LMFDB - Embedded newform 1925.4.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,4,Mod(1,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, 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("1925.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1925.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:	$113.578676761$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.522072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 5x + 1 $ x^4 - x^3 - 12x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.79597$ of defining polynomial
Character	$\chi$	$=$	1925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.53253 q^{2} -10.1459 q^{3} -5.65135 q^{4} +15.5490 q^{6} +7.00000 q^{7} +20.9211 q^{8} +75.9402 q^{9} -11.0000 q^{11} +57.3382 q^{12} +76.3572 q^{13} -10.7277 q^{14} +13.1485 q^{16} -39.7278 q^{17} -116.381 q^{18} -27.9876 q^{19} -71.0216 q^{21} +16.8579 q^{22} -87.2055 q^{23} -212.265 q^{24} -117.020 q^{26} -496.545 q^{27} -39.5594 q^{28} -38.3019 q^{29} -186.071 q^{31} -187.519 q^{32} +111.605 q^{33} +60.8842 q^{34} -429.164 q^{36} +218.781 q^{37} +42.8919 q^{38} -774.716 q^{39} +80.1687 q^{41} +108.843 q^{42} +35.1155 q^{43} +62.1648 q^{44} +133.645 q^{46} +282.620 q^{47} -133.404 q^{48} +49.0000 q^{49} +403.077 q^{51} -431.521 q^{52} -145.296 q^{53} +760.971 q^{54} +146.448 q^{56} +283.961 q^{57} +58.6989 q^{58} +91.0461 q^{59} +808.142 q^{61} +285.160 q^{62} +531.582 q^{63} +182.192 q^{64} -171.039 q^{66} -794.222 q^{67} +224.516 q^{68} +884.783 q^{69} +946.901 q^{71} +1588.75 q^{72} -801.324 q^{73} -335.288 q^{74} +158.168 q^{76} -77.0000 q^{77} +1187.28 q^{78} -890.737 q^{79} +2987.53 q^{81} -122.861 q^{82} +559.333 q^{83} +401.368 q^{84} -53.8156 q^{86} +388.609 q^{87} -230.132 q^{88} -1523.75 q^{89} +534.501 q^{91} +492.829 q^{92} +1887.87 q^{93} -433.125 q^{94} +1902.56 q^{96} -664.651 q^{97} -75.0941 q^{98} -835.342 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 14 q^{3} + 26 q^{4} + 14 q^{6} + 28 q^{7} + 18 q^{8} + 76 q^{9} - 44 q^{11} - 70 q^{12} - 58 q^{13} + 14 q^{14} + 2 q^{16} - 4 q^{17} + 62 q^{18} + 258 q^{19} - 98 q^{21} - 22 q^{22} - 8 q^{23}+ \cdots - 836 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 - 14 * q^3 + 26 * q^4 + 14 * q^6 + 28 * q^7 + 18 * q^8 + 76 * q^9 - 44 * q^11 - 70 * q^12 - 58 * q^13 + 14 * q^14 + 2 * q^16 - 4 * q^17 + 62 * q^18 + 258 * q^19 - 98 * q^21 - 22 * q^22 - 8 * q^23 - 498 * q^24 - 482 * q^26 - 428 * q^27 + 182 * q^28 - 396 * q^29 - 56 * q^31 - 134 * q^32 + 154 * q^33 + 472 * q^34 - 418 * q^36 - 84 * q^37 + 942 * q^38 - 412 * q^39 + 52 * q^41 + 98 * q^42 - 408 * q^43 - 286 * q^44 + 368 * q^46 - 8 * q^47 - 982 * q^48 + 196 * q^49 - 388 * q^51 - 2030 * q^52 - 624 * q^53 + 92 * q^54 + 126 * q^56 - 48 * q^57 - 864 * q^58 - 238 * q^59 - 162 * q^61 - 688 * q^62 + 532 * q^63 - 902 * q^64 - 154 * q^66 - 1340 * q^67 + 1384 * q^68 + 2416 * q^69 + 1788 * q^71 + 2622 * q^72 - 1456 * q^73 + 996 * q^74 + 3042 * q^76 - 308 * q^77 + 2632 * q^78 - 1324 * q^79 + 1444 * q^81 - 1984 * q^82 - 450 * q^83 - 490 * q^84 - 4380 * q^86 - 588 * q^87 - 198 * q^88 - 3072 * q^89 - 406 * q^91 - 544 * q^92 + 1264 * q^93 - 1696 * q^94 + 862 * q^96 + 652 * q^97 + 98 * q^98 - 836 * q^99

Coefficient data

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

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

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

$2$	−1.53253	−0.541832	−0.270916	−	0.962603i	$-0.587327\pi$
$2$	−1.53253	−0.541832	−0.270916	+	0.962603i	$0.587327\pi$
$3$	−10.1459	−1.95259	−0.976294	−	0.216448i	$-0.930553\pi$
$3$	−10.1459	−1.95259	−0.976294	+	0.216448i	$0.930553\pi$
$4$	−5.65135	−0.706418
$5$	0	0
$5$	0	0
$6$	15.5490	1.05797
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	20.9211	0.924592
$9$	75.9402	2.81260
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	57.3382	1.37934
$13$	76.3572	1.62905	0.814526	−	0.580127i	$-0.196997\pi$
$13$	76.3572	1.62905	0.814526	+	0.580127i	$0.196997\pi$
$14$	−10.7277	−0.204793
$15$	0	0
$16$	13.1485	0.205445
$17$	−39.7278	−0.566789	−0.283395	−	0.959003i	$-0.591461\pi$
$17$	−39.7278	−0.566789	−0.283395	+	0.959003i	$0.591461\pi$
$18$	−116.381	−1.52396
$19$	−27.9876	−0.337937	−0.168968	−	0.985621i	$-0.554044\pi$
$19$	−27.9876	−0.337937	−0.168968	+	0.985621i	$0.554044\pi$
$20$	0	0
$21$	−71.0216	−0.738009
$22$	16.8579	0.163368
$23$	−87.2055	−0.790592	−0.395296	−	0.918554i	$-0.629358\pi$
$23$	−87.2055	−0.790592	−0.395296	+	0.918554i	$0.629358\pi$
$24$	−212.265	−1.80535
$25$	0	0
$26$	−117.020	−0.882673
$27$	−496.545	−3.53926
$28$	−39.5594	−0.267001
$29$	−38.3019	−0.245258	−0.122629	−	0.992453i	$-0.539133\pi$
$29$	−38.3019	−0.245258	−0.122629	+	0.992453i	$0.539133\pi$
$30$	0	0
$31$	−186.071	−1.07804	−0.539021	−	0.842292i	$-0.681206\pi$
$31$	−186.071	−1.07804	−0.539021	+	0.842292i	$0.681206\pi$
$32$	−187.519	−1.03591
$33$	111.605	0.588728
$34$	60.8842	0.307105
$35$	0	0
$36$	−429.164	−1.98687
$37$	218.781	0.972090	0.486045	−	0.873934i	$-0.338439\pi$
$37$	218.781	0.972090	0.486045	+	0.873934i	$0.338439\pi$
$38$	42.8919	0.183105
$39$	−774.716	−3.18087
$40$	0	0
$41$	80.1687	0.305372	0.152686	−	0.988275i	$-0.451208\pi$
$41$	80.1687	0.305372	0.152686	+	0.988275i	$0.451208\pi$
$42$	108.843	0.399877
$43$	35.1155	0.124536	0.0622681	−	0.998059i	$-0.480167\pi$
$43$	35.1155	0.124536	0.0622681	+	0.998059i	$0.480167\pi$
$44$	62.1648	0.212993
$45$	0	0
$46$	133.645	0.428368
$47$	282.620	0.877116	0.438558	−	0.898703i	$-0.355489\pi$
