LMFDB - Embedded newform 1560.4.a.n.1.4

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} - 41x^{2} - 30x - 4 $ x^4 - x^3 - 41x^2 - 30x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.574597$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} +11.0618 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +5.00000 q^{5} +11.0618 q^{7} +9.00000 q^{9} -35.6586 q^{11} +13.0000 q^{13} +15.0000 q^{15} +9.34984 q^{17} -77.5269 q^{19} +33.1855 q^{21} -204.486 q^{23} +25.0000 q^{25} +27.0000 q^{27} +176.219 q^{29} -216.206 q^{31} -106.976 q^{33} +55.3092 q^{35} -190.820 q^{37} +39.0000 q^{39} -315.241 q^{41} +33.6302 q^{43} +45.0000 q^{45} -203.252 q^{47} -220.636 q^{49} +28.0495 q^{51} +487.637 q^{53} -178.293 q^{55} -232.581 q^{57} +176.742 q^{59} -234.540 q^{61} +99.5565 q^{63} +65.0000 q^{65} -608.529 q^{67} -613.459 q^{69} -454.705 q^{71} -427.982 q^{73} +75.0000 q^{75} -394.450 q^{77} +1177.90 q^{79} +81.0000 q^{81} +821.981 q^{83} +46.7492 q^{85} +528.658 q^{87} -549.649 q^{89} +143.804 q^{91} -648.619 q^{93} -387.635 q^{95} +213.703 q^{97} -320.928 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9	$ 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9} - 38 q^{11} + 52 q^{13} + 60 q^{15} - 50 q^{17} - 180 q^{19} - 102 q^{21} - 170 q^{23} + 100 q^{25} + 108 q^{27} - 84 q^{29} - 408 q^{31} - 114 q^{33} - 170 q^{35} - 38 q^{37} + 156 q^{39} - 90 q^{41} - 732 q^{43} + 180 q^{45} - 520 q^{47} + 230 q^{49} - 150 q^{51} - 338 q^{53} - 190 q^{55} - 540 q^{57} - 232 q^{59} + 326 q^{61} - 306 q^{63} + 260 q^{65} - 1040 q^{67} - 510 q^{69} - 702 q^{71} - 368 q^{73} + 300 q^{75} + 14 q^{77} - 678 q^{79} + 324 q^{81} - 1888 q^{83} - 250 q^{85} - 252 q^{87} - 2070 q^{89} - 442 q^{91} - 1224 q^{93} - 900 q^{95} + 1126 q^{97} - 342 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9 - 38 * q^11 + 52 * q^13 + 60 * q^15 - 50 * q^17 - 180 * q^19 - 102 * q^21 - 170 * q^23 + 100 * q^25 + 108 * q^27 - 84 * q^29 - 408 * q^31 - 114 * q^33 - 170 * q^35 - 38 * q^37 + 156 * q^39 - 90 * q^41 - 732 * q^43 + 180 * q^45 - 520 * q^47 + 230 * q^49 - 150 * q^51 - 338 * q^53 - 190 * q^55 - 540 * q^57 - 232 * q^59 + 326 * q^61 - 306 * q^63 + 260 * q^65 - 1040 * q^67 - 510 * q^69 - 702 * q^71 - 368 * q^73 + 300 * q^75 + 14 * q^77 - 678 * q^79 + 324 * q^81 - 1888 * q^83 - 250 * q^85 - 252 * q^87 - 2070 * q^89 - 442 * q^91 - 1224 * q^93 - 900 * q^95 + 1126 * q^97 - 342 * 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
$7$	11.0618	0.597283	0.298642	−	0.954365i	$-0.403467\pi$
$7$	11.0618	0.597283	0.298642	+	0.954365i	$0.403467\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−35.6586	−0.977407	−0.488703	−	0.872450i	$-0.662530\pi$
$11$	−35.6586	−0.977407	−0.488703	+	0.872450i	$0.662530\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	9.34984	0.133392	0.0666962	−	0.997773i	$-0.478754\pi$
$17$	9.34984	0.133392	0.0666962	+	0.997773i	$0.478754\pi$
$18$	0	0
$19$	−77.5269	−0.936100	−0.468050	−	0.883702i	$-0.655043\pi$
$19$	−77.5269	−0.936100	−0.468050	+	0.883702i	$0.655043\pi$
$20$	0	0
$21$	33.1855	0.344842
$22$	0	0
$23$	−204.486	−1.85384	−0.926920	−	0.375259i	$-0.877554\pi$
$23$	−204.486	−1.85384	−0.926920	+	0.375259i	$0.877554\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	176.219	1.12838	0.564192	−	0.825644i	$-0.309188\pi$
$29$	176.219	1.12838	0.564192	+	0.825644i	$0.309188\pi$
$30$	0	0
$31$	−216.206	−1.25264	−0.626319	−	0.779567i	$-0.715439\pi$
$31$	−216.206	−1.25264	−0.626319	+	0.779567i	$0.715439\pi$
$32$	0	0
$33$	−106.976	−0.564306
$34$	0	0
$35$	55.3092	0.267113
$36$	0	0
$37$	−190.820	−0.847853	−0.423926	−	0.905697i	$-0.639348\pi$
$37$	−190.820	−0.847853	−0.423926	+	0.905697i	$0.639348\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−315.241	−1.20079	−0.600394	−	0.799704i	$-0.704990\pi$
$41$	−315.241	−1.20079	−0.600394	+	0.799704i	$0.704990\pi$
$42$	0	0
$43$	33.6302	0.119269	0.0596344	−	0.998220i	$-0.481007\pi$
$43$	33.6302	0.119269	0.0596344	+	0.998220i	$0.481007\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−203.252	−0.630793	−0.315397	−	0.948960i	$-0.602138\pi$
$47$	−203.252	−0.630793	−0.315397	+	0.948960i	$0.602138\pi$
$48$	0	0
$49$	−220.636	−0.643253
$50$	0	0
$51$	28.0495	0.0770141
$52$	0	0
$53$	487.637	1.26381	0.631906	−	0.775045i	$-0.282273\pi$
$53$	487.637	1.26381	0.631906	+	0.775045i	$0.282273\pi$
$54$	0	0
$55$	−178.293	−0.437110
$56$	0	0
$57$	−232.581	−0.540457
$58$	0	0
$59$	176.742	0.389998	0.194999	−	0.980803i	$-0.437530\pi$
$59$	176.742	0.389998	0.194999	+	0.980803i	$0.437530\pi$
$60$	0	0
$61$	−234.540	−0.492292	−0.246146	−	0.969233i	$-0.579164\pi$
$61$	−234.540	−0.492292	−0.246146	+	0.969233i	$0.579164\pi$
$62$	0	0
$63$	99.5565	0.199094
$64$	0	0
