LMFDB - Embedded newform 177.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,2,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("177.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	177.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:	$1.41335211578$
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	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +2.23607 q^{5} +1.61803 q^{6} -4.61803 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +2.23607 q^{5} +1.61803 q^{6} -4.61803 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.61803 q^{10} -2.23607 q^{11} -0.618034 q^{12} -1.76393 q^{13} +7.47214 q^{14} -2.23607 q^{15} -4.85410 q^{16} -4.85410 q^{17} -1.61803 q^{18} -8.09017 q^{19} +1.38197 q^{20} +4.61803 q^{21} +3.61803 q^{22} -2.38197 q^{23} -2.23607 q^{24} +2.85410 q^{26} -1.00000 q^{27} -2.85410 q^{28} +8.61803 q^{29} +3.61803 q^{30} +9.56231 q^{31} +3.38197 q^{32} +2.23607 q^{33} +7.85410 q^{34} -10.3262 q^{35} +0.618034 q^{36} -6.85410 q^{37} +13.0902 q^{38} +1.76393 q^{39} +5.00000 q^{40} -3.09017 q^{41} -7.47214 q^{42} +4.70820 q^{43} -1.38197 q^{44} +2.23607 q^{45} +3.85410 q^{46} -4.14590 q^{47} +4.85410 q^{48} +14.3262 q^{49} +4.85410 q^{51} -1.09017 q^{52} +1.76393 q^{53} +1.61803 q^{54} -5.00000 q^{55} -10.3262 q^{56} +8.09017 q^{57} -13.9443 q^{58} -1.00000 q^{59} -1.38197 q^{60} -9.85410 q^{61} -15.4721 q^{62} -4.61803 q^{63} +4.23607 q^{64} -3.94427 q^{65} -3.61803 q^{66} -2.70820 q^{67} -3.00000 q^{68} +2.38197 q^{69} +16.7082 q^{70} +9.94427 q^{71} +2.23607 q^{72} -5.85410 q^{73} +11.0902 q^{74} -5.00000 q^{76} +10.3262 q^{77} -2.85410 q^{78} -3.00000 q^{79} -10.8541 q^{80} +1.00000 q^{81} +5.00000 q^{82} +0.618034 q^{83} +2.85410 q^{84} -10.8541 q^{85} -7.61803 q^{86} -8.61803 q^{87} -5.00000 q^{88} +10.7984 q^{89} -3.61803 q^{90} +8.14590 q^{91} -1.47214 q^{92} -9.56231 q^{93} +6.70820 q^{94} -18.0902 q^{95} -3.38197 q^{96} +3.00000 q^{97} -23.1803 q^{98} -2.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 7 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 - 2 * q^3 - q^4 + q^6 - 7 * q^7 + 2 * q^9	$ 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 7 q^{7} + 2 q^{9} - 5 q^{10} + q^{12} - 8 q^{13} + 6 q^{14} - 3 q^{16} - 3 q^{17} - q^{18} - 5 q^{19} + 5 q^{20} + 7 q^{21} + 5 q^{22} - 7 q^{23} - q^{26} - 2 q^{27} + q^{28} + 15 q^{29} + 5 q^{30} - q^{31} + 9 q^{32} + 9 q^{34} - 5 q^{35} - q^{36} - 7 q^{37} + 15 q^{38} + 8 q^{39} + 10 q^{40} + 5 q^{41} - 6 q^{42} - 4 q^{43} - 5 q^{44} + q^{46} - 15 q^{47} + 3 q^{48} + 13 q^{49} + 3 q^{51} + 9 q^{52} + 8 q^{53} + q^{54} - 10 q^{55} - 5 q^{56} + 5 q^{57} - 10 q^{58} - 2 q^{59} - 5 q^{60} - 13 q^{61} - 22 q^{62} - 7 q^{63} + 4 q^{64} + 10 q^{65} - 5 q^{66} + 8 q^{67} - 6 q^{68} + 7 q^{69} + 20 q^{70} + 2 q^{71} - 5 q^{73} + 11 q^{74} - 10 q^{76} + 5 q^{77} + q^{78} - 6 q^{79} - 15 q^{80} + 2 q^{81} + 10 q^{82} - q^{83} - q^{84} - 15 q^{85} - 13 q^{86} - 15 q^{87} - 10 q^{88} - 3 q^{89} - 5 q^{90} + 23 q^{91} + 6 q^{92} + q^{93} - 25 q^{95} - 9 q^{96} + 6 q^{97} - 24 q^{98}+O(q^{100}) $ 2 * q - q^2 - 2 * q^3 - q^4 + q^6 - 7 * q^7 + 2 * q^9 - 5 * q^10 + q^12 - 8 * q^13 + 6 * q^14 - 3 * q^16 - 3 * q^17 - q^18 - 5 * q^19 + 5 * q^20 + 7 * q^21 + 5 * q^22 - 7 * q^23 - q^26 - 2 * q^27 + q^28 + 15 * q^29 + 5 * q^30 - q^31 + 9 * q^32 + 9 * q^34 - 5 * q^35 - q^36 - 7 * q^37 + 15 * q^38 + 8 * q^39 + 10 * q^40 + 5 * q^41 - 6 * q^42 - 4 * q^43 - 5 * q^44 + q^46 - 15 * q^47 + 3 * q^48 + 13 * q^49 + 3 * q^51 + 9 * q^52 + 8 * q^53 + q^54 - 10 * q^55 - 5 * q^56 + 5 * q^57 - 10 * q^58 - 2 * q^59 - 5 * q^60 - 13 * q^61 - 22 * q^62 - 7 * q^63 + 4 * q^64 + 10 * q^65 - 5 * q^66 + 8 * q^67 - 6 * q^68 + 7 * q^69 + 20 * q^70 + 2 * q^71 - 5 * q^73 + 11 * q^74 - 10 * q^76 + 5 * q^77 + q^78 - 6 * q^79 - 15 * q^80 + 2 * q^81 + 10 * q^82 - q^83 - q^84 - 15 * q^85 - 13 * q^86 - 15 * q^87 - 10 * q^88 - 3 * q^89 - 5 * q^90 + 23 * q^91 + 6 * q^92 + q^93 - 25 * q^95 - 9 * q^96 + 6 * q^97 - 24 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.618034	0.309017
$5$	2.23607	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	2.23607	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	1.61803	0.660560
$7$	−4.61803	−1.74545	−0.872726	−	0.488210i	$-0.837650\pi$
$7$	−4.61803	−1.74545	−0.872726	+	0.488210i	$0.837650\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	−3.61803	−1.14412
$11$	−2.23607	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	−0.618034	−0.178411
$13$	−1.76393	−0.489227	−0.244613	−	0.969621i	$-0.578661\pi$
$13$	−1.76393	−0.489227	−0.244613	+	0.969621i	$0.578661\pi$
$14$	7.47214	1.99701
$15$	−2.23607	−0.577350
$16$	−4.85410	−1.21353
$17$	−4.85410	−1.17729	−0.588646	−	0.808391i	$-0.700339\pi$
$17$	−4.85410	−1.17729	−0.588646	+	0.808391i	$0.700339\pi$
$18$	−1.61803	−0.381374
$19$	−8.09017	−1.85601	−0.928006	−	0.372565i	$-0.878478\pi$
$19$	−8.09017	−1.85601	−0.928006	+	0.372565i	$0.878478\pi$
$20$	1.38197	0.309017
$21$	4.61803	1.00774
$22$	3.61803	0.771367
$23$	−2.38197	−0.496674	−0.248337	−	0.968674i	$-0.579884\pi$
$23$	−2.38197	−0.496674	−0.248337	+	0.968674i	$0.579884\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	2.85410	0.559735
$27$	−1.00000	−0.192450
$28$	−2.85410	−0.539375
$29$	8.61803	1.60033	0.800164	−	0.599781i	$-0.204746\pi$
$29$	8.61803	1.60033	0.800164	+	0.599781i	$0.204746\pi$
$30$	3.61803	0.660560
$31$	9.56231	1.71744	0.858720	−	0.512444i	$-0.171260\pi$
$31$	9.56231	1.71744	0.858720	+	0.512444i	$0.171260\pi$
$32$	3.38197	0.597853
$33$	2.23607	0.389249
$34$	7.85410	1.34697
$35$	−10.3262	−1.74545
$36$	0.618034	0.103006
$37$	−6.85410	−1.12681	−0.563404	−	0.826182i	$-0.690508\pi$
$37$	−6.85410	−1.12681	−0.563404	+	0.826182i	$0.690508\pi$
$38$	13.0902	2.12351
$39$	1.76393	0.282455
$40$	5.00000	0.790569
$41$	−3.09017	−0.482603	−0.241302	−	0.970450i	$-0.577574\pi$
$41$	−3.09017	−0.482603	−0.241302	+	0.970450i	$0.577574\pi$
