LMFDB - Embedded newform 6171.2.a.bs.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6171,2,Mod(1,6171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6171, 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("6171.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6171 = 3 \cdot 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6171.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:	$49.2756830873$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3 x^{19} - 28 x^{18} + 85 x^{17} + 320 x^{16} - 989 x^{15} - 1923 x^{14} + 6124 x^{13} + \cdots + 16 $ x^20 - 3x^19 - 28x^18 + 85x^17 + 320x^16 - 989x^15 - 1923x^14 + 6124x^13 + 6496x^12 - 21860x^11 - 12084x^10 + 45502x^9 + 10780x^8 - 53060x^7 - 2295x^6 + 31131x^5 - 1511x^4 - 7302x^3 + 356x^2 + 264*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 561)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Root			$2.78024$ of defining polynomial
Character	$\chi$	$=$	6171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.78024 q^{2} -1.00000 q^{3} +5.72972 q^{4} -2.98671 q^{5} -2.78024 q^{6} -0.971995 q^{7} +10.3695 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.78024 q^{2} -1.00000 q^{3} +5.72972 q^{4} -2.98671 q^{5} -2.78024 q^{6} -0.971995 q^{7} +10.3695 q^{8} +1.00000 q^{9} -8.30376 q^{10} -5.72972 q^{12} +3.86468 q^{13} -2.70238 q^{14} +2.98671 q^{15} +17.3703 q^{16} +1.00000 q^{17} +2.78024 q^{18} -0.486698 q^{19} -17.1130 q^{20} +0.971995 q^{21} -2.94701 q^{23} -10.3695 q^{24} +3.92042 q^{25} +10.7447 q^{26} -1.00000 q^{27} -5.56926 q^{28} -5.61400 q^{29} +8.30376 q^{30} +10.7749 q^{31} +27.5545 q^{32} +2.78024 q^{34} +2.90306 q^{35} +5.72972 q^{36} -3.47934 q^{37} -1.35314 q^{38} -3.86468 q^{39} -30.9707 q^{40} +12.2834 q^{41} +2.70238 q^{42} -0.330564 q^{43} -2.98671 q^{45} -8.19338 q^{46} -2.04545 q^{47} -17.3703 q^{48} -6.05523 q^{49} +10.8997 q^{50} -1.00000 q^{51} +22.1436 q^{52} -4.26110 q^{53} -2.78024 q^{54} -10.0791 q^{56} +0.486698 q^{57} -15.6083 q^{58} +12.2107 q^{59} +17.1130 q^{60} +6.07107 q^{61} +29.9567 q^{62} -0.971995 q^{63} +41.8675 q^{64} -11.5427 q^{65} +2.98566 q^{67} +5.72972 q^{68} +2.94701 q^{69} +8.07121 q^{70} +8.00676 q^{71} +10.3695 q^{72} +2.88454 q^{73} -9.67339 q^{74} -3.92042 q^{75} -2.78864 q^{76} -10.7447 q^{78} +9.90177 q^{79} -51.8800 q^{80} +1.00000 q^{81} +34.1508 q^{82} +3.25245 q^{83} +5.56926 q^{84} -2.98671 q^{85} -0.919047 q^{86} +5.61400 q^{87} +6.09862 q^{89} -8.30376 q^{90} -3.75645 q^{91} -16.8855 q^{92} -10.7749 q^{93} -5.68683 q^{94} +1.45362 q^{95} -27.5545 q^{96} -6.17135 q^{97} -16.8350 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 3 q^{2} - 20 q^{3} + 25 q^{4} + 7 q^{5} - 3 q^{6} - 5 q^{7} + 12 q^{8} + 20 q^{9}+O(q^{10}) $ 20 * q + 3 * q^2 - 20 * q^3 + 25 * q^4 + 7 * q^5 - 3 * q^6 - 5 * q^7 + 12 * q^8 + 20 * q^9	$ 20 q + 3 q^{2} - 20 q^{3} + 25 q^{4} + 7 q^{5} - 3 q^{6} - 5 q^{7} + 12 q^{8} + 20 q^{9} + 4 q^{10} - 25 q^{12} + 8 q^{13} + 3 q^{14} - 7 q^{15} + 47 q^{16} + 20 q^{17} + 3 q^{18} + 6 q^{19} + 26 q^{20} + 5 q^{21} - 16 q^{23} - 12 q^{24} + 31 q^{25} + 15 q^{26} - 20 q^{27} - 15 q^{28} + 4 q^{29} - 4 q^{30} + 27 q^{31} + 47 q^{32} + 3 q^{34} - 2 q^{35} + 25 q^{36} + 36 q^{37} - 8 q^{39} - 31 q^{40} + 6 q^{41} - 3 q^{42} + 46 q^{43} + 7 q^{45} - 25 q^{46} - 24 q^{47} - 47 q^{48} + 73 q^{49} - 20 q^{51} + 8 q^{52} + 6 q^{53} - 3 q^{54} + 39 q^{56} - 6 q^{57} - 25 q^{58} + 4 q^{59} - 26 q^{60} - 38 q^{61} - 9 q^{62} - 5 q^{63} + 100 q^{64} + 26 q^{65} + 21 q^{67} + 25 q^{68} + 16 q^{69} + 9 q^{70} - 24 q^{71} + 12 q^{72} + 5 q^{73} - 94 q^{74} - 31 q^{75} + 81 q^{76} - 15 q^{78} - 47 q^{79} + 17 q^{80} + 20 q^{81} + 75 q^{82} + 6 q^{83} + 15 q^{84} + 7 q^{85} + 54 q^{86} - 4 q^{87} + 44 q^{89} + 4 q^{90} + 37 q^{91} - 28 q^{92} - 27 q^{93} - 36 q^{94} + 18 q^{95} - 47 q^{96} + 19 q^{97} + 81 q^{98}+O(q^{100}) $ 20 * q + 3 * q^2 - 20 * q^3 + 25 * q^4 + 7 * q^5 - 3 * q^6 - 5 * q^7 + 12 * q^8 + 20 * q^9 + 4 * q^10 - 25 * q^12 + 8 * q^13 + 3 * q^14 - 7 * q^15 + 47 * q^16 + 20 * q^17 + 3 * q^18 + 6 * q^19 + 26 * q^20 + 5 * q^21 - 16 * q^23 - 12 * q^24 + 31 * q^25 + 15 * q^26 - 20 * q^27 - 15 * q^28 + 4 * q^29 - 4 * q^30 + 27 * q^31 + 47 * q^32 + 3 * q^34 - 2 * q^35 + 25 * q^36 + 36 * q^37 - 8 * q^39 - 31 * q^40 + 6 * q^41 - 3 * q^42 + 46 * q^43 + 7 * q^45 - 25 * q^46 - 24 * q^47 - 47 * q^48 + 73 * q^49 - 20 * q^51 + 8 * q^52 + 6 * q^53 - 3 * q^54 + 39 * q^56 - 6 * q^57 - 25 * q^58 + 4 * q^59 - 26 * q^60 - 38 * q^61 - 9 * q^62 - 5 * q^63 + 100 * q^64 + 26 * q^65 + 21 * q^67 + 25 * q^68 + 16 * q^69 + 9 * q^70 - 24 * q^71 + 12 * q^72 + 5 * q^73 - 94 * q^74 - 31 * q^75 + 81 * q^76 - 15 * q^78 - 47 * q^79 + 17 * q^80 + 20 * q^81 + 75 * q^82 + 6 * q^83 + 15 * q^84 + 7 * q^85 + 54 * q^86 - 4 * q^87 + 44 * q^89 + 4 * q^90 + 37 * q^91 - 28 * q^92 - 27 * q^93 - 36 * q^94 + 18 * q^95 - 47 * q^96 + 19 * q^97 + 81 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.78024	1.96593	0.982963	−	0.183806i	$-0.0588417\pi$
$2$	2.78024	1.96593	0.982963	+	0.183806i	$0.0588417\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.72972	2.86486
$5$	−2.98671	−1.33570	−0.667848	−	0.744298i	$-0.732784\pi$
$5$	−2.98671	−1.33570	−0.667848	+	0.744298i	$0.732784\pi$
$6$	−2.78024	−1.13503
$7$	−0.971995	−0.367380	−0.183690	−	0.982984i	$-0.558804\pi$
$7$	−0.971995	−0.367380	−0.183690	+	0.982984i	$0.558804\pi$
$8$	10.3695	3.66618
$9$	1.00000	0.333333
$10$	−8.30376	−2.62588
$11$	0	0
$11$	0	0
$12$	−5.72972	−1.65403
$13$	3.86468	1.07187	0.535935	−	0.844259i	$-0.319959\pi$
$13$	3.86468	1.07187	0.535935	+	0.844259i	$0.319959\pi$
$14$	−2.70238	−0.722241
$15$	2.98671	0.771164
$16$	17.3703	4.34257
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	2.78024	0.655308
$19$	−0.486698	−0.111656	−0.0558281	−	0.998440i	$-0.517780\pi$
$19$	−0.486698	−0.111656	−0.0558281	+	0.998440i	$0.517780\pi$
$20$	−17.1130	−3.82658
$21$	0.971995	0.212107
$22$	0	0
$23$	−2.94701	−0.614494	−0.307247	−	0.951630i	$-0.599408\pi$
