966186i	$-0.416987\pi$
$3$	2.67963	0.515695	0.257847	+	0.966186i	$0.416987\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−24.9473	−1.34703	−0.673514	−	0.739174i	$-0.735216\pi$
$7$	−24.9473	−1.34703	−0.673514	+	0.739174i	$0.735216\pi$
$8$	0	0
$9$	−19.8196	−0.734059
$10$	0	0
$11$	26.0279	0.713428	0.356714	−	0.934214i	$-0.383897\pi$
$11$	26.0279	0.713428	0.356714	+	0.934214i	$0.383897\pi$
$12$	0	0
$13$	54.5603	1.16402	0.582012	−	0.813180i	$-0.302266\pi$
$13$	54.5603	1.16402	0.582012	+	0.813180i	$0.302266\pi$
$14$	0	0
$15$	13.3981	0.230626
$16$	0	0
$17$	−70.5481	−1.00650	−0.503248	−	0.864142i	$-0.667862\pi$
$17$	−70.5481	−1.00650	−0.503248	+	0.864142i	$0.667862\pi$
$18$	0	0
$19$	86.5444	1.04498	0.522491	−	0.852645i	$-0.325003\pi$
$19$	86.5444	1.04498	0.522491	+	0.852645i	$0.325003\pi$
$20$	0	0
$21$	−66.8495	−0.694655
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−125.459	−0.894245
$28$	0	0
$29$	−150.814	−0.965704	−0.482852	−	0.875702i	$-0.660399\pi$
$29$	−150.814	−0.965704	−0.482852	+	0.875702i	$0.660399\pi$
$30$	0	0
$31$	197.127	1.14210	0.571050	−	0.820916i	$-0.306536\pi$
$31$	197.127	1.14210	0.571050	+	0.820916i	$0.306536\pi$
$32$	0	0
$33$	69.7451	0.367911
$34$	0	0
$35$	−124.737	−0.602410
$36$	0	0
$37$	159.729	0.709711	0.354856	−	0.934921i	$-0.384530\pi$
$37$	159.729	0.709711	0.354856	+	0.934921i	$0.384530\pi$
$38$	0	0
$39$	146.201	0.600281
$40$	0	0
$41$	−139.725	−0.532227	−0.266114	−	0.963942i	$-0.585740\pi$
$41$	−139.725	−0.532227	−0.266114	+	0.963942i	$0.585740\pi$
$42$	0	0
$43$	191.106	0.677755	0.338877	−	0.940831i	$-0.389953\pi$
$43$	191.106	0.677755	0.338877	+	0.940831i	$0.389953\pi$
$44$	0	0
$45$	−99.0980	−0.328281
$46$	0	0
$47$	−411.237	−1.27628	−0.638138	−	0.769922i	$-0.720295\pi$
$47$	−411.237	−1.27628	−0.638138	+	0.769922i	$0.720295\pi$
$48$	0	0
$49$	279.369	0.814486
$50$	0	0
$51$	−189.043	−0.519044
$52$	0	0
$53$	100.118	0.259476	0.129738	−	0.991548i	$-0.458586\pi$
$53$	100.118	0.259476	0.129738	+	0.991548i	$0.458586\pi$
$54$	0	0
$55$	130.140	0.319055
$56$	0	0
$57$	231.907	0.538891
$58$	0	0
$59$	−295.192	−0.651367	−0.325684	−	0.945479i	$-0.605594\pi$
$59$	−295.192	−0.651367	−0.325684	+	0.945479i	$0.605594\pi$
$60$	0	0
$61$	−789.233	−1.65657	−0.828286	−	0.560306i	$-0.810684\pi$
$61$	−789.233	−1.65657	−0.828286	+	0.560306i	$0.810684\pi$
$62$	0	0
$63$	494.446	0.988799
$64$	0	0
$65$	272.802	0.520567
$66$	0	0
$67$	670.195	1.22205	0.611025	−	0.791611i	$-0.290758\pi$
$67$	670.195	1.22205	0.611025	+	0.791611i	$0.290758\pi$
$68$	0	0
$69$	−61.6314	−0.107530
$70$	0	0
$71$	−598.061	−0.999673	−0.499837	−	0.866120i	$-0.666607\pi$
$71$	−598.061	−0.999673	−0.499837	+	0.866120i	$0.666607\pi$
$72$	0	0
$73$	535.992	0.859358	0.429679	−	0.902982i	$-0.358627\pi$
$73$	535.992	0.859358	0.429679	+	0.902982i	$0.358627\pi$
$74$	0	0
$75$	66.9907	0.103139
$76$	0	0
$77$	−649.326	−0.961008
$78$	0	0
$79$	1133.10	1.61371	0.806857	−	0.590747i	$-0.201167\pi$
$79$	1133.10	1.61371	0.806857	+	0.590747i	$0.201167\pi$
$80$	0	0
$81$	198.945	0.272902
$82$	0	0
$83$	−806.388	−1.06642	−0.533209	−	0.845984i	$-0.679014\pi$
$83$	−806.388	−1.06642	−0.533209	+	0.845984i	$0.679014\pi$
$84$	0	0
$85$	−352.740	−0.450118
$86$	0	0
$87$	−404.125	−0.498009
$88$	0	0
$89$	−874.184	−1.04116	−0.520580	−	0.853813i	$-0.674284\pi$
$89$	−874.184	−1.04116	−0.520580	+	0.853813i	$0.674284\pi$
$90$	0	0
$91$	−1361.13	−1.56797
$92$	0	0
$93$	528.227	0.588974
$94$	0	0
$95$	432.722	0.467330
$96$	0	0
$97$	−892.297	−0.934010	−0.467005	−	0.884255i	$-0.654667\pi$
$97$	−892.297	−0.934010	−0.467005	+	0.884255i	$0.654667\pi$
$98$	0	0
$99$	−515.863	−0.523698
$100$	0	0
$101$	−510.327	−0.502766	−0.251383	−	0.967888i	$-0.580885\pi$
$101$	−510.327	−0.502766	−0.251383	+	0.967888i	$0.580885\pi$
$102$	0	0
$103$	396.912	0.379698	0.189849	−	0.981813i	$-0.439200\pi$
$103$	396.912	0.379698	0.189849	+	0.981813i	$0.439200\pi$
$104$	0	0
$105$	−334.248	−0.310659
$106$	0	0
$107$	−1723.26	−1.55695	−0.778477	−	0.627673i	$-0.784007\pi$
$107$	−1723.26	−1.55695	−0.778477	+	0.627673i	$0.784007\pi$
$108$	0	0
$109$	−2105.35	−1.85006	−0.925028	−	0.379900i	$-0.875958\pi$
$109$	−2105.35	−1.85006	−0.925028	+	0.379900i	$0.875958\pi$
$110$	0	0
$111$	428.015	0.365994
$112$	0	0
$113$	−997.420	−0.830349	−0.415174	−	0.909742i	$-0.636279\pi$
$113$	−997.420	−0.830349	−0.415174	+	0.909742i	$0.636279\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−1081.36	−0.854462
$118$	0	0
$119$	1759.99	1.35578