$42$	−7.47214	−1.15298
$43$	4.70820	0.717994	0.358997	−	0.933339i	$-0.383119\pi$
$43$	4.70820	0.717994	0.358997	+	0.933339i	$0.383119\pi$
$44$	−1.38197	−0.208339
$45$	2.23607	0.333333
$46$	3.85410	0.568256
$47$	−4.14590	−0.604741	−0.302371	−	0.953190i	$-0.597778\pi$
$47$	−4.14590	−0.604741	−0.302371	+	0.953190i	$0.597778\pi$
$48$	4.85410	0.700629
$49$	14.3262	2.04661
$50$	0	0
$51$	4.85410	0.679710
$52$	−1.09017	−0.151179
$53$	1.76393	0.242295	0.121147	−	0.992635i	$-0.461343\pi$
$53$	1.76393	0.242295	0.121147	+	0.992635i	$0.461343\pi$
$54$	1.61803	0.220187
$55$	−5.00000	−0.674200
$56$	−10.3262	−1.37990
$57$	8.09017	1.07157
$58$	−13.9443	−1.83097
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	−1.38197	−0.178411
$61$	−9.85410	−1.26169	−0.630844	−	0.775909i	$-0.717291\pi$
$61$	−9.85410	−1.26169	−0.630844	+	0.775909i	$0.717291\pi$
$62$	−15.4721	−1.96496
$63$	−4.61803	−0.581818
$64$	4.23607	0.529508
$65$	−3.94427	−0.489227
$66$	−3.61803	−0.445349
$67$	−2.70820	−0.330860	−0.165430	−	0.986222i	$-0.552901\pi$
$67$	−2.70820	−0.330860	−0.165430	+	0.986222i	$0.552901\pi$
$68$	−3.00000	−0.363803
$69$	2.38197	0.286755
$70$	16.7082	1.99701
$71$	9.94427	1.18017	0.590084	−	0.807342i	$-0.299095\pi$
$71$	9.94427	1.18017	0.590084	+	0.807342i	$0.299095\pi$
$72$	2.23607	0.263523
$73$	−5.85410	−0.685171	−0.342585	−	0.939487i	$-0.611303\pi$
$73$	−5.85410	−0.685171	−0.342585	+	0.939487i	$0.611303\pi$
$74$	11.0902	1.28921
$75$	0	0
$76$	−5.00000	−0.573539
$77$	10.3262	1.17678
$78$	−2.85410	−0.323163
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	−10.8541	−1.21353
$81$	1.00000	0.111111
$82$	5.00000	0.552158
$83$	0.618034	0.0678380	0.0339190	−	0.999425i	$-0.489201\pi$
$83$	0.618034	0.0678380	0.0339190	+	0.999425i	$0.489201\pi$
$84$	2.85410	0.311408
$85$	−10.8541	−1.17729
$86$	−7.61803	−0.821474
$87$	−8.61803	−0.923950
$88$	−5.00000	−0.533002
$89$	10.7984	1.14463	0.572313	−	0.820035i	$-0.306046\pi$
$89$	10.7984	1.14463	0.572313	+	0.820035i	$0.306046\pi$
$90$	−3.61803	−0.381374
$91$	8.14590	0.853922
$92$	−1.47214	−0.153481
$93$	−9.56231	−0.991565
$94$	6.70820	0.691898
$95$	−18.0902	−1.85601
$96$	−3.38197	−0.345170
$97$	3.00000	0.304604	0.152302	−	0.988334i	$-0.451331\pi$
$97$	3.00000	0.304604	0.152302	+	0.988334i	$0.451331\pi$
$98$	−23.1803	−2.34157
$99$	−2.23607	−0.224733
$100$	0	0
$101$	−9.70820	−0.966002	−0.483001	−	0.875620i	$-0.660453\pi$
$101$	−9.70820	−0.966002	−0.483001	+	0.875620i	$0.660453\pi$
$102$	−7.85410	−0.777672
$103$	1.23607	0.121793	0.0608967	−	0.998144i	$-0.480604\pi$
$103$	1.23607	0.121793	0.0608967	+	0.998144i	$0.480604\pi$
$104$	−3.94427	−0.386768
$105$	10.3262	1.00774
$106$	−2.85410	−0.277215
$107$	12.0902	1.16880	0.584400	−	0.811465i	$-0.301330\pi$
$107$	12.0902	1.16880	0.584400	+	0.811465i	$0.301330\pi$
$108$	−0.618034	−0.0594703
$109$	−10.8541	−1.03963	−0.519817	−	0.854278i	$-0.674000\pi$
$109$	−10.8541	−1.03963	−0.519817	+	0.854278i	$0.674000\pi$
$110$	8.09017	0.771367
$111$	6.85410	0.650563
$112$	22.4164	2.11815
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	−13.0902	−1.22601
$115$	−5.32624	−0.496674
$116$	5.32624	0.494529
$117$	−1.76393	−0.163076
$118$	1.61803	0.148952
$119$	22.4164	2.05491
$120$	−5.00000	−0.456435
$121$	−6.00000	−0.545455
$122$	15.9443	1.44353
$123$	3.09017	0.278631
$124$	5.90983	0.530718
$125$	−11.1803	−1.00000
$126$	7.47214	0.665671
$127$	−1.94427	−0.172526	−0.0862631	−	0.996272i	$-0.527493\pi$
$127$	−1.94427	−0.172526	−0.0862631	+	0.996272i	$0.527493\pi$
$128$	−13.6180	−1.20368
$129$	−4.70820	−0.414534
$130$	6.38197	0.559735
$131$	10.6525	0.930711	0.465356	−	0.885124i	$-0.345926\pi$
$131$	10.6525	0.930711	0.465356	+	0.885124i	$0.345926\pi$
$132$	1.38197	0.120285
$133$	37.3607	3.23958
$134$	4.38197	0.378544
$135$	−2.23607	−0.192450
$136$	−10.8541	−0.930732
$137$	7.76393	0.663317	0.331659	−	0.943399i	$-0.392392\pi$
$137$	7.76393	0.663317	0.331659	+	0.943399i	$0.392392\pi$
$138$	−3.85410	−0.328083
$139$	16.2361	1.37713	0.688563	−	0.725177i	$-0.258242\pi$
$139$	16.2361	1.37713	0.688563	+	0.725177i	$0.258242\pi$
$140$	−6.38197	−0.539375
$141$	4.14590	0.349148
$142$	−16.0902	−1.35026
$143$	3.94427	0.329837
$144$	−4.85410	−0.404508
$145$	19.2705	1.60033
$146$	9.47214	0.783920
$147$	−14.3262	−1.18161
$148$	−4.23607	−0.348203
$149$	7.90983	0.647999	0.323999	−	0.946057i	$-0.394972\pi$
$149$	7.90983	0.647999	0.323999	+	0.946057i	$0.394972\pi$
$150$	0	0
$151$	−17.5623	−1.42920	−0.714600	−	0.699533i	$-0.753391\pi$
$151$	−17.5623	−1.42920	−0.714600	+	0.699533i	$0.753391\pi$
$152$	−18.0902	−1.46731
$153$	−4.85410	−0.392431
$154$	−16.7082	−1.34639
$155$	21.3820	1.71744
$156$	1.09017	0.0872835
$157$	9.00000	0.718278	0.359139	−	0.933284i	$-0.383070\pi$
$157$	9.00000	0.718278	0.359139	+	0.933284i	$0.383070\pi$
$158$	4.85410	0.386172
$159$	−1.76393	−0.139889
$160$	7.56231	0.597853
$161$	11.0000	0.866921
$162$	−1.61803	−0.127125
$163$	1.56231	0.122369	0.0611846	−	0.998126i	$-0.480512\pi$
$163$	1.56231	0.122369	0.0611846	+	0.998126i	$0.480512\pi$
$164$	−1.90983	−0.149133
$165$	5.00000	0.389249
$166$	−1.00000	−0.0776151
$167$	−22.0344	−1.70508	−0.852538	−	0.522665i	$-0.824938\pi$
$167$	−22.0344	−1.70508	−0.852538	+	0.522665i	$0.824938\pi$
$168$	10.3262	0.796687
$169$	−9.88854	−0.760657
$170$	17.5623	1.34697
$171$	−8.09017	−0.618671
$172$	2.90983	0.221872
$173$	14.6180	1.11139	0.555694	−	0.831387i	$-0.312453\pi$
$173$	14.6180	1.11139	0.555694	+	0.831387i	$0.312453\pi$
$174$	13.9443	1.05711
$175$	0	0
$176$	10.8541	0.818159
$177$	1.00000	0.0751646
$178$	−17.4721	−1.30959
$179$	−9.47214	−0.707981	−0.353990	−	0.935249i	$-0.615175\pi$
$179$	−9.47214	−0.707981	−0.353990	+	0.935249i	$0.615175\pi$
$180$	1.38197	0.103006
$181$	−11.2705	−0.837730	−0.418865	−	0.908048i	$-0.637572\pi$
$181$	−11.2705	−0.837730	−0.418865	+	0.908048i	$0.637572\pi$
$182$	−13.1803	−0.976992
$183$	9.85410	0.728436
$184$	−5.32624	−0.392655
$185$	−15.3262	−1.12681
