LMFDB - Embedded newform 8043.2.a.t.1.39

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$52$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.39
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.64346 q^{2} -1.00000 q^{3} +0.700959 q^{4} +0.804426 q^{5} -1.64346 q^{6} +1.00000 q^{7} -2.13492 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.64346 q^{2} -1.00000 q^{3} +0.700959 q^{4} +0.804426 q^{5} -1.64346 q^{6} +1.00000 q^{7} -2.13492 q^{8} +1.00000 q^{9} +1.32204 q^{10} -1.88038 q^{11} -0.700959 q^{12} +2.51940 q^{13} +1.64346 q^{14} -0.804426 q^{15} -4.91057 q^{16} -4.57039 q^{17} +1.64346 q^{18} -7.09831 q^{19} +0.563870 q^{20} -1.00000 q^{21} -3.09033 q^{22} +2.60322 q^{23} +2.13492 q^{24} -4.35290 q^{25} +4.14053 q^{26} -1.00000 q^{27} +0.700959 q^{28} +0.0852448 q^{29} -1.32204 q^{30} +0.782856 q^{31} -3.80049 q^{32} +1.88038 q^{33} -7.51125 q^{34} +0.804426 q^{35} +0.700959 q^{36} +11.3629 q^{37} -11.6658 q^{38} -2.51940 q^{39} -1.71739 q^{40} +9.26216 q^{41} -1.64346 q^{42} -5.43088 q^{43} -1.31807 q^{44} +0.804426 q^{45} +4.27829 q^{46} +11.3681 q^{47} +4.91057 q^{48} +1.00000 q^{49} -7.15381 q^{50} +4.57039 q^{51} +1.76600 q^{52} +5.27445 q^{53} -1.64346 q^{54} -1.51263 q^{55} -2.13492 q^{56} +7.09831 q^{57} +0.140096 q^{58} -0.612801 q^{59} -0.563870 q^{60} +12.2202 q^{61} +1.28659 q^{62} +1.00000 q^{63} +3.57520 q^{64} +2.02667 q^{65} +3.09033 q^{66} +6.90132 q^{67} -3.20365 q^{68} -2.60322 q^{69} +1.32204 q^{70} +14.6650 q^{71} -2.13492 q^{72} -15.1800 q^{73} +18.6744 q^{74} +4.35290 q^{75} -4.97562 q^{76} -1.88038 q^{77} -4.14053 q^{78} -4.45397 q^{79} -3.95020 q^{80} +1.00000 q^{81} +15.2220 q^{82} +2.88291 q^{83} -0.700959 q^{84} -3.67654 q^{85} -8.92544 q^{86} -0.0852448 q^{87} +4.01446 q^{88} +7.98720 q^{89} +1.32204 q^{90} +2.51940 q^{91} +1.82475 q^{92} -0.782856 q^{93} +18.6829 q^{94} -5.71006 q^{95} +3.80049 q^{96} +9.68210 q^{97} +1.64346 q^{98} -1.88038 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) $ 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * 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 $

$2$	1.64346	1.16210	0.581051	−	0.813867i	$-0.302642\pi$
$2$	1.64346	1.16210	0.581051	+	0.813867i	$0.302642\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.700959	0.350479
$5$	0.804426	0.359750	0.179875	−	0.983689i	$-0.442431\pi$
$5$	0.804426	0.359750	0.179875	+	0.983689i	$0.442431\pi$
$6$	−1.64346	−0.670940
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.13492	−0.754809
$9$	1.00000	0.333333
$10$	1.32204	0.418066
$11$	−1.88038	−0.566956	−0.283478	−	0.958979i	$-0.591488\pi$
$11$	−1.88038	−0.566956	−0.283478	+	0.958979i	$0.591488\pi$
$12$	−0.700959	−0.202349
$13$	2.51940	0.698756	0.349378	−	0.936982i	$-0.386393\pi$
$13$	2.51940	0.698756	0.349378	+	0.936982i	$0.386393\pi$
$14$	1.64346	0.439233
$15$	−0.804426	−0.207702
$16$	−4.91057	−1.22764
$17$	−4.57039	−1.10848	−0.554241	−	0.832356i	$-0.686991\pi$
$17$	−4.57039	−1.10848	−0.554241	+	0.832356i	$0.686991\pi$
$18$	1.64346	0.387367
$19$	−7.09831	−1.62846	−0.814232	−	0.580540i	$-0.802841\pi$
$19$	−7.09831	−1.62846	−0.814232	+	0.580540i	$0.802841\pi$
$20$	0.563870	0.126085
$21$	−1.00000	−0.218218
$22$	−3.09033	−0.658860
$23$	2.60322	0.542809	0.271405	−	0.962465i	$-0.412512\pi$
$23$	2.60322	0.542809	0.271405	+	0.962465i	$0.412512\pi$
$24$	2.13492	0.435789
$25$	−4.35290	−0.870580
$26$	4.14053	0.812025
$27$	−1.00000	−0.192450
$28$	0.700959	0.132469
$29$	0.0852448	0.0158296	0.00791479	−	0.999969i	$-0.497481\pi$
$29$	0.0852448	0.0158296	0.00791479	+	0.999969i	$0.497481\pi$
$30$	−1.32204	−0.241371
$31$	0.782856	0.140605	0.0703026	−	0.997526i	$-0.477604\pi$
$31$	0.782856	0.140605	0.0703026	+	0.997526i	$0.477604\pi$
$32$	−3.80049	−0.671838
$33$	1.88038	0.327332
$34$	−7.51125	−1.28817
$35$	0.804426	0.135973
$36$	0.700959	0.116826
$37$	11.3629	1.86805	0.934024	−	0.357211i	$-0.116272\pi$
$37$	11.3629	1.86805	0.934024	+	0.357211i	$0.116272\pi$
$38$	−11.6658	−1.89244
$39$	−2.51940	−0.403427
$40$	−1.71739	−0.271543
$41$	9.26216	1.44651	0.723253	−	0.690583i	$-0.242646\pi$
$41$	9.26216	1.44651	0.723253	+	0.690583i	$0.242646\pi$
$42$	−1.64346	−0.253591
$43$	−5.43088	−0.828202	−0.414101	−	0.910231i	$-0.635904\pi$
$43$	−5.43088	−0.828202	−0.414101	+	0.910231i	$0.635904\pi$
$44$	−1.31807	−0.198706
$45$	0.804426	0.119917
$46$	4.27829	0.630799
$47$	11.3681	1.65820	0.829101	−	0.559099i	$-0.188853\pi$
$47$	11.3681	1.65820	0.829101	+	0.559099i	$0.188853\pi$
$48$	4.91057	0.708780
$49$	1.00000	0.142857
$50$	−7.15381	−1.01170
$51$	4.57039	0.639982
$52$	1.76600	0.244899
$53$	5.27445	0.724502	0.362251	−	0.932081i	$-0.382008\pi$
$53$	5.27445	0.724502	0.362251	+	0.932081i	$0.382008\pi$
$54$	−1.64346	−0.223647
$55$	−1.51263	−0.203963
$56$	−2.13492	−0.285291
$57$	7.09831	0.940194
$58$	0.140096	0.0183956
$59$	−0.612801	−0.0797799	−0.0398899	−	0.999204i	$-0.512701\pi$
$59$	−0.612801	−0.0797799	−0.0398899	+	0.999204i	$0.512701\pi$
$60$	−0.563870	−0.0727953
$61$	12.2202	1.56464	0.782320	−	0.622876i	$-0.214036\pi$
$61$	12.2202	1.56464	0.782320	+	0.622876i	$0.214036\pi$
$62$	1.28659	0.163397
$63$	1.00000	0.125988
$64$	3.57520	0.446900
$65$	2.02667	0.251378
$66$	3.09033	0.380393
$67$	6.90132	0.843131	0.421566	−	0.906798i	$-0.361481\pi$
$67$	6.90132	0.843131	0.421566	+	0.906798i	$0.361481\pi$
$68$	−3.20365	−0.388500
$69$	−2.60322	−0.313391
$70$	1.32204	0.158014
$71$	14.6650	1.74041	0.870206	−	0.492689i	$-0.163986\pi$
$71$	14.6650	1.74041	0.870206	+	0.492689i	$0.163986\pi$
$72$	−2.13492	−0.251603