$120$	0	0
$121$	−653.548	−0.491020
$122$	0	0
$123$	−374.410	−0.274467
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	277.738	0.194057	0.0970287	−	0.995282i	$-0.469066\pi$
$127$	277.738	0.194057	0.0970287	+	0.995282i	$0.469066\pi$
$128$	0	0
$129$	512.094	0.349514
$130$	0	0
$131$	−1069.43	−0.713253	−0.356627	−	0.934247i	$-0.616073\pi$
$131$	−1069.43	−0.713253	−0.356627	+	0.934247i	$0.616073\pi$
$132$	0	0
$133$	−2159.05	−1.40762
$134$	0	0
$135$	−627.295	−0.399918
$136$	0	0
$137$	−1690.30	−1.05410	−0.527050	−	0.849834i	$-0.676702\pi$
$137$	−1690.30	−1.05410	−0.527050	+	0.849834i	$0.676702\pi$
$138$	0	0
$139$	2272.80	1.38688	0.693441	−	0.720513i	$-0.256094\pi$
$139$	2272.80	1.38688	0.693441	+	0.720513i	$0.256094\pi$
$140$	0	0
$141$	−1101.96	−0.658169
$142$	0	0
$143$	1420.09	0.830447
$144$	0	0
$145$	−754.069	−0.431876
$146$	0	0
$147$	748.604	0.420026
$148$	0	0
$149$	−3616.12	−1.98822	−0.994108	−	0.108398i	$-0.965428\pi$
$149$	−3616.12	−1.98822	−0.994108	+	0.108398i	$0.965428\pi$
$150$	0	0
$151$	738.867	0.398199	0.199100	−	0.979979i	$-0.436198\pi$
$151$	738.867	0.398199	0.199100	+	0.979979i	$0.436198\pi$
$152$	0	0
$153$	1398.23	0.738827
$154$	0	0
$155$	985.635	0.510762
$156$	0	0
$157$	−3096.60	−1.57411	−0.787055	−	0.616882i	$-0.788395\pi$
$157$	−3096.60	−1.57411	−0.787055	+	0.616882i	$0.788395\pi$
$158$	0	0
$159$	268.278	0.133810
$160$	0	0
$161$	573.788	0.280875
$162$	0	0
$163$	1551.84	0.745702	0.372851	−	0.927891i	$-0.378380\pi$
$163$	1551.84	0.745702	0.372851	+	0.927891i	$0.378380\pi$
$164$	0	0
$165$	348.725	0.164535
$166$	0	0
$167$	2074.00	0.961024	0.480512	−	0.876988i	$-0.340451\pi$
$167$	2074.00	0.961024	0.480512	+	0.876988i	$0.340451\pi$
$168$	0	0
$169$	779.827	0.354951
$170$	0	0
$171$	−1715.28	−0.767078
$172$	0	0
$173$	1276.84	0.561133	0.280567	−	0.959835i	$-0.409478\pi$
$173$	1276.84	0.561133	0.280567	+	0.959835i	$0.409478\pi$
$174$	0	0
$175$	−623.683	−0.269406
$176$	0	0
$177$	−791.004	−0.335907
$178$	0	0
$179$	−2496.95	−1.04263	−0.521315	−	0.853364i	$-0.674558\pi$
$179$	−2496.95	−1.04263	−0.521315	+	0.853364i	$0.674558\pi$
$180$	0	0
$181$	711.484	0.292178	0.146089	−	0.989271i	$-0.453331\pi$
$181$	711.484	0.292178	0.146089	+	0.989271i	$0.453331\pi$
$182$	0	0
$183$	−2114.85	−0.854285
$184$	0	0
$185$	798.646	0.317393
$186$	0	0
$187$	−1836.22	−0.718062
$188$	0	0
$189$	3129.87	1.20457
$190$	0	0
$191$	−4660.08	−1.76540	−0.882701	−	0.469936i	$-0.844277\pi$
$191$	−4660.08	−1.76540	−0.882701	+	0.469936i	$0.844277\pi$
$192$	0	0
$193$	−1690.53	−0.630502	−0.315251	−	0.949008i	$-0.602089\pi$
$193$	−1690.53	−0.630502	−0.315251	+	0.949008i	$0.602089\pi$
$194$	0	0
$195$	731.007	0.268454
$196$	0	0
$197$	5313.74	1.92177	0.960885	−	0.276949i	$-0.0893234\pi$
$197$	5313.74	1.92177	0.960885	+	0.276949i	$0.0893234\pi$
$198$	0	0
$199$	3766.40	1.34167	0.670837	−	0.741605i	$-0.265935\pi$
$199$	3766.40	1.34167	0.670837	+	0.741605i	$0.265935\pi$
$200$	0	0
$201$	1795.87	0.630204
$202$	0	0
$203$	3762.40	1.30083
$204$	0	0
$205$	−698.623	−0.238019
$206$	0	0
$207$	455.851	0.153062
$208$	0	0
$209$	2252.57	0.745519
$210$	0	0
$211$	−3804.11	−1.24116	−0.620582	−	0.784142i	$-0.713103\pi$
$211$	−3804.11	−1.24116	−0.620582	+	0.784142i	$0.713103\pi$
$212$	0	0
$213$	−1602.58	−0.515526
$214$	0	0
$215$	955.532	0.303101
$216$	0	0
$217$	−4917.79	−1.53844
$218$	0	0
$219$	1436.26	0.443166
$220$	0	0
$221$	−3849.12	−1.17158
$222$	0	0
$223$	1849.77	0.555469	0.277735	−	0.960658i	$-0.410416\pi$
$223$	1849.77	0.555469	0.277735	+	0.960658i	$0.410416\pi$
$224$	0	0
$225$	−495.490	−0.146812
$226$	0	0
$227$	−608.708	−0.177980	−0.0889899	−	0.996033i	$-0.528364\pi$
$227$	−608.708	−0.177980	−0.0889899	+	0.996033i	$0.528364\pi$
$228$	0	0
$229$	6033.29	1.74101	0.870505	−	0.492160i	$-0.163792\pi$
$229$	6033.29	1.74101	0.870505	+	0.492160i	$0.163792\pi$
$230$	0	0
$231$	−1739.95	−0.495587
$232$	0	0
$233$	−3249.22	−0.913577	−0.456789	−	0.889575i	$-0.651000\pi$
$233$	−3249.22	−0.913577	−0.456789	+	0.889575i	$0.651000\pi$
$234$	0	0
$235$	−2056.18	−0.570768
$236$	0	0
$237$	3036.28	0.832183
$238$	0	0
$239$	−2164.91	−0.585926	−0.292963	−	0.956124i	$-0.594641\pi$
$239$	−2164.91	−0.585926	−0.292963	+	0.956124i	$0.594641\pi$
$240$	0	0
$241$	−949.929	−0.253902	−0.126951	−	0.991909i	$-0.540519\pi$
$241$	−949.929	−0.253902	−0.126951	+	0.991909i	$0.540519\pi$
$242$	0	0
$243$	3920.49	1.03498
$244$	0	0
$245$	1396.84	0.364249
$246$	0	0
$247$	4721.89	1.21638
$248$	0	0
