LMFDB - Embedded newform 1560.4.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,4,Mod(1,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1560.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1560.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:	$92.0429796090$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 112x^{2} - 28x + 1648 $ x^4 - x^3 - 112x^2 - 28x + 1648
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.51296$ of defining polynomial
Character	$\chi$	$=$	1560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} +8.04714 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -5.00000 q^{5} +8.04714 q^{7} +9.00000 q^{9} -31.4750 q^{11} -13.0000 q^{13} +15.0000 q^{15} +32.7870 q^{17} +131.742 q^{19} -24.1414 q^{21} -175.173 q^{23} +25.0000 q^{25} -27.0000 q^{27} -96.4478 q^{29} +146.962 q^{31} +94.4249 q^{33} -40.2357 q^{35} +10.6167 q^{37} +39.0000 q^{39} +159.027 q^{41} +216.543 q^{43} -45.0000 q^{45} -16.2154 q^{47} -278.244 q^{49} -98.3609 q^{51} +181.670 q^{53} +157.375 q^{55} -395.225 q^{57} -614.668 q^{59} +712.726 q^{61} +72.4242 q^{63} +65.0000 q^{65} +696.863 q^{67} +525.520 q^{69} +961.508 q^{71} -1074.31 q^{73} -75.0000 q^{75} -253.283 q^{77} -922.879 q^{79} +81.0000 q^{81} -1223.78 q^{83} -163.935 q^{85} +289.343 q^{87} -433.003 q^{89} -104.613 q^{91} -440.886 q^{93} -658.708 q^{95} -774.186 q^{97} -283.275 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9	$ 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9} + 35 q^{11} - 52 q^{13} + 60 q^{15} - 9 q^{17} + 32 q^{19} + 33 q^{21} - 149 q^{23} + 100 q^{25} - 108 q^{27} + 168 q^{29} + 280 q^{31} - 105 q^{33} + 55 q^{35} + 37 q^{37} + 156 q^{39} + 247 q^{41} + 132 q^{43} - 180 q^{45} + 217 q^{49} + 27 q^{51} + 247 q^{53} - 175 q^{55} - 96 q^{57} + 614 q^{59} + 719 q^{61} - 99 q^{63} + 260 q^{65} + 658 q^{67} + 447 q^{69} + 939 q^{71} - 1452 q^{73} - 300 q^{75} + 199 q^{77} + 289 q^{79} + 324 q^{81} - 118 q^{83} + 45 q^{85} - 504 q^{87} - 1319 q^{89} + 143 q^{91} - 840 q^{93} - 160 q^{95} - 993 q^{97} + 315 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9 + 35 * q^11 - 52 * q^13 + 60 * q^15 - 9 * q^17 + 32 * q^19 + 33 * q^21 - 149 * q^23 + 100 * q^25 - 108 * q^27 + 168 * q^29 + 280 * q^31 - 105 * q^33 + 55 * q^35 + 37 * q^37 + 156 * q^39 + 247 * q^41 + 132 * q^43 - 180 * q^45 + 217 * q^49 + 27 * q^51 + 247 * q^53 - 175 * q^55 - 96 * q^57 + 614 * q^59 + 719 * q^61 - 99 * q^63 + 260 * q^65 + 658 * q^67 + 447 * q^69 + 939 * q^71 - 1452 * q^73 - 300 * q^75 + 199 * q^77 + 289 * q^79 + 324 * q^81 - 118 * q^83 + 45 * q^85 - 504 * q^87 - 1319 * q^89 + 143 * q^91 - 840 * q^93 - 160 * q^95 - 993 * q^97 + 315 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	8.04714	0.434505	0.217252	−	0.976115i	$-0.430291\pi$
$7$	8.04714	0.434505	0.217252	+	0.976115i	$0.430291\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−31.4750	−0.862732	−0.431366	−	0.902177i	$-0.641968\pi$
$11$	−31.4750	−0.862732	−0.431366	+	0.902177i	$0.641968\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	32.7870	0.467765	0.233883	−	0.972265i	$-0.424857\pi$
$17$	32.7870	0.467765	0.233883	+	0.972265i	$0.424857\pi$
$18$	0	0
$19$	131.742	1.59072	0.795358	−	0.606140i	$-0.207283\pi$
$19$	131.742	1.59072	0.795358	+	0.606140i	$0.207283\pi$
$20$	0	0
$21$	−24.1414	−0.250861
$22$	0	0
$23$	−175.173	−1.58810	−0.794048	−	0.607855i	$-0.792030\pi$
$23$	−175.173	−1.58810	−0.794048	+	0.607855i	$0.792030\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−96.4478	−0.617583	−0.308792	−	0.951130i	$-0.599925\pi$
$29$	−96.4478	−0.617583	−0.308792	+	0.951130i	$0.599925\pi$
$30$	0	0
$31$	146.962	0.851457	0.425728	−	0.904851i	$-0.360018\pi$
$31$	146.962	0.851457	0.425728	+	0.904851i	$0.360018\pi$
$32$	0	0
$33$	94.4249	0.498099
$34$	0	0
$35$	−40.2357	−0.194316
$36$	0	0
$37$	10.6167	0.0471721	0.0235861	−	0.999722i	$-0.492492\pi$
$37$	10.6167	0.0471721	0.0235861	+	0.999722i	$0.492492\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	159.027	0.605754	0.302877	−	0.953030i	$-0.402053\pi$
$41$	159.027	0.605754	0.302877	+	0.953030i	$0.402053\pi$
$42$	0	0
$43$	216.543	0.767964	0.383982	−	0.923341i	$-0.374552\pi$
$43$	216.543	0.767964	0.383982	+	0.923341i	$0.374552\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−16.2154	−0.0503246	−0.0251623	−	0.999683i	$-0.508010\pi$
$47$	−16.2154	−0.0503246	−0.0251623	+	0.999683i	$0.508010\pi$
$48$	0	0
$49$	−278.244	−0.811206
$50$	0	0
$51$	−98.3609	−0.270064
$52$	0	0
$53$	181.670	0.470836	0.235418	−	0.971894i	$-0.424354\pi$
$53$	181.670	0.470836	0.235418	+	0.971894i	$0.424354\pi$
$54$	0	0
$55$	157.375	0.385826
$56$	0	0
$57$	−395.225	−0.918400
$58$	0	0
$59$	−614.668	−1.35632	−0.678160	−	0.734914i	$-0.737223\pi$
$59$	−614.668	−1.35632	−0.678160	+	0.734914i	$0.737223\pi$
