LMFDB - Embedded newform 8043.2.a.q.1.3

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:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-2.55412 q^{2} +1.00000 q^{3} +4.52354 q^{4} -2.76084 q^{5} -2.55412 q^{6} -1.00000 q^{7} -6.44543 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.55412 q^{2} +1.00000 q^{3} +4.52354 q^{4} -2.76084 q^{5} -2.55412 q^{6} -1.00000 q^{7} -6.44543 q^{8} +1.00000 q^{9} +7.05153 q^{10} -5.11531 q^{11} +4.52354 q^{12} +1.44487 q^{13} +2.55412 q^{14} -2.76084 q^{15} +7.41535 q^{16} +2.55078 q^{17} -2.55412 q^{18} +2.24739 q^{19} -12.4888 q^{20} -1.00000 q^{21} +13.0651 q^{22} -0.475677 q^{23} -6.44543 q^{24} +2.62225 q^{25} -3.69036 q^{26} +1.00000 q^{27} -4.52354 q^{28} -6.82497 q^{29} +7.05153 q^{30} +2.73799 q^{31} -6.04883 q^{32} -5.11531 q^{33} -6.51501 q^{34} +2.76084 q^{35} +4.52354 q^{36} +4.97641 q^{37} -5.74012 q^{38} +1.44487 q^{39} +17.7948 q^{40} +4.10776 q^{41} +2.55412 q^{42} -0.613198 q^{43} -23.1393 q^{44} -2.76084 q^{45} +1.21494 q^{46} +8.69844 q^{47} +7.41535 q^{48} +1.00000 q^{49} -6.69754 q^{50} +2.55078 q^{51} +6.53591 q^{52} -9.83916 q^{53} -2.55412 q^{54} +14.1226 q^{55} +6.44543 q^{56} +2.24739 q^{57} +17.4318 q^{58} -0.0375418 q^{59} -12.4888 q^{60} -13.8594 q^{61} -6.99316 q^{62} -1.00000 q^{63} +0.618766 q^{64} -3.98905 q^{65} +13.0651 q^{66} -4.09919 q^{67} +11.5386 q^{68} -0.475677 q^{69} -7.05153 q^{70} -6.35516 q^{71} -6.44543 q^{72} +9.02133 q^{73} -12.7104 q^{74} +2.62225 q^{75} +10.1662 q^{76} +5.11531 q^{77} -3.69036 q^{78} +5.91314 q^{79} -20.4726 q^{80} +1.00000 q^{81} -10.4917 q^{82} +3.68918 q^{83} -4.52354 q^{84} -7.04231 q^{85} +1.56618 q^{86} -6.82497 q^{87} +32.9704 q^{88} +14.3975 q^{89} +7.05153 q^{90} -1.44487 q^{91} -2.15174 q^{92} +2.73799 q^{93} -22.2169 q^{94} -6.20470 q^{95} -6.04883 q^{96} +14.7034 q^{97} -2.55412 q^{98} -5.11531 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{2} + 44 q^{3} + 44 q^{4} - 16 q^{5} - 4 q^{6} - 44 q^{7} - 15 q^{8} + 44 q^{9}+O(q^{10}) $ 44 * q - 4 * q^2 + 44 * q^3 + 44 * q^4 - 16 * q^5 - 4 * q^6 - 44 * q^7 - 15 * q^8 + 44 * q^9	\( 44 q - 4 q^{2} + 44 q^{3} + 44 q^{4} - 16 q^{5} - 4 q^{6} - 44 q^{7} - 15 q^{8} + 44 q^{9} - 16 q^{10} - 2 q^{11} + 44 q^{12} - 34 q^{13} + 4 q^{14} - 16 q^{15} + 24 q^{16} - 4 q^{17} - 4 q^{18} - 22 q^{19} - 39 q^{20} - 44 q^{21} - 23 q^{22} - 56 q^{23} - 15 q^{24} + 32 q^{25} - 17 q^{26} + 44 q^{27} - 44 q^{28} - 33 q^{29} - 16 q^{30} - 32 q^{31} - 34 q^{32} - 2 q^{33} - 25 q^{34} + 16 q^{35} + 44 q^{36} - 47 q^{37} - 40 q^{38} - 34 q^{39} - 50 q^{40} + 2 q^{41} + 4 q^{42} - 12 q^{43} - 22 q^{44} - 16 q^{45} + 8 q^{46} - 27 q^{47} + 24 q^{48} + 44 q^{49} - 21 q^{50} - 4 q^{51} - 82 q^{52} - 114 q^{53} - 4 q^{54} - 29 q^{55} + 15 q^{56} - 22 q^{57} - 26 q^{58} - 40 q^{59} - 39 q^{60} - 47 q^{61} - 37 q^{62} - 44 q^{63} - 5 q^{64} - 20 q^{65} - 23 q^{66} - 14 q^{67} - 72 q^{68} - 56 q^{69} + 16 q^{70} - 65 q^{71} - 15 q^{72} - 21 q^{73} - 26 q^{74} + 32 q^{75} - 15 q^{76} + 2 q^{77} - 17 q^{78} + 6 q^{79} - 77 q^{80} + 44 q^{81} - 51 q^{82} - 30 q^{83} - 44 q^{84} - 26 q^{85} - 65 q^{86} - 33 q^{87} - 84 q^{88} - 32 q^{89} - 16 q^{90} + 34 q^{91} - 140 q^{92} - 32 q^{93} - 35 q^{94} - 50 q^{95} - 34 q^{96} - 83 q^{97} - 4 q^{98} - 2 q^{99}+O(q^{100}) \) 44 * q - 4 * q^2 + 44 * q^3 + 44 * q^4 - 16 * q^5 - 4 * q^6 - 44 * q^7 - 15 * q^8 + 44 * q^9 - 16 * q^10 - 2 * q^11 + 44 * q^12 - 34 * q^13 + 4 * q^14 - 16 * q^15 + 24 * q^16 - 4 * q^17 - 4 * q^18 - 22 * q^19 - 39 * q^20 - 44 * q^21 - 23 * q^22 - 56 * q^23 - 15 * q^24 + 32 * q^25 - 17 * q^26 + 44 * q^27 - 44 * q^28 - 33 * q^29 - 16 * q^30 - 32 * q^31 - 34 * q^32 - 2 * q^33 - 25 * q^34 + 16 * q^35 + 44 * q^36 - 47 * q^37 - 40 * q^38 - 34 * q^39 - 50 * q^40 + 2 * q^41 + 4 * q^42 - 12 * q^43 - 22 * q^44 - 16 * q^45 + 8 * q^46 - 27 * q^47 + 24 * q^48 + 44 * q^49 - 21 * q^50 - 4 * q^51 - 82 * q^52 - 114 * q^53 - 4 * q^54 - 29 * q^55 + 15 * q^56 - 22 * q^57 - 26 * q^58 - 40 * q^59 - 39 * q^60 - 47 * q^61 - 37 * q^62 - 44 * q^63 - 5 * q^64 - 20 * q^65 - 23 * q^66 - 14 * q^67 - 72 * q^68 - 56 * q^69 + 16 * q^70 - 65 * q^71 - 15 * q^72 - 21 * q^73 - 26 * q^74 + 32 * q^75 - 15 * q^76 + 2 * q^77 - 17 * q^78 + 6 * q^79 - 77 * q^80 + 44 * q^81 - 51 * q^82 - 30 * q^83 - 44 * q^84 - 26 * q^85 - 65 * q^86 - 33 * q^87 - 84 * q^88 - 32 * q^89 - 16 * q^90 + 34 * q^91 - 140 * q^92 - 32 * q^93 - 35 * q^94 - 50 * q^95 - 34 * q^96 - 83 * q^97 - 4 * q^98 - 2 * 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 $

$2$	−2.55412	−1.80604	−0.903019	−	0.429601i	$-0.858654\pi$
$2$	−2.55412	−1.80604	−0.903019	+	0.429601i	$0.858654\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.52354	2.26177
$5$	−2.76084	−1.23469	−0.617343	−	0.786694i	$-0.711791\pi$
$5$	−2.76084	−1.23469	−0.617343	+	0.786694i	$0.711791\pi$
$6$	−2.55412	−1.04272
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−6.44543	−2.27881
$9$	1.00000	0.333333
$10$	7.05153	2.22989
$11$	−5.11531	−1.54232	−0.771162	−	0.636639i	$-0.780324\pi$
$11$	−5.11531	−1.54232	−0.771162	+	0.636639i	$0.780324\pi$
$12$	4.52354	1.30583
$13$	1.44487	0.400734	0.200367	−	0.979721i	$-0.435787\pi$
$13$	1.44487	0.400734	0.200367	+	0.979721i	$0.435787\pi$
$14$	2.55412	0.682618
$15$	−2.76084	−0.712846
$16$	7.41535	1.85384
$17$	2.55078	0.618656	0.309328	−	0.950955i	$-0.399896\pi$
$17$	2.55078	0.618656	0.309328	+	0.950955i	$0.399896\pi$
$18$	−2.55412	−0.602012
$19$	2.24739	0.515588	0.257794	−	0.966200i	$-0.417004\pi$
$19$	2.24739	0.515588	0.257794	+	0.966200i	$0.417004\pi$
$20$	−12.4888	−2.79258
$21$	−1.00000	−0.218218
$22$	13.0651	2.78550
$23$	−0.475677	−0.0991855	−0.0495928	−	0.998770i	$-0.515792\pi$
$23$	−0.475677	−0.0991855	−0.0495928	+	0.998770i	$0.515792\pi$
$24$	−6.44543	−1.31567
$25$	2.62225	0.524450
$26$	−3.69036	−0.723740
$27$	1.00000	0.192450
$28$	−4.52354	−0.854869
$29$	−6.82497	−1.26737	−0.633683	−	0.773593i	$-0.718458\pi$
$29$	−6.82497	−1.26737	−0.633683	+	0.773593i	$0.718458\pi$
$30$	7.05153	1.28743
$31$	2.73799	0.491757	0.245879	−	0.969301i	$-0.420923\pi$
$31$	2.73799	0.491757	0.245879	+	0.969301i	$0.420923\pi$
$32$	−6.04883	−1.06929
$33$	−5.11531	−0.890462
