LMFDB - Embedded newform 531.4.a.c.1.1

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:	$1$
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} - 41x^{5} - 7x^{4} + 484x^{3} + 63x^{2} - 1736x - 44 $ x^7 - 41x^5 - 7x^4 + 484x^3 + 63x^2 - 1736*x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.58037$ 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)

$f(q)$	$=$	$q-4.58037 q^{2} +12.9798 q^{4} +17.0129 q^{5} -20.9898 q^{7} -22.8093 q^{8} +O(q^{10})$	\(q-4.58037 q^{2} +12.9798 q^{4} +17.0129 q^{5} -20.9898 q^{7} -22.8093 q^{8} -77.9254 q^{10} +48.1602 q^{11} +29.4612 q^{13} +96.1410 q^{14} +0.636565 q^{16} -108.283 q^{17} -111.228 q^{19} +220.824 q^{20} -220.591 q^{22} +14.3946 q^{23} +164.439 q^{25} -134.943 q^{26} -272.443 q^{28} -292.789 q^{29} -216.676 q^{31} +179.558 q^{32} +495.978 q^{34} -357.097 q^{35} -168.516 q^{37} +509.464 q^{38} -388.052 q^{40} +342.050 q^{41} +325.352 q^{43} +625.109 q^{44} -65.9325 q^{46} +315.587 q^{47} +97.5712 q^{49} -753.191 q^{50} +382.400 q^{52} -256.246 q^{53} +819.344 q^{55} +478.762 q^{56} +1341.08 q^{58} -59.0000 q^{59} -749.020 q^{61} +992.457 q^{62} -827.536 q^{64} +501.221 q^{65} +664.622 q^{67} -1405.50 q^{68} +1635.64 q^{70} -207.420 q^{71} +45.3093 q^{73} +771.868 q^{74} -1443.71 q^{76} -1010.87 q^{77} -813.087 q^{79} +10.8298 q^{80} -1566.71 q^{82} -19.2056 q^{83} -1842.22 q^{85} -1490.23 q^{86} -1098.50 q^{88} +636.087 q^{89} -618.385 q^{91} +186.838 q^{92} -1445.50 q^{94} -1892.31 q^{95} -235.487 q^{97} -446.912 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8}+O(q^{10}) $ 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8	$ 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8} - 71 q^{10} + 5 q^{11} - 67 q^{13} + 65 q^{14} - 94 q^{16} + 23 q^{17} - 176 q^{19} + 207 q^{20} - 704 q^{22} + 218 q^{23} - 183 q^{25} - 58 q^{26} - 938 q^{28} - 168 q^{29} - 604 q^{31} + 448 q^{32} - 610 q^{34} + 336 q^{35} - 505 q^{37} + 453 q^{38} - 1080 q^{40} + 265 q^{41} - 493 q^{43} - 504 q^{44} + 381 q^{46} + 244 q^{47} + 770 q^{49} - 1639 q^{50} + 160 q^{52} - 686 q^{53} - 116 q^{55} - 2190 q^{56} + 1584 q^{58} - 413 q^{59} - 838 q^{61} - 286 q^{62} + 205 q^{64} - 490 q^{65} - 1504 q^{67} - 3047 q^{68} + 1530 q^{70} + 1267 q^{71} - 666 q^{73} - 528 q^{74} - 64 q^{76} - 1109 q^{77} - 2741 q^{79} - 1213 q^{80} + 953 q^{82} + 2025 q^{83} - 1274 q^{85} - 4394 q^{86} - 1639 q^{88} - 616 q^{89} - 2415 q^{91} - 218 q^{92} + 900 q^{94} - 2554 q^{95} - 1298 q^{97} + 172 q^{98}+O(q^{100}) $ 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8 - 71 * q^10 + 5 * q^11 - 67 * q^13 + 65 * q^14 - 94 * q^16 + 23 * q^17 - 176 * q^19 + 207 * q^20 - 704 * q^22 + 218 * q^23 - 183 * q^25 - 58 * q^26 - 938 * q^28 - 168 * q^29 - 604 * q^31 + 448 * q^32 - 610 * q^34 + 336 * q^35 - 505 * q^37 + 453 * q^38 - 1080 * q^40 + 265 * q^41 - 493 * q^43 - 504 * q^44 + 381 * q^46 + 244 * q^47 + 770 * q^49 - 1639 * q^50 + 160 * q^52 - 686 * q^53 - 116 * q^55 - 2190 * q^56 + 1584 * q^58 - 413 * q^59 - 838 * q^61 - 286 * q^62 + 205 * q^64 - 490 * q^65 - 1504 * q^67 - 3047 * q^68 + 1530 * q^70 + 1267 * q^71 - 666 * q^73 - 528 * q^74 - 64 * q^76 - 1109 * q^77 - 2741 * q^79 - 1213 * q^80 + 953 * q^82 + 2025 * q^83 - 1274 * q^85 - 4394 * q^86 - 1639 * q^88 - 616 * q^89 - 2415 * q^91 - 218 * q^92 + 900 * q^94 - 2554 * q^95 - 1298 * q^97 + 172 * 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 $

$2$	−4.58037	−1.61941	−0.809703	−	0.586840i	$-0.800372\pi$
$2$	−4.58037	−1.61941	−0.809703	+	0.586840i	$0.800372\pi$
$3$	0	0
$3$	0	0
$4$	12.9798	1.62247
$5$	17.0129	1.52168	0.760840	−	0.648939i	$-0.224787\pi$
$5$	17.0129	1.52168	0.760840	+	0.648939i	$0.224787\pi$
$6$	0	0
$7$	−20.9898	−1.13334	−0.566671	−	0.823944i	$-0.691769\pi$
$7$	−20.9898	−1.13334	−0.566671	+	0.823944i	$0.691769\pi$
$8$	−22.8093	−1.00804
$9$	0	0
$10$	−77.9254	−2.46422
$11$	48.1602	1.32008	0.660038	−	0.751232i	$-0.270540\pi$
$11$	48.1602	1.32008	0.660038	+	0.751232i	$0.270540\pi$
$12$	0	0
$13$	29.4612	0.628544	0.314272	−	0.949333i	$-0.398240\pi$
$13$	29.4612	0.628544	0.314272	+	0.949333i	$0.398240\pi$
$14$	96.1410	1.83534
$15$	0	0
$16$	0.636565	0.00994633
$17$	−108.283	−1.54486	−0.772429	−	0.635101i	$-0.780958\pi$
$17$	−108.283	−1.54486	−0.772429	+	0.635101i	$0.780958\pi$
$18$	0	0
$19$	−111.228	−1.34302	−0.671510	−	0.740995i	$-0.734354\pi$
$19$	−111.228	−1.34302	−0.671510	+	0.740995i	$0.734354\pi$
$20$	220.824	2.46889
$21$	0	0
$22$	−220.591	−2.13774
$23$	14.3946	0.130499	0.0652495	−	0.997869i	$-0.479216\pi$
$23$	14.3946	0.130499	0.0652495	+	0.997869i	$0.479216\pi$
$24$	0	0
$25$	164.439	1.31551
$26$	−134.943	−1.01787
$27$	0	0
$28$	−272.443	−1.83882
$29$	−292.789	−1.87481	−0.937406	−	0.348238i	$-0.886780\pi$
$29$	−292.789	−1.87481	−0.937406	+	0.348238i	$0.886780\pi$
$30$	0	0
$31$	−216.676	−1.25536	−0.627680	−	0.778471i	$-0.715995\pi$
$31$	−216.676	−1.25536	−0.627680	+	0.778471i	$0.715995\pi$
$32$	179.558	0.991929
$33$	0	0
$34$	495.978	2.50175
$35$	−357.097	−1.72458
$36$	0	0
$37$	−168.516	−0.748755	−0.374378	−	0.927276i	$-0.622144\pi$
$37$	−168.516	−0.748755	−0.374378	+	0.927276i	$0.622144\pi$
$38$	509.464	2.17489
$39$	0	0
$40$	−388.052	−1.53391
$41$	342.050	1.30291	0.651453	−	0.758689i	$-0.274160\pi$