$60$	0	0
$61$	712.726	1.49599	0.747993	−	0.663707i	$-0.231018\pi$
$61$	712.726	1.49599	0.747993	+	0.663707i	$0.231018\pi$
$62$	0	0
$63$	72.4242	0.144835
$64$	0	0
$65$	65.0000	0.124035
$66$	0	0
$67$	696.863	1.27068	0.635339	−	0.772233i	$-0.280860\pi$
$67$	696.863	1.27068	0.635339	+	0.772233i	$0.280860\pi$
$68$	0	0
$69$	525.520	0.916887
$70$	0	0
$71$	961.508	1.60718	0.803592	−	0.595181i	$-0.202919\pi$
$71$	961.508	1.60718	0.803592	+	0.595181i	$0.202919\pi$
$72$	0	0
$73$	−1074.31	−1.72244	−0.861219	−	0.508233i	$-0.830299\pi$
$73$	−1074.31	−1.72244	−0.861219	+	0.508233i	$0.830299\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−253.283	−0.374861
$78$	0	0
$79$	−922.879	−1.31433	−0.657165	−	0.753747i	$-0.728244\pi$
$79$	−922.879	−1.31433	−0.657165	+	0.753747i	$0.728244\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1223.78	−1.61841	−0.809204	−	0.587528i	$-0.800101\pi$
$83$	−1223.78	−1.61841	−0.809204	+	0.587528i	$0.800101\pi$
$84$	0	0
$85$	−163.935	−0.209191
$86$	0	0
$87$	289.343	0.356562
$88$	0	0
$89$	−433.003	−0.515710	−0.257855	−	0.966184i	$-0.583016\pi$
$89$	−433.003	−0.515710	−0.257855	+	0.966184i	$0.583016\pi$
$90$	0	0
$91$	−104.613	−0.120510
$92$	0	0
$93$	−440.886	−0.491589
$94$	0	0
$95$	−658.708	−0.711390
$96$	0	0
$97$	−774.186	−0.810379	−0.405189	−	0.914233i	$-0.632794\pi$
$97$	−774.186	−0.810379	−0.405189	+	0.914233i	$0.632794\pi$
$98$	0	0
$99$	−283.275	−0.287577
$100$	0	0
$101$	871.106	0.858201	0.429101	−	0.903257i	$-0.358831\pi$
$101$	871.106	0.858201	0.429101	+	0.903257i	$0.358831\pi$
$102$	0	0
$103$	1094.88	1.04739	0.523696	−	0.851905i	$-0.324553\pi$
$103$	1094.88	1.04739	0.523696	+	0.851905i	$0.324553\pi$
$104$	0	0
$105$	120.707	0.112189
$106$	0	0
$107$	1380.21	1.24701	0.623506	−	0.781818i	$-0.285708\pi$
$107$	1380.21	1.24701	0.623506	+	0.781818i	$0.285708\pi$
$108$	0	0
$109$	418.279	0.367559	0.183779	−	0.982968i	$-0.441167\pi$
$109$	418.279	0.367559	0.183779	+	0.982968i	$0.441167\pi$
$110$	0	0
$111$	−31.8500	−0.0272348
$112$	0	0
$113$	1016.51	0.846242	0.423121	−	0.906073i	$-0.360935\pi$
$113$	1016.51	0.846242	0.423121	+	0.906073i	$0.360935\pi$
$114$	0	0
$115$	875.867	0.710218
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	263.841	0.203246
$120$	0	0
$121$	−340.327	−0.255693
$122$	0	0
$123$	−477.082	−0.349732
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	443.241	0.309695	0.154848	−	0.987938i	$-0.450511\pi$
$127$	443.241	0.309695	0.154848	+	0.987938i	$0.450511\pi$
$128$	0	0
$129$	−649.628	−0.443384
$130$	0	0
$131$	−983.654	−0.656048	−0.328024	−	0.944669i	$-0.606383\pi$
$131$	−983.654	−0.656048	−0.328024	+	0.944669i	$0.606383\pi$
$132$	0	0
$133$	1060.14	0.691173
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−2124.09	−1.32462	−0.662311	−	0.749229i	$-0.730424\pi$
$137$	−2124.09	−1.32462	−0.662311	+	0.749229i	$0.730424\pi$
$138$	0	0
$139$	−613.023	−0.374071	−0.187036	−	0.982353i	$-0.559888\pi$
$139$	−613.023	−0.374071	−0.187036	+	0.982353i	$0.559888\pi$
$140$	0	0
$141$	48.6461	0.0290549
$142$	0	0
$143$	409.174	0.239279
$144$	0	0
$145$	482.239	0.276192
$146$	0	0
$147$	834.731	0.468350
$148$	0	0
$149$	−1644.75	−0.904316	−0.452158	−	0.891938i	$-0.649346\pi$
$149$	−1644.75	−0.904316	−0.452158	+	0.891938i	$0.649346\pi$
$150$	0	0
$151$	−2571.21	−1.38571	−0.692856	−	0.721076i	$-0.743648\pi$
$151$	−2571.21	−1.38571	−0.692856	+	0.721076i	$0.743648\pi$
$152$	0	0
$153$	295.083	0.155922
$154$	0	0
$155$	−734.810	−0.380783
$156$	0	0
$157$	217.661	0.110645	0.0553223	−	0.998469i	$-0.482381\pi$
$157$	217.661	0.110645	0.0553223	+	0.998469i	$0.482381\pi$
$158$	0	0
$159$	−545.011	−0.271838
$160$	0	0
$161$	−1409.64	−0.690035
$162$	0	0
$163$	1271.64	0.611058	0.305529	−	0.952183i	$-0.401167\pi$
$163$	1271.64	0.611058	0.305529	+	0.952183i	$0.401167\pi$
$164$	0	0
$165$	−472.124	−0.222757
$166$	0	0
$167$	−3513.60	−1.62809	−0.814044	−	0.580803i	$-0.802739\pi$
$167$	−3513.60	−1.62809	−0.814044	+	0.580803i	$0.802739\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	1185.67	0.530238
$172$	0	0
$173$	−1974.87	−0.867898	−0.433949	−	0.900937i	$-0.642880\pi$
$173$	−1974.87	−0.867898	−0.433949	+	0.900937i	$0.642880\pi$
$174$	0	0
$175$	201.178	0.0869009
$176$	0	0
$177$	1844.00	0.783072
$178$	0	0
$179$	−2364.94	−0.987510	−0.493755	−	0.869601i	$-0.664376\pi$
$179$	−2364.94	−0.987510	−0.493755	+	0.869601i	$0.664376\pi$
$180$	0	0
$181$	−4660.55	−1.91390	−0.956949	−	0.290255i	$-0.906260\pi$
$181$	−4660.55	−1.91390	−0.956949	+	0.290255i	$0.906260\pi$
$182$	0	0
$183$	−2138.18	−0.863708
$184$	0	0
$185$	−53.0833	−0.0210960
$186$	0	0
$187$	−1031.97	−0.403556
$188$	0	0
$189$	−217.273	−0.0836204
$190$	0	0
$191$	−1110.30	−0.420620	−0.210310	−	0.977635i	$-0.567447\pi$
$191$	−1110.30	−0.420620	−0.210310	+	0.977635i	$0.567447\pi$
