LMFDB - Embedded newform 538.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,2,Mod(1,538)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, 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("538.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	538.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:	$4.29595162874$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 7x^{4} + 27x^{3} - 15x^{2} - 20x + 12 $ x^7 - x^6 - 10x^5 + 7x^4 + 27x^3 - 15x^2 - 20*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.49844$ of defining polynomial
Character	$\chi$	$=$	538.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.370477 q^{3} +1.00000 q^{4} +2.98038 q^{5} +0.370477 q^{6} -2.56529 q^{7} -1.00000 q^{8} -2.86275 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.370477 q^{3} +1.00000 q^{4} +2.98038 q^{5} +0.370477 q^{6} -2.56529 q^{7} -1.00000 q^{8} -2.86275 q^{9} -2.98038 q^{10} -5.77573 q^{11} -0.370477 q^{12} -0.953732 q^{13} +2.56529 q^{14} -1.10416 q^{15} +1.00000 q^{16} +2.51668 q^{17} +2.86275 q^{18} -7.19922 q^{19} +2.98038 q^{20} +0.950379 q^{21} +5.77573 q^{22} -5.17399 q^{23} +0.370477 q^{24} +3.88265 q^{25} +0.953732 q^{26} +2.17201 q^{27} -2.56529 q^{28} +4.63236 q^{29} +1.10416 q^{30} +8.16921 q^{31} -1.00000 q^{32} +2.13977 q^{33} -2.51668 q^{34} -7.64552 q^{35} -2.86275 q^{36} -9.43775 q^{37} +7.19922 q^{38} +0.353336 q^{39} -2.98038 q^{40} +2.46810 q^{41} -0.950379 q^{42} +5.86496 q^{43} -5.77573 q^{44} -8.53207 q^{45} +5.17399 q^{46} -9.50969 q^{47} -0.370477 q^{48} -0.419304 q^{49} -3.88265 q^{50} -0.932373 q^{51} -0.953732 q^{52} -7.51226 q^{53} -2.17201 q^{54} -17.2139 q^{55} +2.56529 q^{56} +2.66714 q^{57} -4.63236 q^{58} -1.01485 q^{59} -1.10416 q^{60} -5.86716 q^{61} -8.16921 q^{62} +7.34377 q^{63} +1.00000 q^{64} -2.84248 q^{65} -2.13977 q^{66} +12.1288 q^{67} +2.51668 q^{68} +1.91685 q^{69} +7.64552 q^{70} -12.3537 q^{71} +2.86275 q^{72} +0.650878 q^{73} +9.43775 q^{74} -1.43843 q^{75} -7.19922 q^{76} +14.8164 q^{77} -0.353336 q^{78} +15.5158 q^{79} +2.98038 q^{80} +7.78356 q^{81} -2.46810 q^{82} -8.27171 q^{83} +0.950379 q^{84} +7.50067 q^{85} -5.86496 q^{86} -1.71618 q^{87} +5.77573 q^{88} +2.89852 q^{89} +8.53207 q^{90} +2.44660 q^{91} -5.17399 q^{92} -3.02650 q^{93} +9.50969 q^{94} -21.4564 q^{95} +0.370477 q^{96} +0.956247 q^{97} +0.419304 q^{98} +16.5345 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} - 4 q^{3} + 7 q^{4} - 6 q^{5} + 4 q^{6} - 3 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^2 - 4 * q^3 + 7 * q^4 - 6 * q^5 + 4 * q^6 - 3 * q^7 - 7 * q^8 + 7 * q^9	\( 7 q - 7 q^{2} - 4 q^{3} + 7 q^{4} - 6 q^{5} + 4 q^{6} - 3 q^{7} - 7 q^{8} + 7 q^{9} + 6 q^{10} - 12 q^{11} - 4 q^{12} + 3 q^{13} + 3 q^{14} - 6 q^{15} + 7 q^{16} - 8 q^{17} - 7 q^{18} - 7 q^{19} - 6 q^{20} - 5 q^{21} + 12 q^{22} - 22 q^{23} + 4 q^{24} + 9 q^{25} - 3 q^{26} - 22 q^{27} - 3 q^{28} - 7 q^{29} + 6 q^{30} - 3 q^{31} - 7 q^{32} - 19 q^{33} + 8 q^{34} - 25 q^{35} + 7 q^{36} - 13 q^{37} + 7 q^{38} + q^{39} + 6 q^{40} - 11 q^{41} + 5 q^{42} - 14 q^{43} - 12 q^{44} - 17 q^{45} + 22 q^{46} - 23 q^{47} - 4 q^{48} + 6 q^{49} - 9 q^{50} - 21 q^{51} + 3 q^{52} - 21 q^{53} + 22 q^{54} + 15 q^{55} + 3 q^{56} - 14 q^{57} + 7 q^{58} - 20 q^{59} - 6 q^{60} + 13 q^{61} + 3 q^{62} - 12 q^{63} + 7 q^{64} - 29 q^{65} + 19 q^{66} - 30 q^{67} - 8 q^{68} + 13 q^{69} + 25 q^{70} - 23 q^{71} - 7 q^{72} + 9 q^{73} + 13 q^{74} + 3 q^{75} - 7 q^{76} + 8 q^{77} - q^{78} + 11 q^{79} - 6 q^{80} + 43 q^{81} + 11 q^{82} - 3 q^{83} - 5 q^{84} + 21 q^{85} + 14 q^{86} + 28 q^{87} + 12 q^{88} + 16 q^{89} + 17 q^{90} - 10 q^{91} - 22 q^{92} - 13 q^{93} + 23 q^{94} + 4 q^{96} + 21 q^{97} - 6 q^{98} - 9 q^{99}+O(q^{100}) \) 7 * q - 7 * q^2 - 4 * q^3 + 7 * q^4 - 6 * q^5 + 4 * q^6 - 3 * q^7 - 7 * q^8 + 7 * q^9 + 6 * q^10 - 12 * q^11 - 4 * q^12 + 3 * q^13 + 3 * q^14 - 6 * q^15 + 7 * q^16 - 8 * q^17 - 7 * q^18 - 7 * q^19 - 6 * q^20 - 5 * q^21 + 12 * q^22 - 22 * q^23 + 4 * q^24 + 9 * q^25 - 3 * q^26 - 22 * q^27 - 3 * q^28 - 7 * q^29 + 6 * q^30 - 3 * q^31 - 7 * q^32 - 19 * q^33 + 8 * q^34 - 25 * q^35 + 7 * q^36 - 13 * q^37 + 7 * q^38 + q^39 + 6 * q^40 - 11 * q^41 + 5 * q^42 - 14 * q^43 - 12 * q^44 - 17 * q^45 + 22 * q^46 - 23 * q^47 - 4 * q^48 + 6 * q^49 - 9 * q^50 - 21 * q^51 + 3 * q^52 - 21 * q^53 + 22 * q^54 + 15 * q^55 + 3 * q^56 - 14 * q^57 + 7 * q^58 - 20 * q^59 - 6 * q^60 + 13 * q^61 + 3 * q^62 - 12 * q^63 + 7 * q^64 - 29 * q^65 + 19 * q^66 - 30 * q^67 - 8 * q^68 + 13 * q^69 + 25 * q^70 - 23 * q^71 - 7 * q^72 + 9 * q^73 + 13 * q^74 + 3 * q^75 - 7 * q^76 + 8 * q^77 - q^78 + 11 * q^79 - 6 * q^80 + 43 * q^81 + 11 * q^82 - 3 * q^83 - 5 * q^84 + 21 * q^85 + 14 * q^86 + 28 * q^87 + 12 * q^88 + 16 * q^89 + 17 * q^90 - 10 * q^91 - 22 * q^92 - 13 * q^93 + 23 * q^94 + 4 * q^96 + 21 * q^97 - 6 * q^98 - 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.370477	−0.213895	−0.106947	−	0.994265i	$-0.534108\pi$
$3$	−0.370477	−0.213895	−0.106947	+	0.994265i	$0.534108\pi$
$4$	1.00000	0.500000
$5$	2.98038	1.33287	0.666433	−	0.745565i	$-0.267820\pi$
$5$	2.98038	1.33287	0.666433	+	0.745565i	$0.267820\pi$
$6$	0.370477	0.151247
$7$	−2.56529	−0.969587	−0.484794	−	0.874629i	$-0.661105\pi$
$7$	−2.56529	−0.969587	−0.484794	+	0.874629i	$0.661105\pi$
$8$	−1.00000	−0.353553
$9$	−2.86275	−0.954249
$10$	−2.98038	−0.942478
$11$	−5.77573	−1.74145	−0.870724	−	0.491772i	$-0.836349\pi$
$11$	−5.77573	−1.74145	−0.870724	+	0.491772i	$0.836349\pi$
$12$	−0.370477	−0.106947
$13$	−0.953732	−0.264518	−0.132259	−	0.991215i	$-0.542223\pi$
$13$	−0.953732	−0.264518	−0.132259	+	0.991215i	$0.542223\pi$
$14$	2.56529	0.685602
$15$	−1.10416	−0.285093
$16$	1.00000	0.250000
$17$	2.51668	0.610385	0.305193	−	0.952291i	$-0.401279\pi$
$17$	2.51668	0.610385	0.305193	+	0.952291i	$0.401279\pi$
$18$	2.86275	0.674756
$19$	−7.19922	−1.65161	−0.825807	−	0.563953i	$-0.809280\pi$
$19$	−7.19922	−1.65161	−0.825807	+	0.563953i	$0.809280\pi$
$20$	2.98038	0.666433
$21$	0.950379	0.207390
$22$	5.77573	1.23139
$23$	−5.17399	−1.07885	−0.539426	−	0.842033i	$-0.681359\pi$
$23$	−5.17399	−1.07885	−0.539426	+	0.842033i	$0.681359\pi$
$24$	0.370477	0.0756233
$25$	3.88265	0.776530
$26$	0.953732	0.187042
$27$	2.17201	0.418004
