LMFDB - Embedded newform 1334.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1334,2,Mod(1,1334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1334, 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("1334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1334 = 2 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1334.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:	$10.6520436296$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.68554$ of defining polynomial
Character	$\chi$	$=$	1334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -1.38372 q^{5} -2.41421 q^{6} +0.526602 q^{7} +1.00000 q^{8} +2.82843 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -1.38372 q^{5} -2.41421 q^{6} +0.526602 q^{7} +1.00000 q^{8} +2.82843 q^{9} -1.38372 q^{10} -1.14288 q^{11} -2.41421 q^{12} +3.48348 q^{13} +0.526602 q^{14} +3.34059 q^{15} +1.00000 q^{16} -2.88761 q^{17} +2.82843 q^{18} -0.0431257 q^{19} -1.38372 q^{20} -1.27133 q^{21} -1.14288 q^{22} +1.00000 q^{23} -2.41421 q^{24} -3.08532 q^{25} +3.48348 q^{26} +0.414214 q^{27} +0.526602 q^{28} -1.00000 q^{29} +3.34059 q^{30} -8.73875 q^{31} +1.00000 q^{32} +2.75916 q^{33} -2.88761 q^{34} -0.728670 q^{35} +2.82843 q^{36} -2.75138 q^{37} -0.0431257 q^{38} -8.40986 q^{39} -1.38372 q^{40} +6.72612 q^{41} -1.27133 q^{42} -4.28919 q^{43} -1.14288 q^{44} -3.91375 q^{45} +1.00000 q^{46} -12.5140 q^{47} -2.41421 q^{48} -6.72269 q^{49} -3.08532 q^{50} +6.97131 q^{51} +3.48348 q^{52} +5.44471 q^{53} +0.414214 q^{54} +1.58143 q^{55} +0.526602 q^{56} +0.104115 q^{57} -1.00000 q^{58} -9.48690 q^{59} +3.34059 q^{60} -7.38715 q^{61} -8.73875 q^{62} +1.48946 q^{63} +1.00000 q^{64} -4.82015 q^{65} +2.75916 q^{66} -13.8773 q^{67} -2.88761 q^{68} -2.41421 q^{69} -0.728670 q^{70} +13.7954 q^{71} +2.82843 q^{72} -10.8351 q^{73} -2.75138 q^{74} +7.44862 q^{75} -0.0431257 q^{76} -0.601845 q^{77} -8.40986 q^{78} -6.10318 q^{79} -1.38372 q^{80} -9.48528 q^{81} +6.72612 q^{82} +4.94424 q^{83} -1.27133 q^{84} +3.99564 q^{85} -4.28919 q^{86} +2.41421 q^{87} -1.14288 q^{88} +14.3135 q^{89} -3.91375 q^{90} +1.83441 q^{91} +1.00000 q^{92} +21.0972 q^{93} -12.5140 q^{94} +0.0596739 q^{95} -2.41421 q^{96} +0.805717 q^{97} -6.72269 q^{98} -3.23256 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{7} + 4 q^{8} - 4 q^{11} - 4 q^{12} - 8 q^{13} - 4 q^{14} - 8 q^{15} + 4 q^{16} - 12 q^{17} - 16 q^{19} - 4 q^{22} + 4 q^{23} - 4 q^{24} + 8 q^{25} - 8 q^{26} - 4 q^{27} - 4 q^{28} - 4 q^{29} - 8 q^{30} - 12 q^{31} + 4 q^{32} + 16 q^{33} - 12 q^{34} - 8 q^{35} - 4 q^{37} - 16 q^{38} + 4 q^{39} - 12 q^{41} + 4 q^{43} - 4 q^{44} + 16 q^{45} + 4 q^{46} - 28 q^{47} - 4 q^{48} - 8 q^{49} + 8 q^{50} + 16 q^{51} - 8 q^{52} + 16 q^{53} - 4 q^{54} - 20 q^{55} - 4 q^{56} + 16 q^{57} - 4 q^{58} + 4 q^{59} - 8 q^{60} - 4 q^{61} - 12 q^{62} + 8 q^{63} + 4 q^{64} - 24 q^{65} + 16 q^{66} - 12 q^{68} - 4 q^{69} - 8 q^{70} - 24 q^{73} - 4 q^{74} - 24 q^{75} - 16 q^{76} - 4 q^{77} + 4 q^{78} + 12 q^{79} - 4 q^{81} - 12 q^{82} - 12 q^{83} - 16 q^{85} + 4 q^{86} + 4 q^{87} - 4 q^{88} + 16 q^{89} + 16 q^{90} + 20 q^{91} + 4 q^{92} + 24 q^{93} - 28 q^{94} - 16 q^{95} - 4 q^{96} + 4 q^{97} - 8 q^{98} - 24 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^7 + 4 * q^8 - 4 * q^11 - 4 * q^12 - 8 * q^13 - 4 * q^14 - 8 * q^15 + 4 * q^16 - 12 * q^17 - 16 * q^19 - 4 * q^22 + 4 * q^23 - 4 * q^24 + 8 * q^25 - 8 * q^26 - 4 * q^27 - 4 * q^28 - 4 * q^29 - 8 * q^30 - 12 * q^31 + 4 * q^32 + 16 * q^33 - 12 * q^34 - 8 * q^35 - 4 * q^37 - 16 * q^38 + 4 * q^39 - 12 * q^41 + 4 * q^43 - 4 * q^44 + 16 * q^45 + 4 * q^46 - 28 * q^47 - 4 * q^48 - 8 * q^49 + 8 * q^50 + 16 * q^51 - 8 * q^52 + 16 * q^53 - 4 * q^54 - 20 * q^55 - 4 * q^56 + 16 * q^57 - 4 * q^58 + 4 * q^59 - 8 * q^60 - 4 * q^61 - 12 * q^62 + 8 * q^63 + 4 * q^64 - 24 * q^65 + 16 * q^66 - 12 * q^68 - 4 * q^69 - 8 * q^70 - 24 * q^73 - 4 * q^74 - 24 * q^75 - 16 * q^76 - 4 * q^77 + 4 * q^78 + 12 * q^79 - 4 * q^81 - 12 * q^82 - 12 * q^83 - 16 * q^85 + 4 * q^86 + 4 * q^87 - 4 * q^88 + 16 * q^89 + 16 * q^90 + 20 * q^91 + 4 * q^92 + 24 * q^93 - 28 * q^94 - 16 * q^95 - 4 * q^96 + 4 * q^97 - 8 * q^98 - 24 * 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
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	1.00000	0.500000
$5$	−1.38372	−0.618818	−0.309409	−	0.950929i	$-0.600131\pi$
$5$	−1.38372	−0.618818	−0.309409	+	0.950929i	$0.600131\pi$
$6$	−2.41421	−0.985599
$7$	0.526602	0.199037	0.0995185	−	0.995036i	$-0.468270\pi$
$7$	0.526602	0.199037	0.0995185	+	0.995036i	$0.468270\pi$
$8$	1.00000	0.353553
$9$	2.82843	0.942809
$10$	−1.38372	−0.437570
$11$	−1.14288	−0.344592	−0.172296	−	0.985045i	$-0.555119\pi$
$11$	−1.14288	−0.344592	−0.172296	+	0.985045i	$0.555119\pi$
$12$	−2.41421	−0.696923
$13$	3.48348	0.966143	0.483071	−	0.875581i	$-0.339521\pi$
$13$	3.48348	0.966143	0.483071	+	0.875581i	$0.339521\pi$
$14$	0.526602	0.140740
$15$	3.34059	0.862538
$16$	1.00000	0.250000
$17$	−2.88761	−0.700349	−0.350174	−	0.936685i	$-0.613878\pi$
$17$	−2.88761	−0.700349	−0.350174	+	0.936685i	$0.613878\pi$
$18$	2.82843	0.666667
$19$	−0.0431257	−0.00989371	−0.00494686	−	0.999988i	$-0.501575\pi$
$19$	−0.0431257	−0.00989371	−0.00494686	+	0.999988i	$0.501575\pi$
$20$	−1.38372	−0.309409
$21$	−1.27133	−0.277427
$22$	−1.14288	−0.243664
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−2.41421	−0.492799
$25$	−3.08532	−0.617064
$26$	3.48348	0.683166
$27$	0.414214	0.0797154
$28$	0.526602	0.0995185
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	3.34059	0.609906
$31$	−8.73875	−1.56953	−0.784763	−	0.619796i	$-0.787215\pi$
$31$	−8.73875	−1.56953	−0.784763	+	0.619796i	$0.787215\pi$
$32$	1.00000	0.176777
$33$	2.75916	0.480309
$34$	−2.88761	−0.495221
$35$	−0.728670	−0.123168
$36$	2.82843	0.471405
$37$	−2.75138	−0.452324	−0.226162	−	0.974090i	$-0.572618\pi$
$37$	−2.75138	−0.452324	−0.226162	+	0.974090i	$0.572618\pi$
$38$	−0.0431257	−0.00699591
$39$	−8.40986	−1.34665
$40$	−1.38372	−0.218785
$41$	6.72612	1.05044	0.525222	−	0.850966i	$-0.323982\pi$
$41$	6.72612	1.05044	0.525222	+	0.850966i	$0.323982\pi$