$41$	342.050	1.30291	0.651453	+	0.758689i	$0.274160\pi$
$42$	0	0
$43$	325.352	1.15385	0.576927	−	0.816795i	$-0.304252\pi$
$43$	325.352	1.15385	0.576927	+	0.816795i	$0.304252\pi$
$44$	625.109	2.14179
$45$	0	0
$46$	−65.9325	−0.211331
$47$	315.587	0.979426	0.489713	−	0.871884i	$-0.337102\pi$
$47$	315.587	0.979426	0.489713	+	0.871884i	$0.337102\pi$
$48$	0	0
$49$	97.5712	0.284464
$50$	−753.191	−2.13035
$51$	0	0
$52$	382.400	1.01980
$53$	−256.246	−0.664115	−0.332058	−	0.943259i	$-0.607743\pi$
$53$	−256.246	−0.664115	−0.332058	+	0.943259i	$0.607743\pi$
$54$	0	0
$55$	819.344	2.00873
$56$	478.762	1.14245
$57$	0	0
$58$	1341.08	3.03608
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	0	0
$61$	−749.020	−1.57217	−0.786083	−	0.618120i	$-0.787894\pi$
$61$	−749.020	−1.57217	−0.786083	+	0.618120i	$0.787894\pi$
$62$	992.457	2.03294
$63$	0	0
$64$	−827.536	−1.61628
$65$	501.221	0.956443
$66$	0	0
$67$	664.622	1.21189	0.605944	−	0.795507i	$-0.292796\pi$
$67$	664.622	1.21189	0.605944	+	0.795507i	$0.292796\pi$
$68$	−1405.50	−2.50649
$69$	0	0
$70$	1635.64	2.79280
$71$	−207.420	−0.346708	−0.173354	−	0.984860i	$-0.555460\pi$
$71$	−207.420	−0.346708	−0.173354	+	0.984860i	$0.555460\pi$
$72$	0	0
$73$	45.3093	0.0726445	0.0363222	−	0.999340i	$-0.488436\pi$
$73$	45.3093	0.0726445	0.0363222	+	0.999340i	$0.488436\pi$
$74$	771.868	1.21254
$75$	0	0
$76$	−1443.71	−2.17901
$77$	−1010.87	−1.49610
$78$	0	0
$79$	−813.087	−1.15797	−0.578984	−	0.815339i	$-0.696551\pi$
$79$	−813.087	−1.15797	−0.578984	+	0.815339i	$0.696551\pi$
$80$	10.8298	0.0151351
$81$	0	0
$82$	−1566.71	−2.10993
$83$	−19.2056	−0.0253986	−0.0126993	−	0.999919i	$-0.504042\pi$
$83$	−19.2056	−0.0253986	−0.0126993	+	0.999919i	$0.504042\pi$
$84$	0	0
$85$	−1842.22	−2.35078
$86$	−1490.23	−1.86856
$87$	0	0
$88$	−1098.50	−1.33068
$89$	636.087	0.757585	0.378792	−	0.925482i	$-0.376339\pi$
$89$	636.087	0.757585	0.378792	+	0.925482i	$0.376339\pi$
$90$	0	0
$91$	−618.385	−0.712355
$92$	186.838	0.211731
$93$	0	0
$94$	−1445.50	−1.58609
$95$	−1892.31	−2.04365
$96$	0	0
$97$	−235.487	−0.246495	−0.123248	−	0.992376i	$-0.539331\pi$
$97$	−235.487	−0.246495	−0.123248	+	0.992376i	$0.539331\pi$
$98$	−446.912	−0.460663
$99$	0	0
$100$	2134.38	2.13438
$101$	−1807.59	−1.78081	−0.890406	−	0.455167i	$-0.849580\pi$
$101$	−1807.59	−1.78081	−0.890406	+	0.455167i	$0.849580\pi$
$102$	0	0
$103$	−1437.56	−1.37522	−0.687609	−	0.726081i	$-0.741340\pi$
$103$	−1437.56	−1.37522	−0.687609	+	0.726081i	$0.741340\pi$
$104$	−671.989	−0.633595
$105$	0	0
$106$	1173.70	1.07547
$107$	430.727	0.389158	0.194579	−	0.980887i	$-0.437666\pi$
$107$	430.727	0.389158	0.194579	+	0.980887i	$0.437666\pi$
$108$	0	0
$109$	459.238	0.403551	0.201775	−	0.979432i	$-0.435329\pi$
$109$	459.238	0.403551	0.201775	+	0.979432i	$0.435329\pi$
$110$	−3752.90	−3.25295
$111$	0	0
$112$	−13.3614	−0.0112726
$113$	1009.57	0.840460	0.420230	−	0.907418i	$-0.361949\pi$
$113$	1009.57	0.840460	0.420230	+	0.907418i	$0.361949\pi$
$114$	0	0
$115$	244.893	0.198578
$116$	−3800.34	−3.04183
$117$	0	0
$118$	270.242	0.210829
$119$	2272.85	1.75085
$120$	0	0
$121$	988.402	0.742601
$122$	3430.79	2.54598
$123$	0	0
$124$	−2812.41	−2.03679
$125$	670.969	0.480107
$126$	0	0
$127$	392.027	0.273912	0.136956	−	0.990577i	$-0.456268\pi$
$127$	392.027	0.273912	0.136956	+	0.990577i	$0.456268\pi$
$128$	2353.96	1.62549
$129$	0	0
$130$	−2295.78	−1.54887
$131$	−1782.88	−1.18909	−0.594546	−	0.804062i	$-0.702668\pi$
$131$	−1782.88	−1.18909	−0.594546	+	0.804062i	$0.702668\pi$
$132$	0	0
$133$	2334.65	1.52210
$134$	−3044.21	−1.96254
$135$	0	0
$136$	2469.86	1.55727
$137$	776.765	0.484405	0.242203	−	0.970226i	$-0.422130\pi$
$137$	776.765	0.484405	0.242203	+	0.970226i	$0.422130\pi$
$138$	0	0
$139$	−555.020	−0.338678	−0.169339	−	0.985558i	$-0.554163\pi$
$139$	−555.020	−0.338678	−0.169339	+	0.985558i	$0.554163\pi$
$140$	−4635.05	−2.79809
$141$	0	0
$142$	950.062	0.561461
$143$	1418.86	0.829726
$144$	0	0
$145$	−4981.19	−2.85287
$146$	−207.533	−0.117641
$147$	0	0
$148$	−2187.31	−1.21484
$149$	−2066.64	−1.13628	−0.568141	−	0.822932i	$-0.692337\pi$
$149$	−2066.64	−1.13628	−0.568141	+	0.822932i	$0.692337\pi$
$150$	0	0
$151$	−922.182	−0.496994	−0.248497	−	0.968633i	$-0.579937\pi$
$151$	−922.182	−0.496994	−0.248497	+	0.968633i	$0.579937\pi$
$152$	2537.02	1.35381
$153$	0	0
$154$	4630.17	2.42279
$155$	−3686.29	−1.91026
$156$	0	0
$157$	−2147.92	−1.09187	−0.545933	−	0.837829i	$-0.683825\pi$
$157$	−2147.92	−1.09187	−0.545933	+	0.837829i	$0.683825\pi$
$158$	3724.24	1.87522
$159$	0	0
$160$	3054.81	1.50940
$161$	−302.139	−0.147900
$162$	0	0
$163$	202.941	0.0975188	0.0487594	−	0.998811i	$-0.484473\pi$
$163$	202.941	0.0975188	0.0487594	+	0.998811i	$0.484473\pi$
$164$	4439.73	2.11393
$165$	0	0
$166$	87.9686	0.0411307
$167$	1984.16	0.919395	0.459697	−	0.888076i	$-0.347958\pi$
$167$	1984.16	0.919395	0.459697	+	0.888076i	$0.347958\pi$
$168$	0	0
$169$	−1329.04	−0.604933
$170$	8438.03	3.80687
$171$	0	0
$172$	4223.00	1.87210
$173$	1484.23	0.652279	0.326139	−	0.945322i	$-0.394252\pi$
$173$	1484.23	0.652279	0.326139	+	0.945322i	$0.394252\pi$
$174$	0	0
$175$	−3451.54	−1.49092
$176$	30.6571	0.0131299
$177$	0	0
$178$	−2913.51	−1.22684
$179$	546.514	0.228203	0.114102	−	0.993469i	$-0.463601\pi$
