LMFDB - Embedded newform 531.4.a.d.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,4,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("531.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	531.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:	$31.3300142130$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 34x^{5} + 25x^{4} + 315x^{3} - 146x^{2} - 736x + 512 $ x^7 - x^6 - 34x^5 + 25x^4 + 315x^3 - 146x^2 - 736*x + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.58179$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.58179 q^{2} +12.9928 q^{4} +12.2965 q^{5} +15.3493 q^{7} +22.8759 q^{8} +O(q^{10})$	\(q+4.58179 q^{2} +12.9928 q^{4} +12.2965 q^{5} +15.3493 q^{7} +22.8759 q^{8} +56.3401 q^{10} +41.7826 q^{11} -23.4975 q^{13} +70.3273 q^{14} +0.870331 q^{16} -123.900 q^{17} +106.282 q^{19} +159.766 q^{20} +191.439 q^{22} +207.156 q^{23} +26.2045 q^{25} -107.660 q^{26} +199.430 q^{28} +53.4048 q^{29} -252.361 q^{31} -179.020 q^{32} -567.686 q^{34} +188.743 q^{35} -357.560 q^{37} +486.962 q^{38} +281.294 q^{40} +353.220 q^{41} -80.6872 q^{43} +542.873 q^{44} +949.144 q^{46} +5.12471 q^{47} -107.399 q^{49} +120.063 q^{50} -305.298 q^{52} +260.709 q^{53} +513.781 q^{55} +351.129 q^{56} +244.689 q^{58} +59.0000 q^{59} +35.7406 q^{61} -1156.26 q^{62} -827.193 q^{64} -288.937 q^{65} +635.590 q^{67} -1609.81 q^{68} +864.781 q^{70} +644.029 q^{71} -531.586 q^{73} -1638.27 q^{74} +1380.90 q^{76} +641.334 q^{77} +594.860 q^{79} +10.7020 q^{80} +1618.38 q^{82} +95.7821 q^{83} -1523.54 q^{85} -369.692 q^{86} +955.816 q^{88} -1002.86 q^{89} -360.670 q^{91} +2691.53 q^{92} +23.4803 q^{94} +1306.90 q^{95} -964.276 q^{97} -492.080 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8}+O(q^{10}) $ 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8	$ 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8} - 79 q^{10} + 131 q^{11} - 123 q^{13} + 117 q^{14} + 202 q^{16} + 235 q^{17} - 80 q^{19} - 61 q^{20} + 688 q^{22} + 274 q^{23} + 193 q^{25} + 180 q^{26} - 118 q^{28} + 406 q^{29} - 346 q^{31} + 854 q^{32} + 178 q^{34} + 424 q^{35} - 157 q^{37} + 129 q^{38} - 590 q^{40} + 825 q^{41} - 815 q^{43} + 1690 q^{44} + 1457 q^{46} + 1196 q^{47} + 914 q^{49} - 713 q^{50} + 1030 q^{52} + 900 q^{53} - 1044 q^{55} - 2172 q^{56} + 1242 q^{58} + 413 q^{59} + 420 q^{61} - 646 q^{62} + 3541 q^{64} - 190 q^{65} + 1316 q^{67} + 611 q^{68} + 4658 q^{70} + 173 q^{71} - 418 q^{73} - 660 q^{74} + 1540 q^{76} + 753 q^{77} + 2635 q^{79} - 6155 q^{80} - 125 q^{82} - 457 q^{83} + 1270 q^{85} - 3482 q^{86} + 7685 q^{88} - 592 q^{89} + 3179 q^{91} + 3500 q^{92} + 2064 q^{94} + 2250 q^{95} - 1906 q^{97} - 2994 q^{98}+O(q^{100}) $ 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8 - 79 * q^10 + 131 * q^11 - 123 * q^13 + 117 * q^14 + 202 * q^16 + 235 * q^17 - 80 * q^19 - 61 * q^20 + 688 * q^22 + 274 * q^23 + 193 * q^25 + 180 * q^26 - 118 * q^28 + 406 * q^29 - 346 * q^31 + 854 * q^32 + 178 * q^34 + 424 * q^35 - 157 * q^37 + 129 * q^38 - 590 * q^40 + 825 * q^41 - 815 * q^43 + 1690 * q^44 + 1457 * q^46 + 1196 * q^47 + 914 * q^49 - 713 * q^50 + 1030 * q^52 + 900 * q^53 - 1044 * q^55 - 2172 * q^56 + 1242 * q^58 + 413 * q^59 + 420 * q^61 - 646 * q^62 + 3541 * q^64 - 190 * q^65 + 1316 * q^67 + 611 * q^68 + 4658 * q^70 + 173 * q^71 - 418 * q^73 - 660 * q^74 + 1540 * q^76 + 753 * q^77 + 2635 * q^79 - 6155 * q^80 - 125 * q^82 - 457 * q^83 + 1270 * q^85 - 3482 * q^86 + 7685 * q^88 - 592 * q^89 + 3179 * q^91 + 3500 * q^92 + 2064 * q^94 + 2250 * q^95 - 1906 * q^97 - 2994 * q^98

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 $

$2$	4.58179	1.61991	0.809954	−	0.586494i	$-0.199492\pi$
$2$	4.58179	1.61991	0.809954	+	0.586494i	$0.199492\pi$
$3$	0	0
$3$	0	0
$4$	12.9928	1.62410
$5$	12.2965	1.09983	0.549917	−	0.835219i	$-0.314659\pi$
$5$	12.2965	1.09983	0.549917	+	0.835219i	$0.314659\pi$
$6$	0	0
$7$	15.3493	0.828784	0.414392	−	0.910098i	$-0.363994\pi$
$7$	15.3493	0.828784	0.414392	+	0.910098i	$0.363994\pi$
$8$	22.8759	1.01098
$9$	0	0
$10$	56.3401	1.78163
$11$	41.7826	1.14527	0.572633	−	0.819812i	$-0.305922\pi$
$11$	41.7826	1.14527	0.572633	+	0.819812i	$0.305922\pi$
$12$	0	0
$13$	−23.4975	−0.501310	−0.250655	−	0.968077i	$-0.580646\pi$
$13$	−23.4975	−0.501310	−0.250655	+	0.968077i	$0.580646\pi$
$14$	70.3273	1.34255
$15$	0	0
$16$	0.870331	0.0135989
$17$	−123.900	−1.76766	−0.883832	−	0.467805i	$-0.845045\pi$
$17$	−123.900	−1.76766	−0.883832	+	0.467805i	$0.845045\pi$
$18$	0	0
$19$	106.282	1.28330	0.641651	−	0.766996i	$-0.278250\pi$
$19$	106.282	1.28330	0.641651	+	0.766996i	$0.278250\pi$
$20$	159.766	1.78624
$21$	0	0
$22$	191.439	1.85523
$23$	207.156	1.87804	0.939021	−	0.343860i	$-0.111735\pi$
$23$	207.156	1.87804	0.939021	+	0.343860i	$0.111735\pi$
$24$	0	0
$25$	26.2045	0.209636
$26$	−107.660	−0.812075
$27$	0	0
$28$	199.430	1.34603
$29$	53.4048	0.341966	0.170983	−	0.985274i	$-0.445306\pi$
$29$	53.4048	0.341966	0.170983	+	0.985274i	$0.445306\pi$
$30$	0	0
$31$	−252.361	−1.46211	−0.731054	−	0.682319i	$-0.760971\pi$
$31$	−252.361	−1.46211	−0.731054	+	0.682319i	$0.760971\pi$
$32$	−179.020	−0.988954
$33$	0	0
$34$	−567.686	−2.86345
$35$	188.743	0.911525
$36$	0	0
$37$	−357.560	−1.58872	−0.794359	−	0.607449i	$-0.792193\pi$