$73$	−15.1800	−1.77669	−0.888345	−	0.459176i	$-0.848145\pi$
$73$	−15.1800	−1.77669	−0.888345	+	0.459176i	$0.848145\pi$
$74$	18.6744	2.17086
$75$	4.35290	0.502629
$76$	−4.97562	−0.570743
$77$	−1.88038	−0.214289
$78$	−4.14053	−0.468823
$79$	−4.45397	−0.501110	−0.250555	−	0.968102i	$-0.580613\pi$
$79$	−4.45397	−0.501110	−0.250555	+	0.968102i	$0.580613\pi$
$80$	−3.95020	−0.441645
$81$	1.00000	0.111111
$82$	15.2220	1.68099
$83$	2.88291	0.316441	0.158220	−	0.987404i	$-0.449424\pi$
$83$	2.88291	0.316441	0.158220	+	0.987404i	$0.449424\pi$
$84$	−0.700959	−0.0764809
$85$	−3.67654	−0.398777
$86$	−8.92544	−0.962455
$87$	−0.0852448	−0.00913921
$88$	4.01446	0.427943
$89$	7.98720	0.846642	0.423321	−	0.905980i	$-0.360864\pi$
$89$	7.98720	0.846642	0.423321	+	0.905980i	$0.360864\pi$
$90$	1.32204	0.139355
$91$	2.51940	0.264105
$92$	1.82475	0.190244
$93$	−0.782856	−0.0811784
$94$	18.6829	1.92700
$95$	−5.71006	−0.585840
$96$	3.80049	0.387886
$97$	9.68210	0.983069	0.491534	−	0.870858i	$-0.336436\pi$
$97$	9.68210	0.983069	0.491534	+	0.870858i	$0.336436\pi$
$98$	1.64346	0.166014
$99$	−1.88038	−0.188985
$100$	−3.05120	−0.305120
$101$	1.80809	0.179912	0.0899559	−	0.995946i	$-0.471327\pi$
$101$	1.80809	0.179912	0.0899559	+	0.995946i	$0.471327\pi$
$102$	7.51125	0.743724
$103$	−4.91503	−0.484293	−0.242146	−	0.970240i	$-0.577851\pi$
$103$	−4.91503	−0.484293	−0.242146	+	0.970240i	$0.577851\pi$
$104$	−5.37872	−0.527427
$105$	−0.804426	−0.0785040
$106$	8.66835	0.841945
$107$	−12.5918	−1.21730	−0.608649	−	0.793440i	$-0.708288\pi$
$107$	−12.5918	−1.21730	−0.608649	+	0.793440i	$0.708288\pi$
$108$	−0.700959	−0.0674498
$109$	1.05340	0.100898	0.0504488	−	0.998727i	$-0.483935\pi$
$109$	1.05340	0.100898	0.0504488	+	0.998727i	$0.483935\pi$
$110$	−2.48594	−0.237025
$111$	−11.3629	−1.07852
$112$	−4.91057	−0.464006
$113$	3.24129	0.304915	0.152458	−	0.988310i	$-0.451281\pi$
$113$	3.24129	0.304915	0.152458	+	0.988310i	$0.451281\pi$
$114$	11.6658	1.09260
$115$	2.09410	0.195276
$116$	0.0597531	0.00554794
$117$	2.51940	0.232919
$118$	−1.00711	−0.0927123
$119$	−4.57039	−0.418967
$120$	1.71739	0.156775
$121$	−7.46417	−0.678561
$122$	20.0835	1.81827
$123$	−9.26216	−0.835141
$124$	0.548750	0.0492792
$125$	−7.52372	−0.672942
$126$	1.64346	0.146411
$127$	9.75584	0.865691	0.432845	−	0.901468i	$-0.357510\pi$
$127$	9.75584	0.865691	0.432845	+	0.901468i	$0.357510\pi$
$128$	13.4767	1.19118
$129$	5.43088	0.478163
$130$	3.33075	0.292126
$131$	3.57941	0.312734	0.156367	−	0.987699i	$-0.450022\pi$
$131$	3.57941	0.312734	0.156367	+	0.987699i	$0.450022\pi$
$132$	1.31807	0.114723
$133$	−7.09831	−0.615501
$134$	11.3420	0.979804
$135$	−0.804426	−0.0692340
$136$	9.75742	0.836692
$137$	0.683612	0.0584049	0.0292025	−	0.999574i	$-0.490703\pi$
$137$	0.683612	0.0584049	0.0292025	+	0.999574i	$0.490703\pi$
$138$	−4.27829	−0.364192
$139$	−5.97258	−0.506587	−0.253294	−	0.967389i	$-0.581514\pi$
$139$	−5.97258	−0.506587	−0.253294	+	0.967389i	$0.581514\pi$
$140$	0.563870	0.0476557
$141$	−11.3681	−0.957363
$142$	24.1013	2.02253
$143$	−4.73743	−0.396164
$144$	−4.91057	−0.409215
$145$	0.0685732	0.00569469
$146$	−24.9478	−2.06469
$147$	−1.00000	−0.0824786
$148$	7.96492	0.654712
$149$	−5.32119	−0.435929	−0.217964	−	0.975957i	$-0.569942\pi$
$149$	−5.32119	−0.435929	−0.217964	+	0.975957i	$0.569942\pi$
$150$	7.15381	0.584106
$151$	−6.57500	−0.535066	−0.267533	−	0.963549i	$-0.586208\pi$
$151$	−6.57500	−0.535066	−0.267533	+	0.963549i	$0.586208\pi$
$152$	15.1543	1.22918
$153$	−4.57039	−0.369494
$154$	−3.09033	−0.249026
$155$	0.629750	0.0505828
$156$	−1.76600	−0.141393
$157$	−0.877447	−0.0700279	−0.0350139	−	0.999387i	$-0.511148\pi$
$157$	−0.877447	−0.0700279	−0.0350139	+	0.999387i	$0.511148\pi$
$158$	−7.31991	−0.582341
$159$	−5.27445	−0.418291
$160$	−3.05721	−0.241694
$161$	2.60322	0.205163
$162$	1.64346	0.129122
$163$	14.9626	1.17196	0.585979	−	0.810326i	$-0.300710\pi$
$163$	14.9626	1.17196	0.585979	+	0.810326i	$0.300710\pi$
$164$	6.49239	0.506971
$165$	1.51263	0.117758
$166$	4.73795	0.367736
$167$	−17.6151	−1.36310	−0.681549	−	0.731772i	$-0.738693\pi$
$167$	−17.6151	−1.36310	−0.681549	+	0.731772i	$0.738693\pi$
$168$	2.13492	0.164713
$169$	−6.65263	−0.511741
$170$	−6.04224	−0.463419
$171$	−7.09831	−0.542821
$172$	−3.80683	−0.290268
$173$	2.80740	0.213443	0.106721	−	0.994289i	$-0.465965\pi$
$173$	2.80740	0.213443	0.106721	+	0.994289i	$0.465965\pi$
$174$	−0.140096	−0.0106207
$175$	−4.35290	−0.329048
$176$	9.23375	0.696020
$177$	0.612801	0.0460609
$178$	13.1266	0.983884
$179$	−20.0461	−1.49832	−0.749158	−	0.662391i	$-0.769542\pi$
$179$	−20.0461	−1.49832	−0.749158	+	0.662391i	$0.769542\pi$
$180$	0.563870	0.0420284
$181$	19.7810	1.47031	0.735153	−	0.677901i	$-0.237110\pi$
$181$	19.7810	1.47031	0.735153	+	0.677901i	$0.237110\pi$
$182$	4.14053	0.306917
$183$	−12.2202	−0.903346
$184$	−5.55767	−0.409717
$185$	9.14061	0.672031
$186$	−1.28659	−0.0943376
$187$	8.59407	0.628460
$188$	7.96854	0.581166
$189$	−1.00000	−0.0727393
$190$	−9.38426	−0.680806
$191$	21.7398	1.57303	0.786517	−	0.617569i	$-0.211882\pi$
$191$	21.7398	1.57303	0.786517	+	0.617569i	$0.211882\pi$
$192$	−3.57520	−0.258018
$193$	15.3118	1.10217	0.551083	−	0.834451i	$-0.314215\pi$
$193$	15.3118	1.10217	0.551083	+	0.834451i	$0.314215\pi$
$194$	15.9121	1.14243
$195$	−2.02667	−0.145133
$196$	0.700959	0.0500685
$197$	20.3981	1.45330	0.726652	−	0.687005i	$-0.241075\pi$
$197$	20.3981	1.45330	0.726652	+	0.687005i	$0.241075\pi$
$198$	−3.09033	−0.219620
$199$	4.17553	0.295995	0.147998	−	0.988988i	$-0.452717\pi$
$199$	4.17553	0.295995	0.147998	+	0.988988i	$0.452717\pi$
