LMFDB - Embedded newform 67.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [67,2,Mod(1,67)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(67, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("67.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	67.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:	$0.534997693543$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	67.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61803 q^{2} -0.381966 q^{3} +4.85410 q^{4} -3.00000 q^{5} +1.00000 q^{6} -3.85410 q^{7} -7.47214 q^{8} -2.85410 q^{9} +O(q^{10})$	\(q-2.61803 q^{2} -0.381966 q^{3} +4.85410 q^{4} -3.00000 q^{5} +1.00000 q^{6} -3.85410 q^{7} -7.47214 q^{8} -2.85410 q^{9} +7.85410 q^{10} +2.23607 q^{11} -1.85410 q^{12} -0.145898 q^{13} +10.0902 q^{14} +1.14590 q^{15} +9.85410 q^{16} -0.763932 q^{17} +7.47214 q^{18} -2.85410 q^{19} -14.5623 q^{20} +1.47214 q^{21} -5.85410 q^{22} +7.47214 q^{23} +2.85410 q^{24} +4.00000 q^{25} +0.381966 q^{26} +2.23607 q^{27} -18.7082 q^{28} -7.47214 q^{29} -3.00000 q^{30} -1.00000 q^{31} -10.8541 q^{32} -0.854102 q^{33} +2.00000 q^{34} +11.5623 q^{35} -13.8541 q^{36} -3.85410 q^{37} +7.47214 q^{38} +0.0557281 q^{39} +22.4164 q^{40} -0.381966 q^{41} -3.85410 q^{42} +4.85410 q^{43} +10.8541 q^{44} +8.56231 q^{45} -19.5623 q^{46} -8.61803 q^{47} -3.76393 q^{48} +7.85410 q^{49} -10.4721 q^{50} +0.291796 q^{51} -0.708204 q^{52} -9.00000 q^{53} -5.85410 q^{54} -6.70820 q^{55} +28.7984 q^{56} +1.09017 q^{57} +19.5623 q^{58} +6.00000 q^{59} +5.56231 q^{60} -13.5623 q^{61} +2.61803 q^{62} +11.0000 q^{63} +8.70820 q^{64} +0.437694 q^{65} +2.23607 q^{66} -1.00000 q^{67} -3.70820 q^{68} -2.85410 q^{69} -30.2705 q^{70} +3.76393 q^{71} +21.3262 q^{72} -4.00000 q^{73} +10.0902 q^{74} -1.52786 q^{75} -13.8541 q^{76} -8.61803 q^{77} -0.145898 q^{78} +6.56231 q^{79} -29.5623 q^{80} +7.70820 q^{81} +1.00000 q^{82} -15.3262 q^{83} +7.14590 q^{84} +2.29180 q^{85} -12.7082 q^{86} +2.85410 q^{87} -16.7082 q^{88} -2.23607 q^{89} -22.4164 q^{90} +0.562306 q^{91} +36.2705 q^{92} +0.381966 q^{93} +22.5623 q^{94} +8.56231 q^{95} +4.14590 q^{96} +14.4164 q^{97} -20.5623 q^{98} -6.38197 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} + 2 q^{6} - q^{7} - 6 q^{8} + q^{9}+O(q^{10}) $ 2 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 6 * q^5 + 2 * q^6 - q^7 - 6 * q^8 + q^9	\( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} + 2 q^{6} - q^{7} - 6 q^{8} + q^{9} + 9 q^{10} + 3 q^{12} - 7 q^{13} + 9 q^{14} + 9 q^{15} + 13 q^{16} - 6 q^{17} + 6 q^{18} + q^{19} - 9 q^{20} - 6 q^{21} - 5 q^{22} + 6 q^{23} - q^{24} + 8 q^{25} + 3 q^{26} - 24 q^{28} - 6 q^{29} - 6 q^{30} - 2 q^{31} - 15 q^{32} + 5 q^{33} + 4 q^{34} + 3 q^{35} - 21 q^{36} - q^{37} + 6 q^{38} + 18 q^{39} + 18 q^{40} - 3 q^{41} - q^{42} + 3 q^{43} + 15 q^{44} - 3 q^{45} - 19 q^{46} - 15 q^{47} - 12 q^{48} + 9 q^{49} - 12 q^{50} + 14 q^{51} + 12 q^{52} - 18 q^{53} - 5 q^{54} + 33 q^{56} - 9 q^{57} + 19 q^{58} + 12 q^{59} - 9 q^{60} - 7 q^{61} + 3 q^{62} + 22 q^{63} + 4 q^{64} + 21 q^{65} - 2 q^{67} + 6 q^{68} + q^{69} - 27 q^{70} + 12 q^{71} + 27 q^{72} - 8 q^{73} + 9 q^{74} - 12 q^{75} - 21 q^{76} - 15 q^{77} - 7 q^{78} - 7 q^{79} - 39 q^{80} + 2 q^{81} + 2 q^{82} - 15 q^{83} + 21 q^{84} + 18 q^{85} - 12 q^{86} - q^{87} - 20 q^{88} - 18 q^{90} - 19 q^{91} + 39 q^{92} + 3 q^{93} + 25 q^{94} - 3 q^{95} + 15 q^{96} + 2 q^{97} - 21 q^{98} - 15 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 6 * q^5 + 2 * q^6 - q^7 - 6 * q^8 + q^9 + 9 * q^10 + 3 * q^12 - 7 * q^13 + 9 * q^14 + 9 * q^15 + 13 * q^16 - 6 * q^17 + 6 * q^18 + q^19 - 9 * q^20 - 6 * q^21 - 5 * q^22 + 6 * q^23 - q^24 + 8 * q^25 + 3 * q^26 - 24 * q^28 - 6 * q^29 - 6 * q^30 - 2 * q^31 - 15 * q^32 + 5 * q^33 + 4 * q^34 + 3 * q^35 - 21 * q^36 - q^37 + 6 * q^38 + 18 * q^39 + 18 * q^40 - 3 * q^41 - q^42 + 3 * q^43 + 15 * q^44 - 3 * q^45 - 19 * q^46 - 15 * q^47 - 12 * q^48 + 9 * q^49 - 12 * q^50 + 14 * q^51 + 12 * q^52 - 18 * q^53 - 5 * q^54 + 33 * q^56 - 9 * q^57 + 19 * q^58 + 12 * q^59 - 9 * q^60 - 7 * q^61 + 3 * q^62 + 22 * q^63 + 4 * q^64 + 21 * q^65 - 2 * q^67 + 6 * q^68 + q^69 - 27 * q^70 + 12 * q^71 + 27 * q^72 - 8 * q^73 + 9 * q^74 - 12 * q^75 - 21 * q^76 - 15 * q^77 - 7 * q^78 - 7 * q^79 - 39 * q^80 + 2 * q^81 + 2 * q^82 - 15 * q^83 + 21 * q^84 + 18 * q^85 - 12 * q^86 - q^87 - 20 * q^88 - 18 * q^90 - 19 * q^91 + 39 * q^92 + 3 * q^93 + 25 * q^94 - 3 * q^95 + 15 * q^96 + 2 * q^97 - 21 * q^98 - 15 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.61803	−1.85123	−0.925615	−	0.378467i	$-0.876451\pi$
$2$	−2.61803	−1.85123	−0.925615	+	0.378467i	$0.876451\pi$
$3$	−0.381966	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$3$	−0.381966	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$4$	4.85410	2.42705
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	1.00000	0.408248
$7$	−3.85410	−1.45671	−0.728357	−	0.685198i	$-0.759716\pi$
$7$	−3.85410	−1.45671	−0.728357	+	0.685198i	$0.759716\pi$
$8$	−7.47214	−2.64180
$9$	−2.85410	−0.951367
$10$	7.85410	2.48369
$11$	2.23607	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$11$	2.23607	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$12$	−1.85410	−0.535233
$13$	−0.145898	−0.0404648	−0.0202324	−	0.999795i	$-0.506441\pi$
$13$	−0.145898	−0.0404648	−0.0202324	+	0.999795i	$0.506441\pi$
$14$	10.0902	2.69671
$15$	1.14590	0.295870
$16$	9.85410	2.46353
$17$	−0.763932	−0.185281	−0.0926404	−	0.995700i	$-0.529531\pi$
$17$	−0.763932	−0.185281	−0.0926404	+	0.995700i	$0.529531\pi$
$18$	7.47214	1.76120
$19$	−2.85410	−0.654776	−0.327388	−	0.944890i	$-0.606168\pi$
$19$	−2.85410	−0.654776	−0.327388	+	0.944890i	$0.606168\pi$
$20$	−14.5623	−3.25623
$21$	1.47214	0.321246
$22$	−5.85410	−1.24810
$23$	7.47214	1.55805	0.779024	−	0.626994i	$-0.215715\pi$
$23$	7.47214	1.55805	0.779024	+	0.626994i	$0.215715\pi$
$24$	2.85410	0.582591
$25$	4.00000	0.800000
$26$	0.381966	0.0749097
$27$	2.23607	0.430331
$28$	−18.7082	−3.53552
$29$	−7.47214	−1.38754	−0.693770	−	0.720196i	$-0.744052\pi$
$29$	−7.47214	−1.38754	−0.693770	+	0.720196i	$0.744052\pi$