$179$	546.514	0.228203	0.114102	+	0.993469i	$0.463601\pi$
$180$	0	0
$181$	−4561.98	−1.87342	−0.936710	−	0.350106i	$-0.886145\pi$
$181$	−4561.98	−1.87342	−0.936710	+	0.350106i	$0.886145\pi$
$182$	2832.43	1.15359
$183$	0	0
$184$	−328.330	−0.131548
$185$	−2866.95	−1.13937
$186$	0	0
$187$	−5214.95	−2.03933
$188$	4096.25	1.58909
$189$	0	0
$190$	8667.46	3.30949
$191$	837.383	0.317230	0.158615	−	0.987341i	$-0.449297\pi$
$191$	837.383	0.317230	0.158615	+	0.987341i	$0.449297\pi$
$192$	0	0
$193$	−3423.57	−1.27686	−0.638430	−	0.769680i	$-0.720416\pi$
$193$	−3423.57	−1.27686	−0.638430	+	0.769680i	$0.720416\pi$
$194$	1078.62	0.399176
$195$	0	0
$196$	1266.45	0.461535
$197$	−3978.23	−1.43877	−0.719383	−	0.694613i	$-0.755575\pi$
$197$	−3978.23	−1.43877	−0.719383	+	0.694613i	$0.755575\pi$
$198$	0	0
$199$	2224.15	0.792290	0.396145	−	0.918188i	$-0.370348\pi$
$199$	2224.15	0.792290	0.396145	+	0.918188i	$0.370348\pi$
$200$	−3750.73	−1.32608
$201$	0	0
$202$	8279.44	2.88386
$203$	6145.58	2.12480
$204$	0	0
$205$	5819.26	1.98261
$206$	6584.58	2.22703
$207$	0	0
$208$	18.7540	0.00625171
$209$	−5356.74	−1.77289
$210$	0	0
$211$	1783.56	0.581921	0.290961	−	0.956735i	$-0.406025\pi$
$211$	1783.56	0.581921	0.290961	+	0.956735i	$0.406025\pi$
$212$	−3326.02	−1.07751
$213$	0	0
$214$	−1972.89	−0.630205
$215$	5535.19	1.75580
$216$	0	0
$217$	4547.98	1.42275
$218$	−2103.48	−0.653513
$219$	0	0
$220$	10634.9	3.25912
$221$	−3190.16	−0.971011
$222$	0	0
$223$	477.801	0.143479	0.0717397	−	0.997423i	$-0.477145\pi$
$223$	477.801	0.143479	0.0717397	+	0.997423i	$0.477145\pi$
$224$	−3768.89	−1.12420
$225$	0	0
$226$	−4624.18	−1.36105
$227$	−2763.61	−0.808050	−0.404025	−	0.914748i	$-0.632389\pi$
$227$	−2763.61	−0.808050	−0.404025	+	0.914748i	$0.632389\pi$
$228$	0	0
$229$	2794.53	0.806408	0.403204	−	0.915110i	$-0.367896\pi$
$229$	2794.53	0.806408	0.403204	+	0.915110i	$0.367896\pi$
$230$	−1121.70	−0.321578
$231$	0	0
$232$	6678.30	1.88988
$233$	−6224.12	−1.75002	−0.875012	−	0.484102i	$-0.839146\pi$
$233$	−6224.12	−1.75002	−0.875012	+	0.484102i	$0.839146\pi$
$234$	0	0
$235$	5369.04	1.49037
$236$	−765.807	−0.211228
$237$	0	0
$238$	−10410.5	−2.83534
$239$	2721.14	0.736468	0.368234	−	0.929733i	$-0.379963\pi$
$239$	2721.14	0.736468	0.368234	+	0.929733i	$0.379963\pi$
$240$	0	0
$241$	2695.03	0.720340	0.360170	−	0.932887i	$-0.382719\pi$
$241$	2695.03	0.720340	0.360170	+	0.932887i	$0.382719\pi$
$242$	−4527.24	−1.20257
$243$	0	0
$244$	−9722.12	−2.55080
$245$	1659.97	0.432863
$246$	0	0
$247$	−3276.90	−0.844147
$248$	4942.22	1.26545
$249$	0	0
$250$	−3073.29	−0.777487
$251$	1146.02	0.288191	0.144096	−	0.989564i	$-0.453973\pi$
$251$	1146.02	0.288191	0.144096	+	0.989564i	$0.453973\pi$
$252$	0	0
$253$	693.245	0.172269
$254$	−1795.63	−0.443574
$255$	0	0
$256$	−4161.69	−1.01604
$257$	964.525	0.234107	0.117053	−	0.993126i	$-0.462655\pi$
$257$	964.525	0.234107	0.117053	+	0.993126i	$0.462655\pi$
$258$	0	0
$259$	3537.12	0.848596
$260$	6505.74	1.55180
$261$	0	0
$262$	8166.26	1.92562
$263$	5111.64	1.19847	0.599234	−	0.800574i	$-0.295472\pi$
$263$	5111.64	1.19847	0.599234	+	0.800574i	$0.295472\pi$
$264$	0	0
$265$	−4359.49	−1.01057
$266$	−10693.5	−2.46490
$267$	0	0
$268$	8626.65	1.96626
$269$	4488.46	1.01735	0.508673	−	0.860960i	$-0.330136\pi$
$269$	4488.46	1.01735	0.508673	+	0.860960i	$0.330136\pi$
$270$	0	0
$271$	4399.29	0.986118	0.493059	−	0.869996i	$-0.335879\pi$
$271$	4399.29	0.986118	0.493059	+	0.869996i	$0.335879\pi$
$272$	−68.9295	−0.0153657
$273$	0	0
$274$	−3557.87	−0.784448
$275$	7919.40	1.73657
$276$	0	0
$277$	−4750.92	−1.03052	−0.515262	−	0.857033i	$-0.672305\pi$
$277$	−4750.92	−1.03052	−0.515262	+	0.857033i	$0.672305\pi$
$278$	2542.20	0.548456
$279$	0	0
$280$	8145.12	1.73844
$281$	−8074.21	−1.71412	−0.857058	−	0.515219i	$-0.827711\pi$
$281$	−8074.21	−1.71412	−0.857058	+	0.515219i	$0.827711\pi$
$282$	0	0
$283$	1950.54	0.409708	0.204854	−	0.978793i	$-0.434328\pi$
$283$	1950.54	0.409708	0.204854	+	0.978793i	$0.434328\pi$
$284$	−2692.27	−0.562525
$285$	0	0
$286$	−6498.89	−1.34366
$287$	−7179.55	−1.47664
$288$	0	0
$289$	6812.30	1.38659
$290$	22815.7	4.61995
$291$	0	0
$292$	588.105	0.117864
$293$	7662.67	1.52784	0.763921	−	0.645309i	$-0.223272\pi$
$293$	7662.67	1.52784	0.763921	+	0.645309i	$0.223272\pi$
$294$	0	0
$295$	−1003.76	−0.198106
$296$	3843.74	0.754772
$297$	0	0
$298$	9465.98	1.84010
$299$	424.082	0.0820243
$300$	0	0
$301$	−6829.07	−1.30771
$302$	4223.93	0.804835
$303$	0	0
$304$	−70.8037	−0.0133581
$305$	−12743.0	−2.39234
$306$	0	0
$307$	−1433.79	−0.266550	−0.133275	−	0.991079i	$-0.542549\pi$
$307$	−1433.79	−0.266550	−0.133275	+	0.991079i	$0.542549\pi$
$308$	−13120.9	−2.42738
$309$	0	0
$310$	16884.6	3.09348
$311$	−8645.17	−1.57628	−0.788140	−	0.615497i	$-0.788955\pi$
$311$	−8645.17	−1.57628	−0.788140	+	0.615497i	$0.788955\pi$
$312$	0	0
$313$	6474.17	1.16914	0.584571	−	0.811342i	$-0.301263\pi$
$313$	6474.17	1.16914	0.584571	+	0.811342i	$0.301263\pi$
$314$	9838.29	1.76817
$315$	0	0
$316$	−10553.7	−1.87877
$317$	−5278.69	−0.935270	−0.467635	−	0.883922i	$-0.654894\pi$
$317$	−5278.69	−0.935270	−0.467635	+	0.883922i	$0.654894\pi$
$318$	0	0
$319$	−14100.8	−2.47490
$320$	−14078.8	−2.45946
$321$	0	0
$322$	1383.91	0.239510
$323$	12044.1	2.07478