$200$	9.29310	0.657121
$201$	−6.90132	−0.486782
$202$	2.97153	0.209076
$203$	0.0852448	0.00598302
$204$	3.20365	0.224301
$205$	7.45073	0.520381
$206$	−8.07766	−0.562797
$207$	2.60322	0.180936
$208$	−12.3717	−0.857823
$209$	13.3475	0.923267
$210$	−1.32204	−0.0912296
$211$	20.9863	1.44476	0.722380	−	0.691497i	$-0.243048\pi$
$211$	20.9863	1.44476	0.722380	+	0.691497i	$0.243048\pi$
$212$	3.69718	0.253923
$213$	−14.6650	−1.00483
$214$	−20.6942	−1.41462
$215$	−4.36875	−0.297946
$216$	2.13492	0.145263
$217$	0.782856	0.0531438
$218$	1.73122	0.117253
$219$	15.1800	1.02577
$220$	−1.06029	−0.0714847
$221$	−11.5146	−0.774558
$222$	−18.6744	−1.25335
$223$	11.7786	0.788752	0.394376	−	0.918949i	$-0.370961\pi$
$223$	11.7786	0.788752	0.394376	+	0.918949i	$0.370961\pi$
$224$	−3.80049	−0.253931
$225$	−4.35290	−0.290193
$226$	5.32693	0.354342
$227$	−15.4921	−1.02825	−0.514123	−	0.857716i	$-0.671883\pi$
$227$	−15.4921	−1.02825	−0.514123	+	0.857716i	$0.671883\pi$
$228$	4.97562	0.329519
$229$	4.42407	0.292351	0.146175	−	0.989259i	$-0.453304\pi$
$229$	4.42407	0.292351	0.146175	+	0.989259i	$0.453304\pi$
$230$	3.44157	0.226930
$231$	1.88038	0.123720
$232$	−0.181991	−0.0119483
$233$	−3.83593	−0.251300	−0.125650	−	0.992075i	$-0.540102\pi$
$233$	−3.83593	−0.251300	−0.125650	+	0.992075i	$0.540102\pi$
$234$	4.14053	0.270675
$235$	9.14477	0.596539
$236$	−0.429548	−0.0279612
$237$	4.45397	0.289316
$238$	−7.51125	−0.486882
$239$	24.5483	1.58790	0.793949	−	0.607984i	$-0.208022\pi$
$239$	24.5483	1.58790	0.793949	+	0.607984i	$0.208022\pi$
$240$	3.95020	0.254984
$241$	12.7990	0.824457	0.412229	−	0.911080i	$-0.364750\pi$
$241$	12.7990	0.824457	0.412229	+	0.911080i	$0.364750\pi$
$242$	−12.2671	−0.788557
$243$	−1.00000	−0.0641500
$244$	8.56588	0.548375
$245$	0.804426	0.0513929
$246$	−15.2220	−0.970518
$247$	−17.8835	−1.13790
$248$	−1.67134	−0.106130
$249$	−2.88291	−0.182697
$250$	−12.3649	−0.782026
$251$	−14.4542	−0.912339	−0.456170	−	0.889893i	$-0.650779\pi$
$251$	−14.4542	−0.912339	−0.456170	+	0.889893i	$0.650779\pi$
$252$	0.700959	0.0441563
$253$	−4.89505	−0.307749
$254$	16.0333	1.00602
$255$	3.67654	0.230234
$256$	14.9980	0.937373
$257$	−21.7395	−1.35607	−0.678035	−	0.735029i	$-0.737168\pi$
$257$	−21.7395	−1.35607	−0.678035	+	0.735029i	$0.737168\pi$
$258$	8.92544	0.555673
$259$	11.3629	0.706056
$260$	1.42061	0.0881027
$261$	0.0852448	0.00527652
$262$	5.88261	0.363429
$263$	−24.9411	−1.53794	−0.768969	−	0.639287i	$-0.779230\pi$
$263$	−24.9411	−1.53794	−0.768969	+	0.639287i	$0.779230\pi$
$264$	−4.01446	−0.247073
$265$	4.24291	0.260640
$266$	−11.6658	−0.715275
$267$	−7.98720	−0.488809
$268$	4.83755	0.295500
$269$	−10.0289	−0.611473	−0.305736	−	0.952116i	$-0.598903\pi$
$269$	−10.0289	−0.611473	−0.305736	+	0.952116i	$0.598903\pi$
$270$	−1.32204	−0.0804569
$271$	4.20862	0.255655	0.127828	−	0.991796i	$-0.459200\pi$
$271$	4.20862	0.255655	0.127828	+	0.991796i	$0.459200\pi$
$272$	22.4432	1.36082
$273$	−2.51940	−0.152481
$274$	1.12349	0.0678725
$275$	8.18510	0.493580
$276$	−1.82475	−0.109837
$277$	28.0722	1.68670	0.843348	−	0.537368i	$-0.180581\pi$
$277$	28.0722	1.68670	0.843348	+	0.537368i	$0.180581\pi$
$278$	−9.81569	−0.588706
$279$	0.782856	0.0468684
$280$	−1.71739	−0.102633
$281$	15.2510	0.909800	0.454900	−	0.890542i	$-0.349675\pi$
$281$	15.2510	0.909800	0.454900	+	0.890542i	$0.349675\pi$
$282$	−18.6829	−1.11255
$283$	28.7853	1.71111	0.855555	−	0.517712i	$-0.173216\pi$
$283$	28.7853	1.71111	0.855555	+	0.517712i	$0.173216\pi$
$284$	10.2795	0.609978
$285$	5.71006	0.338235
$286$	−7.78577	−0.460382
$287$	9.26216	0.546728
$288$	−3.80049	−0.223946
$289$	3.88844	0.228732
$290$	0.112697	0.00661781
$291$	−9.68210	−0.567575
$292$	−10.6406	−0.622693
$293$	16.9194	0.988440	0.494220	−	0.869337i	$-0.335454\pi$
$293$	16.9194	0.988440	0.494220	+	0.869337i	$0.335454\pi$
$294$	−1.64346	−0.0958485
$295$	−0.492953	−0.0287008
$296$	−24.2589	−1.41002
$297$	1.88038	0.109111
$298$	−8.74516	−0.506594
$299$	6.55855	0.379291
$300$	3.05120	0.176161
$301$	−5.43088	−0.313031
$302$	−10.8057	−0.621801
$303$	−1.80809	−0.103872
$304$	34.8568	1.99917
$305$	9.83028	0.562880
$306$	−7.51125	−0.429389
$307$	17.1650	0.979659	0.489830	−	0.871818i	$-0.337059\pi$
$307$	17.1650	0.979659	0.489830	+	0.871818i	$0.337059\pi$
$308$	−1.31807	−0.0751040
$309$	4.91503	0.279606
$310$	1.03497	0.0587823
$311$	−15.3888	−0.872620	−0.436310	−	0.899796i	$-0.643715\pi$
$311$	−15.3888	−0.872620	−0.436310	+	0.899796i	$0.643715\pi$
$312$	5.37872	0.304510
$313$	15.5690	0.880013	0.440007	−	0.897994i	$-0.354976\pi$
$313$	15.5690	0.880013	0.440007	+	0.897994i	$0.354976\pi$
$314$	−1.44205	−0.0813795
$315$	0.804426	0.0453243
$316$	−3.12205	−0.175629
$317$	−11.2296	−0.630715	−0.315357	−	0.948973i	$-0.602124\pi$
$317$	−11.2296	−0.630715	−0.315357	+	0.948973i	$0.602124\pi$
$318$	−8.66835	−0.486097
$319$	−0.160293	−0.00897467
$320$	2.87599	0.160773
$321$	12.5918	0.702807
$322$	4.27829	0.238420
$323$	32.4420	1.80512
$324$	0.700959	0.0389422
$325$	−10.9667	−0.608322
$326$	24.5904	1.36193
$327$	−1.05340	−0.0582533
$328$	−19.7740	−1.09184
$329$	11.3681	0.626741
$330$	2.48594	0.136847
$331$	−7.22811	−0.397293	−0.198647	−	0.980071i	$-0.563655\pi$
$331$	−7.22811	−0.397293	−0.198647	+	0.980071i	$0.563655\pi$
$332$	2.02080	0.110906
$333$	11.3629	0.622683
$334$	−28.9497	−1.58406
$335$	5.55161	0.303317
$336$	4.91057	0.267894
$337$	1.57551	0.0858238	0.0429119	−	0.999079i	$-0.486337\pi$
$337$	1.57551	0.0858238	0.0429119	+	0.999079i	$0.486337\pi$
$338$	−10.9333	−0.594695