$30$	−3.00000	−0.547723
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	−10.8541	−1.91875
$33$	−0.854102	−0.148680
$34$	2.00000	0.342997
$35$	11.5623	1.95439
$36$	−13.8541	−2.30902
$37$	−3.85410	−0.633610	−0.316805	−	0.948491i	$-0.602610\pi$
$37$	−3.85410	−0.633610	−0.316805	+	0.948491i	$0.602610\pi$
$38$	7.47214	1.21214
$39$	0.0557281	0.00892364
$40$	22.4164	3.54435
$41$	−0.381966	−0.0596531	−0.0298265	−	0.999555i	$-0.509495\pi$
$41$	−0.381966	−0.0596531	−0.0298265	+	0.999555i	$0.509495\pi$
$42$	−3.85410	−0.594701
$43$	4.85410	0.740244	0.370122	−	0.928983i	$-0.379316\pi$
$43$	4.85410	0.740244	0.370122	+	0.928983i	$0.379316\pi$
$44$	10.8541	1.63632
$45$	8.56231	1.27639
$46$	−19.5623	−2.88430
$47$	−8.61803	−1.25707	−0.628535	−	0.777782i	$-0.716345\pi$
$47$	−8.61803	−1.25707	−0.628535	+	0.777782i	$0.716345\pi$
$48$	−3.76393	−0.543277
$49$	7.85410	1.12201
$50$	−10.4721	−1.48098
$51$	0.291796	0.0408596
$52$	−0.708204	−0.0982102
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	−5.85410	−0.796642
$55$	−6.70820	−0.904534
$56$	28.7984	3.84834
$57$	1.09017	0.144397
$58$	19.5623	2.56866
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	5.56231	0.718091
$61$	−13.5623	−1.73648	−0.868238	−	0.496149i	$-0.834747\pi$
$61$	−13.5623	−1.73648	−0.868238	+	0.496149i	$0.834747\pi$
$62$	2.61803	0.332491
$63$	11.0000	1.38587
$64$	8.70820	1.08853
$65$	0.437694	0.0542893
$66$	2.23607	0.275241
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	−3.70820	−0.449686
$69$	−2.85410	−0.343594
$70$	−30.2705	−3.61802
$71$	3.76393	0.446697	0.223348	−	0.974739i	$-0.428301\pi$
$71$	3.76393	0.446697	0.223348	+	0.974739i	$0.428301\pi$
$72$	21.3262	2.51332
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	10.0902	1.17296
$75$	−1.52786	−0.176423
$76$	−13.8541	−1.58917
$77$	−8.61803	−0.982116
$78$	−0.145898	−0.0165197
$79$	6.56231	0.738317	0.369159	−	0.929366i	$-0.379646\pi$
$79$	6.56231	0.738317	0.369159	+	0.929366i	$0.379646\pi$
$80$	−29.5623	−3.30517
$81$	7.70820	0.856467
$82$	1.00000	0.110432
$83$	−15.3262	−1.68227	−0.841137	−	0.540823i	$-0.818113\pi$
$83$	−15.3262	−1.68227	−0.841137	+	0.540823i	$0.818113\pi$
$84$	7.14590	0.779681
$85$	2.29180	0.248580
$86$	−12.7082	−1.37036
$87$	2.85410	0.305992
$88$	−16.7082	−1.78110
$89$	−2.23607	−0.237023	−0.118511	−	0.992953i	$-0.537812\pi$
$89$	−2.23607	−0.237023	−0.118511	+	0.992953i	$0.537812\pi$
$90$	−22.4164	−2.36290
$91$	0.562306	0.0589457
$92$	36.2705	3.78146
$93$	0.381966	0.0396080
$94$	22.5623	2.32712
$95$	8.56231	0.878474
$96$	4.14590	0.423139
$97$	14.4164	1.46376	0.731882	−	0.681431i	$-0.238642\pi$
$97$	14.4164	1.46376	0.731882	+	0.681431i	$0.238642\pi$
$98$	−20.5623	−2.07711
$99$	−6.38197	−0.641412
$100$	19.4164	1.94164
$101$	−10.0902	−1.00401	−0.502005	−	0.864865i	$-0.667404\pi$
$101$	−10.0902	−1.00401	−0.502005	+	0.864865i	$0.667404\pi$
$102$	−0.763932	−0.0756405
$103$	7.56231	0.745136	0.372568	−	0.928005i	$-0.378477\pi$
$103$	7.56231	0.745136	0.372568	+	0.928005i	$0.378477\pi$
$104$	1.09017	0.106900
$105$	−4.41641	−0.430997
$106$	23.5623	2.28857
$107$	5.29180	0.511577	0.255789	−	0.966733i	$-0.417665\pi$
$107$	5.29180	0.511577	0.255789	+	0.966733i	$0.417665\pi$
$108$	10.8541	1.04444
$109$	−4.85410	−0.464939	−0.232469	−	0.972604i	$-0.574681\pi$
$109$	−4.85410	−0.464939	−0.232469	+	0.972604i	$0.574681\pi$
$110$	17.5623	1.67450
$111$	1.47214	0.139729
$112$	−37.9787	−3.58865
$113$	15.3820	1.44701	0.723507	−	0.690317i	$-0.242529\pi$
$113$	15.3820	1.44701	0.723507	+	0.690317i	$0.242529\pi$
$114$	−2.85410	−0.267311
$115$	−22.4164	−2.09034
$116$	−36.2705	−3.36763
$117$	0.416408	0.0384969
$118$	−15.7082	−1.44606
$119$	2.94427	0.269901
$120$	−8.56231	−0.781628
$121$	−6.00000	−0.545455
$122$	35.5066	3.21461
$123$	0.145898	0.0131552
$124$	−4.85410	−0.435911
$125$	3.00000	0.268328
$126$	−28.7984	−2.56556
$127$	−5.85410	−0.519468	−0.259734	−	0.965680i	$-0.583635\pi$
$127$	−5.85410	−0.519468	−0.259734	+	0.965680i	$0.583635\pi$
$128$	−1.09017	−0.0963583
$129$	−1.85410	−0.163245
$130$	−1.14590	−0.100502
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	−4.14590	−0.360854
$133$	11.0000	0.953821
$134$	2.61803	0.226164
$135$	−6.70820	−0.577350
$136$	5.70820	0.489474
$137$	0.708204	0.0605059	0.0302530	−	0.999542i	$-0.490369\pi$
$137$	0.708204	0.0605059	0.0302530	+	0.999542i	$0.490369\pi$
$138$	7.47214	0.636070
$139$	−3.00000	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$139$	−3.00000	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$140$	56.1246	4.74340
$141$	3.29180	0.277219
$142$	−9.85410	−0.826938
$143$	−0.326238	−0.0272814
$144$	−28.1246	−2.34372
$145$	22.4164	1.86158
$146$	10.4721	0.866680
$147$	−3.00000	−0.247436
$148$	−18.7082	−1.53780
$149$	19.0344	1.55936	0.779681	−	0.626177i	$-0.215381\pi$
$149$	19.0344	1.55936	0.779681	+	0.626177i	$0.215381\pi$
$150$	4.00000	0.326599
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	21.3262	1.72979
$153$	2.18034	0.176270
$154$	22.5623	1.81812
$155$	3.00000	0.240966
$156$	0.270510	0.0216581
$157$	−3.56231	−0.284303	−0.142151	−	0.989845i	$-0.545402\pi$
$157$	−3.56231	−0.284303	−0.142151	+	0.989845i	$0.545402\pi$
$158$	−17.1803	−1.36679
$159$	3.43769	0.272627
$160$	32.5623	2.57428
$161$	−28.7984	−2.26963
$162$	−20.1803	−1.58552
$163$	−6.85410	−0.536855	−0.268427	−	0.963300i	$-0.586504\pi$
$163$	−6.85410	−0.536855	−0.268427	+	0.963300i	$0.586504\pi$
$164$	−1.85410	−0.144781
$165$	2.56231	0.199475
$166$	40.1246	3.11427
$167$	−11.8885	−0.919963	−0.459982	−	0.887928i	$-0.652144\pi$
$167$	−11.8885	−0.919963	−0.459982	+	0.887928i	$0.652144\pi$
$168$	−11.0000	−0.848668
$169$	−12.9787	−0.998363
$170$	−6.00000	−0.460179
$171$	8.14590	0.622932
$172$	23.5623	1.79661
$173$	−13.9098	−1.05754	−0.528772	−	0.848764i	$-0.677347\pi$
$173$	−13.9098	−1.05754	−0.528772	+	0.848764i	$0.677347\pi$
$174$	−7.47214	−0.566461
$175$	−15.4164	−1.16537
$176$	22.0344	1.66091
$177$	−2.29180	−0.172262
$178$	5.85410	0.438783
$179$	−6.76393	−0.505560	−0.252780	−	0.967524i	$-0.581345\pi$
$179$	−6.76393	−0.505560	−0.252780	+	0.967524i	$0.581345\pi$