$249$	−2160.82	−0.549946
$250$	0	0
$251$	7026.74	1.76703	0.883514	−	0.468405i	$-0.155171\pi$
$251$	7026.74	1.76703	0.883514	+	0.468405i	$0.155171\pi$
$252$	0	0
$253$	−598.642	−0.148760
$254$	0	0
$255$	−945.213	−0.232124
$256$	0	0
$257$	818.213	0.198594	0.0992971	−	0.995058i	$-0.468341\pi$
$257$	818.213	0.198594	0.0992971	+	0.995058i	$0.468341\pi$
$258$	0	0
$259$	−3984.81	−0.956001
$260$	0	0
$261$	2989.07	0.708884
$262$	0	0
$263$	2111.34	0.495023	0.247511	−	0.968885i	$-0.420387\pi$
$263$	2111.34	0.495023	0.247511	+	0.968885i	$0.420387\pi$
$264$	0	0
$265$	500.589	0.116041
$266$	0	0
$267$	−2342.49	−0.536921
$268$	0	0
$269$	−145.404	−0.0329570	−0.0164785	−	0.999864i	$-0.505246\pi$
$269$	−145.404	−0.0329570	−0.0164785	+	0.999864i	$0.505246\pi$
$270$	0	0
$271$	−5228.54	−1.17200	−0.585998	−	0.810312i	$-0.699298\pi$
$271$	−5228.54	−1.17200	−0.585998	+	0.810312i	$0.699298\pi$
$272$	0	0
$273$	−3647.33	−0.808595
$274$	0	0
$275$	650.698	0.142686
$276$	0	0
$277$	−323.595	−0.0701912	−0.0350956	−	0.999384i	$-0.511174\pi$
$277$	−323.595	−0.0701912	−0.0350956	+	0.999384i	$0.511174\pi$
$278$	0	0
$279$	−3906.98	−0.838368
$280$	0	0
$281$	−1232.74	−0.261705	−0.130852	−	0.991402i	$-0.541771\pi$
$281$	−1232.74	−0.261705	−0.130852	+	0.991402i	$0.541771\pi$
$282$	0	0
$283$	−4705.42	−0.988367	−0.494184	−	0.869358i	$-0.664533\pi$
$283$	−4705.42	−0.988367	−0.494184	+	0.869358i	$0.664533\pi$
$284$	0	0
$285$	1159.53	0.240999
$286$	0	0
$287$	3485.75	0.716925
$288$	0	0
$289$	64.0294	0.0130326
$290$	0	0
$291$	−2391.02	−0.481664
$292$	0	0
$293$	579.655	0.115576	0.0577881	−	0.998329i	$-0.481595\pi$
$293$	579.655	0.115576	0.0577881	+	0.998329i	$0.481595\pi$
$294$	0	0
$295$	−1475.96	−0.291300
$296$	0	0
$297$	−3265.44	−0.637979
$298$	0	0
$299$	−1254.89	−0.242716
$300$	0	0
$301$	−4767.59	−0.912955
$302$	0	0
$303$	−1367.49	−0.259274
$304$	0	0
$305$	−3946.16	−0.740841
$306$	0	0
$307$	−2967.43	−0.551661	−0.275831	−	0.961206i	$-0.588953\pi$
$307$	−2967.43	−0.551661	−0.275831	+	0.961206i	$0.588953\pi$
$308$	0	0
$309$	1063.58	0.195808
$310$	0	0
$311$	2955.80	0.538933	0.269466	−	0.963010i	$-0.413153\pi$
$311$	2955.80	0.538933	0.269466	+	0.963010i	$0.413153\pi$
$312$	0	0
$313$	−9242.13	−1.66900	−0.834498	−	0.551011i	$-0.814242\pi$
$313$	−9242.13	−1.66900	−0.834498	+	0.551011i	$0.814242\pi$
$314$	0	0
$315$	2472.23	0.442204
$316$	0	0
$317$	−5480.40	−0.971010	−0.485505	−	0.874234i	$-0.661364\pi$
$317$	−5480.40	−0.971010	−0.485505	+	0.874234i	$0.661364\pi$
$318$	0	0
$319$	−3925.37	−0.688961
$320$	0	0
$321$	−4617.70	−0.802913
$322$	0	0
$323$	−6105.54	−1.05177
$324$	0	0
$325$	1364.01	0.232805
$326$	0	0
$327$	−5641.56	−0.954063
$328$	0	0
$329$	10259.2	1.71918
$330$	0	0
$331$	−8227.72	−1.36627	−0.683137	−	0.730290i	$-0.739385\pi$
$331$	−8227.72	−1.36627	−0.683137	+	0.730290i	$0.739385\pi$
$332$	0	0
$333$	−3165.77	−0.520970
$334$	0	0
$335$	3350.97	0.546517
$336$	0	0
$337$	7660.74	1.23830	0.619150	−	0.785273i	$-0.287477\pi$
$337$	7660.74	1.23830	0.619150	+	0.785273i	$0.287477\pi$
$338$	0	0
$339$	−2672.71	−0.428206
$340$	0	0
$341$	5130.80	0.814805
$342$	0	0
$343$	1587.43	0.249893
$344$	0	0
$345$	−308.157	−0.0480888
$346$	0	0
$347$	−10180.0	−1.57490	−0.787451	−	0.616377i	$-0.788600\pi$
$347$	−10180.0	−1.57490	−0.787451	+	0.616377i	$0.788600\pi$
$348$	0	0
$349$	−8593.37	−1.31803	−0.659015	−	0.752130i	$-0.729027\pi$
$349$	−8593.37	−1.31803	−0.659015	+	0.752130i	$0.729027\pi$
$350$	0	0
$351$	−6845.09	−1.04092
$352$	0	0
$353$	6295.48	0.949220	0.474610	−	0.880196i	$-0.342589\pi$
$353$	6295.48	0.949220	0.474610	+	0.880196i	$0.342589\pi$
$354$	0	0
$355$	−2990.31	−0.447067
$356$	0	0
$357$	4716.10	0.699167
$358$	0	0
$359$	1558.89	0.229178	0.114589	−	0.993413i	$-0.463445\pi$
$359$	1558.89	0.229178	0.114589	+	0.993413i	$0.463445\pi$
$360$	0	0
$361$	630.934	0.0919862
$362$	0	0
$363$	−1751.27	−0.253217
$364$	0	0
$365$	2679.96	0.384316
$366$	0	0
$367$	−8813.67	−1.25360	−0.626798	−	0.779182i	$-0.715635\pi$
$367$	−8813.67	−1.25360	−0.626798	+	0.779182i	$0.715635\pi$
$368$	0	0
$369$	2769.29	0.390686
$370$	0	0
$371$	−2497.67	−0.349522
$372$	0	0
$373$	5762.38	0.799905	0.399953	−	0.916536i	$-0.369027\pi$
$373$	5762.38	0.799905	0.399953	+	0.916536i	$0.369027\pi$
$374$	0	0
$375$	334.953	0.0461251
$376$	0	0
$377$	−8228.45	−1.12410
$378$	0	0
$379$	−3788.94	−0.513522	−0.256761	−	0.966475i	$-0.582655\pi$
$379$	−3788.94	−0.513522	−0.256761	+	0.966475i	$0.582655\pi$
$380$	0	0
$381$	744.235	0.100074