$65$	65.0000	0.124035
$66$	0	0
$67$	−608.529	−1.10961	−0.554803	−	0.831982i	$-0.687206\pi$
$67$	−608.529	−1.10961	−0.554803	+	0.831982i	$0.687206\pi$
$68$	0	0
$69$	−613.459	−1.07032
$70$	0	0
$71$	−454.705	−0.760050	−0.380025	−	0.924976i	$-0.624085\pi$
$71$	−454.705	−0.760050	−0.380025	+	0.924976i	$0.624085\pi$
$72$	0	0
$73$	−427.982	−0.686184	−0.343092	−	0.939302i	$-0.611474\pi$
$73$	−427.982	−0.686184	−0.343092	+	0.939302i	$0.611474\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	−394.450	−0.583789
$78$	0	0
$79$	1177.90	1.67753	0.838763	−	0.544497i	$-0.183279\pi$
$79$	1177.90	1.67753	0.838763	+	0.544497i	$0.183279\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	821.981	1.08704	0.543519	−	0.839397i	$-0.317092\pi$
$83$	821.981	1.08704	0.543519	+	0.839397i	$0.317092\pi$
$84$	0	0
$85$	46.7492	0.0596549
$86$	0	0
$87$	528.658	0.651472
$88$	0	0
$89$	−549.649	−0.654637	−0.327318	−	0.944914i	$-0.606145\pi$
$89$	−549.649	−0.654637	−0.327318	+	0.944914i	$0.606145\pi$
$90$	0	0
$91$	143.804	0.165657
$92$	0	0
$93$	−648.619	−0.723211
$94$	0	0
$95$	−387.635	−0.418636
$96$	0	0
$97$	213.703	0.223693	0.111847	−	0.993725i	$-0.464323\pi$
$97$	213.703	0.223693	0.111847	+	0.993725i	$0.464323\pi$
$98$	0	0
$99$	−320.928	−0.325802
$100$	0	0
$101$	1368.56	1.34829	0.674144	−	0.738600i	$-0.264513\pi$
$101$	1368.56	1.34829	0.674144	+	0.738600i	$0.264513\pi$
$102$	0	0
$103$	−2084.25	−1.99385	−0.996926	−	0.0783455i	$-0.975036\pi$
$103$	−2084.25	−1.99385	−0.996926	+	0.0783455i	$0.975036\pi$
$104$	0	0
$105$	165.928	0.154218
$106$	0	0
$107$	−1837.09	−1.65980	−0.829899	−	0.557913i	$-0.811602\pi$
$107$	−1837.09	−1.65980	−0.829899	+	0.557913i	$0.811602\pi$
$108$	0	0
$109$	−173.527	−0.152485	−0.0762425	−	0.997089i	$-0.524292\pi$
$109$	−173.527	−0.152485	−0.0762425	+	0.997089i	$0.524292\pi$
$110$	0	0
$111$	−572.459	−0.489508
$112$	0	0
$113$	377.475	0.314247	0.157123	−	0.987579i	$-0.449778\pi$
$113$	377.475	0.314247	0.157123	+	0.987579i	$0.449778\pi$
$114$	0	0
$115$	−1022.43	−0.829063
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	103.426	0.0796730
$120$	0	0
$121$	−59.4634	−0.0446757
$122$	0	0
$123$	−945.722	−0.693275
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1453.58	1.01562	0.507812	−	0.861468i	$-0.330455\pi$
$127$	1453.58	1.01562	0.507812	+	0.861468i	$0.330455\pi$
$128$	0	0
$129$	100.891	0.0688599
$130$	0	0
$131$	1106.58	0.738034	0.369017	−	0.929423i	$-0.379694\pi$
$131$	1106.58	0.738034	0.369017	+	0.929423i	$0.379694\pi$
$132$	0	0
$133$	−857.590	−0.559116
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	677.130	0.422271	0.211135	−	0.977457i	$-0.432284\pi$
$137$	677.130	0.422271	0.211135	+	0.977457i	$0.432284\pi$
$138$	0	0
$139$	1536.91	0.937834	0.468917	−	0.883242i	$-0.344644\pi$
$139$	1536.91	0.937834	0.468917	+	0.883242i	$0.344644\pi$
$140$	0	0
$141$	−609.755	−0.364189
$142$	0	0
$143$	−463.562	−0.271084
$144$	0	0
$145$	881.097	0.504628
$146$	0	0
$147$	−661.907	−0.371382
$148$	0	0
$149$	−948.631	−0.521576	−0.260788	−	0.965396i	$-0.583982\pi$
$149$	−948.631	−0.521576	−0.260788	+	0.965396i	$0.583982\pi$
$150$	0	0
$151$	3315.66	1.78692	0.893460	−	0.449143i	$-0.148271\pi$
$151$	3315.66	1.78692	0.893460	+	0.449143i	$0.148271\pi$
$152$	0	0
$153$	84.1486	0.0444641
$154$	0	0
$155$	−1081.03	−0.560197
$156$	0	0
$157$	−1419.72	−0.721697	−0.360848	−	0.932625i	$-0.617513\pi$
$157$	−1419.72	−0.721697	−0.360848	+	0.932625i	$0.617513\pi$
$158$	0	0
$159$	1462.91	0.729662
$160$	0	0
$161$	−2261.99	−1.10727
$162$	0	0
$163$	−3144.63	−1.51108	−0.755541	−	0.655101i	$-0.772626\pi$
$163$	−3144.63	−1.51108	−0.755541	+	0.655101i	$0.772626\pi$
$164$	0	0
$165$	−534.879	−0.252365
$166$	0	0
$167$	1290.24	0.597854	0.298927	−	0.954276i	$-0.403371\pi$
$167$	1290.24	0.597854	0.298927	+	0.954276i	$0.403371\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−697.742	−0.312033
$172$	0	0
$173$	671.469	0.295092	0.147546	−	0.989055i	$-0.452863\pi$
$173$	671.469	0.295092	0.147546	+	0.989055i	$0.452863\pi$
$174$	0	0
$175$	276.546	0.119457
$176$	0	0
$177$	530.226	0.225165
$178$	0	0
$179$	−3002.08	−1.25355	−0.626777	−	0.779199i	$-0.715626\pi$
$179$	−3002.08	−1.25355	−0.626777	+	0.779199i	$0.715626\pi$
$180$	0	0
$181$	−232.735	−0.0955750	−0.0477875	−	0.998858i	$-0.515217\pi$
$181$	−232.735	−0.0955750	−0.0477875	+	0.998858i	$0.515217\pi$
$182$	0	0
$183$	−703.621	−0.284225
$184$	0	0
$185$	−954.098	−0.379171
$186$	0	0
$187$	−333.402	−0.130379
$188$	0	0
$189$	298.670	0.114947
$190$	0	0
$191$	−1741.88	−0.659886	−0.329943	−	0.944001i	$-0.607029\pi$
$191$	−1741.88	−0.659886	−0.329943	+	0.944001i	$0.607029\pi$
$192$	0	0
$193$	−659.276	−0.245885	−0.122942	−	0.992414i	$-0.539233\pi$
$193$	−659.276	−0.245885	−0.122942	+	0.992414i	$0.539233\pi$