$339$	−3.24129	−0.176043
$340$	−2.57710	−0.139763
$341$	−1.47207	−0.0797169
$342$	−11.6658	−0.630813
$343$	1.00000	0.0539949
$344$	11.5945	0.625134
$345$	−2.09410	−0.112743
$346$	4.61385	0.248042
$347$	0.718827	0.0385886	0.0192943	−	0.999814i	$-0.493858\pi$
$347$	0.718827	0.0385886	0.0192943	+	0.999814i	$0.493858\pi$
$348$	−0.0597531	−0.00320310
$349$	−22.8260	−1.22185	−0.610925	−	0.791689i	$-0.709202\pi$
$349$	−22.8260	−1.22185	−0.610925	+	0.791689i	$0.709202\pi$
$350$	−7.15381	−0.382387
$351$	−2.51940	−0.134476
$352$	7.14636	0.380902
$353$	−10.7678	−0.573113	−0.286557	−	0.958063i	$-0.592511\pi$
$353$	−10.7678	−0.573113	−0.286557	+	0.958063i	$0.592511\pi$
$354$	1.00711	0.0535275
$355$	11.7969	0.626114
$356$	5.59870	0.296731
$357$	4.57039	0.241891
$358$	−32.9450	−1.74120
$359$	−36.5181	−1.92735	−0.963675	−	0.267078i	$-0.913942\pi$
$359$	−36.5181	−1.92735	−0.963675	+	0.267078i	$0.913942\pi$
$360$	−1.71739	−0.0905142
$361$	31.3860	1.65189
$362$	32.5092	1.70865
$363$	7.46417	0.391767
$364$	1.76600	0.0925633
$365$	−12.2112	−0.639165
$366$	−20.0835	−1.04978
$367$	−6.88285	−0.359282	−0.179641	−	0.983732i	$-0.557494\pi$
$367$	−6.88285	−0.359282	−0.179641	+	0.983732i	$0.557494\pi$
$368$	−12.7833	−0.666376
$369$	9.26216	0.482169
$370$	15.0222	0.780968
$371$	5.27445	0.273836
$372$	−0.548750	−0.0284514
$373$	33.1591	1.71691	0.858457	−	0.512886i	$-0.171424\pi$
$373$	33.1591	1.71691	0.858457	+	0.512886i	$0.171424\pi$
$374$	14.1240	0.730335
$375$	7.52372	0.388523
$376$	−24.2699	−1.25163
$377$	0.214766	0.0110610
$378$	−1.64346	−0.0845304
$379$	−13.0584	−0.670763	−0.335381	−	0.942082i	$-0.608865\pi$
$379$	−13.0584	−0.670763	−0.335381	+	0.942082i	$0.608865\pi$
$380$	−4.00252	−0.205325
$381$	−9.75584	−0.499807
$382$	35.7284	1.82802
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−13.4767	−0.687729
$385$	−1.51263	−0.0770906
$386$	25.1643	1.28083
$387$	−5.43088	−0.276067
$388$	6.78676	0.344545
$389$	6.76185	0.342839	0.171420	−	0.985198i	$-0.445165\pi$
$389$	6.76185	0.342839	0.171420	+	0.985198i	$0.445165\pi$
$390$	−3.33075	−0.168659
$391$	−11.8977	−0.601694
$392$	−2.13492	−0.107830
$393$	−3.57941	−0.180557
$394$	33.5235	1.68889
$395$	−3.58289	−0.180275
$396$	−1.31807	−0.0662355
$397$	0.668432	0.0335476	0.0167738	−	0.999859i	$-0.494660\pi$
$397$	0.668432	0.0335476	0.0167738	+	0.999859i	$0.494660\pi$
$398$	6.86231	0.343977
$399$	7.09831	0.355360
$400$	21.3752	1.06876
$401$	34.5415	1.72492	0.862461	−	0.506123i	$-0.168922\pi$
$401$	34.5415	1.72492	0.862461	+	0.506123i	$0.168922\pi$
$402$	−11.3420	−0.565690
$403$	1.97233	0.0982486
$404$	1.26740	0.0630554
$405$	0.804426	0.0399723
$406$	0.140096	0.00695287
$407$	−21.3665	−1.05910
$408$	−9.75742	−0.483064
$409$	−21.1660	−1.04659	−0.523297	−	0.852151i	$-0.675298\pi$
$409$	−21.1660	−1.04659	−0.523297	+	0.852151i	$0.675298\pi$
$410$	12.2450	0.604736
$411$	−0.683612	−0.0337201
$412$	−3.44524	−0.169735
$413$	−0.612801	−0.0301540
$414$	4.27829	0.210266
$415$	2.31909	0.113840
$416$	−9.57494	−0.469450
$417$	5.97258	0.292478
$418$	21.9361	1.07293
$419$	3.02459	0.147761	0.0738803	−	0.997267i	$-0.476462\pi$
$419$	3.02459	0.147761	0.0738803	+	0.997267i	$0.476462\pi$
$420$	−0.563870	−0.0275140
$421$	−6.45193	−0.314448	−0.157224	−	0.987563i	$-0.550254\pi$
$421$	−6.45193	−0.314448	−0.157224	+	0.987563i	$0.550254\pi$
$422$	34.4902	1.67896
$423$	11.3681	0.552734
$424$	−11.2605	−0.546860
$425$	19.8944	0.965022
$426$	−24.1013	−1.16771
$427$	12.2202	0.591379
$428$	−8.82635	−0.426638
$429$	4.73743	0.228725
$430$	−7.17986	−0.346243
$431$	−1.59878	−0.0770105	−0.0385053	−	0.999258i	$-0.512260\pi$
$431$	−1.59878	−0.0770105	−0.0385053	+	0.999258i	$0.512260\pi$
$432$	4.91057	0.236260
$433$	−8.87437	−0.426475	−0.213237	−	0.977000i	$-0.568401\pi$
$433$	−8.87437	−0.426475	−0.213237	+	0.977000i	$0.568401\pi$
$434$	1.28659	0.0617584
$435$	−0.0685732	−0.00328783
$436$	0.738392	0.0353626
$437$	−18.4785	−0.883945
$438$	24.9478	1.19205
$439$	31.2387	1.49094	0.745472	−	0.666537i	$-0.232224\pi$
$439$	31.2387	1.49094	0.745472	+	0.666537i	$0.232224\pi$
$440$	3.22934	0.153953
$441$	1.00000	0.0476190
$442$	−18.9238	−0.900115
$443$	23.7132	1.12665	0.563323	−	0.826237i	$-0.309523\pi$
$443$	23.7132	1.12665	0.563323	+	0.826237i	$0.309523\pi$
$444$	−7.96492	−0.377998
$445$	6.42512	0.304580
$446$	19.3576	0.916610
$447$	5.32119	0.251684
$448$	3.57520	0.168912
$449$	33.3787	1.57524	0.787618	−	0.616164i	$-0.211314\pi$
$449$	33.3787	1.57524	0.787618	+	0.616164i	$0.211314\pi$
$450$	−7.15381	−0.337234
$451$	−17.4164	−0.820105
$452$	2.27201	0.106867
$453$	6.57500	0.308920
$454$	−25.4606	−1.19493
$455$	2.02667	0.0950118
$456$	−15.1543	−0.709666
$457$	−6.21287	−0.290626	−0.145313	−	0.989386i	$-0.546419\pi$
$457$	−6.21287	−0.290626	−0.145313	+	0.989386i	$0.546419\pi$
$458$	7.27078	0.339741
$459$	4.57039	0.213327
$460$	1.46788	0.0684402
$461$	26.2587	1.22299	0.611495	−	0.791249i	$-0.290569\pi$
$461$	26.2587	1.22299	0.611495	+	0.791249i	$0.290569\pi$
$462$	3.09033	0.143775
$463$	31.0479	1.44292	0.721458	−	0.692458i	$-0.243472\pi$
$463$	31.0479	1.44292	0.721458	+	0.692458i	$0.243472\pi$
$464$	−0.418601	−0.0194331
$465$	−0.629750	−0.0292040
$466$	−6.30419	−0.292036
$467$	−41.4889	−1.91988	−0.959940	−	0.280207i	$-0.909597\pi$
$467$	−41.4889	−1.91988	−0.959940	+	0.280207i	$0.909597\pi$
$468$	1.76600	0.0816332
$469$	6.90132	0.318674
$470$	15.0291	0.693238
$471$	0.877447	0.0404306
$472$	1.30828	0.0602185
$473$	10.2121	0.469554
$474$	7.31991	0.336215
$475$	30.8982	1.41771
$476$	−3.20365	−0.146839
$477$	5.27445	0.241501