$382$	0	0
$383$	1921.93	0.256413	0.128206	−	0.991748i	$-0.459078\pi$
$383$	1921.93	0.256413	0.128206	+	0.991748i	$0.459078\pi$
$384$	0	0
$385$	−3246.63	−0.429776
$386$	0	0
$387$	−3787.65	−0.497512
$388$	0	0
$389$	−11895.7	−1.55048	−0.775242	−	0.631665i	$-0.782372\pi$
$389$	−11895.7	−1.55048	−0.775242	+	0.631665i	$0.782372\pi$
$390$	0	0
$391$	1622.61	0.209869
$392$	0	0
$393$	−2865.66	−0.367821
$394$	0	0
$395$	5665.49	0.721675
$396$	0	0
$397$	−3741.31	−0.472975	−0.236487	−	0.971635i	$-0.575996\pi$
$397$	−3741.31	−0.472975	−0.236487	+	0.971635i	$0.575996\pi$
$398$	0	0
$399$	−5785.45	−0.725902
$400$	0	0
$401$	7909.19	0.984953	0.492476	−	0.870326i	$-0.336092\pi$
$401$	7909.19	0.984953	0.492476	+	0.870326i	$0.336092\pi$
$402$	0	0
$403$	10755.3	1.32943
$404$	0	0
$405$	994.727	0.122045
$406$	0	0
$407$	4157.42	0.506328
$408$	0	0
$409$	−4530.82	−0.547762	−0.273881	−	0.961764i	$-0.588307\pi$
$409$	−4530.82	−0.547762	−0.273881	+	0.961764i	$0.588307\pi$
$410$	0	0
$411$	−4529.36	−0.543594
$412$	0	0
$413$	7364.24	0.877410
$414$	0	0
$415$	−4031.94	−0.476916
$416$	0	0
$417$	6090.27	0.715208
$418$	0	0
$419$	−6661.59	−0.776707	−0.388353	−	0.921511i	$-0.626956\pi$
$419$	−6661.59	−0.776707	−0.388353	+	0.921511i	$0.626956\pi$
$420$	0	0
$421$	15544.5	1.79950	0.899752	−	0.436402i	$-0.143747\pi$
$421$	15544.5	1.79950	0.899752	+	0.436402i	$0.143747\pi$
$422$	0	0
$423$	8150.54	0.936863
$424$	0	0
$425$	−1763.70	−0.201299
$426$	0	0
$427$	19689.2	2.23145
$428$	0	0
$429$	3805.31	0.428257
$430$	0	0
$431$	7165.07	0.800764	0.400382	−	0.916348i	$-0.368877\pi$
$431$	7165.07	0.800764	0.400382	+	0.916348i	$0.368877\pi$
$432$	0	0
$433$	−5028.86	−0.558133	−0.279067	−	0.960272i	$-0.590025\pi$
$433$	−5028.86	−0.558133	−0.279067	+	0.960272i	$0.590025\pi$
$434$	0	0
$435$	−2020.62	−0.222716
$436$	0	0
$437$	−1990.52	−0.217894
$438$	0	0
$439$	−12725.5	−1.38350	−0.691749	−	0.722138i	$-0.743159\pi$
$439$	−12725.5	−1.38350	−0.691749	+	0.722138i	$0.743159\pi$
$440$	0	0
$441$	−5536.98	−0.597881
$442$	0	0
$443$	−7707.44	−0.826617	−0.413309	−	0.910591i	$-0.635627\pi$
$443$	−7707.44	−0.826617	−0.413309	+	0.910591i	$0.635627\pi$
$444$	0	0
$445$	−4370.92	−0.465621
$446$	0	0
$447$	−9689.85	−1.02531
$448$	0	0
$449$	−10077.8	−1.05924	−0.529620	−	0.848235i	$-0.677666\pi$
$449$	−10077.8	−1.05924	−0.529620	+	0.848235i	$0.677666\pi$
$450$	0	0
$451$	−3636.74	−0.379706
$452$	0	0
$453$	1979.89	0.205349
$454$	0	0
$455$	−6805.67	−0.701219
$456$	0	0
$457$	13342.8	1.36576	0.682878	−	0.730532i	$-0.260728\pi$
$457$	13342.8	1.36576	0.682878	+	0.730532i	$0.260728\pi$
$458$	0	0
$459$	8850.90	0.900053
$460$	0	0
$461$	1389.80	0.140411	0.0702054	−	0.997533i	$-0.477635\pi$
$461$	1389.80	0.140411	0.0702054	+	0.997533i	$0.477635\pi$
$462$	0	0
$463$	17735.7	1.78023	0.890116	−	0.455734i	$-0.150623\pi$
$463$	17735.7	1.78023	0.890116	+	0.455734i	$0.150623\pi$
$464$	0	0
$465$	2641.14	0.263397
$466$	0	0
$467$	1.44728	0.000143410 0	7.17048e−5	−	1.00000i	$-0.499977\pi$
$467$	1.44728	0.000143410 0	7.17048e−5		1.00000i	$0.499977\pi$
$468$	0	0
$469$	−16719.6	−1.64614
$470$	0	0
$471$	−8297.73	−0.811760
$472$	0	0
$473$	4974.10	0.483529
$474$	0	0
$475$	2163.61	0.208996
$476$	0	0
$477$	−1984.29	−0.190471
$478$	0	0
$479$	9673.43	0.922735	0.461368	−	0.887209i	$-0.347359\pi$
$479$	9673.43	0.922735	0.461368	+	0.887209i	$0.347359\pi$
$480$	0	0
$481$	8714.87	0.826121
$482$	0	0
$483$	1537.54	0.144846
$484$	0	0
$485$	−4461.48	−0.417702
$486$	0	0
$487$	−14923.0	−1.38855	−0.694275	−	0.719710i	$-0.744275\pi$
$487$	−14923.0	−1.38855	−0.694275	+	0.719710i	$0.744275\pi$
$488$	0	0
$489$	4158.35	0.384555
$490$	0	0
$491$	16146.8	1.48410	0.742051	−	0.670344i	$-0.233853\pi$
$491$	16146.8	1.48410	0.742051	+	0.670344i	$0.233853\pi$
$492$	0	0
$493$	10639.6	0.971977
$494$	0	0
$495$	−2579.31	−0.234205
$496$	0	0
$497$	14920.0	1.34659
$498$	0	0
$499$	6986.87	0.626804	0.313402	−	0.949621i	$-0.398531\pi$
$499$	6986.87	0.626804	0.313402	+	0.949621i	$0.398531\pi$
$500$	0	0
$501$	5557.55	0.495595
$502$	0	0
$503$	−2260.08	−0.200342	−0.100171	−	0.994970i	$-0.531939\pi$
$503$	−2260.08	−0.200342	−0.100171	+	0.994970i	$0.531939\pi$
$504$	0	0
$505$	−2551.63	−0.224844
$506$	0	0
$507$	2089.65	0.183046
$508$	0	0
$509$	14921.7	1.29940	0.649698	−	0.760193i	$-0.274895\pi$
$509$	14921.7	1.29940	0.649698	+	0.760193i	$0.274895\pi$
$510$	0	0
$511$	−13371.6	−1.15758
$512$	0	0
$513$	−10857.8	−0.934469
$514$	0	0
$515$	1984.56	0.169806
$516$	0	0