$194$	0	0
$195$	195.000	0.0716115
$196$	0	0
$197$	−775.667	−0.280528	−0.140264	−	0.990114i	$-0.544795\pi$
$197$	−775.667	−0.280528	−0.140264	+	0.990114i	$0.544795\pi$
$198$	0	0
$199$	2030.26	0.723222	0.361611	−	0.932329i	$-0.382227\pi$
$199$	2030.26	0.723222	0.361611	+	0.932329i	$0.382227\pi$
$200$	0	0
$201$	−1825.59	−0.640631
$202$	0	0
$203$	1949.31	0.673964
$204$	0	0
$205$	−1576.20	−0.537009
$206$	0	0
$207$	−1840.38	−0.617947
$208$	0	0
$209$	2764.50	0.914950
$210$	0	0
$211$	−4994.83	−1.62966	−0.814830	−	0.579701i	$-0.803170\pi$
$211$	−4994.83	−1.62966	−0.814830	+	0.579701i	$0.803170\pi$
$212$	0	0
$213$	−1364.11	−0.438815
$214$	0	0
$215$	168.151	0.0533386
$216$	0	0
$217$	−2391.64	−0.748180
$218$	0	0
$219$	−1283.94	−0.396169
$220$	0	0
$221$	121.548	0.0369964
$222$	0	0
$223$	−3662.17	−1.09972	−0.549859	−	0.835258i	$-0.685318\pi$
$223$	−3662.17	−1.09972	−0.549859	+	0.835258i	$0.685318\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	959.714	0.280610	0.140305	−	0.990108i	$-0.455192\pi$
$227$	959.714	0.280610	0.140305	+	0.990108i	$0.455192\pi$
$228$	0	0
$229$	−2816.91	−0.812867	−0.406433	−	0.913680i	$-0.633228\pi$
$229$	−2816.91	−0.812867	−0.406433	+	0.913680i	$0.633228\pi$
$230$	0	0
$231$	−1183.35	−0.337051
$232$	0	0
$233$	−2411.40	−0.678009	−0.339004	−	0.940785i	$-0.610090\pi$
$233$	−2411.40	−0.678009	−0.339004	+	0.940785i	$0.610090\pi$
$234$	0	0
$235$	−1016.26	−0.282099
$236$	0	0
$237$	3533.71	0.968520
$238$	0	0
$239$	−2702.81	−0.731509	−0.365754	−	0.930711i	$-0.619189\pi$
$239$	−2702.81	−0.731509	−0.365754	+	0.930711i	$0.619189\pi$
$240$	0	0
$241$	−1591.74	−0.425448	−0.212724	−	0.977112i	$-0.568234\pi$
$241$	−1591.74	−0.425448	−0.212724	+	0.977112i	$0.568234\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−1103.18	−0.287671
$246$	0	0
$247$	−1007.85	−0.259627
$248$	0	0
$249$	2465.94	0.627602
$250$	0	0
$251$	−3515.20	−0.883974	−0.441987	−	0.897022i	$-0.645726\pi$
$251$	−3515.20	−0.883974	−0.441987	+	0.897022i	$0.645726\pi$
$252$	0	0
$253$	7291.69	1.81196
$254$	0	0
$255$	140.248	0.0344417
$256$	0	0
$257$	−685.648	−0.166418	−0.0832092	−	0.996532i	$-0.526517\pi$
$257$	−685.648	−0.166418	−0.0832092	+	0.996532i	$0.526517\pi$
$258$	0	0
$259$	−2110.82	−0.506408
$260$	0	0
$261$	1585.97	0.376128
$262$	0	0
$263$	−317.057	−0.0743368	−0.0371684	−	0.999309i	$-0.511834\pi$
$263$	−317.057	−0.0743368	−0.0371684	+	0.999309i	$0.511834\pi$
$264$	0	0
$265$	2438.18	0.565194
$266$	0	0
$267$	−1648.95	−0.377955
$268$	0	0
$269$	3176.23	0.719920	0.359960	−	0.932968i	$-0.382790\pi$
$269$	3176.23	0.719920	0.359960	+	0.932968i	$0.382790\pi$
$270$	0	0
$271$	3891.72	0.872344	0.436172	−	0.899863i	$-0.356334\pi$
$271$	3891.72	0.872344	0.436172	+	0.899863i	$0.356334\pi$
$272$	0	0
$273$	431.412	0.0956418
$274$	0	0
$275$	−891.465	−0.195481
$276$	0	0
$277$	2397.76	0.520100	0.260050	−	0.965595i	$-0.416261\pi$
$277$	2397.76	0.520100	0.260050	+	0.965595i	$0.416261\pi$
$278$	0	0
$279$	−1945.86	−0.417546
$280$	0	0
$281$	5736.10	1.21775	0.608874	−	0.793267i	$-0.291622\pi$
$281$	5736.10	1.21775	0.608874	+	0.793267i	$0.291622\pi$
$282$	0	0
$283$	2296.93	0.482467	0.241233	−	0.970467i	$-0.422448\pi$
$283$	2296.93	0.482467	0.241233	+	0.970467i	$0.422448\pi$
$284$	0	0
$285$	−1162.90	−0.241700
$286$	0	0
$287$	−3487.14	−0.717210
$288$	0	0
$289$	−4825.58	−0.982206
$290$	0	0
$291$	641.108	0.129149
$292$	0	0
$293$	3464.41	0.690762	0.345381	−	0.938463i	$-0.387750\pi$
$293$	3464.41	0.690762	0.345381	+	0.938463i	$0.387750\pi$
$294$	0	0
$295$	883.710	0.174412
$296$	0	0
$297$	−962.783	−0.188102
$298$	0	0
$299$	−2658.32	−0.514163
$300$	0	0
$301$	372.012	0.0712372
$302$	0	0
$303$	4105.69	0.778434
$304$	0	0
$305$	−1172.70	−0.220160
$306$	0	0
$307$	1025.10	0.190571	0.0952855	−	0.995450i	$-0.469624\pi$
$307$	1025.10	0.190571	0.0952855	+	0.995450i	$0.469624\pi$
$308$	0	0
$309$	−6252.74	−1.15115
$310$	0	0
$311$	3394.75	0.618967	0.309484	−	0.950905i	$-0.399844\pi$
$311$	3394.75	0.618967	0.309484	+	0.950905i	$0.399844\pi$
$312$	0	0
$313$	3430.63	0.619522	0.309761	−	0.950814i	$-0.399751\pi$
$313$	3430.63	0.619522	0.309761	+	0.950814i	$0.399751\pi$
$314$	0	0
$315$	497.783	0.0890377
$316$	0	0
$317$	−7137.48	−1.26461	−0.632304	−	0.774720i	$-0.717891\pi$
$317$	−7137.48	−1.26461	−0.632304	+	0.774720i	$0.717891\pi$
$318$	0	0
$319$	−6283.74	−1.10289
$320$	0	0
$321$	−5511.28	−0.958285
$322$	0	0
$323$	−724.864	−0.124868
$324$	0	0
$325$	325.000	0.0554700
$326$	0	0
$327$	−520.581	−0.0880372
$328$	0	0
$329$	−2248.34	−0.376762
$330$	0	0
$331$	5543.07	0.920468	0.460234	−	0.887798i	$-0.347766\pi$
$331$	5543.07	0.920468	0.460234	+	0.887798i	$0.347766\pi$
$332$	0	0
$333$	−1717.38	−0.282618
$334$	0	0
$335$	−3042.64	−0.496231
$336$	0	0
$337$	9773.05	1.57974	0.789869	−	0.613275i	$-0.210148\pi$