$324$	0	0
$325$	4844.57	0.826856
$326$	−929.545	−0.157923
$327$	0	0
$328$	−7801.90	−1.31338
$329$	−6624.09	−1.11002
$330$	0	0
$331$	7562.70	1.25584	0.627921	−	0.778277i	$-0.283906\pi$
$331$	7562.70	1.25584	0.627921	+	0.778277i	$0.283906\pi$
$332$	−249.284	−0.0412086
$333$	0	0
$334$	−9088.19	−1.48887
$335$	11307.2	1.84411
$336$	0	0
$337$	1241.33	0.200652	0.100326	−	0.994955i	$-0.468011\pi$
$337$	1241.33	0.200652	0.100326	+	0.994955i	$0.468011\pi$
$338$	6087.48	0.979631
$339$	0	0
$340$	−23911.6	−3.81408
$341$	−10435.2	−1.65717
$342$	0	0
$343$	5151.50	0.810947
$344$	−7421.05	−1.16313
$345$	0	0
$346$	−6798.34	−1.05630
$347$	−10314.8	−1.59576	−0.797879	−	0.602818i	$-0.794044\pi$
$347$	−10314.8	−1.59576	−0.797879	+	0.602818i	$0.794044\pi$
$348$	0	0
$349$	6190.01	0.949409	0.474704	−	0.880145i	$-0.342555\pi$
$349$	6190.01	0.949409	0.474704	+	0.880145i	$0.342555\pi$
$350$	15809.3	2.41441
$351$	0	0
$352$	8647.56	1.30942
$353$	−2309.30	−0.348191	−0.174096	−	0.984729i	$-0.555700\pi$
$353$	−2309.30	−0.348191	−0.174096	+	0.984729i	$0.555700\pi$
$354$	0	0
$355$	−3528.82	−0.527579
$356$	8256.27	1.22916
$357$	0	0
$358$	−2503.24	−0.369554
$359$	−12546.4	−1.84449	−0.922244	−	0.386608i	$-0.873647\pi$
$359$	−12546.4	−1.84449	−0.922244	+	0.386608i	$0.873647\pi$
$360$	0	0
$361$	5512.60	0.803703
$362$	20895.5	3.03383
$363$	0	0
$364$	−8026.50	−1.15578
$365$	770.842	0.110542
$366$	0	0
$367$	−9343.94	−1.32902	−0.664509	−	0.747280i	$-0.731359\pi$
$367$	−9343.94	−1.32902	−0.664509	+	0.747280i	$0.731359\pi$
$368$	9.16309	0.00129799
$369$	0	0
$370$	13131.7	1.84509
$371$	5378.55	0.752670
$372$	0	0
$373$	−8906.58	−1.23637	−0.618184	−	0.786033i	$-0.712131\pi$
$373$	−8906.58	−1.23637	−0.618184	+	0.786033i	$0.712131\pi$
$374$	23886.4	3.30250
$375$	0	0
$376$	−7198.30	−0.987297
$377$	−8625.92	−1.17840
$378$	0	0
$379$	3568.77	0.483681	0.241841	−	0.970316i	$-0.422249\pi$
$379$	3568.77	0.483681	0.241841	+	0.970316i	$0.422249\pi$
$380$	−24561.7	−3.31576
$381$	0	0
$382$	−3835.52	−0.513724
$383$	9195.63	1.22683	0.613414	−	0.789762i	$-0.289796\pi$
$383$	9195.63	1.22683	0.613414	+	0.789762i	$0.289796\pi$
$384$	0	0
$385$	−17197.9	−2.27658
$386$	15681.2	2.06775
$387$	0	0
$388$	−3056.57	−0.399932
$389$	7464.36	0.972900	0.486450	−	0.873708i	$-0.338292\pi$
$389$	7464.36	0.972900	0.486450	+	0.873708i	$0.338292\pi$
$390$	0	0
$391$	−1558.69	−0.201602
$392$	−2225.53	−0.286750
$393$	0	0
$394$	18221.8	2.32995
$395$	−13833.0	−1.76206
$396$	0	0
$397$	9258.84	1.17050	0.585249	−	0.810853i	$-0.300997\pi$
$397$	9258.84	1.17050	0.585249	+	0.810853i	$0.300997\pi$
$398$	−10187.4	−1.28304
$399$	0	0
$400$	104.676	0.0130845
$401$	9944.75	1.23845	0.619224	−	0.785215i	$-0.287447\pi$
$401$	9944.75	1.23845	0.619224	+	0.785215i	$0.287447\pi$
$402$	0	0
$403$	−6383.54	−0.789049
$404$	−23462.1	−2.88932
$405$	0	0
$406$	−28149.0	−3.44092
$407$	−8115.78	−0.988414
$408$	0	0
$409$	−2770.27	−0.334917	−0.167459	−	0.985879i	$-0.553556\pi$
$409$	−2770.27	−0.334917	−0.167459	+	0.985879i	$0.553556\pi$
$410$	−26654.3	−3.21064
$411$	0	0
$412$	−18659.3	−2.23125
$413$	1238.40	0.147549
$414$	0	0
$415$	−326.743	−0.0386486
$416$	5290.01	0.623471
$417$	0	0
$418$	24535.9	2.87103
$419$	10519.7	1.22654	0.613272	−	0.789872i	$-0.289853\pi$
$419$	10519.7	1.22654	0.613272	+	0.789872i	$0.289853\pi$
$420$	0	0
$421$	−15029.7	−1.73991	−0.869957	−	0.493127i	$-0.835854\pi$
$421$	−15029.7	−1.73991	−0.869957	+	0.493127i	$0.835854\pi$
$422$	−8169.37	−0.942366
$423$	0	0
$424$	5844.79	0.669453
$425$	−17806.0	−2.03228
$426$	0	0
$427$	15721.8	1.78180
$428$	5590.74	0.631399
$429$	0	0
$430$	−25353.2	−2.84335
$431$	5056.76	0.565140	0.282570	−	0.959247i	$-0.408813\pi$
$431$	5056.76	0.565140	0.282570	+	0.959247i	$0.408813\pi$
$432$	0	0
$433$	−10275.6	−1.14045	−0.570226	−	0.821488i	$-0.693144\pi$
$433$	−10275.6	−1.14045	−0.570226	+	0.821488i	$0.693144\pi$
$434$	−20831.5	−2.30401
$435$	0	0
$436$	5960.82	0.654751
$437$	−1601.08	−0.175263
$438$	0	0
$439$	−1490.51	−0.162046	−0.0810232	−	0.996712i	$-0.525819\pi$
$439$	−1490.51	−0.162046	−0.0810232	+	0.996712i	$0.525819\pi$
$440$	−18688.6	−2.02488
$441$	0	0
$442$	14612.1	1.57246
$443$	1892.30	0.202948	0.101474	−	0.994838i	$-0.467644\pi$
$443$	1892.30	0.202948	0.101474	+	0.994838i	$0.467644\pi$
$444$	0	0
$445$	10821.7	1.15280
$446$	−2188.50	−0.232351
$447$	0	0
$448$	17369.8	1.83180
$449$	6571.09	0.690665	0.345333	−	0.938480i	$-0.387766\pi$
$449$	6571.09	0.690665	0.345333	+	0.938480i	$0.387766\pi$
$450$	0	0
$451$	16473.2	1.71994
$452$	13103.9	1.36362
$453$	0	0
$454$	12658.4	1.30856
$455$	−10520.5	−1.08398
$456$	0	0
$457$	−5361.76	−0.548824	−0.274412	−	0.961612i	$-0.588483\pi$
$457$	−5361.76	−0.548824	−0.274412	+	0.961612i	$0.588483\pi$
$458$	−12800.0	−1.30590
$459$	0	0
$460$	3178.67	0.322187
$461$	1940.27	0.196025	0.0980125	−	0.995185i	$-0.468751\pi$
$461$	1940.27	0.196025	0.0980125	+	0.995185i	$0.468751\pi$
$462$	0	0
$463$	−9369.84	−0.940504	−0.470252	−	0.882532i	$-0.655837\pi$
$463$	−9369.84	−0.940504	−0.470252	+	0.882532i	$0.655837\pi$
$464$	−186.379	−0.0186475
$465$	0	0
$466$	28508.8	2.83400
$467$	10643.9	1.05469	0.527344	−	0.849652i	$-0.323188\pi$
$467$	10643.9	1.05469	0.527344	+	0.849652i	$0.323188\pi$
$468$	0	0
$469$	−13950.3	−1.37348