$34$	−6.51501	−1.11732
$35$	2.76084	0.466667
$36$	4.52354	0.753924
$37$	4.97641	0.818117	0.409059	−	0.912508i	$-0.365857\pi$
$37$	4.97641	0.818117	0.409059	+	0.912508i	$0.365857\pi$
$38$	−5.74012	−0.931170
$39$	1.44487	0.231364
$40$	17.7948	2.81361
$41$	4.10776	0.641525	0.320762	−	0.947160i	$-0.396061\pi$
$41$	4.10776	0.641525	0.320762	+	0.947160i	$0.396061\pi$
$42$	2.55412	0.394110
$43$	−0.613198	−0.0935117	−0.0467559	−	0.998906i	$-0.514888\pi$
$43$	−0.613198	−0.0935117	−0.0467559	+	0.998906i	$0.514888\pi$
$44$	−23.1393	−3.48838
$45$	−2.76084	−0.411562
$46$	1.21494	0.179133
$47$	8.69844	1.26880	0.634399	−	0.773006i	$-0.281248\pi$
$47$	8.69844	1.26880	0.634399	+	0.773006i	$0.281248\pi$
$48$	7.41535	1.07031
$49$	1.00000	0.142857
$50$	−6.69754	−0.947176
$51$	2.55078	0.357181
$52$	6.53591	0.906368
$53$	−9.83916	−1.35151	−0.675756	−	0.737125i	$-0.736183\pi$
$53$	−9.83916	−1.35151	−0.675756	+	0.737125i	$0.736183\pi$
$54$	−2.55412	−0.347572
$55$	14.1226	1.90429
$56$	6.44543	0.861307
$57$	2.24739	0.297675
$58$	17.4318	2.28891
$59$	−0.0375418	−0.00488753	−0.00244376	−	0.999997i	$-0.500778\pi$
$59$	−0.0375418	−0.00488753	−0.00244376	+	0.999997i	$0.500778\pi$
$60$	−12.4888	−1.61230
$61$	−13.8594	−1.77451	−0.887257	−	0.461276i	$-0.847392\pi$
$61$	−13.8594	−1.77451	−0.887257	+	0.461276i	$0.847392\pi$
$62$	−6.99316	−0.888132
$63$	−1.00000	−0.125988
$64$	0.618766	0.0773458
$65$	−3.98905	−0.494780
$66$	13.0651	1.60821
$67$	−4.09919	−0.500796	−0.250398	−	0.968143i	$-0.580562\pi$
$67$	−4.09919	−0.500796	−0.250398	+	0.968143i	$0.580562\pi$
$68$	11.5386	1.39926
$69$	−0.475677	−0.0572648
$70$	−7.05153	−0.842819
$71$	−6.35516	−0.754219	−0.377109	−	0.926169i	$-0.623082\pi$
$71$	−6.35516	−0.754219	−0.377109	+	0.926169i	$0.623082\pi$
$72$	−6.44543	−0.759602
$73$	9.02133	1.05587	0.527933	−	0.849286i	$-0.322967\pi$
$73$	9.02133	1.05587	0.527933	+	0.849286i	$0.322967\pi$
$74$	−12.7104	−1.47755
$75$	2.62225	0.302791
$76$	10.1662	1.16614
$77$	5.11531	0.582944
$78$	−3.69036	−0.417851
$79$	5.91314	0.665281	0.332640	−	0.943054i	$-0.392061\pi$
$79$	5.91314	0.665281	0.332640	+	0.943054i	$0.392061\pi$
$80$	−20.4726	−2.28891
$81$	1.00000	0.111111
$82$	−10.4917	−1.15862
$83$	3.68918	0.404940	0.202470	−	0.979288i	$-0.435103\pi$
$83$	3.68918	0.404940	0.202470	+	0.979288i	$0.435103\pi$
$84$	−4.52354	−0.493559
$85$	−7.04231	−0.763846
$86$	1.56618	0.168886
$87$	−6.82497	−0.731714
$88$	32.9704	3.51466
$89$	14.3975	1.52613	0.763064	−	0.646322i	$-0.223694\pi$
$89$	14.3975	1.52613	0.763064	+	0.646322i	$0.223694\pi$
$90$	7.05153	0.743296
$91$	−1.44487	−0.151463
$92$	−2.15174	−0.224335
$93$	2.73799	0.283916
$94$	−22.2169	−2.29150
$95$	−6.20470	−0.636589
$96$	−6.04883	−0.617356
$97$	14.7034	1.49290	0.746452	−	0.665439i	$-0.231756\pi$
$97$	14.7034	1.49290	0.746452	+	0.665439i	$0.231756\pi$
$98$	−2.55412	−0.258005
$99$	−5.11531	−0.514108
$100$	11.8619	1.18619
$101$	0.395991	0.0394025	0.0197013	−	0.999806i	$-0.493728\pi$
$101$	0.395991	0.0394025	0.0197013	+	0.999806i	$0.493728\pi$
$102$	−6.51501	−0.645082
$103$	0.535568	0.0527710	0.0263855	−	0.999652i	$-0.491600\pi$
$103$	0.535568	0.0527710	0.0263855	+	0.999652i	$0.491600\pi$
$104$	−9.31279	−0.913194
$105$	2.76084	0.269431
$106$	25.1304	2.44088
$107$	−1.60982	−0.155627	−0.0778135	−	0.996968i	$-0.524794\pi$
$107$	−1.60982	−0.155627	−0.0778135	+	0.996968i	$0.524794\pi$
$108$	4.52354	0.435278
$109$	−10.0232	−0.960044	−0.480022	−	0.877256i	$-0.659371\pi$
$109$	−10.0232	−0.960044	−0.480022	+	0.877256i	$0.659371\pi$
$110$	−36.0708	−3.43921
$111$	4.97641	0.472340
$112$	−7.41535	−0.700684
$113$	5.86923	0.552131	0.276065	−	0.961139i	$-0.410969\pi$
$113$	5.86923	0.552131	0.276065	+	0.961139i	$0.410969\pi$
$114$	−5.74012	−0.537611
$115$	1.31327	0.122463
$116$	−30.8731	−2.86649
$117$	1.44487	0.133578
$118$	0.0958864	0.00882705
$119$	−2.55078	−0.233830
$120$	17.7948	1.62444
$121$	15.1664	1.37877
$122$	35.3986	3.20484
$123$	4.10776	0.370385
$124$	12.3854	1.11224
$125$	6.56460	0.587155
$126$	2.55412	0.227539
$127$	−13.2353	−1.17444	−0.587221	−	0.809426i	$-0.699778\pi$
$127$	−13.2353	−1.17444	−0.587221	+	0.809426i	$0.699778\pi$
$128$	10.5173	0.929603
$129$	−0.613198	−0.0539890
$130$	10.1885	0.893592
$131$	6.04902	0.528505	0.264253	−	0.964454i	$-0.414875\pi$
$131$	6.04902	0.528505	0.264253	+	0.964454i	$0.414875\pi$
$132$	−23.1393	−2.01402
$133$	−2.24739	−0.194874
$134$	10.4698	0.904457
$135$	−2.76084	−0.237615
$136$	−16.4409	−1.40980
$137$	−2.49266	−0.212962	−0.106481	−	0.994315i	$-0.533958\pi$
$137$	−2.49266	−0.212962	−0.106481	+	0.994315i	$0.533958\pi$
$138$	1.21494	0.103422
$139$	19.4826	1.65249	0.826245	−	0.563311i	$-0.190473\pi$
$139$	19.4826	1.65249	0.826245	+	0.563311i	$0.190473\pi$
$140$	12.4888	1.05549
$141$	8.69844	0.732541
$142$	16.2319	1.36215
$143$	−7.39094	−0.618061
$144$	7.41535	0.617945
$145$	18.8427	1.56480
$146$	−23.0416	−1.90693
$147$	1.00000	0.0824786
$148$	22.5110	1.85039
$149$	−8.45034	−0.692279	−0.346139	−	0.938183i	$-0.612508\pi$
$149$	−8.45034	−0.692279	−0.346139	+	0.938183i	$0.612508\pi$
$150$	−6.69754	−0.546852
$151$	10.7555	0.875274	0.437637	−	0.899152i	$-0.355816\pi$
$151$	10.7555	0.875274	0.437637	+	0.899152i	$0.355816\pi$
$152$	−14.4854	−1.17492
$153$	2.55078	0.206219
$154$	−13.0651	−1.05282
$155$	−7.55916	−0.607166
$156$	6.53591	0.523292
$157$	3.21001	0.256187	0.128093	−	0.991762i	$-0.459114\pi$
$157$	3.21001	0.256187	0.128093	+	0.991762i	$0.459114\pi$
$158$	−15.1029	−1.20152
$159$	−9.83916	−0.780296
$160$	16.6999	1.32024
$161$	0.475677	0.0374886
$162$	−2.55412	−0.200671
$163$	−11.3523	−0.889180	−0.444590	−	0.895734i	$-0.646651\pi$
$163$	−11.3523	−0.889180	−0.444590	+	0.895734i	$0.646651\pi$
$164$	18.5816	1.45098
$165$	14.1226	1.09944
$166$	−9.42262	−0.731337
$167$	4.05673	0.313919	0.156960	−	0.987605i	$-0.449831\pi$
$167$	4.05673	0.313919	0.156960	+	0.987605i	$0.449831\pi$
$168$	6.44543	0.497276
$169$	−10.9124	−0.839413
$170$	17.9869	1.37953
$171$	2.24739	0.171863
$172$	−2.77382	−0.211502
$173$	17.9525	1.36490	0.682452	−	0.730931i	$-0.260914\pi$
$173$	17.9525	1.36490	0.682452	+	0.730931i	$0.260914\pi$
$174$	17.4318	1.32150
$175$	−2.62225	−0.198223