$47$	282.620	0.877116	0.438558	+	0.898703i	$0.355489\pi$
$48$	−133.404	−0.401149
$49$	49.0000	0.142857
$50$	0	0
$51$	403.077	1.10671
$52$	−431.521	−1.15079
$53$	−145.296	−0.376566	−0.188283	−	0.982115i	$-0.560292\pi$
$53$	−145.296	−0.376566	−0.188283	+	0.982115i	$0.560292\pi$
$54$	760.971	1.91769
$55$	0	0
$56$	146.448	0.349463
$57$	283.961	0.659851
$58$	58.6989	0.132889
$59$	91.0461	0.200901	0.100451	−	0.994942i	$-0.467972\pi$
$59$	91.0461	0.200901	0.100451	+	0.994942i	$0.467972\pi$
$60$	0	0
$61$	808.142	1.69626	0.848131	−	0.529787i	$-0.177728\pi$
$61$	808.142	1.69626	0.848131	+	0.529787i	$0.177728\pi$
$62$	285.160	0.584118
$63$	531.582	1.06306
$64$	182.192	0.355843
$65$	0	0
$66$	−171.039	−0.318991
$67$	−794.222	−1.44820	−0.724102	−	0.689693i	$-0.757745\pi$
$67$	−794.222	−1.44820	−0.724102	+	0.689693i	$0.757745\pi$
$68$	224.516	0.400390
$69$	884.783	1.54370
$70$	0	0
$71$	946.901	1.58277	0.791384	−	0.611319i	$-0.209361\pi$
$71$	946.901	1.58277	0.791384	+	0.611319i	$0.209361\pi$
$72$	1588.75	2.60051
$73$	−801.324	−1.28477	−0.642383	−	0.766384i	$-0.722054\pi$
$73$	−801.324	−1.28477	−0.642383	+	0.766384i	$0.722054\pi$
$74$	−335.288	−0.526709
$75$	0	0
$76$	158.168	0.238725
$77$	−77.0000	−0.113961
$78$	1187.28	1.72350
$79$	−890.737	−1.26855	−0.634277	−	0.773106i	$-0.718702\pi$
$79$	−890.737	−1.26855	−0.634277	+	0.773106i	$0.718702\pi$
$80$	0	0
$81$	2987.53	4.09812
$82$	−122.861	−0.165460
$83$	559.333	0.739696	0.369848	−	0.929092i	$-0.379410\pi$
$83$	559.333	0.739696	0.369848	+	0.929092i	$0.379410\pi$
$84$	401.368	0.521343
$85$	0	0
$86$	−53.8156	−0.0674777
$87$	388.609	0.478888
$88$	−230.132	−0.278775
$89$	−1523.75	−1.81480	−0.907401	−	0.420265i	$-0.861937\pi$
$89$	−1523.75	−1.81480	−0.907401	+	0.420265i	$0.861937\pi$
$90$	0	0
$91$	534.501	0.615724
$92$	492.829	0.558488
$93$	1887.87	2.10497
$94$	−433.125	−0.475249
$95$	0	0
$96$	1902.56	2.02270
$97$	−664.651	−0.695723	−0.347861	−	0.937546i	$-0.613092\pi$
$97$	−664.651	−0.695723	−0.347861	+	0.937546i	$0.613092\pi$
$98$	−75.0941	−0.0774046
$99$	−835.342	−0.848031
$100$	0	0
$101$	1495.68	1.47352	0.736761	−	0.676153i	$-0.236354\pi$
$101$	1495.68	1.47352	0.736761	+	0.676153i	$0.236354\pi$
$102$	−617.728	−0.599649
$103$	−874.379	−0.836457	−0.418229	−	0.908342i	$-0.637349\pi$
$103$	−874.379	−0.836457	−0.418229	+	0.908342i	$0.637349\pi$
$104$	1597.48	1.50621
$105$	0	0
$106$	222.671	0.204035
$107$	−783.854	−0.708205	−0.354103	−	0.935207i	$-0.615214\pi$
$107$	−783.854	−0.708205	−0.354103	+	0.935207i	$0.615214\pi$
$108$	2806.15	2.50020
$109$	−1351.08	−1.18725	−0.593623	−	0.804743i	$-0.702303\pi$
$109$	−1351.08	−1.18725	−0.593623	+	0.804743i	$0.702303\pi$
$110$	0	0
$111$	−2219.74	−1.89809
$112$	92.0393	0.0776508
$113$	188.362	0.156811	0.0784055	−	0.996922i	$-0.475017\pi$
$113$	188.362	0.156811	0.0784055	+	0.996922i	$0.475017\pi$
$114$	−435.179	−0.357529
$115$	0	0
$116$	216.457	0.173255
$117$	5798.58	4.58187
$118$	−139.531	−0.108855
$119$	−278.095	−0.214226
$120$	0	0
$121$	121.000	0.0909091
$122$	−1238.50	−0.919089
$123$	−813.388	−0.596266
$124$	1051.55	0.761549
$125$	0	0
$126$	−814.666	−0.576002
$127$	−586.957	−0.410110	−0.205055	−	0.978750i	$-0.565737\pi$
$127$	−586.957	−0.410110	−0.205055	+	0.978750i	$0.565737\pi$
$128$	1220.94	0.843101
$129$	−356.280	−0.243168
$130$	0	0
$131$	623.054	0.415546	0.207773	−	0.978177i	$-0.433379\pi$
$131$	623.054	0.415546	0.207773	+	0.978177i	$0.433379\pi$
$132$	−630.721	−0.415888
$133$	−195.913	−0.127728
$134$	1217.17	0.784683
$135$	0	0
$136$	−831.151	−0.524049
$137$	954.859	0.595468	0.297734	−	0.954649i	$-0.403769\pi$
$137$	954.859	0.595468	0.297734	+	0.954649i	$0.403769\pi$
$138$	−1355.96	−0.836426
$139$	1590.62	0.970610	0.485305	−	0.874345i	$-0.338709\pi$
$139$	1590.62	0.970610	0.485305	+	0.874345i	$0.338709\pi$
$140$	0	0
$141$	−2867.45	−1.71265
$142$	−1451.16	−0.857594
$143$	−839.929	−0.491178
$144$	998.498	0.577834
$145$	0	0
$146$	1228.05	0.696127
$147$	−497.151	−0.278941
$148$	−1236.41	−0.686702
$149$	−1415.84	−0.778459	−0.389229	−	0.921141i	$-0.627259\pi$
$149$	−1415.84	−0.778459	−0.389229	+	0.921141i	$0.627259\pi$
$150$	0	0
$151$	411.564	0.221806	0.110903	−	0.993831i	$-0.464626\pi$
$151$	411.564	0.221806	0.110903	+	0.993831i	$0.464626\pi$
$152$	−585.532	−0.312454
$153$	−3016.94	−1.59415
$154$	118.005	0.0617475
$155$	0	0
$156$	4378.19	2.24702
$157$	1417.29	0.720460	0.360230	−	0.932864i	$-0.382698\pi$
$157$	1417.29	0.720460	0.360230	+	0.932864i	$0.382698\pi$
$158$	1365.08	0.687343
$159$	1474.17	0.735278
$160$	0	0
$161$	−610.439	−0.298816
$162$	−4578.49	−2.22049
$163$	441.401	0.212105	0.106053	−	0.994361i	$-0.466179\pi$
$163$	441.401	0.212105	0.106053	+	0.994361i	$0.466179\pi$
$164$	−453.061	−0.215720
$165$	0	0
$166$	−857.196	−0.400791
$167$	1486.39	0.688742	0.344371	−	0.938834i	$-0.388092\pi$
$167$	1486.39	0.688742	0.344371	+	0.938834i	$0.388092\pi$
$168$	−1485.85	−0.682357
$169$	3633.43	1.65381
$170$	0	0
$171$	−2125.39	−0.950481
$172$	−198.450	−0.0879747
$173$	2957.39	1.29969	0.649844	−	0.760068i	$-0.274834\pi$
$173$	2957.39	1.29969	0.649844	+	0.760068i	$0.274834\pi$
$174$	−595.555	−0.259477
$175$	0	0
$176$	−144.633	−0.0619439
$177$	−923.748	−0.392278
$178$	2335.20	0.983318
$179$	−24.4424	−0.0102062	−0.00510310	−	0.999987i	$-0.501624\pi$
$179$	−24.4424	−0.0102062	−0.00510310	+	0.999987i	$0.501624\pi$
$180$	0	0
$181$	−120.126	−0.0493308	−0.0246654	−	0.999696i	$-0.507852\pi$
$181$	−120.126	−0.0493308	−0.0246654	+	0.999696i	$0.507852\pi$