$470$	−24592.2	−2.41352
$471$	0	0
$472$	1345.75	0.131235
$473$	15669.0	1.52318
$474$	0	0
$475$	−18290.2	−1.76676
$476$	29501.1	2.84071
$477$	0	0
$478$	−12463.8	−1.19264
$479$	787.956	0.0751620	0.0375810	−	0.999294i	$-0.488035\pi$
$479$	787.956	0.0751620	0.0375810	+	0.999294i	$0.488035\pi$
$480$	0	0
$481$	−4964.70	−0.470625
$482$	−12344.2	−1.16652
$483$	0	0
$484$	12829.2	1.20485
$485$	−4006.31	−0.375087
$486$	0	0
$487$	−11562.0	−1.07582	−0.537908	−	0.843004i	$-0.680785\pi$
$487$	−11562.0	−1.07582	−0.537908	+	0.843004i	$0.680785\pi$
$488$	17084.6	1.58480
$489$	0	0
$490$	−7603.27	−0.700981
$491$	784.077	0.0720670	0.0360335	−	0.999351i	$-0.488528\pi$
$491$	784.077	0.0720670	0.0360335	+	0.999351i	$0.488528\pi$
$492$	0	0
$493$	31704.2	2.89632
$494$	15009.4	1.36702
$495$	0	0
$496$	−137.928	−0.0124862
$497$	4353.71	0.392939
$498$	0	0
$499$	1378.28	0.123648	0.0618238	−	0.998087i	$-0.480308\pi$
$499$	1378.28	0.123648	0.0618238	+	0.998087i	$0.480308\pi$
$500$	8709.04	0.778960
$501$	0	0
$502$	−5249.18	−0.466698
$503$	1919.94	0.170190	0.0850952	−	0.996373i	$-0.472881\pi$
$503$	1919.94	0.170190	0.0850952	+	0.996373i	$0.472881\pi$
$504$	0	0
$505$	−30752.4	−2.70983
$506$	−3175.32	−0.278973
$507$	0	0
$508$	5088.43	0.444414
$509$	20478.5	1.78329	0.891646	−	0.452734i	$-0.149551\pi$
$509$	20478.5	1.78329	0.891646	+	0.452734i	$0.149551\pi$
$510$	0	0
$511$	−951.032	−0.0823311
$512$	230.457	0.0198923
$513$	0	0
$514$	−4417.88	−0.379113
$515$	−24457.1	−2.09264
$516$	0	0
$517$	15198.7	1.29292
$518$	−16201.3	−1.37422
$519$	0	0
$520$	−11432.5	−0.964129
$521$	−15109.6	−1.27056	−0.635280	−	0.772282i	$-0.719115\pi$
$521$	−15109.6	−1.27056	−0.635280	+	0.772282i	$0.719115\pi$
$522$	0	0
$523$	−1296.32	−0.108383	−0.0541914	−	0.998531i	$-0.517258\pi$
$523$	−1296.32	−0.108383	−0.0541914	+	0.998531i	$0.517258\pi$
$524$	−23141.4	−1.92927
$525$	0	0
$526$	−23413.2	−1.94081
$527$	23462.4	1.93935
$528$	0	0
$529$	−11959.8	−0.982970
$530$	19968.1	1.63652
$531$	0	0
$532$	30303.2	2.46957
$533$	10077.2	0.818934
$534$	0	0
$535$	7327.92	0.592175
$536$	−15159.5	−1.22163
$537$	0	0
$538$	−20558.8	−1.64750
$539$	4699.04	0.375514
$540$	0	0
$541$	−15541.1	−1.23505	−0.617525	−	0.786551i	$-0.711865\pi$
$541$	−15541.1	−1.23505	−0.617525	+	0.786551i	$0.711865\pi$
$542$	−20150.4	−1.59692
$543$	0	0
$544$	−19443.2	−1.53239
$545$	7812.98	0.614076
$546$	0	0
$547$	−10003.4	−0.781926	−0.390963	−	0.920406i	$-0.627858\pi$
$547$	−10003.4	−0.781926	−0.390963	+	0.920406i	$0.627858\pi$
$548$	10082.2	0.785934
$549$	0	0
$550$	−36273.8	−2.81222
$551$	32566.3	2.51791
$552$	0	0
$553$	17066.5	1.31237
$554$	21761.0	1.66883
$555$	0	0
$556$	−7204.04	−0.549495
$557$	8031.77	0.610982	0.305491	−	0.952195i	$-0.401179\pi$
$557$	8031.77	0.610982	0.305491	+	0.952195i	$0.401179\pi$
$558$	0	0
$559$	9585.27	0.725248
$560$	−227.316	−0.0171533
$561$	0	0
$562$	36982.9	2.77585
$563$	8865.46	0.663649	0.331825	−	0.943341i	$-0.392336\pi$
$563$	8865.46	0.663649	0.331825	+	0.943341i	$0.392336\pi$
$564$	0	0
$565$	17175.6	1.27891
$566$	−8934.18	−0.663483
$567$	0	0
$568$	4731.11	0.349494
$569$	−20450.1	−1.50670	−0.753351	−	0.657619i	$-0.771564\pi$
$569$	−20450.1	−1.50670	−0.753351	+	0.657619i	$0.771564\pi$
$570$	0	0
$571$	−10105.3	−0.740618	−0.370309	−	0.928909i	$-0.620748\pi$
$571$	−10105.3	−0.740618	−0.370309	+	0.928909i	$0.620748\pi$
$572$	18416.5	1.34621
$573$	0	0
$574$	32885.0	2.39128
$575$	2367.03	0.171673
$576$	0	0
$577$	17381.4	1.25407	0.627033	−	0.778993i	$-0.284269\pi$
$577$	17381.4	1.25407	0.627033	+	0.778993i	$0.284269\pi$
$578$	−31202.8	−2.24544
$579$	0	0
$580$	−64654.8	−4.62870
$581$	403.121	0.0287853
$582$	0	0
$583$	−12340.9	−0.876683
$584$	−1033.47	−0.0732283
$585$	0	0
$586$	−35097.9	−2.47420
$587$	−19176.4	−1.34837	−0.674186	−	0.738561i	$-0.735505\pi$
$587$	−19176.4	−1.34837	−0.674186	+	0.738561i	$0.735505\pi$
$588$	0	0
$589$	24100.4	1.68597
$590$	4597.60	0.320814
$591$	0	0
$592$	−107.272	−0.00744737
$593$	10703.5	0.741213	0.370606	−	0.928790i	$-0.379150\pi$
$593$	10703.5	0.741213	0.370606	+	0.928790i	$0.379150\pi$
$594$	0	0
$595$	38667.7	2.66424
$596$	−26824.6	−1.84359
$597$	0	0
$598$	−1942.45	−0.132831
$599$	16229.7	1.10705	0.553527	−	0.832831i	$-0.313281\pi$
$599$	16229.7	1.10705	0.553527	+	0.832831i	$0.313281\pi$
$600$	0	0
$601$	5481.06	0.372009	0.186004	−	0.982549i	$-0.440446\pi$
$601$	5481.06	0.372009	0.186004	+	0.982549i	$0.440446\pi$
$602$	31279.7	2.11772
$603$	0	0
$604$	−11969.7	−0.806359
$605$	16815.6	1.13000
$606$	0	0
$607$	−17029.8	−1.13874	−0.569372	−	0.822080i	$-0.692814\pi$
$607$	−17029.8	−1.13874	−0.569372	+	0.822080i	$0.692814\pi$
$608$	−19971.9	−1.33218
$609$	0	0
$610$	58367.7	3.87416
$611$	9297.56	0.615612
$612$	0	0
$613$	7096.79	0.467597	0.233798	−	0.972285i	$-0.424884\pi$
$613$	7096.79	0.467597	0.233798	+	0.972285i	$0.424884\pi$
$614$	6567.29	0.431652
$615$	0	0
$616$	23057.2	1.50812
$617$	9243.83	0.603148	0.301574	−	0.953443i	$-0.402488\pi$
$617$	9243.83	0.603148	0.301574	+	0.953443i	$0.402488\pi$
$618$	0	0
$619$	24576.0	1.59579	0.797893	−	0.602799i	$-0.205948\pi$
$619$	24576.0	1.59579	0.797893	+	0.602799i	$0.205948\pi$
$620$	−47847.2	−3.09934
$621$	0	0
$622$	39598.1	2.55263
$623$	−13351.3	−0.858603
$624$	0	0
$625$	−9139.72	−0.584942