$42$	−1.27133	−0.196171
$43$	−4.28919	−0.654096	−0.327048	−	0.945008i	$-0.606054\pi$
$43$	−4.28919	−0.654096	−0.327048	+	0.945008i	$0.606054\pi$
$44$	−1.14288	−0.172296
$45$	−3.91375	−0.583427
$46$	1.00000	0.147442
$47$	−12.5140	−1.82535	−0.912675	−	0.408686i	$-0.865987\pi$
$47$	−12.5140	−1.82535	−0.912675	+	0.408686i	$0.865987\pi$
$48$	−2.41421	−0.348462
$49$	−6.72269	−0.960384
$50$	−3.08532	−0.436330
$51$	6.97131	0.976179
$52$	3.48348	0.483071
$53$	5.44471	0.747888	0.373944	−	0.927451i	$-0.378005\pi$
$53$	5.44471	0.747888	0.373944	+	0.927451i	$0.378005\pi$
$54$	0.414214	0.0563673
$55$	1.58143	0.213240
$56$	0.526602	0.0703702
$57$	0.104115	0.0137903
$58$	−1.00000	−0.131306
$59$	−9.48690	−1.23509	−0.617545	−	0.786536i	$-0.711873\pi$
$59$	−9.48690	−1.23509	−0.617545	+	0.786536i	$0.711873\pi$
$60$	3.34059	0.431269
$61$	−7.38715	−0.945827	−0.472914	−	0.881109i	$-0.656798\pi$
$61$	−7.38715	−0.945827	−0.472914	+	0.881109i	$0.656798\pi$
$62$	−8.73875	−1.10982
$63$	1.48946	0.187654
$64$	1.00000	0.125000
$65$	−4.82015	−0.597866
$66$	2.75916	0.339630
$67$	−13.8773	−1.69538	−0.847689	−	0.530493i	$-0.822007\pi$
$67$	−13.8773	−1.69538	−0.847689	+	0.530493i	$0.822007\pi$
$68$	−2.88761	−0.350174
$69$	−2.41421	−0.290637
$70$	−0.728670	−0.0870927
$71$	13.7954	1.63721	0.818605	−	0.574357i	$-0.194748\pi$
$71$	13.7954	1.63721	0.818605	+	0.574357i	$0.194748\pi$
$72$	2.82843	0.333333
$73$	−10.8351	−1.26815	−0.634075	−	0.773272i	$-0.718619\pi$
$73$	−10.8351	−1.26815	−0.634075	+	0.773272i	$0.718619\pi$
$74$	−2.75138	−0.319841
$75$	7.44862	0.860093
$76$	−0.0431257	−0.00494686
$77$	−0.601845	−0.0685866
$78$	−8.40986	−0.952229
$79$	−6.10318	−0.686662	−0.343331	−	0.939214i	$-0.611555\pi$
$79$	−6.10318	−0.686662	−0.343331	+	0.939214i	$0.611555\pi$
$80$	−1.38372	−0.154704
$81$	−9.48528	−1.05392
$82$	6.72612	0.742775
$83$	4.94424	0.542701	0.271351	−	0.962481i	$-0.412530\pi$
$83$	4.94424	0.542701	0.271351	+	0.962481i	$0.412530\pi$
$84$	−1.27133	−0.138714
$85$	3.99564	0.433388
$86$	−4.28919	−0.462516
$87$	2.41421	0.258831
$88$	−1.14288	−0.121832
$89$	14.3135	1.51723	0.758615	−	0.651539i	$-0.225876\pi$
$89$	14.3135	1.51723	0.758615	+	0.651539i	$0.225876\pi$
$90$	−3.91375	−0.412545
$91$	1.83441	0.192298
$92$	1.00000	0.104257
$93$	21.0972	2.18768
$94$	−12.5140	−1.29072
$95$	0.0596739	0.00612241
$96$	−2.41421	−0.246400
$97$	0.805717	0.0818082	0.0409041	−	0.999163i	$-0.486976\pi$
$97$	0.805717	0.0818082	0.0409041	+	0.999163i	$0.486976\pi$
$98$	−6.72269	−0.679094
$99$	−3.23256	−0.324885
$100$	−3.08532	−0.308532
$101$	−11.2509	−1.11951	−0.559754	−	0.828659i	$-0.689104\pi$
$101$	−11.2509	−1.11951	−0.559754	+	0.828659i	$0.689104\pi$
$102$	6.97131	0.690263
$103$	−17.1438	−1.68922	−0.844612	−	0.535378i	$-0.820169\pi$
$103$	−17.1438	−1.68922	−0.844612	+	0.535378i	$0.820169\pi$
$104$	3.48348	0.341583
$105$	1.75916	0.171677
$106$	5.44471	0.528837
$107$	4.87385	0.471173	0.235586	−	0.971853i	$-0.424299\pi$
$107$	4.87385	0.471173	0.235586	+	0.971853i	$0.424299\pi$
$108$	0.414214	0.0398577
$109$	13.0775	1.25260	0.626300	−	0.779582i	$-0.284568\pi$
$109$	13.0775	1.25260	0.626300	+	0.779582i	$0.284568\pi$
$110$	1.58143	0.150783
$111$	6.64242	0.630470
$112$	0.526602	0.0497592
$113$	0.661029	0.0621844	0.0310922	−	0.999517i	$-0.490101\pi$
$113$	0.661029	0.0621844	0.0310922	+	0.999517i	$0.490101\pi$
$114$	0.104115	0.00975123
$115$	−1.38372	−0.129032
$116$	−1.00000	−0.0928477
$117$	9.85276	0.910888
$118$	−9.48690	−0.873340
$119$	−1.52062	−0.139395
$120$	3.34059	0.304953
$121$	−9.69382	−0.881256
$122$	−7.38715	−0.668801
$123$	−16.2383	−1.46416
$124$	−8.73875	−0.784763
$125$	11.1878	1.00067
$126$	1.48946	0.132691
$127$	−8.93484	−0.792839	−0.396419	−	0.918070i	$-0.629747\pi$
$127$	−8.93484	−0.792839	−0.396419	+	0.918070i	$0.629747\pi$
$128$	1.00000	0.0883883
$129$	10.3550	0.911710
$130$	−4.82015	−0.422755
$131$	−16.9942	−1.48479	−0.742396	−	0.669961i	$-0.766311\pi$
$131$	−16.9942	−1.48479	−0.742396	+	0.669961i	$0.766311\pi$
$132$	2.75916	0.240154
$133$	−0.0227101	−0.00196921
$134$	−13.8773	−1.19881
$135$	−0.573155	−0.0493293
$136$	−2.88761	−0.247611
$137$	21.5208	1.83865	0.919324	−	0.393501i	$-0.128736\pi$
$137$	21.5208	1.83865	0.919324	+	0.393501i	$0.128736\pi$
$138$	−2.41421	−0.205512
$139$	7.37774	0.625772	0.312886	−	0.949791i	$-0.398704\pi$
$139$	7.37774	0.625772	0.312886	+	0.949791i	$0.398704\pi$
$140$	−0.728670	−0.0615838
$141$	30.2114	2.54426
$142$	13.7954	1.15768
$143$	−3.98121	−0.332925
$144$	2.82843	0.235702
$145$	1.38372	0.114912
$146$	−10.8351	−0.896717
$147$	16.2300	1.33863
$148$	−2.75138	−0.226162
$149$	13.2044	1.08174	0.540872	−	0.841105i	$-0.318094\pi$
$149$	13.2044	1.08174	0.540872	+	0.841105i	$0.318094\pi$
$150$	7.44862	0.608178
$151$	6.34420	0.516284	0.258142	−	0.966107i	$-0.416890\pi$
$151$	6.34420	0.516284	0.258142	+	0.966107i	$0.416890\pi$
$152$	−0.0431257	−0.00349796
$153$	−8.16740	−0.660295
$154$	−0.601845	−0.0484980
$155$	12.0920	0.971251
$156$	−8.40986	−0.673327
$157$	7.09814	0.566493	0.283246	−	0.959047i	$-0.408589\pi$
$157$	7.09814	0.566493	0.283246	+	0.959047i	$0.408589\pi$
$158$	−6.10318	−0.485543
$159$	−13.1447	−1.04244
$160$	−1.38372	−0.109393
$161$	0.526602	0.0415021
$162$	−9.48528	−0.745234
$163$	6.05776	0.474481	0.237240	−	0.971451i	$-0.423757\pi$
$163$	6.05776	0.474481	0.237240	+	0.971451i	$0.423757\pi$
$164$	6.72612	0.525222
$165$	−3.81791	−0.297224
$166$	4.94424	0.383748
$167$	6.19011	0.479005	0.239502	−	0.970896i	$-0.423016\pi$
$167$	6.19011	0.479005	0.239502	+	0.970896i	$0.423016\pi$
$168$	−1.27133	−0.0980853
$169$	−0.865391	−0.0665685
$170$	3.99564	0.306452
$171$	−0.121978	−0.00932788
$172$	−4.28919	−0.327048
$173$	−25.3949	−1.93074	−0.965368	−	0.260894i	$-0.915983\pi$
$173$	−25.3949	−1.93074	−0.965368	+	0.260894i	$0.915983\pi$
$174$	2.41421	0.183021
$175$	−1.62474	−0.122819
$176$	−1.14288	−0.0861481
$177$	22.9034	1.72153
$178$	14.3135	1.07284
$179$	−2.08645	−0.155949	−0.0779745	−	0.996955i	$-0.524845\pi$
$179$	−2.08645	−0.155949	−0.0779745	+	0.996955i	$0.524845\pi$