$28$	−2.56529	−0.484794
$29$	4.63236	0.860208	0.430104	−	0.902779i	$-0.358477\pi$
$29$	4.63236	0.860208	0.430104	+	0.902779i	$0.358477\pi$
$30$	1.10416	0.201591
$31$	8.16921	1.46723	0.733617	−	0.679563i	$-0.237831\pi$
$31$	8.16921	1.46723	0.733617	+	0.679563i	$0.237831\pi$
$32$	−1.00000	−0.176777
$33$	2.13977	0.372487
$34$	−2.51668	−0.431608
$35$	−7.64552	−1.29233
$36$	−2.86275	−0.477124
$37$	−9.43775	−1.55156	−0.775778	−	0.631006i	$-0.782642\pi$
$37$	−9.43775	−1.55156	−0.775778	+	0.631006i	$0.782642\pi$
$38$	7.19922	1.16787
$39$	0.353336	0.0565790
$40$	−2.98038	−0.471239
$41$	2.46810	0.385453	0.192726	−	0.981253i	$-0.438267\pi$
$41$	2.46810	0.385453	0.192726	+	0.981253i	$0.438267\pi$
$42$	−0.950379	−0.146647
$43$	5.86496	0.894398	0.447199	−	0.894435i	$-0.352422\pi$
$43$	5.86496	0.894398	0.447199	+	0.894435i	$0.352422\pi$
$44$	−5.77573	−0.870724
$45$	−8.53207	−1.27189
$46$	5.17399	0.762864
$47$	−9.50969	−1.38713	−0.693565	−	0.720394i	$-0.743961\pi$
$47$	−9.50969	−1.38713	−0.693565	+	0.720394i	$0.743961\pi$
$48$	−0.370477	−0.0534737
$49$	−0.419304	−0.0599006
$50$	−3.88265	−0.549089
$51$	−0.932373	−0.130558
$52$	−0.953732	−0.132259
$53$	−7.51226	−1.03189	−0.515944	−	0.856622i	$-0.672559\pi$
$53$	−7.51226	−1.03189	−0.515944	+	0.856622i	$0.672559\pi$
$54$	−2.17201	−0.295573
$55$	−17.2139	−2.32112
$56$	2.56529	0.342801
$57$	2.66714	0.353272
$58$	−4.63236	−0.608259
$59$	−1.01485	−0.132122	−0.0660610	−	0.997816i	$-0.521043\pi$
$59$	−1.01485	−0.132122	−0.0660610	+	0.997816i	$0.521043\pi$
$60$	−1.10416	−0.142547
$61$	−5.86716	−0.751212	−0.375606	−	0.926779i	$-0.622565\pi$
$61$	−5.86716	−0.751212	−0.375606	+	0.926779i	$0.622565\pi$
$62$	−8.16921	−1.03749
$63$	7.34377	0.925228
$64$	1.00000	0.125000
$65$	−2.84248	−0.352566
$66$	−2.13977	−0.263388
$67$	12.1288	1.48177	0.740883	−	0.671634i	$-0.234407\pi$
$67$	12.1288	1.48177	0.740883	+	0.671634i	$0.234407\pi$
$68$	2.51668	0.305193
$69$	1.91685	0.230761
$70$	7.64552	0.913815
$71$	−12.3537	−1.46611	−0.733057	−	0.680167i	$-0.761907\pi$
$71$	−12.3537	−1.46611	−0.733057	+	0.680167i	$0.761907\pi$
$72$	2.86275	0.337378
$73$	0.650878	0.0761795	0.0380897	−	0.999274i	$-0.487873\pi$
$73$	0.650878	0.0761795	0.0380897	+	0.999274i	$0.487873\pi$
$74$	9.43775	1.09712
$75$	−1.43843	−0.166096
$76$	−7.19922	−0.825807
$77$	14.8164	1.68849
$78$	−0.353336	−0.0400074
$79$	15.5158	1.74567	0.872834	−	0.488018i	$-0.162280\pi$
$79$	15.5158	1.74567	0.872834	+	0.488018i	$0.162280\pi$
$80$	2.98038	0.333216
$81$	7.78356	0.864840
$82$	−2.46810	−0.272556
$83$	−8.27171	−0.907939	−0.453969	−	0.891017i	$-0.649992\pi$
$83$	−8.27171	−0.907939	−0.453969	+	0.891017i	$0.649992\pi$
$84$	0.950379	0.103695
$85$	7.50067	0.813562
$86$	−5.86496	−0.632435
$87$	−1.71618	−0.183994
$88$	5.77573	0.615695
$89$	2.89852	0.307242	0.153621	−	0.988130i	$-0.450907\pi$
$89$	2.89852	0.307242	0.153621	+	0.988130i	$0.450907\pi$
$90$	8.53207	0.899359
$91$	2.44660	0.256473
$92$	−5.17399	−0.539426
$93$	−3.02650	−0.313834
$94$	9.50969	0.980850
$95$	−21.4564	−2.20138
$96$	0.370477	0.0378116
$97$	0.956247	0.0970922	0.0485461	−	0.998821i	$-0.484541\pi$
$97$	0.956247	0.0970922	0.0485461	+	0.998821i	$0.484541\pi$
$98$	0.419304	0.0423561
$99$	16.5345	1.66178
$100$	3.88265	0.388265
$101$	5.92549	0.589608	0.294804	−	0.955558i	$-0.404746\pi$
$101$	5.92549	0.589608	0.294804	+	0.955558i	$0.404746\pi$
$102$	0.932373	0.0923187
$103$	−7.57015	−0.745909	−0.372955	−	0.927850i	$-0.621655\pi$
$103$	−7.57015	−0.745909	−0.372955	+	0.927850i	$0.621655\pi$
$104$	0.953732	0.0935211
$105$	2.83249	0.276423
$106$	7.51226	0.729655
$107$	16.0455	1.55118	0.775589	−	0.631238i	$-0.217453\pi$
$107$	16.0455	1.55118	0.775589	+	0.631238i	$0.217453\pi$
$108$	2.17201	0.209002
$109$	2.50942	0.240359	0.120179	−	0.992752i	$-0.461653\pi$
$109$	2.50942	0.240359	0.120179	+	0.992752i	$0.461653\pi$
$110$	17.2139	1.64128
$111$	3.49647	0.331870
$112$	−2.56529	−0.242397
$113$	−13.4589	−1.26611	−0.633055	−	0.774107i	$-0.718199\pi$
$113$	−13.4589	−1.26611	−0.633055	+	0.774107i	$0.718199\pi$
$114$	−2.66714	−0.249801
$115$	−15.4205	−1.43796
$116$	4.63236	0.430104
$117$	2.73029	0.252416
$118$	1.01485	0.0934244
$119$	−6.45602	−0.591822
$120$	1.10416	0.100796
$121$	22.3591	2.03264
$122$	5.86716	0.531187
$123$	−0.914375	−0.0824464
$124$	8.16921	0.733617
$125$	−3.33013	−0.297856
$126$	−7.34377	−0.654235
$127$	2.35435	0.208915	0.104458	−	0.994529i	$-0.466689\pi$
$127$	2.35435	0.208915	0.104458	+	0.994529i	$0.466689\pi$
$128$	−1.00000	−0.0883883
$129$	−2.17283	−0.191307
$130$	2.84248	0.249302
$131$	8.14252	0.711415	0.355708	−	0.934597i	$-0.384240\pi$
$131$	8.14252	0.711415	0.355708	+	0.934597i	$0.384240\pi$
$132$	2.13977	0.186243
$133$	18.4681	1.60138
$134$	−12.1288	−1.04777
$135$	6.47342	0.557143
$136$	−2.51668	−0.215804
$137$	−13.5630	−1.15876	−0.579382	−	0.815056i	$-0.696706\pi$
$137$	−13.5630	−1.15876	−0.579382	+	0.815056i	$0.696706\pi$
$138$	−1.91685	−0.163173
$139$	−17.6145	−1.49405	−0.747023	−	0.664798i	$-0.768518\pi$
$139$	−17.6145	−1.49405	−0.747023	+	0.664798i	$0.768518\pi$
$140$	−7.64552	−0.646165
$141$	3.52312	0.296700
$142$	12.3537	1.03670
$143$	5.50850	0.460644
$144$	−2.86275	−0.238562
$145$	13.8062	1.14654
$146$	−0.650878	−0.0538670
$147$	0.155342	0.0128124
$148$	−9.43775	−0.775778
$149$	4.73354	0.387787	0.193893	−	0.981023i	$-0.437888\pi$
$149$	4.73354	0.387787	0.193893	+	0.981023i	$0.437888\pi$
$150$	1.43843	0.117447
$151$	18.5426	1.50897	0.754486	−	0.656316i	$-0.227886\pi$
$151$	18.5426	1.50897	0.754486	+	0.656316i	$0.227886\pi$
$152$	7.19922	0.583934
$153$	−7.20463	−0.582460
$154$	−14.8164	−1.19394
$155$	24.3473	1.95562
$156$	0.353336	0.0282895
$157$	6.62583	0.528799	0.264399	−	0.964413i	$-0.414826\pi$
$157$	6.62583	0.528799	0.264399	+	0.964413i	$0.414826\pi$
$158$	−15.5158	−1.23437
$159$	2.78312	0.220716
$160$	−2.98038	−0.235620
$161$	13.2728	1.04604
$162$	−7.78356	−0.611534
$163$	−7.35217	−0.575867	−0.287933	−	0.957650i	$-0.592968\pi$
$163$	−7.35217	−0.575867	−0.287933	+	0.957650i	$0.592968\pi$
$164$	2.46810	0.192726
$165$	6.37734	0.496475
$166$	8.27171	0.642010
$167$	−18.4854	−1.43045	−0.715223	−	0.698896i	$-0.753675\pi$
$167$	−18.4854	−1.43045	−0.715223	+	0.698896i	$0.753675\pi$