$180$	41.5623	3.09787
$181$	6.56231	0.487772	0.243886	−	0.969804i	$-0.421578\pi$
$181$	6.56231	0.487772	0.243886	+	0.969804i	$0.421578\pi$
$182$	−1.47214	−0.109122
$183$	5.18034	0.382942
$184$	−55.8328	−4.11605
$185$	11.5623	0.850078
$186$	−1.00000	−0.0733236
$187$	−1.70820	−0.124916
$188$	−41.8328	−3.05097
$189$	−8.61803	−0.626870
$190$	−22.4164	−1.62626
$191$	1.47214	0.106520	0.0532600	−	0.998581i	$-0.483039\pi$
$191$	1.47214	0.106520	0.0532600	+	0.998581i	$0.483039\pi$
$192$	−3.32624	−0.240051
$193$	16.8541	1.21318	0.606592	−	0.795013i	$-0.292536\pi$
$193$	16.8541	1.21318	0.606592	+	0.795013i	$0.292536\pi$
$194$	−37.7426	−2.70976
$195$	−0.167184	−0.0119723
$196$	38.1246	2.72319
$197$	−4.52786	−0.322597	−0.161298	−	0.986906i	$-0.551568\pi$
$197$	−4.52786	−0.322597	−0.161298	+	0.986906i	$0.551568\pi$
$198$	16.7082	1.18740
$199$	10.2705	0.728057	0.364029	−	0.931388i	$-0.381401\pi$
$199$	10.2705	0.728057	0.364029	+	0.931388i	$0.381401\pi$
$200$	−29.8885	−2.11344
$201$	0.381966	0.0269418
$202$	26.4164	1.85865
$203$	28.7984	2.02125
$204$	1.41641	0.0991684
$205$	1.14590	0.0800330
$206$	−19.7984	−1.37942
$207$	−21.3262	−1.48228
$208$	−1.43769	−0.0996862
$209$	−6.38197	−0.441450
$210$	11.5623	0.797875
$211$	−2.14590	−0.147730	−0.0738649	−	0.997268i	$-0.523533\pi$
$211$	−2.14590	−0.147730	−0.0738649	+	0.997268i	$0.523533\pi$
$212$	−43.6869	−3.00043
$213$	−1.43769	−0.0985092
$214$	−13.8541	−0.947047
$215$	−14.5623	−0.993141
$216$	−16.7082	−1.13685
$217$	3.85410	0.261633
$218$	12.7082	0.860708
$219$	1.52786	0.103243
$220$	−32.5623	−2.19535
$221$	0.111456	0.00749735
$222$	−3.85410	−0.258670
$223$	15.7082	1.05190	0.525950	−	0.850516i	$-0.323710\pi$
$223$	15.7082	1.05190	0.525950	+	0.850516i	$0.323710\pi$
$224$	41.8328	2.79507
$225$	−11.4164	−0.761094
$226$	−40.2705	−2.67875
$227$	−20.5066	−1.36107	−0.680535	−	0.732716i	$-0.738252\pi$
$227$	−20.5066	−1.36107	−0.680535	+	0.732716i	$0.738252\pi$
$228$	5.29180	0.350458
$229$	−18.8541	−1.24591	−0.622957	−	0.782256i	$-0.714069\pi$
$229$	−18.8541	−1.24591	−0.622957	+	0.782256i	$0.714069\pi$
$230$	58.6869	3.86970
$231$	3.29180	0.216584
$232$	55.8328	3.66560
$233$	1.52786	0.100094	0.0500469	−	0.998747i	$-0.484063\pi$
$233$	1.52786	0.100094	0.0500469	+	0.998747i	$0.484063\pi$
$234$	−1.09017	−0.0712666
$235$	25.8541	1.68654
$236$	29.1246	1.89585
$237$	−2.50658	−0.162820
$238$	−7.70820	−0.499649
$239$	−19.0902	−1.23484	−0.617420	−	0.786634i	$-0.711822\pi$
$239$	−19.0902	−1.23484	−0.617420	+	0.786634i	$0.711822\pi$
$240$	11.2918	0.728882
$241$	25.8541	1.66541	0.832705	−	0.553718i	$-0.186791\pi$
$241$	25.8541	1.66541	0.832705	+	0.553718i	$0.186791\pi$
$242$	15.7082	1.00976
$243$	−9.65248	−0.619207
$244$	−65.8328	−4.21451
$245$	−23.5623	−1.50534
$246$	−0.381966	−0.0243533
$247$	0.416408	0.0264954
$248$	7.47214	0.474481
$249$	5.85410	0.370989
$250$	−7.85410	−0.496737
$251$	11.2361	0.709214	0.354607	−	0.935015i	$-0.384615\pi$
$251$	11.2361	0.709214	0.354607	+	0.935015i	$0.384615\pi$
$252$	53.3951	3.36358
$253$	16.7082	1.05044
$254$	15.3262	0.961654
$255$	−0.875388	−0.0548189
$256$	−14.5623	−0.910144
$257$	−26.8885	−1.67726	−0.838631	−	0.544701i	$-0.816643\pi$
$257$	−26.8885	−1.67726	−0.838631	+	0.544701i	$0.816643\pi$
$258$	4.85410	0.302203
$259$	14.8541	0.922989
$260$	2.12461	0.131763
$261$	21.3262	1.32006
$262$	7.85410	0.485228
$263$	8.56231	0.527974	0.263987	−	0.964526i	$-0.414962\pi$
$263$	8.56231	0.527974	0.263987	+	0.964526i	$0.414962\pi$
$264$	6.38197	0.392783
$265$	27.0000	1.65860
$266$	−28.7984	−1.76574
$267$	0.854102	0.0522702
$268$	−4.85410	−0.296511
$269$	16.7984	1.02421	0.512107	−	0.858921i	$-0.328865\pi$
$269$	16.7984	1.02421	0.512107	+	0.858921i	$0.328865\pi$
$270$	17.5623	1.06881
$271$	24.7082	1.50092	0.750458	−	0.660918i	$-0.229833\pi$
$271$	24.7082	1.50092	0.750458	+	0.660918i	$0.229833\pi$
$272$	−7.52786	−0.456444
$273$	−0.214782	−0.0129992
$274$	−1.85410	−0.112010
$275$	8.94427	0.539360
$276$	−13.8541	−0.833919
$277$	32.8328	1.97273	0.986366	−	0.164564i	$-0.0526218\pi$
$277$	32.8328	1.97273	0.986366	+	0.164564i	$0.0526218\pi$
$278$	7.85410	0.471058
$279$	2.85410	0.170871
$280$	−86.3951	−5.16310
$281$	8.61803	0.514109	0.257054	−	0.966397i	$-0.417248\pi$
$281$	8.61803	0.514109	0.257054	+	0.966397i	$0.417248\pi$
$282$	−8.61803	−0.513196
$283$	−22.4164	−1.33252	−0.666259	−	0.745721i	$-0.732105\pi$
$283$	−22.4164	−1.33252	−0.666259	+	0.745721i	$0.732105\pi$
$284$	18.2705	1.08416
$285$	−3.27051	−0.193728
$286$	0.854102	0.0505041
$287$	1.47214	0.0868974
$288$	30.9787	1.82544
$289$	−16.4164	−0.965671
$290$	−58.6869	−3.44621
$291$	−5.50658	−0.322801
$292$	−19.4164	−1.13626
$293$	4.41641	0.258009	0.129005	−	0.991644i	$-0.458822\pi$
$293$	4.41641	0.258009	0.129005	+	0.991644i	$0.458822\pi$
$294$	7.85410	0.458061
$295$	−18.0000	−1.04800
$296$	28.7984	1.67387
$297$	5.00000	0.290129
$298$	−49.8328	−2.88674
$299$	−1.09017	−0.0630462
$300$	−7.41641	−0.428187
$301$	−18.7082	−1.07832
$302$	2.61803	0.150651
$303$	3.85410	0.221412
$304$	−28.1246	−1.61306
$305$	40.6869	2.32973
$306$	−5.70820	−0.326316
$307$	17.4164	0.994007	0.497003	−	0.867749i	$-0.334434\pi$
$307$	17.4164	0.994007	0.497003	+	0.867749i	$0.334434\pi$
$308$	−41.8328	−2.38365
$309$	−2.88854	−0.164324
$310$	−7.85410	−0.446083
$311$	−26.8328	−1.52155	−0.760775	−	0.649016i	$-0.775181\pi$
$311$	−26.8328	−1.52155	−0.760775	+	0.649016i	$0.775181\pi$
$312$	−0.416408	−0.0235745
$313$	−13.8541	−0.783080	−0.391540	−	0.920161i	$-0.628058\pi$
$313$	−13.8541	−0.783080	−0.391540	+	0.920161i	$0.628058\pi$
$314$	9.32624	0.526310
$315$	−33.0000	−1.85934
$316$	31.8541	1.79193
$317$	18.5967	1.04450	0.522249	−	0.852793i	$-0.325093\pi$
$317$	18.5967	1.04450	0.522249	+	0.852793i	$0.325093\pi$
$318$	−9.00000	−0.504695
$319$	−16.7082	−0.935480
$320$	−26.1246	−1.46041
$321$	−2.02129	−0.112817
$322$	75.3951	4.20161
$323$	2.18034	0.121317
$324$	37.4164	2.07869
$325$	−0.583592	−0.0323719
$326$	17.9443	0.993841
$327$	1.85410	0.102532
$328$	2.85410	0.157591
$329$	33.2148	1.83119
$330$	−6.70820	−0.369274
