LMFDB - Embedded newform 6042.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, 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("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.56155 q^{5} +1.00000 q^{6} -2.56155 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.56155 q^{5} +1.00000 q^{6} -2.56155 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.56155 q^{10} -4.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -2.56155 q^{14} +2.56155 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +2.56155 q^{20} -2.56155 q^{21} -4.00000 q^{22} -4.56155 q^{23} +1.00000 q^{24} +1.56155 q^{25} -4.00000 q^{26} +1.00000 q^{27} -2.56155 q^{28} -5.68466 q^{29} +2.56155 q^{30} -4.56155 q^{31} +1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} -6.56155 q^{35} +1.00000 q^{36} -1.12311 q^{37} +1.00000 q^{38} -4.00000 q^{39} +2.56155 q^{40} +3.12311 q^{41} -2.56155 q^{42} -5.43845 q^{43} -4.00000 q^{44} +2.56155 q^{45} -4.56155 q^{46} -8.24621 q^{47} +1.00000 q^{48} -0.438447 q^{49} +1.56155 q^{50} -2.00000 q^{51} -4.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} -10.2462 q^{55} -2.56155 q^{56} +1.00000 q^{57} -5.68466 q^{58} +12.8078 q^{59} +2.56155 q^{60} -4.87689 q^{61} -4.56155 q^{62} -2.56155 q^{63} +1.00000 q^{64} -10.2462 q^{65} -4.00000 q^{66} +11.6847 q^{67} -2.00000 q^{68} -4.56155 q^{69} -6.56155 q^{70} +10.2462 q^{71} +1.00000 q^{72} -10.0000 q^{73} -1.12311 q^{74} +1.56155 q^{75} +1.00000 q^{76} +10.2462 q^{77} -4.00000 q^{78} -4.87689 q^{79} +2.56155 q^{80} +1.00000 q^{81} +3.12311 q^{82} +4.00000 q^{83} -2.56155 q^{84} -5.12311 q^{85} -5.43845 q^{86} -5.68466 q^{87} -4.00000 q^{88} -12.5616 q^{89} +2.56155 q^{90} +10.2462 q^{91} -4.56155 q^{92} -4.56155 q^{93} -8.24621 q^{94} +2.56155 q^{95} +1.00000 q^{96} +8.24621 q^{97} -0.438447 q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9} + q^{10} - 8 q^{11} + 2 q^{12} - 8 q^{13} - q^{14} + q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{19} + q^{20} - q^{21} - 8 q^{22} - 5 q^{23} + 2 q^{24} - q^{25} - 8 q^{26} + 2 q^{27} - q^{28} + q^{29} + q^{30} - 5 q^{31} + 2 q^{32} - 8 q^{33} - 4 q^{34} - 9 q^{35} + 2 q^{36} + 6 q^{37} + 2 q^{38} - 8 q^{39} + q^{40} - 2 q^{41} - q^{42} - 15 q^{43} - 8 q^{44} + q^{45} - 5 q^{46} + 2 q^{48} - 5 q^{49} - q^{50} - 4 q^{51} - 8 q^{52} + 2 q^{53} + 2 q^{54} - 4 q^{55} - q^{56} + 2 q^{57} + q^{58} + 5 q^{59} + q^{60} - 18 q^{61} - 5 q^{62} - q^{63} + 2 q^{64} - 4 q^{65} - 8 q^{66} + 11 q^{67} - 4 q^{68} - 5 q^{69} - 9 q^{70} + 4 q^{71} + 2 q^{72} - 20 q^{73} + 6 q^{74} - q^{75} + 2 q^{76} + 4 q^{77} - 8 q^{78} - 18 q^{79} + q^{80} + 2 q^{81} - 2 q^{82} + 8 q^{83} - q^{84} - 2 q^{85} - 15 q^{86} + q^{87} - 8 q^{88} - 21 q^{89} + q^{90} + 4 q^{91} - 5 q^{92} - 5 q^{93} + q^{95} + 2 q^{96} - 5 q^{98} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9 + q^10 - 8 * q^11 + 2 * q^12 - 8 * q^13 - q^14 + q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 2 * q^19 + q^20 - q^21 - 8 * q^22 - 5 * q^23 + 2 * q^24 - q^25 - 8 * q^26 + 2 * q^27 - q^28 + q^29 + q^30 - 5 * q^31 + 2 * q^32 - 8 * q^33 - 4 * q^34 - 9 * q^35 + 2 * q^36 + 6 * q^37 + 2 * q^38 - 8 * q^39 + q^40 - 2 * q^41 - q^42 - 15 * q^43 - 8 * q^44 + q^45 - 5 * q^46 + 2 * q^48 - 5 * q^49 - q^50 - 4 * q^51 - 8 * q^52 + 2 * q^53 + 2 * q^54 - 4 * q^55 - q^56 + 2 * q^57 + q^58 + 5 * q^59 + q^60 - 18 * q^61 - 5 * q^62 - q^63 + 2 * q^64 - 4 * q^65 - 8 * q^66 + 11 * q^67 - 4 * q^68 - 5 * q^69 - 9 * q^70 + 4 * q^71 + 2 * q^72 - 20 * q^73 + 6 * q^74 - q^75 + 2 * q^76 + 4 * q^77 - 8 * q^78 - 18 * q^79 + q^80 + 2 * q^81 - 2 * q^82 + 8 * q^83 - q^84 - 2 * q^85 - 15 * q^86 + q^87 - 8 * q^88 - 21 * q^89 + q^90 + 4 * q^91 - 5 * q^92 - 5 * q^93 + q^95 + 2 * q^96 - 5 * q^98 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	2.56155	1.14556	0.572781	−	0.819709i	$-0.305865\pi$
$5$	2.56155	1.14556	0.572781	+	0.819709i	$0.305865\pi$
$6$	1.00000	0.408248
$7$	−2.56155	−0.968176	−0.484088	−	0.875019i	$-0.660849\pi$
$7$	−2.56155	−0.968176	−0.484088	+	0.875019i	$0.660849\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.56155	0.810034
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−2.56155	−0.684604
$15$	2.56155	0.661390
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.56155	0.572781
$21$	−2.56155	−0.558977
$22$	−4.00000	−0.852803
$23$	−4.56155	−0.951150	−0.475575	−	0.879675i	$-0.657760\pi$
$23$	−4.56155	−0.951150	−0.475575	+	0.879675i	$0.657760\pi$
$24$	1.00000	0.204124
$25$	1.56155	0.312311
$26$	−4.00000	−0.784465
$27$	1.00000	0.192450
$28$	−2.56155	−0.484088
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	2.56155	0.467673
$31$	−4.56155	−0.819279	−0.409640	−	0.912247i	$-0.634346\pi$
$31$	−4.56155	−0.819279	−0.409640	+	0.912247i	$0.634346\pi$
$32$	1.00000	0.176777
$33$	−4.00000	−0.696311
$34$	−2.00000	−0.342997
$35$	−6.56155	−1.10910
$36$	1.00000	0.166667
$37$	−1.12311	−0.184637	−0.0923187	−	0.995730i	$-0.529428\pi$
$37$	−1.12311	−0.184637	−0.0923187	+	0.995730i	$0.529428\pi$
$38$	1.00000	0.162221
$39$	−4.00000	−0.640513
$40$	2.56155	0.405017
$41$	3.12311	0.487747	0.243874	−	0.969807i	$-0.421582\pi$
$41$	3.12311	0.487747	0.243874	+	0.969807i	$0.421582\pi$
$42$	−2.56155	−0.395256
$43$	−5.43845	−0.829355	−0.414678	−	0.909968i	$-0.636106\pi$
$43$	−5.43845	−0.829355	−0.414678	+	0.909968i	$0.636106\pi$
$44$	−4.00000	−0.603023
$45$	2.56155	0.381854
$46$	−4.56155	−0.672564
$47$	−8.24621	−1.20283	−0.601417	−	0.798935i	$-0.705397\pi$
$47$	−8.24621	−1.20283	−0.601417	+	0.798935i	$0.705397\pi$
$48$	1.00000	0.144338
$49$	−0.438447	−0.0626353
$50$	1.56155	0.220837
$51$	−2.00000	−0.280056
$52$	−4.00000	−0.554700
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	1.00000	0.136083
$55$	−10.2462	−1.38160
$56$	−2.56155	−0.342302
$57$	1.00000	0.132453
$58$	−5.68466	−0.746432
$59$	12.8078	1.66743	0.833714	−	0.552196i	$-0.186210\pi$