$168$	−0.950379	−0.0733234
$169$	−12.0904	−0.930030
$170$	−7.50067	−0.575275
$171$	20.6095	1.57605
$172$	5.86496	0.447199
$173$	−15.2035	−1.15590	−0.577951	−	0.816071i	$-0.696148\pi$
$173$	−15.2035	−1.15590	−0.577951	+	0.816071i	$0.696148\pi$
$174$	1.71618	0.130103
$175$	−9.96011	−0.752913
$176$	−5.77573	−0.435362
$177$	0.375978	0.0282602
$178$	−2.89852	−0.217253
$179$	−8.92937	−0.667412	−0.333706	−	0.942677i	$-0.608299\pi$
$179$	−8.92937	−0.667412	−0.333706	+	0.942677i	$0.608299\pi$
$180$	−8.53207	−0.635943
$181$	8.96265	0.666189	0.333094	−	0.942894i	$-0.391907\pi$
$181$	8.96265	0.666189	0.333094	+	0.942894i	$0.391907\pi$
$182$	−2.44660	−0.181354
$183$	2.17365	0.160680
$184$	5.17399	0.381432
$185$	−28.1280	−2.06802
$186$	3.02650	0.221914
$187$	−14.5357	−1.06295
$188$	−9.50969	−0.693565
$189$	−5.57183	−0.405291
$190$	21.4564	1.55661
$191$	−2.71179	−0.196218	−0.0981091	−	0.995176i	$-0.531279\pi$
$191$	−2.71179	−0.196218	−0.0981091	+	0.995176i	$0.531279\pi$
$192$	−0.370477	−0.0267369
$193$	−10.4616	−0.753042	−0.376521	−	0.926408i	$-0.622880\pi$
$193$	−10.4616	−0.753042	−0.376521	+	0.926408i	$0.622880\pi$
$194$	−0.956247	−0.0686545
$195$	1.05307	0.0754122
$196$	−0.419304	−0.0299503
$197$	−8.17270	−0.582280	−0.291140	−	0.956680i	$-0.594035\pi$
$197$	−8.17270	−0.582280	−0.291140	+	0.956680i	$0.594035\pi$
$198$	−16.5345	−1.17505
$199$	23.4576	1.66287	0.831433	−	0.555625i	$-0.187521\pi$
$199$	23.4576	1.66287	0.831433	+	0.555625i	$0.187521\pi$
$200$	−3.88265	−0.274545
$201$	−4.49343	−0.316942
$202$	−5.92549	−0.416916
$203$	−11.8833	−0.834047
$204$	−0.932373	−0.0652792
$205$	7.35587	0.513757
$206$	7.57015	0.527437
$207$	14.8118	1.02949
$208$	−0.953732	−0.0661294
$209$	41.5807	2.87620
$210$	−2.83249	−0.195460
$211$	23.7236	1.63320	0.816600	−	0.577203i	$-0.195856\pi$
$211$	23.7236	1.63320	0.816600	+	0.577203i	$0.195856\pi$
$212$	−7.51226	−0.515944
$213$	4.57676	0.313594
$214$	−16.0455	−1.09685
$215$	17.4798	1.19211
$216$	−2.17201	−0.147787
$217$	−20.9564	−1.42261
$218$	−2.50942	−0.169959
$219$	−0.241135	−0.0162944
$220$	−17.2139	−1.16056
$221$	−2.40024	−0.161458
$222$	−3.49647	−0.234667
$223$	−4.63802	−0.310585	−0.155292	−	0.987869i	$-0.549632\pi$
$223$	−4.63802	−0.310585	−0.155292	+	0.987869i	$0.549632\pi$
$224$	2.56529	0.171400
$225$	−11.1150	−0.741003
$226$	13.4589	0.895274
$227$	2.30295	0.152852	0.0764261	−	0.997075i	$-0.475649\pi$
$227$	2.30295	0.152852	0.0764261	+	0.997075i	$0.475649\pi$
$228$	2.66714	0.176636
$229$	1.98890	0.131430	0.0657150	−	0.997838i	$-0.479067\pi$
$229$	1.98890	0.131430	0.0657150	+	0.997838i	$0.479067\pi$
$230$	15.4205	1.01679
$231$	−5.48913	−0.361159
$232$	−4.63236	−0.304129
$233$	5.58928	0.366166	0.183083	−	0.983097i	$-0.441392\pi$
$233$	5.58928	0.366166	0.183083	+	0.983097i	$0.441392\pi$
$234$	−2.73029	−0.178485
$235$	−28.3425	−1.84886
$236$	−1.01485	−0.0660610
$237$	−5.74826	−0.373389
$238$	6.45602	0.418481
$239$	−3.34553	−0.216404	−0.108202	−	0.994129i	$-0.534509\pi$
$239$	−3.34553	−0.216404	−0.108202	+	0.994129i	$0.534509\pi$
$240$	−1.10416	−0.0712733
$241$	26.7139	1.72079	0.860395	−	0.509627i	$-0.170217\pi$
$241$	26.7139	1.72079	0.860395	+	0.509627i	$0.170217\pi$
$242$	−22.3591	−1.43729
$243$	−9.39967	−0.602989
$244$	−5.86716	−0.375606
$245$	−1.24968	−0.0798394
$246$	0.914375	0.0582984
$247$	6.86613	0.436881
$248$	−8.16921	−0.518745
$249$	3.06448	0.194203
$250$	3.33013	0.210616
$251$	−29.3076	−1.84988	−0.924940	−	0.380112i	$-0.875885\pi$
$251$	−29.3076	−1.84988	−0.924940	+	0.380112i	$0.875885\pi$
$252$	7.34377	0.462614
$253$	29.8836	1.87877
$254$	−2.35435	−0.147725
$255$	−2.77882	−0.174017
$256$	1.00000	0.0625000
$257$	10.3803	0.647504	0.323752	−	0.946142i	$-0.395056\pi$
$257$	10.3803	0.647504	0.323752	+	0.946142i	$0.395056\pi$
$258$	2.17283	0.135275
$259$	24.2105	1.50437
$260$	−2.84248	−0.176283
$261$	−13.2613	−0.820853
$262$	−8.14252	−0.503046
$263$	−8.83916	−0.545046	−0.272523	−	0.962149i	$-0.587858\pi$
$263$	−8.83916	−0.545046	−0.272523	+	0.962149i	$0.587858\pi$
$264$	−2.13977	−0.131694
$265$	−22.3894	−1.37537
$266$	−18.4681	−1.13235
$267$	−1.07383	−0.0657175
$268$	12.1288	0.740883
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	−6.47342	−0.393960
$271$	7.39604	0.449277	0.224639	−	0.974442i	$-0.427880\pi$
$271$	7.39604	0.449277	0.224639	+	0.974442i	$0.427880\pi$
$272$	2.51668	0.152596
$273$	−0.906407	−0.0548583
$274$	13.5630	0.819370
$275$	−22.4251	−1.35229
$276$	1.91685	0.115381
$277$	13.5773	0.815779	0.407890	−	0.913031i	$-0.366265\pi$
$277$	13.5773	0.815779	0.407890	+	0.913031i	$0.366265\pi$
$278$	17.6145	1.05645
$279$	−23.3864	−1.40011
$280$	7.64552	0.456907
$281$	13.5723	0.809653	0.404826	−	0.914394i	$-0.367332\pi$
$281$	13.5723	0.809653	0.404826	+	0.914394i	$0.367332\pi$
$282$	−3.52312	−0.209799
$283$	20.5026	1.21875	0.609377	−	0.792880i	$-0.291419\pi$
$283$	20.5026	1.21875	0.609377	+	0.792880i	$0.291419\pi$
$284$	−12.3537	−0.733057
$285$	7.94909	0.470864
$286$	−5.50850	−0.325724
$287$	−6.33139	−0.373730
$288$	2.86275	0.168689
$289$	−10.6663	−0.627430
$290$	−13.8062	−0.810727
$291$	−0.354267	−0.0207675
$292$	0.650878	0.0380897
$293$	−26.0133	−1.51971	−0.759857	−	0.650090i	$-0.774731\pi$
$293$	−26.0133	−1.51971	−0.759857	+	0.650090i	$0.774731\pi$
$294$	−0.155342	−0.00905976
$295$	−3.02463	−0.176101
$296$	9.43775	0.548558
$297$	−12.5450	−0.727932
$298$	−4.73354	−0.274207
$299$	4.93460	0.285376
$300$	−1.43843	−0.0830479
$301$	−15.0453	−0.867197
$302$	−18.5426	−1.06700
$303$	−2.19526	−0.126114
$304$	−7.19922	−0.412903
$305$	−17.4863	−1.00126
$306$	7.20463	0.411861
$307$	−17.3993	−0.993031	−0.496516	−	0.868028i	$-0.665387\pi$
$307$	−17.3993	−0.993031	−0.496516	+	0.868028i	$0.665387\pi$
$308$	14.8164	0.844243
$309$	2.80457	0.159546
$310$	−24.3473	−1.38284
$311$	−13.4740	−0.764038	−0.382019	−	0.924155i	$-0.624771\pi$
$311$	−13.4740	−0.764038	−0.382019	+	0.924155i	$0.624771\pi$
$312$	−0.353336	−0.0200037
$313$	−9.89195	−0.559126	−0.279563	−	0.960127i	$-0.590190\pi$
$313$	−9.89195	−0.559126	−0.279563	+	0.960127i	$0.590190\pi$
$314$	−6.62583	−0.373917
$315$	21.8872	1.23320
$316$	15.5158	0.872834
$317$	19.2200	1.07950	0.539751	−	0.841825i	$-0.318518\pi$
$317$	19.2200	1.07950	0.539751	+	0.841825i	$0.318518\pi$