$192$	0	0
$193$	−2506.53	−0.934840	−0.467420	−	0.884035i	$-0.654816\pi$
$193$	−2506.53	−0.934840	−0.467420	+	0.884035i	$0.654816\pi$
$194$	0	0
$195$	−195.000	−0.0716115
$196$	0	0
$197$	−2843.58	−1.02841	−0.514205	−	0.857667i	$-0.671913\pi$
$197$	−2843.58	−1.02841	−0.514205	+	0.857667i	$0.671913\pi$
$198$	0	0
$199$	2700.98	0.962149	0.481074	−	0.876680i	$-0.340247\pi$
$199$	2700.98	0.962149	0.481074	+	0.876680i	$0.340247\pi$
$200$	0	0
$201$	−2090.59	−0.733626
$202$	0	0
$203$	−776.129	−0.268343
$204$	0	0
$205$	−795.137	−0.270901
$206$	0	0
$207$	−1576.56	−0.529365
$208$	0	0
$209$	−4146.56	−1.37236
$210$	0	0
$211$	491.500	0.160362	0.0801808	−	0.996780i	$-0.474450\pi$
$211$	491.500	0.160362	0.0801808	+	0.996780i	$0.474450\pi$
$212$	0	0
$213$	−2884.53	−0.927908
$214$	0	0
$215$	−1082.71	−0.343444
$216$	0	0
$217$	1182.62	0.369962
$218$	0	0
$219$	3222.92	0.994451
$220$	0	0
$221$	−426.231	−0.129735
$222$	0	0
$223$	−5864.84	−1.76116	−0.880580	−	0.473897i	$-0.842847\pi$
$223$	−5864.84	−1.76116	−0.880580	+	0.473897i	$0.842847\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	5392.34	1.57666	0.788331	−	0.615252i	$-0.210946\pi$
$227$	5392.34	1.57666	0.788331	+	0.615252i	$0.210946\pi$
$228$	0	0
$229$	−4261.88	−1.22984	−0.614919	−	0.788590i	$-0.710811\pi$
$229$	−4261.88	−1.22984	−0.614919	+	0.788590i	$0.710811\pi$
$230$	0	0
$231$	759.850	0.216426
$232$	0	0
$233$	−3726.18	−1.04768	−0.523842	−	0.851816i	$-0.675502\pi$
$233$	−3726.18	−1.04768	−0.523842	+	0.851816i	$0.675502\pi$
$234$	0	0
$235$	81.0769	0.0225058
$236$	0	0
$237$	2768.64	0.758829
$238$	0	0
$239$	5839.92	1.58056	0.790278	−	0.612748i	$-0.209936\pi$
$239$	5839.92	1.58056	0.790278	+	0.612748i	$0.209936\pi$
$240$	0	0
$241$	−1478.57	−0.395200	−0.197600	−	0.980283i	$-0.563315\pi$
$241$	−1478.57	−0.395200	−0.197600	+	0.980283i	$0.563315\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1391.22	0.362782
$246$	0	0
$247$	−1712.64	−0.441185
$248$	0	0
$249$	3671.35	0.934388
$250$	0	0
$251$	6859.47	1.72496	0.862482	−	0.506087i	$-0.168909\pi$
$251$	6859.47	1.72496	0.862482	+	0.506087i	$0.168909\pi$
$252$	0	0
$253$	5513.58	1.37010
$254$	0	0
$255$	491.805	0.120776
$256$	0	0
$257$	−5497.48	−1.33433	−0.667167	−	0.744909i	$-0.732493\pi$
$257$	−5497.48	−1.33433	−0.667167	+	0.744909i	$0.732493\pi$
$258$	0	0
$259$	85.4338	0.0204965
$260$	0	0
$261$	−868.030	−0.205861
$262$	0	0
$263$	−3618.12	−0.848299	−0.424150	−	0.905592i	$-0.639427\pi$
$263$	−3618.12	−0.848299	−0.424150	+	0.905592i	$0.639427\pi$
$264$	0	0
$265$	−908.351	−0.210564
$266$	0	0
$267$	1299.01	0.297745
$268$	0	0
$269$	2039.35	0.462235	0.231117	−	0.972926i	$-0.425762\pi$
$269$	2039.35	0.462235	0.231117	+	0.972926i	$0.425762\pi$
$270$	0	0
$271$	−220.548	−0.0494367	−0.0247183	−	0.999694i	$-0.507869\pi$
$271$	−220.548	−0.0494367	−0.0247183	+	0.999694i	$0.507869\pi$
$272$	0	0
$273$	313.838	0.0695764
$274$	0	0
$275$	−786.874	−0.172546
$276$	0	0
$277$	−2715.30	−0.588977	−0.294489	−	0.955655i	$-0.595149\pi$
$277$	−2715.30	−0.588977	−0.294489	+	0.955655i	$0.595149\pi$
$278$	0	0
$279$	1322.66	0.283819
$280$	0	0
$281$	−6718.55	−1.42632	−0.713158	−	0.701003i	$-0.752736\pi$
$281$	−6718.55	−1.42632	−0.713158	+	0.701003i	$0.752736\pi$
$282$	0	0
$283$	−1628.80	−0.342127	−0.171064	−	0.985260i	$-0.554720\pi$
$283$	−1628.80	−0.342127	−0.171064	+	0.985260i	$0.554720\pi$
$284$	0	0
$285$	1976.12	0.410721
$286$	0	0
$287$	1279.71	0.263203
$288$	0	0
$289$	−3838.01	−0.781196
$290$	0	0
$291$	2322.56	0.467872
$292$	0	0
$293$	−5515.81	−1.09979	−0.549893	−	0.835235i	$-0.685332\pi$
$293$	−5515.81	−1.09979	−0.549893	+	0.835235i	$0.685332\pi$
$294$	0	0
$295$	3073.34	0.606565
$296$	0	0
$297$	849.824	0.166033
$298$	0	0
$299$	2277.26	0.440458
$300$	0	0
$301$	1742.55	0.333684
$302$	0	0
$303$	−2613.32	−0.495483
$304$	0	0
$305$	−3563.63	−0.669025
$306$	0	0
$307$	3114.98	0.579092	0.289546	−	0.957164i	$-0.406496\pi$
$307$	3114.98	0.579092	0.289546	+	0.957164i	$0.406496\pi$
$308$	0	0
$309$	−3284.63	−0.604712
$310$	0	0
$311$	−2260.53	−0.412163	−0.206082	−	0.978535i	$-0.566071\pi$
$311$	−2260.53	−0.412163	−0.206082	+	0.978535i	$0.566071\pi$
$312$	0	0
$313$	−1928.70	−0.348296	−0.174148	−	0.984720i	$-0.555717\pi$
$313$	−1928.70	−0.348296	−0.174148	+	0.984720i	$0.555717\pi$
$314$	0	0
$315$	−362.121	−0.0647721
$316$	0	0
$317$	2039.86	0.361420	0.180710	−	0.983536i	$-0.442161\pi$
$317$	2039.86	0.361420	0.180710	+	0.983536i	$0.442161\pi$
$318$	0	0
$319$	3035.69	0.532809
$320$	0	0
$321$	−4140.64	−0.719963
$322$	0	0
$323$	4319.41	0.744081
$324$	0	0
$325$	−325.000	−0.0554700
$326$	0	0
$327$	−1254.84	−0.212210
$328$	0	0
$329$	−130.487	−0.0218663
$330$	0	0
$331$	9077.27	1.50735	0.753673	−	0.657249i	$-0.228280\pi$
$331$	9077.27	1.50735	0.753673	+	0.657249i	$0.228280\pi$