$182$	−819.139	−0.333619
$183$	−8199.37	−3.31210
$184$	−1824.44	−0.730975
$185$	0	0
$186$	−2893.22	−1.14054
$187$	437.006	0.170893
$188$	−1597.19	−0.619610
$189$	−3475.81	−1.33772
$190$	0	0
$191$	−2059.27	−0.780123	−0.390061	−	0.920789i	$-0.627546\pi$
$191$	−2059.27	−0.780123	−0.390061	+	0.920789i	$0.627546\pi$
$192$	−1848.51	−0.694816
$193$	4371.31	1.63033	0.815165	−	0.579229i	$-0.196646\pi$
$193$	4371.31	1.63033	0.815165	+	0.579229i	$0.196646\pi$
$194$	1018.60	0.376965
$195$	0	0
$196$	−276.916	−0.100917
$197$	2185.70	0.790479	0.395240	−	0.918578i	$-0.370662\pi$
$197$	2185.70	0.790479	0.395240	+	0.918578i	$0.370662\pi$
$198$	1280.19	0.459490
$199$	−2420.84	−0.862356	−0.431178	−	0.902267i	$-0.641902\pi$
$199$	−2420.84	−0.862356	−0.431178	+	0.902267i	$0.641902\pi$
$200$	0	0
$201$	8058.13	2.82774
$202$	−2292.18	−0.798401
$203$	−268.113	−0.0926988
$204$	−2277.93	−0.781797
$205$	0	0
$206$	1340.01	0.453219
$207$	−6622.41	−2.22362
$208$	1003.98	0.334680
$209$	307.864	0.101892
$210$	0	0
$211$	−3888.39	−1.26866	−0.634331	−	0.773062i	$-0.718724\pi$
$211$	−3888.39	−1.26866	−0.634331	+	0.773062i	$0.718724\pi$
$212$	821.120	0.266013
$213$	−9607.21	−3.09049
$214$	1201.28	0.383728
$215$	0	0
$216$	−10388.3	−3.27237
$217$	−1302.50	−0.407462
$218$	2070.57	0.643288
$219$	8130.19	2.50862
$220$	0	0
$221$	−3033.51	−0.923330
$222$	3401.82	1.02845
$223$	−641.467	−0.192627	−0.0963135	−	0.995351i	$-0.530705\pi$
$223$	−641.467	−0.192627	−0.0963135	+	0.995351i	$0.530705\pi$
$224$	−1312.64	−0.391537
$225$	0	0
$226$	−288.672	−0.0849652
$227$	−3619.11	−1.05819	−0.529094	−	0.848563i	$-0.677468\pi$
$227$	−3619.11	−1.05819	−0.529094	+	0.848563i	$0.677468\pi$
$228$	−1604.76	−0.466131
$229$	2518.14	0.726652	0.363326	−	0.931662i	$-0.381641\pi$
$229$	2518.14	0.726652	0.363326	+	0.931662i	$0.381641\pi$
$230$	0	0
$231$	781.238	0.222518
$232$	−801.318	−0.226763
$233$	4187.89	1.17750	0.588751	−	0.808315i	$-0.299620\pi$
$233$	4187.89	1.17750	0.588751	+	0.808315i	$0.299620\pi$
$234$	−8886.52	−2.48261
$235$	0	0
$236$	−514.533	−0.141920
$237$	9037.37	2.47696
$238$	426.189	0.116075
$239$	2582.63	0.698981	0.349490	−	0.936940i	$-0.386355\pi$
$239$	2582.63	0.698981	0.349490	+	0.936940i	$0.386355\pi$
$240$	0	0
$241$	1522.09	0.406832	0.203416	−	0.979092i	$-0.434796\pi$
$241$	1522.09	0.406832	0.203416	+	0.979092i	$0.434796\pi$
$242$	−185.436	−0.0492574
$243$	−16904.6	−4.46268
$244$	−4567.09	−1.19827
$245$	0	0
$246$	1246.54	0.323076
$247$	−2137.06	−0.550517
$248$	−3892.81	−0.996750
$249$	−5674.96	−1.44432
$250$	0	0
$251$	1463.19	0.367952	0.183976	−	0.982931i	$-0.441103\pi$
$251$	1463.19	0.367952	0.183976	+	0.982931i	$0.441103\pi$
$252$	−3004.15	−0.750967
$253$	959.261	0.238372
$254$	899.531	0.222211
$255$	0	0
$256$	−3328.67	−0.812662
$257$	1183.89	0.287350	0.143675	−	0.989625i	$-0.454108\pi$
$257$	1183.89	0.287350	0.143675	+	0.989625i	$0.454108\pi$
$258$	546.010	0.131756
$259$	1531.46	0.367415
$260$	0	0
$261$	−2908.65	−0.689813
$262$	−954.850	−0.225156
$263$	−151.973	−0.0356315	−0.0178158	−	0.999841i	$-0.505671\pi$
$263$	−151.973	−0.0356315	−0.0178158	+	0.999841i	$0.505671\pi$
$264$	2334.91	0.544333
$265$	0	0
$266$	300.243	0.0692072
$267$	15459.9	3.54356
$268$	4488.42	1.02304
$269$	−255.543	−0.0579209	−0.0289604	−	0.999581i	$-0.509220\pi$
$269$	−255.543	−0.0579209	−0.0289604	+	0.999581i	$0.509220\pi$
$270$	0	0
$271$	2589.73	0.580497	0.290248	−	0.956951i	$-0.406262\pi$
$271$	2589.73	0.580497	0.290248	+	0.956951i	$0.406262\pi$
$272$	−522.360	−0.116444
$273$	−5423.01	−1.20226
$274$	−1463.35	−0.322644
$275$	0	0
$276$	−5000.21	−1.09050
$277$	4441.49	0.963404	0.481702	−	0.876335i	$-0.340019\pi$
$277$	4441.49	0.963404	0.481702	+	0.876335i	$0.340019\pi$
$278$	−2437.68	−0.525907
$279$	−14130.3	−3.03210
$280$	0	0
$281$	1577.34	0.334861	0.167431	−	0.985884i	$-0.446453\pi$
$281$	1577.34	0.334861	0.167431	+	0.985884i	$0.446453\pi$
$282$	4394.46	0.927966
$283$	−3429.29	−0.720317	−0.360159	−	0.932891i	$-0.617277\pi$
$283$	−3429.29	−0.720317	−0.360159	+	0.932891i	$0.617277\pi$
$284$	−5351.27	−1.11810
$285$	0	0
$286$	1287.22	0.266136
$287$	561.181	0.115420
$288$	−14240.3	−2.91360
$289$	−3334.70	−0.678750
$290$	0	0
$291$	6743.51	1.35846
$292$	4528.56	0.907581
$293$	−4601.37	−0.917458	−0.458729	−	0.888576i	$-0.651695\pi$
$293$	−4601.37	−0.917458	−0.458729	+	0.888576i	$0.651695\pi$
$294$	761.900	0.151139
$295$	0	0
$296$	4577.14	0.898786
$297$	5461.99	1.06713
$298$	2169.82	0.421794
$299$	−6658.77	−1.28792
$300$	0	0
$301$	245.808	0.0470703
$302$	−630.736	−0.120181
$303$	−15175.1	−2.87718
$304$	−367.994	−0.0694274
$305$	0	0
$306$	4623.56	0.863762
$307$	−4990.18	−0.927702	−0.463851	−	0.885913i	$-0.653533\pi$
$307$	−4990.18	−0.927702	−0.463851	+	0.885913i	$0.653533\pi$
$308$	435.154	0.0805038
$309$	8871.40	1.63326
$310$	0	0
$311$	3139.78	0.572478	0.286239	−	0.958158i	$-0.407595\pi$
$311$	3139.78	0.572478	0.286239	+	0.958158i	$0.407595\pi$
$312$	−16207.9	−2.94101
$313$	4723.12	0.852929	0.426464	−	0.904504i	$-0.359759\pi$
$313$	4723.12	0.852929	0.426464	+	0.904504i	$0.359759\pi$
$314$	−2172.04	−0.390368
$315$	0	0
$316$	5033.86	0.896130
$317$	−5935.83	−1.05170	−0.525851	−	0.850577i	$-0.676253\pi$
$317$	−5935.83	−1.05170	−0.525851	+	0.850577i	$0.676253\pi$
$318$	−2259.21	−0.398397
$319$	421.321	0.0739481
$320$	0	0
$321$	7952.94	1.38283
$322$	935.517	0.161908
$323$	1111.89	0.191539
$324$	−16883.6	−2.89499
$325$	0	0
$326$	−676.461	−0.114925
$327$	13708.0	2.31820
$328$	1677.22	0.282344