$331$	−3.41641	−0.187783	−0.0938914	−	0.995582i	$-0.529931\pi$
$331$	−3.41641	−0.187783	−0.0938914	+	0.995582i	$0.529931\pi$
$332$	−74.3951	−4.08296
$333$	11.0000	0.602796
$334$	31.1246	1.70306
$335$	3.00000	0.163908
$336$	14.5066	0.791399
$337$	−8.27051	−0.450523	−0.225262	−	0.974298i	$-0.572324\pi$
$337$	−8.27051	−0.450523	−0.225262	+	0.974298i	$0.572324\pi$
$338$	33.9787	1.84820
$339$	−5.87539	−0.319107
$340$	11.1246	0.603317
$341$	−2.23607	−0.121090
$342$	−21.3262	−1.15319
$343$	−3.29180	−0.177740
$344$	−36.2705	−1.95557
$345$	8.56231	0.460979
$346$	36.4164	1.95776
$347$	−11.1803	−0.600192	−0.300096	−	0.953909i	$-0.597019\pi$
$347$	−11.1803	−0.600192	−0.300096	+	0.953909i	$0.597019\pi$
$348$	13.8541	0.742658
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	40.3607	2.15737
$351$	−0.326238	−0.0174133
$352$	−24.2705	−1.29362
$353$	−17.5066	−0.931781	−0.465891	−	0.884842i	$-0.654266\pi$
$353$	−17.5066	−0.931781	−0.465891	+	0.884842i	$0.654266\pi$
$354$	6.00000	0.318896
$355$	−11.2918	−0.599306
$356$	−10.8541	−0.575266
$357$	−1.12461	−0.0595208
$358$	17.7082	0.935908
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	−63.9787	−3.37197
$361$	−10.8541	−0.571269
$362$	−17.1803	−0.902979
$363$	2.29180	0.120288
$364$	2.72949	0.143064
$365$	12.0000	0.628109
$366$	−13.5623	−0.708913
$367$	−19.1246	−0.998297	−0.499148	−	0.866517i	$-0.666354\pi$
$367$	−19.1246	−0.998297	−0.499148	+	0.866517i	$0.666354\pi$
$368$	73.6312	3.83829
$369$	1.09017	0.0567520
$370$	−30.2705	−1.57369
$371$	34.6869	1.80086
$372$	1.85410	0.0961307
$373$	−18.5623	−0.961120	−0.480560	−	0.876962i	$-0.659567\pi$
$373$	−18.5623	−0.961120	−0.480560	+	0.876962i	$0.659567\pi$
$374$	4.47214	0.231249
$375$	−1.14590	−0.0591739
$376$	64.3951	3.32092
$377$	1.09017	0.0561466
$378$	22.5623	1.16048
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	41.5623	2.13210
$381$	2.23607	0.114557
$382$	−3.85410	−0.197193
$383$	−7.20163	−0.367986	−0.183993	−	0.982928i	$-0.558902\pi$
$383$	−7.20163	−0.367986	−0.183993	+	0.982928i	$0.558902\pi$
$384$	0.416408	0.0212497
$385$	25.8541	1.31765
$386$	−44.1246	−2.24588
$387$	−13.8541	−0.704244
$388$	69.9787	3.55263
$389$	−18.6525	−0.945718	−0.472859	−	0.881138i	$-0.656778\pi$
$389$	−18.6525	−0.945718	−0.472859	+	0.881138i	$0.656778\pi$
$390$	0.437694	0.0221635
$391$	−5.70820	−0.288676
$392$	−58.6869	−2.96414
$393$	1.14590	0.0578029
$394$	11.8541	0.597201
$395$	−19.6869	−0.990556
$396$	−30.9787	−1.55674
$397$	−35.5623	−1.78482	−0.892410	−	0.451224i	$-0.850987\pi$
$397$	−35.5623	−1.78482	−0.892410	+	0.451224i	$0.850987\pi$
$398$	−26.8885	−1.34780
$399$	−4.20163	−0.210344
$400$	39.4164	1.97082
$401$	7.85410	0.392215	0.196108	−	0.980582i	$-0.437170\pi$
$401$	7.85410	0.392215	0.196108	+	0.980582i	$0.437170\pi$
$402$	−1.00000	−0.0498755
$403$	0.145898	0.00726770
$404$	−48.9787	−2.43678
$405$	−23.1246	−1.14907
$406$	−75.3951	−3.74180
$407$	−8.61803	−0.427180
$408$	−2.18034	−0.107943
$409$	13.1246	0.648970	0.324485	−	0.945891i	$-0.394809\pi$
$409$	13.1246	0.648970	0.324485	+	0.945891i	$0.394809\pi$
$410$	−3.00000	−0.148159
$411$	−0.270510	−0.0133433
$412$	36.7082	1.80848
$413$	−23.1246	−1.13789
$414$	55.8328	2.74403
$415$	45.9787	2.25701
$416$	1.58359	0.0776420
$417$	1.14590	0.0561149
$418$	16.7082	0.817225
$419$	22.3607	1.09239	0.546195	−	0.837658i	$-0.316076\pi$
$419$	22.3607	1.09239	0.546195	+	0.837658i	$0.316076\pi$
$420$	−21.4377	−1.04605
$421$	−30.8541	−1.50374	−0.751868	−	0.659313i	$-0.770847\pi$
$421$	−30.8541	−1.50374	−0.751868	+	0.659313i	$0.770847\pi$
$422$	5.61803	0.273482
$423$	24.5967	1.19593
$424$	67.2492	3.26591
$425$	−3.05573	−0.148225
$426$	3.76393	0.182363
$427$	52.2705	2.52955
$428$	25.6869	1.24162
$429$	0.124612	0.00601631
$430$	38.1246	1.83853
$431$	27.7639	1.33734	0.668671	−	0.743559i	$-0.266864\pi$
$431$	27.7639	1.33734	0.668671	+	0.743559i	$0.266864\pi$
$432$	22.0344	1.06013
$433$	−40.3951	−1.94127	−0.970633	−	0.240566i	$-0.922667\pi$
$433$	−40.3951	−1.94127	−0.970633	+	0.240566i	$0.922667\pi$
$434$	−10.0902	−0.484344
$435$	−8.56231	−0.410531
$436$	−23.5623	−1.12843
$437$	−21.3262	−1.02017
$438$	−4.00000	−0.191127
$439$	−38.6869	−1.84643	−0.923213	−	0.384289i	$-0.874447\pi$
$439$	−38.6869	−1.84643	−0.923213	+	0.384289i	$0.874447\pi$
$440$	50.1246	2.38960
$441$	−22.4164	−1.06745
$442$	−0.291796	−0.0138793
$443$	−10.5279	−0.500194	−0.250097	−	0.968221i	$-0.580462\pi$
$443$	−10.5279	−0.500194	−0.250097	+	0.968221i	$0.580462\pi$
$444$	7.14590	0.339129
$445$	6.70820	0.317999
$446$	−41.1246	−1.94731
$447$	−7.27051	−0.343883
$448$	−33.5623	−1.58567
$449$	8.67376	0.409340	0.204670	−	0.978831i	$-0.434388\pi$
$449$	8.67376	0.409340	0.204670	+	0.978831i	$0.434388\pi$
$450$	29.8885	1.40896
$451$	−0.854102	−0.0402181
$452$	74.6656	3.51198
$453$	0.381966	0.0179463
$454$	53.6869	2.51965
$455$	−1.68692	−0.0790839
$456$	−8.14590	−0.381467
$457$	−9.58359	−0.448302	−0.224151	−	0.974554i	$-0.571961\pi$
$457$	−9.58359	−0.448302	−0.224151	+	0.974554i	$0.571961\pi$
$458$	49.3607	2.30647
$459$	−1.70820	−0.0797321
$460$	−108.812	−5.07336
$461$	−13.8541	−0.645250	−0.322625	−	0.946527i	$-0.604565\pi$
$461$	−13.8541	−0.645250	−0.322625	+	0.946527i	$0.604565\pi$
$462$	−8.61803	−0.400947
$463$	−27.5623	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$463$	−27.5623	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$464$	−73.6312	−3.41824
$465$	−1.14590	−0.0531397
$466$	−4.00000	−0.185296
$467$	23.1803	1.07266	0.536329	−	0.844009i	$-0.319811\pi$
$467$	23.1803	1.07266	0.536329	+	0.844009i	$0.319811\pi$
$468$	2.02129	0.0934340
$469$	3.85410	0.177966
$470$	−67.6869	−3.12216
$471$	1.36068	0.0626968
$472$	−44.8328	−2.06360
$473$	10.8541	0.499072
$474$	6.56231	0.301417
$475$	−11.4164	−0.523821
$476$	14.2918	0.655063
$477$	25.6869	1.17612
$478$	49.9787	2.28597
$479$	30.6525	1.40055	0.700274	−	0.713874i	$-0.253061\pi$
$479$	30.6525	1.40055	0.700274	+	0.713874i	$0.253061\pi$
$480$	−12.4377	−0.567700
$481$	0.562306	0.0256389
$482$	−67.6869	−3.08305
$483$	11.0000	0.500517
$484$	−29.1246	−1.32385
$485$	−43.2492	−1.96385
$486$	25.2705	1.14629