$332$	0	0
$333$	95.5500	0.0157240
$334$	0	0
$335$	−3484.32	−0.568264
$336$	0	0
$337$	−4337.74	−0.701162	−0.350581	−	0.936532i	$-0.614016\pi$
$337$	−4337.74	−0.701162	−0.350581	+	0.936532i	$0.614016\pi$
$338$	0	0
$339$	−3049.53	−0.488578
$340$	0	0
$341$	−4625.62	−0.734579
$342$	0	0
$343$	−4999.23	−0.786977
$344$	0	0
$345$	−2627.60	−0.410045
$346$	0	0
$347$	−6698.91	−1.03636	−0.518179	−	0.855272i	$-0.673390\pi$
$347$	−6698.91	−1.03636	−0.518179	+	0.855272i	$0.673390\pi$
$348$	0	0
$349$	−1977.41	−0.303290	−0.151645	−	0.988435i	$-0.548457\pi$
$349$	−1977.41	−0.303290	−0.151645	+	0.988435i	$0.548457\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	−4077.25	−0.614759	−0.307379	−	0.951587i	$-0.599452\pi$
$353$	−4077.25	−0.614759	−0.307379	+	0.951587i	$0.599452\pi$
$354$	0	0
$355$	−4807.54	−0.718755
$356$	0	0
$357$	−791.524	−0.117344
$358$	0	0
$359$	2846.58	0.418486	0.209243	−	0.977864i	$-0.432900\pi$
$359$	2846.58	0.418486	0.209243	+	0.977864i	$0.432900\pi$
$360$	0	0
$361$	10496.8	1.53038
$362$	0	0
$363$	1020.98	0.147624
$364$	0	0
$365$	5371.53	0.770298
$366$	0	0
$367$	−7181.63	−1.02147	−0.510733	−	0.859739i	$-0.670626\pi$
$367$	−7181.63	−1.02147	−0.510733	+	0.859739i	$0.670626\pi$
$368$	0	0
$369$	1431.25	0.201918
$370$	0	0
$371$	1461.93	0.204581
$372$	0	0
$373$	−6341.32	−0.880270	−0.440135	−	0.897932i	$-0.645069\pi$
$373$	−6341.32	−0.880270	−0.440135	+	0.897932i	$0.645069\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	1253.82	0.171287
$378$	0	0
$379$	−3868.43	−0.524295	−0.262148	−	0.965028i	$-0.584431\pi$
$379$	−3868.43	−0.524295	−0.262148	+	0.965028i	$0.584431\pi$
$380$	0	0
$381$	−1329.72	−0.178803
$382$	0	0
$383$	−6517.06	−0.869468	−0.434734	−	0.900559i	$-0.643158\pi$
$383$	−6517.06	−0.869468	−0.434734	+	0.900559i	$0.643158\pi$
$384$	0	0
$385$	1266.42	0.167643
$386$	0	0
$387$	1948.88	0.255988
$388$	0	0
$389$	9142.50	1.19163	0.595814	−	0.803123i	$-0.296830\pi$
$389$	9142.50	1.19163	0.595814	+	0.803123i	$0.296830\pi$
$390$	0	0
$391$	−5743.41	−0.742856
$392$	0	0
$393$	2950.96	0.378769
$394$	0	0
$395$	4614.40	0.587786
$396$	0	0
$397$	15593.1	1.97127	0.985635	−	0.168890i	$-0.0540184\pi$
$397$	15593.1	1.97127	0.985635	+	0.168890i	$0.0540184\pi$
$398$	0	0
$399$	−3180.43	−0.399049
$400$	0	0
$401$	12089.7	1.50556	0.752782	−	0.658270i	$-0.228711\pi$
$401$	12089.7	1.50556	0.752782	+	0.658270i	$0.228711\pi$
$402$	0	0
$403$	−1910.51	−0.236152
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−334.159	−0.0406969
$408$	0	0
$409$	−8893.70	−1.07522	−0.537610	−	0.843193i	$-0.680673\pi$
$409$	−8893.70	−1.07522	−0.537610	+	0.843193i	$0.680673\pi$
$410$	0	0
$411$	6372.27	0.764771
$412$	0	0
$413$	−4946.32	−0.589328
$414$	0	0
$415$	6118.92	0.723774
$416$	0	0
$417$	1839.07	0.215970
$418$	0	0
$419$	2919.95	0.340450	0.170225	−	0.985405i	$-0.445550\pi$
$419$	2919.95	0.340450	0.170225	+	0.985405i	$0.445550\pi$
$420$	0	0
$421$	9868.14	1.14238	0.571192	−	0.820816i	$-0.306481\pi$
$421$	9868.14	1.14238	0.571192	+	0.820816i	$0.306481\pi$
$422$	0	0
$423$	−145.938	−0.0167749
$424$	0	0
$425$	819.674	0.0935530
$426$	0	0
$427$	5735.40	0.650013
$428$	0	0
$429$	−1227.52	−0.138148
$430$	0	0
$431$	−8855.77	−0.989716	−0.494858	−	0.868974i	$-0.664780\pi$
$431$	−8855.77	−0.989716	−0.494858	+	0.868974i	$0.664780\pi$
$432$	0	0
$433$	−4684.12	−0.519872	−0.259936	−	0.965626i	$-0.583701\pi$
$433$	−4684.12	−0.519872	−0.259936	+	0.965626i	$0.583701\pi$
$434$	0	0
$435$	−1446.72	−0.159459
$436$	0	0
$437$	−23077.6	−2.52621
$438$	0	0
$439$	14544.7	1.58127	0.790637	−	0.612286i	$-0.209750\pi$
$439$	14544.7	1.58127	0.790637	+	0.612286i	$0.209750\pi$
$440$	0	0
$441$	−2504.19	−0.270402
$442$	0	0
$443$	−15590.1	−1.67203	−0.836014	−	0.548708i	$-0.815120\pi$
$443$	−15590.1	−1.67203	−0.836014	+	0.548708i	$0.815120\pi$
$444$	0	0
$445$	2165.01	0.230632
$446$	0	0
$447$	4934.25	0.522107
$448$	0	0
$449$	−1748.33	−0.183761	−0.0918807	−	0.995770i	$-0.529288\pi$
$449$	−1748.33	−0.183761	−0.0918807	+	0.995770i	$0.529288\pi$
$450$	0	0
$451$	−5005.38	−0.522603
$452$	0	0
$453$	7713.64	0.800041
$454$	0	0
$455$	523.064	0.0538937
$456$	0	0
$457$	−3671.93	−0.375855	−0.187928	−	0.982183i	$-0.560177\pi$
$457$	−3671.93	−0.375855	−0.187928	+	0.982183i	$0.560177\pi$
$458$	0	0
$459$	−885.248	−0.0900215
$460$	0	0
$461$	−17803.1	−1.79864	−0.899319	−	0.437293i	$-0.855937\pi$
$461$	−17803.1	−1.79864	−0.899319	+	0.437293i	$0.855937\pi$
$462$	0	0
$463$	13775.0	1.38268	0.691339	−	0.722531i	$-0.257021\pi$
$463$	13775.0	1.38268	0.691339	+	0.722531i	$0.257021\pi$
$464$	0	0
$465$	2204.43	0.219845
$466$	0	0
$467$	−8847.85	−0.876723	−0.438362	−	0.898799i	$-0.644441\pi$
$467$	−8847.85	−0.876723	−0.438362	+	0.898799i	$0.644441\pi$
$468$	0	0
$469$	5607.75	0.552115
$470$	0	0
$471$	−652.982	−0.0638807
$472$	0	0