$318$	−2.78312	−0.156070
$319$	−26.7553	−1.49801
$320$	2.98038	0.166608
$321$	−5.94449	−0.331789
$322$	−13.2728	−0.739663
$323$	−18.1182	−1.00812
$324$	7.78356	0.432420
$325$	−3.70301	−0.205406
$326$	7.35217	0.407199
$327$	−0.929682	−0.0514115
$328$	−2.46810	−0.136278
$329$	24.3951	1.34494
$330$	−6.37734	−0.351061
$331$	−10.6168	−0.583550	−0.291775	−	0.956487i	$-0.594246\pi$
$331$	−10.6168	−0.583550	−0.291775	+	0.956487i	$0.594246\pi$
$332$	−8.27171	−0.453969
$333$	27.0179	1.48057
$334$	18.4854	1.01148
$335$	36.1483	1.97499
$336$	0.950379	0.0518474
$337$	−4.73621	−0.257998	−0.128999	−	0.991645i	$-0.541176\pi$
$337$	−4.73621	−0.257998	−0.128999	+	0.991645i	$0.541176\pi$
$338$	12.0904	0.657631
$339$	4.98622	0.270814
$340$	7.50067	0.406781
$341$	−47.1832	−2.55511
$342$	−20.6095	−1.11444
$343$	19.0326	1.02767
$344$	−5.86496	−0.316217
$345$	5.71292	0.307573
$346$	15.2035	0.817346
$347$	−9.70333	−0.520902	−0.260451	−	0.965487i	$-0.583871\pi$
$347$	−9.70333	−0.520902	−0.260451	+	0.965487i	$0.583871\pi$
$348$	−1.71618	−0.0919971
$349$	−34.9874	−1.87283	−0.936416	−	0.350893i	$-0.885878\pi$
$349$	−34.9874	−1.87283	−0.936416	+	0.350893i	$0.885878\pi$
$350$	9.96011	0.532390
$351$	−2.07152	−0.110569
$352$	5.77573	0.307847
$353$	17.9950	0.957777	0.478888	−	0.877876i	$-0.341040\pi$
$353$	17.9950	0.957777	0.478888	+	0.877876i	$0.341040\pi$
$354$	−0.375978	−0.0199830
$355$	−36.8187	−1.95413
$356$	2.89852	0.153621
$357$	2.39180	0.126588
$358$	8.92937	0.471932
$359$	−8.64349	−0.456186	−0.228093	−	0.973639i	$-0.573249\pi$
$359$	−8.64349	−0.456186	−0.228093	+	0.973639i	$0.573249\pi$
$360$	8.53207	0.449679
$361$	32.8287	1.72783
$362$	−8.96265	−0.471067
$363$	−8.28351	−0.434772
$364$	2.44660	0.128236
$365$	1.93986	0.101537
$366$	−2.17365	−0.113618
$367$	−25.4599	−1.32899	−0.664497	−	0.747291i	$-0.731354\pi$
$367$	−25.4599	−1.32899	−0.664497	+	0.747291i	$0.731354\pi$
$368$	−5.17399	−0.269713
$369$	−7.06555	−0.367818
$370$	28.1280	1.46231
$371$	19.2711	1.00051
$372$	−3.02650	−0.156917
$373$	−5.63673	−0.291859	−0.145929	−	0.989295i	$-0.546617\pi$
$373$	−5.63673	−0.291859	−0.145929	+	0.989295i	$0.546617\pi$
$374$	14.5357	0.751622
$375$	1.23374	0.0637098
$376$	9.50969	0.490425
$377$	−4.41803	−0.227540
$378$	5.57183	0.286584
$379$	−16.2741	−0.835944	−0.417972	−	0.908460i	$-0.637259\pi$
$379$	−16.2741	−0.835944	−0.417972	+	0.908460i	$0.637259\pi$
$380$	−21.4564	−1.10069
$381$	−0.872233	−0.0446859
$382$	2.71179	0.138747
$383$	17.3292	0.885483	0.442742	−	0.896649i	$-0.354006\pi$
$383$	17.3292	0.885483	0.442742	+	0.896649i	$0.354006\pi$
$384$	0.370477	0.0189058
$385$	44.1585	2.25052
$386$	10.4616	0.532481
$387$	−16.7899	−0.853478
$388$	0.956247	0.0485461
$389$	32.7406	1.66002	0.830008	−	0.557752i	$-0.188336\pi$
$389$	32.7406	1.66002	0.830008	+	0.557752i	$0.188336\pi$
$390$	−1.05307	−0.0533245
$391$	−13.0213	−0.658516
$392$	0.419304	0.0211781
$393$	−3.01661	−0.152168
$394$	8.17270	0.411734
$395$	46.2430	2.32674
$396$	16.5345	0.830888
$397$	−10.1894	−0.511392	−0.255696	−	0.966757i	$-0.582305\pi$
$397$	−10.1894	−0.511392	−0.255696	+	0.966757i	$0.582305\pi$
$398$	−23.4576	−1.17582
$399$	−6.84199	−0.342528
$400$	3.88265	0.194132
$401$	27.6214	1.37935	0.689673	−	0.724121i	$-0.257754\pi$
$401$	27.6214	1.37935	0.689673	+	0.724121i	$0.257754\pi$
$402$	4.49343	0.224112
$403$	−7.79124	−0.388109
$404$	5.92549	0.294804
$405$	23.1979	1.15272
$406$	11.8833	0.589760
$407$	54.5099	2.70195
$408$	0.932373	0.0461593
$409$	−32.3617	−1.60018	−0.800092	−	0.599877i	$-0.795216\pi$
$409$	−32.3617	−1.60018	−0.800092	+	0.599877i	$0.795216\pi$
$410$	−7.35587	−0.363281
$411$	5.02477	0.247854
$412$	−7.57015	−0.372955
$413$	2.60338	0.128104
$414$	−14.8118	−0.727962
$415$	−24.6528	−1.21016
$416$	0.953732	0.0467606
$417$	6.52578	0.319569
$418$	−41.5807	−2.03378
$419$	−20.7391	−1.01317	−0.506586	−	0.862189i	$-0.669093\pi$
$419$	−20.7391	−1.01317	−0.506586	+	0.862189i	$0.669093\pi$
$420$	2.83249	0.138211
$421$	−2.74188	−0.133631	−0.0668155	−	0.997765i	$-0.521284\pi$
$421$	−2.74188	−0.133631	−0.0668155	+	0.997765i	$0.521284\pi$
$422$	−23.7236	−1.15485
$423$	27.2238	1.32367
$424$	7.51226	0.364828
$425$	9.77140	0.473983
$426$	−4.57676	−0.221745
$427$	15.0509	0.728366
$428$	16.0455	0.775589
$429$	−2.04077	−0.0985294
$430$	−17.4798	−0.842950
$431$	−1.93294	−0.0931065	−0.0465533	−	0.998916i	$-0.514824\pi$
$431$	−1.93294	−0.0931065	−0.0465533	+	0.998916i	$0.514824\pi$
$432$	2.17201	0.104501
$433$	4.35794	0.209429	0.104715	−	0.994502i	$-0.466607\pi$
$433$	4.35794	0.209429	0.104715	+	0.994502i	$0.466607\pi$
$434$	20.9564	1.00594
$435$	−5.11487	−0.245239
$436$	2.50942	0.120179
$437$	37.2487	1.78185
$438$	0.241135	0.0115219
$439$	8.19465	0.391109	0.195555	−	0.980693i	$-0.437349\pi$
$439$	8.19465	0.391109	0.195555	+	0.980693i	$0.437349\pi$
$440$	17.2139	0.820638
$441$	1.20036	0.0571601
$442$	2.40024	0.114168
$443$	−5.51425	−0.261990	−0.130995	−	0.991383i	$-0.541817\pi$
$443$	−5.51425	−0.261990	−0.130995	+	0.991383i	$0.541817\pi$
$444$	3.49647	0.165935
$445$	8.63867	0.409512
$446$	4.63802	0.219616
$447$	−1.75367	−0.0829456
$448$	−2.56529	−0.121198
$449$	−19.1545	−0.903956	−0.451978	−	0.892029i	$-0.649281\pi$
$449$	−19.1545	−0.903956	−0.451978	+	0.892029i	$0.649281\pi$
$450$	11.1150	0.523968
$451$	−14.2551	−0.671246
$452$	−13.4589	−0.633055
$453$	−6.86959	−0.322762
$454$	−2.30295	−0.108083
$455$	7.29178	0.341844
$456$	−2.66714	−0.124900
$457$	−41.6610	−1.94882	−0.974409	−	0.224783i	$-0.927833\pi$
$457$	−41.6610	−1.94882	−0.974409	+	0.224783i	$0.927833\pi$
$458$	−1.98890	−0.0929350
$459$	5.46627	0.255144
$460$	−15.4205	−0.718982
$461$	28.7942	1.34108	0.670539	−	0.741874i	$-0.266063\pi$
$461$	28.7942	1.34108	0.670539	+	0.741874i	$0.266063\pi$
$462$	5.48913	0.255378
$463$	1.89638	0.0881321	0.0440660	−	0.999029i	$-0.485969\pi$
$463$	1.89638	0.0881321	0.0440660	+	0.999029i	$0.485969\pi$
$464$	4.63236	0.215052
$465$	−9.02012	−0.418298
$466$	−5.58928	−0.258918
$467$	−9.53499	−0.441227	−0.220613	−	0.975361i	$-0.570806\pi$
$467$	−9.53499	−0.441227	−0.220613	+	0.975361i	$0.570806\pi$
$468$	2.73029	0.126208
$469$	−31.1138	−1.43670
$470$	28.3425	1.30734
$471$	−2.45472	−0.113107
$472$	1.01485	0.0467122
$473$	−33.8744	−1.55755