$37$	−357.560	−1.58872	−0.794359	+	0.607449i	$0.792193\pi$
$38$	486.962	2.07883
$39$	0	0
$40$	281.294	1.11191
$41$	353.220	1.34545	0.672727	−	0.739891i	$-0.265123\pi$
$41$	353.220	1.34545	0.672727	+	0.739891i	$0.265123\pi$
$42$	0	0
$43$	−80.6872	−0.286155	−0.143078	−	0.989711i	$-0.545700\pi$
$43$	−80.6872	−0.286155	−0.143078	+	0.989711i	$0.545700\pi$
$44$	542.873	1.86003
$45$	0	0
$46$	949.144	3.04225
$47$	5.12471	0.0159046	0.00795229	−	0.999968i	$-0.497469\pi$
$47$	5.12471	0.0159046	0.00795229	+	0.999968i	$0.497469\pi$
$48$	0	0
$49$	−107.399	−0.313117
$50$	120.063	0.339591
$51$	0	0
$52$	−305.298	−0.814176
$53$	260.709	0.675681	0.337841	−	0.941203i	$-0.390303\pi$
$53$	260.709	0.675681	0.337841	+	0.941203i	$0.390303\pi$
$54$	0	0
$55$	513.781	1.25960
$56$	351.129	0.837887
$57$	0	0
$58$	244.689	0.553953
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	0	0
$61$	35.7406	0.0750182	0.0375091	−	0.999296i	$-0.488058\pi$
$61$	35.7406	0.0750182	0.0375091	+	0.999296i	$0.488058\pi$
$62$	−1156.26	−2.36848
$63$	0	0
$64$	−827.193	−1.61561
$65$	−288.937	−0.551358
$66$	0	0
$67$	635.590	1.15895	0.579475	−	0.814990i	$-0.303258\pi$
$67$	635.590	1.15895	0.579475	+	0.814990i	$0.303258\pi$
$68$	−1609.81	−2.87086
$69$	0	0
$70$	864.781	1.47659
$71$	644.029	1.07651	0.538255	−	0.842782i	$-0.319084\pi$
$71$	644.029	1.07651	0.538255	+	0.842782i	$0.319084\pi$
$72$	0	0
$73$	−531.586	−0.852293	−0.426147	−	0.904654i	$-0.640129\pi$
$73$	−531.586	−0.852293	−0.426147	+	0.904654i	$0.640129\pi$
$74$	−1638.27	−2.57357
$75$	0	0
$76$	1380.90	2.08421
$77$	641.334	0.949179
$78$	0	0
$79$	594.860	0.847176	0.423588	−	0.905855i	$-0.360770\pi$
$79$	594.860	0.847176	0.423588	+	0.905855i	$0.360770\pi$
$80$	10.7020	0.0149566
$81$	0	0
$82$	1618.38	2.17951
$83$	95.7821	0.126668	0.0633340	−	0.997992i	$-0.479827\pi$
$83$	95.7821	0.126668	0.0633340	+	0.997992i	$0.479827\pi$
$84$	0	0
$85$	−1523.54	−1.94414
$86$	−369.692	−0.463545
$87$	0	0
$88$	955.816	1.15784
$89$	−1002.86	−1.19442	−0.597208	−	0.802086i	$-0.703723\pi$
$89$	−1002.86	−1.19442	−0.597208	+	0.802086i	$0.703723\pi$
$90$	0	0
$91$	−360.670	−0.415477
$92$	2691.53	3.05013
$93$	0	0
$94$	23.4803	0.0257639
$95$	1306.90	1.41142
$96$	0	0
$97$	−964.276	−1.00935	−0.504677	−	0.863308i	$-0.668388\pi$
$97$	−964.276	−1.00935	−0.504677	+	0.863308i	$0.668388\pi$
$98$	−492.080	−0.507220
$99$	0	0
$100$	340.469	0.340469
$101$	−1294.85	−1.27567	−0.637835	−	0.770173i	$-0.720170\pi$
$101$	−1294.85	−1.27567	−0.637835	+	0.770173i	$0.720170\pi$
$102$	0	0
$103$	17.3037	0.0165533	0.00827663	−	0.999966i	$-0.497365\pi$
$103$	17.3037	0.0165533	0.00827663	+	0.999966i	$0.497365\pi$
$104$	−537.526	−0.506815
$105$	0	0
$106$	1194.51	1.09454
$107$	−267.634	−0.241805	−0.120903	−	0.992664i	$-0.538579\pi$
$107$	−267.634	−0.241805	−0.120903	+	0.992664i	$0.538579\pi$
$108$	0	0
$109$	−179.027	−0.157318	−0.0786591	−	0.996902i	$-0.525064\pi$
$109$	−179.027	−0.157318	−0.0786591	+	0.996902i	$0.525064\pi$
$110$	2354.04	2.04044
$111$	0	0
$112$	13.3590	0.0112706
$113$	694.415	0.578098	0.289049	−	0.957314i	$-0.406661\pi$
$113$	694.415	0.578098	0.289049	+	0.957314i	$0.406661\pi$
$114$	0	0
$115$	2547.30	2.06553
$116$	693.877	0.555387
$117$	0	0
$118$	270.326	0.210894
$119$	−1901.78	−1.46501
$120$	0	0
$121$	414.788	0.311636
$122$	163.756	0.121523
$123$	0	0
$124$	−3278.87	−2.37461
$125$	−1214.84	−0.869270
$126$	0	0
$127$	−1929.31	−1.34802	−0.674010	−	0.738722i	$-0.735430\pi$
$127$	−1929.31	−1.34802	−0.674010	+	0.738722i	$0.735430\pi$
$128$	−2357.87	−1.62819
$129$	0	0
$130$	−1323.85	−0.893148
$131$	−2118.62	−1.41301	−0.706506	−	0.707707i	$-0.749730\pi$
$131$	−2118.62	−1.41301	−0.706506	+	0.707707i	$0.749730\pi$
$132$	0	0
$133$	1631.35	1.06358
$134$	2912.14	1.87739
$135$	0	0
$136$	−2834.34	−1.78708
$137$	−318.864	−0.198850	−0.0994249	−	0.995045i	$-0.531700\pi$
$137$	−318.864	−0.198850	−0.0994249	+	0.995045i	$0.531700\pi$
$138$	0	0
$139$	2310.92	1.41014	0.705070	−	0.709138i	$-0.250916\pi$
$139$	2310.92	1.41014	0.705070	+	0.709138i	$0.250916\pi$
$140$	2452.30	1.48041
$141$	0	0
$142$	2950.81	1.74385
$143$	−981.786	−0.574133
$144$	0	0
$145$	656.693	0.376106
$146$	−2435.61	−1.38064
$147$	0	0
$148$	−4645.71	−2.58023
$149$	1168.54	0.642485	0.321243	−	0.946997i	$-0.395899\pi$
$149$	1168.54	0.642485	0.321243	+	0.946997i	$0.395899\pi$
$150$	0	0
$151$	−1537.12	−0.828402	−0.414201	−	0.910186i	$-0.635939\pi$
$151$	−1537.12	−0.828402	−0.414201	+	0.910186i	$0.635939\pi$
$152$	2431.30	1.29740
$153$	0	0
$154$	2938.46	1.53758
$155$	−3103.16	−1.60808
$156$	0	0
$157$	−2351.07	−1.19513	−0.597565	−	0.801820i	$-0.703865\pi$
$157$	−2351.07	−1.19513	−0.597565	+	0.801820i	$0.703865\pi$
$158$	2725.52	1.37235
$159$	0	0
$160$	−2201.32	−1.08769
$161$	3179.70	1.55649
$162$	0	0
$163$	−2755.11	−1.32391	−0.661954	−	0.749545i	$-0.730272\pi$
$163$	−2755.11	−1.32391	−0.661954	+	0.749545i	$0.730272\pi$
$164$	4589.31	2.18515
$165$	0	0
$166$	438.853	0.205190
$167$	3741.97	1.73390	0.866952	−	0.498391i	$-0.166076\pi$
$167$	3741.97	1.73390	0.866952	+	0.498391i	$0.166076\pi$
$168$	0	0
$169$	−1644.87	−0.748689
$170$	−6980.56	−3.14932
$171$	0	0
$172$	−1048.35	−0.464745
$173$	−3970.75	−1.74503	−0.872515	−	0.488587i	$-0.837513\pi$
$173$	−3970.75	−1.74503	−0.872515	+	0.488587i	$0.837513\pi$
$174$	0	0
$175$	402.220	0.173743