$473$	−6815.67	−0.662547
$474$	0	0
$475$	3293.54	0.318143
$476$	0	0
$477$	1635.03	0.156945
$478$	0	0
$479$	10474.1	0.999114	0.499557	−	0.866281i	$-0.333496\pi$
$479$	10474.1	0.999114	0.499557	+	0.866281i	$0.333496\pi$
$480$	0	0
$481$	−138.017	−0.0130832
$482$	0	0
$483$	4228.93	0.398392
$484$	0	0
$485$	3870.93	0.362412
$486$	0	0
$487$	1436.15	0.133631	0.0668153	−	0.997765i	$-0.478716\pi$
$487$	1436.15	0.133631	0.0668153	+	0.997765i	$0.478716\pi$
$488$	0	0
$489$	−3814.92	−0.352795
$490$	0	0
$491$	−3065.83	−0.281790	−0.140895	−	0.990025i	$-0.544998\pi$
$491$	−3065.83	−0.281790	−0.140895	+	0.990025i	$0.544998\pi$
$492$	0	0
$493$	−3162.23	−0.288884
$494$	0	0
$495$	1416.37	0.128609
$496$	0	0
$497$	7737.39	0.698329
$498$	0	0
$499$	17556.2	1.57500	0.787499	−	0.616316i	$-0.211376\pi$
$499$	17556.2	1.57500	0.787499	+	0.616316i	$0.211376\pi$
$500$	0	0
$501$	10540.8	0.939978
$502$	0	0
$503$	21156.2	1.87536	0.937680	−	0.347499i	$-0.112969\pi$
$503$	21156.2	1.87536	0.937680	+	0.347499i	$0.112969\pi$
$504$	0	0
$505$	−4355.53	−0.383799
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−11047.7	−0.962048	−0.481024	−	0.876707i	$-0.659735\pi$
$509$	−11047.7	−0.962048	−0.481024	+	0.876707i	$0.659735\pi$
$510$	0	0
$511$	−8645.09	−0.748407
$512$	0	0
$513$	−3557.02	−0.306133
$514$	0	0
$515$	−5474.38	−0.468408
$516$	0	0
$517$	510.378	0.0434167
$518$	0	0
$519$	5924.60	0.501081
$520$	0	0
$521$	−236.255	−0.0198667	−0.00993334	−	0.999951i	$-0.503162\pi$
$521$	−236.255	−0.0198667	−0.00993334	+	0.999951i	$0.503162\pi$
$522$	0	0
$523$	−14127.1	−1.18114	−0.590568	−	0.806988i	$-0.701096\pi$
$523$	−14127.1	−1.18114	−0.590568	+	0.806988i	$0.701096\pi$
$524$	0	0
$525$	−603.535	−0.0501723
$526$	0	0
$527$	4818.44	0.398282
$528$	0	0
$529$	18518.8	1.52205
$530$	0	0
$531$	−5532.01	−0.452107
$532$	0	0
$533$	−2067.36	−0.168006
$534$	0	0
$535$	−6901.07	−0.557681
$536$	0	0
$537$	7094.83	0.570139
$538$	0	0
$539$	8757.70	0.699853
$540$	0	0
$541$	10827.0	0.860424	0.430212	−	0.902728i	$-0.358439\pi$
$541$	10827.0	0.860424	0.430212	+	0.902728i	$0.358439\pi$
$542$	0	0
$543$	13981.6	1.10499
$544$	0	0
$545$	−2091.40	−0.164377
$546$	0	0
$547$	11404.1	0.891418	0.445709	−	0.895178i	$-0.352952\pi$
$547$	11404.1	0.891418	0.445709	+	0.895178i	$0.352952\pi$
$548$	0	0
$549$	6414.53	0.498662
$550$	0	0
$551$	−12706.2	−0.982399
$552$	0	0
$553$	−7426.54	−0.571082
$554$	0	0
$555$	159.250	0.0121798
$556$	0	0
$557$	6413.38	0.487870	0.243935	−	0.969792i	$-0.421562\pi$
$557$	6413.38	0.487870	0.243935	+	0.969792i	$0.421562\pi$
$558$	0	0
$559$	−2815.06	−0.212995
$560$	0	0
$561$	3095.90	0.232993
$562$	0	0
$563$	−4104.03	−0.307219	−0.153609	−	0.988132i	$-0.549090\pi$
$563$	−4104.03	−0.307219	−0.153609	+	0.988132i	$0.549090\pi$
$564$	0	0
$565$	−5082.56	−0.378451
$566$	0	0
$567$	651.818	0.0482783
$568$	0	0
$569$	−12016.0	−0.885305	−0.442653	−	0.896693i	$-0.645962\pi$
$569$	−12016.0	−0.885305	−0.442653	+	0.896693i	$0.645962\pi$
$570$	0	0
$571$	2426.47	0.177836	0.0889182	−	0.996039i	$-0.471659\pi$
$571$	2426.47	0.177836	0.0889182	+	0.996039i	$0.471659\pi$
$572$	0	0
$573$	3330.90	0.242845
$574$	0	0
$575$	−4379.34	−0.317619
$576$	0	0
$577$	−9824.05	−0.708805	−0.354403	−	0.935093i	$-0.615316\pi$
$577$	−9824.05	−0.708805	−0.354403	+	0.935093i	$0.615316\pi$
$578$	0	0
$579$	7519.59	0.539730
$580$	0	0
$581$	−9847.96	−0.703205
$582$	0	0
$583$	−5718.06	−0.406206
$584$	0	0
$585$	585.000	0.0413449
$586$	0	0
$587$	17204.8	1.20974	0.604871	−	0.796323i	$-0.293225\pi$
$587$	17204.8	1.20974	0.604871	+	0.796323i	$0.293225\pi$
$588$	0	0
$589$	19361.0	1.35443
$590$	0	0
$591$	8530.75	0.593753
$592$	0	0
$593$	20751.7	1.43705	0.718526	−	0.695500i	$-0.244817\pi$
$593$	20751.7	1.43705	0.718526	+	0.695500i	$0.244817\pi$
$594$	0	0
$595$	−1319.21	−0.0908944
$596$	0	0
$597$	−8102.95	−0.555497
$598$	0	0
$599$	21872.0	1.49193	0.745966	−	0.665984i	$-0.231988\pi$
$599$	21872.0	1.49193	0.745966	+	0.665984i	$0.231988\pi$
$600$	0	0
$601$	−9735.69	−0.660777	−0.330388	−	0.943845i	$-0.607180\pi$
$601$	−9735.69	−0.660777	−0.330388	+	0.943845i	$0.607180\pi$
$602$	0	0
$603$	6271.77	0.423559
$604$	0	0
$605$	1701.64	0.114349
$606$	0	0
$607$	13258.6	0.886575	0.443287	−	0.896380i	$-0.353812\pi$
$607$	13258.6	0.886575	0.443287	+	0.896380i	$0.353812\pi$
$608$	0	0
$609$	2328.39	0.154928
$610$	0	0
$611$	210.800	0.0139575
$612$	0	0
$613$	7309.90	0.481638	0.240819	−	0.970570i	$-0.422584\pi$
$613$	7309.90	0.481638	0.240819	+	0.970570i	$0.422584\pi$
$614$	0	0
$615$	2385.41	0.156405
$616$	0	0
$617$	−15920.2	−1.03877	−0.519386	−	0.854540i	$-0.673839\pi$
$617$	−15920.2	−1.03877	−0.519386	+	0.854540i	$0.673839\pi$
$618$	0	0
$619$	−27088.6	−1.75894	−0.879471	−	0.475953i	$-0.842103\pi$
$619$	−27088.6	−1.75894	−0.879471	+	0.475953i	$0.842103\pi$
$620$	0	0