$487$	39.8328	1.80500	0.902499	−	0.430693i	$-0.141731\pi$
$487$	39.8328	1.80500	0.902499	+	0.430693i	$0.141731\pi$
$488$	101.339	4.58742
$489$	2.61803	0.118392
$490$	61.6869	2.78673
$491$	28.4164	1.28241	0.641207	−	0.767368i	$-0.278434\pi$
$491$	28.4164	1.28241	0.641207	+	0.767368i	$0.278434\pi$
$492$	0.708204	0.0319283
$493$	5.70820	0.257085
$494$	−1.09017	−0.0490491
$495$	19.1459	0.860544
$496$	−9.85410	−0.442462
$497$	−14.5066	−0.650709
$498$	−15.3262	−0.686785
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	14.5623	0.651246
$501$	4.54102	0.202878
$502$	−29.4164	−1.31292
$503$	41.9443	1.87020	0.935101	−	0.354380i	$-0.115308\pi$
$503$	41.9443	1.87020	0.935101	+	0.354380i	$0.115308\pi$
$504$	−82.1935	−3.66119
$505$	30.2705	1.34702
$506$	−43.7426	−1.94460
$507$	4.95743	0.220167
$508$	−28.4164	−1.26077
$509$	7.41641	0.328726	0.164363	−	0.986400i	$-0.447443\pi$
$509$	7.41641	0.328726	0.164363	+	0.986400i	$0.447443\pi$
$510$	2.29180	0.101482
$511$	15.4164	0.681982
$512$	40.3050	1.78124
$513$	−6.38197	−0.281771
$514$	70.3951	3.10500
$515$	−22.6869	−0.999705
$516$	−9.00000	−0.396203
$517$	−19.2705	−0.847516
$518$	−38.8885	−1.70866
$519$	5.31308	0.233218
$520$	−3.27051	−0.143421
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−55.8328	−2.44374
$523$	−1.56231	−0.0683149	−0.0341574	−	0.999416i	$-0.510875\pi$
$523$	−1.56231	−0.0683149	−0.0341574	+	0.999416i	$0.510875\pi$
$524$	−14.5623	−0.636157
$525$	5.88854	0.256997
$526$	−22.4164	−0.977402
$527$	0.763932	0.0332774
$528$	−8.41641	−0.366277
$529$	32.8328	1.42751
$530$	−70.6869	−3.07044
$531$	−17.1246	−0.743145
$532$	53.3951	2.31497
$533$	0.0557281	0.00241385
$534$	−2.23607	−0.0967641
$535$	−15.8754	−0.686353
$536$	7.47214	0.322747
$537$	2.58359	0.111490
$538$	−43.9787	−1.89606
$539$	17.5623	0.756462
$540$	−32.5623	−1.40126
$541$	31.4164	1.35070	0.675348	−	0.737499i	$-0.263993\pi$
$541$	31.4164	1.35070	0.675348	+	0.737499i	$0.263993\pi$
$542$	−64.6869	−2.77854
$543$	−2.50658	−0.107568
$544$	8.29180	0.355508
$545$	14.5623	0.623781
$546$	0.562306	0.0240645
$547$	5.43769	0.232499	0.116250	−	0.993220i	$-0.462913\pi$
$547$	5.43769	0.232499	0.116250	+	0.993220i	$0.462913\pi$
$548$	3.43769	0.146851
$549$	38.7082	1.65203
$550$	−23.4164	−0.998479
$551$	21.3262	0.908528
$552$	21.3262	0.907705
$553$	−25.2918	−1.07552
$554$	−85.9574	−3.65198
$555$	−4.41641	−0.187466
$556$	−14.5623	−0.617579
$557$	−30.3820	−1.28733	−0.643663	−	0.765309i	$-0.722586\pi$
$557$	−30.3820	−1.28733	−0.643663	+	0.765309i	$0.722586\pi$
$558$	−7.47214	−0.316321
$559$	−0.708204	−0.0299538
$560$	113.936	4.81468
$561$	0.652476	0.0275476
$562$	−22.5623	−0.951733
$563$	−9.70820	−0.409152	−0.204576	−	0.978851i	$-0.565582\pi$
$563$	−9.70820	−0.409152	−0.204576	+	0.978851i	$0.565582\pi$
$564$	15.9787	0.672825
$565$	−46.1459	−1.94137
$566$	58.6869	2.46680
$567$	−29.7082	−1.24763
$568$	−28.1246	−1.18008
$569$	20.1803	0.846004	0.423002	−	0.906129i	$-0.360976\pi$
$569$	20.1803	0.846004	0.423002	+	0.906129i	$0.360976\pi$
$570$	8.56231	0.358636
$571$	25.9787	1.08718	0.543588	−	0.839352i	$-0.317066\pi$
$571$	25.9787	1.08718	0.543588	+	0.839352i	$0.317066\pi$
$572$	−1.58359	−0.0662133
$573$	−0.562306	−0.0234907
$574$	−3.85410	−0.160867
$575$	29.8885	1.24644
$576$	−24.8541	−1.03559
$577$	9.29180	0.386823	0.193411	−	0.981118i	$-0.438045\pi$
$577$	9.29180	0.386823	0.193411	+	0.981118i	$0.438045\pi$
$578$	42.9787	1.78768
$579$	−6.43769	−0.267541
$580$	108.812	4.51815
$581$	59.0689	2.45059
$582$	14.4164	0.597579
$583$	−20.1246	−0.833476
$584$	29.8885	1.23680
$585$	−1.24922	−0.0516490
$586$	−11.5623	−0.477634
$587$	22.9098	0.945590	0.472795	−	0.881172i	$-0.343245\pi$
$587$	22.9098	0.945590	0.472795	+	0.881172i	$0.343245\pi$
$588$	−14.5623	−0.600539
$589$	2.85410	0.117601
$590$	47.1246	1.94009
$591$	1.72949	0.0711417
$592$	−37.9787	−1.56092
$593$	−34.4721	−1.41560	−0.707800	−	0.706412i	$-0.750312\pi$
$593$	−34.4721	−1.41560	−0.707800	+	0.706412i	$0.750312\pi$
$594$	−13.0902	−0.537096
$595$	−8.83282	−0.362110
$596$	92.3951	3.78465
$597$	−3.92299	−0.160557
$598$	2.85410	0.116713
$599$	10.4721	0.427880	0.213940	−	0.976847i	$-0.431370\pi$
$599$	10.4721	0.427880	0.213940	+	0.976847i	$0.431370\pi$
$600$	11.4164	0.466073
$601$	−7.12461	−0.290619	−0.145309	−	0.989386i	$-0.546418\pi$
$601$	−7.12461	−0.290619	−0.145309	+	0.989386i	$0.546418\pi$
$602$	48.9787	1.99622
$603$	2.85410	0.116228
$604$	−4.85410	−0.197511
$605$	18.0000	0.731804
$606$	−10.0902	−0.409885
$607$	23.4377	0.951307	0.475653	−	0.879633i	$-0.342212\pi$
$607$	23.4377	0.951307	0.475653	+	0.879633i	$0.342212\pi$
$608$	30.9787	1.25635
$609$	−11.0000	−0.445742
$610$	−106.520	−4.31286
$611$	1.25735	0.0508671
$612$	10.5836	0.427816
$613$	−14.4164	−0.582273	−0.291137	−	0.956681i	$-0.594033\pi$
$613$	−14.4164	−0.582273	−0.291137	+	0.956681i	$0.594033\pi$
$614$	−45.5967	−1.84013
$615$	−0.437694	−0.0176495
$616$	64.3951	2.59455
$617$	0.978714	0.0394015	0.0197008	−	0.999806i	$-0.493729\pi$
$617$	0.978714	0.0394015	0.0197008	+	0.999806i	$0.493729\pi$
$618$	7.56231	0.304201
$619$	−34.2918	−1.37830	−0.689152	−	0.724617i	$-0.742017\pi$
$619$	−34.2918	−1.37830	−0.689152	+	0.724617i	$0.742017\pi$
$620$	14.5623	0.584836
$621$	16.7082	0.670477
$622$	70.2492	2.81674
$623$	8.61803	0.345274
$624$	0.549150	0.0219836
$625$	−29.0000	−1.16000
$626$	36.2705	1.44966
$627$	2.43769	0.0973521
$628$	−17.2918	−0.690018
$629$	2.94427	0.117396
$630$	86.3951	3.44206
$631$	−4.43769	−0.176662	−0.0883309	−	0.996091i	$-0.528153\pi$
$631$	−4.43769	−0.176662	−0.0883309	+	0.996091i	$0.528153\pi$
$632$	−49.0344	−1.95049
$633$	0.819660	0.0325786
$634$	−48.6869	−1.93360
$635$	17.5623	0.696939
$636$	16.6869	0.661679
$637$	−1.14590	−0.0454021
$638$	43.7426	1.73179
$639$	−10.7426	−0.424972
$640$	3.27051	0.129278
$641$	−17.0689	−0.674180	−0.337090	−	0.941472i	$-0.609443\pi$
$641$	−17.0689	−0.674180	−0.337090	+	0.941472i	$0.609443\pi$
$642$	5.29180	0.208851
$643$	−20.5623	−0.810898	−0.405449	−	0.914118i	$-0.632885\pi$
$643$	−20.5623	−0.810898	−0.405449	+	0.914118i	$0.632885\pi$
$644$	−139.790	−5.50851