$186$	15.4721	1.13447
$187$	10.8541	0.793731
$188$	−2.56231	−0.186875
$189$	4.61803	0.335913
$190$	29.2705	2.12351
$191$	−10.4164	−0.753705	−0.376852	−	0.926273i	$-0.622994\pi$
$191$	−10.4164	−0.753705	−0.376852	+	0.926273i	$0.622994\pi$
$192$	−4.23607	−0.305712
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	−4.85410	−0.348504
$195$	3.94427	0.282455
$196$	8.85410	0.632436
$197$	10.6525	0.758957	0.379479	−	0.925200i	$-0.376103\pi$
$197$	10.6525	0.758957	0.379479	+	0.925200i	$0.376103\pi$
$198$	3.61803	0.257122
$199$	3.56231	0.252525	0.126263	−	0.991997i	$-0.459702\pi$
$199$	3.56231	0.252525	0.126263	+	0.991997i	$0.459702\pi$
$200$	0	0
$201$	2.70820	0.191022
$202$	15.7082	1.10523
$203$	−39.7984	−2.79330
$204$	3.00000	0.210042
$205$	−6.90983	−0.482603
$206$	−2.00000	−0.139347
$207$	−2.38197	−0.165558
$208$	8.56231	0.593689
$209$	18.0902	1.25132
$210$	−16.7082	−1.15298
$211$	−8.85410	−0.609542	−0.304771	−	0.952426i	$-0.598580\pi$
$211$	−8.85410	−0.609542	−0.304771	+	0.952426i	$0.598580\pi$
$212$	1.09017	0.0748732
$213$	−9.94427	−0.681370
$214$	−19.5623	−1.33725
$215$	10.5279	0.717994
$216$	−2.23607	−0.152145
$217$	−44.1591	−2.99771
$218$	17.5623	1.18947
$219$	5.85410	0.395584
$220$	−3.09017	−0.208339
$221$	8.56231	0.575963
$222$	−11.0902	−0.744323
$223$	−18.4721	−1.23699	−0.618493	−	0.785790i	$-0.712256\pi$
$223$	−18.4721	−1.23699	−0.618493	+	0.785790i	$0.712256\pi$
$224$	−15.6180	−1.04352
$225$	0	0
$226$	14.5623	0.968670
$227$	−28.8541	−1.91511	−0.957557	−	0.288244i	$-0.906929\pi$
$227$	−28.8541	−1.91511	−0.957557	+	0.288244i	$0.906929\pi$
$228$	5.00000	0.331133
$229$	−15.8541	−1.04767	−0.523834	−	0.851820i	$-0.675499\pi$
$229$	−15.8541	−1.04767	−0.523834	+	0.851820i	$0.675499\pi$
$230$	8.61803	0.568256
$231$	−10.3262	−0.679417
$232$	19.2705	1.26517
$233$	−21.7082	−1.42215	−0.711076	−	0.703115i	$-0.751792\pi$
$233$	−21.7082	−1.42215	−0.711076	+	0.703115i	$0.751792\pi$
$234$	2.85410	0.186578
$235$	−9.27051	−0.604741
$236$	−0.618034	−0.0402306
$237$	3.00000	0.194871
$238$	−36.2705	−2.35107
$239$	7.47214	0.483332	0.241666	−	0.970359i	$-0.422306\pi$
$239$	7.47214	0.483332	0.241666	+	0.970359i	$0.422306\pi$
$240$	10.8541	0.700629
$241$	3.41641	0.220070	0.110035	−	0.993928i	$-0.464904\pi$
$241$	3.41641	0.220070	0.110035	+	0.993928i	$0.464904\pi$
$242$	9.70820	0.624067
$243$	−1.00000	−0.0641500
$244$	−6.09017	−0.389883
$245$	32.0344	2.04661
$246$	−5.00000	−0.318788
$247$	14.2705	0.908011
$248$	21.3820	1.35776
$249$	−0.618034	−0.0391663
$250$	18.0902	1.14412
$251$	−7.18034	−0.453219	−0.226610	−	0.973986i	$-0.572764\pi$
$251$	−7.18034	−0.453219	−0.226610	+	0.973986i	$0.572764\pi$
$252$	−2.85410	−0.179792
$253$	5.32624	0.334858
$254$	3.14590	0.197391
$255$	10.8541	0.679710
$256$	13.5623	0.847644
$257$	7.41641	0.462623	0.231311	−	0.972880i	$-0.425698\pi$
$257$	7.41641	0.462623	0.231311	+	0.972880i	$0.425698\pi$
$258$	7.61803	0.474278
$259$	31.6525	1.96679
$260$	−2.43769	−0.151179
$261$	8.61803	0.533443
$262$	−17.2361	−1.06485
$263$	−0.618034	−0.0381096	−0.0190548	−	0.999818i	$-0.506066\pi$
$263$	−0.618034	−0.0381096	−0.0190548	+	0.999818i	$0.506066\pi$
$264$	5.00000	0.307729
$265$	3.94427	0.242295
$266$	−60.4508	−3.70648
$267$	−10.7984	−0.660850
$268$	−1.67376	−0.102241
$269$	−2.52786	−0.154127	−0.0770633	−	0.997026i	$-0.524554\pi$
$269$	−2.52786	−0.154127	−0.0770633	+	0.997026i	$0.524554\pi$
$270$	3.61803	0.220187
$271$	−12.2361	−0.743288	−0.371644	−	0.928375i	$-0.621206\pi$
$271$	−12.2361	−0.743288	−0.371644	+	0.928375i	$0.621206\pi$
$272$	23.5623	1.42867
$273$	−8.14590	−0.493012
$274$	−12.5623	−0.758917
$275$	0	0
$276$	1.47214	0.0886122
$277$	3.47214	0.208620	0.104310	−	0.994545i	$-0.466737\pi$
$277$	3.47214	0.208620	0.104310	+	0.994545i	$0.466737\pi$
$278$	−26.2705	−1.57560
$279$	9.56231	0.572480
$280$	−23.0902	−1.37990
$281$	−3.70820	−0.221213	−0.110606	−	0.993864i	$-0.535279\pi$
$281$	−3.70820	−0.221213	−0.110606	+	0.993864i	$0.535279\pi$
$282$	−6.70820	−0.399468
$283$	10.2705	0.610518	0.305259	−	0.952269i	$-0.401257\pi$
$283$	10.2705	0.610518	0.305259	+	0.952269i	$0.401257\pi$
$284$	6.14590	0.364692
$285$	18.0902	1.07157
$286$	−6.38197	−0.377374
$287$	14.2705	0.842362
$288$	3.38197	0.199284
$289$	6.56231	0.386018
$290$	−31.1803	−1.83097
$291$	−3.00000	−0.175863
$292$	−3.61803	−0.211729
$293$	21.6180	1.26294	0.631470	−	0.775401i	$-0.282452\pi$
$293$	21.6180	1.26294	0.631470	+	0.775401i	$0.282452\pi$
$294$	23.1803	1.35190
$295$	−2.23607	−0.130189
$296$	−15.3262	−0.890819
$297$	2.23607	0.129750
$298$	−12.7984	−0.741390
$299$	4.20163	0.242986
$300$	0	0
$301$	−21.7426	−1.25323
$302$	28.4164	1.63518
$303$	9.70820	0.557722
$304$	39.2705	2.25232
$305$	−22.0344	−1.26169
$306$	7.85410	0.448989
$307$	9.88854	0.564369	0.282185	−	0.959360i	$-0.408941\pi$
$307$	9.88854	0.564369	0.282185	+	0.959360i	$0.408941\pi$
$308$	6.38197	0.363646
$309$	−1.23607	−0.0703175
$310$	−34.5967	−1.96496
$311$	−28.4508	−1.61330	−0.806650	−	0.591030i	$-0.798722\pi$
$311$	−28.4508	−1.61330	−0.806650	+	0.591030i	$0.798722\pi$
$312$	3.94427	0.223300
$313$	3.79837	0.214697	0.107348	−	0.994221i	$-0.465764\pi$
$313$	3.79837	0.214697	0.107348	+	0.994221i	$0.465764\pi$
$314$	−14.5623	−0.821798
$315$	−10.3262	−0.581818
$316$	−1.85410	−0.104301
$317$	−6.81966	−0.383030	−0.191515	−	0.981490i	$-0.561340\pi$
$317$	−6.81966	−0.383030	−0.191515	+	0.981490i	$0.561340\pi$
$318$	2.85410	0.160050
$319$	−19.2705	−1.07894
$320$	9.47214	0.529508
$321$	−12.0902	−0.674807
$322$	−17.7984	−0.991865
$323$	39.2705	2.18507
$324$	0.618034	0.0343352
$325$	0	0
$326$	−2.52786	−0.140005
$327$	10.8541	0.600233
$328$	−6.90983	−0.381532
$329$	19.1459	1.05555
$330$	−8.09017	−0.445349
$331$	29.1246	1.60083	0.800417	−	0.599444i	$-0.204612\pi$
$331$	29.1246	1.60083	0.800417	+	0.599444i	$0.204612\pi$
$332$	0.381966	0.0209631
$333$	−6.85410	−0.375602
$334$	35.6525	1.95082
$335$	−6.05573	−0.330860