$329$	1978.34	0.331519
$330$	0	0
$331$	−6390.75	−1.06123	−0.530615	−	0.847613i	$-0.678039\pi$
$331$	−6390.75	−1.06123	−0.530615	+	0.847613i	$0.678039\pi$
$332$	−3160.98	−0.522535
$333$	16614.3	2.73410
$334$	−2277.93	−0.373183
$335$	0	0
$336$	−933.826	−0.151620
$337$	−8916.41	−1.44127	−0.720635	−	0.693315i	$-0.756149\pi$
$337$	−8916.41	−1.44127	−0.720635	+	0.693315i	$0.756149\pi$
$338$	−5568.34	−0.896088
$339$	−1911.12	−0.306187
$340$	0	0
$341$	2046.78	0.325042
$342$	3257.22	0.515001
$343$	343.000	0.0539949
$344$	734.655	0.115145
$345$	0	0
$346$	−4532.29	−0.704212
$347$	−1400.52	−0.216668	−0.108334	−	0.994115i	$-0.534552\pi$
$347$	−1400.52	−0.216668	−0.108334	+	0.994115i	$0.534552\pi$
$348$	−2196.16	−0.338295
$349$	−8985.39	−1.37816	−0.689079	−	0.724686i	$-0.741985\pi$
$349$	−8985.39	−1.37816	−0.689079	+	0.724686i	$0.741985\pi$
$350$	0	0
$351$	−37914.8	−5.76565
$352$	2062.71	0.312338
$353$	−1767.91	−0.266562	−0.133281	−	0.991078i	$-0.542551\pi$
$353$	−1767.91	−0.266562	−0.133281	+	0.991078i	$0.542551\pi$
$354$	1415.67	0.212549
$355$	0	0
$356$	8611.25	1.28201
$357$	2821.54	0.418296
$358$	37.4587	0.00553004
$359$	−1110.48	−0.163256	−0.0816278	−	0.996663i	$-0.526012\pi$
$359$	−1110.48	−0.163256	−0.0816278	+	0.996663i	$0.526012\pi$
$360$	0	0
$361$	−6075.69	−0.885799
$362$	184.096	0.0267290
$363$	−1227.66	−0.177508
$364$	−3020.65	−0.434959
$365$	0	0
$366$	12565.8	1.79460
$367$	5342.61	0.759896	0.379948	−	0.925008i	$-0.375942\pi$
$367$	5342.61	0.759896	0.379948	+	0.925008i	$0.375942\pi$
$368$	−1146.62	−0.162423
$369$	6088.03	0.858890
$370$	0	0
$371$	−1017.07	−0.142329
$372$	−10669.0	−1.48699
$373$	−6829.95	−0.948101	−0.474050	−	0.880498i	$-0.657209\pi$
$373$	−6829.95	−0.948101	−0.474050	+	0.880498i	$0.657209\pi$
$374$	−669.726	−0.0925955
$375$	0	0
$376$	5912.74	0.810974
$377$	−2924.63	−0.399538
$378$	5326.80	0.724817
$379$	7826.62	1.06076	0.530378	−	0.847761i	$-0.322050\pi$
$379$	7826.62	1.06076	0.530378	+	0.847761i	$0.322050\pi$
$380$	0	0
$381$	5955.24	0.800777
$382$	3155.90	0.422695
$383$	7534.69	1.00523	0.502617	−	0.864509i	$-0.332371\pi$
$383$	7534.69	1.00523	0.502617	+	0.864509i	$0.332371\pi$
$384$	−12387.6	−1.64623
$385$	0	0
$386$	−6699.17	−0.883364
$387$	2666.68	0.350271
$388$	3756.17	0.491471
$389$	−13071.8	−1.70376	−0.851882	−	0.523734i	$-0.824539\pi$
$389$	−13071.8	−1.70376	−0.851882	+	0.523734i	$0.824539\pi$
$390$	0	0
$391$	3464.49	0.448099
$392$	1025.14	0.132085
$393$	−6321.47	−0.811390
$394$	−3349.65	−0.428307
$395$	0	0
$396$	4720.81	0.599065
$397$	3692.51	0.466806	0.233403	−	0.972380i	$-0.425014\pi$
$397$	3692.51	0.466806	0.233403	+	0.972380i	$0.425014\pi$
$398$	3710.02	0.467252
$399$	1987.73	0.249400
$400$	0	0
$401$	8223.85	1.02414	0.512069	−	0.858944i	$-0.328879\pi$
$401$	8223.85	1.02414	0.512069	+	0.858944i	$0.328879\pi$
$402$	−12349.3	−1.53216
$403$	−14207.9	−1.75619
$404$	−8452.61	−1.04092
$405$	0	0
$406$	410.892	0.0502272
$407$	−2406.59	−0.293096
$408$	8432.82	1.02325
$409$	3689.61	0.446062	0.223031	−	0.974811i	$-0.428405\pi$
$409$	3689.61	0.446062	0.223031	+	0.974811i	$0.428405\pi$
$410$	0	0
$411$	−9687.95	−1.16270
$412$	4941.42	0.590889
$413$	637.322	0.0759336
$414$	10149.1	1.20483
$415$	0	0
$416$	−14318.5	−1.68755
$417$	−16138.4	−1.89520
$418$	−471.811	−0.0552082
$419$	15657.0	1.82552	0.912762	−	0.408492i	$-0.133945\pi$
$419$	15657.0	1.82552	0.912762	+	0.408492i	$0.133945\pi$
$420$	0	0
$421$	−6007.30	−0.695435	−0.347717	−	0.937599i	$-0.613043\pi$
$421$	−6007.30	−0.695435	−0.347717	+	0.937599i	$0.613043\pi$
$422$	5959.08	0.687402
$423$	21462.3	2.46698
$424$	−3039.76	−0.348170
$425$	0	0
$426$	14723.4	1.67453
$427$	5656.99	0.641127
$428$	4429.83	0.500289
$429$	8521.88	0.959068
$430$	0	0
$431$	13139.0	1.46841	0.734203	−	0.678931i	$-0.237556\pi$
$431$	13139.0	1.46841	0.734203	+	0.678931i	$0.237556\pi$
$432$	−6528.81	−0.727123
$433$	4392.47	0.487502	0.243751	−	0.969838i	$-0.421622\pi$
$433$	4392.47	0.487502	0.243751	+	0.969838i	$0.421622\pi$
$434$	1996.12	0.220776
$435$	0	0
$436$	7635.41	0.838692
$437$	2440.67	0.267170
$438$	−12459.8	−1.35925
$439$	12676.2	1.37813	0.689066	−	0.724699i	$-0.258021\pi$
$439$	12676.2	1.37813	0.689066	+	0.724699i	$0.258021\pi$
$440$	0	0
$441$	3721.07	0.401800
$442$	4648.95	0.500289
$443$	17489.2	1.87571	0.937855	−	0.347028i	$-0.112809\pi$
$443$	17489.2	1.87571	0.937855	+	0.347028i	$0.112809\pi$
$444$	12544.5	1.34085
$445$	0	0
$446$	983.069	0.104371
$447$	14365.1	1.52001
$448$	1275.34	0.134496
$449$	−15491.8	−1.62830	−0.814148	−	0.580658i	$-0.802796\pi$
$449$	−15491.8	−1.62830	−0.814148	+	0.580658i	$0.802796\pi$
$450$	0	0
$451$	−881.856	−0.0920731
$452$	−1064.50	−0.110774
$453$	−4175.71	−0.433095
$454$	5546.40	0.573360
$455$	0	0
$456$	5940.78	0.610093
$457$	7789.99	0.797375	0.398688	−	0.917087i	$-0.369466\pi$
$457$	7789.99	0.797375	0.398688	+	0.917087i	$0.369466\pi$
$458$	−3859.13	−0.393723
$459$	19726.7	2.00602
$460$	0	0
$461$	−8497.44	−0.858493	−0.429247	−	0.903187i	$-0.641221\pi$
$461$	−8497.44	−0.858493	−0.429247	+	0.903187i	$0.641221\pi$
$462$	−1197.27	−0.120567
$463$	−875.113	−0.0878401	−0.0439200	−	0.999035i	$-0.513985\pi$
$463$	−875.113	−0.0878401	−0.0439200	+	0.999035i	$0.513985\pi$
$464$	−503.611	−0.0503870
$465$	0	0
$466$	−6418.08	−0.638008
$467$	−17652.5	−1.74917	−0.874584	−	0.484874i	$-0.838866\pi$
$467$	−17652.5	−1.74917	−0.874584	+	0.484874i	$0.838866\pi$
$468$	−32769.8	−3.23672
$469$	−5559.55	−0.547369
$470$	0	0
$471$	−14379.8	−1.40676
$472$	1904.79	0.185752
$473$	−386.270	−0.0375491