$59$	12.8078	1.66743	0.833714	+	0.552196i	$0.186210\pi$
$60$	2.56155	0.330695
$61$	−4.87689	−0.624422	−0.312211	−	0.950013i	$-0.601070\pi$
$61$	−4.87689	−0.624422	−0.312211	+	0.950013i	$0.601070\pi$
$62$	−4.56155	−0.579318
$63$	−2.56155	−0.322725
$64$	1.00000	0.125000
$65$	−10.2462	−1.27089
$66$	−4.00000	−0.492366
$67$	11.6847	1.42751	0.713754	−	0.700396i	$-0.246993\pi$
$67$	11.6847	1.42751	0.713754	+	0.700396i	$0.246993\pi$
$68$	−2.00000	−0.242536
$69$	−4.56155	−0.549146
$70$	−6.56155	−0.784256
$71$	10.2462	1.21600	0.608001	−	0.793936i	$-0.291972\pi$
$71$	10.2462	1.21600	0.608001	+	0.793936i	$0.291972\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−1.12311	−0.130558
$75$	1.56155	0.180313
$76$	1.00000	0.114708
$77$	10.2462	1.16766
$78$	−4.00000	−0.452911
$79$	−4.87689	−0.548693	−0.274347	−	0.961631i	$-0.588462\pi$
$79$	−4.87689	−0.548693	−0.274347	+	0.961631i	$0.588462\pi$
$80$	2.56155	0.286390
$81$	1.00000	0.111111
$82$	3.12311	0.344889
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	−2.56155	−0.279488
$85$	−5.12311	−0.555679
$86$	−5.43845	−0.586443
$87$	−5.68466	−0.609459
$88$	−4.00000	−0.426401
$89$	−12.5616	−1.33152	−0.665761	−	0.746165i	$-0.731893\pi$
$89$	−12.5616	−1.33152	−0.665761	+	0.746165i	$0.731893\pi$
$90$	2.56155	0.270011
$91$	10.2462	1.07409
$92$	−4.56155	−0.475575
$93$	−4.56155	−0.473011
$94$	−8.24621	−0.850532
$95$	2.56155	0.262810
$96$	1.00000	0.102062
$97$	8.24621	0.837276	0.418638	−	0.908153i	$-0.362508\pi$
$97$	8.24621	0.837276	0.418638	+	0.908153i	$0.362508\pi$
$98$	−0.438447	−0.0442899
$99$	−4.00000	−0.402015
$100$	1.56155	0.156155
$101$	−7.68466	−0.764652	−0.382326	−	0.924027i	$-0.624877\pi$
$101$	−7.68466	−0.764652	−0.382326	+	0.924027i	$0.624877\pi$
$102$	−2.00000	−0.198030
$103$	17.6847	1.74252	0.871261	−	0.490821i	$-0.163303\pi$
$103$	17.6847	1.74252	0.871261	+	0.490821i	$0.163303\pi$
$104$	−4.00000	−0.392232
$105$	−6.56155	−0.640342
$106$	1.00000	0.0971286
$107$	7.68466	0.742904	0.371452	−	0.928452i	$-0.378860\pi$
$107$	7.68466	0.742904	0.371452	+	0.928452i	$0.378860\pi$
$108$	1.00000	0.0962250
$109$	−15.6847	−1.50232	−0.751159	−	0.660121i	$-0.770505\pi$
$109$	−15.6847	−1.50232	−0.751159	+	0.660121i	$0.770505\pi$
$110$	−10.2462	−0.976938
$111$	−1.12311	−0.106600
$112$	−2.56155	−0.242044
$113$	4.56155	0.429115	0.214557	−	0.976711i	$-0.431169\pi$
$113$	4.56155	0.429115	0.214557	+	0.976711i	$0.431169\pi$
$114$	1.00000	0.0936586
$115$	−11.6847	−1.08960
$116$	−5.68466	−0.527807
$117$	−4.00000	−0.369800
$118$	12.8078	1.17905
$119$	5.12311	0.469634
$120$	2.56155	0.233837
$121$	5.00000	0.454545
$122$	−4.87689	−0.441533
$123$	3.12311	0.281601
$124$	−4.56155	−0.409640
$125$	−8.80776	−0.787790
$126$	−2.56155	−0.228201
$127$	10.8078	0.959034	0.479517	−	0.877533i	$-0.340812\pi$
$127$	10.8078	0.959034	0.479517	+	0.877533i	$0.340812\pi$
$128$	1.00000	0.0883883
$129$	−5.43845	−0.478829
$130$	−10.2462	−0.898652
$131$	−20.4924	−1.79043	−0.895216	−	0.445633i	$-0.852979\pi$
$131$	−20.4924	−1.79043	−0.895216	+	0.445633i	$0.852979\pi$
$132$	−4.00000	−0.348155
$133$	−2.56155	−0.222115
$134$	11.6847	1.00940
$135$	2.56155	0.220463
$136$	−2.00000	−0.171499
$137$	−4.56155	−0.389720	−0.194860	−	0.980831i	$-0.562425\pi$
$137$	−4.56155	−0.389720	−0.194860	+	0.980831i	$0.562425\pi$
$138$	−4.56155	−0.388305
$139$	3.36932	0.285782	0.142891	−	0.989738i	$-0.454360\pi$
$139$	3.36932	0.285782	0.142891	+	0.989738i	$0.454360\pi$
$140$	−6.56155	−0.554552
$141$	−8.24621	−0.694456
$142$	10.2462	0.859843
$143$	16.0000	1.33799
$144$	1.00000	0.0833333
$145$	−14.5616	−1.20927
$146$	−10.0000	−0.827606
$147$	−0.438447	−0.0361625
$148$	−1.12311	−0.0923187
$149$	−1.75379	−0.143676	−0.0718380	−	0.997416i	$-0.522886\pi$
$149$	−1.75379	−0.143676	−0.0718380	+	0.997416i	$0.522886\pi$
$150$	1.56155	0.127500
$151$	14.8078	1.20504	0.602519	−	0.798104i	$-0.294164\pi$
$151$	14.8078	1.20504	0.602519	+	0.798104i	$0.294164\pi$
$152$	1.00000	0.0811107
$153$	−2.00000	−0.161690
$154$	10.2462	0.825663
$155$	−11.6847	−0.938534
$156$	−4.00000	−0.320256
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−4.87689	−0.387985
$159$	1.00000	0.0793052
$160$	2.56155	0.202509
$161$	11.6847	0.920880
$162$	1.00000	0.0785674
$163$	−10.5616	−0.827245	−0.413622	−	0.910449i	$-0.635737\pi$
$163$	−10.5616	−0.827245	−0.413622	+	0.910449i	$0.635737\pi$
$164$	3.12311	0.243874
$165$	−10.2462	−0.797666
$166$	4.00000	0.310460
$167$	−9.12311	−0.705967	−0.352984	−	0.935630i	$-0.614833\pi$
$167$	−9.12311	−0.705967	−0.352984	+	0.935630i	$0.614833\pi$
$168$	−2.56155	−0.197628
$169$	3.00000	0.230769
$170$	−5.12311	−0.392924
$171$	1.00000	0.0764719
$172$	−5.43845	−0.414678
$173$	−21.3693	−1.62468	−0.812340	−	0.583185i	$-0.801806\pi$
$173$	−21.3693	−1.62468	−0.812340	+	0.583185i	$0.801806\pi$
$174$	−5.68466	−0.430953
$175$	−4.00000	−0.302372
$176$	−4.00000	−0.301511
$177$	12.8078	0.962690
$178$	−12.5616	−0.941528
$179$	17.1231	1.27984	0.639921	−	0.768441i	$-0.278967\pi$
$179$	17.1231	1.27984	0.639921	+	0.768441i	$0.278967\pi$
$180$	2.56155	0.190927
$181$	10.5616	0.785034	0.392517	−	0.919745i	$-0.371604\pi$
$181$	10.5616	0.785034	0.392517	+	0.919745i	$0.371604\pi$
$182$	10.2462	0.759500
$183$	−4.87689	−0.360510
$184$	−4.56155	−0.336282
$185$	−2.87689	−0.211513
$186$	−4.56155	−0.334469
$187$	8.00000	0.585018
$188$	−8.24621	−0.601417
$189$	−2.56155	−0.186326
$190$	2.56155	0.185835
$191$	−5.36932	−0.388510	−0.194255	−	0.980951i	$-0.562229\pi$
$191$	−5.36932	−0.388510	−0.194255	+	0.980951i	$0.562229\pi$
$192$	1.00000	0.0721688
$193$	7.75379	0.558130	0.279065	−	0.960272i	$-0.409976\pi$
$193$	7.75379	0.558130	0.279065	+	0.960272i	$0.409976\pi$
$194$	8.24621	0.592043
$195$	−10.2462	−0.733746
$196$	−0.438447	−0.0313177
$197$	−14.2462	−1.01500	−0.507500	−	0.861652i	$-0.669430\pi$
$197$	−14.2462	−1.01500	−0.507500	+	0.861652i	$0.669430\pi$