$336$	−22.4164	−1.22292
$337$	−26.0902	−1.42122	−0.710611	−	0.703585i	$-0.751581\pi$
$337$	−26.0902	−1.42122	−0.710611	+	0.703585i	$0.751581\pi$
$338$	16.0000	0.870285
$339$	9.00000	0.488813
$340$	−6.70820	−0.363803
$341$	−21.3820	−1.15790
$342$	13.0902	0.707835
$343$	−33.8328	−1.82680
$344$	10.5279	0.567624
$345$	5.32624	0.286755
$346$	−23.6525	−1.27156
$347$	1.96556	0.105517	0.0527583	−	0.998607i	$-0.483199\pi$
$347$	1.96556	0.105517	0.0527583	+	0.998607i	$0.483199\pi$
$348$	−5.32624	−0.285516
$349$	−27.0000	−1.44528	−0.722638	−	0.691226i	$-0.757071\pi$
$349$	−27.0000	−1.44528	−0.722638	+	0.691226i	$0.757071\pi$
$350$	0	0
$351$	1.76393	0.0941517
$352$	−7.56231	−0.403072
$353$	13.0344	0.693753	0.346877	−	0.937911i	$-0.387242\pi$
$353$	13.0344	0.693753	0.346877	+	0.937911i	$0.387242\pi$
$354$	−1.61803	−0.0859975
$355$	22.2361	1.18017
$356$	6.67376	0.353709
$357$	−22.4164	−1.18640
$358$	15.3262	0.810017
$359$	13.8885	0.733009	0.366505	−	0.930416i	$-0.380554\pi$
$359$	13.8885	0.733009	0.366505	+	0.930416i	$0.380554\pi$
$360$	5.00000	0.263523
$361$	46.4508	2.44478
$362$	18.2361	0.958466
$363$	6.00000	0.314918
$364$	5.03444	0.263876
$365$	−13.0902	−0.685171
$366$	−15.9443	−0.833420
$367$	−23.8328	−1.24406	−0.622031	−	0.782992i	$-0.713692\pi$
$367$	−23.8328	−1.24406	−0.622031	+	0.782992i	$0.713692\pi$
$368$	11.5623	0.602727
$369$	−3.09017	−0.160868
$370$	24.7984	1.28921
$371$	−8.14590	−0.422914
$372$	−5.90983	−0.306410
$373$	18.6738	0.966891	0.483445	−	0.875375i	$-0.339385\pi$
$373$	18.6738	0.966891	0.483445	+	0.875375i	$0.339385\pi$
$374$	−17.5623	−0.908125
$375$	11.1803	0.577350
$376$	−9.27051	−0.478090
$377$	−15.2016	−0.782924
$378$	−7.47214	−0.384325
$379$	−4.41641	−0.226856	−0.113428	−	0.993546i	$-0.536183\pi$
$379$	−4.41641	−0.226856	−0.113428	+	0.993546i	$0.536183\pi$
$380$	−11.1803	−0.573539
$381$	1.94427	0.0996081
$382$	16.8541	0.862331
$383$	−13.7639	−0.703304	−0.351652	−	0.936131i	$-0.614380\pi$
$383$	−13.7639	−0.703304	−0.351652	+	0.936131i	$0.614380\pi$
$384$	13.6180	0.694942
$385$	23.0902	1.17678
$386$	−12.9443	−0.658846
$387$	4.70820	0.239331
$388$	1.85410	0.0941278
$389$	23.5279	1.19291	0.596455	−	0.802646i	$-0.296575\pi$
$389$	23.5279	1.19291	0.596455	+	0.802646i	$0.296575\pi$
$390$	−6.38197	−0.323163
$391$	11.5623	0.584731
$392$	32.0344	1.61798
$393$	−10.6525	−0.537346
$394$	−17.2361	−0.868341
$395$	−6.70820	−0.337526
$396$	−1.38197	−0.0694464
$397$	3.00000	0.150566	0.0752828	−	0.997162i	$-0.476014\pi$
$397$	3.00000	0.150566	0.0752828	+	0.997162i	$0.476014\pi$
$398$	−5.76393	−0.288920
$399$	−37.3607	−1.87037
$400$	0	0
$401$	−5.90983	−0.295123	−0.147561	−	0.989053i	$-0.547142\pi$
$401$	−5.90983	−0.295123	−0.147561	+	0.989053i	$0.547142\pi$
$402$	−4.38197	−0.218553
$403$	−16.8673	−0.840218
$404$	−6.00000	−0.298511
$405$	2.23607	0.111111
$406$	64.3951	3.19588
$407$	15.3262	0.759693
$408$	10.8541	0.537358
$409$	−37.4164	−1.85012	−0.925061	−	0.379818i	$-0.875987\pi$
$409$	−37.4164	−1.85012	−0.925061	+	0.379818i	$0.875987\pi$
$410$	11.1803	0.552158
$411$	−7.76393	−0.382967
$412$	0.763932	0.0376362
$413$	4.61803	0.227239
$414$	3.85410	0.189419
$415$	1.38197	0.0678380
$416$	−5.96556	−0.292486
$417$	−16.2361	−0.795084
$418$	−29.2705	−1.43167
$419$	31.3050	1.52935	0.764673	−	0.644418i	$-0.222900\pi$
$419$	31.3050	1.52935	0.764673	+	0.644418i	$0.222900\pi$
$420$	6.38197	0.311408
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	14.3262	0.697390
$423$	−4.14590	−0.201580
$424$	3.94427	0.191551
$425$	0	0
$426$	16.0902	0.779571
$427$	45.5066	2.20222
$428$	7.47214	0.361179
$429$	−3.94427	−0.190431
$430$	−17.0344	−0.821474
$431$	14.6180	0.704126	0.352063	−	0.935976i	$-0.385480\pi$
$431$	14.6180	0.704126	0.352063	+	0.935976i	$0.385480\pi$
$432$	4.85410	0.233543
$433$	19.3262	0.928760	0.464380	−	0.885636i	$-0.346277\pi$
$433$	19.3262	0.928760	0.464380	+	0.885636i	$0.346277\pi$
$434$	71.4508	3.42975
$435$	−19.2705	−0.923950
$436$	−6.70820	−0.321265
$437$	19.2705	0.921833
$438$	−9.47214	−0.452596
$439$	−32.3820	−1.54551	−0.772753	−	0.634706i	$-0.781121\pi$
$439$	−32.3820	−1.54551	−0.772753	+	0.634706i	$0.781121\pi$
$440$	−11.1803	−0.533002
$441$	14.3262	0.682202
$442$	−13.8541	−0.658972
$443$	5.61803	0.266921	0.133460	−	0.991054i	$-0.457391\pi$
$443$	5.61803	0.266921	0.133460	+	0.991054i	$0.457391\pi$
$444$	4.23607	0.201035
$445$	24.1459	1.14463
$446$	29.8885	1.41526
$447$	−7.90983	−0.374122
$448$	−19.5623	−0.924232
$449$	−28.8885	−1.36333	−0.681667	−	0.731662i	$-0.738745\pi$
$449$	−28.8885	−1.36333	−0.681667	+	0.731662i	$0.738745\pi$
$450$	0	0
$451$	6.90983	0.325371
$452$	−5.56231	−0.261629
$453$	17.5623	0.825149
$454$	46.6869	2.19113
$455$	18.2148	0.853922
$456$	18.0902	0.847150
$457$	−26.6869	−1.24836	−0.624181	−	0.781280i	$-0.714567\pi$
$457$	−26.6869	−1.24836	−0.624181	+	0.781280i	$0.714567\pi$
$458$	25.6525	1.19866
$459$	4.85410	0.226570
$460$	−3.29180	−0.153481
$461$	−18.7426	−0.872932	−0.436466	−	0.899721i	$-0.643770\pi$
$461$	−18.7426	−0.872932	−0.436466	+	0.899721i	$0.643770\pi$
$462$	16.7082	0.777336
$463$	10.8541	0.504433	0.252216	−	0.967671i	$-0.418841\pi$
$463$	10.8541	0.504433	0.252216	+	0.967671i	$0.418841\pi$
$464$	−41.8328	−1.94204
$465$	−21.3820	−0.991565
$466$	35.1246	1.62712
$467$	−38.8328	−1.79697	−0.898484	−	0.439006i	$-0.855331\pi$
$467$	−38.8328	−1.79697	−0.898484	+	0.439006i	$0.855331\pi$
$468$	−1.09017	−0.0503931
$469$	12.5066	0.577500
$470$	15.0000	0.691898
$471$	−9.00000	−0.414698
$472$	−2.23607	−0.102923
$473$	−10.5279	−0.484072
$474$	−4.85410	−0.222956
$475$	0	0
$476$	13.8541	0.635002
$477$	1.76393	0.0807649
$478$	−12.0902	−0.552992
$479$	−26.0902	−1.19209	−0.596045	−	0.802951i	$-0.703262\pi$
$479$	−26.0902	−1.19209	−0.596045	+	0.802951i	$0.703262\pi$
$480$	−7.56231	−0.345170
$481$	12.0902	0.551264
$482$	−5.52786	−0.251787
$483$	−11.0000	−0.500517
$484$	−3.70820	−0.168555
$485$	6.70820	0.304604
$486$	1.61803	0.0733955