$180$	−3.91375	−0.291714
$181$	14.9949	1.11456	0.557281	−	0.830324i	$-0.311845\pi$
$181$	14.9949	1.11456	0.557281	+	0.830324i	$0.311845\pi$
$182$	1.83441	0.135975
$183$	17.8341	1.31834
$184$	1.00000	0.0737210
$185$	3.80714	0.279906
$186$	21.0972	1.54692
$187$	3.30020	0.241335
$188$	−12.5140	−0.912675
$189$	0.218126	0.0158663
$190$	0.0596739	0.00432920
$191$	−13.9137	−1.00676	−0.503382	−	0.864064i	$-0.667911\pi$
$191$	−13.9137	−1.00676	−0.503382	+	0.864064i	$0.667911\pi$
$192$	−2.41421	−0.174231
$193$	12.6397	0.909823	0.454911	−	0.890537i	$-0.349671\pi$
$193$	12.6397	0.909823	0.454911	+	0.890537i	$0.349671\pi$
$194$	0.805717	0.0578471
$195$	11.6369	0.833334
$196$	−6.72269	−0.480192
$197$	7.59587	0.541183	0.270591	−	0.962694i	$-0.412781\pi$
$197$	7.59587	0.541183	0.270591	+	0.962694i	$0.412781\pi$
$198$	−3.23256	−0.229728
$199$	−25.9495	−1.83951	−0.919755	−	0.392494i	$-0.871613\pi$
$199$	−25.9495	−1.83951	−0.919755	+	0.392494i	$0.871613\pi$
$200$	−3.08532	−0.218165
$201$	33.5027	2.36310
$202$	−11.2509	−0.791612
$203$	−0.526602	−0.0369602
$204$	6.97131	0.488089
$205$	−9.30706	−0.650033
$206$	−17.1438	−1.19446
$207$	2.82843	0.196589
$208$	3.48348	0.241536
$209$	0.0492876	0.00340930
$210$	1.75916	0.121394
$211$	13.9848	0.962754	0.481377	−	0.876514i	$-0.340137\pi$
$211$	13.9848	0.962754	0.481377	+	0.876514i	$0.340137\pi$
$212$	5.44471	0.373944
$213$	−33.3050	−2.28202
$214$	4.87385	0.333169
$215$	5.93504	0.404766
$216$	0.414214	0.0281837
$217$	−4.60184	−0.312394
$218$	13.0775	0.885723
$219$	26.1582	1.76761
$220$	1.58143	0.106620
$221$	−10.0589	−0.676637
$222$	6.64242	0.445810
$223$	−3.41582	−0.228740	−0.114370	−	0.993438i	$-0.536485\pi$
$223$	−3.41582	−0.228740	−0.114370	+	0.993438i	$0.536485\pi$
$224$	0.526602	0.0351851
$225$	−8.72661	−0.581774
$226$	0.661029	0.0439710
$227$	27.1229	1.80021	0.900103	−	0.435676i	$-0.143491\pi$
$227$	27.1229	1.80021	0.900103	+	0.435676i	$0.143491\pi$
$228$	0.104115	0.00689516
$229$	−0.703407	−0.0464825	−0.0232412	−	0.999730i	$-0.507399\pi$
$229$	−0.703407	−0.0464825	−0.0232412	+	0.999730i	$0.507399\pi$
$230$	−1.38372	−0.0912397
$231$	1.45298	0.0955992
$232$	−1.00000	−0.0656532
$233$	−3.57839	−0.234428	−0.117214	−	0.993107i	$-0.537396\pi$
$233$	−3.57839	−0.234428	−0.117214	+	0.993107i	$0.537396\pi$
$234$	9.85276	0.644095
$235$	17.3158	1.12956
$236$	−9.48690	−0.617545
$237$	14.7344	0.957102
$238$	−1.52062	−0.0985673
$239$	−15.0394	−0.972821	−0.486410	−	0.873731i	$-0.661694\pi$
$239$	−15.0394	−0.972821	−0.486410	+	0.873731i	$0.661694\pi$
$240$	3.34059	0.215634
$241$	−3.98717	−0.256836	−0.128418	−	0.991720i	$-0.540990\pi$
$241$	−3.98717	−0.256836	−0.128418	+	0.991720i	$0.540990\pi$
$242$	−9.69382	−0.623142
$243$	21.6569	1.38929
$244$	−7.38715	−0.472914
$245$	9.30231	0.594303
$246$	−16.2383	−1.03532
$247$	−0.150227	−0.00955874
$248$	−8.73875	−0.554911
$249$	−11.9365	−0.756442
$250$	11.1878	0.707579
$251$	−31.1057	−1.96337	−0.981686	−	0.190506i	$-0.938987\pi$
$251$	−31.1057	−1.96337	−0.981686	+	0.190506i	$0.938987\pi$
$252$	1.48946	0.0938269
$253$	−1.14288	−0.0718525
$254$	−8.93484	−0.560621
$255$	−9.64634	−0.604077
$256$	1.00000	0.0625000
$257$	6.05805	0.377891	0.188945	−	0.981988i	$-0.439493\pi$
$257$	6.05805	0.377891	0.188945	+	0.981988i	$0.439493\pi$
$258$	10.3550	0.644676
$259$	−1.44888	−0.0900292
$260$	−4.82015	−0.298933
$261$	−2.82843	−0.175075
$262$	−16.9942	−1.04991
$263$	15.9559	0.983886	0.491943	−	0.870627i	$-0.336287\pi$
$263$	15.9559	0.983886	0.491943	+	0.870627i	$0.336287\pi$
$264$	2.75916	0.169815
$265$	−7.53395	−0.462807
$266$	−0.0227101	−0.00139244
$267$	−34.5559	−2.11479
$268$	−13.8773	−0.847689
$269$	−1.22135	−0.0744670	−0.0372335	−	0.999307i	$-0.511855\pi$
$269$	−1.22135	−0.0744670	−0.0372335	+	0.999307i	$0.511855\pi$
$270$	−0.573155	−0.0348811
$271$	−13.1883	−0.801132	−0.400566	−	0.916268i	$-0.631187\pi$
$271$	−13.1883	−0.801132	−0.400566	+	0.916268i	$0.631187\pi$
$272$	−2.88761	−0.175087
$273$	−4.42865	−0.268034
$274$	21.5208	1.30012
$275$	3.52616	0.212636
$276$	−2.41421	−0.145319
$277$	16.3357	0.981520	0.490760	−	0.871295i	$-0.336719\pi$
$277$	16.3357	0.981520	0.490760	+	0.871295i	$0.336719\pi$
$278$	7.37774	0.442487
$279$	−24.7169	−1.47976
$280$	−0.728670	−0.0435463
$281$	11.4105	0.680695	0.340348	−	0.940300i	$-0.389455\pi$
$281$	11.4105	0.680695	0.340348	+	0.940300i	$0.389455\pi$
$282$	30.2114	1.79906
$283$	22.9435	1.36385	0.681925	−	0.731422i	$-0.261143\pi$
$283$	22.9435	1.36385	0.681925	+	0.731422i	$0.261143\pi$
$284$	13.7954	0.818605
$285$	−0.144065	−0.00853370
$286$	−3.98121	−0.235414
$287$	3.54199	0.209077
$288$	2.82843	0.166667
$289$	−8.66170	−0.509512
$290$	1.38372	0.0812548
$291$	−1.94517	−0.114028
$292$	−10.8351	−0.634075
$293$	17.7592	1.03750	0.518751	−	0.854926i	$-0.326397\pi$
$293$	17.7592	1.03750	0.518751	+	0.854926i	$0.326397\pi$
$294$	16.2300	0.946553
$295$	13.1272	0.764296
$296$	−2.75138	−0.159921
$297$	−0.473398	−0.0274693
$298$	13.2044	0.764908
$299$	3.48348	0.201455
$300$	7.44862	0.430047
$301$	−2.25870	−0.130189
$302$	6.34420	0.365068
$303$	27.1621	1.56042
$304$	−0.0431257	−0.00247343
$305$	10.2217	0.585295
$306$	−8.16740	−0.466899
$307$	5.94585	0.339347	0.169674	−	0.985500i	$-0.445729\pi$
$307$	5.94585	0.339347	0.169674	+	0.985500i	$0.445729\pi$
$308$	−0.601845	−0.0342933
$309$	41.3887	2.35452
$310$	12.0920	0.686778
$311$	−14.4532	−0.819564	−0.409782	−	0.912184i	$-0.634395\pi$
$311$	−14.4532	−0.819564	−0.409782	+	0.912184i	$0.634395\pi$
$312$	−8.40986	−0.476114
$313$	−16.9119	−0.955920	−0.477960	−	0.878382i	$-0.658624\pi$
$313$	−16.9119	−0.955920	−0.477960	+	0.878382i	$0.658624\pi$
$314$	7.09814	0.400571
$315$	−2.06099	−0.116124
$316$	−6.10318	−0.343331
$317$	23.9729	1.34645	0.673227	−	0.739436i	$-0.264908\pi$
$317$	23.9729	1.34645	0.673227	+	0.739436i	$0.264908\pi$
$318$	−13.1447	−0.737117
$319$	1.14288	0.0639892
$320$	−1.38372	−0.0773522
$321$	−11.7665	−0.656742
$322$	0.526602	0.0293464
$323$	0.124530	0.00692905
$324$	−9.48528	−0.526960
$325$	−10.7476	−0.596172
$326$	6.05776	0.335509
$327$	−31.5720	−1.74593
$328$	6.72612	0.371388