$474$	5.74826	0.264026
$475$	−27.9520	−1.28253
$476$	−6.45602	−0.295911
$477$	21.5057	0.984678
$478$	3.34553	0.153021
$479$	11.2845	0.515602	0.257801	−	0.966198i	$-0.417002\pi$
$479$	11.2845	0.515602	0.257801	+	0.966198i	$0.417002\pi$
$480$	1.10416	0.0503978
$481$	9.00108	0.410414
$482$	−26.7139	−1.21678
$483$	−4.91726	−0.223743
$484$	22.3591	1.01632
$485$	2.84998	0.129411
$486$	9.39967	0.426377
$487$	13.3515	0.605014	0.302507	−	0.953147i	$-0.402176\pi$
$487$	13.3515	0.605014	0.302507	+	0.953147i	$0.402176\pi$
$488$	5.86716	0.265594
$489$	2.72381	0.123175
$490$	1.24968	0.0564550
$491$	37.9516	1.71273	0.856365	−	0.516370i	$-0.172717\pi$
$491$	37.9516	1.71273	0.856365	+	0.516370i	$0.172717\pi$
$492$	−0.914375	−0.0412232
$493$	11.6582	0.525058
$494$	−6.86613	−0.308922
$495$	49.2789	2.21492
$496$	8.16921	0.366808
$497$	31.6908	1.42153
$498$	−3.06448	−0.137323
$499$	−27.6365	−1.23718	−0.618590	−	0.785714i	$-0.712296\pi$
$499$	−27.6365	−1.23718	−0.618590	+	0.785714i	$0.712296\pi$
$500$	−3.33013	−0.148928
$501$	6.84842	0.305965
$502$	29.3076	1.30806
$503$	20.9442	0.933855	0.466927	−	0.884296i	$-0.345361\pi$
$503$	20.9442	0.933855	0.466927	+	0.884296i	$0.345361\pi$
$504$	−7.34377	−0.327117
$505$	17.6602	0.785868
$506$	−29.8836	−1.32849
$507$	4.47921	0.198929
$508$	2.35435	0.104458
$509$	−25.5377	−1.13194	−0.565969	−	0.824426i	$-0.691498\pi$
$509$	−25.5377	−1.13194	−0.565969	+	0.824426i	$0.691498\pi$
$510$	2.77882	0.123048
$511$	−1.66969	−0.0738627
$512$	−1.00000	−0.0441942
$513$	−15.6368	−0.690381
$514$	−10.3803	−0.457854
$515$	−22.5619	−0.994197
$516$	−2.17283	−0.0956536
$517$	54.9254	2.41562
$518$	−24.2105	−1.06375
$519$	5.63255	0.247242
$520$	2.84248	0.124651
$521$	−38.8849	−1.70358	−0.851788	−	0.523886i	$-0.824482\pi$
$521$	−38.8849	−1.70358	−0.851788	+	0.523886i	$0.824482\pi$
$522$	13.2613	0.580430
$523$	−25.0299	−1.09448	−0.547240	−	0.836976i	$-0.684321\pi$
$523$	−25.0299	−1.09448	−0.547240	+	0.836976i	$0.684321\pi$
$524$	8.14252	0.355708
$525$	3.68999	0.161044
$526$	8.83916	0.385406
$527$	20.5593	0.895578
$528$	2.13977	0.0931217
$529$	3.77022	0.163922
$530$	22.3894	0.972532
$531$	2.90525	0.126077
$532$	18.4681	0.800692
$533$	−2.35391	−0.101959
$534$	1.07383	0.0464693
$535$	47.8217	2.06751
$536$	−12.1288	−0.523883
$537$	3.30812	0.142756
$538$	1.00000	0.0431131
$539$	2.42179	0.104314
$540$	6.47342	0.278571
$541$	0.997803	0.0428989	0.0214494	−	0.999770i	$-0.493172\pi$
$541$	0.997803	0.0428989	0.0214494	+	0.999770i	$0.493172\pi$
$542$	−7.39604	−0.317687
$543$	−3.32046	−0.142494
$544$	−2.51668	−0.107902
$545$	7.47902	0.320366
$546$	0.906407	0.0387907
$547$	−15.8011	−0.675608	−0.337804	−	0.941217i	$-0.609684\pi$
$547$	−15.8011	−0.675608	−0.337804	+	0.941217i	$0.609684\pi$
$548$	−13.5630	−0.579382
$549$	16.7962	0.716843
$550$	22.4251	0.956211
$551$	−33.3494	−1.42073
$552$	−1.91685	−0.0815863
$553$	−39.8026	−1.69258
$554$	−13.5773	−0.576843
$555$	10.4208	0.442338
$556$	−17.6145	−0.747023
$557$	−36.3525	−1.54031	−0.770153	−	0.637860i	$-0.779820\pi$
$557$	−36.3525	−1.54031	−0.770153	+	0.637860i	$0.779820\pi$
$558$	23.3864	0.990025
$559$	−5.59360	−0.236584
$560$	−7.64552	−0.323082
$561$	5.38514	0.227361
$562$	−13.5723	−0.572511
$563$	−20.1622	−0.849736	−0.424868	−	0.905255i	$-0.639680\pi$
$563$	−20.1622	−0.849736	−0.424868	+	0.905255i	$0.639680\pi$
$564$	3.52312	0.148350
$565$	−40.1127	−1.68755
$566$	−20.5026	−0.861790
$567$	−19.9671	−0.838538
$568$	12.3537	0.518349
$569$	−16.0424	−0.672530	−0.336265	−	0.941767i	$-0.609164\pi$
$569$	−16.0424	−0.672530	−0.336265	+	0.941767i	$0.609164\pi$
$570$	−7.94909	−0.332951
$571$	−32.2187	−1.34831	−0.674156	−	0.738589i	$-0.735493\pi$
$571$	−32.2187	−1.34831	−0.674156	+	0.738589i	$0.735493\pi$
$572$	5.50850	0.230322
$573$	1.00466	0.0419701
$574$	6.33139	0.264267
$575$	−20.0888	−0.837761
$576$	−2.86275	−0.119281
$577$	42.6582	1.77589	0.887943	−	0.459954i	$-0.152134\pi$
$577$	42.6582	1.77589	0.887943	+	0.459954i	$0.152134\pi$
$578$	10.6663	0.443660
$579$	3.87578	0.161072
$580$	13.8062	0.573271
$581$	21.2193	0.880326
$582$	0.354267	0.0146849
$583$	43.3888	1.79698
$584$	−0.650878	−0.0269335
$585$	8.13731	0.336436
$586$	26.0133	1.07460
$587$	−13.8741	−0.572646	−0.286323	−	0.958133i	$-0.592433\pi$
$587$	−13.8741	−0.572646	−0.286323	+	0.958133i	$0.592433\pi$
$588$	0.155342	0.00640622
$589$	−58.8119	−2.42330
$590$	3.02463	0.124522
$591$	3.02779	0.124547
$592$	−9.43775	−0.387889
$593$	33.6435	1.38157	0.690786	−	0.723059i	$-0.257265\pi$
$593$	33.6435	1.38157	0.690786	+	0.723059i	$0.257265\pi$
$594$	12.5450	0.514726
$595$	−19.2414	−0.788819
$596$	4.73354	0.193893
$597$	−8.69051	−0.355679
$598$	−4.93460	−0.201791
$599$	12.8677	0.525761	0.262881	−	0.964828i	$-0.415327\pi$
$599$	12.8677	0.525761	0.262881	+	0.964828i	$0.415327\pi$
$600$	1.43843	0.0587237
$601$	17.2785	0.704804	0.352402	−	0.935849i	$-0.385365\pi$
$601$	17.2785	0.704804	0.352402	+	0.935849i	$0.385365\pi$
$602$	15.0453	0.613201
$603$	−34.7216	−1.41397
$604$	18.5426	0.754486
$605$	66.6384	2.70924
$606$	2.19526	0.0891762
$607$	−36.7366	−1.49109	−0.745547	−	0.666453i	$-0.767812\pi$
$607$	−36.7366	−1.49109	−0.745547	+	0.666453i	$0.767812\pi$
$608$	7.19922	0.291967
$609$	4.40250	0.178398
$610$	17.4863	0.708001
$611$	9.06970	0.366921
$612$	−7.20463	−0.291230
$613$	−35.5029	−1.43395	−0.716975	−	0.697099i	$-0.754474\pi$
$613$	−35.5029	−1.43395	−0.716975	+	0.697099i	$0.754474\pi$
$614$	17.3993	0.702179
$615$	−2.72518	−0.109890
$616$	−14.8164	−0.596970
$617$	5.97403	0.240505	0.120253	−	0.992743i	$-0.461630\pi$
$617$	5.97403	0.240505	0.120253	+	0.992743i	$0.461630\pi$
$618$	−2.80457	−0.112816
$619$	1.54631	0.0621513	0.0310757	−	0.999517i	$-0.490107\pi$
$619$	1.54631	0.0621513	0.0310757	+	0.999517i	$0.490107\pi$
$620$	24.3473	0.977812
$621$	−11.2380	−0.450965
$622$	13.4740	0.540256
$623$	−7.43552	−0.297898
$624$	0.353336	0.0141447
$625$	−29.3383	−1.17353
$626$	9.89195	0.395362
$627$	−15.4047	−0.615205
$628$	6.62583	0.264399
$629$	−23.7518	−0.947047
$630$	−21.8872	−0.872007
$631$	33.5953	1.33741	0.668705	−	0.743528i	$-0.266849\pi$
$631$	33.5953	1.33741	0.668705	+	0.743528i	$0.266849\pi$
$632$	−15.5158	−0.617187
$633$	−8.78905	−0.349333
$634$	−19.2200	−0.763324