$487$	36.7426	1.66497	0.832484	−	0.554049i	$-0.186918\pi$
$487$	36.7426	1.66497	0.832484	+	0.554049i	$0.186918\pi$
$488$	−22.0344	−0.997452
$489$	−1.56231	−0.0706499
$490$	−51.8328	−2.34157
$491$	26.5066	1.19623	0.598113	−	0.801412i	$-0.295918\pi$
$491$	26.5066	1.19623	0.598113	+	0.801412i	$0.295918\pi$
$492$	1.90983	0.0861018
$493$	−41.8328	−1.88406
$494$	−23.0902	−1.03888
$495$	−5.00000	−0.224733
$496$	−46.4164	−2.08416
$497$	−45.9230	−2.05993
$498$	1.00000	0.0448111
$499$	12.4164	0.555835	0.277917	−	0.960605i	$-0.410356\pi$
$499$	12.4164	0.555835	0.277917	+	0.960605i	$0.410356\pi$
$500$	−6.90983	−0.309017
$501$	22.0344	0.984426
$502$	11.6180	0.518538
$503$	−20.7984	−0.927354	−0.463677	−	0.886004i	$-0.653470\pi$
$503$	−20.7984	−0.927354	−0.463677	+	0.886004i	$0.653470\pi$
$504$	−10.3262	−0.459967
$505$	−21.7082	−0.966002
$506$	−8.61803	−0.383118
$507$	9.88854	0.439166
$508$	−1.20163	−0.0533135
$509$	−29.9230	−1.32631	−0.663157	−	0.748481i	$-0.730784\pi$
$509$	−29.9230	−1.32631	−0.663157	+	0.748481i	$0.730784\pi$
$510$	−17.5623	−0.777672
$511$	27.0344	1.19593
$512$	5.29180	0.233867
$513$	8.09017	0.357190
$514$	−12.0000	−0.529297
$515$	2.76393	0.121793
$516$	−2.90983	−0.128098
$517$	9.27051	0.407717
$518$	−51.2148	−2.25025
$519$	−14.6180	−0.641660
$520$	−8.81966	−0.386768
$521$	−14.5066	−0.635545	−0.317772	−	0.948167i	$-0.602935\pi$
$521$	−14.5066	−0.635545	−0.317772	+	0.948167i	$0.602935\pi$
$522$	−13.9443	−0.610324
$523$	31.9443	1.39683	0.698413	−	0.715695i	$-0.253890\pi$
$523$	31.9443	1.39683	0.698413	+	0.715695i	$0.253890\pi$
$524$	6.58359	0.287606
$525$	0	0
$526$	1.00000	0.0436021
$527$	−46.4164	−2.02193
$528$	−10.8541	−0.472364
$529$	−17.3262	−0.753315
$530$	−6.38197	−0.277215
$531$	−1.00000	−0.0433963
$532$	23.0902	1.00109
$533$	5.45085	0.236103
$534$	17.4721	0.756093
$535$	27.0344	1.16880
$536$	−6.05573	−0.261568
$537$	9.47214	0.408753
$538$	4.09017	0.176340
$539$	−32.0344	−1.37982
$540$	−1.38197	−0.0594703
$541$	14.1246	0.607264	0.303632	−	0.952789i	$-0.401801\pi$
$541$	14.1246	0.607264	0.303632	+	0.952789i	$0.401801\pi$
$542$	19.7984	0.850413
$543$	11.2705	0.483664
$544$	−16.4164	−0.703848
$545$	−24.2705	−1.03963
$546$	13.1803	0.564066
$547$	32.5623	1.39226	0.696132	−	0.717914i	$-0.254903\pi$
$547$	32.5623	1.39226	0.696132	+	0.717914i	$0.254903\pi$
$548$	4.79837	0.204976
$549$	−9.85410	−0.420563
$550$	0	0
$551$	−69.7214	−2.97023
$552$	5.32624	0.226700
$553$	13.8541	0.589136
$554$	−5.61803	−0.238687
$555$	15.3262	0.650563
$556$	10.0344	0.425555
$557$	1.41641	0.0600151	0.0300076	−	0.999550i	$-0.490447\pi$
$557$	1.41641	0.0600151	0.0300076	+	0.999550i	$0.490447\pi$
$558$	−15.4721	−0.654988
$559$	−8.30495	−0.351262
$560$	50.1246	2.11815
$561$	−10.8541	−0.458261
$562$	6.00000	0.253095
$563$	28.5967	1.20521	0.602605	−	0.798040i	$-0.294130\pi$
$563$	28.5967	1.20521	0.602605	+	0.798040i	$0.294130\pi$
$564$	2.56231	0.107893
$565$	−20.1246	−0.846649
$566$	−16.6180	−0.698508
$567$	−4.61803	−0.193939
$568$	22.2361	0.933005
$569$	26.5066	1.11121	0.555607	−	0.831445i	$-0.312486\pi$
$569$	26.5066	1.11121	0.555607	+	0.831445i	$0.312486\pi$
$570$	−29.2705	−1.22601
$571$	28.4164	1.18919	0.594595	−	0.804025i	$-0.297312\pi$
$571$	28.4164	1.18919	0.594595	+	0.804025i	$0.297312\pi$
$572$	2.43769	0.101925
$573$	10.4164	0.435152
$574$	−23.0902	−0.963765
$575$	0	0
$576$	4.23607	0.176503
$577$	21.4721	0.893897	0.446948	−	0.894560i	$-0.352511\pi$
$577$	21.4721	0.893897	0.446948	+	0.894560i	$0.352511\pi$
$578$	−10.6180	−0.441652
$579$	−8.00000	−0.332469
$580$	11.9098	0.494529
$581$	−2.85410	−0.118408
$582$	4.85410	0.201209
$583$	−3.94427	−0.163355
$584$	−13.0902	−0.541675
$585$	−3.94427	−0.163076
$586$	−34.9787	−1.44496
$587$	29.3607	1.21184	0.605922	−	0.795524i	$-0.292804\pi$
$587$	29.3607	1.21184	0.605922	+	0.795524i	$0.292804\pi$
$588$	−8.85410	−0.365137
$589$	−77.3607	−3.18759
$590$	3.61803	0.148952
$591$	−10.6525	−0.438184
$592$	33.2705	1.36741
$593$	4.90983	0.201623	0.100811	−	0.994906i	$-0.467856\pi$
$593$	4.90983	0.201623	0.100811	+	0.994906i	$0.467856\pi$
$594$	−3.61803	−0.148450
$595$	50.1246	2.05491
$596$	4.88854	0.200243
$597$	−3.56231	−0.145795
$598$	−6.79837	−0.278006
$599$	−28.6525	−1.17071	−0.585354	−	0.810778i	$-0.699045\pi$
$599$	−28.6525	−1.17071	−0.585354	+	0.810778i	$0.699045\pi$
$600$	0	0
$601$	25.6180	1.04498	0.522491	−	0.852645i	$-0.325003\pi$
$601$	25.6180	1.04498	0.522491	+	0.852645i	$0.325003\pi$
$602$	35.1803	1.43384
$603$	−2.70820	−0.110287
$604$	−10.8541	−0.441647
$605$	−13.4164	−0.545455
$606$	−15.7082	−0.638102
$607$	17.6525	0.716492	0.358246	−	0.933627i	$-0.383375\pi$
$607$	17.6525	0.716492	0.358246	+	0.933627i	$0.383375\pi$
$608$	−27.3607	−1.10962
$609$	39.7984	1.61271
$610$	35.6525	1.44353
$611$	7.31308	0.295856
$612$	−3.00000	−0.121268
$613$	−7.34752	−0.296764	−0.148382	−	0.988930i	$-0.547406\pi$
$613$	−7.34752	−0.296764	−0.148382	+	0.988930i	$0.547406\pi$
$614$	−16.0000	−0.645707
$615$	6.90983	0.278631
$616$	23.0902	0.930329
$617$	28.8541	1.16162	0.580811	−	0.814038i	$-0.302735\pi$
$617$	28.8541	1.16162	0.580811	+	0.814038i	$0.302735\pi$
$618$	2.00000	0.0804518
$619$	−28.1246	−1.13042	−0.565212	−	0.824946i	$-0.691206\pi$
$619$	−28.1246	−1.13042	−0.565212	+	0.824946i	$0.691206\pi$
$620$	13.2148	0.530718
$621$	2.38197	0.0955850
$622$	46.0344	1.84581
$623$	−49.8673	−1.99789
$624$	−8.56231	−0.342767
$625$	−25.0000	−1.00000
$626$	−6.14590	−0.245639
$627$	−18.0902	−0.722452
$628$	5.56231	0.221960
$629$	33.2705	1.32658
$630$	16.7082	0.665671
$631$	−13.5836	−0.540754	−0.270377	−	0.962754i	$-0.587148\pi$
$631$	−13.5836	−0.540754	−0.270377	+	0.962754i	$0.587148\pi$
$632$	−6.70820	−0.266838
$633$	8.85410	0.351919
$634$	11.0344	0.438234
$635$	−4.34752	−0.172526
$636$	−1.09017	−0.0432281
$637$	−25.2705	−1.00125
$638$	31.1803	1.23444
$639$	9.94427	0.393389
$640$	−30.4508	−1.20368
$641$	−10.0557	−0.397177	−0.198589	−	0.980083i	$-0.563636\pi$