$645$	5.56231	0.219016
$646$	−5.70820	−0.224586
$647$	−24.7639	−0.973571	−0.486785	−	0.873522i	$-0.661831\pi$
$647$	−24.7639	−0.973571	−0.486785	+	0.873522i	$0.661831\pi$
$648$	−57.5967	−2.26261
$649$	13.4164	0.526640
$650$	1.52786	0.0599278
$651$	−1.47214	−0.0576976
$652$	−33.2705	−1.30297
$653$	4.47214	0.175008	0.0875041	−	0.996164i	$-0.472111\pi$
$653$	4.47214	0.175008	0.0875041	+	0.996164i	$0.472111\pi$
$654$	−4.85410	−0.189810
$655$	9.00000	0.351659
$656$	−3.76393	−0.146957
$657$	11.4164	0.445396
$658$	−86.9574	−3.38995
$659$	−29.9443	−1.16646	−0.583232	−	0.812306i	$-0.698212\pi$
$659$	−29.9443	−1.16646	−0.583232	+	0.812306i	$0.698212\pi$
$660$	12.4377	0.484137
$661$	50.1246	1.94962	0.974811	−	0.223034i	$-0.0715960\pi$
$661$	50.1246	1.94962	0.974811	+	0.223034i	$0.0715960\pi$
$662$	8.94427	0.347629
$663$	−0.0425725	−0.00165338
$664$	114.520	4.44423
$665$	−33.0000	−1.27969
$666$	−28.7984	−1.11591
$667$	−55.8328	−2.16186
$668$	−57.7082	−2.23280
$669$	−6.00000	−0.231973
$670$	−7.85410	−0.303430
$671$	−30.3262	−1.17073
$672$	−15.9787	−0.616392
$673$	−35.6869	−1.37563	−0.687815	−	0.725886i	$-0.741430\pi$
$673$	−35.6869	−1.37563	−0.687815	+	0.725886i	$0.741430\pi$
$674$	21.6525	0.834022
$675$	8.94427	0.344265
$676$	−63.0000	−2.42308
$677$	11.1803	0.429695	0.214848	−	0.976648i	$-0.431074\pi$
$677$	11.1803	0.429695	0.214848	+	0.976648i	$0.431074\pi$
$678$	15.3820	0.590741
$679$	−55.5623	−2.13229
$680$	−17.1246	−0.656699
$681$	7.83282	0.300154
$682$	5.85410	0.224165
$683$	9.38197	0.358991	0.179495	−	0.983759i	$-0.442553\pi$
$683$	9.38197	0.358991	0.179495	+	0.983759i	$0.442553\pi$
$684$	39.5410	1.51189
$685$	−2.12461	−0.0811772
$686$	8.61803	0.329038
$687$	7.20163	0.274759
$688$	47.8328	1.82361
$689$	1.31308	0.0500245
$690$	−22.4164	−0.853378
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	−67.5197	−2.56672
$693$	24.5967	0.934353
$694$	29.2705	1.11109
$695$	9.00000	0.341389
$696$	−21.3262	−0.808369
$697$	0.291796	0.0110526
$698$	15.7082	0.594564
$699$	−0.583592	−0.0220735
$700$	−74.8328	−2.82841
$701$	40.3607	1.52440	0.762201	−	0.647341i	$-0.224119\pi$
$701$	40.3607	1.52440	0.762201	+	0.647341i	$0.224119\pi$
$702$	0.854102	0.0322360
$703$	11.0000	0.414873
$704$	19.4721	0.733884
$705$	−9.87539	−0.371929
$706$	45.8328	1.72494
$707$	38.8885	1.46255
$708$	−11.1246	−0.418089
$709$	−5.41641	−0.203417	−0.101709	−	0.994814i	$-0.532431\pi$
$709$	−5.41641	−0.203417	−0.101709	+	0.994814i	$0.532431\pi$
$710$	29.5623	1.10945
$711$	−18.7295	−0.702411
$712$	16.7082	0.626166
$713$	−7.47214	−0.279834
$714$	2.94427	0.110187
$715$	0.978714	0.0366018
$716$	−32.8328	−1.22702
$717$	7.29180	0.272317
$718$	47.1246	1.75867
$719$	40.5279	1.51143	0.755717	−	0.654898i	$-0.227288\pi$
$719$	40.5279	1.51143	0.755717	+	0.654898i	$0.227288\pi$
$720$	84.3738	3.14443
$721$	−29.1459	−1.08545
$722$	28.4164	1.05755
$723$	−9.87539	−0.367270
$724$	31.8541	1.18385
$725$	−29.8885	−1.11003
$726$	−6.00000	−0.222681
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	−4.20163	−0.155723
$729$	−19.4377	−0.719915
$730$	−31.4164	−1.16277
$731$	−3.70820	−0.137153
$732$	25.1459	0.929419
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	50.0689	1.84808
$735$	9.00000	0.331970
$736$	−81.1033	−2.98951
$737$	−2.23607	−0.0823666
$738$	−2.85410	−0.105061
$739$	−11.2705	−0.414592	−0.207296	−	0.978278i	$-0.566466\pi$
$739$	−11.2705	−0.414592	−0.207296	+	0.978278i	$0.566466\pi$
$740$	56.1246	2.06318
$741$	−0.159054	−0.00584298
$742$	−90.8115	−3.33380
$743$	24.5967	0.902367	0.451184	−	0.892431i	$-0.351002\pi$
$743$	24.5967	0.902367	0.451184	+	0.892431i	$0.351002\pi$
$744$	−2.85410	−0.104636
$745$	−57.1033	−2.09210
$746$	48.5967	1.77925
$747$	43.7426	1.60046
$748$	−8.29180	−0.303178
$749$	−20.3951	−0.745222
$750$	3.00000	0.109545
$751$	44.1246	1.61013	0.805065	−	0.593187i	$-0.202130\pi$
$751$	44.1246	1.61013	0.805065	+	0.593187i	$0.202130\pi$
$752$	−84.9230	−3.09682
$753$	−4.29180	−0.156402
$754$	−2.85410	−0.103940
$755$	3.00000	0.109181
$756$	−41.8328	−1.52144
$757$	38.4164	1.39627	0.698134	−	0.715967i	$-0.254014\pi$
$757$	38.4164	1.39627	0.698134	+	0.715967i	$0.254014\pi$
$758$	−10.4721	−0.380365
$759$	−6.38197	−0.231651
$760$	−63.9787	−2.32075
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	−5.85410	−0.212072
$763$	18.7082	0.677282
$764$	7.14590	0.258530
$765$	−6.54102	−0.236491
$766$	18.8541	0.681226
$767$	−0.875388	−0.0316084
$768$	5.56231	0.200712
$769$	−10.7082	−0.386148	−0.193074	−	0.981184i	$-0.561846\pi$
$769$	−10.7082	−0.386148	−0.193074	+	0.981184i	$0.561846\pi$
$770$	−67.6869	−2.43927
$771$	10.2705	0.369883
$772$	81.8115	2.94446
$773$	−25.3607	−0.912160	−0.456080	−	0.889939i	$-0.650747\pi$
$773$	−25.3607	−0.912160	−0.456080	+	0.889939i	$0.650747\pi$
$774$	36.2705	1.30372
$775$	−4.00000	−0.143684
$776$	−107.721	−3.86697
$777$	−5.67376	−0.203545
$778$	48.8328	1.75074
$779$	1.09017	0.0390594
$780$	−0.811529	−0.0290574
$781$	8.41641	0.301163
$782$	14.9443	0.534406
$783$	−16.7082	−0.597102
$784$	77.3951	2.76411
$785$	10.6869	0.381432
$786$	−3.00000	−0.107006
$787$	6.41641	0.228720	0.114360	−	0.993439i	$-0.463518\pi$
$787$	6.41641	0.228720	0.114360	+	0.993439i	$0.463518\pi$
$788$	−21.9787	−0.782959
$789$	−3.27051	−0.116433
$790$	51.5410	1.83375
$791$	−59.2837	−2.10788
$792$	47.6869	1.69448
$793$	1.97871	0.0702662
$794$	93.1033	3.30411
$795$	−10.3131	−0.365767
$796$	49.8541	1.76703
$797$	−34.6312	−1.22670	−0.613350	−	0.789811i	$-0.710178\pi$
$797$	−34.6312	−1.22670	−0.613350	+	0.789811i	$0.710178\pi$
$798$	11.0000	0.389396
$799$	6.58359	0.232911
$800$	−43.4164	−1.53500
$801$	6.38197	0.225496
$802$	−20.5623	−0.726080
$803$	−8.94427	−0.315637
$804$	1.85410	0.0653891
$805$	86.3951	3.04503
$806$	−0.381966	−0.0134542
$807$	−6.41641	−0.225868
$808$	75.3951	2.65239
$809$	2.56231	0.0900859	0.0450429	−	0.998985i	$-0.485658\pi$
$809$	2.56231	0.0900859	0.0450429	+	0.998985i	$0.485658\pi$
$810$	60.5410	2.12719
$811$	−48.1246	−1.68988	−0.844942	−	0.534858i	$-0.820365\pi$
$811$	−48.1246	−1.68988	−0.844942	+	0.534858i	$0.820365\pi$
$812$	139.790	4.90568