$635$	7.01686	0.278456
$636$	2.78312	0.110358
$637$	0.399904	0.0158448
$638$	26.7553	1.05925
$639$	35.3655	1.39904
$640$	−2.98038	−0.117810
$641$	22.6820	0.895884	0.447942	−	0.894063i	$-0.352157\pi$
$641$	22.6820	0.895884	0.447942	+	0.894063i	$0.352157\pi$
$642$	5.94449	0.234610
$643$	35.9742	1.41868	0.709341	−	0.704865i	$-0.248993\pi$
$643$	35.9742	1.41868	0.709341	+	0.704865i	$0.248993\pi$
$644$	13.2728	0.523021
$645$	−6.47586	−0.254987
$646$	18.1182	0.712849
$647$	−29.8868	−1.17497	−0.587485	−	0.809235i	$-0.699882\pi$
$647$	−29.8868	−1.17497	−0.587485	+	0.809235i	$0.699882\pi$
$648$	−7.78356	−0.305767
$649$	5.86149	0.230084
$650$	3.70301	0.145244
$651$	7.76385	0.304289
$652$	−7.35217	−0.287933
$653$	−48.1187	−1.88303	−0.941516	−	0.336968i	$-0.890598\pi$
$653$	−48.1187	−1.88303	−0.941516	+	0.336968i	$0.890598\pi$
$654$	0.929682	0.0363534
$655$	24.2678	0.948220
$656$	2.46810	0.0963632
$657$	−1.86330	−0.0726942
$658$	−24.3951	−0.951019
$659$	−27.3372	−1.06491	−0.532454	−	0.846459i	$-0.678730\pi$
$659$	−27.3372	−1.06491	−0.532454	+	0.846459i	$0.678730\pi$
$660$	6.37734	0.248237
$661$	−20.6868	−0.804623	−0.402311	−	0.915503i	$-0.631793\pi$
$661$	−20.6868	−0.804623	−0.402311	+	0.915503i	$0.631793\pi$
$662$	10.6168	0.412632
$663$	0.889234	0.0345350
$664$	8.27171	0.321005
$665$	55.0418	2.13443
$666$	−27.0179	−1.04692
$667$	−23.9678	−0.928037
$668$	−18.4854	−0.715223
$669$	1.71828	0.0664325
$670$	−36.1483	−1.39653
$671$	33.8871	1.30820
$672$	−0.950379	−0.0366617
$673$	−3.68048	−0.141872	−0.0709360	−	0.997481i	$-0.522599\pi$
$673$	−3.68048	−0.141872	−0.0709360	+	0.997481i	$0.522599\pi$
$674$	4.73621	0.182432
$675$	8.43316	0.324593
$676$	−12.0904	−0.465015
$677$	−17.3358	−0.666268	−0.333134	−	0.942880i	$-0.608106\pi$
$677$	−17.3358	−0.666268	−0.333134	+	0.942880i	$0.608106\pi$
$678$	−4.98622	−0.191495
$679$	−2.45305	−0.0941393
$680$	−7.50067	−0.287637
$681$	−0.853190	−0.0326943
$682$	47.1832	1.80674
$683$	2.26489	0.0866636	0.0433318	−	0.999061i	$-0.486203\pi$
$683$	2.26489	0.0866636	0.0433318	+	0.999061i	$0.486203\pi$
$684$	20.6095	0.788025
$685$	−40.4228	−1.54448
$686$	−19.0326	−0.726670
$687$	−0.736840	−0.0281122
$688$	5.86496	0.223599
$689$	7.16468	0.272953
$690$	−5.71292	−0.217487
$691$	−34.9262	−1.32865	−0.664327	−	0.747442i	$-0.731282\pi$
$691$	−34.9262	−1.32865	−0.664327	+	0.747442i	$0.731282\pi$
$692$	−15.2035	−0.577951
$693$	−42.4156	−1.61124
$694$	9.70333	0.368333
$695$	−52.4980	−1.99136
$696$	1.71618	0.0650517
$697$	6.21143	0.235275
$698$	34.9874	1.32429
$699$	−2.07070	−0.0783210
$700$	−9.96011	−0.376457
$701$	−8.67601	−0.327688	−0.163844	−	0.986486i	$-0.552389\pi$
$701$	−8.67601	−0.327688	−0.163844	+	0.986486i	$0.552389\pi$
$702$	2.07152	0.0781844
$703$	67.9444	2.56257
$704$	−5.77573	−0.217681
$705$	10.5002	0.395461
$706$	−17.9950	−0.677251
$707$	−15.2006	−0.571676
$708$	0.375978	0.0141301
$709$	−52.4850	−1.97111	−0.985557	−	0.169342i	$-0.945836\pi$
$709$	−52.4850	−1.97111	−0.985557	+	0.169342i	$0.945836\pi$
$710$	36.8187	1.38178
$711$	−44.4179	−1.66580
$712$	−2.89852	−0.108626
$713$	−42.2675	−1.58293
$714$	−2.39180	−0.0895110
$715$	16.4174	0.613976
$716$	−8.92937	−0.333706
$717$	1.23944	0.0462878
$718$	8.64349	0.322572
$719$	12.2225	0.455822	0.227911	−	0.973682i	$-0.426810\pi$
$719$	12.2225	0.455822	0.227911	+	0.973682i	$0.426810\pi$
$720$	−8.53207	−0.317971
$721$	19.4196	0.723224
$722$	−32.8287	−1.22176
$723$	−9.89686	−0.368068
$724$	8.96265	0.333094
$725$	17.9858	0.667977
$726$	8.28351	0.307430
$727$	32.8117	1.21692	0.608460	−	0.793585i	$-0.291788\pi$
$727$	32.8117	1.21692	0.608460	+	0.793585i	$0.291788\pi$
$728$	−2.44660	−0.0906769
$729$	−19.8683	−0.735864
$730$	−1.93986	−0.0717975
$731$	14.7602	0.545927
$732$	2.17365	0.0803402
$733$	−11.8567	−0.437937	−0.218968	−	0.975732i	$-0.570269\pi$
$733$	−11.8567	−0.437937	−0.218968	+	0.975732i	$0.570269\pi$
$734$	25.4599	0.939741
$735$	0.462979	0.0170772
$736$	5.17399	0.190716
$737$	−70.0525	−2.58042
$738$	7.06555	0.260087
$739$	23.7096	0.872173	0.436086	−	0.899905i	$-0.356364\pi$
$739$	23.7096	0.872173	0.436086	+	0.899905i	$0.356364\pi$
$740$	−28.1280	−1.03401
$741$	−2.54374	−0.0934466
$742$	−19.2711	−0.707464
$743$	16.3166	0.598600	0.299300	−	0.954159i	$-0.403247\pi$
$743$	16.3166	0.598600	0.299300	+	0.954159i	$0.403247\pi$
$744$	3.02650	0.110957
$745$	14.1077	0.516867
$746$	5.63673	0.206375
$747$	23.6798	0.866399
$748$	−14.5357	−0.531477
$749$	−41.1613	−1.50400
$750$	−1.23374	−0.0450496
$751$	30.5188	1.11365	0.556824	−	0.830630i	$-0.312020\pi$
$751$	30.5188	1.11365	0.556824	+	0.830630i	$0.312020\pi$
$752$	−9.50969	−0.346783
$753$	10.8578	0.395680
$754$	4.41803	0.160895
$755$	55.2639	2.01126
$756$	−5.57183	−0.202646
$757$	21.0756	0.766007	0.383003	−	0.923747i	$-0.374890\pi$
$757$	21.0756	0.766007	0.383003	+	0.923747i	$0.374890\pi$
$758$	16.2741	0.591101
$759$	−11.0712	−0.401858
$760$	21.4564	0.778305
$761$	−9.37156	−0.339719	−0.169859	−	0.985468i	$-0.554331\pi$
$761$	−9.37156	−0.339719	−0.169859	+	0.985468i	$0.554331\pi$
$762$	0.872233	0.0315977
$763$	−6.43738	−0.233049
$764$	−2.71179	−0.0981091
$765$	−21.4725	−0.776340
$766$	−17.3292	−0.626131
$767$	0.967894	0.0349486
$768$	−0.370477	−0.0133684
$769$	26.9367	0.971363	0.485681	−	0.874136i	$-0.338572\pi$
$769$	26.9367	0.971363	0.485681	+	0.874136i	$0.338572\pi$
$770$	−44.1585	−1.59136
$771$	−3.84565	−0.138498
$772$	−10.4616	−0.376521
$773$	−14.7111	−0.529121	−0.264560	−	0.964369i	$-0.585227\pi$
$773$	−14.7111	−0.529121	−0.264560	+	0.964369i	$0.585227\pi$
$774$	16.7899	0.603500
$775$	31.7182	1.13935
$776$	−0.956247	−0.0343273
$777$	−8.96944	−0.321777
$778$	−32.7406	−1.17381
$779$	−17.7684	−0.636619
$780$	1.05307	0.0377061
$781$	71.3516	2.55316
$782$	13.0213	0.465641
$783$	10.0615	0.359570
$784$	−0.419304	−0.0149751
$785$	19.7475	0.704818
$786$	3.01661	0.107599
$787$	3.73905	0.133283	0.0666414	−	0.997777i	$-0.478772\pi$
$787$	3.73905	0.133283	0.0666414	+	0.997777i	$0.478772\pi$
$788$	−8.17270	−0.291140
$789$	3.27471	0.116583
$790$	−46.2430	−1.64525
$791$	34.5260	1.22760
$792$	−16.5345	−0.587526
$793$	5.59569	0.198709
$794$	10.1894	0.361609
$795$	8.29474	0.294184
$796$	23.4576	0.831433
$797$	7.11196	0.251919	0.125959	−	0.992035i	$-0.459799\pi$