$626$	−29654.1	−1.89332
$627$	0	0
$628$	−27879.6	−1.77152
$629$	18247.5	1.15672
$630$	0	0
$631$	16376.8	1.03320	0.516602	−	0.856226i	$-0.327197\pi$
$631$	16376.8	1.03320	0.516602	+	0.856226i	$0.327197\pi$
$632$	18545.9	1.16727
$633$	0	0
$634$	24178.3	1.51458
$635$	6669.52	0.416806
$636$	0	0
$637$	2874.56	0.178798
$638$	64586.7	4.00786
$639$	0	0
$640$	40047.6	2.47347
$641$	6614.72	0.407591	0.203795	−	0.979013i	$-0.434672\pi$
$641$	6614.72	0.407591	0.203795	+	0.979013i	$0.434672\pi$
$642$	0	0
$643$	−18544.9	−1.13739	−0.568693	−	0.822550i	$-0.692551\pi$
$643$	−18544.9	−1.13739	−0.568693	+	0.822550i	$0.692551\pi$
$644$	−3921.70	−0.239964
$645$	0	0
$646$	−55166.5	−3.35990
$647$	−28164.5	−1.71137	−0.855687	−	0.517493i	$-0.826865\pi$
$647$	−28164.5	−1.71137	−0.855687	+	0.517493i	$0.826865\pi$
$648$	0	0
$649$	−2841.45	−0.171859
$650$	−22189.9	−1.33902
$651$	0	0
$652$	2634.13	0.158222
$653$	11520.0	0.690370	0.345185	−	0.938535i	$-0.387816\pi$
$653$	11520.0	0.690370	0.345185	+	0.938535i	$0.387816\pi$
$654$	0	0
$655$	−30332.0	−1.80942
$656$	217.737	0.0129591
$657$	0	0
$658$	30340.8	1.79758
$659$	16586.2	0.980434	0.490217	−	0.871600i	$-0.336917\pi$
$659$	16586.2	0.980434	0.490217	+	0.871600i	$0.336917\pi$
$660$	0	0
$661$	−15191.8	−0.893935	−0.446967	−	0.894550i	$-0.647496\pi$
$661$	−15191.8	−0.893935	−0.446967	+	0.894550i	$0.647496\pi$
$662$	−34640.0	−2.03372
$663$	0	0
$664$	438.065	0.0256027
$665$	39719.1	2.31615
$666$	0	0
$667$	−4214.57	−0.244661
$668$	25754.0	1.49169
$669$	0	0
$670$	−51790.9	−2.98636
$671$	−36072.9	−2.07538
$672$	0	0
$673$	22990.5	1.31682	0.658410	−	0.752660i	$-0.271229\pi$
$673$	22990.5	1.31682	0.658410	+	0.752660i	$0.271229\pi$
$674$	−5685.77	−0.324937
$675$	0	0
$676$	−17250.6	−0.981487
$677$	18654.2	1.05899	0.529496	−	0.848313i	$-0.322381\pi$
$677$	18654.2	1.05899	0.529496	+	0.848313i	$0.322381\pi$
$678$	0	0
$679$	4942.82	0.279364
$680$	42019.6	2.36967
$681$	0	0
$682$	47796.9	2.68363
$683$	13970.1	0.782649	0.391325	−	0.920253i	$-0.372017\pi$
$683$	13970.1	0.782649	0.391325	+	0.920253i	$0.372017\pi$
$684$	0	0
$685$	13215.0	0.737110
$686$	−23595.8	−1.31325
$687$	0	0
$688$	207.108	0.0114766
$689$	−7549.32	−0.417426
$690$	0	0
$691$	−4273.34	−0.235261	−0.117631	−	0.993057i	$-0.537530\pi$
$691$	−4273.34	−0.235261	−0.117631	+	0.993057i	$0.537530\pi$
$692$	19265.0	1.05830
$693$	0	0
$694$	47245.6	2.58418
$695$	−9442.50	−0.515359
$696$	0	0
$697$	−37038.3	−2.01281
$698$	−28352.5	−1.53748
$699$	0	0
$700$	−44800.2	−2.41898
$701$	−13904.0	−0.749141	−0.374570	−	0.927199i	$-0.622210\pi$
$701$	−13904.0	−0.749141	−0.374570	+	0.927199i	$0.622210\pi$
$702$	0	0
$703$	18743.7	1.00559
$704$	−39854.3	−2.13362
$705$	0	0
$706$	10577.4	0.563863
$707$	37941.0	2.01827
$708$	0	0
$709$	13659.3	0.723533	0.361766	−	0.932269i	$-0.382174\pi$
$709$	13659.3	0.723533	0.361766	+	0.932269i	$0.382174\pi$
$710$	16163.3	0.854364
$711$	0	0
$712$	−14508.7	−0.763673
$713$	−3118.96	−0.163823
$714$	0	0
$715$	24138.9	1.26258
$716$	7093.64	0.370254
$717$	0	0
$718$	57466.9	2.98697
$719$	−10745.5	−0.557358	−0.278679	−	0.960384i	$-0.589897\pi$
$719$	−10745.5	−0.557358	−0.278679	+	0.960384i	$0.589897\pi$
$720$	0	0
$721$	30174.2	1.55859
$722$	−25249.7	−1.30152
$723$	0	0
$724$	−59213.5	−3.03957
$725$	−48145.9	−2.46634
$726$	0	0
$727$	23351.2	1.19126	0.595632	−	0.803257i	$-0.296902\pi$
$727$	23351.2	1.19126	0.595632	+	0.803257i	$0.296902\pi$
$728$	14104.9	0.718080
$729$	0	0
$730$	−3530.74	−0.179012
$731$	−35230.3	−1.78254
$732$	0	0
$733$	17996.9	0.906865	0.453432	−	0.891291i	$-0.350199\pi$
$733$	17996.9	0.906865	0.453432	+	0.891291i	$0.350199\pi$
$734$	42798.7	2.15222
$735$	0	0
$736$	2584.67	0.129446
$737$	32008.3	1.59978
$738$	0	0
$739$	13155.1	0.654829	0.327415	−	0.944881i	$-0.393823\pi$
$739$	13155.1	0.654829	0.327415	+	0.944881i	$0.393823\pi$
$740$	−37212.5	−1.84859
$741$	0	0
$742$	−24635.8	−1.21888
$743$	25720.3	1.26997	0.634984	−	0.772525i	$-0.281007\pi$
$743$	25720.3	1.26997	0.634984	+	0.772525i	$0.281007\pi$
$744$	0	0
$745$	−35159.6	−1.72906
$746$	40795.4	2.00218
$747$	0	0
$748$	−67688.9	−3.30876
$749$	−9040.87	−0.441050
$750$	0	0
$751$	−35919.3	−1.74529	−0.872645	−	0.488356i	$-0.837597\pi$
$751$	−35919.3	−1.74529	−0.872645	+	0.488356i	$0.837597\pi$
$752$	200.891	0.00974170
$753$	0	0
$754$	39509.9	1.90831
$755$	−15689.0	−0.756266
$756$	0	0
$757$	−29818.5	−1.43167	−0.715834	−	0.698271i	$-0.753953\pi$
$757$	−29818.5	−1.43167	−0.715834	+	0.698271i	$0.753953\pi$
$758$	−16346.3	−0.783276
$759$	0	0
$760$	43162.1	2.06007
$761$	−35322.2	−1.68256	−0.841280	−	0.540600i	$-0.818197\pi$
$761$	−35322.2	−1.68256	−0.841280	+	0.540600i	$0.818197\pi$
$762$	0	0
$763$	−9639.32	−0.457361
$764$	10869.1	0.514697
$765$	0	0
$766$	−42119.4	−1.98673
$767$	−1738.21	−0.0818294
$768$	0	0
$769$	−6704.62	−0.314402	−0.157201	−	0.987567i	$-0.550247\pi$
$769$	−6704.62	−0.314402	−0.157201	+	0.987567i	$0.550247\pi$
$770$	78772.6	3.68671
$771$	0	0
$772$	−44437.2	−2.07167
$773$	−10246.4	−0.476762	−0.238381	−	0.971172i	$-0.576617\pi$
$773$	−10246.4	−0.476762	−0.238381	+	0.971172i	$0.576617\pi$
$774$	0	0
$775$	−35630.0	−1.65144
$776$	5371.28	0.248476
$777$	0	0
$778$	−34189.5	−1.57552
$779$	−38045.4	−1.74983
$780$	0	0
$781$	−9989.40	−0.457681
$782$	7139.39	0.326476