$621$	4729.68	0.305629
$622$	0	0
$623$	−3484.43	−0.224078
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	12439.7	0.792333
$628$	0	0
$629$	348.088	0.0220655
$630$	0	0
$631$	27606.1	1.74165	0.870825	−	0.491593i	$-0.163585\pi$
$631$	27606.1	1.74165	0.870825	+	0.491593i	$0.163585\pi$
$632$	0	0
$633$	−1474.50	−0.0925848
$634$	0	0
$635$	−2216.21	−0.138500
$636$	0	0
$637$	3617.17	0.224988
$638$	0	0
$639$	8653.58	0.535728
$640$	0	0
$641$	7025.18	0.432883	0.216441	−	0.976296i	$-0.430555\pi$
$641$	7025.18	0.432883	0.216441	+	0.976296i	$0.430555\pi$
$642$	0	0
$643$	−12833.3	−0.787085	−0.393542	−	0.919306i	$-0.628751\pi$
$643$	−12833.3	−0.787085	−0.393542	+	0.919306i	$0.628751\pi$
$644$	0	0
$645$	3248.14	0.198287
$646$	0	0
$647$	7606.91	0.462224	0.231112	−	0.972927i	$-0.425764\pi$
$647$	7606.91	0.462224	0.231112	+	0.972927i	$0.425764\pi$
$648$	0	0
$649$	19346.6	1.17014
$650$	0	0
$651$	−3547.87	−0.213597
$652$	0	0
$653$	−1233.23	−0.0739048	−0.0369524	−	0.999317i	$-0.511765\pi$
$653$	−1233.23	−0.0739048	−0.0369524	+	0.999317i	$0.511765\pi$
$654$	0	0
$655$	4918.27	0.293393
$656$	0	0
$657$	−9668.76	−0.574146
$658$	0	0
$659$	3168.19	0.187276	0.0936381	−	0.995606i	$-0.470150\pi$
$659$	3168.19	0.187276	0.0936381	+	0.995606i	$0.470150\pi$
$660$	0	0
$661$	32831.9	1.93194	0.965971	−	0.258650i	$-0.0832777\pi$
$661$	32831.9	1.93194	0.965971	+	0.258650i	$0.0832777\pi$
$662$	0	0
$663$	1278.69	0.0749024
$664$	0	0
$665$	−5300.71	−0.309102
$666$	0	0
$667$	16895.1	0.980781
$668$	0	0
$669$	17594.5	1.01681
$670$	0	0
$671$	−22433.0	−1.29064
$672$	0	0
$673$	6736.91	0.385868	0.192934	−	0.981212i	$-0.438200\pi$
$673$	6736.91	0.385868	0.192934	+	0.981212i	$0.438200\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−4074.80	−0.231325	−0.115663	−	0.993289i	$-0.536899\pi$
$677$	−4074.80	−0.231325	−0.115663	+	0.993289i	$0.536899\pi$
$678$	0	0
$679$	−6229.98	−0.352113
$680$	0	0
$681$	−16177.0	−0.910286
$682$	0	0
$683$	−35020.5	−1.96197	−0.980983	−	0.194095i	$-0.937823\pi$
$683$	−35020.5	−1.96197	−0.980983	+	0.194095i	$0.937823\pi$
$684$	0	0
$685$	10620.5	0.592389
$686$	0	0
$687$	12785.6	0.710047
$688$	0	0
$689$	−2361.71	−0.130587
$690$	0	0
$691$	9113.71	0.501739	0.250870	−	0.968021i	$-0.419283\pi$
$691$	9113.71	0.501739	0.250870	+	0.968021i	$0.419283\pi$
$692$	0	0
$693$	−2279.55	−0.124954
$694$	0	0
$695$	3065.11	0.167290
$696$	0	0
$697$	5214.03	0.283351
$698$	0	0
$699$	11178.5	0.604880
$700$	0	0
$701$	8357.52	0.450298	0.225149	−	0.974324i	$-0.427713\pi$
$701$	8357.52	0.450298	0.225149	+	0.974324i	$0.427713\pi$
$702$	0	0
$703$	1398.66	0.0750374
$704$	0	0
$705$	−243.231	−0.0129938
$706$	0	0
$707$	7009.91	0.372892
$708$	0	0
$709$	−32942.3	−1.74496	−0.872479	−	0.488652i	$-0.837489\pi$
$709$	−32942.3	−1.74496	−0.872479	+	0.488652i	$0.837489\pi$
$710$	0	0
$711$	−8305.91	−0.438110
$712$	0	0
$713$	−25743.8	−1.35219
$714$	0	0
$715$	−2045.87	−0.107009
$716$	0	0
$717$	−17519.8	−0.912535
$718$	0	0
$719$	3521.73	0.182668	0.0913342	−	0.995820i	$-0.470887\pi$
$719$	3521.73	0.182668	0.0913342	+	0.995820i	$0.470887\pi$
$720$	0	0
$721$	8810.62	0.455097
$722$	0	0
$723$	4435.72	0.228169
$724$	0	0
$725$	−2411.20	−0.123517
$726$	0	0
$727$	−21267.9	−1.08499	−0.542493	−	0.840061i	$-0.682519\pi$
$727$	−21267.9	−1.08499	−0.542493	+	0.840061i	$0.682519\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	7099.78	0.359227
$732$	0	0
$733$	−4810.86	−0.242419	−0.121210	−	0.992627i	$-0.538677\pi$
$733$	−4810.86	−0.242419	−0.121210	+	0.992627i	$0.538677\pi$
$734$	0	0
$735$	−4173.65	−0.209452
$736$	0	0
$737$	−21933.7	−1.09625
$738$	0	0
$739$	6151.36	0.306200	0.153100	−	0.988211i	$-0.451074\pi$
$739$	6151.36	0.306200	0.153100	+	0.988211i	$0.451074\pi$
$740$	0	0
$741$	5137.92	0.254718
$742$	0	0
$743$	20563.6	1.01535	0.507676	−	0.861548i	$-0.330505\pi$
$743$	20563.6	1.01535	0.507676	+	0.861548i	$0.330505\pi$
$744$	0	0
$745$	8223.74	0.404422
$746$	0	0
$747$	−11014.1	−0.539469
$748$	0	0
$749$	11106.8	0.541832
$750$	0	0
$751$	−3812.00	−0.185222	−0.0926110	−	0.995702i	$-0.529521\pi$
$751$	−3812.00	−0.185222	−0.0926110	+	0.995702i	$0.529521\pi$
$752$	0	0
$753$	−20578.4	−0.995909
$754$	0	0
$755$	12856.1	0.619709
$756$	0	0
$757$	−30202.8	−1.45012	−0.725058	−	0.688688i	$-0.758187\pi$
$757$	−30202.8	−1.45012	−0.725058	+	0.688688i	$0.758187\pi$
$758$	0	0
$759$	−16540.7	−0.791028
$760$	0	0
$761$	445.500	0.0212212	0.0106106	−	0.999944i	$-0.496622\pi$
$761$	445.500	0.0212212	0.0106106	+	0.999944i	$0.496622\pi$
$762$	0	0
$763$	3365.95	0.159706
$764$	0	0
$765$	−1475.41	−0.0697303
$766$	0	0
$767$	7990.68	0.376176
$768$	0	0
$769$	31398.1	1.47236	0.736180	−	0.676786i	$-0.236628\pi$
$769$	31398.1	1.47236	0.736180	+	0.676786i	$0.236628\pi$
$770$	0	0
$771$	16492.5	0.770378
$772$	0	0