$813$	−9.43769	−0.330994
$814$	22.5623	0.790808
$815$	20.5623	0.720266
$816$	2.87539	0.100659
$817$	−13.8541	−0.484694
$818$	−34.3607	−1.20139
$819$	−1.60488	−0.0560790
$820$	5.56231	0.194244
$821$	−15.7639	−0.550165	−0.275083	−	0.961421i	$-0.588705\pi$
$821$	−15.7639	−0.550165	−0.275083	+	0.961421i	$0.588705\pi$
$822$	0.708204	0.0247014
$823$	−33.6869	−1.17425	−0.587126	−	0.809496i	$-0.699741\pi$
$823$	−33.6869	−1.17425	−0.587126	+	0.809496i	$0.699741\pi$
$824$	−56.5066	−1.96850
$825$	−3.41641	−0.118944
$826$	60.5410	2.10649
$827$	12.7639	0.443845	0.221923	−	0.975064i	$-0.428767\pi$
$827$	12.7639	0.443845	0.221923	+	0.975064i	$0.428767\pi$
$828$	−103.520	−3.59756
$829$	−4.12461	−0.143254	−0.0716268	−	0.997431i	$-0.522819\pi$
$829$	−4.12461	−0.143254	−0.0716268	+	0.997431i	$0.522819\pi$
$830$	−120.374	−4.17824
$831$	−12.5410	−0.435043
$832$	−1.27051	−0.0440470
$833$	−6.00000	−0.207888
$834$	−3.00000	−0.103882
$835$	35.6656	1.23426
$836$	−30.9787	−1.07142
$837$	−2.23607	−0.0772898
$838$	−58.5410	−2.02227
$839$	−19.3050	−0.666481	−0.333240	−	0.942842i	$-0.608142\pi$
$839$	−19.3050	−0.666481	−0.333240	+	0.942842i	$0.608142\pi$
$840$	33.0000	1.13861
$841$	26.8328	0.925270
$842$	80.7771	2.78376
$843$	−3.29180	−0.113375
$844$	−10.4164	−0.358548
$845$	38.9361	1.33944
$846$	−64.3951	−2.21395
$847$	23.1246	0.794571
$848$	−88.6869	−3.04552
$849$	8.56231	0.293858
$850$	8.00000	0.274398
$851$	−28.7984	−0.987196
$852$	−6.97871	−0.239087
$853$	39.6869	1.35885	0.679427	−	0.733743i	$-0.262228\pi$
$853$	39.6869	1.35885	0.679427	+	0.733743i	$0.262228\pi$
$854$	−136.846	−4.68277
$855$	−24.4377	−0.835752
$856$	−39.5410	−1.35148
$857$	−22.4164	−0.765730	−0.382865	−	0.923804i	$-0.625063\pi$
$857$	−22.4164	−0.765730	−0.382865	+	0.923804i	$0.625063\pi$
$858$	−0.326238	−0.0111376
$859$	−1.97871	−0.0675128	−0.0337564	−	0.999430i	$-0.510747\pi$
$859$	−1.97871	−0.0675128	−0.0337564	+	0.999430i	$0.510747\pi$
$860$	−70.6869	−2.41040
$861$	−0.562306	−0.0191633
$862$	−72.6869	−2.47573
$863$	35.5623	1.21055	0.605277	−	0.796015i	$-0.293062\pi$
$863$	35.5623	1.21055	0.605277	+	0.796015i	$0.293062\pi$
$864$	−24.2705	−0.825700
$865$	41.7295	1.41885
$866$	105.756	3.59373
$867$	6.27051	0.212958
$868$	18.7082	0.634998
$869$	14.6738	0.497773
$870$	22.4164	0.759987
$871$	0.145898	0.00494357
$872$	36.2705	1.22827
$873$	−41.1459	−1.39258
$874$	55.8328	1.88857
$875$	−11.5623	−0.390877
$876$	7.41641	0.250577
$877$	−12.1246	−0.409419	−0.204710	−	0.978823i	$-0.565625\pi$
$877$	−12.1246	−0.409419	−0.204710	+	0.978823i	$0.565625\pi$
$878$	101.284	3.41816
$879$	−1.68692	−0.0568983
$880$	−66.1033	−2.22834
$881$	−26.2361	−0.883916	−0.441958	−	0.897036i	$-0.645716\pi$
$881$	−26.2361	−0.883916	−0.441958	+	0.897036i	$0.645716\pi$
$882$	58.6869	1.97609
$883$	20.7082	0.696887	0.348443	−	0.937330i	$-0.386710\pi$
$883$	20.7082	0.696887	0.348443	+	0.937330i	$0.386710\pi$
$884$	0.541020	0.0181965
$885$	6.87539	0.231114
$886$	27.5623	0.925974
$887$	23.0689	0.774577	0.387289	−	0.921959i	$-0.373412\pi$
$887$	23.0689	0.774577	0.387289	+	0.921959i	$0.373412\pi$
$888$	−11.0000	−0.369136
$889$	22.5623	0.756715
$890$	−17.5623	−0.588690
$891$	17.2361	0.577430
$892$	76.2492	2.55301
$893$	24.5967	0.823099
$894$	19.0344	0.636607
$895$	20.2918	0.678280
$896$	4.20163	0.140366
$897$	0.416408	0.0139035
$898$	−22.7082	−0.757783
$899$	7.47214	0.249210
$900$	−55.4164	−1.84721
$901$	6.87539	0.229052
$902$	2.23607	0.0744529
$903$	7.14590	0.237801
$904$	−114.936	−3.82272
$905$	−19.6869	−0.654415
$906$	−1.00000	−0.0332228
$907$	−21.1459	−0.702138	−0.351069	−	0.936350i	$-0.614182\pi$
$907$	−21.1459	−0.702138	−0.351069	+	0.936350i	$0.614182\pi$
$908$	−99.5410	−3.30338
$909$	28.7984	0.955182
$910$	4.41641	0.146402
$911$	−43.1459	−1.42949	−0.714744	−	0.699386i	$-0.753457\pi$
$911$	−43.1459	−1.42949	−0.714744	+	0.699386i	$0.753457\pi$
$912$	10.7426	0.355725
$913$	−34.2705	−1.13419
$914$	25.0902	0.829909
$915$	−15.5410	−0.513770
$916$	−91.5197	−3.02390
$917$	11.5623	0.381821
$918$	4.47214	0.147602
$919$	−50.4164	−1.66308	−0.831542	−	0.555462i	$-0.812541\pi$
$919$	−50.4164	−1.66308	−0.831542	+	0.555462i	$0.812541\pi$
$920$	167.498	5.52226
$921$	−6.65248	−0.219207
$922$	36.2705	1.19451
$923$	−0.549150	−0.0180755
$924$	15.9787	0.525661
$925$	−15.4164	−0.506888
$926$	72.1591	2.37129
$927$	−21.5836	−0.708898
$928$	81.1033	2.66235
$929$	17.9443	0.588732	0.294366	−	0.955693i	$-0.404891\pi$
$929$	17.9443	0.588732	0.294366	+	0.955693i	$0.404891\pi$
$930$	3.00000	0.0983739
$931$	−22.4164	−0.734668
$932$	7.41641	0.242933
$933$	10.2492	0.335545
$934$	−60.6869	−1.98574
$935$	5.12461	0.167593
$936$	−3.11146	−0.101701
$937$	11.2918	0.368887	0.184443	−	0.982843i	$-0.440952\pi$
$937$	11.2918	0.368887	0.184443	+	0.982843i	$0.440952\pi$
$938$	−10.0902	−0.329456
$939$	5.29180	0.172691
$940$	125.498	4.09331
$941$	−0.167184	−0.00545005	−0.00272503	−	0.999996i	$-0.500867\pi$
$941$	−0.167184	−0.00545005	−0.00272503	+	0.999996i	$0.500867\pi$
$942$	−3.56231	−0.116066
$943$	−2.85410	−0.0929423
$944$	59.1246	1.92434
$945$	25.8541	0.841034
$946$	−28.4164	−0.923897
$947$	−2.72949	−0.0886965	−0.0443483	−	0.999016i	$-0.514121\pi$
$947$	−2.72949	−0.0886965	−0.0443483	+	0.999016i	$0.514121\pi$
$948$	−12.1672	−0.395172
$949$	0.583592	0.0189442
$950$	29.8885	0.969712
$951$	−7.10333	−0.230341
$952$	−22.0000	−0.713024
$953$	31.3607	1.01587	0.507936	−	0.861395i	$-0.330409\pi$
$953$	31.3607	1.01587	0.507936	+	0.861395i	$0.330409\pi$
$954$	−67.2492	−2.17727
$955$	−4.41641	−0.142912
$956$	−92.6656	−2.99702
$957$	6.38197	0.206300
$958$	−80.2492	−2.59273
$959$	−2.72949	−0.0881398
$960$	9.97871	0.322062
$961$	−30.0000	−0.967742
$962$	−1.47214	−0.0474636
$963$	−15.1033	−0.486698
$964$	125.498	4.04203
$965$	−50.5623	−1.62766
$966$	−28.7984	−0.926572
$967$	−4.16718	−0.134008	−0.0670038	−	0.997753i	$-0.521344\pi$
$967$	−4.16718	−0.134008	−0.0670038	+	0.997753i	$0.521344\pi$
$968$	44.8328	1.44098
$969$	−0.832816	−0.0267539
$970$	113.228	3.63553
$971$	50.7771	1.62951	0.814757	−	0.579802i	$-0.196870\pi$
$971$	50.7771	1.62951	0.814757	+	0.579802i	$0.196870\pi$