$176$	−37.9318	−2.85922
$177$	−0.0375418	−0.00282181
$178$	−36.7729	−2.75625
$179$	−23.5988	−1.76385	−0.881927	−	0.471385i	$-0.843754\pi$
$179$	−23.5988	−1.76385	−0.881927	+	0.471385i	$0.843754\pi$
$180$	−12.4888	−0.930859
$181$	3.77766	0.280792	0.140396	−	0.990095i	$-0.455163\pi$
$181$	3.77766	0.280792	0.140396	+	0.990095i	$0.455163\pi$
$182$	3.69036	0.273548
$183$	−13.8594	−1.02452
$184$	3.06594	0.226024
$185$	−13.7391	−1.01012
$186$	−6.99316	−0.512763
$187$	−13.0480	−0.954168
$188$	39.3478	2.86973
$189$	−1.00000	−0.0727393
$190$	15.8476	1.14970
$191$	17.4512	1.26273	0.631363	−	0.775487i	$-0.282496\pi$
$191$	17.4512	1.26273	0.631363	+	0.775487i	$0.282496\pi$
$192$	0.618766	0.0446556
$193$	2.82442	0.203306	0.101653	−	0.994820i	$-0.467587\pi$
$193$	2.82442	0.203306	0.101653	+	0.994820i	$0.467587\pi$
$194$	−37.5543	−2.69624
$195$	−3.98905	−0.285662
$196$	4.52354	0.323110
$197$	1.88329	0.134179	0.0670893	−	0.997747i	$-0.478629\pi$
$197$	1.88329	0.134179	0.0670893	+	0.997747i	$0.478629\pi$
$198$	13.0651	0.928499
$199$	−13.8668	−0.982993	−0.491496	−	0.870880i	$-0.663550\pi$
$199$	−13.8668	−0.982993	−0.491496	+	0.870880i	$0.663550\pi$
$200$	−16.9015	−1.19512
$201$	−4.09919	−0.289135
$202$	−1.01141	−0.0711625
$203$	6.82497	0.479019
$204$	11.5386	0.807862
$205$	−11.3409	−0.792082
$206$	−1.36791	−0.0953065
$207$	−0.475677	−0.0330618
$208$	10.7142	0.742895
$209$	−11.4961	−0.795203
$210$	−7.05153	−0.486602
$211$	20.5703	1.41612	0.708060	−	0.706152i	$-0.249571\pi$
$211$	20.5703	1.41612	0.708060	+	0.706152i	$0.249571\pi$
$212$	−44.5079	−3.05681
$213$	−6.35516	−0.435448
$214$	4.11167	0.281068
$215$	1.69294	0.115458
$216$	−6.44543	−0.438556
$217$	−2.73799	−0.185867
$218$	25.6004	1.73388
$219$	9.02133	0.609605
$220$	63.8840	4.30706
$221$	3.68554	0.247916
$222$	−12.7104	−0.853064
$223$	−11.7886	−0.789420	−0.394710	−	0.918806i	$-0.629155\pi$
$223$	−11.7886	−0.789420	−0.394710	+	0.918806i	$0.629155\pi$
$224$	6.04883	0.404155
$225$	2.62225	0.174817
$226$	−14.9907	−0.997169
$227$	−12.4986	−0.829558	−0.414779	−	0.909922i	$-0.636141\pi$
$227$	−12.4986	−0.829558	−0.414779	+	0.909922i	$0.636141\pi$
$228$	10.1662	0.673272
$229$	8.97048	0.592786	0.296393	−	0.955066i	$-0.404216\pi$
$229$	8.97048	0.592786	0.296393	+	0.955066i	$0.404216\pi$
$230$	−3.35425	−0.221173
$231$	5.11531	0.336563
$232$	43.9899	2.88808
$233$	8.96532	0.587338	0.293669	−	0.955907i	$-0.405124\pi$
$233$	8.96532	0.587338	0.293669	+	0.955907i	$0.405124\pi$
$234$	−3.69036	−0.241247
$235$	−24.0150	−1.56657
$236$	−0.169822	−0.0110545
$237$	5.91314	0.384100
$238$	6.51501	0.422305
$239$	23.7750	1.53788	0.768939	−	0.639322i	$-0.220785\pi$
$239$	23.7750	1.53788	0.768939	+	0.639322i	$0.220785\pi$
$240$	−20.4726	−1.32150
$241$	−0.00638292	−0.000411160 0	−0.000205580	−	1.00000i	$-0.500065\pi$
$241$	−0.00638292	−0.000411160 0	−0.000205580		1.00000i	$0.500065\pi$
$242$	−38.7369	−2.49010
$243$	1.00000	0.0641500
$244$	−62.6935	−4.01354
$245$	−2.76084	−0.176384
$246$	−10.4917	−0.668928
$247$	3.24718	0.206613
$248$	−17.6475	−1.12062
$249$	3.68918	0.233792
$250$	−16.7668	−1.06042
$251$	−9.29008	−0.586385	−0.293192	−	0.956053i	$-0.594718\pi$
$251$	−9.29008	−0.586385	−0.293192	+	0.956053i	$0.594718\pi$
$252$	−4.52354	−0.284956
$253$	2.43324	0.152976
$254$	33.8046	2.12109
$255$	−7.04231	−0.441006
$256$	−28.0999	−1.75624
$257$	−15.1589	−0.945587	−0.472793	−	0.881173i	$-0.656754\pi$
$257$	−15.1589	−0.945587	−0.472793	+	0.881173i	$0.656754\pi$
$258$	1.56618	0.0975062
$259$	−4.97641	−0.309219
$260$	−18.0446	−1.11908
$261$	−6.82497	−0.422455
$262$	−15.4499	−0.954500
$263$	−7.05145	−0.434811	−0.217406	−	0.976081i	$-0.569759\pi$
$263$	−7.05145	−0.434811	−0.217406	+	0.976081i	$0.569759\pi$
$264$	32.9704	2.02919
$265$	27.1644	1.66869
$266$	5.74012	0.351949
$267$	14.3975	0.881111
$268$	−18.5429	−1.13269
$269$	−14.9630	−0.912312	−0.456156	−	0.889900i	$-0.650774\pi$
$269$	−14.9630	−0.912312	−0.456156	+	0.889900i	$0.650774\pi$
$270$	7.05153	0.429142
$271$	5.64918	0.343163	0.171582	−	0.985170i	$-0.445112\pi$
$271$	5.64918	0.343163	0.171582	+	0.985170i	$0.445112\pi$
$272$	18.9149	1.14689
$273$	−1.44487	−0.0874472
$274$	6.36655	0.384617
$275$	−13.4136	−0.808872
$276$	−2.15174	−0.129520
$277$	−13.6828	−0.822120	−0.411060	−	0.911608i	$-0.634841\pi$
$277$	−13.6828	−0.822120	−0.411060	+	0.911608i	$0.634841\pi$
$278$	−49.7609	−2.98446
$279$	2.73799	0.163919
$280$	−17.7948	−1.06344
$281$	−3.22550	−0.192417	−0.0962085	−	0.995361i	$-0.530672\pi$
$281$	−3.22550	−0.192417	−0.0962085	+	0.995361i	$0.530672\pi$
$282$	−22.2169	−1.32300
$283$	−16.4600	−0.978443	−0.489221	−	0.872160i	$-0.662719\pi$
$283$	−16.4600	−0.978443	−0.489221	+	0.872160i	$0.662719\pi$
$284$	−28.7478	−1.70587
$285$	−6.20470	−0.367535
$286$	18.8774	1.11624
$287$	−4.10776	−0.242474
$288$	−6.04883	−0.356431
$289$	−10.4935	−0.617265
$290$	−48.1265	−2.82609
$291$	14.7034	0.861929
$292$	40.8083	2.38813
$293$	−0.819734	−0.0478894	−0.0239447	−	0.999713i	$-0.507623\pi$
$293$	−0.819734	−0.0478894	−0.0239447	+	0.999713i	$0.507623\pi$
$294$	−2.55412	−0.148959
$295$	0.103647	0.00603456
$296$	−32.0751	−1.86433
$297$	−5.11531	−0.296821
$298$	21.5832	1.25028
$299$	−0.687289	−0.0397470
$300$	11.8619	0.684844
$301$	0.613198	0.0353441
$302$	−27.4710	−1.58078
$303$	0.395991	0.0227491
$304$	16.6652	0.955815
$305$	38.2636	2.19097
$306$	−6.51501	−0.372438
$307$	−31.5261	−1.79929	−0.899646	−	0.436620i	$-0.856175\pi$
$307$	−31.5261	−1.79929	−0.899646	+	0.436620i	$0.856175\pi$
$308$	23.1393	1.31849
$309$	0.535568	0.0304674
$310$	19.3070	1.09656
$311$	−12.1579	−0.689410	−0.344705	−	0.938711i	$-0.612021\pi$
$311$	−12.1579	−0.689410	−0.344705	+	0.938711i	$0.612021\pi$
$312$	−9.31279	−0.527233
$313$	−1.29175	−0.0730141	−0.0365070	−	0.999333i	$-0.511623\pi$
$313$	−1.29175	−0.0730141	−0.0365070	+	0.999333i	$0.511623\pi$
$314$	−8.19876	−0.462683
$315$	2.76084	0.155556
$316$	26.7483	1.50471
$317$	−16.9724	−0.953267	−0.476634	−	0.879102i	$-0.658143\pi$
$317$	−16.9724	−0.953267	−0.476634	+	0.879102i	$0.658143\pi$
$318$	25.1304	1.40924
$319$	34.9119	1.95469
$320$	−1.70832	−0.0954978
$321$	−1.60982	−0.0898513
$322$	−1.21494	−0.0677058
$323$	5.73261	0.318971