$337$	9773.05	1.57974	0.789869	+	0.613275i	$0.210148\pi$
$338$	0	0
$339$	1132.43	0.181430
$340$	0	0
$341$	7709.61	1.22434
$342$	0	0
$343$	−6234.85	−0.981487
$344$	0	0
$345$	−3067.29	−0.478659
$346$	0	0
$347$	−3741.52	−0.578834	−0.289417	−	0.957203i	$-0.593461\pi$
$347$	−3741.52	−0.578834	−0.289417	+	0.957203i	$0.593461\pi$
$348$	0	0
$349$	−4116.43	−0.631368	−0.315684	−	0.948864i	$-0.602234\pi$
$349$	−4116.43	−0.631368	−0.315684	+	0.948864i	$0.602234\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	10581.9	1.59552	0.797762	−	0.602973i	$-0.206017\pi$
$353$	10581.9	1.59552	0.797762	+	0.602973i	$0.206017\pi$
$354$	0	0
$355$	−2273.52	−0.339904
$356$	0	0
$357$	310.279	0.0459992
$358$	0	0
$359$	1887.32	0.277462	0.138731	−	0.990330i	$-0.455698\pi$
$359$	1887.32	0.277462	0.138731	+	0.990330i	$0.455698\pi$
$360$	0	0
$361$	−848.580	−0.123718
$362$	0	0
$363$	−178.390	−0.0257935
$364$	0	0
$365$	−2139.91	−0.306871
$366$	0	0
$367$	−13302.3	−1.89203	−0.946015	−	0.324121i	$-0.894931\pi$
$367$	−13302.3	−1.89203	−0.946015	+	0.324121i	$0.894931\pi$
$368$	0	0
$369$	−2837.16	−0.400263
$370$	0	0
$371$	5394.16	0.754853
$372$	0	0
$373$	−9798.85	−1.36023	−0.680114	−	0.733106i	$-0.738070\pi$
$373$	−9798.85	−1.36023	−0.680114	+	0.733106i	$0.738070\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	2290.85	0.312957
$378$	0	0
$379$	−10426.1	−1.41307	−0.706534	−	0.707679i	$-0.749742\pi$
$379$	−10426.1	−1.41307	−0.706534	+	0.707679i	$0.749742\pi$
$380$	0	0
$381$	4360.73	0.586371
$382$	0	0
$383$	−9010.04	−1.20207	−0.601033	−	0.799224i	$-0.705244\pi$
$383$	−9010.04	−1.20207	−0.601033	+	0.799224i	$0.705244\pi$
$384$	0	0
$385$	−1972.25	−0.261078
$386$	0	0
$387$	302.672	0.0397563
$388$	0	0
$389$	8329.22	1.08563	0.542813	−	0.839854i	$-0.317359\pi$
$389$	8329.22	1.08563	0.542813	+	0.839854i	$0.317359\pi$
$390$	0	0
$391$	−1911.91	−0.247288
$392$	0	0
$393$	3319.74	0.426104
$394$	0	0
$395$	5889.52	0.750212
$396$	0	0
$397$	−9157.34	−1.15767	−0.578833	−	0.815446i	$-0.696492\pi$
$397$	−9157.34	−1.15767	−0.578833	+	0.815446i	$0.696492\pi$
$398$	0	0
$399$	−2572.77	−0.322806
$400$	0	0
$401$	−1373.73	−0.171075	−0.0855374	−	0.996335i	$-0.527261\pi$
$401$	−1373.73	−0.171075	−0.0855374	+	0.996335i	$0.527261\pi$
$402$	0	0
$403$	−2810.68	−0.347419
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	6804.36	0.828697
$408$	0	0
$409$	−5743.14	−0.694328	−0.347164	−	0.937804i	$-0.612855\pi$
$409$	−5743.14	−0.694328	−0.347164	+	0.937804i	$0.612855\pi$
$410$	0	0
$411$	2031.39	0.243798
$412$	0	0
$413$	1955.09	0.232939
$414$	0	0
$415$	4109.91	0.486138
$416$	0	0
$417$	4610.73	0.541459
$418$	0	0
$419$	1765.62	0.205862	0.102931	−	0.994689i	$-0.467178\pi$
$419$	1765.62	0.205862	0.102931	+	0.994689i	$0.467178\pi$
$420$	0	0
$421$	3626.15	0.419781	0.209890	−	0.977725i	$-0.432689\pi$
$421$	3626.15	0.419781	0.209890	+	0.977725i	$0.432689\pi$
$422$	0	0
$423$	−1829.26	−0.210264
$424$	0	0
$425$	233.746	0.0266785
$426$	0	0
$427$	−2594.45	−0.294038
$428$	0	0
$429$	−1390.69	−0.156510
$430$	0	0
$431$	3219.85	0.359848	0.179924	−	0.983680i	$-0.442415\pi$
$431$	3219.85	0.359848	0.179924	+	0.983680i	$0.442415\pi$
$432$	0	0
$433$	−3020.19	−0.335199	−0.167599	−	0.985855i	$-0.553601\pi$
$433$	−3020.19	−0.335199	−0.167599	+	0.985855i	$0.553601\pi$
$434$	0	0
$435$	2643.29	0.291347
$436$	0	0
$437$	15853.2	1.73538
$438$	0	0
$439$	−16314.0	−1.77363	−0.886815	−	0.462124i	$-0.847088\pi$
$439$	−16314.0	−1.77363	−0.886815	+	0.462124i	$0.847088\pi$
$440$	0	0
$441$	−1985.72	−0.214418
$442$	0	0
$443$	−948.348	−0.101710	−0.0508548	−	0.998706i	$-0.516195\pi$
$443$	−948.348	−0.101710	−0.0508548	+	0.998706i	$0.516195\pi$
$444$	0	0
$445$	−2748.25	−0.292762
$446$	0	0
$447$	−2845.89	−0.301132
$448$	0	0
$449$	−3371.66	−0.354384	−0.177192	−	0.984176i	$-0.556701\pi$
$449$	−3371.66	−0.354384	−0.177192	+	0.984176i	$0.556701\pi$
$450$	0	0
$451$	11241.0	1.17366
$452$	0	0
$453$	9946.99	1.03168
$454$	0	0
$455$	719.019	0.0740838
$456$	0	0
$457$	5898.46	0.603760	0.301880	−	0.953346i	$-0.402386\pi$
$457$	5898.46	0.603760	0.301880	+	0.953346i	$0.402386\pi$
$458$	0	0
$459$	252.446	0.0256714
$460$	0	0
$461$	−16370.6	−1.65391	−0.826955	−	0.562268i	$-0.809929\pi$
$461$	−16370.6	−1.65391	−0.826955	+	0.562268i	$0.809929\pi$
$462$	0	0
$463$	−13334.3	−1.33844	−0.669221	−	0.743063i	$-0.733372\pi$
$463$	−13334.3	−1.33844	−0.669221	+	0.743063i	$0.733372\pi$
$464$	0	0
$465$	−3243.09	−0.323430
$466$	0	0
$467$	6584.38	0.652439	0.326219	−	0.945294i	$-0.394225\pi$
$467$	6584.38	0.652439	0.326219	+	0.945294i	$0.394225\pi$
$468$	0	0
$469$	−6731.44	−0.662749
$470$	0	0
$471$	−4259.17	−0.416672
$472$	0	0
$473$	−1199.21	−0.116574
$474$	0	0
$475$	−1938.17	−0.187220
$476$	0	0
$477$	4388.73	0.421271