$329$	−6.58989	−0.363312
$330$	−3.81791	−0.210169
$331$	6.55759	0.360438	0.180219	−	0.983627i	$-0.442319\pi$
$331$	6.55759	0.360438	0.180219	+	0.983627i	$0.442319\pi$
$332$	4.94424	0.271351
$333$	−7.78208	−0.426455
$334$	6.19011	0.338708
$335$	19.2022	1.04913
$336$	−1.27133	−0.0693568
$337$	−8.22223	−0.447893	−0.223947	−	0.974601i	$-0.571894\pi$
$337$	−8.22223	−0.447893	−0.223947	+	0.974601i	$0.571894\pi$
$338$	−0.865391	−0.0470711
$339$	−1.59587	−0.0866755
$340$	3.99564	0.216694
$341$	9.98737	0.540846
$342$	−0.121978	−0.00659581
$343$	−7.22640	−0.390189
$344$	−4.28919	−0.231258
$345$	3.34059	0.179852
$346$	−25.3949	−1.36524
$347$	28.2171	1.51477	0.757387	−	0.652966i	$-0.226476\pi$
$347$	28.2171	1.51477	0.757387	+	0.652966i	$0.226476\pi$
$348$	2.41421	0.129415
$349$	−13.6197	−0.729046	−0.364523	−	0.931194i	$-0.618768\pi$
$349$	−13.6197	−0.729046	−0.364523	+	0.931194i	$0.618768\pi$
$350$	−1.62474	−0.0868459
$351$	1.44290	0.0770165
$352$	−1.14288	−0.0609159
$353$	−7.39150	−0.393410	−0.196705	−	0.980463i	$-0.563024\pi$
$353$	−7.39150	−0.393410	−0.196705	+	0.980463i	$0.563024\pi$
$354$	22.9034	1.21730
$355$	−19.0889	−1.01314
$356$	14.3135	0.758615
$357$	3.67111	0.194296
$358$	−2.08645	−0.110273
$359$	−14.4497	−0.762625	−0.381313	−	0.924446i	$-0.624528\pi$
$359$	−14.4497	−0.762625	−0.381313	+	0.924446i	$0.624528\pi$
$360$	−3.91375	−0.206273
$361$	−18.9981	−0.999902
$362$	14.9949	0.788114
$363$	23.4029	1.22834
$364$	1.83441	0.0961490
$365$	14.9927	0.784754
$366$	17.8341	0.932206
$367$	12.3227	0.643241	0.321621	−	0.946869i	$-0.395772\pi$
$367$	12.3227	0.643241	0.321621	+	0.946869i	$0.395772\pi$
$368$	1.00000	0.0521286
$369$	19.0243	0.990367
$370$	3.80714	0.197924
$371$	2.86720	0.148857
$372$	21.0972	1.09384
$373$	13.8470	0.716970	0.358485	−	0.933536i	$-0.383293\pi$
$373$	13.8470	0.716970	0.358485	+	0.933536i	$0.383293\pi$
$374$	3.30020	0.170649
$375$	−27.0098	−1.39478
$376$	−12.5140	−0.645359
$377$	−3.48348	−0.179408
$378$	0.218126	0.0112192
$379$	−6.37526	−0.327475	−0.163738	−	0.986504i	$-0.552355\pi$
$379$	−6.37526	−0.327475	−0.163738	+	0.986504i	$0.552355\pi$
$380$	0.0596739	0.00306120
$381$	21.5706	1.10510
$382$	−13.9137	−0.711889
$383$	−35.2601	−1.80171	−0.900853	−	0.434125i	$-0.857058\pi$
$383$	−35.2601	−1.80171	−0.900853	+	0.434125i	$0.857058\pi$
$384$	−2.41421	−0.123200
$385$	0.832784	0.0424426
$386$	12.6397	0.643342
$387$	−12.1317	−0.616688
$388$	0.805717	0.0409041
$389$	−9.74055	−0.493866	−0.246933	−	0.969033i	$-0.579423\pi$
$389$	−9.74055	−0.493866	−0.246933	+	0.969033i	$0.579423\pi$
$390$	11.6369	0.589256
$391$	−2.88761	−0.146033
$392$	−6.72269	−0.339547
$393$	41.0277	2.06957
$394$	7.59587	0.382674
$395$	8.44509	0.424919
$396$	−3.23256	−0.162442
$397$	11.2646	0.565354	0.282677	−	0.959215i	$-0.408778\pi$
$397$	11.2646	0.565354	0.282677	+	0.959215i	$0.408778\pi$
$398$	−25.9495	−1.30073
$399$	0.0548270	0.00274478
$400$	−3.08532	−0.154266
$401$	0.326157	0.0162875	0.00814376	−	0.999967i	$-0.497408\pi$
$401$	0.326157	0.0162875	0.00814376	+	0.999967i	$0.497408\pi$
$402$	33.5027	1.67096
$403$	−30.4412	−1.51639
$404$	−11.2509	−0.559754
$405$	13.1250	0.652185
$406$	−0.526602	−0.0261348
$407$	3.14451	0.155867
$408$	6.97131	0.345131
$409$	−2.81492	−0.139189	−0.0695944	−	0.997575i	$-0.522171\pi$
$409$	−2.81492	−0.139189	−0.0695944	+	0.997575i	$0.522171\pi$
$410$	−9.30706	−0.459643
$411$	−51.9559	−2.56279
$412$	−17.1438	−0.844612
$413$	−4.99583	−0.245828
$414$	2.82843	0.139010
$415$	−6.84144	−0.335833
$416$	3.48348	0.170791
$417$	−17.8114	−0.872230
$418$	0.0492876	0.00241074
$419$	27.4540	1.34122	0.670609	−	0.741811i	$-0.266033\pi$
$419$	27.4540	1.34122	0.670609	+	0.741811i	$0.266033\pi$
$420$	1.75916	0.0858384
$421$	−24.2378	−1.18128	−0.590639	−	0.806936i	$-0.701124\pi$
$421$	−24.2378	−1.18128	−0.590639	+	0.806936i	$0.701124\pi$
$422$	13.9848	0.680770
$423$	−35.3949	−1.72096
$424$	5.44471	0.264418
$425$	8.90921	0.432160
$426$	−33.3050	−1.61363
$427$	−3.89009	−0.188255
$428$	4.87385	0.235586
$429$	9.61148	0.464047
$430$	5.93504	0.286213
$431$	−25.6839	−1.23715	−0.618574	−	0.785726i	$-0.712289\pi$
$431$	−25.6839	−1.23715	−0.618574	+	0.785726i	$0.712289\pi$
$432$	0.414214	0.0199289
$433$	3.97293	0.190927	0.0954635	−	0.995433i	$-0.469567\pi$
$433$	3.97293	0.190927	0.0954635	+	0.995433i	$0.469567\pi$
$434$	−4.60184	−0.220896
$435$	−3.34059	−0.160169
$436$	13.0775	0.626300
$437$	−0.0431257	−0.00206298
$438$	26.1582	1.24989
$439$	31.3522	1.49636	0.748180	−	0.663496i	$-0.230928\pi$
$439$	31.3522	1.49636	0.748180	+	0.663496i	$0.230928\pi$
$440$	1.58143	0.0753917
$441$	−19.0146	−0.905459
$442$	−10.0589	−0.478454
$443$	−20.4613	−0.972144	−0.486072	−	0.873919i	$-0.661571\pi$
$443$	−20.4613	−0.972144	−0.486072	+	0.873919i	$0.661571\pi$
$444$	6.64242	0.315235
$445$	−19.8059	−0.938890
$446$	−3.41582	−0.161744
$447$	−31.8781	−1.50778
$448$	0.526602	0.0248796
$449$	−18.8062	−0.887520	−0.443760	−	0.896146i	$-0.646356\pi$
$449$	−18.8062	−0.887520	−0.443760	+	0.896146i	$0.646356\pi$
$450$	−8.72661	−0.411376
$451$	−7.68717	−0.361975
$452$	0.661029	0.0310922
$453$	−15.3163	−0.719621
$454$	27.1229	1.27294
$455$	−2.53830	−0.118998
$456$	0.104115	0.00487561
$457$	−8.52064	−0.398579	−0.199289	−	0.979941i	$-0.563863\pi$
$457$	−8.52064	−0.398579	−0.199289	+	0.979941i	$0.563863\pi$
$458$	−0.703407	−0.0328681
$459$	−1.19609	−0.0558286
$460$	−1.38372	−0.0645162
$461$	−12.2656	−0.571264	−0.285632	−	0.958339i	$-0.592203\pi$
$461$	−12.2656	−0.571264	−0.285632	+	0.958339i	$0.592203\pi$
$462$	1.45298	0.0675988
$463$	3.59947	0.167282	0.0836409	−	0.996496i	$-0.473345\pi$
$463$	3.59947	0.167282	0.0836409	+	0.996496i	$0.473345\pi$
$464$	−1.00000	−0.0464238
$465$	−29.1926	−1.35377
$466$	−3.57839	−0.165766
$467$	0.386454	0.0178829	0.00894147	−	0.999960i	$-0.497154\pi$
$467$	0.386454	0.0178829	0.00894147	+	0.999960i	$0.497154\pi$
$468$	9.85276	0.455444
$469$	−7.30780	−0.337443
$470$	17.3158	0.798719
$471$	−17.1364	−0.789604
$472$	−9.48690	−0.436670
$473$	4.90205	0.225396
$474$	14.7344	0.676773
$475$	0.133057	0.00610506
$476$	−1.52062	−0.0696976