$198$	−4.00000	−0.284268
$199$	−13.4384	−0.952626	−0.476313	−	0.879276i	$-0.658027\pi$
$199$	−13.4384	−0.952626	−0.476313	+	0.879276i	$0.658027\pi$
$200$	1.56155	0.110418
$201$	11.6847	0.824172
$202$	−7.68466	−0.540691
$203$	14.5616	1.02202
$204$	−2.00000	−0.140028
$205$	8.00000	0.558744
$206$	17.6847	1.23215
$207$	−4.56155	−0.317050
$208$	−4.00000	−0.277350
$209$	−4.00000	−0.276686
$210$	−6.56155	−0.452790
$211$	2.87689	0.198054	0.0990268	−	0.995085i	$-0.468427\pi$
$211$	2.87689	0.198054	0.0990268	+	0.995085i	$0.468427\pi$
$212$	1.00000	0.0686803
$213$	10.2462	0.702059
$214$	7.68466	0.525312
$215$	−13.9309	−0.950077
$216$	1.00000	0.0680414
$217$	11.6847	0.793206
$218$	−15.6847	−1.06230
$219$	−10.0000	−0.675737
$220$	−10.2462	−0.690799
$221$	8.00000	0.538138
$222$	−1.12311	−0.0753779
$223$	−4.24621	−0.284347	−0.142174	−	0.989842i	$-0.545409\pi$
$223$	−4.24621	−0.284347	−0.142174	+	0.989842i	$0.545409\pi$
$224$	−2.56155	−0.171151
$225$	1.56155	0.104104
$226$	4.56155	0.303430
$227$	−4.80776	−0.319103	−0.159551	−	0.987190i	$-0.551005\pi$
$227$	−4.80776	−0.319103	−0.159551	+	0.987190i	$0.551005\pi$
$228$	1.00000	0.0662266
$229$	11.9309	0.788414	0.394207	−	0.919022i	$-0.371019\pi$
$229$	11.9309	0.788414	0.394207	+	0.919022i	$0.371019\pi$
$230$	−11.6847	−0.770464
$231$	10.2462	0.674151
$232$	−5.68466	−0.373216
$233$	16.2462	1.06432	0.532162	−	0.846642i	$-0.321380\pi$
$233$	16.2462	1.06432	0.532162	+	0.846642i	$0.321380\pi$
$234$	−4.00000	−0.261488
$235$	−21.1231	−1.37792
$236$	12.8078	0.833714
$237$	−4.87689	−0.316788
$238$	5.12311	0.332082
$239$	−9.36932	−0.606051	−0.303025	−	0.952982i	$-0.597997\pi$
$239$	−9.36932	−0.606051	−0.303025	+	0.952982i	$0.597997\pi$
$240$	2.56155	0.165348
$241$	16.2462	1.04651	0.523255	−	0.852176i	$-0.324717\pi$
$241$	16.2462	1.04651	0.523255	+	0.852176i	$0.324717\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	−4.87689	−0.312211
$245$	−1.12311	−0.0717526
$246$	3.12311	0.199122
$247$	−4.00000	−0.254514
$248$	−4.56155	−0.289659
$249$	4.00000	0.253490
$250$	−8.80776	−0.557052
$251$	16.4924	1.04099	0.520496	−	0.853864i	$-0.325747\pi$
$251$	16.4924	1.04099	0.520496	+	0.853864i	$0.325747\pi$
$252$	−2.56155	−0.161363
$253$	18.2462	1.14713
$254$	10.8078	0.678139
$255$	−5.12311	−0.320821
$256$	1.00000	0.0625000
$257$	4.87689	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$257$	4.87689	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$258$	−5.43845	−0.338583
$259$	2.87689	0.178762
$260$	−10.2462	−0.635443
$261$	−5.68466	−0.351872
$262$	−20.4924	−1.26603
$263$	11.9309	0.735689	0.367844	−	0.929887i	$-0.380096\pi$
$263$	11.9309	0.735689	0.367844	+	0.929887i	$0.380096\pi$
$264$	−4.00000	−0.246183
$265$	2.56155	0.157355
$266$	−2.56155	−0.157059
$267$	−12.5616	−0.768755
$268$	11.6847	0.713754
$269$	13.6847	0.834368	0.417184	−	0.908822i	$-0.363017\pi$
$269$	13.6847	0.834368	0.417184	+	0.908822i	$0.363017\pi$
$270$	2.56155	0.155891
$271$	5.43845	0.330362	0.165181	−	0.986263i	$-0.447179\pi$
$271$	5.43845	0.330362	0.165181	+	0.986263i	$0.447179\pi$
$272$	−2.00000	−0.121268
$273$	10.2462	0.620129
$274$	−4.56155	−0.275573
$275$	−6.24621	−0.376661
$276$	−4.56155	−0.274573
$277$	−27.6155	−1.65926	−0.829628	−	0.558316i	$-0.811448\pi$
$277$	−27.6155	−1.65926	−0.829628	+	0.558316i	$0.811448\pi$
$278$	3.36932	0.202078
$279$	−4.56155	−0.273093
$280$	−6.56155	−0.392128
$281$	0.0691303	0.00412397	0.00206198	−	0.999998i	$-0.499344\pi$
$281$	0.0691303	0.00412397	0.00206198	+	0.999998i	$0.499344\pi$
$282$	−8.24621	−0.491055
$283$	26.7386	1.58945	0.794723	−	0.606972i	$-0.207616\pi$
$283$	26.7386	1.58945	0.794723	+	0.606972i	$0.207616\pi$
$284$	10.2462	0.608001
$285$	2.56155	0.151733
$286$	16.0000	0.946100
$287$	−8.00000	−0.472225
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−14.5616	−0.855084
$291$	8.24621	0.483401
$292$	−10.0000	−0.585206
$293$	−24.2462	−1.41648	−0.708239	−	0.705972i	$-0.750510\pi$
$293$	−24.2462	−1.41648	−0.708239	+	0.705972i	$0.750510\pi$
$294$	−0.438447	−0.0255708
$295$	32.8078	1.91014
$296$	−1.12311	−0.0652792
$297$	−4.00000	−0.232104
$298$	−1.75379	−0.101594
$299$	18.2462	1.05521
$300$	1.56155	0.0901563
$301$	13.9309	0.802962
$302$	14.8078	0.852091
$303$	−7.68466	−0.441472
$304$	1.00000	0.0573539
$305$	−12.4924	−0.715314
$306$	−2.00000	−0.114332
$307$	6.24621	0.356490	0.178245	−	0.983986i	$-0.442958\pi$
$307$	6.24621	0.356490	0.178245	+	0.983986i	$0.442958\pi$
$308$	10.2462	0.583832
$309$	17.6847	1.00605
$310$	−11.6847	−0.663644
$311$	7.12311	0.403914	0.201957	−	0.979394i	$-0.435270\pi$
$311$	7.12311	0.403914	0.201957	+	0.979394i	$0.435270\pi$
$312$	−4.00000	−0.226455
$313$	0.876894	0.0495650	0.0247825	−	0.999693i	$-0.492111\pi$
$313$	0.876894	0.0495650	0.0247825	+	0.999693i	$0.492111\pi$
$314$	18.0000	1.01580
$315$	−6.56155	−0.369702
$316$	−4.87689	−0.274347
$317$	−18.4924	−1.03864	−0.519319	−	0.854580i	$-0.673814\pi$
$317$	−18.4924	−1.03864	−0.519319	+	0.854580i	$0.673814\pi$
$318$	1.00000	0.0560772
$319$	22.7386	1.27312
$320$	2.56155	0.143195
$321$	7.68466	0.428916
$322$	11.6847	0.651161
$323$	−2.00000	−0.111283
$324$	1.00000	0.0555556
$325$	−6.24621	−0.346477
$326$	−10.5616	−0.584950
$327$	−15.6847	−0.867364
$328$	3.12311	0.172445
$329$	21.1231	1.16455
$330$	−10.2462	−0.564035
$331$	−29.6155	−1.62782	−0.813908	−	0.580993i	$-0.802664\pi$
$331$	−29.6155	−1.62782	−0.813908	+	0.580993i	$0.802664\pi$
$332$	4.00000	0.219529
$333$	−1.12311	−0.0615458
$334$	−9.12311	−0.499194
$335$	29.9309	1.63530
$336$	−2.56155	−0.139744
$337$	−18.8078	−1.02452	−0.512262	−	0.858829i	$-0.671192\pi$
$337$	−18.8078	−1.02452	−0.512262	+	0.858829i	$0.671192\pi$
$338$	3.00000	0.163178
$339$	4.56155	0.247750
$340$	−5.12311	−0.277839
$341$	18.2462	0.988088
$342$	1.00000	0.0540738
$343$	19.0540	1.02882
$344$	−5.43845	−0.293221
$345$	−11.6847	−0.629081