$23$	−2.94701	−0.614494	−0.307247	+	0.951630i	$0.599408\pi$
$24$	−10.3695	−2.11667
$25$	3.92042	0.784084
$26$	10.7447	2.10722
$27$	−1.00000	−0.192450
$28$	−5.56926	−1.05249
$29$	−5.61400	−1.04249	−0.521247	−	0.853406i	$-0.674533\pi$
$29$	−5.61400	−1.04249	−0.521247	+	0.853406i	$0.674533\pi$
$30$	8.30376	1.51605
$31$	10.7749	1.93522	0.967611	−	0.252446i	$-0.0812349\pi$
$31$	10.7749	1.93522	0.967611	+	0.252446i	$0.0812349\pi$
$32$	27.5545	4.87099
$33$	0	0
$34$	2.78024	0.476807
$35$	2.90306	0.490708
$36$	5.72972	0.954954
$37$	−3.47934	−0.572000	−0.286000	−	0.958230i	$-0.592326\pi$
$37$	−3.47934	−0.572000	−0.286000	+	0.958230i	$0.592326\pi$
$38$	−1.35314	−0.219508
$39$	−3.86468	−0.618845
$40$	−30.9707	−4.89690
$41$	12.2834	1.91834	0.959172	−	0.282822i	$-0.0912706\pi$
$41$	12.2834	1.91834	0.959172	+	0.282822i	$0.0912706\pi$
$42$	2.70238	0.416986
$43$	−0.330564	−0.0504106	−0.0252053	−	0.999682i	$-0.508024\pi$
$43$	−0.330564	−0.0504106	−0.0252053	+	0.999682i	$0.508024\pi$
$44$	0	0
$45$	−2.98671	−0.445232
$46$	−8.19338	−1.20805
$47$	−2.04545	−0.298359	−0.149180	−	0.988810i	$-0.547663\pi$
$47$	−2.04545	−0.298359	−0.149180	+	0.988810i	$0.547663\pi$
$48$	−17.3703	−2.50719
$49$	−6.05523	−0.865032
$50$	10.8997	1.54145
$51$	−1.00000	−0.140028
$52$	22.1436	3.07076
$53$	−4.26110	−0.585306	−0.292653	−	0.956219i	$-0.594538\pi$
$53$	−4.26110	−0.585306	−0.292653	+	0.956219i	$0.594538\pi$
$54$	−2.78024	−0.378342
$55$	0	0
$56$	−10.0791	−1.34688
$57$	0.486698	0.0644647
$58$	−15.6083	−2.04947
$59$	12.2107	1.58970	0.794850	−	0.606807i	$-0.207550\pi$
$59$	12.2107	1.58970	0.794850	+	0.606807i	$0.207550\pi$
$60$	17.1130	2.20928
$61$	6.07107	0.777321	0.388661	−	0.921381i	$-0.372938\pi$
$61$	6.07107	0.777321	0.388661	+	0.921381i	$0.372938\pi$
$62$	29.9567	3.80450
$63$	−0.971995	−0.122460
$64$	41.8675	5.23344
$65$	−11.5427	−1.43169
$66$	0	0
$67$	2.98566	0.364757	0.182378	−	0.983228i	$-0.441620\pi$
$67$	2.98566	0.364757	0.182378	+	0.983228i	$0.441620\pi$
$68$	5.72972	0.694831
$69$	2.94701	0.354778
$70$	8.07121	0.964694
$71$	8.00676	0.950228	0.475114	−	0.879924i	$-0.342407\pi$
$71$	8.00676	0.950228	0.475114	+	0.879924i	$0.342407\pi$
$72$	10.3695	1.22206
$73$	2.88454	0.337610	0.168805	−	0.985649i	$-0.446009\pi$
$73$	2.88454	0.337610	0.168805	+	0.985649i	$0.446009\pi$
$74$	−9.67339	−1.12451
$75$	−3.92042	−0.452691
$76$	−2.78864	−0.319879
$77$	0	0
$78$	−10.7447	−1.21660
$79$	9.90177	1.11404	0.557018	−	0.830500i	$-0.311945\pi$
$79$	9.90177	1.11404	0.557018	+	0.830500i	$0.311945\pi$
$80$	−51.8800	−5.80036
$81$	1.00000	0.111111
$82$	34.1508	3.77132
$83$	3.25245	0.357003	0.178501	−	0.983940i	$-0.442875\pi$
$83$	3.25245	0.357003	0.178501	+	0.983940i	$0.442875\pi$
$84$	5.56926	0.607657
$85$	−2.98671	−0.323954
$86$	−0.919047	−0.0991034
$87$	5.61400	0.601884
$88$	0	0
$89$	6.09862	0.646452	0.323226	−	0.946322i	$-0.395233\pi$
$89$	6.09862	0.646452	0.323226	+	0.946322i	$0.395233\pi$
$90$	−8.30376	−0.875293
$91$	−3.75645	−0.393783
$92$	−16.8855	−1.76044
$93$	−10.7749	−1.11730
$94$	−5.68683	−0.586552
$95$	1.45362	0.149139
$96$	−27.5545	−2.81227
$97$	−6.17135	−0.626606	−0.313303	−	0.949653i	$-0.601436\pi$
$97$	−6.17135	−0.626606	−0.313303	+	0.949653i	$0.601436\pi$
$98$	−16.8350	−1.70059
$99$	0	0
$100$	22.4629	2.24629
$101$	5.46420	0.543708	0.271854	−	0.962338i	$-0.412363\pi$
$101$	5.46420	0.543708	0.271854	+	0.962338i	$0.412363\pi$
$102$	−2.78024	−0.275285
$103$	4.46977	0.440420	0.220210	−	0.975453i	$-0.429326\pi$
$103$	4.46977	0.440420	0.220210	+	0.975453i	$0.429326\pi$
$104$	40.0749	3.92967
$105$	−2.90306	−0.283310
$106$	−11.8469	−1.15067
$107$	−13.7239	−1.32674	−0.663372	−	0.748290i	$-0.730875\pi$
$107$	−13.7239	−1.32674	−0.663372	+	0.748290i	$0.730875\pi$
$108$	−5.72972	−0.551343
$109$	14.5393	1.39262	0.696308	−	0.717743i	$-0.254825\pi$
$109$	14.5393	1.39262	0.696308	+	0.717743i	$0.254825\pi$
$110$	0	0
$111$	3.47934	0.330244
$112$	−16.8838	−1.59537
$113$	−17.3353	−1.63077	−0.815385	−	0.578919i	$-0.803475\pi$
$113$	−17.3353	−1.63077	−0.815385	+	0.578919i	$0.803475\pi$
$114$	1.35314	0.126733
$115$	8.80185	0.820777
$116$	−32.1667	−2.98660
$117$	3.86468	0.357290
$118$	33.9487	3.12523
$119$	−0.971995	−0.0891027
$120$	30.9707	2.82723
$121$	0	0
$122$	16.8790	1.52816
$123$	−12.2834	−1.10756
$124$	61.7370	5.54414
$125$	3.22439	0.288398
$126$	−2.70238	−0.240747
$127$	11.7286	1.04074	0.520370	−	0.853941i	$-0.325794\pi$
$127$	11.7286	1.04074	0.520370	+	0.853941i	$0.325794\pi$
$128$	61.2926	5.41755
$129$	0.330564	0.0291046
$130$	−32.0914	−2.81460
$131$	−3.02398	−0.264206	−0.132103	−	0.991236i	$-0.542173\pi$
$131$	−3.02398	−0.264206	−0.132103	+	0.991236i	$0.542173\pi$
$132$	0	0
$133$	0.473068	0.0410202
$134$	8.30085	0.717085
$135$	2.98671	0.257055
$136$	10.3695	0.889179
$137$	−9.98452	−0.853035	−0.426518	−	0.904479i	$-0.640260\pi$
$137$	−9.98452	−0.853035	−0.426518	+	0.904479i	$0.640260\pi$
$138$	8.19338	0.697467
$139$	9.04644	0.767309	0.383655	−	0.923477i	$-0.374665\pi$
$139$	9.04644	0.767309	0.383655	+	0.923477i	$0.374665\pi$
$140$	16.6338	1.40581
$141$	2.04545	0.172258
$142$	22.2607	1.86808
$143$	0	0
$144$	17.3703	1.44752
$145$	16.7674	1.39246
$146$	8.01971	0.663716
$147$	6.05523	0.499427
$148$	−19.9357	−1.63870
$149$	21.5740	1.76741	0.883707	−	0.468041i	$-0.155040\pi$
$149$	21.5740	1.76741	0.883707	+	0.468041i	$0.155040\pi$
$150$	−10.8997	−0.889957
$151$	−10.1048	−0.822315	−0.411158	−	0.911564i	$-0.634875\pi$
$151$	−10.1048	−0.822315	−0.411158	+	0.911564i	$0.634875\pi$
$152$	−5.04682	−0.409351
$153$	1.00000	0.0808452
$154$	0	0
$155$	−32.1814	−2.58487
$156$	−22.1436	−1.77290
$157$	15.7805	1.25942	0.629712	−	0.776828i	$-0.283173\pi$
$157$	15.7805	1.25942	0.629712	+	0.776828i	$0.283173\pi$
$158$	27.5293	2.19011
$159$	4.26110	0.337927
$160$	−82.2972	−6.50617
$161$	2.86448	0.225752
$162$	2.78024	0.218436
$163$	2.39457	0.187557	0.0937785	−	0.995593i	$-0.470105\pi$
$163$	2.39457	0.187557	0.0937785	+	0.995593i	$0.470105\pi$
$164$	70.3805	5.49579
$165$	0	0
$166$	9.04258	0.701841
$167$	−9.28068	−0.718160	−0.359080	−	0.933307i	$-0.616910\pi$