$474$	−13850.1	−1.34210
$475$	0	0
$476$	1571.61	0.151333
$477$	−11033.8	−1.05913
$478$	−3957.96	−0.378730
$479$	3129.76	0.298544	0.149272	−	0.988796i	$-0.452307\pi$
$479$	3129.76	0.298544	0.149272	+	0.988796i	$0.452307\pi$
$480$	0	0
$481$	16705.5	1.58359
$482$	−2332.66	−0.220435
$483$	6193.48	0.583464
$484$	−683.813	−0.0642198
$485$	0	0
$486$	25906.9	2.41802
$487$	366.055	0.0340606	0.0170303	−	0.999855i	$-0.494579\pi$
$487$	366.055	0.0340606	0.0170303	+	0.999855i	$0.494579\pi$
$488$	16907.2	1.56835
$489$	−4478.43	−0.414154
$490$	0	0
$491$	−14577.2	−1.33984	−0.669919	−	0.742434i	$-0.733671\pi$
$491$	−14577.2	−1.33984	−0.669919	+	0.742434i	$0.733671\pi$
$492$	4596.73	0.421213
$493$	1521.65	0.139010
$494$	3275.11	0.298288
$495$	0	0
$496$	−2446.55	−0.221478
$497$	6628.31	0.598230
$498$	8697.06	0.782580
$499$	9504.05	0.852624	0.426312	−	0.904576i	$-0.359813\pi$
$499$	9504.05	0.852624	0.426312	+	0.904576i	$0.359813\pi$
$500$	0	0
$501$	−15080.8	−1.34483
$502$	−2242.39	−0.199368
$503$	6149.90	0.545150	0.272575	−	0.962134i	$-0.412125\pi$
$503$	6149.90	0.545150	0.272575	+	0.962134i	$0.412125\pi$
$504$	11121.3	0.982900
$505$	0	0
$506$	−1470.10	−0.129158
$507$	−36864.5	−3.22921
$508$	3317.10	0.289709
$509$	−16132.0	−1.40479	−0.702396	−	0.711787i	$-0.747886\pi$
$509$	−16132.0	−1.40479	−0.702396	+	0.711787i	$0.747886\pi$
$510$	0	0
$511$	−5609.27	−0.485596
$512$	−4666.24	−0.402775
$513$	13897.1	1.19605
$514$	−1814.35	−0.155695
$515$	0	0
$516$	2013.46	0.171778
$517$	−3108.83	−0.264460
$518$	−2347.02	−0.199077
$519$	−30005.5	−2.53775
$520$	0	0
$521$	7654.31	0.643650	0.321825	−	0.946799i	$-0.395704\pi$
$521$	7654.31	0.643650	0.321825	+	0.946799i	$0.395704\pi$
$522$	4457.60	0.373763
$523$	−21493.7	−1.79705	−0.898523	−	0.438926i	$-0.855359\pi$
$523$	−21493.7	−1.79705	−0.898523	+	0.438926i	$0.855359\pi$
$524$	−3521.09	−0.293549
$525$	0	0
$526$	232.904	0.0193063
$527$	7392.20	0.611023
$528$	1467.44	0.120951
$529$	−4562.20	−0.374965
$530$	0	0
$531$	6914.06	0.565056
$532$	1107.17	0.0902294
$533$	6121.46	0.497467
$534$	−23692.8	−1.92001
$535$	0	0
$536$	−16616.0	−1.33900
$537$	247.991	0.0199285
$538$	391.628	0.0313834
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−8661.03	−0.688293	−0.344147	−	0.938916i	$-0.611832\pi$
$541$	−8661.03	−0.688293	−0.344147	+	0.938916i	$0.611832\pi$
$542$	−3968.84	−0.314532
$543$	1218.79	0.0963227
$544$	7449.74	0.587142
$545$	0	0
$546$	8310.94	0.651420
$547$	−21372.9	−1.67064	−0.835321	−	0.549763i	$-0.814718\pi$
$547$	−21372.9	−1.67064	−0.835321	+	0.549763i	$0.814718\pi$
$548$	−5396.24	−0.420650
$549$	61370.5	4.77091
$550$	0	0
$551$	1071.98	0.0828817
$552$	18510.6	1.42729
$553$	−6235.16	−0.479468
$554$	−6806.72	−0.522003
$555$	0	0
$556$	−8989.15	−0.685656
$557$	−18062.1	−1.37399	−0.686997	−	0.726660i	$-0.741071\pi$
$557$	−18062.1	−1.37399	−0.686997	+	0.726660i	$0.741071\pi$
$558$	21655.1	1.64289
$559$	2681.32	0.202876
$560$	0	0
$561$	−4433.84	−0.333684
$562$	−2417.32	−0.181438
$563$	−962.299	−0.0720356	−0.0360178	−	0.999351i	$-0.511467\pi$
$563$	−962.299	−0.0720356	−0.0360178	+	0.999351i	$0.511467\pi$
$564$	16205.0	1.20984
$565$	0	0
$566$	5255.49	0.390291
$567$	20912.7	1.54894
$568$	19810.2	1.46341
$569$	−25409.7	−1.87211	−0.936055	−	0.351853i	$-0.885552\pi$
$569$	−25409.7	−1.87211	−0.936055	+	0.351853i	$0.885552\pi$
$570$	0	0
$571$	−5211.13	−0.381925	−0.190962	−	0.981597i	$-0.561161\pi$
$571$	−5211.13	−0.381925	−0.190962	+	0.981597i	$0.561161\pi$
$572$	4746.73	0.346977
$573$	20893.2	1.52326
$574$	−860.028	−0.0625381
$575$	0	0
$576$	13835.7	1.00085
$577$	−409.463	−0.0295428	−0.0147714	−	0.999891i	$-0.504702\pi$
$577$	−409.463	−0.0295428	−0.0147714	+	0.999891i	$0.504702\pi$
$578$	5110.53	0.367768
$579$	−44351.0	−3.18336
$580$	0	0
$581$	3915.33	0.279579
$582$	−10334.7	−0.736057
$583$	1598.26	0.113539
$584$	−16764.6	−1.18788
$585$	0	0
$586$	7051.75	0.497108
$587$	11756.7	0.826661	0.413331	−	0.910581i	$-0.364365\pi$
$587$	11756.7	0.826661	0.413331	+	0.910581i	$0.364365\pi$
$588$	2809.57	0.197049
$589$	5207.68	0.364310
$590$	0	0
$591$	−22176.0	−1.54348
$592$	2876.63	0.199711
$593$	312.172	0.0216178	0.0108089	−	0.999942i	$-0.496559\pi$
$593$	312.172	0.0216178	0.0108089	+	0.999942i	$0.496559\pi$
$594$	−8370.68	−0.578204
$595$	0	0
$596$	8001.42	0.549918
$597$	24561.7	1.68383
$598$	10204.8	0.697834
$599$	22486.4	1.53384	0.766918	−	0.641745i	$-0.221789\pi$
$599$	22486.4	1.53384	0.766918	+	0.641745i	$0.221789\pi$
$600$	0	0
$601$	25019.6	1.69812	0.849061	−	0.528295i	$-0.177168\pi$
$601$	25019.6	1.69812	0.849061	+	0.528295i	$0.177168\pi$
$602$	−376.709	−0.0255042
$603$	−60313.4	−4.07322
$604$	−2325.89	−0.156687
$605$	0	0
$606$	23256.3	1.55895
$607$	−7074.22	−0.473037	−0.236519	−	0.971627i	$-0.576006\pi$
$607$	−7074.22	−0.473037	−0.236519	+	0.971627i	$0.576006\pi$
$608$	5248.22	0.350072
$609$	2720.26	0.181003
$610$	0	0
$611$	21580.1	1.42887
$612$	17049.8	1.12614
$613$	−4934.94	−0.325155	−0.162578	−	0.986696i	$-0.551981\pi$
$613$	−4934.94	−0.325155	−0.162578	+	0.986696i	$0.551981\pi$
$614$	7647.61	0.502658
$615$	0	0
$616$	−1610.93	−0.105367
$617$	−2125.51	−0.138687	−0.0693434	−	0.997593i	$-0.522090\pi$
$617$	−2125.51	−0.138687	−0.0693434	+	0.997593i	$0.522090\pi$
$618$	−13595.7	−0.884951
$619$	8168.09	0.530377	0.265188	−	0.964197i	$-0.414566\pi$
$619$	8168.09	0.530377	0.265188	+	0.964197i	$0.414566\pi$
$620$	0	0
$621$	43301.5	2.79811
$622$	−4811.81	−0.310187
$623$	−10666.3	−0.685931
$624$	−10186.3	−0.653493