$324$	4.52354	0.251308
$325$	3.78880	0.210165
$326$	28.9951	1.60589
$327$	−10.0232	−0.554282
$328$	−26.4763	−1.46191
$329$	−8.69844	−0.479561
$330$	−36.0708	−1.98563
$331$	12.4470	0.684148	0.342074	−	0.939673i	$-0.388871\pi$
$331$	12.4470	0.684148	0.342074	+	0.939673i	$0.388871\pi$
$332$	16.6882	0.915882
$333$	4.97641	0.272706
$334$	−10.3614	−0.566950
$335$	11.3172	0.618326
$336$	−7.41535	−0.404540
$337$	−23.9101	−1.30246	−0.651232	−	0.758879i	$-0.725747\pi$
$337$	−23.9101	−1.30246	−0.651232	+	0.758879i	$0.725747\pi$
$338$	27.8715	1.51601
$339$	5.86923	0.318773
$340$	−31.8562	−1.72764
$341$	−14.0057	−0.758450
$342$	−5.74012	−0.310390
$343$	−1.00000	−0.0539949
$344$	3.95232	0.213095
$345$	1.31327	0.0707040
$346$	−45.8529	−2.46507
$347$	−15.5442	−0.834459	−0.417229	−	0.908801i	$-0.636999\pi$
$347$	−15.5442	−0.834459	−0.417229	+	0.908801i	$0.636999\pi$
$348$	−30.8731	−1.65497
$349$	11.5353	0.617470	0.308735	−	0.951148i	$-0.400094\pi$
$349$	11.5353	0.617470	0.308735	+	0.951148i	$0.400094\pi$
$350$	6.69754	0.357999
$351$	1.44487	0.0771212
$352$	30.9417	1.64920
$353$	−7.00555	−0.372868	−0.186434	−	0.982468i	$-0.559693\pi$
$353$	−7.00555	−0.372868	−0.186434	+	0.982468i	$0.559693\pi$
$354$	0.0958864	0.00509630
$355$	17.5456	0.931223
$356$	65.1276	3.45175
$357$	−2.55078	−0.135002
$358$	60.2741	3.18559
$359$	−36.0085	−1.90046	−0.950228	−	0.311556i	$-0.899150\pi$
$359$	−36.0085	−1.90046	−0.950228	+	0.311556i	$0.899150\pi$
$360$	17.7948	0.937870
$361$	−13.9492	−0.734169
$362$	−9.64862	−0.507120
$363$	15.1664	0.796030
$364$	−6.53591	−0.342575
$365$	−24.9065	−1.30366
$366$	35.3986	1.85031
$367$	3.70639	0.193472	0.0967359	−	0.995310i	$-0.469160\pi$
$367$	3.70639	0.193472	0.0967359	+	0.995310i	$0.469160\pi$
$368$	−3.52731	−0.183874
$369$	4.10776	0.213842
$370$	35.0913	1.82431
$371$	9.83916	0.510824
$372$	12.3854	0.642154
$373$	−34.3585	−1.77902	−0.889509	−	0.456918i	$-0.848953\pi$
$373$	−34.3585	−1.77902	−0.889509	+	0.456918i	$0.848953\pi$
$374$	33.3263	1.72326
$375$	6.56460	0.338994
$376$	−56.0652	−2.89134
$377$	−9.86117	−0.507876
$378$	2.55412	0.131370
$379$	−7.56426	−0.388550	−0.194275	−	0.980947i	$-0.562235\pi$
$379$	−7.56426	−0.388550	−0.194275	+	0.980947i	$0.562235\pi$
$380$	−28.0672	−1.43982
$381$	−13.2353	−0.678065
$382$	−44.5726	−2.28053
$383$	1.00000	0.0510976
$383$	1.00000	0.0510976
$384$	10.5173	0.536707
$385$	−14.1226	−0.719753
$386$	−7.21391	−0.367178
$387$	−0.613198	−0.0311706
$388$	66.5115	3.37661
$389$	−29.3607	−1.48865	−0.744324	−	0.667819i	$-0.767228\pi$
$389$	−29.3607	−1.48865	−0.744324	+	0.667819i	$0.767228\pi$
$390$	10.1885	0.515915
$391$	−1.21335	−0.0613617
$392$	−6.44543	−0.325544
$393$	6.04902	0.305133
$394$	−4.81015	−0.242332
$395$	−16.3253	−0.821413
$396$	−23.1393	−1.16279
$397$	−13.4020	−0.672626	−0.336313	−	0.941750i	$-0.609180\pi$
$397$	−13.4020	−0.672626	−0.336313	+	0.941750i	$0.609180\pi$
$398$	35.4175	1.77532
$399$	−2.24739	−0.112510
$400$	19.4449	0.972244
$401$	12.7151	0.634964	0.317482	−	0.948264i	$-0.397163\pi$
$401$	12.7151	0.634964	0.317482	+	0.948264i	$0.397163\pi$
$402$	10.4698	0.522188
$403$	3.95603	0.197064
$404$	1.79128	0.0891195
$405$	−2.76084	−0.137187
$406$	−17.4318	−0.865127
$407$	−25.4559	−1.26180
$408$	−16.4409	−0.813946
$409$	30.0895	1.48783	0.743916	−	0.668274i	$-0.232966\pi$
$409$	30.0895	1.48783	0.743916	+	0.668274i	$0.232966\pi$
$410$	28.9660	1.43053
$411$	−2.49266	−0.122954
$412$	2.42266	0.119356
$413$	0.0375418	0.00184731
$414$	1.21494	0.0597109
$415$	−10.1852	−0.499974
$416$	−8.73975	−0.428501
$417$	19.4826	0.954065
$418$	29.3625	1.43617
$419$	30.8948	1.50931	0.754656	−	0.656121i	$-0.227804\pi$
$419$	30.8948	1.50931	0.754656	+	0.656121i	$0.227804\pi$
$420$	12.4888	0.609390
$421$	23.1419	1.12787	0.563934	−	0.825820i	$-0.309287\pi$
$421$	23.1419	1.12787	0.563934	+	0.825820i	$0.309287\pi$
$422$	−52.5391	−2.55757
$423$	8.69844	0.422933
$424$	63.4177	3.07983
$425$	6.68879	0.324454
$426$	16.2319	0.786436
$427$	13.8594	0.670703
$428$	−7.28208	−0.351993
$429$	−7.39094	−0.356838
$430$	−4.32398	−0.208521
$431$	−11.9855	−0.577320	−0.288660	−	0.957432i	$-0.593210\pi$
$431$	−11.9855	−0.577320	−0.288660	+	0.957432i	$0.593210\pi$
$432$	7.41535	0.356771
$433$	−1.85521	−0.0891556	−0.0445778	−	0.999006i	$-0.514194\pi$
$433$	−1.85521	−0.0891556	−0.0445778	+	0.999006i	$0.514194\pi$
$434$	6.99316	0.335682
$435$	18.8427	0.903437
$436$	−45.3402	−2.17140
$437$	−1.06903	−0.0511388
$438$	−23.0416	−1.10097
$439$	35.4053	1.68980	0.844901	−	0.534923i	$-0.179659\pi$
$439$	35.4053	1.68980	0.844901	+	0.534923i	$0.179659\pi$
$440$	−91.0261	−4.33950
$441$	1.00000	0.0476190
$442$	−9.41332	−0.447746
$443$	13.5624	0.644368	0.322184	−	0.946677i	$-0.395583\pi$
$443$	13.5624	0.644368	0.322184	+	0.946677i	$0.395583\pi$
$444$	22.5110	1.06833
$445$	−39.7491	−1.88429
$446$	30.1094	1.42572
$447$	−8.45034	−0.399687
$448$	−0.618766	−0.0292340
$449$	3.95316	0.186561	0.0932805	−	0.995640i	$-0.470265\pi$
$449$	3.95316	0.186561	0.0932805	+	0.995640i	$0.470265\pi$
$450$	−6.69754	−0.315725
$451$	−21.0125	−0.989440
$452$	26.5497	1.24879
$453$	10.7555	0.505340
$454$	31.9228	1.49821
$455$	3.98905	0.187009
$456$	−14.4854	−0.678342
$457$	9.15850	0.428417	0.214208	−	0.976788i	$-0.431283\pi$
$457$	9.15850	0.428417	0.214208	+	0.976788i	$0.431283\pi$
$458$	−22.9117	−1.07059
$459$	2.55078	0.119060
$460$	5.94063	0.276983
$461$	−13.4947	−0.628513	−0.314257	−	0.949338i	$-0.601755\pi$
$461$	−13.4947	−0.628513	−0.314257	+	0.949338i	$0.601755\pi$
$462$	−13.0651	−0.607845
$463$	−20.5692	−0.955932	−0.477966	−	0.878378i	$-0.658626\pi$
$463$	−20.5692	−0.955932	−0.477966	+	0.878378i	$0.658626\pi$
$464$	−50.6095	−2.34949
$465$	−7.55916	−0.350548
$466$	−22.8985	−1.06075
$467$	29.9350	1.38522	0.692612	−	0.721310i	$-0.256460\pi$
$467$	29.9350	1.38522	0.692612	+	0.721310i	$0.256460\pi$
$468$	6.53591	0.302123
$469$	4.09919	0.189283
$470$	61.3373	2.82928
$471$	3.21001	0.147909
$472$	0.241973	0.0111377
$473$	3.13670	0.144225
$474$	−15.1029	−0.693699
$475$	5.89323	0.270400
$476$	−11.5386	−0.528870
$477$	−9.83916	−0.450504
$478$	−60.7243	−2.77746
$479$	11.3317	0.517758	0.258879	−	0.965910i	$-0.416647\pi$