$167$	−9.28068	−0.718160	−0.359080	+	0.933307i	$0.616910\pi$
$168$	10.0791	0.777621
$169$	1.93579	0.148907
$170$	−8.30376	−0.636869
$171$	−0.486698	−0.0372187
$172$	−1.89404	−0.144419
$173$	−21.9777	−1.67093	−0.835467	−	0.549540i	$-0.814803\pi$
$173$	−21.9777	−1.67093	−0.835467	+	0.549540i	$0.814803\pi$
$174$	15.6083	1.18326
$175$	−3.81063	−0.288056
$176$	0	0
$177$	−12.2107	−0.917813
$178$	16.9556	1.27088
$179$	22.7129	1.69765	0.848823	−	0.528678i	$-0.177312\pi$
$179$	22.7129	1.69765	0.848823	+	0.528678i	$0.177312\pi$
$180$	−17.1130	−1.27553
$181$	13.9255	1.03508	0.517539	−	0.855660i	$-0.326848\pi$
$181$	13.9255	1.03508	0.517539	+	0.855660i	$0.326848\pi$
$182$	−10.4438	−0.774149
$183$	−6.07107	−0.448787
$184$	−30.5591	−2.25284
$185$	10.3918	0.764018
$186$	−29.9567	−2.19653
$187$	0	0
$188$	−11.7199	−0.854758
$189$	0.971995	0.0707022
$190$	4.04142	0.293196
$191$	−26.2086	−1.89639	−0.948194	−	0.317691i	$-0.897093\pi$
$191$	−26.2086	−1.89639	−0.948194	+	0.317691i	$0.897093\pi$
$192$	−41.8675	−3.02153
$193$	−8.70855	−0.626855	−0.313427	−	0.949612i	$-0.601477\pi$
$193$	−8.70855	−0.626855	−0.313427	+	0.949612i	$0.601477\pi$
$194$	−17.1578	−1.23186
$195$	11.5427	0.826588
$196$	−34.6948	−2.47820
$197$	3.32716	0.237050	0.118525	−	0.992951i	$-0.462183\pi$
$197$	3.32716	0.237050	0.118525	+	0.992951i	$0.462183\pi$
$198$	0	0
$199$	−4.15741	−0.294711	−0.147355	−	0.989084i	$-0.547076\pi$
$199$	−4.15741	−0.294711	−0.147355	+	0.989084i	$0.547076\pi$
$200$	40.6529	2.87459
$201$	−2.98566	−0.210592
$202$	15.1918	1.06889
$203$	5.45679	0.382991
$204$	−5.72972	−0.401161
$205$	−36.6869	−2.56233
$206$	12.4270	0.865832
$207$	−2.94701	−0.204831
$208$	67.1307	4.65468
$209$	0	0
$210$	−8.07121	−0.556967
$211$	2.46075	0.169405	0.0847025	−	0.996406i	$-0.473006\pi$
$211$	2.46075	0.169405	0.0847025	+	0.996406i	$0.473006\pi$
$212$	−24.4149	−1.67682
$213$	−8.00676	−0.548614
$214$	−38.1558	−2.60828
$215$	0.987298	0.0673332
$216$	−10.3695	−0.705556
$217$	−10.4731	−0.710961
$218$	40.4228	2.73778
$219$	−2.88454	−0.194919
$220$	0	0
$221$	3.86468	0.259967
$222$	9.67339	0.649236
$223$	−19.3895	−1.29841	−0.649207	−	0.760612i	$-0.724899\pi$
$223$	−19.3895	−1.29841	−0.649207	+	0.760612i	$0.724899\pi$
$224$	−26.7828	−1.78950
$225$	3.92042	0.261361
$226$	−48.1963	−3.20597
$227$	16.4562	1.09224	0.546120	−	0.837707i	$-0.316104\pi$
$227$	16.4562	1.09224	0.546120	+	0.837707i	$0.316104\pi$
$228$	2.78864	0.184683
$229$	−14.2382	−0.940887	−0.470444	−	0.882430i	$-0.655906\pi$
$229$	−14.2382	−0.940887	−0.470444	+	0.882430i	$0.655906\pi$
$230$	24.4712	1.61359
$231$	0	0
$232$	−58.2145	−3.82197
$233$	−10.0178	−0.656290	−0.328145	−	0.944627i	$-0.606423\pi$
$233$	−10.0178	−0.656290	−0.328145	+	0.944627i	$0.606423\pi$
$234$	10.7447	0.702406
$235$	6.10916	0.398517
$236$	69.9640	4.55427
$237$	−9.90177	−0.643189
$238$	−2.70238	−0.175169
$239$	5.49305	0.355316	0.177658	−	0.984092i	$-0.443148\pi$
$239$	5.49305	0.355316	0.177658	+	0.984092i	$0.443148\pi$
$240$	51.8800	3.34884
$241$	11.1122	0.715799	0.357900	−	0.933760i	$-0.383493\pi$
$241$	11.1122	0.715799	0.357900	+	0.933760i	$0.383493\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	34.7856	2.22692
$245$	18.0852	1.15542
$246$	−34.1508	−2.17737
$247$	−1.88093	−0.119681
$248$	111.730	7.09487
$249$	−3.25245	−0.206116
$250$	8.96458	0.566970
$251$	−8.58540	−0.541906	−0.270953	−	0.962593i	$-0.587339\pi$
$251$	−8.58540	−0.541906	−0.270953	+	0.962593i	$0.587339\pi$
$252$	−5.56926	−0.350831
$253$	0	0
$254$	32.6082	2.04602
$255$	2.98671	0.187035
$256$	86.6730	5.41706
$257$	−11.9396	−0.744775	−0.372387	−	0.928077i	$-0.621461\pi$
$257$	−11.9396	−0.744775	−0.372387	+	0.928077i	$0.621461\pi$
$258$	0.919047	0.0572174
$259$	3.38190	0.210141
$260$	−66.1364	−4.10160
$261$	−5.61400	−0.347498
$262$	−8.40738	−0.519409
$263$	−26.2356	−1.61776	−0.808879	−	0.587975i	$-0.799925\pi$
$263$	−26.2356	−1.61776	−0.808879	+	0.587975i	$0.799925\pi$
$264$	0	0
$265$	12.7266	0.781791
$266$	1.31524	0.0806427
$267$	−6.09862	−0.373229
$268$	17.1070	1.04498
$269$	−8.04917	−0.490766	−0.245383	−	0.969426i	$-0.578914\pi$
$269$	−8.04917	−0.490766	−0.245383	+	0.969426i	$0.578914\pi$
$270$	8.30376	0.505351
$271$	−17.4338	−1.05903	−0.529513	−	0.848302i	$-0.677625\pi$
$271$	−17.4338	−1.05903	−0.529513	+	0.848302i	$0.677625\pi$
$272$	17.3703	1.05323
$273$	3.75645	0.227351
$274$	−27.7594	−1.67700
$275$	0	0
$276$	16.8855	1.01639
$277$	13.9544	0.838437	0.419218	−	0.907885i	$-0.362304\pi$
$277$	13.9544	0.838437	0.419218	+	0.907885i	$0.362304\pi$
$278$	25.1513	1.50847
$279$	10.7749	0.645074
$280$	30.1034	1.79902
$281$	23.1097	1.37861	0.689304	−	0.724472i	$-0.257916\pi$
$281$	23.1097	1.37861	0.689304	+	0.724472i	$0.257916\pi$
$282$	5.68683	0.338646
$283$	−18.7895	−1.11692	−0.558461	−	0.829531i	$-0.688608\pi$
$283$	−18.7895	−1.11692	−0.558461	+	0.829531i	$0.688608\pi$
$284$	45.8765	2.72227
$285$	−1.45362	−0.0861053
$286$	0	0
$287$	−11.9394	−0.704761
$288$	27.5545	1.62366
$289$	1.00000	0.0588235
$290$	46.6173	2.73746
$291$	6.17135	0.361771
$292$	16.5276	0.967206
$293$	13.4465	0.785555	0.392777	−	0.919634i	$-0.371514\pi$
$293$	13.4465	0.785555	0.392777	+	0.919634i	$0.371514\pi$
$294$	16.8350	0.981835
$295$	−36.4698	−2.12335
$296$	−36.0791	−2.09705
$297$	0	0
$298$	59.9809	3.47460
$299$	−11.3893	−0.658658
$300$	−22.4629	−1.29690
$301$	0.321307	0.0185198
$302$	−28.0937	−1.61661
$303$	−5.46420	−0.313910
$304$	−8.45408	−0.484875
$305$	−18.1325	−1.03826
$306$	2.78024	0.158936
$307$	16.7041	0.953353	0.476676	−	0.879079i	$-0.341841\pi$
$307$	16.7041	0.953353	0.476676	+	0.879079i	$0.341841\pi$
$308$	0	0
$309$	−4.46977	−0.254276
$310$	−89.4718	−5.08166
$311$	−4.21405	−0.238957	−0.119479	−	0.992837i	$-0.538122\pi$
$311$	−4.21405	−0.238957	−0.119479	+	0.992837i	$0.538122\pi$
$312$	−40.0749	−2.26880
$313$	25.9403	1.46623	0.733116	−	0.680104i	$-0.238065\pi$
$313$	25.9403	1.46623	0.733116	+	0.680104i	$0.238065\pi$
$314$	43.8737	2.47594
$315$	2.90306	0.163569
$316$	56.7344	3.19156