$517$	−10703.6	−0.910532
$518$	0	0
$519$	3421.45	0.289373
$520$	0	0
$521$	−2469.64	−0.207671	−0.103836	−	0.994594i	$-0.533112\pi$
$521$	−2469.64	−0.207671	−0.103836	+	0.994594i	$0.533112\pi$
$522$	0	0
$523$	−900.726	−0.0753079	−0.0376539	−	0.999291i	$-0.511988\pi$
$523$	−900.726	−0.0753079	−0.0376539	+	0.999291i	$0.511988\pi$
$524$	0	0
$525$	−1671.24	−0.138931
$526$	0	0
$527$	−13906.9	−1.14952
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	5850.58	0.478142
$532$	0	0
$533$	−7623.42	−0.619525
$534$	0	0
$535$	−8616.31	−0.696291
$536$	0	0
$537$	−6690.90	−0.537679
$538$	0	0
$539$	7271.38	0.581077
$540$	0	0
$541$	9015.96	0.716500	0.358250	−	0.933626i	$-0.383374\pi$
$541$	9015.96	0.716500	0.358250	+	0.933626i	$0.383374\pi$
$542$	0	0
$543$	1906.51	0.150674
$544$	0	0
$545$	−10526.8	−0.827370
$546$	0	0
$547$	989.245	0.0773256	0.0386628	−	0.999252i	$-0.487690\pi$
$547$	989.245	0.0773256	0.0386628	+	0.999252i	$0.487690\pi$
$548$	0	0
$549$	15642.3	1.21602
$550$	0	0
$551$	−13052.1	−1.00914
$552$	0	0
$553$	−28267.7	−2.17372
$554$	0	0
$555$	2140.07	0.163678
$556$	0	0
$557$	20163.4	1.53385	0.766923	−	0.641739i	$-0.221787\pi$
$557$	20163.4	1.53385	0.766923	+	0.641739i	$0.221787\pi$
$558$	0	0
$559$	10426.8	0.788922
$560$	0	0
$561$	−4920.38	−0.370301
$562$	0	0
$563$	−3123.29	−0.233803	−0.116901	−	0.993144i	$-0.537296\pi$
$563$	−3123.29	−0.233803	−0.116901	+	0.993144i	$0.537296\pi$
$564$	0	0
$565$	−4987.10	−0.371343
$566$	0	0
$567$	−4963.16	−0.367607
$568$	0	0
$569$	7228.12	0.532546	0.266273	−	0.963898i	$-0.414208\pi$
$569$	7228.12	0.532546	0.266273	+	0.963898i	$0.414208\pi$
$570$	0	0
$571$	−17674.5	−1.29537	−0.647685	−	0.761908i	$-0.724263\pi$
$571$	−17674.5	−1.29537	−0.647685	+	0.761908i	$0.724263\pi$
$572$	0	0
$573$	−12487.3	−0.910408
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−12850.4	−0.927156	−0.463578	−	0.886056i	$-0.653435\pi$
$577$	−12850.4	−0.927156	−0.463578	+	0.886056i	$0.653435\pi$
$578$	0	0
$579$	−4529.98	−0.325146
$580$	0	0
$581$	20117.2	1.43649
$582$	0	0
$583$	2605.86	0.185118
$584$	0	0
$585$	−5406.82	−0.382127
$586$	0	0
$587$	2218.39	0.155984	0.0779922	−	0.996954i	$-0.475149\pi$
$587$	2218.39	0.155984	0.0779922	+	0.996954i	$0.475149\pi$
$588$	0	0
$589$	17060.2	1.19347
$590$	0	0
$591$	14238.9	0.991046
$592$	0	0
$593$	6899.92	0.477817	0.238909	−	0.971042i	$-0.423210\pi$
$593$	6899.92	0.477817	0.238909	+	0.971042i	$0.423210\pi$
$594$	0	0
$595$	8799.93	0.606322
$596$	0	0
$597$	10092.5	0.691894
$598$	0	0
$599$	5961.31	0.406632	0.203316	−	0.979113i	$-0.434828\pi$
$599$	5961.31	0.406632	0.203316	+	0.979113i	$0.434828\pi$
$600$	0	0
$601$	2564.19	0.174035	0.0870177	−	0.996207i	$-0.472266\pi$
$601$	2564.19	0.174035	0.0870177	+	0.996207i	$0.472266\pi$
$602$	0	0
$603$	−13283.0	−0.897057
$604$	0	0
$605$	−3267.74	−0.219591
$606$	0	0
$607$	4767.15	0.318769	0.159384	−	0.987217i	$-0.449049\pi$
$607$	4767.15	0.318769	0.159384	+	0.987217i	$0.449049\pi$
$608$	0	0
$609$	10081.8	0.670832
$610$	0	0
$611$	−22437.2	−1.48562
$612$	0	0
$613$	−5490.98	−0.361792	−0.180896	−	0.983502i	$-0.557900\pi$
$613$	−5490.98	−0.361792	−0.180896	+	0.983502i	$0.557900\pi$
$614$	0	0
$615$	−1872.05	−0.122745
$616$	0	0
$617$	21938.1	1.43144	0.715718	−	0.698389i	$-0.246099\pi$
$617$	21938.1	1.43144	0.715718	+	0.698389i	$0.246099\pi$
$618$	0	0
$619$	7647.56	0.496577	0.248289	−	0.968686i	$-0.420132\pi$
$619$	7647.56	0.496577	0.248289	+	0.968686i	$0.420132\pi$
$620$	0	0
$621$	2885.56	0.186463
$622$	0	0
$623$	21808.5	1.40247
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	6036.05	0.384460
$628$	0	0
$629$	−11268.6	−0.714321
$630$	0	0
$631$	19385.3	1.22301	0.611503	−	0.791242i	$-0.290565\pi$
$631$	19385.3	1.22301	0.611503	+	0.791242i	$0.290565\pi$
$632$	0	0
$633$	−10193.6	−0.640061
$634$	0	0
$635$	1388.69	0.0867851
$636$	0	0
$637$	15242.4	0.948081
$638$	0	0
$639$	11853.3	0.733819
$640$	0	0
$641$	3931.24	0.242238	0.121119	−	0.992638i	$-0.461352\pi$
$641$	3931.24	0.242238	0.121119	+	0.992638i	$0.461352\pi$
$642$	0	0
$643$	−10500.5	−0.644009	−0.322004	−	0.946738i	$-0.604357\pi$
$643$	−10500.5	−0.644009	−0.322004	+	0.946738i	$0.604357\pi$
$644$	0	0
$645$	2560.47	0.156308
$646$	0	0
$647$	−3869.43	−0.235120	−0.117560	−	0.993066i	$-0.537507\pi$
$647$	−3869.43	−0.235120	−0.117560	+	0.993066i	$0.537507\pi$
$648$	0	0
$649$	−7683.22	−0.464704
$650$	0	0
$651$	−13177.9	−0.793365
$652$	0	0
$653$	18718.0	1.12173	0.560866	−	0.827906i	$-0.310468\pi$