$176$	36.3647	0.0155744
$177$	0	0
$178$	−4594.90	−1.93484
$179$	−1322.29	−0.552139	−0.276070	−	0.961138i	$-0.589032\pi$
$179$	−1322.29	−0.552139	−0.276070	+	0.961138i	$0.589032\pi$
$180$	0	0
$181$	3244.93	1.33256	0.666280	−	0.745701i	$-0.267885\pi$
$181$	3244.93	1.33256	0.666280	+	0.745701i	$0.267885\pi$
$182$	−1652.51	−0.673035
$183$	0	0
$184$	4738.88	1.89867
$185$	−4396.75	−1.74733
$186$	0	0
$187$	−5176.89	−2.02445
$188$	66.5842	0.0258306
$189$	0	0
$190$	5987.93	2.28637
$191$	−1496.99	−0.567110	−0.283555	−	0.958956i	$-0.591514\pi$
$191$	−1496.99	−0.567110	−0.283555	+	0.958956i	$0.591514\pi$
$192$	0	0
$193$	219.504	0.0818665	0.0409333	−	0.999162i	$-0.486967\pi$
$193$	219.504	0.0818665	0.0409333	+	0.999162i	$0.486967\pi$
$194$	−4418.11	−1.63506
$195$	0	0
$196$	−1395.41	−0.508533
$197$	−740.027	−0.267638	−0.133819	−	0.991006i	$-0.542724\pi$
$197$	−740.027	−0.267638	−0.133819	+	0.991006i	$0.542724\pi$
$198$	0	0
$199$	2138.04	0.761617	0.380808	−	0.924654i	$-0.375646\pi$
$199$	2138.04	0.761617	0.380808	+	0.924654i	$0.375646\pi$
$200$	599.452	0.211938
$201$	0	0
$202$	−5932.74	−2.06647
$203$	819.726	0.283416
$204$	0	0
$205$	4343.37	1.47978
$206$	79.2820	0.0268147
$207$	0	0
$208$	−20.4506	−0.00681727
$209$	4440.74	1.46972
$210$	0	0
$211$	−510.304	−0.166497	−0.0832483	−	0.996529i	$-0.526529\pi$
$211$	−510.304	−0.166497	−0.0832483	+	0.996529i	$0.526529\pi$
$212$	3387.34	1.09737
$213$	0	0
$214$	−1226.24	−0.391702
$215$	−992.172	−0.314723
$216$	0	0
$217$	−3873.56	−1.21177
$218$	−820.264	−0.254841
$219$	0	0
$220$	6675.45	2.04572
$221$	2911.35	0.886146
$222$	0	0
$223$	−2449.19	−0.735469	−0.367735	−	0.929931i	$-0.619867\pi$
$223$	−2449.19	−0.735469	−0.367735	+	0.929931i	$0.619867\pi$
$224$	−2747.83	−0.819629
$225$	0	0
$226$	3181.66	0.936465
$227$	185.296	0.0541786	0.0270893	−	0.999633i	$-0.491376\pi$
$227$	185.296	0.0541786	0.0270893	+	0.999633i	$0.491376\pi$
$228$	0	0
$229$	6185.66	1.78498	0.892488	−	0.451071i	$-0.148958\pi$
$229$	6185.66	1.78498	0.892488	+	0.451071i	$0.148958\pi$
$230$	11671.2	3.34597
$231$	0	0
$232$	1221.68	0.345722
$233$	4695.37	1.32019	0.660094	−	0.751183i	$-0.270516\pi$
$233$	4695.37	1.32019	0.660094	+	0.751183i	$0.270516\pi$
$234$	0	0
$235$	63.0161	0.0174924
$236$	766.575	0.211440
$237$	0	0
$238$	−8713.58	−2.37318
$239$	3690.23	0.998748	0.499374	−	0.866386i	$-0.333563\pi$
$239$	3690.23	0.998748	0.499374	+	0.866386i	$0.333563\pi$
$240$	0	0
$241$	−129.939	−0.0347307	−0.0173654	−	0.999849i	$-0.505528\pi$
$241$	−129.939	−0.0347307	−0.0173654	+	0.999849i	$0.505528\pi$
$242$	1900.47	0.504822
$243$	0	0
$244$	464.370	0.121837
$245$	−1320.63	−0.344377
$246$	0	0
$247$	−2497.36	−0.643332
$248$	−5772.99	−1.47817
$249$	0	0
$250$	−5566.15	−1.40814
$251$	921.958	0.231846	0.115923	−	0.993258i	$-0.463017\pi$
$251$	921.958	0.231846	0.115923	+	0.993258i	$0.463017\pi$
$252$	0	0
$253$	8655.51	2.15086
$254$	−8839.69	−2.18367
$255$	0	0
$256$	−4185.71	−1.02190
$257$	−2067.03	−0.501704	−0.250852	−	0.968026i	$-0.580711\pi$
$257$	−2067.03	−0.501704	−0.250852	+	0.968026i	$0.580711\pi$
$258$	0	0
$259$	−5488.30	−1.31670
$260$	−3754.10	−0.895459
$261$	0	0
$262$	−9707.07	−2.28895
$263$	−4434.33	−1.03967	−0.519833	−	0.854268i	$-0.674006\pi$
$263$	−4434.33	−1.03967	−0.519833	+	0.854268i	$0.674006\pi$
$264$	0	0
$265$	3205.81	0.743138
$266$	7474.52	1.72290
$267$	0	0
$268$	8258.09	1.88225
$269$	−2401.90	−0.544410	−0.272205	−	0.962239i	$-0.587753\pi$
$269$	−2401.90	−0.544410	−0.272205	+	0.962239i	$0.587753\pi$
$270$	0	0
$271$	3740.90	0.838537	0.419268	−	0.907862i	$-0.362287\pi$
$271$	3740.90	0.838537	0.419268	+	0.907862i	$0.362287\pi$
$272$	−107.834	−0.0240383
$273$	0	0
$274$	−1460.97	−0.322118
$275$	1094.89	0.240089
$276$	0	0
$277$	−1443.22	−0.313049	−0.156525	−	0.987674i	$-0.550029\pi$
$277$	−1443.22	−0.313049	−0.156525	+	0.987674i	$0.550029\pi$
$278$	10588.1	2.28430
$279$	0	0
$280$	4317.67	0.921536
$281$	3497.86	0.742580	0.371290	−	0.928517i	$-0.378916\pi$
$281$	3497.86	0.742580	0.371290	+	0.928517i	$0.378916\pi$
$282$	0	0
$283$	−396.792	−0.0833457	−0.0416728	−	0.999131i	$-0.513269\pi$
$283$	−396.792	−0.0833457	−0.0416728	+	0.999131i	$0.513269\pi$
$284$	8367.74	1.74836
$285$	0	0
$286$	−4498.34	−0.930043
$287$	5421.67	1.11509
$288$	0	0
$289$	10438.3	2.12463
$290$	3008.83	0.609257
$291$	0	0
$292$	−6906.79	−1.38421
$293$	−9247.65	−1.84387	−0.921934	−	0.387347i	$-0.873391\pi$
$293$	−9247.65	−1.84387	−0.921934	+	0.387347i	$0.873391\pi$
$294$	0	0
$295$	725.495	0.143186
$296$	−8179.52	−1.60617
$297$	0	0
$298$	5353.99	1.04077
$299$	−4867.63	−0.941480
$300$	0	0
$301$	−1238.49	−0.237161
$302$	−7042.74	−1.34193
$303$	0	0
$304$	92.5005	0.0174515
$305$	439.485	0.0825076
$306$	0	0
$307$	5013.01	0.931946	0.465973	−	0.884799i	$-0.345704\pi$
$307$	5013.01	0.931946	0.465973	+	0.884799i	$0.345704\pi$
$308$	8332.72	1.54156
$309$	0	0
$310$	−14218.0	−2.60494
$311$	6356.34	1.15896	0.579478	−	0.814988i	$-0.303257\pi$
$311$	6356.34	1.15896	0.579478	+	0.814988i	$0.303257\pi$
$312$	0	0
$313$	1664.15	0.300521	0.150261	−	0.988646i	$-0.451989\pi$
$313$	1664.15	0.300521	0.150261	+	0.988646i	$0.451989\pi$
$314$	−10772.1	−1.93600
$315$	0	0
$316$	7728.89	1.37590
$317$	9467.95	1.67752	0.838759	−	0.544503i	$-0.183282\pi$
$317$	9467.95	1.67752	0.838759	+	0.544503i	$0.183282\pi$