$625$	0	0
$626$	−7238.34	−0.462144
$627$	−3123.57	−0.198953
$628$	−8009.60	−0.508946
$629$	−8691.69	−0.550970
$630$	0	0
$631$	−8419.88	−0.531205	−0.265602	−	0.964083i	$-0.585571\pi$
$631$	−8419.88	−0.531205	−0.265602	+	0.964083i	$0.585571\pi$
$632$	−18635.2	−1.17289
$633$	39451.4	2.47717
$634$	9096.85	0.569846
$635$	0	0
$636$	−8331.04	−0.519414
$637$	3741.50	0.232722
$638$	−645.687	−0.0400674
$639$	71907.9	4.45169
$640$	0	0
$641$	27238.9	1.67843	0.839213	−	0.543803i	$-0.183016\pi$
$641$	27238.9	1.67843	0.839213	+	0.543803i	$0.183016\pi$
$642$	−12188.1	−0.749263
$643$	12438.7	0.762882	0.381441	−	0.924393i	$-0.375428\pi$
$643$	12438.7	0.762882	0.381441	+	0.924393i	$0.375428\pi$
$644$	3449.80	0.211089
$645$	0	0
$646$	−1704.00	−0.103782
$647$	9788.76	0.594801	0.297400	−	0.954753i	$-0.403880\pi$
$647$	9788.76	0.594801	0.297400	+	0.954753i	$0.403880\pi$
$648$	62502.5	3.78909
$649$	−1001.51	−0.0605741
$650$	0	0
$651$	13215.1	0.795605
$652$	−2494.51	−0.149835
$653$	−5539.90	−0.331996	−0.165998	−	0.986126i	$-0.553084\pi$
$653$	−5539.90	−0.331996	−0.165998	+	0.986126i	$0.553084\pi$
$654$	−21007.9	−1.25608
$655$	0	0
$656$	1054.10	0.0627371
$657$	−60852.7	−3.61353
$658$	−3031.87	−0.179627
$659$	−18751.7	−1.10844	−0.554221	−	0.832370i	$-0.686984\pi$
$659$	−18751.7	−1.10844	−0.554221	+	0.832370i	$0.686984\pi$
$660$	0	0
$661$	24849.3	1.46222	0.731108	−	0.682262i	$-0.239004\pi$
$661$	24849.3	1.46222	0.731108	+	0.682262i	$0.239004\pi$
$662$	9794.03	0.575009
$663$	30777.8	1.80288
$664$	11701.9	0.683917
$665$	0	0
$666$	−25461.9	−1.48142
$667$	3340.14	0.193899
$668$	−8400.08	−0.486540
$669$	6508.29	0.376121
$670$	0	0
$671$	−8889.56	−0.511442
$672$	13317.9	0.764510
$673$	9532.72	0.546002	0.273001	−	0.962014i	$-0.411984\pi$
$673$	9532.72	0.546002	0.273001	+	0.962014i	$0.411984\pi$
$674$	13664.7	0.780926
$675$	0	0
$676$	−20533.7	−1.16828
$677$	6544.90	0.371552	0.185776	−	0.982592i	$-0.440520\pi$
$677$	6544.90	0.371552	0.185776	+	0.982592i	$0.440520\pi$
$678$	2928.85	0.165902
$679$	−4652.56	−0.262958
$680$	0	0
$681$	36719.3	2.06621
$682$	−3136.76	−0.176118
$683$	−8438.16	−0.472734	−0.236367	−	0.971664i	$-0.575957\pi$
$683$	−8438.16	−0.472734	−0.236367	+	0.971664i	$0.575957\pi$
$684$	12011.3	0.671437
$685$	0	0
$686$	−525.658	−0.0292562
$687$	−25548.9	−1.41885
$688$	461.715	0.0255853
$689$	−11094.4	−0.613446
$690$	0	0
$691$	−8196.16	−0.451225	−0.225613	−	0.974217i	$-0.572438\pi$
$691$	−8196.16	−0.451225	−0.225613	+	0.974217i	$0.572438\pi$
$692$	−16713.2	−0.918123
$693$	−5847.40	−0.320526
$694$	2146.34	0.117398
$695$	0	0
$696$	8130.13	0.442776
$697$	−3184.93	−0.173082
$698$	13770.4	0.746730
$699$	−42490.1	−2.29918
$700$	0	0
$701$	−7172.05	−0.386426	−0.193213	−	0.981157i	$-0.561891\pi$
$701$	−7172.05	−0.386426	−0.193213	+	0.981157i	$0.561891\pi$
$702$	58105.6	3.12401
$703$	−6123.15	−0.328505
$704$	−2004.11	−0.107291
$705$	0	0
$706$	2709.38	0.144432
$707$	10469.8	0.556939
$708$	5220.42	0.277112
$709$	−16766.6	−0.888127	−0.444063	−	0.895995i	$-0.646464\pi$
$709$	−16766.6	−0.888127	−0.444063	+	0.895995i	$0.646464\pi$
$710$	0	0
$711$	−67642.8	−3.56794
$712$	−31878.6	−1.67795
$713$	16226.4	0.852292
$714$	−4324.09	−0.226646
$715$	0	0
$716$	138.132	0.00720984
$717$	−26203.2	−1.36482
$718$	1701.84	0.0884571
$719$	−5923.31	−0.307235	−0.153618	−	0.988130i	$-0.549092\pi$
$719$	−5923.31	−0.307235	−0.153618	+	0.988130i	$0.549092\pi$
$720$	0	0
$721$	−6120.65	−0.316151
$722$	9311.20	0.479954
$723$	−15443.1	−0.794376
$724$	678.872	0.0348482
$725$	0	0
$726$	1881.43	0.0961795
$727$	4256.80	0.217161	0.108581	−	0.994088i	$-0.465369\pi$
$727$	4256.80	0.217161	0.108581	+	0.994088i	$0.465369\pi$
$728$	11182.4	0.569293
$729$	90850.1	4.61566
$730$	0	0
$731$	−1395.06	−0.0705858
$732$	46337.4	2.33973
$733$	−24556.4	−1.23740	−0.618698	−	0.785629i	$-0.712340\pi$
$733$	−24556.4	−1.23740	−0.618698	+	0.785629i	$0.712340\pi$
$734$	−8187.72	−0.411736
$735$	0	0
$736$	16352.7	0.818981
$737$	8736.44	0.436650
$738$	−9330.10	−0.465374
$739$	27603.8	1.37405	0.687025	−	0.726634i	$-0.258916\pi$
$739$	27603.8	1.37405	0.687025	+	0.726634i	$0.258916\pi$
$740$	0	0
$741$	21682.5	1.07493
$742$	1558.70	0.0771181
$743$	−19808.8	−0.978082	−0.489041	−	0.872261i	$-0.662653\pi$
$743$	−19808.8	−0.978082	−0.489041	+	0.872261i	$0.662653\pi$
$744$	39496.3	1.94624
$745$	0	0
$746$	10467.1	0.513711
$747$	42475.9	2.08047
$748$	−2469.67	−0.120722
$749$	−5486.98	−0.267677
$750$	0	0
$751$	596.125	0.0289653	0.0144826	−	0.999895i	$-0.495390\pi$
$751$	596.125	0.0289653	0.0144826	+	0.999895i	$0.495390\pi$
$752$	3716.03	0.180199
$753$	−14845.5	−0.718459
$754$	4482.08	0.216482
$755$	0	0
$756$	19643.0	0.944987
$757$	1845.87	0.0886253	0.0443127	−	0.999018i	$-0.485890\pi$
$757$	1845.87	0.0886253	0.0443127	+	0.999018i	$0.485890\pi$
$758$	−11994.6	−0.574752
$759$	−9732.61	−0.465443
$760$	0	0
$761$	−13034.2	−0.620877	−0.310439	−	0.950593i	$-0.600476\pi$
$761$	−13034.2	−0.620877	−0.310439	+	0.950593i	$0.600476\pi$
$762$	−9126.59	−0.433886
$763$	−9457.55	−0.448737
$764$	11637.6	0.551093
$765$	0	0
$766$	−11547.2	−0.544668
$767$	6952.02	0.327279
$768$	33772.5	1.58680
$769$	−38530.6	−1.80683	−0.903413	−	0.428770i	$-0.858947\pi$
$769$	−38530.6	−1.80683	−0.903413	+	0.428770i	$0.858947\pi$
$770$	0	0
$771$	−12011.7	−0.561076
$772$	−24703.8	−1.15169
$773$	−13835.1	−0.643745	−0.321873	−	0.946783i	$-0.604312\pi$
$773$	−13835.1	−0.643745	−0.321873	+	0.946783i	$0.604312\pi$
$774$	−4086.77	−0.189788
$775$	0	0
$776$	−13905.2	−0.643259