$346$	−21.3693	−1.14882
$347$	14.2462	0.764777	0.382388	−	0.924002i	$-0.375102\pi$
$347$	14.2462	0.764777	0.382388	+	0.924002i	$0.375102\pi$
$348$	−5.68466	−0.304730
$349$	−3.75379	−0.200936	−0.100468	−	0.994940i	$-0.532034\pi$
$349$	−3.75379	−0.200936	−0.100468	+	0.994940i	$0.532034\pi$
$350$	−4.00000	−0.213809
$351$	−4.00000	−0.213504
$352$	−4.00000	−0.213201
$353$	28.7386	1.52960	0.764802	−	0.644266i	$-0.222837\pi$
$353$	28.7386	1.52960	0.764802	+	0.644266i	$0.222837\pi$
$354$	12.8078	0.680725
$355$	26.2462	1.39300
$356$	−12.5616	−0.665761
$357$	5.12311	0.271144
$358$	17.1231	0.904984
$359$	18.8078	0.992636	0.496318	−	0.868141i	$-0.334685\pi$
$359$	18.8078	0.992636	0.496318	+	0.868141i	$0.334685\pi$
$360$	2.56155	0.135006
$361$	1.00000	0.0526316
$362$	10.5616	0.555103
$363$	5.00000	0.262432
$364$	10.2462	0.537047
$365$	−25.6155	−1.34078
$366$	−4.87689	−0.254919
$367$	−26.7386	−1.39575	−0.697873	−	0.716222i	$-0.745870\pi$
$367$	−26.7386	−1.39575	−0.697873	+	0.716222i	$0.745870\pi$
$368$	−4.56155	−0.237787
$369$	3.12311	0.162582
$370$	−2.87689	−0.149563
$371$	−2.56155	−0.132989
$372$	−4.56155	−0.236505
$373$	−19.0540	−0.986577	−0.493289	−	0.869866i	$-0.664205\pi$
$373$	−19.0540	−0.986577	−0.493289	+	0.869866i	$0.664205\pi$
$374$	8.00000	0.413670
$375$	−8.80776	−0.454831
$376$	−8.24621	−0.425266
$377$	22.7386	1.17110
$378$	−2.56155	−0.131752
$379$	38.7386	1.98987	0.994935	−	0.100521i	$-0.0320508\pi$
$379$	38.7386	1.98987	0.994935	+	0.100521i	$0.0320508\pi$
$380$	2.56155	0.131405
$381$	10.8078	0.553699
$382$	−5.36932	−0.274718
$383$	−31.3693	−1.60290	−0.801449	−	0.598064i	$-0.795937\pi$
$383$	−31.3693	−1.60290	−0.801449	+	0.598064i	$0.795937\pi$
$384$	1.00000	0.0510310
$385$	26.2462	1.33763
$386$	7.75379	0.394657
$387$	−5.43845	−0.276452
$388$	8.24621	0.418638
$389$	−31.0540	−1.57450	−0.787250	−	0.616635i	$-0.788496\pi$
$389$	−31.0540	−1.57450	−0.787250	+	0.616635i	$0.788496\pi$
$390$	−10.2462	−0.518837
$391$	9.12311	0.461375
$392$	−0.438447	−0.0221449
$393$	−20.4924	−1.03371
$394$	−14.2462	−0.717714
$395$	−12.4924	−0.628562
$396$	−4.00000	−0.201008
$397$	−13.3693	−0.670987	−0.335493	−	0.942043i	$-0.608903\pi$
$397$	−13.3693	−0.670987	−0.335493	+	0.942043i	$0.608903\pi$
$398$	−13.4384	−0.673608
$399$	−2.56155	−0.128238
$400$	1.56155	0.0780776
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	11.6847	0.582778
$403$	18.2462	0.908909
$404$	−7.68466	−0.382326
$405$	2.56155	0.127285
$406$	14.5616	0.722678
$407$	4.49242	0.222681
$408$	−2.00000	−0.0990148
$409$	−4.87689	−0.241147	−0.120573	−	0.992704i	$-0.538473\pi$
$409$	−4.87689	−0.241147	−0.120573	+	0.992704i	$0.538473\pi$
$410$	8.00000	0.395092
$411$	−4.56155	−0.225005
$412$	17.6847	0.871261
$413$	−32.8078	−1.61436
$414$	−4.56155	−0.224188
$415$	10.2462	0.502967
$416$	−4.00000	−0.196116
$417$	3.36932	0.164996
$418$	−4.00000	−0.195646
$419$	10.2462	0.500560	0.250280	−	0.968173i	$-0.419477\pi$
$419$	10.2462	0.500560	0.250280	+	0.968173i	$0.419477\pi$
$420$	−6.56155	−0.320171
$421$	29.9309	1.45874	0.729371	−	0.684119i	$-0.239813\pi$
$421$	29.9309	1.45874	0.729371	+	0.684119i	$0.239813\pi$
$422$	2.87689	0.140045
$423$	−8.24621	−0.400945
$424$	1.00000	0.0485643
$425$	−3.12311	−0.151493
$426$	10.2462	0.496431
$427$	12.4924	0.604551
$428$	7.68466	0.371452
$429$	16.0000	0.772487
$430$	−13.9309	−0.671806
$431$	−23.6847	−1.14085	−0.570425	−	0.821350i	$-0.693221\pi$
$431$	−23.6847	−1.14085	−0.570425	+	0.821350i	$0.693221\pi$
$432$	1.00000	0.0481125
$433$	−32.7386	−1.57332	−0.786659	−	0.617388i	$-0.788191\pi$
$433$	−32.7386	−1.57332	−0.786659	+	0.617388i	$0.788191\pi$
$434$	11.6847	0.560882
$435$	−14.5616	−0.698173
$436$	−15.6847	−0.751159
$437$	−4.56155	−0.218209
$438$	−10.0000	−0.477818
$439$	−20.2462	−0.966299	−0.483149	−	0.875538i	$-0.660507\pi$
$439$	−20.2462	−0.966299	−0.483149	+	0.875538i	$0.660507\pi$
$440$	−10.2462	−0.488469
$441$	−0.438447	−0.0208784
$442$	8.00000	0.380521
$443$	17.4384	0.828526	0.414263	−	0.910157i	$-0.364039\pi$
$443$	17.4384	0.828526	0.414263	+	0.910157i	$0.364039\pi$
$444$	−1.12311	−0.0533002
$445$	−32.1771	−1.52534
$446$	−4.24621	−0.201064
$447$	−1.75379	−0.0829514
$448$	−2.56155	−0.121022
$449$	−24.2462	−1.14425	−0.572125	−	0.820167i	$-0.693881\pi$
$449$	−24.2462	−1.14425	−0.572125	+	0.820167i	$0.693881\pi$
$450$	1.56155	0.0736123
$451$	−12.4924	−0.588245
$452$	4.56155	0.214557
$453$	14.8078	0.695729
$454$	−4.80776	−0.225640
$455$	26.2462	1.23044
$456$	1.00000	0.0468293
$457$	7.12311	0.333205	0.166602	−	0.986024i	$-0.446720\pi$
$457$	7.12311	0.333205	0.166602	+	0.986024i	$0.446720\pi$
$458$	11.9309	0.557493
$459$	−2.00000	−0.0933520
$460$	−11.6847	−0.544800
$461$	27.3693	1.27472	0.637358	−	0.770568i	$-0.280027\pi$
$461$	27.3693	1.27472	0.637358	+	0.770568i	$0.280027\pi$
$462$	10.2462	0.476697
$463$	−24.4924	−1.13826	−0.569130	−	0.822248i	$-0.692720\pi$
$463$	−24.4924	−1.13826	−0.569130	+	0.822248i	$0.692720\pi$
$464$	−5.68466	−0.263904
$465$	−11.6847	−0.541863
$466$	16.2462	0.752591
$467$	13.1231	0.607265	0.303632	−	0.952789i	$-0.401801\pi$
$467$	13.1231	0.607265	0.303632	+	0.952789i	$0.401801\pi$
$468$	−4.00000	−0.184900
$469$	−29.9309	−1.38208
$470$	−21.1231	−0.974336
$471$	18.0000	0.829396
$472$	12.8078	0.589525
$473$	21.7538	1.00024
$474$	−4.87689	−0.224003
$475$	1.56155	0.0716490
$476$	5.12311	0.234817
$477$	1.00000	0.0457869
$478$	−9.36932	−0.428543
$479$	21.0540	0.961981	0.480990	−	0.876726i	$-0.340277\pi$
$479$	21.0540	0.961981	0.480990	+	0.876726i	$0.340277\pi$
$480$	2.56155	0.116918
$481$	4.49242	0.204837
$482$	16.2462	0.739995
$483$	11.6847	0.531670
$484$	5.00000	0.227273
$485$	21.1231	0.959151
$486$	1.00000	0.0453609
$487$	−1.50758	−0.0683149	−0.0341574	−	0.999416i	$-0.510875\pi$
$487$	−1.50758	−0.0683149	−0.0341574	+	0.999416i	$0.510875\pi$
$488$	−4.87689	−0.220767
$489$	−10.5616	−0.477610