$318$	0	0
$319$	2231.39	0.391642
$320$	−10171.6	−1.77691
$321$	0	0
$322$	14568.7	2.52137
$323$	−13168.4	−2.26845
$324$	0	0
$325$	−615.739	−0.105092
$326$	−12623.3	−2.14461
$327$	0	0
$328$	8080.23	1.36023
$329$	78.6606	0.0131815
$330$	0	0
$331$	−5398.35	−0.896436	−0.448218	−	0.893924i	$-0.647941\pi$
$331$	−5398.35	−0.896436	−0.448218	+	0.893924i	$0.647941\pi$
$332$	1244.48	0.205721
$333$	0	0
$334$	17144.9	2.80876
$335$	7815.54	1.27465
$336$	0	0
$337$	4793.04	0.774759	0.387379	−	0.921920i	$-0.373380\pi$
$337$	4793.04	0.774759	0.387379	+	0.921920i	$0.373380\pi$
$338$	−7536.44	−1.21281
$339$	0	0
$340$	−19795.1	−3.15747
$341$	−10544.3	−1.67450
$342$	0	0
$343$	−6913.31	−1.08829
$344$	−1845.79	−0.289298
$345$	0	0
$346$	−18193.1	−2.82679
$347$	10721.0	1.65860	0.829298	−	0.558806i	$-0.188740\pi$
$347$	10721.0	1.65860	0.829298	+	0.558806i	$0.188740\pi$
$348$	0	0
$349$	1043.92	0.160114	0.0800571	−	0.996790i	$-0.474490\pi$
$349$	1043.92	0.160114	0.0800571	+	0.996790i	$0.474490\pi$
$350$	1842.89	0.281447
$351$	0	0
$352$	−7479.91	−1.13262
$353$	9834.72	1.48286	0.741429	−	0.671031i	$-0.234148\pi$
$353$	9834.72	1.48286	0.741429	+	0.671031i	$0.234148\pi$
$354$	0	0
$355$	7919.32	1.18398
$356$	−13030.0	−1.93985
$357$	0	0
$358$	−6058.48	−0.894414
$359$	−9944.23	−1.46194	−0.730970	−	0.682409i	$-0.760932\pi$
$359$	−9944.23	−1.46194	−0.730970	+	0.682409i	$0.760932\pi$
$360$	0	0
$361$	4436.85	0.646866
$362$	14867.6	2.15862
$363$	0	0
$364$	−4686.11	−0.674777
$365$	−6536.66	−0.937382
$366$	0	0
$367$	8177.88	1.16317	0.581583	−	0.813487i	$-0.302433\pi$
$367$	8177.88	1.16317	0.581583	+	0.813487i	$0.302433\pi$
$368$	180.294	0.0255393
$369$	0	0
$370$	−20145.0	−2.83051
$371$	4001.70	0.559994
$372$	0	0
$373$	−5263.36	−0.730633	−0.365317	−	0.930883i	$-0.619039\pi$
$373$	−5263.36	−0.730633	−0.365317	+	0.930883i	$0.619039\pi$
$374$	−23719.4	−3.27941
$375$	0	0
$376$	117.232	0.0160793
$377$	−1254.88	−0.171431
$378$	0	0
$379$	−1581.26	−0.214310	−0.107155	−	0.994242i	$-0.534174\pi$
$379$	−1581.26	−0.214310	−0.107155	+	0.994242i	$0.534174\pi$
$380$	16980.3	2.29229
$381$	0	0
$382$	−6858.87	−0.918666
$383$	−18.1202	−0.00241749	−0.00120875	−	0.999999i	$-0.500385\pi$
$383$	−18.1202	−0.00241749	−0.00120875	+	0.999999i	$0.500385\pi$
$384$	0	0
$385$	7886.18	1.04394
$386$	1005.72	0.132616
$387$	0	0
$388$	−12528.6	−1.63929
$389$	7475.52	0.974354	0.487177	−	0.873303i	$-0.338027\pi$
$389$	7475.52	0.974354	0.487177	+	0.873303i	$0.338027\pi$
$390$	0	0
$391$	−25666.7	−3.31974
$392$	−2456.85	−0.316556
$393$	0	0
$394$	−3390.65	−0.433549
$395$	7314.70	0.931754
$396$	0	0
$397$	−8078.60	−1.02129	−0.510647	−	0.859791i	$-0.670594\pi$
$397$	−8078.60	−1.02129	−0.510647	+	0.859791i	$0.670594\pi$
$398$	9796.06	1.23375
$399$	0	0
$400$	22.8066	0.00285082
$401$	−1634.60	−0.203561	−0.101781	−	0.994807i	$-0.532454\pi$
$401$	−1634.60	−0.203561	−0.101781	+	0.994807i	$0.532454\pi$
$402$	0	0
$403$	5929.84	0.732969
$404$	−16823.7	−2.07181
$405$	0	0
$406$	3755.81	0.459108
$407$	−14939.8	−1.81951
$408$	0	0
$409$	−2121.07	−0.256430	−0.128215	−	0.991746i	$-0.540925\pi$
$409$	−2121.07	−0.256430	−0.128215	+	0.991746i	$0.540925\pi$
$410$	19900.4	2.39710
$411$	0	0
$412$	224.824	0.0268841
$413$	905.609	0.107899
$414$	0	0
$415$	1177.79	0.139314
$416$	4206.51	0.495772
$417$	0	0
$418$	20346.5	2.38082
$419$	14819.7	1.72790	0.863950	−	0.503577i	$-0.167983\pi$
$419$	14819.7	1.72790	0.863950	+	0.503577i	$0.167983\pi$
$420$	0	0
$421$	5226.64	0.605061	0.302531	−	0.953140i	$-0.402169\pi$
$421$	5226.64	0.605061	0.302531	+	0.953140i	$0.402169\pi$
$422$	−2338.10	−0.269709
$423$	0	0
$424$	5963.96	0.683102
$425$	−3246.75	−0.370565
$426$	0	0
$427$	548.593	0.0621739
$428$	−3477.31	−0.392715
$429$	0	0
$430$	−4545.92	−0.509823
$431$	10926.6	1.22115	0.610577	−	0.791957i	$-0.290937\pi$
$431$	10926.6	1.22115	0.610577	+	0.791957i	$0.290937\pi$
$432$	0	0
$433$	3172.98	0.352157	0.176078	−	0.984376i	$-0.443659\pi$
$433$	3172.98	0.352157	0.176078	+	0.984376i	$0.443659\pi$
$434$	−17747.9	−1.96296
$435$	0	0
$436$	−2326.06	−0.255500
$437$	22016.9	2.41010
$438$	0	0
$439$	2162.68	0.235123	0.117562	−	0.993066i	$-0.462492\pi$
$439$	2162.68	0.235123	0.117562	+	0.993066i	$0.462492\pi$
$440$	11753.2	1.27344
$441$	0	0
$442$	13339.2	1.43547
$443$	10270.6	1.10152	0.550760	−	0.834664i	$-0.314338\pi$
$443$	10270.6	1.10152	0.550760	+	0.834664i	$0.314338\pi$
$444$	0	0
$445$	−12331.7	−1.31366
$446$	−11221.7	−1.19139
$447$	0	0
$448$	−12696.8	−1.33899
$449$	3822.43	0.401763	0.200882	−	0.979616i	$-0.435619\pi$
$449$	3822.43	0.401763	0.200882	+	0.979616i	$0.435619\pi$
$450$	0	0
$451$	14758.4	1.54090
$452$	9022.39	0.938889
$453$	0	0
$454$	848.989	0.0877644
$455$	−4434.98	−0.456956
$456$	0	0
$457$	16654.3	1.70471	0.852357	−	0.522961i	$-0.175173\pi$
$457$	16654.3	1.70471	0.852357	+	0.522961i	$0.175173\pi$
$458$	28341.4	2.89150
$459$	0	0
$460$	33096.5	3.35463
$461$	−14478.5	−1.46276	−0.731378	−	0.681972i	$-0.761122\pi$
$461$	−14478.5	−1.46276	−0.731378	+	0.681972i	$0.761122\pi$
$462$	0	0
$463$	−11106.8	−1.11485	−0.557425	−	0.830227i	$-0.688211\pi$
$463$	−11106.8	−1.11485	−0.557425	+	0.830227i	$0.688211\pi$
$464$	46.4798	0.00465037
$465$	0	0
$466$	21513.2	2.13858
$467$	−18099.7	−1.79347	−0.896737	−	0.442563i	$-0.854069\pi$