$479$	11.3317	0.517758	0.258879	+	0.965910i	$0.416647\pi$
$480$	16.6999	0.762241
$481$	7.19025	0.327847
$482$	0.0163028	0.000742570 0
$483$	0.475677	0.0216441
$484$	68.6059	3.11845
$485$	−40.5938	−1.84327
$486$	−2.55412	−0.115857
$487$	17.0311	0.771754	0.385877	−	0.922550i	$-0.373899\pi$
$487$	17.0311	0.771754	0.385877	+	0.922550i	$0.373899\pi$
$488$	89.3298	4.04377
$489$	−11.3523	−0.513368
$490$	7.05153	0.318556
$491$	−3.21319	−0.145009	−0.0725047	−	0.997368i	$-0.523099\pi$
$491$	−3.21319	−0.145009	−0.0725047	+	0.997368i	$0.523099\pi$
$492$	18.5816	0.837725
$493$	−17.4090	−0.784063
$494$	−8.29370	−0.373151
$495$	14.1226	0.634762
$496$	20.3031	0.911638
$497$	6.35516	0.285068
$498$	−9.42262	−0.422238
$499$	−27.4697	−1.22971	−0.614855	−	0.788640i	$-0.710786\pi$
$499$	−27.4697	−1.22971	−0.614855	+	0.788640i	$0.710786\pi$
$500$	29.6952	1.32801
$501$	4.05673	0.181241
$502$	23.7280	1.05903
$503$	20.1191	0.897069	0.448534	−	0.893766i	$-0.351946\pi$
$503$	20.1191	0.897069	0.448534	+	0.893766i	$0.351946\pi$
$504$	6.44543	0.287102
$505$	−1.09327	−0.0486498
$506$	−6.21478	−0.276281
$507$	−10.9124	−0.484635
$508$	−59.8704	−2.65632
$509$	17.5713	0.778834	0.389417	−	0.921061i	$-0.372676\pi$
$509$	17.5713	0.778834	0.389417	+	0.921061i	$0.372676\pi$
$510$	17.9869	0.796474
$511$	−9.02133	−0.399080
$512$	50.7361	2.24224
$513$	2.24739	0.0992249
$514$	38.7177	1.70776
$515$	−1.47862	−0.0651557
$516$	−2.77382	−0.122111
$517$	−44.4952	−1.95690
$518$	12.7104	0.558462
$519$	17.9525	0.788027
$520$	25.7111	1.12751
$521$	1.00567	0.0440592	0.0220296	−	0.999757i	$-0.492987\pi$
$521$	1.00567	0.0440592	0.0220296	+	0.999757i	$0.492987\pi$
$522$	17.4318	0.762970
$523$	−32.0766	−1.40261	−0.701305	−	0.712861i	$-0.747399\pi$
$523$	−32.0766	−1.40261	−0.701305	+	0.712861i	$0.747399\pi$
$524$	27.3630	1.19536
$525$	−2.62225	−0.114444
$526$	18.0103	0.785285
$527$	6.98402	0.304229
$528$	−37.9318	−1.65077
$529$	−22.7737	−0.990162
$530$	−69.3811	−3.01372
$531$	−0.0375418	−0.00162918
$532$	−10.1662	−0.440760
$533$	5.93517	0.257081
$534$	−36.7729	−1.59132
$535$	4.44445	0.192151
$536$	26.4211	1.14122
$537$	−23.5988	−1.01836
$538$	38.2174	1.64767
$539$	−5.11531	−0.220332
$540$	−12.4888	−0.537432
$541$	−11.6527	−0.500988	−0.250494	−	0.968118i	$-0.580593\pi$
$541$	−11.6527	−0.500988	−0.250494	+	0.968118i	$0.580593\pi$
$542$	−14.4287	−0.619765
$543$	3.77766	0.162115
$544$	−15.4293	−0.661524
$545$	27.6724	1.18535
$546$	3.69036	0.157933
$547$	−1.79432	−0.0767194	−0.0383597	−	0.999264i	$-0.512213\pi$
$547$	−1.79432	−0.0767194	−0.0383597	+	0.999264i	$0.512213\pi$
$548$	−11.2756	−0.481671
$549$	−13.8594	−0.591504
$550$	34.2600	1.46085
$551$	−15.3384	−0.653438
$552$	3.06594	0.130495
$553$	−5.91314	−0.251452
$554$	34.9475	1.48478
$555$	−13.7391	−0.583192
$556$	88.1302	3.73755
$557$	−17.9641	−0.761165	−0.380583	−	0.924747i	$-0.624277\pi$
$557$	−17.9641	−0.761165	−0.380583	+	0.924747i	$0.624277\pi$
$558$	−6.99316	−0.296044
$559$	−0.885988	−0.0374733
$560$	20.4726	0.865125
$561$	−13.0480	−0.550889
$562$	8.23831	0.347512
$563$	−19.3669	−0.816218	−0.408109	−	0.912933i	$-0.633812\pi$
$563$	−19.3669	−0.816218	−0.408109	+	0.912933i	$0.633812\pi$
$564$	39.3478	1.65684
$565$	−16.2040	−0.681708
$566$	42.0407	1.76710
$567$	−1.00000	−0.0419961
$568$	40.9618	1.71872
$569$	−5.17147	−0.216799	−0.108400	−	0.994107i	$-0.534573\pi$
$569$	−5.17147	−0.216799	−0.108400	+	0.994107i	$0.534573\pi$
$570$	15.8476	0.663781
$571$	37.1824	1.55604	0.778018	−	0.628242i	$-0.216225\pi$
$571$	37.1824	1.55604	0.778018	+	0.628242i	$0.216225\pi$
$572$	−33.4332	−1.39791
$573$	17.4512	0.729035
$574$	10.4917	0.437916
$575$	−1.24734	−0.0520178
$576$	0.618766	0.0257819
$577$	20.4404	0.850943	0.425472	−	0.904972i	$-0.360108\pi$
$577$	20.4404	0.850943	0.425472	+	0.904972i	$0.360108\pi$
$578$	26.8017	1.11480
$579$	2.82442	0.117379
$580$	85.2356	3.53922
$581$	−3.68918	−0.153053
$582$	−37.5543	−1.55668
$583$	50.3304	2.08447
$584$	−58.1464	−2.40611
$585$	−3.98905	−0.164927
$586$	2.09370	0.0864900
$587$	7.67730	0.316876	0.158438	−	0.987369i	$-0.449354\pi$
$587$	7.67730	0.316876	0.158438	+	0.987369i	$0.449354\pi$
$588$	4.52354	0.186548
$589$	6.15334	0.253544
$590$	−0.264727	−0.0108986
$591$	1.88329	0.0774681
$592$	36.9018	1.51666
$593$	31.6897	1.30134	0.650669	−	0.759361i	$-0.274488\pi$
$593$	31.6897	1.30134	0.650669	+	0.759361i	$0.274488\pi$
$594$	13.0651	0.536069
$595$	7.04231	0.288707
$596$	−38.2255	−1.56578
$597$	−13.8668	−0.567531
$598$	1.75542	0.0717845
$599$	3.29929	0.134805	0.0674027	−	0.997726i	$-0.478529\pi$
$599$	3.29929	0.134805	0.0674027	+	0.997726i	$0.478529\pi$
$600$	−16.9015	−0.690002
$601$	4.45055	0.181542	0.0907709	−	0.995872i	$-0.471067\pi$
$601$	4.45055	0.181542	0.0907709	+	0.995872i	$0.471067\pi$
$602$	−1.56618	−0.0638328
$603$	−4.09919	−0.166932
$604$	48.6532	1.97967
$605$	−41.8721	−1.70234
$606$	−1.01141	−0.0410857
$607$	−46.2011	−1.87524	−0.937622	−	0.347657i	$-0.886977\pi$
$607$	−46.2011	−1.87524	−0.937622	+	0.347657i	$0.886977\pi$
$608$	−13.5941	−0.551314
$609$	6.82497	0.276562
$610$	−97.7299	−3.95697
$611$	12.5681	0.508450
$612$	11.5386	0.466419
$613$	23.9353	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$613$	23.9353	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$614$	80.5216	3.24959
$615$	−11.3409	−0.457309
$616$	−32.9704	−1.32842
$617$	−19.6832	−0.792417	−0.396209	−	0.918161i	$-0.629674\pi$
$617$	−19.6832	−0.792417	−0.396209	+	0.918161i	$0.629674\pi$
$618$	−1.36791	−0.0550252
$619$	−28.0143	−1.12599	−0.562994	−	0.826461i	$-0.690351\pi$
$619$	−28.0143	−1.12599	−0.562994	+	0.826461i	$0.690351\pi$
$620$	−34.1942	−1.37327
$621$	−0.475677	−0.0190883
$622$	31.0527	1.24510
$623$	−14.3975	−0.576823
$624$	10.7142	0.428910
$625$	−31.2351	−1.24940
$626$	3.29929	0.131866
$627$	−11.4961	−0.459111
$628$	14.5206	0.579436
$629$	12.6937	0.506133
$630$	−7.05153	−0.280940
$631$	−1.53725	−0.0611968	−0.0305984	−	0.999532i	$-0.509741\pi$
$631$	−1.53725	−0.0611968	−0.0305984	+	0.999532i	$0.509741\pi$
$632$	−38.1128	−1.51604
$633$	20.5703	0.817597
$634$	43.3497	1.72164
$635$	36.5406	1.45007
$636$	−44.5079	−1.76485
$637$	1.44487	0.0572477
$638$	−89.1692	−3.53024
$639$	−6.35516	−0.251406