$317$	−22.2737	−1.25101	−0.625507	−	0.780219i	$-0.715108\pi$
$317$	−22.2737	−1.25101	−0.625507	+	0.780219i	$0.715108\pi$
$318$	11.8469	0.664339
$319$	0	0
$320$	−125.046	−6.99028
$321$	13.7239	0.765996
$322$	7.96393	0.443812
$323$	−0.486698	−0.0270806
$324$	5.72972	0.318318
$325$	15.1512	0.840436
$326$	6.65747	0.368723
$327$	−14.5393	−0.804027
$328$	127.373	7.03300
$329$	1.98817	0.109611
$330$	0	0
$331$	−9.60040	−0.527686	−0.263843	−	0.964566i	$-0.584990\pi$
$331$	−9.60040	−0.527686	−0.263843	+	0.964566i	$0.584990\pi$
$332$	18.6356	1.02276
$333$	−3.47934	−0.190667
$334$	−25.8025	−1.41185
$335$	−8.91730	−0.487204
$336$	16.8838	0.921089
$337$	−4.67883	−0.254872	−0.127436	−	0.991847i	$-0.540675\pi$
$337$	−4.67883	−0.254872	−0.127436	+	0.991847i	$0.540675\pi$
$338$	5.38194	0.292739
$339$	17.3353	0.941525
$340$	−17.1130	−0.928083
$341$	0	0
$342$	−1.35314	−0.0731692
$343$	12.6896	0.685175
$344$	−3.42779	−0.184814
$345$	−8.80185	−0.473876
$346$	−61.1033	−3.28493
$347$	−5.82698	−0.312809	−0.156404	−	0.987693i	$-0.549990\pi$
$347$	−5.82698	−0.312809	−0.156404	+	0.987693i	$0.549990\pi$
$348$	32.1667	1.72432
$349$	−5.15706	−0.276051	−0.138026	−	0.990429i	$-0.544076\pi$
$349$	−5.15706	−0.276051	−0.138026	+	0.990429i	$0.544076\pi$
$350$	−10.5945	−0.566297
$351$	−3.86468	−0.206282
$352$	0	0
$353$	3.38117	0.179961	0.0899807	−	0.995944i	$-0.471319\pi$
$353$	3.38117	0.179961	0.0899807	+	0.995944i	$0.471319\pi$
$354$	−33.9487	−1.80435
$355$	−23.9139	−1.26922
$356$	34.9434	1.85200
$357$	0.971995	0.0514434
$358$	63.1474	3.33744
$359$	−15.1745	−0.800881	−0.400441	−	0.916323i	$-0.631143\pi$
$359$	−15.1745	−0.800881	−0.400441	+	0.916323i	$0.631143\pi$
$360$	−30.9707	−1.63230
$361$	−18.7631	−0.987533
$362$	38.7163	2.03489
$363$	0	0
$364$	−21.5234	−1.12814
$365$	−8.61528	−0.450944
$366$	−16.8790	−0.882281
$367$	−3.11798	−0.162757	−0.0813786	−	0.996683i	$-0.525932\pi$
$367$	−3.11798	−0.162757	−0.0813786	+	0.996683i	$0.525932\pi$
$368$	−51.1904	−2.66848
$369$	12.2834	0.639448
$370$	28.8916	1.50200
$371$	4.14176	0.215030
$372$	−61.7370	−3.20091
$373$	−3.59917	−0.186358	−0.0931791	−	0.995649i	$-0.529703\pi$
$373$	−3.59917	−0.186358	−0.0931791	+	0.995649i	$0.529703\pi$
$374$	0	0
$375$	−3.22439	−0.166507
$376$	−21.2103	−1.09384
$377$	−21.6964	−1.11742
$378$	2.70238	0.138995
$379$	−0.979498	−0.0503134	−0.0251567	−	0.999684i	$-0.508008\pi$
$379$	−0.979498	−0.0503134	−0.0251567	+	0.999684i	$0.508008\pi$
$380$	8.32886	0.427262
$381$	−11.7286	−0.600872
$382$	−72.8662	−3.72816
$383$	−14.2517	−0.728228	−0.364114	−	0.931354i	$-0.618628\pi$
$383$	−14.2517	−0.728228	−0.364114	+	0.931354i	$0.618628\pi$
$384$	−61.2926	−3.12782
$385$	0	0
$386$	−24.2118	−1.23235
$387$	−0.330564	−0.0168035
$388$	−35.3602	−1.79514
$389$	10.6219	0.538554	0.269277	−	0.963063i	$-0.413215\pi$
$389$	10.6219	0.538554	0.269277	+	0.963063i	$0.413215\pi$
$390$	32.0914	1.62501
$391$	−2.94701	−0.149037
$392$	−62.7898	−3.17136
$393$	3.02398	0.152539
$394$	9.25030	0.466023
$395$	−29.5737	−1.48801
$396$	0	0
$397$	−16.7854	−0.842433	−0.421217	−	0.906960i	$-0.638397\pi$
$397$	−16.7854	−0.842433	−0.421217	+	0.906960i	$0.638397\pi$
$398$	−11.5586	−0.579379
$399$	−0.473068	−0.0236830
$400$	68.0988	3.40494
$401$	−25.5455	−1.27568	−0.637841	−	0.770168i	$-0.720172\pi$
$401$	−25.5455	−1.27568	−0.637841	+	0.770168i	$0.720172\pi$
$402$	−8.30085	−0.414009
$403$	41.6414	2.07431
$404$	31.3083	1.55765
$405$	−2.98671	−0.148411
$406$	15.1712	0.752932
$407$	0	0
$408$	−10.3695	−0.513368
$409$	25.1020	1.24121	0.620607	−	0.784122i	$-0.286886\pi$
$409$	25.1020	1.24121	0.620607	+	0.784122i	$0.286886\pi$
$410$	−101.998	−5.03734
$411$	9.98452	0.492500
$412$	25.6105	1.26174
$413$	−11.8688	−0.584023
$414$	−8.19338	−0.402683
$415$	−9.71411	−0.476847
$416$	106.489	5.22107
$417$	−9.04644	−0.443006
$418$	0	0
$419$	−24.9859	−1.22064	−0.610322	−	0.792154i	$-0.708960\pi$
$419$	−24.9859	−1.22064	−0.610322	+	0.792154i	$0.708960\pi$
$420$	−16.6338	−0.811644
$421$	6.51463	0.317504	0.158752	−	0.987319i	$-0.449253\pi$
$421$	6.51463	0.317504	0.158752	+	0.987319i	$0.449253\pi$
$422$	6.84147	0.333038
$423$	−2.04545	−0.0994531
$424$	−44.1855	−2.14584
$425$	3.92042	0.190168
$426$	−22.2607	−1.07853
$427$	−5.90105	−0.285572
$428$	−78.6344	−3.80094
$429$	0	0
$430$	2.74492	0.132372
$431$	−11.2042	−0.539686	−0.269843	−	0.962904i	$-0.586972\pi$
$431$	−11.2042	−0.539686	−0.269843	+	0.962904i	$0.586972\pi$
$432$	−17.3703	−0.835728
$433$	−15.8733	−0.762824	−0.381412	−	0.924405i	$-0.624562\pi$
$433$	−15.8733	−0.762824	−0.381412	+	0.924405i	$0.624562\pi$
$434$	−29.1177	−1.39770
$435$	−16.7674	−0.803935
$436$	83.3064	3.98965
$437$	1.43430	0.0686120
$438$	−8.01971	−0.383197
$439$	−22.5697	−1.07719	−0.538596	−	0.842564i	$-0.681045\pi$
$439$	−22.5697	−1.07719	−0.538596	+	0.842564i	$0.681045\pi$
$440$	0	0
$441$	−6.05523	−0.288344
$442$	10.7447	0.511075
$443$	18.1909	0.864278	0.432139	−	0.901807i	$-0.357759\pi$
$443$	18.1909	0.864278	0.432139	+	0.901807i	$0.357759\pi$
$444$	19.9357	0.946104
$445$	−18.2148	−0.863464
$446$	−53.9073	−2.55258
$447$	−21.5740	−1.02042
$448$	−40.6950	−1.92266
$449$	4.87376	0.230007	0.115003	−	0.993365i	$-0.463312\pi$
$449$	4.87376	0.230007	0.115003	+	0.993365i	$0.463312\pi$
$450$	10.8997	0.513817
$451$	0	0
$452$	−99.3266	−4.67193
$453$	10.1048	0.474764
$454$	45.7523	2.14726
$455$	11.2194	0.525975
$456$	5.04682	0.236339
$457$	6.87255	0.321484	0.160742	−	0.986996i	$-0.448611\pi$
$457$	6.87255	0.321484	0.160742	+	0.986996i	$0.448611\pi$
$458$	−39.5856	−1.84971
$459$	−1.00000	−0.0466760
$460$	50.4322	2.35141
$461$	−9.98634	−0.465110	−0.232555	−	0.972583i	$-0.574709\pi$
$461$	−9.98634	−0.465110	−0.232555	+	0.972583i	$0.574709\pi$
$462$	0	0
$463$	−17.0411	−0.791969	−0.395985	−	0.918257i	$-0.629597\pi$
$463$	−17.0411	−0.791969	−0.395985	+	0.918257i	$0.629597\pi$
$464$	−97.5169	−4.52711
$465$	32.1814	1.49237
$466$	−27.8520	−1.29022
$467$	11.8703	0.549293	0.274646	−	0.961545i	$-0.411439\pi$
$467$	11.8703	0.549293	0.274646	+	0.961545i	$0.411439\pi$
$468$	22.1436	1.02359
$469$	−2.90205	−0.134004