$773$	−25143.7	−1.16993	−0.584964	−	0.811059i	$-0.698892\pi$
$773$	−25143.7	−1.16993	−0.584964	+	0.811059i	$0.698892\pi$
$774$	0	0
$775$	3674.05	0.170291
$776$	0	0
$777$	−256.301	−0.0118337
$778$	0	0
$779$	20950.5	0.963582
$780$	0	0
$781$	−30263.4	−1.38657
$782$	0	0
$783$	2604.09	0.118854
$784$	0	0
$785$	−1088.30	−0.0494818
$786$	0	0
$787$	26828.4	1.21516	0.607579	−	0.794259i	$-0.292141\pi$
$787$	26828.4	1.21516	0.607579	+	0.794259i	$0.292141\pi$
$788$	0	0
$789$	10854.4	0.489766
$790$	0	0
$791$	8180.00	0.367696
$792$	0	0
$793$	−9265.43	−0.414912
$794$	0	0
$795$	2725.05	0.121569
$796$	0	0
$797$	4185.54	0.186022	0.0930109	−	0.995665i	$-0.470351\pi$
$797$	4185.54	0.186022	0.0930109	+	0.995665i	$0.470351\pi$
$798$	0	0
$799$	−531.653	−0.0235401
$800$	0	0
$801$	−3897.02	−0.171903
$802$	0	0
$803$	33813.7	1.48600
$804$	0	0
$805$	7048.22	0.308593
$806$	0	0
$807$	−6118.04	−0.266871
$808$	0	0
$809$	−17044.2	−0.740718	−0.370359	−	0.928889i	$-0.620765\pi$
$809$	−17044.2	−0.740718	−0.370359	+	0.928889i	$0.620765\pi$
$810$	0	0
$811$	35399.9	1.53275	0.766375	−	0.642394i	$-0.222059\pi$
$811$	35399.9	1.53275	0.766375	+	0.642394i	$0.222059\pi$
$812$	0	0
$813$	661.644	0.0285423
$814$	0	0
$815$	−6358.20	−0.273274
$816$	0	0
$817$	28527.7	1.22161
$818$	0	0
$819$	−941.515	−0.0401700
$820$	0	0
$821$	23905.0	1.01619	0.508093	−	0.861302i	$-0.330350\pi$
$821$	23905.0	1.01619	0.508093	+	0.861302i	$0.330350\pi$
$822$	0	0
$823$	2580.47	0.109295	0.0546473	−	0.998506i	$-0.482597\pi$
$823$	2580.47	0.109295	0.0546473	+	0.998506i	$0.482597\pi$
$824$	0	0
$825$	2360.62	0.0996197
$826$	0	0
$827$	6640.95	0.279236	0.139618	−	0.990205i	$-0.455413\pi$
$827$	6640.95	0.279236	0.139618	+	0.990205i	$0.455413\pi$
$828$	0	0
$829$	4475.00	0.187483	0.0937413	−	0.995597i	$-0.470117\pi$
$829$	4475.00	0.187483	0.0937413	+	0.995597i	$0.470117\pi$
$830$	0	0
$831$	8145.90	0.340046
$832$	0	0
$833$	−9122.76	−0.379454
$834$	0	0
$835$	17568.0	0.728103
$836$	0	0
$837$	−3967.97	−0.163863
$838$	0	0
$839$	−46234.6	−1.90250	−0.951249	−	0.308425i	$-0.900198\pi$
$839$	−46234.6	−1.90250	−0.951249	+	0.308425i	$0.900198\pi$
$840$	0	0
$841$	−15086.8	−0.618591
$842$	0	0
$843$	20155.6	0.823484
$844$	0	0
$845$	−845.000	−0.0344010
$846$	0	0
$847$	−2738.66	−0.111100
$848$	0	0
$849$	4886.39	0.197527
$850$	0	0
$851$	−1859.76	−0.0749139
$852$	0	0
$853$	48245.9	1.93659	0.968293	−	0.249819i	$-0.0803710\pi$
$853$	48245.9	1.93659	0.968293	+	0.249819i	$0.0803710\pi$
$854$	0	0
$855$	−5928.37	−0.237130
$856$	0	0
$857$	3719.99	0.148276	0.0741380	−	0.997248i	$-0.476379\pi$
$857$	3719.99	0.148276	0.0741380	+	0.997248i	$0.476379\pi$
$858$	0	0
$859$	33852.1	1.34461	0.672305	−	0.740274i	$-0.265304\pi$
$859$	33852.1	1.34461	0.672305	+	0.740274i	$0.265304\pi$
$860$	0	0
$861$	−3839.14	−0.151960
$862$	0	0
$863$	−47599.7	−1.87753	−0.938767	−	0.344552i	$-0.888031\pi$
$863$	−47599.7	−1.87753	−0.938767	+	0.344552i	$0.888031\pi$
$864$	0	0
$865$	9874.34	0.388136
$866$	0	0
$867$	11514.0	0.451024
$868$	0	0
$869$	29047.6	1.13391
$870$	0	0
$871$	−9059.22	−0.352423
$872$	0	0
$873$	−6967.68	−0.270126
$874$	0	0
$875$	−1005.89	−0.0388633
$876$	0	0
$877$	31549.8	1.21478	0.607390	−	0.794404i	$-0.292217\pi$
$877$	31549.8	1.21478	0.607390	+	0.794404i	$0.292217\pi$
$878$	0	0
$879$	16547.4	0.634962
$880$	0	0
$881$	−25807.8	−0.986931	−0.493465	−	0.869765i	$-0.664270\pi$
$881$	−25807.8	−0.986931	−0.493465	+	0.869765i	$0.664270\pi$
$882$	0	0
$883$	−34353.0	−1.30925	−0.654626	−	0.755953i	$-0.727174\pi$
$883$	−34353.0	−1.30925	−0.654626	+	0.755953i	$0.727174\pi$
$884$	0	0
$885$	−9220.02	−0.350201
$886$	0	0
$887$	−48620.1	−1.84048	−0.920238	−	0.391359i	$-0.872005\pi$
$887$	−48620.1	−1.84048	−0.920238	+	0.391359i	$0.872005\pi$
$888$	0	0
$889$	3566.82	0.134564
$890$	0	0
$891$	−2549.47	−0.0958591
$892$	0	0
$893$	−2136.24	−0.0800521
$894$	0	0
$895$	11824.7	0.441628
$896$	0	0
$897$	−6831.77	−0.254299
$898$	0	0
$899$	−14174.2	−0.525845
$900$	0	0
$901$	5956.42	0.220241
$902$	0	0
$903$	−5227.65	−0.192652
$904$	0	0
$905$	23302.7	0.855921
$906$	0	0
$907$	−5763.67	−0.211003	−0.105501	−	0.994419i	$-0.533645\pi$
$907$	−5763.67	−0.211003	−0.105501	+	0.994419i	$0.533645\pi$
$908$	0	0
$909$	7839.96	0.286067
$910$	0	0
$911$	10640.8	0.386988	0.193494	−	0.981101i	$-0.438018\pi$
$911$	10640.8	0.386988	0.193494	+	0.981101i	$0.438018\pi$
$912$	0	0
$913$	38518.6	1.39625
$914$	0	0
$915$	10690.9	0.386262
$916$	0	0
$917$	−7915.60	−0.285056
$918$	0	0
$919$	44905.4	1.61185	0.805926	−	0.592017i	$-0.201668\pi$
$919$	44905.4	1.61185	0.805926	+	0.592017i	$0.201668\pi$
$920$	0	0
$921$	−9344.93	−0.334339
$922$	0	0
$923$	−12499.6	−0.445753
$924$	0	0
$925$	265.417	0.00943443
$926$	0	0
$927$	9853.89	0.349131
$928$	0	0
$929$	14049.6	0.496180	0.248090	−	0.968737i	$-0.420197\pi$
$929$	14049.6	0.496180	0.248090	+	0.968737i	$0.420197\pi$