$467$	−18099.7	−1.79347	−0.896737	+	0.442563i	$0.854069\pi$
$468$	0	0
$469$	9755.86	0.960519
$470$	288.726	0.0283361
$471$	0	0
$472$	1349.68	0.131619
$473$	−3371.32	−0.327724
$474$	0	0
$475$	2785.06	0.269026
$476$	−24709.5	−2.37932
$477$	0	0
$478$	16907.8	1.61788
$479$	15597.9	1.48787	0.743933	−	0.668255i	$-0.232958\pi$
$479$	15597.9	1.48787	0.743933	+	0.668255i	$0.232958\pi$
$480$	0	0
$481$	8401.76	0.796439
$482$	−595.353	−0.0562606
$483$	0	0
$484$	5389.25	0.506128
$485$	−11857.2	−1.11012
$486$	0	0
$487$	14116.3	1.31349	0.656745	−	0.754113i	$-0.271933\pi$
$487$	14116.3	1.31349	0.656745	+	0.754113i	$0.271933\pi$
$488$	817.598	0.0758421
$489$	0	0
$490$	−6050.87	−0.557858
$491$	−14694.8	−1.35064	−0.675321	−	0.737524i	$-0.735995\pi$
$491$	−14694.8	−1.35064	−0.675321	+	0.737524i	$0.735995\pi$
$492$	0	0
$493$	−6616.88	−0.604481
$494$	−11442.4	−1.04214
$495$	0	0
$496$	−219.638	−0.0198831
$497$	9885.40	0.892195
$498$	0	0
$499$	4395.99	0.394372	0.197186	−	0.980366i	$-0.436820\pi$
$499$	4395.99	0.394372	0.197186	+	0.980366i	$0.436820\pi$
$500$	−15784.2	−1.41178
$501$	0	0
$502$	4224.22	0.375570
$503$	2345.33	0.207899	0.103950	−	0.994583i	$-0.466852\pi$
$503$	2345.33	0.207899	0.103950	+	0.994583i	$0.466852\pi$
$504$	0	0
$505$	−15922.2	−1.40303
$506$	39657.7	3.48419
$507$	0	0
$508$	−25067.1	−2.18932
$509$	7711.84	0.671554	0.335777	−	0.941941i	$-0.391001\pi$
$509$	7711.84	0.671554	0.335777	+	0.941941i	$0.391001\pi$
$510$	0	0
$511$	−8159.47	−0.706367
$512$	−315.083	−0.0271970
$513$	0	0
$514$	−9470.70	−0.812713
$515$	212.776	0.0182058
$516$	0	0
$517$	214.124	0.0182150
$518$	−25146.2	−2.13294
$519$	0	0
$520$	−6609.70	−0.557413
$521$	−5616.57	−0.472296	−0.236148	−	0.971717i	$-0.575885\pi$
$521$	−5616.57	−0.472296	−0.236148	+	0.971717i	$0.575885\pi$
$522$	0	0
$523$	−14763.5	−1.23435	−0.617173	−	0.786828i	$-0.711722\pi$
$523$	−14763.5	−1.23435	−0.617173	+	0.786828i	$0.711722\pi$
$524$	−27526.8	−2.29487
$525$	0	0
$526$	−20317.1	−1.68416
$527$	31267.6	2.58452
$528$	0	0
$529$	30746.5	2.52704
$530$	14688.4	1.20381
$531$	0	0
$532$	21195.8	1.72736
$533$	−8299.77	−0.674489
$534$	0	0
$535$	−3290.97	−0.265945
$536$	14539.7	1.17168
$537$	0	0
$538$	−11005.0	−0.881894
$539$	−4487.41	−0.358602
$540$	0	0
$541$	15882.7	1.26220	0.631102	−	0.775700i	$-0.282603\pi$
$541$	15882.7	1.26220	0.631102	+	0.775700i	$0.282603\pi$
$542$	17140.0	1.35835
$543$	0	0
$544$	22180.6	1.74814
$545$	−2201.41	−0.173024
$546$	0	0
$547$	−3258.45	−0.254701	−0.127350	−	0.991858i	$-0.540647\pi$
$547$	−3258.45	−0.254701	−0.127350	+	0.991858i	$0.540647\pi$
$548$	−4142.94	−0.322952
$549$	0	0
$550$	5016.56	0.388922
$551$	5675.96	0.438846
$552$	0	0
$553$	9130.68	0.702126
$554$	−6612.53	−0.507111
$555$	0	0
$556$	30025.3	2.29021
$557$	8065.81	0.613572	0.306786	−	0.951779i	$-0.400746\pi$
$557$	8065.81	0.613572	0.306786	+	0.951779i	$0.400746\pi$
$558$	0	0
$559$	1895.94	0.143452
$560$	164.269	0.0123958
$561$	0	0
$562$	16026.5	1.20291
$563$	−19540.5	−1.46276	−0.731379	−	0.681971i	$-0.761123\pi$
$563$	−19540.5	−1.46276	−0.731379	+	0.681971i	$0.761123\pi$
$564$	0	0
$565$	8538.89	0.635812
$566$	−1818.02	−0.135012
$567$	0	0
$568$	14732.8	1.08833
$569$	−13811.1	−1.01756	−0.508780	−	0.860897i	$-0.669903\pi$
$569$	−13811.1	−1.01756	−0.508780	+	0.860897i	$0.669903\pi$
$570$	0	0
$571$	9120.73	0.668460	0.334230	−	0.942492i	$-0.391524\pi$
$571$	9120.73	0.668460	0.334230	+	0.942492i	$0.391524\pi$
$572$	−12756.1	−0.932449
$573$	0	0
$574$	24841.0	1.80634
$575$	5428.41	0.393705
$576$	0	0
$577$	19680.3	1.41993	0.709965	−	0.704237i	$-0.248711\pi$
$577$	19680.3	1.41993	0.709965	+	0.704237i	$0.248711\pi$
$578$	47826.2	3.44171
$579$	0	0
$580$	8532.28	0.610834
$581$	1470.19	0.104980
$582$	0	0
$583$	10893.1	0.773835
$584$	−12160.5	−0.861654
$585$	0	0
$586$	−42370.8	−2.98689
$587$	26509.2	1.86398	0.931988	−	0.362489i	$-0.118073\pi$
$587$	26509.2	1.86398	0.931988	+	0.362489i	$0.118073\pi$
$588$	0	0
$589$	−26821.4	−1.87633
$590$	3324.06	0.231948
$591$	0	0
$592$	−311.196	−0.0216048
$593$	1515.35	0.104937	0.0524686	−	0.998623i	$-0.483291\pi$
$593$	1515.35	0.104937	0.0524686	+	0.998623i	$0.483291\pi$
$594$	0	0
$595$	−23385.3	−1.61127
$596$	15182.6	1.04346
$597$	0	0
$598$	−22302.5	−1.52511
$599$	4721.57	0.322067	0.161033	−	0.986949i	$-0.448517\pi$
$599$	4721.57	0.322067	0.161033	+	0.986949i	$0.448517\pi$
$600$	0	0
$601$	14051.0	0.953663	0.476831	−	0.878995i	$-0.341785\pi$
$601$	14051.0	0.953663	0.476831	+	0.878995i	$0.341785\pi$
$602$	−5674.51	−0.384179
$603$	0	0
$604$	−19971.4	−1.34541
$605$	5100.45	0.342748
$606$	0	0
$607$	−8076.32	−0.540046	−0.270023	−	0.962854i	$-0.587031\pi$
$607$	−8076.32	−0.540046	−0.270023	+	0.962854i	$0.587031\pi$
$608$	−19026.6	−1.26913
$609$	0	0
$610$	2013.63	0.133655
$611$	−120.418	−0.00797312
$612$	0	0
$613$	−3386.14	−0.223107	−0.111554	−	0.993758i	$-0.535583\pi$
$613$	−3386.14	−0.223107	−0.111554	+	0.993758i	$0.535583\pi$
$614$	22968.6	1.50967
$615$	0	0
$616$	14671.1	0.959604
$617$	−26131.0	−1.70502	−0.852508	−	0.522714i	$-0.824920\pi$
$617$	−26131.0	−1.70502	−0.852508	+	0.522714i	$0.824920\pi$
$618$	0	0
$619$	7409.73	0.481134	0.240567	−	0.970633i	$-0.422667\pi$
$619$	7409.73	0.481134	0.240567	+	0.970633i	$0.422667\pi$
$620$	−40318.7	−2.61168
$621$	0	0
$622$	29123.4	1.87740
$623$	−15393.2	−0.989913
$624$	0	0