$470$	16.9849	0.783455
$471$	−15.7805	−0.727129
$472$	126.619	5.82812
$473$	0	0
$474$	−27.5293	−1.26446
$475$	−1.90806	−0.0875478
$476$	−5.56926	−0.255267
$477$	−4.26110	−0.195102
$478$	15.2720	0.698524
$479$	−10.8009	−0.493504	−0.246752	−	0.969079i	$-0.579363\pi$
$479$	−10.8009	−0.493504	−0.246752	+	0.969079i	$0.579363\pi$
$480$	82.2972	3.75634
$481$	−13.4465	−0.613110
$482$	30.8945	1.40721
$483$	−2.86448	−0.130338
$484$	0	0
$485$	18.4320	0.836955
$486$	−2.78024	−0.126114
$487$	40.3186	1.82701	0.913506	−	0.406825i	$-0.133364\pi$
$487$	40.3186	1.82701	0.913506	+	0.406825i	$0.133364\pi$
$488$	62.9541	2.84980
$489$	−2.39457	−0.108286
$490$	50.2811	2.27147
$491$	−25.7880	−1.16379	−0.581897	−	0.813262i	$-0.697689\pi$
$491$	−25.7880	−1.16379	−0.581897	+	0.813262i	$0.697689\pi$
$492$	−70.3805	−3.17300
$493$	−5.61400	−0.252842
$494$	−5.22944	−0.235284
$495$	0	0
$496$	187.162	8.40384
$497$	−7.78254	−0.349094
$498$	−9.04258	−0.405208
$499$	12.0648	0.540095	0.270048	−	0.962847i	$-0.412961\pi$
$499$	12.0648	0.540095	0.270048	+	0.962847i	$0.412961\pi$
$500$	18.4749	0.826222
$501$	9.28068	0.414630
$502$	−23.8695	−1.06535
$503$	−34.7064	−1.54748	−0.773742	−	0.633501i	$-0.781617\pi$
$503$	−34.7064	−1.54748	−0.773742	+	0.633501i	$0.781617\pi$
$504$	−10.0791	−0.448960
$505$	−16.3200	−0.726229
$506$	0	0
$507$	−1.93579	−0.0859713
$508$	67.2014	2.98158
$509$	39.2596	1.74015	0.870076	−	0.492918i	$-0.164070\pi$
$509$	39.2596	1.74015	0.870076	+	0.492918i	$0.164070\pi$
$510$	8.30376	0.367697
$511$	−2.80376	−0.124031
$512$	118.386	5.23199
$513$	0.486698	0.0214882
$514$	−33.1951	−1.46417
$515$	−13.3499	−0.588267
$516$	1.89404	0.0833805
$517$	0	0
$518$	9.40249	0.413122
$519$	21.9777	0.964715
$520$	−119.692	−5.24884
$521$	2.00219	0.0877174	0.0438587	−	0.999038i	$-0.486035\pi$
$521$	2.00219	0.0877174	0.0438587	+	0.999038i	$0.486035\pi$
$522$	−15.6083	−0.683155
$523$	−21.4956	−0.939936	−0.469968	−	0.882683i	$-0.655735\pi$
$523$	−21.4956	−0.939936	−0.469968	+	0.882683i	$0.655735\pi$
$524$	−17.3266	−0.756914
$525$	3.81063	0.166309
$526$	−72.9413	−3.18039
$527$	10.7749	0.469360
$528$	0	0
$529$	−14.3151	−0.622398
$530$	35.3831	1.53694
$531$	12.2107	0.529900
$532$	2.71055	0.117517
$533$	47.4715	2.05622
$534$	−16.9556	−0.733741
$535$	40.9894	1.77213
$536$	30.9599	1.33726
$537$	−22.7129	−0.980136
$538$	−22.3786	−0.964810
$539$	0	0
$540$	17.1130	0.736427
$541$	2.68381	0.115386	0.0576929	−	0.998334i	$-0.481626\pi$
$541$	2.68381	0.115386	0.0576929	+	0.998334i	$0.481626\pi$
$542$	−48.4700	−2.08197
$543$	−13.9255	−0.597602
$544$	27.5545	1.18139
$545$	−43.4247	−1.86011
$546$	10.4438	0.446955
$547$	−32.5517	−1.39181	−0.695906	−	0.718133i	$-0.744997\pi$
$547$	−32.5517	−1.39181	−0.695906	+	0.718133i	$0.744997\pi$
$548$	−57.2086	−2.44383
$549$	6.07107	0.259107
$550$	0	0
$551$	2.73232	0.116401
$552$	30.5591	1.30068
$553$	−9.62447	−0.409274
$554$	38.7965	1.64830
$555$	−10.3918	−0.441106
$556$	51.8336	2.19823
$557$	−11.2409	−0.476294	−0.238147	−	0.971229i	$-0.576540\pi$
$557$	−11.2409	−0.476294	−0.238147	+	0.971229i	$0.576540\pi$
$558$	29.9567	1.26817
$559$	−1.27753	−0.0540336
$560$	50.4271	2.13093
$561$	0	0
$562$	64.2504	2.71024
$563$	−43.5426	−1.83510	−0.917550	−	0.397620i	$-0.869836\pi$
$563$	−43.5426	−1.83510	−0.917550	+	0.397620i	$0.869836\pi$
$564$	11.7199	0.493495
$565$	51.7755	2.17821
$566$	−52.2394	−2.19578
$567$	−0.971995	−0.0408200
$568$	83.0263	3.48371
$569$	−1.38183	−0.0579294	−0.0289647	−	0.999580i	$-0.509221\pi$
$569$	−1.38183	−0.0579294	−0.0289647	+	0.999580i	$0.509221\pi$
$570$	−4.04142	−0.169277
$571$	−9.36432	−0.391885	−0.195942	−	0.980615i	$-0.562777\pi$
$571$	−9.36432	−0.391885	−0.195942	+	0.980615i	$0.562777\pi$
$572$	0	0
$573$	26.2086	1.09488
$574$	−33.1944	−1.38551
$575$	−11.5535	−0.481814
$576$	41.8675	1.74448
$577$	−18.6919	−0.778155	−0.389077	−	0.921205i	$-0.627206\pi$
$577$	−18.6919	−0.778155	−0.389077	+	0.921205i	$0.627206\pi$
$578$	2.78024	0.115643
$579$	8.70855	0.361915
$580$	96.0725	3.98919
$581$	−3.16137	−0.131156
$582$	17.1578	0.711215
$583$	0	0
$584$	29.9113	1.23774
$585$	−11.5427	−0.477231
$586$	37.3846	1.54434
$587$	23.2289	0.958758	0.479379	−	0.877608i	$-0.340862\pi$
$587$	23.2289	0.958758	0.479379	+	0.877608i	$0.340862\pi$
$588$	34.6948	1.43079
$589$	−5.24410	−0.216079
$590$	−101.395	−4.17436
$591$	−3.32716	−0.136861
$592$	−60.4371	−2.48395
$593$	−27.7087	−1.13786	−0.568930	−	0.822386i	$-0.692643\pi$
$593$	−27.7087	−1.13786	−0.568930	+	0.822386i	$0.692643\pi$
$594$	0	0
$595$	2.90306	0.119014
$596$	123.613	5.06340
$597$	4.15741	0.170151
$598$	−31.6648	−1.29487
$599$	−11.3137	−0.462264	−0.231132	−	0.972922i	$-0.574243\pi$
$599$	−11.3137	−0.462264	−0.231132	+	0.972922i	$0.574243\pi$
$600$	−40.6529	−1.65965
$601$	−37.1107	−1.51378	−0.756888	−	0.653545i	$-0.773281\pi$
$601$	−37.1107	−1.51378	−0.756888	+	0.653545i	$0.773281\pi$
$602$	0.893309	0.0364086
$603$	2.98566	0.121586
$604$	−57.8976	−2.35582
$605$	0	0
$606$	−15.1918	−0.617124
$607$	20.3230	0.824884	0.412442	−	0.910984i	$-0.364676\pi$
$607$	20.3230	0.824884	0.412442	+	0.910984i	$0.364676\pi$
$608$	−13.4107	−0.543876
$609$	−5.45679	−0.221120
$610$	−50.4127	−2.04115
$611$	−7.90501	−0.319803
$612$	5.72972	0.231610
$613$	15.0517	0.607932	0.303966	−	0.952683i	$-0.401689\pi$
$613$	15.0517	0.607932	0.303966	+	0.952683i	$0.401689\pi$
$614$	46.4413	1.87422
$615$	36.6869	1.47936
$616$	0	0
$617$	32.9806	1.32775	0.663874	−	0.747845i	$-0.268911\pi$
$617$	32.9806	1.32775	0.663874	+	0.747845i	$0.268911\pi$
$618$	−12.4270	−0.499888
$619$	17.6598	0.709807	0.354903	−	0.934903i	$-0.384514\pi$
$619$	17.6598	0.709807	0.354903	+	0.934903i	$0.384514\pi$
$620$	−184.390	−7.40529
$621$	2.94701	0.118259
$622$	−11.7161	−0.469772
$623$	−5.92783	−0.237493
$624$	−67.1307	−2.68738
$625$	−29.2324	−1.16930
$626$	72.1202	2.88250
$627$	0	0
$628$	90.4182	3.60808
$629$	−3.47934	−0.138730
$630$	8.07121	0.321565
$631$	5.41561	0.215592	0.107796	−	0.994173i	$-0.465621\pi$
$631$	5.41561	0.215592	0.107796	+	0.994173i	$0.465621\pi$