$777$	−15538.2	−0.717411
$778$	20032.9	0.923154
$779$	−2243.73	−0.103196
$780$	0	0
$781$	−10415.9	−0.477222
$782$	−5309.44	−0.242794
$783$	19018.6	0.868032
$784$	644.275	0.0293493
$785$	0	0
$786$	9687.86	0.439637
$787$	−11704.2	−0.530125	−0.265062	−	0.964231i	$-0.585393\pi$
$787$	−11704.2	−0.530125	−0.265062	+	0.964231i	$0.585393\pi$
$788$	−12352.1	−0.558409
$789$	1541.91	0.0695737
$790$	0	0
$791$	1318.54	0.0592690
$792$	−17476.3	−0.784083
$793$	61707.5	2.76330
$794$	−5658.89	−0.252930
$795$	0	0
$796$	13681.0	0.609184
$797$	3367.27	0.149655	0.0748274	−	0.997196i	$-0.476159\pi$
$797$	3367.27	0.149655	0.0748274	+	0.997196i	$0.476159\pi$
$798$	−3046.25	−0.135133
$799$	−11227.9	−0.497140
$800$	0	0
$801$	−115714.	−5.10432
$802$	−12603.3	−0.554911
$803$	8814.56	0.387371
$804$	−45539.3	−1.99757
$805$	0	0
$806$	21774.0	0.951559
$807$	2592.72	0.113096
$808$	31291.3	1.36241
$809$	20869.5	0.906961	0.453481	−	0.891266i	$-0.350182\pi$
$809$	20869.5	0.906961	0.453481	+	0.891266i	$0.350182\pi$
$810$	0	0
$811$	−22445.9	−0.971863	−0.485931	−	0.873997i	$-0.661520\pi$
$811$	−22445.9	−0.971863	−0.485931	+	0.873997i	$0.661520\pi$
$812$	1515.20	0.0654841
$813$	−26275.2	−1.13347
$814$	3688.17	0.158809
$815$	0	0
$816$	5299.84	0.227367
$817$	−982.798	−0.0420854
$818$	−5654.45	−0.241691
$819$	40590.1	1.73179
$820$	0	0
$821$	−25516.4	−1.08469	−0.542343	−	0.840157i	$-0.682463\pi$
$821$	−25516.4	−1.08469	−0.542343	+	0.840157i	$0.682463\pi$
$822$	14847.1	0.629990
$823$	−36376.7	−1.54072	−0.770360	−	0.637609i	$-0.779924\pi$
$823$	−36376.7	−1.54072	−0.770360	+	0.637609i	$0.779924\pi$
$824$	−18293.0	−0.773382
$825$	0	0
$826$	−976.717	−0.0411433
$827$	−25520.1	−1.07306	−0.536530	−	0.843881i	$-0.680265\pi$
$827$	−25520.1	−1.07306	−0.536530	+	0.843881i	$0.680265\pi$
$828$	37425.5	1.57080
$829$	23202.5	0.972084	0.486042	−	0.873936i	$-0.338440\pi$
$829$	23202.5	0.972084	0.486042	+	0.873936i	$0.338440\pi$
$830$	0	0
$831$	−45063.1	−1.88113
$832$	13911.7	0.579688
$833$	−1946.66	−0.0809699
$834$	24732.6	1.02688
$835$	0	0
$836$	−1739.84	−0.0719782
$837$	92392.6	3.81548
$838$	−23994.9	−0.989127
$839$	−10538.6	−0.433649	−0.216824	−	0.976211i	$-0.569570\pi$
$839$	−10538.6	−0.433649	−0.216824	+	0.976211i	$0.569570\pi$
$840$	0	0
$841$	−22922.0	−0.939849
$842$	9206.38	0.376809
$843$	−16003.6	−0.653846
$844$	21974.6	0.896206
$845$	0	0
$846$	−32891.6	−1.33669
$847$	847.000	0.0343604
$848$	−1910.42	−0.0773635
$849$	34793.3	1.40648
$850$	0	0
$851$	−19078.9	−0.768526
$852$	54293.7	2.18318
$853$	−40061.0	−1.60805	−0.804023	−	0.594598i	$-0.797311\pi$
$853$	−40061.0	−1.60805	−0.804023	+	0.594598i	$0.797311\pi$
$854$	−8669.52	−0.347383
$855$	0	0
$856$	−16399.1	−0.654801
$857$	−775.719	−0.0309195	−0.0154598	−	0.999880i	$-0.504921\pi$
$857$	−775.719	−0.0309195	−0.0154598	+	0.999880i	$0.504921\pi$
$858$	−13060.1	−0.519654
$859$	−10241.5	−0.406792	−0.203396	−	0.979097i	$-0.565198\pi$
$859$	−10241.5	−0.406792	−0.203396	+	0.979097i	$0.565198\pi$
$860$	0	0
$861$	−5693.71	−0.225367
$862$	−20135.9	−0.795629
$863$	1268.51	0.0500356	0.0250178	−	0.999687i	$-0.492036\pi$
$863$	1268.51	0.0500356	0.0250178	+	0.999687i	$0.492036\pi$
$864$	93111.8	3.66635
$865$	0	0
$866$	−6731.60	−0.264144
$867$	33833.7	1.32532
$868$	7360.86	0.287839
$869$	9798.11	0.382483
$870$	0	0
$871$	−60644.6	−2.35920
$872$	−28266.1	−1.09772
$873$	−50473.8	−1.95679
$874$	−3740.41	−0.144761
$875$	0	0
$876$	−45946.5	−1.77213
$877$	−6691.33	−0.257640	−0.128820	−	0.991668i	$-0.541119\pi$
$877$	−6691.33	−0.257640	−0.128820	+	0.991668i	$0.541119\pi$
$878$	−19426.6	−0.746716
$879$	46685.3	1.79142
$880$	0	0
$881$	14514.6	0.555063	0.277531	−	0.960717i	$-0.410484\pi$
$881$	14514.6	0.555063	0.277531	+	0.960717i	$0.410484\pi$
$882$	−5702.66	−0.217708
$883$	−10335.2	−0.393891	−0.196946	−	0.980414i	$-0.563102\pi$
$883$	−10335.2	−0.393891	−0.196946	+	0.980414i	$0.563102\pi$
$884$	17143.4	0.652257
$885$	0	0
$886$	−26802.8	−1.01632
$887$	−42946.3	−1.62570	−0.812851	−	0.582472i	$-0.802086\pi$
$887$	−42946.3	−1.62570	−0.812851	+	0.582472i	$0.802086\pi$
$888$	−46439.4	−1.75496
$889$	−4108.70	−0.155007
$890$	0	0
$891$	−32862.8	−1.23563
$892$	3625.15	0.136075
$893$	−7909.87	−0.296410
$894$	−22014.9	−0.823590
$895$	0	0
$896$	8546.59	0.318662
$897$	67559.5	2.51477
$898$	23741.7	0.882263
$899$	7126.87	0.264399
$900$	0	0
$901$	5772.31	0.213434
$902$	1351.47	0.0498882
$903$	−2493.96	−0.0919089
$904$	3940.75	0.144986
$905$	0	0
$906$	6399.41	0.234665
$907$	−8376.30	−0.306649	−0.153324	−	0.988176i	$-0.548998\pi$
$907$	−8376.30	−0.306649	−0.153324	+	0.988176i	$0.548998\pi$
$908$	20452.8	0.747524
$909$	113582.	4.14443
$910$	0	0
$911$	17335.4	0.630460	0.315230	−	0.949015i	$-0.397918\pi$
$911$	17335.4	0.630460	0.315230	+	0.949015i	$0.397918\pi$
$912$	3733.65	0.135563
$913$	−6152.66	−0.223027
$914$	−11938.4	−0.432043
$915$	0	0
$916$	−14230.9	−0.513321
$917$	4361.38	0.157061
$918$	−30231.7	−1.08692
$919$	−34995.3	−1.25613	−0.628067	−	0.778159i	$-0.716154\pi$
$919$	−34995.3	−1.25613	−0.628067	+	0.778159i	$0.716154\pi$
$920$	0	0
$921$	50630.1	1.81142
$922$	13022.6	0.465159
$923$	72302.8	2.57841
$924$	−4415.05	−0.157191
$925$	0	0
$926$	1341.14	0.0475945
$927$	−66400.5	−2.35262
$928$	7182.35	0.254065
$929$	−10671.6	−0.376882	−0.188441	−	0.982084i	$-0.560343\pi$
$929$	−10671.6	−0.376882	−0.188441	+	0.982084i	$0.560343\pi$
$930$	0	0
$931$	−1371.39	−0.0482767
$932$	−23667.2	−0.831808
$933$	−31856.0	−1.11781
$934$	27053.1	0.947755
$935$	0	0
$936$	121313.	4.23636