$640$	−29.0365	−1.14777
$641$	−13.3143	−0.525882	−0.262941	−	0.964812i	$-0.584693\pi$
$641$	−13.3143	−0.525882	−0.262941	+	0.964812i	$0.584693\pi$
$642$	4.11167	0.162275
$643$	35.1229	1.38511	0.692557	−	0.721363i	$-0.256484\pi$
$643$	35.1229	1.38511	0.692557	+	0.721363i	$0.256484\pi$
$644$	2.15174	0.0847906
$645$	1.69294	0.0666595
$646$	−14.6418	−0.576074
$647$	−13.1024	−0.515108	−0.257554	−	0.966264i	$-0.582917\pi$
$647$	−13.1024	−0.515108	−0.257554	+	0.966264i	$0.582917\pi$
$648$	−6.44543	−0.253201
$649$	0.192038	0.00753815
$650$	−9.67705	−0.379565
$651$	−2.73799	−0.107310
$652$	−51.3526	−2.01112
$653$	−15.1532	−0.592989	−0.296495	−	0.955035i	$-0.595818\pi$
$653$	−15.1532	−0.592989	−0.296495	+	0.955035i	$0.595818\pi$
$654$	25.6004	1.00105
$655$	−16.7004	−0.652538
$656$	30.4605	1.18928
$657$	9.02133	0.351955
$658$	22.2169	0.866104
$659$	9.57446	0.372968	0.186484	−	0.982458i	$-0.440291\pi$
$659$	9.57446	0.372968	0.186484	+	0.982458i	$0.440291\pi$
$660$	63.8840	2.48668
$661$	−46.3729	−1.80370	−0.901849	−	0.432052i	$-0.857790\pi$
$661$	−46.3729	−1.80370	−0.901849	+	0.432052i	$0.857790\pi$
$662$	−31.7911	−1.23560
$663$	3.68554	0.143134
$664$	−23.7784	−0.922780
$665$	6.20470	0.240608
$666$	−12.7104	−0.492517
$667$	3.24648	0.125704
$668$	18.3508	0.710014
$669$	−11.7886	−0.455772
$670$	−28.9056	−1.11672
$671$	70.8951	2.73688
$672$	6.04883	0.233339
$673$	−6.24932	−0.240894	−0.120447	−	0.992720i	$-0.538433\pi$
$673$	−6.24932	−0.240894	−0.120447	+	0.992720i	$0.538433\pi$
$674$	61.0692	2.35230
$675$	2.62225	0.100930
$676$	−49.3625	−1.89856
$677$	−12.1093	−0.465398	−0.232699	−	0.972549i	$-0.574756\pi$
$677$	−12.1093	−0.465398	−0.232699	+	0.972549i	$0.574756\pi$
$678$	−14.9907	−0.575716
$679$	−14.7034	−0.564265
$680$	45.3907	1.74066
$681$	−12.4986	−0.478946
$682$	35.7722	1.36979
$683$	26.7970	1.02536	0.512680	−	0.858580i	$-0.328653\pi$
$683$	26.7970	1.02536	0.512680	+	0.858580i	$0.328653\pi$
$684$	10.1662	0.388714
$685$	6.88183	0.262941
$686$	2.55412	0.0975168
$687$	8.97048	0.342245
$688$	−4.54707	−0.173355
$689$	−14.2163	−0.541597
$690$	−3.35425	−0.127694
$691$	−49.2064	−1.87190	−0.935949	−	0.352135i	$-0.885456\pi$
$691$	−49.2064	−1.87190	−0.935949	+	0.352135i	$0.885456\pi$
$692$	81.2089	3.08710
$693$	5.11531	0.194315
$694$	39.7019	1.50706
$695$	−53.7883	−2.04031
$696$	43.9899	1.66743
$697$	10.4780	0.396883
$698$	−29.4626	−1.11517
$699$	8.96532	0.339100
$700$	−11.8619	−0.448336
$701$	−44.6538	−1.68655	−0.843275	−	0.537482i	$-0.819376\pi$
$701$	−44.6538	−1.68655	−0.843275	+	0.537482i	$0.819376\pi$
$702$	−3.69036	−0.139284
$703$	11.1840	0.421811
$704$	−3.16518	−0.119292
$705$	−24.0150	−0.904458
$706$	17.8930	0.673413
$707$	−0.395991	−0.0148928
$708$	−0.169822	−0.00638230
$709$	−32.2114	−1.20972	−0.604861	−	0.796331i	$-0.706771\pi$
$709$	−32.2114	−1.20972	−0.604861	+	0.796331i	$0.706771\pi$
$710$	−44.8136	−1.68182
$711$	5.91314	0.221760
$712$	−92.7980	−3.47775
$713$	−1.30240	−0.0487752
$714$	6.51501	0.243818
$715$	20.4052	0.763112
$716$	−106.750	−3.98943
$717$	23.7750	0.887894
$718$	91.9701	3.43229
$719$	−18.8711	−0.703773	−0.351886	−	0.936043i	$-0.614460\pi$
$719$	−18.8711	−0.703773	−0.351886	+	0.936043i	$0.614460\pi$
$720$	−20.4726	−0.762969
$721$	−0.535568	−0.0199456
$722$	35.6280	1.32594
$723$	−0.00638292	−0.000237383 0
$724$	17.0884	0.635086
$725$	−17.8968	−0.664670
$726$	−38.7369	−1.43766
$727$	20.8930	0.774880	0.387440	−	0.921895i	$-0.373359\pi$
$727$	20.8930	0.774880	0.387440	+	0.921895i	$0.373359\pi$
$728$	9.31279	0.345155
$729$	1.00000	0.0370370
$730$	63.6141	2.35447
$731$	−1.56413	−0.0578516
$732$	−62.6935	−2.31722
$733$	−11.3624	−0.419679	−0.209839	−	0.977736i	$-0.567294\pi$
$733$	−11.3624	−0.419679	−0.209839	+	0.977736i	$0.567294\pi$
$734$	−9.46657	−0.349417
$735$	−2.76084	−0.101835
$736$	2.87729	0.106058
$737$	20.9687	0.772390
$738$	−10.4917	−0.386206
$739$	16.5743	0.609695	0.304847	−	0.952401i	$-0.401395\pi$
$739$	16.5743	0.609695	0.304847	+	0.952401i	$0.401395\pi$
$740$	−62.1494	−2.28466
$741$	3.24718	0.119288
$742$	−25.1304	−0.922567
$743$	6.97758	0.255983	0.127991	−	0.991775i	$-0.459147\pi$
$743$	6.97758	0.255983	0.127991	+	0.991775i	$0.459147\pi$
$744$	−17.6475	−0.646990
$745$	23.3301	0.854747
$746$	87.7559	3.21297
$747$	3.68918	0.134980
$748$	−59.0234	−2.15811
$749$	1.60982	0.0588215
$750$	−16.7668	−0.612236
$751$	32.8784	1.19975	0.599875	−	0.800094i	$-0.295217\pi$
$751$	32.8784	1.19975	0.599875	+	0.800094i	$0.295217\pi$
$752$	64.5019	2.35214
$753$	−9.29008	−0.338549
$754$	25.1866	0.917243
$755$	−29.6944	−1.08069
$756$	−4.52354	−0.164520
$757$	−4.82300	−0.175295	−0.0876474	−	0.996152i	$-0.527935\pi$
$757$	−4.82300	−0.175295	−0.0876474	+	0.996152i	$0.527935\pi$
$758$	19.3201	0.701736
$759$	2.43324	0.0883209
$760$	39.9920	1.45066
$761$	−41.0965	−1.48975	−0.744873	−	0.667206i	$-0.767490\pi$
$761$	−41.0965	−1.48975	−0.744873	+	0.667206i	$0.767490\pi$
$762$	33.8046	1.22461
$763$	10.0232	0.362863
$764$	78.9413	2.85600
$765$	−7.04231	−0.254615
$766$	−2.55412	−0.0922842
$767$	−0.0542429	−0.00195860
$768$	−28.0999	−1.01397
$769$	2.93199	0.105730	0.0528651	−	0.998602i	$-0.483165\pi$
$769$	2.93199	0.105730	0.0528651	+	0.998602i	$0.483165\pi$
$770$	36.0708	1.29990
$771$	−15.1589	−0.545935
$772$	12.7764	0.459832
$773$	−23.2948	−0.837854	−0.418927	−	0.908020i	$-0.637594\pi$
$773$	−23.2948	−0.837854	−0.418927	+	0.908020i	$0.637594\pi$
$774$	1.56618	0.0562952
$775$	7.17969	0.257902
$776$	−94.7699	−3.40204
$777$	−4.97641	−0.178528
$778$	74.9909	2.68855
$779$	9.23177	0.330762
$780$	−18.0446	−0.646101
$781$	32.5086	1.16325
$782$	3.09904	0.110821
$783$	−6.82497	−0.243905
$784$	7.41535	0.264834
$785$	−8.86233	−0.316310
$786$	−15.4499	−0.551081
$787$	−20.2751	−0.722731	−0.361365	−	0.932424i	$-0.617689\pi$
$787$	−20.2751	−0.722731	−0.361365	+	0.932424i	$0.617689\pi$
$788$	8.51913	0.303481
$789$	−7.05145	−0.251038
$790$	41.6967	1.48350
$791$	−5.86923	−0.208686
$792$	32.9704	1.17155
$793$	−20.0250	−0.711107
$794$	34.2303	1.21479
$795$	27.1644	0.963421
$796$	−62.7271	−2.22330
$797$	34.9021	1.23630	0.618148	−	0.786061i	$-0.287883\pi$
$797$	34.9021	1.23630	0.618148	+	0.786061i	$0.287883\pi$