$490$	−1.12311	−0.0507367
$491$	−7.68466	−0.346804	−0.173402	−	0.984851i	$-0.555476\pi$
$491$	−7.68466	−0.346804	−0.173402	+	0.984851i	$0.555476\pi$
$492$	3.12311	0.140800
$493$	11.3693	0.512048
$494$	−4.00000	−0.179969
$495$	−10.2462	−0.460533
$496$	−4.56155	−0.204820
$497$	−26.2462	−1.17730
$498$	4.00000	0.179244
$499$	−26.7386	−1.19699	−0.598493	−	0.801128i	$-0.704233\pi$
$499$	−26.7386	−1.19699	−0.598493	+	0.801128i	$0.704233\pi$
$500$	−8.80776	−0.393895
$501$	−9.12311	−0.407590
$502$	16.4924	0.736093
$503$	−16.5616	−0.738443	−0.369222	−	0.929341i	$-0.620376\pi$
$503$	−16.5616	−0.738443	−0.369222	+	0.929341i	$0.620376\pi$
$504$	−2.56155	−0.114101
$505$	−19.6847	−0.875956
$506$	18.2462	0.811143
$507$	3.00000	0.133235
$508$	10.8078	0.479517
$509$	4.24621	0.188210	0.0941050	−	0.995562i	$-0.470001\pi$
$509$	4.24621	0.188210	0.0941050	+	0.995562i	$0.470001\pi$
$510$	−5.12311	−0.226855
$511$	25.6155	1.13316
$512$	1.00000	0.0441942
$513$	1.00000	0.0441511
$514$	4.87689	0.215111
$515$	45.3002	1.99616
$516$	−5.43845	−0.239414
$517$	32.9848	1.45067
$518$	2.87689	0.126403
$519$	−21.3693	−0.938009
$520$	−10.2462	−0.449326
$521$	−24.2462	−1.06225	−0.531123	−	0.847295i	$-0.678230\pi$
$521$	−24.2462	−1.06225	−0.531123	+	0.847295i	$0.678230\pi$
$522$	−5.68466	−0.248811
$523$	−27.3693	−1.19678	−0.598388	−	0.801206i	$-0.704192\pi$
$523$	−27.3693	−1.19678	−0.598388	+	0.801206i	$0.704192\pi$
$524$	−20.4924	−0.895216
$525$	−4.00000	−0.174574
$526$	11.9309	0.520211
$527$	9.12311	0.397409
$528$	−4.00000	−0.174078
$529$	−2.19224	−0.0953146
$530$	2.56155	0.111267
$531$	12.8078	0.555810
$532$	−2.56155	−0.111057
$533$	−12.4924	−0.541107
$534$	−12.5616	−0.543592
$535$	19.6847	0.851042
$536$	11.6847	0.504700
$537$	17.1231	0.738917
$538$	13.6847	0.589988
$539$	1.75379	0.0755410
$540$	2.56155	0.110232
$541$	−31.4384	−1.35164	−0.675822	−	0.737065i	$-0.736211\pi$
$541$	−31.4384	−1.35164	−0.675822	+	0.737065i	$0.736211\pi$
$542$	5.43845	0.233601
$543$	10.5616	0.453240
$544$	−2.00000	−0.0857493
$545$	−40.1771	−1.72100
$546$	10.2462	0.438497
$547$	−16.4924	−0.705165	−0.352583	−	0.935781i	$-0.614696\pi$
$547$	−16.4924	−0.705165	−0.352583	+	0.935781i	$0.614696\pi$
$548$	−4.56155	−0.194860
$549$	−4.87689	−0.208141
$550$	−6.24621	−0.266339
$551$	−5.68466	−0.242175
$552$	−4.56155	−0.194153
$553$	12.4924	0.531232
$554$	−27.6155	−1.17327
$555$	−2.87689	−0.122117
$556$	3.36932	0.142891
$557$	−17.4384	−0.738891	−0.369445	−	0.929252i	$-0.620452\pi$
$557$	−17.4384	−0.738891	−0.369445	+	0.929252i	$0.620452\pi$
$558$	−4.56155	−0.193106
$559$	21.7538	0.920087
$560$	−6.56155	−0.277276
$561$	8.00000	0.337760
$562$	0.0691303	0.00291609
$563$	−1.75379	−0.0739134	−0.0369567	−	0.999317i	$-0.511766\pi$
$563$	−1.75379	−0.0739134	−0.0369567	+	0.999317i	$0.511766\pi$
$564$	−8.24621	−0.347228
$565$	11.6847	0.491577
$566$	26.7386	1.12391
$567$	−2.56155	−0.107575
$568$	10.2462	0.429921
$569$	12.8769	0.539827	0.269914	−	0.962885i	$-0.413005\pi$
$569$	12.8769	0.539827	0.269914	+	0.962885i	$0.413005\pi$
$570$	2.56155	0.107292
$571$	−21.6155	−0.904582	−0.452291	−	0.891870i	$-0.649393\pi$
$571$	−21.6155	−0.904582	−0.452291	+	0.891870i	$0.649393\pi$
$572$	16.0000	0.668994
$573$	−5.36932	−0.224306
$574$	−8.00000	−0.333914
$575$	−7.12311	−0.297054
$576$	1.00000	0.0416667
$577$	−38.9848	−1.62296	−0.811480	−	0.584380i	$-0.801338\pi$
$577$	−38.9848	−1.62296	−0.811480	+	0.584380i	$0.801338\pi$
$578$	−13.0000	−0.540729
$579$	7.75379	0.322236
$580$	−14.5616	−0.604636
$581$	−10.2462	−0.425084
$582$	8.24621	0.341816
$583$	−4.00000	−0.165663
$584$	−10.0000	−0.413803
$585$	−10.2462	−0.423629
$586$	−24.2462	−1.00160
$587$	26.7386	1.10362	0.551811	−	0.833969i	$-0.313937\pi$
$587$	26.7386	1.10362	0.551811	+	0.833969i	$0.313937\pi$
$588$	−0.438447	−0.0180813
$589$	−4.56155	−0.187956
$590$	32.8078	1.35067
$591$	−14.2462	−0.586011
$592$	−1.12311	−0.0461594
$593$	20.8769	0.857311	0.428656	−	0.903468i	$-0.358987\pi$
$593$	20.8769	0.857311	0.428656	+	0.903468i	$0.358987\pi$
$594$	−4.00000	−0.164122
$595$	13.1231	0.537995
$596$	−1.75379	−0.0718380
$597$	−13.4384	−0.549999
$598$	18.2462	0.746143
$599$	48.4924	1.98135	0.990673	−	0.136258i	$-0.0435077\pi$
$599$	48.4924	1.98135	0.990673	+	0.136258i	$0.0435077\pi$
$600$	1.56155	0.0637501
$601$	−42.1771	−1.72044	−0.860220	−	0.509924i	$-0.829674\pi$
$601$	−42.1771	−1.72044	−0.860220	+	0.509924i	$0.829674\pi$
$602$	13.9309	0.567780
$603$	11.6847	0.475836
$604$	14.8078	0.602519
$605$	12.8078	0.520710
$606$	−7.68466	−0.312168
$607$	19.6155	0.796170	0.398085	−	0.917349i	$-0.369675\pi$
$607$	19.6155	0.796170	0.398085	+	0.917349i	$0.369675\pi$
$608$	1.00000	0.0405554
$609$	14.5616	0.590064
$610$	−12.4924	−0.505803
$611$	32.9848	1.33442
$612$	−2.00000	−0.0808452
$613$	19.6155	0.792264	0.396132	−	0.918194i	$-0.370352\pi$
$613$	19.6155	0.792264	0.396132	+	0.918194i	$0.370352\pi$
$614$	6.24621	0.252077
$615$	8.00000	0.322591
$616$	10.2462	0.412832
$617$	−38.4924	−1.54965	−0.774823	−	0.632178i	$-0.782161\pi$
$617$	−38.4924	−1.54965	−0.774823	+	0.632178i	$0.782161\pi$
$618$	17.6847	0.711381
$619$	−4.80776	−0.193240	−0.0966202	−	0.995321i	$-0.530803\pi$
$619$	−4.80776	−0.193240	−0.0966202	+	0.995321i	$0.530803\pi$
$620$	−11.6847	−0.469267
$621$	−4.56155	−0.183049
$622$	7.12311	0.285611
$623$	32.1771	1.28915
$624$	−4.00000	−0.160128
$625$	−30.3693	−1.21477
$626$	0.876894	0.0350477
$627$	−4.00000	−0.159745
$628$	18.0000	0.718278
$629$	2.24621	0.0895623
$630$	−6.56155	−0.261419
$631$	−27.8617	−1.10916	−0.554579	−	0.832131i	$-0.687121\pi$
$631$	−27.8617	−1.10916	−0.554579	+	0.832131i	$0.687121\pi$
$632$	−4.87689	−0.193992
$633$	2.87689	0.114346
$634$	−18.4924	−0.734428
$635$	27.6847	1.09863
$636$	1.00000	0.0396526
$637$	1.75379	0.0694876
$638$	22.7386	0.900231
$639$	10.2462	0.405334
$640$	2.56155	0.101254