$930$	0	0
$931$	−36656.3	−1.29040
$932$	0	0
$933$	6781.59	0.237963
$934$	0	0
$935$	5159.84	0.180476
$936$	0	0
$937$	−28797.0	−1.00401	−0.502004	−	0.864865i	$-0.667404\pi$
$937$	−28797.0	−1.00401	−0.502004	+	0.864865i	$0.667404\pi$
$938$	0	0
$939$	5786.10	0.201089
$940$	0	0
$941$	−37299.6	−1.29217	−0.646086	−	0.763265i	$-0.723595\pi$
$941$	−37299.6	−1.29217	−0.646086	+	0.763265i	$0.723595\pi$
$942$	0	0
$943$	−27857.4	−0.961995
$944$	0	0
$945$	1086.36	0.0373962
$946$	0	0
$947$	−33078.3	−1.13506	−0.567530	−	0.823353i	$-0.692101\pi$
$947$	−33078.3	−1.13506	−0.567530	+	0.823353i	$0.692101\pi$
$948$	0	0
$949$	13966.0	0.477719
$950$	0	0
$951$	−6119.58	−0.208666
$952$	0	0
$953$	−5734.20	−0.194910	−0.0974549	−	0.995240i	$-0.531070\pi$
$953$	−5734.20	−0.194910	−0.0974549	+	0.995240i	$0.531070\pi$
$954$	0	0
$955$	5551.49	0.188107
$956$	0	0
$957$	−9107.07	−0.307617
$958$	0	0
$959$	−17092.8	−0.575554
$960$	0	0
$961$	−8193.17	−0.275022
$962$	0	0
$963$	12421.9	0.415671
$964$	0	0
$965$	12532.7	0.418073
$966$	0	0
$967$	−30993.0	−1.03068	−0.515340	−	0.856986i	$-0.672334\pi$
$967$	−30993.0	−1.03068	−0.515340	+	0.856986i	$0.672334\pi$
$968$	0	0
$969$	−12958.2	−0.429596
$970$	0	0
$971$	9506.65	0.314195	0.157097	−	0.987583i	$-0.449786\pi$
$971$	9506.65	0.314195	0.157097	+	0.987583i	$0.449786\pi$
$972$	0	0
$973$	−4933.08	−0.162536
$974$	0	0
$975$	975.000	0.0320256
$976$	0	0
$977$	−21343.7	−0.698921	−0.349461	−	0.936951i	$-0.613635\pi$
$977$	−21343.7	−0.698921	−0.349461	+	0.936951i	$0.613635\pi$
$978$	0	0
$979$	13628.7	0.444919
$980$	0	0
$981$	3764.51	0.122520
$982$	0	0
$983$	−19769.6	−0.641457	−0.320728	−	0.947171i	$-0.603928\pi$
$983$	−19769.6	−0.641457	−0.320728	+	0.947171i	$0.603928\pi$
$984$	0	0
$985$	14217.9	0.459919
$986$	0	0
$987$	391.462	0.0126245
$988$	0	0
$989$	−37932.5	−1.21960
$990$	0	0
$991$	−12184.1	−0.390555	−0.195278	−	0.980748i	$-0.562561\pi$
$991$	−12184.1	−0.390555	−0.195278	+	0.980748i	$0.562561\pi$
$992$	0	0
$993$	−27231.8	−0.870267
$994$	0	0
$995$	−13504.9	−0.430286
$996$	0	0
$997$	48653.6	1.54551	0.772756	−	0.634703i	$-0.218878\pi$
$997$	48653.6	1.54551	0.772756	+	0.634703i	$0.218878\pi$
$998$	0	0
$999$	−286.650	−0.00907828

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.4.a.l.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.4.a.l.1.3	✓	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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1560.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} +8.04714 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -5.00000 q^{5} +8.04714 q^{7} +9.00000 q^{9} -31.4750 q^{11} -13.0000 q^{13} +15.0000 q^{15} +32.7870 q^{17} +131.742 q^{19} -24.1414 q^{21} -175.173 q^{23} +25.0000 q^{25} -27.0000 q^{27} -96.4478 q^{29} +146.962 q^{31} +94.4249 q^{33} -40.2357 q^{35} +10.6167 q^{37} +39.0000 q^{39} +159.027 q^{41} +216.543 q^{43} -45.0000 q^{45} -16.2154 q^{47} -278.244 q^{49} -98.3609 q^{51} +181.670 q^{53} +157.375 q^{55} -395.225 q^{57} -614.668 q^{59} +712.726 q^{61} +72.4242 q^{63} +65.0000 q^{65} +696.863 q^{67} +525.520 q^{69} +961.508 q^{71} -1074.31 q^{73} -75.0000 q^{75} -253.283 q^{77} -922.879 q^{79} +81.0000 q^{81} -1223.78 q^{83} -163.935 q^{85} +289.343 q^{87} -433.003 q^{89} -104.613 q^{91} -440.886 q^{93} -658.708 q^{95} -774.186 q^{97} -283.275 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9	\( 4 q - 12 q^{3} - 20 q^{5} - 11 q^{7} + 36 q^{9} + 35 q^{11} - 52 q^{13} + 60 q^{15} - 9 q^{17} + 32 q^{19} + 33 q^{21} - 149 q^{23} + 100 q^{25} - 108 q^{27} + 168 q^{29} + 280 q^{31} - 105 q^{33} + 55 q^{35} + 37 q^{37} + 156 q^{39} + 247 q^{41} + 132 q^{43} - 180 q^{45} + 217 q^{49} + 27 q^{51} + 247 q^{53} - 175 q^{55} - 96 q^{57} + 614 q^{59} + 719 q^{61} - 99 q^{63} + 260 q^{65} + 658 q^{67} + 447 q^{69} + 939 q^{71} - 1452 q^{73} - 300 q^{75} + 199 q^{77} + 289 q^{79} + 324 q^{81} - 118 q^{83} + 45 q^{85} - 504 q^{87} - 1319 q^{89} + 143 q^{91} - 840 q^{93} - 160 q^{95} - 993 q^{97} + 315 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 - 20 * q^5 - 11 * q^7 + 36 * q^9 + 35 * q^11 - 52 * q^13 + 60 * q^15 - 9 * q^17 + 32 * q^19 + 33 * q^21 - 149 * q^23 + 100 * q^25 - 108 * q^27 + 168 * q^29 + 280 * q^31 - 105 * q^33 + 55 * q^35 + 37 * q^37 + 156 * q^39 + 247 * q^41 + 132 * q^43 - 180 * q^45 + 217 * q^49 + 27 * q^51 + 247 * q^53 - 175 * q^55 - 96 * q^57 + 614 * q^59 + 719 * q^61 - 99 * q^63 + 260 * q^65 + 658 * q^67 + 447 * q^69 + 939 * q^71 - 1452 * q^73 - 300 * q^75 + 199 * q^77 + 289 * q^79 + 324 * q^81 - 118 * q^83 + 45 * q^85 - 504 * q^87 - 1319 * q^89 + 143 * q^91 - 840 * q^93 - 160 * q^95 - 993 * q^97 + 315 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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