$798$	5.74012	0.203198
$799$	22.1878	0.784949
$800$	−15.8615	−0.560790
$801$	14.3975	0.508710
$802$	−32.4760	−1.14677
$803$	−46.1469	−1.62849
$804$	−18.5429	−0.653957
$805$	−1.31327	−0.0462866
$806$	−10.1042	−0.355905
$807$	−14.9630	−0.526724
$808$	−2.55233	−0.0897907
$809$	39.4221	1.38601	0.693004	−	0.720934i	$-0.256287\pi$
$809$	39.4221	1.38601	0.693004	+	0.720934i	$0.256287\pi$
$810$	7.05153	0.247765
$811$	35.0588	1.23108	0.615540	−	0.788106i	$-0.288938\pi$
$811$	35.0588	1.23108	0.615540	+	0.788106i	$0.288938\pi$
$812$	30.8731	1.08343
$813$	5.64918	0.198125
$814$	65.0175	2.27886
$815$	31.3419	1.09786
$816$	18.9149	0.662155
$817$	−1.37810	−0.0482135
$818$	−76.8523	−2.68708
$819$	−1.44487	−0.0504877
$820$	−51.3010	−1.79151
$821$	−7.75589	−0.270682	−0.135341	−	0.990799i	$-0.543213\pi$
$821$	−7.75589	−0.270682	−0.135341	+	0.990799i	$0.543213\pi$
$822$	6.36655	0.222059
$823$	15.4966	0.540178	0.270089	−	0.962835i	$-0.412947\pi$
$823$	15.4966	0.540178	0.270089	+	0.962835i	$0.412947\pi$
$824$	−3.45197	−0.120255
$825$	−13.4136	−0.467002
$826$	−0.0958864	−0.00333631
$827$	−53.8349	−1.87202	−0.936011	−	0.351971i	$-0.885512\pi$
$827$	−53.8349	−1.87202	−0.936011	+	0.351971i	$0.885512\pi$
$828$	−2.15174	−0.0747783
$829$	−5.72631	−0.198883	−0.0994414	−	0.995043i	$-0.531706\pi$
$829$	−5.72631	−0.198883	−0.0994414	+	0.995043i	$0.531706\pi$
$830$	26.0144	0.902972
$831$	−13.6828	−0.474651
$832$	0.894034	0.0309951
$833$	2.55078	0.0883794
$834$	−49.7609	−1.72308
$835$	−11.2000	−0.387592
$836$	−52.0032	−1.79857
$837$	2.73799	0.0946388
$838$	−78.9092	−2.72587
$839$	1.65706	0.0572081	0.0286040	−	0.999591i	$-0.490894\pi$
$839$	1.65706	0.0572081	0.0286040	+	0.999591i	$0.490894\pi$
$840$	−17.7948	−0.613980
$841$	17.5803	0.606217
$842$	−59.1073	−2.03697
$843$	−3.22550	−0.111092
$844$	93.0507	3.20294
$845$	30.1273	1.03641
$846$	−22.2169	−0.763832
$847$	−15.1664	−0.521124
$848$	−72.9608	−2.50548
$849$	−16.4600	−0.564904
$850$	−17.0840	−0.585976
$851$	−2.36717	−0.0811454
$852$	−28.7478	−0.984884
$853$	−38.5082	−1.31850	−0.659248	−	0.751926i	$-0.729125\pi$
$853$	−38.5082	−1.31850	−0.659248	+	0.751926i	$0.729125\pi$
$854$	−35.3986	−1.21131
$855$	−6.20470	−0.212196
$856$	10.3760	0.354644
$857$	13.4331	0.458865	0.229432	−	0.973325i	$-0.426313\pi$
$857$	13.4331	0.458865	0.229432	+	0.973325i	$0.426313\pi$
$858$	18.8774	0.644463
$859$	19.2760	0.657690	0.328845	−	0.944384i	$-0.393341\pi$
$859$	19.2760	0.657690	0.328845	+	0.944384i	$0.393341\pi$
$860$	7.65809	0.261139
$861$	−4.10776	−0.139992
$862$	30.6124	1.04266
$863$	−32.2296	−1.09711	−0.548554	−	0.836115i	$-0.684821\pi$
$863$	−32.2296	−1.09711	−0.548554	+	0.836115i	$0.684821\pi$
$864$	−6.04883	−0.205785
$865$	−49.5640	−1.68523
$866$	4.73843	0.161018
$867$	−10.4935	−0.356378
$868$	−12.3854	−0.420388
$869$	−30.2476	−1.02608
$870$	−48.1265	−1.63164
$871$	−5.92278	−0.200686
$872$	64.6036	2.18775
$873$	14.7034	0.497635
$874$	2.73044	0.0923586
$875$	−6.56460	−0.221924
$876$	40.8083	1.37879
$877$	−18.5881	−0.627676	−0.313838	−	0.949477i	$-0.601615\pi$
$877$	−18.5881	−0.627676	−0.313838	+	0.949477i	$0.601615\pi$
$878$	−90.4294	−3.05184
$879$	−0.819734	−0.0276490
$880$	104.724	3.53024
$881$	35.4592	1.19465	0.597325	−	0.801999i	$-0.296230\pi$
$881$	35.4592	1.19465	0.597325	+	0.801999i	$0.296230\pi$
$882$	−2.55412	−0.0860018
$883$	−42.6295	−1.43460	−0.717298	−	0.696767i	$-0.754621\pi$
$883$	−42.6295	−1.43460	−0.717298	+	0.696767i	$0.754621\pi$
$884$	16.6717	0.560730
$885$	0.103647	0.00348406
$886$	−34.6400	−1.16375
$887$	0.550030	0.0184682	0.00923410	−	0.999957i	$-0.497061\pi$
$887$	0.550030	0.0184682	0.00923410	+	0.999957i	$0.497061\pi$
$888$	−32.0751	−1.07637
$889$	13.2353	0.443898
$890$	101.524	3.40310
$891$	−5.11531	−0.171369
$892$	−53.3260	−1.78549
$893$	19.5488	0.654176
$894$	21.5832	0.721850
$895$	65.1525	2.17781
$896$	−10.5173	−0.351357
$897$	−0.687289	−0.0229479
$898$	−10.0968	−0.336936
$899$	−18.6867	−0.623237
$900$	11.8619	0.395395
$901$	−25.0976	−0.836121
$902$	53.6685	1.78697
$903$	0.613198	0.0204059
$904$	−37.8298	−1.25820
$905$	−10.4295	−0.346689
$906$	−27.4710	−0.912662
$907$	−16.3174	−0.541811	−0.270905	−	0.962606i	$-0.587323\pi$
$907$	−16.3174	−0.541811	−0.270905	+	0.962606i	$0.587323\pi$
$908$	−56.5377	−1.87627
$909$	0.395991	0.0131342
$910$	−10.1885	−0.337746
$911$	−35.1753	−1.16541	−0.582706	−	0.812683i	$-0.698006\pi$
$911$	−35.1753	−1.16541	−0.582706	+	0.812683i	$0.698006\pi$
$912$	16.6652	0.551840
$913$	−18.8713	−0.624549
$914$	−23.3919	−0.773737
$915$	38.2636	1.26496
$916$	40.5783	1.34075
$917$	−6.04902	−0.199756
$918$	−6.51501	−0.215027
$919$	−25.0341	−0.825799	−0.412899	−	0.910777i	$-0.635484\pi$
$919$	−25.0341	−0.825799	−0.412899	+	0.910777i	$0.635484\pi$
$920$	−8.46459	−0.279069
$921$	−31.5261	−1.03882
$922$	34.4672	1.13512
$923$	−9.18235	−0.302241
$924$	23.1393	0.761228
$925$	13.0494	0.429061
$926$	52.5363	1.72645
$927$	0.535568	0.0175903
$928$	41.2831	1.35518
$929$	4.04753	0.132795	0.0663976	−	0.997793i	$-0.478849\pi$
$929$	4.04753	0.132795	0.0663976	+	0.997793i	$0.478849\pi$
$930$	19.3070	0.633102
$931$	2.24739	0.0736554
$932$	40.5550	1.32842
$933$	−12.1579	−0.398031
$934$	−76.4576	−2.50177
$935$	36.0236	1.17810
$936$	−9.31279	−0.304398
$937$	−19.4147	−0.634249	−0.317125	−	0.948384i	$-0.602717\pi$
$937$	−19.4147	−0.634249	−0.317125	+	0.948384i	$0.602717\pi$
$938$	−10.4698	−0.341852
$939$	−1.29175	−0.0421547
$940$	−108.633	−3.54322
$941$	0.959736	0.0312865	0.0156432	−	0.999878i	$-0.495020\pi$
$941$	0.959736	0.0312865	0.0156432	+	0.999878i	$0.495020\pi$
$942$	−8.19876	−0.267130
$943$	−1.95397	−0.0636300
$944$	−0.278385	−0.00906067
$945$	2.76084	0.0898102
$946$	−8.01151	−0.260477
$947$	0.837814	0.0272253	0.0136127	−	0.999907i	$-0.495667\pi$
$947$	0.837814	0.0272253	0.0136127	+	0.999907i	$0.495667\pi$
$948$	26.7483	0.868746
$949$	13.0346	0.423121
$950$	−15.0520	−0.488352
$951$	−16.9724	−0.550369
$952$	16.4409	0.532853
$953$	−32.5770	−1.05527	−0.527636	−	0.849471i	$-0.676921\pi$
$953$	−32.5770	−1.05527	−0.527636	+	0.849471i	$0.676921\pi$
$954$	25.1304	0.813628
$955$	−48.1801	−1.55907
$956$	107.547	3.47833
$957$	34.9119	1.12854
$958$	−28.9425	−0.935091