$632$	102.677	4.08426
$633$	−2.46075	−0.0978061
$634$	−61.9261	−2.45940
$635$	−35.0298	−1.39011
$636$	24.4149	0.968114
$637$	−23.4015	−0.927203
$638$	0	0
$639$	8.00676	0.316743
$640$	−183.063	−7.23620
$641$	−22.0041	−0.869109	−0.434554	−	0.900646i	$-0.643094\pi$
$641$	−22.0041	−0.869109	−0.434554	+	0.900646i	$0.643094\pi$
$642$	38.1558	1.50589
$643$	14.6263	0.576805	0.288402	−	0.957509i	$-0.406876\pi$
$643$	14.6263	0.576805	0.288402	+	0.957509i	$0.406876\pi$
$644$	16.4127	0.646750
$645$	−0.987298	−0.0388748
$646$	−1.35314	−0.0532384
$647$	29.4690	1.15854	0.579272	−	0.815134i	$-0.303337\pi$
$647$	29.4690	1.15854	0.579272	+	0.815134i	$0.303337\pi$
$648$	10.3695	0.407353
$649$	0	0
$650$	42.1239	1.65223
$651$	10.4731	0.410474
$652$	13.7202	0.537325
$653$	11.1382	0.435870	0.217935	−	0.975963i	$-0.430068\pi$
$653$	11.1382	0.435870	0.217935	+	0.975963i	$0.430068\pi$
$654$	−40.4228	−1.58066
$655$	9.03173	0.352899
$656$	213.366	8.33055
$657$	2.88454	0.112537
$658$	5.52758	0.215487
$659$	−35.5308	−1.38408	−0.692042	−	0.721857i	$-0.743289\pi$
$659$	−35.5308	−1.38408	−0.692042	+	0.721857i	$0.743289\pi$
$660$	0	0
$661$	−11.7380	−0.456556	−0.228278	−	0.973596i	$-0.573309\pi$
$661$	−11.7380	−0.456556	−0.228278	+	0.973596i	$0.573309\pi$
$662$	−26.6914	−1.03739
$663$	−3.86468	−0.150092
$664$	33.7263	1.30884
$665$	−1.41292	−0.0547905
$666$	−9.67339	−0.374836
$667$	16.5445	0.640606
$668$	−53.1757	−2.05743
$669$	19.3895	0.749640
$670$	−24.7922	−0.957807
$671$	0	0
$672$	26.7828	1.03317
$673$	7.79474	0.300465	0.150233	−	0.988651i	$-0.451998\pi$
$673$	7.79474	0.300465	0.150233	+	0.988651i	$0.451998\pi$
$674$	−13.0083	−0.501059
$675$	−3.92042	−0.150897
$676$	11.0915	0.426597
$677$	−32.0808	−1.23297	−0.616483	−	0.787368i	$-0.711443\pi$
$677$	−32.0808	−1.23297	−0.616483	+	0.787368i	$0.711443\pi$
$678$	48.1963	1.85097
$679$	5.99853	0.230202
$680$	−30.9707	−1.18767
$681$	−16.4562	−0.630605
$682$	0	0
$683$	−3.29441	−0.126057	−0.0630285	−	0.998012i	$-0.520076\pi$
$683$	−3.29441	−0.126057	−0.0630285	+	0.998012i	$0.520076\pi$
$684$	−2.78864	−0.106626
$685$	29.8208	1.13940
$686$	35.2802	1.34700
$687$	14.2382	0.543222
$688$	−5.74199	−0.218911
$689$	−16.4678	−0.627373
$690$	−24.4712	−0.931604
$691$	49.1917	1.87134	0.935671	−	0.352874i	$-0.114796\pi$
$691$	49.1917	1.87134	0.935671	+	0.352874i	$0.114796\pi$
$692$	−125.926	−4.78700
$693$	0	0
$694$	−16.2004	−0.614959
$695$	−27.0191	−1.02489
$696$	58.2145	2.20662
$697$	12.2834	0.465267
$698$	−14.3379	−0.542696
$699$	10.0178	0.378909
$700$	−21.8338	−0.825242
$701$	−5.02164	−0.189665	−0.0948323	−	0.995493i	$-0.530231\pi$
$701$	−5.02164	−0.189665	−0.0948323	+	0.995493i	$0.530231\pi$
$702$	−10.7447	−0.405534
$703$	1.69339	0.0638673
$704$	0	0
$705$	−6.10916	−0.230084
$706$	9.40045	0.353791
$707$	−5.31117	−0.199747
$708$	−69.9640	−2.62941
$709$	40.0595	1.50447	0.752233	−	0.658897i	$-0.228977\pi$
$709$	40.0595	1.50447	0.752233	+	0.658897i	$0.228977\pi$
$710$	−66.4862	−2.49518
$711$	9.90177	0.371345
$712$	63.2397	2.37001
$713$	−31.7536	−1.18918
$714$	2.70238	0.101134
$715$	0	0
$716$	130.139	4.86352
$717$	−5.49305	−0.205142
$718$	−42.1888	−1.57447
$719$	17.8476	0.665604	0.332802	−	0.942997i	$-0.392006\pi$
$719$	17.8476	0.665604	0.332802	+	0.942997i	$0.392006\pi$
$720$	−51.8800	−1.93345
$721$	−4.34460	−0.161801
$722$	−52.1660	−1.94142
$723$	−11.1122	−0.413267
$724$	79.7895	2.96535
$725$	−22.0092	−0.817403
$726$	0	0
$727$	−23.7898	−0.882315	−0.441158	−	0.897430i	$-0.645432\pi$
$727$	−23.7898	−0.882315	−0.441158	+	0.897430i	$0.645432\pi$
$728$	−38.9526	−1.44368
$729$	1.00000	0.0370370
$730$	−23.9525	−0.886523
$731$	−0.330564	−0.0122264
$732$	−34.7856	−1.28571
$733$	−5.34985	−0.197601	−0.0988006	−	0.995107i	$-0.531501\pi$
$733$	−5.34985	−0.197601	−0.0988006	+	0.995107i	$0.531501\pi$
$734$	−8.66873	−0.319969
$735$	−18.0852	−0.667082
$736$	−81.2033	−2.99319
$737$	0	0
$738$	34.1508	1.25711
$739$	−48.1640	−1.77174	−0.885870	−	0.463933i	$-0.846438\pi$
$739$	−48.1640	−1.77174	−0.885870	+	0.463933i	$0.846438\pi$
$740$	59.5420	2.18881
$741$	1.88093	0.0690978
$742$	11.5151	0.422732
$743$	11.8392	0.434337	0.217169	−	0.976134i	$-0.430318\pi$
$743$	11.8392	0.434337	0.217169	+	0.976134i	$0.430318\pi$
$744$	−111.730	−4.09623
$745$	−64.4353	−2.36073
$746$	−10.0066	−0.366366
$747$	3.25245	0.119001
$748$	0	0
$749$	13.3396	0.487419
$750$	−8.96458	−0.327340
$751$	−2.00645	−0.0732166	−0.0366083	−	0.999330i	$-0.511655\pi$
$751$	−2.00645	−0.0732166	−0.0366083	+	0.999330i	$0.511655\pi$
$752$	−35.5300	−1.29565
$753$	8.58540	0.312869
$754$	−60.3210	−2.19676
$755$	30.1800	1.09836
$756$	5.56926	0.202552
$757$	37.1187	1.34910	0.674551	−	0.738229i	$-0.264337\pi$
$757$	37.1187	1.34910	0.674551	+	0.738229i	$0.264337\pi$
$758$	−2.72324	−0.0989124
$759$	0	0
$760$	15.0734	0.546769
$761$	−28.2329	−1.02344	−0.511720	−	0.859152i	$-0.670992\pi$
$761$	−28.2329	−1.02344	−0.511720	+	0.859152i	$0.670992\pi$
$762$	−32.6082	−1.18127
$763$	−14.1322	−0.511619
$764$	−150.168	−5.43289
$765$	−2.98671	−0.107985
$766$	−39.6231	−1.43164
$767$	47.1905	1.70395
$768$	−86.6730	−3.12754
$769$	−23.0513	−0.831250	−0.415625	−	0.909536i	$-0.636437\pi$
$769$	−23.0513	−0.831250	−0.415625	+	0.909536i	$0.636437\pi$
$770$	0	0
$771$	11.9396	0.429996
$772$	−49.8976	−1.79585
$773$	−47.6375	−1.71340	−0.856702	−	0.515812i	$-0.827490\pi$
$773$	−47.6375	−1.71340	−0.856702	+	0.515812i	$0.827490\pi$
$774$	−0.919047	−0.0330345
$775$	42.2420	1.51738
$776$	−63.9940	−2.29725
$777$	−3.38190	−0.121325
$778$	29.5315	1.05876
$779$	−5.97831	−0.214195
$780$	66.1364	2.36806
$781$	0	0
$782$	−8.19338	−0.292995
$783$	5.61400	0.200628
$784$	−105.181	−3.75646
$785$	−47.1319	−1.68221
$786$	8.40738	0.299881
$787$	−1.61849	−0.0576930	−0.0288465	−	0.999584i	$-0.509183\pi$
$787$	−1.61849	−0.0576930	−0.0288465	+	0.999584i	$0.509183\pi$
$788$	19.0637	0.679116
$789$	26.2356	0.934013
$790$	−82.2219	−2.92532
$791$	16.8499	0.599112
$792$	0	0
$793$	23.4628	0.833188
$794$	−46.6673	−1.65616
$795$	−12.7266	−0.451367
$796$	−23.8208	−0.844306