$797$	7.11196	0.251919	0.125959	+	0.992035i	$0.459799\pi$
$798$	6.84199	0.242204
$799$	−23.9329	−0.846685
$800$	−3.88265	−0.137272
$801$	−8.29772	−0.293185
$802$	−27.6214	−0.975345
$803$	−3.75929	−0.132663
$804$	−4.49343	−0.158471
$805$	39.5579	1.39423
$806$	7.79124	0.274435
$807$	0.370477	0.0130414
$808$	−5.92549	−0.208458
$809$	−10.2508	−0.360398	−0.180199	−	0.983630i	$-0.557674\pi$
$809$	−10.2508	−0.360398	−0.180199	+	0.983630i	$0.557674\pi$
$810$	−23.1979	−0.815093
$811$	−17.6750	−0.620655	−0.310327	−	0.950630i	$-0.600439\pi$
$811$	−17.6750	−0.620655	−0.310327	+	0.950630i	$0.600439\pi$
$812$	−11.8833	−0.417023
$813$	−2.74006	−0.0960981
$814$	−54.5099	−1.91057
$815$	−21.9123	−0.767553
$816$	−0.932373	−0.0326396
$817$	−42.2231	−1.47720
$818$	32.3617	1.13150
$819$	−7.00399	−0.244739
$820$	7.35587	0.256878
$821$	14.4456	0.504153	0.252077	−	0.967707i	$-0.418886\pi$
$821$	14.4456	0.504153	0.252077	+	0.967707i	$0.418886\pi$
$822$	−5.02477	−0.175259
$823$	−5.39437	−0.188036	−0.0940179	−	0.995571i	$-0.529971\pi$
$823$	−5.39437	−0.188036	−0.0940179	+	0.995571i	$0.529971\pi$
$824$	7.57015	0.263719
$825$	8.30799	0.289247
$826$	−2.60338	−0.0905831
$827$	−40.7336	−1.41645	−0.708223	−	0.705989i	$-0.750503\pi$
$827$	−40.7336	−1.41645	−0.708223	+	0.705989i	$0.750503\pi$
$828$	14.8118	0.514747
$829$	−15.9379	−0.553547	−0.276773	−	0.960935i	$-0.589265\pi$
$829$	−15.9379	−0.553547	−0.276773	+	0.960935i	$0.589265\pi$
$830$	24.6528	0.855712
$831$	−5.03007	−0.174491
$832$	−0.953732	−0.0330647
$833$	−1.05526	−0.0365625
$834$	−6.52578	−0.225969
$835$	−55.0936	−1.90659
$836$	41.5807	1.43810
$837$	17.7436	0.613309
$838$	20.7391	0.716421
$839$	47.0290	1.62362	0.811811	−	0.583921i	$-0.198482\pi$
$839$	47.0290	1.62362	0.811811	+	0.583921i	$0.198482\pi$
$840$	−2.83249	−0.0977302
$841$	−7.54122	−0.260042
$842$	2.74188	0.0944914
$843$	−5.02821	−0.173181
$844$	23.7236	0.816600
$845$	−36.0339	−1.23961
$846$	−27.2238	−0.935975
$847$	−57.3574	−1.97082
$848$	−7.51226	−0.257972
$849$	−7.59575	−0.260685
$850$	−9.77140	−0.335156
$851$	48.8308	1.67390
$852$	4.57676	0.156797
$853$	5.50121	0.188358	0.0941790	−	0.995555i	$-0.469977\pi$
$853$	5.50121	0.188358	0.0941790	+	0.995555i	$0.469977\pi$
$854$	−15.0509	−0.515032
$855$	61.4242	2.10066
$856$	−16.0455	−0.548424
$857$	−6.74343	−0.230351	−0.115176	−	0.993345i	$-0.536743\pi$
$857$	−6.74343	−0.230351	−0.115176	+	0.993345i	$0.536743\pi$
$858$	2.04077	0.0696708
$859$	38.3733	1.30928	0.654641	−	0.755940i	$-0.272820\pi$
$859$	38.3733	1.30928	0.654641	+	0.755940i	$0.272820\pi$
$860$	17.4798	0.596056
$861$	2.34563	0.0799390
$862$	1.93294	0.0658363
$863$	27.5099	0.936449	0.468224	−	0.883610i	$-0.344894\pi$
$863$	27.5099	0.936449	0.468224	+	0.883610i	$0.344894\pi$
$864$	−2.17201	−0.0738934
$865$	−45.3122	−1.54066
$866$	−4.35794	−0.148089
$867$	3.95162	0.134204
$868$	−20.9564	−0.711306
$869$	−89.6153	−3.03999
$870$	5.11487	0.173410
$871$	−11.5676	−0.391953
$872$	−2.50942	−0.0849797
$873$	−2.73749	−0.0926501
$874$	−37.2487	−1.25996
$875$	8.54273	0.288797
$876$	−0.241135	−0.00814720
$877$	40.4307	1.36525	0.682625	−	0.730769i	$-0.260838\pi$
$877$	40.4307	1.36525	0.682625	+	0.730769i	$0.260838\pi$
$878$	−8.19465	−0.276556
$879$	9.63733	0.325059
$880$	−17.2139	−0.580279
$881$	14.9852	0.504866	0.252433	−	0.967614i	$-0.418769\pi$
$881$	14.9852	0.504866	0.252433	+	0.967614i	$0.418769\pi$
$882$	−1.20036	−0.0404183
$883$	45.8998	1.54465	0.772325	−	0.635227i	$-0.219094\pi$
$883$	45.8998	1.54465	0.772325	+	0.635227i	$0.219094\pi$
$884$	−2.40024	−0.0807289
$885$	1.12056	0.0376671
$886$	5.51425	0.185255
$887$	38.4897	1.29236	0.646179	−	0.763186i	$-0.276366\pi$
$887$	38.4897	1.29236	0.646179	+	0.763186i	$0.276366\pi$
$888$	−3.49647	−0.117334
$889$	−6.03959	−0.202561
$890$	−8.63867	−0.289569
$891$	−44.9557	−1.50607
$892$	−4.63802	−0.155292
$893$	68.4623	2.29100
$894$	1.75367	0.0586514
$895$	−26.6129	−0.889570
$896$	2.56529	0.0857002
$897$	−1.82816	−0.0610404
$898$	19.1545	0.639193
$899$	37.8427	1.26213
$900$	−11.1150	−0.370501
$901$	−18.9060	−0.629849
$902$	14.2551	0.474643
$903$	5.57393	0.185489
$904$	13.4589	0.447637
$905$	26.7121	0.887940
$906$	6.86959	0.228227
$907$	−3.30242	−0.109655	−0.0548275	−	0.998496i	$-0.517461\pi$
$907$	−3.30242	−0.109655	−0.0548275	+	0.998496i	$0.517461\pi$
$908$	2.30295	0.0764261
$909$	−16.9632	−0.562633
$910$	−7.29178	−0.241720
$911$	0.593599	0.0196668	0.00983341	−	0.999952i	$-0.496870\pi$
$911$	0.593599	0.0196668	0.00983341	+	0.999952i	$0.496870\pi$
$912$	2.66714	0.0883180
$913$	47.7752	1.58113
$914$	41.6610	1.37802
$915$	6.47828	0.214165
$916$	1.98890	0.0657150
$917$	−20.8879	−0.689779
$918$	−5.46627	−0.180414
$919$	7.37035	0.243125	0.121563	−	0.992584i	$-0.461209\pi$
$919$	7.37035	0.243125	0.121563	+	0.992584i	$0.461209\pi$
$920$	15.4205	0.508397
$921$	6.44604	0.212404
$922$	−28.7942	−0.948285
$923$	11.7821	0.387813
$924$	−5.48913	−0.180579
$925$	−36.6435	−1.20483
$926$	−1.89638	−0.0623188
$927$	21.6714	0.711783
$928$	−4.63236	−0.152065
$929$	0.386422	0.0126781	0.00633905	−	0.999980i	$-0.497982\pi$
$929$	0.386422	0.0126781	0.00633905	+	0.999980i	$0.497982\pi$
$930$	9.02012	0.295782
$931$	3.01866	0.0989327
$932$	5.58928	0.183083
$933$	4.99179	0.163424
$934$	9.53499	0.311994
$935$	−43.3218	−1.41678
$936$	−2.73029	−0.0892424
$937$	−29.0617	−0.949405	−0.474702	−	0.880146i	$-0.657444\pi$
$937$	−29.0617	−0.949405	−0.474702	+	0.880146i	$0.657444\pi$
$938$	31.1138	1.01590
$939$	3.66474	0.119594
$940$	−28.3425	−0.924429
$941$	−22.0276	−0.718081	−0.359040	−	0.933322i	$-0.616896\pi$
$941$	−22.0276	−0.718081	−0.359040	+	0.933322i	$0.616896\pi$
$942$	2.45472	0.0799790
$943$	−12.7699	−0.415847
$944$	−1.01485	−0.0330305
$945$	−16.6062	−0.540199
$946$	33.8744	1.10135
$947$	−6.64851	−0.216047	−0.108024	−	0.994148i	$-0.534452\pi$
$947$	−6.64851	−0.216047	−0.108024	+	0.994148i	$0.534452\pi$
$948$	−5.74826	−0.186695
$949$	−0.620763	−0.0201508
$950$	27.9520	0.906884
$951$	−7.12056	−0.230900
$952$	6.45602	0.209241
$953$	−57.4933	−1.86239	−0.931195	−	0.364521i	$-0.881233\pi$
$953$	−57.4933	−1.86239	−0.931195	+	0.364521i	$0.881233\pi$
$954$	−21.5057	−0.696273
$955$	−8.08216	−0.261532
$956$	−3.34553	−0.108202
$957$	9.91221	0.320416