$478$	0	0
$479$	18224.3	1.73839	0.869196	−	0.494467i	$-0.164637\pi$
$479$	18224.3	1.73839	0.869196	+	0.494467i	$0.164637\pi$
$480$	0	0
$481$	−2480.65	−0.235152
$482$	0	0
$483$	−6785.98	−0.639281
$484$	0	0
$485$	1068.51	0.100039
$486$	0	0
$487$	−6708.02	−0.624167	−0.312084	−	0.950055i	$-0.601027\pi$
$487$	−6708.02	−0.624167	−0.312084	+	0.950055i	$0.601027\pi$
$488$	0	0
$489$	−9433.89	−0.872424
$490$	0	0
$491$	12035.2	1.10620	0.553099	−	0.833116i	$-0.313445\pi$
$491$	12035.2	1.10620	0.553099	+	0.833116i	$0.313445\pi$
$492$	0	0
$493$	1647.62	0.150518
$494$	0	0
$495$	−1604.64	−0.145703
$496$	0	0
$497$	−5029.87	−0.453965
$498$	0	0
$499$	−16094.3	−1.44384	−0.721922	−	0.691974i	$-0.756741\pi$
$499$	−16094.3	−1.44384	−0.721922	+	0.691974i	$0.756741\pi$
$500$	0	0
$501$	3870.71	0.345171
$502$	0	0
$503$	−11793.3	−1.04541	−0.522703	−	0.852515i	$-0.675076\pi$
$503$	−11793.3	−1.04541	−0.522703	+	0.852515i	$0.675076\pi$
$504$	0	0
$505$	6842.81	0.602973
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	8249.74	0.718396	0.359198	−	0.933261i	$-0.383050\pi$
$509$	8249.74	0.718396	0.359198	+	0.933261i	$0.383050\pi$
$510$	0	0
$511$	−4734.26	−0.409846
$512$	0	0
$513$	−2093.23	−0.180152
$514$	0	0
$515$	−10421.2	−0.891678
$516$	0	0
$517$	7247.67	0.616542
$518$	0	0
$519$	2014.41	0.170371
$520$	0	0
$521$	21345.2	1.79491	0.897455	−	0.441105i	$-0.145413\pi$
$521$	21345.2	1.79491	0.897455	+	0.441105i	$0.145413\pi$
$522$	0	0
$523$	16607.7	1.38853	0.694266	−	0.719718i	$-0.255729\pi$
$523$	16607.7	1.38853	0.694266	+	0.719718i	$0.255729\pi$
$524$	0	0
$525$	829.638	0.0689683
$526$	0	0
$527$	−2021.49	−0.167092
$528$	0	0
$529$	29647.6	2.43672
$530$	0	0
$531$	1590.68	0.129999
$532$	0	0
$533$	−4098.13	−0.333039
$534$	0	0
$535$	−9185.46	−0.742285
$536$	0	0
$537$	−9006.25	−0.723740
$538$	0	0
$539$	7867.56	0.628720
$540$	0	0
$541$	16180.6	1.28588	0.642939	−	0.765917i	$-0.277715\pi$
$541$	16180.6	1.28588	0.642939	+	0.765917i	$0.277715\pi$
$542$	0	0
$543$	−698.206	−0.0551803
$544$	0	0
$545$	−867.634	−0.0681934
$546$	0	0
$547$	12935.7	1.01114	0.505568	−	0.862786i	$-0.331283\pi$
$547$	12935.7	1.01114	0.505568	+	0.862786i	$0.331283\pi$
$548$	0	0
$549$	−2110.86	−0.164097
$550$	0	0
$551$	−13661.7	−1.05628
$552$	0	0
$553$	13029.8	1.00196
$554$	0	0
$555$	−2862.29	−0.218915
$556$	0	0
$557$	11049.1	0.840510	0.420255	−	0.907406i	$-0.361941\pi$
$557$	11049.1	0.840510	0.420255	+	0.907406i	$0.361941\pi$
$558$	0	0
$559$	437.193	0.0330792
$560$	0	0
$561$	−1000.21	−0.0752741
$562$	0	0
$563$	6681.42	0.500157	0.250078	−	0.968226i	$-0.419544\pi$
$563$	6681.42	0.500157	0.250078	+	0.968226i	$0.419544\pi$
$564$	0	0
$565$	1887.38	0.140535
$566$	0	0
$567$	896.009	0.0663648
$568$	0	0
$569$	−11299.5	−0.832510	−0.416255	−	0.909248i	$-0.636658\pi$
$569$	−11299.5	−0.832510	−0.416255	+	0.909248i	$0.636658\pi$
$570$	0	0
$571$	−11280.1	−0.826724	−0.413362	−	0.910567i	$-0.635646\pi$
$571$	−11280.1	−0.826724	−0.413362	+	0.910567i	$0.635646\pi$
$572$	0	0
$573$	−5225.65	−0.380985
$574$	0	0
$575$	−5112.16	−0.370768
$576$	0	0
$577$	21169.9	1.52741	0.763703	−	0.645567i	$-0.223379\pi$
$577$	21169.9	1.52741	0.763703	+	0.645567i	$0.223379\pi$
$578$	0	0
$579$	−1977.83	−0.141962
$580$	0	0
$581$	9092.62	0.649269
$582$	0	0
$583$	−17388.4	−1.23526
$584$	0	0
$585$	585.000	0.0413449
$586$	0	0
$587$	−4014.61	−0.282284	−0.141142	−	0.989989i	$-0.545077\pi$
$587$	−4014.61	−0.282284	−0.141142	+	0.989989i	$0.545077\pi$
$588$	0	0
$589$	16761.8	1.17259
$590$	0	0
$591$	−2327.00	−0.161963
$592$	0	0
$593$	6045.14	0.418624	0.209312	−	0.977849i	$-0.432878\pi$
$593$	6045.14	0.418624	0.209312	+	0.977849i	$0.432878\pi$
$594$	0	0
$595$	517.132	0.0356308
$596$	0	0
$597$	6090.78	0.417553
$598$	0	0
$599$	26952.7	1.83849	0.919246	−	0.393684i	$-0.128800\pi$
$599$	26952.7	1.83849	0.919246	+	0.393684i	$0.128800\pi$
$600$	0	0
$601$	4064.24	0.275846	0.137923	−	0.990443i	$-0.455957\pi$
$601$	4064.24	0.275846	0.137923	+	0.990443i	$0.455957\pi$
$602$	0	0
$603$	−5476.76	−0.369869
$604$	0	0
$605$	−297.317	−0.0199796
$606$	0	0
$607$	17221.0	1.15153	0.575765	−	0.817615i	$-0.304704\pi$
$607$	17221.0	1.15153	0.575765	+	0.817615i	$0.304704\pi$
$608$	0	0
$609$	5847.93	0.389113
$610$	0	0
$611$	−2642.27	−0.174951
$612$	0	0
$613$	14810.5	0.975838	0.487919	−	0.872889i	$-0.337756\pi$
$613$	14810.5	0.975838	0.487919	+	0.872889i	$0.337756\pi$
$614$	0	0
$615$	−4728.61	−0.310042
$616$	0	0
$617$	21995.4	1.43517	0.717585	−	0.696470i	$-0.245247\pi$
$617$	21995.4	1.43517	0.717585	+	0.696470i	$0.245247\pi$
$618$	0	0
$619$	−28215.2	−1.83209	−0.916045	−	0.401077i	$-0.868636\pi$
$619$	−28215.2	−1.83209	−0.916045	+	0.401077i	$0.868636\pi$
$620$	0	0
$621$	−5521.13	−0.356772
$622$	0	0
$623$	−6080.13	−0.391004