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	7.68466	0.303289
$643$	−14.0691	−0.554832	−0.277416	−	0.960750i	$-0.589478\pi$
$643$	−14.0691	−0.554832	−0.277416	+	0.960750i	$0.589478\pi$
$644$	11.6847	0.460440
$645$	−13.9309	−0.548527
$646$	−2.00000	−0.0786889
$647$	−2.63068	−0.103423	−0.0517114	−	0.998662i	$-0.516468\pi$
$647$	−2.63068	−0.103423	−0.0517114	+	0.998662i	$0.516468\pi$
$648$	1.00000	0.0392837
$649$	−51.2311	−2.01099
$650$	−6.24621	−0.244997
$651$	11.6847	0.457958
$652$	−10.5616	−0.413622
$653$	−5.61553	−0.219753	−0.109876	−	0.993945i	$-0.535045\pi$
$653$	−5.61553	−0.219753	−0.109876	+	0.993945i	$0.535045\pi$
$654$	−15.6847	−0.613319
$655$	−52.4924	−2.05105
$656$	3.12311	0.121937
$657$	−10.0000	−0.390137
$658$	21.1231	0.823464
$659$	17.7538	0.691589	0.345795	−	0.938310i	$-0.387609\pi$
$659$	17.7538	0.691589	0.345795	+	0.938310i	$0.387609\pi$
$660$	−10.2462	−0.398833
$661$	49.1231	1.91067	0.955334	−	0.295529i	$-0.0954960\pi$
$661$	49.1231	1.91067	0.955334	+	0.295529i	$0.0954960\pi$
$662$	−29.6155	−1.15104
$663$	8.00000	0.310694
$664$	4.00000	0.155230
$665$	−6.56155	−0.254446
$666$	−1.12311	−0.0435195
$667$	25.9309	1.00405
$668$	−9.12311	−0.352984
$669$	−4.24621	−0.164168
$670$	29.9309	1.15633
$671$	19.5076	0.753082
$672$	−2.56155	−0.0988140
$673$	19.1231	0.737142	0.368571	−	0.929600i	$-0.379847\pi$
$673$	19.1231	0.737142	0.368571	+	0.929600i	$0.379847\pi$
$674$	−18.8078	−0.724448
$675$	1.56155	0.0601042
$676$	3.00000	0.115385
$677$	−28.2462	−1.08559	−0.542795	−	0.839865i	$-0.682634\pi$
$677$	−28.2462	−1.08559	−0.542795	+	0.839865i	$0.682634\pi$
$678$	4.56155	0.175185
$679$	−21.1231	−0.810630
$680$	−5.12311	−0.196462
$681$	−4.80776	−0.184234
$682$	18.2462	0.698684
$683$	−42.7386	−1.63535	−0.817674	−	0.575681i	$-0.804737\pi$
$683$	−42.7386	−1.63535	−0.817674	+	0.575681i	$0.804737\pi$
$684$	1.00000	0.0382360
$685$	−11.6847	−0.446448
$686$	19.0540	0.727484
$687$	11.9309	0.455191
$688$	−5.43845	−0.207339
$689$	−4.00000	−0.152388
$690$	−11.6847	−0.444827
$691$	−13.6155	−0.517959	−0.258980	−	0.965883i	$-0.583386\pi$
$691$	−13.6155	−0.517959	−0.258980	+	0.965883i	$0.583386\pi$
$692$	−21.3693	−0.812340
$693$	10.2462	0.389221
$694$	14.2462	0.540779
$695$	8.63068	0.327380
$696$	−5.68466	−0.215476
$697$	−6.24621	−0.236592
$698$	−3.75379	−0.142083
$699$	16.2462	0.614488
$700$	−4.00000	−0.151186
$701$	28.1771	1.06423	0.532117	−	0.846671i	$-0.321397\pi$
$701$	28.1771	1.06423	0.532117	+	0.846671i	$0.321397\pi$
$702$	−4.00000	−0.150970
$703$	−1.12311	−0.0423587
$704$	−4.00000	−0.150756
$705$	−21.1231	−0.795542
$706$	28.7386	1.08159
$707$	19.6847	0.740318
$708$	12.8078	0.481345
$709$	40.8769	1.53516	0.767582	−	0.640951i	$-0.221460\pi$
$709$	40.8769	1.53516	0.767582	+	0.640951i	$0.221460\pi$
$710$	26.2462	0.985003
$711$	−4.87689	−0.182898
$712$	−12.5616	−0.470764
$713$	20.8078	0.779257
$714$	5.12311	0.191727
$715$	40.9848	1.53275
$716$	17.1231	0.639921
$717$	−9.36932	−0.349904
$718$	18.8078	0.701900
$719$	10.9460	0.408218	0.204109	−	0.978948i	$-0.434570\pi$
$719$	10.9460	0.408218	0.204109	+	0.978948i	$0.434570\pi$
$720$	2.56155	0.0954634
$721$	−45.3002	−1.68707
$722$	1.00000	0.0372161
$723$	16.2462	0.604203
$724$	10.5616	0.392517
$725$	−8.87689	−0.329680
$726$	5.00000	0.185567
$727$	24.0000	0.890111	0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000	0.890111	0.445055	+	0.895503i	$0.353184\pi$
$728$	10.2462	0.379750
$729$	1.00000	0.0370370
$730$	−25.6155	−0.948073
$731$	10.8769	0.402296
$732$	−4.87689	−0.180255
$733$	7.93087	0.292933	0.146467	−	0.989216i	$-0.453210\pi$
$733$	7.93087	0.292933	0.146467	+	0.989216i	$0.453210\pi$
$734$	−26.7386	−0.986941
$735$	−1.12311	−0.0414264
$736$	−4.56155	−0.168141
$737$	−46.7386	−1.72164
$738$	3.12311	0.114963
$739$	−30.2462	−1.11262	−0.556312	−	0.830973i	$-0.687784\pi$
$739$	−30.2462	−1.11262	−0.556312	+	0.830973i	$0.687784\pi$
$740$	−2.87689	−0.105757
$741$	−4.00000	−0.146944
$742$	−2.56155	−0.0940376
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	−4.56155	−0.167235
$745$	−4.49242	−0.164590
$746$	−19.0540	−0.697616
$747$	4.00000	0.146352
$748$	8.00000	0.292509
$749$	−19.6847	−0.719262
$750$	−8.80776	−0.321614
$751$	−22.9848	−0.838729	−0.419364	−	0.907818i	$-0.637747\pi$
$751$	−22.9848	−0.838729	−0.419364	+	0.907818i	$0.637747\pi$
$752$	−8.24621	−0.300708
$753$	16.4924	0.601017
$754$	22.7386	0.828092
$755$	37.9309	1.38045
$756$	−2.56155	−0.0931628
$757$	−6.17708	−0.224510	−0.112255	−	0.993679i	$-0.535807\pi$
$757$	−6.17708	−0.224510	−0.112255	+	0.993679i	$0.535807\pi$
$758$	38.7386	1.40705
$759$	18.2462	0.662296
$760$	2.56155	0.0929173
$761$	24.7386	0.896775	0.448387	−	0.893839i	$-0.351999\pi$
$761$	24.7386	0.896775	0.448387	+	0.893839i	$0.351999\pi$
$762$	10.8078	0.391524
$763$	40.1771	1.45451
$764$	−5.36932	−0.194255
$765$	−5.12311	−0.185226
$766$	−31.3693	−1.13342
$767$	−51.2311	−1.84985
$768$	1.00000	0.0360844
$769$	33.8617	1.22109	0.610543	−	0.791983i	$-0.290951\pi$
$769$	33.8617	1.22109	0.610543	+	0.791983i	$0.290951\pi$
$770$	26.2462	0.945848
$771$	4.87689	0.175637
$772$	7.75379	0.279065
$773$	−41.8617	−1.50566	−0.752831	−	0.658214i	$-0.771312\pi$
$773$	−41.8617	−1.50566	−0.752831	+	0.658214i	$0.771312\pi$
$774$	−5.43845	−0.195481
$775$	−7.12311	−0.255870
$776$	8.24621	0.296022
$777$	2.87689	0.103208
$778$	−31.0540	−1.11334
$779$	3.12311	0.111897
$780$	−10.2462	−0.366873
$781$	−40.9848	−1.46655
$782$	9.12311	0.326242
$783$	−5.68466	−0.203153
$784$	−0.438447	−0.0156588
$785$	46.1080	1.64566
$786$	−20.4924	−0.730941
$787$	−49.9309	−1.77984	−0.889922	−	0.456113i	$-0.849241\pi$
$787$	−49.9309	−1.77984	−0.889922	+	0.456113i	$0.849241\pi$
$788$	−14.2462	−0.507500
$789$	11.9309	0.424750
$790$	−12.4924	−0.444460
$791$	−11.6847	−0.415459
$792$	−4.00000	−0.142134
$793$	19.5076	0.692734
$794$	−13.3693	−0.474459
$795$	2.56155	0.0908489