$958$	−11.2845	−0.364586
$959$	34.7929	1.12352
$960$	−1.10416	−0.0356366
$961$	35.7360	1.15277
$962$	−9.00108	−0.290207
$963$	−45.9342	−1.48021
$964$	26.7139	0.860395
$965$	−31.1795	−1.00370
$966$	4.91726	0.158210
$967$	44.0592	1.41685	0.708424	−	0.705788i	$-0.249407\pi$
$967$	44.0592	1.41685	0.708424	+	0.705788i	$0.249407\pi$
$968$	−22.3591	−0.718647
$969$	6.71236	0.215632
$970$	−2.84998	−0.0915072
$971$	25.8900	0.830850	0.415425	−	0.909627i	$-0.363633\pi$
$971$	25.8900	0.830850	0.415425	+	0.909627i	$0.363633\pi$
$972$	−9.39967	−0.301494
$973$	45.1864	1.44861
$974$	−13.3515	−0.427810
$975$	1.37188	0.0439353
$976$	−5.86716	−0.187803
$977$	16.1799	0.517642	0.258821	−	0.965925i	$-0.416666\pi$
$977$	16.1799	0.517642	0.258821	+	0.965925i	$0.416666\pi$
$978$	−2.72381	−0.0870979
$979$	−16.7410	−0.535046
$980$	−1.24968	−0.0399197
$981$	−7.18383	−0.229362
$982$	−37.9516	−1.21108
$983$	−27.7004	−0.883504	−0.441752	−	0.897137i	$-0.645643\pi$
$983$	−27.7004	−0.883504	−0.441752	+	0.897137i	$0.645643\pi$
$984$	0.914375	0.0291492
$985$	−24.3577	−0.776101
$986$	−11.6582	−0.371272
$987$	−9.03781	−0.287677
$988$	6.86613	0.218441
$989$	−30.3453	−0.964923
$990$	−49.2789	−1.56619
$991$	−15.6306	−0.496523	−0.248262	−	0.968693i	$-0.579859\pi$
$991$	−15.6306	−0.496523	−0.248262	+	0.968693i	$0.579859\pi$
$992$	−8.16921	−0.259373
$993$	3.93326	0.124818
$994$	−31.6908	−1.00517
$995$	69.9126	2.21638
$996$	3.06448	0.0971017
$997$	59.0197	1.86917	0.934586	−	0.355737i	$-0.115770\pi$
$997$	59.0197	1.86917	0.934586	+	0.355737i	$0.115770\pi$
$998$	27.6365	0.874819
$999$	−20.4989	−0.648557

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.2.a.d.1.4	✓	7
3.2	odd	2		4842.2.a.o.1.1		7
4.3	odd	2		4304.2.a.i.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.d.1.4	✓	7	1.1	even	1	trivial
4304.2.a.i.1.4		7	4.3	odd	2
4842.2.a.o.1.1		7	3.2	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:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.370477 q^{3} +1.00000 q^{4} +2.98038 q^{5} +0.370477 q^{6} -2.56529 q^{7} -1.00000 q^{8} -2.86275 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.370477 q^{3} +1.00000 q^{4} +2.98038 q^{5} +0.370477 q^{6} -2.56529 q^{7} -1.00000 q^{8} -2.86275 q^{9} -2.98038 q^{10} -5.77573 q^{11} -0.370477 q^{12} -0.953732 q^{13} +2.56529 q^{14} -1.10416 q^{15} +1.00000 q^{16} +2.51668 q^{17} +2.86275 q^{18} -7.19922 q^{19} +2.98038 q^{20} +0.950379 q^{21} +5.77573 q^{22} -5.17399 q^{23} +0.370477 q^{24} +3.88265 q^{25} +0.953732 q^{26} +2.17201 q^{27} -2.56529 q^{28} +4.63236 q^{29} +1.10416 q^{30} +8.16921 q^{31} -1.00000 q^{32} +2.13977 q^{33} -2.51668 q^{34} -7.64552 q^{35} -2.86275 q^{36} -9.43775 q^{37} +7.19922 q^{38} +0.353336 q^{39} -2.98038 q^{40} +2.46810 q^{41} -0.950379 q^{42} +5.86496 q^{43} -5.77573 q^{44} -8.53207 q^{45} +5.17399 q^{46} -9.50969 q^{47} -0.370477 q^{48} -0.419304 q^{49} -3.88265 q^{50} -0.932373 q^{51} -0.953732 q^{52} -7.51226 q^{53} -2.17201 q^{54} -17.2139 q^{55} +2.56529 q^{56} +2.66714 q^{57} -4.63236 q^{58} -1.01485 q^{59} -1.10416 q^{60} -5.86716 q^{61} -8.16921 q^{62} +7.34377 q^{63} +1.00000 q^{64} -2.84248 q^{65} -2.13977 q^{66} +12.1288 q^{67} +2.51668 q^{68} +1.91685 q^{69} +7.64552 q^{70} -12.3537 q^{71} +2.86275 q^{72} +0.650878 q^{73} +9.43775 q^{74} -1.43843 q^{75} -7.19922 q^{76} +14.8164 q^{77} -0.353336 q^{78} +15.5158 q^{79} +2.98038 q^{80} +7.78356 q^{81} -2.46810 q^{82} -8.27171 q^{83} +0.950379 q^{84} +7.50067 q^{85} -5.86496 q^{86} -1.71618 q^{87} +5.77573 q^{88} +2.89852 q^{89} +8.53207 q^{90} +2.44660 q^{91} -5.17399 q^{92} -3.02650 q^{93} +9.50969 q^{94} -21.4564 q^{95} +0.370477 q^{96} +0.956247 q^{97} +0.419304 q^{98} +16.5345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} - 4 q^{3} + 7 q^{4} - 6 q^{5} + 4 q^{6} - 3 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^2 - 4 * q^3 + 7 * q^4 - 6 * q^5 + 4 * q^6 - 3 * q^7 - 7 * q^8 + 7 * q^9	\( 7 q - 7 q^{2} - 4 q^{3} + 7 q^{4} - 6 q^{5} + 4 q^{6} - 3 q^{7} - 7 q^{8} + 7 q^{9} + 6 q^{10} - 12 q^{11} - 4 q^{12} + 3 q^{13} + 3 q^{14} - 6 q^{15} + 7 q^{16} - 8 q^{17} - 7 q^{18} - 7 q^{19} - 6 q^{20} - 5 q^{21} + 12 q^{22} - 22 q^{23} + 4 q^{24} + 9 q^{25} - 3 q^{26} - 22 q^{27} - 3 q^{28} - 7 q^{29} + 6 q^{30} - 3 q^{31} - 7 q^{32} - 19 q^{33} + 8 q^{34} - 25 q^{35} + 7 q^{36} - 13 q^{37} + 7 q^{38} + q^{39} + 6 q^{40} - 11 q^{41} + 5 q^{42} - 14 q^{43} - 12 q^{44} - 17 q^{45} + 22 q^{46} - 23 q^{47} - 4 q^{48} + 6 q^{49} - 9 q^{50} - 21 q^{51} + 3 q^{52} - 21 q^{53} + 22 q^{54} + 15 q^{55} + 3 q^{56} - 14 q^{57} + 7 q^{58} - 20 q^{59} - 6 q^{60} + 13 q^{61} + 3 q^{62} - 12 q^{63} + 7 q^{64} - 29 q^{65} + 19 q^{66} - 30 q^{67} - 8 q^{68} + 13 q^{69} + 25 q^{70} - 23 q^{71} - 7 q^{72} + 9 q^{73} + 13 q^{74} + 3 q^{75} - 7 q^{76} + 8 q^{77} - q^{78} + 11 q^{79} - 6 q^{80} + 43 q^{81} + 11 q^{82} - 3 q^{83} - 5 q^{84} + 21 q^{85} + 14 q^{86} + 28 q^{87} + 12 q^{88} + 16 q^{89} + 17 q^{90} - 10 q^{91} - 22 q^{92} - 13 q^{93} + 23 q^{94} + 4 q^{96} + 21 q^{97} - 6 q^{98} - 9 q^{99}+O(q^{100}) \) 7 * q - 7 * q^2 - 4 * q^3 + 7 * q^4 - 6 * q^5 + 4 * q^6 - 3 * q^7 - 7 * q^8 + 7 * q^9 + 6 * q^10 - 12 * q^11 - 4 * q^12 + 3 * q^13 + 3 * q^14 - 6 * q^15 + 7 * q^16 - 8 * q^17 - 7 * q^18 - 7 * q^19 - 6 * q^20 - 5 * q^21 + 12 * q^22 - 22 * q^23 + 4 * q^24 + 9 * q^25 - 3 * q^26 - 22 * q^27 - 3 * q^28 - 7 * q^29 + 6 * q^30 - 3 * q^31 - 7 * q^32 - 19 * q^33 + 8 * q^34 - 25 * q^35 + 7 * q^36 - 13 * q^37 + 7 * q^38 + q^39 + 6 * q^40 - 11 * q^41 + 5 * q^42 - 14 * q^43 - 12 * q^44 - 17 * q^45 + 22 * q^46 - 23 * q^47 - 4 * q^48 + 6 * q^49 - 9 * q^50 - 21 * q^51 + 3 * q^52 - 21 * q^53 + 22 * q^54 + 15 * q^55 + 3 * q^56 - 14 * q^57 + 7 * q^58 - 20 * q^59 - 6 * q^60 + 13 * q^61 + 3 * q^62 - 12 * q^63 + 7 * q^64 - 29 * q^65 + 19 * q^66 - 30 * q^67 - 8 * q^68 + 13 * q^69 + 25 * q^70 - 23 * q^71 - 7 * q^72 + 9 * q^73 + 13 * q^74 + 3 * q^75 - 7 * q^76 + 8 * q^77 - q^78 + 11 * q^79 - 6 * q^80 + 43 * q^81 + 11 * q^82 - 3 * q^83 - 5 * q^84 + 21 * q^85 + 14 * q^86 + 28 * q^87 + 12 * q^88 + 16 * q^89 + 17 * q^90 - 10 * q^91 - 22 * q^92 - 13 * q^93 + 23 * q^94 + 4 * q^96 + 21 * q^97 - 6 * q^98 - 9 * q^99

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