$641$	−10.0557	−0.397177	−0.198589	+	0.980083i	$0.563636\pi$
$642$	19.5623	0.772063
$643$	−13.1459	−0.518424	−0.259212	−	0.965821i	$-0.583463\pi$
$643$	−13.1459	−0.518424	−0.259212	+	0.965821i	$0.583463\pi$
$644$	6.79837	0.267893
$645$	−10.5279	−0.414534
$646$	−63.5410	−2.49999
$647$	27.0557	1.06367	0.531835	−	0.846848i	$-0.321503\pi$
$647$	27.0557	1.06367	0.531835	+	0.846848i	$0.321503\pi$
$648$	2.23607	0.0878410
$649$	2.23607	0.0877733
$650$	0	0
$651$	44.1591	1.73073
$652$	0.965558	0.0378142
$653$	28.9098	1.13133	0.565665	−	0.824635i	$-0.308620\pi$
$653$	28.9098	1.13133	0.565665	+	0.824635i	$0.308620\pi$
$654$	−17.5623	−0.686741
$655$	23.8197	0.930711
$656$	15.0000	0.585652
$657$	−5.85410	−0.228390
$658$	−30.9787	−1.20768
$659$	−4.85410	−0.189089	−0.0945445	−	0.995521i	$-0.530139\pi$
$659$	−4.85410	−0.189089	−0.0945445	+	0.995521i	$0.530139\pi$
$660$	3.09017	0.120285
$661$	−11.7426	−0.456736	−0.228368	−	0.973575i	$-0.573339\pi$
$661$	−11.7426	−0.456736	−0.228368	+	0.973575i	$0.573339\pi$
$662$	−47.1246	−1.83155
$663$	−8.56231	−0.332532
$664$	1.38197	0.0536307
$665$	83.5410	3.23958
$666$	11.0902	0.429735
$667$	−20.5279	−0.794842
$668$	−13.6180	−0.526898
$669$	18.4721	0.714174
$670$	9.79837	0.378544
$671$	22.0344	0.850630
$672$	15.6180	0.602479
$673$	0.472136	0.0181995	0.00909975	−	0.999959i	$-0.497103\pi$
$673$	0.472136	0.0181995	0.00909975	+	0.999959i	$0.497103\pi$
$674$	42.2148	1.62605
$675$	0	0
$676$	−6.11146	−0.235056
$677$	−50.1803	−1.92859	−0.964294	−	0.264836i	$-0.914682\pi$
$677$	−50.1803	−1.92859	−0.964294	+	0.264836i	$0.914682\pi$
$678$	−14.5623	−0.559262
$679$	−13.8541	−0.531672
$680$	−24.2705	−0.930732
$681$	28.8541	1.10569
$682$	34.5967	1.32478
$683$	39.0000	1.49229	0.746147	−	0.665782i	$-0.231902\pi$
$683$	39.0000	1.49229	0.746147	+	0.665782i	$0.231902\pi$
$684$	−5.00000	−0.191180
$685$	17.3607	0.663317
$686$	54.7426	2.09008
$687$	15.8541	0.604872
$688$	−22.8541	−0.871304
$689$	−3.11146	−0.118537
$690$	−8.61803	−0.328083
$691$	−5.12461	−0.194949	−0.0974747	−	0.995238i	$-0.531077\pi$
$691$	−5.12461	−0.194949	−0.0974747	+	0.995238i	$0.531077\pi$
$692$	9.03444	0.343438
$693$	10.3262	0.392261
$694$	−3.18034	−0.120724
$695$	36.3050	1.37713
$696$	−19.2705	−0.730447
$697$	15.0000	0.568166
$698$	43.6869	1.65357
$699$	21.7082	0.821080
$700$	0	0
$701$	42.7984	1.61647	0.808236	−	0.588859i	$-0.200422\pi$
$701$	42.7984	1.61647	0.808236	+	0.588859i	$0.200422\pi$
$702$	−2.85410	−0.107721
$703$	55.4508	2.09137
$704$	−9.47214	−0.356995
$705$	9.27051	0.349148
$706$	−21.0902	−0.793739
$707$	44.8328	1.68611
$708$	0.618034	0.0232271
$709$	−14.2918	−0.536740	−0.268370	−	0.963316i	$-0.586485\pi$
$709$	−14.2918	−0.536740	−0.268370	+	0.963316i	$0.586485\pi$
$710$	−35.9787	−1.35026
$711$	−3.00000	−0.112509
$712$	24.1459	0.904906
$713$	−22.7771	−0.853009
$714$	36.2705	1.35739
$715$	8.81966	0.329837
$716$	−5.85410	−0.218778
$717$	−7.47214	−0.279052
$718$	−22.4721	−0.838653
$719$	12.3820	0.461769	0.230885	−	0.972981i	$-0.425838\pi$
$719$	12.3820	0.461769	0.230885	+	0.972981i	$0.425838\pi$
$720$	−10.8541	−0.404508
$721$	−5.70820	−0.212585
$722$	−75.1591	−2.79713
$723$	−3.41641	−0.127058
$724$	−6.96556	−0.258873
$725$	0	0
$726$	−9.70820	−0.360305
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	18.2148	0.675085
$729$	1.00000	0.0370370
$730$	21.1803	0.783920
$731$	−22.8541	−0.845289
$732$	6.09017	0.225099
$733$	29.4721	1.08858	0.544289	−	0.838898i	$-0.316799\pi$
$733$	29.4721	1.08858	0.544289	+	0.838898i	$0.316799\pi$
$734$	38.5623	1.42336
$735$	−32.0344	−1.18161
$736$	−8.05573	−0.296938
$737$	6.05573	0.223066
$738$	5.00000	0.184053
$739$	−39.1033	−1.43844	−0.719220	−	0.694783i	$-0.755500\pi$
$739$	−39.1033	−1.43844	−0.719220	+	0.694783i	$0.755500\pi$
$740$	−9.47214	−0.348203
$741$	−14.2705	−0.524240
$742$	13.1803	0.483865
$743$	−22.3820	−0.821115	−0.410557	−	0.911835i	$-0.634666\pi$
$743$	−22.3820	−0.821115	−0.410557	+	0.911835i	$0.634666\pi$
$744$	−21.3820	−0.783901
$745$	17.6869	0.647999
$746$	−30.2148	−1.10624
$747$	0.618034	0.0226127
$748$	6.70820	0.245276
$749$	−55.8328	−2.04009
$750$	−18.0902	−0.660560
$751$	−45.1246	−1.64662	−0.823310	−	0.567592i	$-0.807875\pi$
$751$	−45.1246	−1.64662	−0.823310	+	0.567592i	$0.807875\pi$
$752$	20.1246	0.733869
$753$	7.18034	0.261666
$754$	24.5967	0.895761
$755$	−39.2705	−1.42920
$756$	2.85410	0.103803
$757$	−43.6180	−1.58532	−0.792662	−	0.609661i	$-0.791306\pi$
$757$	−43.6180	−1.58532	−0.792662	+	0.609661i	$0.791306\pi$
$758$	7.14590	0.259551
$759$	−5.32624	−0.193330
$760$	−40.4508	−1.46731
$761$	17.2918	0.626827	0.313414	−	0.949617i	$-0.398527\pi$
$761$	17.2918	0.626827	0.313414	+	0.949617i	$0.398527\pi$
$762$	−3.14590	−0.113964
$763$	50.1246	1.81463
$764$	−6.43769	−0.232908
$765$	−10.8541	−0.392431
$766$	22.2705	0.804666
$767$	1.76393	0.0636919
$768$	−13.5623	−0.489388
$769$	−35.9787	−1.29743	−0.648713	−	0.761033i	$-0.724692\pi$
$769$	−35.9787	−1.29743	−0.648713	+	0.761033i	$0.724692\pi$
$770$	−37.3607	−1.34639
$771$	−7.41641	−0.267095
$772$	4.94427	0.177948
$773$	−2.65248	−0.0954029	−0.0477015	−	0.998862i	$-0.515190\pi$
$773$	−2.65248	−0.0954029	−0.0477015	+	0.998862i	$0.515190\pi$
$774$	−7.61803	−0.273825
$775$	0	0
$776$	6.70820	0.240810
$777$	−31.6525	−1.13553
$778$	−38.0689	−1.36484
$779$	25.0000	0.895718
$780$	2.43769	0.0872835
$781$	−22.2361	−0.795669
$782$	−18.7082	−0.669004
$783$	−8.61803	−0.307983
$784$	−69.5410	−2.48361
$785$	20.1246	0.718278
$786$	17.2361	0.614790
$787$	−17.7082	−0.631229	−0.315615	−	0.948887i	$-0.602211\pi$
$787$	−17.7082	−0.631229	−0.315615	+	0.948887i	$0.602211\pi$
$788$	6.58359	0.234531
$789$	0.618034	0.0220026
$790$	10.8541	0.386172
$791$	41.5623	1.47779
$792$	−5.00000	−0.177667
$793$	17.3820	0.617252
$794$	−4.85410	−0.172266
$795$	−3.94427	−0.139889
$796$	2.20163	0.0780346
$797$	3.23607	0.114627	0.0573137	−	0.998356i	$-0.481746\pi$
$797$	3.23607	0.114627	0.0573137	+	0.998356i	$0.481746\pi$