$477$	15.4000	0.705116
$478$	−15.0394	−0.687888
$479$	−34.8045	−1.59026	−0.795130	−	0.606439i	$-0.792597\pi$
$479$	−34.8045	−1.59026	−0.795130	+	0.606439i	$0.792597\pi$
$480$	3.34059	0.152477
$481$	−9.58437	−0.437010
$482$	−3.98717	−0.181610
$483$	−1.27133	−0.0578475
$484$	−9.69382	−0.440628
$485$	−1.11489	−0.0506244
$486$	21.6569	0.982375
$487$	10.2168	0.462968	0.231484	−	0.972839i	$-0.425642\pi$
$487$	10.2168	0.462968	0.231484	+	0.972839i	$0.425642\pi$
$488$	−7.38715	−0.334400
$489$	−14.6247	−0.661353
$490$	9.30231	0.420236
$491$	−30.5440	−1.37843	−0.689216	−	0.724556i	$-0.742045\pi$
$491$	−30.5440	−1.37843	−0.689216	+	0.724556i	$0.742045\pi$
$492$	−16.2383	−0.732078
$493$	2.88761	0.130051
$494$	−0.150227	−0.00675905
$495$	4.47296	0.201044
$496$	−8.73875	−0.392381
$497$	7.26468	0.325865
$498$	−11.9365	−0.534886
$499$	0.603650	0.0270231	0.0135115	−	0.999909i	$-0.495699\pi$
$499$	0.603650	0.0270231	0.0135115	+	0.999909i	$0.495699\pi$
$500$	11.1878	0.500334
$501$	−14.9442	−0.667660
$502$	−31.1057	−1.38831
$503$	−23.5942	−1.05202	−0.526008	−	0.850480i	$-0.676312\pi$
$503$	−23.5942	−1.05202	−0.526008	+	0.850480i	$0.676312\pi$
$504$	1.48946	0.0663457
$505$	15.5681	0.692772
$506$	−1.14288	−0.0508074
$507$	2.08924	0.0927863
$508$	−8.93484	−0.396419
$509$	15.2298	0.675050	0.337525	−	0.941317i	$-0.390410\pi$
$509$	15.2298	0.675050	0.337525	+	0.941317i	$0.390410\pi$
$510$	−9.64634	−0.427147
$511$	−5.70578	−0.252409
$512$	1.00000	0.0441942
$513$	−0.0178632	−0.000788682 0
$514$	6.05805	0.267209
$515$	23.7221	1.04532
$516$	10.3550	0.455855
$517$	14.3020	0.629002
$518$	−1.44888	−0.0636603
$519$	61.3086	2.69115
$520$	−4.82015	−0.211378
$521$	−26.4757	−1.15992	−0.579961	−	0.814645i	$-0.696932\pi$
$521$	−26.4757	−1.15992	−0.579961	+	0.814645i	$0.696932\pi$
$522$	−2.82843	−0.123797
$523$	24.4048	1.06715	0.533574	−	0.845753i	$-0.320848\pi$
$523$	24.4048	1.06715	0.533574	+	0.845753i	$0.320848\pi$
$524$	−16.9942	−0.742396
$525$	3.92246	0.171190
$526$	15.9559	0.695712
$527$	25.2341	1.09922
$528$	2.75916	0.120077
$529$	1.00000	0.0434783
$530$	−7.53395	−0.327254
$531$	−26.8330	−1.16445
$532$	−0.0227101	−0.000984607 0
$533$	23.4303	1.01488
$534$	−34.5559	−1.49538
$535$	−6.74404	−0.291570
$536$	−13.8773	−0.599407
$537$	5.03715	0.217369
$538$	−1.22135	−0.0526561
$539$	7.68325	0.330941
$540$	−0.573155	−0.0246647
$541$	12.0577	0.518401	0.259200	−	0.965824i	$-0.416541\pi$
$541$	12.0577	0.518401	0.259200	+	0.965824i	$0.416541\pi$
$542$	−13.1883	−0.566486
$543$	−36.2009	−1.55353
$544$	−2.88761	−0.123805
$545$	−18.0956	−0.775132
$546$	−4.42865	−0.189529
$547$	−38.1401	−1.63075	−0.815376	−	0.578932i	$-0.803470\pi$
$547$	−38.1401	−1.63075	−0.815376	+	0.578932i	$0.803470\pi$
$548$	21.5208	0.919324
$549$	−20.8940	−0.891734
$550$	3.52616	0.150356
$551$	0.0431257	0.00183722
$552$	−2.41421	−0.102756
$553$	−3.21395	−0.136671
$554$	16.3357	0.694039
$555$	−9.19124	−0.390146
$556$	7.37774	0.312886
$557$	22.7896	0.965626	0.482813	−	0.875723i	$-0.339615\pi$
$557$	22.7896	0.965626	0.482813	+	0.875723i	$0.339615\pi$
$558$	−24.7169	−1.04635
$559$	−14.9413	−0.631950
$560$	−0.728670	−0.0307919
$561$	−7.96739	−0.336384
$562$	11.4105	0.481324
$563$	−39.5653	−1.66748	−0.833740	−	0.552158i	$-0.813805\pi$
$563$	−39.5653	−1.66748	−0.833740	+	0.552158i	$0.813805\pi$
$564$	30.2114	1.27213
$565$	−0.914679	−0.0384808
$566$	22.9435	0.964387
$567$	−4.99497	−0.209769
$568$	13.7954	0.578841
$569$	26.6206	1.11599	0.557996	−	0.829844i	$-0.311570\pi$
$569$	26.6206	1.11599	0.557996	+	0.829844i	$0.311570\pi$
$570$	−0.144065	−0.00603424
$571$	29.7834	1.24640	0.623199	−	0.782064i	$-0.285833\pi$
$571$	29.7834	1.24640	0.623199	+	0.782064i	$0.285833\pi$
$572$	−3.98121	−0.166463
$573$	33.5908	1.40327
$574$	3.54199	0.147840
$575$	−3.08532	−0.128667
$576$	2.82843	0.117851
$577$	16.3924	0.682424	0.341212	−	0.939986i	$-0.389163\pi$
$577$	16.3924	0.682424	0.341212	+	0.939986i	$0.389163\pi$
$578$	−8.66170	−0.360279
$579$	−30.5148	−1.26815
$580$	1.38372	0.0574558
$581$	2.60365	0.108018
$582$	−1.94517	−0.0806300
$583$	−6.22267	−0.257716
$584$	−10.8351	−0.448359
$585$	−13.6335	−0.563674
$586$	17.7592	0.733624
$587$	−17.2838	−0.713378	−0.356689	−	0.934223i	$-0.616094\pi$
$587$	−17.2838	−0.713378	−0.356689	+	0.934223i	$0.616094\pi$
$588$	16.2300	0.669314
$589$	0.376865	0.0155284
$590$	13.1272	0.540439
$591$	−18.3380	−0.754326
$592$	−2.75138	−0.113081
$593$	13.0474	0.535794	0.267897	−	0.963448i	$-0.413671\pi$
$593$	13.0474	0.535794	0.267897	+	0.963448i	$0.413671\pi$
$594$	−0.473398	−0.0194237
$595$	2.10411	0.0862603
$596$	13.2044	0.540872
$597$	62.6476	2.56399
$598$	3.48348	0.142450
$599$	29.6231	1.21037	0.605183	−	0.796087i	$-0.293100\pi$
$599$	29.6231	1.21037	0.605183	+	0.796087i	$0.293100\pi$
$600$	7.44862	0.304089
$601$	−22.9002	−0.934117	−0.467059	−	0.884226i	$-0.654686\pi$
$601$	−22.9002	−0.934117	−0.467059	+	0.884226i	$0.654686\pi$
$602$	−2.25870	−0.0920577
$603$	−39.2509	−1.59842
$604$	6.34420	0.258142
$605$	13.4135	0.545337
$606$	27.1621	1.10339
$607$	30.6259	1.24307	0.621534	−	0.783387i	$-0.286510\pi$
$607$	30.6259	1.24307	0.621534	+	0.783387i	$0.286510\pi$
$608$	−0.0431257	−0.00174898
$609$	1.27133	0.0515169
$610$	10.2217	0.413866
$611$	−43.5921	−1.76355
$612$	−8.16740	−0.330147
$613$	39.9715	1.61444	0.807218	−	0.590254i	$-0.200972\pi$
$613$	39.9715	1.61444	0.807218	+	0.590254i	$0.200972\pi$
$614$	5.94585	0.239955
$615$	22.4692	0.906047
$616$	−0.601845	−0.0242490
$617$	−10.9702	−0.441644	−0.220822	−	0.975314i	$-0.570874\pi$
$617$	−10.9702	−0.441644	−0.220822	+	0.975314i	$0.570874\pi$
$618$	41.3887	1.66490
$619$	38.4181	1.54415	0.772076	−	0.635530i	$-0.219218\pi$
$619$	38.4181	1.54415	0.772076	+	0.635530i	$0.219218\pi$
$620$	12.0920	0.485625
$621$	0.414214	0.0166218
$622$	−14.4532	−0.579519
$623$	7.53754	0.301985
$624$	−8.40986	−0.336664
$625$	−0.0541840	−0.00216736
$626$	−16.9119	−0.675937
$627$	−0.118991	−0.00475204
$628$	7.09814	0.283246
$629$	7.94492	0.316785
$630$	−2.06099	−0.0821118
$631$	−12.7172	−0.506264	−0.253132	−	0.967432i	$-0.581461\pi$