$478$	40.3442	1.84530
$479$	8.39831	0.383728	0.191864	−	0.981421i	$-0.438547\pi$
$479$	8.39831	0.383728	0.191864	+	0.981421i	$0.438547\pi$
$480$	3.05721	0.139542
$481$	28.6277	1.30531
$482$	21.0347	0.958103
$483$	−2.60322	−0.118451
$484$	−5.23208	−0.237822
$485$	7.78854	0.353659
$486$	−1.64346	−0.0745488
$487$	−25.0974	−1.13727	−0.568636	−	0.822589i	$-0.692529\pi$
$487$	−25.0974	−1.13727	−0.568636	+	0.822589i	$0.692529\pi$
$488$	−26.0892	−1.18100
$489$	−14.9626	−0.676630
$490$	1.32204	0.0597238
$491$	−12.8522	−0.580011	−0.290005	−	0.957025i	$-0.593657\pi$
$491$	−12.8522	−0.580011	−0.290005	+	0.957025i	$0.593657\pi$
$492$	−6.49239	−0.292700
$493$	−0.389602	−0.0175468
$494$	−29.3908	−1.32235
$495$	−1.51263	−0.0679875
$496$	−3.84427	−0.172613
$497$	14.6650	0.657814
$498$	−4.73795	−0.212313
$499$	−11.8470	−0.530345	−0.265172	−	0.964201i	$-0.585429\pi$
$499$	−11.8470	−0.530345	−0.265172	+	0.964201i	$0.585429\pi$
$500$	−5.27382	−0.235852
$501$	17.6151	0.786985
$502$	−23.7548	−1.06023
$503$	22.5111	1.00372	0.501861	−	0.864949i	$-0.332649\pi$
$503$	22.5111	1.00372	0.501861	+	0.864949i	$0.332649\pi$
$504$	−2.13492	−0.0950970
$505$	1.45448	0.0647234
$506$	−8.04481	−0.357635
$507$	6.65263	0.295454
$508$	6.83844	0.303407
$509$	−21.1179	−0.936034	−0.468017	−	0.883719i	$-0.655031\pi$
$509$	−21.1179	−0.936034	−0.468017	+	0.883719i	$0.655031\pi$
$510$	6.04224	0.267555
$511$	−15.1800	−0.671526
$512$	−2.30480	−0.101859
$513$	7.09831	0.313398
$514$	−35.7279	−1.57589
$515$	−3.95378	−0.174224
$516$	3.80683	0.167586
$517$	−21.3763	−0.940127
$518$	18.6744	0.820508
$519$	−2.80740	−0.123231
$520$	−4.32678	−0.189742
$521$	−23.5530	−1.03187	−0.515937	−	0.856626i	$-0.672556\pi$
$521$	−23.5530	−1.03187	−0.515937	+	0.856626i	$0.672556\pi$
$522$	0.140096	0.00613186
$523$	−0.718468	−0.0314164	−0.0157082	−	0.999877i	$-0.505000\pi$
$523$	−0.718468	−0.0314164	−0.0157082	+	0.999877i	$0.505000\pi$
$524$	2.50902	0.109607
$525$	4.35290	0.189976
$526$	−40.9898	−1.78724
$527$	−3.57796	−0.155858
$528$	−9.23375	−0.401847
$529$	−16.2232	−0.705358
$530$	6.97305	0.302890
$531$	−0.612801	−0.0265933
$532$	−4.97562	−0.215721
$533$	23.3351	1.01075
$534$	−13.1266	−0.568045
$535$	−10.1292	−0.437923
$536$	−14.7338	−0.636403
$537$	20.0461	0.865054
$538$	−16.4821	−0.710593
$539$	−1.88038	−0.0809937
$540$	−0.563870	−0.0242651
$541$	−11.2764	−0.484808	−0.242404	−	0.970175i	$-0.577936\pi$
$541$	−11.2764	−0.484808	−0.242404	+	0.970175i	$0.577936\pi$
$542$	6.91670	0.297098
$543$	−19.7810	−0.848882
$544$	17.3697	0.744720
$545$	0.847385	0.0362980
$546$	−4.14053	−0.177198
$547$	−6.54833	−0.279986	−0.139993	−	0.990152i	$-0.544708\pi$
$547$	−6.54833	−0.279986	−0.139993	+	0.990152i	$0.544708\pi$
$548$	0.479184	0.0204697
$549$	12.2202	0.521547
$550$	13.4519	0.573590
$551$	−0.605094	−0.0257779
$552$	5.55767	0.236550
$553$	−4.45397	−0.189402
$554$	46.1356	1.96011
$555$	−9.14061	−0.387997
$556$	−4.18653	−0.177548
$557$	34.8509	1.47668	0.738340	−	0.674429i	$-0.235610\pi$
$557$	34.8509	1.47668	0.738340	+	0.674429i	$0.235610\pi$
$558$	1.28659	0.0544658
$559$	−13.6826	−0.578711
$560$	−3.95020	−0.166926
$561$	−8.59407	−0.362842
$562$	25.0645	1.05728
$563$	−6.27482	−0.264452	−0.132226	−	0.991220i	$-0.542212\pi$
$563$	−6.27482	−0.264452	−0.132226	+	0.991220i	$0.542212\pi$
$564$	−7.96854	−0.335536
$565$	2.60738	0.109693
$566$	47.3075	1.98848
$567$	1.00000	0.0419961
$568$	−31.3085	−1.31368
$569$	34.9058	1.46333	0.731664	−	0.681665i	$-0.238744\pi$
$569$	34.9058	1.46333	0.731664	+	0.681665i	$0.238744\pi$
$570$	9.38426	0.393063
$571$	40.9798	1.71495	0.857476	−	0.514524i	$-0.172031\pi$
$571$	40.9798	1.71495	0.857476	+	0.514524i	$0.172031\pi$
$572$	−3.32074	−0.138847
$573$	−21.7398	−0.908191
$574$	15.2220	0.635353
$575$	−11.3316	−0.472559
$576$	3.57520	0.148967
$577$	−35.7007	−1.48624	−0.743119	−	0.669159i	$-0.766654\pi$
$577$	−35.7007	−1.48624	−0.743119	+	0.669159i	$0.766654\pi$
$578$	6.39050	0.265810
$579$	−15.3118	−0.636335
$580$	0.0480670	0.00199587
$581$	2.88291	0.119603
$582$	−15.9121	−0.659580
$583$	−9.91798	−0.410761
$584$	32.4082	1.34106
$585$	2.02667	0.0837925
$586$	27.8063	1.14867
$587$	−22.5524	−0.930836	−0.465418	−	0.885091i	$-0.654096\pi$
$587$	−22.5524	−0.930836	−0.465418	+	0.885091i	$0.654096\pi$
$588$	−0.700959	−0.0289071
$589$	−5.55696	−0.228970
$590$	−0.810149	−0.0333533
$591$	−20.3981	−0.839066
$592$	−55.7983	−2.29330
$593$	−4.50812	−0.185126	−0.0925632	−	0.995707i	$-0.529506\pi$
$593$	−4.50812	−0.185126	−0.0925632	+	0.995707i	$0.529506\pi$
$594$	3.09033	0.126798
$595$	−3.67654	−0.150723
$596$	−3.72994	−0.152784
$597$	−4.17553	−0.170893
$598$	10.7787	0.440775
$599$	7.25825	0.296564	0.148282	−	0.988945i	$-0.452626\pi$
$599$	7.25825	0.296564	0.148282	+	0.988945i	$0.452626\pi$
$600$	−9.29310	−0.379389
$601$	−48.6338	−1.98381	−0.991907	−	0.126963i	$-0.959477\pi$
$601$	−48.6338	−1.98381	−0.991907	+	0.126963i	$0.959477\pi$
$602$	−8.92544	−0.363774
$603$	6.90132	0.281044
$604$	−4.60880	−0.187530
$605$	−6.00438	−0.244113
$606$	−2.97153	−0.120710
$607$	−40.8738	−1.65902	−0.829508	−	0.558495i	$-0.811379\pi$
$607$	−40.8738	−1.65902	−0.829508	+	0.558495i	$0.811379\pi$
$608$	26.9770	1.09406
$609$	−0.0852448	−0.00345430
$610$	16.1557	0.654124
$611$	28.6407	1.15868
$612$	−3.20365	−0.129500
$613$	−42.4161	−1.71317	−0.856584	−	0.516007i	$-0.827418\pi$
$613$	−42.4161	−1.71317	−0.856584	+	0.516007i	$0.827418\pi$
$614$	28.2100	1.13846
$615$	−7.45073	−0.300442
$616$	4.01446	0.161747
$617$	−0.244192	−0.00983079	−0.00491540	−	0.999988i	$-0.501565\pi$