$796$	−13.4384	−0.476313
$797$	47.4773	1.68173	0.840866	−	0.541244i	$-0.182046\pi$
$797$	47.4773	1.68173	0.840866	+	0.541244i	$0.182046\pi$
$798$	−2.56155	−0.0906780
$799$	16.4924	0.583460
$800$	1.56155	0.0552092
$801$	−12.5616	−0.443841
$802$	−2.00000	−0.0706225
$803$	40.0000	1.41157
$804$	11.6847	0.412086
$805$	29.9309	1.05492
$806$	18.2462	0.642695
$807$	13.6847	0.481723
$808$	−7.68466	−0.270345
$809$	−11.9309	−0.419467	−0.209734	−	0.977759i	$-0.567260\pi$
$809$	−11.9309	−0.419467	−0.209734	+	0.977759i	$0.567260\pi$
$810$	2.56155	0.0900038
$811$	−20.9848	−0.736878	−0.368439	−	0.929652i	$-0.620108\pi$
$811$	−20.9848	−0.736878	−0.368439	+	0.929652i	$0.620108\pi$
$812$	14.5616	0.511010
$813$	5.43845	0.190735
$814$	4.49242	0.157459
$815$	−27.0540	−0.947659
$816$	−2.00000	−0.0700140
$817$	−5.43845	−0.190267
$818$	−4.87689	−0.170517
$819$	10.2462	0.358032
$820$	8.00000	0.279372
$821$	35.5464	1.24058	0.620289	−	0.784373i	$-0.287015\pi$
$821$	35.5464	1.24058	0.620289	+	0.784373i	$0.287015\pi$
$822$	−4.56155	−0.159102
$823$	40.9848	1.42864	0.714321	−	0.699818i	$-0.246736\pi$
$823$	40.9848	1.42864	0.714321	+	0.699818i	$0.246736\pi$
$824$	17.6847	0.616074
$825$	−6.24621	−0.217465
$826$	−32.8078	−1.14153
$827$	17.7538	0.617360	0.308680	−	0.951166i	$-0.400113\pi$
$827$	17.7538	0.617360	0.308680	+	0.951166i	$0.400113\pi$
$828$	−4.56155	−0.158525
$829$	−0.315342	−0.0109523	−0.00547613	−	0.999985i	$-0.501743\pi$
$829$	−0.315342	−0.0109523	−0.00547613	+	0.999985i	$0.501743\pi$
$830$	10.2462	0.355651
$831$	−27.6155	−0.957972
$832$	−4.00000	−0.138675
$833$	0.876894	0.0303826
$834$	3.36932	0.116670
$835$	−23.3693	−0.808729
$836$	−4.00000	−0.138343
$837$	−4.56155	−0.157670
$838$	10.2462	0.353949
$839$	−33.9309	−1.17142	−0.585712	−	0.810519i	$-0.699185\pi$
$839$	−33.9309	−1.17142	−0.585712	+	0.810519i	$0.699185\pi$
$840$	−6.56155	−0.226395
$841$	3.31534	0.114322
$842$	29.9309	1.03149
$843$	0.0691303	0.00238097
$844$	2.87689	0.0990268
$845$	7.68466	0.264360
$846$	−8.24621	−0.283511
$847$	−12.8078	−0.440080
$848$	1.00000	0.0343401
$849$	26.7386	0.917668
$850$	−3.12311	−0.107122
$851$	5.12311	0.175618
$852$	10.2462	0.351029
$853$	−38.4924	−1.31796	−0.658978	−	0.752162i	$-0.729011\pi$
$853$	−38.4924	−1.31796	−0.658978	+	0.752162i	$0.729011\pi$
$854$	12.4924	0.427482
$855$	2.56155	0.0876033
$856$	7.68466	0.262656
$857$	20.4233	0.697646	0.348823	−	0.937189i	$-0.386581\pi$
$857$	20.4233	0.697646	0.348823	+	0.937189i	$0.386581\pi$
$858$	16.0000	0.546231
$859$	−28.9848	−0.988950	−0.494475	−	0.869192i	$-0.664640\pi$
$859$	−28.9848	−0.988950	−0.494475	+	0.869192i	$0.664640\pi$
$860$	−13.9309	−0.475039
$861$	−8.00000	−0.272639
$862$	−23.6847	−0.806703
$863$	22.0691	0.751242	0.375621	−	0.926773i	$-0.377429\pi$
$863$	22.0691	0.751242	0.375621	+	0.926773i	$0.377429\pi$
$864$	1.00000	0.0340207
$865$	−54.7386	−1.86117
$866$	−32.7386	−1.11250
$867$	−13.0000	−0.441503
$868$	11.6847	0.396603
$869$	19.5076	0.661749
$870$	−14.5616	−0.493683
$871$	−46.7386	−1.58368
$872$	−15.6847	−0.531150
$873$	8.24621	0.279092
$874$	−4.56155	−0.154297
$875$	22.5616	0.762720
$876$	−10.0000	−0.337869
$877$	45.6155	1.54033	0.770163	−	0.637847i	$-0.220175\pi$
$877$	45.6155	1.54033	0.770163	+	0.637847i	$0.220175\pi$
$878$	−20.2462	−0.683277
$879$	−24.2462	−0.817804
$880$	−10.2462	−0.345400
$881$	−21.0540	−0.709326	−0.354663	−	0.934994i	$-0.615404\pi$
$881$	−21.0540	−0.709326	−0.354663	+	0.934994i	$0.615404\pi$
$882$	−0.438447	−0.0147633
$883$	20.9848	0.706196	0.353098	−	0.935586i	$-0.385128\pi$
$883$	20.9848	0.706196	0.353098	+	0.935586i	$0.385128\pi$
$884$	8.00000	0.269069
$885$	32.8078	1.10282
$886$	17.4384	0.585856
$887$	−2.38447	−0.0800627	−0.0400314	−	0.999198i	$-0.512746\pi$
$887$	−2.38447	−0.0800627	−0.0400314	+	0.999198i	$0.512746\pi$
$888$	−1.12311	−0.0376890
$889$	−27.6847	−0.928514
$890$	−32.1771	−1.07858
$891$	−4.00000	−0.134005
$892$	−4.24621	−0.142174
$893$	−8.24621	−0.275949
$894$	−1.75379	−0.0586555
$895$	43.8617	1.46614
$896$	−2.56155	−0.0855755
$897$	18.2462	0.609223
$898$	−24.2462	−0.809107
$899$	25.9309	0.864843
$900$	1.56155	0.0520518
$901$	−2.00000	−0.0666297
$902$	−12.4924	−0.415952
$903$	13.9309	0.463590
$904$	4.56155	0.151715
$905$	27.0540	0.899305
$906$	14.8078	0.491955
$907$	−36.9848	−1.22806	−0.614031	−	0.789282i	$-0.710453\pi$
$907$	−36.9848	−1.22806	−0.614031	+	0.789282i	$0.710453\pi$
$908$	−4.80776	−0.159551
$909$	−7.68466	−0.254884
$910$	26.2462	0.870053
$911$	9.30019	0.308129	0.154064	−	0.988061i	$-0.450764\pi$
$911$	9.30019	0.308129	0.154064	+	0.988061i	$0.450764\pi$
$912$	1.00000	0.0331133
$913$	−16.0000	−0.529523
$914$	7.12311	0.235611
$915$	−12.4924	−0.412987
$916$	11.9309	0.394207
$917$	52.4924	1.73345
$918$	−2.00000	−0.0660098
$919$	4.49242	0.148191	0.0740957	−	0.997251i	$-0.476393\pi$
$919$	4.49242	0.148191	0.0740957	+	0.997251i	$0.476393\pi$
$920$	−11.6847	−0.385232
$921$	6.24621	0.205820
$922$	27.3693	0.901360
$923$	−40.9848	−1.34903
$924$	10.2462	0.337076
$925$	−1.75379	−0.0576642
$926$	−24.4924	−0.804871
$927$	17.6847	0.580840
$928$	−5.68466	−0.186608
$929$	−8.73863	−0.286705	−0.143353	−	0.989672i	$-0.545788\pi$
$929$	−8.73863	−0.286705	−0.143353	+	0.989672i	$0.545788\pi$
$930$	−11.6847	−0.383155
$931$	−0.438447	−0.0143695
$932$	16.2462	0.532162
$933$	7.12311	0.233200
$934$	13.1231	0.429401
$935$	20.4924	0.670174
$936$	−4.00000	−0.130744
$937$	−18.1771	−0.593819	−0.296910	−	0.954906i	$-0.595956\pi$
$937$	−18.1771	−0.593819	−0.296910	+	0.954906i	$0.595956\pi$
$938$	−29.9309	−0.977278
$939$	0.876894	0.0286164
$940$	−21.1231	−0.688960
$941$	12.0691	0.393442	0.196721	−	0.980459i	$-0.436971\pi$
$941$	12.0691	0.393442	0.196721	+	0.980459i	$0.436971\pi$
$942$	18.0000	0.586472
$943$	−14.2462	−0.463920
$944$	12.8078	0.416857
$945$	−6.56155	−0.213447
$946$	21.7538	0.707277
$947$	5.12311	0.166479	0.0832393	−	0.996530i	$-0.473473\pi$