$625$	−18213.9	−1.16569
$626$	7624.77	0.486816
$627$	0	0
$628$	−30546.9	−1.94101
$629$	44301.9	2.80832
$630$	0	0
$631$	−18236.5	−1.15053	−0.575264	−	0.817968i	$-0.695101\pi$
$631$	−18236.5	−1.15053	−0.575264	+	0.817968i	$0.695101\pi$
$632$	13608.0	0.856481
$633$	0	0
$634$	43380.2	2.71742
$635$	−23723.8	−1.48260
$636$	0	0
$637$	2523.61	0.156968
$638$	10223.8	0.634424
$639$	0	0
$640$	−28993.6	−1.79074
$641$	27792.6	1.71255	0.856273	−	0.516524i	$-0.172774\pi$
$641$	27792.6	1.71255	0.856273	+	0.516524i	$0.172774\pi$
$642$	0	0
$643$	−19061.5	−1.16907	−0.584534	−	0.811369i	$-0.698723\pi$
$643$	−19061.5	−1.16907	−0.584534	+	0.811369i	$0.698723\pi$
$644$	41313.1	2.52790
$645$	0	0
$646$	−60334.7	−3.67467
$647$	6267.83	0.380856	0.190428	−	0.981701i	$-0.439012\pi$
$647$	6267.83	0.380856	0.190428	+	0.981701i	$0.439012\pi$
$648$	0	0
$649$	2465.17	0.149101
$650$	−2821.19	−0.170240
$651$	0	0
$652$	−35796.6	−2.15016
$653$	−21401.4	−1.28254	−0.641271	−	0.767314i	$-0.721593\pi$
$653$	−21401.4	−1.28254	−0.641271	+	0.767314i	$0.721593\pi$
$654$	0	0
$655$	−26051.7	−1.55408
$656$	307.418	0.0182967
$657$	0	0
$658$	360.406	0.0213527
$659$	−17027.7	−1.00653	−0.503267	−	0.864131i	$-0.667869\pi$
$659$	−17027.7	−1.00653	−0.503267	+	0.864131i	$0.667869\pi$
$660$	0	0
$661$	26007.0	1.53034	0.765170	−	0.643828i	$-0.222655\pi$
$661$	26007.0	1.53034	0.765170	+	0.643828i	$0.222655\pi$
$662$	−24734.1	−1.45214
$663$	0	0
$664$	2191.10	0.128059
$665$	20060.0	1.16976
$666$	0	0
$667$	11063.1	0.642227
$668$	48618.6	2.81603
$669$	0	0
$670$	35809.2	2.06482
$671$	1493.33	0.0859159
$672$	0	0
$673$	11947.8	0.684329	0.342164	−	0.939640i	$-0.388840\pi$
$673$	11947.8	0.684329	0.342164	+	0.939640i	$0.388840\pi$
$674$	21960.7	1.25504
$675$	0	0
$676$	−21371.4	−1.21594
$677$	6173.05	0.350442	0.175221	−	0.984529i	$-0.443936\pi$
$677$	6173.05	0.350442	0.175221	+	0.984529i	$0.443936\pi$
$678$	0	0
$679$	−14801.0	−0.836537
$680$	−34852.5	−1.96549
$681$	0	0
$682$	−48311.8	−2.71254
$683$	31454.6	1.76219	0.881097	−	0.472936i	$-0.156806\pi$
$683$	31454.6	1.76219	0.881097	+	0.472936i	$0.156806\pi$
$684$	0	0
$685$	−3920.92	−0.218702
$686$	−31675.3	−1.76293
$687$	0	0
$688$	−70.2246	−0.00389140
$689$	−6126.00	−0.338726
$690$	0	0
$691$	5719.61	0.314883	0.157441	−	0.987528i	$-0.449675\pi$
$691$	5719.61	0.314883	0.157441	+	0.987528i	$0.449675\pi$
$692$	−51591.1	−2.83410
$693$	0	0
$694$	49121.3	2.68677
$695$	28416.3	1.55092
$696$	0	0
$697$	−43764.1	−2.37831
$698$	4783.03	0.259370
$699$	0	0
$700$	5225.97	0.282176
$701$	1237.43	0.0666719	0.0333360	−	0.999444i	$-0.489387\pi$
$701$	1237.43	0.0666719	0.0333360	+	0.999444i	$0.489387\pi$
$702$	0	0
$703$	−38002.2	−2.03881
$704$	−34562.3	−1.85031
$705$	0	0
$706$	45060.6	2.40209
$707$	−19875.1	−1.05725
$708$	0	0
$709$	−14978.1	−0.793393	−0.396697	−	0.917950i	$-0.629844\pi$
$709$	−14978.1	−0.793393	−0.396697	+	0.917950i	$0.629844\pi$
$710$	36284.7	1.91794
$711$	0	0
$712$	−22941.4	−1.20753
$713$	−52278.0	−2.74590
$714$	0	0
$715$	−12072.6	−0.631452
$716$	−17180.3	−0.896729
$717$	0	0
$718$	−45562.4	−2.36821
$719$	−1416.69	−0.0734822	−0.0367411	−	0.999325i	$-0.511698\pi$
$719$	−1416.69	−0.0734822	−0.0367411	+	0.999325i	$0.511698\pi$
$720$	0	0
$721$	265.600	0.0137191
$722$	20328.7	1.04786
$723$	0	0
$724$	42160.7	2.16421
$725$	1399.44	0.0716884
$726$	0	0
$727$	−15069.5	−0.768773	−0.384387	−	0.923172i	$-0.625587\pi$
$727$	−15069.5	−0.768773	−0.384387	+	0.923172i	$0.625587\pi$
$728$	−8250.65	−0.420041
$729$	0	0
$730$	−29949.6	−1.51847
$731$	9997.18	0.505826
$732$	0	0
$733$	17103.5	0.861844	0.430922	−	0.902389i	$-0.358188\pi$
$733$	17103.5	0.861844	0.430922	+	0.902389i	$0.358188\pi$
$734$	37469.3	1.88422
$735$	0	0
$736$	−37085.0	−1.85730
$737$	26556.6	1.32731
$738$	0	0
$739$	9749.49	0.485306	0.242653	−	0.970113i	$-0.421982\pi$
$739$	9749.49	0.485306	0.242653	+	0.970113i	$0.421982\pi$
$740$	−57126.0	−2.83783
$741$	0	0
$742$	18334.9	0.907138
$743$	13906.1	0.686630	0.343315	−	0.939220i	$-0.388450\pi$
$743$	13906.1	0.686630	0.343315	+	0.939220i	$0.388450\pi$
$744$	0	0
$745$	14369.0	0.706628
$746$	−24115.6	−1.18356
$747$	0	0
$748$	−67262.2	−3.28790
$749$	−4107.99	−0.200404
$750$	0	0
$751$	−11178.8	−0.543170	−0.271585	−	0.962415i	$-0.587548\pi$
$751$	−11178.8	−0.543170	−0.271585	+	0.962415i	$0.587548\pi$
$752$	4.46019	0.000216285 0
$753$	0	0
$754$	−5749.58	−0.277702
$755$	−18901.2	−0.911105
$756$	0	0
$757$	4697.99	0.225563	0.112782	−	0.993620i	$-0.464024\pi$
$757$	4697.99	0.225563	0.112782	+	0.993620i	$0.464024\pi$
$758$	−7244.98	−0.347163
$759$	0	0
$760$	29896.5	1.42692
$761$	−15776.9	−0.751526	−0.375763	−	0.926716i	$-0.622619\pi$
$761$	−15776.9	−0.751526	−0.375763	+	0.926716i	$0.622619\pi$
$762$	0	0
$763$	−2747.94	−0.130383
$764$	−19450.0	−0.921044
$765$	0	0
$766$	−83.0230	−0.00391611
$767$	−1386.35	−0.0652649
$768$	0	0
$769$	−25079.3	−1.17605	−0.588024	−	0.808843i	$-0.700094\pi$
$769$	−25079.3	−1.17605	−0.588024	+	0.808843i	$0.700094\pi$
$770$	36132.8	1.69109
$771$	0	0
$772$	2851.97	0.132959
$773$	−15991.3	−0.744071	−0.372036	−	0.928218i	$-0.621340\pi$
$773$	−15991.3	−0.744071	−0.372036	+	0.928218i	$0.621340\pi$
$774$	0	0
$775$	−6612.99	−0.306510
$776$	−22058.7	−1.02044
$777$	0	0
$778$	34251.2	1.57836
$779$	37540.9	1.72663
$780$	0	0
$781$	26909.2	1.23289