$798$	60.4508	2.13994
$799$	20.1246	0.711958
$800$	0	0
$801$	10.7984	0.381542
$802$	9.56231	0.337657
$803$	13.0902	0.461942
$804$	1.67376	0.0590290
$805$	24.5967	0.866921
$806$	27.2918	0.961313
$807$	2.52786	0.0889850
$808$	−21.7082	−0.763692
$809$	23.3262	0.820107	0.410053	−	0.912062i	$-0.365510\pi$
$809$	23.3262	0.820107	0.410053	+	0.912062i	$0.365510\pi$
$810$	−3.61803	−0.127125
$811$	−16.6525	−0.584748	−0.292374	−	0.956304i	$-0.594445\pi$
$811$	−16.6525	−0.584748	−0.292374	+	0.956304i	$0.594445\pi$
$812$	−24.5967	−0.863177
$813$	12.2361	0.429138
$814$	−24.7984	−0.869183
$815$	3.49342	0.122369
$816$	−23.5623	−0.824846
$817$	−38.0902	−1.33261
$818$	60.5410	2.11677
$819$	8.14590	0.284641
$820$	−4.27051	−0.149133
$821$	23.9230	0.834918	0.417459	−	0.908696i	$-0.362921\pi$
$821$	23.9230	0.834918	0.417459	+	0.908696i	$0.362921\pi$
$822$	12.5623	0.438161
$823$	24.7082	0.861274	0.430637	−	0.902525i	$-0.358289\pi$
$823$	24.7082	0.861274	0.430637	+	0.902525i	$0.358289\pi$
$824$	2.76393	0.0962861
$825$	0	0
$826$	−7.47214	−0.259989
$827$	−25.1803	−0.875606	−0.437803	−	0.899071i	$-0.644243\pi$
$827$	−25.1803	−0.875606	−0.437803	+	0.899071i	$0.644243\pi$
$828$	−1.47214	−0.0511603
$829$	14.6869	0.510098	0.255049	−	0.966928i	$-0.417908\pi$
$829$	14.6869	0.510098	0.255049	+	0.966928i	$0.417908\pi$
$830$	−2.23607	−0.0776151
$831$	−3.47214	−0.120447
$832$	−7.47214	−0.259050
$833$	−69.5410	−2.40945
$834$	26.2705	0.909673
$835$	−49.2705	−1.70508
$836$	11.1803	0.386680
$837$	−9.56231	−0.330522
$838$	−50.6525	−1.74976
$839$	−47.1803	−1.62885	−0.814423	−	0.580271i	$-0.802946\pi$
$839$	−47.1803	−1.62885	−0.814423	+	0.580271i	$0.802946\pi$
$840$	23.0902	0.796687
$841$	45.2705	1.56105
$842$	−1.61803	−0.0557611
$843$	3.70820	0.127717
$844$	−5.47214	−0.188359
$845$	−22.1115	−0.760657
$846$	6.70820	0.230633
$847$	27.7082	0.952065
$848$	−8.56231	−0.294031
$849$	−10.2705	−0.352483
$850$	0	0
$851$	16.3262	0.559656
$852$	−6.14590	−0.210555
$853$	−33.0344	−1.13108	−0.565539	−	0.824722i	$-0.691332\pi$
$853$	−33.0344	−1.13108	−0.565539	+	0.824722i	$0.691332\pi$
$854$	−73.6312	−2.51961
$855$	−18.0902	−0.618671
$856$	27.0344	0.924018
$857$	42.7771	1.46124	0.730619	−	0.682786i	$-0.239232\pi$
$857$	42.7771	1.46124	0.730619	+	0.682786i	$0.239232\pi$
$858$	6.38197	0.217877
$859$	24.4721	0.834979	0.417489	−	0.908682i	$-0.362910\pi$
$859$	24.4721	0.834979	0.417489	+	0.908682i	$0.362910\pi$
$860$	6.50658	0.221872
$861$	−14.2705	−0.486338
$862$	−23.6525	−0.805607
$863$	43.7984	1.49091	0.745457	−	0.666554i	$-0.232231\pi$
$863$	43.7984	1.49091	0.745457	+	0.666554i	$0.232231\pi$
$864$	−3.38197	−0.115057
$865$	32.6869	1.11139
$866$	−31.2705	−1.06262
$867$	−6.56231	−0.222868
$868$	−27.2918	−0.926344
$869$	6.70820	0.227560
$870$	31.1803	1.05711
$871$	4.77709	0.161865
$872$	−24.2705	−0.821903
$873$	3.00000	0.101535
$874$	−31.1803	−1.05469
$875$	51.6312	1.74545
$876$	3.61803	0.122242
$877$	−51.8885	−1.75215	−0.876076	−	0.482173i	$-0.839848\pi$
$877$	−51.8885	−1.75215	−0.876076	+	0.482173i	$0.839848\pi$
$878$	52.3951	1.76825
$879$	−21.6180	−0.729158
$880$	24.2705	0.818159
$881$	−2.29180	−0.0772126	−0.0386063	−	0.999254i	$-0.512292\pi$
$881$	−2.29180	−0.0772126	−0.0386063	+	0.999254i	$0.512292\pi$
$882$	−23.1803	−0.780523
$883$	3.41641	0.114971	0.0574856	−	0.998346i	$-0.481692\pi$
$883$	3.41641	0.114971	0.0574856	+	0.998346i	$0.481692\pi$
$884$	5.29180	0.177982
$885$	2.23607	0.0751646
$886$	−9.09017	−0.305390
$887$	−31.4721	−1.05673	−0.528365	−	0.849017i	$-0.677195\pi$
$887$	−31.4721	−1.05673	−0.528365	+	0.849017i	$0.677195\pi$
$888$	15.3262	0.514315
$889$	8.97871	0.301136
$890$	−39.0689	−1.30959
$891$	−2.23607	−0.0749111
$892$	−11.4164	−0.382250
$893$	33.5410	1.12241
$894$	12.7984	0.428042
$895$	−21.1803	−0.707981
$896$	62.8885	2.10096
$897$	−4.20163	−0.140288
$898$	46.7426	1.55982
$899$	82.4083	2.74847
$900$	0	0
$901$	−8.56231	−0.285252
$902$	−11.1803	−0.372265
$903$	21.7426	0.723550
$904$	−20.1246	−0.669335
$905$	−25.2016	−0.837730
$906$	−28.4164	−0.944072
$907$	−2.05573	−0.0682593	−0.0341297	−	0.999417i	$-0.510866\pi$
$907$	−2.05573	−0.0682593	−0.0341297	+	0.999417i	$0.510866\pi$
$908$	−17.8328	−0.591803
$909$	−9.70820	−0.322001
$910$	−29.4721	−0.976992
$911$	42.1033	1.39495	0.697473	−	0.716611i	$-0.254308\pi$
$911$	42.1033	1.39495	0.697473	+	0.716611i	$0.254308\pi$
$912$	−39.2705	−1.30038
$913$	−1.38197	−0.0457364
$914$	43.1803	1.42828
$915$	22.0344	0.728436
$916$	−9.79837	−0.323747
$917$	−49.1935	−1.62451
$918$	−7.85410	−0.259224
$919$	35.5967	1.17423	0.587114	−	0.809504i	$-0.300264\pi$
$919$	35.5967	1.17423	0.587114	+	0.809504i	$0.300264\pi$
$920$	−11.9098	−0.392655
$921$	−9.88854	−0.325839
$922$	30.3262	0.998741
$923$	−17.5410	−0.577370
$924$	−6.38197	−0.209951
$925$	0	0
$926$	−17.5623	−0.577133
$927$	1.23607	0.0405978
$928$	29.1459	0.956761
$929$	−35.2918	−1.15789	−0.578943	−	0.815368i	$-0.696535\pi$
$929$	−35.2918	−1.15789	−0.578943	+	0.815368i	$0.696535\pi$
$930$	34.5967	1.13447
$931$	−115.902	−3.79852
$932$	−13.4164	−0.439469
$933$	28.4508	0.931439
$934$	62.8328	2.05595
$935$	24.2705	0.793731
$936$	−3.94427	−0.128923
$937$	−37.7214	−1.23230	−0.616152	−	0.787628i	$-0.711309\pi$
$937$	−37.7214	−1.23230	−0.616152	+	0.787628i	$0.711309\pi$
$938$	−20.2361	−0.660731
$939$	−3.79837	−0.123955
$940$	−5.72949	−0.186875
$941$	−22.3050	−0.727121	−0.363560	−	0.931571i	$-0.618439\pi$
$941$	−22.3050	−0.727121	−0.363560	+	0.931571i	$0.618439\pi$
$942$	14.5623	0.474466
$943$	7.36068	0.239697
$944$	4.85410	0.157988
$945$	10.3262	0.335913
$946$	17.0344	0.553837
$947$	−4.03444	−0.131102	−0.0655509	−	0.997849i	$-0.520880\pi$
$947$	−4.03444	−0.131102	−0.0655509	+	0.997849i	$0.520880\pi$
$948$	1.85410	0.0602184
$949$	10.3262	0.335204
$950$	0	0
$951$	6.81966	0.221143
$952$	50.1246	1.62455
$953$	−12.9443	−0.419306	−0.209653	−	0.977776i	$-0.567233\pi$