$972$	−46.8541	−1.50285
$973$	11.5623	0.370671
$974$	−104.284	−3.34146
$975$	0.222912	0.00713891
$976$	−133.644	−4.27785
$977$	3.65248	0.116853	0.0584265	−	0.998292i	$-0.481392\pi$
$977$	3.65248	0.116853	0.0584265	+	0.998292i	$0.481392\pi$
$978$	−6.85410	−0.219170
$979$	−5.00000	−0.159801
$980$	−114.374	−3.65354
$981$	13.8541	0.442327
$982$	−74.3951	−2.37404
$983$	−29.8328	−0.951519	−0.475760	−	0.879575i	$-0.657827\pi$
$983$	−29.8328	−0.951519	−0.475760	+	0.879575i	$0.657827\pi$
$984$	−1.09017	−0.0347533
$985$	13.5836	0.432809
$986$	−14.9443	−0.475923
$987$	−12.6869	−0.403829
$988$	2.02129	0.0643057
$989$	36.2705	1.15334
$990$	−50.1246	−1.59306
$991$	50.5623	1.60616	0.803082	−	0.595868i	$-0.203192\pi$
$991$	50.5623	1.60616	0.803082	+	0.595868i	$0.203192\pi$
$992$	10.8541	0.344618
$993$	1.30495	0.0414114
$994$	37.9787	1.20461
$995$	−30.8115	−0.976791
$996$	28.4164	0.900408
$997$	27.5410	0.872233	0.436116	−	0.899890i	$-0.356354\pi$
$997$	27.5410	0.872233	0.436116	+	0.899890i	$0.356354\pi$
$998$	65.4508	2.07181
$999$	−8.61803	−0.272663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	67.2.a.b.1.1	✓	2
3.2	odd	2		603.2.a.h.1.2		2
4.3	odd	2		1072.2.a.f.1.1		2
5.2	odd	4		1675.2.c.d.1274.1		4
5.3	odd	4		1675.2.c.d.1274.4		4
5.4	even	2		1675.2.a.h.1.2		2
7.6	odd	2		3283.2.a.f.1.1		2
8.3	odd	2		4288.2.a.h.1.2		2
8.5	even	2		4288.2.a.p.1.1		2
11.10	odd	2		8107.2.a.i.1.2		2
12.11	even	2		9648.2.a.bi.1.2		2
67.66	odd	2		4489.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
67.2.a.b.1.1	✓	2	1.1	even	1	trivial
603.2.a.h.1.2		2	3.2	odd	2
1072.2.a.f.1.1		2	4.3	odd	2
1675.2.a.h.1.2		2	5.4	even	2
1675.2.c.d.1274.1		4	5.2	odd	4
1675.2.c.d.1274.4		4	5.3	odd	4
3283.2.a.f.1.1		2	7.6	odd	2
4288.2.a.h.1.2		2	8.3	odd	2
4288.2.a.p.1.1		2	8.5	even	2
4489.2.a.d.1.2		2	67.66	odd	2
8107.2.a.i.1.2		2	11.10	odd	2
9648.2.a.bi.1.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	67.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61803 q^{2} -0.381966 q^{3} +4.85410 q^{4} -3.00000 q^{5} +1.00000 q^{6} -3.85410 q^{7} -7.47214 q^{8} -2.85410 q^{9} +O(q^{10})\)	\(q-2.61803 q^{2} -0.381966 q^{3} +4.85410 q^{4} -3.00000 q^{5} +1.00000 q^{6} -3.85410 q^{7} -7.47214 q^{8} -2.85410 q^{9} +7.85410 q^{10} +2.23607 q^{11} -1.85410 q^{12} -0.145898 q^{13} +10.0902 q^{14} +1.14590 q^{15} +9.85410 q^{16} -0.763932 q^{17} +7.47214 q^{18} -2.85410 q^{19} -14.5623 q^{20} +1.47214 q^{21} -5.85410 q^{22} +7.47214 q^{23} +2.85410 q^{24} +4.00000 q^{25} +0.381966 q^{26} +2.23607 q^{27} -18.7082 q^{28} -7.47214 q^{29} -3.00000 q^{30} -1.00000 q^{31} -10.8541 q^{32} -0.854102 q^{33} +2.00000 q^{34} +11.5623 q^{35} -13.8541 q^{36} -3.85410 q^{37} +7.47214 q^{38} +0.0557281 q^{39} +22.4164 q^{40} -0.381966 q^{41} -3.85410 q^{42} +4.85410 q^{43} +10.8541 q^{44} +8.56231 q^{45} -19.5623 q^{46} -8.61803 q^{47} -3.76393 q^{48} +7.85410 q^{49} -10.4721 q^{50} +0.291796 q^{51} -0.708204 q^{52} -9.00000 q^{53} -5.85410 q^{54} -6.70820 q^{55} +28.7984 q^{56} +1.09017 q^{57} +19.5623 q^{58} +6.00000 q^{59} +5.56231 q^{60} -13.5623 q^{61} +2.61803 q^{62} +11.0000 q^{63} +8.70820 q^{64} +0.437694 q^{65} +2.23607 q^{66} -1.00000 q^{67} -3.70820 q^{68} -2.85410 q^{69} -30.2705 q^{70} +3.76393 q^{71} +21.3262 q^{72} -4.00000 q^{73} +10.0902 q^{74} -1.52786 q^{75} -13.8541 q^{76} -8.61803 q^{77} -0.145898 q^{78} +6.56231 q^{79} -29.5623 q^{80} +7.70820 q^{81} +1.00000 q^{82} -15.3262 q^{83} +7.14590 q^{84} +2.29180 q^{85} -12.7082 q^{86} +2.85410 q^{87} -16.7082 q^{88} -2.23607 q^{89} -22.4164 q^{90} +0.562306 q^{91} +36.2705 q^{92} +0.381966 q^{93} +22.5623 q^{94} +8.56231 q^{95} +4.14590 q^{96} +14.4164 q^{97} -20.5623 q^{98} -6.38197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} + 2 q^{6} - q^{7} - 6 q^{8} + q^{9}+O(q^{10}) \) 2 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 6 * q^5 + 2 * q^6 - q^7 - 6 * q^8 + q^9	\( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 6 q^{5} + 2 q^{6} - q^{7} - 6 q^{8} + q^{9} + 9 q^{10} + 3 q^{12} - 7 q^{13} + 9 q^{14} + 9 q^{15} + 13 q^{16} - 6 q^{17} + 6 q^{18} + q^{19} - 9 q^{20} - 6 q^{21} - 5 q^{22} + 6 q^{23} - q^{24} + 8 q^{25} + 3 q^{26} - 24 q^{28} - 6 q^{29} - 6 q^{30} - 2 q^{31} - 15 q^{32} + 5 q^{33} + 4 q^{34} + 3 q^{35} - 21 q^{36} - q^{37} + 6 q^{38} + 18 q^{39} + 18 q^{40} - 3 q^{41} - q^{42} + 3 q^{43} + 15 q^{44} - 3 q^{45} - 19 q^{46} - 15 q^{47} - 12 q^{48} + 9 q^{49} - 12 q^{50} + 14 q^{51} + 12 q^{52} - 18 q^{53} - 5 q^{54} + 33 q^{56} - 9 q^{57} + 19 q^{58} + 12 q^{59} - 9 q^{60} - 7 q^{61} + 3 q^{62} + 22 q^{63} + 4 q^{64} + 21 q^{65} - 2 q^{67} + 6 q^{68} + q^{69} - 27 q^{70} + 12 q^{71} + 27 q^{72} - 8 q^{73} + 9 q^{74} - 12 q^{75} - 21 q^{76} - 15 q^{77} - 7 q^{78} - 7 q^{79} - 39 q^{80} + 2 q^{81} + 2 q^{82} - 15 q^{83} + 21 q^{84} + 18 q^{85} - 12 q^{86} - q^{87} - 20 q^{88} - 18 q^{90} - 19 q^{91} + 39 q^{92} + 3 q^{93} + 25 q^{94} - 3 q^{95} + 15 q^{96} + 2 q^{97} - 21 q^{98} - 15 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - 6 * q^5 + 2 * q^6 - q^7 - 6 * q^8 + q^9 + 9 * q^10 + 3 * q^12 - 7 * q^13 + 9 * q^14 + 9 * q^15 + 13 * q^16 - 6 * q^17 + 6 * q^18 + q^19 - 9 * q^20 - 6 * q^21 - 5 * q^22 + 6 * q^23 - q^24 + 8 * q^25 + 3 * q^26 - 24 * q^28 - 6 * q^29 - 6 * q^30 - 2 * q^31 - 15 * q^32 + 5 * q^33 + 4 * q^34 + 3 * q^35 - 21 * q^36 - q^37 + 6 * q^38 + 18 * q^39 + 18 * q^40 - 3 * q^41 - q^42 + 3 * q^43 + 15 * q^44 - 3 * q^45 - 19 * q^46 - 15 * q^47 - 12 * q^48 + 9 * q^49 - 12 * q^50 + 14 * q^51 + 12 * q^52 - 18 * q^53 - 5 * q^54 + 33 * q^56 - 9 * q^57 + 19 * q^58 + 12 * q^59 - 9 * q^60 - 7 * q^61 + 3 * q^62 + 22 * q^63 + 4 * q^64 + 21 * q^65 - 2 * q^67 + 6 * q^68 + q^69 - 27 * q^70 + 12 * q^71 + 27 * q^72 - 8 * q^73 + 9 * q^74 - 12 * q^75 - 21 * q^76 - 15 * q^77 - 7 * q^78 - 7 * q^79 - 39 * q^80 + 2 * q^81 + 2 * q^82 - 15 * q^83 + 21 * q^84 + 18 * q^85 - 12 * q^86 - q^87 - 20 * q^88 - 18 * q^90 - 19 * q^91 + 39 * q^92 + 3 * q^93 + 25 * q^94 - 3 * q^95 + 15 * q^96 + 2 * q^97 - 21 * q^98 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more