$617$	−0.244192	−0.00983079	−0.00491540	+	0.999988i	$0.501565\pi$
$618$	8.07766	0.324931
$619$	46.2721	1.85983	0.929916	−	0.367773i	$-0.119880\pi$
$619$	46.2721	1.85983	0.929916	+	0.367773i	$0.119880\pi$
$620$	0.441429	0.0177282
$621$	−2.60322	−0.104464
$622$	−25.2909	−1.01407
$623$	7.98720	0.320001
$624$	12.3717	0.495264
$625$	15.7122	0.628489
$626$	25.5871	1.02266
$627$	−13.3475	−0.533048
$628$	−0.615054	−0.0245433
$629$	−51.9328	−2.07070
$630$	1.32204	0.0526714
$631$	−5.28833	−0.210525	−0.105263	−	0.994444i	$-0.533568\pi$
$631$	−5.28833	−0.210525	−0.105263	+	0.994444i	$0.533568\pi$
$632$	9.50887	0.378242
$633$	−20.9863	−0.834132
$634$	−18.4553	−0.732954
$635$	7.84785	0.311432
$636$	−3.69718	−0.146603
$637$	2.51940	0.0998222
$638$	−0.263435	−0.0104295
$639$	14.6650	0.580137
$640$	10.8410	0.428528
$641$	−3.01742	−0.119181	−0.0595904	−	0.998223i	$-0.518979\pi$
$641$	−3.01742	−0.119181	−0.0595904	+	0.998223i	$0.518979\pi$
$642$	20.6942	0.816733
$643$	−14.5877	−0.575282	−0.287641	−	0.957738i	$-0.592871\pi$
$643$	−14.5877	−0.575282	−0.287641	+	0.957738i	$0.592871\pi$
$644$	1.82475	0.0719053
$645$	4.36875	0.172019
$646$	53.3171	2.09773
$647$	−42.2355	−1.66045	−0.830224	−	0.557429i	$-0.811788\pi$
$647$	−42.2355	−1.66045	−0.830224	+	0.557429i	$0.811788\pi$
$648$	−2.13492	−0.0838676
$649$	1.15230	0.0452317
$650$	−18.0233	−0.706932
$651$	−0.782856	−0.0306826
$652$	10.4881	0.410747
$653$	9.32890	0.365068	0.182534	−	0.983200i	$-0.441570\pi$
$653$	9.32890	0.365068	0.182534	+	0.983200i	$0.441570\pi$
$654$	−1.73122	−0.0676962
$655$	2.87937	0.112506
$656$	−45.4825	−1.77579
$657$	−15.1800	−0.592230
$658$	18.6829	0.728337
$659$	1.75841	0.0684978	0.0342489	−	0.999413i	$-0.489096\pi$
$659$	1.75841	0.0684978	0.0342489	+	0.999413i	$0.489096\pi$
$660$	1.06029	0.0412717
$661$	14.7294	0.572908	0.286454	−	0.958094i	$-0.407523\pi$
$661$	14.7294	0.572908	0.286454	+	0.958094i	$0.407523\pi$
$662$	−11.8791	−0.461695
$663$	11.5146	0.447191
$664$	−6.15479	−0.238852
$665$	−5.71006	−0.221427
$666$	18.6744	0.723620
$667$	0.221911	0.00859244
$668$	−12.3475	−0.477738
$669$	−11.7786	−0.455386
$670$	9.12384	0.352485
$671$	−22.9787	−0.887082
$672$	3.80049	0.146607
$673$	24.6391	0.949767	0.474883	−	0.880049i	$-0.342490\pi$
$673$	24.6391	0.949767	0.474883	+	0.880049i	$0.342490\pi$
$674$	2.58929	0.0997359
$675$	4.35290	0.167543
$676$	−4.66322	−0.179355
$677$	−25.2288	−0.969621	−0.484811	−	0.874619i	$-0.661111\pi$
$677$	−25.2288	−0.969621	−0.484811	+	0.874619i	$0.661111\pi$
$678$	−5.32693	−0.204580
$679$	9.68210	0.371565
$680$	7.84912	0.301000
$681$	15.4921	0.593658
$682$	−2.41928	−0.0926391
$683$	−17.6829	−0.676618	−0.338309	−	0.941035i	$-0.609855\pi$
$683$	−17.6829	−0.676618	−0.338309	+	0.941035i	$0.609855\pi$
$684$	−4.97562	−0.190248
$685$	0.549916	0.0210112
$686$	1.64346	0.0627476
$687$	−4.42407	−0.168789
$688$	26.6688	1.01674
$689$	13.2885	0.506250
$690$	−3.44157	−0.131018
$691$	21.7712	0.828216	0.414108	−	0.910228i	$-0.364094\pi$
$691$	21.7712	0.828216	0.414108	+	0.910228i	$0.364094\pi$
$692$	1.96787	0.0748072
$693$	−1.88038	−0.0714297
$694$	1.18136	0.0448439
$695$	−4.80450	−0.182245
$696$	0.181991	0.00689835
$697$	−42.3317	−1.60343
$698$	−37.5137	−1.41991
$699$	3.83593	0.145088
$700$	−3.05120	−0.115325
$701$	−37.1024	−1.40134	−0.700669	−	0.713486i	$-0.747115\pi$
$701$	−37.1024	−1.40134	−0.700669	+	0.713486i	$0.747115\pi$
$702$	−4.14053	−0.156274
$703$	−80.6573	−3.04205
$704$	−6.72274	−0.253373
$705$	−9.14477	−0.344412
$706$	−17.6965	−0.666015
$707$	1.80809	0.0680003
$708$	0.429548	0.0161434
$709$	13.0727	0.490957	0.245479	−	0.969402i	$-0.421055\pi$
$709$	13.0727	0.490957	0.245479	+	0.969402i	$0.421055\pi$
$710$	19.3877	0.727607
$711$	−4.45397	−0.167037
$712$	−17.0520	−0.639053
$713$	2.03795	0.0763218
$714$	7.51125	0.281101
$715$	−3.81091	−0.142520
$716$	−14.0515	−0.525129
$717$	−24.5483	−0.916774
$718$	−60.0160	−2.23978
$719$	7.12759	0.265814	0.132907	−	0.991128i	$-0.457569\pi$
$719$	7.12759	0.265814	0.132907	+	0.991128i	$0.457569\pi$
$720$	−3.95020	−0.147215
$721$	−4.91503	−0.183045
$722$	51.5816	1.91967
$723$	−12.7990	−0.476001
$724$	13.8656	0.515312
$725$	−0.371062	−0.0137809
$726$	12.2671	0.455273
$727$	7.59623	0.281729	0.140864	−	0.990029i	$-0.455012\pi$
$727$	7.59623	0.281729	0.140864	+	0.990029i	$0.455012\pi$
$728$	−5.37872	−0.199349
$729$	1.00000	0.0370370
$730$	−20.0687	−0.742774
$731$	24.8212	0.918047
$732$	−8.56588	−0.316604
$733$	0.0653045	0.00241208	0.00120604	−	0.999999i	$-0.499616\pi$
$733$	0.0653045	0.00241208	0.00120604	+	0.999999i	$0.499616\pi$
$734$	−11.3117	−0.417522
$735$	−0.804426	−0.0296717
$736$	−9.89351	−0.364680
$737$	−12.9771	−0.478018
$738$	15.2220	0.560329
$739$	45.6415	1.67895	0.839475	−	0.543398i	$-0.182862\pi$
$739$	45.6415	1.67895	0.839475	+	0.543398i	$0.182862\pi$
$740$	6.40719	0.235533
$741$	17.8835	0.656966
$742$	8.66835	0.318225
$743$	−47.3422	−1.73682	−0.868409	−	0.495849i	$-0.834857\pi$
$743$	−47.3422	−1.73682	−0.868409	+	0.495849i	$0.834857\pi$
$744$	1.67134	0.0612742
$745$	−4.28051	−0.156826
$746$	54.4956	1.99523
$747$	2.88291	0.105480
$748$	6.02409	0.220262
$749$	−12.5918	−0.460095
$750$	12.3649	0.451503
$751$	31.9429	1.16561	0.582807	−	0.812611i	$-0.301954\pi$
$751$	31.9429	1.16561	0.582807	+	0.812611i	$0.301954\pi$
$752$	−55.8237	−2.03568
$753$	14.4542	0.526739
$754$	0.352959	0.0128540
$755$	−5.28910	−0.192490
$756$	−0.700959	−0.0254936
$757$	−34.7696	−1.26372	−0.631862	−	0.775081i	$-0.717709\pi$
$757$	−34.7696	−1.26372	−0.631862	+	0.775081i	$0.717709\pi$
$758$	−21.4609	−0.779494
$759$	4.89505	0.177679