$953$	−12.9443	−0.419306	−0.209653	+	0.977776i	$0.567233\pi$
$954$	−2.85410	−0.0924050
$955$	−23.2918	−0.753705
$956$	4.61803	0.149358
$957$	19.2705	0.622927
$958$	42.2148	1.36390
$959$	−35.8541	−1.15779
$960$	−9.47214	−0.305712
$961$	60.4377	1.94960
$962$	−19.5623	−0.630714
$963$	12.0902	0.389600
$964$	2.11146	0.0680054
$965$	17.8885	0.575853
$966$	17.7984	0.572653
$967$	−18.2705	−0.587540	−0.293770	−	0.955876i	$-0.594910\pi$
$967$	−18.2705	−0.587540	−0.293770	+	0.955876i	$0.594910\pi$
$968$	−13.4164	−0.431220
$969$	−39.2705	−1.26155
$970$	−10.8541	−0.348504
$971$	22.2016	0.712484	0.356242	−	0.934394i	$-0.384058\pi$
$971$	22.2016	0.712484	0.356242	+	0.934394i	$0.384058\pi$
$972$	−0.618034	−0.0198234
$973$	−74.9787	−2.40371
$974$	−59.4508	−1.90493
$975$	0	0
$976$	47.8328	1.53109
$977$	14.8885	0.476327	0.238163	−	0.971225i	$-0.423455\pi$
$977$	14.8885	0.476327	0.238163	+	0.971225i	$0.423455\pi$
$978$	2.52786	0.0808322
$979$	−24.1459	−0.771706
$980$	19.7984	0.632436
$981$	−10.8541	−0.346545
$982$	−42.8885	−1.36863
$983$	−58.5967	−1.86895	−0.934473	−	0.356034i	$-0.884129\pi$
$983$	−58.5967	−1.86895	−0.934473	+	0.356034i	$0.884129\pi$
$984$	6.90983	0.220277
$985$	23.8197	0.758957
$986$	67.6869	2.15559
$987$	−19.1459	−0.609421
$988$	8.81966	0.280591
$989$	−11.2148	−0.356609
$990$	8.09017	0.257122
$991$	−18.7426	−0.595380	−0.297690	−	0.954663i	$-0.596216\pi$
$991$	−18.7426	−0.595380	−0.297690	+	0.954663i	$0.596216\pi$
$992$	32.3394	1.02678
$993$	−29.1246	−0.924242
$994$	74.3050	2.35681
$995$	7.96556	0.252525
$996$	−0.381966	−0.0121031
$997$	1.78522	0.0565384	0.0282692	−	0.999600i	$-0.491000\pi$
$997$	1.78522	0.0565384	0.0282692	+	0.999600i	$0.491000\pi$
$998$	−20.0902	−0.635943
$999$	6.85410	0.216854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.a.b.1.1	✓	2
3.2	odd	2		531.2.a.b.1.2		2
4.3	odd	2		2832.2.a.o.1.2		2
5.4	even	2		4425.2.a.t.1.2		2
7.6	odd	2		8673.2.a.k.1.1		2
12.11	even	2		8496.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.a.b.1.1	✓	2	1.1	even	1	trivial
531.2.a.b.1.2		2	3.2	odd	2
2832.2.a.o.1.2		2	4.3	odd	2
4425.2.a.t.1.2		2	5.4	even	2
8496.2.a.bb.1.1		2	12.11	even	2
8673.2.a.k.1.1		2	7.6	odd	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +2.23607 q^{5} +1.61803 q^{6} -4.61803 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +2.23607 q^{5} +1.61803 q^{6} -4.61803 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.61803 q^{10} -2.23607 q^{11} -0.618034 q^{12} -1.76393 q^{13} +7.47214 q^{14} -2.23607 q^{15} -4.85410 q^{16} -4.85410 q^{17} -1.61803 q^{18} -8.09017 q^{19} +1.38197 q^{20} +4.61803 q^{21} +3.61803 q^{22} -2.38197 q^{23} -2.23607 q^{24} +2.85410 q^{26} -1.00000 q^{27} -2.85410 q^{28} +8.61803 q^{29} +3.61803 q^{30} +9.56231 q^{31} +3.38197 q^{32} +2.23607 q^{33} +7.85410 q^{34} -10.3262 q^{35} +0.618034 q^{36} -6.85410 q^{37} +13.0902 q^{38} +1.76393 q^{39} +5.00000 q^{40} -3.09017 q^{41} -7.47214 q^{42} +4.70820 q^{43} -1.38197 q^{44} +2.23607 q^{45} +3.85410 q^{46} -4.14590 q^{47} +4.85410 q^{48} +14.3262 q^{49} +4.85410 q^{51} -1.09017 q^{52} +1.76393 q^{53} +1.61803 q^{54} -5.00000 q^{55} -10.3262 q^{56} +8.09017 q^{57} -13.9443 q^{58} -1.00000 q^{59} -1.38197 q^{60} -9.85410 q^{61} -15.4721 q^{62} -4.61803 q^{63} +4.23607 q^{64} -3.94427 q^{65} -3.61803 q^{66} -2.70820 q^{67} -3.00000 q^{68} +2.38197 q^{69} +16.7082 q^{70} +9.94427 q^{71} +2.23607 q^{72} -5.85410 q^{73} +11.0902 q^{74} -5.00000 q^{76} +10.3262 q^{77} -2.85410 q^{78} -3.00000 q^{79} -10.8541 q^{80} +1.00000 q^{81} +5.00000 q^{82} +0.618034 q^{83} +2.85410 q^{84} -10.8541 q^{85} -7.61803 q^{86} -8.61803 q^{87} -5.00000 q^{88} +10.7984 q^{89} -3.61803 q^{90} +8.14590 q^{91} -1.47214 q^{92} -9.56231 q^{93} +6.70820 q^{94} -18.0902 q^{95} -3.38197 q^{96} +3.00000 q^{97} -23.1803 q^{98} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 7 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 - 2 * q^3 - q^4 + q^6 - 7 * q^7 + 2 * q^9	\( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 7 q^{7} + 2 q^{9} - 5 q^{10} + q^{12} - 8 q^{13} + 6 q^{14} - 3 q^{16} - 3 q^{17} - q^{18} - 5 q^{19} + 5 q^{20} + 7 q^{21} + 5 q^{22} - 7 q^{23} - q^{26} - 2 q^{27} + q^{28} + 15 q^{29} + 5 q^{30} - q^{31} + 9 q^{32} + 9 q^{34} - 5 q^{35} - q^{36} - 7 q^{37} + 15 q^{38} + 8 q^{39} + 10 q^{40} + 5 q^{41} - 6 q^{42} - 4 q^{43} - 5 q^{44} + q^{46} - 15 q^{47} + 3 q^{48} + 13 q^{49} + 3 q^{51} + 9 q^{52} + 8 q^{53} + q^{54} - 10 q^{55} - 5 q^{56} + 5 q^{57} - 10 q^{58} - 2 q^{59} - 5 q^{60} - 13 q^{61} - 22 q^{62} - 7 q^{63} + 4 q^{64} + 10 q^{65} - 5 q^{66} + 8 q^{67} - 6 q^{68} + 7 q^{69} + 20 q^{70} + 2 q^{71} - 5 q^{73} + 11 q^{74} - 10 q^{76} + 5 q^{77} + q^{78} - 6 q^{79} - 15 q^{80} + 2 q^{81} + 10 q^{82} - q^{83} - q^{84} - 15 q^{85} - 13 q^{86} - 15 q^{87} - 10 q^{88} - 3 q^{89} - 5 q^{90} + 23 q^{91} + 6 q^{92} + q^{93} - 25 q^{95} - 9 q^{96} + 6 q^{97} - 24 q^{98}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 + q^6 - 7 * q^7 + 2 * q^9 - 5 * q^10 + q^12 - 8 * q^13 + 6 * q^14 - 3 * q^16 - 3 * q^17 - q^18 - 5 * q^19 + 5 * q^20 + 7 * q^21 + 5 * q^22 - 7 * q^23 - q^26 - 2 * q^27 + q^28 + 15 * q^29 + 5 * q^30 - q^31 + 9 * q^32 + 9 * q^34 - 5 * q^35 - q^36 - 7 * q^37 + 15 * q^38 + 8 * q^39 + 10 * q^40 + 5 * q^41 - 6 * q^42 - 4 * q^43 - 5 * q^44 + q^46 - 15 * q^47 + 3 * q^48 + 13 * q^49 + 3 * q^51 + 9 * q^52 + 8 * q^53 + q^54 - 10 * q^55 - 5 * q^56 + 5 * q^57 - 10 * q^58 - 2 * q^59 - 5 * q^60 - 13 * q^61 - 22 * q^62 - 7 * q^63 + 4 * q^64 + 10 * q^65 - 5 * q^66 + 8 * q^67 - 6 * q^68 + 7 * q^69 + 20 * q^70 + 2 * q^71 - 5 * q^73 + 11 * q^74 - 10 * q^76 + 5 * q^77 + q^78 - 6 * q^79 - 15 * q^80 + 2 * q^81 + 10 * q^82 - q^83 - q^84 - 15 * q^85 - 13 * q^86 - 15 * q^87 - 10 * q^88 - 3 * q^89 - 5 * q^90 + 23 * q^91 + 6 * q^92 + q^93 - 25 * q^95 - 9 * q^96 + 6 * q^97 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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