$783$	0	0
$784$	62.1104	0.00282937
$785$	−36542.4	−1.66147
$786$	0	0
$787$	25804.8	1.16880	0.584398	−	0.811467i	$-0.301331\pi$
$787$	25804.8	1.16880	0.584398	+	0.811467i	$0.301331\pi$
$788$	−51636.5	−2.33436
$789$	0	0
$790$	63360.1	2.85348
$791$	−21190.6	−0.952528
$792$	0	0
$793$	−22067.0	−0.988176
$794$	−42408.9	−1.89551
$795$	0	0
$796$	28869.0	1.28547
$797$	−7971.15	−0.354269	−0.177135	−	0.984187i	$-0.556683\pi$
$797$	−7971.15	−0.354269	−0.177135	+	0.984187i	$0.556683\pi$
$798$	0	0
$799$	−34172.8	−1.51307
$800$	29526.4	1.30489
$801$	0	0
$802$	−45550.7	−2.00555
$803$	2182.10	0.0958963
$804$	0	0
$805$	−5140.26	−0.225056
$806$	29239.0	1.27779
$807$	0	0
$808$	41229.8	1.79512
$809$	1869.81	0.0812595	0.0406297	−	0.999174i	$-0.487064\pi$
$809$	1869.81	0.0812595	0.0406297	+	0.999174i	$0.487064\pi$
$810$	0	0
$811$	−4096.49	−0.177370	−0.0886851	−	0.996060i	$-0.528266\pi$
$811$	−4096.49	−0.177370	−0.0886851	+	0.996060i	$0.528266\pi$
$812$	79768.3	3.44744
$813$	0	0
$814$	37173.3	1.60064
$815$	3452.62	0.148393
$816$	0	0
$817$	−36188.2	−1.54965
$818$	12688.9	0.542367
$819$	0	0
$820$	75532.7	3.21673
$821$	−32856.3	−1.39670	−0.698350	−	0.715756i	$-0.746082\pi$
$821$	−32856.3	−1.39670	−0.698350	+	0.715756i	$0.746082\pi$
$822$	0	0
$823$	−1727.41	−0.0731638	−0.0365819	−	0.999331i	$-0.511647\pi$
$823$	−1727.41	−0.0731638	−0.0365819	+	0.999331i	$0.511647\pi$
$824$	32789.8	1.38627
$825$	0	0
$826$	−5672.32	−0.238941
$827$	38654.4	1.62533	0.812664	−	0.582733i	$-0.198017\pi$
$827$	38654.4	1.62533	0.812664	+	0.582733i	$0.198017\pi$
$828$	0	0
$829$	−9211.90	−0.385938	−0.192969	−	0.981205i	$-0.561812\pi$
$829$	−9211.90	−0.385938	−0.192969	+	0.981205i	$0.561812\pi$
$830$	1496.60	0.0625877
$831$	0	0
$832$	−24380.2	−1.01590
$833$	−10565.3	−0.439457
$834$	0	0
$835$	33756.3	1.39902
$836$	−69529.4	−2.87646
$837$	0	0
$838$	−48184.2	−1.98627
$839$	44020.8	1.81140	0.905701	−	0.423917i	$-0.139345\pi$
$839$	44020.8	1.81140	0.905701	+	0.423917i	$0.139345\pi$
$840$	0	0
$841$	61336.4	2.51492
$842$	68841.7	2.81763
$843$	0	0
$844$	23150.2	0.944152
$845$	−22610.8	−0.920514
$846$	0	0
$847$	−20746.3	−0.841621
$848$	−163.117	−0.00660551
$849$	0	0
$850$	81558.1	3.29108
$851$	−2425.72	−0.0977118
$852$	0	0
$853$	14157.1	0.568265	0.284133	−	0.958785i	$-0.408294\pi$
$853$	14157.1	0.568265	0.284133	+	0.958785i	$0.408294\pi$
$854$	−72011.5	−2.88546
$855$	0	0
$856$	−9824.56	−0.392286
$857$	21094.5	0.840810	0.420405	−	0.907337i	$-0.361888\pi$
$857$	21094.5	0.840810	0.420405	+	0.907337i	$0.361888\pi$
$858$	0	0
$859$	−37442.2	−1.48721	−0.743603	−	0.668622i	$-0.766885\pi$
$859$	−37442.2	−1.48721	−0.743603	+	0.668622i	$0.766885\pi$
$860$	71845.5	2.84874
$861$	0	0
$862$	−23161.8	−0.915191
$863$	−15063.2	−0.594156	−0.297078	−	0.954853i	$-0.596012\pi$
$863$	−15063.2	−0.594156	−0.297078	+	0.954853i	$0.596012\pi$
$864$	0	0
$865$	25251.1	0.992560
$866$	47066.2	1.84685
$867$	0	0
$868$	59031.9	2.30838
$869$	−39158.4	−1.52861
$870$	0	0
$871$	19580.6	0.761725
$872$	−10474.9	−0.406794
$873$	0	0
$874$	7333.52	0.283821
$875$	−14083.5	−0.544125
$876$	0	0
$877$	32525.7	1.25235	0.626177	−	0.779681i	$-0.284619\pi$
$877$	32525.7	1.25235	0.626177	+	0.779681i	$0.284619\pi$
$878$	6827.11	0.262419
$879$	0	0
$880$	521.566	0.0199795
$881$	17706.4	0.677123	0.338561	−	0.940944i	$-0.390060\pi$
$881$	17706.4	0.677123	0.338561	+	0.940944i	$0.390060\pi$
$882$	0	0
$883$	24205.7	0.922522	0.461261	−	0.887264i	$-0.347397\pi$
$883$	24205.7	0.922522	0.461261	+	0.887264i	$0.347397\pi$
$884$	−41407.6	−1.57544
$885$	0	0
$886$	−8667.43	−0.328655
$887$	−36290.4	−1.37374	−0.686872	−	0.726778i	$-0.741017\pi$
$887$	−36290.4	−1.37374	−0.686872	+	0.726778i	$0.741017\pi$
$888$	0	0
$889$	−8228.56	−0.310435
$890$	−49567.3	−1.86685
$891$	0	0
$892$	6201.75	0.232791
$893$	−35102.0	−1.31539
$894$	0	0
$895$	9297.80	0.347253
$896$	−49409.0	−1.84223
$897$	0	0
$898$	−30098.0	−1.11847
$899$	63440.4	2.35357
$900$	0	0
$901$	27747.2	1.02596
$902$	−75453.2	−2.78527
$903$	0	0
$904$	−23027.5	−0.847214
$905$	−77612.5	−2.85075
$906$	0	0
$907$	−1572.57	−0.0575705	−0.0287852	−	0.999586i	$-0.509164\pi$
$907$	−1572.57	−0.0575705	−0.0287852	+	0.999586i	$0.509164\pi$
$908$	−35871.1	−1.31104
$909$	0	0
$910$	48187.9	1.75540
$911$	23504.6	0.854820	0.427410	−	0.904058i	$-0.359426\pi$
$911$	23504.6	0.854820	0.427410	+	0.904058i	$0.359426\pi$
$912$	0	0
$913$	−924.944	−0.0335281
$914$	24558.8	0.888768
$915$	0	0
$916$	36272.4	1.30838
$917$	37422.3	1.34765
$918$	0	0
$919$	−28818.0	−1.03440	−0.517202	−	0.855863i	$-0.673026\pi$
$919$	−28818.0	−1.03440	−0.517202	+	0.855863i	$0.673026\pi$
$920$	−5585.84	−0.200174
$921$	0	0
$922$	−8887.17	−0.317444
$923$	−6110.86	−0.217921
$924$	0	0
$925$	−27710.7	−0.984995
$926$	42917.3	1.52306
$927$	0	0
$928$	−52572.7	−1.85968
$929$	40908.5	1.44474	0.722370	−	0.691506i	$-0.243053\pi$
$929$	40908.5	1.44474	0.722370	+	0.691506i	$0.243053\pi$
$930$	0	0
$931$	−10852.6	−0.382041
$932$	−80787.7	−2.83937
$933$	0	0
$934$	−48752.9	−1.70797
$935$	−88721.4	−3.10321
$936$	0	0
$937$	42875.5	1.49486	0.747428	−	0.664343i	$-0.231289\pi$
$937$	42875.5	1.49486	0.747428	+	0.664343i	$0.231289\pi$
$938$	63897.4	2.22423
$939$	0	0
$940$	69689.0	2.41809
$941$	−12952.0	−0.448697	−0.224348	−	0.974509i	$-0.572025\pi$