$631$	−12.7172	−0.506264	−0.253132	+	0.967432i	$0.581461\pi$
$632$	−6.10318	−0.242772
$633$	−33.7623	−1.34193
$634$	23.9729	0.952087
$635$	12.3633	0.490623
$636$	−13.1447	−0.521221
$637$	−23.4183	−0.927868
$638$	1.14288	0.0452472
$639$	39.0192	1.54358
$640$	−1.38372	−0.0546963
$641$	36.7490	1.45150	0.725750	−	0.687959i	$-0.241493\pi$
$641$	36.7490	1.45150	0.725750	+	0.687959i	$0.241493\pi$
$642$	−11.7665	−0.464387
$643$	−23.3057	−0.919089	−0.459544	−	0.888155i	$-0.651987\pi$
$643$	−23.3057	−0.919089	−0.459544	+	0.888155i	$0.651987\pi$
$644$	0.526602	0.0207510
$645$	−14.3285	−0.564182
$646$	0.124530	0.00489958
$647$	27.3931	1.07693	0.538466	−	0.842647i	$-0.319004\pi$
$647$	27.3931	1.07693	0.538466	+	0.842647i	$0.319004\pi$
$648$	−9.48528	−0.372617
$649$	10.8424	0.425602
$650$	−10.7476	−0.421557
$651$	11.1098	0.435429
$652$	6.05776	0.237240
$653$	1.41877	0.0555209	0.0277604	−	0.999615i	$-0.491162\pi$
$653$	1.41877	0.0555209	0.0277604	+	0.999615i	$0.491162\pi$
$654$	−31.5720	−1.23456
$655$	23.5152	0.918816
$656$	6.72612	0.262611
$657$	−30.6462	−1.19562
$658$	−6.58989	−0.256900
$659$	46.9698	1.82969	0.914843	−	0.403811i	$-0.132315\pi$
$659$	46.9698	1.82969	0.914843	+	0.403811i	$0.132315\pi$
$660$	−3.81791	−0.148612
$661$	−8.48528	−0.330039	−0.165020	−	0.986290i	$-0.552769\pi$
$661$	−8.48528	−0.330039	−0.165020	+	0.986290i	$0.552769\pi$
$662$	6.55759	0.254868
$663$	24.2844	0.943128
$664$	4.94424	0.191874
$665$	0.0314244	0.00121859
$666$	−7.78208	−0.301549
$667$	−1.00000	−0.0387202
$668$	6.19011	0.239502
$669$	8.24651	0.318828
$670$	19.2022	0.741848
$671$	8.44265	0.325925
$672$	−1.27133	−0.0490426
$673$	−21.0165	−0.810128	−0.405064	−	0.914288i	$-0.632751\pi$
$673$	−21.0165	−0.810128	−0.405064	+	0.914288i	$0.632751\pi$
$674$	−8.22223	−0.316708
$675$	−1.27798	−0.0491896
$676$	−0.865391	−0.0332843
$677$	2.89960	0.111441	0.0557203	−	0.998446i	$-0.482254\pi$
$677$	2.89960	0.111441	0.0557203	+	0.998446i	$0.482254\pi$
$678$	−1.59587	−0.0612888
$679$	0.424292	0.0162828
$680$	3.99564	0.153226
$681$	−65.4804	−2.50921
$682$	9.98737	0.382436
$683$	33.0614	1.26506	0.632530	−	0.774536i	$-0.282017\pi$
$683$	33.0614	1.26506	0.632530	+	0.774536i	$0.282017\pi$
$684$	−0.121978	−0.00466394
$685$	−29.7788	−1.13779
$686$	−7.22640	−0.275905
$687$	1.69818	0.0647894
$688$	−4.28919	−0.163524
$689$	18.9665	0.722567
$690$	3.34059	0.127174
$691$	−9.91581	−0.377215	−0.188608	−	0.982053i	$-0.560397\pi$
$691$	−9.91581	−0.377215	−0.188608	+	0.982053i	$0.560397\pi$
$692$	−25.3949	−0.965368
$693$	−1.70227	−0.0646641
$694$	28.2171	1.07111
$695$	−10.2087	−0.387239
$696$	2.41421	0.0915105
$697$	−19.4224	−0.735676
$698$	−13.6197	−0.515514
$699$	8.63899	0.326757
$700$	−1.62474	−0.0614093
$701$	6.58617	0.248756	0.124378	−	0.992235i	$-0.460306\pi$
$701$	6.58617	0.248756	0.124378	+	0.992235i	$0.460306\pi$
$702$	1.44290	0.0544589
$703$	0.118655	0.00447516
$704$	−1.14288	−0.0430740
$705$	−41.8041	−1.57443
$706$	−7.39150	−0.278183
$707$	−5.92476	−0.222823
$708$	22.9034	0.860763
$709$	−41.9973	−1.57724	−0.788621	−	0.614879i	$-0.789205\pi$
$709$	−41.9973	−1.57724	−0.788621	+	0.614879i	$0.789205\pi$
$710$	−19.0889	−0.716395
$711$	−17.2624	−0.647391
$712$	14.3135	0.536422
$713$	−8.73875	−0.327269
$714$	3.67111	0.137388
$715$	5.50887	0.206020
$716$	−2.08645	−0.0779745
$717$	36.3084	1.35596
$718$	−14.4497	−0.539257
$719$	−33.2408	−1.23967	−0.619836	−	0.784732i	$-0.712801\pi$
$719$	−33.2408	−1.23967	−0.619836	+	0.784732i	$0.712801\pi$
$720$	−3.91375	−0.145857
$721$	−9.02794	−0.336218
$722$	−18.9981	−0.707038
$723$	9.62587	0.357990
$724$	14.9949	0.557281
$725$	3.08532	0.114586
$726$	23.4029	0.868565
$727$	12.2722	0.455151	0.227575	−	0.973760i	$-0.426920\pi$
$727$	12.2722	0.455151	0.227575	+	0.973760i	$0.426920\pi$
$728$	1.83441	0.0679876
$729$	−23.8284	−0.882534
$730$	14.9927	0.554905
$731$	12.3855	0.458095
$732$	17.8341	0.659169
$733$	−20.9532	−0.773923	−0.386961	−	0.922096i	$-0.626475\pi$
$733$	−20.9532	−0.773923	−0.386961	+	0.922096i	$0.626475\pi$
$734$	12.3227	0.454840
$735$	−22.4578	−0.828367
$736$	1.00000	0.0368605
$737$	15.8601	0.584214
$738$	19.0243	0.700295
$739$	−14.0103	−0.515379	−0.257689	−	0.966228i	$-0.582961\pi$
$739$	−14.0103	−0.515379	−0.257689	+	0.966228i	$0.582961\pi$
$740$	3.80714	0.139953
$741$	0.362681	0.0133234
$742$	2.86720	0.105258
$743$	0.0369638	0.00135607	0.000678034	−	1.00000i	$-0.499784\pi$
$743$	0.0369638	0.00135607	0.000678034		1.00000i	$0.499784\pi$
$744$	21.0972	0.773461
$745$	−18.2711	−0.669402
$746$	13.8470	0.506974
$747$	13.9844	0.511664
$748$	3.30020	0.120667
$749$	2.56658	0.0937808
$750$	−27.0098	−0.986257
$751$	−29.8495	−1.08922	−0.544612	−	0.838688i	$-0.683323\pi$
$751$	−29.8495	−1.08922	−0.544612	+	0.838688i	$0.683323\pi$
$752$	−12.5140	−0.456338
$753$	75.0957	2.73664
$754$	−3.48348	−0.126861
$755$	−8.77859	−0.319486
$756$	0.218126	0.00793316
$757$	52.4740	1.90720	0.953600	−	0.301076i	$-0.0973457\pi$
$757$	52.4740	1.90720	0.953600	+	0.301076i	$0.0973457\pi$
$758$	−6.37526	−0.231560
$759$	2.75916	0.100151
$760$	0.0596739	0.00216460
$761$	−30.6254	−1.11017	−0.555085	−	0.831794i	$-0.687314\pi$
$761$	−30.6254	−1.11017	−0.555085	+	0.831794i	$0.687314\pi$
$762$	21.5706	0.781420
$763$	6.88666	0.249314
$764$	−13.9137	−0.503382
$765$	11.3014	0.408602
$766$	−35.2601	−1.27400
$767$	−33.0474	−1.19327
$768$	−2.41421	−0.0871154
$769$	36.5768	1.31899	0.659497	−	0.751707i	$-0.270769\pi$
$769$	36.5768	1.31899	0.659497	+	0.751707i	$0.270769\pi$
$770$	0.832784	0.0300115
$771$	−14.6254	−0.526722
$772$	12.6397	0.454911
$773$	3.73743	0.134426	0.0672131	−	0.997739i	$-0.478589\pi$
$773$	3.73743	0.134426	0.0672131	+	0.997739i	$0.478589\pi$
$774$	−12.1317	−0.436064
$775$	26.9618	0.968498
$776$	0.805717	0.0289236
$777$	3.49791	0.125487
$778$	−9.74055	−0.349216
$779$	−0.290068	−0.0103928
$780$	11.6369	0.416667
$781$	−15.7665	−0.564170
$782$	−2.88761	−0.103261
$783$	−0.414214	−0.0148028
$784$	−6.72269	−0.240096
$785$	−9.82182	−0.350556
$786$	41.0277	1.46341
$787$	23.8809	0.851262	0.425631	−	0.904897i	$-0.360052\pi$