$959$	2.49266	0.0804920
$960$	−1.70832	−0.0551357
$961$	−23.5034	−0.758175
$962$	−18.3648	−0.592104
$963$	−1.60982	−0.0518757
$964$	−0.0288734	−0.000929949 0
$965$	−7.79777	−0.251019
$966$	−1.21494	−0.0390900
$967$	11.9863	0.385453	0.192727	−	0.981252i	$-0.438267\pi$
$967$	11.9863	0.385453	0.192727	+	0.981252i	$0.438267\pi$
$968$	−97.7541	−3.14194
$969$	5.73261	0.184158
$970$	103.682	3.32901
$971$	−50.2914	−1.61393	−0.806963	−	0.590602i	$-0.798890\pi$
$971$	−50.2914	−1.61393	−0.806963	+	0.590602i	$0.798890\pi$
$972$	4.52354	0.145093
$973$	−19.4826	−0.624582
$974$	−43.4996	−1.39382
$975$	3.78880	0.121339
$976$	−102.772	−3.28966
$977$	−34.5751	−1.10616	−0.553078	−	0.833130i	$-0.686547\pi$
$977$	−34.5751	−1.10616	−0.553078	+	0.833130i	$0.686547\pi$
$978$	28.9951	0.927163
$979$	−73.6476	−2.35379
$980$	−12.4888	−0.398940
$981$	−10.0232	−0.320015
$982$	8.20689	0.261892
$983$	−56.6802	−1.80782	−0.903909	−	0.427725i	$-0.859315\pi$
$983$	−56.6802	−1.80782	−0.903909	+	0.427725i	$0.859315\pi$
$984$	−26.4763	−0.844034
$985$	−5.19946	−0.165669
$986$	44.4648	1.41605
$987$	−8.69844	−0.276874
$988$	14.6888	0.467312
$989$	0.291684	0.00927501
$990$	−36.0708	−1.14640
$991$	−33.8774	−1.07615	−0.538075	−	0.842897i	$-0.680848\pi$
$991$	−33.8774	−1.07615	−0.538075	+	0.842897i	$0.680848\pi$
$992$	−16.5616	−0.525833
$993$	12.4470	0.394993
$994$	−16.2319	−0.514843
$995$	38.2841	1.21369
$996$	16.6882	0.528785
$997$	−36.8434	−1.16684	−0.583421	−	0.812170i	$-0.698286\pi$
$997$	−36.8434	−1.16684	−0.583421	+	0.812170i	$0.698286\pi$
$998$	70.1609	2.22090
$999$	4.97641	0.157447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.q.1.3	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.q.1.3	✓	44	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)

\(f(q)\)	\(=\)	\(q-2.55412 q^{2} +1.00000 q^{3} +4.52354 q^{4} -2.76084 q^{5} -2.55412 q^{6} -1.00000 q^{7} -6.44543 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.55412 q^{2} +1.00000 q^{3} +4.52354 q^{4} -2.76084 q^{5} -2.55412 q^{6} -1.00000 q^{7} -6.44543 q^{8} +1.00000 q^{9} +7.05153 q^{10} -5.11531 q^{11} +4.52354 q^{12} +1.44487 q^{13} +2.55412 q^{14} -2.76084 q^{15} +7.41535 q^{16} +2.55078 q^{17} -2.55412 q^{18} +2.24739 q^{19} -12.4888 q^{20} -1.00000 q^{21} +13.0651 q^{22} -0.475677 q^{23} -6.44543 q^{24} +2.62225 q^{25} -3.69036 q^{26} +1.00000 q^{27} -4.52354 q^{28} -6.82497 q^{29} +7.05153 q^{30} +2.73799 q^{31} -6.04883 q^{32} -5.11531 q^{33} -6.51501 q^{34} +2.76084 q^{35} +4.52354 q^{36} +4.97641 q^{37} -5.74012 q^{38} +1.44487 q^{39} +17.7948 q^{40} +4.10776 q^{41} +2.55412 q^{42} -0.613198 q^{43} -23.1393 q^{44} -2.76084 q^{45} +1.21494 q^{46} +8.69844 q^{47} +7.41535 q^{48} +1.00000 q^{49} -6.69754 q^{50} +2.55078 q^{51} +6.53591 q^{52} -9.83916 q^{53} -2.55412 q^{54} +14.1226 q^{55} +6.44543 q^{56} +2.24739 q^{57} +17.4318 q^{58} -0.0375418 q^{59} -12.4888 q^{60} -13.8594 q^{61} -6.99316 q^{62} -1.00000 q^{63} +0.618766 q^{64} -3.98905 q^{65} +13.0651 q^{66} -4.09919 q^{67} +11.5386 q^{68} -0.475677 q^{69} -7.05153 q^{70} -6.35516 q^{71} -6.44543 q^{72} +9.02133 q^{73} -12.7104 q^{74} +2.62225 q^{75} +10.1662 q^{76} +5.11531 q^{77} -3.69036 q^{78} +5.91314 q^{79} -20.4726 q^{80} +1.00000 q^{81} -10.4917 q^{82} +3.68918 q^{83} -4.52354 q^{84} -7.04231 q^{85} +1.56618 q^{86} -6.82497 q^{87} +32.9704 q^{88} +14.3975 q^{89} +7.05153 q^{90} -1.44487 q^{91} -2.15174 q^{92} +2.73799 q^{93} -22.2169 q^{94} -6.20470 q^{95} -6.04883 q^{96} +14.7034 q^{97} -2.55412 q^{98} -5.11531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{2} + 44 q^{3} + 44 q^{4} - 16 q^{5} - 4 q^{6} - 44 q^{7} - 15 q^{8} + 44 q^{9}+O(q^{10}) \) 44 * q - 4 * q^2 + 44 * q^3 + 44 * q^4 - 16 * q^5 - 4 * q^6 - 44 * q^7 - 15 * q^8 + 44 * q^9	\( 44 q - 4 q^{2} + 44 q^{3} + 44 q^{4} - 16 q^{5} - 4 q^{6} - 44 q^{7} - 15 q^{8} + 44 q^{9} - 16 q^{10} - 2 q^{11} + 44 q^{12} - 34 q^{13} + 4 q^{14} - 16 q^{15} + 24 q^{16} - 4 q^{17} - 4 q^{18} - 22 q^{19} - 39 q^{20} - 44 q^{21} - 23 q^{22} - 56 q^{23} - 15 q^{24} + 32 q^{25} - 17 q^{26} + 44 q^{27} - 44 q^{28} - 33 q^{29} - 16 q^{30} - 32 q^{31} - 34 q^{32} - 2 q^{33} - 25 q^{34} + 16 q^{35} + 44 q^{36} - 47 q^{37} - 40 q^{38} - 34 q^{39} - 50 q^{40} + 2 q^{41} + 4 q^{42} - 12 q^{43} - 22 q^{44} - 16 q^{45} + 8 q^{46} - 27 q^{47} + 24 q^{48} + 44 q^{49} - 21 q^{50} - 4 q^{51} - 82 q^{52} - 114 q^{53} - 4 q^{54} - 29 q^{55} + 15 q^{56} - 22 q^{57} - 26 q^{58} - 40 q^{59} - 39 q^{60} - 47 q^{61} - 37 q^{62} - 44 q^{63} - 5 q^{64} - 20 q^{65} - 23 q^{66} - 14 q^{67} - 72 q^{68} - 56 q^{69} + 16 q^{70} - 65 q^{71} - 15 q^{72} - 21 q^{73} - 26 q^{74} + 32 q^{75} - 15 q^{76} + 2 q^{77} - 17 q^{78} + 6 q^{79} - 77 q^{80} + 44 q^{81} - 51 q^{82} - 30 q^{83} - 44 q^{84} - 26 q^{85} - 65 q^{86} - 33 q^{87} - 84 q^{88} - 32 q^{89} - 16 q^{90} + 34 q^{91} - 140 q^{92} - 32 q^{93} - 35 q^{94} - 50 q^{95} - 34 q^{96} - 83 q^{97} - 4 q^{98} - 2 q^{99}+O(q^{100}) \) 44 * q - 4 * q^2 + 44 * q^3 + 44 * q^4 - 16 * q^5 - 4 * q^6 - 44 * q^7 - 15 * q^8 + 44 * q^9 - 16 * q^10 - 2 * q^11 + 44 * q^12 - 34 * q^13 + 4 * q^14 - 16 * q^15 + 24 * q^16 - 4 * q^17 - 4 * q^18 - 22 * q^19 - 39 * q^20 - 44 * q^21 - 23 * q^22 - 56 * q^23 - 15 * q^24 + 32 * q^25 - 17 * q^26 + 44 * q^27 - 44 * q^28 - 33 * q^29 - 16 * q^30 - 32 * q^31 - 34 * q^32 - 2 * q^33 - 25 * q^34 + 16 * q^35 + 44 * q^36 - 47 * q^37 - 40 * q^38 - 34 * q^39 - 50 * q^40 + 2 * q^41 + 4 * q^42 - 12 * q^43 - 22 * q^44 - 16 * q^45 + 8 * q^46 - 27 * q^47 + 24 * q^48 + 44 * q^49 - 21 * q^50 - 4 * q^51 - 82 * q^52 - 114 * q^53 - 4 * q^54 - 29 * q^55 + 15 * q^56 - 22 * q^57 - 26 * q^58 - 40 * q^59 - 39 * q^60 - 47 * q^61 - 37 * q^62 - 44 * q^63 - 5 * q^64 - 20 * q^65 - 23 * q^66 - 14 * q^67 - 72 * q^68 - 56 * q^69 + 16 * q^70 - 65 * q^71 - 15 * q^72 - 21 * q^73 - 26 * q^74 + 32 * q^75 - 15 * q^76 + 2 * q^77 - 17 * q^78 + 6 * q^79 - 77 * q^80 + 44 * q^81 - 51 * q^82 - 30 * q^83 - 44 * q^84 - 26 * q^85 - 65 * q^86 - 33 * q^87 - 84 * q^88 - 32 * q^89 - 16 * q^90 + 34 * q^91 - 140 * q^92 - 32 * q^93 - 35 * q^94 - 50 * q^95 - 34 * q^96 - 83 * q^97 - 4 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more