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	8293.50	0.528247
$628$	0	0
$629$	−1784.13	−0.113097
$630$	0	0
$631$	−9709.56	−0.612569	−0.306285	−	0.951940i	$-0.599086\pi$
$631$	−9709.56	−0.612569	−0.306285	+	0.951940i	$0.599086\pi$
$632$	0	0
$633$	−14984.5	−0.940884
$634$	0	0
$635$	7267.89	0.454201
$636$	0	0
$637$	−2868.26	−0.178406
$638$	0	0
$639$	−4092.34	−0.253350
$640$	0	0
$641$	27122.9	1.67128	0.835641	−	0.549276i	$-0.185097\pi$
$641$	27122.9	1.67128	0.835641	+	0.549276i	$0.185097\pi$
$642$	0	0
$643$	23741.6	1.45611	0.728055	−	0.685519i	$-0.240425\pi$
$643$	23741.6	1.45611	0.728055	+	0.685519i	$0.240425\pi$
$644$	0	0
$645$	504.453	0.0307951
$646$	0	0
$647$	13313.5	0.808974	0.404487	−	0.914544i	$-0.367450\pi$
$647$	13313.5	0.808974	0.404487	+	0.914544i	$0.367450\pi$
$648$	0	0
$649$	−6302.38	−0.381186
$650$	0	0
$651$	−7174.92	−0.431962
$652$	0	0
$653$	12314.4	0.737976	0.368988	−	0.929434i	$-0.379704\pi$
$653$	12314.4	0.737976	0.368988	+	0.929434i	$0.379704\pi$
$654$	0	0
$655$	5532.91	0.330059
$656$	0	0
$657$	−3851.83	−0.228728
$658$	0	0
$659$	−22580.7	−1.33478	−0.667389	−	0.744709i	$-0.732588\pi$
$659$	−22580.7	−1.33478	−0.667389	+	0.744709i	$0.732588\pi$
$660$	0	0
$661$	−5125.08	−0.301577	−0.150789	−	0.988566i	$-0.548181\pi$
$661$	−5125.08	−0.301577	−0.150789	+	0.988566i	$0.548181\pi$
$662$	0	0
$663$	364.644	0.0213599
$664$	0	0
$665$	−4287.95	−0.250044
$666$	0	0
$667$	−36034.4	−2.09184
$668$	0	0
$669$	−10986.5	−0.634922
$670$	0	0
$671$	8363.38	0.481170
$672$	0	0
$673$	6059.06	0.347043	0.173521	−	0.984830i	$-0.444485\pi$
$673$	6059.06	0.347043	0.173521	+	0.984830i	$0.444485\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−16337.2	−0.927461	−0.463731	−	0.885976i	$-0.653489\pi$
$677$	−16337.2	−0.927461	−0.463731	+	0.885976i	$0.653489\pi$
$678$	0	0
$679$	2363.95	0.133608
$680$	0	0
$681$	2879.14	0.162010
$682$	0	0
$683$	28871.6	1.61749	0.808743	−	0.588162i	$-0.200148\pi$
$683$	28871.6	1.61749	0.808743	+	0.588162i	$0.200148\pi$
$684$	0	0
$685$	3385.65	0.188845
$686$	0	0
$687$	−8450.72	−0.469309
$688$	0	0
$689$	6339.27	0.350518
$690$	0	0
$691$	−3611.03	−0.198799	−0.0993993	−	0.995048i	$-0.531692\pi$
$691$	−3611.03	−0.198799	−0.0993993	+	0.995048i	$0.531692\pi$
$692$	0	0
$693$	−3550.05	−0.194596
$694$	0	0
$695$	7684.55	0.419412
$696$	0	0
$697$	−2947.45	−0.160176
$698$	0	0
$699$	−7234.20	−0.391449
$700$	0	0
$701$	−19383.1	−1.04435	−0.522176	−	0.852837i	$-0.674880\pi$
$701$	−19383.1	−1.04435	−0.522176	+	0.852837i	$0.674880\pi$
$702$	0	0
$703$	14793.7	0.793674
$704$	0	0
$705$	−3048.77	−0.162870
$706$	0	0
$707$	15138.8	0.805310
$708$	0	0
$709$	15298.3	0.810355	0.405177	−	0.914238i	$-0.367210\pi$
$709$	15298.3	0.810355	0.405177	+	0.914238i	$0.367210\pi$
$710$	0	0
$711$	10601.1	0.559175
$712$	0	0
$713$	44211.2	2.32219
$714$	0	0
$715$	−2317.81	−0.121232
$716$	0	0
$717$	−8108.44	−0.422337
$718$	0	0
$719$	−19824.5	−1.02827	−0.514137	−	0.857708i	$-0.671888\pi$
$719$	−19824.5	−1.02827	−0.514137	+	0.857708i	$0.671888\pi$
$720$	0	0
$721$	−23055.6	−1.19089
$722$	0	0
$723$	−4775.22	−0.245633
$724$	0	0
$725$	4405.48	0.225677
$726$	0	0
$727$	−26379.1	−1.34573	−0.672866	−	0.739764i	$-0.734937\pi$
$727$	−26379.1	−1.34573	−0.672866	+	0.739764i	$0.734937\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	314.437	0.0159095
$732$	0	0
$733$	−5790.62	−0.291789	−0.145895	−	0.989300i	$-0.546606\pi$
$733$	−5790.62	−0.291789	−0.145895	+	0.989300i	$0.546606\pi$
$734$	0	0
$735$	−3309.54	−0.166087
$736$	0	0
$737$	21699.3	1.08454
$738$	0	0
$739$	695.082	0.0345995	0.0172997	−	0.999850i	$-0.494493\pi$
$739$	695.082	0.0345995	0.0172997	+	0.999850i	$0.494493\pi$
$740$	0	0
$741$	−3023.55	−0.149896
$742$	0	0
$743$	20134.8	0.994179	0.497089	−	0.867699i	$-0.334402\pi$
$743$	20134.8	0.994179	0.497089	+	0.867699i	$0.334402\pi$
$744$	0	0
$745$	−4743.16	−0.233256
$746$	0	0
$747$	7397.83	0.362346
$748$	0	0
$749$	−20321.6	−0.991370
$750$	0	0
$751$	22248.7	1.08105	0.540523	−	0.841329i	$-0.318226\pi$
$751$	22248.7	1.08105	0.540523	+	0.841329i	$0.318226\pi$
$752$	0	0
$753$	−10545.6	−0.510362
$754$	0	0
$755$	16578.3	0.799135
$756$	0	0
$757$	−1351.98	−0.0649124	−0.0324562	−	0.999473i	$-0.510333\pi$
$757$	−1351.98	−0.0649124	−0.0324562	+	0.999473i	$0.510333\pi$
$758$	0	0
$759$	21875.1	1.04613
$760$	0	0
$761$	35726.0	1.70179	0.850897	−	0.525332i	$-0.176059\pi$
$761$	35726.0	1.70179	0.850897	+	0.525332i	$0.176059\pi$
$762$	0	0
$763$	−1919.53	−0.0910767
$764$	0	0
$765$	420.743	0.0198850
$766$	0	0
$767$	2297.65	0.108166
$768$	0	0
$769$	−4867.72	−0.228263	−0.114132	−	0.993466i	$-0.536409\pi$
$769$	−4867.72	−0.228263	−0.114132	+	0.993466i	$0.536409\pi$
$770$	0	0
$771$	−2056.94	−0.0960817
$772$	0	0
$773$	18247.3	0.849044	0.424522	−	0.905418i	$-0.360442\pi$