$653$	18718.0	1.12173	0.560866	+	0.827906i	$0.310468\pi$
$654$	0	0
$655$	−5347.13	−0.318976
$656$	0	0
$657$	−10623.1	−0.630819
$658$	0	0
$659$	−380.115	−0.0224692	−0.0112346	−	0.999937i	$-0.503576\pi$
$659$	−380.115	−0.0224692	−0.0112346	+	0.999937i	$0.503576\pi$
$660$	0	0
$661$	−7893.91	−0.464505	−0.232252	−	0.972656i	$-0.574610\pi$
$661$	−7893.91	−0.464505	−0.232252	+	0.972656i	$0.574610\pi$
$662$	0	0
$663$	−10314.2	−0.604180
$664$	0	0
$665$	−10795.3	−0.629507
$666$	0	0
$667$	3468.72	0.201363
$668$	0	0
$669$	4956.69	0.286453
$670$	0	0
$671$	−20542.1	−1.18184
$672$	0	0
$673$	−11023.7	−0.631400	−0.315700	−	0.948859i	$-0.602239\pi$
$673$	−11023.7	−0.631400	−0.315700	+	0.948859i	$0.602239\pi$
$674$	0	0
$675$	−3136.48	−0.178849
$676$	0	0
$677$	1650.93	0.0937227	0.0468614	−	0.998901i	$-0.485078\pi$
$677$	1650.93	0.0937227	0.0468614	+	0.998901i	$0.485078\pi$
$678$	0	0
$679$	22260.4	1.25814
$680$	0	0
$681$	−1631.11	−0.0917832
$682$	0	0
$683$	16873.7	0.945320	0.472660	−	0.881245i	$-0.343294\pi$
$683$	16873.7	0.945320	0.472660	+	0.881245i	$0.343294\pi$
$684$	0	0
$685$	−8451.48	−0.471408
$686$	0	0
$687$	16167.0	0.897829
$688$	0	0
$689$	5462.46	0.302036
$690$	0	0
$691$	−14788.2	−0.814139	−0.407070	−	0.913397i	$-0.633449\pi$
$691$	−14788.2	−0.814139	−0.407070	+	0.913397i	$0.633449\pi$
$692$	0	0
$693$	12869.4	0.705437
$694$	0	0
$695$	11364.0	0.620233
$696$	0	0
$697$	9857.30	0.535684
$698$	0	0
$699$	−8706.70	−0.471127
$700$	0	0
$701$	−33601.9	−1.81045	−0.905226	−	0.424930i	$-0.860299\pi$
$701$	−33601.9	−1.81045	−0.905226	+	0.424930i	$0.860299\pi$
$702$	0	0
$703$	13823.7	0.741635
$704$	0	0
$705$	−5509.80	−0.294342
$706$	0	0
$707$	12731.3	0.677241
$708$	0	0
$709$	9746.80	0.516289	0.258144	−	0.966106i	$-0.416889\pi$
$709$	9746.80	0.516289	0.258144	+	0.966106i	$0.416889\pi$
$710$	0	0
$711$	−22457.5	−1.18456
$712$	0	0
$713$	−4533.92	−0.238144
$714$	0	0
$715$	7100.45	0.371387
$716$	0	0
$717$	−5801.15	−0.302159
$718$	0	0
$719$	14005.1	0.726427	0.363214	−	0.931706i	$-0.381680\pi$
$719$	14005.1	0.726427	0.363214	+	0.931706i	$0.381680\pi$
$720$	0	0
$721$	−9901.88	−0.511464
$722$	0	0
$723$	−2545.46	−0.130936
$724$	0	0
$725$	−3770.35	−0.193141
$726$	0	0
$727$	−7291.15	−0.371958	−0.185979	−	0.982554i	$-0.559546\pi$
$727$	−7291.15	−0.371958	−0.185979	+	0.982554i	$0.559546\pi$
$728$	0	0
$729$	5133.94	0.260831
$730$	0	0
$731$	−13482.2	−0.682157
$732$	0	0
$733$	−14901.4	−0.750881	−0.375440	−	0.926847i	$-0.622509\pi$
$733$	−14901.4	−0.750881	−0.375440	+	0.926847i	$0.622509\pi$
$734$	0	0
$735$	3743.02	0.187841
$736$	0	0
$737$	17443.8	0.871844
$738$	0	0
$739$	−2906.81	−0.144694	−0.0723469	−	0.997380i	$-0.523049\pi$
$739$	−2906.81	−0.144694	−0.0723469	+	0.997380i	$0.523049\pi$
$740$	0	0
$741$	12652.9	0.627282
$742$	0	0
$743$	−13205.3	−0.652027	−0.326014	−	0.945365i	$-0.605706\pi$
$743$	−13205.3	−0.652027	−0.326014	+	0.945365i	$0.605706\pi$
$744$	0	0
$745$	−18080.6	−0.889157
$746$	0	0
$747$	15982.3	0.782813
$748$	0	0
$749$	42990.8	2.09726
$750$	0	0
$751$	38262.5	1.85914	0.929572	−	0.368640i	$-0.120177\pi$
$751$	38262.5	1.85914	0.929572	+	0.368640i	$0.120177\pi$
$752$	0	0
$753$	18829.1	0.911247
$754$	0	0
$755$	3694.33	0.178080
$756$	0	0
$757$	35789.8	1.71836	0.859182	−	0.511670i	$-0.170973\pi$
$757$	35789.8	1.71836	0.859182	+	0.511670i	$0.170973\pi$
$758$	0	0
$759$	−1604.14	−0.0767147
$760$	0	0
$761$	−36230.3	−1.72582	−0.862910	−	0.505358i	$-0.831360\pi$
$761$	−36230.3	−1.72582	−0.862910	+	0.505358i	$0.831360\pi$
$762$	0	0
$763$	52522.9	2.49208
$764$	0	0
$765$	6991.17	0.330413
$766$	0	0
$767$	−16105.7	−0.758207
$768$	0	0
$769$	27258.0	1.27822	0.639109	−	0.769116i	$-0.279303\pi$
$769$	27258.0	1.27822	0.639109	+	0.769116i	$0.279303\pi$
$770$	0	0
$771$	2192.51	0.102414
$772$	0	0
$773$	−9879.71	−0.459700	−0.229850	−	0.973226i	$-0.573824\pi$
$773$	−9879.71	−0.459700	−0.229850	+	0.973226i	$0.573824\pi$
$774$	0	0
$775$	4928.18	0.228420
$776$	0	0
$777$	−10677.8	−0.493005
$778$	0	0
$779$	−12092.4	−0.556168
$780$	0	0
$781$	−15566.3	−0.713195
$782$	0	0
$783$	18921.0	0.863576
$784$	0	0
$785$	−15483.0	−0.703964
$786$	0	0
$787$	28632.0	1.29685	0.648426	−	0.761278i	$-0.275428\pi$
$787$	28632.0	1.29685	0.648426	+	0.761278i	$0.275428\pi$
$788$	0	0
$789$	5657.61	0.255281
$790$	0	0
$791$	24883.0	1.11850
$792$	0	0
$793$	−43060.8	−1.92829
$794$	0	0
$795$	1341.39	0.0598418
$796$	0	0
$797$	−25254.5	−1.12241	−0.561205	−	0.827677i	$-0.689662\pi$
$797$	−25254.5	−1.12241	−0.561205	+	0.827677i	$0.689662\pi$