$760$	12.1905	0.442197
$761$	23.6148	0.856034	0.428017	−	0.903771i	$-0.359212\pi$
$761$	23.6148	0.856034	0.428017	+	0.903771i	$0.359212\pi$
$762$	−16.0333	−0.580826
$763$	1.05340	0.0381357
$764$	15.2387	0.551316
$765$	−3.67654	−0.132926
$766$	−1.64346	−0.0593806
$767$	−1.54389	−0.0557466
$768$	−14.9980	−0.541192
$769$	−15.1014	−0.544572	−0.272286	−	0.962216i	$-0.587780\pi$
$769$	−15.1014	−0.544572	−0.272286	+	0.962216i	$0.587780\pi$
$770$	−2.48594	−0.0895871
$771$	21.7395	0.782928
$772$	10.7329	0.386286
$773$	−46.2247	−1.66259	−0.831294	−	0.555833i	$-0.812399\pi$
$773$	−46.2247	−1.66259	−0.831294	+	0.555833i	$0.812399\pi$
$774$	−8.92544	−0.320818
$775$	−3.40769	−0.122408
$776$	−20.6705	−0.742029
$777$	−11.3629	−0.407641
$778$	11.1128	0.398414
$779$	−65.7457	−2.35558
$780$	−1.42061	−0.0508661
$781$	−27.5757	−0.986736
$782$	−19.5534	−0.699230
$783$	−0.0852448	−0.00304640
$784$	−4.91057	−0.175378
$785$	−0.705841	−0.0251926
$786$	−5.88261	−0.209826
$787$	4.75697	0.169568	0.0847839	−	0.996399i	$-0.472980\pi$
$787$	4.75697	0.169568	0.0847839	+	0.996399i	$0.472980\pi$
$788$	14.2982	0.509353
$789$	24.9411	0.887928
$790$	−5.88833	−0.209497
$791$	3.24129	0.115247
$792$	4.01446	0.142648
$793$	30.7877	1.09330
$794$	1.09854	0.0389857
$795$	−4.24291	−0.150480
$796$	2.92687	0.103740
$797$	33.6598	1.19229	0.596145	−	0.802877i	$-0.296698\pi$
$797$	33.6598	1.19229	0.596145	+	0.802877i	$0.296698\pi$
$798$	11.6658	0.412964
$799$	−51.9564	−1.83809
$800$	16.5431	0.584888
$801$	7.98720	0.282214
$802$	56.7676	2.00453
$803$	28.5443	1.00730
$804$	−4.83755	−0.170607
$805$	2.09410	0.0738073
$806$	3.24144	0.114175
$807$	10.0289	0.353034
$808$	−3.86013	−0.135799
$809$	−16.6057	−0.583827	−0.291913	−	0.956445i	$-0.594292\pi$
$809$	−16.6057	−0.583827	−0.291913	+	0.956445i	$0.594292\pi$
$810$	1.32204	0.0464518
$811$	−42.6812	−1.49874	−0.749370	−	0.662151i	$-0.769644\pi$
$811$	−42.6812	−1.49874	−0.749370	+	0.662151i	$0.769644\pi$
$812$	0.0597531	0.00209692
$813$	−4.20862	−0.147603
$814$	−35.1151	−1.23078
$815$	12.0363	0.421612
$816$	−22.4432	−0.785670
$817$	38.5501	1.34870
$818$	−34.7855	−1.21625
$819$	2.51940	0.0880349
$820$	5.22265	0.182383
$821$	47.1267	1.64473	0.822366	−	0.568959i	$-0.192654\pi$
$821$	47.1267	1.64473	0.822366	+	0.568959i	$0.192654\pi$
$822$	−1.12349	−0.0391862
$823$	34.5884	1.20568	0.602838	−	0.797864i	$-0.294037\pi$
$823$	34.5884	1.20568	0.602838	+	0.797864i	$0.294037\pi$
$824$	10.4932	0.365548
$825$	−8.18510	−0.284969
$826$	−1.00711	−0.0350420
$827$	−41.6268	−1.44750	−0.723752	−	0.690060i	$-0.757584\pi$
$827$	−41.6268	−1.44750	−0.723752	+	0.690060i	$0.757584\pi$
$828$	1.82475	0.0634145
$829$	4.83682	0.167990	0.0839948	−	0.996466i	$-0.473232\pi$
$829$	4.83682	0.167990	0.0839948	+	0.996466i	$0.473232\pi$
$830$	3.81133	0.132293
$831$	−28.0722	−0.973815
$832$	9.00736	0.312274
$833$	−4.57039	−0.158355
$834$	9.81569	0.339889
$835$	−14.1701	−0.490375
$836$	9.35606	0.323586
$837$	−0.782856	−0.0270595
$838$	4.97078	0.171713
$839$	−6.97306	−0.240737	−0.120368	−	0.992729i	$-0.538408\pi$
$839$	−6.97306	−0.240737	−0.120368	+	0.992729i	$0.538408\pi$
$840$	1.71739	0.0592555
$841$	−28.9927	−0.999749
$842$	−10.6035	−0.365420
$843$	−15.2510	−0.525274
$844$	14.7106	0.506358
$845$	−5.35155	−0.184099
$846$	18.6829	0.642333
$847$	−7.46417	−0.256472
$848$	−25.9006	−0.889430
$849$	−28.7853	−0.987910
$850$	32.6957	1.12145
$851$	29.5801	1.01399
$852$	−10.2795	−0.352171
$853$	1.30603	0.0447178	0.0223589	−	0.999750i	$-0.492882\pi$
$853$	1.30603	0.0447178	0.0223589	+	0.999750i	$0.492882\pi$
$854$	20.0835	0.687242
$855$	−5.71006	−0.195280
$856$	26.8826	0.918827
$857$	53.3776	1.82334	0.911672	−	0.410919i	$-0.134792\pi$
$857$	53.3776	1.82334	0.911672	+	0.410919i	$0.134792\pi$
$858$	7.78577	0.265802
$859$	−11.7351	−0.400398	−0.200199	−	0.979755i	$-0.564159\pi$
$859$	−11.7351	−0.400398	−0.200199	+	0.979755i	$0.564159\pi$
$860$	−3.06231	−0.104424
$861$	−9.26216	−0.315654
$862$	−2.62753	−0.0894940
$863$	18.3469	0.624537	0.312268	−	0.949994i	$-0.398911\pi$
$863$	18.3469	0.624537	0.312268	+	0.949994i	$0.398911\pi$
$864$	3.80049	0.129295
$865$	2.25835	0.0767860
$866$	−14.5847	−0.495607
$867$	−3.88844	−0.132059
$868$	0.548750	0.0186258
$869$	8.37515	0.284107
$870$	−0.112697	−0.00382080
$871$	17.3872	0.589142
$872$	−2.24893	−0.0761584
$873$	9.68210	0.327690
$874$	−30.3686	−1.02723
$875$	−7.52372	−0.254348
$876$	10.6406	0.359512
$877$	38.9597	1.31557	0.657787	−	0.753204i	$-0.271493\pi$
$877$	38.9597	1.31557	0.657787	+	0.753204i	$0.271493\pi$
$878$	51.3396	1.73263
$879$	−16.9194	−0.570676
$880$	7.42787	0.250393
$881$	43.4496	1.46385	0.731926	−	0.681384i	$-0.238622\pi$
$881$	43.4496	1.46385	0.731926	+	0.681384i	$0.238622\pi$
$882$	1.64346	0.0553382
$883$	43.8290	1.47496	0.737482	−	0.675367i	$-0.236015\pi$
$883$	43.8290	1.47496	0.737482	+	0.675367i	$0.236015\pi$
$884$	−8.07128	−0.271467
$885$	0.492953	0.0165704
$886$	38.9716	1.30928
$887$	−55.3011	−1.85683	−0.928415	−	0.371544i	$-0.878829\pi$
$887$	−55.3011	−1.85683	−0.928415	+	0.371544i	$0.878829\pi$
$888$	24.2589	0.814075
$889$	9.75584	0.327200
$890$	10.5594	0.353952
$891$	−1.88038	−0.0629951
$892$	8.25630	0.276441
$893$	−80.6940	−2.70032
$894$	8.74516	0.292482
$895$	−16.1256	−0.539020
$896$	13.4767	0.450224
$897$	−6.55855	−0.218984
$898$	54.8565	1.83058
$899$	0.0667345	0.00222572
$900$	−3.05120	−0.101707
$901$	−24.1063	−0.803097
$902$	−28.6231	−0.953046
$903$	5.43088	0.180728
$904$	−6.91991	−0.230153
$905$	15.9123	0.528943
$906$	10.8057	0.358997
$907$	36.5690	1.21425	0.607127	−	0.794605i	$-0.292322\pi$