$773$	18247.3	0.849044	0.424522	+	0.905418i	$0.360442\pi$
$774$	0	0
$775$	−5405.16	−0.250528
$776$	0	0
$777$	−6332.45	−0.292375
$778$	0	0
$779$	24439.6	1.12406
$780$	0	0
$781$	16214.1	0.742878
$782$	0	0
$783$	4757.92	0.217157
$784$	0	0
$785$	−7098.62	−0.322752
$786$	0	0
$787$	18326.4	0.830072	0.415036	−	0.909805i	$-0.363769\pi$
$787$	18326.4	0.830072	0.415036	+	0.909805i	$0.363769\pi$
$788$	0	0
$789$	−951.171	−0.0429184
$790$	0	0
$791$	4175.57	0.187694
$792$	0	0
$793$	−3049.02	−0.136537
$794$	0	0
$795$	7314.55	0.326315
$796$	0	0
$797$	27446.9	1.21985	0.609924	−	0.792460i	$-0.291200\pi$
$797$	27446.9	1.21985	0.609924	+	0.792460i	$0.291200\pi$
$798$	0	0
$799$	−1900.37	−0.0841430
$800$	0	0
$801$	−4946.84	−0.218212
$802$	0	0
$803$	15261.2	0.670681
$804$	0	0
$805$	−11310.0	−0.495185
$806$	0	0
$807$	9528.70	0.415646
$808$	0	0
$809$	39886.6	1.73342	0.866711	−	0.498810i	$-0.166229\pi$
$809$	39886.6	1.73342	0.866711	+	0.498810i	$0.166229\pi$
$810$	0	0
$811$	35333.6	1.52988	0.764938	−	0.644104i	$-0.222769\pi$
$811$	35333.6	1.52988	0.764938	+	0.644104i	$0.222769\pi$
$812$	0	0
$813$	11675.2	0.503648
$814$	0	0
$815$	−15723.1	−0.675777
$816$	0	0
$817$	−2607.24	−0.111647
$818$	0	0
$819$	1294.23	0.0552188
$820$	0	0
$821$	−18360.4	−0.780491	−0.390245	−	0.920711i	$-0.627610\pi$
$821$	−18360.4	−0.780491	−0.390245	+	0.920711i	$0.627610\pi$
$822$	0	0
$823$	20535.7	0.869779	0.434889	−	0.900484i	$-0.356787\pi$
$823$	20535.7	0.869779	0.434889	+	0.900484i	$0.356787\pi$
$824$	0	0
$825$	−2674.40	−0.112861
$826$	0	0
$827$	26977.3	1.13433	0.567166	−	0.823603i	$-0.308040\pi$
$827$	26977.3	1.13433	0.567166	+	0.823603i	$0.308040\pi$
$828$	0	0
$829$	−29482.0	−1.23517	−0.617583	−	0.786506i	$-0.711888\pi$
$829$	−29482.0	−1.23517	−0.617583	+	0.786506i	$0.711888\pi$
$830$	0	0
$831$	7193.29	0.300280
$832$	0	0
$833$	−2062.91	−0.0858050
$834$	0	0
$835$	6451.19	0.267368
$836$	0	0
$837$	−5837.57	−0.241070
$838$	0	0
$839$	−48464.4	−1.99425	−0.997126	−	0.0757571i	$-0.975863\pi$
$839$	−48464.4	−1.99425	−0.997126	+	0.0757571i	$0.975863\pi$
$840$	0	0
$841$	6664.26	0.273249
$842$	0	0
$843$	17208.3	0.703067
$844$	0	0
$845$	845.000	0.0344010
$846$	0	0
$847$	−657.775	−0.0266841
$848$	0	0
$849$	6890.78	0.278552
$850$	0	0
$851$	39020.0	1.57178
$852$	0	0
$853$	21104.6	0.847135	0.423568	−	0.905864i	$-0.360778\pi$
$853$	21104.6	0.847135	0.423568	+	0.905864i	$0.360778\pi$
$854$	0	0
$855$	−3488.71	−0.139545
$856$	0	0
$857$	−36507.4	−1.45515	−0.727577	−	0.686026i	$-0.759354\pi$
$857$	−36507.4	−1.45515	−0.727577	+	0.686026i	$0.759354\pi$
$858$	0	0
$859$	−25174.6	−0.999938	−0.499969	−	0.866043i	$-0.666655\pi$
$859$	−25174.6	−0.999938	−0.499969	+	0.866043i	$0.666655\pi$
$860$	0	0
$861$	−10461.4	−0.414082
$862$	0	0
$863$	21109.2	0.832639	0.416319	−	0.909218i	$-0.363320\pi$
$863$	21109.2	0.832639	0.416319	+	0.909218i	$0.363320\pi$
$864$	0	0
$865$	3357.34	0.131969
$866$	0	0
$867$	−14476.7	−0.567077
$868$	0	0
$869$	−42002.4	−1.63963
$870$	0	0
$871$	−7910.87	−0.307749
$872$	0	0
$873$	1923.33	0.0745644
$874$	0	0
$875$	1382.73	0.0534226
$876$	0	0
$877$	25911.4	0.997679	0.498839	−	0.866694i	$-0.333760\pi$
$877$	25911.4	0.997679	0.498839	+	0.866694i	$0.333760\pi$
$878$	0	0
$879$	10393.2	0.398812
$880$	0	0
$881$	−28772.5	−1.10031	−0.550153	−	0.835064i	$-0.685431\pi$
$881$	−28772.5	−1.10031	−0.550153	+	0.835064i	$0.685431\pi$
$882$	0	0
$883$	−35094.1	−1.33750	−0.668749	−	0.743488i	$-0.733170\pi$
$883$	−35094.1	−1.33750	−0.668749	+	0.743488i	$0.733170\pi$
$884$	0	0
$885$	2651.13	0.100697
$886$	0	0
$887$	10041.4	0.380109	0.190054	−	0.981774i	$-0.439134\pi$
$887$	10041.4	0.380109	0.190054	+	0.981774i	$0.439134\pi$
$888$	0	0
$889$	16079.2	0.606615
$890$	0	0
$891$	−2888.35	−0.108601
$892$	0	0
$893$	15757.5	0.590485
$894$	0	0
$895$	−15010.4	−0.560606
$896$	0	0
$897$	−7974.96	−0.296852
$898$	0	0
$899$	−38099.7	−1.41346
$900$	0	0
$901$	4559.32	0.168583
$902$	0	0
$903$	1116.04	0.0411288
$904$	0	0
$905$	−1163.68	−0.0427425
$906$	0	0
$907$	−22706.0	−0.831245	−0.415622	−	0.909537i	$-0.636436\pi$
$907$	−22706.0	−0.831245	−0.415622	+	0.909537i	$0.636436\pi$
$908$	0	0
$909$	12317.1	0.449429
$910$	0	0
$911$	24072.6	0.875479	0.437739	−	0.899102i	$-0.355779\pi$
$911$	24072.6	0.875479	0.437739	+	0.899102i	$0.355779\pi$
$912$	0	0
$913$	−29310.7	−1.06248
$914$	0	0
$915$	−3518.11	−0.127109
$916$	0	0
$917$	12240.8	0.440815
$918$	0	0
$919$	−13036.6	−0.467941	−0.233970	−	0.972244i	$-0.575172\pi$
$919$	−13036.6	−0.467941	−0.233970	+	0.972244i	$0.575172\pi$
$920$	0	0
$921$	3075.29	0.110026
$922$	0	0
$923$	−5911.16	−0.210800
$924$	0	0
$925$	−4770.49	−0.169571
$926$	0	0
$927$	−18758.2	−0.664618
$928$	0	0
$929$	2923.55	0.103249	0.0516246	−	0.998667i	$-0.483560\pi$