$782$	−117599.	−5.37768
$783$	0	0
$784$	−93.4727	−0.00425805
$785$	−28909.9	−1.31445
$786$	0	0
$787$	32468.4	1.47061	0.735307	−	0.677734i	$-0.237038\pi$
$787$	32468.4	1.47061	0.735307	+	0.677734i	$0.237038\pi$
$788$	−9615.02	−0.434671
$789$	0	0
$790$	33514.4	1.50935
$791$	10658.8	0.479119
$792$	0	0
$793$	−839.813	−0.0376073
$794$	−37014.4	−1.65440
$795$	0	0
$796$	27779.1	1.23694
$797$	−16858.3	−0.749248	−0.374624	−	0.927177i	$-0.622228\pi$
$797$	−16858.3	−0.749248	−0.374624	+	0.927177i	$0.622228\pi$
$798$	0	0
$799$	−634.953	−0.0281139
$800$	−4691.12	−0.207320
$801$	0	0
$802$	−7489.40	−0.329751
$803$	−22211.1	−0.976103
$804$	0	0
$805$	39099.2	1.71188
$806$	27169.3	1.18734
$807$	0	0
$808$	−29620.9	−1.28968
$809$	9132.42	0.396884	0.198442	−	0.980113i	$-0.436412\pi$
$809$	9132.42	0.396884	0.198442	+	0.980113i	$0.436412\pi$
$810$	0	0
$811$	−10235.5	−0.443179	−0.221589	−	0.975140i	$-0.571124\pi$
$811$	−10235.5	−0.443179	−0.221589	+	0.975140i	$0.571124\pi$
$812$	10650.5	0.460296
$813$	0	0
$814$	−68451.0	−2.94743
$815$	−33878.3	−1.45608
$816$	0	0
$817$	−8575.59	−0.367224
$818$	−9718.28	−0.415393
$819$	0	0
$820$	56432.5	2.40330
$821$	32830.2	1.39559	0.697797	−	0.716295i	$-0.254164\pi$
$821$	32830.2	1.39559	0.697797	+	0.716295i	$0.254164\pi$
$822$	0	0
$823$	27497.4	1.16464	0.582320	−	0.812960i	$-0.302145\pi$
$823$	27497.4	1.16464	0.582320	+	0.812960i	$0.302145\pi$
$824$	395.839	0.0167351
$825$	0	0
$826$	4149.31	0.174786
$827$	39886.2	1.67712	0.838560	−	0.544809i	$-0.183398\pi$
$827$	39886.2	1.67712	0.838560	+	0.544809i	$0.183398\pi$
$828$	0	0
$829$	−42518.9	−1.78135	−0.890676	−	0.454638i	$-0.849769\pi$
$829$	−42518.9	−1.78135	−0.890676	+	0.454638i	$0.849769\pi$
$830$	5396.37	0.225676
$831$	0	0
$832$	19436.9	0.809922
$833$	13306.8	0.553485
$834$	0	0
$835$	46013.2	1.90701
$836$	57697.6	2.38698
$837$	0	0
$838$	67900.8	2.79904
$839$	−22032.0	−0.906592	−0.453296	−	0.891360i	$-0.649752\pi$
$839$	−22032.0	−0.906592	−0.453296	+	0.891360i	$0.649752\pi$
$840$	0	0
$841$	−21536.9	−0.883059
$842$	23947.4	0.980143
$843$	0	0
$844$	−6630.27	−0.270407
$845$	−20226.2	−0.823434
$846$	0	0
$847$	6366.70	0.258279
$848$	226.903	0.00918854
$849$	0	0
$850$	−14875.9	−0.600282
$851$	−74070.7	−2.98368
$852$	0	0
$853$	−43140.4	−1.73165	−0.865825	−	0.500346i	$-0.833206\pi$
$853$	−43140.4	−1.73165	−0.865825	+	0.500346i	$0.833206\pi$
$854$	2513.54	0.100716
$855$	0	0
$856$	−6122.37	−0.244461
$857$	−8131.01	−0.324096	−0.162048	−	0.986783i	$-0.551810\pi$
$857$	−8131.01	−0.324096	−0.162048	+	0.986783i	$0.551810\pi$
$858$	0	0
$859$	−25210.4	−1.00136	−0.500680	−	0.865632i	$-0.666917\pi$
$859$	−25210.4	−1.00136	−0.500680	+	0.865632i	$0.666917\pi$
$860$	−12891.1	−0.511142
$861$	0	0
$862$	50063.6	1.97816
$863$	−23.0095	−0.000907593 0	−0.000453797	−	1.00000i	$-0.500144\pi$
$863$	−23.0095	−0.000907593 0	−0.000453797		1.00000i	$0.500144\pi$
$864$	0	0
$865$	−48826.4	−1.91924
$866$	14537.9	0.570461
$867$	0	0
$868$	−50328.4	−1.96804
$869$	24854.8	0.970243
$870$	0	0
$871$	−14934.7	−0.580993
$872$	−4095.41	−0.159046
$873$	0	0
$874$	100877.	3.90413
$875$	−18647.0	−0.720437
$876$	0	0
$877$	−3988.93	−0.153588	−0.0767939	−	0.997047i	$-0.524468\pi$
$877$	−3988.93	−0.153588	−0.0767939	+	0.997047i	$0.524468\pi$
$878$	9908.94	0.380877
$879$	0	0
$880$	447.160	0.0171293
$881$	−42436.5	−1.62284	−0.811420	−	0.584464i	$-0.801305\pi$
$881$	−42436.5	−1.62284	−0.811420	+	0.584464i	$0.801305\pi$
$882$	0	0
$883$	−9854.79	−0.375583	−0.187792	−	0.982209i	$-0.560133\pi$
$883$	−9854.79	−0.375583	−0.187792	+	0.982209i	$0.560133\pi$
$884$	37826.5	1.43919
$885$	0	0
$886$	47057.9	1.78436
$887$	−631.042	−0.0238876	−0.0119438	−	0.999929i	$-0.503802\pi$
$887$	−631.042	−0.0238876	−0.0119438	+	0.999929i	$0.503802\pi$
$888$	0	0
$889$	−29613.6	−1.11722
$890$	−56501.2	−2.12801
$891$	0	0
$892$	−31821.8	−1.19447
$893$	544.664	0.0204104
$894$	0	0
$895$	−16259.6	−0.607262
$896$	−36191.6	−1.34942
$897$	0	0
$898$	17513.6	0.650819
$899$	−13477.3	−0.499992
$900$	0	0
$901$	−32301.9	−1.19438
$902$	67620.1	2.49612
$903$	0	0
$904$	15885.4	0.584447
$905$	39901.3	1.46560
$906$	0	0
$907$	18816.6	0.688858	0.344429	−	0.938812i	$-0.388072\pi$
$907$	18816.6	0.688858	0.344429	+	0.938812i	$0.388072\pi$
$908$	2407.52	0.0879915
$909$	0	0
$910$	−20320.2	−0.740227
$911$	−3772.72	−0.137207	−0.0686036	−	0.997644i	$-0.521854\pi$
$911$	−3772.72	−0.137207	−0.0686036	+	0.997644i	$0.521854\pi$
$912$	0	0
$913$	4002.03	0.145069
$914$	76306.4	2.76148
$915$	0	0
$916$	80368.9	2.89898
$917$	−32519.3	−1.17108
$918$	0	0
$919$	−28498.4	−1.02293	−0.511467	−	0.859303i	$-0.670898\pi$
$919$	−28498.4	−1.02293	−0.511467	+	0.859303i	$0.670898\pi$
$920$	58271.7	2.08822
$921$	0	0
$922$	−66337.4	−2.36953
$923$	−15133.1	−0.539665
$924$	0	0
$925$	−9369.68	−0.333052
$926$	−50888.9	−1.80595
$927$	0	0
$928$	−9560.51	−0.338189
$929$	15966.5	0.563879	0.281940	−	0.959432i	$-0.409022\pi$
$929$	15966.5	0.563879	0.281940	+	0.959432i	$0.409022\pi$
$930$	0	0
$931$	−11414.6	−0.401824
$932$	61006.0	2.14412
$933$	0	0
$934$	−82928.9	−2.90526
$935$	−63657.7	−2.22656
$936$	0	0
$937$	3986.01	0.138973	0.0694863	−	0.997583i	$-0.477864\pi$
$937$	3986.01	0.138973	0.0694863	+	0.997583i	$0.477864\pi$
$938$	44699.3	1.55595
$939$	0	0
$940$	818.755	0.0284094