$787$	23.8809	0.851262	0.425631	+	0.904897i	$0.360052\pi$
$788$	7.59587	0.270591
$789$	−38.5211	−1.37139
$790$	8.44509	0.300463
$791$	0.348099	0.0123770
$792$	−3.23256	−0.114864
$793$	−25.7330	−0.913804
$794$	11.2646	0.399766
$795$	18.1886	0.645082
$796$	−25.9495	−0.919755
$797$	−27.2621	−0.965674	−0.482837	−	0.875710i	$-0.660394\pi$
$797$	−27.2621	−0.965674	−0.482837	+	0.875710i	$0.660394\pi$
$798$	0.0548270	0.00194086
$799$	36.1355	1.27838
$800$	−3.08532	−0.109083
$801$	40.4848	1.43046
$802$	0.326157	0.0115170
$803$	12.3832	0.436995
$804$	33.5027	1.18155
$805$	−0.728670	−0.0256822
$806$	−30.4412	−1.07225
$807$	2.94860	0.103796
$808$	−11.2509	−0.395806
$809$	−27.2555	−0.958251	−0.479126	−	0.877746i	$-0.659046\pi$
$809$	−27.2555	−0.958251	−0.479126	+	0.877746i	$0.659046\pi$
$810$	13.1250	0.461164
$811$	−5.62693	−0.197588	−0.0987940	−	0.995108i	$-0.531498\pi$
$811$	−5.62693	−0.197588	−0.0987940	+	0.995108i	$0.531498\pi$
$812$	−0.526602	−0.0184801
$813$	31.8394	1.11666
$814$	3.14451	0.110215
$815$	−8.38224	−0.293617
$816$	6.97131	0.244045
$817$	0.184974	0.00647144
$818$	−2.81492	−0.0984214
$819$	5.18849	0.181300
$820$	−9.30706	−0.325017
$821$	−1.50495	−0.0525231	−0.0262615	−	0.999655i	$-0.508360\pi$
$821$	−1.50495	−0.0525231	−0.0262615	+	0.999655i	$0.508360\pi$
$822$	−51.9559	−1.81217
$823$	−39.7338	−1.38503	−0.692517	−	0.721401i	$-0.743498\pi$
$823$	−39.7338	−1.38503	−0.692517	+	0.721401i	$0.743498\pi$
$824$	−17.1438	−0.597231
$825$	−8.51291	−0.296381
$826$	−4.99583	−0.173827
$827$	−32.0139	−1.11323	−0.556617	−	0.830770i	$-0.687901\pi$
$827$	−32.0139	−1.11323	−0.556617	+	0.830770i	$0.687901\pi$
$828$	2.82843	0.0982946
$829$	−31.5947	−1.09733	−0.548664	−	0.836043i	$-0.684863\pi$
$829$	−31.5947	−1.09733	−0.548664	+	0.836043i	$0.684863\pi$
$830$	−6.84144	−0.237470
$831$	−39.4380	−1.36809
$832$	3.48348	0.120768
$833$	19.4125	0.672604
$834$	−17.8114	−0.616760
$835$	−8.56537	−0.296417
$836$	0.0492876	0.00170465
$837$	−3.61971	−0.125115
$838$	27.4540	0.948384
$839$	38.0086	1.31220	0.656101	−	0.754673i	$-0.272205\pi$
$839$	38.0086	1.31220	0.656101	+	0.754673i	$0.272205\pi$
$840$	1.75916	0.0606969
$841$	1.00000	0.0344828
$842$	−24.2378	−0.835290
$843$	−27.5475	−0.948785
$844$	13.9848	0.481377
$845$	1.19746	0.0411938
$846$	−35.3949	−1.21690
$847$	−5.10479	−0.175403
$848$	5.44471	0.186972
$849$	−55.3905	−1.90100
$850$	8.90921	0.305583
$851$	−2.75138	−0.0943161
$852$	−33.3050	−1.14101
$853$	−13.9783	−0.478607	−0.239303	−	0.970945i	$-0.576919\pi$
$853$	−13.9783	−0.478607	−0.239303	+	0.970945i	$0.576919\pi$
$854$	−3.89009	−0.133116
$855$	0.168783	0.00577226
$856$	4.87385	0.166585
$857$	−7.14822	−0.244179	−0.122089	−	0.992519i	$-0.538959\pi$
$857$	−7.14822	−0.244179	−0.122089	+	0.992519i	$0.538959\pi$
$858$	9.61148	0.328131
$859$	−5.24424	−0.178931	−0.0894656	−	0.995990i	$-0.528516\pi$
$859$	−5.24424	−0.178931	−0.0894656	+	0.995990i	$0.528516\pi$
$860$	5.93504	0.202383
$861$	−8.55112	−0.291421
$862$	−25.6839	−0.874796
$863$	50.2235	1.70963	0.854814	−	0.518934i	$-0.173671\pi$
$863$	50.2235	1.70963	0.854814	+	0.518934i	$0.173671\pi$
$864$	0.414214	0.0140918
$865$	35.1393	1.19477
$866$	3.97293	0.135006
$867$	20.9112	0.710181
$868$	−4.60184	−0.156197
$869$	6.97523	0.236618
$870$	−3.34059	−0.113257
$871$	−48.3412	−1.63798
$872$	13.0775	0.442861
$873$	2.27891	0.0771295
$874$	−0.0431257	−0.00145875
$875$	5.89153	0.199170
$876$	26.1582	0.883803
$877$	34.9270	1.17940	0.589700	−	0.807622i	$-0.299246\pi$
$877$	34.9270	1.17940	0.589700	+	0.807622i	$0.299246\pi$
$878$	31.3522	1.05809
$879$	−42.8744	−1.44612
$880$	1.58143	0.0533100
$881$	47.7701	1.60941	0.804707	−	0.593672i	$-0.202322\pi$
$881$	47.7701	1.60941	0.804707	+	0.593672i	$0.202322\pi$
$882$	−19.0146	−0.640256
$883$	8.43556	0.283879	0.141940	−	0.989875i	$-0.454666\pi$
$883$	8.43556	0.283879	0.141940	+	0.989875i	$0.454666\pi$
$884$	−10.0589	−0.338318
$885$	−31.6919	−1.06531
$886$	−20.4613	−0.687409
$887$	41.5987	1.39675	0.698375	−	0.715732i	$-0.253907\pi$
$887$	41.5987	1.39675	0.698375	+	0.715732i	$0.253907\pi$
$888$	6.64242	0.222905
$889$	−4.70511	−0.157804
$890$	−19.8059	−0.663895
$891$	10.8406	0.363173
$892$	−3.41582	−0.114370
$893$	0.539674	0.0180595
$894$	−31.8781	−1.06616
$895$	2.88707	0.0965040
$896$	0.526602	0.0175925
$897$	−8.40986	−0.280797
$898$	−18.8062	−0.627571
$899$	8.73875	0.291454
$900$	−8.72661	−0.290887
$901$	−15.7222	−0.523782
$902$	−7.68717	−0.255955
$903$	5.45298	0.181464
$904$	0.661029	0.0219855
$905$	−20.7487	−0.689711
$906$	−15.3163	−0.508849
$907$	19.9898	0.663750	0.331875	−	0.943323i	$-0.392319\pi$
$907$	19.9898	0.663750	0.331875	+	0.943323i	$0.392319\pi$
$908$	27.1229	0.900103
$909$	−31.8224	−1.05548
$910$	−2.53830	−0.0841439
$911$	16.0539	0.531889	0.265945	−	0.963988i	$-0.414316\pi$
$911$	16.0539	0.531889	0.265945	+	0.963988i	$0.414316\pi$
$912$	0.104115	0.00344758
$913$	−5.65069	−0.187011
$914$	−8.52064	−0.281838
$915$	−24.6775	−0.815811
$916$	−0.703407	−0.0232412
$917$	−8.94920	−0.295529
$918$	−1.19609	−0.0394768
$919$	24.4241	0.805676	0.402838	−	0.915271i	$-0.368024\pi$
$919$	24.4241	0.805676	0.402838	+	0.915271i	$0.368024\pi$
$920$	−1.38372	−0.0456199
$921$	−14.3545	−0.472998
$922$	−12.2656	−0.403944
$923$	48.0559	1.58178
$924$	1.45298	0.0477996
$925$	8.48889	0.279113
$926$	3.59947	0.118286
$927$	−48.4899	−1.59262
$928$	−1.00000	−0.0328266
$929$	22.1686	0.727329	0.363665	−	0.931530i	$-0.381525\pi$
$929$	22.1686	0.727329	0.363665	+	0.931530i	$0.381525\pi$
$930$	−29.1926	−0.957263
$931$	0.289921	0.00950177
$932$	−3.57839	−0.117214
$933$	34.8930	1.14235
$934$	0.386454	0.0126452
$935$	−4.56655	−0.149342
$936$	9.85276	0.322048
$937$	28.8078	0.941108	0.470554	−	0.882371i	$-0.344054\pi$
$937$	28.8078	0.941108	0.470554	+	0.882371i	$0.344054\pi$
$938$	−7.30780	−0.238608
$939$	40.8290	1.33241
$940$	17.3158	0.564780
$941$	−18.2155	−0.593808	−0.296904	−	0.954907i	$-0.595954\pi$
$941$	−18.2155	−0.593808	−0.296904	+	0.954907i	$0.595954\pi$
$942$	−17.1364	−0.558335
$943$	6.72612	0.219033
$944$	−9.48690	−0.308772
$945$	−0.301825	−0.00981836
$946$	4.90205	0.159379