$798$	0	0
$799$	29011.9	1.28457
$800$	0	0
$801$	17326.0	0.764274
$802$	0	0
$803$	13950.7	0.613090
$804$	0	0
$805$	2868.94	0.125611
$806$	0	0
$807$	−389.629	−0.0169958
$808$	0	0
$809$	−36214.7	−1.57385	−0.786923	−	0.617051i	$-0.788327\pi$
$809$	−36214.7	−1.57385	−0.786923	+	0.617051i	$0.788327\pi$
$810$	0	0
$811$	−4251.64	−0.184088	−0.0920439	−	0.995755i	$-0.529340\pi$
$811$	−4251.64	−0.184088	−0.0920439	+	0.995755i	$0.529340\pi$
$812$	0	0
$813$	−14010.5	−0.604392
$814$	0	0
$815$	7759.20	0.333488
$816$	0	0
$817$	16539.2	0.708241
$818$	0	0
$819$	26977.1	1.15098
$820$	0	0
$821$	17861.6	0.759286	0.379643	−	0.925133i	$-0.376047\pi$
$821$	17861.6	0.759286	0.379643	+	0.925133i	$0.376047\pi$
$822$	0	0
$823$	−22164.7	−0.938777	−0.469388	−	0.882992i	$-0.655526\pi$
$823$	−22164.7	−0.938777	−0.469388	+	0.882992i	$0.655526\pi$
$824$	0	0
$825$	1743.63	0.0735822
$826$	0	0
$827$	−18380.1	−0.772841	−0.386420	−	0.922323i	$-0.626289\pi$
$827$	−18380.1	−0.772841	−0.386420	+	0.922323i	$0.626289\pi$
$828$	0	0
$829$	−19659.6	−0.823648	−0.411824	−	0.911263i	$-0.635108\pi$
$829$	−19659.6	−0.823648	−0.411824	+	0.911263i	$0.635108\pi$
$830$	0	0
$831$	−867.115	−0.0361972
$832$	0	0
$833$	−19708.9	−0.819776
$834$	0	0
$835$	10370.0	0.429783
$836$	0	0
$837$	−24731.4	−1.02132
$838$	0	0
$839$	−39673.0	−1.63249	−0.816247	−	0.577703i	$-0.803949\pi$
$839$	−39673.0	−1.63249	−0.816247	+	0.577703i	$0.803949\pi$
$840$	0	0
$841$	−1644.18	−0.0674149
$842$	0	0
$843$	−3303.28	−0.134960
$844$	0	0
$845$	3899.14	0.158739
$846$	0	0
$847$	16304.3	0.661419
$848$	0	0
$849$	−12608.8	−0.509696
$850$	0	0
$851$	−3673.77	−0.147985
$852$	0	0
$853$	28310.3	1.13637	0.568187	−	0.822899i	$-0.307645\pi$
$853$	28310.3	1.13637	0.568187	+	0.822899i	$0.307645\pi$
$854$	0	0
$855$	−8576.38	−0.343048
$856$	0	0
$857$	−29612.3	−1.18032	−0.590161	−	0.807286i	$-0.700936\pi$
$857$	−29612.3	−1.18032	−0.590161	+	0.807286i	$0.700936\pi$
$858$	0	0
$859$	20727.6	0.823300	0.411650	−	0.911342i	$-0.364953\pi$
$859$	20727.6	0.823300	0.411650	+	0.911342i	$0.364953\pi$
$860$	0	0
$861$	9340.52	0.369715
$862$	0	0
$863$	26972.6	1.06392	0.531958	−	0.846771i	$-0.321457\pi$
$863$	26972.6	1.06392	0.531958	+	0.846771i	$0.321457\pi$
$864$	0	0
$865$	6384.18	0.250946
$866$	0	0
$867$	171.575	0.00672087
$868$	0	0
$869$	29492.1	1.15127
$870$	0	0
$871$	36566.0	1.42249
$872$	0	0
$873$	17685.0	0.685619
$874$	0	0
$875$	−3118.41	−0.120482
$876$	0	0
$877$	28224.7	1.08675	0.543376	−	0.839489i	$-0.317146\pi$
$877$	28224.7	1.08675	0.543376	+	0.839489i	$0.317146\pi$
$878$	0	0
$879$	1553.26	0.0596020
$880$	0	0
$881$	−27130.6	−1.03752	−0.518758	−	0.854921i	$-0.673606\pi$
$881$	−27130.6	−1.03752	−0.518758	+	0.854921i	$0.673606\pi$
$882$	0	0
$883$	−7425.32	−0.282992	−0.141496	−	0.989939i	$-0.545191\pi$
$883$	−7425.32	−0.282992	−0.141496	+	0.989939i	$0.545191\pi$
$884$	0	0
$885$	−3955.02	−0.150222
$886$	0	0
$887$	15439.4	0.584448	0.292224	−	0.956350i	$-0.405605\pi$
$887$	15439.4	0.584448	0.292224	+	0.956350i	$0.405605\pi$
$888$	0	0
$889$	−6928.83	−0.261401
$890$	0	0
$891$	5178.13	0.194696
$892$	0	0
$893$	−35590.2	−1.33369
$894$	0	0
$895$	−12484.7	−0.466278
$896$	0	0
$897$	−3362.63	−0.125167
$898$	0	0
$899$	−29729.5	−1.10293
$900$	0	0
$901$	−7063.12	−0.261161
$902$	0	0
$903$	−12775.4	−0.470806
$904$	0	0
$905$	3557.42	0.130666
$906$	0	0
$907$	49471.3	1.81110	0.905550	−	0.424240i	$-0.139459\pi$
$907$	49471.3	1.81110	0.905550	+	0.424240i	$0.139459\pi$
$908$	0	0
$909$	10114.5	0.369060
$910$	0	0
$911$	18748.9	0.681863	0.340931	−	0.940088i	$-0.389258\pi$
$911$	18748.9	0.681863	0.340931	+	0.940088i	$0.389258\pi$
$912$	0	0
$913$	−20988.6	−0.760812
$914$	0	0
$915$	−10574.2	−0.382048
$916$	0	0
$917$	26679.3	0.960772
$918$	0	0
$919$	28370.7	1.01835	0.509174	−	0.860664i	$-0.329951\pi$
$919$	28370.7	1.01835	0.509174	+	0.860664i	$0.329951\pi$
$920$	0	0
$921$	−7951.60	−0.284489
$922$	0	0
$923$	−32630.4	−1.16364
$924$	0	0
$925$	3993.23	0.141942
$926$	0	0
$927$	−7866.63	−0.278721
$928$	0	0
$929$	10605.1	0.374534	0.187267	−	0.982309i	$-0.440037\pi$
$929$	10605.1	0.374534	0.187267	+	0.982309i	$0.440037\pi$
$930$	0	0
$931$	24177.8	0.851123
$932$	0	0
$933$	7920.45	0.277925
$934$	0	0
$935$	−9181.09	−0.321127
$936$	0	0
$937$	48074.3	1.67611	0.838057	−	0.545583i	$-0.183692\pi$
$937$	48074.3	1.67611	0.838057	+	0.545583i	$0.183692\pi$
$938$	0	0
$939$	−24765.5	−0.860692
$940$	0	0
$941$	30780.7	1.06634	0.533168	−	0.846009i	$-0.321001\pi$