$907$	36.5690	1.21425	0.607127	+	0.794605i	$0.292322\pi$
$908$	−10.8593	−0.360379
$909$	1.80809	0.0599706
$910$	3.33075	0.110413
$911$	26.5426	0.879396	0.439698	−	0.898146i	$-0.355085\pi$
$911$	26.5426	0.879396	0.439698	+	0.898146i	$0.355085\pi$
$912$	−34.8568	−1.15422
$913$	−5.42097	−0.179408
$914$	−10.2106	−0.337736
$915$	−9.83028	−0.324979
$916$	3.10109	0.102463
$917$	3.57941	0.118202
$918$	7.51125	0.247908
$919$	−22.8585	−0.754033	−0.377017	−	0.926206i	$-0.623050\pi$
$919$	−22.8585	−0.754033	−0.377017	+	0.926206i	$0.623050\pi$
$920$	−4.47074	−0.147396
$921$	−17.1650	−0.565607
$922$	43.1551	1.42124
$923$	36.9469	1.21612
$924$	1.31807	0.0433613
$925$	−49.4615	−1.62628
$926$	51.0259	1.67681
$927$	−4.91503	−0.161431
$928$	−0.323972	−0.0106349
$929$	5.72440	0.187812	0.0939058	−	0.995581i	$-0.470065\pi$
$929$	5.72440	0.187812	0.0939058	+	0.995581i	$0.470065\pi$
$930$	−1.03497	−0.0339380
$931$	−7.09831	−0.232638
$932$	−2.68883	−0.0880755
$933$	15.3888	0.503807
$934$	−68.1854	−2.23109
$935$	6.91329	0.226089
$936$	−5.37872	−0.175809
$937$	14.4075	0.470674	0.235337	−	0.971914i	$-0.424381\pi$
$937$	14.4075	0.470674	0.235337	+	0.971914i	$0.424381\pi$
$938$	11.3420	0.370331
$939$	−15.5690	−0.508076
$940$	6.41011	0.209075
$941$	20.5494	0.669892	0.334946	−	0.942237i	$-0.391282\pi$
$941$	20.5494	0.669892	0.334946	+	0.942237i	$0.391282\pi$
$942$	1.44205	0.0469845
$943$	24.1115	0.785177
$944$	3.00920	0.0979413
$945$	−0.804426	−0.0261680
$946$	16.7832	0.545669
$947$	39.9871	1.29941	0.649703	−	0.760188i	$-0.274893\pi$
$947$	39.9871	1.29941	0.649703	+	0.760188i	$0.274893\pi$
$948$	3.12205	0.101399
$949$	−38.2446	−1.24147
$950$	50.7800	1.64752
$951$	11.2296	0.364143
$952$	9.75742	0.316240
$953$	31.7866	1.02967	0.514834	−	0.857290i	$-0.327854\pi$
$953$	31.7866	1.02967	0.514834	+	0.857290i	$0.327854\pi$
$954$	8.66835	0.280648
$955$	17.4880	0.565899
$956$	17.2074	0.556526
$957$	0.160293	0.00518153
$958$	13.8023	0.445931
$959$	0.683612	0.0220750
$960$	−2.87599	−0.0928221
$961$	−30.3871	−0.980230
$962$	47.0484	1.51690
$963$	−12.5918	−0.405766
$964$	8.97159	0.288955
$965$	12.3172	0.396504
$966$	−4.27829	−0.137652
$967$	38.1747	1.22762	0.613808	−	0.789455i	$-0.289637\pi$
$967$	38.1747	1.22762	0.613808	+	0.789455i	$0.289637\pi$
$968$	15.9354	0.512184
$969$	−32.4420	−1.04219
$970$	12.8001	0.410988
$971$	1.25327	0.0402192	0.0201096	−	0.999798i	$-0.493598\pi$
$971$	1.25327	0.0402192	0.0201096	+	0.999798i	$0.493598\pi$
$972$	−0.700959	−0.0224833
$973$	−5.97258	−0.191472
$974$	−41.2466	−1.32163
$975$	10.9667	0.351215
$976$	−60.0084	−1.92082
$977$	−61.8841	−1.97985	−0.989924	−	0.141598i	$-0.954776\pi$
$977$	−61.8841	−1.97985	−0.989924	+	0.141598i	$0.954776\pi$
$978$	−24.5904	−0.786313
$979$	−15.0190	−0.480009
$980$	0.563870	0.0180122
$981$	1.05340	0.0336326
$982$	−21.1220	−0.674031
$983$	−32.8876	−1.04895	−0.524475	−	0.851426i	$-0.675739\pi$
$983$	−32.8876	−1.04895	−0.524475	+	0.851426i	$0.675739\pi$
$984$	19.7740	0.630372
$985$	16.4088	0.522827
$986$	−0.640295	−0.0203912
$987$	−11.3681	−0.361849
$988$	−12.5356	−0.398810
$989$	−14.1378	−0.449556
$990$	−2.48594	−0.0790084
$991$	51.3762	1.63202	0.816009	−	0.578040i	$-0.196182\pi$
$991$	51.3762	1.63202	0.816009	+	0.578040i	$0.196182\pi$
$992$	−2.97524	−0.0944638
$993$	7.22811	0.229377
$994$	24.1013	0.764446
$995$	3.35890	0.106484
$996$	−2.02080	−0.0640316
$997$	−37.9624	−1.20228	−0.601140	−	0.799144i	$-0.705287\pi$
$997$	−37.9624	−1.20228	−0.601140	+	0.799144i	$0.705287\pi$
$998$	−19.4701	−0.616315
$999$	−11.3629	−0.359506

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.39	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.39	✓	52	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q+1.64346 q^{2} -1.00000 q^{3} +0.700959 q^{4} +0.804426 q^{5} -1.64346 q^{6} +1.00000 q^{7} -2.13492 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.64346 q^{2} -1.00000 q^{3} +0.700959 q^{4} +0.804426 q^{5} -1.64346 q^{6} +1.00000 q^{7} -2.13492 q^{8} +1.00000 q^{9} +1.32204 q^{10} -1.88038 q^{11} -0.700959 q^{12} +2.51940 q^{13} +1.64346 q^{14} -0.804426 q^{15} -4.91057 q^{16} -4.57039 q^{17} +1.64346 q^{18} -7.09831 q^{19} +0.563870 q^{20} -1.00000 q^{21} -3.09033 q^{22} +2.60322 q^{23} +2.13492 q^{24} -4.35290 q^{25} +4.14053 q^{26} -1.00000 q^{27} +0.700959 q^{28} +0.0852448 q^{29} -1.32204 q^{30} +0.782856 q^{31} -3.80049 q^{32} +1.88038 q^{33} -7.51125 q^{34} +0.804426 q^{35} +0.700959 q^{36} +11.3629 q^{37} -11.6658 q^{38} -2.51940 q^{39} -1.71739 q^{40} +9.26216 q^{41} -1.64346 q^{42} -5.43088 q^{43} -1.31807 q^{44} +0.804426 q^{45} +4.27829 q^{46} +11.3681 q^{47} +4.91057 q^{48} +1.00000 q^{49} -7.15381 q^{50} +4.57039 q^{51} +1.76600 q^{52} +5.27445 q^{53} -1.64346 q^{54} -1.51263 q^{55} -2.13492 q^{56} +7.09831 q^{57} +0.140096 q^{58} -0.612801 q^{59} -0.563870 q^{60} +12.2202 q^{61} +1.28659 q^{62} +1.00000 q^{63} +3.57520 q^{64} +2.02667 q^{65} +3.09033 q^{66} +6.90132 q^{67} -3.20365 q^{68} -2.60322 q^{69} +1.32204 q^{70} +14.6650 q^{71} -2.13492 q^{72} -15.1800 q^{73} +18.6744 q^{74} +4.35290 q^{75} -4.97562 q^{76} -1.88038 q^{77} -4.14053 q^{78} -4.45397 q^{79} -3.95020 q^{80} +1.00000 q^{81} +15.2220 q^{82} +2.88291 q^{83} -0.700959 q^{84} -3.67654 q^{85} -8.92544 q^{86} -0.0852448 q^{87} +4.01446 q^{88} +7.98720 q^{89} +1.32204 q^{90} +2.51940 q^{91} +1.82475 q^{92} -0.782856 q^{93} +18.6829 q^{94} -5.71006 q^{95} +3.80049 q^{96} +9.68210 q^{97} +1.64346 q^{98} -1.88038 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more