$797$	17.5604	0.622022	0.311011	−	0.950406i	$-0.399332\pi$
$797$	17.5604	0.622022	0.311011	+	0.950406i	$0.399332\pi$
$798$	−1.31524	−0.0465591
$799$	−2.04545	−0.0723628
$800$	108.025	3.81927
$801$	6.09862	0.215484
$802$	−71.0226	−2.50790
$803$	0	0
$804$	−17.1070	−0.603318
$805$	−8.55535	−0.301537
$806$	115.773	4.07793
$807$	8.04917	0.283344
$808$	56.6611	1.99333
$809$	18.6689	0.656362	0.328181	−	0.944615i	$-0.393564\pi$
$809$	18.6689	0.656362	0.328181	+	0.944615i	$0.393564\pi$
$810$	−8.30376	−0.291764
$811$	30.2294	1.06150	0.530748	−	0.847529i	$-0.321911\pi$
$811$	30.2294	1.06150	0.530748	+	0.847529i	$0.321911\pi$
$812$	31.2659	1.09722
$813$	17.4338	0.611429
$814$	0	0
$815$	−7.15187	−0.250519
$816$	−17.3703	−0.608082
$817$	0.160885	0.00562865
$818$	69.7895	2.44013
$819$	−3.75645	−0.131261
$820$	−210.206	−7.34071
$821$	21.5071	0.750604	0.375302	−	0.926903i	$-0.377539\pi$
$821$	21.5071	0.750604	0.375302	+	0.926903i	$0.377539\pi$
$822$	27.7594	0.968219
$823$	−5.19918	−0.181232	−0.0906161	−	0.995886i	$-0.528884\pi$
$823$	−5.19918	−0.181232	−0.0906161	+	0.995886i	$0.528884\pi$
$824$	46.3494	1.61466
$825$	0	0
$826$	−32.9980	−1.14815
$827$	9.15024	0.318185	0.159092	−	0.987264i	$-0.449143\pi$
$827$	9.15024	0.318185	0.159092	+	0.987264i	$0.449143\pi$
$828$	−16.8855	−0.586813
$829$	−9.25152	−0.321319	−0.160659	−	0.987010i	$-0.551362\pi$
$829$	−9.25152	−0.321319	−0.160659	+	0.987010i	$0.551362\pi$
$830$	−27.0076	−0.937446
$831$	−13.9544	−0.484072
$832$	161.805	5.60957
$833$	−6.05523	−0.209801
$834$	−25.1513	−0.870917
$835$	27.7187	0.959244
$836$	0	0
$837$	−10.7749	−0.372434
$838$	−69.4669	−2.39969
$839$	10.1118	0.349099	0.174549	−	0.984648i	$-0.444153\pi$
$839$	10.1118	0.349099	0.174549	+	0.984648i	$0.444153\pi$
$840$	−30.1034	−1.03867
$841$	2.51704	0.0867946
$842$	18.1122	0.624189
$843$	−23.1097	−0.795940
$844$	14.0994	0.485322
$845$	−5.78162	−0.198894
$846$	−5.68683	−0.195517
$847$	0	0
$848$	−74.0164	−2.54174
$849$	18.7895	0.644855
$850$	10.8997	0.373857
$851$	10.2536	0.351490
$852$	−45.8765	−1.57170
$853$	2.75153	0.0942105	0.0471052	−	0.998890i	$-0.485000\pi$
$853$	2.75153	0.0942105	0.0471052	+	0.998890i	$0.485000\pi$
$854$	−16.4063	−0.561413
$855$	1.45362	0.0497129
$856$	−142.311	−4.86408
$857$	9.70809	0.331622	0.165811	−	0.986158i	$-0.446976\pi$
$857$	9.70809	0.331622	0.165811	+	0.986158i	$0.446976\pi$
$858$	0	0
$859$	−42.9503	−1.46544	−0.732722	−	0.680528i	$-0.761750\pi$
$859$	−42.9503	−1.46544	−0.732722	+	0.680528i	$0.761750\pi$
$860$	5.65695	0.192900
$861$	11.9394	0.406894
$862$	−31.1502	−1.06098
$863$	20.4956	0.697677	0.348838	−	0.937183i	$-0.386576\pi$
$863$	20.4956	0.697677	0.348838	+	0.937183i	$0.386576\pi$
$864$	−27.5545	−0.937423
$865$	65.6410	2.23186
$866$	−44.1317	−1.49966
$867$	−1.00000	−0.0339618
$868$	−60.0080	−2.03681
$869$	0	0
$870$	−46.6173	−1.58048
$871$	11.5386	0.390972
$872$	150.766	5.10558
$873$	−6.17135	−0.208869
$874$	3.98770	0.134886
$875$	−3.13409	−0.105952
$876$	−16.5276	−0.558417
$877$	7.76113	0.262075	0.131037	−	0.991377i	$-0.458169\pi$
$877$	7.76113	0.262075	0.131037	+	0.991377i	$0.458169\pi$
$878$	−62.7490	−2.11768
$879$	−13.4465	−0.453540
$880$	0	0
$881$	27.5755	0.929041	0.464521	−	0.885562i	$-0.346227\pi$
$881$	27.5755	0.929041	0.464521	+	0.885562i	$0.346227\pi$
$882$	−16.8350	−0.566863
$883$	−5.13840	−0.172921	−0.0864604	−	0.996255i	$-0.527556\pi$
$883$	−5.13840	−0.172921	−0.0864604	+	0.996255i	$0.527556\pi$
$884$	22.1436	0.744769
$885$	36.4698	1.22592
$886$	50.5752	1.69911
$887$	42.4159	1.42419	0.712093	−	0.702085i	$-0.247747\pi$
$887$	42.4159	1.42419	0.712093	+	0.702085i	$0.247747\pi$
$888$	36.0791	1.21073
$889$	−11.4001	−0.382347
$890$	−50.6414	−1.69750
$891$	0	0
$892$	−111.096	−3.71978
$893$	0.995516	0.0333137
$894$	−59.9809	−2.00606
$895$	−67.8369	−2.26754
$896$	−59.5761	−1.99030
$897$	11.3893	0.380276
$898$	13.5502	0.452176
$899$	−60.4901	−2.01746
$900$	22.4629	0.748764
$901$	−4.26110	−0.141958
$902$	0	0
$903$	−0.321307	−0.0106924
$904$	−179.759	−5.97869
$905$	−41.5915	−1.38255
$906$	28.0937	0.933350
$907$	14.6752	0.487283	0.243642	−	0.969865i	$-0.421658\pi$
$907$	14.6752	0.487283	0.243642	+	0.969865i	$0.421658\pi$
$908$	94.2897	3.12911
$909$	5.46420	0.181236
$910$	31.1927	1.03403
$911$	−5.28310	−0.175037	−0.0875185	−	0.996163i	$-0.527894\pi$
$911$	−5.28310	−0.175037	−0.0875185	+	0.996163i	$0.527894\pi$
$912$	8.45408	0.279943
$913$	0	0
$914$	19.1073	0.632014
$915$	18.1325	0.599442
$916$	−81.5810	−2.69551
$917$	2.93929	0.0970640
$918$	−2.78024	−0.0917615
$919$	−37.5383	−1.23827	−0.619137	−	0.785283i	$-0.712517\pi$
$919$	−37.5383	−1.23827	−0.619137	+	0.785283i	$0.712517\pi$
$920$	91.2709	3.00911
$921$	−16.7041	−0.550418
$922$	−27.7644	−0.914372
$923$	30.9436	1.01852
$924$	0	0
$925$	−13.6405	−0.448496
$926$	−47.3784	−1.55695
$927$	4.46977	0.146807
$928$	−154.691	−5.07798
$929$	50.2078	1.64726	0.823632	−	0.567125i	$-0.191944\pi$
$929$	50.2078	1.64726	0.823632	+	0.567125i	$0.191944\pi$
$930$	89.4718	2.93390
$931$	2.94707	0.0965862
$932$	−57.3994	−1.88018
$933$	4.21405	0.137962
$934$	33.0023	1.07987
$935$	0	0
$936$	40.0749	1.30989
$937$	33.1573	1.08320	0.541602	−	0.840635i	$-0.317818\pi$
$937$	33.1573	1.08320	0.541602	+	0.840635i	$0.317818\pi$
$938$	−8.06839	−0.263442
$939$	−25.9403	−0.846529
$940$	35.0038	1.14170
$941$	37.8004	1.23226	0.616129	−	0.787646i	$-0.288700\pi$
$941$	37.8004	1.23226	0.616129	+	0.787646i	$0.288700\pi$
$942$	−43.8737	−1.42948
$943$	−36.1993	−1.17881
$944$	212.104	6.90338
$945$	−2.90306	−0.0944367
$946$	0	0
$947$	−38.2948	−1.24441	−0.622207	−	0.782853i	$-0.713764\pi$
$947$	−38.2948	−1.24441	−0.622207	+	0.782853i	$0.713764\pi$
$948$	−56.7344	−1.84265
$949$	11.1478	0.361874
$950$	−5.30486	−0.172112
$951$	22.2737	0.722273
$952$	−10.0791	−0.326666
$953$	−8.44008	−0.273401	−0.136701	−	0.990612i	$-0.543650\pi$
$953$	−8.44008	−0.273401	−0.136701	+	0.990612i	$0.543650\pi$
$954$	−11.8469	−0.383556
$955$	78.2775	2.53300
$956$	31.4736	1.01793
$957$	0	0
$958$	−30.0290	−0.970193
$959$	9.70491	0.313388