$947$	−40.3277	−1.31047	−0.655237	−	0.755423i	$-0.727431\pi$
$947$	−40.3277	−1.31047	−0.655237	+	0.755423i	$0.727431\pi$
$948$	14.7344	0.478551
$949$	−37.7437	−1.22521
$950$	0.133057	0.00431693
$951$	−57.8758	−1.87675
$952$	−1.52062	−0.0492837
$953$	−15.1949	−0.492211	−0.246106	−	0.969243i	$-0.579151\pi$
$953$	−15.1949	−0.492211	−0.246106	+	0.969243i	$0.579151\pi$
$954$	15.4000	0.498592
$955$	19.2527	0.623003
$956$	−15.0394	−0.486410
$957$	−2.75916	−0.0891911
$958$	−34.8045	−1.12448
$959$	11.3329	0.365959
$960$	3.34059	0.107817
$961$	45.3657	1.46341
$962$	−9.58437	−0.309012
$963$	13.7853	0.444226
$964$	−3.98717	−0.128418
$965$	−17.4897	−0.563015
$966$	−1.27133	−0.0409044
$967$	−41.0813	−1.32109	−0.660544	−	0.750788i	$-0.729674\pi$
$967$	−41.0813	−1.32109	−0.660544	+	0.750788i	$0.729674\pi$
$968$	−9.69382	−0.311571
$969$	−0.300643	−0.00965803
$970$	−1.11489	−0.0357968
$971$	−1.56524	−0.0502311	−0.0251155	−	0.999685i	$-0.507995\pi$
$971$	−1.56524	−0.0502311	−0.0251155	+	0.999685i	$0.507995\pi$
$972$	21.6569	0.694644
$973$	3.88513	0.124552
$974$	10.2168	0.327368
$975$	25.9471	0.830973
$976$	−7.38715	−0.236457
$977$	−55.5462	−1.77708	−0.888540	−	0.458798i	$-0.848280\pi$
$977$	−55.5462	−1.77708	−0.888540	+	0.458798i	$0.848280\pi$
$978$	−14.6247	−0.467648
$979$	−16.3587	−0.522826
$980$	9.30231	0.297152
$981$	36.9889	1.18096
$982$	−30.5440	−0.974699
$983$	34.4854	1.09991	0.549957	−	0.835193i	$-0.314644\pi$
$983$	34.4854	1.09991	0.549957	+	0.835193i	$0.314644\pi$
$984$	−16.2383	−0.517658
$985$	−10.5105	−0.334894
$986$	2.88761	0.0919603
$987$	15.9094	0.506402
$988$	−0.150227	−0.00477937
$989$	−4.28919	−0.136388
$990$	4.47296	0.142160
$991$	−46.3283	−1.47167	−0.735833	−	0.677163i	$-0.763209\pi$
$991$	−46.3283	−1.47167	−0.735833	+	0.677163i	$0.763209\pi$
$992$	−8.73875	−0.277456
$993$	−15.8314	−0.502395
$994$	7.26468	0.230422
$995$	35.9068	1.13832
$996$	−11.9365	−0.378221
$997$	−1.82569	−0.0578203	−0.0289101	−	0.999582i	$-0.509204\pi$
$997$	−1.82569	−0.0578203	−0.0289101	+	0.999582i	$0.509204\pi$
$998$	0.603650	0.0191082
$999$	−1.13966	−0.0360572

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1334.2.a.e.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1334.2.a.e.1.1	✓	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1334 = 2 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1334.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -1.38372 q^{5} -2.41421 q^{6} +0.526602 q^{7} +1.00000 q^{8} +2.82843 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -1.38372 q^{5} -2.41421 q^{6} +0.526602 q^{7} +1.00000 q^{8} +2.82843 q^{9} -1.38372 q^{10} -1.14288 q^{11} -2.41421 q^{12} +3.48348 q^{13} +0.526602 q^{14} +3.34059 q^{15} +1.00000 q^{16} -2.88761 q^{17} +2.82843 q^{18} -0.0431257 q^{19} -1.38372 q^{20} -1.27133 q^{21} -1.14288 q^{22} +1.00000 q^{23} -2.41421 q^{24} -3.08532 q^{25} +3.48348 q^{26} +0.414214 q^{27} +0.526602 q^{28} -1.00000 q^{29} +3.34059 q^{30} -8.73875 q^{31} +1.00000 q^{32} +2.75916 q^{33} -2.88761 q^{34} -0.728670 q^{35} +2.82843 q^{36} -2.75138 q^{37} -0.0431257 q^{38} -8.40986 q^{39} -1.38372 q^{40} +6.72612 q^{41} -1.27133 q^{42} -4.28919 q^{43} -1.14288 q^{44} -3.91375 q^{45} +1.00000 q^{46} -12.5140 q^{47} -2.41421 q^{48} -6.72269 q^{49} -3.08532 q^{50} +6.97131 q^{51} +3.48348 q^{52} +5.44471 q^{53} +0.414214 q^{54} +1.58143 q^{55} +0.526602 q^{56} +0.104115 q^{57} -1.00000 q^{58} -9.48690 q^{59} +3.34059 q^{60} -7.38715 q^{61} -8.73875 q^{62} +1.48946 q^{63} +1.00000 q^{64} -4.82015 q^{65} +2.75916 q^{66} -13.8773 q^{67} -2.88761 q^{68} -2.41421 q^{69} -0.728670 q^{70} +13.7954 q^{71} +2.82843 q^{72} -10.8351 q^{73} -2.75138 q^{74} +7.44862 q^{75} -0.0431257 q^{76} -0.601845 q^{77} -8.40986 q^{78} -6.10318 q^{79} -1.38372 q^{80} -9.48528 q^{81} +6.72612 q^{82} +4.94424 q^{83} -1.27133 q^{84} +3.99564 q^{85} -4.28919 q^{86} +2.41421 q^{87} -1.14288 q^{88} +14.3135 q^{89} -3.91375 q^{90} +1.83441 q^{91} +1.00000 q^{92} +21.0972 q^{93} -12.5140 q^{94} +0.0596739 q^{95} -2.41421 q^{96} +0.805717 q^{97} -6.72269 q^{98} -3.23256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{7} + 4 q^{8} - 4 q^{11} - 4 q^{12} - 8 q^{13} - 4 q^{14} - 8 q^{15} + 4 q^{16} - 12 q^{17} - 16 q^{19} - 4 q^{22} + 4 q^{23} - 4 q^{24} + 8 q^{25} - 8 q^{26} - 4 q^{27} - 4 q^{28} - 4 q^{29} - 8 q^{30} - 12 q^{31} + 4 q^{32} + 16 q^{33} - 12 q^{34} - 8 q^{35} - 4 q^{37} - 16 q^{38} + 4 q^{39} - 12 q^{41} + 4 q^{43} - 4 q^{44} + 16 q^{45} + 4 q^{46} - 28 q^{47} - 4 q^{48} - 8 q^{49} + 8 q^{50} + 16 q^{51} - 8 q^{52} + 16 q^{53} - 4 q^{54} - 20 q^{55} - 4 q^{56} + 16 q^{57} - 4 q^{58} + 4 q^{59} - 8 q^{60} - 4 q^{61} - 12 q^{62} + 8 q^{63} + 4 q^{64} - 24 q^{65} + 16 q^{66} - 12 q^{68} - 4 q^{69} - 8 q^{70} - 24 q^{73} - 4 q^{74} - 24 q^{75} - 16 q^{76} - 4 q^{77} + 4 q^{78} + 12 q^{79} - 4 q^{81} - 12 q^{82} - 12 q^{83} - 16 q^{85} + 4 q^{86} + 4 q^{87} - 4 q^{88} + 16 q^{89} + 16 q^{90} + 20 q^{91} + 4 q^{92} + 24 q^{93} - 28 q^{94} - 16 q^{95} - 4 q^{96} + 4 q^{97} - 8 q^{98} - 24 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^7 + 4 * q^8 - 4 * q^11 - 4 * q^12 - 8 * q^13 - 4 * q^14 - 8 * q^15 + 4 * q^16 - 12 * q^17 - 16 * q^19 - 4 * q^22 + 4 * q^23 - 4 * q^24 + 8 * q^25 - 8 * q^26 - 4 * q^27 - 4 * q^28 - 4 * q^29 - 8 * q^30 - 12 * q^31 + 4 * q^32 + 16 * q^33 - 12 * q^34 - 8 * q^35 - 4 * q^37 - 16 * q^38 + 4 * q^39 - 12 * q^41 + 4 * q^43 - 4 * q^44 + 16 * q^45 + 4 * q^46 - 28 * q^47 - 4 * q^48 - 8 * q^49 + 8 * q^50 + 16 * q^51 - 8 * q^52 + 16 * q^53 - 4 * q^54 - 20 * q^55 - 4 * q^56 + 16 * q^57 - 4 * q^58 + 4 * q^59 - 8 * q^60 - 4 * q^61 - 12 * q^62 + 8 * q^63 + 4 * q^64 - 24 * q^65 + 16 * q^66 - 12 * q^68 - 4 * q^69 - 8 * q^70 - 24 * q^73 - 4 * q^74 - 24 * q^75 - 16 * q^76 - 4 * q^77 + 4 * q^78 + 12 * q^79 - 4 * q^81 - 12 * q^82 - 12 * q^83 - 16 * q^85 + 4 * q^86 + 4 * q^87 - 4 * q^88 + 16 * q^89 + 16 * q^90 + 20 * q^91 + 4 * q^92 + 24 * q^93 - 28 * q^94 - 16 * q^95 - 4 * q^96 + 4 * q^97 - 8 * q^98 - 24 * q^99

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