$941$	−30284.1	−1.04913	−0.524566	−	0.851370i	$-0.675773\pi$
$941$	−30284.1	−1.04913	−0.524566	+	0.851370i	$0.675773\pi$
$942$	0	0
$943$	73171.5	2.52682
$944$	51.3495	0.00177043
$945$	0	0
$946$	−15446.7	−0.530883
$947$	30270.4	1.03871	0.519354	−	0.854559i	$-0.326173\pi$
$947$	30270.4	1.03871	0.519354	+	0.854559i	$0.326173\pi$
$948$	0	0
$949$	12490.9	0.427263
$950$	12760.6	0.435798
$951$	0	0
$952$	−43505.1	−1.48110
$953$	−19399.4	−0.659401	−0.329700	−	0.944086i	$-0.606948\pi$
$953$	−19399.4	−0.659401	−0.329700	+	0.944086i	$0.606948\pi$
$954$	0	0
$955$	−18407.7	−0.623728
$956$	47946.3	1.62207
$957$	0	0
$958$	71466.4	2.41020
$959$	−4894.35	−0.164804
$960$	0	0
$961$	33895.0	1.13776
$962$	38495.1	1.29016
$963$	0	0
$964$	−1688.27	−0.0564062
$965$	2699.14	0.0900396
$966$	0	0
$967$	20800.3	0.691720	0.345860	−	0.938286i	$-0.387587\pi$
$967$	20800.3	0.691720	0.345860	+	0.938286i	$0.387587\pi$
$968$	9488.65	0.315059
$969$	0	0
$970$	−54327.4	−1.79830
$971$	27717.3	0.916055	0.458028	−	0.888938i	$-0.348556\pi$
$971$	27717.3	0.916055	0.458028	+	0.888938i	$0.348556\pi$
$972$	0	0
$973$	35471.0	1.16870
$974$	64677.8	2.12773
$975$	0	0
$976$	31.1061	0.00102017
$977$	19576.1	0.641039	0.320519	−	0.947242i	$-0.396143\pi$
$977$	19576.1	0.641039	0.320519	+	0.947242i	$0.396143\pi$
$978$	0	0
$979$	−41902.1	−1.36792
$980$	−17158.7	−0.559302
$981$	0	0
$982$	−67328.3	−2.18792
$983$	11769.7	0.381888	0.190944	−	0.981601i	$-0.438845\pi$
$983$	11769.7	0.381888	0.190944	+	0.981601i	$0.438845\pi$
$984$	0	0
$985$	−9099.76	−0.294358
$986$	−30317.1	−0.979203
$987$	0	0
$988$	−32447.6	−1.04483
$989$	−16714.8	−0.537412
$990$	0	0
$991$	52861.1	1.69444	0.847219	−	0.531245i	$-0.178275\pi$
$991$	52861.1	1.69444	0.847219	+	0.531245i	$0.178275\pi$
$992$	45177.6	1.44596
$993$	0	0
$994$	45292.8	1.44527
$995$	26290.5	0.837652
$996$	0	0
$997$	−38949.2	−1.23725	−0.618623	−	0.785688i	$-0.712309\pi$
$997$	−38949.2	−1.23725	−0.618623	+	0.785688i	$0.712309\pi$
$998$	20141.5	0.638846
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.d.1.6		7
3.2	odd	2		177.4.a.a.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.a.1.2	✓	7	3.2	odd	2
531.4.a.d.1.6		7	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+4.58179 q^{2} +12.9928 q^{4} +12.2965 q^{5} +15.3493 q^{7} +22.8759 q^{8} +O(q^{10})\)	\(q+4.58179 q^{2} +12.9928 q^{4} +12.2965 q^{5} +15.3493 q^{7} +22.8759 q^{8} +56.3401 q^{10} +41.7826 q^{11} -23.4975 q^{13} +70.3273 q^{14} +0.870331 q^{16} -123.900 q^{17} +106.282 q^{19} +159.766 q^{20} +191.439 q^{22} +207.156 q^{23} +26.2045 q^{25} -107.660 q^{26} +199.430 q^{28} +53.4048 q^{29} -252.361 q^{31} -179.020 q^{32} -567.686 q^{34} +188.743 q^{35} -357.560 q^{37} +486.962 q^{38} +281.294 q^{40} +353.220 q^{41} -80.6872 q^{43} +542.873 q^{44} +949.144 q^{46} +5.12471 q^{47} -107.399 q^{49} +120.063 q^{50} -305.298 q^{52} +260.709 q^{53} +513.781 q^{55} +351.129 q^{56} +244.689 q^{58} +59.0000 q^{59} +35.7406 q^{61} -1156.26 q^{62} -827.193 q^{64} -288.937 q^{65} +635.590 q^{67} -1609.81 q^{68} +864.781 q^{70} +644.029 q^{71} -531.586 q^{73} -1638.27 q^{74} +1380.90 q^{76} +641.334 q^{77} +594.860 q^{79} +10.7020 q^{80} +1618.38 q^{82} +95.7821 q^{83} -1523.54 q^{85} -369.692 q^{86} +955.816 q^{88} -1002.86 q^{89} -360.670 q^{91} +2691.53 q^{92} +23.4803 q^{94} +1306.90 q^{95} -964.276 q^{97} -492.080 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8}+O(q^{10}) \) 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8	\( 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8} - 79 q^{10} + 131 q^{11} - 123 q^{13} + 117 q^{14} + 202 q^{16} + 235 q^{17} - 80 q^{19} - 61 q^{20} + 688 q^{22} + 274 q^{23} + 193 q^{25} + 180 q^{26} - 118 q^{28} + 406 q^{29} - 346 q^{31} + 854 q^{32} + 178 q^{34} + 424 q^{35} - 157 q^{37} + 129 q^{38} - 590 q^{40} + 825 q^{41} - 815 q^{43} + 1690 q^{44} + 1457 q^{46} + 1196 q^{47} + 914 q^{49} - 713 q^{50} + 1030 q^{52} + 900 q^{53} - 1044 q^{55} - 2172 q^{56} + 1242 q^{58} + 413 q^{59} + 420 q^{61} - 646 q^{62} + 3541 q^{64} - 190 q^{65} + 1316 q^{67} + 611 q^{68} + 4658 q^{70} + 173 q^{71} - 418 q^{73} - 660 q^{74} + 1540 q^{76} + 753 q^{77} + 2635 q^{79} - 6155 q^{80} - 125 q^{82} - 457 q^{83} + 1270 q^{85} - 3482 q^{86} + 7685 q^{88} - 592 q^{89} + 3179 q^{91} + 3500 q^{92} + 2064 q^{94} + 2250 q^{95} - 1906 q^{97} - 2994 q^{98}+O(q^{100}) \) 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8 - 79 * q^10 + 131 * q^11 - 123 * q^13 + 117 * q^14 + 202 * q^16 + 235 * q^17 - 80 * q^19 - 61 * q^20 + 688 * q^22 + 274 * q^23 + 193 * q^25 + 180 * q^26 - 118 * q^28 + 406 * q^29 - 346 * q^31 + 854 * q^32 + 178 * q^34 + 424 * q^35 - 157 * q^37 + 129 * q^38 - 590 * q^40 + 825 * q^41 - 815 * q^43 + 1690 * q^44 + 1457 * q^46 + 1196 * q^47 + 914 * q^49 - 713 * q^50 + 1030 * q^52 + 900 * q^53 - 1044 * q^55 - 2172 * q^56 + 1242 * q^58 + 413 * q^59 + 420 * q^61 - 646 * q^62 + 3541 * q^64 - 190 * q^65 + 1316 * q^67 + 611 * q^68 + 4658 * q^70 + 173 * q^71 - 418 * q^73 - 660 * q^74 + 1540 * q^76 + 753 * q^77 + 2635 * q^79 - 6155 * q^80 - 125 * q^82 - 457 * q^83 + 1270 * q^85 - 3482 * q^86 + 7685 * q^88 - 592 * q^89 + 3179 * q^91 + 3500 * q^92 + 2064 * q^94 + 2250 * q^95 - 1906 * q^97 - 2994 * q^98

Introduction

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