$960$	125.046	4.03584
$961$	85.0976	2.74509
$962$	−37.3846	−1.20533
$963$	−13.7239	−0.442248
$964$	63.6698	2.05067
$965$	26.0099	0.837288
$966$	−7.96393	−0.256235
$967$	34.4440	1.10764	0.553822	−	0.832635i	$-0.313169\pi$
$967$	34.4440	1.10764	0.553822	+	0.832635i	$0.313169\pi$
$968$	0	0
$969$	0.486698	0.0156350
$970$	51.2454	1.64539
$971$	−8.13734	−0.261140	−0.130570	−	0.991439i	$-0.541681\pi$
$971$	−8.13734	−0.261140	−0.130570	+	0.991439i	$0.541681\pi$
$972$	−5.72972	−0.183781
$973$	−8.79310	−0.281894
$974$	112.095	3.59177
$975$	−15.1512	−0.485226
$976$	105.456	3.37557
$977$	−11.4930	−0.367693	−0.183846	−	0.982955i	$-0.558855\pi$
$977$	−11.4930	−0.367693	−0.183846	+	0.982955i	$0.558855\pi$
$978$	−6.65747	−0.212882
$979$	0	0
$980$	103.623	3.31012
$981$	14.5393	0.464205
$982$	−71.6967	−2.28793
$983$	48.0033	1.53107	0.765534	−	0.643395i	$-0.222475\pi$
$983$	48.0033	1.53107	0.765534	+	0.643395i	$0.222475\pi$
$984$	−127.373	−4.06050
$985$	−9.93725	−0.316627
$986$	−15.6083	−0.497069
$987$	−1.98817	−0.0632840
$988$	−10.7772	−0.342869
$989$	0.974175	0.0309770
$990$	0	0
$991$	44.4851	1.41312	0.706558	−	0.707655i	$-0.250247\pi$
$991$	44.4851	1.41312	0.706558	+	0.707655i	$0.250247\pi$
$992$	296.896	9.42645
$993$	9.60040	0.304659
$994$	−21.6373	−0.686294
$995$	12.4170	0.393644
$996$	−18.6356	−0.590493
$997$	−38.3567	−1.21477	−0.607384	−	0.794409i	$-0.707781\pi$
$997$	−38.3567	−1.21477	−0.607384	+	0.794409i	$0.707781\pi$
$998$	33.5431	1.06179
$999$	3.47934	0.110081

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6171.2.a.bs.1.20		20
11.3	even	5		561.2.m.e.460.10	✓	40
11.4	even	5		561.2.m.e.511.10	yes	40
11.10	odd	2		6171.2.a.bp.1.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
561.2.m.e.460.10	✓	40	11.3	even	5
561.2.m.e.511.10	yes	40	11.4	even	5
6171.2.a.bp.1.1		20	11.10	odd	2
6171.2.a.bs.1.20		20	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:	\( N \)	\(=\)	\( 6171 = 3 \cdot 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6171.a (trivial)

\(f(q)\)	\(=\)	\(q+2.78024 q^{2} -1.00000 q^{3} +5.72972 q^{4} -2.98671 q^{5} -2.78024 q^{6} -0.971995 q^{7} +10.3695 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.78024 q^{2} -1.00000 q^{3} +5.72972 q^{4} -2.98671 q^{5} -2.78024 q^{6} -0.971995 q^{7} +10.3695 q^{8} +1.00000 q^{9} -8.30376 q^{10} -5.72972 q^{12} +3.86468 q^{13} -2.70238 q^{14} +2.98671 q^{15} +17.3703 q^{16} +1.00000 q^{17} +2.78024 q^{18} -0.486698 q^{19} -17.1130 q^{20} +0.971995 q^{21} -2.94701 q^{23} -10.3695 q^{24} +3.92042 q^{25} +10.7447 q^{26} -1.00000 q^{27} -5.56926 q^{28} -5.61400 q^{29} +8.30376 q^{30} +10.7749 q^{31} +27.5545 q^{32} +2.78024 q^{34} +2.90306 q^{35} +5.72972 q^{36} -3.47934 q^{37} -1.35314 q^{38} -3.86468 q^{39} -30.9707 q^{40} +12.2834 q^{41} +2.70238 q^{42} -0.330564 q^{43} -2.98671 q^{45} -8.19338 q^{46} -2.04545 q^{47} -17.3703 q^{48} -6.05523 q^{49} +10.8997 q^{50} -1.00000 q^{51} +22.1436 q^{52} -4.26110 q^{53} -2.78024 q^{54} -10.0791 q^{56} +0.486698 q^{57} -15.6083 q^{58} +12.2107 q^{59} +17.1130 q^{60} +6.07107 q^{61} +29.9567 q^{62} -0.971995 q^{63} +41.8675 q^{64} -11.5427 q^{65} +2.98566 q^{67} +5.72972 q^{68} +2.94701 q^{69} +8.07121 q^{70} +8.00676 q^{71} +10.3695 q^{72} +2.88454 q^{73} -9.67339 q^{74} -3.92042 q^{75} -2.78864 q^{76} -10.7447 q^{78} +9.90177 q^{79} -51.8800 q^{80} +1.00000 q^{81} +34.1508 q^{82} +3.25245 q^{83} +5.56926 q^{84} -2.98671 q^{85} -0.919047 q^{86} +5.61400 q^{87} +6.09862 q^{89} -8.30376 q^{90} -3.75645 q^{91} -16.8855 q^{92} -10.7749 q^{93} -5.68683 q^{94} +1.45362 q^{95} -27.5545 q^{96} -6.17135 q^{97} -16.8350 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 3 q^{2} - 20 q^{3} + 25 q^{4} + 7 q^{5} - 3 q^{6} - 5 q^{7} + 12 q^{8} + 20 q^{9}+O(q^{10}) \) 20 * q + 3 * q^2 - 20 * q^3 + 25 * q^4 + 7 * q^5 - 3 * q^6 - 5 * q^7 + 12 * q^8 + 20 * q^9	\( 20 q + 3 q^{2} - 20 q^{3} + 25 q^{4} + 7 q^{5} - 3 q^{6} - 5 q^{7} + 12 q^{8} + 20 q^{9} + 4 q^{10} - 25 q^{12} + 8 q^{13} + 3 q^{14} - 7 q^{15} + 47 q^{16} + 20 q^{17} + 3 q^{18} + 6 q^{19} + 26 q^{20} + 5 q^{21} - 16 q^{23} - 12 q^{24} + 31 q^{25} + 15 q^{26} - 20 q^{27} - 15 q^{28} + 4 q^{29} - 4 q^{30} + 27 q^{31} + 47 q^{32} + 3 q^{34} - 2 q^{35} + 25 q^{36} + 36 q^{37} - 8 q^{39} - 31 q^{40} + 6 q^{41} - 3 q^{42} + 46 q^{43} + 7 q^{45} - 25 q^{46} - 24 q^{47} - 47 q^{48} + 73 q^{49} - 20 q^{51} + 8 q^{52} + 6 q^{53} - 3 q^{54} + 39 q^{56} - 6 q^{57} - 25 q^{58} + 4 q^{59} - 26 q^{60} - 38 q^{61} - 9 q^{62} - 5 q^{63} + 100 q^{64} + 26 q^{65} + 21 q^{67} + 25 q^{68} + 16 q^{69} + 9 q^{70} - 24 q^{71} + 12 q^{72} + 5 q^{73} - 94 q^{74} - 31 q^{75} + 81 q^{76} - 15 q^{78} - 47 q^{79} + 17 q^{80} + 20 q^{81} + 75 q^{82} + 6 q^{83} + 15 q^{84} + 7 q^{85} + 54 q^{86} - 4 q^{87} + 44 q^{89} + 4 q^{90} + 37 q^{91} - 28 q^{92} - 27 q^{93} - 36 q^{94} + 18 q^{95} - 47 q^{96} + 19 q^{97} + 81 q^{98}+O(q^{100}) \) 20 * q + 3 * q^2 - 20 * q^3 + 25 * q^4 + 7 * q^5 - 3 * q^6 - 5 * q^7 + 12 * q^8 + 20 * q^9 + 4 * q^10 - 25 * q^12 + 8 * q^13 + 3 * q^14 - 7 * q^15 + 47 * q^16 + 20 * q^17 + 3 * q^18 + 6 * q^19 + 26 * q^20 + 5 * q^21 - 16 * q^23 - 12 * q^24 + 31 * q^25 + 15 * q^26 - 20 * q^27 - 15 * q^28 + 4 * q^29 - 4 * q^30 + 27 * q^31 + 47 * q^32 + 3 * q^34 - 2 * q^35 + 25 * q^36 + 36 * q^37 - 8 * q^39 - 31 * q^40 + 6 * q^41 - 3 * q^42 + 46 * q^43 + 7 * q^45 - 25 * q^46 - 24 * q^47 - 47 * q^48 + 73 * q^49 - 20 * q^51 + 8 * q^52 + 6 * q^53 - 3 * q^54 + 39 * q^56 - 6 * q^57 - 25 * q^58 + 4 * q^59 - 26 * q^60 - 38 * q^61 - 9 * q^62 - 5 * q^63 + 100 * q^64 + 26 * q^65 + 21 * q^67 + 25 * q^68 + 16 * q^69 + 9 * q^70 - 24 * q^71 + 12 * q^72 + 5 * q^73 - 94 * q^74 - 31 * q^75 + 81 * q^76 - 15 * q^78 - 47 * q^79 + 17 * q^80 + 20 * q^81 + 75 * q^82 + 6 * q^83 + 15 * q^84 + 7 * q^85 + 54 * q^86 - 4 * q^87 + 44 * q^89 + 4 * q^90 + 37 * q^91 - 28 * q^92 - 27 * q^93 - 36 * q^94 + 18 * q^95 - 47 * q^96 + 19 * q^97 + 81 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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