$941$	−12952.0	−0.448697	−0.224348	+	0.974509i	$0.572025\pi$
$942$	0	0
$943$	4923.66	0.170028
$944$	−37.5574	−0.00129490
$945$	0	0
$946$	−71769.9	−2.46664
$947$	11075.5	0.380046	0.190023	−	0.981780i	$-0.439144\pi$
$947$	11075.5	0.380046	0.190023	+	0.981780i	$0.439144\pi$
$948$	0	0
$949$	1334.87	0.0456603
$950$	83775.7	2.86110
$951$	0	0
$952$	−51841.9	−1.76492
$953$	16254.9	0.552515	0.276257	−	0.961084i	$-0.410906\pi$
$953$	16254.9	0.552515	0.276257	+	0.961084i	$0.410906\pi$
$954$	0	0
$955$	14246.3	0.482722
$956$	35319.8	1.19490
$957$	0	0
$958$	−3609.13	−0.121718
$959$	−16304.1	−0.548997
$960$	0	0
$961$	17157.5	0.575930
$962$	22740.2	0.762133
$963$	0	0
$964$	34980.9	1.16873
$965$	−58244.9	−1.94297
$966$	0	0
$967$	17700.6	0.588637	0.294318	−	0.955707i	$-0.404907\pi$
$967$	17700.6	0.588637	0.294318	+	0.955707i	$0.404907\pi$
$968$	−22544.7	−0.748569
$969$	0	0
$970$	18350.4	0.607418
$971$	−37930.1	−1.25359	−0.626794	−	0.779185i	$-0.715633\pi$
$971$	−37930.1	−1.25359	−0.626794	+	0.779185i	$0.715633\pi$
$972$	0	0
$973$	11649.8	0.383837
$974$	52958.0	1.74218
$975$	0	0
$976$	−476.800	−0.0156373
$977$	14003.7	0.458565	0.229283	−	0.973360i	$-0.426362\pi$
$977$	14003.7	0.458565	0.229283	+	0.973360i	$0.426362\pi$
$978$	0	0
$979$	30634.0	1.00007
$980$	21546.0	0.702309
$981$	0	0
$982$	−3591.36	−0.116706
$983$	34728.6	1.12683	0.563413	−	0.826175i	$-0.309488\pi$
$983$	34728.6	1.12683	0.563413	+	0.826175i	$0.309488\pi$
$984$	0	0
$985$	−67681.2	−2.18934
$986$	−145217.	−4.69031
$987$	0	0
$988$	−42533.5	−1.36961
$989$	4683.31	0.150577
$990$	0	0
$991$	48169.3	1.54405	0.772023	−	0.635595i	$-0.219245\pi$
$991$	48169.3	1.54405	0.772023	+	0.635595i	$0.219245\pi$
$992$	−38906.0	−1.24523
$993$	0	0
$994$	−19941.6	−0.636327
$995$	37839.2	1.20561
$996$	0	0
$997$	−15159.6	−0.481553	−0.240776	−	0.970581i	$-0.577402\pi$
$997$	−15159.6	−0.481553	−0.240776	+	0.970581i	$0.577402\pi$
$998$	−6313.01	−0.200235
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.c.1.1		7
3.2	odd	2		177.4.a.b.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.b.1.7	✓	7	3.2	odd	2
531.4.a.c.1.1		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)

\(f(q)\)	\(=\)	\(q-4.58037 q^{2} +12.9798 q^{4} +17.0129 q^{5} -20.9898 q^{7} -22.8093 q^{8} +O(q^{10})\)	\(q-4.58037 q^{2} +12.9798 q^{4} +17.0129 q^{5} -20.9898 q^{7} -22.8093 q^{8} -77.9254 q^{10} +48.1602 q^{11} +29.4612 q^{13} +96.1410 q^{14} +0.636565 q^{16} -108.283 q^{17} -111.228 q^{19} +220.824 q^{20} -220.591 q^{22} +14.3946 q^{23} +164.439 q^{25} -134.943 q^{26} -272.443 q^{28} -292.789 q^{29} -216.676 q^{31} +179.558 q^{32} +495.978 q^{34} -357.097 q^{35} -168.516 q^{37} +509.464 q^{38} -388.052 q^{40} +342.050 q^{41} +325.352 q^{43} +625.109 q^{44} -65.9325 q^{46} +315.587 q^{47} +97.5712 q^{49} -753.191 q^{50} +382.400 q^{52} -256.246 q^{53} +819.344 q^{55} +478.762 q^{56} +1341.08 q^{58} -59.0000 q^{59} -749.020 q^{61} +992.457 q^{62} -827.536 q^{64} +501.221 q^{65} +664.622 q^{67} -1405.50 q^{68} +1635.64 q^{70} -207.420 q^{71} +45.3093 q^{73} +771.868 q^{74} -1443.71 q^{76} -1010.87 q^{77} -813.087 q^{79} +10.8298 q^{80} -1566.71 q^{82} -19.2056 q^{83} -1842.22 q^{85} -1490.23 q^{86} -1098.50 q^{88} +636.087 q^{89} -618.385 q^{91} +186.838 q^{92} -1445.50 q^{94} -1892.31 q^{95} -235.487 q^{97} -446.912 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8}+O(q^{10}) \) 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8	\( 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8} - 71 q^{10} + 5 q^{11} - 67 q^{13} + 65 q^{14} - 94 q^{16} + 23 q^{17} - 176 q^{19} + 207 q^{20} - 704 q^{22} + 218 q^{23} - 183 q^{25} - 58 q^{26} - 938 q^{28} - 168 q^{29} - 604 q^{31} + 448 q^{32} - 610 q^{34} + 336 q^{35} - 505 q^{37} + 453 q^{38} - 1080 q^{40} + 265 q^{41} - 493 q^{43} - 504 q^{44} + 381 q^{46} + 244 q^{47} + 770 q^{49} - 1639 q^{50} + 160 q^{52} - 686 q^{53} - 116 q^{55} - 2190 q^{56} + 1584 q^{58} - 413 q^{59} - 838 q^{61} - 286 q^{62} + 205 q^{64} - 490 q^{65} - 1504 q^{67} - 3047 q^{68} + 1530 q^{70} + 1267 q^{71} - 666 q^{73} - 528 q^{74} - 64 q^{76} - 1109 q^{77} - 2741 q^{79} - 1213 q^{80} + 953 q^{82} + 2025 q^{83} - 1274 q^{85} - 4394 q^{86} - 1639 q^{88} - 616 q^{89} - 2415 q^{91} - 218 q^{92} + 900 q^{94} - 2554 q^{95} - 1298 q^{97} + 172 q^{98}+O(q^{100}) \) 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8 - 71 * q^10 + 5 * q^11 - 67 * q^13 + 65 * q^14 - 94 * q^16 + 23 * q^17 - 176 * q^19 + 207 * q^20 - 704 * q^22 + 218 * q^23 - 183 * q^25 - 58 * q^26 - 938 * q^28 - 168 * q^29 - 604 * q^31 + 448 * q^32 - 610 * q^34 + 336 * q^35 - 505 * q^37 + 453 * q^38 - 1080 * q^40 + 265 * q^41 - 493 * q^43 - 504 * q^44 + 381 * q^46 + 244 * q^47 + 770 * q^49 - 1639 * q^50 + 160 * q^52 - 686 * q^53 - 116 * q^55 - 2190 * q^56 + 1584 * q^58 - 413 * q^59 - 838 * q^61 - 286 * q^62 + 205 * q^64 - 490 * q^65 - 1504 * q^67 - 3047 * q^68 + 1530 * q^70 + 1267 * q^71 - 666 * q^73 - 528 * q^74 - 64 * q^76 - 1109 * q^77 - 2741 * q^79 - 1213 * q^80 + 953 * q^82 + 2025 * q^83 - 1274 * q^85 - 4394 * q^86 - 1639 * q^88 - 616 * q^89 - 2415 * q^91 - 218 * q^92 + 900 * q^94 - 2554 * q^95 - 1298 * q^97 + 172 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more