$947$	5.12311	0.166479	0.0832393	+	0.996530i	$0.473473\pi$
$948$	−4.87689	−0.158394
$949$	40.0000	1.29845
$950$	1.56155	0.0506635
$951$	−18.4924	−0.599658
$952$	5.12311	0.166041
$953$	40.4233	1.30944	0.654719	−	0.755872i	$-0.272787\pi$
$953$	40.4233	1.30944	0.654719	+	0.755872i	$0.272787\pi$
$954$	1.00000	0.0323762
$955$	−13.7538	−0.445062
$956$	−9.36932	−0.303025
$957$	22.7386	0.735036
$958$	21.0540	0.680223
$959$	11.6847	0.377317
$960$	2.56155	0.0826738
$961$	−10.1922	−0.328782
$962$	4.49242	0.144842
$963$	7.68466	0.247635
$964$	16.2462	0.523255
$965$	19.8617	0.639372
$966$	11.6847	0.375948
$967$	−50.7386	−1.63164	−0.815822	−	0.578303i	$-0.803715\pi$
$967$	−50.7386	−1.63164	−0.815822	+	0.578303i	$0.803715\pi$
$968$	5.00000	0.160706
$969$	−2.00000	−0.0642493
$970$	21.1231	0.678222
$971$	12.8078	0.411021	0.205510	−	0.978655i	$-0.434115\pi$
$971$	12.8078	0.411021	0.205510	+	0.978655i	$0.434115\pi$
$972$	1.00000	0.0320750
$973$	−8.63068	−0.276687
$974$	−1.50758	−0.0483059
$975$	−6.24621	−0.200039
$976$	−4.87689	−0.156106
$977$	−44.2462	−1.41556	−0.707781	−	0.706432i	$-0.750304\pi$
$977$	−44.2462	−1.41556	−0.707781	+	0.706432i	$0.750304\pi$
$978$	−10.5616	−0.337721
$979$	50.2462	1.60588
$980$	−1.12311	−0.0358763
$981$	−15.6847	−0.500773
$982$	−7.68466	−0.245227
$983$	40.1771	1.28145	0.640725	−	0.767771i	$-0.278634\pi$
$983$	40.1771	1.28145	0.640725	+	0.767771i	$0.278634\pi$
$984$	3.12311	0.0995610
$985$	−36.4924	−1.16275
$986$	11.3693	0.362073
$987$	21.1231	0.672356
$988$	−4.00000	−0.127257
$989$	24.8078	0.788841
$990$	−10.2462	−0.325646
$991$	22.9848	0.730138	0.365069	−	0.930981i	$-0.381045\pi$
$991$	22.9848	0.730138	0.365069	+	0.930981i	$0.381045\pi$
$992$	−4.56155	−0.144829
$993$	−29.6155	−0.939820
$994$	−26.2462	−0.832479
$995$	−34.4233	−1.09129
$996$	4.00000	0.126745
$997$	27.4384	0.868984	0.434492	−	0.900676i	$-0.356928\pi$
$997$	27.4384	0.868984	0.434492	+	0.900676i	$0.356928\pi$
$998$	−26.7386	−0.846397
$999$	−1.12311	−0.0355335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.p.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.p.1.2	✓	2	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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.56155 q^{5} +1.00000 q^{6} -2.56155 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.56155 q^{5} +1.00000 q^{6} -2.56155 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.56155 q^{10} -4.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -2.56155 q^{14} +2.56155 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +2.56155 q^{20} -2.56155 q^{21} -4.00000 q^{22} -4.56155 q^{23} +1.00000 q^{24} +1.56155 q^{25} -4.00000 q^{26} +1.00000 q^{27} -2.56155 q^{28} -5.68466 q^{29} +2.56155 q^{30} -4.56155 q^{31} +1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} -6.56155 q^{35} +1.00000 q^{36} -1.12311 q^{37} +1.00000 q^{38} -4.00000 q^{39} +2.56155 q^{40} +3.12311 q^{41} -2.56155 q^{42} -5.43845 q^{43} -4.00000 q^{44} +2.56155 q^{45} -4.56155 q^{46} -8.24621 q^{47} +1.00000 q^{48} -0.438447 q^{49} +1.56155 q^{50} -2.00000 q^{51} -4.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} -10.2462 q^{55} -2.56155 q^{56} +1.00000 q^{57} -5.68466 q^{58} +12.8078 q^{59} +2.56155 q^{60} -4.87689 q^{61} -4.56155 q^{62} -2.56155 q^{63} +1.00000 q^{64} -10.2462 q^{65} -4.00000 q^{66} +11.6847 q^{67} -2.00000 q^{68} -4.56155 q^{69} -6.56155 q^{70} +10.2462 q^{71} +1.00000 q^{72} -10.0000 q^{73} -1.12311 q^{74} +1.56155 q^{75} +1.00000 q^{76} +10.2462 q^{77} -4.00000 q^{78} -4.87689 q^{79} +2.56155 q^{80} +1.00000 q^{81} +3.12311 q^{82} +4.00000 q^{83} -2.56155 q^{84} -5.12311 q^{85} -5.43845 q^{86} -5.68466 q^{87} -4.00000 q^{88} -12.5616 q^{89} +2.56155 q^{90} +10.2462 q^{91} -4.56155 q^{92} -4.56155 q^{93} -8.24621 q^{94} +2.56155 q^{95} +1.00000 q^{96} +8.24621 q^{97} -0.438447 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9} + q^{10} - 8 q^{11} + 2 q^{12} - 8 q^{13} - q^{14} + q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{19} + q^{20} - q^{21} - 8 q^{22} - 5 q^{23} + 2 q^{24} - q^{25} - 8 q^{26} + 2 q^{27} - q^{28} + q^{29} + q^{30} - 5 q^{31} + 2 q^{32} - 8 q^{33} - 4 q^{34} - 9 q^{35} + 2 q^{36} + 6 q^{37} + 2 q^{38} - 8 q^{39} + q^{40} - 2 q^{41} - q^{42} - 15 q^{43} - 8 q^{44} + q^{45} - 5 q^{46} + 2 q^{48} - 5 q^{49} - q^{50} - 4 q^{51} - 8 q^{52} + 2 q^{53} + 2 q^{54} - 4 q^{55} - q^{56} + 2 q^{57} + q^{58} + 5 q^{59} + q^{60} - 18 q^{61} - 5 q^{62} - q^{63} + 2 q^{64} - 4 q^{65} - 8 q^{66} + 11 q^{67} - 4 q^{68} - 5 q^{69} - 9 q^{70} + 4 q^{71} + 2 q^{72} - 20 q^{73} + 6 q^{74} - q^{75} + 2 q^{76} + 4 q^{77} - 8 q^{78} - 18 q^{79} + q^{80} + 2 q^{81} - 2 q^{82} + 8 q^{83} - q^{84} - 2 q^{85} - 15 q^{86} + q^{87} - 8 q^{88} - 21 q^{89} + q^{90} + 4 q^{91} - 5 q^{92} - 5 q^{93} + q^{95} + 2 q^{96} - 5 q^{98} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9 + q^10 - 8 * q^11 + 2 * q^12 - 8 * q^13 - q^14 + q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 2 * q^19 + q^20 - q^21 - 8 * q^22 - 5 * q^23 + 2 * q^24 - q^25 - 8 * q^26 + 2 * q^27 - q^28 + q^29 + q^30 - 5 * q^31 + 2 * q^32 - 8 * q^33 - 4 * q^34 - 9 * q^35 + 2 * q^36 + 6 * q^37 + 2 * q^38 - 8 * q^39 + q^40 - 2 * q^41 - q^42 - 15 * q^43 - 8 * q^44 + q^45 - 5 * q^46 + 2 * q^48 - 5 * q^49 - q^50 - 4 * q^51 - 8 * q^52 + 2 * q^53 + 2 * q^54 - 4 * q^55 - q^56 + 2 * q^57 + q^58 + 5 * q^59 + q^60 - 18 * q^61 - 5 * q^62 - q^63 + 2 * q^64 - 4 * q^65 - 8 * q^66 + 11 * q^67 - 4 * q^68 - 5 * q^69 - 9 * q^70 + 4 * q^71 + 2 * q^72 - 20 * q^73 + 6 * q^74 - q^75 + 2 * q^76 + 4 * q^77 - 8 * q^78 - 18 * q^79 + q^80 + 2 * q^81 - 2 * q^82 + 8 * q^83 - q^84 - 2 * q^85 - 15 * q^86 + q^87 - 8 * q^88 - 21 * q^89 + q^90 + 4 * q^91 - 5 * q^92 - 5 * q^93 + q^95 + 2 * q^96 - 5 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more