$937$	1855.85	0.0647045	0.0323522	−	0.999477i	$-0.489700\pi$
$937$	1855.85	0.0647045	0.0323522	+	0.999477i	$0.489700\pi$
$938$	8520.19	0.296582
$939$	−47920.6	−1.66542
$940$	0	0
$941$	15378.8	0.532766	0.266383	−	0.963867i	$-0.414171\pi$
$941$	15378.8	0.532766	0.266383	+	0.963867i	$0.414171\pi$
$942$	22037.4	0.762228
$943$	−6991.16	−0.241425
$944$	1197.12	0.0412742
$945$	0	0
$946$	591.972	0.0203453
$947$	14600.9	0.501020	0.250510	−	0.968114i	$-0.419402\pi$
$947$	14600.9	0.501020	0.250510	+	0.968114i	$0.419402\pi$
$948$	−51073.3	−1.74977
$949$	−61186.9	−2.09295
$950$	0	0
$951$	60224.6	2.05354
$952$	−5818.06	−0.198072
$953$	2114.27	0.0718658	0.0359329	−	0.999354i	$-0.488560\pi$
$953$	2114.27	0.0718658	0.0359329	+	0.999354i	$0.488560\pi$
$954$	16909.7	0.573870
$955$	0	0
$956$	−14595.3	−0.493773
$957$	−4274.70	−0.144390
$958$	−4796.46	−0.161761
$959$	6684.02	0.225066
$960$	0	0
$961$	4831.40	0.162177
$962$	−25601.7	−0.858037
$963$	−59526.0	−1.99190
$964$	−8601.87	−0.287394
$965$	0	0
$966$	−9491.70	−0.316139
$967$	−7251.75	−0.241159	−0.120579	−	0.992704i	$-0.538475\pi$
$967$	−7251.75	−0.241159	−0.120579	+	0.992704i	$0.538475\pi$
$968$	2531.46	0.0840538
$969$	−11281.2	−0.373997
$970$	0	0
$971$	2742.57	0.0906420	0.0453210	−	0.998972i	$-0.485569\pi$
$971$	2742.57	0.0906420	0.0453210	+	0.998972i	$0.485569\pi$
$972$	95533.9	3.15252
$973$	11134.4	0.366856
$974$	−560.991	−0.0184551
$975$	0	0
$976$	10625.8	0.348488
$977$	−23770.4	−0.778384	−0.389192	−	0.921157i	$-0.627246\pi$
$977$	−23770.4	−0.778384	−0.389192	+	0.921157i	$0.627246\pi$
$978$	6863.34	0.224402
$979$	16761.3	0.547184
$980$	0	0
$981$	−102601.	−3.33925
$982$	22340.1	0.725967
$983$	49265.4	1.59850	0.799249	−	0.601000i	$-0.205231\pi$
$983$	49265.4	1.59850	0.799249	+	0.601000i	$0.205231\pi$
$984$	−17017.0	−0.551302
$985$	0	0
$986$	−2331.98	−0.0753198
$987$	−20072.2	−0.647319
$988$	12077.2	0.388895
$989$	−3062.26	−0.0984574
$990$	0	0
$991$	−17123.5	−0.548884	−0.274442	−	0.961604i	$-0.588493\pi$
$991$	−17123.5	−0.548884	−0.274442	+	0.961604i	$0.588493\pi$
$992$	34891.9	1.11675
$993$	64840.2	2.07215
$994$	−10158.1	−0.324140
$995$	0	0
$996$	32071.2	1.02030
$997$	41275.5	1.31114	0.655571	−	0.755134i	$-0.272428\pi$
$997$	41275.5	1.31114	0.655571	+	0.755134i	$0.272428\pi$
$998$	−14565.3	−0.461979
$999$	−108634.	−3.44048

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.4.a.p.1.2		4
5.4	even	2		77.4.a.d.1.3	✓	4
15.14	odd	2		693.4.a.l.1.2		4
20.19	odd	2		1232.4.a.s.1.1		4
35.34	odd	2		539.4.a.g.1.3		4
55.54	odd	2		847.4.a.d.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.3	✓	4	5.4	even	2
539.4.a.g.1.3		4	35.34	odd	2
693.4.a.l.1.2		4	15.14	odd	2
847.4.a.d.1.2		4	55.54	odd	2
1232.4.a.s.1.1		4	20.19	odd	2
1925.4.a.p.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53253 q^{2} -10.1459 q^{3} -5.65135 q^{4} +15.5490 q^{6} +7.00000 q^{7} +20.9211 q^{8} +75.9402 q^{9} -11.0000 q^{11} +57.3382 q^{12} +76.3572 q^{13} -10.7277 q^{14} +13.1485 q^{16} -39.7278 q^{17} -116.381 q^{18} -27.9876 q^{19} -71.0216 q^{21} +16.8579 q^{22} -87.2055 q^{23} -212.265 q^{24} -117.020 q^{26} -496.545 q^{27} -39.5594 q^{28} -38.3019 q^{29} -186.071 q^{31} -187.519 q^{32} +111.605 q^{33} +60.8842 q^{34} -429.164 q^{36} +218.781 q^{37} +42.8919 q^{38} -774.716 q^{39} +80.1687 q^{41} +108.843 q^{42} +35.1155 q^{43} +62.1648 q^{44} +133.645 q^{46} +282.620 q^{47} -133.404 q^{48} +49.0000 q^{49} +403.077 q^{51} -431.521 q^{52} -145.296 q^{53} +760.971 q^{54} +146.448 q^{56} +283.961 q^{57} +58.6989 q^{58} +91.0461 q^{59} +808.142 q^{61} +285.160 q^{62} +531.582 q^{63} +182.192 q^{64} -171.039 q^{66} -794.222 q^{67} +224.516 q^{68} +884.783 q^{69} +946.901 q^{71} +1588.75 q^{72} -801.324 q^{73} -335.288 q^{74} +158.168 q^{76} -77.0000 q^{77} +1187.28 q^{78} -890.737 q^{79} +2987.53 q^{81} -122.861 q^{82} +559.333 q^{83} +401.368 q^{84} -53.8156 q^{86} +388.609 q^{87} -230.132 q^{88} -1523.75 q^{89} +534.501 q^{91} +492.829 q^{92} +1887.87 q^{93} -433.125 q^{94} +1902.56 q^{96} -664.651 q^{97} -75.0941 q^{98} -835.342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 14 q^{3} + 26 q^{4} + 14 q^{6} + 28 q^{7} + 18 q^{8} + 76 q^{9} - 44 q^{11} - 70 q^{12} - 58 q^{13} + 14 q^{14} + 2 q^{16} - 4 q^{17} + 62 q^{18} + 258 q^{19} - 98 q^{21} - 22 q^{22} - 8 q^{23}+ \cdots - 836 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 14 * q^3 + 26 * q^4 + 14 * q^6 + 28 * q^7 + 18 * q^8 + 76 * q^9 - 44 * q^11 - 70 * q^12 - 58 * q^13 + 14 * q^14 + 2 * q^16 - 4 * q^17 + 62 * q^18 + 258 * q^19 - 98 * q^21 - 22 * q^22 - 8 * q^23 - 498 * q^24 - 482 * q^26 - 428 * q^27 + 182 * q^28 - 396 * q^29 - 56 * q^31 - 134 * q^32 + 154 * q^33 + 472 * q^34 - 418 * q^36 - 84 * q^37 + 942 * q^38 - 412 * q^39 + 52 * q^41 + 98 * q^42 - 408 * q^43 - 286 * q^44 + 368 * q^46 - 8 * q^47 - 982 * q^48 + 196 * q^49 - 388 * q^51 - 2030 * q^52 - 624 * q^53 + 92 * q^54 + 126 * q^56 - 48 * q^57 - 864 * q^58 - 238 * q^59 - 162 * q^61 - 688 * q^62 + 532 * q^63 - 902 * q^64 - 154 * q^66 - 1340 * q^67 + 1384 * q^68 + 2416 * q^69 + 1788 * q^71 + 2622 * q^72 - 1456 * q^73 + 996 * q^74 + 3042 * q^76 - 308 * q^77 + 2632 * q^78 - 1324 * q^79 + 1444 * q^81 - 1984 * q^82 - 450 * q^83 - 490 * q^84 - 4380 * q^86 - 588 * q^87 - 198 * q^88 - 3072 * q^89 - 406 * q^91 - 544 * q^92 + 1264 * q^93 - 1696 * q^94 + 862 * q^96 + 652 * q^97 + 98 * q^98 - 836 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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