$941$	30780.7	1.06634	0.533168	+	0.846009i	$0.321001\pi$
$942$	0	0
$943$	3213.67	0.110977
$944$	0	0
$945$	15649.3	0.538702
$946$	0	0
$947$	−15490.7	−0.531551	−0.265776	−	0.964035i	$-0.585628\pi$
$947$	−15490.7	−0.531551	−0.265776	+	0.964035i	$0.585628\pi$
$948$	0	0
$949$	29243.9	1.00031
$950$	0	0
$951$	−14685.4	−0.500744
$952$	0	0
$953$	41111.4	1.39741	0.698703	−	0.715411i	$-0.253761\pi$
$953$	41111.4	1.39741	0.698703	+	0.715411i	$0.253761\pi$
$954$	0	0
$955$	−23300.4	−0.789511
$956$	0	0
$957$	−10518.5	−0.355293
$958$	0	0
$959$	42168.4	1.41990
$960$	0	0
$961$	9068.09	0.304390
$962$	0	0
$963$	34154.4	1.14290
$964$	0	0
$965$	−8452.64	−0.281969
$966$	0	0
$967$	−4520.69	−0.150337	−0.0751683	−	0.997171i	$-0.523949\pi$
$967$	−4520.69	−0.150337	−0.0751683	+	0.997171i	$0.523949\pi$
$968$	0	0
$969$	−16360.6	−0.542391
$970$	0	0
$971$	−35766.9	−1.18210	−0.591048	−	0.806636i	$-0.701286\pi$
$971$	−35766.9	−1.18210	−0.591048	+	0.806636i	$0.701286\pi$
$972$	0	0
$973$	−56700.3	−1.86817
$974$	0	0
$975$	3655.03	0.120056
$976$	0	0
$977$	14528.9	0.475764	0.237882	−	0.971294i	$-0.423547\pi$
$977$	14528.9	0.475764	0.237882	+	0.971294i	$0.423547\pi$
$978$	0	0
$979$	−22753.2	−0.742793
$980$	0	0
$981$	41727.2	1.35805
$982$	0	0
$983$	46543.7	1.51019	0.755094	−	0.655617i	$-0.227591\pi$
$983$	46543.7	1.51019	0.755094	+	0.655617i	$0.227591\pi$
$984$	0	0
$985$	26568.7	0.859441
$986$	0	0
$987$	27491.0	0.886573
$988$	0	0
$989$	−4395.45	−0.141322
$990$	0	0
$991$	−44749.4	−1.43442	−0.717211	−	0.696856i	$-0.754581\pi$
$991$	−44749.4	−1.43442	−0.717211	+	0.696856i	$0.754581\pi$
$992$	0	0
$993$	−22047.2	−0.704580
$994$	0	0
$995$	18832.0	0.600015
$996$	0	0
$997$	55681.2	1.76875	0.884373	−	0.466780i	$-0.154586\pi$
$997$	55681.2	1.76875	0.884373	+	0.466780i	$0.154586\pi$
$998$	0	0
$999$	−20039.5	−0.634656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.u.1.4		6
4.3	odd	2		920.4.a.a.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.a.1.3	✓	6	4.3	odd	2
1840.4.a.u.1.4		6	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)

\(f(q)\)	\(=\)	\(q+2.67963 q^{3} +5.00000 q^{5} -24.9473 q^{7} -19.8196 q^{9} +O(q^{10})\)	\(q+2.67963 q^{3} +5.00000 q^{5} -24.9473 q^{7} -19.8196 q^{9} +26.0279 q^{11} +54.5603 q^{13} +13.3981 q^{15} -70.5481 q^{17} +86.5444 q^{19} -66.8495 q^{21} -23.0000 q^{23} +25.0000 q^{25} -125.459 q^{27} -150.814 q^{29} +197.127 q^{31} +69.7451 q^{33} -124.737 q^{35} +159.729 q^{37} +146.201 q^{39} -139.725 q^{41} +191.106 q^{43} -99.0980 q^{45} -411.237 q^{47} +279.369 q^{49} -189.043 q^{51} +100.118 q^{53} +130.140 q^{55} +231.907 q^{57} -295.192 q^{59} -789.233 q^{61} +494.446 q^{63} +272.802 q^{65} +670.195 q^{67} -61.6314 q^{69} -598.061 q^{71} +535.992 q^{73} +66.9907 q^{75} -649.326 q^{77} +1133.10 q^{79} +198.945 q^{81} -806.388 q^{83} -352.740 q^{85} -404.125 q^{87} -874.184 q^{89} -1361.13 q^{91} +528.227 q^{93} +432.722 q^{95} -892.297 q^{97} -515.863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9	\( 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9} + 23 q^{11} - 42 q^{13} + 20 q^{15} - 124 q^{17} - 53 q^{19} - 114 q^{21} - 138 q^{23} + 150 q^{25} + 103 q^{27} - 320 q^{29} + 229 q^{31} - 583 q^{33} + 70 q^{35} - 377 q^{37} - 37 q^{39} - 683 q^{41} + 168 q^{43} + 20 q^{45} - 211 q^{47} - 374 q^{49} - 777 q^{51} - 613 q^{53} + 115 q^{55} - 316 q^{57} - 1029 q^{59} - 1169 q^{61} - 183 q^{63} - 210 q^{65} + 1227 q^{67} - 92 q^{69} - 237 q^{71} - 1001 q^{73} + 100 q^{75} - 1498 q^{77} + 898 q^{79} - 838 q^{81} + 1281 q^{83} - 620 q^{85} + 695 q^{87} - 2780 q^{89} - 857 q^{91} - 1569 q^{93} - 265 q^{95} + 91 q^{97} + 1015 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9 + 23 * q^11 - 42 * q^13 + 20 * q^15 - 124 * q^17 - 53 * q^19 - 114 * q^21 - 138 * q^23 + 150 * q^25 + 103 * q^27 - 320 * q^29 + 229 * q^31 - 583 * q^33 + 70 * q^35 - 377 * q^37 - 37 * q^39 - 683 * q^41 + 168 * q^43 + 20 * q^45 - 211 * q^47 - 374 * q^49 - 777 * q^51 - 613 * q^53 + 115 * q^55 - 316 * q^57 - 1029 * q^59 - 1169 * q^61 - 183 * q^63 - 210 * q^65 + 1227 * q^67 - 92 * q^69 - 237 * q^71 - 1001 * q^73 + 100 * q^75 - 1498 * q^77 + 898 * q^79 - 838 * q^81 + 1281 * q^83 - 620 * q^85 + 695 * q^87 - 2780 * q^89 - 857 * q^91 - 1569 * q^93 - 265 * q^95 + 91 * q^97 + 1015 * q^99

Introduction

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Modular forms

Varieties

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