$929$	2923.55	0.103249	0.0516246	+	0.998667i	$0.483560\pi$
$930$	0	0
$931$	17105.2	0.602149
$932$	0	0
$933$	10184.3	0.357361
$934$	0	0
$935$	−1667.01	−0.0583071
$936$	0	0
$937$	38569.5	1.34473	0.672365	−	0.740220i	$-0.265279\pi$
$937$	38569.5	1.34473	0.672365	+	0.740220i	$0.265279\pi$
$938$	0	0
$939$	10291.9	0.357681
$940$	0	0
$941$	−39933.9	−1.38343	−0.691715	−	0.722170i	$-0.743145\pi$
$941$	−39933.9	−1.38343	−0.691715	+	0.722170i	$0.743145\pi$
$942$	0	0
$943$	64462.3	2.22607
$944$	0	0
$945$	1493.35	0.0514059
$946$	0	0
$947$	51099.1	1.75343	0.876715	−	0.481010i	$-0.159730\pi$
$947$	51099.1	1.75343	0.876715	+	0.481010i	$0.159730\pi$
$948$	0	0
$949$	−5563.76	−0.190313
$950$	0	0
$951$	−21412.5	−0.730122
$952$	0	0
$953$	38862.5	1.32097	0.660483	−	0.750841i	$-0.270351\pi$
$953$	38862.5	1.32097	0.660483	+	0.750841i	$0.270351\pi$
$954$	0	0
$955$	−8709.41	−0.295110
$956$	0	0
$957$	−18851.2	−0.636754
$958$	0	0
$959$	7490.30	0.252215
$960$	0	0
$961$	16954.1	0.569103
$962$	0	0
$963$	−16533.8	−0.553266
$964$	0	0
$965$	−3296.38	−0.109963
$966$	0	0
$967$	−28744.4	−0.955903	−0.477952	−	0.878386i	$-0.658621\pi$
$967$	−28744.4	−0.955903	−0.477952	+	0.878386i	$0.658621\pi$
$968$	0	0
$969$	−2174.59	−0.0720929
$970$	0	0
$971$	34207.9	1.13057	0.565285	−	0.824895i	$-0.308766\pi$
$971$	34207.9	1.13057	0.565285	+	0.824895i	$0.308766\pi$
$972$	0	0
$973$	17001.0	0.560152
$974$	0	0
$975$	975.000	0.0320256
$976$	0	0
$977$	−54148.0	−1.77313	−0.886566	−	0.462603i	$-0.846916\pi$
$977$	−54148.0	−1.77313	−0.886566	+	0.462603i	$0.846916\pi$
$978$	0	0
$979$	19599.7	0.639847
$980$	0	0
$981$	−1561.74	−0.0508283
$982$	0	0
$983$	20490.3	0.664841	0.332420	−	0.943131i	$-0.392135\pi$
$983$	20490.3	0.664841	0.332420	+	0.943131i	$0.392135\pi$
$984$	0	0
$985$	−3878.34	−0.125456
$986$	0	0
$987$	−6745.01	−0.217524
$988$	0	0
$989$	−6876.91	−0.221105
$990$	0	0
$991$	44944.0	1.44066	0.720330	−	0.693631i	$-0.243990\pi$
$991$	44944.0	1.44066	0.720330	+	0.693631i	$0.243990\pi$
$992$	0	0
$993$	16629.2	0.531432
$994$	0	0
$995$	10151.3	0.323435
$996$	0	0
$997$	−28863.0	−0.916852	−0.458426	−	0.888733i	$-0.651587\pi$
$997$	−28863.0	−0.916852	−0.458426	+	0.888733i	$0.651587\pi$
$998$	0	0
$999$	−5152.13	−0.163169

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.4.a.n.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.4.a.n.1.4	✓	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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} +11.0618 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +5.00000 q^{5} +11.0618 q^{7} +9.00000 q^{9} -35.6586 q^{11} +13.0000 q^{13} +15.0000 q^{15} +9.34984 q^{17} -77.5269 q^{19} +33.1855 q^{21} -204.486 q^{23} +25.0000 q^{25} +27.0000 q^{27} +176.219 q^{29} -216.206 q^{31} -106.976 q^{33} +55.3092 q^{35} -190.820 q^{37} +39.0000 q^{39} -315.241 q^{41} +33.6302 q^{43} +45.0000 q^{45} -203.252 q^{47} -220.636 q^{49} +28.0495 q^{51} +487.637 q^{53} -178.293 q^{55} -232.581 q^{57} +176.742 q^{59} -234.540 q^{61} +99.5565 q^{63} +65.0000 q^{65} -608.529 q^{67} -613.459 q^{69} -454.705 q^{71} -427.982 q^{73} +75.0000 q^{75} -394.450 q^{77} +1177.90 q^{79} +81.0000 q^{81} +821.981 q^{83} +46.7492 q^{85} +528.658 q^{87} -549.649 q^{89} +143.804 q^{91} -648.619 q^{93} -387.635 q^{95} +213.703 q^{97} -320.928 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9	\( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9} - 38 q^{11} + 52 q^{13} + 60 q^{15} - 50 q^{17} - 180 q^{19} - 102 q^{21} - 170 q^{23} + 100 q^{25} + 108 q^{27} - 84 q^{29} - 408 q^{31} - 114 q^{33} - 170 q^{35} - 38 q^{37} + 156 q^{39} - 90 q^{41} - 732 q^{43} + 180 q^{45} - 520 q^{47} + 230 q^{49} - 150 q^{51} - 338 q^{53} - 190 q^{55} - 540 q^{57} - 232 q^{59} + 326 q^{61} - 306 q^{63} + 260 q^{65} - 1040 q^{67} - 510 q^{69} - 702 q^{71} - 368 q^{73} + 300 q^{75} + 14 q^{77} - 678 q^{79} + 324 q^{81} - 1888 q^{83} - 250 q^{85} - 252 q^{87} - 2070 q^{89} - 442 q^{91} - 1224 q^{93} - 900 q^{95} + 1126 q^{97} - 342 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9 - 38 * q^11 + 52 * q^13 + 60 * q^15 - 50 * q^17 - 180 * q^19 - 102 * q^21 - 170 * q^23 + 100 * q^25 + 108 * q^27 - 84 * q^29 - 408 * q^31 - 114 * q^33 - 170 * q^35 - 38 * q^37 + 156 * q^39 - 90 * q^41 - 732 * q^43 + 180 * q^45 - 520 * q^47 + 230 * q^49 - 150 * q^51 - 338 * q^53 - 190 * q^55 - 540 * q^57 - 232 * q^59 + 326 * q^61 - 306 * q^63 + 260 * q^65 - 1040 * q^67 - 510 * q^69 - 702 * q^71 - 368 * q^73 + 300 * q^75 + 14 * q^77 - 678 * q^79 + 324 * q^81 - 1888 * q^83 - 250 * q^85 - 252 * q^87 - 2070 * q^89 - 442 * q^91 - 1224 * q^93 - 900 * q^95 + 1126 * q^97 - 342 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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